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CK-12 Geometry - Basic,
Teacher's Edition
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Dirga Jordan
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Printed: May 10, 2012
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Authors
Kathryn Dirga, Lori Jordan
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Contents
1 Basic Geometry TE - Teaching
Tips 1
1.1 Basics of Geometry 1
1.2 Reasoning and Proof 7
1.3 Parallel and Perpendicular Lines 11
1.4 Triangles and Congruence 19
1.5 Relationships with Triangles 24
1.6 Polygons and Quadrilaterals 29
1.7 Similarity 34
1.8 Right Triangle Trigonometry 41
1.9 Circles 47
1.10 Perimeter and Area 54
1.11 Surface Area and Volume 58
1.12 Rigid Transformations 65
2 Basic Geometry TE - Common
Errors 71
2.1 Basics of Geometry 71
2.2 Reasoning and Proof 76
2.3 Parallel and Perpendicular Lines 82
2.4 Triangles and Congruence 88
2.5 Relationships with Triangles 93
2.6 Polygons and Quadrilaterals 98
2.7 Similarity 104
2.8 Right Triangle Trigonometry 110
2.9 Circles 115
2.10 Perimeter and Area 120
2.11 Surface Area and Volume 124
2.12 Rigid Transformations 131
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3 Basic Geometry TE - Enrichment 135
3.1 Basics of Geometry 135
3.2 Reasoning and Proof 143
3.3 Parallel and Perpendicular Lines 148
3.4 Triangles and Congruence 160
3.5 Relationships with Triangles 167
3.6 Polygons and Quadrilaterals 178
3.7 Similarity 188
3.8 Right Triangle Trigonometry 197
3.9 Circles 205
3.10 Perimeter and Area 217
3.11 Surface Area and Volume 226
3.12 Rigid Transformations 238
4 Basic Geometry TE -
Differentiated Instruction 246
4.1 Basics of Geometry 247
4.2 Reasoning and Proof 253
4.3 Parallel and Perpendicular Lines 258
4.4 Triangles and Congruence 268
4.5 Relationships with Triangles 273
4.6 Polygons and Quadrilaterals 278
4.7 Similarity 284
4.8 Right Triangle Trigonometry 290
4.9 Circles 296
4.10 Perimeter and Area 305
4.11 Surface Area and Volume 310
4.12 Rigid Transformations 317
5 Basic Geometry TE - Problem
Solving 322
5.1 Basics of Geometry 323
5.2 Reasoning and Proof 327
5.3 Parallel and Perpendicular Lines 332
5.4 Triangles and Congruence 338
5.5 Relationships with Triangles 341
5.6 Polygons and Quadrilaterals 348
5.7 Similarity 353
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5.8 Right Triangle Trigonometry 358
5.9 Circles 363
5.10 Perimeter and Area 370
5.11 Surface Area and Volume 375
5.12 Rigid Transformations 381
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Chapter 1
Basic Geometry TE - Teaching
Tips
1.1 Basics of Geometry
Author's Note
This component of the Teacher's Edition for the Basic Geometry FlexBook is designed to help teacher's
lesson plan. Suggestions for block planning, daily supplemental activities, and study skills tips are also
included. It is recommended to hand out the Study Guides (at the end of each chapter and print-ready)
at the beginning of each chapter and fill it out as the chapter progresses.
The Review Queue at the beginning of each section in the FlexBook is designed to be a warm-up for the
beginning of each lesson and is intended to be done at the beginning of the period. Answers are at the end
of each section.
The Know What? at the beginning of each section in the FlexBook is designed as a discussion point for
the beginning of a lesson and then answered at the conclusion of the lesson. It can be added to homework
or done as an end-of-the-lesson "quiz" to assess how students are progressing.
Throughout the text there are investigations pertaining to theorems or concepts within a lesson. These
investigations may be constructions or detailed drawings that are designed to steer students towards discov-
ering a theorem or concept on their own. This is a hands-on approach to learning the material and usually
received well by low-level students. It provides them an opportunity to gain ownership of the material
without being told to accept something as truth. These investigations may use: a ruler (or straightedge),
compass, protractor, pencil/pen, colored pencils, construction paper, patty paper, or scissors. They can
be done as a teacher-led activity, as a group, in pairs, or as an individual activity. If you decide to make
an investigation teacher-led, have the students follow along, answer the questions in the text, and then
draw their own conclusions. In a block period setting, these activities could be done as a group (because
activities seem to take longer when students work in groups) with each group member owning a particular
role. One or two students can do the investigation, one can record the group's conclusions, and one can
report back to the class.
At the beginning of the Review Questions, there is a bulleted list with the examples that are similar to
which review questions. Encourage students to use this list to and then reference the examples to help
them with their homework.
In the Pacing sections for each chapter, consider each "Day" a traditional 50-minute period. For block
scheduling, group two days together.
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Pacing
Table 1.1:
Day 1
Day 2
Day 3
Day 4
Day 5
Points, Lines,
Planes
and
Day 6
Midpoints and Bi-
sectors
Investigation 1-4
Investigation 1-5
Day 11
Continue Review
Continue Points,
Lines, and Planes
Investigation 1-1
Day 7
Quiz 2
Start Angle Pairs
Day 12
Chapter 1
Test
Segments and Dis-
tance
Quiz 1
Start Angles
Measurement
and
Day 8 Day 9
Finish Angle Pairs Classifying Poly-
Investigation 1-6 gons
Day 13
Continue testing
(if needed)
Start Chapter 2
Finish Angles and
Measurement
Investigation 1-2
Investigation 1-3
Day 10
Quiz 3
Start Review
Chapter 1
of
Points, Lines, and Planes
Goal
This lesson introduces students to the basic principles of geometry. Students will become familiar with
the terms points, lines, and planes and how these terms are used to define other geometric vocabulary.
Students will also be expected to correctly draw and label geometric figures.
Study Skills Tip
Geometry is very vocabulary-intensive, unlike Algebra. Devote 5-10 minutes of each class period to thor-
oughly defining and describing vocabulary. Use the Study Guides at the end of each chapter to assist
you with this. Also make sure that students know how to correctly label diagrams. You can use personal
whiteboards to perform quick vocabulary checks. Or, visit Discovery School's puzzle maker to make word
searches and crosswords (http://puzzlemaker.discoveryeducation.com/).
Real World Connection
Have a class discussion to identify real-life examples of points, lines, planes in the classroom, as well as
sets of collinear and coplanar. For example, points could be chairs, lines could be the intersection of the
ceiling and wall, and the floor is a plane. If your chairs are four-legged, this is a fantastic example of why 3
points determine a plane, not four. Four legged chairs tend to wobble, while 3-legged stools remain stable.
During this discussion, have students fill out the following table:
Table 1.2:
Dimensions
Description
Geometry Represen- Real-Life
tation ple(s)
Exam-
Zero
One
n/a
Length
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Table 1.2: (continued)
Dimensions Description Geometry Represen- Real-Life Exam-
tation ple(s)
Two Length and width
Three Length, width, and
height
Students may have difficulty distinguishing the difference between a postulate and a theorem. Use real-
world examples like "my eyes are blue," for a postulate. That cannot be proven true, but we know that it
is by looking (fill in your eye color). A theorem would be something like, "If the refrigerator is not working,
then it is unplugged." We can go through steps to prove (or disprove) that the fridge is unplugged. Students
may conclude other reasons that would make the fridge not work (it could be broken, the fan could have
gone out, it is old, etc), making this a statement that needs to be proven and cannot be accepted as true.
Segments and Distance
Goal
Students should be familiar with using rulers to measure distances. This lesson incorporates geometric
postulates and properties to measurement, such as the Segment Addition Property. There is also an
algebraic tie-in, finding the distance of vertical and horizontal lines on the coordinate plane.
Notation Note
Double (and triple) check that students understand the difference between the labeling of a line segment,
AZ?, and its distance, AB. Even though the alternative notation, mAB, is introduced in this section, this
text primarily uses AB.
Relevant Review
Students may need to review how to plot points and count the squares for the horizontal and vertical
distances between two points. It might also be helpful add a few algebraic equations to the Review
Queue. Problems involving the Segment Addition Postulate can be similar to solving an algebraic equation
(Example 9).
Real World Connection
To review the concept of measurement, use an enlarged map of your community. Label several things on
your map important to students - high school, grocery store, movie theatre, etc. Have students practice
finding the distances between landmarks "as the crow flies" and using different street routes to determine
the shortest distance between the two.
Teaching Strategy
The Segment Addition Postulate can seem simple to students at first. Start with basic examples, like
Examples 5 and 6 and then progress to more complicated examples, like 7 and 8. Finally, introduce
problems like Example 9. For more examples, see the Differentiated Instruction component. With the
Segment Addition Postulate, you can start to introduce the concept of a proof. Use Example 7 and have
students write out an explanation of their drawing. Tell students to use language such that the person
reading their explanation knows nothing about math.
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Angles and Measurement
Goal
This lesson introduces students to angles and how to use a protractor to measure them. Then, we will
apply the Angle Addition Postulate in the same way as the Segment Addition Postulate.
Notation Note
Beginning geometry students may get confused regarding the ray notation. Draw rays in different directions
so students become comfortable with the concept that ray notation always has the non-arrow end over the
endpoint (regardless of the direction the ray points). Reinforce that AB and BA represent the same ray.
Real World Connection
Have students Think-Pair-Share their answers to the opening question, "Can you think of real-life examples
of rays?" Then, open up the discussion to the whole class.
Teaching Strategies
Using a classroom sized protractor will allow students to check to make sure their drawings are the same
as yours. An overhead projector or digital imager is also a great way to demonstrate the proper way to
use a protractor.
In this section, we only tell students that they can use three letters (and always three letters) to label and
angle. In Chapter 2 we introduce the shortcut. We did not want the confusion that so commonly occurs
where students will name any angle by only its vertex.
Stress the similarities between the Segment Addition Property and Angle Addition Property. Students will
discover that many geometrical theorems and properties are quite similar.
Have students take a piece of paper and fold it at any angle of their choosing from the corner of the
paper. Open the fold and refold the paper at a different angle, forming two "rays" and three angles.
Show how the angle addition property can be used by asking students to measure their created angles and
finding the sum. You can also use this opportunity to explain how angles can also be labeled as numbers,
mZl+raZ2 + mZ3 = 90°
Student may need additional practice drawing and copying angles. This is the first time they have used
a compass (in this course). Encourage students to play with the compass and show them how to use it
to draw a circle and arcs. Once they are familiar with the compass (after 5-10 minutes), then go into
Investigation 1-3. In addition to copying a 50° angle, it might be helpful to walk students through copying
a 90° angle and an obtuse angle.
Midpoints and Bisectors
Goal
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The lesson introduces students to the concept of congruency, midpoints, and bisectors. The difference
between congruence and equality will also be stressed. Students will use algebra to write equivalence
statements and solve for unknown variables.
Teaching Strategies
This is a great lesson for students to create a "dictionary" of all the notations learned thus far. In addition
to the Study Guide, the dictionary provides an invaluable reference before assessments.
When teaching the Midpoint Postulate, reiterate to students that this really is the arithmetic average
of the endpoints, incorporating algebra and statistics into the lesson. Explain the average between two
numbers, is the sum divided by 2. The midpoint of two points is the exact same idea.
Ask students to define "bisector" on their own, before discussing a perpendicular bisector (Example 4).
Hopefully students will construct multiple bisectors. This will help students visualize that there are an
infinite amount of bisectors, and lead them to the fact there is only one perpendicular bisector and the
Perpendicular Bisector Postulate.
With Investigations 1-4 and 1-5, students may need to repeat the construction a few times. Copy a handout
with several line segments and different angle measures and have them practice the construction on their
own or in pairs.
In this lesson and the previous lesson, we have introduced how to make drawings. Encourage students to
redraw any pictures that are in the homework so they can mark congruent segments and angles. Also, let
students know that it is ok to mark on quizzes and tests (depending on your preference).
Angle Pairs
Goal
This lesson introduces students to common angle pairs, the Linear Angle Postulate and the Vertical Angles
Theorem.
Teaching Strategies
Students can get complementary and supplementary confused. A way to help them remember:
• C in Complementary also stands for Corner (in a right angle)
a.
• S in Supplementary also stands for Straight (in a straight angle)
O
To illustrate the concept of the Linear Pair Postulate, offer several examples of linear pairs. Have students
measure each angle and find the sum of the linear pair. Students should discover any linear pair is
supplementary. Also explain that a linear pair must be adjacent. Discuss the difference between adjacent
supplementary angles (a linear pair) and non-adjacent supplementary angles (same side interior angles,
consecutive angles, or two angles in a drawing that are not next to each other).
To further illustrate the idea of vertical angles, repeatlnvestigation 1-6 with two different intersecting lines.
Also, encourage students two draw their intersecting lines at different angles than yours. This way, they
will see that no matter the angle measures the vertical angles are always equal and the linear pairs are
always supplementary. Draw this investigation on a piece of white paper and have students use the whole
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page. Then, when they are done, have them exchange papers with the students around them to reinforce
that the angle measures do not matter.
• V in Vertical angles also stands for V in Vertex. Vertical angles do not have to be "vertical" (one on
top of the other). Students might get the definitions confused.
In this section there are a lot of Algebra tie-ins (Example 5, Review Questions 17-25). Students might need
additional examples showing linear pairs and vertical angles with algebraic expression representations.
Additional Example: Find the value of y.
(14/-42)
Solution: Because these are vertical angles, set the two expressions equal to each other.
(14y-42)° = (lly + 6)°
3y = 48°
y = 16°
Classifying Polygons
Goal
In this lesson, we will explore the different types of triangles and polygons. Students will learn how to
classify triangles by their sides and angles, as well as classify polygons by the number of sides. The
definitions of convex and concave polygons will also be explored.
Teaching Strategy
Divide students into pairs. Give each pair three raw pieces of spaghetti. Instruct one partner to break one
piece of spaghetti into three pieces and attempt to construct a triangle using these segments. Students will
reach the conclusion that the sum of two segments must always be larger than the third if a triangle is to
be formed. The Triangle Inequality Theorem is introduced in Chapter 4-
Next, have students create right, obtuse, acute, scalene, isosceles, and equilateral triangles with their pieces
of spaghetti. Show them that if the spaghetti pieces' endpoints are not touching, the polygon is not closed,
and therefore not a polygon. You can use pieces of spaghetti on an overhead projector.
To show the difference between line segments and curves, introduce cooked spaghetti. The flexibility of the
spaghetti demonstrates to students that segments must be straight in order to provide rigidity and follow
the definitions of polygons.
After playing with the spaghetti, brainstorm the qualities of polygons and write them on the board (or
overhead) and develop a definition. From here, you can compare and contrast convex and concave polygons.
Use a Venn diagram to show the properties that overlap and those that are different.
Review
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At the end of this chapter there is a Symbol Toolbox with all the labels and ways to mark drawings. Have
students make flash cards with the symbols and markings on one side, and what they represent on the
other. Students may also want to make flash cards for the definitions for the other words in the (and
future) chapters.
In addition to the Study Guide, it might be helpful to go over the constructions from this chapter. You
might want to have a Construction Toolbox, where students have one example of each construction they
have learned. These construction pages can supplement the Study Guide and should be added to from
chapter to chapter. As an added incentive, you might want to grade students' Study Guides at the end of
the chapter. Another option could be to allow students to use their Study Guide on tests and/or allow it
to be extra credit. These options can change from test to test or at the teacher's discretion.
1.2 Reasoning and Proof
Pacing
Table 1.3:
Day 1
Day 2
Day 3
Day 4
Day 5
Inductive Reason-
ing
Conditional State-
ments
Continue Condi-
tional Statements
Start Deductive
Reasoning
Quiz 1
Finish Deductive
Reasoning
Algebraic and
Congruence Prop-
erties
Day 6
Quiz 2
Start Proofs about
Angle Pairs and
Segments
Day 7
Finish Proofs
about Angle Pairs
and Segments
Day 8
Quiz 3
Review for
ter 2 Test
Day 9
More Review
Chap-
Bay 10
Chapter 2 Test
(May need to con-
tinue testing on
Day 11)
Inductive Reasoning
Goal
This lesson introduces students to inductive reasoning, which applies to algebraic patterns and integrates
algebra with geometry.
Teaching Strategies
After the Review Queue do a Round Robin with difference sequences. Call on one student to say a number,
call on a second student to say another number. The third student needs to distinguish the pattern and say
the correct number. The fourth, fifth, sixth, etc. students need to say the correct numbers that follow the
pattern. Start over whenever you feel is appropriate and repeat. To make the patterns more challenging,
you can interject at the third spot (to introduce geometric sequences, Fibonacci, and squared patterns).
Now, take one of the sequences that was created by the class and ask students to try to find the rule. Ask
students to recognize the pattern and write the generalization in words.
Additional Example: Find the next three terms of the sequence 14, 10, 15, 11, 16, 12, ...
Solution: Students can look at this sequence in two different ways. One is to subtract 4 and then add 5.
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Another way is to take the odd terms as one sequence (14, 15, 16, ...) and then the even terms as another
sequence (10, 11, 12, ...). Either way, the next three terms will be 17, 13, and 18.
Know What? Suggestion
When going over this Know What? (the locker problem) draw lockers on the front board (as many that
will fit). Have students come up to the board and mark x's to close the appropriate doors, as if they are
acting out the problem. This will help students see the pattern.
Real Life Connection
Apply the idea of counterexample to real life situations. Begin by devising a statement, such as, "If the
sun is shining, then you can wear shorts." While this is true for warm weather states such as Florida and
California, for those living in the Midwest or Northern states, it is quite common to be sunny and 12° F.
Have students create their own statements and encourage other students to find counterexamples.
Conditional Statements
Goal
This lesson introduces conditional statements. Students will gain an understanding of how converses,
inverses, and contrapositives are formed from a conditional.
Teaching Strategies
The first portion of this lesson may be best taught using direct instruction and visual aids. Design phrases
you can laminate, such as "you are sixteen" and "you can drive." Adhere magnets to the back of the
phrases (to stick to the white board), or you can use a SMART board. Begin by writing the words "IF"
and "THEN," giving ample space to place your phrases. When discussing each type of conditional, show
students how each is constructed by rearranging your phrases, yet leaving the words "IF" and "THEN"
intact.
Have students create a chart listing the type of statement, its symbolic form and an example. This allows
students anquick reference sheet when trying to decipher between converse, conditional, contrapositive,
and inverse. The chart can be added to the Study Guide or place in class notes.
Table 1.4:
Symbolic Form Example True or False?
Conditional State- p — > q
ment
Converse
Inverse
Contrapositive Logically equivalent to
original.
Spend time reviewing the definition of a counterexample (from the previous section). A counterexample
is a quick way to disprove the converse and inverse. Explain to students that the same counterexample
should work for both the converse and inverse (if they are false, see Examples 2, 3, and 7).
Use the same setup as the opening activity when discussing biconditionals. Begin with a definition, such
as Example 4. Set up your magnetic phrases in if and only if form, then illustrate to students how the
biconditional can be separated into its conditional and converse.
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Deductive Reasoning
Goal
This lesson introduces deductive reasoning. Different than inductive reasoning, deductive reasoning begins
with a generalized statement, and assuming the hypothesis is true, specific examples are deduced.
Teaching Strategies
Start this lesson by writing the Know What? on the board (or copy it onto a transparency). Have the
students read each door (either out loud or to themselves) and try to reason which door the peasant should
pick. This discussion can lend itself to the definitions of logic and deductive reasoning.
Students may or may not realize that they do deductive reasoning every day. Explain that solving an
equation is an example of deductive reasoning. Try to brainstorm, as a class, other examples of deductive
reasoning and inductive reasoning.
The best way for students to understand the Laws of Detachment, Contrapositive, and Syllogism is to do
lots of practice. Make sure to include problems that do not have a logical conclusion. Like in Examples
7 and 8, it might be helpful for students to put the statements in symbolic form. This will make it easier
for them to find the logical conclusion.
Additional Example: Is the following argument logical? Why or why not?
Any student that likes math must have a logical mind.
Lily is logical.
Conclusion: Lily likes math.
Solution: Change this argument into symbols.
q
:.p
If we were to combine the last statement of the argument and the conclusion, it would be q — > p or the
inverse. We know that the inverse is not logically equivalent to the original statement, so this is not a
logical (or valid) argument.
Algebraic and Congruence Properties
Goal
Students should have some familiarity with these properties. Here we can extend algebraic properties to
geometric logic and congruence.
Teaching Strategy
Use personal whiteboards to do a spot check. Write down examples of each property (4 + a = a + 4, for
example) either on the board or overhead. Then have students "race" to see who writes the correct answer
on their whiteboard the fastest. If you do not want to make it a competition, just have students show you
the answer quickly, 2-3 seconds, and then put their whiteboard down to erase. This could also be done as
a competition in groups.
Stress to students that the properties of congruence can only be used with a = symbol and properties of
equality with an = sign. Remind students of the difference between congruence and equality that was
discussed in Chapter 1.
Have students expand on the properties mentioned in this lesson. Students may come up with the multi-
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plying fractions property, reciprocals, or cross-multiplication.
Prove Move
This lesson introduces proofs. In this text, we will primarily use two-column proofs. Because of the nature
of this text, all homework questions and assessment relating to proofs will be fill-in-the-blank. Feel free to
explain the concept of a paragraph proof and flow-chart proof, if you feel it would help your students.
Proofs can be very difficult for students to understand. They might ask "why" they have to give a reason
for every step. Explain that not everyone reading their proof understands math as well as they or you do.
Also, apply proofs (and logical arguments) to the real world. Lawyers use logical arguments all the time.
Tell them it might help them they are trying to rationalize something with their parents; a new video
game, longer curfew, etc. If they have a logical, fluid "proof" to present to their parents, the parents may
be more apt to agree and give them what they are asking for.
Example 4 in the text outlines the basic steps of how to start and complete a proof. Encourage students
to draw their own diagrams and mark on them. The bullet list after this example should be gone over
several times and addressed as you present Example 5.
Diagrams
The best way to describe what you can and cannot assume is "looks are deceiving." Reiterate to students
that nothing can be assumed. The picture must be marked with notation such as tic marks, angle arcs,
arrows, etc. in order for it to be used in a proof. If an angle looks like it is a right angle, it might not be.
It needs to be explicitly stated in the Given or the angle must be marked.
Additional Example: Use the diagram to list everything that can be determined from the drawing and
those things that cannot. For the latter list, what additional information is needed to clarify the drawing?
Given: AG bisects LBGF 4
i
-+
A B/
-|
-D ^
V c
+-
* F
G H
>
t
1 '
Solution: All vertical angles are congruent, AB = FG, and EH J_ AD are given from the markings. From
the given statement, we know that lAGF = ABGA, which can be marked on the drawing.
Things that cannot be assumed are: DC = HI, ~EH _L FI, AD || F/, GE bisects LAGH, GB = BE. To
conclude that these things are true, you must be told them or it needs to be marked. (There may be more,
this list will get you started)
In the above example, put the drawing on the overhead or whiteboard. After brainstorming what can
and cannot be concluded from the diagram, ask students to correctly mark those things that could not be
assumed true. Once they are marked by students, then the statement is validated. This example can lead
into a discussion of the different ways you can interpret two perpendicular lines (lines are perpendicular,
four right angles, congruent linear pairs, etc). Let students know, in this instance, they only need to be
told one of these pieces of information and the others can be concluded from it.
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10
Proofs about Angle Pairs and Segments
Goal
Students will become familiar with two-column proofs and be able to fill out a short proof on their own.
Teaching Strategy
Like with the definitions of complementary and supplementary, the Same Angle Supplements Theorem
and the Same Angle Complements Theorem can be confused by students. Remind them of the mnemonic
in the Teaching Tips from Chapter 1 (C is for Complementary and Corner, S is for Supplementary and a
Straight angle).
Prove Move
There are several ways to approach the same proof. The order does not always matter, and sometimes
different reasons can be used. For example, the Midpoint Postulate (every line segment has exactly one
point that divides it equally in half) and the Definition of a Midpoint (a point that splits a line segment
equally in half) can be used interchangeably.
Students may also get stuck on the reasons. Encourage students to not worry about getting the name
quite right. If they can't remember the name of a proof, tell them to write it all out. Depending on your
preference, you can also let students use abbreviations for names of theorems as well. For example, the
Vertical Angles Theorem can be shortened to the VA Thm. Establish these abbreviations for the entire
class so there is no confusion.
In the Proof of the Vertical Angles Theorem, steps 2-4 might seem redundant to students. Explain that
they need to completely explain everything they know about linear pairs.
Additional Example: Complete the proof by matching each statement with its corresponding reason.
Given : AG bisects LBGF
Prove : mAAGF = ^mLBGF
Statements
1. AG bisects LBGF
2. £AGF=LAGB
3. m£AGF=m£AGB
4. mlBGF = m/-AGF + mLAGB
5. mlBGF = m^AGF + mLAGF
6. m£BGF = 2-m£AGF
7. ±m/.BGF=m/.AGF
2
A. Distributive Property
B. Substitution Property of Equality
C. Division Property of Equality
D. Definition of an Angle Bisector
E. Given
F. Angle Addition Postulate
G * Definition of Congruent Angles
Solution: The order of the reasons is E, D, G, F, B, A, D
1.3 Parallel and Perpendicular Lines
Pacing
li
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Table 1.5:
Day 1
Day 2
Day 3
Day 4
Day 5
Lines and Angles
Investigation 3-1
Investigation 3-2
Finish Lines and
Angles
Investigation 3-3
Start Properties of
Parallel Lines
Finish Properties
of Parallel Lines
Investigation 3-4
Quiz 1
Start Proving
Lines Parallel
Finish Proving
Lines Parallel
Investigation 3-5
Day 6
Properties of Per-
pendicular Lines
Day 7
Quiz 2
Start Parallel
and Perpendicu-
lar Lines in the
Coordinate Plane
Day 8
Finish Parallel
and Perpendicu-
lar Lines in the
Coordinate Plane
Day 9
The Distance For-
mula
Day 10
Quiz 3
Start Review
Chapter 3
of
Day 11
Review Chapter 3
Day 12
Chapter 3 Test
Day 13
Finish testing
needed)
Start Chapter 4
(if
Lines and Angles
Goal
Students will be introduced to parallel, perpendicular, and skew lines in this lesson. Transversals and the
angles formed by such are also introduced.
Teaching Strategies
To introduce skew lines, use two pencils and hold them in the air, like skew lines. This will help students
visualize that skew lines are in different planes. Use Example 1 as a jumping off point and find more
parallel, skew, and perpendicular lines, other than those listed in the solution.
Investigation 3-1 is a useful tool to help students visualize the Parallel Line Postulate. You can decide
whether you want to do this activity individually or teacher-led. If you decide to make it a teacher-led
demonstration, consider using the overhead and folding a transparency rather than patty paper.
Explore the similarities and differences between the Parallel Line Postulate and the Perpendicular Line
Postulate. You can use a Venn diagram to aid in this discussion.
Investigations 3-2 and 3-3 demonstrate the Perpendicular Line Postulate. Guide students through these
constructions using a whiteboard compass. If you do not have access to a whiteboard compass, tie a piece
of string around your marker and use your finger as the pointer. Make the string at least 8 inches long. If
you have access to an LCD screen or computer in the classroom, the website listed in these investigations
(www.mathisfun.com) has a great animation of these constructions.
When introducing the different angle pairs, discuss other ways that students can identify the relationships.
For example, corresponding angles are in the "same place" on lines / and m. Draw a large diagram, like the
ones to the left, and find all the linear pairs, vertical angles, corresponding angles, alternate interior angles,
alternate exterior angles, and same side interior angles. Use two different pictures to show the different
orientations and that the lines do not have to be parallel to have these angle relationships. Explain that
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12
vertical angles and linear pairs only use two lines; however these new angle relationships require three lines
to be defined. Use Examples 4 and 5. You can also expand on Example 5 and ask:
d) What is a same side interior angle to Z6? (z7)
e) What is a corresponding angle to Z8? (z6)
f) What is an alternate exterior angle to Z8? (zl)
Students might wonder why there is no same side exterior relationship. You can explain that it does exist
(zl and Z4 in the second picture), but not explicitly defined.
Real Life Connection
Discuss examples of parallel, skew, and perpendicular lines and planes in the real world. Examples could
be: a table top and the floor (parallel planes), the legs of the table and the table top or floor (perpendicular
planes), or the cables in the Brooklyn Bridge (skew lines).
Properties of Parallel Lines
Goal
In this section we will extend the notion of transversals and parallel lines to illustrate the corresponding
angles postulate and the alternate interior angles postulate. Additional theorems and postulates are proven
13
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in this lesson.
Teaching Strategies
If you discuss the Know What? at the beginning of the lesson, students will only know how to find angle
measures that are vertical or a linear pair with /.FTS and iSQV. Revisit this at the end of the lesson
and use the new-found postulates and theorems to find corresponding angles, alternate interior angles,
alternate exterior angles, and same side interior angles. You could also test that the angles in AFST add
up to 180°.
Discuss Example 1 as a refresher on where the corresponding angles are and now, if the lines are parallel,
which angles are congruent. Then, guide students through Investigation 3-4. If you prefer, you can do the
investigation before introducing the Corresponding Angles Postulate and Example 1, so that students can
discover this postulate on their own.
When introducing the alternate interior angles, alternate exterior angles, and same side angles use the
results that the students found in Investigation 3-4. Let them draw their own conclusions about all the
angles and angle measures. They already know the names of the relationships, so then ask students if any
other relationships that they learned in the previous lesson are equal. This will allow you to explain the
Alternate Interior Angles Theorem and Alternate Exterior Angles Theorem.
For the Same Side Interior Theorem, ask students which angles are same side interior and then ask what
the relationship is. Students should notice that the two angles add up to 180°.
Students may notice that there are other angles that are supplementary or congruent. Encourage students
to make these observations even though there are no explicit theorems.
Reinforce to students that these theorems do not apply to parallel lines. Demonstrate this by drawing two
non-parallel lines and a transversal. Measure all angles. Students will see the alternate interior angles,
corresponding angles, and alternate interior angles are not congruent, nor are the consecutive interior
angles supplementary.
Proving Lines Parallel
Goal
The converse of the previous lesson's theorems and postulates are provided in this lesson. Students are
encouraged to read through this lesson and follow along with the proofs.
Vocabulary
Let students rewrite theorems, postulates, and properties symbolically and using pictures. The converses
and the Parallel Lines Property in this section are written in this way to help students understand them
better. Continue to use this strategy throughout the text.
Teaching Strategies
Review the concept of a converse from Chapter 2. Then, introduce the converse of the Corresponding
Angles Postulate and ask students if they think it is true. Investigation 3-5 is one way to show students
that converse of the Corresponding Angle Postulate must be true. Remind students that Postulates do
not need to be proven true. However, it is always nice to show students why.
Decide if you would like Investigation 3-5 to be student driven or teacher-led. As a teacher-led investigation,
this activity will show students that the converse of the Corresponding Angles Postulate is true. As a
student driven activity, encourage students to work in pairs. Before starting, demonstrate how to copy an
angle (Investigation 2-2) and then allow students to work through the investigation. Expect it to take 15
minutes.
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Investigation 3-5 can also be redone such that students copy the angle and place it in the location of the
alternate interior or alternate exterior angle location.
Additional Example: Put the reasons for the proof in the correct order.
Given: Z2 and Z3 are supplementary.
Prove: I II m
Statement
1. Z2 and Z3 are
supplementary.
2. mZ2 + mZ3 = 180°
3. Z1 andZ2area
linear pair.
4.Z1 and Z2 are
supplementary.
5. mZ1 +mZ2 = 180°
6. mZ1 = mZ3
7. Z1 =Z3
8. I|| m
Reason
A. Linear Pair Postulate
B- Substitution Property of Equality
C. Given
P. Definition of Congruent Angles
E. Converse of the Corresponding
Angles Postulate
R Definition of Supplementary Angles
Q. Definition of a Linear Pair
H. Definition of Supplementary Angles
Solution: The correct order is C, F or H, G, A, F or H, B, D, E.
Properties of Perpendicular Lines
Goal
This section further explains the properties of perpendicular lines and how they affect transversals.
Perpendicular Lines Investigation
On the whiteboard, draw a linear pair such that the shared side is perpendicular to the non-adjacent sides
(see picture). Ask students what the angle measure of each angle is and what they add up to. Once
students arrive at the correct conclusion, reiterate that a congruent linear pairs is the same as a linear pair
formed by perpendicular lines and the angles will always be 90°.
Cl
A B C
Second, extend BD to make a line and add a parallel line to AC (see picture). Now, discuss the effects of a
perpendicular transversal. Steer this discussion towards Theorems 3-1 and 3-2 and see if the converses of
either of these theorems are true. Again, reiterate that all eight angles in this picture will be 90°.
15
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4-+
B
-— ►
<-—
-^-*
H
Prove Move
In a proof involving perpendicular lines, the following three steps must be included to say that the angles
are 90°.
• Two lines are perpendicular (usually the given)
• The angles formed are right angles (definition of perpendicular lines)
• The angles formed are 90° (definition of right angles)
To say that two right angles are congruent, the following steps must be included:
• Two lines are perpendicular (usually the given)
• The angles formed are right angles (definition of perpendicular lines)
• The right angles are congruent/equal (all right angles are congruent or congruent linear pairs)
This can seem repetitive to students because many of them will feel that it should be inferred that if two
lines are perpendicular, then all the angles will be equal/right/90°. This is not the case. Explain that they
are writing a proof to someone who knows nothing about math or the definitions of perpendicular lines or
right angles. They cannot assume that it is the math teacher that is reading each proof. See Example 3
in this section as an example of these steps.
Additional Example: Algebra Connection Solve for x.
(5x - 6Pyy^4x +15 )°
Solution: The three angles add up to 180°, or (5jc - 6)° and (4* + 15)° add up to 90°.
(5x-6)° + (4jc+15)° = 90°
9x + 9° = 90°
9x = 81°
x = 9°
Parallel and Perpendicular Lines in the Coordinate Plane
Goal
Students should feel comfortable with slopes and lines. Use this lesson as a review of key concepts needed
to determine parallel and perpendicular lines in the coordinate plane. Then, we will apply the concepts
learned in this chapter to the coordinate plane.
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16
Real Life Connection
Ask students to brainstorm the many different interpretations of the word slope. Apply these to real world
situations such as the slope of a mountain, or the part of a continent draining into a particular ocean
(Alaska's North Slope), the slope of a wheelchair ramp, etc. Discuss synonyms for slope: grade, slant,
incline. Then, have students brainstorm further. Relate this back to the Know What? for this section.
Explain how the slope and the grade are related. For example, in the Know What? the slope of the
California Incline is ^ (see FlexBook). The grade of this incline is a percentage, so ^ • 100% = 12%.
Relevant Review
Before discussing standard form for a linear equation, make sure students can clear fractions.
Additional Example: Solve the following equations for x.
a) |jc = 30
b) |x + 3 = 9
c ) Z ■ l = l
Solution: Multiply each number by what the lowest common denominator would be.
a) 6 • (|jc = 30)
5x = 180
x = 36
b) 3 • (|jc + 3 = 9)
2x + 9 = 27
2x= 18
x = 9
c) 12- (!*+£ = §)
4x + 3 = 18
4x= 15
x = 3.75
Of course, there are other ways to approach these problems, but this method of clearing fractions will help
students change slope-intercept form into standard form. Show students these alternate ways of solving
these problems and let them decide which is easier. Then, apply both to changing a slope-intercept equation
into standard form.
Additional Example: Change y = | x - \ into standard form using two different methods.
Solution: Method #1: Clear fractions
4-(y=|x-
5)
Ay = 3x - 2
- 3x + Ay =
-2
or 3x - Ay ■
= 2
Method #2: Find a common denominator
A y= A X -A
3 4 2
-A X+ A y = -A
-3x + Ay = -2
3x - Ay = 2
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Students generally want to avoid fractions, so Method #1 should seem for desirable to them.
Teaching Strategies
When discussing the rise over run triangles, begin making the right triangle connection to students, demon-
strating that every rise/run triangle will form a 90° angle. When students are asked to find the distance
between two points, they can use the Pythagorean Theorem.
Have students trace the top and bottom edges of a ruler onto a coordinate plane (use graph paper). Ask
students to determine the equations for each line and compare the results. Students should notice that, if
done correctly, the slopes will be equal. Recall that this is an easy way to draw parallel lines (Investigation
3-4).
The Distance Formula
Goal
Students are introduced to the Distance Formula and its applications.
Teaching Strategies
Students should be familiar with the Pythagorean Theorem and possibly even the Distance Formula from
previous classes. A quick review of the Pythagorean Theorem might be helpful. The reason neither are
derived at this time is because we have not yet introduced triangles or the properties of right triangles,
which is in Chapter 9. At this point, students can accept both formulas as true and they will be proven
later.
Even though the Distance Formula is written d = y[{x2 - x\) 2 + (j2 ~ Ji) 2 , where (xi,yi) is the first point
and (x2,y2) is the second point the order does not matter as long as the same point's coordinate is first.
So, if students prefer, they can use d = yj(x\ - X2) 2 + (yi - J2) 2 . Using Example 1, show students that
they can use d = y(xi - X2) 2 + {y[ - J2) 2 and the answer will be the same.
A /(4-(-10)) 2 + (-2-3) 2 = ^14 2 + (-5) 2 = V196 + 25
Finding the distance between two parallel lines can be quite complicated for some students because there
are so many steps to remember. For this text, we have simplified this subsection to only use lines with a
slope of 1 or -1. Reinforce the steps used to find the distance between two parallel lines from Example 5.
Additional Example: Find the shortest distance between y = x + 4 and y = x - 4.
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18
Solution: First, graph the two lines and find the y-intercept of the top line, which is (0, 4).
The perpendicular slope is -1. From (0, 4) draw a straight line with a slope of -1 towards y = x - 4. This
perpendicular line intersects y = x - 4 at (4, 0). Use these two points to determine how far apart the lines
are.
d= A /(0-4) 2 + (4-0) 2
= ^(-4) 2 + 4 2 = V16 + 16
v32 « 5.66 units
1.4 Triangles and Congruence
Pacing
Table 1.6:
Day 1
Day 2
Day 3
Day 4
Day 5
Triangle Sums
Investigation 4-1
Congruent Figures
Quiz 1
Start Triangle
Congruence using
SSS and SAS
Finish Triangle
Congruence using
SSS and SAS
Investigation 4-2
Investigation 4-3
Triangle Congru-
ence using ASA,
AAS, and HL
Investigation 4-4
Day 6
More Triangle
Congruence using
ASA, AAS, and
HL
Day 7
Quiz 2
Day 8
Isosceles and Equi-
lateral Triangles
Investigation 4-5
Investigation 4-6
Day 9
Quiz 3
Day 10
Review Chapter 4
Day 11
Finish Review of
Chapter 4
Day 12
Chapter
4
Test
Triangle Sums
Goal
First, this lesson reviews the types of triangles. The Triangle Sum Theorem will be introduced and proven
followed by the Exterior Angle Theorem.
Notation Note
A new symbol, A, is introduced to label a triangle. The order of the vertices do not matter for a triangle
(unlike when labeling an angle). Usually the vertices are written in alphabetical order.
Teaching Strategies
To review finding angle measures, give them the six triangles at the beginning of the section and have
them use their protractors to measure all the angles. Then, discuss their results. Students should notice
that all the angles add up to 180°, all the angles in an equilateral triangle are 60°, and two of the angles
in an isosceles triangle are equal.
19
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Investigation 4-1 is a version of a proof of the Triangle Sum Theorem. One approach to this investigation
is to demonstrate for the students. You can do the activity on the overhead and have students discover the
sum of all the angles. You may need to remind students that a straight angle is 180°. You could also have
the students perform this investigation in pairs. After completing this investigation, go over the traditional
proof in the text. Ask students how the traditional proof is similar to the investigation. This will make
the traditional proof easier to understand.
Guide students through Example 5 before showing them the answer. Once all the exterior angles are
found, ask students to find their sum. This will lead into the Exterior Angle Sum Theorem. Students
might need a little clarification with this theorem. Explain that each set of exterior angles add up to 360°.
This theorem will be addressed again in Chapter 6.
The Exterior Angle Theorem can be hard for students to remember. Present it like a shortcut. If students
forget the shortcut, they can still use the Triangle Sum Theorem and the Linear Pair Postulate. See
Examples 7 and 8.
Congruent Figures
Goal
The goal of this lesson is to prepare students for the five triangle congruency theorems and the definition
of congruent triangles.
Notation Note
Revisit congruence notation from earlier lessons. This is the first time students will apply congruence to
a shape. Remind them that figures are congruent and measurements are equal. So, two triangles can be
congruent and the measurements of their sides would be equal. Stress the importance of labeling each
congruency statement such that the congruent vertices match.
Stress the tic mark notation in relation to the congruency statement. Simply because the letters used
are in alphabetical order does not necessarily mean they will line up this way in a congruency statement.
Students must follow the tic marks around the figure when writing congruency statements.
Teaching Strategies
When writing congruence statements, have students put the first triangle's vertices in alphabetical order.
Then, match up the second triangle's vertices so that the congruent angles are lined up. Remember that
it is very common to use letters in alphabetical order, however they might not always line up so that the
congruent triangles vertices will be in alphabetical order. For example, AABC might not be congruent to
ADEF, but it could be AABC = AFDE.
Rather than needing to know all three pairs of angles and sides are congruent, the Third Angle Theorem
eliminates one set of angles. Now, students need to know that two sets of angles and three sets of sides
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are congruent to show that two triangles are congruent. Ask students if they think there are any other
shortcuts to finding out if two triangles are congruent. Can they show that two triangles are congruent
using 4 pieces of information? 3 pieces? This could be a discussion for the end of the lesson and lead into
the next.
Prove Move
In this lesson, we introduce CPCTC (corresponding parts of congruent triangles are congruent). Even
though this is not a theorem, it will be used in proving that parts of triangles are congruent. CPCTC can
only be used after two triangles are stated and proven congruent in a proof.
The Reflexive Property of Congruence is commonly used in proofs to say that a shared side or angle is
congruent to itself. We will discuss this more in the next section.
Triangle Congruence using SSS and SAS
Goal
This lesson introduces students to the formal concept of triangle congruency through the SSS and SAS
Congruence Theorems.
Teaching Strategies
When introducing SSS Congruence Postulate let students do Investigation 4-2 individually. Walk through
the classroom and assist students with the steps. Once they reach Step 5, ask if they can make another
triangle with these three measurements. Every student should have a 3-4-5 right triangle and have them
show each other their constructions. Have students rotate and flip their triangles, but demonstrate that
they still have the same shape.
You can also use the Distance Formula to show that two triangles are congruent using SSS (Examples 5
and 6). In Example 5 the two triangles are congruent. Show students that they are in different places,
flipped and rotated. Put this example on a transparency and cut out AABC. Then, place it over ADEF
so that they are lined up. This also shows that the two triangles are congruent.
To introduce the SAS Congruence Theorem, you can either let students do Investigation 4-3 individually
or in pairs. Like with the previous investigation, as students to compare their triangles to the triangles
drawn by other students in the class. Again, they will see that all the triangles have the same shape and
are congruent.
The concept of an included angle can be confusing for some students. Draw triangles to show students the
difference. See picture.
Reinforce that the angle must be between the two sides to be a valid congruence theorem. The way the
letters are written, SAS, also should remind students that the angle is between the two sides. SSA (or
ASS) implies that the angle is not between the two sides.
Angle
Angle
Side Side
SAS SSA
Students may ask if SSA is a valid congruence theorem; it is not. There is an explanation of this in the next
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section. Also, students will realize that SSA and ASS are the same thing. You can address this however
you seem fit. Some teachers approach it straight on while others may choose to avoid it and refer to this
combination as SSA only.
Prove Move
When using SSS and SAS in a proof, students must present each piece as a step. For SSS, there needs to
be three steps, one for each set of congruent sides. For SAS, there needs to be two steps for the two sets
of congruent sides and one step for the included angles. Then, students can list the congruence statement
and reason.
The Reflexive Property of Congruence can be used in triangle proofs. If two triangles share a side or
an angle, the Reflexive Property is the reason this piece is congruent to itself. Students might feel as
though this is an unnecessary step, but just remind them that they must right all three sets of congruent
sides/angles in order to state that two triangles are congruent.
Triangle Congruence using ASA, AAS, and HL
Goal
Students will learn the ASA, A AS and HL Congruence Theorems and how to complete proofs using all
five of the congruence theorems.
Teaching Strategies
Investigation 4-4 should be done individually and then students can compare their triangles with the
students around them. Like with Investigations 4-2 and 4-3, students should realize that no other triangle
can be drawn. Rotation and reflection do not change the shape of the triangle (Chapter 12).
ASA and AAS can be hard for students to distinguish between. Draw the two theorems side by side and
compare. See picture.
Here we introduce the concept of an included side. The definition of an included side is very similar to
that of an included angle. Ask students to compare the differences and similarities.
.Angle *Angle
Side.
included
side
Angle H Angle
ASA AAS
The proof of the AAS Congruence Theorem may help students better understand the difference between it
and ASA. Explain that because of the Third Angle Theorem, AAS is also a congruence theorem. Example
3 shows which sides or angles are needed to show that the same two triangles are congruent using SAS,
ASA, and AAS. Go over this example thoroughly with students.
Hypotenuse-Leg (HL) is the only congruence theorem that is triangle-specific. This theorem can only be
used with right triangles. So, in order to use this congruence theorem in a proof, the student must know
that the two triangles are right triangles (or be able to show it in the proof).
AAA and SSA are introduced at the end of the section as the other possible side-angle relationships that
we have yet to explore. Neither of these relationships can prove that two triangles are congruent, but it is
useful to show students why they do not work. AAA shows that two triangles are similar, as in Chapter 7
and SSA can actually create two different triangles.
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Have students copy the Recap Chart into their notes. This chart will be a very helpful study guide for the
chapter test.
Example 7 is the only example that touches on CPCTC, even though there are proofs that use it in the
homework. Explain to students that they can only use CPCTC after they have proven two triangles are
congruent.
This is a very challenging lesson for students. If you feel as though not everyone is grasping the concept
of proofs or all the different triangle congruence theorems, slow down and go back over this lesson.
Additional Example: Put the reasons to the proof below in the correct order.
Given: EF || HG, EF = HG
Prove: EH II FG
Statement
Reason
1. EF || HG
A. CPCTC
2. ZFHG=ZEFH
3. EF = HG
4. HF = HF
5. AEFH^AGHF
B> Given
C. Given
P. Alternate Interior Angles Theorem
6. ZEHF=ZFGH
7. EH || FG
E Converse of the Alternate Interior
Angles Theorem
F. SAS Congruence Theorem
Q. Reflexive Property of Congruence
Solution: The correct order is B, D, C, G, F, A, E
Isosceles and Equilateral Triangles
Goal
This lesson illustrates the special properties that arise from isosceles and equilateral triangles.
Teaching Strategies
Investigation 4-5 can be done individually or teacher-led. As a teacher-led activity, this investigation
should be done as a group discovery. You should ask questions of the students to keep them engaged while
you are performing the construction (on the overhead or whiteboard). Then use a protractor to measure
the angles. Or, you could have a student come up and measure the angles. Ask them to generalize this
construction into the Base Angles Theorem.
This investigation also leads into the Isosceles Triangle Theorem. Have students duplicate ADEF (just
before Example 1) in their notes. They should write down all markings and all corresponding congruence
statements (for angles and sides) and any perpendicular statements. Stress that this theorem is only true
at the vertex angle.
Investigation 4-6 can also be done individually or teacher-led. This investigation allows students to come
to their own conclusion about equilateral triangles. They should discover that an equilateral triangle is
also an equiangular triangle in Step 4.
Additional Example: Algebra Connection Solve for x and y.
23
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(6y -7) e
(4x+2) c
Solution: It does not matter which variable you solve for first.
(6v - 7)° = 65°
6j = 72°
y = 12°
(4x + 2) o + 65° + 65 o = 180°
4x + 132° = 180°
4x = 48°
x=12°
1.5 Relationships with Triangles
Pacing
Table 1.7:
Day 1
Day 2
Day 3
Day 4
Day 5
Midsegments
Start Perpendicu-
Finish Perpendic-
Quiz 1
Finish Medians
lar Bisectors and
ular Bisectors and
Start Medians and
and Altitudes in
Angle Bisectors in
Angle Bisectors in
Altitudes in Trian-
Triangles
Triangles
Triangles
gles
Investigation 5-6
Investigation 5-1
Investigation 5-3
Investigation 5-5
Investigation 5-2
Investigation 5-4
Day 6
Day 7
Day 8
Day 9
Day 10
Inequalities in
Tri-
Quiz 2
* Finish Extension:
*Quiz 3
Review Chapter 5
angles
* Start Extension:
Indirect Proof
Indirect Proof
Day 11
Day 12
Day 13
Review Chapter 5
Chapter 5 Test
Finish Chapter 5
Test (if needed)
Start Chapter 6
Midsegments
Goal
This lesson introduces students to midsegments and the properties they hold.
Vocabulary .
This lesson begins a chapter that is full of vocabulary and new types of line segments. As a new line
segment is learned, have students write each one with its definition and a picture in a self-made table. By
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24
the end of this chapter, students should have: midsegment , perpendicular bisector, angle bisector, median,
and altitude. You can also have students draw these line segments in acute, right and obtuse triangles.
Make sure to include the appropriate labeling and congruence statements for each line segment within each
triangle as well.
Notation Note
Review with students the difference between a line segment, NM and its distance NM. These notations
will be used frequently in this chapter.
Teaching Strategies
Stress the properties of midsegments to students and make sure they understand the definition of a mid-
segment before moving on to the next section. Each segment in a triangle is very similar, so students tend
to get them mixed up. A midsegment is unique because it connects two midpoints.
Examples 3-5 investigate the properties of a midsegment in the coordinate plane. Give students these
examples without the solutions and have them work in pairs to arrive at the Midsegment Theorem on their
own. At the completion of Example 5, ask students if they notice any similarities between the slopes of
NM and QO and the lengths of NM and QO. Explain that their findings are the Midpoint Theorem.
Discuss that a midsegment is both parallel and half the length of the third side. Stress to students that
if a line is parallel to a side in a triangle that does not make it a midsegment. The parallel line must also
connect the midpoints, pass through the midpoints, or cut the sides it passes through in half. Go over all
of these different ways to state what a midpoint and midsegment are.
If you have access to an LCD display screen (in the classroom) or a computer lab, use the website http:
//www.mathopenref . com/trianglemidsegment .html (in the FlexBook) to play with midsegments within
a triangle. It is a great resource to help students to better understand the Midsegment Theorem.
Additional Example: B,D,F, and H are midpoints of AACG and ACGE. BH = 10 Find CG and DF.
Solution: BH is the midsegment of AACG that is parallel to CG. DF is the midsegment of ACGE that
is parallel to CG. Therefore, BH = DF = 10 and CG = 2 • 10 = 20.
Perpendicular Bisectors and Angle Bisectors in Triangles
Goal
Students will apply perpendicular bisectors and angle bisectors to triangles and investigate their properties.
Teaching Strategies
Review the constructions of a perpendicular bisector and angle bisector (Review Queue #1 and #2). When
going over #3a ask students if the line that bisects the line segment is a perpendicular bisector (it is not, it
is just a segment bisector). Explain that the markings must look like the ones in #4 to be a perpendicular
bisector.
25 www.ckl2.org
Investigation 5-1 guides students through the properties of a perpendicular bisector before placing it in a
triangle. Show students that AACB is an isosceles triangle (in the description of the Perpendicular Bisector
Theorem) which reinforces the fact that C is equidistant from the endpoints of AB. Explain the difference
between the Perpendicular Bisector Theorem and its converse. You can also have the students put the
theorems into a biconditional statement.
Investigation 5-2 places the perpendicular bisectors in a triangle. This activity should be done individually,
while you show students what to do. You will need to circle around the classroom to make sure students
understand step 2. Then, students should be able to do step 3 on their own. Do step 4 as a class to make
sure that every student understands that the circle drawn will pass through every vertex of the triangle.
Here, two new words are introduced: circumscribe and inscribe. If students have hard time remembering
their definitions use their Latin roots. Circum = around and In = inside or interior. Scribe = draw or
write.
The angle bisectors are also first introduced with one angle. Investigation 5-3 explores the property of
one angle bisector and its relationship to the sides of the angle. This activity should be teacher-led while
students are encouraged to follow along. In step 2, the folded line does not have to be a perpendicular
bisector, but just a perpendicular line through D.
After Investigation 5-3 compare the properties of an angle bisector and a perpendicular bisector. Draw
them next to each other with markings and draw a Venn diagram with the similarities and differences. If
this lesson takes more than one day, this could be a good warm-up.
A
B
S
X
After analyzing the pictures, lead students towards the conclusion that B is equidistant from the endpoints
of AC (in picture 1) and B is equidistant from the sides of lADC. Then, to translate a perpendicular
bisector into a triangle, the point where they all intersect would be equidistant from the endpoints of all
the line segments, which are the vertices of the triangle. The point of intersection of the angle bisectors
would be equidistant from all the sides of the angles which are also the sides of the triangle.
Just like with Investigation 5-2, lead students through this activity. Make sure every student understands
how to fold the patty paper to create an angle bisector (step 2). Once they make all the angle bisectors,
do step 4 together to ensure that every student understands that the circle will pass through the sides of
the circle.
Students can get the properties of perpendicular bisectors and angle bisectors within triangles confused.
One way to help students remember which is which show them that the point of intersection of the perpen-
dicular bisectors can outside the triangle so the circle would go around the outside the circle (circumscribe).
The point of intersection of the angle bisectors is always inside the triangle so the circle will always be
inside the triangles (inscribed).
The points of concurrency of the perpendicular bisectors, angle bisectors and altitudes were intentionally
left out of the Basic Geometry FlexBook to avoid confusion and to encourage students to focus on the
theorems and properties of these lines. The Enrichment Teacher's Edition FlexBook discusses the names
of these points of concurrency if you would like to include them in your curriculum.
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Medians and Altitudes in Triangles
Goal
Students will be introduced to medians and their point of intersection, the centroid. They will explore the
properties of a centroid as well as learn how to construct an altitude.
Teaching Strategies
The median is now the third segment that passes through at least one midpoint. Make sure students
understand the difference between a median, midsegment and a perpendicular bisector. Also, students
may get the angle bisector confused with a median because sometimes it "looks like" (a fatal flaw in
geometry) the angle bisector will pass through the opposite side's midpoint. Students can never assume
from a picture that the angle bisector and a median are the same. Discuss the cases when they are the
same, this may alleviate some confusion. When a triangle is an isosceles triangle, the line segments are all
the same when drawn from the vertex. Also when a triangle is equilateral, the line segments are all the
same regardless of which vertex they are drawn from. The following picture might better illustrate this
point:
Points F and D are the midpoints of the sides they are on.
FD is a midsegment
ED is a perpendicular bisector
BG is an angle bisector
BD is a median
BH is an altitude
With Investigation 5-6, encourage students to construct more than one altitude on the given obtuse triangle.
While we did not explore the point of intersection for the altitudes, there is one. Have students arrive at this
conclusion on their own, while constructing the other altitudes in AABC. While students are performing
the constructions, encourage them to turn their paper around so that the side they are making the altitude
perpendicular to is horizontal. This will make the process easier.
At the end of this lesson, compare all of the line segments within triangles. Use the table below. The
answers are filled in, but draw on the board without answers and generate the answers as a class. You can
also use the picture above to help students determine the properties of the line segments.
Table 1.8:
Pass through
the mid-
point (s)?
Pass through a
vertex?
Perpendicular?
Properties
the point
intersection
of
of
Midsegment
Yes, two.
No
No
No point of inter-
section
27
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Table 1.8: (continued)
Pass through
Pass through a
Perpendicular?
Properties
of
the mid-
vertex?
the point
of
point (s)?
intersection
Angle Bisector No
Perpendicular
Bisector
Median
Yes
Yes
Yes
No
Yes
No
Yes
No
Altitude
No
Yes
Yes
Equidistant from
the sides of a trian-
gle; inscribed cir-
cle.
Equidistant from
the vertices of a
triangle; circum-
scribed circle.
The centroid; the
"center of gravity"
of a triangle. Also
it splits the me-
dians into | - |
pieces.
None
Check and recheck that students understand these five line segments before moving on. It is very common
for students to get the definitions and properties confused.
Inequalities in Triangles
Goal
The purpose of this lesson is to familiarize students with the angle inequality theorems and the Triangle
Inequality Theorem and the SAS and SSS Inequality Theorems.
Teaching Strategies
Students have probably figured out the Triangle Inequality Theorem but not actually put it into words.
Ask the class if they can make a triangle out of the lengths 3 in, 5 in, and 9 in. You can give each student
a few pieces of dry spaghetti and have them break the pieces so that they are the lengths above and then
attempt to make a physical model. They will discover that it is impossible. Then, tell the class to break
off 1 in of the 9 in piece and try again. Again, this will not work. Finally, tell them to break off another
^-inch and try a third time. This time it will work. Analyze each set of numbers. You could also have
students do this a fourth time and make the longest piece either 6 or 7 inches. They will still be able to
make a triangle.
3, 5, 9 — > No triangle
3, 5, 8 — > No triangle
3, 5, 7.5 -> Yes!
3, 5, 6 -> Yes!
Guide students towards the Triangle Inequality Theorem. Example 4 explores the possible range of the
third side, given two sides. Explain to students that this third side can be the shortest side, the longest
side or somewhere in-between. We have no idea, so we have to propose a range of lengths that the third
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28
side could be. Have students shout out possible lengths of the third side and place them in a table. Then,
show them the way to write the lengths as a compound inequality.
Example 4 leads students into the SAS Inequality Theorem, which compares two triangles where two sides
are the same length and the included angles are different measurements. We know, from Chapter 4, that
if the included angles are congruent, then the triangles would be congruent, but in this case, we know that
one is bigger than the other. Logically, it follows that the triangle with the bigger included angle will have
the longer opposite side. This is a very wordy theorem; it might help to explain using the symbols and
picture in the text. This theorem is also called the Hinge Theorem.
The SSS Inequality Theorem is the converse of the SAS Inequality Theorem. Now we know that two sides
are congruent and the third sides are not. It follows that the angle opposite the longer side is going to be
larger than the same angle in the other triangle. Example 6 is a good example of how this theorem works.
You can also reverse the question and ask: li mil > m/2, what can we say about XY and XZ1
Extension: Indirect Proof
Goal
Students should be able to understand how an indirect proof is organized and executed.
Teaching Strategy
An indirect proof is a powerful reasoning tool that students might find useful outside of mathematics. Ask
students what professions they think would use indirect proofs (also called proof by contradiction). Ex-
amples are lawyers (disproving innocence/guilt), doctors (disproving diagnosis), crime scene investigators
(collecting evidence and trying to prove or disprove).
Additional Example: Prove Vl5 ^ 4.
Solution: Assume Vl5 = 4
Squaring both sides, we get 15 = 16.
But 15 gfc 16, therefore, Vl5 * 4.
1.6 Polygons and Quadrilaterals
Pacing
Table 1.9:
Day 1
Day 2
Day 3
Day 4
Day 5
Angles in Polygons Properties of Par- Quiz 1 Finish Proving Rectangles, Rhom-
Investigation 6-1 allelograms Start Proving Quadrilaterals are buses, and Squares
Investigation 6-2 Investigation 6-3 Quadrilaterals are Parallelograms Investigation 6-4
Parallelograms Investigation 6-5
Day 6 Day 7
Quiz 2 Finish Trapezoids
Start Trapezoids and Kites
and Kites Investigation 6-6
Day 8 Day 9 Day 10
Quiz 3 Finish Review of Chapter 6 Test
Start Review of Chapter 6
Chapter 6
29
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Angles in Polygons
Goal
Students will use the Triangle Sum Theorem to derive the Polygonal Sum Theorem by dividing a convex
polygon into triangles. Students will also be reintroduced to the Exterior Angle Sum Theorem, but now it
will be applied to any polygon.
Teaching Strategies
Using the Know What? at the beginning of this lesson, discuss where someone might see polygons in
nature and the real world. Determine if any of these polygons are regular polygons or not. Students might
need a review of the definition of a regular polygon.
Investigation 6-1 is intended to be a student-driven activity while the teacher monitors and leads or answers
questions. Students should know what a quadrilateral, pentagon, and hexagon are from Chapter 1; however
they may need a little review. Diagonals were also addressed in Chapter 1. Make sure that students only
draw the diagonals in step 2 from one vertex so that none of the triangles overlap. In step 3 it might
be helpful to write out the last column as a list, including the triangle; 180°, 360°, 540°, 720°. Then, ask
students what the next number in the pattern should be. Continue this for a few more terms and then
generalize into the Polygon Sum Formula.
Students might think that the Polygon Sum Formula and the Equiangular Polygon Formula are two
different formulas for them to memorize. This is not the case. Tell students that they need to memorize
the Polygon Sum Formula and then the Equiangular Polygon Formula simply divides the Polygon Sum
Formula by the number of angles in the polygon. Stress to students that the Equiangular Polygon Formula
can be used on equiangular polygons as well as regular polygons.
To introduce exterior angles for polygons, draw a triangle with its exterior angles. Students should remem-
ber that each set of exterior angles of a triangle add up to 360°. Then, show students the exterior angles
for a square. Each exterior angle is 90°, so their sum would be 360° as well. Now, have students complete
Investigation 6-2 in pairs.
The first question in the review questions is a table with angle sums and individual angles in a regular
n-gon. Complete this table at the end of the lesson so that students see the relationship between all the
angles and their sums. If you would like, add a final column labeled "Each exterior angle in a regular
n-gon." Students can either use linear pairs with column 4 or divide 360° by the number of sides.
Properties of Parallelograms
Goal
The purpose of this lesson is to familiarize students with properties special to parallelograms.
Notation Note
The notation for any quadrilateral or is the list of vertices, usually clockwise, such as ABCD (quadrilateral)
or LMNOP (pentagon). Students can start at any vertex they would like. The only requirement is that
the vertices are listed such that they are next to each other in the picture. Quadrilateral ABCD could be:
tj' rr rr ty
D l f c c* 'b eT 'a fir 'd
Notice that the first vertex listed and the last vertex listed are next to each other.
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Teaching Strategies
Investigation 6-3 enables students to discover the properties of parallelograms on their own. Encourage
students to label the vertices of the parallelogram (ABCD, for example) and then they can write equal-
ity/congruence statements for the angles and sides. Once they have all the equality/congruence statements
written (step 3), have students write down all their conclusions and see if they can generate the parallelo-
gram theorems on their own.
Students might wonder if there is a difference between consecutive angles and same side interior angles.
Discuss this with your students. One could argue they are the same. Another could argue they are different
because consecutive refers to two angles that are next to each other in a polygon. Same side interior angles
refer to two angles that are formed by parallel lines and one transversal.
Encourage students to make as many connections as possible. For example, students have learned parallel
lines are equidistant from each other. Make this connection to a parallelogram. If students drawn in the
diagonals of a parallelogram; review that alternate interior angles are congruent.
In Example 3, we place the parallelogram in a coordinate plane. In this chapter we will show that certain
quadrilaterals are parallelograms (and rhombuses, squares, and rectangles) when they are in the coordinate
plane. In this section we introduce one method. Because the diagonals of a parallelogram bisect each other,
their point of intersection should be the midpoint of each. Therefore, the midpoint of each diagonal will
be the same point.
Proving Quadrilaterals are Parallelograms
Goal
Students will use triangle congruence postulates and theorems to prove quadrilaterals are parallelograms.
Students will also determine if a quadrilateral is a parallelogram when placed in the coordinate plane.
Teaching Strategies
There are two main ways to prove that a quadrilateral is a parallelogram: formal proof and using the coor-
dinate plane. If students are given a formal proof, they must prove that the two halves of a parallelogram
(split by one of the diagonals) are congruent and then use of the converses used in this lesson. They will
also have to use CPCTC somehow.
If students are doing a problem with a quadrilateral in the coordinate plane, then they must use the distance
formula, slope formula, or midpoint formula. The distance formula would lend itself to the Opposite
Sides Theorem Converse, and finding the slopes of all four sides is the definition of a parallelogram and
the midpoint formula is the Parallelogram Diagonals Theorem Converse. Discuss all these options with
students before allowing them to start class work or homework.
As a way to introduce the proof of the Opposite Sides Theorem Converse and the proof of Theorem 6-10
(Example 1), you can copy these theorems and then cut up the statements and reasons and put them in an
envelope. Given an envelope to either pairs or groups of students and have them match up the statements
with the corresponding reasons and put them in the correct order. On one side of the envelope, put the
Given and Prove as well as the picture. This technique can be done for any proof.
Rectangles, Rhombuses and Squares
Goal
This lesson introduces rectangles, rhombuses, and squares. These are more specific types of parallelograms.
Teaching Strategies
31 www.ckl2.org
Make sure students understand that everything that falls within a rhombus possess the same characteristics
and properties of a parallelogram. A rectangle also has all the properties of a parallelogram as well as its
properties. A square has all the properties of a rhombus, rectangle and parallelogram. Squares do not
have any of its own unique properties.
Investigation 6-4 shows us that the diagonals of a rectangle are congruent. So, if the diagonals of a
parallelogram are congruent, then the parallelogram is a rectangle. Investigation 6-5 explores the properties
of the diagonals of a rhombus. Here, the diagonals are perpendicular and they bisect each angle in the
rhombus. Note that students do not need to show both to be a rhombus. However, if they decide to use
Theorem 6-16, they do need to show that the diagonals bisect all four angles of the rhombus. Therefore,
it is much easier for students to use Theorem 6-15 (showing that the diagonals are perpendicular) to show
a parallelogram is a rhombus.
Because there are no theorems regarding squares, make sure you go over Example 4 thoroughly. This is
the only place we discuss all the properties of squares. Stress that a square holds all the properties of a
parallelogram, rectangle, and rhombus. Ask students if they think it has any of its own properties. A
square is a specific version of a rhombus, so the angles are bisected. Because a square is also a rectangle,
the angles are all 90°, so the bisected angles are all 45°. In addition to the properties listed in Example 4,
show students other ways to write some of these properties. For example, a rectangle has four congruent
angles. Students can also write this as the sides are all perpendicular to each other.
At this point, students might have the different types of parallelograms confused. There are a lot of
different properties and students might have them all mixed up in their heads. One way to help is to
draw a hierarchy diagram with QUADRILATERALS at the top and then two arrows. One points to
PARALLELOGRAMS and the other points to OTHER TYPES. Then, ask students if they can determine
how rectangles, squares, and rhombuses fit into this diagram. The final diagram should look like:
Quadrilaterals
Parallelograms Other Types
Rectangles Rhombuses
Squares
Students can add to this diagram in the next lesson.
The steps after Example 5 are very helpful for students to determine if a parallelogram is a rectangle,
rhombus or square. Here is an additional example, using these steps.
Additional Example: The vertices of WXYZ are below. Determine the type of quadrilateral it is.
Solution: Let's follow the steps.
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>s
tA
1
HI
i
I
J
1
t
5
\
LJ
J
1
\
V „
1
1
i
){
tf
\
■M
-1,
M
I
i ;
1
4
e
s
1?
4^
j
f ^
f i
*
I
-
Y
40-
u
1. The quadrilateral is graphed.
2. Do the diagonals bisect each other?
+ 2 3-7
) = (-3,-2)
midpointwY —
\ Zj Zj
/0-6 l-5\ . „ „.
midpointxz = I ~ o - ' ~ o - ) = *- -3 ' ~ 2 )
The diagonals bisect each other. This is a parallelogram.
3. Are the diagonals equal?
WY
XZ
= A /(-8-2) 2 + (3-(-7)) 2 = VlO 2 + 10 2 = V200
= A /(0-(-6)) 2 + (l-(-5)) 2 = V6 2 + 6 2 = V72
The diagonals are not equal. This is not a rectangle.
4. At this step, we know this figure is a rhombus. It cannot be a square because the diagonals are not
equal, from step 3. So, to prove that it is a rhombus, we can either show that all the sides are equal or
that the diagonals are perpendicular. It is easier to find the slopes of the diagonals than do the distance
formula four times.
m WY
3 - (-7)
-8-2
10
^10
-1
mxz
l-(-5) = 6
- (-6) 6
1
The diagonals are perpendicular, so this reaffirms our earlier conclusion that WXYZ is a rhombus.
Trapezoids and Kites
Goal
This lesson introduces students to the special properties of kites, trapezoids, and isosceles trapezoids.
Teaching Strategies
33
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After going over the definition of a trapezoid, discuss the difference between a trapezoid and a parallelogram.
Students need to realize that a trapezoid has exactly one pair of parallel sides. It is not like the definition
of an isosceles triangle (at least two congruent angles). Therefore, a parallelogram is not a trapezoid.
For a trapezoid to be isosceles, the non-parallel sides must be congruent. Describe an isosceles trapezoid as
cutting off the top of an isosceles triangle. Therefore, Theorem 6-17 is a form of the Base Angles Theorem
for isosceles triangles. Isosceles trapezoids also have congruent diagonals, like a rectangle. Ask students
if isosceles trapezoids are rectangles. This is a great example of two different quadrilaterals that have the
same property. Of course, rectangles are not isosceles trapezoids because rectangles have four congruent
angles and two sets of parallel sides.
Just like triangles, a trapezoid also has a midsegment. Trapezoids only have one midsegment because it
connects the non-parallel sides. For this reason, the midsegment is also parallel with the parallel sides.
Stress that the length of the midsegment is the average of the lengths of the parallel sides. You could also
say that the midsegment is halfway between the parallel sides, so its length is halfway between the lengths
of the parallel sides.
Kites are very similar to rhombuses, but a rhombus is not a kite. The definition of a kite says "a quadri-
lateral with two sets of adjacent congruent sides." All sides are congruent in a rhombus. Because a kite
has two sets of congruent adjacent sides, it has some properties of rhombuses. Go over the similarities and
differences between kites and rhombuses. At the end of this lesson, complete the hierarchy diagram that
was started in the previous lesson.
Quadrilaterals
Parallelograms
Other Types
Rectangles Rhombuses Trapezoids Kites
isosceles
Squares
Review Game
Trapezoids
Have your students create flashcards with quadrilateral names on one side and important information or
properties on the reverse. Have various types of quadrilaterals, both abstract and real world, ready to show
students. Once students believe they have classified the quadrilateral, they are to hold up the appropriate
name. Have students do this in pairs and keep score. You can give the winners bonus points on the test,
candy, or a homework pass.
1.7 Similarity
Pacing
Table 1.10:
Day 1
Day 2
Day 3
Day 4
Day 5
Ratios & Propor- Similar Polygons
tions
Quiz 1 Finish Similarity Similarity by SSS
Start Similarity by by AA and SAS
A A Investigation 7-1 Investigation 7-2
Day 6
Day 7
Day 8
Day 9
Day 10
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34
Table 1.10: (continued)
Day 1
Day 2
Day 3
Day 4
Day 5
Finish Similarity
by SSS and SAS
Investigation 7-3
Quiz 2
Start Proportion-
ality Relationships
Finish Proportion-
ality Relationships
Similarity
formations
Trans-
Quiz 3
* Extension:
Similarity
Self-
Day 11
^Finish Extension:
Self- Similarity
Start Review of
Chapter 7
Day 12
* Extension Quiz
More Review of
Chapter 7
Day 13
Finish Review
Chapter 7
of
Day 14
Chapter 7 Test
Day 15
Start Chapter 8
^Without the Extension lesson, this chapter will take 12-13 days.
Ratios and Proportions
Goal
The purpose of this lesson is to reinforce the algebraic concept of ratios and proportions. Proportions are
necessary when discussing similarity of geometric objects.
Teaching Strategies
The Know What? for this lesson is a little different than the ones in in previous lessons. Have students take
this activity one step further and apply the concept of a scale drawing to their own room. Students can
draw scale representations of their rooms (or even their whole house) and the furniture for an alternative
assessment or a chapter project.
Ratios can be written three different ways. This text primarily uses fractions or the colon notation. Remind
students that a ratio has no units and can be reduced just like a fraction. With rates, to contrast to with
ratios, there are units. Rates are also fractions. Have students compare the similarities and differences
between ratios and rates. Ask them to brainstorm types of ratios and rates.
Proportions are two ratios that are set equal to each other. The most common way to solve a proportion is
cross-multiplication. Show students the proof of the Cross-Multiplication Theorem in the FlexBook. This
proof makes the denominators the same, so that the numerators can be set equal to each other. Students
can go through this whole process each time or they can use cross-multiplication, which can be considered
a shortcut.
All of the corollaries in this lesson are considered the same as |
2 because when cross-multiplied you
would end up with ad = be. Therefore, the placement of the a,b,c, and d is important. When in doubt
about if two proportions are the same, always have students cross-multiply.
Corollaries 7-4 and 7-5 are a little different from the previous three. To show that these proportions are
true, it might be helpful for students to see the proofs. Each corollary is proven true when the last step is
ad = be because this is the same as the cross-multiplication of | = |.
Proof of Corollary 7-4
35
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Proof of Corollary 7-5
a + b c + d
b = d
d(a + b) = b(c + d)
ad + bd = be + bd
ad = be
a - b c - d
b d
d(a - b) = fc(c - d)
ad - bd = be - bd
ad = be
Brainstorm other possible reconfigurations of | = ^ with students to see if there are any other possible
"corollaries." Have students cross-multiply their "true" proportions to check to see if they are valid.
Similar Polygons
Goal
This lesson connects the properties of proportions to similar polygons. An introduction to scale factors is
also presented within this lesson.
Teaching Strategies
To see if students understand the definition of similar polygons, have students come up to the front board
and draw pictures of two similar polygons or two non-similar polygons. You could also do this several
times, before you give students the formal definition, and then students can generate one as a class.
After doing Examples 1 and 2, brainstorm with the class as to which specific types of triangles, quadrilater-
als or polygons are always similar. For example, the text cites equilateral triangles and squares. This leads
to the fact that all regular polygons are similar. Explore why just equilateral (rhombuses) or equiangular
(rectangles) polygons are not always similar.
Students might wonder which value goes on top when finding the scale factor. Remind students that
the scale factor is a ratio. In Example 4, it says U AABC to AXYZT This is a ratio too. So, the side of
AABC should go in the numerator and the sides of AXYZ should go in the denominator. The triangle or
figure listed first in the written-out ratio goes in the numerator and the figure listed second goes in the
denominator.
In Example 5 we show that the scale factor could be | or |. It is useful for students to know that either
can technically be the scale factor. The important thing is that they know when to use which fraction.
Ask students which fraction is larger. Then explain that to find the sides of the larger rectangle we would
multiply the sides of the smaller triangle by the larger ratio. To find the sides of the smaller rectangle, we
would multiply the sides of the larger rectangle by the smaller ratio.
Similarity by AA
Goal
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The purpose of this lesson is to enable students to see the relationship between triangle similarity and
proportions. Here we discuss the A A Similarity Postulate and show how it can prove that two triangles
are similar.
Teaching Strategies
Investigation 7-1 is designed to be teacher-led while the student also does the activity and follows along.
Students can work individually or in pairs. One option is to have half the students make the triangle with
a 3 inch side, like in step 1, and the other half make a triangle with a 4 inch side, like in step 3. Then,
have each pair of students compare their two different triangles and complete step 4 together.
Students can find the corresponding sides in two similar triangles by identifying the congruent angles in
the two triangles. Then, the sides that are opposite the congruent angles are corresponding. Students can
also place the two largest sides together as corresponding, the shortest sides are corresponding and the
middle sides are corresponding. You must be careful with this option, however. Students can only use this
method when they know that the triangles are similar.
Another option for indirect measurement (other than Example 5) utilizes the Law of Reflection. It states
that the angle at which a ray of light (ray of incidence) approaches a mirror will be the same angle in
which the light bounces off (ray of reflection). This method is the basis of reflecting points in real world
applications such as billiards and miniature golf. See the additional example below.
Additional Example: You want to shoot the red ball into the corner pocket, as shown below. How far
must the cue ball travel in order to do this?
48 in.
Solution: The answer is the total amount traveled by the cue ball, which is 36 + x. First we need to find
x. Set up a proportion using similar triangles.
48 _ 32
x 36
32x = 1728
x = 54 in
The cue ball must travel 54 + 36 = 100 inches to sink the red ball.
Similarity by SSS and SAS
Goal
The purpose of this lesson is to extend the SSS and SAS Congruence Theorems to include similarity.
Teaching Strategies
After completing the Review Queue, introduce the lesson by asking students, "How can triangles be
congruent and similar simultaneously?" Have a discussion with students about the answer to this question,
including how we can prove that triangles are congruent or similar. This will lead into Investigation 7-2.
37 www.ckl2.org
Investigation 7-2 uses the construction from Chapter 4 for SSS Congruence. Here, students will construct
two similar 3-4-5 triangles and then determine if they actually are similar. Have students work in pairs for
this activity and circulate to answer questions. Students might need a review of Investigation 4-2 before
beginning this investigation. You can show students the provided link in the FlexBook as a review of this
construction.
One could say that that SSS Congruence Theorem is a more specific version of the SSS Similarity Theorem.
Discuss this point with your students.
As with the previous lesson, make sure students understand which sides are corresponding. If students do
not match up corresponding sides, they may get the wrong answer to a homework problem or on a test.
Review the points discussed in the previous lesson about how to match up corresponding sides.
Like with Investigation 7-2, you can split the class in half and have one half draw the triangle in step 1
and the other half draw the triangle in step 2 for Investigation 7-3. Then, have students from each half,
pair up with someone from the other half and they can do steps 3 and 4 together.
Proportionality Relationships
Goal
In this lesson, students will learn about proportionality relationships when two parallel lines are split by
two transversals.
Teaching Strategies
Example 1 can be done as a mini-investigation before Investigation 7-4. Discuss the properties of a
midsegment and how it splits the two sides that it intersects. Find the ratio of the split sides (1:1) and the
ratio of the similar triangles (1:2). Point out that these two ratios are always different.
Investigation 7-4 should be teacher-led and student can follow along, writing down any important infor-
mation in their notes. They can sketch your drawings and write down your measurements from steps 3
and 6.
To make the proof of Triangle Proportionality Theorem easier to understand, you can utilize the technique
presented earlier in this text where you would copy the entire proof, cut out the statements and reasons
(separately) and place it in an envelope. Then, students (in pairs or groups) can put the proof in the
correct order.
Before introducing Theorem 7-7, show students an example, like Example 4. See if they can figure out the
answer. In actuality, Example 4 really is not any different from the examples within triangles, just that
the transversals do not intersect. Rather than being sides of triangles, now these lines are the transversals
passing through parallel lines. Show students several different orientations (like Example 5), so they not
confused by the homework problems.
Proportions with Angle Bisectors can be a little tricky for some students. Tell them to set up the proportion
like the picture below:
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whole side it is
adjacent to
segment of split
side OVER
segment of split
side OVER
whole side it is
adjacent to
Notice that this set-up is different from the proportion given in Theorem 7-8. Using the letters from the
theorem, the proportion would be ^ = ^, which is corollary 7-1.
Encourage students to use this set-up if they are having difficulties with the proportion given in the text.
Similarity Transformations
Goal
Dilations produce similar figures. This lesson introduces the algorithm to produce similar figures using
measurements and a scale factor.
Teaching Strategies
To help students with this lesson's Know What? show them pictures of different types of perspective
drawings. Here are three very different examples to discuss.
An easy way to remember enlargements versus reductions is a rhyme. Have your students repeat the
rhyme, "A reduction is a proper fraction." Improper fractions are mixed numbers, and greater than 1, thus
creating enlargements. Other texts might use other words for enlargements and reductions. Brainstorm
with students synonyms for these words. Possibilities are: stretch and shrink or expansion or contraction.
Dilations can also be clarified using a photograph. School pictures are great examples of dilations. Suppose
a typical photograph is 4 x 6. An 8 x 10 enlargement does not alter the appearance. This is also true for
shrinking photos for a 2 x 3. Using a base picture, bring in several enlargements and reductions to further
illustrate this concept.
In Examples 1 and 2, a point is dilated. Recall that a point has no dimension, so it cannot be enlarged
or reduced. Therefore, the distance from the center of the dilation is stretched and shrunk. Point out to
students that the original distance is 6 units and the distance from P to Q' is three times 6 units, or 18
units.
Anytime a figure is dilated, the distance from the center of the dilation is also stretched or shrunk according
to the scale factor. The dilated point will always be collinear with the original point as well. Even though
the terminology image and preimage (the original image) are not introduced until Chapter 12, feel free to
39
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introduce it now.
Extension: Self-Similarity
Goal
This lesson introduces students to popular fractals. Fractals possess self-similarity and maintain the prop-
erties of similarity.
Teaching Strategies
This optional lesson is a great mathematical connection to art and nature. Show students examples of
fractals in art and graphic design. See the examples below. The last picture is a type of broccoli called
romanesco broccoli. You can sometimes find it in specialty grocery stores.
Mathematician Benoit Mandelbrot derived the term "fractal" from the Latin word frangere, meaning to
fragment. A fractal is a geometric figure in which its branches are smaller versions of the "parent" figure.
Most fractals are explained using higher level mathematics; however, students can create their own fractal
patterns easily. The Mandelbrot set is illustrated in the middle picture above.
Additional Example: Follow these instructions to create a cauliflower fractal.
1. Hold your paper in landscape format.
2. Draw a horizontal segment AB, such that AB = 8 m, in the center of the paper.
3. Find and mark C, the midpoint of AB.
4. Find the midpoint from A to C. This is ^AB.
5. Use this quarter- length to be the length of DC. Draw DC such that it is perpendicular to AB at C.
6. Draw lines AD and BD forming AADB.
7. Repeat steps 3-6 for AD and BD
8. Continue to repeat this process for the legs of each new smaller triangle.
Final image:
The surface of this fractal (as you continue with smaller and smaller triangles) looks like the outside of a
head of cauliflower.
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* f k ■
i
1.8 Right Triangle Trigonometry
Pacing
Table 1.11:
Day 1
Day 2
Day 3
Day 4
Day 5
The Pythagorean
Theorem
Investigation 8-1
Day 6
Finish Special
Right Triangles
Investigation 8-3
The Pythagorean Quiz 1
Theorem Converse Start Similar
Right Triangles
Day 7
Quiz 2
Start Tangent,
Sine, and Cosine
Ratios
Day 8
Finish Tangent,
Sine, and Cosine
Ratios
Finish Similar
Right Triangles
Day 9
Solving Right Tri-
angles
Special Right Tri-
angles
Investigation 8-2
Day 10
Finish Solving
Right Triangles
Day 11
Quiz 3
Start Review
Chapter 8
of
Day 12
Review Chapter 8
Day 13
Chapter 8 Test
Start Chapter 9
Be sure to take your time with this chapter. Remember that the Pacing Guide is merely a suggestion.
Students can get caught up with vocabulary and theorems in this lesson.
The Pythagorean Theorem
Goal
This lesson introduces the Pythagorean Theorem. It has several applications which we will explore through-
out this chapter.
Relevant Review
Thoroughly review the Simplifying and Reducing Radicals subsection in this lesson. In this and subsequent
lessons, answers will be given in simplest radical form. Make sure students know how to add, subtract,
multiply, divide, and reduce radicals before moving on. Present anything under the radical like a variable.
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Students know they cannot add 2x + 5y. Therefore, they cannot combine 2 V3 + 5 y2. This should make
it easier for students to understand.
Teaching Strategies
Investigation 8-1 provides one proof of the Pythagorean Theorem. First, lead students through steps 1-3,
then have them finish step 4 individually while you circle around to answer questions. Once students are
done with this investigation, either take students to the computer lab or if you have an LCD screen, go to
the site in the FlexBook to see two additional proofs of the Pythagorean Theorem. Both of these proofs
are animated and provide another viewpoint.
There are several applications of the Pythagorean Theorem. Students need to be familiar with as many
as possible. Go over each example to make sure students understand the range of questions that can be
asked. Each of the examples, even though worded differently, are completed the same way. The directions
for a given problem can be one place where students can have confusion. Going over the different types of
directions for the same type of problem can be very helpful.
Finally the Distance Formula is proven in this lesson. Students might already have an idea of how it
works. Explain that it is a variation on the Pythagorean Theorem, especially the before we solve for d,
when the equation looks like d 2 = (xi - X2) 2 + (yi -V2) 2 . Notice in this section, the Distance Formula looks
different than it did when it was introduced in Chapter 3. Recall that order does not matter, as long as
the corresponding x or y value is first {x\ and y\ are first or X2 and V2 are first).
Converse of the Pythagorean Theorem
Goal
This lesson applies the converse of Pythagorean's Theorem to determine whether triangles are right, acute,
or obtuse.
Relevant Review
Make sure students are comfortable squaring and simplifying radical numbers.
Teaching Strategies
The Converse of the Pythagorean Theorem basically says if the sides of a triangle do not satisfy the
Pythagorean Theorem, then the triangle is not a right triangle. Theorems 8-3 and 8-4 extend this concept
to determine if the non-right triangle is acute or obtuse. To help students remember which is which, tell
them to think opposite. When a 2 + b 2 > c 2 (a 2 + b 2 is greater than c 2 ) the triangle is acute (all angles are
less than 90°). When a 2 + b 2 < c 2 (a 2 + b 2 is less than c 2 ) the triangle is obtuse (one angle is greater than
90°). Also, when in doubt, have them draw the triangle, as best they can, to scale. Then, they can see
what the triangle should be and they can do the Pythagorean Theorem to confirm or deny.
Remind students that the Triangle Inequality Theorem still holds. So, if they are given lengths like 5, 7,
and 15, they need to be able to recognize that these lengths do not make a triangle at all. 5 + 7 > 15, so no
triangle can be formed. If students do not see this, they will be doing unnecessary work, not to mention
think that the triangle is obtuse. Review the Triangle Inequality Theorem as you are going through the
examples in the text. No one likes to do extra work, if they do not have to.
Additional Example: Do the following lengths form a triangle? If so, is it acute, right, or obtuse?
a) 10, 15, 20
b) 7, 14, 21
c) 8 V2, 4 V6, 4 Vli
Solution: First, check all the lengths to see if they make a triangle, b) does not, 7 + 14 = 21, so those
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c)
lengths cannot make a triangle. Let's see what type of triangles a) and c) are.
a) 10 2 + 15 2 = 20 2
100 + 225 < 400
obtuse triangle
(8V2) 2 + (4V6) 2 = (4Vl4) 2
64 • 2 + 16 • 6 = 16 • 14
128 + 96 = 224
right triangle
Additional Example: Find an integer such that 9, 12, represent an acute or obtuse triangle.
Solution: 9, 12, 15 would be a right triangle (this is a multiple of the Pythagorean triple, 3-4-5). So 8, 9,
10, 11, 12, 13, or 14 would work. If the integer is less than 8, then the triangle would be obtuse, with 12
as the longest side.
For an obtuse triangle, the third side could be less than 8, but greater than 3(9 + 3 = 12). And, it could
also be greater than 15, but less than 21(9 + 12 = 21). So, the possibilities are 4, 5, 6, 7, 16, 17, 18, 19, or
20.
Using Similar Right Triangles
Goal
Students will review similar triangles, primarily right triangles. Then, the concept of the geometric mean
is introduced and applied to right triangles.
Relevant Review
Review similar triangles and their properties briefly before introducing Theorem 8-5. The two triangles
formed by the altitude from the right angle in a right triangle are similar to the larger right triangle by
AA (see Example 1). Show students how the three triangles fit together and which angles are congruent
to each other and which sides are proportional (Examples 1 and 2).
Also, remind students that answers should be in simplest radical form. You may need to add a few
simplifying and reducing radical questions in the Review Queue.
Teaching Strategies
Example 3 introduces the geometric mean as it applies to right triangles. You can choose to use this
example before discussing the geometric mean or after it, after Example 7. It might be easier for students
to see the geometric in its literal, algebraic form (without being applied to triangles) and practice it that
way for a few examples, and then apply it to a right triangle.
In Examples 5 and 6, it might be helpful to show students the corresponding proportions for the geometric
mean, ^- = ^ and ^ = ^. Students will like the short cut, x = yab^ but they should be shown the
proportion first. The proportion directly relates to its application to right triangles.
Examples 3 and 7 apply the geometric mean to the right triangle. The set-up of these proportions, using
similar triangles, is the same as the geometric mean. So, students can always use similar triangles, rather
than memorize the geometric mean.
So often in math, there are two or even three ways to solve one problem. In Example 8, there are two
ways presented to solve for y. Encourage students to try the geometric mean, but they can also use
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the Pythagorean Theorem. Show students both methods and then brainstorm why one method could be
preferred over the other. Students should be allowed to use which ever method they feel more comfortable
with.
Special Right Triangles
Goal
The purpose of this lesson is to encourage the use of ratios to find values of the sides of special right
triangles. These triangles are extremely useful in trigonometry.
Relevant Review
Special right triangle ratios are extended ratios. Here is an additional Review Queue question:
Additional Review Queue: The sides of a triangle are in the extended ratio 3:7:9. If the shortest side
is 21, find the length of the other two sides.
Solution: We will rewrite the ratio as 3x : 7x : 9x. So, 3x = 21, which means x = 7. Therefore, the other
two sides are 7 • 7 = 49 and 9 • 7 = 63.
Teaching Strategies
Students should complete Investigation 8-2 independently. Lead students through step 1, but then allow
them to complete the Pythagorean Theorem in steps 2 and 3 on their own. Ask students if they see a
pattern among the hypotenuses. Then, go over how to solve an isosceles triangle given various sides and
with various square roots. They should know how to solve an isosceles right triangle when the legs are
not whole numbers. To find the legs, always divide the hypotenuse by V2 and then simplify the radical.
(Students might also notice there is a pattern here too. The leg will be the hypotenuse divided by 2 and
multiplied by V2. For example, if the hypotenuse is 20, the legs will be 20 4- 2 • a/2 = 10 V2. This is a little
short cut.) To find the hypotenuse, multiply the leg by V2. Students may need to simplify the square
root. Also, the diagonal of a square will always split the square into two 45-45-90 triangles. Make sure
students notice this connection.
With 30-60-90 triangles, students must make the connection that it is half an equilateral triangle. In
Investigation 8-3, lead students through steps 1-3, then allow the students complete steps 4 and 5 on their
own. Students will have a hard time remembering this extended ratio. Some students will think that x V3
should be the hypotenuse because 3 is bigger than 2. However, explain that a/3 « 1.73, which is smaller
than 2, so 2x will always be the longest side, which is the hypotenuse. To solve 30-60-90 triangles, students
will need to find the shortest leg, if they are not given it. If they are given the hypotenuse, divide by 2,
then they can multiply by V3 to get the longer leg. If they are given the longer leg, they will need to divide
by V3 (or use the shortcut as applied to the 45-45-90 triangle above, use 3 and V3 where appropriate) to
get the shorter leg, then multiply that by 2.
Students will also get these two ratios confused. One way to help them remember is for a 45-45-90 triangle
there are 2 45° angles, so the hypotenuse is xV2. For a 30-60-90 triangle all the angles are divisible by 3,
so the V3 is in the radical for this ratio.
The best way for students to become comfortable with these ratios is to have them do lots of problems.
Students can also make flashcards for these ratios. If students have trouble remembering these special
shortcuts, encourage them to use Pythagorean's Theorem and simplify the answer. The resulting answer
will equal the shortcut.
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Tangent, Sine and Cosine
Goal
This lesson introduces the trigonometric functions; sine, cosine and tangent.
Teaching Strategies
Before introducing the trig ratios, make sure students understand what adjacent and opposite mean and
which angles they are in reference to. c will always be the hypotenuse, but a and b can be either opposite
or adjacent, depending on which acute angle we are using. At this point, do not overwhelm students with
the fact that the trig functions can be applied to any angle; focus on acute angles in triangles.
Encourage students to make flash cards for the sine, cosine and tangent ratios and to use the pneumonic
SOH-CAH-TOA (in FlexBook). Both of these things will help students internalize the ratios.
In the types of problems in this lesson, it will be very common that two of the three sides are given and
students will need to use the Pythagorean Theorem to find the third side. This should always be done
first, and then they can apply the ratios. Students will also need to reduce ratios and simplify any radicals.
Show students several different orientations of the triangles (rotated, flipped, etc) so they are familiar with
where an angle is and which sides are adjacent and opposite.
Have each student check to ensure their calculator is set to degrees (DEG), not radians (RAD). Having
a calculator in radians will provide incorrect answers. When checking homework at the beginning of the
class period, check the mode of students' calculators as well.
30
When introducing how to find the sides of a right triangle, using the trig ratios, draw the triangle from
Example 5 on the board with only one variable, a. This will isolate a and students should be able to see
that the cosine ratio will solve for a. Use the arrow to help illustrate that a is adjacent to 22° and 30 is the
hypotenuse. After a is found, redraw the triangle so b is the only variable. Now, b is isolated and students
will be able to recognize that the sine ratio will solve for it. Again, use arrows, if needed.
30
Stress to students that they should only use information that they are given in the problem. Using "solved
for" information will not give them the most accurate answer or it could be completely wrong (if the "solved
for" answer used is incorrect).
To help illustrate the angles of elevation and depression, see the picture below.
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Show this to students and ask what the angle of elevation from the cow to the goat is. Then, ask what the
angle of depression from the sheep down to the cow is. Students should notice it is the same measurement
and alternate interior angles. Fill in the angle of depression/elevations with any measurement.
Inverse Trigonometric Ratios
Goal
In the previous lesson, students used the special trigonometric values to determine approximate angle
measurements. This lesson enables students to "cancel" a trigonometric function by applying its inverse
to accurately find an angle measurement.
Relevant Review
Begin by listing several mathematical operations on the board in one column. In a second column, title
it "Inverse." Be sure students understand what an inverse means (an inverse cancels an operation, leaving
the original value undisturbed).
Table 1.12:
Operation
Inverse
Example
Addition
Multiplication
Squaring
Sine
Cosine
Tangent
The first four are typically easy for students (Subtraction, square root, multiplication, and addition). You
may have to lead students a little more on the last two (inverse tangent and inverse sine). Students may say,
"Un-tangent it." Use the correct terminology here, but also use their wording, if at all possible. Students
will be able to cancel the trigonometric function using the inverse of that function, even though they may
use incorrect terminology.
Go over Example 1 thoroughly. Make sure every student understands how to input inverse trig functions
into their calculator. Remind students that the set-up for inverse problems is the same as those from the
previous lesson. However, instead of being given an angle measure, we leave it as a variable. Students need
to solve for the angle and then put everything into the calculator at the same time. Get them in this habit
so they will produce the most accurate answers. For example:
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46
Yes
= --(§)
No ism -1 (0.666)
Example 5 is a special right triangle. Students will probably go through the motions and not notice that
they can use the ratios learned in the Special Right Triangles lesson. Either way, students will still arrive
at the correct answer, but point out to them to not go blindly into each question. Read a problem, re-read,
and then decide how to answer.
Like the last lesson, real-life situations are a major application. At the end of this lesson, create a word
problem as a class. Use the names of students or find the height of a local building. Make the problem
personal. Then, add it to the test as an extra question or a bonus.
1.9 Circles
Pacing
Table 1.13:
Day 1
Day 2
Day 3
Day 4
Day 5
Parts of Circles &
Finish Parts of
Finish Properties
Quiz 1
Finish Properties
Tangent Lines
Circles & Tangent
of Arcs
Start Properties of
of Chords
Investigation 9-1
Lines
Start Properties of
Arcs
Chords
Investigation 9-2
Investigation 9-3
Day 6
Day 7
Day 8
Day 9
Day 10
Inscribed Angles
Finish Inscribed
Quiz 2
Finish Angles from
Segments from Se-
Investigation 9-4
Angles
Start Angles from
Chords, Secants
cants and Tan-
Investigation 9-5
Chords, Secants,
and Tangents
Investigation 9-6
and Tangents
Investigation 9-7
Investigation 9-8
gents
Day 11
Day 12
Day 13
Day 14
Day 15
Quiz 3
^Finish Extension:
Review for Chap-
Chapter 9 Test
Start Chapter
*Start Extension:
Equations of Cir-
ter 9 Test
10
Equations of Cir-
cles
cles
Parts of Circles & Tangent Lines
Goal
This lesson introduces circles and several parts of circles. Then, we will explore the properties of tangent
lines.
Notation Note
O is the symbol for a circle. A circle is labeled using this symbol and its center. Some textbooks place a
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dot in the center, some do not. Do not confuse this symbol with an uppercase O-
Teaching Strategies
There is a lot of vocabulary in this chapter. Encourage students to make flash cards for each word and the
theorems. Give students index cards at the end of each class period so they can make flash cards in the
last 5-10 minutes of all the terms they learned.
Discuss with students where one could find tangent circles, tangent lines, concentric circles, and intersecting
circles in real life. Concentric circles can also be formed when raindrops hit a body of water, such as a
lake or puddle. An example of intersecting circles can be a Venn diagram. Common external tangent lines
could be the gears of a bicycle.
Investigation 9-1 can be done as a teacher- led activity or individually. For step 2, it can be difficult for
students to draw a perfectly tangent line at point B. You may need to circulate to ensure they each student
has the correct drawing, to ensure that they get the correct measurement of /.ABC. After measuring lABC,
ask students if they think this will happen at angle drawn to the point of tangency. Feel free to repeat this
investigation with a larger or smaller circle so students can see two different cases. Lead them towards the
Tangent to a Circle Theorem.
In this lesson and chapter, students will need to apply the Pythagorean Theorem. Having a tangent line
and a radius that meet at the same point will always produce a 90° angle. Therefore, students will need
to recall all the information they learned in the last chapter (special right triangles, Pythagorean Theorem
and its converse, and the trig ratios). Review these points, if needed, and proceed with Example 4.
Example 6 is an important example because it uses the converse of the Pythagorean Theorem to show that
B is not a point of tangency. Make sure students understand this point. In Example 7, students will need
to draw an additional line, EB. Encourage students to draw in additional lines if ever needed.
If students wonder the reasoning behind Theorem 9-2, draw in radii AD and AB and segment AC. You can
give students this additional example.
Additional Example: Put the reasons for the proof of Theorem 9-2 in the correct order.
Given:
Prove:
Radii AD and DB, tangent lines DC and BC
DC = BC
Statement
Reason
1. Radii AD and DB,
A. CPCTC
tangent lines DC
and BC
2. AD = DB
8. Given
C^ HL Congruence Theorem
3. ZCDAandZCBA
P. Tangent to a Circle Theorem
are right angles.
4. AC=AC
5. A ADC = A ABC
E. Aft radii are congruent
R Reflexive Property of Congruence
6, DC SBC
Solution: B,E,D,F,C, and A.
Properties of Arcs
Goal
This lesson introduces arcs, central angles, their properties and how to measure them.
Notation Note
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48
Use a curved line above the endpoints of the arc to label it. Some arcs, such as major arcs, require three
letters to label it. Explain to students that they must use three letters for some arcs to distinguish between
the two different arcs that have the same endpoints. If only two letters are used to label an arc, it is
assumed that the arc is less than 180°.
Teaching Strategies
When introducing arcs, discuss with students where they might see arcs and degrees in real life. Examples
are pizza crust, pie crust, a basketball hoop (the rim), among others. An arc could be a piece of pizza
crust or pie crust. Some students might be familiar with skateboarding or snowboarding. A 360° jump is
one rotation, a 540° is one-and-a-half rotations, and a 720° is two full rotations.
Make sure that students are comfortable with finding arc measures using central angles. This is the
beginning of arc measures and corresponding angles in circles, so having a solid foundation is important.
In Example 3, there are more congruent arcs than the ones that are listed. See if students can find other
possibilities: DAE = DBE = ADB = AEB (all semicircles), ADE = DAB, and DBA = EDB This would also
be a good time to discuss different ways to label the same major arc. For example, ADE from above could
also have been labeled ABE.
The Arc Addition Postulate should be familiar to students; it is very similar to the Segment Addition
Postulate and the Angle Addition Postulate. Rather than adding segments or angles, we are now adding
arcs. This is a very useful postulate when trying to find all the arcs in a circle.
Properties of Chords
Goal
Students will find the lengths and learn the properties of chords in a circle.
Notation Note
There is no explicit way to mark that two arcs are congruent in a picture. Students will have to infer from
other information if two arcs are congruent or not. Ways that they can tell are: if corresponding chords
are congruent or if the central angles are congruent.
Teaching Strategies
Make sure students understand all the possible ways to interpret Theorem 9-3. First, if the chords are
congruent, then the arcs are congruent. Second, the converse is also true (you may need to review "if and
only if" and biconditional statements). Finally, you can also say that if the central angles are congruent,
then the arcs are congruent AND the chords are congruent. The converse of this statement is also true.
In Investigation 9-2, the construction tells us that a diameter bisects a chord. From this, students should
also see that the diameter is perpendicular. Therefore, the perpendicular bisector of a chord is also a
diameter (Theorem 9-4). Stress to students that other chords can be perpendicular to a chord and other
chords can bisect a chord, but only the diameter is both. Remind students that every line segment has
exactly one perpendicular bisector. Also, students should understand that not every diameter drawn that
intersects a given chord will be its perpendicular bisector. This investigation is best done as a teacher-led
activity.
Like Theorem 9-3, make sure students understand all the possible ways to interpret Theorems 9-4 and
9-5. Be careful, though. Write the converse of Theorem 9-4; if a line is a diameter, then it is also the
perpendicular bisector of a chord. At first glance, students might think this is a true statement. Show
students a counterexample or have a student come up and draw one (see Example 3). However, the converse
of Theorem 9-5 is true; if the diameter bisects a chord and its corresponding arc, then the diameter is also
perpendicular to the chord. When applying these theorems to a diagram, two of the following three things
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must be marked: chord is bisected, diameter passes through the center (to ensure that this chord is actually
a diameter), or diameter is perpendicular to the chord. If two of these are marked, then it can be inferred
that the third is also true.
Investigation 9-3 and Theorem 9-6 apply Theorems 9-3, 9-4, and 9-5 to two congruent chords in the same
circle (or congruent circles). If you want, you can continue Investigation 9-3 on the same circle from
Investigation 9-2. Draw a second chord that is the same length as BC (from Investigation 9-2) somewhere
else in the circle. You can repeat step 2 from Investigation 9-2, rather than using step 2 from Investigation
9-3 as well. Both steps will produce the same result.
Review with students the definition of equidistant and why the shortest distance between a point and a
line is the perpendicular line between them. In order for two chords to be congruent, using Theorem 9-6,
the segments from the center to the chords must be marked congruent and perpendicular. Ask students if
they notice any other properties of these segments. Students should notice that these segments are part
of a diameter and that they also bisect each chord.
Finally, you can show students the following picture and see if they can find all the congruent chords,
segments and arcs. Tell them that BE = IF. This diagram applies all the theorems learned in this lesson.
BJ^JE = IM = MF BC = CE = HF = HI
CA = AD = AG = AH CD = HG
CJ = MH DE = GF
JK = ML CH = DG
KE = LF
7I = AM
AL = AK
DK = LG
*There are semicircles and major arcs that are also congruent.
And, AJKA = AMLA by HL, SSS, or ASA
Inscribed Angles
Goal
This lesson will demonstrate how to find measures of inscribed angles and intercepted arcs.
Relevant Review
At this point, it would be very helpful to review all the theorems and vocabulary learned in this chapter.
Make sure students have a firm grasp on the chapter up to this point. The chapter gets continually harder,
so it is important that they have a strong foundation.
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Double-check that every student has his/her flash cards of vocabulary and theorems and make sure they
are using them regularly. One way to spot-test students on vocabulary is to have them lie out all their
flash cards with the vocab or the name of the theorem side up. Then, you read the definition or theorem
out loud. The first student to raise the correct card gets some sort of reward; either candy or an extra
credit point. You can also tally the points and if any student reaches 5 (or any number of your choosing),
then they will receive extra credit or a homework pass.
Teaching Strategies
It is very important that students understand how an inscribed angle and intercepted arc are defined and
relate to each other. Any angle in a circle can (and usually does) have an intercepted arc, even though it
is defined in terms of an inscribed angle. The important point to note is that the intercepted arc is the
interior arc with the given endpoints on the circle. The blue arcs below are considered intercepted arcs for
the inscribed angles below.
For Investigation 9-4, it might be helpful to already have three (or more) drawn inscribed angles on a
handout for students. Pass these out and they still should draw in the corresponding central angle. Do
not rush through this activity. Walk around to answer questions and let students arrive at the Inscribed
Angle Theorem on their own. Then, proceed with lots of practice problems and examples to get students
used to using this theorem.
Draw the pictures for Theorems 9-8 and 9-9 before telling the students what exactly the theorems are. For
Theorem 9-8, ask students if they can conclude anything about lADB and lACB. For Theorem 9-9, ask
students what the measure of the intercepted arc is, then they should be able to determine the measure
of the inscribed angle. Here, the wording might be a little off from what is in the theorem. Make sure
students can infer everything from these two theorems. First, in 9-8, the triangles created are only similar,
not necessarily congruent. In 9-9, the endpoints of the inscribed angles are on a diameter and the diameter
would be the hypotenuse of a right triangle.
Investigation 9-5 should be a teacher-led demonstration that can easily be done on an overhead projector
(pre-cut a transparency into an inscribed quadrilateral and color the angles). Guide students towards the
next theorem. The word "cyclic" is not used to describe these types of quadrilaterals. You can choose
whether or not you would like to introduce this vocabulary or use what is in the text.
Angles of Chords, Secants, and Tangents
Goal
This lesson further explores angles in circles. Students will be able to find the measures of angles formed
by chords, secants, and tangents.
Teaching Strategies
This lesson divides angles in circles into three different categories: angles with the vertex ON the circle,
angles with the vertex IN the circle, and angles with the vertex OUTSIDE the circle.
central angle = intercepted arc
angle inside = half the sum of the intercepted arcs
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angle on circle = half intercept arc
angle outside = half the difference of the intercepted arcs
Use the pictures above to help students generate formulas for each case. Then, draw the other options for
angles on circles and angles outside circles.
Example 2 shows students that Theorem 9-8 (from the previous lesson) works for any angle where the
vertex is on the circle.
Investigations 9-6, 9-7, and 9-8 can all be teacher- led investigations where students follow along and the
class discovers the formulas together. Allow students to guess the possibilities for the formulas, even if
they are wrong. Developing the correct formula will give students ownership over the material and it will
help them retain the information.
Make sure you do plenty of practice problems in class to ensure that students are using the correct formula
in the appropriate place. Give students a handout with several problems. You can choose to put the
formulas on the board, let them use notes, or use nothing at all. Include problems from the previous
lesson(s) as well.
Segments of Chords, Secants, and Tangents
Goal
This goal of this lesson is to explain the formulas for determining segment lengths formed by intersecting
secants and tangents.
Relevant Review
Students might need a little algebraic review with solving quadratic equations. Problems involving tangents
and secants can become a factoring problem or use the quadratic formula. Students might also need a
review of square roots and simplifying square roots.
Teaching Strategies
Get students organized with all the information from this lesson and the previous two lessons. Have
students draw and complete the following table.
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Table 1.14:
Picture
Angle Formula
Segment Formula
x = a
x ° = \{ a ° +b°)
l^o
o 1 o
x° = \{a° -b°)
x° = \{a° -b°)
l(„o
(a° - b°
Sides of angle are radii. No for-
mula
pq = sr
Sides are chords. No formula.
One side is a chord, other is a ray.
No formula.
s{s + p) =q{q + r)
s 2 = q(q + r)
Extension: Writing and Graphing the Equations of Circles
Show students how a point can be on a circle, using the Pythagorean Theorem. For example, if the equation
of a circle is x 2 +y 2 = 25, is (3, -4) on the circle? Yes. If students plug in 3 for x and -4 for y, they will see
that the Pythagorean Theorem holds true. Conversely, if we tested (-6, 2), we would find that it is not on
the circle because 36 + 4 ^ 25.
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In Example 3, students might have problems finding the diameter. Remind them that the diameter is the
longest chord in a circle. Students would need to count the squares vertically and horizontally to see where
the longest segment would be (within the circle).
1.10 Perimeter and Area
Pacing
Table 1.15:
Day 1
Day 2
Day 3
Day 4
Day 5
Triangles and Par-
allelograms
Day 6
Finish Circum-
ference and Arc
Trapezoids,
Rhombi,
Kites
Day 7
Area of Circles
Sectors
and
and
More Trapezoids,
Rhombi, and Kites
Start Area of Sim-
ilar Polygons
Day 8
Quiz 2
Start Review of
Finish Area of
Similar Polygons
Day 9
Review of Chapter
10
Quiz 1
Start Circum-
ference and Arc
Length
Day 10
Chapter 10 Test
Length Chapter 10
Investigation 10-1
Triangles and Parallelograms
Goal
This lesson introduces students to the area and perimeter formulas for triangles, parallelograms and rect-
angles.
Relevant Review
Most of this lesson should be review for students. They have learned about area and perimeter of triangles
and rectangles in a previous math class (Math 6, Pre- Algebra, or equivalent).
Notation Note
In this chapter, students need to use square units. If no specific units are given, students can write units 2
or u 2 .
Teaching Strategies
If students are having a hard time with the formulas for area and perimeter of a rectangle, place Example 3
on a piece of graph paper or transparency. Then, students can could the squares for the area and perimeter
and you can generate the formula together.
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9 cm
4 5 6
4 cm
26
25
24
23
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
2 8
29
30
31
32
33
34
35
36
10
11
12
13
22 21 20 19 18 17 16 15 14
If you count all the squares, there are 36 squares in the area, or square centimeters (red numbers). Counting
around the rectangle (blue numbers), we see there are 26 squares. Therefore, the perimeter of this square
is 26 cm.
This technique will also work for squares.
An important note, each problem will have some sort of units. Remind students that the shapes might not
always be drawn to scale.
Example 5 is a counterexample for the converse of the Congruent Areas Postulate. Therefore, the converse
is false. An additional counterexample would to have them draw all the possible rectangles with an area
of 20 in 2 . Use graph paper so students will see that each rectangle has 20 squares. Possible answers are:
20x1,10x2, and 5x4.
The Area Addition Postulate encourages students to separate a figure into smaller shapes. Always divide
the larger shape into smaller shapes that students know how to find the area of.
To show students the area of a parallelogram, cut out the picture (or draw a similar picture to cut out)
of the parallelogram and then cut the side off and move it over so that the parallelogram is transformed
into a rectangle. Explain to students that the line that you cut is the height of the parallelogram, which is
not a side of the parallelogram. Then, cut this parallelogram along a diagonal to create a triangle. Here,
students will see that the area of a triangle is half the area of a parallelogram. You may need to rotate
the halves (triangles) so that they overlap perfectly. This will show the students that the triangles are
congruent and each is exactly half of the parallelogram.
Create another set of flashcards for the area formulas in this chapter. These flashcards should be double-
sided. The blank side should be a sketch of the figure and its name. The flip side should have the formula
for its area and the formula for its perimeter. Students should create flashcards as the chapter progresses.
Trapezoids, Rhombi, and Kites
Goal
This lesson further expands upon area formulas to include trapezoids, rhombi, and kites.
Relevant Review
Students might need a quick review of the definitions of trapezoids, rhombi, and kites. Go over their
properties (especially that the diagonals of rhombi and kites are perpendicular) and theorems. Students
may know the area formula of a trapezoid from a previous math class.
Review the Pythagorean Theorem and special right triangles. There are several examples and review
questions that will use these properties. If students do not remember the special right triangle ratios, they
can use the Pythagorean Theorem.
Teaching Strategies
Use the same technique discussed in the previous lesson for the area of a parallelogram and triangle. Cut
55
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out two congruent trapezoids and demonstrate the explanation at the beginning of the lesson explaining
the area of a trapezoid. Going over this with students (rather than just giving them a handout or reading
it) will enable them to understand the formula better. These activities are done best on an overhead
projector.
Conveniently, the area formula of the rhombus and kite are the same. Again, you can cut out a rhombus
and kite, then cut them on the diagonals and piece each together to form a rectangle. Generate the formula
with students. Another way to write the formula of a rhombus is to say that it has 4 congruent triangles,
with area \ (\d\\\\d2j = \d\di- Multiplying this by 4, we get \d\d2 = \d\d2- This process is not as easily
done with a kite because one of the diagonals is not bisected.
Additional Example: Find two different rhombi that have an area of 48 units 2 .
Solution: The diagonals are used to find the area, so when solving this problem, we are going to be finding
the diagonals' lengths, \d\d2 = 48, so d\d2 = 96. This means that the product of the diagonals is double
the area.
The diagonals can be: 1 and 96, 2 and 48, 3 and 32, 4 and 24, 6 and 16, 8 and 12.
As an extension, you can students draw the rhombi. The diagonals bisect each other, so have the diagonals
cut each other in half and then connect the endpoints of the diagonals to form the rhombus. Three examples
are below.
Areas of Similar Polygons
Goal
Students will learn about the relationship between the scale factor of similar polygons and their areas.
Students should also be able to apply area ratios to solving problems.
Relevant Review
Review the properties of similar shapes, primarily triangles and quadrilaterals, from Chapter 7. Remind
students that the perimeter, sides, diagonals, etc. have the same ratio as the scale factor. The Review
Queue reviews similar squares. As an additional question, ask students to find the perimeter of both
squares and then reduce the ratio (smaller square = 40, larger square = 100, ratio is 2:5, the same as
the ratio of the sides). Ask students why they think the ratio of the sides is the same as the ratio of the
perimeters.
Teaching Strategies
Examples 1 and 2 lead students towards the Area of Similar Polygons Theorem. As an additional example
(before introducing the Area of Similar Polygons Theorem), ask students to find the area of two more
similar shapes. Having students repeat problems like Example 2, they should see a pattern and arrive at
the theorem on their own.
Additional Example: Two similar triangles are below. Find their areas and the ratio of the areas. How
does the ratio of the areas relate to the scale factor?
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LL
Solution: Each half of the isosceles triangles are 3-4-5 triangles. The smaller triangle has a height of 3
and the larger triangle has a height of 9 (because 12 is 3 time 4, so this triangle is three time larger than
the smaller triangle). The areas are: A\ arger a = \ • 24 • 9 = 108 and A sma u er a = \ * 8 • 3 = 12. The ratio of
the area is ^ = g. The ratios of the scale factor and areas relate by squaring the scale factor, \ = \\) •
Circumference and Arc Length
Goal
The purpose of this lesson is to review the circumference formula and then derive a formula for arc length.
Relevant Review
The Review Queue is a necessary review of circles. Students need to be able to apply central angles, find
intercepted arcs and inscribed angles. They also need to know that there are 360° in a circle.
Teaching Strategies
Students may already know the formula for circumference, but probably do not remember where n comes
from. Investigation 10-1 is a useful activity so that students can see how n was developed and why it is
necessary to find the circumference and area of circles. You can decide to make this investigation teacher-
led or allow students to work in pairs or groups. From this investigation, we see that the circumference is
dependent upon n.
When introducing arc length, first have students find the circumference of a circle with radius of 6(127r).
Then, see what the length of the arc of a semicircle (67r). Students should make the connection that the
arc length of the semicircle will be half of the circumference. Then ask students what the arc length of half
of the semicircle is (37r). Ask what the corresponding angle measure for this arc length would be (90°). See
if students can reduce ^ and if they make the correlation that the measure of this arc is a quarter of the
total circumference, just like 90° is a quarter of 360°. Using this same circle, see if students can find the
arc length of a 30° portion of the circle I y = n\. Then, as students what portion of the total circumference
7i is. 7r is Y2 of 127r, just like 30° is ^ °f 360°. This should lead students towards the Arc Length Formula.
Students may wonder why it is necessary to leave answer in exact value, in terms of 7r, instead of approxi-
mate (multiplying by 3.14). This is usually a teacher preference. By using the approximate value for 7r, the
answer automatically has a rounding error. Rounding the decimal too short will cause a much larger error
than using the decimal to the hundred-thousandths place. Whatever your preference, be sure to explain
both methods to your students. The review questions request that answers be left in terms of 7r, but this
can be easily changed, depending on your decision.
Area of Circles and Sectors
Goal
This lesson reviews the formula for the area of a circle and introduces the formula for the area of a sector
and segment of a circle.
Teaching Strategies
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If you have access to an LCD display or a computer lab, show students the animation of the area of a circle
formula (link is in the FlexBook).
The formula for the area of a sector is very similar to the formula for arc length. Ask students to compare
the two formulas. Stress to students that the angle fraction in the sector formula is the same as it is for
the arc length formula. Therefore, students do not need to memorize a new formula; they just need to
remember the angle fraction for both.
To find the area of the shaded regions (like Example 8), students will need to add or subtract areas of
circles, triangles, rectangles, or squares in order to find the correct area. Encourage students to identify
the shapes in these types of problems before they begin to solve it. At the end of this lesson, quickly go
over problems 23-25, so that students know how to solve the problems that evening. Remind students to
use the examples in the lesson to help them with homework problems.
Finding the area of a segment can be quite challenging for students. This text keeps the angles fairly
simple, using special right triangle ratios. Depending on your level of student, you may decide to omit this
portion of this lesson. If so, skip Example 9 and review questions 26-31.
Additional Example: Find the area of the blue shaded region below.
Solution: The triangle that is inscribed in the circle is a 45-45-90 triangle and its hypotenuse is on the
diameter of the circle. Therefore, the hypotenuse is 24 V2 and the radius is 12 V2. The area of the shaded
region is the area of the circle minus the area of the triangle.
Aq = tt(12 V2) 2 = 71 • 144 • 2 = 288;r
A A = - • 24 • 24 = 288
The area of the shaded region is 2887T - 288 « 616.78 units 2
1.11 Surface Area and Volume
Pacing
Table 1.16:
Day 1
Day 2
Day 3
Day 4
Day 5
Exploring Solids
Surface Area of
Prisms & Cylin-
ders
Finish Surface
Area of Prisms &
Cylinders
Quiz 1
Start Surface Area
of Pyramids and
Cones
Finish Surface
Area of Pyramids
and Cones
Day 6
Day 7
Day 8
Day 9
Day 10
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Table 1.16: (continued)
Day 1
Day 2
Day 3
Day 4
Day 5
Volume of Prisms
Quiz 2
Finish Volume of
Surface Area and
Quiz 3
& Cylinders
Start Volume of
Pyramids & Cones
Volume of Spheres
*Start Extension:
Pyramids & Cones
Investigation 11-1
Exploring Similar
Solids
Day 11
Day 12
Day 13
Day 14
Day 15
*Finish Extension:
* Extension Quiz
Review of Chapter
Chapter 11 Test
Start Chapter
Exploring Similar
Start Review of
11
12
Solids
Chapter 11
Exploring Solids
Goal
The purpose of this lesson is to introduce students to three-dimensional figures. Polyhedral figures are
presented in this lesson and common terms such as edge, vertex, and face are explained, as well as how to
name polyhedra.
Relevant Review
Students should know the definition of a regular polygon and how to find the area of various triangles and
quadrilaterals.
Teaching Strategies
Like with previous chapters, it might be helpful for students to make flash cards of the vocabulary and
theorems learned.
Discuss with students where they would see polyhedra, prisms, pyramids, cylinders, and cones in real life.
Tell students to visualize themselves in a grocery store; there are several examples of these solids there.
Examples could be: soup can (cylinder), Toblerone chocolate bar (triangular prism), oranges (sphere), or
waffle cones (cone). Write down the items students come up with and draw a representation.
After going over Example 1 and putting the faces, vertices, and edges into a table, see if students can come
up with Euler's Theorem on their own. Students might wonder why they need to know Euler's Theorem,
when they can just count the number of vertices, edges, and faces. One application could be when the
prism has so many sides that it becomes difficult to count the edges (E is always the largest of F, V, and
E in the formula). Other applications are Examples 3 and 4, when you are not given a picture in the
problem.
One way to "view" a three-dimensional solid is to use cross-sections. You might need to use physical models
to help students understand this concept. For example, you could mold Play-doh into a cylinder or cube
and "cut" them in different ways to show the different cross-sections. A cylinder can have a circle, oval (or
portion of an oval), or rectangle as a cross-section.
Another way to "view a three-dimensional solid is to use a net. A net takes all the faces of a solid and lies
them out flat and adjoining. Students need to be careful with nets. There can be several answers for one
solid and depending on how the faces are laid out, the net might not even work. Be sure to show students
nets that do not work. The activity in the link, http://illuminations.nctm.org/activitydetail.
aspx?ID=84, gives students 20 possible nets of a cube and they need to find the 11 that work.
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Surface Area of Prisms and Cylinders
Goal
Students will learn how to find the surface area of prisms and cylinders.
Relevant Review
Students should know how to find a net of a three-dimensional solid. They will also need to be able to find
the area of triangles and quadrilaterals.
Teaching Strategies
The surface area can be difficult for students to visualize. Make sure they have a firm grasp on how to find
nets of a three-dimension figure. Students need to find all the lengths and heights of the faces of a prism
before finding the total surface area. For example, in Example 2, students need to find the length of the
hypotenuse of the base because it is also the length of a rectangle.
Additional Example: Find the surface area of the trapezoidal prism. The bases are isosceles trapezoids.
Solution: First find the length of the legs of the isosceles trapezoid bases.
19
j 15 j\V8 2 +15 2 =17
h
8 19 8
Here is the net for this prism.
19
40
17
35
17
19
Surface Area:
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Trapezoids A = -(35 + 19)15 = 405
Rectangles A = 17 • 40 = 680
A = 35 • 40 = 1400
A = 19 • 40 = 760
Total = 405 + 405 + 680 + 680 + 1400 + 760
= 4330 units 2
The surface area of a cylinder can be difficult for students to visualize. As the FlexBook suggests, take
the label off of a soup can so that it opens up to a rectangle. Then, students will see that the width of the
rectangle is the circumference of the circular base.
Surface Area of Pyramids and Cones
Goal
Students will learn the formulas for the surface area of a regular pyramid and cone.
Relevant Review
Students should review nets and the formula for the area of triangles and quadrilaterals. Students should
also be able to apply the Pythagorean Theorem, Pythagorean triples, and special right triangle ratios.
Teaching Strategies
The slant height is a new term that applies to the lateral faces. It is only used to find the surface area
of pyramids and cones. Students may need to use the actual height to find the slant height through the
Pythagorean Theorem. Help students develop a formula for finding the slant height for different types of
pyramids and cones. In this lesson, we will find the surface area of equilateral triangle based pyramids,
square based pyramids, and right cones.
/3 the length
of the altitude
of the base
half the
base edge
(radius)
The most difficult of the three is the equilateral based pyramid. For the review questions in this section,
the slant height for these pyramids is given. However, students will still need to find the altitude of the
! V3
for the area
base in order to find the area. Or, if students remember, they can use the formula, A
of any equilateral triangle, where s is the length of the sides.
Since students only need to worry about two types of pyramids (for surface area), then you could generate
more specific surface area formulas for each one. They are:
Equilateral triangle based pyramid :
Square based pyramid :
SA = B + -nbl =
s 2 V3
s 2 V3
4 + 2^ /= 4
+ \ Sl
SA = B+ -nbl = s 2 + -{A)sl = s 2 + 2sl
Where s is the edge length and / is the slant height. Once students know the basic surface area, derive
these with the class for n = 3 and n = 4.
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Compared to the pyramid, the surface area of a cone is relatively simple. Students will always be given
two of the three pieces of information needed; h, r, or / (in the picture above). If the problem does not give
/, the slant height, then students will have to use the Pythagorean Theorem to solve for it.
Volume of Prisms and Cylinders
Goal
Students will learn how to find the volume of prisms and cylinders. They will also discover that the volume
of an oblique prism or cylinder is the same as a right prism or cylinder with the same height.
Relevant Review
Students will still need to know how to find the area of triangles and quadrilaterals so they can properly
apply the volume formulas in this lesson. Also, review with students how to "cube" and "cube root" a
number. Lastly, review the definitions of prisms and cylinders, so students are clear about which faces are
bases. This will be very helpful when using the volume formulas.
Teaching Strategies
Begin with a discussion of volume. Students might get volume, mass, and density confused. Volume is
simply the amount of physical space a three-dimension takes up. Think of two different objects that have
the same shape; a 12-pound bowling ball and a plastic empty sphere of the same size. Both of these solids
have the same volume, but very different masses or densities. Therefore, an empty solid has the same
volume as a full solid of the same size.
Volume is measured in cubic units, or units 3 . This is because the formula for volume always comes back
to length X width X height. Each of these have a unit associated with them. So, the answer would be
unit X unit x unit = units 3 .
Example 1 explores the concept of placing cubes (of whatever unit) into a prism to find its volume. Students
can count all these cubes and find that there are 60 within the prism. Another option would be to show
students what the area of each face is and then multiply that by the relative height. This will demonstrate
that the order of multiplication does not matter.
3 blocks
high
4 blocks
deep
wide
V= (3x5) x4 = 15x4 = 60
V = (3 x 4) x 5 = 12 x 5 = 60
V = (5 x 4) x 3 = 20 x 3 = 60
Regardless of the order, the formula for the volume of a rectangular prism will be length x width x height.
It does not matter which face is the base for a rectangular prism.
Students will take the words "base" and "height" literally. However, in the formula for volume, the "height"
might not always be the vertical length. As in Example 3, the apparent height of the solid is only the
height of the base. The "height" in the formula is actually 7 ft, which looks like the length of the base.
Students will think that the base of this tent is the rectangle on the bottom. However, we know that
the bases are actually the triangles at the front and back of the tent. Be very careful when discussing
the formula for volume and the definition of a prism. Review with students that the "bases" are the two
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62
congruent parallel faces. Before starting on a volume problem, students should examine the solid so they
know which faces are the bases.
As with the case of an oblique prism, the sides are all parallelograms, however the bases will still be parallel.
Students should know that the height of an oblique prism is not going to be an edge (like it is in a right
prism), but a vertical length that is outside the solid. Review with students that the base of a cylinder is
a circle.
Volume of Pyramids and Cones
Goal
In this lesson, students will discover that the volume of a prism is one-third the volume of a prism with
the same base. This property also applies to cones. Then, they will find the volume of composite solids.
Relevant Review
Make sure students are comfortable finding the volume of prisms and cylinders from the previous lesson.
Teaching Strategies
Investigation 11-1 should be a teacher- led activity. You should pre-make the open nets of a cube and
pyramid and then demonstrate that filling the pyramid three times will completely fill the cube.
If you decide to allow students to do the investigation, be prepared for it to take 20-30 minutes. You should
make the nets with students in case they have questions. Have students complete this activity in pairs and
each student in the pair can make one net.
In this lesson, students should be able to find the volume of any triangle or quadrilateral based pyramid
and any type of cone. If students are ever given the slant height, they will need to solve for the overall
height of the pyramid or cone. If a cone is not a right cone, it will not have a slant height and students
will need to be given the height in order to find the volume.
Like with surface area of pyramids, you can generate more specific formulas for the volume of an equilateral
triangle based pyramid and a square based pyramid. They would be:
1 l(s 2 ^) s 2 h^
Equilateral triangle based pyramid : V = -Bh = - — - — h =
3 3^ 4 ) 12
1 1 2
Square based pyramid : V = -Bh = -s h
s and h are the base edge length and the height. You can also generate formulas for the volume of a
rectangular based pyramid and a right triangle based pyramid.
Problems 3 and 6 will be quite difficult for students to complete (in the Review Questions). This is because
the bases are equilateral triangles and they are given the slant height. If you desire, you can change the
values given as the slant height to be the height of the pyramids. If you do this, the answers will be — ^—
and 240 V3, respectively. Students will still need to find the area of the equilateral triangle bases (^-4 )•
If you do not alter the problems, students will have to find the height of the pyramids, using the slant
height. This will be a difficult process, because students will have to recall the properties of the centroid
(the point where the vertical height hits the base). Therefore, the distance from the bottom of the slant
height to the centroid will be one-third the length of the entire altitude. (See the picture in the teaching
tips for Surface Area of Pyramids and Cones.)
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Surface Area and Volume of Spheres
Goal
Students will learn how to find the surface area and volume of spheres and hemispheres.
Relevant Review
Make sure students are comfortable with the formulas for circumference and area of a circle. Also, students
will apply the formulas for the surface area and volume of cylinders and cones to composite solids.
Teaching Strategies
Ask students where they have seen spheres and hemispheres in real life. Discuss the parts of a sphere
that are the same as a circle and those parts that are different. Be sure to show students the animated
derivations of the surface area and volume by Russell Knightley.
http : //www . rkm . com . au/ANIMATIONS/animation- Sphere-Surf ace-Area-Derivation . html
http : //www . rkm . com . au/ANIMATIONS/animation- Sphere- Volume-Derivation . html
When finding the surface area or volume of composite solids discuss with students what the parts are before
starting each problem This will make it easier for students to complete the problem. Also, make sure they
understand when to not include the top or bottom of cylinders or hemispheres in the total surface area
(see Example 5).
Extension: Exploring Similar Solids
Goal
Students will understand the relationship between similar solids, their surface areas, and their volumes.
Relevant Review
The areas of similar polygons should be reviewed before starting this extension. Students should know
that the square of the scale factor is the ratio of the areas of two similar shapes.
Teaching Strategies
Help students make the connection between the ratios of the areas of two similar shapes is the same as the
ratio of the surface areas of two similar solids. Both are area, so both ratios will be the square of the scale
factor. Following this pattern, ask students if they have an idea as to what the ratio of the volumes of two
similar solids would be.
Additional Example: The two square based pyramids below are similar. Find the surface area and
volume of both solids.
Solution: The scale factor is ^ = |. The slant height of the smaller pyramid is / = V4 2 + 8 2 =
Vl6 + 64 = v80 = 4 yb. Using the scale factor, the slant height of the larger pyramid is | • 4 V5 = 6 v5.
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SAsmaiUr = 8 2 + 4 (i • 8 • 4 VJi) = 64 + 64 V5
5A /arger = 12 2 + 4 (- • 12 • 6 Vo") = 144 + 144 Vo"
v smaller — ^
^smaller
, 9x 512
'12 2 )12 = 576
Encourage students to find the ratios of the surface areas and volumes above to reinforce what was learned
in this lesson.
1.12 Rigid Transformations
Pacing
Table 1.17:
Day 1
Day 2
Day 3
Day 4
Day 5
Exploring Symme-
try
Translations
Reflections
Quiz 1
Start Rotations
Investigation 12-1
Finish Rotations
Day 6 Day 7 Day 8 Day 9 Day 10
Compositions of Quiz 2 * Extension Quiz Finish Review of Chapter 12 Test
Transformations ^Extension: Tes- Start Review of Chapter 12
Investigation 12-2 sellating Polygons Chapter 12
Exploring Symmetry
Goal
This lesson introduces line symmetry and rotational symmetry.
Teaching Strategies
The Know What? for this lesson provides a good starting point for a discussion about symmetry in nature.
After defining line symmetry and rotational symmetry, discuss how both are found in nature and the real
world. Then, go over the symmetry in the starfish.
Have your students write the alphabet in uppercase letters. Using one colored pencil, show which letters
possess horizontal or vertical symmetry by drawing in the line. For example, B, E, and K have a line of
horizontal symmetry. Using a second color, draw in the vertical lines of symmetry. Have a contest to
determine who can write the longest word possessing one type of symmetry. For example, MAXIMUM is
a word where all the letters have vertical symmetry. KICKBOXED has a horizontal line of symmetry. You
could even do words like TOT. The entire word has vertical symmetry.
When discussing rotational symmetry, some textbooks may refer to the rotations as n-fold rotational
symmetry. This simply means that the n is the number of times the figure can rotate onto itself. For
example, a regular pentagon has 5-fold rotation symmetry, because it can be rotated 5 times of 108° before
returning to its original position.
65
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Translations
Goal
The purpose of this lesson is to introduce the concept of translations in the coordinate plane.
Relevant Review
Students should be comfortable with the distance formula and finding the slope between two points.
Teaching Strategies
All the transformations in this chapter are rigid transformations or isometries. Students need to know that
these transformations never change the size or shape of the preimage and, therefore, will always create
congruent images.
A "translation rule" is not property defined in this lesson. A translation rule is the amount an image
is translated (or moved) from the preimage. A rule can only be applied when the translation is in the
coordinate plane. The horizontal movement is added or subtracted from x and the vertical movement is
added or subtracted from y. The horizontal and vertical change will always be the same for every point
in a figure. For example, if the translation rule is (x,y) — > (x + 1, y - 2) for a triangle, each vertex of the
triangle will be moved to the right one unit and down two units.
Additional Example:
a) If AABC is the preimage, find the translation rule for image AXYZ.
b) If AXYZ is the preimage, find the translation rule for image AABC.
Solution:
a) From A to X, the triangle is translated to the left 8 units and down 5 units. (x,y) — > (x - 8,y - 5).
b) From X to A, the triangle is translated to the right 8 units and up 5 units. (x,y) — > (x + 8, v + 5).
Reflections
Goal
Students will reflect a figure over a given line and find the rules for reflections over vertical and horizontal
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lines in the coordinate plane.
Teaching Strategies
Using patty paper (or tracing paper), have students draw a small scalene triangle AABC on the right side
of the paper. Fold the paper so that AABC is covered and then trace it. Unfold the patty paper and label
the vertices as A\B\ and C, the images of A, 5, and C. Darken the fold line; this is the line of reflection.
Use a ruler to draw AA f . Mark the intersection of the reflecting line and AA f point M. Find AM and A'M.
Ask students what they notice about the distances and how the line AA' intersects the line of reflection.
Students might have a hard time visualizing where a reflection should be placed. Tell students to use the
activity described above to help them. They can fold their graph paper on the appropriate line (the y-axis,
for example) and then trace the figure on the other side. Until they get used to using the rules, this can
be one way for students to apply a reflection.
One way to help students remember the rules for reflections over vertical or horizontal is that the other
coordinate is changed. For a reflection over the x-axis (or horizontal line), the x- value will stay the same
and the y-value will change. For a reflection over the y-axis (or vertical line), the y-value will stay the
same and the x- value will change. The new coordinates of an image depend on how far away the preimage
points are. If a point is 5 units to the left of a line of reflection, then the image will be 5 units to the right
of the line of reflection.
The only diagonal lines that we will reflect over in this chapter are y = x and y = -x. Again, encourage
students to fold their graph paper when starting rotations over these lines and completing homework or
class work problems.
Rotations
Goal
In this lesson, students will learn about general rotations and in the coordinate plane. By the end of the
lesson, students should be able to apply the rules of rotation for 90°, 180°, and 270°, as well as draw a
rotation (using a protractor) of any degree.
Relevant Review
Make sure students remember how to draw and measure an angle, using a protractor. Practice this before
starting Investigation 12-1.
Teaching Strategies
Investigation 12-1 should be done individually by each student. The teacher can also lead the students in
the activity, on the overhead projector, if desired. Students need to be comfortable rotating a figure around
a fixed point. Every student will need a protractor for this activity. After completing the investigation,
have students repeat it with another figure of their choosing. Encourage students to pick a figure that has
straight sides, such as a quadrilateral or the letters H, T, or E.
Unless otherwise stated, rotations are always done in a counterclockwise direction. Tell students this is
because the quadrants are numbered in a counterclockwise direction. In the coordinate plane, the origin
is always the center of rotation.
After going over the rules for the rotations of 90°, 180°, and 270°, compare the rules learned in the previous
lesson to these (reflections over the x and y axis and y = x and y = -x). At this point, students know seven
different reflection and rotation rules that are all very similar.
Reflection over x-axis: (x,y) —> (x,-y)
Reflection over y-axis: (x,y) — > (—x,y)
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Reflection over y = x: (x, y) — > (y, x)
Reflection over y = -x: (x,y) — > (-y, -x)
Rotation of 90°: (x,y) — > (-y, jc)
Rotation of 180°: (x,y) -> (-x 9 -y)
Rotation of 270°: (x,y) -> (y,-x)
All of these rules are different, but very similar. It is encouraged that students make flash cards for this
rules to help them memorize each one. Ask students which ones have the x and y values switched and why
they think that is. Also, see if they can make a correlation between the reflections over the x and y axis
and the rotation of 180°. These three are the only rules where there is just a sign change. As it will be seen
in the next chapter, a rotation of 180° is a composition of the double reflection over the axes. Students
might be able to notice this by looking at the rules now. The reflection over the x-axis has a negative y
and the reflection over the y-axis has a negative x. In the rotation of 180°, both x and y are negative.
Composition of Transformations
Goal
This lesson introduces students to the concept of composition. Composition is the process of applying two
(or more) operations to an object. In this lesson, we will only apply two transformations to an object and
then determine what one transformation this double-translation is the same as.
Teaching Strategies
Students can get easily confused when applying compositions. They may attempt to perform the composi-
tion from left to right, as in reading a sentence. Point out to the students they must begin with the object
and, according to the order of operations, should perform the operation occurring within the parentheses
first. A glide reflection is the only composition where order does not matter.
Have students do Example 6 and see if they can come up with the Reflection over the Axes Theorem on
their own. You may need to do an additional example so students see the pattern.
Additional Example: Reflect AXYZ over the y-axis and the x-axis. Find the coordinates of AX"Y"Z"
and the one transformation this double reflection is the same is.
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Solution: The coordinates of AXYZ and AX"Y"Z" are:
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From these coordinates, we see that a double reflection over the x and y axes is the same as a rotation of
180°.
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Investigation 12-2 should be a teacher-led activity. You can either have students perform the investigation
along with you or you can just do the activity and have student write down the necessary information.
Extension: Tessellating Polygons
Goal
This lesson defines tessellations and shows students how to create a tessellation from regular polygons.
Relevant Review
Review with students the definition of a regular polygon and how many degrees are in a quadrilateral,
pentagon, hexagon, octagon, etc.
Teaching Strategies
Take students to the computer lab and let them play with the Tessellation Artist, from the website given in
the FlexBook (http://www.mathisfun.com/geometry/tessellation-artist.html). This website does
not create true tessellations, but it is fun for students to see if or how their drawing would tessellate.
Students may wonder if shapes other than regular polygons tessellate. You can show them the example
below and then see if they can tessellate any quadrilateral. When tessellating this quadrilateral, make sure
that every student has a different quadrilateral. When everyone is done, have the students hold up their
tessellations or share them with each other.
Additional Example: Tessellate the quadrilateral below.
69
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Solution: To tessellate any image you will need to reflect and rotate the image so that the sides all fit
together. First, start by matching up the each side with itself around the quadrilateral.
Now, continue to fill in around the figures with either the original or the rotation.
This is the final tessellation. You can continue to tessellate this shape forever.
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70
Chapter 2
Basic Geometry TE - Common
Errors
2.1 Basics of Geometry
Points, Lines and Planes
Naming Lines - Students often want to use all the labeled points on a line in its name, especially if there
are exactly three points labeled. Tell them they get to pick two, any two, to use in the name. This means
there are often many possible correct names for a single line.
Practice Exercise: How many different names can be written for a line that has four labeled points?
Answer : 12, Students can get to this answer by listing all the combinations of two letters. Recommend
that they make the list in an orderly way so they do not leave out any possibilities. This exercise is good
practice for counting techniques learned in probability.
Naming Rays - There is so much freedom in naming lines that students often struggle with the precise
way in which rays must be named. It is helpful to think of the name of a ray as a starting point and
direction. There is only one possible starting point, but often several points that can indicate direction.
Any point on the ray other than the endpoint can be the second point in the name. The "ray" symbol
drawn above the two points should be drawn such that the endpoint is over the endpoint of the ray and
the arrow is over the second point which indicates the direction.
Example: The ray to the right could be named AB or AC
ABC
There is only one point B - English is an ambiguous language. Two people can have the same name;
one word can have two separate meanings. Math is also a language, but is different from other languages
in that there can be no ambiguity. In a particular figure there can be only one point B. A point marks
a location and a diagram is like a map. If you have multiple streets with the same name, it is impossible
to distinguish between them and find a particular address. Students also need to be reminded that points
should always be labeled with capitol letters.
Coplanar Points - Students are often confused by the phrase "three non-collinear points" are necessary
to define a plane. They think that none of the points can be on the same line but in reality there is a line
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through any two points. It must be made clear that the phrase "three non-collinear points" implies that
all three of the points are not on the same line but that any two of them may be collinear.
Intersections - Visualizing geometric figures and their intersections can be very difficult for students.
Sometimes it helps to use a pencil and paper to illustrate the difference between a line intersecting a plane
and a line in a plane. Objects in the room or parts of the building can be used to visualize two planes
intersecting in a line (a wall and the ceiling) or three planes intersecting in a point (two walls and the
floor). Many high school students still struggle with visualizing these abstract concepts, the more they can
be realized in everyday objects, the better students will understand them.
Segments and Distance
Using a Ruler - Many Geometry students need to be taught how to use a ruler. The problems stems
from students not truly understanding fractions and decimals. This is a good practical application and
an important life skill. Measuring in centimeters will be learned quickly. Give a brief explanation of
how centimeters and millimeters are marked on the ruler. Since a millimeter is a tenth of a centimeter,
both fractions and decimals of centimeters are easily written. Students need to practice using a ruler and
recognizing centimeters, millimeters and inches. It is very common for students to measure incorrectly,
particularly when using inches because they have difficulty interpreting the fractional divisions of inches
on a ruler. Some may need to be shown how an inch is divided using the marks for \,\,\ and j^. These
fractions often need to be added and reduced to get a measurement in inches.
Another difficulty students encounter is inaccurate measuring because they used the edge of the ruler rather
than the mark at zero. Example 1 in the textbook illustrates the correct way to line up the ruler with a
segment and the diagram which follows this example shows how to read the partial inches and centimeters
on the ruler.
Number or Object - The measure of a segment is a number that can be added, subtracted and combined
arithmetically with other numbers. The segment itself is an object to which postulates and theorems can
be applied. Using the correct notation may not seem important to the students, but is a good habit that
will work to their benefit as they progress in their study of mathematics. For example, in calculus whether
a variable represents a scalar or a vector is critical. To be sure students can differentiate between the two
concepts, emphasize the notation for the measure of AB - Students really struggle with the difference in
meaning of AB,mAB and AB. The first one, AZ?, refers to the object, the segment with endpoints A and B.
The latter two expressions refer to the length of the segment and/or the distance from A to B and may be
used interchangeably.
Segment Addition - Students should be encouraged to always make a sketch of the segment with
particular endpoints and the point between them. Having this concrete diagram will help them avoid
setting up an incorrect equation. The process of going from a description to a picture also helps them
review their vocabulary. One common mistake with these problems is that once students have been
presented with the example in which the point on the segment is the midpoint they sometimes think that
the point on the segment in all subsequent examples is also a midpoint. It is important to help them
read the questions carefully and note terms such as "in the middle" which indicates a midpoint and "in
between" which does not indicate a midpoint.
Review the Coordinate Plane and Horizontal and Vertical Distance - Some students will have
forgotten how to graph an ordered pair on the coordinate plane, or will get the words vertical and horizontal
confused. A reminder that the x-coordinate is first, and measures horizontal distance from the origin,
and that the v-coordinate is second and measures vertical distance from the origin will be helpful. The
coordinates are listed in alphabetical order. When counting these distances on the coordinate plane students
occasionally will count the starting point as one and thus end up with a length that is one unit more than
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the actual distance. Comparing this to when they are playing a board game and counting off squares
as they move their token (you don't count the square you start on as one), or counting laps as they run
around the track (you don't count one until you've completed the first lap) may give them a couple of real
world example to which they can relate an otherwise abstract concept.
Angles and Measurement
Naming Angles with Three Points - Naming and identifying angles named with three points is often
challenging for students when they first learn it. The middle letter of the angle name, the vertex of the
angle, is the most important point. Instruct the students to start by identifying this point and working
from there. Remind students that an angle is made up of two rays and that the three points used to
identify the angle come directly from these two rays. With practice students will become adept at seeing
and naming different angles in a complex picture. Review of this concept is also important. Every few
months give the students a problem that requires using this important skill. It can be difficult for students
to learn all of the different notations and labeling practices, especially in the beginning, but practicing
these skills will help build a strong foundation for students in geometry. It is crucial that they can read,
correctly interpret and correctly communicate in this language in order to be successful in this course.
Using a Protractor - Students need to be shown how to line up the vertex and side of the angle correctly
with the protractors available for their use. Not all protractors are the same and students often struggle
with this procedure, particularly if the protractor the teacher is using is different from theirs. It is worth
taking the time to check with each student to make sure they know how to use their own protractor.
The two sets of numbers on a protractor are convenient for measuring angles oriented in many different
directions, but often lead to errors on the part of the students. There is a simple way for students to check
their work when measuring an angle with a protractor. Visual inspection of an angle usually can be used
to tell if an angle is acute or obtuse. After the measurement is taken, students should notice if their answer
matches with the classification. In fact, encouraging students to always consider whether their answers are
reasonable is a good practice to encourage. Students don't think to do this and often make a small error
in calculation that leads to an answer that is clearly wrong. They can help themselves be more accurate
in their work if they learn to embrace this habit.
Marking Segments and Angles - Students need to be able to interpret these markings and use them
to communicate which angles and segments are congruent and which are not in their own diagrams. It is
imperative to practice these skills with students to avoid confusion later in the course.
Midpoints and Bisectors
Congruent or Equal - Frequently students interchange the words congruent and equal. Stress that
equal is a word that describes two numbers, and congruent is a word that describes two geometric objects.
Equality of measure is often one of the conditions for congruence. If the students have been correctly using
the naming conventions for a segment and its measure and an angle and its measure in previous lessons
they will be less likely to confuse the words congruent and equal now.
The Number of Tick Marks or Arcs Does Not Give Relative Length - A common misconception
is that a pair of segments marked with one tick, are longer than a pair of segments marked with two ticks in
the same figure. Clarify that the number of ticks just groups the segments; it does not give any relationship
in measure between the groups. An analogous problem occurs for angles.
Midpoint or Bisector - Midpoint is a location, a noun, and bisect is an action, a verb. One geometric
object can bisect another by passing through its midpoint. This link to English grammar often helps
students differentiate between these similar terms.
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Intersects vs. Bisects - Many students replace the word intersects with bisects. Remind the students
that if a segment or angle is bisected it is intersected, and it is know that the intersection takes place at
the exact middle.
Orientation Does Not Affect Congruence - The only stipulation for segments or angles to be congruent
is that they have the same measure. How they are twisted or turned on the page does not matter. This
becomes more important when considering congruent polygons later, so it is worth making a point of now.
Labeling a Bisector or Midpoint - Creating a well-labeled picture is an important step in solving many
Geometry problems. How to label a midpoint or a bisector is not obvious to many students. It is often
best to explicitly explain that in these situations, one marks the congruent segments or angles created by
the bisector.
Midpoint Formula - Students often have a hard time remembering this formula. It helps to make the
connection between a midpoint and the "averages" of the x and y coordinates. Students frequently use
subtraction rather than addition in this formula and connecting the formula to finding an average helps
them to remember that it should be addition. It is also important to remind students that the result is a
point, a pair of coordinates and thus it should be written this way.
Angle Pairs
Complementary or Supplementary - The quantity of vocabulary in Geometry is frequently challenging
for students. It is common for students to interchange the words complementary and supplementary. There
are several ways to help students remember which is which. One is to tell students that it is always right
to compliment someone and thus complementary angles add up to 90 degrees or could form a right angle.
Another way to remember that complementary angles add up to 90 degrees is to connect the first letter of
the word complementary to the first letter of the term corner-typically a corner of a piece of paper is a 90
degree angle. It is also important to present students with examples of each of these pairs of these angles
that are separate and adjacent. It is very easy for students to get in the habit of expecting these pairs of
angles to occur one way or the other.
Linear Pair and Supplementary - All linear pairs have supplementary angles, but not all supplementary
angles form linear pairs. Linear pairs are always adjacent pairs of supplementary angles but not all pairs
of supplementary angles are adjacent. Understanding how Geometry terms are related helps students
remember them. Linear pairs are a subset of pairs of supplementary angles.
Angles formed by Two Intersecting Lines - Students frequently have to determine the measures of the
four angles formed by intersecting lines. They can check their results quickly when they realize that there
will always be two sets of congruent angles, and that angles that are not congruent must be supplementary.
They can also check that all four angles measures have a sum of 360 degrees.
Write on the Picture - In a complex picture that contains many angle measures which need to be found,
students should write angle measures on the figure as they find them. Once they know an angle they can
use it to find other angles. When students don't write each angle measure on the diagram they often
overlook a relationship between angles that helps them find another measure. It is easy for them to think
that they should only find the measures of the angles which are asked for in the problem, when in fact it
may be helpful or even necessary to find other, unmarked, angles in the process. This may require them
to draw or trace the picture on their paper. It is worth taking the time to do this. The act of drawing the
picture will help them gain a deeper understanding of the angle relationships.
Proofs - The word proof strikes fear into the heart of many Geometry students. It is important to define
what a mathematical proof is, and let the students know what is expected of them regarding each proof.
Definition: A mathematical proof is a mathematical argument that begins with a truth and proceeds by
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logical steps to a conclusion which then must be true.
The students' responsibilities regarding each proof depend on the proof, the ability level of the students,
and where in the course the proof occurs. Some options are (1) The student should understand the logical
progression of the steps in the proof. (2) The student should be able to reproduce the proof. (3) The
student should be able to create proofs using similar arguments.
Classifying Polygons
Vocabulary Overload - Students frequently interchange the words isosceles and scalene. This would be
a good time to make flashcards. Each flashcard should have the definition in words and a marked and
labeled figure. Just making the flashcards will help the students organize the material in their brains. The
flashcards can also be arranged and grouped physically to help students remember the words and how they
are related. For example, have the students separate out all the flashcards that describe angles. The cards
could also be arranged in a tree diagram to show subsets, for instance equilateral would go under isosceles,
and all the triangle words would go under the triangle card.
Angle or Triangle - Both angles and triangles can be named with three letters. The symbol in front
of the letters determines which object is being referred to. Remind the students that the language of
Geometry is extremely precise and little changes can make a big difference.
Acute Triangles need all Three - A student may see one acute angle in a triangle and immediately
classify it as an acute triangle. Remind the students that unlike the classifications of right and obtuse, for
a triangle to be acute all three angles must be acute.
Equilateral Subset of Isosceles - In many instances one term is a subset of another term. A Venn
diagram is a good way to illustrate this relationship. Having the students practice with this simple instance
of subsets will make it easier for the students to understand the more complex situation when classifying
quadrilaterals. It is also important to point out that these subsets are determined by properties exhibited
in a figure. In this case, and Equilateral triangle possesses all of the characteristics or properties of an
isosceles triangle plus additional properties which make it a subset of the Isosceles triangle category.
Additional Exercises:
1. Draw and mark an isosceles right and an isosceles obtuse triangle.
Answer: The congruent sides of the triangles must be the sides of the right or obtuse angle.
This exercise lays the groundwork for studying the relationship between the sides and angles of a triangle
in later chapters. It is important that students take the time to use a straightedge and mark the picture.
Using and reading the tick marks correctly helps the students think more clearly about the concepts.
Vocab, Vocab, Vocab - If the students do not know the vocabulary well, they will have no chance
at leaning the concepts and doing the exercises. Remind them that the first step is to memorize the
vocabulary. This will take considerable effort and time. The student edition gives a good mnemonic
device, "caving in" for remembering the word concave. Ask the students to create tricks to memorize other
words and have them share their ideas.
Side or Diagonal - A side of a polygon is formed by a segment connecting consecutive vertices, and a
diagonal connects nonconsecutive vertices. This distinction is important when student are working out the
pattern between the number of sides and the number of vertices of a polygon. It is also worth noting that
all diagonals in a convex polygon are inside the figure and at least one diagonal in a concave polygon lies
outside the polygon. This is an additional method to distinguish between concave and convex polygons.
Squaring in the Distance Formula - After subtracting in the distance formula, students will need to
square the result. This result is often a negative number. Remind them that the square of a negative
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number is a positive number. After the squaring step there should be no negatives or subtraction. If they
have a negative in the square root, they have made a mistake.
2.2 Reasoning and Proof
Inductive Reasoning
Process of Inductive Reasoning - Students often struggle with the concept of inductive reasoning. It
is important to emphasize the three steps of the process: making observations, recognizing a pattern and
forming a conjecture. Use real life examples, such as the following:
On Monday the principal decided to play classical music in the cafeteria during lunch. He measured the
volume of noise in during lunch and determined it was 80 decibels. On Tuesday he played no music and
the volume was 90 decibels. On Wednesday he played pop music and the volume was 85 decibels. The
principal repeated this pattern of alternating classical music, no music and pop music a couple more times
and noticed similar results. He concludes that classical music reduces the volume of noise in the cafeteria
the most during lunch.
This example is very much like a science experiment, which are also typically examples of inductive rea-
soning. Finding a pattern in a sequence of numbers or figures and continuing the pattern is also inductive
reasoning. Having students come up with their own examples of inductive reasoning- either made up ex-
amples or examples based on their actually experiences) often helps them to really internalize the concept.
The nth Term - Students enjoy using inductive reasoning to find missing terms in a pattern. They are
good at finding the next term, or the tenth term, but have trouble finding a generic term or rule for the
number sequence. If the sequence is linear (the difference between terms is constant), they can use methods
they learned in Algebra for writing the equation of a line.
Sample: Find a rule for the nth term in the following sequence: 13, 9, 5, 1,...
Sometimes making an input/output table helps students see that the term numbers are the x values and
the terms are the y values.
x 1 2 3 4 ... n
y 13 9 5 1
Since this sequence is arithmetic (each term is found by adding or subtracting the same value, or common
difference, from the previous term), these points lie on a line. From the table, students can identify two
points on the line such as (1, 13) and (2, 9). It may be helpful to do an example where students work in
a group and each student in the group chooses a different pair of points. Then they can find the slope
between the points and write the equation of the line. Each student should come up with the same equation
regardless of which pair of points was chosen. In this case the equation of the line is y = -Ax + 17 and the
rule is -An +17.
True Means Always True - In mathematics a statement is said to be true if it is always true, no
exceptions. Sometimes students will think that a statement only has to hold once, or a few times to be
considered true. Explain to them that just one counterexample makes a statement false, even if there are
a thousand cases where the statement holds. Truth is a hard criterion to meet, proving a statement false
is much less burdensome.
Sequences - A list of numbers is called a sequence. If the students are doing well with the number of
vocabulary words in the class, the term sequence can be introduced.
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Conditional Statements
The Advantages and Disadvantages of Non-Math Examples - When first working with conditional
statements, using examples outside of mathematics can be very helpful for the students. Statements about
the students' daily lives can be easily broken down into parts and evaluated for veracity. This gives the
students a chance to work with the logic, without having to use any mathematical knowledge. The problem
is that there is almost always some crazy exception or grey area that students will love to point out. This
is a good time to remind students of how much more precise math is compared to our daily language. Ask
the students to look for the idea of what you are saying in the non-math examples, and use their powerful
minds to critically evaluate the math examples that will follow. One way to help students avoid confusion
is to use Euler (pronounced "Oiler") diagrams to show the relationship between the hypothesis and the
conclusion. Here is an example based on the conditional statement, "If today is Saturday, then I will go
to the park."
Figure 1: In this example, the hypothesis is, "today is Saturday." The hypothesis is written in the inner
circle. The conclusion, "I will go to the park," is written in the outer circle. This diagram is interpreted
like a Venn Diagram- if the statement in the inner circle is true, then the statement in the outer circle is
also true.
Figure 2: It may help to make an "jc" in the inner circle as shown in the figure and say, "You are here,
where this statement is true. Does this indicate that you are also inside the circle where it is true that you
will go to the park?"
Figure 3: This figure shows an x in the outer circle. Now students should understand that just because
you are inside the circle of "going to the park", that doesn't not necessary require that you are in the circle
of, "it is Saturday." It could be Saturday, but it doesn't have to be Saturday.
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Figure 4: This final figure illustrates the scenario in which we do not go to the park. If the X is outside
of the outer circle, then it is outside the "it is Saturday" circle.
Converse and Contrapositive - The most important variations of a conditional statement are the
converse and the contrapositive. Unfortunately, these two sound similar, and students often confuse them.
Emphasize the converse and contrapositive in this lesson. Ask the students to compare and contrast them.
It is helpful to use the diagrams above to verify the validity of these statements. The Contrapositive can be
shown to be true using the last diagram. The statement would be, "If I do not go to the park, then today
is not Saturday." The converse can be shown to be inconclusive using figure 3. The converse statement is,
"It I go to the park, then today is Saturday". Is this true? The correct answer is no, it is not necessarily
true.
Converse and Biconditional - The converse of a true statement is not necessarily true! The important
concept of implication is prevalent in Geometry and all of mathematics. It takes some time for students to
completely understand the direction of the implication. Daily life examples where the converse is obviously
not true is a good place to start and making the Euler diagrams should help as well. A good question for
students would be, "What would the Euler diagram look like if the converse is true?" They should come
to the conclusion that the two circles would completely overlap. In other words there would be one circle
with two statements inside as in the example below of the conditional statement, "If an angle is a right
angle, then its measure is 90°.
Both of these statements are equivalent because the definition of a right angle is, "an angle which measures
90°. In fact, all definitions can be written as true biconditional statements. The students will spend
considerable time deciding what theorems have true converses (are bicondtional) in subsequent lessons.
Practice, Practice, Practice - Students are going to need a lot of practice working with conditional
statements. It is fun to have the students write and share conditional statements that meet certain
conditions. For example, have them write a statement that is true, but that has an inverse that is false.
There will be some creative, funny answers that will help all the members of the class remember the
material. Encourage students who are struggling to draw the Euler diagram for each conditional statement
to help interpret whether or not the other statements are true.
Deductive Reasoning
Inductive or Deductive Reasoning - Students frequently struggle with the uses of inductive and
deductive reasoning. It is harder for them to see the strengths and weaknesses of each type of thinking,
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and understand how inductive and deductive reasoning work together to form conclusions. Use situations
that the students are familiar with where either inductive or deductive reasoning is being used to familiarize
them with the different types of logic. The side by side comparison of the two types of thinking will cement
the students' understanding of the concepts. It would also be beneficial to have the students write their
own examples. Some examples follow.
Is inductive or deductive reasoning being used in the following paragraph? Why did you come to this
conclusion?
1. The rules of Checkers state that a piece will be crowned when it reaches the last row of the opponent's
side of the board. Susan jumped Tony's piece and landed in the last row, so Tony put a crown on her
piece.
Answer: This is an example of detachment, a form of deductive reasoning. The conclusion follows from an
agreed upon rule.
2. For the last three days a boy has walked by Ana's house at 5 pm with a cute puppy. Today Ana decides
to take her little sister outside at 5 pm to show her the dog.
Answer: Ana used inductive reasoning. She is assuming that the pattern she observed will continue.
3. Paul finds the n th term rule for the arithmetic sequence: 5, 9, 13, 17,... to be 4n + 1. He then uses this
rule to determine that the 100 f/z term is 401.
Answer: This example uses both forms of reasoning. First, Paul uses inductive reasoning to determine the
n th term rule. He then uses deductive reasoning when he uses the rule to find the 100^ term.
A good rule of thumb for establishing which type of reasoning is being used is to think about what part
of the process is occurring. If you are coming up with a conjecture or hypothesis, then it is more likely
inductive reasoning. If you are using a known rule, formula or type of argument then it is most likely
deductive reasoning. As shown in the previous example, the two work together to form and then prove
conjectures or "guesses" about the observed patterns.
Valid Arguments - Students need lots of practice recognizing the valid arguments. The Euler diagrams
in the previous section can be used here as well to show that the Law of the Contrapositive and the Law
of Detachment are valid. By adding a third circle, the Law of Syllogism can also be illustrated in an Euler
diagram.
Converse/Inverse Errors - Students often make the following false conclusions in logical reasoning:
Converse Error: If it rains, then I will bring my umbrella. I bring my umbrella. Therefore, it is raining.
Inverse Error: If it rains, then I will bring my umbrella. It does not rain. Therefore, I do not bring my
umbrella.
In these examples, the conclusion is made by assuming that the converse or inverse of the statement is
true. We learned in the previous section that they are not necessarily true. This is a good opportunity to
review these statements and revisit the Euler diagrams for them again as well.
Algebraic and Congruence Properties
Commutative or Associate - Students sometimes have trouble distinguishing between the commutative
and associative properties. It may help to put these properties into words. The associative property is
about the order in which multiple operations are done. The commutative is about the first and second
operand having different roles in the operation. In subtraction the first operand is the starting amount
and the second is the amount of change. Often student will just look for parenthesis; if the statement has
parenthesis they will choose associate, and they will usually be correct. Expose them to an exercise like
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the one below to help break them of this habit.
What property of addition is demonstrated in each of the following statements?
a. (x + y) + z = z + (x + y)
b. (x + y)+z = x+(y + z)
Answer: For example a, it is the commutative property that ensures these two quantities are equal. On
the left-hand side of the equation the first operand is the sum of x and y, and on the right-hand side of the
equation the sum if x and y is the second operand. In example /?, the parentheses are grouping different
variables so this is an example of the associative property.
Sometimes it helps students to come up with an expression to remember which is which. One example
is: Your group of friends is the people you associate with. This indicates that the associative property
refers to a change in grouping. For commutative, think of the word commute-you move from one place to
another such as going from home to work. When the variables change position in the expression then it is
the commutative property.
Transitive or Substitution - The transitive property is actually a special case of the substitution
property. The transitive property has the additional requirement that the first statement ends with the
same number or object with which the second statement begins. Acknowledging this to the students helps
avoid confusion, and will help them see how the properties fit together. The following statement is true
due to the substitution property of equality. How can the statement be changed so that the transitive
property of equality would also ensure the statement's validity?
If ab = cd, and ab = /, then cd = /.
Answer: The equality ab = cd can be changed to cd = ab due to the symmetric property of equality. Then
the statement would read:
If cd = ab, and ab = /, then cd = f.
This is justified by the transitive property of equality.
Keeping It All Straight - At this point in the class the students have been introduced to an incredible
amount of material that they will need to use in proofs. Laying out a logic argument in proof form is, at
first, a hard task. Searching their memories for terms at the same time makes it near impossible for many
students. A notebook that serves as a "tool cabinet" full of the definitions, properties, postulates, and
later theorems that they will need, will free the students' minds to concentrate on the logic of the proof.
After the students have gained some experience, they will no longer need to refer to their notebook. The
act of making the book itself will help the students collect and organize the material in their heads. It is
their collection; every time they learn something new, they can add to it.
All Those Symbols - In the back of many math books there is a page that lists all of the symbols
and their meanings. The use of symbols is not always consistent between texts and instructors. Students
should know this in case they refer to other materials. It is a good idea for students to keep a page in
their notebooks where they list symbols, and their agreed upon meanings, as they learn them in class.
Some of the symbols they should know at this point in are the ones for equal, congruent, angle, triangle,
perpendicular, and parallel.
Don't Assume Congruence! - When looking at a figure students have a hard time adjusting to the idea
that even if two segments or angles look congruent they cannot be assumed to be congruent unless they
are marked. A triangle is not isosceles unless at least two of the sides are marked congruent, no matter
how much it looks like an isosceles triangle. Maybe one side is a millimeter longer, but the picture is too
small to show the difference. Congruent means exactly the same. It is helpful to remind the students that
they are learning a new, extremely precise language. In geometry congruence must be communicated with
the proper marks if it is known to exist.
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Communicate with Figures - A good way to have the students practice communicating by drawing
and marking figures is with a small group activity. One person in a group of two or three draws and
marks a figure, and then the other members of the group tell the artist what if anything is congruent,
perpendicular, parallel, intersecting, and so on. They take turns drawing and interpreting. Have them use
as much vocabulary as possible in their descriptions of the figures.
Two-Column Proofs
Diagram and Plan - Students frequently want to skip over the diagramming and planning stage of
writing a proof. They think it is a waste of time because it is not part of the end result. Diagramming
and marking the given information enables the writer of the proof to think and plan. It is analogous to
making an outline before writing an essay. It is possible that the student will be able to muddle through
without a diagram, but in the end it will probably have taken longer, and the proof will not be written as
clearly or beautifully as it could have been if a diagram and some thinking time had been used. Inform
students that as proofs get more complicated, mathematicians pride themselves in writing simple, clear,
and elegant proofs. They want to make an argument that is undeniably true.
Teacher Encouragement - When talking about proofs and demonstrating the writing of proofs in class,
take time to make a well-drawn, well-marked diagram. After the diagram is complete, pause, pretend like
you are considering the situation, and ask students for ideas of how they would go about writing this proof.
Assign exercises where students only have to draw and mark a diagram. Use a proof that is beyond their
ability at this point in the class and just make the diagram the assignment.
When grading proofs, use a rubric that assigns a certain number of points to the diagram. The diagram
should be almost as important as the proof itself.
Refer students to the tips for proof writing that appear between Examples 4 and 5 of this section in the
text. They should keep going back to those tips periodically until they become second nature.
Start with "Given", but don't end with "Prove" - After a student divides the statement to be
proved into a given and prove statements he or she will enjoy writing the givens into the proof. It is like
a free start. Sometimes they get a little carried away with this and when they get to the end of the proof
write "prove" for the last reason. Remind them that the last step has to have a definition, postulate,
property, or theorem to show why it follows from the previous steps. Perhaps reminding them the "prove"
is a command statement, not a reason will help them remember that it is just part of the question.
Scaffolding - Proofs are challenging for many students. Many students have a hard time reading proofs.
They are just not used to this kind of writing; it is very specialized, like a poem. One strategy for making
students accustom to the form of the proof is to give them incomplete proofs and have them fill in the
missing statements and reason. There should be a progression where each proof has less already written
in, and before they know it, they will be writing proofs by themselves.
Number or Geometric Object - The difference between equality of numbers and congruence of geomet-
ric objects was addressed earlier in the class. Before starting this lesson, a short review of this distinction to
remind students is worthwhile. If the difference between equality and congruence is not clear in students'
heads, the proofs in this section will seem pointless to them.
Theorems - The concept of a theorem and how it differs from a postulate has been briefly addressed
several times in the course, but this is the first time theorems have been the focus of the section. Now
would be a good time for students to start a theorem section in their notebook. As they prove, or read a
proof of each theorem it can be added to the notebook to be used in other proofs.
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Proofs about Angle Pairs and Segments
Mark-Up That Picture - Angles are sometimes hard to see in a complex picture because they are not
really written on the page; they are the amount of rotation between two rays that are directly written on
the page. It is helpful for students to copy diagram onto their papers and mark all the angles of interest.
They can use highlighters and different colored pens and pencils. Each pair of vertical angles or linear
pairs can be marked in a different color. Using colors is fun, and gives the students the opportunity to
really analyze the angle relationships.
Add New Information to the Diagram - It is common in geometry to have multiple questions about
the same diagram. The questions build on each other leading the student though a difficult exercise. As
new information is found it should be added to the diagram so that it is readily available to use in answering
the next question.
Try a Numerical Example - Sometimes students have trouble understanding a theorem because they
get lost in all the symbols and abstraction. When this happens, advise the students to assign a plausible
number to the measures of the angles in question and work from there to understand the relationships.
Make sure the student understands that this does not prove anything. When numbers are assigned, they
are looking at an example, using inductive reasoning to get a better understanding of the situation. The
abstract reasoning of deductive reasoning must be used to write a proof.
Inductive vs. Deductive Again - The last six sections have given the students a good amount of
practice drawing diagrams, using deductive reasoning, and writing proofs, skills which are closely related.
Before moving on to chapter three, take some time to review the first two sections of this chapter. It is
quite possible that students have forgotten all about inductive reasoning. Now that they have had practice
with deductive reasoning they can compare it to inductive reasoning and gain a deeper understanding of
both. They should understand that inductive reasoning often helps a mathematician decide what should
be attempted to be proved, and deductive reasoning proves it.
Review - The second section of chapter two contains information about conditional statements that will
be used in the more complex proofs in later chapters. Continue to review these variations of the conditional
statement in verbal and symbolic form so that students do not forget them.
2.3 Parallel and Perpendicular Lines
Lines and Angles
Marking the Diagram - Sometimes students confuse the marks for parallel and congruent. When
introducing them to the arrows that represent parallel lines, review the ticks that represent congruent
segments. Seeing the two at the same time helps avoid confusion. Also, explaining that the arrows show
that the lines or segments are going in the same direction or have the same slope may help them understand
why arrows are used and thus help them remember.
When given the information that two lines or segments are perpendicular, students don't always imme-
diately see how to mark the diagram accordingly. They need to use the definition of perpendicular and
mark one of the right angles created by the lines with a box.
Students also struggle with marking angle bisectors and midpoints (or segments bisectors). Just like the
perpendicular lines, they aren't marking the object directly. They need to mark the results, which are
congruent angles or congruent segments in these cases.
Symbol Update - Students should be keeping a list of symbols and how they will be used in this class in
their notebooks. Remind them to update this page with the symbols for parallel, ||, and perpendicular, J_.
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Construction - The parallel and perpendicular line postulates are used in construction. Constructing
parallel and perpendicular lines with a compass and straightedge is a good way to give students kinesthetic
experience with these concepts. Construction can also be done with computer software. To construct
a parallel or perpendicular line the student will select the line they want the new line to be parallel or
perpendicular to, and the point they want the new line to pass through, and chose construct. The way the
programs have the students select the line and then the point reinforces the postulates.
Parallel vs Skew Lines - Students often struggle with the difference between these two scenarios. It is
difficult for them to picture the skew lines in three dimensions and even harder for them to draw them.
Use concrete examples of skew and perpendicular lines (objects in the classroom) to help them visualize
the difference. Also, reinforce the requirement for the lines to be coplanar in the definition of parallel lines.
If it is not specifically stated that the lines are coplanar, then the lines may be skew.
Naming the Angle Pairs Formed when a Transversal Intersects Two Lines - Students often
struggle with memorizing the names of the angle pairs. It helps to go through the names and explain how
the names truly describe the angle pairs. Using colored pencils or highlighters to indicate the space in the
interior of the parallel lines and the space in the exterior of the parallel lines helps distinguish between
alternate exterior and alternate interior. Discussing what alternate vs same side means will help distinguish
between these pair of interior angles. Corresponding angles are in the same location with respect to the
transversal and the line. Using words like above or below and to the right or left helps students locate
the pairs of corresponding angles, particularly in situations where the lines are not parallel and the angles
don't "match".
Properties of Parallel Lines
The Parallel Hypothesis - So far seven different pairs of angles that may be supplementary or congruent
have been introduced. All seven of these pairs are used in the situation where two lines are being crossed
by a transversal forming eight angles. Some of these pairs require the two lines to be parallel and some do
not. Students sometimes get confused about when they need parallel lines to apply a postulate or theorem,
and if a specific pair is congruent or supplementary. A chart like the one below will help them sort it out.
Table 2.1:
Type of Angle Pair
Relationship
Do Not Require Parallel Linear Pairs
Lines Vertical Angles
Supplementary
Congruent
Parallel Lines Required
Corresponding Angles
Alternate Interior Angles
Alternate Exterior Angles
Consecutive Interior Angles
Consecutive Exterior Angles
Congruent
Congruent
Congruent
Supplementary
Supplementary
Encourage students who are really struggling to use common sense when deciding whether a pair of angles
is supplementary or congruent. When parallel lines are intersected by a transversal, any pair of two angles
will be either congruent or supplementary. For students at this level, it is advisable to make these drawings
accurate (i.e. make the lines look parallel if they are parallel) so that students can practice using common
sense. If the angles look congruent, they are congruent and if one is clearly obtuse and the other is clearly
acute, then they are supplementary. This goes against the idea that it is not advisable to encourage
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students to rely on the appearance of a diagram but for very low level students, this can really help them.
In addition, practicing common sense will help them in the real world and in application problems.
Patty Paper Activity - When two lines are intersected by a transversal eight angles are formed in two
sets of four. When the lines are parallel, the two sets of four angles are exactly the same. To help students
see this relationship, have them darken a set of parallel lines on their binder paper a few inches apart (or
they can use the two sides of a ruler to make the parallel lines) and draw a transversal through the parallel
lines. Next, they should trace one set of four angles on some thin paper (tracing paper or patty paper).
When they slide the set of four angles along the transversal they will coincide with the other set of four
angles. Have them try the same thing with a set of lines that are not parallel. This will help students find
missing angle measures quickly and remember when they can transfer numbers down the transversal. It
does not help them learn the names of the different pairs of angles which is important for communicating
with others about mathematical concepts and for writing proofs.
Proving Lines Parallel
When to Use the Converse - It takes some experience before most students truly understand the
difference between a statement and its converse. They will be able to write and recognize the converse of
a statement, but then will have a hard time deciding which one applies in a specific situation. Tell them
when you know the lines are parallel and are looking for angles, you are using the original statements;
when you are trying to decide if the lines are parallel or not, you are using the converse.
Proofs of Converse Theorems - The textbook includes a proof of the Converse of the Alternate Interior
Angles Theorem. It would be helpful to prove each of the other theorems to help students see when the
converse theorems are used.
Example Proof of the Converse of the Alternate Exterior Angle Theorem:
Given: L\ = Z8
Prove: / II m
Table 2.2:
Statements
Reasons
1. L\ = 18
2. L\ = z4
3. z4 = z8
4. / || m
1. Given
2. Vertical Angles Theorem
3. Transitive Property of Angle Congruence
4. Converse of the Corresponding Angles Postulate
The Converse of the Alternate Interior Angle Theorem could also be proved using the Converse of the
Alternate Exterior Angle Theorem. This would demonstrate to the students that once a theorem has been
proved, it can be used in the proof of other theorems. This demonstrates the building block nature of
math. Here is one way to do this using the same diagram from above.
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Given: L\ = Z8
Prove: / II m
Table 2.3:
Statements Reasons
1. L\ = Z8 1. Given
2. L\ = Z4 2. Vertical Angles Theorem
3. Z5 = Z8 3. Vertical Angles Theorem
4. Z4 = Z5 4. Substitution Property of Angle Congruence
5. / || m 5. Converse of the Alternate Interior Angles The-
orem
Proving the theorem in several ways gives students a chance to practice with the concepts and their proof
writing skills. Similar proofs can be assigned for the other theorems in this section.
Properties of Perpendicular Lines
Complementary, Supplementary, or Congruent - When finding angle measures, students generally
need to decide between three possible relationships: complementary, supplementary, and congruent. A good
way for them to practice with these and review their equation solving skills, is to assign variable expressions
to angle measures, state the relationship of the angles, and have the students use this information to write
an equation that when solved will lead to a numerical measurement for the angle. Encourage students to
take the time to write out and solve the equation neatly. This process helps them avoid errors. Many times
students will find the value of jc, and then stop without plugging in the value to the expression for the
angle measures. Have the students find the actual angles measures by plugging their x value in to verify
that their final answers are angle measures that have the desired relationship.
Interpreting "Word" Questions — Students often have difficulty translating the words in a statement
into an algebraic equation. Here are a couple examples that might help students interpret verbal questions
that are not accompanied by a diagram.
Example 1: Two vertical angles have measures (2x - 30)° and (x + 60)°. Find the measures of the two
angles.
Students may wish to make a diagram of vertical angles, then label them with these measures before
setting up the equation: 2x-30 = x + 60. Once they solve for x, students need to plug this value into both
expressions to get the measures of the two angles.
Remind students that these angles should have equal measures. Model for students the process of checking
your answers for reasonableness by asking them if the values make sense and how they know. At first
they may have trouble answering these questions but if you consistently model this process, it may become
second nature to them as well and help them to identify and correct mistakes as they solve problems
throughout the course.
Example 2: The outer rays of two adjacent angle with measures (4x+10)° and (5x-10)° are perpendicular.
Find the measures of each angle.
This example contains a lot of information that students will need to sort out. In this case a diagram is
especially useful. Break down the information with students and help them diagram the angles to discover
that the sum of the two angles must be 90°. Here is the equation and value of x:
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Ax + 10 + 5x - 10 = 90
x= 10
Using this value of x, the two angles are 50° and 40°, respectively. Do these measures make sense? How do
you know? Practice asking and answering these questions yourself so you can help students answer them
correctly.
Example 3: The angles of a linear pair have measures (3x + 45)° and (2x + 35)°. Find the measure of each
angle.
Example 4: Perpendicular lines form an angle with measure (8x + 10)°. What is the value of x?
Again, help students interpret the given information to make a diagram and set up an equation to solve.
Then remind students to plug in their x value to find the actual angle measures. Finally, prompt students
to check their work for reasonableness and accuracy.
Answers:
Example 3: x = 20° and the angles are 105° and 75°.
Example 4: x = 10°
Proof Tip - In a proof, students must first state which lines are perpendicular and why, then they can say
that all four angles formed by those perpendicular lines are right angles, then right angles are congruent,
etc. Students are apt to just jump to the final conclusion because the lines are perpendicular. In a complete
proof, these middle steps are important to show understanding of the thought process. Refer to the proof
in question 26 of the review problems for an example.
Parallel and Perpendicular Lines in the Coordinate Plane
Order of Subtraction - When calculating the slope of a line using two points it is important to keep
straight which point was made point one and which one was point two. It does not matter how these labels
are assigned, but the order of subtraction has to stay the same in the numerator and the denominator
of the slope ratio. If students switch the order they will get the opposite of the correct answer. If they
have a graph of the line, ask them to compare the sign of the slope to the direction of the line. Is the line
increasing or decreasing? Does that match the slope? This will give them another opportunity to practice
checking their results for reasonableness.
Another strategy to help students with this is to have them write the points vertically and subtract as
shown in the diagram below:
/ (4, -3) \
W-2,5)/
-3-5 _ ^8 _ z±
4-(-2) — 6 — 3
Graphing Lines with Integer Slopes - The slope of a line is the ratio of two numbers. When students
are asked to graph a line with an integer slope they often fail to realize what and where the second number
is. Frequently they will make the "run" of the line zero and graph a vertical line. It is helpful to have
them write the slope as a ratio over one before they do any graphing. They may only need to do this a
few times on paper before they are able to graph the lines correctly.
Zero or Undefined - Students need to make these associations:
Zero in numerator - slope is zero - line is horizontal
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Zero in denominator - slope is undefended - line is vertical
Students frequently switch these around. Try to connect these concepts to their experiences. For example
explain that when they are walking on a flat surface, they are not going up or down so the slope is zero.
Ask them if they can walk up a vertical wall, hopefully they will say, "No." Then you can explain that
the slope of this wall is undefined. After the relationships are explained in class, remind them frequently,
maybe have a poster up in the room that shows lines with a positive slope, negative slope, zero slope
(horizontal) and undefined slope (vertical). You could also write the relationship on a corner of the board
that does not get erased.
Students also struggle with realizing that a line perpendicular to a horizontal line is vertical and vice versa.
When they look at these equations, the slope is not evident and so they don't know what to do. Practice
lots of examples with horizontal and vertical lines.
Use Graph Paper - Making a connection between the numbers that describe a line and the line itself is
an important skill. Requiring that the students use graph paper encourages them to make nice, thoughtful
graphs, and helps them make this connection.
The y-axis is Vertical - When using the slope-intercept form to graph a line or write an equation, it
is common for students to use the x-intercept instead of the y-intercept. Remind them that they want
to use the vertical axis, y-intercept, to begin the graph. Requiring that the y-intercept be written as a
point, say (0, 3) instead of just 3, helps to alleviate this problem.
Where's the Slope - Students are quickly able to identify the slope as the coefficient of the x-variable
when a line is in slope-intercept form. Unfortunately they sometimes extend this to standard form. Remind
the students that if the equation of a line is in standard form, or any other form, they must first algebraically
convert it to slope-intercept form before they can easily read off the slope. It is wise to do several examples
which require changing the equation from standard form into slope-intercept form.
Why Use Standard Form - The slope-intercept form of the line holds so much valuable information
about the graph of a line that students probably won't understand why any other form would ever be
used. Mention to them that standard form is convenient when solving systems and putting equations into
matrices, things they will be doing in their second year of algebra, to motivate them to learn and remember
the standard form.
Organizing Work in Multi-Step Problems - Part of the struggle with these problems is that students
get lost in the process. They get wrapped up in a particular step and forget where they are going. It may
be helpful to have students write out the steps they will need to follow in the beginning so that they will
have a road map to follow. They can even leave some space between each step to go back and fill in with
their work. Here is an example of this:
Find the equation of the line that is perpendicular to the line passing through the points (5, 7) and (12,
3) and passes through the second point.
1) Find the slope between the given two points.
3-7 -4
m = = —
12-5 7
2) Find the opposite reciprocal (perpendicular) slope.
7
J_ m = -
4
3) Use the perpendicular slope and the given point (12, 3) to find the y-intercept.
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4) Write the new equation.
4
3 = 21 + /?
-18 = *
7
y= -X-18
Many students forget that at the end they need to write an equation with variables x and y. They don't
quite understand that this is an equation that relates all possible pairs of x and y coordinates. Explain
to them that they need to use the given pair to determine what the y-intercept is, but not in the final
equation.
The Distance Formula
The Perpendicular Distance - In theory, measuring along a perpendicular line makes sense to the
students, but in practice, when lining up the ruler or deciding which points to put in the distance formula,
there are many distractions. Students can evaluate their decision by taking a second look to see if the path
they chose was the shortest one possible. This is another opportunity to practice checking their work for
reasonableness.
Where to Measure? - Now that the students know to measure along a line that is perpendicular to
both parallel lines, they might wonder where along the lines to measure. When working on a coordinate
plane it is best to start with a point that has integer coordinates, just to keep the problem simple and
accurate. They will get the same distance no matter where they measure though. You may wish to share
the alternate definition of parallel lines: Two lines that are a constant distance apart.
Order of Operations - Students might want to cancel the squares with the square root in the distance
formula, even though they cannot. See example below:
v
(xi - x 2 ) 2 + (yi - y 2 ) 2 ± (xi - x 2 ) + (ji - y 2 )
Remind students that the square root is like parenthesis. The operations contained inside the square root
must be performed before the square root can be taken. Give students lots of opportunity to practice
finding the distance between two points to practice this formula.
2.4 Triangles and Congruence
Triangle Sums
Interior vs. Exterior Angles - Students frequently have trouble keeping interior and exterior angles
straight. They may fail to identify to which category a specific angle belongs and include an exterior angle
in a sum with two interior. They also sometimes use the wrong total, 360°, verses 180°. Encourage the
students to draw the figure on their papers and color code it. They can highlight or use a specific color of
pencil to label all the exterior angle measures and another color for the interior angle measures. Then it
is easy to do some checks on their work. Each interior/exterior adjacent (linear) pair should have a sum
of 180°, all of the interior angles should add to 180°, and the measures of the exterior angles total 360°.
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Find All the Angles You Can - When a student is asked to find a specific angle in a complex figure
and they do not immediately see how they can do it, they can become stuck, and not see how to proceed.
A good strategy is to find any angle they can and write the measures in the diagram, even if it is not
the one they are after. Finding other angles keeps their brains active and working, they practice using
angle relationships, and the new information will often help they find the target angle. Students often miss
possible connections when they don't write down all of the angles as they find them. It is important that
students know that many exercises are not designed to do in one step.
Congruent Angles in a Triangle - In later sections students will study different ways of determining if
two or more angles in a triangle are congruent, and will then have to use this information to find missing
angles in a triangle. To start them on this process it is good to have them work with triangles in which
two angles are stated to be congruent. A few example problems follow:
Example 1: An acute triangle has two congruent angles each measuring 70°. What is the measure of the
third angle?
Encourage students to make a triangle diagram and label two angles 70°. This helps review making and
marking diagrams and gives them a concrete visual to work with. Next, students should set up an equation
showing that the two given angles plus the third, unknown angle, will add up to 180°. Finally, students
should solve and find that the third angle is 40°.
Example 2: An obtuse triangle has two congruent angles. One angle of the triangle measures 130°. What
are the measures of the other two angles?
Again, drawing a diagram will help students organize their thoughts. They will recognize that the obtuse
angle cannot be one of the congruent angles. They should arrive at the conclusion that each of the congruent
angles should be 25°.
Congruent Figures
Rotation Difficulties - When congruent triangles are shown with different orientations, many students
find it difficult to rotate the figures in their head to align corresponding sides and angles. Remind them
that the angles and sides marked congruent are the corresponding pairs of angles and sides. Have students
practice listing the corresponding pairs of congruent sides and angles. Another recommendation is to
redraw the figures on paper so that they have the same orientation. It may be necessary for students to
physically rotate the paper at first. After students have had some time to practice this skill, most will be
able to skip this step.
Stress the Definition - The definition of congruent triangles requires six congruencies, three pairs of
angles and three pairs of sides. If students understand what a large requirement this is, they will be more
motivated to develop the congruence shortcuts in subsequent lessons.
The Language of Math - Many students fail to see that math is a language, a form of communication,
which is extremely dense. Just a few symbols hold great amounts of information. The congruence state-
ments for example, not only tell the reader which triangles are congruent, but which parts of the triangle
correspond. When put in terms of communication students have an easier time understanding why they
must put the corresponding vertices in the same order when writing the congruence statement.
Third Angle Theorem by Proof - In the text an example is given to demonstrate the Third Angle
Theorem, this is inductive reasoning. A deeper understanding of the theorem, and different types of
reasoning, can be gained by using deductive reasoning to write a proof. It will also reinforce the idea that
theorems must be proved, and shows how inductive and deductive reasoning work together. Use review
question 23 as a template for a proof of the Third Angle Theorem.
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Triangle Congruence Using SSS and SAS
One Triangle or Two - In previous chapters, students learned to classify a single triangle by its sides.
Now students are comparing two triangles by looking for corresponding pairs of congruent sides. Evaluating
the same triangle in both of these ways helps the students remember the difference, and is a good way
to review previous material. For instance, students could be asked to draw a pair of isosceles triangles
that are not congruent, and a pair of scalene triangles that can be shown to be congruent with the SSS
postulate.
Congruent Segments - This is a good time to remind students that overlapping (shared) segments will
be congruent. Also, remind students that when they are given a midpoint, they can mark the two halves
of the segment congruent. Practice this with them and illustrate marking the diagram appropriately.
Correct Congruence Statements - Determining which vertices of congruent triangles correspond is
more difficult when no congruent angles are marked. Once the students have determined that the triangles
are in fact congruent using the SSS Congruence postulate, it is advisable for them to mark congruent
angles before writing the congruence statement. Corresponding congruent angles are found by matching
up side markings. The angle made by the sides marked with one and two tick marks corresponds to the
angle made by the corresponding sides in the other triangle, and so on.
The Included Angle - Students often have a hard time differentiating between SAS and SSA. It helps
to have students practice identifying the angle included between two particular sides in a triangle. This
should be practiced using a diagram and using the name of the triangle. This will help students identify
which triangle congruence theorem is being used are help them identify correct pairs of corresponding
triangle parts.
Example: What is the included angle between sides AB and BC in AABC7
Answer: The included angle is LB. Note that it is the point that is an endpoint in both segments.
Tricky Problems - When looking to see which triangle congruence theorem is being used, students are
liable to only look at one of the triangles and make a mistake as shown in the example below. This is
problem #3 from the review questions for this section in the textbook.
If a student were to only look at AAZ?C, he or she might say that these triangles are congruent by SAS.
The second triangle, though it does have a pair of sides and an angle congruent to a pair of sides and an
angle in the first triangle, does not have the angle included between the sides.
SSA - Students are apt to try to use this as a triangle congruence theorem. It is helpful to have them
attempt to construct two different triangles with the same two side lengths and non-included angle. This
will only be possible in certain cases (think back to the ambiguous case for the Law of Sines- this is the
connection). Here are two side lengths and a non-included angle for which two distinct triangles can be
formed. AABC with AB = 12 cm, BC = 5 cm and mlA = 35°.
Student diagrams should look something like this:
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7 err
C
This diagram is scaled down but should reflect the general "shape" of student work.
Using the Distance Formula to Prove Triangles Congruent - It may be necessary to review the
distance formula with students again and remind them that they cannot "distribute" the square root. In
other words, remind them that:
4
(xi - x 2 f + (yi - y 2 f * (xi - x 2 ) + (ji - y 2 )
Triangle Congruence Using ASA, AAS and HL
AAA - Students sometimes have to think for a bit to realize that AAA does not prove triangle congruence.
Ask them to think back to the definition of triangle. Congruent triangles have the same size and shape.
Most students intuitively see that AAA guarantees that the triangles will have the same shape. To see
that triangles can have AAA and be different sizes ask them to consider a triangle they are familiar with,
the equiangular triangle. They can draw an equiangular triangle on their paper, and you can draw an
equiangular triangle on the board. The triangles have AAA, but are definitely different in size. This is a
counterexample to AAA congruence. Have the students note that the triangles are the same shape; this
relationship is called similar and will be studied in later chapters.
Why Not LL? - Some students may wonder why there is not a LL shortcut for the congruence of right
triangles. It also leads to SSS when the Pythagorean theorem is applied. Have the students explore the
situation with a drawing. They can draw out two congruent right triangles and mark sides so that the
triangles have LL. There is already a congruence guarantee for this, SAS. What would the non-right triangle
congruence be for HL? Is this a guarantee? (It would be SSA, and no, this does not work in triangles that
are not right.)
Importance of Right Triangles - When using math to model situations that occur in the world around
us the right triangle is used frequently. Have the students think of right angles that they see every day:
walls with the ceilings and the floors, widows, desks, and many more constructed objects. Right triangles
are also important in trigonometry which they will be studying soon. Stressing the usefulness of right
triangles will motivate them to think about why HL guarantees triangle congruence but SSA, in general,
does not.
Congruent Angle/ Segment Pairs - It is important to review what pairs of angles will be congruent
in diagrams. Students may forget that they can mark vertical angles and shared (overlapping) angles
congruent. They may also need a reminder that if segments are parallel, then alternate interior angles will
be congruent. Remind them as well that shared sides can be marked congruent.
Marking the Diagram - Once again, students should be encouraged to mark all given information and
all deduced congruencies in their diagrams. Seeing the markings will help them determine which triangle
congruence theorem is being utilized.
An Important Distinction - At first students may not see why it is important to identify whether ASA
or AAS is the correct tool to use for a specific set of triangles. They both lead to congruent triangles, right?
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Sometimes either can be used to prove triangles are congruent, but this will not always be the case, as they
will see in the next lesson. Sometimes the configuration of the corresponding congruent sides and angles in
the triangles determines if the triangles can be proved to be congruent or not. Knowing this will motivate
students to study the difference between ASA and A AS. This is a good time to practice identifying the
"included" side between two angles. This will help students see when a side is included and when it is not.
AAS or SAA - Sometimes students try to list the congruent sides and angles in a circle as they move
around the triangle. This could result in AAS or SAA when there are two pairs of congruent angles and
one pair of congruent sides that is not between the angles. They know AAS proves congruence and want
to know if SAA does as well. When this occurs it is best to redirect their thinking process. With two sets
of angles and one set of sides there are only two possibilities, the side is between the angles or it is another
side. When it is between the angles we have ASA, if it is either of the other two sides we use SAA. This
same situation occurs with SSA, but is even more important since SSA is not a test for congruence. A good
way for the students to remember this is that when the order of SSA is reversed it makes an inappropriate
word. This word should not be used in class or in proofs, even if it is spelled backwards.
Patterns and Structure - All of the shortcuts to triangle congruence require three pieces of information.
Reinforce this concept with students are you complete proofs. Students often wonder why we need to
include statements such as BC = BC when it is so obvious to them that it is the same segment and that it
has to be congruent to itself. It is important to the structure of the proof that we include exactly which
segment and/or angle pairs we are using in order to conclude the triangles are congruent by a particular
triangle congruence theorem.
Isosceles and Equilateral Triangles
The Useful Definition of Congruent Triangles - The arguments used in the proof of the Base Angle
Theorem apply what the students have learned about triangles and congruent figures in this chapter, and
what they learned about reasoning and implication in the second chapter. It is a lot of information to
bring together and students may need to review before they can fully understand the proof.
This is a good point to summarize what the students have learned in this chapter about congruent triangles
and demonstrate how it can be put to use. To understand this proof, students need to remember that the
definition of congruent triangles requires three pairs of congruent sides and three pairs of congruent angles,
but realize that not all six pieces of information need to be verified before it is certain that the triangles
are congruent. There are shortcuts. The proof of the Base Angle Theorem uses one of these shortcuts
and jumps to congruence which implies that the base angles, a pair of corresponding angles of congruent
triangles, are congruent.
To a student new to geometry this argument is not as straightforward as it may seem to an instructor
experienced in mathematical proofs. Plan to take some time explaining this important proof.
A Proved Theorem Can Be Used - Now that the students have the proof of the Base Angle Theorem
they can use it as opportunities present themselves. They should be on the lookout for isosceles triangles
in the proofs of other theorems, in complex figures, and in all other situations. When they spot them, they
need to immediately apply the Base Angle Theorem and mark those base angles congruent. This is true for
the converse as well. When they spot a triangle with congruent angles, they should mark the appropriate
sides congruent. Students sometimes do not realize what a powerful tool this theorem is and that they will
be using it extensively throughout this class, and in math classes they will take in the future.
The Process of Writing a Proof - When students first start examining pairs of triangles to determine
congruence it is difficult for them to sort out all the sides and angles.
The first step is for them to copy the figure onto their paper. It is helpful to color code the sides and
angles, congruent sides marked in one color and the congruent angles in another. Some congruent parts
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will not be marked in the original figure that is given to the students in the text. For example, there could
be an overlapping side that is congruent to itself, due to the reflexive property; mark it as well. Then they
should do a final check to ensure that the congruent parts do correspond.
The next step is for them to count how many pairs of congruent corresponding sides and how many pairs
of congruent corresponding angles there are. With this information they can eliminate some possibilities
from the list of way to prove triangles congruent. If there is no right angle they can eliminate HL, or if
they only have one set of corresponding congruent angles, they can eliminate both ASA and A AS.
If at this point there is still more than one possibility, they are going to need to decide if an angle is
between two sides or if a side is between two angles. Remind them that both ASA and A AS can be used to
guarantee triangle congruence, and that SAS works, but that SSA cannot be used to prove two triangles
are congruent.
If all postulates and theorems have been eliminated, then it is not possible to determine if the triangles
are congruent.
2.5 Relationships with Triangles
Midsegments
Don't Forget the \ — In this section there are two types of relationships that the students need to
keep in mind when writing equations with variable expressions. The first involves the midpoint. When
the expressions represent the two parts of a segment separated by the midpoint they just have to set the
expressions equal to each other. The second is when comparing the length of a side of the triangle with
the midsegment parallel to it. In this case they need to multiply the expression representing the side of the
triangle by ^, and then set it equal to the expression representing the midsegment. They may forget the \
or forget to use parenthesis and distribute. Remind them that they need to multiply the entire expression
by ?p not just the first term. Similarly, they may mess up the distribution going the other way. When
given the midsegment, they must multiply the whole expression by 2 to get the third side of the triangle.
Midpoint Formula, Distance Formula and Slope Formula - You may need to review these formulas
again. Remind students that the Midpoint Formula produces a point. Also, remind students that the
slopes of parallel lines are the same.
Parallel - Students may forget that a midsegment is also parallel to the third side of a triangle. They are
apt to focus most on the length relationship since it is used most in the problem sets.
Perpendicular Bisectors and Angle Bisectors in Triangles
Construction Frustrations - Using a compass and straightedge to make clean, accurate constructions
takes a bit of practice. Some students will pick up the skill quickly and others will struggle. Practice,
practice, practice. Help the students individually to make nice arcs. A few minutes of practicing just
making circles will help them to get more comfortable and accurate with the compass. They will know
right away if they are changing the size of the arc mid circle because the "ends" will not meet up. What is
nice about doing construction in the classroom is that it is often the students that typically struggle with
mathematics, the more artistically minded students, that excel and learn from constructing figures.
Here are some other tips for good construction: (1) Hold the compass at an angle to the paper rather than
perpendicularly. Suggest that students try not to press down very hard- a light arc is sufficient and will
be easier to make. (2) Try rotating the paper while holding the compass steady. (3) Work on a stack of a
few papers so that the needle of the compass can really dig into the paper and will not slip. (4) Suggest
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that students hold the compass by the "circle" or vertex where the two radii meet- often students will try
to hold the compass by the needle and pencil and they grip it so tightly that they change the angle in mid
arc.
Perpendicular Bisector Quirks - There are two key ways in which the perpendicular bisector of a
triangle is different from the other segments in the triangle that the students will learn about in subsequent
sections. Since they are learning about the perpendicular bisector first these differences do not become
apparent until the end of the chapter.
The perpendicular bisector of the side of a triangle does not have to pass through a vertex. Have the
students explore in what situations the perpendicular bisector does pass through the vertex. They should
discover that this is true for equilateral triangles and for the vertex angle of isosceles triangles.
The point of concurrency of the three perpendiculars of a triangle, the circumcenter, can be located outside
the triangle. This is true for obtuse triangles. The circumcenter will be on the hypotenuse of a right triangle.
This is also true for the orthocenter, the point of concurrency of the altitudes.
Circumcenter and Perpendicular Bisector Relationship - Stress the relationship between the points
on a perpendicular bisector and the center of a circle. Remind students that the definition of the center
of a circle is the point equidistant from every point on the circle. Remind students that every point on
a perpendicular bisector is equidistant from the endpoints of a segment. Make the connection that this
means that the point of concurrency of the perpendicular bisectors is equidistant from all three vertices of
the triangle. This means that the vertices all lay on a circle with the center at this point of concurrency-
the circumcenter.
Same Construction for Midpoint and Perpendicular Bisector - The Perpendicular Bisector The-
orem is used to construct the perpendicular bisector of a segment and to find the midpoint of a segment.
When finding the midpoint, the students should make the arcs, one from each endpoint with the same
compass setting, to find two equidistant points, but instead of drawing in the perpendicular bisector, they
can just line up their ruler and mark the midpoint. This will keep the drawing from getting overcrowded
and confusing.
Incenter and Angle Bisector Relationship - Stress the relationship between the incenter and the
angle bisectors of a triangle. Again, discuss the definition of the center of a circle. Also, remind students
that all the points on an angle bisector are equidistant from the sides of the angles. In this case, the
point of concurrency of the angle bisectors will be equidistant from the three sides of the triangle. Remind
students that the distance measured from a point to a line (or segment in this case) is measured along a
perpendicular segment. You may wish to demonstrate constructing the perpendiculars from the incenter
to each of the sides. This segment is the radius of the inscribed circle. Students may not be able to do this
accurately themselves but doing this in a demonstration for them may help make the concept stick.
Adaptation - For practical purposes, you may wish to skip having students construct the perpendiculars
to determine the correct compass setting for the inscribed circle. In this case, have students to place the
center of the compass at the incenter, choose one side, and adjust the compass setting until the compass
brushes by that side of the triangle, without passing through it. The word tangent does not have to be
introduced at this point if the students already have enough vocabulary to learn. When the incenter is
correctly placed, the compass should also hit the other two sides of the triangle once, creating the inscribed
circle.
Check with a Third - When constructing the point of concurrency of the perpendicular bisectors or
angle bisectors of a triangle, it is strictly necessary to construct only two of the three segments. The
theorems proved in the texts ensure that all three segments meet in one point. It is advisable to construct
the third segment as a check of accuracy. Sometimes the compass will slip a bit while the student is doing
the construction. If the three segments form a little triangle, instead of meeting at a single point, the
student will know that their drawing is not accurate and can go back and check their marks.
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Special Triangles - In some special triangles, these segments overlap. The following examples may be
used to have students explore these cases.
Example 1: Construct an equilateral triangle. Now construct the perpendicular bisector of one of the sides.
Construct the angle bisector from the angle opposite of the side with the perpendicular bisector. What
do you notice about these two segments? Will this be true of a scalene triangle? Consider what would
happen if you found the circumcenter and the incenter. Where would they be located?
Answer: The segments should coincide on the equilateral triangle, but not on the scalene triangle. The
incenter and the circumcenter will be located in the same place. You could extend this to an isosceles
triangle to show that the angle bisector of the vertex angle overlaps with the perpendicular bisector of the
base.
Example 2: Construct an equilateral triangle. Now construct one of the angle bisectors. This will create
two right triangles. Label the measures of the angles of the right triangles. With your compass compare
the lengths of the shorter leg to the hypotenuse of either right triangle. What do you notice?
Answer: The hypotenuse should be twice the length of the shorter leg. You may wish to refer back to
this example in the next lesson to show that the angle bisector is the same segment as the median in an
equilateral triangle.
Medians and Altitudes in Triangles
Vocabulary Overload - So far this chapter has introduced to a large number of vocabulary words, and
there will be more to come. This is a good time to stop and review the new words before the students
become overwhelmed. Have them make flashcards, or play a vocabulary game in class.
Label the Picture - When using the Concurrency of Medians Theorem to find the measure of segments,
it is helpful for the students to copy the figure onto their paper and write the given measures by the appro-
priate segments. When they see the number in place, it allows them to concentrate on the relationships
between the lengths since they no longer have to work on remembering the specific numbers.
Median or Perpendicular Bisector - Students sometimes confuse the median and the perpendicular
bisector since they both involve the midpoint of a side of the triangle. The difference is that the perpen-
dicular bisector must be perpendicular to the side of the triangle, and the median must end at the opposite
vertex. Show lots of examples of the two of these on the same triangle so students have a visual memory
of the difference. Discuss with students (or have them explore and then discuss) that these segments will
be the same for each vertex and opposite side in equilateral triangles and for the median drawn from the
vertex angle of an isosceles triangle and the perpendicular bisector of the base.
Applications - Students are much more willing to spend time and effort learning about topics when they
know of their applications. Questions like the ones below improve student motivation.
In the following situations would it be best to find the circumcenter, incenter, or centroid?
Example 1: The drama club is building a triangular stage. They have supports on all three corners and
want to put one in the middle of the triangle.
Answer: Centroid, because it is the center of mass or the balancing point of the triangle
Example 2: A designer wants to fit the largest circular sink possible into a triangular countertop.
Answer: Incenter, because it is equidistant from the sides of the triangle.
Extending the Side - Many students have trouble knowing when and how to extend the sides of a
triangle when drawing in an altitude. First, this only needs to be done with obtuse triangles when drawing
the altitude that intersects the vertex of one of the acute angles. It is the sides of the triangle that form
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the obtuse angle that need to be extended. The students should rotate their paper so that the vertex of
the acute angle they want to start an altitude from is above the other two, and the segment opposite of
this vertex is horizontal. Now they just need to extend the horizontal side until it passes underneath the
raised vertex.
The Altitude and Distance - The distance between a point and a line is defined to be the shortest
segment with one endpoint on the point and the other on the line. It has been shown that the shortest
segment is the one that is perpendicular to the line. So, the altitude is the segment along which the distance
between a vertex and the opposite side is measured. Seeing this connection will help students remember
and understand why the length of the altitude is the height of a triangle when calculating the triangle's
area using the formula A = - bh.
Orthocenter - Students need to connect this term to something to help them remember that it is the
point of concurrency of the altitudes in a triangle. They are likely unfamiliar with the term orthogonal -
which means perpendicular. Telling them this may help them connect the term orthocenter to the altitudes
(or perpendicular segments from vertices to the opposite side in triangles).
Explorations - When students discover a property or relationship themselves it will be much more
meaningful. They will have an easier time remembering the fact because they remember the process that
resulted in it. They will also have a better understanding of why it is true now that they have experience
with the situation. Unfortunately, students sometimes become frustrated with explorations. They may not
understand the instruction, or they may not be carefully enough and the results are unclear. Some of the
difficulties can be alleviated by having the students work in groups. They can work together to understand
the directions and interpret the results. Students strong in one area, like construction, can take on that
part of the task and help the others with their technique.
Some guidelines for successful group work.
• Groups of three work best.
• The instructor should choose the groups before class.
• Students should work with new groups as often as possible.
• Desks or tables should be arranged so that the members of the group are physically facing each other.
• The first task of the group is to assign jobs: person one reads the directions, person two performs
the construction, person three records the results. Students should regularly trade tasks.
• Teachers need to circulate and provide additional assistance so that groups do not get frustrated or
go off task.
Inequalities in Triangles
The Opposite Side/Angle - At first it may be difficult for students to recognize what side is opposite
a given angle or what angle is opposite a given side. If it is not obvious to them from the picture, obtuse,
scalene triangles can be confusing, they should use the names. For AAZ?C, the letters are divided up by
the opposite relationship, the angle with vertex A is opposite the side with endpoints B and C. Being able
to determine these relationships without a figure is important when studying trigonometry.
Small, Medium, and Large - When working with the relationship between the sides and angles of a
triangle, students will summarize the theorem to "largest side is opposite largest angle". They sometimes
forget that this comparison only works within one triangle. There can be a small obtuse triangle in the
same figure as a large acute triangle. Just because the obtuse angle is the largest in the figure, does not
mean the side opposite of it is the longest among all the segments in the figure, just that it is the longest in
that obtuse triangle. If the triangles are connected or information is given about the sides of both triangles,
a comparison between triangles could be made. See exercise #29 in the text.
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Add the Two Smallest - The triangle inequality says that the sum of the lengths of any two sides of a
triangle is greater than the length of the third side. In practice it is enough to check that the sum of the
lengths of the smaller two sides is larger than the length of the longest side. When given the three sides
lengths for a triangle, students who do not fully understand the theorem will add the first two numbers
instead of the smallest two. When writing exercises it is easy to always put the numbers in ascending order
without thinking much about it. Have the students try to draw a picture of the triangle. After making a
few sketches they will understand what they are doing, instead of just blindly following a pattern.
Range of Possible Lengths - Students are so used to finding a single "answer" to a problem that they
often struggle with the idea that the third side of a triangle could take on multiple possible lengths and
how to figure out this range of values. A shortcut is that the length of the third side is always between
the difference and sum of the two know lengths. Students also do not immediately understand why the
inequality shows values that the third side cannot assume. For example in a triangle with known sides,
5 and 8, the range of possible values for the third side is 3 < x < 13. They may question why it isn't
4 < x < 12 since the third side cannot be 3 or 13. Remind them that the sides don't have to have whole
number lengths.
SSS and SAS Use Color - When the figures have two triangles instead of just one, they become more
complex. The students may need some help sorting out the shapes. A good way for them to begin this
process is to draw the figure on their paper and use highlighters to color code the information.
Both of the theorems presented in this section require two pairs of congruent sides. The first step is for
student to highlight these four sides in a common color, let's say yellow. Once they have identified the two
pairs of congruent sides, they know the hypothesis of the theorem has been filled and they can apply the
conclusion.
The conclusions of these theorems involve the third side and the angle between the two congruent sides.
These parts of the triangles can be highlighted in a different color, let's say pink.
Now the students need to determine if they need to use the SAS Inequality theorem or the SSS inequality
theorem. If they know one of the pink angles is bigger than the other, than they will use the SAS Inequality
theorem and write an inequality involving the pink sides. If they know that one of the pink sides is bigger
than the other, they will apply the SSS Inequality theorem, and write an inequality involving the two pink
angles.
Having a step-by-step process is good scaffolding for students as they begin working with new types of
problems. After the students have gained some experience, they will no longer need to go through all the
steps.
Solving Inequalities - Students learned to solve inequalities in algebra, but a short review may be in
order. Solving inequalities involves the same process as solving equations except the equal sign is replaced
with an inequality, and there is the added rule that if both sides of the inequality are multiplied or divided
by a negative number the direction of the inequality changes. Students frequently want to change the
direction of the inequality when it is not required. They might mistakenly change the inequality if they
subtract from both sides, or if result of multiplication or division is a negative even if the number used to
change the inequality was not negative. In geometry it is most common to be working with all positive
numbers, but depending on how the students apply the Properties of Inequalities, they may create some
negative values.
Extension: Indirect Proof
Why Learn Indirect Proof - For a statement to be mathematically true it must always be true, no
exceptions. This frequently makes it easier to prove that a statement is false than to prove it is true.
Indirect proof gives mathematician the choice between proving a statement true or proving a statement
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false and can therefore greatly simplify some proofs. Letting the students know that indirect proof can be
a potential shortcut will motivate them to learn to use this type of logic.
Indirect Proof - Students will not really understand the method of indirect proof the first time they see
it. Let them know that this is just the first introduction, and that in subsequent lessons they will be given
more examples and opportunities to learn this new method of proof. If students think they are supposed
to understand something perfectly the first time they see it, and they don't, they will become frustrated
with themselves and mathematics. Let them know that the brain needs time and multiple exposures to
master these challenging concepts.
Review the Contrapositive - Proving a statement using indirect proof is equivalent to proving the
contrapositive of the statement. If students are having trouble setting up indirect proofs, or even if they
are not, it is a good idea to have them review conditional statements and the contrapositive. The first step
to writing an indirect proof, can be to have them write out the contrapositive of the statement they want
to prove. This will reduce confusion about what statement to start with, and what statement concludes
the proof.
Does This Really Prove Anything? - Even after students have become adept with the mechanics of
indirect proof, they may not be convinced that what they are doing really proves the original statement.
This is the same as asking if the contrapositive is equivalent to the original statement. Using examples
outside the field of mathematics can help students concentrate on the logic.
Start with the equivalence of the contrapositive. Does statement (1) have the same meaning as statement
(2)?
1. If you attend St. Peter Academy, you must wear a blue uniform.
2. If you don't wear a blue uniform, you don't attend St. Peter Academy.
Let the students discuss the logic, and have them create and share their own examples.
If a good class discussion ensues, and the students provide many statements on a single topic, it may be
possible to write some indirect proofs of statements not concerned with mathematics. This could be a
good bonus assignment or project that when presented to the class will make the logic of indirect proof
clearer for other students.
2.6 Polygons and Quadrilaterals
Angles in Polygons
All Those Polygons - Although they have probably been taught it before, not all students will remember
the names of the different polygons. There are not very many opportunities in life to use the word heptagon.
Add these words to their vocabulary list.
This is most likely the first time they have been introduced to a polygon with a variable number of sides,
an n-gon. This notation can be used when referring to a polygon that does not have a special name in
common use, like a 19-gon. It can also be used when the number of sides of the polygon is unknown.
Clockwise or Counterclockwise But Not Both - At this point in the class, student are usually good
at recognizing vertical angles. They will understand that the exterior angles made by extending the sides of
the polygon in a clockwise rotation are congruent, at each vertex, to the exterior angle formed by extending
the sides counterclockwise. What they will sometimes do is include both of these angles when using the
Exterior Angle Sum theorem. Reinforce that the number of exterior angles is the same as the number of
interior angles and sides, one at each vertex.
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Interior or Exterior - Interior and exterior angles come in linear pairs. If one of these angles is known
at a particular vertex, it is simple to find the other. When finding missing angles in a polygon, students
need to decide from the beginning if they are going to use the interior or exterior sum. Most likely, if
the majority of the known angle measures are from interior angles they will use the interior sum. They
need to convert the exterior angle measures to interior angle measures before including them in the sum.
If there are more exterior angle measures given, they can convert the interior angle measures and use the
sum of 360°. It is important that they make a clear choice. They may mix the two types of angles in one
summation if they are not careful.
Do a Double Check - Students often do not take the time to think about their answers. Going over the
arithmetic and logic is one way to check work, but it is common to not recognize the error the second time
either. A better strategy is to use other relationships to do the checking. In this lesson if the exterior sum
was used, the work can be checked with the interior sum.
What's the Interior Sum of a Nonagon Again? - If students do not remember the interior sum for a
specific polygon, and do not remember the formula, they can always convert to the exterior angle measures
using the linear pair relationship. The sum of the exterior angles is always 360°. This strategy will work
just as well as using the interior sum. Remind the students to be creative. When taking a test, they may
not know an answer directly, but many times they can figure out the answer in an alternative way.
It is important to practice lots of problems like the following examples and to help student devise strategies
to solve them.
Example 1: What is the measure of each interior angle of a regular n-gon if the sum of the interior angles
is 1080°?
Answer: 135° First the number of sides needs to be found, so set the sum equal to (n - 2)180° and solve to
find n = 8. Now the total of 1080° needs to be divided into 8 congruent angles: 1Q g° = 135°.
Example 2: If three angles in a pentagon have measures 115°, 100° and 125° what are the measures of the
two remaining angles if they are congruent to each other?
Answer: 100° First we must determine the sum of the 5 angles in a pentagon: (5 - 2)180° = 540°. Next,
subtract the three known angles from the sum: 540° - 115° - 100° - 125° = 200°. Finally, divide the
remaining measure by 2 to get 100°.
Example 3: If the measure of one interior angle in a regular polygon is 168°, how many sides does it have?
Answer: 30 The easiest way to solve this problem is to use the exterior angle sum. If the interior angle
measure is 168°, then the measure of each exterior angle is the supplement of this, or 12°. Now simply
divide the exterior sum, 360° by 12° to get 30, the number of sides.
Sketchpad Alternatives - Many students become particularly engaged in a topic when they are able
to investigate it while playing around with the computer. Here are a couple of ways to use Geometers'
Sketchpad in the classroom as an alternative or supplement to direct instruction.
Angle Sum Conjecture - Have student make different convex polygons and measure the sum of their interior
angles.
1. The students should observe that for each type of polygon, no matter how many were drawn, they all
have the same interior angle sum. They also like to see that if they change the shape of the convex
polygon (by dragging a vertex- making sure it stays convex) that the sum remains the same.
2. The students should drag a vertex of each polygon toward the center to create a concave polygon,
and notice if the sum stays the same. (It won't.)
3. Put the sums in order on the board: 180°, 360°, 540°, . . . Ask the students to find the pattern in this
sequence of numbers. Lead them to discovering the Angle Sum formula, (n — 2)180° from the pattern.
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Exterior Angle Sum Conjecture - Have student make different convex polygons and measure the sum of their
exterior angles. Using Sketchpad to extend the sides of the polygon helps students gain an understanding
of where the exterior angle is in relation to the polygon.
1. Students should observe that the sum is always 360°, regardless of the number of sides. Doing this
for polygons with different numbers of sides will help them see why the exterior angle sum remains
the same.
2. Students can again drag the vertex of a polygon and see that although the angles may change, the
sum does not.
Properties of Parallelograms
Tree Diagram - Most students will need practice working with the classification of quadrilaterals before
they completely understand and remember all of the relationships. The Venn diagram is an important
mathematical tool and should definitely be used to display the relationships among the different types of
quadrilaterals. A tree diagram will also make an informative visual. Using both methods will reinforce the
students' understanding of quadrilaterals, and their ability to make good diagrams.
Parallel Line Properties - In Chapter Three: Parallel and Perpendicular Lines, the students learn about
the relationships between the measures of the angles formed by parallel lines and a transversal. Many of
the quadrilaterals studied in this section have parallel sides. The students can apply what they learned
in chapter three to the quadrilaterals in this chapter. They may have trouble seeing the relationships
because instead of lines the quadrilaterals are made of segments. Recommend that the students draw the
figures on their papers and extend the sides of the quadrilaterals so they can see all four angles made by
the intersection of the lines. These angles will be useful when looking for specific information about the
quadrilateral.
Show Clear, Organized Work - When using the distance or slope formula to verify information about
a quadrilateral on the coordinate plane, students will often do messy scratch work as if they are the only
ones that will need to read it. In this situation, the work is a major part of the answer. They need to
communicate their thoughts on the situation. They should write as if they are trying to convince the
reader that they are correct. As students progress in their study of mathematics, this is more often the
case than the need for a single numerical answer. They should start developing good habits now.
Symmetry - Most students have already studied symmetry at some point in their education. A review
here may be in order. When studying quadrilaterals, symmetry is a good property to consider. Symmetry
is also important when discussing the graphs of key functions that the students will be studying in the
next few years. It will serve the students well to be adept in recognizing different types of symmetry.
Proving Quadrilaterals are Parallelograms
Proofs Using Congruent Triangles - The majority of the proofs in this section use congruent triangles.
The quadrilateral of interest is somehow divided into triangles that can be proved congruent with the
theorems and postulates of the previous chapters. Once the triangles are known to be congruent, the
definition of congruent triangles ensures that certain parts of the quadrilateral are also congruent. Students
should be made aware of this pattern if they are having difficulty writing or understanding the proofs of
the properties of various quadrilaterals. If they are still struggling they should spend some time reviewing
Chapter Four: Triangles and Congruence.
The Diagonals of Parallelograms - The properties concerning the sides and angles of parallelograms
are fairly intuitive, and students pick them up quickly. More emphasis should be placed on what is known,
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and not known about the diagonals. Students frequently try to use the incorrect fact that the diagonally
of a parallelogram are congruent. Rectangles are the focus of an upcoming lesson, but demonstrating to
students that the diagonals of a quadrilateral are only congruent in the special case where all the angles
of the parallelogram are congruent. For a general parallelogram, the diagonals bisect each other. This can
be shown nicely using a sketch in Geometer's Sketchpad.
Proof Practice - The proofs in this section may seem a bit repetitive, but students will benefit from
practicing these proofs since they review important concepts learned earlier in the course. To avoid loosing
the students' attention, find different ways of presenting the proofs. One idea is to divide the students into
groups, and have each group demonstrate a different proof to the class.
Parallel or Congruent - When looking at a marked figure students will sometimes see the arrows
that designate parallel segments and take that the segments to be congruent. This could be due to the
misreading of the marks, or mistakenly thinking parallel always implies congruence. Warn students not
to make this error. The last method of proof in this section which utilizes that one pair of sides are both
congruent and parallel, along with an example of a trapezoid where the parallel sides are not congruent,
will help students remember the difference.
Below are some additional examples to be shown in class.
Example 1: Quadrilateral ABCD is a parallelogram. AB = 2x + 5, BC = x— 3 and DC — 3x — 10. Find the
measures of all four sides of the quadrilateral.
Answer: AB = CD = 35 and BC = AD = 12.
Encourage students to draw and label a diagram for problems such as these. Remind them that the name
of a polygon lists the vertices in a circular order. This will help ensure that they match up correct pairs of
congruent sides. Set the congruent sides equal to each other and solve for x.
2x + 5 = 3x - 10
jc=15
Example 2: JACK is a parallelogram. mlA = (lOx - 60)° and mlC = (2x + 45)°. Find the measures of all
four angles.
Answer: mlC = mlJ = 77.5° and mlA = mlK = 102.5°
Again, encourage students to sketch and label a diagram so they can see that LA and LC are consecutive
angles in the parallelogram and therefore supplementary. Now, they can set the sum of the expressions for
the measures of these two angles equal to 180° and solve for x.
10x-60 + 2x + 45 = 180°
x= I67
4
Example 3: KATE is a parallelogram with a perimeter of 40 cm. KA = 3x + 8 and AT = x + 4. Find the
length of each side.
Answer: KA = ET = 14 cm and AT = KE = 6 cm
After sketching a diagram, students should recognize that KA and AT are adjacent sides and therefore not
necessarily congruent. Remind them that they were given the perimeter and help them figure out that they
need to add two times each of the given side length expressions and set this sum equal to the perimeter of
the parallelogram as shown below.
2(3jc + 8) + 2(x + 4) = 40
x = 2
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Example 4: SAMY is a parallelogram with diagonals intersecting at point X. SX = x-\- 5,XM = 2x — 7 and
AX = 12x. Find the length of each diagonal.
Answer: SM = 34 cm and AY = 288 cm
Students should use the fact that diagonals bisect each other in a parallelogram to solve this one. By-
setting the two parts of diagonal SM each to each other they can find x and then solve for the lengths of
the diagonals.
x + 5 = 2x - 7
x= 12
Example 5: JEDI is a parallelogram. mlJ = 2x + 60 and mlD = 3x + 45. Find the measures of the four
angles of the parallelogram. Does this parallelogram have a more specific categorization?
Answer: All four angles measure 90°. JEDI is a rectangle.
The angles given here are opposite angles. By setting them equal we can solve for x and find the angle
measures.
2x + 60 = 3jc + 45
x= 15
Rectangles, Rhombuses and Squares
The Power of the Square - Students should know by the classification of quadrilaterals that all the
theorems for parallelograms, rectangles, and rhombuses, also apply to squares. It is a good idea to talk
about this in class though in case they have not put it together on their own. These theorems and the
definition of a square can be combined to from some interesting exercises.
A Venn diagram can be used to show the relationships between the figures and their properties. This is also
an excellent opportunity to review some logic and practice making conditional statements and checking
their validity based on the Venn diagram. For example: If the quadrilateral is a square, then it is a
parallelogram. This statement is true, but its converse is false.
Information Overload - Quite a few theorems are presented in this chapter. Remembering them all and
which quadrilaterals they apply to can be a challenge for students. If they are unsure, and cannot check
reference material, a test case can be drawn. For example: Do the diagonals of a parallelogram bisect the
interior angles of that parallelogram? First they need to draw a parallelogram that clearly does not fit
into any subcategory. It should be long and skinny, so no rhombi properties are mistakenly attributed to
it. It should also be well slanted over, so as not to be mistaken for a rectangle. Now they can draw in the
diagonals. It will be obvious that the diagonals are not bisecting the interior angles. They could also try
to recreate the proof, but that will probably be more time consuming and it requires a bit of skill.
Also, students need to be reminded that they must show that the quadrilateral is a parallelogram before it
can be determined that it is one of the special parallelograms. For example, just because the diagonals of
the quadrilateral are congruent, it is not sufficient information to conclude the quadrilateral is a rectangle-
it could be an isosceles trapezoid. If students can also show that one of the properties of parallelograms is
true for the quadrilateral as well, such as opposite sides are parallel, then they can correctly conclude that
the quadrilateral is a rectangle.
Example 1: S QUR is a square and X is the point where the diagonals meet. QX = 3x - 9 and SX = 2x.
Find the length of both diagonals.
Answer: SU = QR = 36.
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Since the diagonals in a square are equal and bisect each other, we can set 3x - 9 = 2x and solve to get
x = 9. Then QX = 18 and both diagonals are 36.
Example 2: DAVE is a rhombus with diagonals that intersect at point X. DX = 3 cm and AX = 4 cm. How
long is each side of the rhombus?
Answer: DA = AV = VE = ED = 5 cm
The diagonals of a rhombus are perpendicular bisectors of each other so DX and AX are the legs of a
right triangle. The hypotenuse of this right triangle is a side of the rhombus and is 5 cm by Pythagorean
Theorem.
Trapezoids and Kites
Average for the Median - Students who have trouble memorizing formulas may be intimidated by the
formula for the length of the median of a trapezoid. Inform them that they already know this formula;
it is just the average. The application of the formula makes since, the location of the median is directly
between the two bases, and the length of the median is exactly between the lengths of the bases. They
will have no problem finding values involving the median.
Where Are We? - It is easy for students to forget how what they are learning today relates to the
chapter and to the class. Use the Venn diagram of the classification of quadrilaterals to orient them in the
chapter. They are no longer learning about parallelograms, but have moved over to the separate trapezoid
area. When student are able to organize their new knowledge, they are better able to retain and apply it.
Does it have to be Isosceles? - Students may have trouble remembering which theorems in this section
apply only to isosceles trapezoids. Note that base angle, and diagonal congruence apply only to isosceles
trapezoids, but the relationship of the length of the median to the bases is the same for all trapezoids.
Example 1: TRAP is a trapezoid. The median has length 4 cm, and one of the bases has length 7 cm.
What is the length of the other base?
Answer: 1 cm
There are two ways to approach this problem. One is to compare differences - seven is three more than
four, so the other base must be three less than four. The second way is to set up and solve an equation as
shown below.
x= 1
Example 2: WXYZ is a trapezoid. The length of one base is twice the length of the other base, and the
median is 9 cm. How long is each base?
Answer: The bases are 6 cm and 12 cm.
This problem can be solved by allowing one base to be x and the other base to be 2x. Then solve the
following equation:
x + 2x _ n
x = 6
Only One Congruent Set - It is important to note that in a kite, only one set of interior angles are
congruent, and only one of the diagonals is bisected. Sometimes students struggle with identifying where
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these properties hold. It is the nonvertex angles that are congruent, and the diagonal connecting the
nonvertex angles that is bisected. The single line of symmetry of a kite shows both these relationships.
Remind students to think of this line of symmetry and what it tells them about congruent pairs of angles
and segments.
Break it Up - When working with a kite, it is sometimes easier to think of it as two isosceles triangles,
or four right triangles, instead of one quadrilateral.
At this point in the class, students have had extensive experience working with isosceles triangles, and can
easily apply the Base Angle theorem to see that the nonvertex angles of the kite are congruent. They have
also seen that the diagonal segment between the vertex angles creates many symmetries in the triangle.
If students think of the symmetries in the triangle, it will make sense to them that the vertex angles are
bisected and that the diagonal connecting the nonvertex angles is bisected.
They can also think of a kite as four right triangles. This will help them remember that the diagonals
are perpendicular, and remind them that the Pythagorean Theorem can be used to find missing segment
measures. Noticing that the right triangles are in two congruent sets will help them identify congruent
segments and angles.
Example 1: KITE is a kite with KE = ~K~L Prove that AKSE = AKSL
Answer:
Table 2.4:
Statement
Reason
KE = KI
ES =Sl
IKS E and IKS I are right angles
IKSE = IKS I
AKSE = AKSI
Given
Kite Diagonal Theorem
Definition of Perpendicular
Right Angle Theorem
HL
2.7 Similarity
Ratios and Proportions
Keep it in Order - When writing a ratio, the order of the numbers is important. When the ratio is
written in fraction form the amount mentioned first goes in the numerator, and the second number goes
in the denominator. Remind the students it is important to keep the values straight to avoid confusion or
misunderstandings. For example, if they are looking at a college and see that the male to female ratio is
11 to 12, it is important to know which one comes first to interpret the ratio correctly.
To Reduce or Not to Reduce - When a ratio is written in fraction form it can be reduced like any
other fraction. This will often make the arithmetic simpler and is frequently required by instructors for
fractions in general. But when reducing a ratio, useful information can be lost. If the ratio of girls to
boys in a classroom isl6 to 14, it may be best to use the fraction || because it gives the total number of
students in the class where the reduced ratio | does not.
Consistent Proportions - A proportion can be correctly written in many ways. As long as the student
sets up the ratios in a consistent, orderly fashion, they will most likely have written a correct proportion.
There should be a common tie between the two numerators, the two denominators, the numbers in the
first ratio, and the numbers in the second ratio. They should think about what the numbers represent, and
not just use them in the order given in the exercise, although the numbers are often given in the correct
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order.
Example: Victor got a new hybrid. He went 525 gallons on the first five gallons that came with the car.
He just put 12 gallons in the tank. How far can he expect to go on that amount of gas?
Answer: | = ^,soi= 12*25 an( ^ yj c ^ or can expect to go 1,260 miles.
Note: Students may be tempted to put the 12 in the numerator of the second ratio because it was the
third number given in the exercise, but it should go in the denominator with the other amount of gas.
The Fraction Bar is a Grouping Symbol - Students know that parenthesis are a grouping symbol
and that they need to distribute when multiplying a number with a sum or difference. A fraction bar is a
more subtle grouping symbol that students frequently overlook, causing them to forget to distribute. To
help them remember have them put parenthesis around sums and differences in proportions before they
cross-multiply.
Example: *& = *=§ becomes ^1 = M
Everybody Loves to Cross-Multiply - There is something satisfying about cross-multiplying and
students are prone to overusing this method. Remind them that cross-multiplication can only be used in
proportions, when two ratios are equal to each other. It is not appropriate to cross-multiply when two
fractions are being added, subtracted, multiplied or divided. It might be helpful to do some example of
these to illustrate the difference and discuss the difference between "cross multiplication" and "multiplying
across."
Examples: Cross multiply here: | = ^ or ^±- = |
Don't cross multiply here: § + ™ or ^ -f |
Only Cancel Common Factors - When reducing a fraction or putting a ratio in simplest terms, students
often try to cancel over an addition or subtraction sign. This problem occurs most frequently when students
work with fractions that contain variable expressions. To combat this error, go back to numerical examples.
Students will see that what they are doing does not make sense when the variables are removed. Then go
back to example with variables. Hopefully the students will be able to carry over the concept.
Examples: Can be reduced: |^| = | and 3 2 = ^-^
Can't be reduced: |^| ^ | and ^l £ x - 1
Similar Polygons
A Common Vocabulary Error - Students frequently interchange the words proportional and similar.
Remind them that proportional describes a relationship between numbers, and similar describes a rela-
tionship between figures. You can relate this difference in definition back to the difference between the
terms equal and congruent.
Compare and Contrast Similar with Congruent - If your students have already learned about
congruent figures, now would be a good time to review. The definitions of congruent and similar are very
close. Ask the students if they can identify the difference; it's only one word. You can also point out
that congruent is a subset of similar like square is a subset of rectangle, or mother is a subset of women.
Understanding the differences between congruent and similar will be important in upcoming lessons when
proving triangles similar.
Use that Similarity Statement - In some figures, which sides of similar polygons correspond is obvious,
but when the polygons are almost congruent, or oriented differently, the figure can be misleading. Students
usually begin by using the figure and then forget to use the similarity statement when necessary. Remind
them about this information as they start working on more complicated problems. The similarity statement
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is particularly useful for students that have a hard time with visual-spatial processing. It is a good idea
to do several examples in which students are "forced" to use the similarity statement to align the correct
sides and angles to get them in the habit of using the similarity statement rather than their "eyes."
Who's in the Numerator - When writing a proportion students sometimes carelessly switch which
polygon's measurements are in the numerator. To help students avoid this pitfall, I tell the students to
choose right from the beginning and BE CONSISTENT throughout the problem. When it comes to writing
proportions if the students focus on being orderly and consistent, they will usually come up with a correct
setup.
Bigger or Smaller - After completing a problem it is always a good idea to take a minute to decide if the
answer makes sense. This is hard to get students to do. When using a scale factor, a good way to check
that the correct ratio was used is to notice if the number got bigger or smaller. Is that what we expected
to happen?
Update the List of Symbols - In previous lessons it has been recommended that students create a
reference page in their note books that contains a list of all the symbols and how they are being used in
this class. Students should add the symbol for similar to the list, and take few minutes to compare it to
the symbols they already know. Sometimes students will read the similarity symbol as "approximately
equal". It is standard to use two wavy lines for approximately equal and one wavy line for similar, but this
is not always the case.
Similarity by AA
Definition of Similar Triangles vs. AA Shortcut - Let the students know what a deal they are
getting with the A A Triangle Similarity Postulate. The definition of similar polygons requires that all
three corresponding pairs of angles be congruent, and that all three pairs of corresponding sides are
proportional. This is a significant amount of information to verify, especially when writing a proof. The
A A postulate is a significant shortcut; only two piece of information need to be verified and all the rest
comes for free. When students see how much this reduces the work, they will be motivated to understand
the proof and will enjoy using the postulate. Everybody likes to use a tricky shortcut. A fun activity to
explore this concept would be to have students create a triangle with three given angle measures. Then,
in pairs, compare the lengths of corresponding sides to discover that they are indeed similar. Then ask
students if they really needed to be told the third angle measure? This process should help them remember
A A Similarity and help reinforce the idea that A A does not ensure congruent triangles.
Get Some Sun - It is always a good idea to create some variety in the class. It will keep students' minds
active. Although it is time consuming, get some yard sticks and take the students outside to measure a
tree or a flagpole using their shadows and similar triangles. Have them evaluate their accuracy. They will
have to measure carefully if they are to get a reasonable numbers. This will give them some practice using
a ruler and converting units. The experience will also help them put what they are learning about similar
triangles into their long term memory.
Trigonometry - Let the students know that the next chapter is all about trigonometry, and that the AA
Triangle Similarity Postulate is what make trigonometry possible. Mentioning what is to come will start
to prepare their minds and make learning the material in the next chapter that much easier. Here are some
problems that involve similar right triangles to accustom the students to this new branch of mathematics.
Example 1: AABC is a right triangle with right angle C and AABC ~ AXYZ. Which angle in AXYZ is the
right angle?
Answer: LZ
Example 2: ACAT ~ ADOG with right angle at A. If CA = 5 cm, CT = 13 cm and DO = 15 cm, what is
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the length of OG?
Answer: 36 cm
There is more than one step required to solve this problem. Students must use the Pythagorean theorem
and the definition of similar polygons. First, the ratio between the figures is determined by ^ = jz = I-
Next, we need the length of AT since it corresponds to OG. Using the Pythagorean Theorem gives us
AT = 12 cm. Now we can set up the proportion | = ^ and solve it to get x = 36.
Similarity by SSS and SAS
The S of a Triangle Similarity Postulate - At this point in the class, students have shown that a
significant number of triangles are congruent. They have learned the process well. When teaching them to
show that triangles are similar, it is helpful to build on what they have learned. The similarity postulates
have S's and A's just like the congruence postulates and theorems. The A's are treated exactly the same
in similarity postulates as they were in congruence theorems. Each A in a similarity shortcut stands for
one pair of congruent corresponding angles in the triangles.
The S's represent a different requirement in similarity postulates than they did in congruence postulates
and theorems. Congruent triangles have congruent sides, but similar triangles have proportional sides.
Each S in a similarity postulate represents a ratio of corresponding sides. Once the ratios (two for SAS
and three for SSS) are written, equality of the ratios must be verified. If the ratios are equal, the sides in
question are proportional, and the postulate can be applied.
It is sometimes hard for student to adjust to this new side requirement. They have done so much work
with congruent triangles that it is easy for them to slip back into congruent mode. Warn them not to fall
into the old way of thinking.
Triangle Congruence Postulates and Theorems Triangle Similarity Postulates
S — > congruent sides S — > proportional sides
SSS AA
SAS SSS
ASA SAS
AAS
HL
Only Three Similarity Postulates - Students will sometimes try to use ASA, or other congruence
theorems to show that two triangles are similar. Bring it to their attention that there are only three
postulates for similarity, and that they do not all have the same side and angle combinations as congruence
postulates or theorems. It may help them to show them that the Congruence Postulates ASA and AAS
are both "represented" by the Similarity Postulate AA.
Proportionality Relationships
Similar Triangles Formed by an Interior Parallel Segment - Students frequently are presented with
a triangle that contains a segment that is parallel to one side of the triangle and intersects the other two
sides. This segment creates a smaller triangle in the tip of the original triangle. There are two ways to
consider this situation. The two triangles can be considered separately, or the Triangle Proportionality
Theorem can be applied.
(1) Consider the two triangles separately.
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The original triangle and the smaller triangle created by the parallel segment are similar as seen in the
proof of the Triangle Proportionality theorem. One way students can tackle this situation is to draw the
triangles separately and use proportions to solve for missing sides. The strength of this method is that it
can be used for all three sides of the triangles. Students need to be careful when labeling the sides of the
larger triangle; often the lengths will be labeled as two separate segments and the students will have to
add to get the total length.
(2) Use the Triangle Proportionality theorem.
When using this theorem it is much easier to setup the proportions, but there is the limitation that the
theorem cannot be used to find the lengths of the parallel segments.
Ideally, students will be able to identify the situations where each method is the most efficient, and apply
it. This may not happen until the students have had some experience with these types of problems. It is
best to have students use method (1) at first, then after they have worked a few exercises on their own,
they can use (2) as a shortcut in the appropriate situations.
Make sure to do many examples where students need to use the similar triangles to solve for one of the
parallel segments. Combining this requirement in the same problem where the shortcut can be utilized is
especially helpful for them to begin recognizing the difference.
Example: AABC has point E on AB and F on BC such that EF \\ AC. Given AE = 5 cm, EB = 3 cm, BF =
4 cm and AC = 10 cm, find EF and EC.
Answer: § = f£,£F = 3§ cm
- = — or - = 4 FC = 6- cm
5 FC U1 8 FC+4' ru u 3 cm
Encourage students to draw a diagram and label it with the given lengths. Next, they may wish to draw
the two similar triangles separately to better visualize which parts correspond.
Proportions with Angle Bisectors - Students have a hard time with this one because the two triangles
formed by the angle bisector are not similar. Students need to see lots of examples to get this one straight.
Remind them that this proportion is true only when the segment is an angle bisector- it is not necessarily
true for altitudes or medians in the triangle.
Similarity Transformations
Apostrophe vs Prime - When a geometric figure is transformed (by translation, rotation, dilation, etc.)
the image is denoted using the "prime" marking. For example, if AABC is transformed, the image is
denoted by AA'B'C.
Scale Factor Compared to Segment and Area Ratios - When a polygon is dilated using scale
factor, k, the ratio of the image of the segment to the original segment is k. This is true for the sides of the
polygon, all the special segments of triangles studied in chapter five, and the perimeter of the polygon. The
relationship holds for any linear measurement. Area is not a linear measurement and has a different scale
factor. The ratio of the area of the image to the area of the original polygons is k 2 . Students frequently
forget to square the scale factor when working with the ratios of a figure and its image. This concept will
be explored further in Chapter 10: Perimeter and Area, but the idea can be introduced here.
Example: AABC has coordinates A(l, 13), 5(6, 1) and C(l, 1). Complete the following:
1. Graph AABC.
2. Use the distance formula to find the length of each side of AABC.
3. Calculate the perimeter of AABC.
4. Calculate the area of AABC.
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5. AA'B'C is the image of AABC under a dilation centered at the origin with scale factor 3. Graph
AA'B'C.
6. Calculate the perimeter of AABC.
7. Calculate the area of AA'B'C .
8. Compare AA'B'C to AABC. What is the ratio of each set of corresponding side lengths, the perime-
ters and the areas? What do you notice when these ratios are compared to the scale factor?
Answers:
1. Graph
2. AC = 12, BC = 5,AB= 13
3. 30
4. 30
5. Graph
6. AC = 36, BC = 15, AB = 39
7. 90
8. 270
9. The ratios of the side lengths and the perimeter are 3:1, the same ratio as the scale factor. The ratio
of the areas is 9:1, the square of the scale factor.
Extension: Self-Similarity
More Complex Fractals — Students need to begin learning about fractals with the simple examples given
in the text. Once they have taken some time to work with, and understand the self-similar relationship,
it is amazing to see how complex and beautiful fractals can become. Numerous examples of exquisite
fractals can be found on-line. If you are lucky enough to have access to computers and a projector, have
the students search for fractals and choose their favorite to share with the class. Student will begin to
realize the importance of what there are learning when they see what a huge ocean they are dipping their
toe into.
Applications - Many students need to know how a subject is useful before they are motivated to spend
time and energy learning about it. Throughout the text there have been references to modeling and how
mathematical concepts often need to be adjusted to fit the world around us. Fractals are used to model
many aspects of nature including tree branches, shells, and the coast line. Knowing of the applications of
fractals motivates students. If time permits give a more in-depth explanation, or use this topic to assign
research projects.
Video Time - Self-similarity and fractals make up an extremely complex visual topic. There are many
videos in common use that can give a much more exciting and attention grabbing explanation than most
teachers can deliver while standing in front of the classroom. These videos are not hard to come by, and
they give an excellent explanation of the material. It is a nice change of pace for the students, and make
help them develop some genuine interest in the subject of mathematics.
Create Your Own Fractal - Having the students create their own fractal outside of class is a fun,
creative project. This gives the more artistically minded students an opportunity to shine in the class, and
the products make beautiful wall decorations. Here are some guidelines for the assignment.
1. The fractal should fill the top half of a piece of 8^ x 11 inch plain white paper turned vertically. To
give them more space, provide them with legal size paper. Be aware that each student will probably
require more than one piece before they create their final product.
2. The fractal should be boldly colored to accentuate the self-similarity.
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3. The students should be encouraged to be creative and original in their design.
4. The bottom half of the paper will have a paragraph explaining the self-similarity in the fractal. They
should explain why their design is a fractal.
5. Create a rubric to give to the students at the time the project is assigned so that they will feel like
they are being graded fairly. It is hard to evaluate artwork in a way that everyone feels is objective.
2.8 Right Triangle Trigonometry
The Pythagorean Theorem
Reducing Radicals - Students may or may not have spent much time reducing radicals in previous math
courses. You may need to review how to do this and how to perform operations with radicals. Students
may need to see more than one way to reduce radicals. The most common is to take out the greatest factor
which is a perfect square, but this can be difficult for many students with weak mental math skills. Another
method is to make a factor tree and find the prime factorization of the number and identify doubles that
can be taken out. There is not necessarily a "best" way, it is more important to figure out which way your
students can best reduce radicals correctly.
Method 1: V72 = V36 x 2 = 6 V2
Method 2: V72 = V9 x 8 = V3x3x2x2x2 = 3x2V2 = 6a/2
The following are some examples that will help students review the basic properties of radicals and practice
reducing radicals.
Example 1: Vll2
Answer: Vl6 X 7 = 4 V7
Students may not recognize right away that 16 is a factor of 112. This problem can also be solved by
completely factoring 112 (method 2).
Example 2: 4Vl92
Answer: 4 V64 X 3 = 4x8V3 = 32V3
Example 3: 2 V5 + V45
Answer: 2V5 + 3V5 = 5V5
Review with students that in order to add or subtract radicals, the radical must be identical. Sometimes
they can meet this condition by simplifying one or both terms and sometimes they just won't be able to
add them together.
Example 4: V6 x Vl8
Answer: Vl08 = V36 X 3 = 6 V3
Students may have forgotten that radicals can be multiplied or divided to form a new radical which they
may then be able to reduce.
Example 5: ( VH)
Answer: 11
Review the definition of a square root with students to help them understand this one. The square root
and the square cancel each other out - they are inverse operations.
Skipping Around - Not all texts present material in the same order, and many instructors have a preferred
way to develop concepts that is not always the same as the one used in the text. The Pythagorean theorem
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is frequently moved from place to place. You should follow the link given in the text to see additional
proofs of the Pythagorean Theorem. Proofs are hard for most students to understand. It is important to
choose one that the students can feel good about. Don't limit the possibilities to these two, research other
methods, and pick the one that is most appropriate for your class. Or better yet, pick the best two or
three. Different proofs will appeal to different students.
The Height Must be Measured Along a Segment That is Perpendicular to the Base - When
given an isosceles triangle where the altitude is not explicitly shown, student will frequently try to use
the length of one of the sides of the triangle for the height. Tell them that they must find the length of
the altitude that is perpendicular to the segment that's length is being used for the base in the formula
A = \bh. Sometimes they do not know what to do, and are just trying something, which is, in a way,
admirable. The more common explanation though is that they forget. The students have been using this
formula for years, they think they know this material, so they just plug and chug, not realizing that the
given information has changed. Remind the students that now that they are in Geometry class, there is
an extra step. The new challenge is to find the height, and then they can do the easy part and plug it into
the formula.
Derive the Distance Formula - After doing an example with numbers to show how the distance formula
is basically just the Pythagorean theorem, use variables to derive the distance formula. Most students will
understand the proof if they have seen a number example first. Point out to the students that the number
example was inductive reasoning, and that the proof was deductive reasoning. Taking the time to do this
is a good review of logic and algebra as well as great proof practice.
Squaring a Negative - One of the most common errors students make when using the distance formula
is that they use their calculator incorrectly when squaring negative numbers. Remind students that when
they square a negative number, the result is positive. Describe the difference between -8 2 and (-8) 2 and
review the correct of order of operations. You may also want to stress the connection between the distance
formula and the Pythagorean theorem and encourage them to think of the difference between the jc's and
y's as lengths and therefore they can just use the absolute value of the difference (i.e. ignore the negative).
Converse of the Pythagorean Theorem
Acute and Obtuse Triangles - Many students have trouble remembering that the inequality with the
greater than is true when the triangle is acute, and that the equation with the less than is true for obtuse
triangles. It seems backwards to them. One way to present this relationship is to compare the longest
side and the angle opposite of it. In a right triangle, the equation has an equal sign; the hypotenuse is the
perfect size. When the longest side of the triangle is shorter than what it would be in a right triangle, the
angle opposite that side is also smaller, and the triangle is acute. When the longest side of the triangle is
longer than what is would be in a right triangle, the angle opposite that side is also larger, and the triangle
is obtuse.
Is It Really a Triangle? - I have found that once students start using the Pythagorean theorem to
determine whether lengths form a right, acute or obtuse triangle that they forget completely that the sum
of two sides must be greater than the remaining side in order for a triangle to exist at all. The following
example illustrates this misunderstanding. You may want to put it on the board and ask your students
what kind of triangle is formed.
Example: What kind of triangle is formed by lengths 3, 4, 7?
Answer: None! There is no triangle at all. 3 + 4 = 7.
If students used the Pythagorean theorem and didn't check to make sure there was a triangle at all then
they would have said that the triangle is obtuse. This in incorrect.
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Using Similar Right Triangles
Separate the Three Triangles - The altitude from the right angle of a triangle divides the triangle into
two smaller right triangles that are similar to each other, and to the original triangle. All the relationships
among the segments in this figure are based on the similarity of the three triangles. Many students have
trouble rotating shapes in their minds, or seeing individual polygons when they are overlapping. It is
helpful for these students to draw the triangles separately and oriented in the same direction. After going
through the process of turning and redrawing the triangles a few times, they will remember how the
triangles fit together, and this step will no longer be necessary.
Color-Coded Flashcards - It is difficult to describe in words which segments to use in the geometric
mean to find the desired segment. Labeling the figure with variables and using a formula is the standard
method. The relationship is easier to remember if the labeling of the triangles is kept the same every time
the figure is drawn. The students need to remember the location of the segments relative to each other.
Making color-coded pictures or flashcards will be helpful. For each relationship the figure should be drawn
on both sides of the card. The segment whose measure is to be found should be highlighted in one color
on the front, and on the back, the two segments that need to be used in the geometric mean should be
highlighted with two different colors. Using two colors on the back is important because the segments often
overlap. Making these cards will be helpful even if the students never use them. Those that have trouble
remembering the relationship will use these cards frequently as a reference.
Add a Step and Find the Areas - The exercises in this section have the students find the base or
height of triangles. They have all the information that they need to also calculate the areas of these
triangles. Students need practice with multi-step problems. Having them find the area will help them
think through a more complex problem, and give them practice laying out organized work for calculations
that are more complex. Chose to extend the assignment or not based on how well the students are doing
with the material, and how much time there is to work on this section.
Special Right Triangles
Memorize These Ratios - There are some prevalent relationships and formulas in mathematics that
need to be committed to long term memory, and the ratios made by the sides of these two special right
triangles are definitely among them. Students will use these relationships not only in the rest of this
class, but also in trigonometry, and in other future math classes. Students are expected to know these
relationships, so the sooner they learn to use them and commit them to memory, the better off they will
be.
Two is Greater Than the Square Root of Three - One way that students can remember the ratios of
the sides of these special right triangles, is to use the fact that in a triangle, the longest side is opposite the
largest angle, and the shortest side is opposite the smallest angle. At this point in the class, students know
that the hypotenuse is the longest side in a right triangle. What sometimes confuses them is that in the
30-60-90 triangle, the ratio of the sides is 1 : 2 : v3, and if they do not really think about it, they sometimes
put the V3 as the hypotenuse because it might seem bigger than 2. Using the opposite relationship is a
good method to use when working with these triangles. Just bring to the students' attention that 2 > V3.
How Do I Find the Short Leg Again? - While students may quickly memorize the two special right
triangle ratios, they may have trouble applying the ratios to find the unknown sides. One way to help
students with this process is to have them write the ratios with a variable. For example the 30-60-90
triangle ratio would be x : 2x : xV3. Next, have them identify which side they are given and use the
appropriate part of the ratio to determine x and the other side.
Example 1: Find the other two sides in a 30-60-90 triangle given that the hypotenuse is 8.
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Answer: First set 2x = 8 and solve to get x = 4 which is the short leg. The long leg is then 4 v3.
Example 2: Find the length of a leg in an isosceles right triangle with hypotenuse 3.
Answer: The ratio for an isosceles right triangle or 45-45-90 triangle is x : x : x V2. Since we are given the
hypotenuse here, x"v2 = 3. Now we must solve for x as shown below.
xV2 _ 3
V2 V2
_ _3_ V2 _ 3V2
Rationalizing the Denominator - Sometimes student will not recognize that -y= and -^ are equivalent.
Most likely, they learned how to rationalize denominators in algebra, but it is nice to do a short review
before using these types of ratios in special right triangles.
Derive with Variables - The beginning of the last chapter offers students a good amount of experience
with ratios. If they did well on those sections, it would benefit them to see the derivation of the ratios
done with variable expressions. It would give them practice with a rigorous derivation, review and apply
the algebra they have learned, and help them see how the triangles can change in size.
Exact vs. Decimal Approximation - Many students do not realize that when they enter V2 into a
calculator and get 1.414213562, that this decimal is only an approximation of v2. They also do not realize
that when arithmetic is done with an approximation, that the error usually grown. If 3.2 is rounded to
3, the error is only 0.2, but if the three is now multiplied by five, the result is 15, instead of the 16 it
would have been if original the original number had not been rounded. The error has grown to 1.0. Most
students find it more difficult to do operations with radical expressions than to put the numbers into their
calculator. Making them aware of error magnification will motivate them to learn how to do operations
with radicals. In the last step, it may be nice to have a decimal approximation so that the number can be
easily compared with other numbers. It is always good to have an exact form for the answer so that the
person using your work can round the number to the desired degree of accuracy.
Tangent, Sine and Cosine
Trig Thinking - Students sometimes have a difficult time understanding trigonometry when they are
first introduced to this new branch of mathematics. It is quite a different way of thinking when compared
to algebra or even geometry. Let them know that as they begin their study of trigonometry in the next
few sections the calculations won't be difficult, the challenge will be to understand what is being asked.
Sometimes students have trouble because they think it must be more difficult than it appears to be. Most
students find they like trigonometry once they get the feel of it.
Ratios for a Right Angle - Students will sometimes try to take the sine, cosine or tangent of the
right angle in a right triangle. They should soon see that something is amiss since the opposite leg is the
hypotenuse. Let them know that there are other methods of finding the tangent of angles 90° or more.
The triangle based definitions of the trigonometric functions that the students are learning in this chapter
only apply to angles in the interval 0° < m < 90°.
The Ratios of an Angle - The sine, cosine, and tangent are ratios that are associated with a specific
angle. Emphasize that there is a pairing between an acute angle measure, and a ratio of side lengths.
Sine, cosine, and tangent is best described as functions. If the students' grasp of functions is such that
introducing the concept will only confuse matters, the one-to-one correspondence between acute angle and
ratio can be taught without getting into the full function definition. When students understand this, they
will have an easier time using the notation and understanding that the sine, cosine, and tangent for a
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specific angle are the same, no matter what right triangle it is being used because all right triangles with
that angle will be similar.
Use Similar Triangles - Many students have trouble understanding that the sine, cosine, and tangent
of a specific angle measure do not depend on the size of the right triangle used to take the ratio. Take
some time to go back and explain why this is true using what the students know about similar triangle.
It will be a great review and application. Remind the students that if the right triangles have one set of
congruent acute angles, then they are similar by the AA Triangle Similarly Postulate. Once the triangles
are known to be similar it follows that their sides are proportional. It may be a good activity to have
students make right triangles with a particular acute angle measure and compare the ratios of the sides.
They should see that no matter how big or small they made their triangle, they get the same ratios as
their classmates. You could also refer to the special right triangles to make this connection. Explaining
that the names sine, cosine and tangent were given to these ratios and the values were recorded in tables
by angle measure may help them understand the idea a little better. You may even want to show them
a trig table of values and explain that their calculator is simply looking up a value in a table when they
type in the trig function and a particular angle measure.
Here is a Sketchpad activity that may further enhance student understanding:
1. Students can construct similar right triangles using dilation from the transformation menu.
2. After choosing a specific angle they should measure the corresponding angle in all the triangles. Each
of these measurements should be equal.
3. The legs of all the right triangles can be measured.
4. Then the tangents can be calculated.
5. Student should observe that all of ratios are the same.
Trig Errors are Hard to Catch - The math of trigonometry is, at this point, not difficult. Not much
computation is necessary to chose two numbers and put them in a ratio. What students need to be aware
of is how easy it is to make a little mistake and not realize that there is an error. When solving an equation
the answer can be substituted back into the original equation to be checked. The sine and cosine for acute
angles do not have a wide range. It is extremely easy to mistakenly use the sine instead of the cosine in an
application and the difference often is small enough to seem reasonable, but still definitely wrong. Ask the
student to focus on accuracy as they work with these new concepts. Remind them to be slow and careful.
SOHCAHTOA - This pneumonic device has been around for a while because it helps students keep the
ratios straight. Another way to write it that makes it even more clear is: S^ Cjj Tj.
Something to Consider - Ask the students to combine their knowledge of side-angle relationships in a
triangle with the definition of sine. How does the length of the hypotenuse compare to the lengths of the
legs of a right triangle? What does that mean about the types of numbers that can be sine ratios? With
leading questions like these students should be able to see that the sine and cosine ratios for an acute angle
will always be less than one. This type of analysis will prepare them for future math classes and increase
their analytical thinking skills. It will also be a good review of previous material and help them check
there work when they first start writing sine and cosine ratios.
Two-Step Problems - Having the students write sine, cosine, and tangent ratios as part of two-step
problems will help them connect the new material that they have learned to other geometry they know.
They will remember it longer, and be better able to see where it can be applied.
Example: AABC is a right triangle with the right angle at vertex C. AC = 3 cm, BC = 4 cm. What is the
sine of LAI
Answer: AB = 5 cm by the Pythagorean theorem, therefore sin A = f.
Note: The sine of an angle does not have units. The units will cancel out in the ratio.
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Inverse Trigonometric Ratios
Regular or Arc - Students will sometimes be confused about when to use the regular trigonometric
function and when to use the inverse. They understand to concepts, but do not want to go through the
entire thought process each time they must make the decision. I give them this short rule of thumb to help
them remember: When looking for a ratio or side length, use regular and when looking for an angle use
arc. They can associate "angle" and "arc" in their minds. Use the alliteration.
It may also help to explain that the inverse or "arc" trigonometric function does the reverse of the original.
Each of the original three trig functions essentially go to a table, look up an angle measure and find the
corresponding ratio. The "arc" functions will look up the ratio in the table and give back the corresponding
angle measure. Understanding what it is that the calculator does when they use these functions may help
reinforce student understanding.
Which Trig Ratio - A common mistake students make when using the inverse trigonometric functions
to find angles in right triangles is to use the wrong function. They may use arcsine instead of arccosine for
example. There is a process that students can use to reduce the number of these kinds of errors.
1. First, the students should mark the angle whose measure is to be found. With the angle in question
highlighted, it is easier for the students to see the relationship the sides have to that angle. It is fun
for the students to use colored pencils, pens, or highlighters.
2. Next, the students should look at the sides with known side measures and determine their relationship
to the angle. They can make notes on the triangle, labeling the hypotenuse, the adjacent leg and
the opposite let. If they are having trouble with this I have them look for the hypotenuse first and
always highlight it green, then they can decide between opposite and adjacent for the remaining two
sides.
3. Now, they need to look at the two sides they have chosen, and decide if they need to use sine, cosine,
or tangent.
2.9 Circles
Parts of Circles & Tangent Lines
Circle Vocabulary - This section has quite a few vocabulary words. Some the students will already know,
like radius, and some, like secant, will be new. Encourage the students to make flashcards or a vocabulary
list. They should know the word definition and have pictures drawn and labeled. It is also important for
students to know the relationships between the words. The radius is half the length of the diameter and
the diameter is the longest chard in a circle. Make knowing the vocabulary a specific assignment, otherwise
many students will forget to take the time to learn the vocabulary well.
Circle or Disk - The phrase "a point on the circle" is commonly used. This will confuses the students
that do not realize that the circle is the set of points exactly some set distance from the center, and not
the points less than that distance. What is happening is that they are confusing the definition of a disk
and a circle. Emphasize to the students that a circle is one dimensional; it only contains the points on the
edge. Another option is to give them the definition of a disc along with that of a circle, so that they can
compare and contrast the two definitions.
Inscribed or Circumscribed - An inscribed circle can also be described as a circumscribed polygon. The
different ways that these vocabulary words can be used can make learning the relationships complicated.
As a guide, tell the students that the object inscribed is on the inside. Starting with that, they can work
out the rest. For practice, ask the students to draw different figures that are described in words, like a
circumscribed hexagon, or a circle inscribed in an octagon.
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All the Radii of a Circle Are Congruent - It may seem obvious, but frequently students forget to
use the fact that all the radii of a circle are congruent. This follows directly from the definition of a circle.
Remind students to use this fact when setting up equations and assigning variables to different radii in the
same circle.
Tangency - Initially students get very confused by the different tangencies. There is a lot here that they
need to digest. Keep reviewing the differences between internally and externally tangent circles and tangent
lines that are internal and external so that students have a chance to practice identifying the differences.
Congruent Tangents - In this section the Tangent Segment Theorems is proved and applied. Remind
student that this is only true for tangents and does not extend to secants. Sometimes student will see a
secant enter a circle and think the distance from the exterior point to where the secant intersects the circle
is the same as a tangent or another secant from that same point.
Hidden Tangent Segments - Sometimes it is difficult for students to recognize tangent segments because
they are imbedded in a more complex figure, or the tangent segment is extended in some way. A common
situation where this occurs is when there is an inscribed circle. Tell the students to be on the lookout for
tangent segments. They should look at segments individually and as part of the whole.
Using the Pythagorean Theorem to Find Side Measurers in Right Triangles - Using the
Pythagorean Theorem to find the measures of sides in a right triangle is a common practice in this section.
Students should be on the lookout for right triangles formed by a radius and tangent segment. Later
in this chapter they will also find right triangles formed by radii (and diameters) and chords that they
bisect. Recognizing these perpendicular segments and the right triangles they form will help students solve
problems.
Properties of Arcs
Naming Major Arcs and Semicircles - When naming and reading the names of major arcs and
semicircles, the three letter system is sometimes confusing for students. When naming an angle with three
letters, the first place to look is to the middle letter, the vertex. It is just the opposite for a three letter arc
name. First, the students should locate the endpoints of the arc at the ends of the name. For a major arc
they have two arcs to choose from. The major arc uses three letters and is the long way around. Any of
the other points on the major arc can be used to designate that the long path is being taken. A semicircle
divides the circle into two congruent arcs. A third letter is needed to designate which half of the circle is
being named.
Look For Diameters - When working exercises that call for students to find the measures of arcs by
adding and subtracting arc and angle measures in a circle, students often forget that a diameter divides
the circle in half, or into two 180 degree arcs. Remind the students to be on the lookout for diameters
when finding arc measures.
Sum is 360° - Remind students that the entire arc (all the way around the circle) is 360°. This may seem
obvious to some students, but others don't pick up on this right away and need reminding.
Central Angles - Students may grasp the idea immediately that central angles are equal in measure
to the arc they intercept, but they often forget this property as they add additional angle theorems for
circles. There are so many different angles that can be formed in circles and this is just the first to be
explored. It is imperative that students focus on the fact that central angles have a vertex at the center of
the circle (not on the circle or in the circle). Keep saying this every time you talk about a central angle so
that students really internalize this definition. If they continually go back to where the vertex is located
they will be better able to distinguish between the different types of angles formed by segments in, on and
around a circle.
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Properties of Chords
Update the Theorem List - Students should be keeping a notebook full of all the theorems they have
learned in geometry class. The outlines of keywords, definitions and theorems provided at the beginning go
each chapter is an excellent basis for this. These theorems are like tools that can be used to work exercises
and write proofs. This section has quite a few different theorems about the relationships or chords and
angles that need to be included in their notebook. Each entry should have the name of the theorem, the
written statement of the theorem, and a picture to illustrate the relationship. Not only will this be good
reference material, making the notebook will help the students to remember the material.
Tips and Suggestions - There are a few strategies that students should keep in mind when working on
the exercises in this section.
1. Draw in segments to create right triangles, central angles, and any other useful geometric objects.
2. Remember to split the length of the chord in half if only half of it is used in a right triangle. Don't
just use the numbers that are given. The theorems must be applied to get the correct number, and
multiple steps will usually be necessary.
3. Use trigonometry of right triangles to find the angles and segment lengths needed to complete the
exercise.
4. Don't forget that all radii are congruent. If you have the length of one radius, you have them all,
including the ones you add to the figure.
5. Employ the Pythagorean theorem and any other tool you have from previous lessons that might be
useful.
Inscribed Angles
Inscribed Angle or Central Angle - When students spot an arc/angle pair to use in solving a complex
circle exercise, the first step is to identify the angle as a central angle, an inscribed angle, or possibly
neither. If necessary, they can trace the sides of the angle back from the arc to see where the vertex is
located. If the vertex is at the center of the circle, it is a central angle, and the measure of the arc and the
angle are equal. If the vertex is on the circle, it is an inscribed angle, and the students must remember to
double the angle measure to get the arc measure (or divide the arc measure by 2 to get the angle measure).
A good mnemonic device is to think of the arc of an inscribed angle being farther away from the vertex
than the arc of a central angle. Therefore the measure of the arc will be larger. If the vertex is at neither
the exact center or on the circle, no arc/angle relationship can be determined with only one arc.
What to Look For - Students can be overwhelmed by the number of different relationships that need to
be used to solve circle exercises. Sometimes they can just get paralyzed and not know where to start. In
small groups, or as a class, have them create a list of possible tools that are commonly used in these types
of situations.
Does the figure contain?
1. A triangle with a sum of 180°.
2. A convex quadrilateral with a sum of 360°.
3. A right triangle formed with a tangent and radius
4. An isosceles triangle formed with two radii
5. A diameter creating a semicircle
6. Arcs covering the entire circle
7. Central or Inscribed angles
8. Congruent tangents
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9. A right angle inscribed in a semicircle
10. Perpendicular segments (radius and tangent, radius and chord that is bisected)
11. Similar triangle with proportional sides
12. Congruent triangles with congruent corresponding parts
Any New Information is Good - If students can not immediately see how to find the measure they are
after, advise them to find any measure they can. This keeps their mind active and working. Frequently,
they will be able to use the new information to find other measures, and will eventually work their way
around to the desired answer. This might not be the most efficient method, but the students' technique
will improve with practice.
Angles of Chords, Secants, and Tangents
Where's the Vertex? - When determining the relationships between angles and arcs in a circle the
location of the vertex of the angle is the determining factor. There are four possibilities.
1. The vertex of the angle is at the center of the circle, it is a central angle, and the arc and angle have
the same measure.
2. The vertex of the angle is on the circle. The angle could be made by two cords, an inscribed angle,
or by a chord and a tangent. In either situation, the measure of the arc is twice that of the angle.
3. The vertex of the angle is inside the circle, but not at the center. In this case two arcs are necessary,
and the angle measure is the average of the measures of the arcs cut off by the chords that form the
vertical angles.
4. The vertex of the angle is outside the circle. Then the two intersected arcs have to be subtracted
and the difference divided by two. Note the similarity to an average.
Students often need help organizing information in this way. It is best to do this with them, as a class
activity so that in the future they will be able to do it for themselves. After that, lots of practice is
advised. This unit is often very difficult for even the strongest math students simply because of the
amount of information they must absorb and apply.
Use the Arcs - It is typical to have more than one angle intercepting a specific arc. In this case a measure
can be moved to an arc and then back out to another angle. Another situation students should look for
is when a circle is divided into two arcs. One arc can be represented as 360°- (an expression for the other
arc). Students sometimes miss these kinds of moves. It may be beneficial to have students share with the
class the different strategies and patterns they see when working on these exercises.
Example 1: Two tangent segments with a common endpoint intercept a circle dividing it into two arcs, one
of which is twice as big as the other. What is the measure of the angle formed by the by the two tangents?
Answer: 60°.
Make a sketch to illustrate the problem as shown here.
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x + 2x = 360
x= 120
angle measure = (240 - 120) -f 2 = 60°
Example 2: Two intersecting chords intercept congruent arcs. What kind of angles do the chords form?
Answer: Central angles. The chords must intersect at the center of the circle in order to intercept congruent
arcs, which makes the angles formed central angles.
Segments of Chords, Secants, and Tangents
Chapter Study Sheet - This chapter contains many relationships for students to remember. It would
be helpful for them to summarize all of these relationships on a single sheet of paper to use when studying.
Some instructors allow students to use these sheets on the exam in order to encourage students to make the
sheets. The value of a study sheet is in its making. Students should know this and make them regardless
of whether they can be used on the exams. Sometimes if students know that they will be able to use the
study sheet, they will not work to remember all of the relationships, and their ability to learn the material
is compromised. It is a hard issue to work around and each instructor needs to deal with it as he or she
feels best with their particular classes.
When to Add - When writing proportions involving secants, students will have a difficult time remem-
bering to add the two segments together to form the second factor. A careful study of the proof will help
them remember this detail. When they see secants, have them picture the similar triangles that could be
drawn. Remind them, and give them ample opportunity to practice.
Go Through the Proportions - Take the time, if you can, to go through the process of proving these
relationships between the lengths of secants, chords and tangents using similar triangles. It is not necessary
for students to memorize these proofs, but sometimes they have an easier time remembering the properties
if they understand their origins.
Have Them Subtract - One way to give students more practice with the lengths of secants in circles is
to give them exercises where the entire length of the secant is given, and they have to setup an expression
using subtraction to use in the proportion.
Example 1: A secant and a tangent segment have a common exterior endpoint. The secant has a total
length of 12 cm and the tangent has length 7 cm. What is the measure of both segments of the secant?
Answer: The secant is composed of two segments with approximate lengths 4.1 cm and 7.9 cm.
Let one segment of the secant be jc, so the other can be represented by 20 - x.
7 2 = (12-jc)x12
x^7.9
Example 2: Two secant segments have a common endpoint outside of a circle. One has interior and exterior
segments of lengths 10 ft and 12 ft respectively and the other has a total measure of 18 ft. What is the
measure of the two segments composing the other secant?
Answer: The secant is composed of two segments with lengths 3^ ft and 14 | ft.
12(10 + 12) = (18-x)xl8
1
X = 3 3
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Practice, Practice, Practice - Perhaps the most effective method to get students to internalize the
concepts in this chapter is to provide extensive practice. There is just so much information that students
are going to have a hard time remembering it all unless they practice it repeatedly. Mixed up the different
concepts in a practice assignment so that they get in the habit of switching gears repeatedly and seeing
different types of problems on the same page. Too often, students will just assume that all the problems in
a particular assignment should be solve the same way and they don't stop to think about how each problem
might be different from the previous one. Mixing problems up will help them practice this important skill.
Extension: Writing and Graphing the Equations of Circles
Square the Radius - When working with the equation of a circle, students frequently forget that the
radius is squared in the equation, especially when the radius is an irrational number. Explaining the
equation of the circle in terms of the Pythagorean theorem will help the students remember and understand
how to graph this conic section.
Remembering the Equation - Refer back to the definition of a circle- set of all point in the plane
equidistant from a given point. This implies that the equation requires this "given" point, or center, and
the distance, or radius, of the circle. Show students that the formula is derived by taking a point on the
circle, (x,v), the center, (/z, £), and setting the distance between them equal to the radius as shown below.
i
x-h) 2 + (y-k) 2 = r
Now, square both sides to get the equation: (x - h) + (y - k) = r
The more students know and understand about the origins of particular equations and properties, the more
likely they are to remember them in the future. Going through this process also helps students realize
that all these equations are not just "made up", they have roots in concepts they may already know and
understand.
Completing the Square - Completing the square to put the equation of a conic section in standard form
is a nice little math trick. It exemplifies the kinds of moves mathematicians use to manipulate expressions
and equations. Students find it difficult to do especially when fractions are involved and they have trouble
retaining the process for more than a few days. Give them many opportunities to practice.
2.10 Perimeter and Area
Triangles and Parallelograms
The Importance of Units - Students will give answers that do not include the proper units, unless it
is required by the instructor. When stating an area, square units should be included, and when referring
to a length, linear units should be used. Using proper units helps reinforce the basic concepts. With
these first simple area problems including the units seems like a small detail, but as the students move to
more complex situations combining length, area, and volume, units can be a helpful guide. In physics and
chemistry dimensional analysis is an important tool.
The Power of Labeling - When doing an exercise where a figure needs to be broken into polynomials with
known area formulas, it is important for the student to draw on and label the figure well. Each polygon,
so far only parallelograms and triangles, should have their base and height labeled and the individual area
should be in the center of each. By solving these exercises in a neat, orderly way student will avoid errors
like using the wrong values in the formulas, overlapping polygons, or leaving out some of the total area.
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Subtracting Areas - Another way of finding the area of a figure that is not a standard polygon is to
calculate a larger known area and then subtracting off the areas of polygons that are not included in the
target area. This can often result in fewer calculations than adding areas. Different minds work in different
ways, and this method might appeal to some students. It is nice to give them as many options as possible
so they feel they have the freedom to be creative.
The Height Must Be Perpendicular to the Base - Students will frequently take the numbers from a
polygon and plug them into the area formula without really thinking about what the numbers represent.
In geometry there will frequently be more steps. The students will have to use what they have learned to
find the correct base and height and then use those numbers in an area formula. Remind students that
they already know how to use a formula; many exercises in this class will require more conceptual work.
Write Out the Formula - When using an area formula, it is a good idea to have students first write
out the formula they are using, substitute numbers in the next step, and then solve the resulting equation.
Writing the formula helps them memorize it and also reduces error when substituting and solving. It
is especially important when the area is given and the student is solving for a length measurement in
the polygon. Students will be able to do these calculations in their heads for parallelograms, and maybe
triangles as well, but it is important to start good habits for the more complex polygons to come.
Trapezoids, Rhombi, and Kites
It's Arts and Crafts Time - Student have trouble remembering how to derive the area formulas. At
this level it is required that they understand the nature of the formulas and why the formulas work so
they can modify and apply them in less straightforward situations. An activity where student follow the
explanation by illustrating it with shapes that they cut out and manipulate is much more powerful than
just listening and taking notes. It will engage the students, keep their attention, and make them remember
the lesson longer. Here are examples of how students can do this for each figure in this section.
Trapezoid - This procedure models the description in the text. Actually have students physically cut out
the figures and do this will deepen understanding. Afterwards, the pictures in the lesson in the text will
have much more meaning to them as they will have a more concrete understanding of what is going on.
1. Have student use the parallel lines on binder paper to draw a trapezoid. They should draw in the
height and label it h. They should also label the two bases b\ and Z?2-
2. Now they can trace and cut out a second congruent trapezoid and label it as they did the first.
3. The two trapezoids can be arranged into a parallelogram and glued down to another piece of paper.
4. Identify the base and height of the parallelogram in terms of the trapezoid variables. Then substitute
these expressions into the area formula of a parallelogram to derive the area formula for a trapezoid.
5. Remember that two congruent trapezoids were used in the parallelogram, and the formula should
only find the area of one trapezoid.
Kite - This particular method is different than the description in the text. You can choose to do this
either way, but it helps to improve understanding sometimes to show different methods to achieve the same
results. Some students may have an easier time grasping the concept one way or the other.
1. Have the students draw a kite. They should start by making perpendicular diagonals, one of which
is bisecting the other. Then they can connect the vertices to form a kite.
2. Now they can draw in the rectangle around the kite.
3. Identify the base and height of the parallelogram in terms of d\ and ^2, and then substitute into the
parallelogram area formula to derive the kite area formula.
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4. Now have the students cut off the four triangles that are not part of the kite and arrange them over
the congruent triangle in the kite to demonstrate that the area of the kite is half the area of the
rectangle.
Rhombus - You could allow students to decide which way they would rather do this one or just let them
come up with their own method. The more they can do this on their own, the more they have gleaned
from this activity.
Also, the area of a rhombus can be found using either the kite or parallelogram area formulas. Use this as
an opportunity to review subsets and what they mean in terms of applying formulas and theorems.
Area of Similar Polygons
Reducing Fractions - It is helpful in this section to have ratios in reduced form. Once students start
square-rooting ratios (in fraction form), however, they may start messing up their reductions. For example,
they may start thinking that - reduces to |. Just be aware of this common error and point out the difference
to students as needed.
Adjust the Scale Factor - It is difficult for students to remember to square and cube the scale factor
when writing proportions involving area and volume. Writing and solving a proportion is a skill they know
well and have used frequently. Once the process is started, it is hard to remember to add that extra step
of checking and adjusting the scale factor in the middle of the process. Here are some ways to reinforce
this step in the students' minds.
1. Inform students that this material is frequently used on the SAT and other standardized tests in
some of the more difficult problems.
2. Play with graph paper. Have students draw similar shape on graph paper. They can estimate the
area by counting squares, and then compare the ratio of the areas to the ratio of the side lengths.
Creating the shapes on graph paper will give the students a good visual impression of the areas.
3. Write out steps, or have the students write out the process they will use to tackle these problems.
(1) Write a ratio comparing the two polygons. (2) Identify the type of ratio: linear, area, or volume.
(3) Adjust the ratio using powers or roots to get the desired ratio. (4) Write and solve a proportion.
4. Mix-up the exercises so that students will have to square the ratio in one problem and not in the
next. Keep them on the lookout. Make them analyze the situation instead of falling into a habit.
Example 1: The ratio of the lengths of the sides of two squares is 2:3. What is the ratio of their areas?
Answer: 4:9, The ratio of areas is the ratio of the lengths squared.
Example 2: The area of a small triangle is 15 cm 2 , and it has a height of 5 cm. A larger similar triangle
has an area of 60 cm 2 . What is the corresponding height of the larger triangle?
Answer: 10 cm
The area ratio is 15:60 or 1:4. The length ratio is then the square root of this, or 1:2. Now set up a
proportion to solve for the height of the smaller triangle.
15
2 ~ x
Example 3: The ratio of the lengths of two similar rectangles is 4:5. The larger rectangle has a width of
45 cm. What is the width of the smaller rectangle?
Answer: 36 cm.
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In this problem, we did not have to adjust the ratio since we are given a ratio of lengths and a length is
what we are trying to find. Just set up the proportion and solve.
1 _ ±
5 " 45
Example 4: The ratio of the areas of two regular pentagons is 25:64. What is the ratio of their corresponding
sides?
Answer: 5:8
This time students have to square root the area ratio to find the ratio of the lengths of the sides.
Circumference and Arc Length
Pi is an Irrational Number - Many students can give the definition of an irrational number. They
know that an irrational number has an infinite decimal that has no pattern, but they have not really
internalized what this means. Infinity is a difficult concept. A fun way to help the students develop this
concept is to have a pi contest. The students can chose to compete by memorizing digits of pi. They can
be given points, possible extra credit, for ever ten digits or so, and the winner gets a pie of their choice.
The students can also research records for memorizing digits of pi. The competition can be done on March
14^, pi day. When the contest is introduced, there is always a student who asks "How many points do I
get if I memorize it all?" It is a fun way to reinforce the concept of irrational numbers and generate a little
excitement in math class.
There Are Two Values That Describe an Arc - The measure of an arc describes how curved the
arc is, and the length describes the size of the arc. Whenever possible, have the students give both values
with units so that they will remember that there are two different numerical descriptions of an arc. Often
student will give the measure of an arc when asked to calculate its length.
Arc Length Fractions - Fractions are a difficult concept for many students even when they have come as
far as geometry. For many of them putting the arc measure over 360 does not obviously give the part of the
circumference included in the arc. It is best to start with easy fractions. Use a semi-circle and show how
||§ reduces to |, then a ninety degree arc, and then a 120° arc. After some practice with fractions they
can easily visualize, the students will be able to work with any arc measure as a fraction of 360 degrees.
Exact or Approximate - When dealing with the circumference of a circle there are often two ways to
express the answer. The students can give exact answers, such as 2n cm or the decimal approximation
6.28 cm. Explain the strengths and weaknesses of both types of answers. It is hard to visualize 137T ft, but
that is the only way to accurately express the circumference of a circle with diameter 13 ft. The decimal
approximations 41, 40.8, 40.84, etc, can be calculated to any degree of accuracy, are easy to understand in
terms of length, but are always slightly wrong. Let the students know if they should give one, the other,
or both forms of the answer.
Areas of Circles and Sectors
Reinforce - This section on area of a circle and the area of a sector is analogous to the previous section
about circumference of a circle and arc length. This gives students another chance to go back over the
arguments and logic to better understand, remember, and apply them. Focus on the same key points and
methods in this section, and compare it to the previous section. Mix-up exercises so students will see the
similarities and learn each more thoroughly.
Don't Forget the Units - Remind students that when they calculate an area the units are squared.
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When an answer contains the pi symbol, students are more likely to leave off the units. In the answer
7n cm 2 , the n is part of the number and the cm 2 are units of area.
Draw a Picture - When applying geometry to the world around us, it is helpful to draw, label, and work
with a picture. Visually organized information is a powerful tool. Remind students to take the time for
this step when calculating the areas of the irregular shapes that surround us.
Example 1: What is the area between two concentric circles with radii 5 cm and 12 cm?
(Hint: Don't subtract the radii.)
Answer: 1447T - 257T = 119ncm 2
Example 2: The area of a sector of a circle with radius 6 cm, is 127T cm 2 . What is the measure of the
central angle that defines the sector?
Answer: Yin = ^ * n * 6 2 ,x = 120°. The central angle measures 120°.
Example 3: A square with side length 5 V2 cm is inscribed in a circle. What is the area of the region
between the square and the circle?
Answer: approx. 28.5 cm 2
First find the diagonal of the square using special right triangles: 5 v2 * v2 = 5*2 = 10.
This makes the radius of the circle 5 cm. Now we can find the area of the circle and subtract the area of
the square as shown below.
5 2 * n - (5 V2) 2
25tt - 50 « 28.5cm 2
2.11 Surface Area and Volume
Exploring Solids
Polygon or Polyhedron - A polyhedron is defined using polygons, so in the beginning students will
understand the difference. After some time has passed though, students tend to get these similar sounding
words confused. Remind them that polygons are two-dimensional and polyhedrons are three-dimensional.
The extra letters in polyhedron represents it spreading out into three-dimensions.
The Limitations of Two Dimensions - It is difficult for students to see the two-dimensional repre-
sentations of three-dimensional figures provided in books and on computer screens. A set of geometric
solids is easily obtained through teacher supply companies, and are extremely helpful for students as they
familiarize themselves with three-dimensional figures. When first counting faces, edges, and vertices most
students need to hold the solid in their hands, turn it around, and see how it is put together. After they
have some experience with these objects, students will be better able to read the figures drawn in the text
to represent three-dimensional objects.
Assemble Solids - A valuable exercise for students as they learn about polyhedrons is to make their own.
Students can cut out polygons from light cardboard and assemble them into polyhedrons. Patterns are
readily available. This hands-on experience with how three-dimensional shapes are put together will help
them develop the visualization skills required to count faces, edges, and vertices of polyhedrons described
to them.
Computer Representations - When shopping on-line it is possible to "grab" and turn merchandise so
that they can be seen from different perspectives. The same can be done with polyhedrons. With a little
poking around students can find sites that will let them virtually manipulate a three-dimensional shape.
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This is another possible option to develop the students' ability to visualize the solids they will be working
with for the remainder of this chapter.
Using the Contrapositive - If students have already learned about conditional statements, point out
to them that Example 4 in this section makes use of the contrapositive. Euler's formula states that if a
solid is a polyhedron, then V + F = E + 2. The contrapositive is that if V + F ^ E + 2, then the solid is
not a polyhedron. Students need periodic review of important concepts in order to transfer them to their
long-term memory. For more review of conditional statements, see the second chapter of this text.
Each Representation Has Its Use - Each of the methods for making two-dimensional representations
of three-dimensional figures was developed for a specific reason and different representations are most
appropriate depending on what aspect of the geometric solid is of interest.
Perspective - used in art, and when one wants to make the representation look realistic
Isometric View - used when finding volume
Orthographic View - used when finding surface area
Cross Section - used when finding volume and the study of conic sections (circles, ellipses, parabolas, and
hyperbolas) is based on the cross sections of a cone
Nets - used when finding surface area or assembling solids
Ask the students to think of other uses for these representations.
When students know and fully understand the options, they will be able to choose the best tool for each
task they undertake.
Isometric Dot Paper - If students are having trouble making isometric drawings, they might benefit
from the use of isometric dot paper. The spacing of the dots allows students to make consistent lengths
and angles on their polyhedrons. After some practice with the dot paper, they should be able to make
decent drawings on any paper. A good drawing will be helpful when calculating volumes and surface areas.
Practice - Most students will need to make quite a few drawings before the result is good enough to be
helpful when making calculations. The process of making these representations provides the student with
an opportunity to contemplate three-dimensional polyhedrons. The better their concept of these solids,
the easier it will be for them to calculate surface areas and volumes in the sections to come.
Additional Exercises:
1. In the next week look around you for polyhedrons. Some example may be a cereal box or a door stop.
Make a two-dimensional representation of the object. Choose four objects and use a different method of
representation for each.
These can make nice decorations for classroom walls and the assignment makes students look for way to
apply what they will learn in this chapter about surface area and volume.
Surface Area of Prisms and Cylinders
The Proper Units - Students will frequently use volume units when reporting a surface area. Because
the number describes a three-dimensional figure, the use of cubic units seems appropriate. This shows a
lack of understanding of what exactly it is that they are calculating. Provide students with some familiar
applications of surface area like wrapping a present or painting a room, to improve their understanding of
the concept. Insist on the use of correct units so the student will have to consider what exactly is being
calculated in each exercise.
Review Area Formulas - Calculating the surface area and volume of polyhedrons requires the students
to find the areas of different polygons. Before starting the new material, take some time to review the area
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formulas for the polygons that will be used in the lesson. When students are comfortable with the basic
area calculations, they can focus their attention on the new skill of working with three-dimensional solids.
A Prism Does Will Not Always Be Sitting On Its Base - When identifying prisms, calculating
volumes, or using the perimeter method for calculating surface area, it is necessary to locate the bases.
Students sometimes have trouble with this when the polyhedron in question is not sitting on its base.
Remind students that the mathematical definition of the bases of a prism is two parallel congruent polygons,
not the common language definition of a base, which is something an object sits on. Once students think
they have identified the bases, they can check that any cross-section taken parallel to the bases is congruent
to the bases. Thinking about the cross-sections will also help them understand why the volume formula
works later in this chapter.
Understand the Formula - Many times students think it is enough to remember and know how to apply
a formula. They do not see why it is necessary to understand how and why it works. The benefit of fully
understanding what the formula is doing is versatility. Substituting and simplifying works wonderfully for
standard cylinders, but what if the surface area of a composite solid needs to be found?
Example 1: Find the surface area of the composite solid shown below.
Answer: 426 u 2
There are a number of ways to "divide" up this figure into numerous rectagular pieces to find the surface
area. One way is to find the areas of the rectangular "steps" and add those to the rectangles formed on
the sides of the figure by drawing vertical lines.
2(4 x 12) + 2(3 x 12) + 2(4 x 12) + 2 {(4 x 11) + (3 x 7) + (4 x 4)} = 426 u 2
Make and Take Apart a Cylinder - Students have a difficult time understanding that the length of
the rectangle that composes the lateral area of a cylinder has length equal to the circumference of the
circular base. First, review the definition of circumference with the students. A good way to help students
visualize this is a soup can label. When the label is removed and placed flat on a table, it is a rectangle.
Another way to describe the circumference is to talk about an ant walking around the circle. Next, let
them play with some paper cylinders. Have them cut out circular bases, and then fit a rectangle to the
circles to make the lateral surface. After some time spent trying to tape the rectangle to the circle, they
will understand that the length of the rectangle matches up with the outside of the circle, and therefore,
must be the same as the circumference of the circle.
Surface Area of Pyramids and Cones
Prism or Pyramid - Some students have trouble deciding if a solid is a prism or a pyramid. Most try to
make the determination by looking for the bases. This is especially tricky if the figure is not sitting on its
base. Another method for differentiating between these solids is to look at the lateral faces. If there are
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a large number of parallelograms, the figure is probably a prism. If there are more triangles, the figure is
most likely a pyramid. Once the student has located the lateral faces, then they can make a more detailed
inspection of the base or bases.
Height, Slant Height, or Edge - A pyramid contains a number of segments with endpoints at the
vertex of the pyramid. There is the altitude which is located inside right pyramids, the slant height of
the pyramid is the height of the triangular lateral faces, and there are lateral edges, where two lateral
faces intersect. Students frequently get these segments confused. To improve their understanding, give
them the opportunity to explore with three-dimensional pyramids. Have the students build pyramids out
of paper or light cardboard. The slant height of the pyramid should be highlighted along each lateral face
in one color, and the edges where the lateral faces come together in another color. A string can be hung
from the vertex to represent the altitude of the pyramid. The lengths of all of these segments should be
carefully measured and compared. They should make detailed observations before and after the pyramid
is assembled. Once the students have gained some familiarity with pyramids and these different segments,
it will make intuitive sense to them to use the height when calculating volume, and the slant height for
surface area.
Example 1: A square pyramid is placed on top of a cube. The cube has side length 3 cm. The slant height
of the triangular lateral faces of the pyramid is 2 cm. What is the surface area of this composite solid?
Answer: 57 cm 2
Five faces of the cubic base are visible and therefore part of the surface area. Their total area is: 5x3x3 =
45 cm 2 . The surface area of the pyramid is just the lateral surface area because the base is not visible.
This lateral surface area is calculated by multiplying the perimeter of the base by the slant height of the
pyramid and then multiplying by one half as shown: ^X4x3x2 = 12 cm 2 . Adding these two areas together
yields the answer shown above.
Example 2: Calculate the surface area of the composite figure shown below.
Answer: 226.19 u 2
The surface area of this figure is the surface area of the cylinder minus the area of the "top" base plus
the lateral surface area of the cone. The area of the "top" of the base and the "base" of the cone are not
visible and therefore not part of the total surface area of the composite solid. Students are prone to just
find the surface areas of the two figures and add them together. This will result in an answer that includes
the areas of these two circular bases.
tt3 2 + 2(3)tt(8) + tt(3)(5) « 226.19 u 2
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Volume of Prisms and Cylinders
The Volume Base - In the past, when students used formulas, they just needed to identify the correct
number to substitute in for each variable. Calculating a volume requires more steps. To find the correct
value to substitute into the B in the formula V = Bh, usually requires an additional calculation with an
area formula. Students will often forget this step, and use the length of the base of the polygon that
is the base of the prism for the B. Emphasize the difference between /?, the linear measurement of the
length of a side of a polygon, and Z?, the area of the two-dimensional polygon that is the base of the prism.
Students can use dimensional analysis to check their work. Volume is measured in cubic units, so three
linear measurements, or a linear unit and a squared unit must be fed into the formula.
Example 1: The volume of a 4 in tall coffee cup is approximately 50 in 3 . What is the radius of the base of
the cup?
Answer: The cup has a radius of approximately 2 inches.
Since we know the volume we can use the volume formula with the given height and solve for the radius
as shown below:
50 = ;rr 2 (4)
3.97887... ~r 2
Example 2: A prism has a base with area 15 cm 2 and a height of 10 cm. What is the volume of the prism?
Answer: V = 15 x 10 = 150 cm 3
Example 3: A triangular prism has a height of 7 cm. Its base is an equilateral triangle with side length 4
cm. What is the volume of the prism?
Answer: V = Bh = ±(4) (2 V3)(7) = 28 V3 * 48.5 cm 3
Given that the sides of the equilateral triangle are 4, then the altitude (height) of the triangle is 2 V3. So,
the area of the base is ^(4) (2 V3). Then we can multiply this by the height, 7.
Example 4: The volume of a cube is 27 cm 3 . What is the cube's surface area?
Answer: 36 cm 2
This is a two step problem. First, students should find the length of one edge of the cube by finding the
cubed root of 27. This is 3. Next, find the area of one square face (3 2 = 9) and multiply it by 6 because
there are 6 faces.
Volume of Pyramids and Cones
Don't Forget the - - The most common mistake students make when calculating the volume of a
pyramid is to forget to divide by three. They also might mistakenly divide by three when trying to find the
volume of a prism. The first step students should take when beginning a volume calculation, is to make
the decision if the solid is a prism or a pyramid. Once they have chosen, they should immediately write
down the correct volume formula.
Prism or Pyramid - Some students have trouble deciding if a solid is a prism or a pyramid. Most try to
make the determination by looking for the bases. This is especially tricky if the figure is not sitting on its
base. Another method for differentiating between these solids is to look at the lateral faces. If there are
a large number of parallelograms, the figure is probably a prism. If there are more triangles, the figure is
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most likely a pyramid. Once the student has located the lateral faces, then they can make a more detailed
inspection of the base or bases.
Mix'em Up - Students have just learned to calculate the surface area and volume of prisms, cylinders,
and cones. Most students do quite well when focused on one type of solid. They remember the formulas
and how to apply them. It is a bit more difficult when students have to choose between the formulas for all
four solids. Take a review day here. Have the students work in small groups during class on a worksheet
or group quiz that has a mixture of volume and surface area exercises for these four solids. The extra day
will greatly help to solidify the material learned in the last few lessons.
Example 1: A square pyramid is placed on top of a cube. The cube has side length 4 cm. The height of
the pyramid is 6 cm. What is the volume of this composite solid?
Answer: 96 cm 3
The volumes of the cube and square pyramid should be calculated separately and then added together to
get the total volume of the composite solid.
Example 2: Calculate the volume of the composite figure shown below.
Answer: 263.89 u 2
The volume of this composite solid can be calculated by finding the volumes of the cylinder and the cone
and adding them together. The Pythagorean theorem (or recognition of a Pythagorean Triple) will be
used to determine the height (4) of the cone.
tt3 2 (8) + -tt(3 2 )(4) * 263.89 u 2
o
Surface Area and Volume of Spheres
Expand on Circles - Students learned about circles earlier in the course. Review and expand on this
knowledge as they learn about spheres. Ask the students what they know about circles. Being able
to demonstrate their knowledge will build their confidence and activate their minds. Now, modify the
definitions that the students have provided to fit the three-dimensional sphere. Students will learn the new
material quickly and will remember it because it is now integrated with their knowledge of circles.
Explore Cross-Sections - One of the goals of this chapter is to develop the students' ability to think
about three-dimensional objects. Most students will need a significant amount of practice before becoming
competent at this skill. Take some time and ask the students to think about what the cross-sections of a
sphere and a plane will look like. Explore trends. What happens to the cross-section as the plane moves
farther away from the center of the circle? A cross-section that passes through the center of the sphere
makes the largest possible circle, or the great circle of the sphere.
Cylinder to Sphere - It would be a good exercise for students to take the formula for the surface area
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of a cylinder and derive the formula for the surface area of a sphere. It is just a matter of switching a few
variables, but it would be a good exercise for them. During the lesson, ask them to do it in their notes,
wait a few minutes and then do it on the board or ask one of them to put their work on the board. It
should look something like this:
^cylinder = bases + lateral area
^cylinder = 27ZT 2 + 2jirh
Now, replace h with r to get:
A sphere = 2flr 2 + 2jir{r)
^sphere = ^Tir 2 + 27ZT 2
^sphere = 47ZT
Point out to students that in the last line the terms could be combined because they both had nr 2 and are
therefore like terms. The coefficients could have been different, but to combine terms using the distributive
property they must have the exact same variable combination. Here the n is being treated as a variable
even though it represents a number. This is a more complex application of like terms than students are
used to seeing.
Limits - Another way to derive the volume of the sphere is to consider a limit. Essentially, the idea is to
sum the volume of an infinite number of pyramids. The base of each pyramid is a regular polygon on the
surface of the sphere and its height is the sphere's radius.
V pyramid = ~Bh = -Br
If we let the areas of each of the infinite number of bases be B\, B2, B3, . . . we get a volume formula for the
sphere of:
1 1 1
V sphere = « B x r + -B 2 r + -B 3 r + . . .
Now factor out the ^ and the r to get:
y sphere = "K^l + B 2 + B 3 , . . .)
Now, we can replace the sum of the infinite base areas with the surface area of the sphere to get:
V sphere = \r(^)
_ 4 3
V sphere — 77^*
You can do this in reverse to go from the formula for the volume of a sphere to the surface area of the
sphere:
y sphere = "^ 3 = -r(47ZT 2 )
-7rr 3 = -r(5 1 +5 2 +5 3 ,...)
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Now, dividing both sides by \r we get:
4;rr 2 = {B 1 +B 2 + B$,...)
Initially, the logic might seem fuzzy to them. The limit is a fundamental concept to all of calculus. It
is worthwhile to give it some attention here and some students are more interested in a formula and
mathematics in general when they understand where it comes from.
Extension: Exploring Similar Solids
Surface Area is Squared - Surface area is a two-dimensional measurement taken of a three-dimensional
object. Students are often distracted by the solid and use cubed units when calculating surface area or
mistakenly cube the ratio of linear measurements of similar solids when trying to find the ratio of the
surface areas. Remind them, and give them many opportunities to practice with exercises where surface
area and volume are both used.
Don't Forget to Adjust the Ratio - There are three distinct ratios that describe the relationship
between similar solids. When the different ratios and their uses are the subject of the lesson, students
usually remember to use the correct ratio for the given situation. In a few weeks when it comes to the
chapter test or on the final at the end of the year, students will frequently forget that the area ratio
is different from the volume ratio and the linear ratio. They enjoy writing proportions and when they
recognize that a proportion will be used, they get right to it without analyzing the ratios. One way to
remind them is to have them use units when writing proportions. The units on both sides of the equal
sign have to match before they can cross-multiply. Give them opportunities to consider the relationship
between the different ratios with questions like the one below.
Example 1: If a fully reduced ratio is raised to a power, will the resulting ratio be fully reduced? Explain
your reasoning.
Answer: Yes, two numbers make a fully reduced ratio if they have no common factors. Raising a number
to a power increases the exponent of each factor already present, but does not introduce new factors.
Therefore, the resulting two numbers will still not have any common factors.
These concepts frequently appear on the SAT. It will serve the students well to practice them from time
to time to keep the knowledge fresh.
Example 2: The ratio of the surface areas of two cubes is 25:49. What is the ratio of their volumes?
Answer: In order to find the ratio of the volumes, we need to square root the ratio of the areas and then
cube the resulting ratio:
(V^5) 3
5 3
125
(V49) 3
7 3
343
2.12 Rigid Transformations
Exploring Symmetry
360° Doesn't Count - When looking for rotational symmetries students will often list 360° rotational
symmetry. When a figure is rotated 360° the result is not congruent to the original figure, it is the original
figure itself. This does not fit the definition of rotational symmetry. This misconception can cause error
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when counting the numbers of symmetries a figure has or deciding if a figure has symmetry or not. Another
important note here is that sometimes rotational symmetry is referred to as point symmetry. The center
"point" of the figure is the center of rotation in a figure with rotational symmetry. Students might be
thrown by this new term which might appear on a standardized test if it isn't introduced here.
Review Quadrilateral Classifications - Earlier in the course students learned to classify quadrilaterals.
Now would be a good time to break out that Venn diagram. Students will have trouble understanding that
some parallelograms have line symmetry if they do not remember that squares and rectangles are types
of parallelograms. As the course draws to an end, reviewing helps students retain what they have learned
past the final. It is possible to redefine the classes of quadrilaterals based on symmetry. This pursuit
will make the student use and combine knowledge in different ways making what they have learned more
flexible and useful. You may also want to look at the symmetry of regular polygons.
Applications - Symmetry has numerous applications both in and outside of mathematics. Knowing some
of the uses for symmetry will motivate student, especially those who are not inspired by pure mathematics,
to spend their time and energy learning this material.
Biology - Most higher level animals have bilateral symmetry, starfish and flowers often have 72° rota-
tional symmetry. Naturally formed nonliving structures like honeycomb and crystals have 60° rotational
symmetry. These patterns are fascinating and can be used for classification and study.
Trigonometry - Many identities of trigonometry are based on the symmetry of a circle. In the next few
years of mathematics the students will see how to simplify extremely complex expressions using these
identities.
Advertising - Many company logos make use of symmetry. Ask the students to bring in examples of logos
with particular types of symmetry and create a class collection. Analyze the trends. Are certain products
more appropriately represented by logos that contain a specific type of symmetry? Does the symmetry
make the logo more pleasing to the eye or more easily remembered?
Functions - A function can be classified as even or odd based on the symmetry of its graph. Even functions
have symmetry around the y-axis, and odd functions have 180° rotational symmetry about the origin. Once
a function is classified as even or odd, properties and theorems can be applied to it.
Draw - Have students be creative and create their own logos or designs with specific types of symmetry.
Using these concepts in many ways will build a deeper understanding and the ability to apply the new
knowledge in different situations.
Translations
Translation or Transformation - The words translation and transformation look and sound quite similar
to students at first. Emphasis their relationship: A translation is just one of the many transformations the
students will be learning about in this chapter.
Image vs Pre-image - Students often get these two terms mixed up. Help students focus on the prefix,
"pre" which means "before". The pre-image is the image before a transformation is performed.
Mapping Notation - An ordered pair is use to represent a location on the coordinate plane, and mapping
notation indicates how a point is "moved" to create an image of a point. Give students ample opportunity
to practice reading and writing with this notation.
The Power of Good Notation - There is a lot going on in these exercises. There are the points that
make the preimage, the corresponding points of the image, and the mapping notation used to describe the
translation. Good notation is the key to keeping all of this straight. The points of the image should be
labeled with capital letters, and the prime marks should be used on the points of the image. In this way
it is easy to see where each point has gone. This will be even more important when working with more
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complex transformations in later sections. Start good habits now.
Use Graph Paper and a Ruler - When making graphs of these translations by hand, insist that the
students use graph paper and a ruler. If students try to graph on lined paper, the result is frequently
messy and inaccurate. It is beneficial for students to see that the pre-image and image are congruent to
reinforce the knowledge that a translation is an isometry. It is also important that students take pride in
producing quality work. They will learn so much more when they take the time to do an assignment well,
instead of just rushing through the work.
Translations of Sketchpad - Geometers' Sketchpad uses vectors to translate figures. The program will
display the pre-image, vector, and image at the same time. Students can type in the vector and can also
drag points on the screen to see how the image moves when the vector is changed. It is a quick and
engaging way to explore the relationships. If the students have access to Sketchpad and there is a little
class time available, it is a worthwhile activity. You will need to explain how students can write vector
from mapping notation in the program.
Reflections
Rules for Reflections in the Coordinate Plane - Students are likely to have a hard time memorizing
these rules. Encourage students to just reflect one point at a time until they notice a pattern. When
reflecting over the y-axis, the x changes sign and when reflecting over the x-axis, the y changes sign.
Reviewing the quadrants and where x and y are positive/negative may also help. They will then see how
the signs change as they cross the particular axes.
Will the Pre-image and Image Match Up? - Encourage students to visualize whether or not the
image and pre-image will "match up" if they fold over the line of reflection. This caheck at the end of the
process may help students avoid making careless errors.
Reflections over y — x - This particular reflection is very important for students to be able to recognize
for future math courses. The inverse of a function is a reflection of the original function over this line.
Understanding the connection between the process of creating this reflection (switching x and y values)
and finding an inverse function later in more advanced algebra courses (students will switch the variables
in the equation) will help students gain a deeper understanding of both concepts.
Rotations
Clockwise vs Counterclockwise Rotations - Students sometimes have difficulty keeping these two
straight. It is also counter-intuitive to them that a counterclockwise angle is considered positive and a
clockwise angle is negative. This text uses all positive angles and indicates clockwise vs counterclockwise but
if you decide to use Geometer's Sketchpad (or other computer programs) to do some additional activities,
you will need to explain this concept to students.
Rotating in the Coordinate Plane - There are mapping rules given in the text for the different rotations
in the coordinate plane. Sometimes, however, it is helpful to have students attempt to "visualize" these on
the coordinate plane. Often students forget the "rules" and cannot complete the assignment or problem on
a quiz or test. It may be helpful to do some examples using patty paper. Students can draw a triangle or
quadrilateral on the coordinate plane, then trace the figure and the axes onto the patty paper. Now they
can rotate the patty paper 90, 180 and 270 degrees using the axes as a reference to see how the figure (and
the coordinates of its vertices) change. This technique is particularly beneficial for visual and kinesthetic
learners.
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Composition of Transformations
Glide Reflections - Students sometimes struggle with the concept that order doesn't matter here. It is
helpful to have students do a problem such as the example below to experience this.
Example: AABC has vertices (2, -3), (5, 3) and (6, 0). Translate this figure 5 units left and reflect it over
the line y = -2. What are the coordinates of the final image? Now try doing the reflection before the
translation. What are the coordinates of the final image?
Answer: The coordinates of the resulting images for both orders is (-3, -1), (0, -7), (1, -4).
Reflections over Parallel Lines - Students may initially struggle with the relationship between this
double reflection and the resulting equivalent translation. Have students practice this on the coordinate
plane where distances between the vertices of the pre-image and image can be easily calculated to verify
the relationship.
Reflections over both Axes - Patty paper can be used again here to help students see that this double
reflection is actually a rotation of 180° about the origin. Students can also compare the rules for reflections
to the rules for rotations to see that the combination of the two reflections results in the rule for the
rotation.
Extension: Tessellating Polygons
Review Interior Angles Measures for Polygons - Earlier in the course students learned how to
calculate the sum of the measures of interior angles of a convex polygon, and how to divide by the number
of angles to find the measures of the interior angle of regular polygons. Now would be a good time to
review this lesson. The students will need this knowledge to see which regular polygons will tessellate and
the final is fast approaching.
Move Them Around - When learning about regular and semi-regular tessellation it is helpful for students
to have a set of regular triangles, squares, pentagons, hexagons, and octagons that they can slide around
and fit together. These shapes can be bought from a mathematics education supply company or made with
paper. Exploring the relationships in this way gives the students a fuller understanding of the concepts.
Use On-Line Resources - A quick search on tessellations will produce many beautiful, artistic examples
like the work of M. C. Escher and cultural examples like Moorish tiling. This bit of research will inspire
students and show them how applicable this knowledge is to many areas.
Tessellation Project - A good long-term project is to have the students create their own tessellations.
This is an artistic endeavor that will appeal to students that typically struggle with mathematics, and the
tessellations make nice decorations for the classroom.
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Chapter 3
Basic Geometry TE - Enrichment
Introduction
This FlexBook is designed to supplement the Basic Geometry FlexBook with enriching and extension
activities. Mostly, the activities will cross into art, technology, architecture, and history. By definition,
enrichment means to "enrich" or enhance something. The activities or problems included will enrich the
lesson and add to the lesson without adding too much of a challenge. There will also be "challenging"
problems that be separately identified. Like with the other FlexBooks, the activities and problems in this
text are suggestions and designed to supplement a lesson. Pick and choose activities as you see fit.
In addition to the supplemental activities in this FlexBook, see the Texas Instruments Geometry FlexBook.
This book has activities that supplement some sections and are designed to utilize a graphing calculator.
See http://www.ckl2.org/flexbook/book/3783/ for the Teacher's Edition and http://www.ckl2.org/
f lexbook/book/3766/ for the Students' Edition.
3.1 Basics of Geometry
Points, Lines, and Planes
Connections to Art
Tangrams are a fun way to introduce Geometry to any student. If you have tangrams, let students play
with them, make designs or animals. If you do not have access to tangrams, take them to the computer
lab to do online tangrams: http://pbskids.org/cyberchase/games/area/tangram.html
Point of points of intersection, lines, triangles, quadrilaterals, and planes to students as they play. You
could even take picture of students creations and put them up in your classroom.
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(To purchase, see: http://www.amazon. com/Learning- Advantage-Tangrams-Classroom-Set/dp/B000XURP14/
ref =sr_l_20?s=toys-and-games& ; ie=UTF8& ; qid=1305226538& ; sr=l-20)
Connections to Map Reading
A practical application of points, lines, and planes are maps. Show students a map of a state and see if
they can determine what represents a city (point), a highway (line), and the state itself (plane).
Taking this a step further, ask students what points (cities) are collinear (on the same highway) and which
are coplanar (all the cities). Have a discussion as to what sort of representation would be "space." Students
should see that it would be a globe.
Challenge
1. Three planes intersect in three different ways. Draw all three.
2. One line can divide a plane into two regions. Two lines can divide a plane into four regions. Three lines
that intersect at a point can divide a plane into six regions, but you can get more regions if the lines do not
all intersect at the same point. If you have six lines, what is the maximum number of regions into which
you can divide a plane? BONUS: What if you used n lines? What is the maximum number of regions into
which you can divide the plane?
Answers
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1. See the pictures below. Notice that we did not include options where all the planes were parallel (no
intersection) or when one plane intersected the other two (two planes parallel).
^c
2. For six lines, the maximum number of regions is 22. You can decide if you would like students to
generate a pattern or draw pictures for 1-6 lines (the pattern is: 2, 4, 7, 11, 16, 22, 29, ...). For n lines,
there will be ^n 2 + \n + 1 regions.
Segments and Distance
Extension
Using the Know What? as a guide, have students measure their own "head" width and length. Then, have
them find their height in heads. Students can also find: the length from the wrist to the elbow, the length
from the top of the neck to the hip, and the width from shoulder to shoulder, and hip height. Some of
these are given in the picture in the text, but have students verify the measurements for their own bodies.
Taking it one step further, you could have students create a scale drawing of him/her based on the head
measurements.
Connections to History
Describe the advantages of using the metric system to measure length over the English system. Use the
examples of the two rulers (one in inches and one in centimeters) to aid in your description.
Research the origins of ancient measurement units such as the cubit. Research the origins of the units
of measure we use today such as: foot, inch, mile, meter. Why are standard units important? Student
answers should include that the cubit was the first recorded unit of measure and it was integral to the
building of the Egyptian pyramids.
Research the facial proportions that da Vinci used to create his Vitruvian man. Write a summary of your
findings. Student answers should comment on the "ideal" proportions found in the human face and how
these correspond to our perception of beauty.
Challenge
1. Draw four points, A, B, C, and D such that AB = BC = AC = AD = BD.
(HINT: A,B,C and D should NOT be collinear)
2. AB = BC and AC = CD. If AD = 20, find the length of all the segments in the diagram.
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2. AC = CD = 10, AB = BC = 5,BD = 15
Angles and Measurement
Connections to Astronomy
Give students pictures of the Big Dipper and Orion (below). For homework, ask students to observer the
night sky and try to find these two constellations. Have them make note of any other constellations. To
see a full list of constellations, visit http://en.wikipedia.org/wiki/List_of_constellations. Ask the
students to find 2-3 constellations on line or in a book and to bring the hard copy of a picture of the
constellation to class.
The next day, in class, begin a whole discussion about the angles in the constellations and the distance
between stars. Ancient astronomers used to measure the degrees between stars using their hands. Here
are the approximations:
• Extend your little finger; its width is approximately 1 degree.
• Extend your three middle fingers; this is about 5 degrees.
• A clenched fist (thumb to little finger) is about 10 degrees.
• From the tip of the little finger to the tip of the thumb, an extended hand with fingers and thumb
splayed is about 20 degrees.
20 c
Now, for homework, have students see if they can find the angle measures between the stars in each constel-
lation they found the night before. Remind students that these hand measurements are approximations.
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138
They only work because the stars are so far away.
Challenge
1. You are measuring LABC with a protractor. When you line up BC with the 10° mark, AB lines up
with the 90° mark. Then you move the protractor so that BC lines up with the 25° mark. What
mark should AB line up with?
2. Why do you think the degree measure of a straight line is 180, the degree measure of a right angle
is 90, etc? In other words, answer the question, "Why is the straight line exactly 180° and not some
other number of degrees?"
Answers
1. It should line up with 115°.
2. Answers will vary. Students should comment about the necessity to have a number of degrees in a
line that is divisible by 30, 45, 60 and 90 degrees because these degree measures are prevalent in the
study of geometrical figures. Basically, setting the measure of a straight line equal to 180° allows us
to have more whole number degree measures in common geometrical figures.
Midpoints and Bisectors
Connections to Construction
This is a picture of a wooden frame for a home. Notice that BD bisects the sides of the roof {/.ABC)
and the other support beams (iFDG and lEDH). It is also perpendicular to EH. Ask students why it is
important why BD bisects these angles. What would happen if BD did not perfectly bisect these angles?
There are also several congruent segments. Ask students to find these. Then, as a classwork (individually
or in pairs) assignment, have students answer the following questions.
1. If mlABC = 150°, find mlABD.
2. If mlBDF = 80°, find mlFDG.
3. What is mlEFDl
4. List an example of an acute, obtuse, right and straight angle.
Answers
1. mlABD = 75°
2. mlFDG = 160°
3. mlEFD = 10°
4. Acute angles: LABD, LDBC, LBDF, LBDG, LFDE, LGDH
Obtuse angles: LABC, iFDG, lEDG, LHDF
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Right angles: iBDE, iBDH
Straight angle: lEDH
Challenge
1. Construct a 90° angle using the constructions learned in this section.
2. Using #1, construct a 45° angle.
3. Using #1, construct a square (all sides are equal and all angles are 90°)
Answers
1. Construct a perpendicular bisector to create two 90° angles.
A
90'
90'
y
2. Bisect one of the angles in #1 to create a 45° angle.
3. The square is a little harder. Students will need to start with #1 and repeat this once more. Then, they
can mark the distance between two created 90° angles and repeat these on the created lines (the green and
purple arc marks).
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Angle Pairs
Connections to Map Reading
Use the image of a street map of Manhattan or Washington DC (www.mapquest.com). Print a copy of this
map for students to work with during the activity. There are several different examples of complementary
angles, supplementary angles, linear pairs, and vertical angles. Have the students work in pairs with a
highlighter, colored pencils or markers and identify examples of each of the types of angles in the map.
Ask the students to make a list of the intersections (by street name) on paper and how each angle fits the
description. You could also enlarge the map and have students use a protractor to measure the angles.
Students can either share the findings as a whole class, as a discussion, or in small groups. An extension
could be to have students repeat this activity for the city they live in.
Extension
Ask students if they think three angles can form a linear pair. Can three angles be supplementary? Com-
plementary? By the definition, only two angles can form a linear pair, be supplementary or complementary.
However, explain to students that this does not mean that three angles cannot add up to 180° (such as in
a triangle) or 90°.
Challenge
1. Find the value of x and y. You may need to factor or use the square root.
2. What is a congruent linear pair?
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Answers
1. x= U° 9 y = 7°
2. A congruent linear pair would be two 90° angles. Congruent linear pairs are created by perpendicular
lines.
Classifying Polygons
Connections to Art
To prepare, you will need an assortment of one or more of the following items: gumdrops, marshmallows,
toothpicks, tinkertoys, or kynex. Be sure that the students understand the different types of triangles and
have an example of each type before beginning. Then have them create an example of each type triangles
using the materials provided. The gumdrops or marshmallows would be the vertices and the toothpicks
would be the sides, for example. Let students play for a while. You could also extend this activity to
polygons (squares and other regular polygons are pretty easy to create). Finally, you could have students
create geometric designs using the materials.
Connections to Architecture
Either in class or as a homework assignment, have students search for "polygon architecture" on the
internet. Tell students to click "images" on the search engine and have them print out 2-3 images to share
with the class. Encourage students to print out pictures of buildings, tile designs, interiors, windows, or
playground equipment.
As a class, brainstorm the different types of polygons used in architecture (write these on the board and
how they are used) and why certain polygons are used over others. As a homework assignment, students
can go home and write down all the polygons they find their home.
Extension
You can introduce students to the concept of an equilateral polygon and an equiangular polygon. Only
in the case of a triangle are equilateral and equiangular the same shape. Also, equilateral polygons can
be concave (see hexagon below), but equiangular polygons cannot. After drawing examples of equilateral
and equiangular quadrilaterals, pentagons, and hexagons, show students regular polygons (when sides and
angles are congruent). Encourage to guess the definitions of these new terms and ask for volunteers to
come up to draw different polygons. Here are a few examples.
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equilateral quadrilateral
{rhombus, students
might say diamond)
equilangular
pentagon
equilateral
hexagon
regular
octagon
Challenge
1. Can an equiangular polygon be concave? Why or why not?
2. Find the pattern for the number of diagonals drawn from one vertex. Determine how many diagonals
can be drawn from one vertex of an n-gon.
3. Find the pattern for the total number of diagonals in a polygon. Determine how many diagonals are
in an ^z-gon.
Answers
1. An equiangular polygon cannot be concave because all the angles must be the same measure. In a
concave polygon, one angle will be larger than 180°.
2. The number of diagonals at one vertex increases by one. For an ^z-gon, the equation would be n - 3.
3. For the total number of diagonals, the pattern is: 0, 2, 5, 9, 14, 20, 27, ... Here, you add one more
than what was previously added. If there are n vertices and n - 3 diagonals from each vertex, then
start with n(n - 3) as the equation for the total number of diagonals. However, if you plug in n = 5
(or any other number), the answer would be double the correct answer (this is because n(n-3) counts
all of the diagonals twice). Therefore, you need to divide by two. The equation is 2 .
3.2 Reasoning and Proof
Inductive Reasoning
Connections to Music
This will need to be prepared ahead of time. Prepare several different examples of repetitive music, such
as rap, classical, folk songs, or children's songs. Make sure that the songs have a refrain (chorus) or a
clear consistent pattern. Examples are: Old MacDonald, Puff the Magic Dragon, Let's Dance (Lady Gaga),
excerpts from the Nutcracker (the Russian Dance, Dance of the Reed Flutes, Arabian Dance, Waltz of the
Flowers), or Gotta Feeling (Black Eyed Peas). For students to really "hear" the repetition, it might be
easiest to start with songs that do not have lyrics.
Play the songs in class and have students develop a rule for each selection. You may have to play the songs
3-5 times before students recognize a pattern. Then, brainstorm a list of possible pattern rules, decide
on one and write it on the board. See if students can come up with a counterexample to the rule. After
doing one or two examples together as a class, split students into groups and have them listen to remaining
selections. Once all the groups are finished, have them share their rules and see if the other groups can
find any counterexamples.
As an extension to this activity, you could survey the class to see if any student has an interest in DJ-ing.
If one does (and has the appropriate equipment), allow he/she to bring in the necessary equipment and
have them give a demonstration on how to mix and dj. Have the class find patterns in their sampling
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and/or mixing. If no student has the necessary equipment, see if your music department, mass media, or
theater classes do.
Connections to Nature
In the computer lab, allow students to search for "nature patterns" in the "images" tab on the search
engine. Have students print out 2-3 pictures of plants, fruit, or trees that they feel show a pattern. Give
them time to come up with an appropriate rule for each picture. Then, pair up students and have them
exchange their pictures with their partner. See if the partner can find any counterexamples.
Challenge
1. Find the next three terms in the sequence and describe the rule.
a) 1, 1, 2, 3, 5, 8, 13, ...
b) 3, 7, 12, 18, 25, ...
2. Plot the points (1, 3), (2, 8), and (3, 13). What do you notice? Can you use algebra to find the rule
that maps x onto y?
Answers
1. a) 21, 34, 55 Each term is made up of the sum of the two numbers before it. This is called a Fibonacci
sequence.
b) 33, 42, 52 To find each term, add one more than what was added to the previous term. In other words,
we added 4, 5, 6, 7, ... each time.
*As an added challenge, have students find the equation for part b. The equation would be " ^ - . A hint
would be to tell students to double every term and see if they can find that pattern, then divide by 2.
2. The three points form a line. The equation of this line is y = 5x— 2. Any set of points that are collinear
will follow a pattern where the rule is the equation of the line.
Conditional Statements
Connections to Literature
Give students a copy of the poem "The Road Not Taken" by Robert Frost. Read the poem with the class
and discuss the meaning of the poem and the thoughts behind it. Then have the students rewrite the poem
in all conditional statements. When finished, ask the students if the meaning of the poem has changed
and how conditional statements can impact the different statements. You can have students work in pairs
or in groups. Also, decide if you want students to read their new poems aloud to the class.
Connections to Cinema
In the Know What? is a cartoon by Rube Goldman. You can show students the "Breakfast Machine"
scene from Pee Wee's Big Adventure (1985). This scene is an acted out series of conditional statements.
Have students attempt to write the series of if-then statements that complete the breakfast machine.
Challenge
1. Rewrite your Know What? answer as a series of conditional statements.
Answers
1. A — > B: If the man raises his spoon, then it pulls a string.
B — > C: If the string is pulled, then it tugs back a spoon.
C — > D: If the spoon is tugged back, then it throws a cracker into the air.
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D — > E: If the cracker is tossed into the air, the bird will eat it.
E — > F: If the bird eats the cracker, then it turns the pedestal.
F — > G: If the bird turns the pedestal, then the water tips over.
G — > //: If the water tips over, it goes into the bucket.
H —> I: If the water goes into the bucket, then it pulls down the string.
/ — > /: If the bucket pulls down the string, then the string opens the box.
J ^ K: If the box is opened, then a fire lights the rocket.
K — > L: If the rocket is lit, then the hook pulls a string.
L — > M: If the hook pulls the string, then the man's face is wiped with the napkin.
Deductive Reasoning
Extension
Draw five different parallelograms on the board. Tell students the definition of a parallelogram. Then,
have each student make 2-3 conjectures about the angles or sides. Allow students to share their conjectures
with the class. As a class, try to find counterexamples or see if the conjectures are true. Then, see if the
conjectures hold for other parallelograms.
Extension
After going over the Laws of Detachment, Contrapositive and Syllogism, students might question why
there are no Law of Converse or Law of Inverse. These two statements lead to the Converse Error and
the Inverse Error. Example 4 would be an example of the Converse Error. You can also extend the use of
symbols. See if students can find the conclusion (if there is one) for each of the following:
v — > g d — > e x — > y v — > g
v — > h r — > d y — > z h —> v
None r — > d x — > z h — > g
Logic Challenge
Logic Puzzles are a great way to use the Law of Syllogism and Contrapositive. Below is an example.
Have students read the hints and see if they can figure out who left work on what day, what their favorite
"vice" is and how they commute. Students should place an "X" where they know something is not a
possibility. If something is true, they can either put an "O" in the square or color the square in. These
are great activities for students to do after a test or for extra credit. If you need more logic puzzles, go to
www.logic-puzzles.org.
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Feb. 3
Feb. 15
Aon 7
Sept. 1
10-speed bike
scooter
seqway
skateboard
nachos
chocolate
pizza
ice cream
Q)
<D
CD
Q
TO
O
_Q
to
HINTS
1 . The commuter who rides a skateboard doesn't enjoy chocolate.
2. Davis will leave before the commuter who rides a scooter
3. Of Davis and the commuter who rides a segway, one tries to hide
their love of pizza and the other is leaving the company on April 7.
4. The employee whose last day will be April 7 doesn't enjoy pizza
and doesn't ride a skateboard.
5. Either the commuter who rides a skateboard or the commuter who
rides a segway tries to hide their love of ice cream.
6. The employee whose last day will be February 3 is not McKenna.
7. The commuter who rides a skateboard is McKenna.
8. The person who secretly enjoys ice cream will leave after the
person who secretly enjoys nachos.
9. The employee whose last day will be February 15 is Alexa.
10. The person who secretly enjoys chocolate is not Alexa.
Answers
Feb 3 rd — > Davis — > pizza — > 10-speed
Feb 15 th — > Alexa — > nachos — > scooter
Apr 7 th — > Noe — > chocolate — > Segway
Sept 1 st — > McKenna — > ice cream — > skateboard
Algebraic and Congruence Properties
Connections to Scales
Bring in several different scales for students to work with. Then, prepare an assortment of items for students
to work with. For example, apples, bananas, bags of flour, bags of rice, oranges, etc. You can use non-food
items too. From here, have students work in groups and come up with collections of items that demonstrate
equality. For example, apples and oranges: How many apples "equal" how many oranges (in weight)? Have
one member of the group be the recorder and list of the items that equal other items. Then ask students to
use the properties from the chapter and write a reflexive statement, a symmetric statement and a transitive
statement about two of their equal statements. Allow time for the students to share their work.
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Extension
In this lesson, we introduce the two-column proof. If students feel confident with the concept, you can take
away the fill-in-the-blank option that is presented throughout this text. See the Differentiated Instruction
FlexBook for modifications and alternate proof options.
Challenge
1. Write a two-column proof.
Given : AC bisects iDAB
Prove: mlBAC = 45°
2. Draw a picture and write a two-column proof.
Given : Zl and Z2 form a linear pair and m/1 = mZ2.
Prove : Zl is a right angle
Answers
1.
Table 3.1:
Statement
Reason
1. /.DAB is a right angle
2. mlDAB = 90°
3. AC bisects LDAB
4. mlDAC = mlBAC
5. mlDAB = mlDAC + mlBAC
6. mlDAB = mlBAC + mlBAC
7. mlDAB = 2mlBAC
8. 90° = 2mlBAC
9. 45° = mlBAC
Given
Definition of a right angle
Given
Definition of an angle bisector
Angle Addition Postulate
Substitution PoE
Combine like terms
Substitution PoE
Division PoE
2.
V
m
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Table 3.2:
Statement
Reason
1. Zl and Z2 form a linear pair mil = m/2
2. Zl and Z2 are supplementary
3. mzl + mz2 = 180°
4. mzl + mzl = 180°
5. 2mzl = 180°
6. mil = 90°
7. Zl is a right angle
Given
Linear Pair Postulate
Definition of Supplementary
Substitution
Simplify
Division PoE
Definition of a right angle
Proofs about Angle Pairs and Segments
Connections to Cooking
In this activity, the students are going to need to prove the following statement: "You must have eggs to
make a chocolate cake."
Assign half of the class the job of proving that this is a true statement. Assign the other half of the class
the job of disproving the statement. Allow students to research on the internet for recipes. Recipes that
would disprove this statement would be dairy- free or vegan cakes. Students will need at least three sources
(websites, cookbooks, or other) to support their ability to prove or disprove. Then, they will need to come
up with an argument (this can be done in groups or pairs) to persuade the rest of the class as to why a
chocolate cake is better with or without eggs (depending on which they were assigned). Once the groups
have shared their proofs with the class, allow time to debate as a whole on which was the best proof.
Challenge
1. Find the measure of the lettered angles in the picture below. Hint: Recall the sum of the three angles
in a triangle is 180°.
Answers
1. a = 25°, b = 75°, c = 105°, d = 90°, e = 50°, / = 40°, g = 25°, h = 130°, j = 155°, k = 130°
3.3 Parallel and Perpendicular Lines
Lines and Angles
Connections to Architecture
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Discuss the lines and angles in the picture of Westminster Abbey. Show students how perpendicular lines
are a major feature in the structure. What would happen if the lines were not perpendicular? Ask students
how geometry plays a role in architecture.
You can extend this activity by taking students to the computer lab and having them find another building
with parallel, perpendicular, and/or skew lines. Have them print out the picture and explain how the
architecture of the building they chose relates to geometry.
i
Connections to Art
This drawing is an example of lines and angles in art. Have students discuss the different lines and angles
in this drawing. Then, use this image as a springboard for students to create their own drawing.
Students will need white paper, colored pencils, markers or crayons, and rulers. Stress that the artwork
should be simple and prominently display intersecting lines. Allow for some time in class to create their
art and then display in the classroom.
If students need additional help getting started, show them http://fineartamerica.com/profiles/
todd-hoover.html, specifically "Coming Together" and "Coming Apart" by Todd Hoover.
Algebra Extension
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Give students the series of problems below to begin to explore the concepts of parallel and perpendicular
lines in the coordinate plane.
1. Write the equations of two lines parallel to y = 3.
2. Write the equations of two lines perpendicular to y = 5.
3. What is the relationship between the two lines you found for number 27?
4. Plot the points A(2, -5), £(-3, 1), C(0, 4), £>(-5, 10). Draw the lines AB and CD. What are the slopes
of these lines? What is the geometric relationship between these lines?
5. Plot the points A(2, 1),£(7,-2),C(2,-2),D(5,3). Draw the lines AB and CD. What are the slopes of
these lines? What is the geometric relationship between these lines?
6. Based on what you discovered in numbers 4 and 5, can you make a conjecture about the slopes of
parallel and perpendicular lines?
Answers
1. Any two equations in the form y = b, where b is a constant.
2. Any two equations in the form x = /?, where b is a constant.
3. These two lines are parallel to each other.
4. slope of AB equals slope of CD = -|; these lines are parallel
5. slope of AB = -|, slope of CD = |; these lines are perpendicular
6. It appears that the slopes of parallel lines are the same and the slopes of perpendicular lines are
opposite reciprocals.
Properties of Parallel Lines
Connections to Map Reading
Give each student a copy of the Tube map of London below. They will be using the map to find an example
of each of the postulates/theorems:
1. Corresponding Angles Postulate
2. Alternate Interior Angles Theorem
3. Alternative Exterior Angles Theorem
4. Same Side Interior Angles Theorem
Students will need to prove that each of their examplesis accurate. Students may need to use a protractor.
Students may also want to work in pairs.
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As an extension, have students research other subway lines, such as the "L" in Chicago, the Metro in
Washington DC and the Subway in New York City.
Know What? Extension
When looking at the street map of Washington DC, point out to students that all the "state" streets are
transversals throughout the city. Ask students why they think these streets do not follow the grid layout.
(Lettered streets run east to west, numbered streets run north to south.)
Algebra Challenge
Find the values of x and y.
1.
Answers
1. x= 15°,y = 21°
2. x = 37°,y = 28°
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Proof Challenge
Use the picture to the right to complete each proof.
1. Given : / || m, s \\ t
Prove : Z4 = ZlO
2. Given : / || m,s \\t
Prove : Z2 = Zl5
3. Given : / || m, s \\ t
Prove : Z4 and Z9 are supplementary
Answers
1.
Table 3.3:
Statement
Reason
1. / || m, s \\t
2. Z4 = Z12
3. Zl2 = ZlO
4. Z4 = ZlO
Given
Corresponding Angles Postulate
Corresponding Angles Postulate
Transitive PoC
2.
Table 3.4:
Statement
Reason
1. l\\m,s \\t
2. Z2 = Z13
3. Zl3 = Z15
4. Z2 = Z15
Given
Alternate Exterior Angles Theorem
Corresponding Angles Postulate
Transitive PoC
3.
Table 3.5:
Statement
Reason
1. l\\m,s\\t
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Given
152
Table 3.5: (continued)
Statement
Reason
2. Z6 = Z9
3. z4 = z7
4. Z6 and Z7 are supplementary
5. Z9 an Z4 are supplementary
Alternate Interior Angles Theorem
Vertical Angles Theorem
Same Side Interior Angles
Same Angle Supplements Theorem
Proving Lines Parallel
Construction Extension
Extend Investigation 3-5 to the Converses of the Alternate Interior Angles Theorem and the Alternate
Exterior Angles Theorem. You can decide to do this as a student or teacher-led activity. The major
difference will be that in Step 3, the copied angle will be in a different location. Ask students how they
can extend this investigation to the Converse of the Same Side Interior Angle Theorem.
Construction Challenge
Draw a straight line. Construct a line perpendicular to this line through a point on the line. Now, construct
a perpendicular line to this new line. What can you conclude about the original line and this final line?
How could you prove this conjecture?
Answers
Construction, the first and last lines are parallel. You might conjecture that two lines perpendicular to
the same line are parallel to each other. You could prove this using any of the converse theorems learned
in this section because all four angles formed where the transversal intersects the two parallel lines are
right angles. Thus, Alternate Interior Angles, Alternate Exterior Angles and Corresponding Angles are all
congruent and the Same Side Interior Angles are supplementary.
Connections to Map Reading
Below is a map of the mall in Washington DC.
Commerce
IRS
Justice
Newseum
nrf $y/^ Embassy
iian
ssy
Agriculture
Sackler African Art
Independence Ave.
There are several different examples of parallel lines and transversals in this map. Have students to write a
series of directions to take someone on a tour of the mall. They must go to the American History Museum,
the National Archives, the Newseum, the West National Gallery of Art, and the Air and Space Museum.
Students can pick the order in which they go to these sites. Working in pairs, their job is to write down
directions with the shortest path. Then, they need to determine if there is another set of directions that is
close to (or the same length) as their original directions. When finished, have the students swap directions
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with a neighboring group and check to be sure that the directions work.
Properties of Perpendicular Lines
Connections to Gymnastics
Gymnasts use the parallel bars and the uneven bars in routines. Ask students if they see any perpendicular
transversals for these pieces of equipment. Extend this discussion to other gymnastic equipment (pommel
horse, rings, balance beam, vault). How would the events be different if this equipment was not parallel
to the ground?
Judges also score gymnasts on their ability to get their body perfectly vertical and legs perpendicular or
vertical. Are there any perpendicular transversals or parallel lines between the gymnast and the equipment
or the ground?
Challenge
Find x and y. Remember that the angles in a triangle add up to 180°.
J
t 4
i
%.
y
J
r
^ 00 *2V^ ^n^
n
*L
>
t
Answers
x = 49°,.y = 111°
Parallel and Perpendicular Lines in the Coordinate Plane
Connections to Road Construction
Typically, the slope of a steep road, also called a grade, is measured in a percentage. If a road has a grade
of 5%, which means in 100 horizontal feet the road will rise vertically 5 feet.
5% grade
100 ft.
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This would be a slope triangle for this particular piece of road. The slope would be Jq = ^. Present
students with the following problem: The Grapevine on Highway 5 rises from 400 feet to 4144 feet over 6
miles. What is the grade of this road? (This is for the southbound side)
Answer
First, convert 6 miles to feet, 6 x 5280 = 31, 680 ft. Now, write a ratio and then turn that into a percent.
41 3i 4 6~8Q QQ = ilMo ~ 0-118 x 100% = 11.8% grade. In general, if a grade is over 10% it is considered steep.
As an extension, students can find the grade of the California Incline in the Know What? at the beginning
of this lesson. (11.6%)
More Connections to Construction
Slope, also called grade or pitch, is a very important part in roof design, wheelchair ramp design, and
skateboard ramp design. Have students research any or all of these online to see why slope is important
to these items.
Challenge
1. A line passes through (5, -2) and (-4, 1). Find the equations of the lines that are perpendicular to
this one, through each point. How do these two lines relate to each other?
Answer
1. The equation of the line is y = -~jc - |. The perpendicular line through (5, -2) is y = 3x - 17 and
through (-4, 1) is y = 3x + 13. These two lines are parallel to each other.
The Distance Formula
Extension: Shortest Distance between a Point and a Line
We know that the shortest distance between two points is a straight line. This distance can be calculated
by using the distance formula. Let's extend this concept to the shortest distance between a point and a
line.
Just by looking at a few line segments from A to line /, we can tell that the shortest distance between a
point and a line is the perpendicular line between them. Therefore, AD is the shortest distance between
A and line /.
Example: Determine the shortest distance between the point (1, 5) and the line y — \x—2.
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Solution: First, graph the line and point. Second determine the equation of the perpendicular line. The
opposite sign and reciprocal of | is -3, so that is the slope. We know the line must go through the given
point, (1, 5), so use that to find the y-intercept.
y = -3x + b
5 = -3(1) + b The equation of the line is y = -3x + 8.
8 = b
Next, we need to find the point of intersection of these two lines. By graphing them on the same axes, we
can see that the point of intersection is (3, -1), the green point.
Finally, plug (1, 5) and (3,-1) into the distance formula to find the shortest distance.
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d=V(3-l) 2 + (-l"5) 2
= ^(2)2 + (-6)2
= V2 + 36
= V38 « 6.16 Mmta
Extension: Shortest Distance between Parallel Lines (slopes other than 1 or -1)
In the text, we went over how to find the distance between two parallel lines where the slope was either 1
or -1. Here, we extend this concept to any two parallel lines. It is roughly the same process, but there is
substantially more math involved. This extension would probably be done best in pairs or in groups.
Example: What is the shortest distance between y = 2x + 4 and y = 2x - 1?
Solution: Graph the two lines and determine the perpendicular slope, which is -^. Find a point on
y = 2x + 4, let's say (-1, 2). From here, use the slope of the perpendicular line to find the corresponding
point on y = 2x - 1. If you move down 1 from 2 and over to the right 2 from -1, you will hit y = 2x — 1 at
(1, 1). Use these two points to determine the distance between the two lines.
d= ^(l + 1) 2 + (1-2)2
= ^ + (-1)2
= V4 + 1
= V5 » 2.24 units
The lines are about 2.24 units apart.
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Notice that you could have used any two points, as long as they are on the same perpendicular line. For
example, you could have also used (-3, -2) and (-1, -3) and you still would have gotten the same answer.
d= ^(-l + 3) 2 + (-3 + 2) 2
= V 22 + (- 1 ) 2
= VITT
= V5 « 2.24 units
Example: Find the distance between the two parallel lines below.
Solution: First you need to find the slope of the two lines. Because they are parallel, they are the same
slope, so if you find the slope of one, you have the slope of both.
Start at the y-intercept of the top line, 7. From there, you would go down 1 and over 3 to reach the line
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again. Therefore the slope is — i and the perpendicular slope would be 3.
Next, find two points on the lines. Let's use the y-intercept of the bottom line, (0, -3). Then, rise 3 and
go over 1 until your reach the second line. Doing this three times, you would hit the top line at (3, 6). Use
these two points in the distance formula to find how far apart the lines are.
d= ^(0 - 3) 2 + (-3 - 6) 2
= V(- 3 ) 2 + (-°) 2
= V9 + 81
= a/90 « 9.49 units
Extension Problems
Determine the shortest distance between the given line and point. Round your answers to the nearest
hundredth.
1. y=i* + 4; (5,-1)
2. y = 2x-4; (-7,-3)
3. y = -4jc+l; (4,2)
4. y = -| x -8; (7,9)
Determine the shortest distance between the each pair of parallel lines. Round your answer to the nearest
hundredth.
5. y = -l x + 2,y = -|jc — 8
6. y = 4jt + 9,y = 4x-8
7. y= \x,y =
Answers
1. 6.32 units
2. 6.71 units
3. 12 units
b -5
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4. 7 units
5. 9.49 units
6. 4.12 units
7. 4.47 units
3.4 Triangles and Congruence
Triangle Sums
Extension
After going over Investigation 4-1, go to the computer lab and use Geometer's Sketchpad. Have students
draw a triangle and find the measure of each angle. Tell them to move around the angles so that they get
4 or 5 different sets of triangle angle measures. Let them play with the angles and sides until they realize
that no matter the size of the sides or angles, the angles will always add up to 180°.
Connections to Hang Gliders
Ask students what they notice about the number of triangles that are in the hang glider. Then have the
students identify all of the different angles of the triangles, also include the interior and exterior angles. There
are angles created by the strings too. Complete this activity as a whole class discussion.
Algebra Challenge
Find the value of x.
1.
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2.
3.
4.
(3x+4)°
(5x-13)°
Answers
1. x = ±8 c
2. x= 17°
3. x= 11°
4. x = 7°
Congruent Figures
Connections to Bridge Construction
For this activity, the students are going to use popsicle sticks or toothpicks to build a truss bridge. Begin
by showing students some truss bridge designs.
Working in pairs, students should draw a design first and get it approved. Then they can move on to
the construction piece of the project. When finished, have students explain the importance of congruent
triangles in building a solid bridge.
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As an extension, have students research "triangles in bridges" online. Then they can explore one or two
different types bridge designs. Ask the students to write congruence statements explaining the congruence
of the triangles in the different bridge designs.
Challenge
1. Draw a 50° angle using a protractor. Make one side of the 50° angle 3 inches and the other side of
the angle 2 inches long. Draw the line segment that connects the end of these two sides, completing the
triangle.
a) Measure the third side of this triangle. What is the length (in inches)?
b) Use a protractor to measure the other two angles.
c) Can you draw a different triangle with the original information? If so, draw it. If not, why do you think
this is?
2. Draw a 40° angle using a protractor. Make one side horizontal and 6 cm long. At the endpoint, draw
a 70° angle (6 cm side has the two angles at each endpoint). Extend the other sides of both angles until
they intersect to form a triangle.
a) What is the measure of the third angle? What type of triangle is this?
b) What is the measure of the other two sides?
c) Can you draw a different triangle with the original information? If so, draw it. If not, why do you think
this is?
Answers
1. a) The triangle to the left is to scale; 2.3 in
b) Using a protractor one angle is 42° and the other is 88°.
c) No, only one triangle can be drawn. These three pieces of information are enough to draw one triangle
because the two sides are fixed. There is no way to manipulate these sides to make the third side a different
measurement.
3 in
2. a) The third angle is 70° because 40° + 70° = 110° and 180° - 110° = 70°. Two angles are equal so the
triangle is isosceles.
b) Using a ruler the other two sides are 6 cm and 4.1 cm
c) No, only one triangle can be drawn. These three pieces of information are enough to draw one triangle
because the one side is fixed, as are the two angles. The other sides of the angles can only intersect in one
way.
2"- side/ \ 3* side
6 cm
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Triangle Congruence Using SSS and SAS
Investigation Extension
For Investigation 4-2, you could have students repeat Steps 1-5, starting with the 4 in. side and then
starting with the 3 in. side. Then, compare these two triangles to the original made (starting with the 5
in. side). See if students think the triangles are congruent. Then discuss this "shortcut" (SSS Congruence
Postulate) with students.
This extension can be repeated with Investigation 4-3. Have students start with the 2 in. side and draw
the 5 in. after creating the 45° angle (switch Steps 1 and 3). Have the same discussion you did above to
determine if students feel that SAS is a congruence shortcut.
Extension: SAS in the Coordinate Plane
Using the SAS Congruence Postulate in the coordinate plane is a bit more challenging than SSS. First, it is
very difficult to measure the angles based on the slope of the sides and/or the distance formula. Therefore,
we must limit using SAS in the coordinate plane to right triangles. The measure of the right angle can
determined if we know the two sides are perpendicular.
Extension Example: Determine if the two triangles are congruent by SAS. If so, write the congruence
statement.
Solution: First, we need to determine which angle can be "measured." It looks like LB and IE are the
right angles, so let's see if the sides are perpendicular.
2-6
m A B =
m E F
-7-
-6
-6)
-7)
7-3
=1=4
-1
1
4
mBC =
m DE
6-4 _
-6-2 ~
2 ~ (-6)
5-7
1
"I
-2
This shows that AB ± BC and EF ± DE and thus mlB = mlE =
formula to see if the two sides around the right angles are equal.
90°. Now, we need to use the distance
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AB = > /(-7-(-6)) 2 + (2-6) 2 = ^(-1)2 + (-4)2 = VTTI6 = Vl7
£C = ^(-6 -2)2 + (6 -4)2 = ^(-8) 2 + 2 2 = V64TI = V68 = 2 Vl7
EF = ^{7-3) 2 + (-6-(-7)) 2 = ^4 2 + (-1)2 = VT6TT = Vl7
D£ = ^(5 - 7) 2 + (2 - (-6)) 2 = ^(-2)2 + 8 2 = V4T64 = V68 = 2 Vl7
From this, we see that AB = EF and BC = DE. So, by SAS AA£C = AFED.
Challenge
1. See if the two triangles are congruent by SAS.
AABC: A(-9,-6),fi(-7,0),C(2,-3) ADEF : D(0,2),£(-6,4),F(2,8)
Answer
1. The two triangles are not congruent. Both have a right angle, but one set of legs is not congruent.
Triangle Congruence Using ASA, AAS, and HL
Construction Extension
By now, students will have noticed that there is no ASS or SSA congruence theorem. Give students the
angle and the two side lengths below. Have them create two different AABCs from these givens. This
should show students that for some cases of SSA there will be two triangles created which is why it cannot
be a congruence theorem.
B
» A
B
Point out to students the vertex labeling above. Remind them that the angle is not included. The two
triangles that are created are below.
Have a class discussion as to why this is possible. Students need to realize that the length of AC will vary
because of the length of BC. Using a compass will best demonstrate this.
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When students perform this construction, it will be easiest for them to draw the two triangles together,
like the ones to the left. Then, when students measure the fixed length of Z?C, they will see that it will
cross AC in two places.
Algebra Challenge
1. Find the values of x and y. Is AABC = ADOG?
5x+ 19
x 2 -5
A -|-| /+ 2 c
Answers
Yes, AABC = ADOG by ASA. Therefore, by CPCTC, AC = DG and BC = OG. Set each side equal to
each other and factor.
x 2 - 5 = 5x + 19
x 2 - 5x - 24 =
(jc-8)(jc + 3) =
x = 8,-3
2/ + 4y -13 = llj + 2
2y 2 - ly - 15 =
(2y + 3)(y-5)=0
x = — , 5
2
Because length is never negative, students should eliminate the negative answers.
Isosceles and Equilateral Triangles
Proof Extension
Have students prove the Base Angles Theorem for equilateral triangles. This is Equilateral Triangle
Theorem in the text.
Given: AABC is equilateral
Prove: AABC is equiangular
A.
Table 3.6:
Statement
Reason
1. AABC is equilateral
2. AABC is isosceles
3. AB = BC,AB = AC
Given
All equilateral triangles are isosceles
Definition of equilateral triangle
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Table 3.6: (continued)
Statement
Reason
4. IC = lA, LC = IB
5. LA = LB
6. AABC is equiangular
Base Angle Theorem
Transitive Property
Definition of equiangular
Connections to Construction
For this Construction Connection, we will analyze geodesic domes. Use Epcot Center as an example.
Here, students will see that the Epcot Center is made up entirely of equilateral triangles. All geodesic
domes are domes made up of equilateral triangles of the same size. Have students make their own geodesic
dome either on paper or using Geometer's Sketchpad. Once they decide on the size of the triangle, the
rest is just a series of repetition and rotation of the equilateral triangle.
As an extension or in-class project, have students turn their designs into a 3-D model using cardboard. You
can decide if you want students to make the entire dome or half a dome. Make sure the for the equilateral
triangles are all the same size.
Construction Challenge
1. Using the construction of an equilateral triangle (investigation 4-6), construct a 30° angle.
Hint: recall how to bisect an angle from investigation 1-4-
2. Use the construction of a 60° angle to construct a 120° angle. Is there any other way to construct a
120° angle?
Answers
1. Bisect a 60° angle as shown.
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2. Construct a 60° angle, then extendone side.
The adjacent angle is 120°.
In investigations 3-2 and 3-3 you learned how to construct perpendiculars (i.e. 90° angles). You could
make a 90° angle and copy your 30° onto it to make 120°. See investigation 1-2 for a review of copying an
angle.
3.5 Relationships with Triangles
Midsegments
Coordinate Plane Extension
Another type of problem that was not covered in this lesson is how to find the vertices of a triangle, given
the midpoints. Here is an example.
Extension Example: If the midpoints of the sides of a triangle are A(1,5),S(4, — 2), and C(— 5, 1), find
the vertices of the triangle.
Solution: The easiest way to solve this problem is to graph the midpoints and then apply what we know
from the Midpoint Theorem.
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Now, find the slopes between the three midpoints.
slope AB =
slope BC =
slope AC =
5 + 2 7
1-4 ~ 3
-2-1 -3
1
4 + 5 9
3
5-1 4 2
1 + 5 ~ 6 ~ 3
Using the slope between two of the points, plot that slope triangle on either side of the third point and
extend the line. Repeat this process for all three midpoints.
The green lines in the graph to the left represent the slope triangles of each midsegment. For example, we
took the slope between A and B and mapped it on either side of point C. The three dotted lines represent
the sides of the triangle. Where they intersect are the vertices of the triangle (the blue points), which are
(-8, 8), (10, 2) and (-2, 6).
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This process can also be done algebraically. Use the slopes and the appropriate midpoint to find the
equation of the side. Then, you would have to find the points of intersection for each pair of sides.
Extension Example: Find the equation of the side with midpoint B from the previous example.
Solution: If B is the midpoint of the side, then we need to use the slope from AC, which is 3. Plug in
what we know to the equation of a line to solve for the y-intercept.
y = mx + b
-2= 2 -(A)+b
14 u
The equation of the line is y = |x - 4- .
Challenge
Given the midpoints of the sides of a triangle, find the vertices of the triangle.
1. (-2, 1), (0, -1) and (-2, -3)
2. (1, 4), (4, 1) and (2, 1)
3. Find the equations of the sides of the triangle from #2.
ACAT has vertices C(xi,yi),A(x2,y2) an d T(xs,ys).
4. Find the midpoints of sides CA and CT. Label them L and M respectively.
5. Find the slopes of LM and AT.
6. Find the lengths of LM and AT.
7. What have you just proven algebraically?
Answers
1. (0, 3), (0, -5) and (-4, -1)
2. (-1, 4), (3, 4) and (5, -2)
3. y = 4, v = 3x - 5, v = -x + 3
n+Z3_)i±Z2 ,
5. slope of LM = ^ ^ = g+glglg = g^ = slope of AT
2 2
6. length of LM
l(xi + *3 *i + *2 \ 2 , Tyi + y 3 yi + y 2 \ 2 l(x 3 -x 2 \ 2 /y 3 -y 2 \ 2 /T^ l
7. We have just proven algebraically that the midsegment (or segment which connects midpoints of sides
in a triangle) is parallel to and half the length of the third side.
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y 2 ) 2
Perpendicular Bisectors and Angle Bisectors in Triangles
Circumcenter Extension
In this lesson, the point of intersection, also called the point of concurrency (when more than two lines
intersect at the same spot), for the perpendicular bisectors was never defined. This point is called the
circumcenter and has a couple of unique properties. First, as was touched on in this text, the circumcenter
is the center of a circle that circumscribes the triangle. What this means is that the circumcenter is
equidistant to the vertices of the triangle. You can also show students this alternative construction to find
the circumcenter, using a compass and straightedge.
Investigation 5-2b: Constructing the Perpendicular Bisectors of the Sides of a Triangle
Tools Needed: paper, pencil, compass, ruler
1. Draw a scalene triangle.
2. Construct the perpendicular bisector (Investigation 1-3) for all three sides.
The three perpendicular bisectors all intersect at the same point, called the circumcenter.
Students do not need to construct all three perpendicular bisectors here. Only two are needed to find the
point of intersection.
3. Erase the arc marks to leave only the perpendicular bisectors. Put the pointer of your compass on the
circumcenter. Open the compass so that the pencil is on one of the vertices. Draw a circle.
The circumcenter is the center of a circle that passes through all the vertices of the triangle. We say that
this circle circumscribes the triangle and is equidistant to the vertices.
Angle Bisector Extension
In this lesson, the point of concurrency for the angle bisectors was never defined. This point is called the
incenter and has a couple of unique properties. First, as was touched on in this text, the incenter is the
center of a circle that inscribes the triangle. What this means is that the incenter is equidistant to the
sides of the triangle. You can also show students this alternative construction to find the incenter, using
a compass and straightedge.
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Investigation 5-4b: Constructing Angle Bisectors in Triangles
Tools Needed: compass, ruler, pencil, paper
1. Draw a scalene triangle. Construct the angle bisector of each angle. Use Investigation 1-4 and #1 from
the Review Queue to help you.
2. Erase the arc marks and the angle bisectors after the incenter. Draw or construct the perpendicular
lines to each side, through the incenter.
3. Erase the arc marks from #2 and the perpendicular lines beyond the sides of the triangle. Place the
pointer of the compass on the incenter. Open the compass to intersect one of the three perpendicular lines
drawn in #2. Draw a circle.
Extension Problems
What are points A and B?
1. J,E, and G are midpoints and KA = AD = AH.
2. The blue lines are congruent. The green lines are angle bisectors.
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3. Both sets of lines are congruent. The green lines are also perpendicular to the sides
Determine if the following descriptions refer to the incenter or circumcenter of the triangle.
4. A lighthouse on a triangular island is equidistant to the three coastlines.
5. A hospital is equidistant to three cities.
6. A circular walking path passes through three historical landmarks.
7. A circular walking path connects three other straight paths.
Answers
1. A is the incenter because KA = AD = AH, which means that it is equidistant to the sides. B is the
circumcenter because JB,BE, and BG are the perpendicular bisectors to the sides.
2. A is the incenter because it is on the angle bisectors. B is the circumcenter because it is equidistant to
the vertices.
3. A is the circumcenter because it is equidistant to the vertices. B is the incenter because it is equidistant
to the sides.
4. Incenter
5. Circumcenter
6. Circumcenter
7. Incenter
Medians and Altitudes in Triangles
Coordinate Plane Extension
Students can find the equation of a median in the coordinate plane. To do this, first they need to find
the midpoint of the opposite side. Then, using this midpoint and the opposite vertex, they can find the
slope and the equation of the median. This process can be repeated for all the medians in a triangle.
Then, students can use these equations and use them to find the centroid. The centroid can be found
algebraically by solving a system of equation (medians) using substitution or elimination (also called the
linear combination method).
Extension Example: Find the equation of the median from B to the midpoint of AC for the triangle in
the x — y plane below.
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1 1 1 1 1 1 1 1 1 lAf 1 1 1 1 1 1 1 1 1
fr
: z zz\i
^tr =
t__» — 1_
^E -L -n %_ <! .
:?S^. i, j 10^
_s — -k^
j
(^
fcL
"J - iff JTI
it 1- 7 A
^5, -(£
-f- 4^-*}
V
Solution: To find the equation of the median, first we need to find the midpoint of AC, using the Midpoint
Formula.
-6 + 6 -4+(-4)\ /0 -8
HI)-*-*
Now, we have two points that make a line, B and the midpoint. Find the slope and y-intercept.
-4-4
m =
- (-2) 2
y = -4x + b
-4 = -4(0) + b
-4 = b
The equation of the median is y = -Ax - 4
Extension Example: Using the picture above, find the equation of the median from A to the midpoint
of BC. Then, find the centroid of AABC.
Solution: First, find the midpoint of BC.
/-2 + 6 4-4\ /4
)=(.)=(«)
Now, find the slope and the y-intercept.
-4-0 -4
m
-6-2
0=i(2) + Z>
The equation of this median is y — \x - 1. To find the centroid, set this equation and the equation from
the previous example equal to each other and solve.
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-4x-4
9
2
3
2
~9
2
3
y
H)
12
y
The centroid is ( _ §> _ f)-
Equilateral Triangle Extension
The circumcenter, incenter, centroid, and orthocenter are the same point for equilateral triangle. To verify
this, you can have students construct an equilateral triangle and an angle bisector from one vertex, a
median from another vertex, and the altitude from the last vertex. All these lines should intersect at the
same point. For an equilateral triangle, the point of concurrency is just called the center. Napoleon's
Theorem uses the centers of equilateral triangles to create more equilateral triangles.
Napoleon's Theorem: If equilateral triangles are constructed on the sides of any triangle, then the
centers of those equilateral triangles will form an equilateral triangle.
Extension Example: Trace the triangle below. Then, create equilateral triangles on each side of the
triangle. Next, find the center of each equilateral triangle and connect the three centers. What type of
triangle is created?
Solution: First, construct the three equilateral triangles with each side. Then, draw the medians or
altitudes (or angle bisectors or perpendicular bisectors) to find the center of each equilateral triangle.
Finally, connect the three centers. Measure the length of each side of the triangle created. This triangle is
equilateral.
Challenge
Write a two-column proof.
1. Given: AABC = ADEF
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AP and DO are altitudes
Prove: AP = DO
c ^^^ —
p B E
Use AABC with A(-2,9),£(6, 1) and C(-4,-7) for questions 21-26.
2. Find the midpoint of AB and label it M.
3. Write the equation of CM.
4. Find the midpoint of BC and label it N.
5. Write the equation of AN.
6. Find the intersection of CM and AN.
7. What is this point called?
Answers
Table 3.7:
Statement
Reason
1. AABC = ADEF,AP and DO are altitudes
2. AB = DE
3. LP and LO are right angles
4. LP = LO
5. LABC = LDEF
6. ZAZ?C and LABP are a linear pair
LDEF and LDEO are a linear pair
7. ZAZ?C and LABP are supplementary
LDEF and LDEO are supplementary
8. ZA£P = ZD£0
9. AAP£ = ADOE
10. AP = DO
Given
CPCTC
Definition of an altitude
All right angles are congruent
CPCTC
Definition of a linear pair
Linear Pair Postulate
Congruent Supplements Theorem
AAS
CPCTC
2. M(2,5)
3. y = 2jc+1
4. #(1,-3)
5. y = -4jc+ 1
6. (0,1)
7. Centroid
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Inequalities in Triangles
Proof Challenge
Have students prove Theorem 5-9.
Given: AC > AB
Prove: mlABC > mlC
Table 3.8:
Statement
Reason
1. AC>AB
2. Locate point P such that AB = AP
3. AABP is an isosceles triangle
4. mil = m/3
5. m/3 = mZ2 + mlC
6. mil = ml2 + mZC
7. mlABC = mll + mZ2
8. mlABC = mZ2 + mZ2 + mZC
9. mlABC > mlC
Given
Ruler Postulate
Definition of an isosceles triangle
Base Angles Theorem
Exterior Angle Theorem
Substitution PoE
Angle Addition Postulate
Substitution PoE
Definition of "greater than" (from step
Connections to Sculpture Design
In this activity, students will design using triangles. The catch is that student will need to show that
the triangles represent an inequality. To do this, students should make a sketch of their sculptures before
creating. This design should have measurements and demonstrate an inequality. When finished with
the design, students can use clay to build their triangles. When finished, have students write a short
explanation of their sculpture, what they designed, how it was created and how it demonstrates the
concept of an inequality.
Challenge
In questions 1-3, compare the measures of a and b.
i- a
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3 - 3.2
In questions 4 and 5, list the measures of the sides in order from least to greatest
4- b
In questions 6 and 7 determine the range of possible values for x.
6- 5
7.
2x+5
Answers
3x+2
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1. a = b
2. a>b
3. a < b
4. d < a < e < c <b
5. a = b<d<e<c
6. x< 18
7. ;t>3
Extension: Indirect Proof
Connections to Sports
Students will have the "job" of being a sports announcer at a basketball game, reporting on the actions of
the game. However, students can only report findings using if-then statements. They will need to prepare
a short broadcast and then present it with a peer to the class. This is meant to be a fun short assignment
to help students to see how to use if then statements in real life. Let students have time to present to the
class. See if the drama department has props or costumes that can be borrowed for the class period.
Challenge
Prove the following statements indirectly.
1. If a collection of nickels and dimes is worth 85 cents, then there must be an odd number of nickels.
2. Hugo is taking a true/false test in his Geometry class. There are five questions on the quiz. The
teacher gives her students the following clues: The last answer on the quiz is not the same as the
fourth answer. The third answer is true. If the fourth answer is true, then the one before it is false.
Use an indirect proof to prove that the last answer on the quiz is true.
3. On a test of 15 questions, Charlie claims that his friend Suzie must have gotten at least 10 questions
right. Another friend, Larry, does not agree and suggests that Suzie could not have gotten that many
correct. Rebecca claims that Suzie certainly got at least one question correct. If only one of these
statements is true, how many questions did Suzie get right?
Hints
1. If we assume that we have an even number of nickels, then the value of the coin collection must be
a multiple of ten and we have a contradiction.
2. Assume that the last answer on the quiz is false. This implies that the fourth answer is true. If the
fourth answer is true, then the one before it (the third answer) is false. However, this contradicts
the fact that the third answer is true.
3. None. To prove this by contradiction, select each statement as the "true" statement and you will see
that at least one of the other statements will also be true. If Charlie is right, then Rebecca is also
right. If Larry is right, then Rebecca is right. If Rebecca is right, then Larry is right.
3.6 Polygons and Quadrilaterals
Angles in Polygons
Connections to Nature
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This is a picture of Devil's Post pile, near Mammoth Lakes, California. These posts are cooled lava
(called columnar basalt) and as the lava pools and cools, it ideally would form regular hexagonal columns.
However, variations in cooling caused some columns to either not be perfect or pentagonal.
Have students see if there are any polygons that naturally occur in nature. Other possibilities are sea stars,
honeycombs, star fruit, and the Giant's Causeway in Ireland.
Proof Challenge
Here is the proof of the Exterior Angle Sum Theorem. Rather than have students solve this proof, split
the class into groups and cut up the proof by statements and reasons. Then, have each group put the proof
in the correct order.
Given: Any n-gon with n sides, n interior angles and n exterior angles.
Prove: n exterior angles add up to 360°
NOTE: The interior angles are x\,X2,...x n .
The exterior angles are Vi,y2, ...y n .
Table 3.9:
Statement
Reason
1. Any ft-gon with n sides, n interior angles and n Given
exterior angles.
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Table 3.9: (continued)
Statement
Reason
2. x„° and y n ° are a linear pair
3. x n ° and y n ° are supplementary
4. x n °+y n o = 180°
5. (xi°+x 2 + ...+x„°) + ();i°+;y2 +...+;y„ ) = 180°«
6. (n - 2)180° = (xi° + x 2 ° + ... + x„°)
7. 180°n = (n - 2)180° + (ji° + y 2 ° + • • • + J«°)
8. 180°n = 180°« - 360° + (yi° + y 2 ° + . . . + y„°)
9. 360° = (yi°+y 2 ° + ...+y„°)
Definition of a linear pair
Linear Pair Postulate
Definition of supplementary angles
Sum of all interior and exterior angles in an n-gon
Polygon Sum Formula
Substitution PoE
Distributive PoE
Subtraction PoE
Challenge
1. Each interior angle forms a linear pair with an exterior angle. In a regular polygon you can use two
0-2)180°
different formulas to find the measure of each exterior angle. One way is ^- and the other is 180°-
(180° minus Equiangular Polygon Formula). Use algebra to show these two expressions are equivalent.
2. Find the measures of the lettered angles below given that m || n.
Answers
1.
180°
(n- 2)180°
180°n - 180°n + 360
360°
360°
n
360°
n
360°
2. a = 120°, b = 60°, c = 48°, d = 60V = 48°,/ = 84°, g = 120°, h = 108°, j = 96°
Properties of Parallelograms
Connections to Architecture
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180
These homes are on www.trendir.com/house-design/. Either print these images and distribute them to
students or have them use computers to look at the images. All of the homes here are constructed using
many different parallelograms. Ask the students to select one of the houses and work with it to identify
the elements of the parallelograms in the designs. Students should look for the relationship between the
opposite sides of a parallelogram, the opposite angles and the consecutive angles. Then, students can design
their own home using parallelograms. Make sure that all the elements of the exterior of the house; walls,
windows, doors, are all parallelograms. Discuss with students the different types of parallelograms. Graph
paper may be helpful for this activity.
Challenge
Plot the points £(-l,3),F(3, 4), G(5, -!),//(!, -2) and use parallelogram EFGH for problems 1-4.
1. Find the coordinates of the point at which the diagonals intersect. How did you do this?
2. Find the slopes of all four sides. What do you notice?
3. Use the distance formula to find the lengths of all four sides. What do you notice?
4. Make a conjecture about how you might determine whether a quadrilateral in the coordinate is a
parallelogram.
Answers
1. (2, 1), Find the midpoint of one of the diagonals since the midpoints are the same for both
2. slope of EF = slope of GH = |; slope of EH = slope of FG = — |; Slopes of opposite sides are the
same, therefore opposite sides are parallel.
3. EF = HG = Vl7; FG = EH = V29; lengths of opposite sides are the same (congruent).
4. A quadrilateral in the coordinate plane can be show to be a parallelogram by showing any one of the
three properties of parallelograms shown in questions 1-3.
Proving Quadrilaterals are Parallelograms
Coordinate Plane Challenge
Suppose that A(-2, 3),5(3, 3) and C(l,-3) are three of four vertices of a parallelogram.
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4*0
-A(-2, 3>
-B
3,3>-
10'
:c(i
■3]:
-10
1. Depending on where you choose to put point D, the name of the parallelogram you draw will change.
Sketch a picture to show all possible parallelograms. How many can you draw?
2. If you know the parallelogram is named ABDC, what is the slope of side parallel to AC?
3. Again, assuming the parallelogram is named ABDC, what is the length of BD1
4. Find the points of intersection of the diagonals of the three parallelograms formed. Label them X in
parallelogram ABCD, Y in parallelogram ADBC and Z in parallelogram ABDC.
5. Connect the points X, Y and Z to form a triangle. What do you notice about this triangle?
Answers
1. See graph below; 3 parallelograms can be drawn. The coordinates of D could be (0, 9), (6, -3), and (-4,
-3). Thus, the parallelograms created are ADBC, ABDC, and ABCD.
2. m AC
3-(-3)
-2-1 — -3 — "
3. d = V(3-6) 2 + (3-(-3) 2 ) = V9T36 = 3 V5
4. See graph; the intersections of the diagonals are (0.5, 3), (2, 0), and (-0.5, 0). They form the yellow
triangle in the center of the graph.
5. The triangle is formed by the midsegments of the triangle formed when the parallelograms overlap.
Four congruent triangles are formed within this center triangle, which is also congruent to the three outer
triangles.
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182
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xntpw. J r
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^— —--——— L—^
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V
Connections to Technology
Students can explore the properties of parallelograms with the following website:
www.mathwarehouse.com/geometry/quadrilaterals/parallelograms/interactive-parallelogram.php
On this website, there is a place where it lists the criteria for proving that a quadrilateral is a parallelogram.
There is also an interactive part where you can click and drag the vertices of the parallelogram to alter
the side lengths, angle measures, etc. The numbers change instantly on the screen as you move the vertices
around. This is a great website for students to explore the properties of parallelograms and familiarize
themselves with parallelograms.
Rectangles, Rhombuses, and Squares
Connections to Quilt Making
Give students the quilt blocks above. You can decide if you want to give each group one quilt block or
all three. Ask the students to work in groupsto identify all of the different quadrilaterals in the drawing.
Encourage students to outline the shapes as they find them. Also, there may be more quadrilaterals than
what are visible. For example, in the second block, two small squares (a blue and yellow square) create a
rectangle. In the third block there are also trapezoids. Students need to identify each quadrilateral and
write explain why.
Next, students can work to draw their own quadrilateral art piece. You can decide if students can include
triangles or not. Students should try to include as many different quadrilaterals as they can. Once they
have completed the design, have the students write a brief description of their work identifying each
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quadrilateral in the design.
Coordinate Plane Challenge
The points (?(— 1, 1), 1/(7, 1),A(1,7) and D(-l,5) are the vertices of quadrilateral QUAD. Plot the points
on graph paper to complete problems 27-30.
1. Find the midpoints of sides QU, UA,AD and DQ. Label them W, X, Y and Z respectively.
2. Connect the midpoints to form quadrilateral WXYZ. What does this quadrilateral appear to be?
3. Use slopes to verify your answer to problem 29.
4. Use midpoints to verify your answer to problem 29.
5. This phenomenon occurs in all quadrilaterals. Describe how you might prove this fact. (Hint: each
side of quadrilateral WXYZ is a midsegment in a triangle formed by two sides of the parallelogram and a
diagonal.)
Answers
1. See graph; W(3, 1),X(4,4), F(0,6),Z(-1,3)
\\
rt
*:Ai1, 7t
Cu
^r > jt -
DM 5^d
lJ \ i \f t °pj
J^\2Z
~ 7 i
T^
tZ* s
*^ t s^:u(7, 1)
DM \X*
-— -Ej-— — — — — -^
^ \ 4 1414 [J
r i i i l r
2. It is a parallelogram.
3. slope of WX = slope of YZ = 3; slope of XF = slope of ZW = -| opposite sides parallel
4. midpoint of diagonal FW is (1.5, 3.5); midpoint of diagonal XZ is (1.5, 3.5); midpoints bisect each other
5. Each side of the parallelogram is parallel to the diagonal. For example, XY \\ DU \\ ZW, so opposite
sides are parallel. They are also half the length of the diagonal so opposite sides are congruent. Either
proves that WXYZ is a parallelogram.
Challenge
This is a proof of a specific case of the investigation above.
A y D
1. Given: ABCD is a rectangle
W, X, Y and Z are midpoints of BC,AB,AD, and CD respectively
Prove: quadrilateral WXYZ is a rhombus
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Table 3.10:
Statements
Reasons
1. ABCD is a rectangle
1. BW = WC , ,
3. BD = AC
4. XY is a midsegment in AABD
1. Given
2. Definition of a midpoint
3.
4. Definition of a midsegment in a triangle
5. XY = \BD = WZ and
6. \BD = \AC
7. XY=WZ=YZ = XW
8. WXYZ is a rhombus
5. Midsegment in a triangle is half the length of
the parallel side.
6.
7.
2. Explain why a quadrilateral formed by the segments connecting the midpoints of the sides in a rhombus
is always a rectangle.
3. Explain why a quadrilateral formed by the segments connecting the midpoints of the sides in a square
is always a square.
For problems 4-6, find the value of each variable in the figures.
4.
Answers
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1.
Table 3.11:
Statements
Reasons
1. ABCD is a rectangle
2. BW= WC,AY= YD,BX = XA,CZ = ZD
3. BD = AC
4. XY is a midsegment in AABD
ZY is a midsegment in AACD
XW is a midsegment in AABC
WZ is a midsegment in ABCD
5. XY
\BD
WZ and XW
\AC
YZ
6. \BD = \AC
7. XY = WZ=YZ = XW
8. WXYZ is a rhombus
1. Given
2. Definition of a midpoint
3. Diagonals are congruent in a rectangle
4. Definition of a midsegment in a triangle
5. Midsegment in a triangle is half the length of
the parallel side.
6. Division POE
7. Substitution
8. Definition of a rhombus
2. Answers may vary. The quadrilateral inscribed in the rhombus will always be a rectangle because
the diagonals of a rhombus are perpendicular and the opposite sides of the inscribed quadrilateral will be
parallel to the diagonals and thus perpendicular to one another.
3. Answers may vary. First, the square is a rhombus, the inscribed quadrilateral will be a rectangle (see
problem 28). Second, the diagonals of the square are congruent so the sides of the inscribed quadrilateral
will be congruent (see problem 27). Since the sides of the inscribed quadrilateral are perpendicular and
congruent the parallelogram is a square.
4. x = 10, w = 53°, y = 37°, z = 37°
5. ;t = 45 o ,y = 90 o ,z = 2V2
6. x = y= 13,w = z = 25°
Trapezoids and Kites
Connections to Kite Design
In this activity, students can design their own kites. They can look at the different kinds of kites that are
possible by looking at the Wikipedia website, www.en.wikipedia.org/wiki/Kite. In the kite construction,
it is very important that the diagonals are perpendicular. Students can also use the dimensions in the
Know What? for this lesson to help them get started. However, they do not have to use these dimensions.
Allow students to be creative with their design.
Proof Challenge
Here is the proof to the Convers of Theorem 6-17.
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Given: Trapezoid ZOID and parallelogram ZOIM LD = Li
Prove: ZD = 01
Table 3.12:
Statement Reason
1. Trapezoid ZOID and parallelogram ZOIM, LD = Given
//
2. ZM = 01 Opposite Sides Theorem
3. Li = LZMD Corresponding Angles Postulate
4. ID = LZMD Transitive PoC
5. ZM = ZD Base Angles Converse
6. ZD = 01 Transitive PoC
Example 3 Extension
Extend the idea of Example 3 to general points. Then, this example will show that the diagonals for any
isosceles trapezoid are congruent.
Example 3 Extension: Show TA = RP.
R(x-d, y)
P(0, 0} A(x, 0)
Solution: This is an example of a coordinate proof. Here, we will use the distance formula to show that
TA = RP, but with letters instead of numbers for the coordinates.
TA = ^j(x-d) 2 + (0-y) 2 RP = A /(x-^/-0) 2 + (y-0) 2
= ^ x -d)* + {-y)* = ^{x-df+y 2
= ^( x -df+y 2
Notice that we end up with the same thing for both diagonals. This means that the diagonals are equal
and we have proved the Isosceles Trapezoid Diagonals Theorem.
Challenge
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Determine what type of quadrilateral ABCD is. ABCD could be any quadrilateral that we have learned in
this chapter. If it is none of these, write none.
1. A(-2,2),fi(0,l),C(2,2),D(l,5)
2. A(-7, 4), B(-4, 4), C(0, 0), D(0, -3)
3. A(3,3),fl(5,-l),C(7,0),Z)(5,4)
4. A(-4, 4), B(-l, 2), C(2, 4), D(-l, 6)
Answers
1. none
2. isosceles trapezoid
3. rectangle
4. rhombus
3.7 Similarity
Ratios and Proportions
Connections to History
The "golden ratio", labeled (Greek letter phi), is when two numbers, a and b are in the ratio ^- = | = 0.
There is one solution for this ratio, = 1+ 2 = 1.6180339887... Since the Renaissance, artists and architects
have used this proportion, believing that it is aesthetically pleasing. For example, many of the proportions
in the Parthenon are in the golden ratio. As a mini research project, you could have students research the
golden ratio and find it in art, architecture, nature, or music. Websites that might help students are:
www.intmath.com/Numbers/mathOfBeauty.php
• This site analyzes the mathematics in beauty and nature,
www. mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fiblnArt. html
• This website has a ton of different links for students to explore when looking at how ratios play into
different topics.
Challenge
1. Explain why ^- = ^- is a valid proportion. HINT: Cross-multiply.
2. Explain why ^y- = ^p is a valid proportion. HINT: Cross-multiply.
3. A recipe for crispy rice treats calls for 6 cups of rice cereal and 40 large marshmallows. You want
to make a larger batch of goodies and have 9 cups of rice cereal. How many large marshmallows do
you need? However, you only have the miniature marshmallows at your house. You find a list of
substitution quantities on the internet that suggests 10 large marshmallows are equivalent to 1 cup
miniatures. How many cups of miniatures do you need?
Answers
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a + b c + d
b = d
d(a + b) = b{c + d)
ad + bd = bc + bd
ad = be
a-b c - d
b d
d(a - b) = b(c - d)
ad - bd = be - bd
ad = be
3. 60 marshmallows; 6 cups miniatures
Similar Polygons
Connections to Sports
In sports, there can be several versions of a court or playing surface for men, women, or different versions
of the same game. In tennis, the court is 78 feet by 36 feet (no, 78 : 36 ^ 9 : 5). In table tennis, the table
is 9 feet by 5 feet. Are these two rectangles similar? Have students research and compare other sporting
venues in this way. Some examples are:
• WNBA court vs. NBA court vs. College (men or women) court vs. international play (students can
also compare the size of the keys and the three-point line)
• Softball infield vs. baseball infield
• Arena football field vs. NFL football field
• Curling triangle vs. shuffleboard triangle
Challenge
1. AABC ~ ADEF
Solve for x and y.
B
D'
2. QUAD ~ KENT
Find the perimeter of QUAD.
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21
T N
3. ACAT ~ ADOG
Solve for x and y.
A
D 28
4. PENT A ~ F/V£fl
Solve for x.
P E
5. AMNO ~ AXNF
Solve for a and /?.
N
12 —
6. Trapezoids #AV£ - /^VOr
Solve for x and y.
H 4 a
E 8 V
^k >
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7. Two similar octagons have a scale factor of yj. If the perimeter of the smaller octagon is 99 meters,
what is the perimeter of the larger octagon?
8. Two right triangles are similar. The legs of one of the triangles are 5 and 12. The second right triangle
has a hypotenuse of length 39. What is the scale factor between the two triangles?
9. What is the area of the smaller triangle in problem 30? What is the area of the larger triangle in
problem 30? What is the ratio of the areas? How does it compare to the ratio of the lengths (or scale
factor)? Recall that the area of a triangle is A = ^ bh.
Answers
1. x = 12, y= 15
2. 31
3. x = 20,y = 7
4. x « 14.6
5. a*7A,b = 9.6
6. X = 6,y = 10.5
7. 121
8. 1:3
9. 30w 2 , 270w 2 , 1 : 9, this is the ratio of the lengths squared or ( ^ J .
Similarity by AA
Connections to History and Pyramids
One of the first ancient Greek philosophers, Thales, is credited with also being one of the first mathemati-
cians. He was one of the first people to truly understand similar triangles and indirect measurement. The
story goes that he measured the height of the pyramids by their shadows at the moment when his own
shadow was equal to his height. We know that a right triangle with two equal legs is a 45-degree right
triangle, all of which are similar (by AA). The length of the pyramid's shadow measured from the center of
the pyramid at that moment must have been equal to its height. Students can continue to research Thales
using the website www.phoenicia.org/thales.html. Make sure to fully discuss Thales' logic and though
process before moving on to the next part of this activity.
Have students investigate pyramids like Thales did. Using sugar cubes, have students create a scale model
of an ancient pyramid. After completing the model, use a darkened room and a flashlight to find the
shadow of the pyramid. Then, see if students can find a way to test and prove Thales' logic. Allow time
for students to share their work when finished.
Challenge
1. Janet wants to measure the height of her apartment building. She places a pocket mirror on the
ground 20 ft from the building and steps backwards until she can see the top of the build in the
mirror. She is 18 in from the mirror and her eyes are 5 ft 3 in above the ground. The angle formed by
her line of sight and the ground is congruent to the angle formed by the reflection of the building and
the ground. You may wish to draw a diagram to illustrate this problem. How tall is the building?
2. Sebastian is curious to know how tall the announcer's box is on his school's football field. On a sunny
day he measures the shadow of the box to be 45 ft and his own shadow is 9 ft. Sebastian is 5 ft 10
in tall. Find the height of the box.
3. Juanita wonders how tall the mast of a ship she spots in the harbor is. The deck of the ship is the
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same height as the pier on which she is standing. The shadow of the mast is on the pier and she
measures it to be 18 ft long. Juanita is 5 ft 4 in tall and her shadow is 4 ft long. How tall is the
ship's mast?
4. Use shadows or a mirror to measure the height of an object in your yard or on the school grounds.
Draw a picture to illustrate your method.
Answers
1. 70 ft
2. 29 ft 2 in
3. 24 ft
4. Answers will vary.
Similarity by SSS and SAS
Connections to Bridge Construction
Similar triangles are used in bridge construction. Have students look at a picture of the Golden Gate
Bridge. Ask them why they think the support of the bridge and the columns are created from triangles.
To help students arrive at the correct answer, give them strips of construction paper (all the same length)
and brads. In groups, have each student create an equilateral polygon: triangle, rhombus, pentagon, and
hexagon. Then, let them determine why a triangle is the best of those polygons to use as the support for
a bridge (a triangle will not change shape when put together, unlike the other 3, which will move around).
Therefore, no matter the shape, a triangle is the sturdiest of the polygons. This is because for any three
points there is exactly one plane.
Coordinate Plane Challenge
For questions 1-3, use AABC with A(-3, 0),£(-1.5, 3) and C(0,0) and ADEF with D(0,2),£(l,4) and
F(2,2).
1. Find AB,BC,AC,DE,EF and DF .
2. Use these values to find the following proportions: ^§, |^ and ^.
3. Are these triangles similar? Justify your answer.
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For questions 4-7, use ACAR with C(-3, 3),A(-3, -1) and R(0,-1) and ALOT with L(5, -2), 0(5, 6) and
r(-i,6).
4. Find the slopes of CA,AR,LO and OT.
5. What are the measures of LA and LO? Explain.
6. Find LO, OT, CA and AT?. Use these values to write the ratios LO : CA and OT : AT?
7. Are the triangles similar? Justify your answer.
Answers
1. AB = BC = Vll.25,AC = 3,£>£ = EF = V5,£>F = 2
9 A£ _ £C _ AC _ 3
Z * £>£ EF DF 2
3. Yes, AABC ~ A£>£F by SSS similarity.
4. slope of CA = slope of LO = undefined (vertical); slope of AT? = slope of OT = (horizontal).
5. 90°, vertical and horizontal lines are perpendicular.
6. TO = 6, OL = 8, CA = 4 and AR = 3; LO : CA = OT : AR = 2 : 1
7. Yes, by SAS similarity.
Proportionality Relationships
Construction Extension
This construction will guide students through constructing a line segment that is divided into four equal
parts.
1. Draw a line segment that is about 3 inches long. Label the endpoints A and B. Draw AC, such that C
is above AB.
2. Using any length, place the pointer on A and make and arc intersecting AC. Without changing the
width of the compass, repeat this measurement on AC three more times. Label the intersections D, E, F,
and G. From our construction, AD = DE = EF = FG.
3. Draw GB. You are going to draw a series of parallel lines to GB. Copy lAGB (Investigation 1-3) on
D,E, and F. Where the second side of the angles intersect AB, should all be equidistant, creating four
congruent segments.
A "> s *^B
Proof Challenge
Write the proof of Theorem 7-8.
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Given: ABAD with AC is the angle bisector of /.BAD Auxiliary lines AX and XD, such that X,A, B are
collinear and AC \\ XD.
p rove . MC = BA
^ iUVU - CD AD
Answer
Table 3.13:
Statement
Reason
1. AC is the angle bisector of LB AD X,A,B are Given
collinear and AC \\ XD
2. LBAC = LCAD
3. LX = LBAC
4. LCAD = LADX
5. LX = LADX
6. AX AD is isosceles
7. AX = AD
8. AX = AD
9.
10.
5A
A ^A _ C l
AD
BC
CD
Definition of an angle bisector
Corresponding Angles Postulate
Alternate Interior Angles Theorem
Transitive PoC
Base Angles Converse
Definition of an Isosceles Triangle
Congruent segments are also equal
Theorem 7-7
Substitution PoE
Challenge
Find the values of the variables below.
1.
Find the unknown lengths.
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Answers
1. a = 1.8,6= |
2. x = 5,y = 7
3. \b or 1.56
4. ^a or 3.2a
Similarity Transformations
Connections to Art
In this activity students are going to take any polygon that they would like to and create an art piece that
shows the dilations of the polygon. The first polygon should be red so that you can tell which polygon is
being transformed. Students should create dilations that are smaller and larger. Students can be creative
and decide what the scale factors will be. It might be easier for students to do this art piece on a coordinate
plane. Students need to make at least three dilations of the original figure.
Extension
As an extension to the Know What? in this lesson, students can also create two-point perspective drawings.
A computer program, such as Geometer's Sketchpad, can be very helpful for activity.
Construction Challenge
We can use a compass and straight edge to construct a dilation as well. Copy the diagram below.
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1. Set your compass to be CG and use this setting to mark off a point 3 times as far from C as G is. Label
this point G'. Repeat this process for CO and CD to find O' and D' .
2. Connect G' \0' and £>' to make AD'O'G'. Find the ratios, ^, ^ and ^.
3. What is the scale factor of this dilation?
4. Describe how you would dilate the figure by a scale factor of 4.
5. Describe how you would dilate the figure by a scale factor of \.
Answers
1.
VQi _ o VGi _ o i GD^_ _ o
Z ' DO ~ °» OG — ° cillQ GD — °'
3. 3
4. To dilate the original figure by a scale factor of 4 make one additional tick mark with your compass.
5. To dilate the original figure by a scale factor of ^, construct the perpendicular bisectors of CD, CO and
CG. The midpoints created by the perpendicular bisectors would be D' ,0' ', and G f respectively.
Extension: Self-Similarity
Connections to Art and Music
Have students do an internet search for fractals. Let students pick one aspect of art, music, nature, or
another topic that interests them and how fractals can be related to it. Have students do a short write-up
on what they find out.
Challenge
Have students research how the golden ratio and fractals relate to each other.
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3.8 Right Triangle Trigonometry
The Pythagorean Theorem
Connections to History
Pythagoras was a Greek philosopher and mathematician. He is credited with discovering the Pythagorean
Theorem. Students can see his Wikipedia page, http://en.wikipedia.org/wiki/Pythagoras and write
a mini-project about his life and accomplishments.
Proof of the Pythagorean Theorem
This proof is "more formal" than the one in the text. Here, we will use letters, a, b, and c to represent the
sides of the right triangle. In this particular proof, we will take four right triangles, with legs a and b and
hypotenuse c and make the areas equal.
A — r A
f^green square — c
^green square
= 4(-ab)+(b-a) 2
= 2ab + b 2 - 2ab + a 2
b 2 + a 2
Pythagorean Proof Challenge
This first proof below is similar to the one above. Use the picture below to answer the following questions.
a b
1. Find the area of the square with sides (a + b).
2. Find the sum of the areas of the square with sides c and the right triangles with legs a and b.
3. The areas found in the previous two problems should be the same value. Set the expressions equal to
each other and simplify to get the Pythagorean Theorem.
Major General James A. Garfield (and former President of the U.S) is credited with deriving this next
proof of the Pythagorean Theorem using a trapezoid.
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4. Find the area of the trapezoid using the trapezoid area formula: A = \[b\+ b<2)h
5. Find the sum of the areas of the three right triangles in the diagram.
6. The areas found in the previous two problems should be the same value. Set the expressions equal to
each other and simplify to get the Pythagorean Theorem.
Answers
1. a 2 + 2ab + b 2
2. c 2 + 4,(\)ab = c 2 + 2ab
3. a 2 + 2ab + b 2 = c 2 + 2ab, which simplifies to a 2 + b 2 = c 2
4. \[a + b){a + b) = \{a 2 + 2ab + b 2 )
5. 2(^)^/7+ ^c = ab+ \c
6. ^(a 2 + 2ab + Z? 2 ) = ab + ^c ^ a 2 + 2ab + b 2 = 2ab + c 2 , which simplifies to a 2 + b 2 = c 2
Converse of the Pythagorean Theorem
Proof Challenge
Prove Theorem 8-3.
Given: In AABC,a 2 + b 2 > c 2 , where c is the longest side.
In ALMN, AN is a right angle.
Prove: AABC is an acute triangle.
(all angles are less than 90°)
Answer
Table 3.14:
Statement
Reason
1. In AABC, a 2 -\-b 2 > c 2 , and c is the longest side.
In ALMN, AN is a right angle.
2. a 2 + b 2 = h 2
3. c 2 < h 2
4. c < h
Given
Pythagorean Theorem
Transitive PoE
Take the square root of both sides
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Table 3.14: (continued)
Statement
Reason
5. AC is the largest angle in AABC.
6. mAN = 90°
7. mAC < mAN
8. mAC < 90°
9. AC is an acute angle.
10. AABC is an acute triangle.
The largest angle is opposite the longest side.
Definition of a right angle
SSS Inequality Theorem
Transitive PoE
Definition of an acute angle
If the largest angle is less than 90°, then all the
angles are less than 90°.
Prove Theorem 8-4.
L
b " N b M
Given: In AABC, a 2 + b 2 < c 2 , where c is the longest side.
In ALMN, AN is a right angle.
Prove: AABC is an obtuse triangle.
(one angle is greater than 90°)
Answer
Table 3.15:
Statement
Reason
1. In AABC, a 2 -\-b 2 < c 2 , and c is the longest side. Given
In ALMN, AN is a right angle.
2. a 2 + b 2 = h 2 Pythagorean Theorem
3. c 2 > h 2 Transitive PoE
4. c > h Take the square root of both sides
5. AC is the largest angle in AABC. The largest angle is opposite the longest side.
6. mAN = 90° Definition of a right angle
7. mAC > mAN SSS Inequality Theorem
8. mAC > 90° Transitive PoE
9. AC is an obtuse angle. Definition of an obtuse angle.
10. AABC is an obtuse triangle. Definition of an obtuse triangle.
Coordinate Plane Challenge
Given AZ?, with A(3, 3) and B{2, -3) determine whether the given point, C, makes an acute, right or obtuse
triangle.
1. C(3,-3)
2. C(4,-l)
3. C(5,-2)
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Given AZ?, with A(-2, 5) and 5(1, -3) find at least two possible points, C, such that AABC is
4. right, with right AC.
5. acute, with acute iC.
6. obtuse, with obtuse iC.
Answers
1. right
2. obtuse
3. acute
4. (1, 5), (-2, -3)
5. Answers will vary, possibilities are (-3, -3) or (-4, 3)
6. Answers will vary, possibilities are (1, 4), (-1, 3), or (4, 3).
Using Similar Right Triangles
Geometric Mean Extension
The true definition of the geometric mean is: the n th root of a product of n numbers. For use with triangles
we limited this to the second root of the product of two numbers. Using the expanded definition, students
can find the geometric and arithmetic mean of any set of numbers and compare the answers.
Extension Example: Find the geometric and arithmetic mean of the set of numbers below.
4, 6, 8, 9, 11, 15, 18, 25, 26
Solution: Arithmetic mean: 4+6+8+9+11+15+18+25+26 = 122 = 13 5
Geometric mean: ^4 x 6 x 8 x 9 x 11 x 15 x 18 x 25 x 26 = ^3, 335, 904, 000 « 11.43
With this expanded definition, the geometric mean can be applied to population growth. It is a better
approximation of the mean for exponential growth and proportional growth. In business, the geometric
mean of growth rates is known as thecompound annual growth rate (CAGR). It is commonly used with
percentages to see how much a population has grown or decreased.
Extension Example: The population of a town is shown from 1980, 1990, 2000 and 2010: 6324, 8571,
9867, 12,108. What is the average rate of growth over the thirty years?
Solution: This is an example where we would use the geometric mean. The percentage increase from year
to year is: 35.5%, 15.1% and 22.7%. Therefore, the amount of increase is 1.355, 1.151 and 1.227. Use these
numbers in the geometric mean. Vl.355 X 1.151 x 1.227 = 1.241. The average rate of growth over the four
years is 24.2%.
Students may wonder why we use this mean in this situation over the arithmetic mean. The arithmetic
mean for this data set is 35 - 5 + 1 5-i+22. _ 24.43%. Using the starting population, the population would
increase by 6324 x 0.2443 = 1545 people each year. The final population in 2010 would be 12,504. Using
the geometric mean in this way, the increase would be 6324 x 0.241 = 1524 people. The final population
in 2010 would be 12,420, which is closer to the actual final population.
The geometric mean was also used to find the aspect ratio for HDTVs (16:9). This ratio was determined by
the Society of Motion Picture and Television Engineers (SMPTE) by finding the geometric ratio between
the widescreen format movies are shot in (2.35:1) and standard TV (4:3). -J2.35 x g = 1.77 This ratio,
1.777:1 is best approximated by 16:9.
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Extension with Geometric Sequences
A geometric sequence is a sequence of numbers in which each successive term is determined by multiplying
the previous term by the common ratio. An example is the sequence 1, 3, 9, 27, ... Here each term is
multiplied by 3 to get the next term in the sequence. Another way to look at this sequence is to compare
the ratios of the consecutive terms.
1. Find the ratio of the 2 nd to 1 st terms and the ratio of the 3 rd to 2 nd terms. What do you notice? Is this
true for the next set (4 th to 3 rd terms)? Use 1, 3, 9, 27, ... from above.
2. Given the sequence 4, 8, 16, ..., if we equate the ratios of the consecutive terms we get: | = ^. This
means that 8 is the of 4 and 16. We can generalize this to say that every
term in a geometric sequence is the of the previous and subsequent terms.
Use what you discovered in problem 2 to find the middle term in the following geometric sequences.
3. 5, , 20
4. 4, , 100
5. 2, ,\
Answers
1. ratios are y and |, which both reduce to the common ratio 3. Yes, this is true for the next pair of terms
since ^ also reduces to 3.
2. geometric mean; geometric mean
3. 10
4. 20
5. 1
Another Proof of the Pythagorean Theorem
We can use what we have learned in this section in another proof of the Pythagorean Theorem. Use the
diagram to fill in the blanks in the proof below.
Table 3.16:
Statement
Reason
1. £ = ? and { = *
a d-\-e b :
-»-" stl A I si\ r\-r\/A ln^
2. a 1 = e(d + e) and b 2 = d{d + e)
3. a 2 + b 2 =?
4. ?
5. c = d + e
6. ?
Theorem 8-7
?
Combine equations from #2.
Distributive Property
?
Substitution PoE
Answer
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Table 3.17:
Statement
Reason
1. e - = -jS- and £ = -£-
2. a 2 = e(d + e) and Z? 2 = d(d + e)
3. a 2 +b 2 = e(d + e) + d[d + <?)
4. a 2 + b 2 = (e + d)(d + e)
5. c = d + £
6. a 2 + Z> 2 = c 2
Theorem 8-7
Cross-Multiplication Property
Combine equations from #2.
Distributive Property
Segment Addition Postulate
Substitution PoE
Special Right Triangles
Know What? Extension/Connection to Baseball
In this Know What? for this lesson, students found the distance from 1 st to 3 rd base for a baseball
diamond. Using the special right triangle ratios, we found that the distance is 90 V2 « 127.27922... As an
extension, have students research what value Major League Baseball actually uses as the distance from
1 st to 3 rd base. The distance cannot be an irrational number because the decimal repeats. Students
should find that the regulation distance is 127 feet, 3| inches. Then, have them compare the amounts
as decimals to see what the discrepancy is. Discuss why MLB uses this distance and not 90 V2 feet.
127/Y.3| in. = 127 ft. 3. 375 in. = 127.28125 ft. Students can also see if this generalization holds true for
softball infields.
Challenge
Use the special right triangle ratios to find the length of x.
1.
.20
120°
M-
2.
M-
10
3. An equilateral triangle is an isosceles triangle. If all the sides of an equilateral triangle are s, find the
area, A = ^ bh, using special right triangles. Leave your answer in simplest radical form.
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4. What is the height of an equilateral triangle with sides of length 3 in?
5. What is the area of an equilateral triangle with sides of length 5 ft?
6. A regular hexagon has sides of length 3 in. What is the area of the hexagon? (Hint: the hexagon is
made up a 6 equilateral triangles.)
7. The area of an equilateral triangle is 36 V3. What is the length of a side?
8. Four isosceles triangles are formed when both diagonals are drawn in a square. If the length of each
side in the square is s, what are the lengths of the legs of the isosceles triangles?
Answers
1. 12V2 + 20 + 4V6-46.77
2. 16 + 6V3-26.39
O. r\ — 7y ' S ' r\ S — a S
4. *f in
5. ^fi ft*
6. %fi in 2
7. 12
8.^
Tangent, Sine and Cosine
Connections to Land Surveying
This activity involves allowing the class going outside and creating a clinometer.
Making a Clinometer
Tools Needed: straw, paper copies of a protractor, string (about 6 inches), washer, tape
1. Tie the washer to the string at one end and then to the straw at the other end. The string should be
at the center of the straw.
2. Tape the straight edge of the protractor to the straw so that its center lines up with where the string
is tied to the straw.
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measured angle
angle of elevation
72*+18 o = 90 Q
The measured angle
from the clinometer
and the angle of
elevation are
complimentary.
Measuring the Height of Objects using a Clinometer
1. Now that students have made a clinometer (probably needs to be done the class before this portion),
go outside and have students measure the height of objects they cannot physically measure.
2. Once an object is selected, have students look through the straw of the clinometer until they see the
top of the object. At this point, they need to find the measured angle (see the picture above) from
the string. This angle is complementary to the angle of elevation.
3. Use the angle of elevation and the tangent ratio to find the height of the object. Have students repeat
this from several distances away (horizontal) and/or for 2-3 different objects.
Inverse Trigonometric Ratios
Problem Set Extension
1. Using the table from questions 25-30, what value is equal to cos 60°?
2. How does the degree answer from #1 relate to 60°?
3. Fill in the blanks:
(a) cos A = sin
(b) sin B = cos
Answers
1. sin 30°
2. they are complements, 60° + 30° = 90°
3. (a) sin(90-A)°
(b) cos(90-£)°
Extensions with Trig Ratios
Students may ask about other trig ratios that exist. Feel free to introduce secant, cosecant and cotangent.
esc A =
sin A =
1
sin A
1
esc A
sec A =
cos A =
cos A
1
sec A
cot A =
tan A =
tan A
1
cot A
Have students apply these ratios to the values in the table from questions 25-30. To enter these values into
a calculator, they will need to use the fractions above.
Another extension would be to explore the trig identities sin 2 A + cos 2 A = 1 and tan A = ^j.
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1. Find sin 2 30° + cos 2 30°.
2. Find sin 2 80° + cos 2 80°.
3. A
C a B
Let's explore why this works. Using the triangle, find:
a) sin A
b) cos A
c) sin 2 A
d) cos 2 A
4. Find sin 2 A + cos 2 A
5. Use the Pythagorean Theorem to simplify.
6. Find ^^
cos 30
7. Find tan 30°.
8. Let's explore why this works. Again, use the triangle from #3. Using the ratios you found in parts a
and b, find ^J±A and simplify.
Answers
1. 0.25 + 0.75 = 1
2. 0.9698 + 0.0302 = 1
3.a)f
4. c 2 -h c 2 - c 2
5. From the Pythagorean Theorem, a 2 + b 2 = c 2 , so ^ + ^ = ^4^ = ^ = 1.
5 Q-5 ~ n 5774
U ' 0.866 u '°' '^
7. 0.5774
8. ^4 = f = 2 + k = a x c = a For tllig triangle, tanA = f therefore tanA = ^4.
c
3.9 Circles
Parts of Circles and Tangent Lines
Connections to Nature
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Use these figures as an introductory activity to circles. Discuss how circles exist in nature and mechanics.
See if students can come up with other examples. Then, after going over the parts of circles, see if students
can identify the different parts of circles within these images. As an extension, have students go to the
computer lab, find an image online and print it out. Then, they should identify all the parts of the circle
that they can find. You can have a little competition to see which student has the image with the most
parts of a circle in it.
Connections to Technology
The following website, www.cut-the-knot.org/Curriculum/Geometry/TangentTwoCirclesI.shtml, is a great
example of tangent circles and lines. In working with this website, students manipulate the center or edge
of one of the circles. When they do this, they will alter the diagram of the two circles and their tangents.
Proof Challenge
Write a proof for Theorem 9-2.
Given: AB and CB with points of tangency at A and C.
AD and DC are radii.
Prove: AB = CB
Answer
Table 3.18:
Statement
Reason
1. AB and CB with points of tangency at A and C. Given
A D nr\c\ DC arp trH ii
AD and DC are radii.
2. AD = DC
3. DA _L AB and DC _L CB
4. mlBAD = 90° and mlBCD = 90°
5. Draw ~BD.
6. AADB and ADCB are right triangles
7.DB = DB
8. AABD = ACBD
9.AB = CB
All radii are congruent.
Tangent to a Circle Theorem
Definition of perpendicular lines
Connecting two existing points
Definition of right triangles (Step 4)
Reflexive PoC
HL
CPCTC
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Challenge
1. Circles tangent at T are centered at M and N. ST is tangent to both circles at T . Find the radius of
the smaller circle if IN _L Jm,SM = 22, TN = 25 and m/5A^r = 40°.
2. Four circles are arranged inside an equilateral triangle as shown. If the triangle has sides equal to 16
cm, what is the radius of the bigger circle? What are the radii of the smaller circles?
HINT: Use 30-60-90 triangles.
Answers
1. 9.23
9 8V3. 8V3
Z ' 3 ' 9
Properties of Arcs
Connections to History
The original reason as to why there are 360° degrees in a circle is uncertain, but there are a few theories.
One theory is that ancient calendars had 360 days. Astronomers notices that the stars revolved around
the North Star and after 360 days, the stars were back in their starting position. This evolved to 1 - 360^
of a rotation each day or 1° each day of the year.
Another theory is that 360 isvery divisible: 360 has 24 divisors, including every number from 1 to 10 except
7. This property has many useful applications, such as dividing the world into 24 time zones, to correlate
with the 24-hour day.
Challenge
Use the diagram below to find the measures of the indicated arcs in problems 1-6. LP and KN are diameters.
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1. mMN
2. mLK
3. mMP
4. mMK
5. mNPL
6. rnLKM
Use the diai
X
ram below to find the measures indicated in problems 7-12. XZ and YV are diameters.
Y
(5x-5) c
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8. 105°
9. 68°
10. 105°
11. 255°
12. 217°
Properties of Chords
Construction Challenge
Trace the arc below onto your paper then follow the steps to locate the center using a compass and
straightedge.
a. Use your straightedge to make a chord in the arc.
b. Use your compass and straightedge to construct the perpendicular bisector of this chord.
c. Repeat steps a and b so that you have two chords and their perpendicular bisectors.
d. What is the significance of the point where the perpendicular bisectors intersect?
e. Verify your answer to part d by using the point and your compass to draw the rest of the circle.
Answer
This construction is not done to scale and your chords might be in different places but it should give you
an idea of what the construction should look like.
Coordinate Plane Challenge
Let's repeat what we did in the Construction Challenge above using coordinate geometry.
Given the points A(-3, 5),fi(5, 5) and C(4, -2) on the circle (an arc could be drawn through these points
from A to C) . The following steps will walk you through the process to find the equation of the perpendicular
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bisector of a chord, and use two of these perpendicular bisectors to locate the center of the circle.
1. Let's first find the perpendicular bisector of chord AB.
(a) Since the perpendicular bisector passes through the midpoint of a segment, find the midpoint
between A and B.
(b) Now the perpendicular line must have a slope that is the opposite reciprocal of the slope of AB.
< — >
Find the slope of AB and then its opposite reciprocal.
(c) Finally, you can write the equation of the perpendicular bisector of AB using the point you
found in part a and the slope you found in part b.
2. Repeat steps a-c for chord BC.
3. Now that we have the two perpendicular bisectors of the chord we can use algebra to find their
intersection. Solve the system of linear equations to find the center of the circle.
4. Find the radius of the circle by finding the distance from the center (point found in part e) to any of
the three given points on the circle.
Answers
1. for AS:
(a) (1, 5)
(b) m = 0, ± m is undefined
(c) x = 1
2. for BC:
(a) (1J)
(b) m = 7, J_ m = -h
3. Point of intersection (center of the circle) is (1, 2).
4. radius is 5 units
Inscribed Angles
Proof Challenge
Use the diagram below to write a proof of Theorem 9-8.
B
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Table 3.19:
Statement
Reason
1. LACB and LADB intercept AB
2. mlACB = \mAB
mlADB = \mAB
3. mlACB = mlADB
4. LACB = LADB
1. Given
2. Inscribed Angle Theorem
3. Transitive Property
4. Definition of Congruence
Challenge
1. Find x and v in the picture below.
Use the diagram below to find the measures of the indicated angles and arcs in problems 2-7.
D
2. mLEBO
3. rnLEOB
4. rnBC
5. mLABO
6. mLA
7. mLEDC
8. Suppose that AZ? is a diameter of a circle centered at 0, and C is any other point on the circle. Draw
the line through O that is parallel to AC, and let D be the point where it meets BC. Prove that D is the
midpoint of BC.
Answers
1. x= 134°, y = 120°
2. 55°
3. 70°
4. 110°
5. 90°
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6. 20°
7. 90°
Since AC
mlCAB
OD,mlCAB = mlDOB by Corresponding Angles Postulate. Also, mlDOB
- 2 mCB. This makes D the midpoint of CB.
mDB and
Angles of Chords, Secants, and Tangents
Connections to Architecture
Either in this lesson or as a review activity, you can have students do an internet search on circles in
architecture. Have students select three different images that best illustrate the content of the chapter.
They can either pick an entire circle, parts of a circle, or use a theorem to help them interpret the buildings
they choose. Each student should write a paragraph explaining how each building illustrates the concept (s)
they decided to highlight. Also encourage students to explain why the circular architecture is important
for the particular building they chose.
This particular building is a bullfighting ring in Malaga, Spain. Sporting arenas are usually a very good
representation of circular seating.
Algebra Challenge
Solve for the variable (s)
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1.
x+15)°
(2x - 30)
3.
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6.
(2x 2 )
(15x-3) c
Answers
1. jk = 75°
2. jc = 45°
3. jt = 35°,;y
4. jc = 60°
5. x= 10°
6. x = 3°
35°
Segments of Chords, Secants, and Tangents
Connections to Technology
Go to the website http: //mathbits . com/mathbits/gsp/CircleSegments .htm and download the Geome-
ter's Sketchpad file MBcirclesegments.gsp. Use this file as a demonstration or go to the computer lab and
have students investigate the circle segments on their own. This is a great tool to help students see the
relationship between the segments and help them develop a formula.
Proof Challenge
Prove Theorem 9-16.
Given: Secant AC and tangent CD.
Prove: z 2 = y(y + jc)
HINT: Show AACD ~ ADCB.
Answer
Table 3.20:
Statement
Reason
1. Secant AC and tangent CD.
2. LC = LC
3. LCAD = LBDC
4. AACD ~ ADCB
Given
Reflexive PoC
Theorem 9-8
AA Similarity Postulate
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Table 3.20: (continued)
Statement
Reason
5.
z _ £±Z
y z
6. z 2 = y(y + x)
Corresponding parts of similar triangles are pro-
portional
Cross multiplication
Challenge
For problems 1-8, solve for x.
1- 3
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8. Find x and y.
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Extension: Writing and Graphing the Equations of Circles
Coordinate Plane Challenge
Let's find the equation of a circle using three points on the circle. Do you remember how we found the
center and radius of a circle given three points on the circle in the Properties of Chords Enrichment? We
used the fact that the perpendicular bisector of any chord in the circle will pass through the center. By
finding the perpendicular bisectors of two different chords and their intersection we can find the center
of the circle. Then we can use the distance formula with the center and a point on the circle to find the
radius. Finally, we will write the equation.
Given the points A(— 12, —21), 5(2, 27) and C(19, 10) on the circle (an arc could be drawn through these
points from A to C), follow the steps below.
a. Since the perpendicular bisector passes through the midpoint of a segment we must first find the
midpoint between A and B.
< — >
b. Now the perpendicular line must have a slope that is the opposite reciprocal of the slope of AB. Find
the slope of AB and then its opposite reciprocal.
c. Finally, you can write the equation of the perpendicular bisector of AB using the point you found in
part a and the slope you found in part b.
d. Repeat steps a-c for chord BC.
e. Now that we have the two perpendicular bisectors of the chord we can find their intersection. Solve the
system of linear equations to find the center of the circle.
f. Find the radius of the circle by finding the distance from the center (point found in part e) to any of
the three given points on the circle.
g. Now, use the center and radius to write the equation of the circle.
Find the equations of the circles which contain three points.
1. A(-2,5),5(5,6) and C(6,-l)
2. A(-ll,-14),£(5,16) andC(12,9)
Answers
a-d. J_ bisector of AB is y = -^x + ||, _L bisector of BC is y = x + 8
e. center of circle (-5, 3)
f. radius 25
g. (x + 5) 2 + (y-3) 2 = 625
1. (jt-2) 2 + (y-2) 2 = 25
2. (x + 3) 2 + (y-l) 2 = 289
3.10 Perimeter and Area
Triangles and Parallelograms
Extension: Heron's Formula
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Another way to find the area of a triangle is to use Heron's Formula. This formula states that the area
can be represented by the formula A = ^s(s - a)(s - b)(s - c) where a, b, and c are the sides of a triangle
and s is the semiperimeter, s = fl+ 2 +c . This formula is useful because you do not need the altitude to find
the area. To derive this formula, you would need to use the Law of Cosines.
Extension Example: Find the area of a triangle with sides 6, 9, and 11.
2
Solution: First find the semiperimeter, s = 6 + 9 + n = 20. = 13. Now, plug everything into Heron's
Formula.
A = ^13(13 -6)(13-9)(13- 11) = Vl3 • 7 • 4 • 2 = V728 = 2 Vl82 « 26.98 u 2
Sample Problems
Find the area of a triangle with the given side lengths. Round answers to the nearest hundredth.
1. 5, 10, 6
2. 20, 16, 22
3. 9, 14, 19
4. 31, 32, 33
Answers
All units below are square units.
1. 11.40
2. 154.11
3. 56.87
4. 442.54
Challenge
Find the dimensions of the rectangles with the given information.
1. A rectangle with a perimeter of 20 units and an area of 24 units 2 .
2. A rectangle with a perimeter of 72 units and an area of 288 units 2 .
For problems 3 and 4 find the height and area of the equilateral triangle with the given perimeter.
3. Perimeter 18 units.
4. Perimeter 30 units.
Answers
1. 6x4
2. 12x24
3. /z = 3V3,A = 9V3
4. /z = 5V3,A = 25V3
Trapezoids, Rhombi, and Kites
Proof Challenge
Using the pictures below, prove the area formulas for a trapezoid, rhombus, and kite.
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1. Use the isosceles trapezoid to show that the area of this trapezoid can also be written as the sum of the
area of the two triangles plus the rectangle in the middle. Write the formula and then reduce it to equal
\h{b 1 + b 2 ) or !(6i+6 2 ).
h
^
2. Use this picture of a rhombus to show that the area of a rhombus is equal to the sum of the areas of
the four congruent triangles. Write a formula and reduce it to equal \d\d2.
3. Use this picture of a kite to show that the area of a kite is equal to the sum of the areas of the two pairs
of congruent triangles. Recall that d\ is bisected by d^- Write a formula and reduce it to equal \d\d2.
Answers
1.
2.
n + 2As
hbi +
h{b 2 -b x )
2hb x + hb 2 - hbi
hbi +hb 2 h
— 2 =2 {bl + b2)
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4As
d\ • d 2
3.
-d\d 2
2As + 2As
2 (^H +2 (r^ 2 " x) )
-di-x+ -d x d 2 - -dix
-d\d 2
Challenge
For problems 1-3, determine what kind of quadrilateral ABCD is and find its area.
1. A(-2, 3), 5(2, 3), C(4, -3), D(-2, -1)
2. A(0,1),£(2,6),C(8,6),£>(13,1)
3. A(-2, 2), 5(5, 6), C(6, -2), D(-l, -6)
4. Given that the lengths of the diagonals of a kite are in the ratio 4:7 and the area of the kite is 56 square
units, find the lengths of the diagonals.
5. Given that the lengths of the diagonals of a rhombus are in the ratio 3:4 and the area of the rhombus
is 54 square units, find the lengths of the diagonals.
6. In the figure to the right, ABCD is a square. AP = PB = BQ and DC = 20ft.
A P
D C
a. What is the area of PBQD?
b. What is the area of ABCD?
c. What fractional part of the area of ABCD is PBQD?
Answers
1. kite, 24 units 2
2. Trapezoid, 47.5 units 2
3. rhombus, 12 V5 units 2
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4.
8, 14
5.
9, 12
6.
a) 200 ft 2
b) 400 ft 2
c)
l
2
Areas of Similar Polygons
Recall that a regular polygon is a polygon with congruent sides and angles. Regular polygons are the only
polygons that have a consistent formula for area and perimeter.
Extension: Perimeter of Regular Polygons
Recall that the perimeter of a square is 4 times the length of a side because each side is congruent. We
can extend this concept to any regular polygon.
Perimeter of a Regular Polygon: If the length of a side is s and there are n sides in a regular polygon,
then the perimeter is P = ns.
Extension Example: What is the perimeter of a regular octagon with 4 inch sides?
Solution: If each side is 4 inches and there are 8 sides, that means the perimeter is 8(4 in) = 32 inches.
4 in
Extension: Area of Regular Polygons
In order to find the area of a regular polygon, we need to define some new terminology. All regular polygons
can be inscribed in a circle, which means regular polygons have a center and radius. Also like a
circle, a regular polygon has a central angle. However, the central angle is the angle formed by two radii
drawn to consecutive vertices of the polygon. In the picture, the central angle is /.BAD. Also, notice
that ABAD is an isosceles triangle. Every regular polygon with n sides is formed by n isosceles
triangles. In a regular hexagon, the triangles are equilateral. The height of these isosceles triangles is
called the apothem. By the Isosceles Triangle Theorem, the apothem is the perpendicular bisector of the
side of the regular polygon. The apothem is also the height, or altitude of the isosceles triangle.
From this information and the picture above, we can derive a formula for the area of a regular polygon.
The area of each triangle is: A a = \bh = \sa^ where s is the length of a side and a is the apothem.
If there are n sides in the regular polygon, then it is made up of n congruent triangles.
/I \ 1
A = nA/\ = n \-sa = -nsa
\2 / 2
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In this formula we can also substitute the perimeter formula, P = ns, for n and s.
1 1
A = -nsa = -Pa
2 2
Area of a Regular Polygon: If there are n sides with length s in a regular polygon and a is the apothem,
then A = ^asn or A = \aP, where P is the perimeter.
Extension Example: Find the area of the regular polygon with radius 4.
Solution: In this problem we need to find the apothem and the length of the side before we can find the
area of the entire polygon. Each central angle for a regular pentagon is ^9- = 72°. So, half of that, to
make a right triangle with the apothem, is 36°. We need to use sine and cosine.
sin 36°
.5n
= T
4 sin 36°
1
8 sin 36°
= n
n
*4.7
cos 36° = -
4
4 cos 36° = a
a « 3.24
Using these two pieces of information, we can now find the area. A = | (3.24) (5) (4.7) ~ 38.07 units 2 .
Extension Example: The area of a regular hexagon is 54 V3 and the perimeter is 36. Find the length of
the sides and the apothem.
Solution: Plug in what you know into both the area and the perimeter formulas to solve for the length
of a side and the apothem.
P = sn
36 = 6s
s = 6
A = -aP
2
54V3 = ^(36)
54V3 = 18a
3V3 = a
Extension Problems
Find the area and perimeter of each of the following regular polygons. Round your answer to the nearest
hundredth.
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1.
2.
6.
7. If the perimeter of a regular decagon is 65, what is the length of each side?
8. A regular polygon has a perimeter of 132 and the sides are 11 units long. How many sides does the
polygon have?
9. The area of a regular pentagon is 440.44 in 2 and the perimeter is 80 in. Find the length of the
apothem of the pentagon.
10. The area of a regular octagon is 695.3 cm 2 and the sides are 12 cm. What is the length of the
apothem?
Answers
1.
2.
A = 384 V3
P = 96
A = 8V2
P = 6.12
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3.
4.
5.
6.
A = 68.26
A = 72
A = 688.19
P= 100
A = 73.47
P = 15.45
A = 68.26
P = 63
7. 6.5
8. 12
9. a = 11.01
10. a = 14.49
Circumference and Arc Length
Pi Extension
Pi is a fascinating number to students. Have a competition in class to see who can memorize the most
digits of 7i. Students can go to the website in the text to see more digits.
March 14^ can also be considered "Pi Day" because it is 3-14. You can celebrate this however you want,
from having students bring in different kinds of pie to doing a project about n. See the Problem Solving
FlexBook for project ideas.
Connections to Playtime
Hula Hoops are a great example of circumference. Have students measure with a string (much like Investi-
gation 10-1) the outline of the hula hoop and then lie of out straight to find its length. Using the formula
for circumference, find the circumference in this way too. Have students compare their measured answers
to the circumference. See if students can count how many times they can spin the hula hoop around their
body and then determine that measurement. For example, the average hula hoop has a circumference of
407T in. « 125.66 in. If a student was to spin the hoop 15 times around themselves, that would be a total
length of 1884.96 in. or 157 ft.
Challenge
1. Mario's Pizza Palace offers a stuffed crust pizza in three sizes (diameter length) for the indicated prices:
The Little Cheese, 8 in, $7.00
The Big Cheese, 10 in, $9.00
The Cheese Monster, 12 in, $12.00
What is the crust (in) to price ($) ratio for each of these pizzas? Michael thinks the cheesy crust is the
best part of the pizza and wants to get the most crust for his money. Which pizza should he buy?
2. Jay is decorating a cake for a friend's birthday. They want to put gumdrops around the edge of the
cake which has a 12 in diameter. Each gumdrop is has a diameter of 1.25 cm. To the nearest gumdrop,
how many will they need?
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3. A spedometer in a car measures the distance travelled by counting the rotations of the tires. The
number of rotations required to travel one tenth of a mile in a particular vehicle is approximately 9.34. To
the nearest inch, find the diameter of the wheel. (1 mile = 5280 feet)
4. Bob wants to put new weather stripping around a semicircular window above his door. The base of the
window (diameter) is 36 inches. How much weather stripping does he need?
5. Each car on a Ferris wheel travels 942.5 ft during the 10 rotations of each ride. How high is each car at
the highest point of each rotation?
Answers
1. The Little Cheese, 3.59:1; The Big Cheese, 3.49:1; The Cheese Monster, 3.14:1; Michael should buy The
Little Cheese
2. 31 gumdrops
3. 18 in
4. 93 in
5. 30 ft
Area of Circles and Sectors
Challenge
Find the area of the shaded region. Round your answer to the nearest hundredth.
1. The quadrilateral is a square.
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6. The area of a sector of a circle is 547T and its arc length is Qti. Find the radius of the circle.
7. The area of a sector of a circle is 23047T and its arc length is 32n. Find the central angle of the sector.
Answers
1. 292.25
2. 13.73
3. 21.21
4. 98.17
5. 24.65
6. 18 units
7. 40°
3.11 Surface Area and Volume
Exploring Solids
Extension: Geometric Probability
We are going to connect the Platonic Solids to probability. A six sided die is the shape of a cube. The
probability of any one side landing face up is 1/6 because each of the six faces is congruent to each other.
We can also relate the other Platonic Solids to probability in this way, however, keep in mind that they
have a different number of faces, and therefore different odds.
1. A cube-shaped die has sides numbered 1 through 6. Determine the probability of:
(a) Rolling an even number.
(b) Rolling a factor of 6.
(c) Rolling a prime number.
2. What shape would we make a die with 12 faces? If we number these faces 1 to 12, and each face has
the same likelihood of landing face up, what is the probability of rolling a multiple of three?
3. I have a die that is a regular octahedron. Each face is labeled with a consecutive prime number
starting with 2. What is the largest prime number on my die?
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226
4. Rebecca wants to design a new die. She wants it to have one red face. The other faces will be yellow,
blue or green. How many faces should her die have and how many of each color does it need so that:
the probability of rolling yellow is eight times the probability of rolling red, the probability of rolling
green is half the probability of rolling yellow and the probability of rolling blue is seven times the
probability of rolling red?
Answers
1. (a) 1/2
(b) 2/3
(c) 2/3
2. regular dodecahedron, 1/3
3. 19
4. 1 red face, 8 yellow faces, 7 blue faces and 4 green faces
Extension: Isometric Views
Isometric views of a solid is similar to perspective drawing. Typically, isometric views are drawn on special
dot paper that looks like this:
We usually draw figures that can be made from cubes on isometric paper. Students can experiment with
the dot paper and draw different views of stairs or prisms.
• m •
• # • •
XrO:
To have students draw something in an isometric view, you will need to give them the map plan, which is
the overhead view.
Extension Example: Draw the isometric view of the map plan below.
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front
Solution: When drawing an isometric view, the front is drawn to the left. The drawing would look
something like this:
Extension Problems
Draw the isometric views of the following map plans.
1.
1
2
4
2
3
fron
1
t
4
1
1
1
2
2
1
front
Answers
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1.
2.
• •
Surface Area of Prisms and Cylinders
Extension: Finding the Surface Area of a Regular Polygon Prism
Regular polygon prisms have the same strategy towards finding the surface area as any other prism. The
catch here is to remember how to find the area of the regular pentagon. See the Chapter 10 Enrichment
for a recap on how to do this.
Extension Example: Find the surface area of the regular pentagonal prism.
Solution: For this prism, each lateral face has an area of 160 units 2 . Then, find the area of the regular
pentagonal bases. Recall that the area of a regular polygon is \ asn. s = 8 and n = 5, so we need to find
a, the apothem.
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Extension Problems
tan 36° = -
a
4
a = « 5.51
tan 36°
SA = 5(160) + 2 (^ ■ 5.51 • 8 • 5) = 1020.4
Find the surface area of the following prisms. All polygons are regular, unless otherwise specified.
1.
10 cm
20 cm
Sin
24 in
Answers
1. 1719.62 cm 2
2. 1809.14 in 2
Challenge
1. An open top box is made by cutting out 2 in by 2 in squares from the corners of a large square piece of
cardboard. Using the picture as a guide, find an expression for the surface area of the box. If the surface
area is 609 m 2 , find the length of x. Remember, there is no top.
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230
X
2 ;
■ 2
: x-4 :
2. Find an expression for the surface area of a cylinder in which the ratio of the height to the diameter is
2:1. If x is the diameter, use your expression to find x if the surface area is 1607T.
Answers
1. x 2 - 16 in 2 ,x = 25 in
2. |jc 2 tt,x = 8
Surface Area of Pyramids and Cones
Extension: Find the Surface Area of a Regular Right Pyramid
See the Chapter 10 Enrichment for a recap on how to find the area of a regular polygon.
Extension Example: Find the area of the regular hexagonal pyramid below.
Solution: To find the area of the base, we need to find the apothem. If the base edges are 10 units, then
the apothem is 5 V3 for a regular hexagon. The area of the base is \ asn = \ (5 V3j (10) (6) = 150 V3. The
total surface area is:
SA = 150V3+-(6)(10)(22)
150 V3 + 660 « 919.81 units 2
Extension Problems
Find the surface area of the regular right pyramids. Round answers to the nearest hundredth.
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1.
4 in
Answers
1. 750.44 in 2
2. 41.57 in 2
Challenge
For questions 1-4, consider the sector of a circle with radius 25 cm and arc length 14n.
1. What is the central angle of this sector?
2. If this sector is rolled into a cone, what are the radius and area of the base of the cone?
3. What is the height of this cone?
4. What is the total surface are of the cone?
For questions 5-7, consider a square with diagonal length 10 V2 in.
5. What is the length of a side of the square?
6. If this square is the base of a right pyramid with height 12, what is the slant height of the pyramid?
7. What is the surface area of the pyramid?
Answers
1. 100.8°
2. 7
3. 24
4. 175tt
5. 10 in
6. 13 in
7. 360 in 2
Volume of Prisms and Cylinders
Extension: Volume of Regular Prisms
Extension Example: Find the volume of the regular hexagonal prism below.
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Solution: Recall that a regular hexagon is divided up into six equilateral triangles. The height of one of
those triangles would be the apothem. If each side is 6, then half of that is 3 and half of an equilateral
triangle is a 30-60-90 triangle. Therefore, the apothem is going to be 3 v3. The area of the base is:
And the volume will be:
Extension Problems
B = - (3 V3) (6) (6) = 54 V3 units 2
V = Bh = (54 V3) (15) = 810 V3 units 3
Find the volume of the following prisms. All polygons are regular, unless otherwise specified.
1.
2.
10 cm
20 cm
8 in
24 in
Answers
1. 5196.21 in 3
2. 5581.69 in 3
Challenge
1. The volume of a cylinder with height to radius ratio of 4:1 is 1087T cm 3 . Find the radius and height
of the cylinder.
2. The length of a side of the base of a hexagonal prism is 8 cm and its volume is 1056 v3 cm 3 . Find
the height of the prism.
3. A cylinder fits tightly inside a rectangular prism with dimensions in the ratio 5:5:7 and volume
1400 in 3 . Find the volume of the space inside the prism that is not contained in the cylinder.
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Answers
1. r = 3 cm,h = 12 cm
2. 11 cm
3. 300.44 m 3
Volume of Pyramids and Cones
Connections to Construction
Teepees are cones. Native Americans would construct teepees as living spaces. Have students construct a
teepee (on paper) that is tall enough for them to stand up inside of. Should their height be the height of
the cone? Point out to students that the height of the teepee should be more than their own height. If
not, students would only be able to stand in the center of the teepee.
Extension: Find the Volume of a Regular Right Pyramid
Extension Example: Find the volume of the regular hexagonal pyramid below.
28 in
16 in
Solution: To find the area of the base, we need to find the apothem. If the base edges are 16 units, then
the apothem is 8 V3 for a regular hexagon. The area of the base is \ asn = | (8 V3J (16) (6) = 384 V3. The
total volume is:
V= \Bh= ^-384 V3-28
3 3
= 3584 V3 « 6207.67 units 3
Extension Problems
Find the surface area of the regular right pyramids. Round answers to the nearest hundredth.
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1.
2.
20 in
Answers
1. 1080 V3~ 1870.61 in 3
2. 6400 V3~ 11,085.13 in 3
Challenge
1. The ratio of the height to radius in a cone is 3:2. If the volume is 1087T m 3 , find the height and radius
of the cone.
2. The ratio of the height to the radius in a cone is 4:3. If the volume is 25927T ft 3 , find the height and
radius of the cone.
3. Derive a generalized formula for the volume of a regular hexagonal pyramid. Write the apothem in
terms of the length of a side.
Answers
1. h = 9m,r = 6m
2. h = 24ft.,r= 18 ft.
3. n = 6,B= \ asn,a= ^s -> V = \Bh = ± U • -^ s • s • 6jh = ^s 2 h
Surface Area and Volume of Spheres
Connections to Sports
One way to find the formula for the surface area of a sphere is to look at a baseball. We can best approximate
the cover of the baseball by 4 circles. The area of a circle is 7ir 2 , so the surface area of a sphere is Anr 2 .
While the covers of a baseball are not four perfect circles, they are stretched and skewed.
^/ ^circle
circle"7> g (back)
/">£ § ^circle
circle
(back)
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To take this investigation further, have students work in pairs or groups and given each group a baseball.
First, they need to measure the circumference around the great circle and record this number. Then, let
them take the baseball apart on the seams and lay the covering flat. Give the students graph paper and
have them trace the covering pieces onto the paper. The two pieces of the covering should be congruent,
so students really only need to trace one piece. Let students count the squares that are within the traced
outline to approximate the surface area. Don't forget to multiply this number by two (depending on
the graph paper, students may also need to divide by 16 because there are usually 4 squares per inch or
16 squares per square inch). Now, have students find the actual surface area of a baseball. Using the
measurement of the great circle from the beginning of the activity, students need to find the surface area.
Students should round their answer to the nearest hundredth and compare with what they found earlier.
Extension: Derivation of the Volume of a Sphere
A sphere can be thought of as a regular polyhedron with an infinite number of congruentregular polygon
faces. As n, the number of faces increases to an infinite number, the figure approaches becoming a sphere. So
a sphere can be thought of as a polyhedron with an infinite number faces. Each of those faces is the base
of a pyramid whose vertex is located at the center of the sphere. Each of the pyramids that make up the
sphere would be congruent to the pyramid shown. The volume of this pyramid is given by V = ^ Bh.
To find the volume of the sphere, you need to add up the volumes of an infinite number of infinitely small
pyramids.
V(all pyramids) = Vi + V 2 + V 3 -\ \-V n
= -{B x h + B 2 h + B 3 h + • • • + B n h)
o
= -h(B 1 + B 2 +Bz + --- + B n )
The sum of all of the bases of the pyramids is the surface area of the sphere. Since you know that the
surface area of the sphere is 47rr 2 , you can substitute this quantity into the equation above.
= l -h{Anr 2 )
In the picture above, we can see that the height of each pyramid is the radius, so h = r.
4
yjir'
4
= -r(nr )
3 V ;
= -7ZT 3
3
Challenge
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1. One hot day at a fair you buy yourself a snow cone. The height of the cone shaped container is 5 in and
its radius is 2 in. The shaved ice is perfectly rounded on top forming a hemisphere. What is the volume
of the ice in your frozen treat? If the ice melts at a rate of 2 err? per minute, how long do you have to eat
your treat before it all melts? Give your answer to the nearest minute.
2. (a) What is the surface area of a cylinder?
(b) Adjust your answer from part a for the case where r = h.
(c) What is the surface area of a sphere?
(d) What is the relationship between your answers to parts b and c? Can you explain this?
For problems 3-5, use the fact that the earth's radius is approximately 4,000 miles.
3. Find the length of the equator.
4. Find the surface area of earth, rounding your answer to the nearest million square miles.
5. Find the volume of the earth, rounding your answer to the nearest billion cubic miles.
Answers
1. 127T cm 3 , 19 minutes
2. a. SA = 2jir 2 + 2nrh
b. SA = 4nr 2
c. SA = 4tzt 2
d. They are the same. Think back to the explanation for the formula for the surface area of a sphere using
the baseball-it is really the sum of the area of four circles. For the cylinder, the SA is the sum of the areas
of the two circular bases and the lateral area. The lateral area is 27rr/z, when we replace h with r this part
of the formula becomes the area of two more circles. That makes the total surface area of the cylinder
equal to the area of four circles, just like the sphere.
3. 25,132.74 miles
4. 201 million square miles
5. 268 billion cubic miles
Extension: Exploring Similar Solids
Connections to Art
A kaleidocycle is a chain of rotating tetrahedra that connect to form a ring. To investigate this further,
see the website: www.mathematische-basteleien.de/kaleidocycles.htm. This website provides pictures and
directions of how to make different kaleidocycles and is a great way for students to see how similar solids
can be combines together. Some of the solids are congruent and some are similar. Become familiar with
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some of the patterns and designs before assigning this to the students.
Give students instructions, by printing or providing websites and let them go to work. Students need to
select two different kaleidocycles to create or design their own unique ones. This activity might be best
done in pairs.
Connections to Science
Animal A and animal B are similar (meaning the size and shape of their bones and bodies are similar) and
the strength of their respective bones are proportional to the cross sectional area of their bones. Answer
the following questions given that the ratio of the height of animal A to the height of animal B is 3:5. You
may assume the lengths of their bones are in the same ratio.
1. Find the ratio of the strengths of the bones. How much stronger are the bones in animal B?
2. If their weights are proportional to their volumes, find the ratio of their weights.
3. Which animal has a skeleton more capable of supporting its own weight? Explain.
Answers
1. 9:25, about 2.78 times as strong
2. 27:125
3. Animal A, Animal B's weight is about 4.63 times the weight of animal A but his bones are only 2.78
times as strong.
3.12 Rigid Transformations
Exploring Symmetry
Connections to Art
Let students search through magazines for a picture of an object that they feel comfortable drawing and
has reflectional symmetry. Once they find a picture they like (students can also bring in a picture as long
as it is ok to cut it), have students cut the picture on the line of symmetry. Then, tape one half on a blank
piece of paper and have students draw the symmetrical half without referring to the part they cut off. As
an extension, students can do this with a picture of their face.
Connections to Science
Identical twins are mirror images of each other. If you know of any students that are twins at your school,
ask if they would come in to demonstrate. The left side of one twin's face will be the exact mirror image
of the other twin's right side and vice versa. If there are no twins at your school, find pictures of twins
online, copy the images and cut their faces vertically along the line of symmetry. Then switch the sides of
the faces and see if they are actually mirror images.
Extension: Point Symmetry
When an object is symmetric about a point, we say that it has point symmetry. Point symmetry is the
same as having exactly 180° rotational symmetry. Most playing cards have point symmetry. Have students
look through a deck of cards to see if they can determine how many cards have point symmetry. The odd
numbered cards do not have point symmetry. See if students can determine why not.
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point
IN^f^ symmetry
Challenge
1. Can an object have point symmetry and reflectional symmetry? If so, draw an example.
2. Point symmetry in the coordinate plane is called origin symmetry, which means that an object is
symmetrical about (0, 0). A line segment has endpoint (4, 2). What must the other endpoint be so
that the line segment has origin symmetry?
Answers
1. Yes, it is possible. A square has point and reflectional symmetry.
2. The line segment must pass through the origin and be 4 units to the left and 2 units down from the
origin. The other endpoint must be (-4, -2).
Translations
Extension: Vectors
Another way to write a translation rule is to use vectors. A vector is a quantity that has direction and
size. In the graph below, the line from A to Z?, or the distance travelled, is the vector. This vector would
be labeled AB because A is the initial point and B is the terminal point. The terminal point always
has the arrow pointing towards it and has the half-arrow over it in the label.
terminal point
The component form of AB combines the horizontal distance travelled and the vertical distance travelled.
We write the component form of AB as (3, 7) because AB travels 3 units to the right and 7 units up. Notice
the brackets are pointed, (), not curved.
Extension Example: Name the vector and write its component form.
a)
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b)
Solution:
a) The vector is DC. From the initial point D to terminal point C, you would move 6 units to the left and
4 units up. The component form of DC is (-6,4).
b) The vector is EF. The component form of EF is (4, 1).
Extension Example: Draw the vector ST with component form (2,-5).
Solution: The graph to the left is the vector ST. From the initial point S it moves down 5 units and to
the right 2 units.
Extension Problems
For questions 1-3, name each vector and find its component form.
1.
3.
M
For questions 4-6, plot and correctly label each vector.
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240
4. AB = (4, -3)
5. CD = (-6,8)
6. FF = (-2,0)
7. The coordinates of ADFF are D(4,-2),F(7,-4) and F(5,3). Translate ADFF using the vector <5, 11)
and find the coordinates of AD'E'F' .
For problems 8 and 9, write the translation rule as a translation vector.
8. (x,y)^(x-3,y + 8)
9. (je,y)-»(je + 9,y-12)
For problems 10 and 11, write the translation vector as a translation rule.
10. (-7,2)
11. (11,25)
Answers
1. G# = (6,3)
2. £/ = (-2,4)
3. LAf = <3,-l>
4.
t*X
5.
iP
H
^^*
m r\
r d
F
7. D'(9,-9),F'(12,7),F'(10,14)
8. (-3,8)
9. (9,-12)
10. (x,y)->(x-7,y + 2)
11. (x,y)^(x+ll,y + 25)
Reflections
Connections to Mirrors
Give each student a small mirror. Have them place the mirror in front of their writing hand and hold it
with the other hand. Then, see if they can write their name so that it can be read in the mirror. Students
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should look only at their hand in the mirror. Once they are done, students can then look at their actual
handwriting. How does this version of their name look different than normal?
Coordinate Plane Challenge
The vertices of AGHI are G(l, l),if(5, 1), and 7(5,4). Use this information to answer questions 1-7.
1. Plot AGHI on the coordinate plane.
2. Reflect AGHI over the x-axis. Find the coordinates of AG'HT .
3. Reflect AG'HT over the y-axis. Find the coordinates of AG"H"I" .
4. What one transformation would be the same as this double reflection?
5. Following the steps to reflect a triangle using a compass and straightedge.
a. Make a triangle on a piece of paper. Label the vertices A, B and C.
b. Make a line next to your triangle (this will be your line of reflection).
c. Construct perpendiculars from each vertex of your triangle through the line of reflection.
d. Use your compass to mark off points on the other side of the line that are the same distance from the
line as the original A, B and C. Label the points A', Z?' and C".
e. Connect the new points to make the image AA'B'C.
6. Describe the relationship between the line of reflection and the segments connecting the pre-image and
image points.
7. Repeat the steps from #5 with a line of reflection that passes through the triangle.
Answers
1-3.
4. A rotation of 180°.
5.
6. Perpendicular Bisector
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7.
Rotations
Connections to Playtime
Several carnival and amusement park rides use rotation. For example, the Tea Cups and Disneyland rotate
around the floor of the ride and individually. Other examples could be a Ferris wheel or the Yo-Yo (the
rotating swing), Have students search online for rotating amusement park rides. Then, they should focus
on 2 or 3 rides. Students should turn in a short paragraph for each ride that consists of the point of
rotation, how the ride fits into the mathematical definition, and a description of the ride. Students need
to include a picture of the ride with their paragraph.
Construction Challenge
1. Draw two lines that intersect, m and n, and AABC. Reflect AABC over line m to make AA'B'C . Reflect
AA'B'C over line n to get AA"B"C". Make sure AABC does not intersect either line.
2. Draw segments from the intersection point of lines m and n to A and A" . Measure the angle between
these segments. This is the angle of rotation between AABC and AA"B n C" .
3. Measure the angle between lines m and n. Make sure it is the angle which contains AA'B'C in the
interior of the angle.
4. What is the relationship between the angle of rotation and the angle between the two lines of reflection?
Answers
1-3.
4. Angle of rotation is double the angle between the lines.
Composition of Transformations
Extension: More Double Reflections
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Through the following examples, we determine what single rotation is double reflections in the coordinate
plane.
Extension Example: Reflect the square over y = jc, followed by a reflection over the x-axis.
Solution: First, reflect the square over y — x. The answer is the red square in the graph to the right.
Second, reflect the red square over the x-axis. The answer is the green square below.
irf^
2 4 6 8 10
Extension Example: Determine the one rotation that is the same as the double reflection from the
previous Example.
y = x
Solution: First, we need to figure out what the angle of intersection is between y = x and the x-axis.
y = x is halfway between the two axes, which are perpendicular, so is 45° from the x-axis. Therefore, the
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angle of rotation is 90° clockwise or 270° counterclockwise. The correct answer is 270° counterclockwise
because we always measure angle of rotation in the coordinate plane in a counterclockwise direction. From
the diagram, we could have also said the two lines are 135° apart, which is supplementary to 45°.
Therefore, a reflection over y = x followed by a reflection over the x-axis is the same as a rotation of 270°
counterclockwise.
Extension Problems
1. What one rotation is the same as a reflection over the y-axis, followed by a reflection over y = -xl
2. What one rotation is the same as a reflection over the x-axis, followed by a reflection over y = xl
3. What one rotation is the same as a reflection over y = jc, followed by a reflection over y = -xl
Answers
1. 90° counterclockwise
2. 90° counterclockwise
3. 180° rotation
Extension: Tessellating Polygons
Connections to Nature
In this activity, students will create a piece of art based on a honeycomb. They can use any shape that
will tessellate as they saw with the honeycomb. The key is that the honeycomb, according to a Wikipedia
definition, is a space filling or close packing of polyhedral or higher - dimensional cells, so that there are
no gaps. Students can use any size that they choose and can incorporate color too.
Connections to Art
Have students study the work of M.C. Escher who was famous for his tessellations, www.en.wikipedia.org/wiki/M.
C._Escher. They can begin with the Wikipedia site, but there are so many other sites to work with as
well. Have the students select one piece of his work as a favorite piece and share in small groups the elements
that tessellate and how they tessellate.
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Chapter 4
Basic Geometry TE -
Differentiated Instruction
Introduction
In this Teacher's Edition, we will help the educator adapt our Basic Geometry FlexBook for all learners.
Differentiated Instruction involves providing students with different avenues to acquiring content, involving
processing, constructing, and making sense of ideas. This text develops teaching materials so that all
students within a classroom can learn effectively, regardless of differences in ability.
Each lesson is split up by the type of learner: visual, kinesthetic, or auditory. Not every lesson will have a
suggestion for every type of learner. There are also modifications for the English language learner (ELL)
and the learning disabled student.
Several students who claim they do not like math really enjoy creative writing. Every couple of lessons
there is an alternate journal entry that can be used as a warm-up. These journey entries are somewhat
math related, but really draw on the creativity of the student. Have students keep these entries in with
their Review Queues and collect all warm-ups at the end of each chapter.
The low-achieving or basic level student needs all the incentives you are willing to give in your classroom.
One suggestion is to grade everything. Give students a participation grade, worth 5-10% of their total
grade. Have a checklist of student names and for each day give them points based on completing notes, the
Review Queue, investigations, and certain class work. Each day is worth 2 points; 2 = they did everything,
1 = did half, = did nothing. Even though it does not seem like a lot, the points will add up. This is a
great incentive for the basic level or EL student who has a hard time understanding math, but are good
kids who want to try. Then, occasionally collect class work and give a separate grade for these assignments.
You can also collect the Review Queues and journal entries at the end of each chapter and give students
a grade or extra credit toward the test.
Collaborative learning can be a very rewarding experience for basic students as well. Allow your class room
to be "flexible." At the beginning of class and during note-taking, organize the desks in rows. Then, when
students are doing group work, allow them to rotate their desks into groups of four. Change seats every
1-2 chapters. On the first day of new groups, have students practice going from rows to groups and back
to make the process more efficient.
Groups can be picked a number of different ways. Three suggestions are: one student from each grade
achievement: 1 A, 1-2 B/C, 1-2 D/F, the students may pick their own groups (this should be used sparingly
and as a reward for students), or picked totally at random by a grade program or Excel. However you
pick the groups, the grading should be uniform. One way to grade group work is to staple all the work
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together and grade one student's work out of each packet; the second paper in each group, for example.
Another option would be to grade everyone's paper and the "group grade" would be the average of the
member's scores. Both methods will encourage students to double-check everyone's work and make sure
that everyone has the same answers, so that everyone gets the best grade. Explain these repercussions to
the class. You can decide if you would like to tell the class which grading method you are going to use
before they begin working or after you have collected the assignment.
Testing can also be a cause of high anxiety for all levels of students. This text will present alternative
assessments as a final chapter. There are options for group testing, projects, alternative set-ups for quizzes
and tests, as well as alternative grading rubrics.
4.1 Basics of Geometry
Points, Lines and Planes
Journal Entry
If math were an animal, what would it be? Explain your answer in 1-2 sentences and draw a picture.
These journal entries are intended to make math more approachable and (hopefully) fun. You can decide
whether or not to have students share with the class upon completion. This is a fun activity for any student
at the beginning of a school year. Give students construction paper and markers to illustrate their animal.
Some examples to get them going: A spider because it is scary, a worm because it wriggles around in my
brain, a unicorn because it is mythical, and a butterfly because while it is beautiful it is also complex
(these are all actual student responses as seen by the author).
Visual Learners
One way to assist visual learners with this lesson is to use the actual objects mentioned in the lesson. This
will assist students with special needs in making a connection with the material.
When discussing the postulates, use physical models to demonstrate what each one says. For example, with
Postulate 1-2, draw three dots on a piece of paper. Then, for Postulate 1-3, connect two of those points
to form a line. For Postulate 1-5, use two pieces of paper (possibly cutting one so they slide together) to
show that they intersect to form a line.
Kinesthetic Learners
Allow move time so that students can walk around the classroom identifying points, lines and planes in
their surroundings. Request that students make a list of the things that they find. After a few minutes,
come back together and write the examples students find on the whiteboard.
The Know What? for this lesson discusses wooden blocks. If possible, it might be helpful for students to
use blocks and identify the geometric terms learned in this lesson. You could also have students use their
prior knowledge by asking them which other geometric terms they know and then finding the appropriate
block.
Collaborative Learning
Have students work in pairs or small groups to discuss their findings from the "walk around" activity. This
engages students who need to talk about their work to gain a better understanding of a concept.
Special Needs/Modifications
Be sure that all of the vocabulary and postulates are written on a board or overhead as they are presented
and discussed. Request that students copy this information into a notebook. Reading the terms, hearing
them discussed, seeing them written again and writing the words themselves assists students in retaining
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information. At the end of the lesson, make sure each student has all the vocabulary and postulates
written in their notes. They should have the definitions and a picture for each term. Give students either
participation points or a small class work grade for their notes as an added incentive so that each student
writes notes. If you do not want to check notes every day, you could also collect them at the end of each
chapter and look through the work while students are taking the test.
Segments and Distance
Visual/Auditory Learners
An alternative way to introduce the Segment Addition Postulate is to draw a line segment with a point on
the line on the white board or overhead projector.
B
Then, set AB = 5 and BC = 15. Ask students what the value of AC is. Once a student says the correct
answer, ask them how they came up with that answer. Then, ask students if they can write an algebraic
equation for the line segment without numbers. This should lead students towards the Segment Addition
Postulate.
Kinesthetic Learners
As an extension for Examples 1 and 2, have students work in pairs. One member in the pair draws a
line segment that s/he has measured to find the distance. The other member also draws a line segment
that he/she has measured. Then they switch drawings. Student A must figure out the length of student
Z?'s line segment, and student B must figure out the length of student A's line segment. Repeat this with
both inches and centimeters. You could even have students measure the same segment in both units and
compare.
After Examples 10 and 11, hand out grid paper. Ask students to draw a coordinate plane and provide
given distances on the board/overhead. Then allow the student time to draw a line segment with this
distance. Provide a time for sharing/feedback from the exercise.
Modifications
Write all vocabulary and names of postulates on the board as it is brought up in the lesson. Leave this
list on the board throughout the lesson so that students can reference and become comfortable with the
new vocabulary.
Be sure that students are given plenty of time to think through their work and be sure that all students
have finished examples before going over the answers. Sometimes, special needs students require more time
to complete tasks and will stop working if the answers to a particular question are given before they have
finished.
Observe students as they work in groups. Make a note of student that may need extra help and set aside
a time to check in with each one. To help you with this task, create an observation checklist of things to
watch for when students are working in a group or individually. This should be done throughout the year.
Angles and Measurement
All Learners
After Example 1, have students work in small groups. Assign one group rays and the other group angles.
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Using rulers, the students need to design a series of either rays or angles. You can use index cards for this
activity. Then have the groups switch cards. The angle group needs to name all of the rays that the other
group has drawn. The ray group needs to name all of the angles that the angle group has drawn. Then
the groups exchange answers and check each other's work. This involves peer discussion and tutoring.
Kinesthetic Learners
The angle formed at a person's elbow is a useful physical model of angles. Have the students put their
arm straight out, illustrating a straight angle. Then have the student gradually turn their arm up (or
down) gradually to demonstrate how the degree changes. Use several students as examples to show that
the length of the forearm and bicep do not change the angle measurement.
After completing Example 7, provide students with drawings of several different angles so that they can
practice measuring more angles. Make sure to have several types of angles. After finding all the measure-
ments, have students classify each angle as acute, right, obtuse, or straight.
With the investigations in this lesson, let students play with the protractor and compass. Most students
have seen these tools, but never used them. After going over Investigation 1-2, tell students to draw 3-4
angles on the back of the handout from above. Then, after going over Investigation 1-3, have students copy
these 3-4 angles. Once these angles are copied by students, walk around and give them a participation
grade and answer questions.
Auditory Learners
For Investigations 1-2 and 1-3, explain each step as you complete it. Answer questions as you go over this
activity. These investigations may take a little longer than you expect to complete, but if you do not take
the time to answer questions now, students will have even more questions the next time they need to do
either of these tasks.
Special Needs/Modifications
Leave the terms on the board from the previous class and add the new terms from this lesson. As the
chapter progresses, continue to add new terms. At the end of the chapter, students should double-check
that they have all the definitions and theorems that are on the board.
Rather than giving students a large class work assignment, break it up. Have students do a few problems
after Investigation 1-2, come back and then do a few more after Investigation 1-3, then a few more after
going over the Angle Addition Postulate.
Midpoints and Bisectors
Journal Entry
What does it mean to bisect something? Can you think of any examples in real life? Write 2-3 sentences.
Visual Learners/ELL
We can differentiate this lesson by organizing the content into a table. This should be done as part of
a class discussion, not ahead of time and then presented. Creating the chart is meant to be interactive.
Since this lesson works with line segments and angles, we can use these as the two columns of our table.
Here is a sample of a table and how to organize it for the students.
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Table 4.1:
Line Segments
Angles
Congruent
(draw a picture with labels and markings)
Equality
(write congruence statement vs. equality)
Segment Midpoint (add to picture above)
Segment Midpoint Postulate
Congruent
(draw a picture with labels and markings)
Equality
(write congruence statement vs. equality)
Angle Bisector (add to picture above)
Angle Bisector Postulate
Be sure to explain each concept and how they are different and similar depending on whether you are
working with line segments or angles. For the Segment Midpoint and Angle Bisector Postulates, write
each out symbolically. The Segment Midpoint Postulate can be written, "For AB there is one point M,
such that AM = MBT Use the same labels that were used in the picture you drew. Especially for EL
Students, use as little language as possible. Rely on the symbols to get the concept across. As an alternate
method to solve Example 3, you can use the slope (triangle) between B and M to and duplicate it on the
other side of M to find A.
Example 3, Alternate Solving Method: Either plot M and B or find the horizontal distance and
vertical distance between the two points. Using the picture, from B to M, we move to the left 4 and up
5. Therefore, we would repeat those distances from M to find A. This means we would need to subtract 4
from the x-value and add 5 to the y-value of M.
(3-4,-1 + 5) =A(-1, 4)
Notice that we get the same answer with this alternate method. Encourage students to use which ever
method they are most comfortable with for these types of problems. You might only want to show this
method to students that are having difficulty with the method used in the text, to avoid confusion.
Be sure to show students the animations listed in the text of the constructions of Investigations 1-4 and 1-5.
Both of these animations are very helpful for visual learners and it is recommended that you show them
to the class at least twice after going over the investigations. The more students can see each construction
(and any construction), the more comfortable they will be with using these tools.
Kinesthetic Learner
When introducing the Midpoint Formula, tell students the order of the coordinates does not matter. They
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can decide which point they would like to put first. To demonstrate this, redo Example 2 with the second
point first.
Example 2, Alternate Solving Method:
/-5 + 9 14 -2\ /4 12
H •¥)="•«>
You could also have half the class do the example (before showing them the answer) using the first point
first and the other half can do the example using the second point first in the Midpoint Formula.
Auditory Learner
Before Example 4, discuss the students' journal entries from the beginning of class. This will help them
to understand the definition of a segment bisector and perpendicular bisector.
Angle Pairs
Visual Leaner
When introducing the concept of a linear pair, ask students what a straight angle is and if they remember
how many degrees it has. Draw one on the board. Then, draw a ray from the vertex of this right angle
and explain that this straight angle is now a linear pair. This will help students visualize the definition
and that the two angles are supplementary.
Before doing Example 5, you may need to do a few easier examples first, so students can get used to seeing
linear pairs and the relationship.
Additional Example: Find mlDBC.
A
Solution: lABD and iDBC are a linear pair, so they are supplementary. Set up an equation.
mlDBC + 128° = 180°
mlDBC = 52°
You may need to do several examples like the one above before students have fully internalized the concept.
Visual/Kinesthetic/EL Leaners
After going over all the material in the lesson, have students create a Venn diagram that compares the
different angle relationships in this lesson. Have students work in groups and pass out a blank diagram. Do
two separate diagrams; one for complementary vs. supplementary and a second for linear pair vs. vertical
angles. Remind students that the things that are the same go in the overlapping portion of the two circles.
Once they are done, draw two large Venn diagrams on the board and have a "reporter" from each group
tell you what they came up with. The successive groups only need to add what is not on the board. Decide
whether you want students to copy these into their notes.
Modifications
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As with any of the investigations, you can decide how to use them in your classroom. Suggestions are:
have the students do them individually or in groups (as you walk around and answer questions), you do
them and students can follow along by also doing the activity or by taking notes, or you can demonstrate
the investigation while the students observe.
To help students, you can take a picture of the above Venn diagrams and either put it on your website (if
you have one), or make copies and pass them out in the following class.
Classifying Polygons
Auditory/Kinesthetic Leaner
After defining the different classifications of triangles, ask students the following questions:
1. Can a scalene triangle be
(a) right?
(b) acute?
(c) obtuse?
2. Can an isosceles triangle be
(a) right?
(b) acute
(c) obtuse?
3. Can an equilateral triangle be
(a) right?
(b) acute
(c) obtuse?
Ask students to draw a picture of the possible combinations and an explanation as to why certain combi-
nations will not work. Discuss the answers as a class.
Visual/Auditory Learner
An alternative definition to a concave polygon is: A polygon where at least one diagonal crosses outside
the figure. Explain this definition by drawing a picture.
Both figures above are hexagons, but the one on the right has three diagonals that cross outside the figure
(in red). That makes this hexagon concave, and the one on the left is convex. Notice that both hexagons
will still have the same number of diagonals.
Kinesthetic Learner/Collaborative Learning
First, tell students to draw either four concave polygons or four convex polygons and they cannot tell
anyone else what they have chosen (this should be done individually). When finished, the students form
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groups and exchange papers. The must show, by drawing lines, whether the figures they have been given
are concave or convex.
Modifications
Provide students with diagrams of a triangle, quadrilateral, pentagon, and other polygons. Have the vertices
labeled and sides labeled. Be sure that students understand where to find the interior angles because they
will need to do this to classify triangles. Help students draw in diagonals for the larger polygons. You
can also copy the table in the text and give that to students as an aid (under the Classifying Polygons
sub-header) .
Continue to write all vocabulary words on the board. It is up to you to leave the words up during the test
or not.
4.2 Reasoning and Proof
Inductive Reasoning
Visual/Kinesthetic Learners
Have students work in pairs. Each student draws a visual pattern (Examples 1-3) and exchanges papers
with their partner. Have each student write down the next two steps and explain the rule of the pattern
on a separate piece of paper. Then, have each pair exchange with another pair of students and repeat.
Each student should look at three unique patterns this way. This process can be repeated with number
patterns as well. When writing out the rule for the number pattern, encourage students to try and find
the equation.
Auditory Learner
When going over a counterexample, use a few statements that students might be familiar with. See if they
can figure out an example that would make the statement false.
• All integers are positive or negative (zero would be the counterexample).
• I only use my calculator in math class (let students be creative with these counterexamples; paying
bills could be one possibility).
• A number that is divisible by 3 is also divisible by 9 (6, 12, 15, etc. are counterexamples).
• If a number is divisible by 2 and 3, it is also divisible by 12. (6 and 18 are counterexamples).
Read these statements aloud to students and see if they can come up with the counterexamples. The first
statement only has one counterexample, but the other three have infinitely many. See if students can come
up with several counterexamples for these.
Modifications
Continue to write vocabulary on the board throughout this chapter. Erase the vocabulary from the first
chapter.
Give students a 100 x 100 grid for the locker problem (Know What?) so they can fill it out and find the
pattern.
Give students a handout with the false statements above (for finding the counterexample). Then, they can
write down 1-2 counterexamples and turn in the handout as a class work assignment. You can also put
several visual and number patterns on this handout such that students can find the next 2-3 terms and
describe the rule.
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Conditional Statements
Journal Entry
How do you use inductive reasoning in your day-to-day life? Give and explain 2-3 examples.
When students are finished with their journal entries have a discussion about their answers and reasoning.
Kinesthetic/Auditory Learners
Working in pairs, have each student create a conditional statement that is not math related. It can be
silly, but should be a true statement. Have the partners switch papers and then they must write down the
converse, inverse, and contrapositive. Then, let students determine which of these three statements are
true. Do this activity after Example 4.
ELL Students
Encourage ELL students to use the symbolic notation when writing the converse, inverse, and contrapos-
itive. This is a very language-intensive chapter and anytime students can use symbols will help them to
understand the concepts better. For example, you could introduce the symbolic notation at the beginning
of the lesson and then students can set the hypotheses and conclusions equal to p and q. Encourage students
to find p and g, then they can write "If /?, then qT Then, when the converse, inverse, and contrapositive
are introduced, students will have an easier time translating the original statement.
There are also words that mean the same thing as "if" in conditional statements, such as "when." Tell
students that "if = when." Also, whatever is after the "if" is the hypothesis, even when "if" is in the middle
of the statement (see Statement 2). "If and only if" can also be replaced with "iff." All students might like
this shortcut so they do not have to write as much.
Visual Learner
Students might get caught up in the symbolic notation of statements. Explain to students they can write
the statements two different ways. Encourage students to use whichever representation they are most
comfortable with.
Original Statement: If/?, then q. p — > q
Converse: If g, then p. q — > p
Inverse: If not /?, then not q. ~ p — >~ q
Contrapositive: If not q 1 then not p. ~ q — >~ p
Biconditional: p if and only if q. p <-> q
Modifications
This is a difficult lesson for special needs students to understand because the language and symbols are so
verbal. Many students with language based learning disabilities will find this challenging. You could alter
these definitions and provide an example (in words not symbols) for each term. For example:
Converse:
Switch the hypothesis and the conclusion of the conditional statement.
Inverse:
Add "nots" to the hypothesis and conclusion to negate the original statement.
Contrapositive:
Add "nots" to the hypothesis and conclusion to negate the converse.
Switch the hypothesis and conclusion and negate both parts.
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Biconditional Statement:
Combine the conditional statement and its converse together.
Deductive Reasoning
Visual/Auditory Learners
Write the "Door A" and "Door B" statements on the board (from the Know What?). Show this to students
at the beginning of the lesson and let the students read each statement for a few minutes. Then, have one
student read each door aloud and have a discussion as to why they think the tiger is behind which door.
This is a great introductory activity that allows students to use deductive reasoning. Students need to
figure out which statement is true.
Ask students what it means to be "logical." See if they can generate a definition on their own that is close
to the one in the text. In this lesson, we introduce the concept of a "logical argument." Arguments can
be sound, or logical, but not necessarily true. This will confuse students. Here are similar examples of a
logical argument and an illogical one. Present each to your students and see if they can determine which
one is which.
Additional Example: Of the two arguments below, one is logical and the other is illogical. See if you
can determine which one is which.
// it is foggy in San Francisco, then you cannot see the Golden Gate Bridge.
I carft see the Bridge. Therefore, it is foggy in San Francisco.
If it is foggy in San Francisco, then you cannot see the Golden Gate Bridge.
I can see the Bridge. Therefore, it is not foggy in San Francisco.
Solution: Use the symbol notation to determine what type of logic or statement is being used.
// p, then ~ q. If p, then ~ q.
~ q. Therefore, p. q. Therefore, ~ p.
The first statement's (blue) conclusion is in the form ~ q — > /?, which is the converse of the original
statement. We know that the converse is false, so this is not a logical conclusion. The second statement's
(red) conclusion is in the form q — >~ p. Here, p and q are switch and they are also both negated, which
means this is the contrapositive. We know that the contrapositive is the logical equivalent of the original
statement, so this is a logical conclusion.
Be careful to explain that even though the red conclusion is logical, it might not be true because the
original statement might not be true. Of course, if it is foggy around the Golden Gate Bridge, we probably
would not be able to see it. However, it could be foggy in a different neighborhood or you could be driving
on the Bridge. In both of these cases, our original statement is false. See if students can determine other
cases when the original statement could be false.
As with the previous two lessons in this chapter, this lesson is also very verbal. Some students might have
trouble with difference between inductive and deductive reasoning. First, inductive reasoning does not
have all these Laws of logic (Detachment, Contrapositive, and Syllogism). So, if students see these types
of statements, they should know that the logic is deductive. Secondly, stress that inductive reasoning uses
patterns. So, any number pattern, shape pattern, or patterns in nature or observation will be inductive
reasoning.
Modifications
Students might want alternative definitions to the Laws of logic.
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Law of Detachment:
Assume a given statement is true. If you are told the hypothesis of this statement, then you can conclude
that the conclusion must be true.
Law of Contrapositive:
Assume a given statement is true. If you are told the negated conclusion of this statement, then you can
conclude that the negated hypothesis must be true.
Law of Syllogism:
You are given at least two true conditional statements that tie together such that the conclusion of the
first is the hypothesis of the second. If you are told the hypothesis of the first statement, you can conclude
that the conclusion of the last statement must be true.
Algebraic and Congruence Properties
Visual Learners
Review the definition of equality and write all the Properties of Equality on one half of the board. Then,
write the statements of congruence on the other half of the board. Show students how to combine these two
together visually. Use different colored chalk or pens (on a whiteboard) to illustrate how combining these
statements together can help to prove the given statement. Practice this with a few examples. Encourage
class participation. Example 5 uses both congruence and equality.
Kinesthetic Learners/Collaborative Learning
There are 12 properties of equality and congruence. After going over these properties, divide students into
12 groups and assign each a property. Give each group a large, poster-sized piece of paper and markers. On
their paper, they need to clearly state the property and draw and give two examples using line segments,
angles or algebra. Let them use the examples in the text to get started. At the end of the activity, allow
groups to quickly present their property to the class. Place these posters around the room. You can decide
if you would like to leave them up for the test or not.
Modifications
If you have many special needs students in the class, you may want to break up this lesson over two days.
Day One: Review properties and statements of congruence. Review basics of geometry.
Day Two: Show students how to use the two and introduce the two-column proof.
Here are some steps for combining and using properties of equality vs. congruence.
1. Look at whether you are working with line segments or angles. This will help you choose a property
of congruence. If the statement is an equation of equality, students will use an equality property.
2. Choose a property that explains the given statement.
3. Students might wonder how to go "back and forth" between equality and congruence. Reasons for
this could be "If the angles are congruent/equal, then they are equal/congruent" or the definition of
equality/congruence.
When introducing proof, go very slowly. Students with special needs will have a very difficult time with
proofs. Be prepared to go over Examples 4 and 5 at least two times each. Either give each student a
print-out of the steps for proofs in the text or discuss them in the notes. Encourage students to have this
list out while they are preforming class work or homework problems. As an added modification, all the
proofs in this text are fill-in-the-blank. A few extra examples in the Teaching Tips FlexBook are placing
the Reasons in the correct order.
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Proofs about Angle Pairs and Segments
Visual Learners
In this lesson we introduce the shortcut to labeling angles. The reason being, we did not want students to
get confused and only use one letter in the wrong instances. You can decide if you want to introduce this
shortcut at all. However, it is recommended that students know this option because it is commonly used
in standardized testing.
Encourage students to use symbols or abbreviations where ever possible when doing proofs. Agree upon
these abbreviations and/or symbols as a class to avoid confusion.
Kinesthetic Learner
To illustrate the Same Angle Supplements (or Complements) Theorem, make several angles of various
measures on construction paper or overhead transparencies and give one to each student. Make sure to
make a few of your angles equal, some supplementary and some complementary. Draw a straight angle (or
right angle) on the front board or overhead and place an angle (100°, for example) on this vertex. Then,
have students come up to the board and place their angle on the vertex of the straight angle. Continue
in this manner until a supplement is found (for example, if the angle is 100°, the supplement is 80°).
Once the supplement is found, remove it and place it somewhere else on the board. A second supplement
(another 80° angle) should also be in a student's possession. At this point ask students to see if they have
a congruent angle to this angle. This student will come up and place their angle on the board. Then, ask
students to draw a conclusion about the two supplemental angles. Repeat this activity with a right angle
and complements.
Angles to give to students (draw these on transparencies or construction paper, with or without measure-
ments):
40° 140° 130° 50°
40° 140° 120° 60°
20° 110° 10° 80°
20° 110° 30° 150°
45° 135° 45° 135°
160° 170° 125° 90°
Angles to draw on the board: 50°, 70°, 45° Draw these in a right angle for complements.
40°, 70°, 135° Draw these in a straight angle for supplements.
Take the proof of #1 in the Review Questions and remove all the statements and reasons so that it is
blank. Write each statement and reason on a separate piece of paper and give each student one (see the
Solution Key FlexBook). There will be 10 statements and 10 reasons. Do not tell students which one
they have. Write the blank proof, with given, prove, and diagram on the board. Starting with the Given
statement, have students place their statements or reasons on the board in the correct order. Encourage
students to work together and have an open discussion about what goes next. This activity can be done
for any proof.
An alternative to the previous proof activity is to give groups of students an envelope with all the statements
and reasons (all written on separate pieces of paper) for a proof. Write the Given, Prove, and diagram
on the front of the envelope. Then, have groups work together to put the statements and reasons in the
correct order. Once groups are done, they can switch envelopes until every group has done all the proofs.
Modifications
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Show students how to draw a diagram to illustrate a given statement. A picture often helps special needs
students. If no picture is given, always encourage students to draw their own.
Review the basic definitions of right angles, supplementary angles, complementary angles, and vertical
angles and draw an example of each on the board/overhead. Also review that a postulate or property does
not need to be proven, but a theorem does.
Explain to students how to move from the basic definition of each angle relationship to the theorem.
Write each of the theorems on the board/overhead and request that students copy this information in their
notebooks.
Be sure that the students have a current list of postulates, properties and vocabulary where they can access
it easily.
When grading proofs, be lenient on the reasoning at first. If students describe the theorem or property,
that should be ok. Also, with proofs involving right angles, students might have a difficult time identifying
which theorem or definition is the correct reason.
4.3 Parallel and Perpendicular Lines
Lines and Angles
Visual Learners/ELL
Continue to write the vocabulary on the board. In this chapter there are new symbols and labels for
parallel and perpendicular lines. Make sure to include these in your definitions. You could also write a
chart, like the one below for these definitions.
Table 4.2:
Word
Definition
Label
Mark it in a Picture
Parallel
Perpendicular
Two lines that are in the AB\\CD
same plane and never
intersect.
< — > < — >
Two lines that intersect AB _L CD
to form a 90° angle.
You can have students continue this chart for definitions for the rest of the chapter.
Visual/Kinesthetic Learners
Have students walk around the room (or school) and make a list of the places where they see parallel,
perpendicular and skew lines. They should draw pictures of the location. If students have access to a
digital camera, they could also take pictures of the real-life examples. Students could work in pairs on this
activity.
Investigations 3-1, 3-2, and 3-3 are excellent visual representations of the Parallel Line and Perpendicular
Line Postulates. If you do not have access to patty paper, wax paper works very well. Just make sure to
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precut squares for each student. Encourage students to draw a line that has a different orientation than
their neighbor. When they are done with each investigation, have students compare their drawings to the
people around them.
When working with transversals, use color to indicate the different angles in a diagram. Use color for
definitions too. This will help students to keep things clear and organized.
Modifications
Decide if you want the investigations to be individual, pair, or teacher-led activities. To help students, you
could give them a handout with Step 1 already completed (the point and line drawn for them) for each
investigation.
Before defining transversals, review the meaning of the following words: adjacent, vertical, interior, exterior,
corresponding and consecutive. Draw pictures to aid in the discussion of transversals.
You could also give students a chart for the different types of angles formed by lines and transversals.
Leave the last three columns blank and fill out as a class. For the last column, have students generate
their own definitions for each angle relationship.
Table 4.3:
Name of Relation- Picture Number of Lines Relationship in
ship Needed Words (Definition)
Vertical Angles
Linear Pair
Corresponding Angles
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Table 4.3: (continued)
Name of Relation- Picture Number of Lines Relationship in
ship Needed Words (Definition)
Alternate Interior An-
gles
Alternate Exterior An-
gles
Same Side Interior An-
gles
Properties of Parallel Lines
Journal Entry
Where do you see parallel lines in real life? Why do you think they are important to day-to-day activities
or certain jobs?
Collaborative Learning/ Auditory Learners
Divide students into groups of four. Assign each of the four students a different topic-Corresponding Angles
Postulate, Alternate Interior Angles Theorem, Alternate Exterior Angles Theorem, Consecutive Interior
Angles Theorem.
Let the students know that their assignment is to prepare a lesson and teach the other students in the
group about their topic. They can use their notes, a diagram, real life examples, a poem, a song, whatever
they would like to make the topic clear. The other students in the groups will let the "teacher" know what
he/she did well and also offer suggestions to improve the presentation. You can decide whether or not the
peers should grade each "teacher" or you do. If you have the students grade each other, give them a clear
rubric with 3-5 criteria and a grade of 1-5 for each. Then, have the students add up the criterion for a
final score.
Visual Learners
For the Algebra Connection examples (Examples 5 and 7), students might need more explanation than
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what is given. To help students, add in the following red steps.
(4x-10)°
= 58°
+10°
+ 10°
4x
= 681
4
4
X
= 17°
(Sy + 53)° =
= (7y - 55
-3y + 55° -
- 3y + 55°
108°
= 4y
4
4
27°
= y
Example 11 introduces a fourth line so that there are two sets of parallel lines. It might be helpful for
students if you put this example on the overhead projector. Place a second transparency over the example
and trace one angle, Z2 for example, and move it over all the angles that it is congruent to. You could
repeat this with Zl for the other angles as well.
Throughout this lesson, it might be helpful to show students different sets of parallel lines cut by transversals
so that students become familiar with several different orientations. As a warm-up activity, you could place
two or three different versions on the board and ask students to find different angle pairs.
Modifications
Review the angles formed by a transversal from the previous lesson.
For Investigation 3-4, give students Steps 1 and 2 on a handout, already draw for them. You can ask them
if the lines are parallel, how do they know, and then find all the angle measures. This will shorten the
investigation, as well as eliminate a lot of questions about how to correctly draw the lines.
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Proving Lines Parallel
Auditory/Visual Learner
As a part of going over the Review Queue, ensure that students remember what a converse is and how to
determine if it is true or not.
Visual/Kinesthetic Learner
Draw two different pictures of the board/overhead, one with parallel lines cut by a transversal, and a
second where the lines are not parallel. Do not mark the first set of lines parallel. Then, fill in sets of
corresponding angles. For the first set of lines, tell students that the corresponding angles are congruent.
Then, see if they can arrive at the correct conclusion. For the second set of lines, ask students if they
think the corresponding angles are congruent. When they arrive at the correct answer, then ask if they
feel those lines are parallel. This will lead to the Converse of the Corresponding Angles Postulate.
This activity can be repeated for the other angle relationships.
Modifications
Review what a conditional statement is and how to write the converse of a conditional statement.
Practice writing converse statements from conditional statements by using real life examples.
Investigation 3-5 is a very powerful construction, but can also be quite challenging for students. You might
want to make this a teacher-led investigation to ease students. Another option is to give students the
picture of completed Steps 1 and 2. Then, they just have to copy the angle at A onto point B.
To make proofs easier for students, you can give them a Statements/Reasons bank, where all the correct
answers are in a box, along with some extras. This way they can pick which answers they want to go and
at what step. This is a nice option to help students start working with proofs. Eventually, the goal would
be to have students complete a proof without the aid of a bank.
Allow students to shorten the names of theorems and properties to make it easier for them to write. You
could also give students a complete list of all the theorems, postulates, and properties for each chapter.
You can decide if you want to let students use the list for tests or quizzes.
Review the Transitive Property before introducing the Parallel Lines Property.
Properties of Perpendicular Lines
Auditory/Kinesthetic Learners
Before defining and going over what congruent linear pairs are, ask students if they can figure out what
they are by the name alone. Have an open discussion and allow students to come up to the board to draw
pictures. At the end of this discussion, fill in any holes that students might have missed.
When discussing perpendicular transversals, have students refer to Review Queue #1. See if students can
come up with Theorems 3-1 and 3-2 on their own as part of a class discussion.
Visual Learners
It might be difficult for students to infer the conclusions of Example s 5-7 without a little help. For Example
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5, remind students that the two angles, 55° and 35°, make up the vertical angle with Zl. Therefore, because
they add up to 90°, mil must be 90° as well. You could make a similar argument with Example 6. Example
7 is the converse of Examples 5 and 6. Students need to determine if / and m are perpendicular. In this
example, you could encourage students to find all the other angle measures. If they find that the three
other angles are 90° (and they should), then they should realize that / and m are perpendicular.
Modifications
Review the following vocabulary prior to beginning the lesson: congruent, perpendicular, complementary,
supplementary, vertical angles, adjacent angles, and linear pair.
Give students a handout with all the pictures and directions for each example. This will give students
an instant place to write notes. Then, you can go over the examples as a class and students can fill in
the solutions and add any other important information. You can decide to give students a grade for their
notes, whether it be for participation or class work.
Parallel and Perpendicular Lines in the Coordinate Plane
Visual/Kinesthetic Learners
After Example 4, students may need additional review of finding the equation of a line. You can use
Examples 1 and 2. Once students have found the slopes between the points, they can also find the
equations of the lines.
Example 1 Continuation: Find the equation of the line between (2, 2) and (4, 6).
Solution: We found that the slope was 2. Plug this into the equation of a line, y = mx + /?, for m. Use
either point for x and y in the equation. You could have half the class choose the first point and the other
half use the second point. Then, reconvene and compare to make sure everyone got the correct answer.
Point (2, 2)
y = 2x + b
2 = 2(2)+Z?
2 = 4 + Z?
-2 = b
Point (4, 6)
y = 2x + b
6 = 2(4)+/?
6 = 8 + Z?
-2 = b
Using either point, we see that the y-intercept is 2. The equation of the line is y = 2x - 2.
Example 2 Continuation: Find the equation of the line between (-8, 3) and (2, -2).
Solution: We found that the slope was -^. Again, have students use both points and compare.
Point (-8, 3)
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y
3
--x + b
4h
+ b
3 = 4 + b
-l = b
Point (2, -2)
y = --x + b
-2 = ~(2)+b
-2 = -l + b
-l = b
Using either point, we see that the y-intercept is -1. The equation of the line is y — -|;
Visual Learners
1.
Another approach to Examples 1-4, would be to plot the points on the board or overhead (on a grid/graph
of some kind) before going over the equation for slope. Have students draw horizontal and vertical lines
from the points to create a slope triangle. Here are Examples 1 and 2 with this alternative approach.
Example 1, Alternative Approach: What is the slope of the line through (2, 2) and (4, 6)?
Solution: Draw a slope triangle between the two points. Then, count the length of the horizontal and
vertical sides.
4
^
(4,6h»rt
(2,2;
§
move up
(positive)
4 units
move to the right
(positive) 2 units
^
=
After counting the sides, make a fraction. We say "rise over run," so the "rise" or the y-distance will be
in the numerator and the "run" will be in the denominator.
m
4
2
Example 2, Alternative Approach: Find the slope between (-8, 3) and (2, -2).
Solution: Draw a slope triangle between the two points. Then, count the length of the horizontal and
vertical sides.
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264
^
Zuj-A ^^—
_1
K,
move down
(negative)
5 units 1
\
s
"■^
L
■
-j
-1
I *
^.
2
.
.
"X
s
-2)
\T
^*
^k
move to the right
(positive) 10 units
J^
After counting the sides, make a fraction. We say "rise over run," so the "rise" or the y-distance will be
in the numerator and the "run" will be in the denominator.
__ -5 _ _1
m ~ To ~~ ~2
When counting the sides, start at the first point. And, always reduce all fractions.
When you get to Example 3, students will see that there is no vertical leg (or y-direction) and cannot
create a slope triangle. Because we cannot rise, this is a horizontal line and will only pass through the
y-axis, making the equation y = b. The opposite occurs with Example 4. There will be no run, so the line
will only pass through the x-axis, making the equation x = b.
With Example 5, you can encourage students to graph the line and plot (9, -5) on a piece of graph paper.
Then, they can use the slope from the line, start at (9, -5) and create the slope triangle (fall 1, run 3).
Doing this, we find that a second point on the line is (12, -6). You can also show students that they can
go backwards by rising 1 and running in the negative direction 3. This would give them a point of (6, -4).
Have students connect these points (see picture). Where this line passes through the y-axis will be the
y-intercept. After going over this approach, show students the algebraic approach.
If you applied this graphical approach to Example 7, the graph would look like the one to the right. Notice
that students would have to draw a very large y-axis in order to find the y-intercept in this way.
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Auditory Learners
The graph to the right lends itself to a discussion regarding the two different methods presented: graphical
vs. algebraic. After going over the algebraic approach to Example 7, ask students which method they
think is "easier." If the parallel or perpendicular line falls within the usual-sized graph (-10 < x < 10, -10 <
y < 10), they will probably say that the graphical approach is easier. However, it is difficult to know where
the y-intercept will be before we draw the graph. Also, some students might realize that the y-intercept
will not always be an integer. This leads to estimation on a graph, which means they might not always
get the correct answer. Therefore, even though the algebraic method might not always be the "easiest," it
is the most consistent and they will always be able to find the correct answer.
Kinesthetic Learners
Some students may have a hard time visualizing the reciprocal of a fraction or number to find the per-
pendicular slope. You can also show students that the product of the perpendicular slopes will always be
-1. This way, students can set up an equation and slope for the perpendicular slope. Using Example 6a,
students can solve the equation, 2 • m ± = -1. Solving for m_ Ll we get -\.
Modifications
Consider splitting this lesson into 2 or 3 days, depending on the comprehension level of the students.
If several students need assistance, consider allowing them to work in pairs.
The Distance Formula
Visual Learners
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266
Rather than just plugging in the points to the Distance Formula, students can use the slope triangle and
the Pythagorean Theorem (which they have probably seen in a previous math class). Here is an alternate
solution to Example 1.
Example 1 Alternate Approach: Find the distance between (4, -2) and (-10, 3).
Solution: Draw the slope triangle and find the lengths of the horizontal and vertical sides.
The hypotenuse of the right triangle is the length between the two points. Using the Pythagorean Theorem,
we have:
5 2 + 14 2 = c 2
25 + 196 = c 2
221 = c 2
^T2\ = c
f
1 n
£h
3)
S ^>
ta ^.
2
13- (-2)| = |5|
5 units
-"H
*-■*.
^-
-
B
o
-
*
-
\ "
->
^
2
■i
"-N.
i
kf
\
|-10
1
-4|
4 |M
- I-
lits
ihi
Explain to students that this approach is the Distance Formula. This technique can also be applied to
Example 2.
Visual/Kinesthetic Learners
For Examples 5 and 6, students might get confused by Step 3. Rather than count down from the y-intercept,
draw a perpendicular line (using a ruler) from the y-intercept. Then, find this point of intersection.
Continue with Step 4.
V
So
ft _i
U A
(0,-e
A ^S
^
(4
t ™
^
V:
^
I
Modifi
cati
Y
ons
U)
The subsection titled "Shortest Distance between Parallel Lines with m = 1 or -1" can be skipped if you
feel your students are struggling with the content of this section. This subsection is not a part of many
states' standards. Make sure to double-check your state's standards before omitting this subsection.
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4.4 Triangles and Congruence
Triangle Sums
Visual/Kinesthetic Learner
As a step before Step 2 in Investigation 4-1, have students measure all three angle in their triangles from
Step 1. Then, ask students to add up the three measurements and see what they get. Possibly discuss
why their measurements may not add up to 180° (human error). Then, review a straight angle and what
it adds up to and move on to Step 2.
There are two corollaries in this lesson: Each angle in an equiangular triangle is 60° and The acute angles
in a right triangle are supplementary (Examples 2 and 3). While it is not explicitly stated that these are
supplemental theorems, students should be aware of these properties of these specific triangles. For both
of these examples, have students draw their own equiangular and right triangles and see if the corollaries
work for their triangles.
Either have students draw the picture below or give them a handout.
Have students measure the interior angles and double-check that they add up to 180°. Then, ask them
about the vertical angles to each interior angle. Ask what those three angles add up to (also 180°). Then,
show students the exterior angles. It might be hard for students to determine which exterior angle is a part
of which set. You can cup your hand around one angle and then rotate around triangle in a clockwise (or
counterclockwise for the second set) motion to show students that all angles in one set of exterior angles
open in the same direction.
Color-coating the angles will help students keep all the relationships straight.
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Modifications
Give students a handout for Step 1 for Investigation 4-1.
Show the additional steps for Examples 9 and 10. Additional steps are in red.
Example 9
(8x - 1)° + (3x + 9)° + (3x + 4)° = 180°
(8x + 3x + 3x)° + (-1 + 9 + 4)° =180°
(14x+12)° = 180°
-12° -12°
14x = 168°
14 14
x=12°
Example 10
(4x + 2)° + (2x - 9)° = (5x + 13)°
(4x + 2x)° + (2 - 9)° = (5x + 13)°
(6x-7)° = (5x+13)°
-5x + 7° - 5x + 7°
x = 20°
Congruent Figures
Visual Learners
Review what it is to be congruent vs. equal. Show students two congruent line segments, congruent angles
and then introduce two congruent triangles. In this lesson, we do not introduce any shortcuts so that
students can get used to the fact that there are six sets of congruent pieces that make up two congruent
triangles.
Copy the triangles from the definition of Congruent Triangles onto an overhead transparency. Cut the
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triangles out to visually show students that the two triangles are congruent. This can also be repeated
with the triangles from Example 1.
Additional Example: Draw AC AT = ADOG (after Example 3).
Solution: These two triangles can be drawn in several different ways. Here is one possible answer.
Students need to make sure that the corresponding sides and angles match up when labeled. The use of
color is very helpful for students to visualize corresponding angles.
Kinesthetic Learners
Before introducing the Third Angle Theorem, have students draw two different congruent triangles (or,
to modify this, give them a handout with two congruent triangles on it, without any markings). Ask
students to measure two angles in the first triangle. Using the Triangle Sum Theorem, have students find
the measure of the third angle. Then, have students measure two angles in the second triangle. Before
finding the measure of the third angle, see if the students can infer what the measure of the third angle
will be. This can lead to a discussion of the Third Angle Theorem.
Modifications
Allow students to have a list of the theorems, postulates, and properties that will be useful for this chapter
and triangle proofs.
Give students handouts of triangles or other figures any time the directions say to "draw" something. This
will be easier on you and ensure accuracy when students are doing self-discovery. To take this a step
further, you can also give students handout for each day's notes. You can list the important theorems and
vocabulary. Then, as you go over the notes, students fill in the information.
Triangle Congruence using SSS and SAS
Auditory/Visual Learners
Use the Know What? as a discussion point to see if the two triangles are congruent. Ask students what
they think. Have students volunteer to represent the fridge, sink and stove in each house and place them
the appropriate distances apart to represent each kitchen. Rotate the second kitchen in the picture (the
neighbor's kitchen) so that the 3 ft. and 2.5 ft. sides match up with the first kitchen. Then, ask students
if the two kitchens are congruent. Students should see that they are not. As an extension, ask students if
they feel the angle between the 3 ft. and 2.5 ft. sides in each triangle are congruent or not. (These angles
are not congruent because the opposite sides are not congruent.)
Visual/Kinesthetic Learners
Rather than using a traditional two-column proof, students can use a paragraph proof or a flow chart
proof.
Example 2, Alternate Approach: Write a paragraph proof to show that the two triangles are congruent.
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Given: AB = DE
C is the midpoint of AE and DB.
Prove: AACB = AECD
Solution: A paragraph proof is a written out version of a two-column proof.
We are given that AB = DE and C is the midpoint of AE and BD. From the definition of a midpoint, we
can conclude that AC = CE and DC = CB. By the SSS Postulate, we can conclude that AACB = AECD.
Example 2, Alternate Approach: Write a paragraph proof to show that the two triangles are congruent.
Given: AB = DE
C is the midpoint of AE and DB.
Prove: AACB = AECD
Solution: A flow chart proof is a more visual representation of a two-column proof. Start with one given
and allow each step to "flow" from one to the next. Each reason is written below the statement.
C is \he midpoint
ofAEand DB
given
AC = CE
definition of a
midpoint
DC = CB
definition of a
midpoint
AB = DE
AACB =AECD
SSS Congruence
Postulate
given
After Example 6 you can do the following activity as an additional example.
1. Each student draws a triangle on the coordinate grid and the passes his/her paper to the right.
2. The next student takes the triangle and uses the distance formula to figure out the lengths of each
side of the triangle. Then he/she passes the paper to the right.
3. The next student takes the measurements and draws a triangle congruent to the first triangle some-
where on the coordinate grid. Then he/she passes the paper to the right. Using slope triangles might
also be helpful here.
4. The final student checks the work of all of the others.
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Modifications
Give students a handout with a 5 in. line segment drawn on it for Step 1 of Investigations 4-2 and 4-3.
You will have to give students one handout for each investigation. On this handout, you can also put 4 in.
and 3 in. segments for students to use as a guide for the other steps.
For both of these investigations, you can also do either as a teacher lead activity and encourage students
to follow along.
Make sure to review the Distance Formula before Example 5.
Triangle Congruence using ASA, AAS, and HL
Visual Learners
When going over the Review Queue, students might have a hard time visualizing the congruent triangles.
Consider separating the two triangles to show that DB = DB by the Reflexive Property.
B
B
Point out to students that when the triangles are separated, DB is repeated. Explain that students must
use the Reflexive Property in this instance. You could even take it one step further and say that anytime
a side or angle is repeated in both triangles, the reason will always be the Reflexive Property.
You can also use this separation technique for Example 2 and 5. Here is the picture for Example 2.
Auditory Learner
After going over ASA and A AS, do Example 5. After completing Example 5, ask students if they think they
could have completed this proof using ASA. Complete the alternate proof as a class. The only difference
is that there would be an added step using the Third Angle Theorem and then the reason why the two
triangles are congruent would be ASA. Discuss why this is also an acceptable proof for Example 5.
Modifications
Give students a handout with a 5 in. line segment drawn on it for Step 1 of Investigations 4-4. You could
also make this a teacher-led investigation and encourage students to follow along.
Give students a copy of the Recap Table for their notes.
At this point, students should be familiar with proofs, but probably not proficient. They should be able
to fill out Step 1 (the Given), but may have a hard time coming up with the reasons. If you sense students
struggling, give them a word bank for each problem in the Review Questions. For example, with #27, the
word bank could be: Definition of a midpoint, Definition of perpendicular lines, Definition of right angles,
SAS, CPCTC, iSTW = lUTV. At first, only give students the necessary answers. In the next section, try
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expanding the word bank to include possibilities that will not be used.
Isosceles and Equilateral Triangles
Auditory/Visual Learners
Prior to teaching the lesson, ask students to recall information about isosceles and equilateral triangles.
Ask them to make a list of the characteristics of each in their notebooks. When finished, use a class
discussion to generate a list of characteristics for both isosceles and equilateral triangles on the board.
Visual/Kinesthetic Learners
After going over the parts of an isosceles triangle, either have students draw an isosceles triangle in their
notes or give them a handout of one. Then, as a class, have students construct the angle bisector of the
vertex angle. Ask students what they know about this triangle (including congruent legs and base angles)
so far. Mark these on the triangle. Then, have students find any other properties of the angle bisector.
Ask students to measure the lengths of the two segments that the angle bisector splits the base into (they
should be equal). Then, measure the angles at that point of intersection (should be 90°). From this,
students should realize that the angle bisector of the vertex angle is also the perpendicular bisector of the
base. You are leading students to the Isosceles Triangle Theorem. Once the theorem has been stated, ask
is if students think the converse is also true (yes). Also, ask if they think this is valid at any other angle in
the isosceles triangle. As a modification, you can do this as a teacher-led activity and have students write
down the important information in their notes.
After students do Investigation 4-6, have them draw perpendicular bisectors for each side in the triangle.
Ask them if the Isosceles Triangle Theorem applies to equilateral triangles. Students should see that it
does and that it applies to every angle in an equilateral triangle.
Modifications
You can give students a handout with a 3 in segment drawn 6 in down the page for Investigation 4-5 and
then have them refer back to Investigation 4-2 (SSS Construction). You can also do this investigation as
a teacher-led activity while students follow along.
Give students pictures/handout of an isosceles triangle and an equilateral triangle. They can label each
triangle with their parts (leg, base angles, etc).
As with Investigation 4-5, you can give students a handout with a 2 in. segment, instead of having them
draw Step 1. Also, this investigation can be teacher-led.
Show the additional algebraic steps for Examples 4 and 5.
4.5 Relationships with Triangles
Midsegments
Visual/Kinesthetic Learner
Give each student a blank piece of paper, ruler and compass. Instruct each student to construct a 3-4-5 right
triangle using Investigation 4-2. After the triangle is constructed, have them each find the midpoint of each
side. Once all the midpoints are marked, students should connect two midpoints to create a midsegment.
Have students measure the midsegment and the side that it does not intersect. See if students can arrive
at the Midsegment Theorem. Then, have them create the other midsegments to see if the same holds true
for all the midsegments in a triangle.
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Kinesthetic Learner
Have students complete Examples 3-5 by finding the slopes of all the sides, midsegments, and all the
lengths. Only do this if you feel that students are still struggling with the properties of a midsegment
and/or if students need review on slope and the distance formula.
Slope of NL~ ^^~ ^~ l
-8-4
-2
-12
NL
^
4) 2 + (3-5) 2
V144 + 4 = Vl48 = 2V37
Slope of LM = \^ = f = 2
Slope of PO = 4^# = Eg = g
LM= V(4-(-2)) 2 + (5-(-7)) 2
= V36 + 144 = Vl80 = 6V5
PO= A /("5-l) 2 + (-2-(-l)) 2
= V36 + 1 = V37
Slope of QP =
-2-4 _ ^6 _ o
-5-(-2) _ -3 _ Z
QP= V(-5-(-2)) 2 + (-2-4) 2
= V9T36= V45 = 3V5
Visual Learner/ELL
This chapter has a lot of new vocabulary and parts of triangles. Have students make a graphic organizer
to help them with the organization. You can either have students draw their own chart or give them a
blank one to fill out as the chapter progresses.
Table 4.4:
Name
Picture (with Properties
markings)
Pass through
the opposite
vertex?
Passes through
the midpoint?
Do they inter-
sect?
Midsegment
Perpendicular
Bisector
Angle Bisector
Median
Altitude
Modifications
Give students handouts of the 3-4-5 right triangle for the activity above. Students can also benefit from
copies of the chart above, possibly with triangles drawn in the "Picture" column.
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Provide students with graph paper for any notes involving coordinate geometry and homework.
Perpendicular Bisectors and Angle Bisectors in Triangles
Visual/Kinesthetic Learners
After doing Investigation 5-2, give students two more pieces of patty paper. Have them repeat this inves-
tigation with an obtuse triangle and a right triangle. If done correctly, the circumcenter for the obtuse
triangle will be on the outside of the triangle. The circumcenter for the right triangle will be on the
hypotenuse.
If students need an additional construction of perpendicular bisectors or angle bisectors in triangles, see
the Enrichment FlexBook for this chapter and lesson. There are traditional straightedge and compass
constructions there.
Auditory/Visual Learners
After going over Investigation 5-4, see if students can reason if point of intersection of the angle bisectors
(the incenter) is ever outside of the triangle. Ask them if they could inscribe a circle if the point was
outside the circle. The answer here is no. In order to inscribe any circle in a polygon, the center of the
circle must be inside the polygon. Therefore, the incenter must always be inside the triangle. You can also
do Investigation 5-4 with obtuse and right triangles to show that this is the case.
Visual Learner/ELL
Don't forget to add perpendicular and angle bisector to the chart created in the previous section.
Modifications
Review how to construct the perpendicular bisector and angle bisector. Then, show how these can be placed
into a triangle. If you or students prefer, you can do the patty paper construction for the perpendicular
bisector or angle bisector. For both of these patty paper constructions, you would only use Step 2 of
Investigations 5-2 and 5-4. However, in Step 1, only draw a line segment for the perpendicular bisector
patty paper construction. For the angle bisector, only draw one angle.
We intentionally left out the terms points of concurrency, circumcenter, and incenter to avoid confusion
for students. If you feel your students can handle this, see the Enrichment Supplemental FlexBook.
Allow students to do flowchart proofs for the fill in the blank proofs in the review questions.
Medians and Altitudes in Triangles
Journal Entry
Where have you heard the words median and altitude? Think of any setting, including math, where you
have heard these words.
Auditory Learners
Begin class with a discussion of the journal entry. Most students should remember that the median of a
set of numbers in the "middle number." Students may also know the term median from freeway driving.
In this case, the median is the middle divider between the two directions of traffic. In both cases, median
refers to the middle. Stress this point with students.
As for the altitude, hopefully students come up with the height of a plane when it's flying, or the height of
a mountain. Remind students that, when measuring altitude, the height is always measured from sea level.
Here, altitude refers to height. When defining these terms in triangles, hopefully this prior knowledge will
help them connect to the definitions.
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Visual Learners
Students may need an additional example for the properties of a median.
Additional Example: If F is the centroid, find DF, FC, and AF.
Solution: When it comes to these types of problems, sometimes students forget common sense. Remind
them that the shorter piece of the centroid is a third of the whole and the longer piece is two-thirds of the
whole. This means that DF = 15 • ~ = 5 and FC = 15 • g = 10.
Notice from these two measurements that DF is half of FC. Therefore, FE will be half of AF, making
AF = 16.
Kinesthetic Learners
Students may need to repeat Investigation 5-6 with a right triangle and an acute triangle. They may
also wonder if the altitudes intersect at one point like all the other lines in triangles. Explore this as an
extension of Investigation 5-6. After constructing one altitude, construct the other two to see if they all
intersect at the same point. (This extension might be easier to do with an acute triangle.) Of course,
they do all intersect at the same point, called the orthocenter. Unlike the other points of concurrency, the
orthocenter does not really have any useful properties. However, like the circumcenter, the orthocenter
can lie outside of a triangle.
Table 4.5:
Acute Triangle
Right Triangle
Obtuse Triangle
orthocenter
The orthocenter is inside the tri-
angle.
.orthocenter
The legs of the triangle are two
of the altitudes. The orthocenter
is the vertex of the right angle.
The orthocenter is outside the
triangle.
Modifications
Students should have the chart started in the first lesson of this chapter completed. This will be an excellent
study guide for the chapter test.
Give students copies of triangles for the investigations in this lesson. It might make the investigations go
smoother.
Assess students' knowledge by walking around and asking questions. This can be a very difficult chapter
for students. Blank papers and looks can indicate that they may be struggling.
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276
Inequalities in Triangles
Visual/Auditory Learners
Give students graph paper and have them construct a 12, 6, 7 triangle using Investigation 4-2. Let them
use each square as a unit. Always have students start with the longest side as the horizontal side. Then,
using a protractor, have students measure each angle and list them from least to greatest. At this point,
have a discussion about the correlation between the sides and the angles. Poke students towards Theorem
5-9 and its converse.
Repeat the above mini-investigation with lengths 4, 5, 6 and 5, 12, 13. After they are confident that these
all make triangles, repeat one last time with lengths 4, 8, 11 and 3, 2, 5. What happens? Discuss why
these lengths do not make triangles. Have them look back at all the lengths to see why some worked and
some do not. If students are struggling, have them look at the sums of the lengths. Let students arrive at
the Triangle Inequality Theorem. Then, go over Example 3.
Kinesthetic Learners
Another approach to the above investigation is to use dry spaghetti. Let students break off the correct
lengths and see if they can create triangles. This approach also works well for problems like Example 4.
You can also demonstrate this using the spaghetti on the overhead. The use of dry spaghetti is also helpful
with the SAS and SSS Inequality Theorems.
Additional Example: Find the length of the third side of a triangle if the other two sides are 4 and 5.
Solution: Using the spaghetti, break off lengths of 4 (inches or centimeters) and 5. Let students play
with the spaghetti until they arrive at the correct answer. It might help them to trace triangles onto a
piece of paper and then they can measure the third side. Students should make the "extremes" of the
possibilities, 4, 5, 1.1 (a very narrow and tall acute triangle) and 4, 5, 8.9 (a long, flat obtuse triangle)
with their spaghetti. 1 < third side < 9.
Modifications
Give students a list of the theorems covered in this lesson. You can make these fill-in-the-blank so that
students have to follow along. Leave spaces in the handout for examples.
Review the congruent triangle theorems. Students may get them confused with the inequality theorems in
this lesson. If that is the case, the SAS Inequality Theorem is also called the Hinge Theorem and the SSS
Inequality Theorem is its converse.
Extension: Indirect Proof
All Learners
Incorporate a real-life example into this lesson. Have students come up with an if-then statement and then
see if they can prove it indirectly. Here is an example.
If-then statement: If it is foggy in San Francisco, then you can't see the Golden Gate Bridge.
Assume the opposite of the conclusion: I can see the Golden Gate Bridge.
Find a contradiction: Let students reason at this point. If we can see the Golden Gate Bridge, then it is
sunny, which means, it cannot be foggy. This is our contradiction.
Our original statement must be true.
Modifications
Pick and choose the parts of this extension that you want to use. Double check your state's standards to
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see if it is included. If not, the best approach may be to skip this extension. Do a quick assessment of your
students to see if they can handle this extension. You could take a poll, decide based on their quiz scores,
or based on their ability to answer your in-class questions.
4.6 Polygons and Quadrilaterals
Angles in Polygons
Visual/Kinesthetic Learners
Rather than tell students to draw the diagonals from one vertex (in Investigation 6-1), give students a
blank octagon and have them divide it into triangle. They can divide octagon however they want. Once
they have it split up into triangles, have students count the triangles and multiply the answer by 180°. Of
course, their answers will probably not be 1080°. Everywhere students have intersecting lines inside the
octagon, students need to subtract 360°. For example,
8 triangles = 8 x 180° = 1440°.
There is one place where all the sides intersect, so subtract 360°.
1440° - 360° = 1080°
You can repeat this process for any polygon.
Additional Example: Algebra Connection Find the measure of x and each angle in the hexagon.
Solution: From the Polygon Sum Theorem a hexagon adds up to 720°. Set up an equation.
6x + 10° + 10jc - 10° + 8jc - 10° + 9x - 5° + 9x - 5° + 7x + 5° = 720°
49x - 15° = 720°
49* = 735°
x= 15°
The angles are: 100°, 140°, 110°, 130°, 130°, and 110°
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Visual Learners
External angles can also be represented by a shutter on a camera lens. Notice how the shutter closes and
as it does it approaches a point. There are 360° around a point. If you have an older camera (or the art
department might), with a shutter, it could be very helpful for students to visualize this concept. You
could also show students the opening credits of a James Bond movie. This has the same visual effect.
Modifications
Review the definitions of diagonals, convex, concave, and regular polygons.
Make Investigation 6-1 teacher-led. Give students a handout with polygons and the chart from Step 3 so
they can follow along.
Make Investigation 6-2 teacher-led. Demonstrate this investigation on the overhead and have students
follow along.
Properties of Parallelograms
Kinesthetic Learners
Have students work in pairs. Give each pair of students four strips of paper, where two are the same length
and the other two are a different length. Have students fasten the four strips together (with brads or
fasteners) so the congruent strips are opposite each other. This is a parallelogram. Have students measure
the angles. Then, have students trace a parallelogram onto a piece of paper, draw in the diagonals and
measure them.
With this activity, review with students the properties of parallel lines and transversals. While students
will discover some new properties of parallelograms, the properties of parallel lines will still apply.
Visual Learners/ELL
Make a bullet list of the properties of a parallelogram. It should be:
Opposite sides are parallel
Opposite sides are congruent
Opposite angles are congruent
Consecutive angles are supplementary
Diagonals bisect each other
If it helps students, include a picture with each bullet point. Make these notes short and concise. Then,
students can add to them for each type of quadrilateral.
Show students a flowchart proof of the Opposite Sides Theorem in the text.
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AB II DC —lABQ=£BDC
ABCD is a
parallelogram
with diagonal 8D
Definition of a
paraflelogram
AlA Theorem
AD = BC
AD || BC
ZADB^ZDBC
AADB^ADBC<
CPCTC
Definition of a
parallelogram
AlA Theorem
ASA
AB = DC
DB = DB
CPCTC
Reflexive PoC
Modifications
Review how to find the midpoint in the coordinate plane.
Give students a handout with a parallelogram on in for Investigation 6-3. Students can then start at Step
3.
Proving Quadrilaterals are Parallelograms
Collaborative Learning
Give each group a theorem from this lesson. Show students the proof of the Opposite Sides Converse that
is in the text as an example. Then, assign each group a theorem to prove: Opposite Angles Converse,
Parallelogram Diagonals Converse, or Theorem 6-10 (some groups will overlap). Then, give each group a
poster-piece of paper and have them write the proofs of their theorems on the paper. Walk around to help
students and answer questions. If there is time, allow groups to present their proofs to the class. Answers
to the other proofs are Example 1 and #23 and #24 in the review questions.
The converse of the Consecutive Angles Theorem is also true, but not commonly used as a reason as to
why a quadrilateral is a parallelogram. In general, it is very tedious to show that the 4 sets of consecutive
angles are supplementary, which makes it hard to use in a proof. If students are feeling up to a challenge,
you can have them prove this converse.
Visual/Kinesthetic Leaners
Show that the parallelogram in Example 4 can be a parallelogram by using a different theorem. Students
need to know that the theorem they choose does not matter. In the example, Theorem 6-10 was used.
In coordinate geometry, it is challenging to show that two angles are congruent (unless they are 90°), so
we will only use the definition of a parallelogram, Opposite Sides Converse, the Parallelogram Diagonals
Converse, and Theorem 6-10.
Example 4, Alternate Solutions: Is the quadrilateral ABCD a parallelogram?
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280
Solution 1: Let's use the definition of a parallelogram.
Find the slopes.
Slope AB =
5-3
-1-3 ~~
2
-4 ~~
1
2
Slope AD =
5-(-2) .
-1-2
7
~~ 3
Slope CD =
-2+4
2-6 ~
2
~ -4 ~~
1
2
Slope BC =
3-(-4) .
3-6 "
7
3
Opposite sides are parallel, ABCD is a parallelogram.
Solution 2: Let's use the Opposite Sides Converse.
AB
2
= a/(-1-3) 2 + (5-3) 2
= V(-4) 2 + 2 2
= V16 + 4
= V20 = 2Vo"
CD = a/(2 - 6) 2 + (-2 + 4) 5
= V(-4) 2 + 2 2
= V16 + 4
= V20 = 2V5
AD = > /(-l-2) 2 + (5-(-2))
= V(- 3 ) 2 + 72
= V9 + 49
= V58
5C = a/(3-6) 2 + (3 + 4) 2
= V(- 3 ) 2 + 72
= V9 + 49
= V58
Opposite sides are equal, ABCD is a parallelogram.
Solution 3: Let's use the Parallelogram Diagonals Converse.
Midpoint of AC = (4±6, ^) = (§, |)
Midpoint of SD = (2±2, 3=2) = (§, ±)
The midpoints of the diagonals are the same, which means that they bisect each other. Therefore, ABCD
is a parallelogram.
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From all these solutions, we see that there are four different ways to show that a quadrilateral is a paral-
lelogram. Students should be able to choose any solution they want. You could discuss as a class, which
methods students like and why.
Modifications
Leave a list of characteristics of a parallelogram on the board for students to reference.
Thoroughly review how to complete a proof and the properties of parallel lines.
Give students graph paper to assist with the coordinate geometry examples.
Rectangles, Rhombuses and Squares
Visual Learners/ELL
Break down the properties of rectangles, rhombuses, and squares in bullet lists, like we did in the Properties
of Parallelogram lesson.
Rectangles
• has all the properties of parallelograms
— opposite sides are parallel
— opposite sides are =
— opposite angles are =
— consecutive angles are supplementary
— diagonals bisect each other
• all angles are = (90°)
• diagonals are =
Rhombuses
• has all the properties of parallelograms
— opposite sides are parallel
— opposite sides are =
— opposite angles are =
— consecutive angles are supplementary
— diagonals bisect each other
• all sides are =
• diagonals are J_
• diagonals bisect each angle
Squares
• has all the properties of parallelograms
— opposite sides are parallel
— opposite sides are =
— opposite angles are =
— consecutive angles are supplementary
— diagonals bisect each other
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• has all the properties of rectangles
— all angles are = (90°)
— diagonals are =
• has all the properties of rhombuses
— all sides =
— diagonals are J_
— diagonals bisect each angle
Have students add these lists to their list from the Properties of a Parallelogram lesson.
Modifications
For Investigations 6-4 and 6-5, consider giving students a handout of a pre-drawn rectangle and rhombus.
If you do this, students can start at Step 3 for 6-4 and Step 4 for 6-5.
Give students a handout of the notes and/or the above chart. Students may also need a copy of the
guidelines for determining if a quadrilateral is a special parallelogram (after Example 5). Students can use
other ways to show that a figure is a special parallelogram that are not listed here. Discuss with students
the other options. For example another algorithm could be:
1. Graph all four points on graph paper.
2. Find the lengths of all four sides.
All sides = : Square or Rhombus, go to #3
Opposite sides = : Parallelogram or Rectangle, go to #3.
No sides =: Quadrilateral (done)
3. Find lengths of diagonals.
Diagonals =: If the shape was a square or rhombus, we now know that it is a square (done).
If the shape was a parallelogram or a rectangle, we now know that it is a rectangle (done).
No = diagonals: If the shape was a square or rhombus, we now know that it is a rhombus (done).
If the shape was a parallelogram or rectangle, we now know that it is a parallelogram (done).
This algorithm does not take trapezoids or kites into account. Make sure if students use this algorithm they
are aware of this.
Trapezoids and Kites
Visual Learners/ELL
Break down the properties of isosceles trapezoids, trapezoids, and kites in bullet lists, like we did for
parallelograms, rectangles, rhombuses, and squares.
Trapezoids
• one pair of parallel sides
Isosceles Trapezoids
• one pair of parallel sides
• non-parallel sides are =
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• base angles are =
• diagonals are =
Kites
• two sets of adjacent sides are =
• non- vertex angles are =
• diagonals are J_
• the diagonal through the vertex angles bisects them
Add these lists to the ones students have created from the previous sections. Students should have all
seven descriptions on one sheet of paper. This can serve as a study guide or a "cheat sheet" for the test.
All Learners/Collaborative Learning
In groups, give each group a different set of four points. Here suggestions for each type of quadrilateral:
(-8, 2), (-4, -3), (-3, -1), (-7, -2) -> Square
(6, -1), (10, -7), (7, -9), (3, -3) -> Rectangle
(-2, 7), (3, 5), (5, -1), (-1, 1) -> Rhombus
(-7, 4), (-2, 6), (-1, -1), (-6, -3) -> Parallelogram
(4, 8), (8, 9), (9, 5), (2, -1) -> Kite
(3, 2), (5, -5), (3, -7), (-4, 5) — > Isosceles Trapezoid
(-3, 7), (7, 5), (6, 1), (1, 2) -> Trapezoid
Once each group has a set of four points (do not give them the type of quadrilateral), give the groups
markers and graph poster paper. Let students work on their own to determine what type of shape they
have. As a part of their poster, students need to explain their steps. Let students use their bullet lists as
well as the algorithm from the previous lesson. Students will need to change this algorithm if they decide
to use it.
If time allows, have students present their shapes and process to the rest of the class. Then, they can see
the different ways each group used to find the type of quadrilateral they had.
Modifications
Write all notes on the board/overhead. Request that students copy this information in their notebooks.
You may also want to give students handouts and/or fill-in-the-blank notes.
In Investigation 6-6, give students graph paper and/or the trapezoid already drawn.
4.7 Similarity
Ratios and Proportions
Visual Learners
With Example 1, show students the other representations of the ratio of girls to boys. The ratio can also
be written j| and 14 to 18. It is easiest for students to see that this ratio reduces when it is written as a
fraction.
Visual/Kinesthetic Learners
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Show students how and why cross-multiplication works. We can solve Example 7a in a different way that
is more like what they were used to from Algebra I.
Example 7a, Alternate Method: Solve the proportion.
^) 5 ~~ 30
Solution: We need to get x by itself, so multiply both sides by 30.
_ 4 30 4 120 _
You can also let students do cross - canceling, but sometime kids can get cross - canceling and cross-
multiplication confused. Be careful.
Collaborative Learning
Divide the students into five groups, one for each corollary of the Cross-Multiplication Theorem. After
teaching the lesson and going through the material, assign each group one of the corollary theorems. Then
put an example of two similar triangles (with measurements) on the board or overhead (see below).
Ask each group to use the example on the board to create an example that illustrates their corollary
theorem. Basically, they will replace the letters in the proportions in each corollary with the correct
sides/lengths from the similar triangles on the board. Allow time for the students to work and then have
each group present their example to the class. When finished, encourage the other students to ask questions
to see how well the students in the group can answer them. The way students answer each other's questions
is a great way to assess understanding.
Modifications
Review how to reduce and multiply fractions.
As you have been doing with previous chapters, continue to write and leave vocabulary and theorems on
the board for the duration of the chapter.
Let students shorten up the corollaries so that they are only the proportions. An example could be:
a
b '
= 2i then
1.
a b
c d
?,.
d c
3.
1 = 1
a c
4.
a+b c+d
b ~ d
5.
a-b c-d
b ~ d
Similar Polygons
All Learners
Let students walk around the school grounds to find examples of similar polygons. Allow students to work
in pairs or groups of three. If they have phones or digital cameras, encourage students to take a picture
of their similar polygons. If not, they need to draw a picture of the item and where it is located. After
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students find 2-3 different examples, regroup in the classroom. Go over the students' findings as a class
discussion. Students need to explain why their polygons are similar. Award bonus points to students that
found similar pentagons or hexagons (or larger).
Kinesthetic Learners/Collaborative Learning
Ask students to draw a polygon and label the lengths of the sides of their polygon. Have students exchange
polygons with someone else. Students may exchange more than once just be sure that everyone has a
different polygon than the one that they started with. Students need to complete the following with this
new polygon.
1. Draw a similar polygon to the one that you have been given.
2. Write proportions to demonstrate that the side lengths are similar.
3. Determine the scale factor.
4. Determine the ratio of the perimeters.
When finished, divide into small groups to share their findings. Use peers to correct any errors in the work
of each individual.
Visual/Auditory Learners
Between Examples 2 and 3, students are told that all equilateral triangles and square are similar. But,
why? Draw pictures of several equilateral triangles on the board. Then, ask students what the scale factor
is for different pairs of equilateral triangles. Students will notice that if the sides of one triangle is a and
the sides of another is /?, then the scale factor is a : b. You can repeat this explanation with squares and
any regular polygon.
Modifications
Write all assignment directions on the board or pass out a handout, so that students can refer back to
what is needed for each step.
Develop a list with students about what ratios are the same as the scale factor.
• ratio of the sides
• ratio of the perimeters
• ratio of the diagonals, medians, altitudes, and midsegments.
Students can use any of these to find the scale factor, as well.
Similarity by AA
Visual Learners
In Examples 1-4, the triangles are not oriented in the same way, making it harder for students to visualize
if the triangles are similar. If students are having a hard time with this, encourage them to use a piece of
patty paper and trace one triangle. Then they can rotate or flip it so that they line up. If two triangles
are similar, the corresponding angles should overlap perfectly. Students can do this with the patty paper,
over the other triangle.
Visual/Kinesthetic Learners
On a sunny day, take your class outside to perform their own indirect measurement problems. Let students
work in pairs. They need to compare their heights to the height of an object they cannot measure, such
as a flagpole, the goal posts on the football field, a tall tree or the school building. Students need to make
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sure that they can easily measure the shadow of whatever object they decide to measure. They will need
to record their height (make students do this here, to ensure accuracy), their shadow and the shadow of
the object. Once they have these measurements, head back to the classroom to finish the calculations.
Modifications
For Investigation 7-1, give students a handout with the triangles from Step 1-3. Then, have students
measure the angles and sides of these two triangles to see if they are similar.
Encourage students to draw in any markings on triangles. Students need to be able to show that two
angles are congruent in two triangles to verify they are similar.
Explain to students that the AA Similarity Postulate could also be the AAA Similarity Postulate. However,
because of the Third Angle Theorem, we only need to show that two angles are congruent. It is inferred
that the third set is congruent because of this theorem.
Similarity by SSS and SAS
Visual/Kinesthetic Learner
Using Geometer's Sketchpad, explore triangles with proportional side lengths. Draw two triangles, AABC
and AMNP, with each side length of AMNP being k times the length of the corresponding side of AABC.
Measure the angles of both triangles and side lengths. Record the results in a chart like the one below.
Repeat steps 1-3 for each value of k in the chart. Keep AABC the same throughout the exploration.
Triangle Data
Table 4.6:
AB BC AC mlA mlB mlC
Measurements
for AABC
MN NP MP mlM mlN mlP
k = 2
k = 5
Now, have students analyze their data. Have students complete the following ratios: ^, ^, jfp and
§§, f^, |§, ^, |§, f^. Do they notice anything? Are the triangles similar?
Kinesthetic/Auditory Learners
Have students work with in pairs. Partner 1 needs to construct a 6-8-10 triangle and Partner 2 needs to
construct a 9-12-15 triangle. They can refer back to Investigation 4-2 to help them with this. Students
will need a ruler, compass and protractor. After constructing, students should compare their triangles.
Discuss how the triangles are different and the same. Are they similar?
From what we are given, the students should notice that the side lengths are proportional. We found
that the angles are congruent, which means that the triangles are similar, by the AA Similarity Postulate.
Discuss with students if the triangles are similar (yes, they are). In this case, students were given three
side lengths that turned out to be proportional. This leads us to the SSS Similarity Postulate. This is a
variation of Investigation 7-2.
Modifications
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Write the directions for the technology activity or the variation of Investigation 7-2 on the board so students
can follow along.
The first five review questions are vocabulary. As a class, go over these questions as a quick review at the
end of the lesson. From this, students will also have five of their homework questions completed.
Proportionality Relationships
Kinesthetic Learners/Collaborative Learning
In groups of 3-5, give each student the same triangle, plus one extra (students may want this extra triangle
later in the investigation, do not let them cut this one). Then, each student needs to draw a line on their
triangle that is parallel to one side. It does not matter where it is, but make sure that every group member
puts the line in a different place or parallel to different sides. Cut the triangle on this line.
"I V
Before this activity, students know that H^ = -^- from similar triangles. Now, we will investigate if ^ = -.
Have each student measure w,x,y, and z and the write these lengths in this ratio. Once everyone in the
group is done, allow time for the group to compare their ratios. Students should see that the proportion
■H^ = -q^ still holds and ^ = y - is a new, valid proportion. However, stress that these two proportions are
not equal to each other. Students should see this, considering they are using numbers and will arrive at
different scale factors for each proportion.
This activity can be repeated with Theorem 7-8 (the angle bisector theorem). However, have students cut
a
d'
the triangle on the angle bisector and measure the sides below to show that - = a
Visual Learners
Another name for Theorem 7-7 is the Notebook Paper Theorem. To demonstrate this for students, you
could have them draw two transversals on their lined paper. Then, the notebook paper lines divide the
two transversals proportionally.
Another way to write the proportion in Theorem 7-8, \ — % (using the triangle above) is | = | or |
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Students might like these two options better than the one that is in the theorem because they group
together the two sides that are on the same side of the triangle together in each ratio.
Modifications
Students will get confused and think that the parallel sides in the Triangle Proportionality Theorem are in
the same ratio as the sides that were cut by the parallel line. In actuality, the parallel sides are in the same
ratio as the similar triangles. Go over that these ratios are not the same. It is helpful to insert numbers
into an example problem so that students see that you do not end up with the same thing.
a b c
d = ~e*~f
a b
c
a-\- d b + e
f
Triangle Proportionality Theorem
Similar Triangle Ratios
Insert a = 6, b = 8,c = 6,d = 3,e = 4,f = 9 as an example for students to help them with these
proportions.
Similarity Transformations
Visual/Kinesthetic Learners
Let students work in pairs. After going over the material from this lesson, have students create a scale
drawing of the classroom. You can either have students do their drawing on poster paper or an 8 X 11
piece of paper. They need to pick an appropriate scale factor, depending on the size of their paper. If a
classroom is 25 ft. by 30 ft., then students could make the scale ^ if they are using poster paper or ^ for
an 8 x 11 in. piece.
After they have scaled down the room, have scale down the student desks, your desk, and chalkboards
or whiteboards, cabinets, or tables. Then, let students be creative and redesign the layout of their desks.
Once all your classes have finished this part of the assignment, allow classes to vote on their favorite and
then redo the classroom in that manner.
Visual/Auditory Learners
When going over Example 7, encourage students to draw rays from the origin to A',Z?', and C . What do
they notice? Discuss why the image and preimage are collinear with the origin. Do they think this will
always happen? Reference Example 6 if needed.
Modifications
Students have not yet seen reflections, rotations or translations. You may want to go over these ideas
quickly to help students understand the concept of a transformation and dilation.
To tie into problems 25-34, you may need to expand on Example 8, to show the distances from the origin
to each vertex of the image and preimage. Here we will find OA, OA\AA\ OB, OB' and BB f for Example 7.
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OA = A /(2-0) 2 + (l-0) 2 = V5 OB = ^(5-0) 2 + (l-0) 2 = V26
OA' = ^(6 - 0) 2 + (3 - 0) 2 = V45 = 3 V5 OB' = J(15 - 0) 2 + (3 - 0) 2 = V234 = 3 V26
AA' = ^(2-6) 2 + (l-3) 2 = V20 = 2 V5 55'= ^(5 - 15) 2 + (1 - 3) 2 = Vl04 = 2 V26
Now, compare the ratios. The ratios of OA : OA f and OB : OB r are still the same as the scale factor.
However, the ratio of OA : AA' and OB : Z?Z?' is 1 : 2. You may want to discuss with your students as to
why this is.
Extension: Self-Similarity
Kinesthetic Learners
Go to the website, http://fractalfoundation.org/resources/fractal-software/, to download free
fractal-creating software. Go to the Help tab — > Tutorials — > An introduction to fractals — > Whole story.
This will show students how to generate several different fractals and define them. Then, let students play
with the software to generate fractals, Fractals tab — > Formulae or Formulae 2.
Modifications
Fractals may or may not be a part of your state's standards for Geometry. Please double check to ensure
that you are not covering unnecessary material. Depending on the time, you may not want to go over this
extension.
Write the steps for each fractal on the board/overhead or hand out directions. Leave these up as students
work on the examples in the text. Encourage students to color in the Sierpinksi Triangle and other fractals.
4.8 Right Triangle Trigonometry
The Pythagorean Theorem
Visual Learners
When going over the Pythagorean Theorem, encourage students to draw a right triangle at first (if there
is none given). This way, students will avoid confusion with the appropriate sides. Remind students that
the hypotenuse is always the longest side in a right triangle. So, it should be the length that is by itself,
c, when being squared in the Pythagorean Theorem.
Visual/Kinesthetic Learners
Students may wonder where a Pythagorean Triple comes from. Euclid developed a formula to generate
Pythagorean Triples: a = m 2 -n 2 ,b = 2mn, and c = m 2 -\-n 2 . Show students this where m = 2 and n = 1 to
see what they come up with. Then, have students generate 2-3 triples on their own. The only restriction
is that m > n, so a is not negative.
Kinesthetic Learners
Students may need an additional example where the three numbers given do not form a Pythagorean
Triple. Do this example after Example 8.
Additional Example: Is 12, 19, 25 a Pythagorean Triple?
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Solution: 12 2 + 19 2 = 144 + 361 = 505 ± 25 2 . 25 2 = 625, so this is not a Pythagorean Triple. However,
from our prior knowledge of triangles, it does still create a triangle.
Visual Learners
To avoid confusion, show students the proof of the Distance Formula with numbers for (x\,yi) and (x2,y2)-
Go through the entire proof, as it is, with (-5, 6) and (3, 2). Showing students the vertical and horizontal
distance is an important part of understanding where the Distance Formula comes from.
Modifications
Review the concept of a perfect square.
Decide if you want students to memorize the Pythagorean Theorem and the Distance Formula. Other
options are leaving them on the board for the chapter or allow students to use a formula sheet.
In Investigation 8-1, you may want to give students pre-cut squares that are 3 in, 4 in, and 5 in. Students
can then start with Step 2. You can also do this investigation as a teacher-led activity and students can
follow along while you fit the areas together. This is one way the green (4 in) and blue (3 in) squares fit
into the red square. There are several possibilities. If done as individually by students, encourage them to
be creative with how they cut up their squares.
I I ! I
Converse of the Pythagorean Theorem
Visual/Kinesthetic Learners
Have three students pretend to be a right triangle on the coordinate plane. Have students A and B be
the hypotenuse and give them a tape measure to find the distance between them. Then, ask student C
where the perfect spot for them to stand would be so that they complete the right triangle. At this point,
transfer the tape measure to find AC and BC. Then, as a class, use the Pythagorean Theorem to see how
close they got to being a right triangle. They can also determine if the triangle created was an acute or
obtuse triangle using the converse. Repeat this process with several different students and placements,
including Pythagorean Triples. Encourage students to move their desks to the edge of the classroom so
students have more room to move.
Visual/Auditory Learners
Write the following three triangles and the variations of the Pythagorean Theorem on the board. Have
students discuss why these theorems are true.
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a* + b* > c*
ACUTE
a 2 + b 2 - c 2
RIGHT
a 2 + b 2 <c 2
OBTUSE
b ACb A^ b A
It might help to insert lengths for a, /?, and c. Use the following numbers:
acute triangle: 20,21,25 (20 2 + 21 2 > 25 2 )
right triangle: 20,21,29 (20 2 + 21 2 = 29 2 )
obtuse triangle: 20,21,35 (20 2 + 21 2 < 35 2 )
If you have access to a computer lab and Geometer's Sketchpad, this activity can be easily done using
technology. Draw a triangle in Geometer's Sketchpad; label the vertices A,Z?, and C. Then, have students
move around B so that they can see when the triangle is acute, right, and obtuse.
Modifications
Review the definition of a converse statement.
Decide if you would like students to memorize Theorems 8-3 and 8-4, add them to their formula sheets, or
write them on the board.
Using Similar Right Triangles
Visual/Kinesthetic Learners
When initially going over Theorem 8-5, have students draw two 6-8-10 right triangle. Then, they need to
cut out both triangles. Label the triangles like the one below. To make this activity go a little smoother,
you may want to hand out pre-cut 6-8-10 triangles.
With one triangle, students need to find the altitude from the right angle and cut this triangle on the
altitude. Students should now have three similar triangles.
Ask them to prove the triangles are similar by the A A Similarity Postulate. Next, have students measure
the lengths of all the sides of all three sides. Students should generate the following proportions: f§ = ^
(short legs to longer legs), f§ = ^f (short legs to hypotenuses), and f§ = ^§ (long legs to hypotenuses).
Point out to students that all of these proportions repeat one value. This is a property of the geometric
mean. Therefore, AC is the geometric mean of BC and DC,AB is the geometric mean of BC and BD, and
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AD is the geometric mean of CD and BD. See if students can find any other geometric means among the
three triangles. Also, stress that this only happens with similar right triangles.
Modifications
Review similarity and how to draw/find the altitude of a triangle.
Students may have a lot of trouble visualizing which sides match up with others in the similar right
triangles. Encourage them to draw out the triangles, like in the examples in the text. If this is not enough,
give students triangles that they can cut (like the activity above, the triangles do not have to be to scale)
on the altitude. They should write the vertex labels and lengths along the legs and hypotenuses before
cutting.
Special Right Triangles
Journal Entry
What makes a triangle "special"? What would be your definition of a special right triangle?
Visual/Kinesthetic Learners
See if students can generate a shortcut for rationalizing the denominator. In Example 2Z?, the leg of the
45-45-90 right triangle is 8 V2 and the hypotenuse is 16. Help students notice the pattern that the leg
would be the hypotenuse divided by 2 with a V2 multiplied by (or next to) it. The same will work for any
time students need to rationalize the denominator. For example:
b)% = f V2 = 13V2
Kinesthetic Learners
Before Investigation 8-2, give each student the table below.
Table 4.7:
a 2 + b 2 c
2
4 4V2
6 6
10 10V2
5V2 5V2
3V6 6V3
Have students fill out the table and then see if they notice any patterns. All of the lengths above correspond
to isosceles right triangles. Once students see the patterns, start Investigation 8-2.
Before Investigation 8-3, have students fill out the following table and find the patterns. Here, all the
lengths will correspond to 30-60-90 triangles. Once students are comfortable with the patterns in this
table, move on to Investigation 8-3.
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Table 4.8:
a 2 +b 2
5V3
3V6
6V3
10 V3
15
20
6V6
Modifications
In Investigation 8-2, give students a copy of the triangle described in Step 1. Then, start with Step 2. Go
over Step 3 together.
In Investigation 8-3, give students a copy of the equilateral triangle described in Step 1 with an altitude.
Then, start with Step 3 and go over steps 4 and 5 as a class.
As a homework assignment, students need to bring in calculators for the next lesson/class period.
Tangent, Sine and Cosine
Visual Learners
When defining the sine, cosine, and tangent ratios, show students several different triangles where variables
are the sides. Students need to see different orientations so that they will be able to answer any type of
question.
Additional Example: Find the sine, cosine and tangent of AG and Lj .
Solution: sinG = |,cosG = ^, and tanG = -.
sin/ = f , cos/ = |, and tan/ = -
Additional Example: Find the sine, cosine, and tangent of IF and IE.
f
Solution: sinF = ^,cosF = |, and tanF =
sinis = ^,cosE = ^, and tan£ = j
Kinesthetic Learners/Collaborative Learning
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In this activity, we are going to have students work backwards. Using the ratios below, have students
design a triangle that matches. Let students work in pairs or groups of three. Also, we know that there
could be several different triangles that will work for each ratio. Encourage students to come up with two
different triangles for each set of ratios. Label all the angles A, Z?, and right angle C.
Problem 1
Problem 2
Problem 3
Problem 4
sin A = :j7j and sin B = j| Answer 5x - Ylx - 13x, where x is any whole number,
cos A = j~ and tanA = ^ Answer 8x - 15x - 17x, where x is any whole number,
sin A = t^t Answer 7x - 24x - 15x, where x is any whole number.*
_9_
40
tan B = Jj Answer 9x - 40x - 41x, where x is any whole number.*
* Students will need to use the Pythagorean Theorem to solve Problems 3 and 4.
Modifications
Give students notes or fill-in-the blank handouts for this lesson. Students might also require triangle
handouts for their notes or the examples in the text. A copy of Example 5 might also be nice for students
have in their notes for the triangle diagram with the opposite and adjacent sides and the hypotenuse
identified in reference to the 22° angle.
Make sure every student has a scientific calculator to work with to find the trig ratios. As students walk
into class or while checking homework, check their calculators to make sure they are in the correct mode.
Also show students the buttons that will be used today.
Inverse Trigonometric Ratios
Visual Learners
After doing the Review Queue, go back over questions 1 and find the other angle in the triangle, which is
54°. Then, look at the complete triangles and set up a sine, cosine, and tangent ratios.
4.11
5.66
From the previous lesson, students know that sin 36° = ^M-. Use algebra to ask students how they would
get 36° by itself? Students might shout out "divide by sine". They are on the right track, but there is a
new way to "undo" the trig functions. Show students the sin" 1 notation.
sin (sin 3d ) = sin I — — I
30== sin- (*f)
The sin and sin cancel each other out and 36° is isolated.
36° = sin
m (—J
We have now introduced the inverse trig functions to students. Ask students to fill in the blanks. Write
the problems below on the board.
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36° = cos" 1 ( - J 36° = tan" 1 i- j 54° = sin" 1 i-
54° = cos- 1 [-) 54° =tan- 1 [-
Then, go over the traditional definition in the text.
Visual Learners/Kinesthetic Learners
In Example 4, we used 62° to find AB and BC. Students might notice that it is more challenging to solve
for these sides when they are in the denominator of the fraction ( sin 62° = || J. Encourage students to find
the third angle measure before finding the lengths of the sides.
Example 4, Alternate Solution: Solve the right triangle.
c
Solution: First, find the third angle using the Triangle Sum Theorem. 180° - 90° - 62° = 28°.
AB : Use cosine.
BC : Use tangent.
cos 28°
25
~~ AB
AB
25
~~ cos 28°
AB
* 28.31
t n28° =
BC
25"
£C =
25 • tan 28°
BC*
13.30
Notice that when students use 28° to find Z?C, it is in the numerator of the tangent ratio. This equation
25
BC
might be easier for students to solve than tan 62° = |^. If students are given two angles, encourage them
to explore the trig ratios using both acute angles.
Modifications
Give students a copy of the Review Queue for this lesson.
When checking homework, double-check students' calculators to make sure they are in the correct mode.
At this time, show students where the inverse trig buttons are.
4.9 Circles
Parts of Circles and Tangent Lines
Visual Learners/ELL
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Allow students to define the circle terms by using pictures rather than words. Here are a few examples:
secant diameter chord tangent line
O*0 Q
P
point of tangency
If students are making flash cards, they should put the name on one side of the card and the picture on
the other. There are lots of pictures in the text for the other definitions.
Kinesthetic Learners
See if you can find physical examples of common internally and externally tangent lines. An easy example
could be gears. The science department might have something or you could probably buy them at a bike
shop. The bike shop might have scrap parts or broken parts that you could purchase at a discount. Thread
the gears using both the internally tangent lines and externally tangent lines. After showing students both
options, ask why they think bike gears do not have common internal tangents.
Modifications
Leave formulas and theorems on the board throughout the chapter. You could also give students a blank
formula sheet and fill it out as the chapter progresses. Decide if you want students to memorize the formulas
in this chapter or allow them to use some sort of cheat sheet on quizzes and tests.
This is a long lesson with a lot of vocabulary and new concepts. Allow 2-3 class periods to thoroughly
go over the material. In addition to a blank formula sheet, you might also want to give students a blank
vocabulary sheet for the chapter.
Give students 3x5 cards at the beginning of each lesson for flash cards. Allow for time so that students
can make their flash cards in class.
Properties of Arcs
Visual/Kinesthetic Learners
In addition to a circle having 360°, this is also the number of degrees around a point. The picture in the
text illustrates this point quite well, using the straight angle as part of the reason as to why there are
360° in a circle. Make sure students understand this picture and use it to explain the degree measure of a
semicircle.
ELL Students
Semi = half, minor = little or small, and major = big or large. These quick definitions of these terms will
help student decipher the vocabulary in this lesson. Also, in Spanish, menor translates to less. Therefore,
a minor arc could also be likened to un arco menor.
Semicircle = half circle, minor arc = small(er) arc, major arc = large(er) arc
Visual Learners
Show students that the arcs in Example 2 will add up to 360° when added together. This is a good way
for students to check their work.
mEF + mFB + mBC + mCD + mDE = 120° + 60° + 52° + 90° + 38° = 360°
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Ask students to draw pictures of congruent arcs that were not shown in Example 4. Remind students that
in order for arcs to be congruent, they either need to be drawn in congruent circles or in the same circles.
They must also have the same central angle measure. Here is an example of congruent arcs that was not
shown in the text.
Auditory Learners
Before going over the Arc Addition Postulate, ask students how they would find mDB from Example 2.
Encourage students to come up to the board and explain how they would find the answer and shout out
how to find the answer. Students will jump to the Arc Addition Postulate without even realizing it. When
someone says, "add mDC and mCB" or "add 90° and 52°," ask the students why? Why can this be
done? We have not done it yet, so why can we do it now? Part of the explanation could be that there
are Segment and Angle Addition Postulates, so it would only make sense that there is an Arc Addition
Postulate. Students might also say, that the angles add up to 142°, so the corresponding arc should too.
Modifications
Like in the previous lesson, let students use pictures to define the terms.
Properties of Chords
Visual/Kinesthetic Learners
As a class investigation, derive Theorem 9-3 together. First, draw a circle. If you do not have a board
compass, tie a string around a piece of chalk or whiteboard marker. Then, at the other end, use your
finger as the pointer. Once your circle is drawn, use two rulers as congruent chords. Either draw congruent
chords on the circle or place the rulers in the circle. Then, ask students if they notice anything about the
corresponding arcs. Draw in the central angles and ask if students can conclude anything about them.
Allow students to shout out their thoughts. Together, discuss and formulate Theorem 9-3. Then, use this
same diagram to go over Example 1 in the text.
When discussing Theorems 9-4 and 9-5, students need to be careful when looking at diagrams. This is
a good opportunity to remind students that nothing can be assumed; they must be told information in
statements or from labeling. To use these theorems, one chord must clearly pass through the center and
be labeled (or written) perpendicular to the other chord, like in Example 2.
For additional practice with congruent chords, have students measure the arcs in Step 3 of Investigation
9-3. Students should draw in the corresponding central angles and measure them. As a quick review of
Theorem 9-3, these central angles should be congruent.
Visual/Kinesthetic Learners
At the conclusion of this lesson, have students draw the follow picture and label everything they have
learned from this lesson.
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B D
Given: E is the center, EF J_ AZ?, and EG _L CD
Students can conclude: AB = CD, F and G are the midpoints of AB and EG, AE, EC, ED and EB are all
radii and congruent, AAEB = ACEB,AAEB and ACEB are isosceles triangles, AB = CD.
Modifications
For Investigation 9-2, give students a handout with Steps 1 and 2 completed. Start at Step 3. You could
also make this investigation teacher-led and just have students follow along.
For Investigation 9-3, give students a handout with Step 1 completed and then start at Step 2. Again, this
investigation could be teacher-led.
Go over the definition of equidistant and how it relates to circles and chords.
Inscribed Angles
Kinesthetic Learners/Collaborative Learning
After going over the Inscribed Angle Theorem and Examples 1 and 2, give students a handout with the
extra practice below. Solve for x° and y°.
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120°
8.
140
Answers
1. x = 55°
2. x = 80 o ,y = 100 o
3. x = y = Q0°
4. x=160°,y = 65°
5. x = 90°,y = 70°
6. x = 50°,y = 100°
7. x = 170°
8. x=180°,y = 90°
9. x = 40°,y = 20 o ,z = 40°
After students complete the problems, have them look back at #5 and #8 to see if they can come up with
Theorem 9-9. Repeat this with #3 and #6 and Theorem 9-8. This activity would be done well in pairs.
With #9, see if students can find all the arc measures and all the angles in the inscribe quadrilateral. Then,
see if students can come up with Theorem 9-10. If they are having problems making the connections, move
on to Investigation 9-5.
Modifications
For Investigation 9-4, give students several different circles (5-7) with inscribed angles and the corresponding
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central angle. Then, have students start with Step 3. This investigation is best done by the students,
individually, to generate their own meaning for the relationship between the inscribed angle, intercepted
arc and central angle.
Investigation 9-5 can be a little challenging for some students. This investigation might be best done as a
teacher-led investigation on the overhead. If you decide to make this a student activity, consider passing
out handouts with Steps 1 and 2 completed.
Angles from Chords, Secants, and Tangents
Visual Learners
For Investigation 9-6, it might help students to draw in the insribed angle, iCFB. This picture shows that
iCFB = lECB because they intercept the same arc. Therefore, any angle with its vertex on the circle will
be half the measure of the intercepted arc.
After going over all the formulas, either give students the following picture or have them draw it in their
notes with the corresponding formulas.
The formulas progress — ^ — , ^-, ^—^ — as the angle moves from the inside to the outside of the circle.
Putting the circles and formulas in this order might help students understand and remember them.
Kinesthetic Learners
Students will need lots of practice with these new formulas. Consider giving them the following problems
as extra practice.
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120°
160°
Answers
1. x = 85°
2. x = 100°
3. x = 45°
4. jc= 110°
5. x = 200°
6. x = 60°
Modifications
Investigations 9-7 and 9-8 might be a little challenging for students. One modification for these would
be to omit them and give students the formulas and several examples. Students might get confused by
all the angles in the investigations and not make the proper connections. Another modification for these
investigations would be to make them teacher-led.
Segments from Chords, Secants, and Tangents
Kinesthetic Learners
Here are a series of investigations for the theorems presented in this lesson.
Investigation 1: Ratios from Chords
Tools Needed: picture in step 1 (below), colored pencils, pencil
1. Give students the following picture.
2. Have students connect the two sets of endpoints to create two triangles. Have a discussion as to
why the two triangles are similar (AA Similarity Postulate). Encourage them to use color to identify the
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corresponding sides.
3. Have students generate a proportion with a,b,c, and d. Students should come up with ~ c = | or | =
4. Lastly, they need to cross-multiply. Then, they will arrive at Theorem 9-14, ab = cd.
Investigation 2: Ratios from Secants
Tools Needed: picture in step 1 (below), colored pencils, pencil
1. Give students the following picture.
2. Draw two chords that intersect inside the circle, to form two triangles. Have a discussion as to why
the two triangles are similar (both triangles share the angle outside the circle). Have students use color to
identify the corresponding sides. You could also have students extract the triangles so that it is easier for
them to see which sides line up.
3. Have students set up a proportion using a,b,c, and d. Students should come up with ~ c
a c
c+d ~ a+b'
4. Lastly, cross-multiply. Students will arrive at Theorem 9-15, a(a + b) = c(c + d).
Investigation 3: Ratios from a Secant and a Tangent
Tools Needed: picture in step 1 (below), colored pencils, pencil
= c+d
a-\-b
or
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1. Give students the following picture.
a
2. Draw two chords from the point of tangency of a so that two triangles are formed. However, we are
not going to use the two triangles that are most obviously visible. We are going to use the overall, large
triangle and the small triangle that is partially outside of the circle (see below). Discuss with students why
these two triangles are similar. Have students use color to identify the corresponding sides. You could also
have students extract the triangles so that it is easier for them to see which sides line up.
3. Have students set up a proportion using a,b, and c. Students should come up with | = -^-.
4. Lastly, cross-multiply. Students will arrive at Theorem 9-16, a 2 = b(b + c).
Extension: Writing and Graphing the Equations of Circles
Kinesthetic Learners
In Geometer's Sketchpad, open a new sketch and go to Graph — > Show Grid. Then select the circle tool
to draw a circle. Then, go to Measure — > Equation. The equation of the circle should appear in the upper
left-hand corner. Move around the center and the point on the circle to see how the equation changes.
Modifications
Consider only showing students the equation of a circle when it is centered at the origin. Students will see
this material in Algebra II.
Print out this extension and go over everything as a class. Students can write down notes in the margins.
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4.10 Perimeter and Area
Triangles and Parallelograms
Visual/Kinesthetic Learners
For students that are having difficulty with the area formulas, give them graph paper and let students
draw the shapes on the graph paper. This way, students can count the squares to find the area.
Give students a piece of graph paper for Example 5. Have students draw all the different rectangles with
an area of 36 cm 2 . Then, compare the perimeters of these rectangles.
Use the examples below so that students are familiar with all placements of the altitude.
Additional Example: Find the altitude of the triangle, if the area is 64 units 2 .
Solution: Plug in what you know to the area formula.
64 = - • 16 • a
2
64 = 8a
a = 8
Additional Example: Find the area of the triangle below.
Solution: This is a right triangle so the height and base are on either side of the right angle. Even though
25 does not look like the height, if we rotate the triangle it would be.
A = - • 25 • 16 = 800 units 2
2
Kinesthetic Learners/Collaborative Learning
To complete this activity, you will need to prepare enough figures for one-half of the students in the class.
The other half of the class will get the area measurements of the figures given to the other half. Pass out all
the figures and measurements. Some students will receive drawings and some will receive measurements.
The students will need to figure out the measure of their figure and find the person in the room who has
the correct area measurement for their figure. The activity is complete when both persons are sure that
they have been matched up correctly. This is a noisy activity, but the students will have a lot of fun doing
it. It also has a lot of movement in it which is excellent for kinesthetic learners.
Collect all the papers and then repeat this activity. Be sure that those who received drawings get area
measurements and those who had measurements receive drawings.
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Modifications
Write down the formulas for this chapter on the board. You need to decide if you would like to leave these
formulas on the board for quizzes and tests.
Trapezoids, Rhombi, and Kites
Kinesthetic Learners/Collaborative Learning
Repeat the activity described in the previous lesson for trapezoids, rhombuses, and kites (matching figures
with their area). This would also be an excellent review game for the end of the chapter.
Visual/Kinesthetic Learners
Another approach to the area of a trapezoid is to complete the trapezoid to make it a rectangle.
_*i b.
K? 2 K> 2
Area of the rectangle: A = hb 2
Area of the triangles: A = \h(b 2 - b\)
Area of the trapezoid:
A = hb 2 - -h{p2 - b\)
= hb 2 - -hb 2 + -hbi
= -hb 2 + -hbi
1
h(b 1 + b 2 )
You can use this same approach to finding the area of a kite.
Area of the rectangle: A = d\d 2
Area of outer triangles: A = 2 • i \\d^\\d\\ = \d\d 2
Area of kite: A = d\d 2 - \d\d 2 = \d\d 2
For the rhombus, all the interior triangles are congruent. The area for one triangle is: A = ^ \k^i) \\^V ~
\d\d 2 . Multiplying this by 4 we get: \d\d 2 .
For these investigations, give students the picture of the trapezoid and kite and see if the students can
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figure out the area of the rectangle by themselves. Finding the area of the outer triangles might be more
complicated. Tell them they will need to move the two triangles, for the trapezoid, next to each other
so that the heights are back-to-back. Then, it will be easier for them to see that the base length of the
triangle is b2 - b\. This idea can be repeated with the outer triangles for a kite.
After students work on this individually for a few minutes, let them work with the person sitting next to
them. After a few more minutes, come together as a class and have a discussion as to what the areas are
and finish the area problems as a class. This entire process could also be done with the area derivations
that are in the text. Decide which approach your students would be most receptive to.
Modifications
Review how to find the area of a triangle, rectangle, and parallelogram. Keep these formulas on the board,
along with the new ones introduced in this section.
For problems in the coordinate plane (Example 5), give students graph paper and/or a copy of the problem.
Then, allow students to draw on the paper to find the diagonals or lengths. Remind students that things
are not always drawn to scale.
Areas of Similar Polygons
All Learners/Collaborative Learning
One of the best ways for the students to understand scale drawings is to complete one. Using the guidelines
from the Know What? in this lesson, break the students off into pairs and have them create the scale
drawing of their hand. Allow students to use chart paper, rulers, tape measures, colored pencils and to
create their own scale for the diagram. After students have finished drawing their enlarged hands, have
them do a scale drawing of a comic strip, something in a magazine, or picture.
Visual/Kinesthetic Learners
For Example 1, give students a piece of graph paper and have them draw the two rectangles, to scale, on
the paper. Then, cut out the smaller rectangle and see how many times it will fit into the larger rectangle.
Students will need three copies of the smaller rectangle.
in -
I u
16
1
15
1
V
Ebdte
-*m
iilt
h
24
H
From the picture above, we see that the smaller rectangle fits into the larger rectangle 2.25 times. This
goes along with the scale factors of the area, because 2.25 is |. Therefore, we can conclude that the larger
rectangle is | times larger than the smaller rectangle. We can also say that the smaller rectangle is |
smaller than the larger rectangle.
Modifications
Review perimeter and the area formulas. Students will probably need review of ratios and proportions as
well.
For problems 19-26, students might require a picture. Give students a handout with pictures designed
to go along with these problems so that they can properly visualize what they need to do in order to
complete the problem. Another option would be to encourage students to draw their own pictures for
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these problems.
Circumference and Arc Length
Auditory Learners
Discuss with students the difference between degree measure of an arc and arc length. Also, ask students
how the degree measure of an arc plays into finding the arc length.
Visual/Kinesthetic Learners
Have students fill out the chart below when they are first introduced to arc length. This will tie the angle
measure to fractions. Below, semicircle, ^ circle, g circle, and g circle refer only to that portion of the
total circumference.
Table 4.9:
Radius
Circumference Semicircle
\ -circle
| -circle
\ -circle
r = 2
r = 4
d= 10
d = 24
Answers
Table 4.10:
Radius
Circumference
Semicircle
\ -circle
l
6
circle
| -circle
r = 2
\n
2tt
n
2n
3
71
2
r = 4
8n
47T
2n
An
3
n
d= 10
IOtt
5n
5tt
2
hn
3
bn
4
d = 24
24tt
Yin
Qn
4tt
3;r
Allow students to draw pictures for these questions if they need to. Once the chart is filled out, see if
students see any patterns between the columns. After students understand all the answers, ask them how
many degrees are in a semicircle, —circle, g -circle, and g -circle. Then, show them the formula for arc
length and plug in the radius or diameter and show how we will arrive at the exact same answer.
Modifications
Do a quick review at the beginning of the period about the parts of circles, primarily radii, diameters,
central angles, inscribed angles, and intercepted arcs. Rather than just telling students all the information
that they need to know, brainstorm the information as a group. Write down what students come up with.
Students might already be familiar with pi, but might not know where it comes from. Investigation 10-1
can be a useful investigation if this is the case. If students are comfortable with pi, feel free to skip the
investigation or save it for the end of the lesson if there is time.
For students that are struggling with the idea of pi, have them treat it as a variable.
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Area of Circles and Sectors
Auditory Learners
Besides pizza (in the Know What?), discuss with students where they might see sectors in real life.
Visual/Kinesthetic Learners
Have students fill out the chart below when they are first introduced to sectors. This will tie the angle
measure to fractions. Below, semicircle, j circle, - circle, and | circle refer only to that portion of the
total area.
Table 4.11:
Radius
Area
Semicircle
\ -circle
| -circle
\ -circle
r = 2
r = 4
d= 10
d = 24
Answers
Table 4.12:
Radius
Area
Semicircle
I -circle
| -circle
-circle
r = 2
4/r
r = 4
16tt
d= 10
25tt
d = 24
1447T
2tt
8n
25tt
2
72tt
71
257T
4
36tt
2n
k
3
25tt
6
24tt
2
2n
25;r
8
18tt
Allow students to draw pictures for these questions if they need to. Once the chart is filled out, see if
students see any patterns between the columns. After students understand all the answers, ask them how
many degrees are in a semicircle, \ -circle, ^-circle, and | -circle. Then, show them the formula for a
sector and plug in the radius or diameter and show how we will arrive at the exact same answer.
All Learners
Discuss with students how to solve the complex area problems in the examples and the review questions.
It might not be so obvious to students which formulas to use. For example, with Example 8, students will
need to subtract the area of the square from the area of the circle. Do this at the end of the lesson; it will
avoid confusion on their homework.
#23
#24
#25
Area of square minus area of circle.
Area of larger circle minus area of smaller circle.
Area of square minus the area of the four circles.
Modifications
The subsection about segments can be challenging for some students. Depending on your state's standards,
you can choose to omit this section or not. Another option would be to present this example instead of or
before the one in the book.
Additional Example: Find the area of the segment below.
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Solution: First, what portion of the circle would the sector be? Because the angle is 90°, the sector would
be | of the circle. The area of the sector is A = \n8 2 = lQn. Then, we need to find the area of the
triangle. A = ^ • 8 • 8 = 32. Finally, to find the sector, we need to subtract the area of the triangle from
the area of the sector to find the blue segment.
A = 16tt - 32 « 18.27 units 2
4.11 Surface Area and Volume
Exploring Solids
Visual/Kinesthetic Learners
Bring in physical representations of prisms and pyramids. Split the students into groups and have them
identify all the faces, edges, and vertices for each of these items. Then, have them verify Euler's Theorem.
Possibilities for solids could be: a shoe box, Toblerone chocolate bar (probably the container only), an
actual triangular prism (for refracting light), or cracker/cereal box. For things other than prisms, look into
purchasing a set of the wooden geometric solids. Die are also a great representation of the Platonic Solids.
Have groups rotate the objects every 5-10 minutes until every group has seen all the solids.
Show students the following website: http: //www. mathsnet .net /geometry/solid/nets .html. This web-
site provides animations of the nets of a cube, tetrahedron, octahedron, dodecahedron, and an icosahedron.
The website http://www.korthalsaltes.com/ has several examples of nets that are print-ready. Having
students fold nets is a great way for them to become familiar with solids.
Kinesthetic Learners
Have students work in groups of two or three. Students will need graph paper, plain paper, rulers, tape
and colored pencils. Either assign a solid or let students choose a solid to work with. You can provide
students with a model of a solid if you have them. Then students are going to create three different things:
1-3 cross-sections of the solid, a net for the solid, and an actual model of the solid using the net. You can
decide if you would like groups to do more than one solid as well or mini-presentations to the class.
Visual Learners
Students might have a hard time with cross-sections. Use the visual models mentioned about to help
students see how cross-sections can change in the same solid. For example, with a rectangular box (prism),
use a piece of paper to represent the cross section. See how many different cross sections students can come
up with. For a rectangular prism, there should be at least three different rectangles as cross-sections. Then,
move on to the triangular prism, which should have two different triangles and two different trapezoids as
cross-sections.
Modifications
Give students pictures of all the polyhedra discussed in this lesson. If there is time, use Euler's Theorem
on a few of these polyhedra so students are more familiar with the theorem. In addition, consider giving
students a vocabulary sheet so that they can fill in necessary words for throughout the chapter.
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The only Platonic Solid that students are probably familiar with is the cube. They are all interesting
solids, but other than just showing them to students, we don't do much with them. Don't dwell on these
solids when presenting them to students.
Have nets pre-cut for students to use.
Surface Area of Prisms and Cylinders
Visual Learners
When showing students the difference between lateral faces and bases of prisms, use a physical model.
Also, make sure that you show students a prism lying down, so that the solid is not standing on one of the
bases. Students need to see that, no matter the orientation, the lateral faces will always be the rectangles.
This can be a common error for students when it comes to finding the volume of prisms.
Visual/Kinesthetic Learners/Collaborative Learning
When introducing surface area and lateral surface area; use nets. You have two options here: give students
paper copies of nets or physical models so that they can make their own net. Either way, once students
have cut out the net, they will need to use their knowledge of area from the previous chapter to find the
total surface area. Repeat this process with several solids. Students can work in groups of 3-4 for this
activity.
Visual Learners
When introducing cylinders, find physical models that you can bring into the classroom. Canned food,
soda cans, or a Quaker Oats container are all great examples of cylinders. Have each student bring in a
soup can or other food can (beans, vegetables) to class. Then, take off the label to show them that the
lateral area of a right cylinder is a rectangle. After this investigation, donate the cans of food to a local
food bank. You should offer some sort of extra credit if you decide to do this. And, make sure to put the
labels back on.
Kinesthetic Learners/Collaborative Learning
Give each group one of the physical models mentioned above. Have them find the circumference of the
bases, the height, and the total surface area. Students may need rulers or string to measure their cylinders.
Have students use the string to measure the circumference of their cylinder. Once they have wrapped the
string around the cylinder, they will need to lie it flat and then measure the distance with a ruler. To
compare, have students find the surface area using the formula in the text and using a "string" measurement
of the circumference. For the "string" measurement of the circumference, students will need to reorganize
the formula for surface area of a cylinder. If C = 2nr 1 then the formula becomes SA = Cr + Ch.
Modifications
Review the area formulas at the beginning of the period. You may want to put any necessary area formulas
on the board for reference.
Divide this lesson into two class periods or two mini-lessons. If doing this within the same class period,
give students a break after going over the surface area of a prism. After 5 minutes, resume with the surface
area of a cylinder and the rest of the lesson.
Make sure students understand the difference between a right prism and an oblique prism. You may need
to use physical models for this. Also, it is important that students understand the difference between
lateral area and total surface area.
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Surface Area of Pyramids and Cones
Auditory/Visual Learners
Discuss the difference between a slant height and the height of a pyramid. When using a net, the slant
height will no longer be slanted, but flat. In the picture below, / is the slant height.
Collaborative Learning
After going over the parts of a pyramid and finding the slant height, break students into groups. Give each
group a different pyramid: triangular based, rectangular based, or square based. (There are no regular
polygon bases, other than triangles and squares, in this chapter.) Have groups create a net for their
pyramid and then find the surface area.
Visual/Kinesthetic Learners
Find snow cone paper cones (online or a party store) and give each student one. Go over the different
parts of a cone. Then, have them cut the cone along the slant height and force the cone flat. This will
become their net. At this point, have students figure out the formula for the surface area. We know that
the base is a circle, so that portion is nr 2 . The "flat" cone will be harder to find.
9/rrr
Notice that the circumference of the base is the same as the red arc of the flattened piece. We will have to
use ratios to find the area of the flattened piece. First, this piece is a sector of a circle with radius /. Let's
set up a ratio.
2nr portion of circumference for cone r
2jii ~
circumference of circle with r = /
/
This is the portion of the whole circle. So, if we multiply the area of the circle by our ratio, we will have
the area of the sector/flattened piece of the cone.
-M)
Tirl
Adding this to the base circle, we get the total surface area of a cone: nr 2 + Tirl.
This derivation might be a little challenging for some students. Decide how much of it you would like to
cover in class.
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Modification
Even though there is a formula for the surface area of a pyramid, students do not need to memorize it. It
might be easier for students to visualize (or draw) the net of the pyramid and then use the formulas for
the area of a parallelogram and triangle that they are already familiar with.
Write the formulas learned in this lesson on the board.
Volume of Prisms and Cylinders
Visual/Kinesthetic Learners/Collaborative Learning
Use the physical models of prisms and cylinders from the previous two lessons. Have students split into
groups and find the volume of one of these solids. Students may need to measure the height and the
dimensions of the base. After groups have found the volume of one solid, have them switch so that every
group has the opportunity to see all the solids.
Visual Learners
To demonstrate Cavalieri's Principle, show students a stack of books, like the one in the text. Then, slowly
slide the books so that the pile is leaning. Try to use books that are all the same size, like textbooks.
Explain to students that no matter how the pile is arranged, it will always have the same volume. You
could even have students come up to rearrange the books.
Collaborative Learning
At the end of the lesson, have students work in groups of two or three to compare a cylinder and square
based prism with the same dimensions. Give students the following picture.
8 in
T
10 in
i
8 in^.,8 in
8 in
f3> t
10 in
10 in
The last prism has diagonals of 8 inches. Ask students which one of these solids has the greatest volume.
Then, have students find the volumes and compare.
V,
cylinder
v,
prisml
n4 z • 10
8-8-10
V"m2 = 4V2-4V2-10
: 160tt ~ 502.65 irf
640 in 3
320 in 3
The first prism has the largest volume, followed by the cylinder, then the last prism. If the bases were all
drawn together, the largest square would be on the outside, the circle would be inscribed in it and then
the base of the other prism would be inscribed in the circle.
If you would like to extend this problem, have students make to scale nets of the three solids and see if
they can fit them together like the picture to the right.
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8 in-
Modifications
Review finding the area and circumference of a circle.
Write out the formulas for volume of prisms and cylinders on the board. Encourage students to focus
memorizing the second formula for a prism, V = Bh. This way, when they are introduced to the volume
of a prism in the next lesson, the only thing they have to add is the o to the formula. Also, if students
thing "area of the base times the height," they will remember that they have to find the area of the base,
no matter what the base is.
Write out the steps for finding the volume of composite solids.
1. Break each composite solid into its smaller solid parts.
2. Select the correct formula for either surface area or volume based on the problem.
3. Find the volume of each smaller solid.
4. Add or subtract the results of the volume based on the question.
Volume of Pyramids and Cones
Visual/Kinesthetic Learners/Collaborative Learning
Use the physical models of pyramids and cones from the previous lessons. Have students split into groups
and find the volume of one of these solids. Students may need to measure the height and the dimensions
of the base. After groups have found the volume of one solid, have them switch so that every group has
the opportunity to see all the solids.
Collaborative Learning
At the end of the lesson, have students work in groups of two or three to compare a equilateral triangle
based pyramid, a square based pyramid, and a cone with the same dimensions. Give students the following
picture.
L/\
1
8 in
8 irh * in
8 in
8 in
Ask students which one of these solids has the greatest volume. Then, have students find the volumes and
compare.
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V rnnp = -tt4 2 • 8 » 134.04 in 3
' cone <-> -
1
3
1 /l
y square pyramid — -8 • 8 • 8 « 170.67 in
3
V equilateral A pyramid - - (- • 8 • 4 V5) • 8 * 73.81 m 3
The square pyramid has the largest volume, followed by the cone, then the equilateral triangle pyramid.
If the bases were all drawn together, the square would be on the outside, the circle would be inscribed in
it and then the equilateral triangle.
If you would like to extend this problem, have students make to scale nets of the three solids and see if
they can fit them together like the picture to the right.
8 in-
Modification
Investigation 11-1 can be a little messy but a very valuable activity. You may want to have students work
in pairs. You can extend this investigation to the volume of cylinders and cones. Give students a snow
cone cup and then have them design a net of a cylinder around that.
Go back over the list for finding the volume of composite solids that was generated in the last lesson. Leave
all the formulas for area and volume on the board for students to reference. Decide whether you want to
leave them on the board for test and quizzes.
Surface Area and Volume of Spheres
Visual Learners
Begin by showing students some spheres of different sizes. You could use a baseball, a globe or a basketball.
Visual/Kinesthetic Learners
Often the measurement of a ball is given according to the diameter. Use a 14-inch basketball, for example,
to show: the center, radius, diameter, a chord, secant line, tangent line, the surface area and volume.
After the students have worked with the surface area and volume formulas, give groups the physical models
that you brought to class. Have them find the surface area and volume of these solids.
Kinesthetic Learners/Collaborative Learning
You will need some Styrofoam balls and aluminum foil for this activity. Students should work in pairs.
Give each pair a ball. Have them measure the circumference and use that to find the radius. Then, give
each pair a piece of aluminum foil. Students need to form the aluminum foil around the ball without
overlap and as little wrinkles as possible. Once it is perfectly covered, students need to carefully remove
the aluminum foil and lay it flat. This is equivalent to the surface area of the Styrofoam ball. See if
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students can cut the aluminum foil into a rectangle with dimensions of 2jit by 2r. This way, the area of
the rectangle would be 47rr 2 , just like the formula for the surface area of the sphere.
Modifications
You may need to review circles with your students. Do this by comparing what is the same and different
between the parts of circle and the parts of spheres.
Give students a handout for their notes and/or examples in the text.
Extension: Exploring Similar Solids
Visual/Kinesthetic Learners/Collaborative Learning
Give students the two nets of the rectangular prisms below.
Have students cut out the nets (you can enlarge or shrink these, the larger net is twice as large as the
smaller net) and put them together. They should measure the lengths of the sides and find the scale factor
and the surface area of each solid. Then, have students find the volume of each solid and compare the
volumes.
You could also have students build similar solids using Legos or other type of building block. Just use the
unit sized building blocks.
Modifications
Decide how much of this lesson you would like to cover. This is an extension, so check with your state's
standards to see how much of the lesson is necessary.
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4.12 Rigid Transformations
Exploring Symmetry
Visual Learners/Collaborative Learning
Ask each group to come up with an example to explain line symmetry and rotational symmetry. When
finished, have each group share their images. Then move on to the next part of the activity. Ask each
group to draw half of an image that has line symmetry and rotational symmetry. Students can use objects
in the room to help them brainstorm which image to draw for each. Then they are going to pass their
papers to a group near them. The next group must finish the drawings according to each description.
When finished, allow time for sharing.
Kinesthetic Learners
Let students get out of the classroom and walk around the school grounds. Have students search for exam-
ples of reflectional and rotational symmetry in nature, construction, or just around the school. Students
should either take pictures of their examples or draw pictures.
Visual/Kinesthetic Learners
Give students handouts with 3-4 pictures of figures with rotational symmetry. Let students cut out the
images and physically have them rotate the images to determine the number of lines of rotational symmetry.
To be consistent with the text, only use images where the center of the object is also the center of rotation.
Here are a few pictures that you can use.
Lines of Rotation: 8, 10, 3, 7
Angle of Rotation between each Line: 45°, 36°, 120°, 51.43°
Modifications
Give students lots of pictures of examples of line and rotational symmetry. The best way for students to
understand symmetry is through examples and real-life.
Make a notes handout for this lesson and the entire chapter. Have students fill in vocabulary and examples.
Write down important information on the board. Leave it up for the duration of the chapter.
Translations
Auditory Learners/ELL
Have students generate synonyms for translation. Possibilities are move and shift.
Kinesthetic Learners/Collaborative Learning
Turn this lesson into a discovery. Give students the following set of questions/directions and graph paper.
These questions are from the examples in the text. This presentation of material is an alternative to what
is in the text and not to be done in conjunction.
1. Plot the points 5(1,2), <2(4, l),/?(5,4) and £"(2,5) on graph paper. Connect the points to form a
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square.
2. Now, move each point to units to the left and three units up. What are the new coordinates? Label
them S',Q',R', and E\
3. Plot the new points from #2 on the same set of axes as #1. What do you notice?
4. A translation rule is written (x,y) — > (x±a,y±b), where a and b are the amount the figure moved.
What is the translation rule for questions 1-3?
5. Draw a new set of axes. Plot A(3,-l),5(7,-5) and C(-2,-2).
6. Move each point to the left 4 units and up 5 units. What are the coordinates of A\B\ and C"?
7. Plot A\B\ and C". Write the translation rule.
8. Draw a new set of axes. Plot T{— 3, 3),/?(2, 6),/(— 2,8). Connect these to form a triangle.
9. Plot r'(3, -l),/?'(8, 2), and /'(4, 4) on the same set of axes. Do these two triangles look congruent?
10. Find TR,RI, TI, T'R f \RT ', and TV using the Distance Formula. Now are these two triangles congru-
ent?
11. What is the translation (movement) from I to T? R to R f ! I to VI Are they all the same? If so,
then we can write a translation rule of this triangle. Write it.
Answers
1-3. See graph. The two squares are congruent.
4. The translation rule is (x,y) — > (x - 2,y + 3)
5-7. The translation rule is (x,y) — > (jc - 4,y + 5)
i±rf<4i
C'
<TX
T- ^s
+b
-^t
SSI _*.
:±^s_ j
J!^[_ir*
-
,± r<? -q
■^^^
_-4^Ll_!^^._
-4
i
8. See graph.
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9. See graph. Yes, these triangles seem congruent.
10.
TR = ^(-3 - 2) 2 + (3 - 6) 2 = V34
T'R' = A /(3-8) 2 + (-l-2) 2 = V34
RI = ^(2-(-2)) 2 + (6-8) 2 = V20
R'l' = ^(8-4) 2 + (2-4) 2 = V20
77 = A /(-3-(-2)) 2 + (3-8) 2 = V26
r /' = ^(3-4) 2 + (-l-4) 2 = V26
The triangles are congruent by SSS .
11. The movement from each to its prime is down 4 and to the right 6. Because all the translations are
the same this is considered a translation. The rule is (x,y) — > (x + 6,y - 4).
Modifications
The alternative lesson presentation above could be considered a modification. Give students a print-out of
the 11 questions. Decide if you would like student to work in pairs or individually. Students will also need
rulers and graph paper.
Reflections
Visual/Kinesthetic Learners
Use the patty paper construction below to create reflections.
Tools Needed: Patty paper, pencil, ruler
1. Draw a picture on the left side of the patty paper.
2. Draw a line of reflection to the right of your image.
3. Fold the paper on the line of reflection.
4. Trace your original image on the other side of the line of reflection.
5. Open the paper to view your transformation.
This construction can be done with the x-axis and y-axis by drawing a set of axes on the patty paper.
Students can also do double reflections on patty paper by repeating the steps above a second time. De-
pending on the thickness of the graph paper that you use, it could be folded and the images traced to
perform reflections on the coordinate plane.
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Modifications
Let students do examples and homework problems on tracing paper.
Print out the homework problems for students to take home. Then, they can do the reflections on this
handout by folding the paper on the lines of reflection.
Rotations
Visual/Kinesthetic Learners
To prepare this activity, you will need to prepare some small squares on 3 x 5 cards. Draw several (enough
for a class set) squares and pass them out to your students. Have them cut out the squares and decorate
them. Now, students will need graph paper. Have them draw a large set of axes that takes up the entire
paper. Then have each student draw their square somewhere on the paper. They can draw the square
wherever and with any orientation.
At this point, you tell each student to rotate their square 90° counterclockwise. If they need to, they can
use their cut-out square to help with the rotation. After the rotation is complete, have them exchange
papers with a neighbor. Make sure everyone has a new paper. Then, have students rotate this new square
180°. If you would like, you can have students exchange one more time. This time, see if students can
figure out the correct rotation needed to place the image back on the original square.
Modifications
For investigation 12-1, give students a cut-out of AABC to work with. Then, on a plain piece of paper, have
them draw a center of rotation, R, and trace AABC on the paper. They can continue with the investigation.
At Step 4, have students use the cut-out and rotate it the 100° so that B' lines up with the angle drawn
in Step 3. When students like the placement of the cut-out triangle, have them trace it to create AA'B'C.
Composition of Transformations
Visual/Kinesthetic Learners/Collaborative Learning
Students need to work in pairs for this "game." Give each pair a plain piece of paper and have them fold
it into 16 congruent squares (three vertical folds, three horizontal folds). Make the START square and the
END square at opposite vertices of the square (see picture below). Place a triangle in the START square.
Then, students will reflect the triangle over the folds. Students will alternate turns. The student whose
reflection lands in the END square wins. Students will need to use strategy for this game. Students can
only reflect over the horizontal and vertical lines.
Start
A
End
Variations on this game: make more squares (32 or 64), reflect over the points of intersection of the folded
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lines, rotations of 90°, 180° or 270° around the closest point of intersection, and combining rotations and
reflections.
Modifications
Decide if the Reflections over Parallel Lines Theorem and the Reflection over Intersecting Lines Theorem
are too challenging for your students. You might want to show your students these theorems through an
example, but not require them to memorize the properties.
Make Investigation 12-2 a teacher-led investigation. Give students a copy of the finished Example 7 and
then start the investigation at Step 2.
Extension: Tessellating Polygons
Collaborative Learning
Do this activity after going over the definition and a few examples of tessellations. Students will work
in groups. You will need to give each group an example of a regular polygon. Give each group their
polygon and tell them that they will need to prove whether each one tessellates or not. Students need to
demonstrate that it has no gaps, no overlapping shapes, that the entire plane is covered in all directions.
Allow students time for this exploration and then have students share their work. From the lesson they
will discover that only equilateral triangles, squares and regular hexagons tessellate.
Visual/Kinesthetic Learners
Have students create a tessellation with regular octagons and squares. There will have to be two regular
octagons and one square around each point. Encourage them to add color and design.
Modifications
Write the steps of how to tessellate a polygon on the board.
Decide if you want to cover this extension in your class. Reference your state's standards to see if it
required part of Geometry.
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Chapter 5
Basic Geometry TE - Problem
Solving
Problem Solving in Geometry
Problem solving and applications are particularly challenging for many students. Let the students know
that this is difficult for most people and that with time they will get better at it. They are probably going
to struggle, have to reread the information several times, and may be confused for a while. It is all part of
the process. This section will give them strategies to work through the difficulties without giving up.
Highlight Important Information - It is nice when students can actually mark up the text of the
exercise, but frequently this is not the case. As they read the paragraph have the students take notes or
organize the information into a chart or diagram. Otherwise the students can just get lost in all the words.
Translating from English to math is often the hardest part.
The Last Sentence - When the students are faced with a sizable paragraph of information the most
important sentence, the one that asks the question, is usually at the end. Advise the students to read the
last sentence first, then as they read the rest of the paragraph they will see how the information they are
being given is important.
Does This Make Sense? - It is so hard to get the students to ask themselves this question at the end
of a word problem or application. I think they are so happy to have an answer they do not want to know
if it is wrong. As you examples in class, model the process of checking the validity of the answer. Give
examples of obviously wrong answers-like a hypotenuse that turns out to be shorter than a leg in a right
triangle or angles in a triangle that don't add up to 180 degrees. Many students really don't know how to
determine whether their answer is reasonable. Keep reminding them and giving examples. Sometimes it
is possible to not accept work with an obviously wrong answer. The paper can be returned to the student
so they can look for their mistake. This is a good argument for the importance of showing clear, organized
work.
Naming Quadrilaterals - When naming a quadrilateral the letter representing the vertices will be listed
in a clockwise or counterclockwise rotation starting from any vertex. Students are accustomed to reading
from left to right and will sometimes continue this pattern when naming a quadrilateral.
The Pythagorean Theorem - Most students have learned to use the Pythagorean Theorem before
Geometry class and will want to use it instead of the distance formula. They are closely related; the
distance formula is derived from the Pythagorean Theorem as will be explained in another chapter. If they
are allowed to use the Pythagorean Theorem remind them that it can only be used for right triangles, and
that the length of the longest side of the right triangle, the hypotenuse, must be substituted into the c
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variable if it is known. If the hypotenuse is the side of the triangle being found, the c stays a variable, and
the other two side lengths are substituted for a and b.
5.1 Basics of Geometry
Points, Lines and Planes
Project-Global Architecture
The objective of this activity is for students to recognize the postulates connected with points, lines and
planes in real life architecture.
Students should find and use pictures of at least two different structures from at least two different countries
to identify an example of each of the following postulates:
• Line Postulate - There is exactly one line through any two points.
• Plane Postulate - There is exactly one plane that contains any three non-collinear points.
• Postulate - A line connecting points in a plane also lies in the plane.
• Postulate - the intersection of two distinct lines will be a single point.
• Postulate - the intersection of two planes is a line.
Students can either print the pictures, use computer displays or slides.
Students should be encouraged to use mathematical language as they describe and write about each
example of a postulate.
Students may also wish to draw sketches or draw and label points or lines on printouts of the pictures.
Some students may require assistance in locating appropriate structures to analyze and in identifying the
postulates. It may be helpful for teachers to use a sample structure and analyze it as a class. The level of
assistance given while students complete their work is entirely up to the individual teacher.
Basic Assessment:
This rubric contains the objectives that students should meet or exceed. Teachers are encouraged to adapt
this to meet the needs of their students and assign their own point values.
Mathematical Process - Students should correctly identify points, lines and planes in both of their
chosen pieces of architecture. All five postulates should be illustrated using their choices with at least
three identifiable in each building.
Mathematical Communication - Students should use the terms points, lines, planes, segments, etc.
correctly in their discourse and all notations should also be appropriate and correct.
Presentation and General Communication - Students should demonstrate an appropriate level of
writing skill. Complete sentences and clearly communicated thoughts are essential.
Meeting the Requirements - Students should analyze two different buildings from two different countries.
The printed photographs should be clear enough for them to identify the postulates.
Going Above and Beyond - Some students are bound to do more than is expected and here are some
ways in which they may achieve this and perhaps earn bonus points.
• Finding all five postulates in each structure.
• Analyzing a third structure.
• Word processing their work and/or creating additional diagrams to illustrate their work.
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Segments and Distance
Project-My Town
The objectives of this project are for students practice measuring distances using different tools, to under-
stand and apply the ruler postulate and segments addition postulate to measurement and to use endpoints
to identify distances on a coordinate grid.
Students are assigned the task of using measurement to design their own town.
Students will need rulers, pencils, colored pencils and chart paper.
Each town needs to have the following buildings in it: a post office, a police station, a bank, a park, a
school and some houses. The students can expand this list if they choose.
Each town map needs a scale to determine the distances from one building to another building. This scale
could be expanded to include standard and metric measurement.
Each students needs to develop a key that shows the measurements from one building to another.
Distances must be actual distances measured with rulers and matched to scale. This could be expanded
to include both standard and metric measurement.
After the design has been completed, the students need to write a series of directions and distances for
someone to travel around their town between the various buildings and locations. The distances should be
clear enough for any other student to follow.
Students pair up with a peer to check and make sure that student directions/measurements are accurate.
Finally, allow students time to share their designs in small groups or with the entire class.
Adapting the Project
Teachers may wish to allow students to work in pairs if the students expand the town appropriately and/or
expand their work to include both metric and standard measurements.
Teachers may choose to award bonus points for extra effort including more locations, more detailed drawing
or inclusion of both standard and metric units.
Basic Assessment:
Observe students as they work on this assignment and provide assistance as needed to make sure that
students are correctly measuring distances using the grid or a ruler.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Scale
• Accurate measurements
• Clear and correct directions
• Inclusion of all essential buildings
Angles and Measurement
Activity- Angle Hunt
The objectives of this activity are for students to understand and identify rays, to understand and classify
angles and to understand and apply the protractor and angle addition postulates.
Print copies of the figure below for students to use in this activity.
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Students will need rulers, colored pencils or markers and protractors.
First, students take the drawing and find ten different angles. They should name each angle with three
letters.
Next, they make a list of each of the ten angles and classify each.
Then, they apply the protractor postulate to measure each of the ten angles.
Finally, they apply the angle addition postulate and create four different combinations of angles to calculate
total measures.
When finished, pair up students and have them check each other's work.
Each student needs to provide their peer partner with verbal and written feedback.
Then allow students time to share feedback in small groups.
Basic Assessment:
Teachers should create a rubric to grade each students work. Key components of the project that should
be included in the rubric are:
• Were ten angles labeled and identified?
• Were the angles correctly classified?
• Are the measurements of each angle accurate?
• Did students successfully use the angle addition postulate?
• How well did students provide feedback to their peers?
Midpoints and Bisectors
The objectives of this project are to understand and identify congruent line segments, identify the midpoint
and bisector of line segments, understand and identify congruent angles and to understand and apply the
Angle Bisector Postulate.
Activity-Revolving Door Design
Students are going to be assigned the task of designing their own revolving door.
Show students the Wikipedia site. http://en.wikipedia.org/wiki/Revolving_door
Discuss the importance of revolving doors and the design used. Point out that the diagram of the revolving
door has four wings to it and four congruent angles. Discuss with students the measures of the angles,
bisectors and midpoints in the diagram on the site.
Students will design a revolving door with at least six wings in it.
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Students will need compasses, rulers, protractors, pencils, and paper.
They can choose to add more wings, but the revolving door needs to have at least six in it.
Here are the specifics of the assignment:
• Design a revolving door with at least six wings.
• Each angle must be congruent.
• Label each angle measure using a protractor.
• Identify line segments that are bisected.
• Identify the midpoint of each line segment.
• Label each part of the revolving door and demonstrate congruency.
Basic Assessment:
Assess student work by thinking about each of the following points.
• Were the students successful in executing a design that matches the specifics of the assignment?
• Are the angles of the wings congruent?
• Are the angle measures labeled?
• Are all midpoints and bisectors identified?
• Is it clear that students understand the concepts discussed in the lesson?
Angle Pairs
Activity- Visualize It
The objectives of this activity are for students to understand and identify complementary and supple-
mentary angles, understand and utilize the Linear Pair Postulate and to understand and identify Vertical
Angles.
Students are going to go on a search for different types of angle pairs. This can be done in the classroom,
but it would be best to expand it to the entire school or outside.
If possible, allow the use of digital cameras. If this is not possible, students can draw sketches of the places
where they locate each type of angle pairs.
They will need rulers, pencils, chart paper, clip boards.
Students need to locate three examples of each of the following:
• Complementary Angles
• Supplementary Angles
• Vertical Angles
Students must write a description of each example pair and explain why it is a complimentary, supple-
mentary or vertical angle pair.
Print and display student work. If possible, display the pictures with "hidden" descriptions so that other
students in the class can circulate the room and test their knowledge trying to identify the pairs and
checking their answers. Another idea is to have students create booklets with the pictures and "lift the
flap" answers like a children's book.
Basic Assessment:
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Have students work in groups to assess each other's work. Request that students read each description of
the angle pair to be sure that it describes each angle pair in mathematical terms.
You want to see that students are using measurements such as 90° for complementary angles, and that
they are demonstrating that vertical angles are congruent.
Students should have three appropriate examples of each type of angle pair in their project.
Students should be clearly expressing themselves in their written explanations and using appropriate
mathematical terminology.
Classifying Polygons
The objective of this activity is to practice identifying and classifying the various types of polygons such as
convex and concave, classifying polygons by the number of sides they have and identifying and classifying
the particular types of triangles based on their angles and side lengths.
Activity-Polygon Sort
This activity requires students to sort polygons in multiple ways. To prepare this activity, you will need to
create or copy different polygons. You want an assortment of polygons and non-polygons, convex polygons,
concave polygons, triangles (acute, obtuse, right, scalene, isosceles and equilateral) and regular polygons
(i.e. quadrilaterals, hexagons, etc.).
There are a number of things you can now do with these figures. Here is one idea:
Make categories on the boards and have students tape up or use magnets to place the figures appropriately.
Possible sorting ideas:
• Polygons vs non-polygons
• Concave vs convex
• Classification based on number of sides-use the names
• Triangle sort (put up the categories of triangles based on angles and sides). This will be a good one
to have students discuss since some triangles may fit in multiple categories. This could lead into
creating a tree or Venn diagram of triangles.
Basic Assessment:
This activity is really more of the formative variety. This will allow you to determine whether or not your
students understand the basics of polygons and help you refine their understanding. With that in mind,
you should do the following during the activity:
Assess student understanding by checking each "sorting exercise"
Also ask different student for feedback about why they "sorted" their polygons the way that they did.
Allow time for feedback and student questions.
5.2 Reasoning and Proof
Inductive Reasoning
Activity-Pascal's Triangle
The objective of this activity is to recognize patterns and number patterns, extend and generalize patterns
and write a counterexample to a pattern rule.
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Students will be using a diagram of Pascal's Triangle like the one below for this activity
1
1 1
1 2 1
13 3 1
14 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
You may also wish to look at other diagrams such as the one on the website: http://en.wikipedia.org/
wiki/Pascal°/ 27s_triangle
Students are going to practice their inductive reasoning skills an make conjectures about patterns they see
in Pascal's Triangle.
Students should work in small groups and may wish to use color on the triangle to identify different
patterns. Allow students time to really explore the patterns in the triangle.
Students should eventually develop a rule for the triangle.
Once students have found the rule, they need to do the following:
1. Write the rule in complete sentences as if they are writing directions for someone to construct the
triangle.
2. Write out the next two rows in the triangle.
3. Demonstrate two ways that you know your rule is accurate. Students may do this by recreating the
diagram and using colored pencils or markers to mark the diagram.
4. Conjecture two other patterns in the triangle.
5. Share your results with the class.
Teachers may extend this project by having students do some research on Pascal and the other pat-
terns/properties of the triangle. The website given above is a great place to start but encourage students
to look further and practice their research skills by going to other websites as well.
Basic Assessment:
It is helpful to do some research on Pascal's Triangle and the many patterns within it before assigning this
project. Students may initially struggle to find a pattern. They may not have done anything quite like this
before and may have a hard time understanding how to approach the problem. Try not to help them too
much, but encouraging them to look at "sums" or "rows" may give them a hint. Throughout the activity
check in with the groups to make sure they are on the right track.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Correct rule found and communicated clearly.
• Next two rows of the triangle were found correctly.
• Two demonstrations verifying the rule were completed.
• Two additional patterns found and clearly described.
• Presentation.
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Conditional Statements
The objective of this activity is to recognize if-then statements; identify the hypothesis and conclusion of an
if-then statement; write the converse, inverse and contrapositive of an if-then statement and to understand
a biconditional statement.
Activity- Advertisements
Students are going to use newspapers, magazines and online advertisements for this problem solving ac-
tivity.
Begin the activity by talking about how advertisers use conditional statements to lure people into pur-
chasing their products. For example, a phone company will often offer a free phone for a cell phone plan.
Provide several examples of advertisements with conditional statements. Ideally, some are explicitly if-then
statements and some must be re-written as such.
Tell students that their assignment is to use newspapers, magazines and the internet to find one such
conditional advertisement.
Then they are to take that advertisement and create a display using it to show the converse, inverse,
contrapositive and biconditional statement of the advertisement.
Students can decorate and design their display.
Allow time for students to share their work when finished and display them in the classroom.
Basic Assessment:
You may wish to have students bring in several advertisements and help them select the best one. Some
students may have trouble picking out the conditional statement and you want to be sure they have selected
something usable before they continue.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Students have selected a conditional statement in an advertisement.
• Correct statements written.
• Creativity in display and presentation.
Deductive Reasoning
The objective of this activity is to recognize and apply some basic rules of logic, including the Law of
Detachment, the Law of the Contrapositive and the Law of Syllogism.
Activity-Logic Puzzles
Students will use the following series of statements in this activity.
If we don't get sand in the car, then we are not sandy.
If we go to the beach, then we build a sandcastle.
If we aren't sandy, then we didn't build a sandcastle.
If it is summer, then we go to the beach.
If we get sand in the car, then Mom will be mad.
It is summer.
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Students will use the Law of the Contrapositive and rearrange the statements in order to use the Law of
Syllogism and the Law of Detachment to make a conclusion.
Students will then write their own logic puzzle:
• In groups, students will create a series of at least 4 statements that can be linked using the Law of
Syllogism
• At least one statement should be written as a contrapositive and then the statements should be
mixed up.
• Students should identify the initial hypothesis and state that it is true.
• Each group will present their puzzle by either writing it on the board, on poster paper or creating a
slide for a Powerpoint presentation for the other groups to solve.
Basic Assessment:
It is a good idea to find other examples of logic puzzles such as the ones written by Lewis Carroll, the author
of Alice in Wonderland. A lesson and many examples can be found online at: http: //www. math. hawaii.
edu/~hile/mathlOO/logice.rrtm It may help students to do a few of these as a class before assigning this
activity.
Students may need help identifying which statements need to be rewritten as a contrapositive and may
need some guidance. Try to let them puzzle it out a while but if they need help a hint is to suggest that
students start with the statement, "It is summer" and then follow the statements, changing them to a
contrapositive as needed.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did all students participate in the process?
• Were students able to reorder the given statements and reach the correct conclusion?
• Did each student contribute at least one conditional statement to the group's puzzle?
• Did students create a solvable puzzle that contained at least four statements and one contrapositive?
• Did groups make a reasonable attempt to solve their classmates' puzzles?
Algebraic and Congruence Properties
The objective of this activity is to identify and apply properties of equality and congruence and use these
properties to solve problems.
Activity-Match It Up
Students are going to create a matching game that they can then play in small groups.
Each small group needs to create a pair for each of the properties. One card will have the name of the
property on it, and the match will be a numerical example, and or a geometric example.
Students should write out an actual example of the property and not just variables as they did in class.
An example is a card that has: if x — 7, then x + 5 = 12 and a second card with: Addition Property of
Equality.
Students will use index or small cards, pens, rulers, etc.
After the cards have all been created, have one group exchange with another group and play that team's
game.
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When finished, ask the teams to give each other feedback on the examples used.
Here are the properties to use:
Reflexive Property
Symmetric Property
Transitive Property
Substitution Property
Distributive Property
Addition Property of Equality
Subtraction Property of Equality
Multiplication Property of Equality
Division Property of Equality
Reflexive Property of Congruence with segments and angles
Symmetric Property of Congruence with segments and angles
Transitive Property of Congruence with segments and angles
Basic Assessment:
This activity has two parts. During the first part, it is very important to watch carefully to make sure that
groups create workable examples for their games cards. During the second part, it is important to observe
whether or not students can recognize the properties and make correct matches. They may also find their
classmates' errors. Allow time for students to provide feedback and corrections to their peers.
Proofs about Angle Pairs and Segments
The objective of this activity is to create, mark and interpret diagrams that represent the postulates learned
thus far in the course.
Activity-Name That Postulate!
This is a game. The students will create the game cards and then a "Jeopardy" kind of game can be played
in the large class or in small groups.
Students are assigned the task of creating an index card with a diagram that represents each postulate.
Students should use diagrams and also standard marks for segments and angles in their examples.
There are twelve postulates, so if there are twenty- four students in the class, each postulate would be
represented by two different diagrams. You need to assign the students the postulates to make sure all are
made and to avoid too many repeats.
Allow time for the students to create their diagrams and then use peers to check each other's work for
accuracy.
When finished, collect the cards and play the game with the students.
The postulates to be represented are:
• Postulate 1-1
• Postulate 1-2
• Postulate 1-3
• Postulate 1-4
• Postulate 1-5
• Ruler Postulate
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• Segment Addition Postulate
• Protractor Postulate
• Angle Addition Postulate
• Right Angle Theorem
• Same Angle Supplements Theorem
• Same Angle Complements Theorem
Remind students that they should have all of these with examples in their notes to reference.
Basic Assessment:
Assessment is made easier with this activity because the students will be playing the game. You will be able
to see who understands the postulates and who doesn't. Having students check each other's work before
playing the game will provide additional opportunity for students to review the material and will help
to catch any errors. You can help add any corrections when playing the game and looking at each game
card. This activity allows opportunity for formative assessment. A rubric could be created to assign points
for the cards created, but the real objective here is to insure understanding of marking and interpreting
diagrams and recognizing the postulates.
5.3 Parallel and Perpendicular Lines
Lines and Angles
Activity-School Map
The objectives of this project are to practice identifying parallel lines, skew lines, and perpendicular lines,
to use the Parallel and Perpendicular Line Postulates and to identify angles made by transversals.
Students are going to use parallel lines, perpendicular lines and skew lines to create a map of the school.
You can begin this lesson by looking at a fire exit map of the school or another building to give them a
concrete example of what it is that they will be creating.
The students are going to work in groups of three to create a map of the school.
Students will need chart paper, rulers, yard sticks, colored pencils.
Take students on a walk around the school to begin taking notes for their design.
Notes: If your school is very large, you can either challenge students with the whole school or select a floor
to design. This could create a complete school map in the end with groups putting their "floors" together.
Also, if you have access to a device that can be used to actually measure the distances in the building,
students could practice creating a "scaled" map by actually measuring the lengths of the hallways. Even
a pedometer could be used to do this.
Have students draw and design their map.
When finished with the map, students must identify two sets of parallel lines and two sets of perpendicular
lines.
Extension- you can extend this activity even further by asking the students to identify an example of the
Parallel Line Postulate and an example of the Perpendicular Line Postulate on their map. You may wish
to have them discuss why there would be no skew lines on their two-dimensional map, even though there
may well be hallways in the building that are skew.
Allow time at the end for the students to share their work.
Create a class display of student maps.
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Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Do the students have a good understanding of parallel lines?
• Of perpendicular lines?
• Of skew lines?
• Are students able to identify an example of each postulate?
• Is the map neatly drawn using a straightedge?
• Is the map an accurate representation of the building (or portion thereof)?
• Were the lengths scaled correctly (this only applies if you have them actually measure distances?)
Properties of Parallel Lines
Activity- Airport Map
The objective of this activity is to identify and use properties of corresponding, alternate interior, alternate
exterior and consecutive interior angles formed when two lines are intersected by a transversal.
For this problem solving activity, students are going to use an aerial map of Logan International Airport
in Boston, Massachusetts. Use the following link to get the map:
http : //en . wikipedia . org/wiki/File : KBOS_Aerial_NGS . j pg
Students need to use the picture to draw their own version of the map. You may wish to project the image
in the classroom for them to "copy."
Then using color, they need to identify the following:
In red, two parallel lines and a non-perpendicular transversal.
In blue-two corresponding angles
In green-two alternate interior angles
In orange-two alternate exterior angles
In purple-two consecutive interior angles
When finished, you can extend this by having the students use a protractor to determine angle measures.
Allow time for students to share their work.
Basic Assessment:
Teachers should begin assessing this activity while students are working on it by observing and assisting
throughout the process. This is an excellent opportunity to assess understanding and help students correct
themselves. Points may be awarded for following directions and completing the assignment correctly. The
following questions may help you decide how to award points.
• Is the diagram complete and labeled correctly?
• Was the student able to present his/her work to the class?
Proving Lines Parallel
Activity-Forest Tower
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The objectives of this activity are to use the Converse of the Corresponding, Alternate Interior, Alternate
Exterior and Consecutive Interior Angles Theorems.
Students are going to design a tower to be used in National Park by a Forest Ranger. You may wish to
begin this activity by showing pictures of forest towers found on the internet and explaining the uses of
forest towers. For example, Forest Rangers use towers to spot forest fires and observe various animals in
the park. The following is a link to a picture of one such lookout tower:
http : //en . wikipedia . org/wiki/File : Unity_Ranger_Stat ion_lookout_tower . jpg
The students are going to need to design the tower so that the four posts of the tower which support the
platform are parallel and are connected or braced by transversals.
They will need to demonstrate how the angles of the transversal prove that the tower poles are parallel.
You must use what you have learned about angles to design the tower and prove that the poles are parallel.
Students will need protractors, rulers, chart paper and pencils.
All angles must be labeled and measured.
• Look at Figure 03.03.01 to get an idea of a possible example.
When finished, allow students time to share their work.
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did the student create a platform atop four parallel poles with transversals connecting the posts for
support?
• Is the diagram completely and correctly labeled?
• Are the angle measures correct?
• Were the students able to demonstrate that the poles are parallel?
• Overall present at ion-both the diagram and the verbal presentation to their classmates.
Properties of Perpendicular Lines
The objectives of this activity are to identify congruent linear pairs of angles, indentify the angles formed
by perpendicular intersecting lines and to identify complementary adjacent angles.
Activity-Rug Angles
In this problem solving activity, students are going to identify the angles formed by perpendicular lines.
Show students the sample activity rug design found at the website:
http : //www . homedec orators . com/detail . php?parent id=26996& ; aid=bzrt
Ask students to use this figure as an example to create and draw their own rug design.
They can use color and different sizes of polygons on the rug. The big point is to be sure that they are
using perpendicular lines on the rug.
In the design, the students need to include the following:
• two linear pairs
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• two sets of perpendicular intersecting angles
• one set of complementary adjacent angles
These rugs can be designed on chart paper or graph paper-you can leave it up to the students on the size
of the rug.
Allow time for the students to share their work when finished.
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Are there two linear pairs?
• Are there two sets of perpendicular intersecting angles?
• Is there one set of complementary adjacent angles?
• Are all of the perpendicular lines accurate (truly perpendicular)?
• Is there anything missing in the design?
• Is the design visually appealing?
Parallel and Perpendicular Lines in the Coordinate Plane
Activity 1-Wheelchair Ramps
The objective of this activity is to use the concept of slope in a real world application problem.
Students are going to use what they have learned about slope to design a wheelchair ramp.
A wheelchair ramp must have a maximum slope of ^ ft- Discuss with students why this is true and refer
to the website: www.newdisability.com/wheelchairramp.htm before beginning the problem below:
A new home has a front door that is 3^ ft off of the ground. A wheelchair ramp needs to have a slope of
Y2 ft. Based on this fact and on the height of the door, design a wheelchair ramp that will work for this
new home. Show all of your work in your diagram.
How far away from the house will the ramp begin? What could you do to meet the maximum slop
requirements if the sidewalk is not far enough away from the door to start the ramp there? In other words,
there isn't enough space for a straight ramp to the door?
Allow time for the students to work on this dilemma.
Then tell students that every unit on a coordinate grid represents one foot. Ask them to draw their
wheelchair ramp on coordinate grid.
When finished, allow time for students to share their work.
An extension to this activity would be to find a wheel chair ramp in the school and determine its slope.
Students will have to work together to figure out a way to measure the ramp slope.
Basic Assessment:
The main purpose of this activity is to help students connect what they are learning in the classroom
to what they see in the world. Before the activity is completed students should have demonstrated the
following:
• Students show that they can use what they have learned about slope to determine the rise and run
of the ramp.
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Students will demonstrate an understanding of how the slope of a line impacts the rise and run of
the line.
Is the rise and run of the ramp accurate-it should be 42 feet?
Is this drawn accurately?
Is it graphed correctly on the coordinate grid?
Allow time for students to share their work and offer feedback and correction when necessary.
Activity 2-The Park Path
The objectives of this activity are to identify and write equations in slope-intercept form, identify equations
of parallel lines, identify equations of perpendicular lines and to identify and write equations in standard
form.
Students are going to be designing a path for a park using Parallel and Perpendicular Lines as described
in the problem below.
You are on a team that is designing paths through a local park. The team has cleared some of the brush
and has created one path through the park. Here is the path graphed on the coordinate grid. Note the
locations of attractions and facilities in the park. Students cannot make their lines go through existing
buildings, playground equipment or into the pond.
It is recommended that the diagram below is reproduced for students to work with directly. They may
wish to use colored pencils or markers to show their new paths.
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The equation for this path isy = -^x + 2.
The team needs to draw in two more paths. The first one will be parallel to this one, and the next one
will be perpendicular to this one. The objective is to have these two additional paths connect the original
path to the attractions.
Use what you have learned to draw these three paths on a coordinate grid. Use your problem solving skills
to name the equation of each line. Be sure to write your equations in slope-intercept form.
Students should present their plans to the class and be ready to explain why they chose to put their paths
in particular places.
Basic Assessment:
There is more than one correct solution to this problem. Students should be able to justify their particular
plans.
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Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Is there a correct equation for a line perpendicular to the original line?
• Is there a correct equation for a line parallel to the original line?
• Have students managed to connect the three attractions using the three paths?
• How well have students solved the problem? In other words, how well did they place their lines to
allow for easy access by path to each of the attractions?
The Distance Formula
Activity-Park Distances
The objective of this activity is to use the distance formula to find distances in the coordinate plane.
Students will revisit the park from the last activity and calculate the distances between the attractions if
the paths are constructed as shown on the map below.
playground
Students should calculate the distances between the attractions based on the map above. Each unit on
the grid represents 10 ft. Find the following distances:
• Refreshments to Playground
• Playground to Pond
• Pond to refreshments
Once students have calculated these distances an extension would be to discuss the two possible ways to
link the attractions in the park as seen in these two activities. Which way might be better? Why? Is there
a third way to make the paths such that they are even more practical (i.e. distance visitor must travel
between attractions is minimized?) This discussion could take many tracks from here to discuss traffic
flow patterns encouraged by path construction and benefits and drawbacks of minimizing distance between
attractions.
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
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• Were the distances calculated correctly?
• Did the student participate meaningfully in the discussion?
5.4 Triangles and Congruence
Triangle Sums
Activity- Triangle Sums
The objective of this activity is to practice identifying interior and exterior angles in a triangle and applying
the Triangle Sum Theorem. Students will also practice justifying each conclusion they make and practice
logical reasoning develop skills required for writing proofs.
Provide students with a copy of the figure below
Students are going to work in pairs with this figure.
Part one-students need to figure out the angle measures for each of the missing angles.
Part two-students are going to write down the theorem that helped them to figure out the measure of each
angle.
• For example: a = 30°, by Triangle Sum Theorem.
Allow time for the students to explain their work in small groups.
Next, have each group make its own triangle angle puzzle. Each group's puzzle must require use of each
of the following (you may wish to add additional requirements):
• Triangle Sum Theorem
• Exterior/Interior Angle pair (linear pair)
• Vertical Angles
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Is the student's work accurate?
• Can the student talk about why each theorem is appropriate for each angle measure?
• Does the group's new puzzle contain the specified angle pairs?
• Is the new puzzle solvable?
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Congruent Figures
Activity-Congruent Triangles Game
The objectives of this game are to demonstrate student understanding of congruence in triangles, congru-
ence statements, the Third Angles Theorem and to explore properties of triangle congruence.
This game involves a bit of prep by the teacher. You first need to create sets of three cards such that each
student will get one card of a set. Students will be using these cards to figure out which other two students
in the class are in their set. Here is a sample set of three cards:
• Card 1- a congruent triangle
• Card 2- it's matching triangle
• Card 3- a congruence statement about the two triangles.
M
AABC = ALMN
Create enough cards so that each student receives one. Be sure that your triangles have tic marks on them.
Add some challenge by using repeated letters in the diagrams and congruence statements. Students will
need to pay close attention to the order of the angles in the congruence statement to find the correct
matches.
Then shuffle and hand out the cards.
Students need to move around the room and find their matches. Each group should have three students
in each group when finished.
This is a fun activity that has a lot of movement in it. You can reshuffle and do multiple rounds if desired.
Basic Assessment:
This is a great formative assessment activity. You will see very quickly which students understand and
which students are struggling. By the end all groups should be correct. Listen to student conversation as
they work, are the students talking about properties of congruence? Allow time for the students to share
the strategies that they used to find each other.
Triangle Congruence using SSS and SAS
Activity- Triangle Measurement
The objective of this activity is to the distance formula to analyze triangles on a coordinate grid and apply
the SSS postulate of triangle congruence.
Students are going to use the distance formula to analyze triangles on a coordinate grid and solve the
following problem.
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You wish to design a triangular garden in your yard. Your neighbor wants an identical triangle garden in
her yard too. You are going to use the same dimensions to build your neighbor a garden. To plan your
work, plot your garden out using a coordinate grid. Here are the coordinates for your triangular garden.
(-3,7), (-1,5), (-4,1)
If you use these dimensions, and move the plot five units to the left and three units down, what will the
coordinates of your neighbor's garden be?
Students can work on this problem in pairs or individually.
Have the students draw out the coordinate grid with the two triangles on them.
Allow time for the students to share their work when finished.
Extension-students can transfer these triangles to measurement. For example, they could use 1 foot for
every one unit on the coordinate grid.
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Correctly plotted triangles
• Correctly calculated lengths of the segments
• Did students explain how their calculations prove that the two triangles will be congruent?
Triangle Congruence using ASA, AAS and HL
Activity-Double Triangles
The objectives of this activity are to understand and apply the SSS, SAS, ASA, AAS and HL Congruence
Postulates. Students will also practice writing two-column proofs in this activity.
For this activity, create single triangles with certain pieces of information indicated. For example one
triangle might have all three side lengths indicated. Another triangle would have two sides and the
included angle measure given. A third might be a right triangle. There should be enough triangles created
for each group of four students to have two triangles to work with. All five of the congruence theorems
should be represented in the selection of triangles.
Each group of four will choose two of the triangles that have been drawn to work with. The students in the
group need to use the given information to draw two triangles that are congruent to the selected triangles.
They should mark their triangles with the information they used to create the congruent triangles (i.e.
SSS, SAS, AAS, ASA or HL) Each group will now have two pairs of Congruent Triangles.
Now change the groups into new teams of two pairs of students. Each pair on the team selects one set of
Congruent Triangles with which to work. You may wish to select the teams or allow students to choose
partners. You may also wish to decide with pairs get which triangles.
Now, the pairs on each team quiz each other. Each pair needs to prove that their triangles are congruent.
The other pair can question and challenge any claims that the teammates use to justify congruency.
When the team has successfully proven that their triangles are congruent, the team can write a two-column
proof with their statements and reasons in it. Each team will have two proofs to complete.
Basic Assessment:
This can be a formative assessment without a rubric or grade. You may also wish to give a grade for
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the final products-the triangles and the proofs. The following components should give an idea of how to
develop rubric for this activity:
• Were the groups able to make congruent triangles?
• Did the teams work together to figure out their proofs?
• Are the proofs accurate?
Isosceles and Equilateral Triangles
Activity- Triangle Proofs
The objective of this activity is to prove the Converse of the Base Angles Theorem. Students will be using
a variety of theorems in the process and develop their proof skills as well.
Provide students with the Base Angles Theorem: If two sides of a triangle are congruent, then the angles
opposite the congruent sides are congruent. Have students write out the converse of this statement.
Next, assign students the task of proving that this converse is true.
The students need to draw a diagram involving isosceles triangles to demonstrate this theorem.
Then, they need to write a two-column proof for the Converse of the Base Angles Theorem. The given
information is that the two base angles are congruent. Students must now work through the proof to show
that the triangle is isosceles. (They may need the hint to add the altitude from the vertex angle of their
isosceles triangle.)
When finished, the students are going to present their work in small groups. Allow time for the students
in other groups to provide feedback. Each group may have done this a little differently. This is a great
way to show students that there is often more than one way to prove something.
Basic Assessment:
Groups may need assistance with this activity. Try to help steer the groups in the right direction without
giving away the steps of the proof.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did the group make an appropriate diagram?
• Does the proof have the correct statements and reasons?
• Are the group members able to explain their thought process and reasoning in words?
5.5 Relationships with Triangles
Midsegments
Activity-Race Day Route
The objective of this activity is to practice using the midpoint and distance formulas and to apply the
Midsegment Theorem to a real world situation.
Students will need colored pencils and rulers for this activity.
Sally, Nick and Luis are helping out at water stations along a race route for a local fundraising race. The
race is taking place in a local park. The race route is shown on the coordinate grid below. Racers will
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begin at A, turn at B and C and finish back at A. Reproduce the image below for students to use.
10
Use the distance formula to determine the perimeter of the triangle. This perimeter is the length of the
race. If each unit on the grid represents 200 m, find the length of the race to the nearest 0.5 km.
Now, there are to be three water stations, X, Y and Z, on the race route. X will be equidistant from points
A and B, Y should be equidistant from points B and C and Z will be equidistant from points A and C. Find
the coordinates of points X, Y and Z. Show all your work and label these points on your diagram.
Sally and Luis are at point X. Nick is at point Z. All of the runners have passed Sally and Luis and they
are closing up their station. They need to take their extra cups to Nick at point Z. Sally and Luis need to
get to Nick but must either go to point A or point Y first to pick up additional cups. Sally thinks that is
faster if they go to Y and then to Z. Luis disagrees and proposes that they go to A and then to Z. Mark
these routes on your diagram in different colors.
Sally and Luis decide to split up and each take their own route. Who travels farther? Support your answer
with algebraic calculations.
Now, prove your answer to the last question using the Midsegment Theorem. You may do a two-column
proof or a paragraph proof.
Basic Assessment:
Students should determine that the same distance will be travelled by both Sally and Luis. They should
use the distance formula to show this algebraically. They will prove this using the Midsegment Theorem.
There are many possible proofs.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Correct race length determined in the correct units (7.5 km)
• Points X, Y and Z found using the midpoint formula. (-4, 2.5), (-1, 6.5) and (2, 0), respectively.
• Correct distance travelled by Sally and Luis-they should be the same (2.4 km)
• Proof is complete (example proof below).
In order to prove that Sally and Luis travelled the same distance, we need to prove that XY-\-YZ = AX+AZ.
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We will start with the fact that X, Y and Z are midpoints of the sides of the triangle.
Table 5.1:
Statements
Reasons
1. X, Y and Z are midpoints of AB,BC and CA 1. Given
respectively
2. AT and YZ are midsegments
3. AX = XB and AZ = ZC
4. A£ = AX + XB and AC = AZ + ZC
5. A£ = AX + AX and AC = AZ + AZ
6. A£ = 2AX and AC = 2AZ
7. AX = lAB and AZ = \AC
8. XF = ±AC and YZ = ±AB
9. XY+YZ= \AB + ±AC
10. XY+YZ = AX + AZ
2. Definition of Triangle Midsegment
3. Definition of Midpoint
4. Segment Addition Postulate
5. Substitution
6. Distributive Property
7. Division Property of Equality
8. Triangle Midsegment Theorem
9. Substitution
10. Substitution
Perpendicular Bisectors and Angle Bisectors in Triangles
Activity-The Garden Dilemma
The objectives of this lesson are to practice using algebra skills (midpoints, slope, equations of lines, solving
systems) and construction tools and connecting them to the practice of finding the centers of circumscribed
and inscribed triangles in a real world problem solving activity.
This project may be difficult for some students with weak algebra skills and they may need significant
assistance in completing the work correctly. Allow time to help students work through this problem and
help them as needed. This problem may also be adapted to solve using only patty paper constructions
rather than algebra.
The environmental club at Triangle High School wants to make a triangular shaped garden in one of the
courtyards. The club begins by plotting the desired triangular garden shape on a grid as shown below.
Students will need two copies of the diagram below (one for each part of the activity), a straightedge, patty
paper and a compass.
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Part 1 — The club needs to figure out where to place a single sprinkler that sprays water in a circle such
that it will water the entire garden. Use algebra skills to locate this point.
Note to teacher - Students should make the connection that this point is the intersection of the perpendic-
ular bisectors of the triangle. You may wish to discuss this problem as a class and come up with a list of
steps that students will follow to locate this point. Essentially, they will need to find the intersection of
the equations of two perpendicular bisectors in the triangles. Then find the intersection of these lines.
Part 2 — The club wants to make a path leading from each side of the garden to a bench in the "center"
of the garden. Each path should be the same length. Find and mark the desired location of the bench and
where the paths will originate on each side of the triangle. Draw the paths on your diagram.
Note to teacher - Students should recognize that this point would be the intersection of the angle bisectors
of the triangle. They may need help constructing the two perpendicular bisectors required to locate this
point and determining where on each side of the of the triangle the path with originate. You may wish
to have them trace the triangle onto patty paper and reference the investigation 5-4: Constructing Angle
Bisectors in Triangles. Once they have drawn the angle bisectors on the patty paper, then they can lay
the patty paper on top or the original triangle and locate the point where they intersect on the grid.
Extensions - (1) Figure out the cost of the pavers for the paths students will put in leading from each edge
of the garden to the bench. (2) Determine the radius of the water spray necessary to water the garden.
Basic Assessment:
Again, students may need significant help with this activity, particularly with the algebra. You may wish
to have each group submit one final piece of work. This will allow students to work together, and compare
answers. This will also help students who struggle with accuracy using patty paper or the construction
tools. This is particularly useful if you group the students such that each group has at least on strong
algebra student and one who is good at making accurate constructions.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
Part 1
• Are all steps included and accurate in the algebra?
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• Is the center of a circumscribed circle (the point of intersection of the perpendicular bisectors in the
correct location?
The midpoint of LM is (1, 2). The slope of LM is -|, so the slope of the line perpendicular to LM is |.
Therefore the perpendicular bisector of LM is y = |x + |.
The midpoint of MiV is (2, -5). The slope of MN is ^, so the slope of the line perpendicular to MTV is -7.
Therefore the perpendicular bisector of MN is y = — 7jc + 9.
Now the point of intersection can be found by solving a system. Since both equations are equal to y we
can conclude: |x + | = -7x + 9. Solving this equation gives us x = 1 and then y = 2. (Note that this
is the midpoint of the hypotenuse since the triangle is a right triangle. Students may have used the third
perpendicular bisector instead of one of these but they should still arrive at the same solution.)
Part 2
• Are all necessary markings visible for the construction of the angle bisectors?
• Is the point of intersection of the angle bisectors in the correct location?
• Did students correctly identify the points on the sides of the triangle where the paths will originate?
The center of the inscribed circle should be at approximately (-1.5, -1.5). The points or origination of the
paths (where the inscribed circle touch the triangle) should be approximately (1, 2), (-5.5, 2), (-1, -5.5).
Keep in mind that these are approximations and that students may have slightly different answers so allow
a little room for variance.
Medians and Altitudes in Triangles
Project-Hanging Triangles
The objectives of this project are to use medians in triangles to locate the centroid and use this point in a
practical application as the center of gravity (or balancing point) of the triangle.
Begin by referring back to the Know What problem in this section. The hanging triangle problem described
there is exactly what students will do in this project. You may wish to talk out the process before setting
students to work on making their triangles and find the circumcenters. Your discussion might help them
avoid problems later-such as, if they cut out their triangles before finding the centroid then they will have
trouble constructing the perpendicular bisectors of the sides to find the midpoints. It might help if you
make a sample ahead of time as well because then you may discover other issues that hinder progress and
then you can help students avoid them as well.
Students will need several pieces of cardstock in a variety of colors, a piece of string, scissors, a straightedge
and a compass.
Students will make at least four different triangles and locate their centroids.
Next, students will punch a hole in the triangle at the centroid and thread the string through the hole.
Students may tie a knot in the string to keep the triangle in place. Students will continue this process
until they have all four (or more) triangles strung on the string.
The triangles should remain relatively parallel to the floor when this is done accurately. These mobiles can
then be hung in the classroom.
Basic Assessment:
This can be a really fun project, particularly for students with an interest or talent in art. Some students
may need significant assistance with the mechanics of the constructions.
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Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did the student make at least four triangles?
• Did the student use correct construction techniques to locate the centroids of the triangles?
• Do the triangles remain horizontal to the floor when the mobile is suspended? You may wish to allow
some room for error here.
Inequalities in Triangles
Activities-Prove the Theorem and Name that Inequality
The objectives of this activity are to determine relationships among the angles and sides of triangles, to
apply the Triangle Inequality Theorem and to apply the SSS and SAS Triangle Inequality Theorems.
Part 1 - Name that Theorem
In this problem, students need to use the information given to draw a diagram and prove the theorem.
Students are going to prove the theorem that states that the angle opposite the longest side of a triangle
with unequal sides will have the greatest measure.
In AABC with AB < BC, the measure of angle A is 80°. The measure of angle C is 40°, and the measure
of angle B is 60°.
Sketch a diagram of this triangle. Given the theorem on side lengths and angle measures, can this be a
true statement? Why or why not?
Provide students time to work on this problem. Students need to provide a diagram, a written explanation
and a verbal explanation to explain their thinking.
When finished, allow students time to present their work to the class.
Part 2 - Name that Inequality
This problem solving activity is a game. The preparation for this game is to prepare two Congruent
Triangles for each group to work with. These triangles should have measurements indicated on them.
Students will work in groups of four-two teams of two. Each pair is a team that plays against each other.
When the students play, their goal is to "stump" the other team. The play begins like this, one team comes
up with a problem for the other team to solve. For example, "If I lengthen side AB what else changes in
the triangle?" or "If the measure of LB is decreased then what else changes in the triangle?"
Then the other team has to answer the question. If they answer it correctly, the team receives a point. If
not, the other team gets a point.
Then they repeat the process by switching team positions. Both teams play until time is up.
Students need to be encouraged to use the SAS Triangle Inequality Theorem and the SSS Triangle In-
equality Theorem as well as the converse Theorems.
Students can create as many different types of questions as they would like. Students can be very creative
in their approach to writing questions.
Basic Assessment:
Teachers may wish to create a rubric where each piece of the assignment is worth points. Exactly how
many points each component should be worth is left up to individual teachers. The following is a list of
possible components:
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Part 1: Students should have a correctly marked sketch and some variation of the following explanation.
The statement is true because the length of BC is longer than the length of AB. The angle opposite BC is
the greatest of the three angles in the triangle. It measures 80°. Therefore, the theorem is accurate and
proven through this statement.
Part 2: Students may need help getting started with this one. They may need a little assistance coming
up with the first few questions until they get the hang of it.
Walk around as students play and assist students when necessary. Offer suggestions and challenge students
to create difficult questions. Notice which students are having difficulty with the assignment and then offer
assistance and coaching.
Extension: Indirect Proof
Activity- Write Your Own Proof
The objective of this activity is to develop an indirect proof.
Assign students the task of creating a situation for which an indirect proof can be used to prove a conjecture.
These could be algebraic, geometric or just situational. Below are two examples you may wish to share
with the class before they begin working on their own.
Here is one possible example for an algebraic problem.
Marcy is selling candy bars for the school band. She starts out selling five bars. But in the end, she sells
three times as many as her friend John does. The band teacher congratulates her on selling over forty
candy bars. If John sold less than 12 bars, prove that Marcy did not sell more than forty bars.
We can use algebra skills to prove this one. Let x represent the number of candy bars Marcy sells. First
we start by assuming that Marcy did sell more than 40 bars or that x > 40. Now, let the number of candy
bars that John sells be represented by y. Since John sells less than 12 bars, y < 12. Now, we know that
Marcy sells three times as many candy bars as John, soi = 3y. Now, 3y < 36 since y < 12. Sox< 36 by
substitution. This contradicts the statement that x > 40, so what we assumed is false and Marcy did not
sell more than 40 candy bars.
Here is one possible example for situational problem.
Teresa is younger than Leslie and Kathryn is younger than Teresa. Prove that Leslie is older than Kathryn.
First, we assume the opposite of what we are trying to prove is true. In this case, we assume that Leslie
is not older than Kathryn. If Leslie is not older than Kathryn, then Leslie is younger than Teresa. This
contradicts the fact that Teresa is younger than Leslie. Therefore, what we assumed is false and Leslie
must be older than Kathryn.
Here is one example for a geometric problem.
Two angles form a linear pair. If the first angle is acute, then the second angle must be obtuse.
Assume that the second angle is not obtuse. This means that m/2 < 90°. If the angles form a linear pair
then mil + m/2 = 180°. If m/2 < 90°, then by subtraction, mil > 90°. This contradicts the fact that the
first angle is acute. Therefore, what we assumed is false and the second angle must be obtuse.
Basic Assessment:
Teachers may wish to create a rubric where each piece of the assignment is worth points. Exactly how
many points each component should be worth is left up to individual teachers. The following is a list of
possible components:
• Do students begin their proof by assuming the opposite of what they are trying to prove?
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• Do students use valid forms of reasoning to reach a contradiction?
• Students should then deny the assumption and determine that what they are trying to prove is
actually true.
• Originality
5.6 Polygons and Quadrilaterals
Angles in Polygons
Activity-Make a Garden
The objective of this activity is to practice using the sum of the interior angles formula and the sum of the
exterior angles for a polygon.
Students will need a copy of the diagram below for this activity.
Label your diagram with the following information:
mlAFE = 108°
mlEDC = 110°
mlFEZ = 70°
mlDCY = 60°
mlFAB = mlABC
Use this information to find all the other angle measures in the diagram.
Now, XY, YZ and ZX to be paths in a park. Your job is to describe to the gardener how to make paths
CD,AB and FE if the points for A, C and E are already marked on the paths. Use the angles measures you
found in your description.
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did students label their diagram correctly?
• Were students able to use the interior sum formula and the exterior angle sum to find the unknown
measures? They may also use the supplementary relationship between adjacent pairs (one interior,
one exterior) of angles.
• Did students accurately describe how to make the paths to the gardener?
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Angles measures should be:
mlFED = 110°
mlCDY = 70°
mlBCD = 120°
mlFAB = mlABC = 136°
mlXAB = mlXBA = 44°
mlZFE = 72°
Directions should read something like this. Standing at point C and looking toward point F, turn 60°
clockwise to make path CD. Standing at point A and looking towards point F, turn 136° to make path
AB. Standing at point E and looking towards point Z, turn 70° clockwise to make path EF.
Properties of Parallelograms
Activity-Show the Properties
The objective of this activity is two-fold. First, students will be reviewing and verifying the properties of
parallelograms. Second, they will be practicing algebra skills and using a protractor to do so.
Students will need graph paper (large graph paper that can be posted in the room is ideal), protractor,
ruler and colored pencils.
Give the students the coordinates of the vertices of the parallelogram THOM, with r(-2,4),//(4, 7), O(2,0),M(-4, -3).
Note to teacher: You may wish to make up additional sets of vertices so that each student/group has a
different parallelogram. In doing so, you could make some of them special parallelograms that could be
revisited later in the chapter to identify other special properties of those figures.
Instruct students to plot these points on their graph. Once the parallelogram is graphed, students should
verify that each of the following properties is true using algebra or a protractor (no rulers except to be
used as straightedges). Students should mark their parallelogram with the information they find using a
different color for each property.
• Opposite angles congruent (protractor)
• Consecutive angles supplementary (addition)
• Opposite sides congruent (distance formula)
• Opposite sides parallel (slope formula)
• Diagonals bisect each other (midpoint formula)
Allow students to share their work with one another
parallelogram with which to work.
Basic Assessment:
especially if each student/group had a different
Students may need significant assistance using the formulas and perhaps a review of the formulas before-
hand. Use this opportunity to observe student work and verify that all students are able to use the formulas
correctly and show that the properties are true for their parallelogram.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
Angle measures correct: 132° and 48° (allow for slight variance)
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• Opposite side lengths correct: V53 and 3 V5
• Side slopes correct: | and \
• Midpoints correct: (0, 2)
Proving Quadrilaterals are Parallelograms
Activity-Is it really a Parallelogram?
The objective of this activity is to prove a quadrilateral is a parallelogram given opposite sides are congruent,
opposite angles are congruent, the diagonals bisect each other or that one pair of opposite sides are both
congruent and parallel.
For this activity, students will work in pairs. One student in the pair will need to cut two strips of paper
that are the same length and two strips that aren't. The other student of the pair will cut two sets of
congruent strips.
Then have the students attach the four strips of paper together at the ends with fasteners. This will form a
quadrilateral. Explain to the students that some quadrilaterals are parallelograms and some aren't. Then
divide the students into groups.
Students need to come up with ways to demonstrate the following points using their moveable figures.
These points will help students to see how to prove that a quadrilateral is a parallelogram or that the
shape that they have created is NOT a parallelogram.
Use a ruler to compare opposite sides.
Use a protractor to compare opposite angles.
Add pairs of adjacent angles to see if the Supplement Theorem applies.
Show how the number of degrees in a quadrilateral is the same as a parallelogram by using the Triangle
Sum Theorem.
When students are finished working in pairs, give them time for each group to demonstrate one property
to the rest of the class.
Ask students to write what they have learned in their notebooks.
Basic Assessment:
Check student work for accuracy as they are working. Listening to them explain their reasoning to their
partner is often a great way to identify misunderstandings. Offer feedback during their discussions in their
pairs and during their presentations as necessary.
Rectangles, Rhombuses and Squares
Activity-Can you prove it?
The objective of this activity is to have students use algebra skills to check whether properties of rhombuses,
rectangles and squares can be shown to exist in quadrilaterals plotted in the coordinate plane. In this
process students will identify the relationships between diagonals in rectangles and rhombuses.
To prepare this activity, you will need to choose sets of four vertices that can be connected to form a
quadrilateral. Some should form rhombuses, some should be rectangles and some should be close to one
of these figures but neither a rhombus nor a rectangle.
The students are going to need to figure out if the figure is a rectangle or a rhombus or does it just look
like one. Students will be using the principles that they learned in the text to determine whether the figure
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is really a rectangle or a rhombus. Students can work in pairs or small groups on this activity.
In a rectangle, the students should:
1. Use the distance formula to prove that the diagonals are or are not congruent.
2. Use the slope formula to prove that the diagonals are or are not perpendicular.
3. Decide whether their figure is a rhombus, a rectangle or a square.
Allow time for the students to investigate and prepare to prove what their figure is or is not.
Then allow time for each group to present their discovery.
Basic Assessment:
This activity could be used as either a formative or summative assessment. If you choose to do this as
a formative assessment, then it is recommended to check on students as they work, provide assistance as
needed and note whether or not students seem to comprehend what it is that they are doing. If you choose
to do this activity as a summative assessment (graded for accuracy), then it is recommended that you
do a couple examples with the whole class, review the formulas needed and review the properties before
assigning the task. One idea would be to do the activity as a formative assessment individually and then
have pairs work on a second figure for a grade. If the work is assessed, the following list of key components
will help to determine a score.
• Was the correct length of each diagonal found?
• Was the correct slope of each diagonal found?
• Did the student make the correct conclusion regarding the lengths of the diagonals?
• Did the student make the right conclusion regarding the angle at which the diagonals intersect?
• Was the student able to assimilate this information and correctly categorize the figure?
• Was the student able to clearly and correctly communicate his/her process and findings to the class?
Trapezoids and Kites
Activity- Trapezoidal Shelters?
The objective of this activity is to use the properties of an isosceles trapezoid to prove that a quadrilateral
is an isosceles trapezoid. Students will be using the properties of the base angles the diagonals.
Veronica, Nick and Santi are on a camping trip. They are sitting outside of the sheltered picnic area and
wondering whether or not the shelter structure (roof and support beams) form an isosceles trapezoid.
Between them they have a roll of string and a piece of chalk to work with. (You may wish to allow their
characters to have other common camping supplies-i.e. pocket knife, compass, etc. This may result in even
more creative problem solving ideas.
They know that the roof line is parallel to the ground, thus ensuring that the figure outlined is a trapezoid.
In pairs, have students think about and describe a way to show:
1. The diagonals are congruent
2. The base angles are congruent
Students should draw a picture of the situation on large paper and illustrate their strategy to measure the
diagonals and the angles.
When pairs are finished, allow time for sharing.
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Basic Assessment:
Students may have a hard time with this problem at first because they don't know where to start. Try
to point them in the right direction by asking questions rather than telling them one way to solve the
problem. This is a great way to get kids to think outside the box and be creative about problem solving.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Were students able to prove that the diagonals are congruent? One possible way to do this would
be to use the string. One end can be held at the base of one of the supports and the other at the
opposite vertex. Mark the length of the string required to stretch this distance and check to see if it
is the same as the length required to connect the other pair of opposite vertices.
• Were students able to determine whether the base angles are congruent? Students could use the
string and the chalk to "construct" congruent angles. If the string is tied around the chalk and the
string is anchored to the point where the support meets the ground, students can mark points on
the support and ground that are equidistant from the intersection (just like they use a compass to
mark points on the sides of an angle that are equidistant from the vertex). Making sure that they
are using the same amount of string, they should make marks on the other support post and ground
(the second base angle). Now, they can use the string to check whether or not the distance between
each pair of marks is the same. If it is, then the angles are congruent.
• Students may come up with other solutions. Use your discretion.
• Did students illustrate the problem and their solution?
• Were students able to communicate their solution to the class?
Activity-Quad Design
The objective of this activity is to have students create, identify and classify quadrilaterals as parallelo-
grams, rectangles, rhombuses, squares, kites and trapezoids.
This is a creative design activity. The students are going to need colored pencils, crayons, markers, rulers,
large blank sheets of paper.
The task is to create a design that has at least one of each of the seven figures in it. Students need to
create a color key to identify the seven figures in the design. They can include any other shapes/color that
they would like, as long as the seven figures are in the design.
Here are the figures that must be in the design:
Parallelogram
Rhombus
Rectangle
Square
Kite
Trapezoid
Isosceles trapezoid
Allow students significant time to work on this activity. Students should be encouraged to be creative in
this activity.
When they finish, have students share their work with the rest of the class and then display the artwork.
Basic Assessment:
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Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Are all seven figures represented?
• Are the figures drawn accurately? How "exact" you require them to be is up to you, but students
should be using rulers, protractors and compasses as needed to make their figures accurate.
• How creative is the picture? This one is rather subjective so you may want to allow "bonus" points
here for exceptional work rather than to deduct points for less original/artistic work.
5.7 Similarity
Ratios and Proportions
Activity-Ratio/Proportion Relay
The objective of this lesson is for students to practice identifying and writing ratios and proportions.
Students will also be using the properties of proportions in this activity.
Since this lesson is mostly review, use this fun game to review the concepts in the lesson.
To prepare, bring in a bunch of assorted items from home. Make these random household items, but have
more than one of each type of items. For example: three hairbrushes or five apples.
Students will play this game in teams of four. For the directions, they will be referred to as players 1,
2, 3 and 4. The activity is timed and will be played in five timed segments. To start, lay out the items
on a table in the front of the room. Each team member needs to do their individual portion of the game
individually, there should be no coaching from teammates.
Segment 1 (20 seconds): Player 1 from each team will come to the table and write down as many proportions
as they can. For example: hairbrushes to apples is 3 to 5.
Segment 2 (20 seconds): Player 2 from each team will come to the table and write down as many proportions
as they can.
Segment 3 (20 seconds): Player 3 from each team will come to the table and write down as many proportions
as they can.
Segment 4 (2 minutes): Player 4 will write down as many correct proportions as he/she can from the ratios
players 1, 2 and 3 wrote down. For example if there were six books and ten forks on the table then player
4 could write: | = yL This is a correct proportion.
Stop at this point to check their proportions and assign points to each group for each correct proportion-two
points per correct proportion.
Segment 5 (2 minutes): All players on the team will take whatever remaining correct proportions they
have and rearrange the proportion using the Corollaries of the Cross-Multiplication Theorem.
Now check their work. Award 1 point for each correct rearrangement. Tally the points and announce the
winner.
This game can be played multiple times with the same groups or new groups. You may wish to adjust the
times according to student ability in subsequent rounds. Students should enjoy this one.
Basic Assessment:
You may wish to assign participation points for this game. You could give the winning team bonus points.
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Similar Polygons
The objective of this activity is for students to recognize similar polygons, identify corresponding angles
and sides of similar polygons from a statement of similarity and to calculate and apply scale factors.
This activity utilizes the jigsaw grouping technique to give students a chance to work in missed groups.
You will need cardstock (or thick paper), pencils, protractors, rulers and scissors.
First, group the students in fours. These groups will make a set of regular polygons: one triangle, one
quadrilateral, one pentagon and one hexagon. Circulate and make sure that students are creating these
correctly. They can be any size, but the angles should be determined using the polygon sum formula
learned in Chapter 6. You may want to review this before beginning. Students should label the vertices of
their polygons in the interior of the figure as they will be cutting them out of the paper.
Now, have all of the triangle creators move to one group, the quadrilaterals in another group, the pentagons
in a third group and the hexagons in a fourth group. If your class is large, have two groups for each figure.
In this second grouping, the students will take two polygons at a time and measure the sides of the figures,
identify corresponding sides and angles and find the scale factor between the figures. Students should
record their findings.
Now have the groups rotate. Students will stay with their group, but the group will move to the next
figure and repeat the last step. Continue until all groups have analyzed the different polygons.
Ask for students to volunteer to share their findings and begin a discussion about regular polygons and
similarity.
Basic Assessment:
This is a great way for students to practice what they know about the measures of the interior angles in
regular polygons, using a protractor and a ruler to measure/draw, identifying ratios and writing propor-
tions. The purpose of this activity is to explore and practice. Use the following guidelines to assess student
learning:
• Students should be noticing that all corresponding angles are congruent in similar polygons.
• Students should be noticing that any pair of sides is proportional since each figure is equilateral.
• Students should realize, either during the activity or during the subsequent discussion, that all regular
triangles are similar, all regular quadrilaterals are similar, all regular pentagons are similar, etc.
Similarity by AA
Activity-Thales and the Pyramids
The objective of this activity is to determine whether triangles are similar and use A A similarity to solve
problems with similar triangles.
This is a great lesson to use Thales and the simple way that Thales measured the pyramids. Essentially,
Thales figured out the height of the pyramids by using his own height. He waited until his shadow equaled
his height. Then he measured the shadow of the pyramid and he knew that the height of the pyramid was
equal to the pyramid's shadow length. It is recommended that you do a little research on Thales and why
he measured the height of the pyramids and share the full story with your class. This is a great way to
make a connection between math and history.
Discuss this technique with the class and how it relates to similar triangles and proportions.
Now have students solve the following problem:
Ralph is 6 feet tall. He goes outside on a sunny day and measures his shadow to be 9 feet long. The
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shadow of a tree in his yard is 18 feet long. Based on these measurements, what is the height of the tree?
Use similar triangles to back up your claim.
For the next part of the activity students will need large paper, rulers and markers.
Now, in pairs or groups, design your own scenario. Write the word problem and draw a picture to illustrate
it on large paper. Work out the solution in your notebook (not on the large paper).
Hang the papers around the room and have each pair/group solve the other problems. Allow a couple
minutes per problem and then have them rotate to the next one. Once all groups have has a chance to
solve all the other groups' problems, have the creators share their solutions.
Basic Assessment:
Students may need help designing the problems. Encourage creativity. You will be able to check their
problems as the other groups go around the room completing them.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Were students are to solve the Ralph problem? Did they use clear and correct reasoning to de-
termine/show that the triangles are similar? They should have identified that the shadow and the
object meet at a right angle and that the angle of the sun is the same in both triangles, therefore,
the triangles are similar by AA similarity. Now, the proportion | = -^ can be solved to determine
that the height of the tree is 12 feet.
• Did students create a solvable problem? Hopefully they all will do so-you may need to check up on
them as they work to make sure this happens.
• Did each group solve their own problem correctly?
• Did all groups attempt to solve all of the other problems? You could have them turn in these solutions
and check them for a part of the assessment.
Similarity by SSS and SAS
Activity- Triangle Jeopardy
The objective of this activity is use SSS and SAS to determine whether triangles are similar and apply SSS
and SAS to solve problems about similar triangles.
To play this game, divide students into small groups. To prepare this game, use a set of index cards and
write one of the ways to prove similarity among triangles on each card. You should have cards that say
SSS, SAS, and AA. Make sure that you have several of each card and then mix them up.
Each student in the group takes a turn. The student selects a card. Then he/she must come up with an
example that illustrates the way to prove similar triangles.
Each team (note that this is not each team member, but each group) can have 1 helpful hint-that is from
you, and 1 lifeline from their group.
If the student completes the challenge correctly, the team receives a point.
You can play this game for quite a while.
Some variations can include scale factor or diagrams on the board and then the group needs to show how
the triangles are similar.
Basic Assessment:
This should be a fun game for the class to play. Assessment comes through the process of the game. Each
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student will be working individually so you will be able to determine their level of understanding quite
readily. Provide coaching/feedback when necessary through "helpful hints".
Proportionality Relationships
Activity-Midsegment Match-up
The objective of this activity is to identify proportional segments when two sides of a triangle are cut by
a segment parallel to the third side.
To prepare this activity, you will need to draw a bunch of triangles and cut them along the midsegment
line.
Then pass out one part of a triangle to each student.
Students need to measure the lengths of the sides of their part of the triangle.
Then, they need to find the student who has their match. Students will walk around the room and find a
match for their triangle. When finished, all of the triangles should be complete.
Once students have found a match, or think that they have, they need to write proportions to justify
their thinking. Each pair of students will then sit down and write the appropriate proportions for their
triangle. These proportions should include the ratios of the sides of the corresponding triangles as well as
the proportions know to be true based on the proportionality theorem.
Allow time for the students to share their work when finished.
After students have finished, you can collect the triangle pieces and repeat the process.
Basic Assessment:
Observe students as they walk around finding the match for their triangle part. Notice if students are
using measurements or not. If not, remind students to look for the proportionality in the measurements of
each section of the triangle. Be sure to listen as students share their work. Are they able to articulate why
the triangles are similar? Is the match that they selected an accurate match? Offer feedback/correction
as needed. Did students write the correct proportionality statements? Did students match up correct
corresponding sides of the similar triangles and use the proportionality theorem correctly?
You may wish to have students record their work (i.e. the proportions) on a sheet of paper that you
can collect at the end to assess. I recommend that you label the triangles and calculate the correct
measurements and write the correct proportions ahead of time to facilitate grading. You may want to
label each part of each triangle differently (maybe use a letter for the "top" and a number for the "bottom"
and record them as pairs so that students can't use your labeling to determine the matches. For example
one triangle might consist of parts A and 6. As long as you record the "pair" you'll know how they go
together.
Similarity Transformations
Activity-Cartoon Dilation
The objective of this activity is to draw a dilation of a given figure. Students will practice plotting the
image of a point when given the center of dilation and a scale factor. Students will also recognize the
significance of the scale factor of a dilation.
Have students select a favorite cartoon character and bring in a small picture of it (the picture should be
no bigger that about 2.5 square inches.
Students will use dilation techniques to enlarge their picture by a scale factor of 3.
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Using a notebook size piece of graph paper, create axes that intersect as close to the center of the page as
possible.
Now, students should center their picture under the origin. Ideally, they will be able to see their picture
through the graph paper (they may need to darken the outline of their character to make this happen.)
Now, students will identify key points on the outline of their character to dilate. They will find the
coordinate of these points and dilate them using a scale factor of 3. For example if the top of the character's
hat is at point (1, 4) originally, then the image would be at (3, 12). If the center of the characters nose is
at (0, 0), then the image is at (0, 0) also.
Students will then connect the dots to make an enlarged version of their character. Students may find
that they need to identify additional key points to make their character look the same as the original but
larger. Students should add color as appropriate.
Once they have finished, have students do at least three measurements and compare them. For example,
they could measure the distance between the eyes of their characters in the original and compare it to the
length in the dilation. They should notice that the distance triples.
Have students put their pictures on the wall.
Basic Assessment:
The results will mostly speak for themselves. If students do this activity correctly the dilation should very
much resemble the original character.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did the students create an image that is three times bigger than the original?
• Is it apparent in their work that students used the technique described above to do so?
• Is the final image similar to the original?
• Did students compare several measurements and illustrate the similarity and scale factor?
• Have the students shown all of their work?
Extension: Self-Similarity
The objective of this activity is for students to develop an appreciation for fractals and to understand the
concept of self - similarity. In this activity students will extend the pattern in a self - similar figure.
If you have access to technology, then use this site to have students watch the different fractals being
formed, http : //en . wikipedia . org/wiki/Fractal
Then tell students that they are going to create their own fractal. They can begin with a triangle, a
pentagon or a hexagon.
Students need to show four levels of the fractal. Remind them of the steps on how the fractal is created.
You may wish to refer back to the example of the Koch snowflake in the text or the animation on the
website.
Then provide them with rulers, colored pencils and paper to create their fractals.
Hint: There are bound to be errors at first. Remind the students to use the text to support their work.
They can also revisit the website to help them to brainstorm ideas and create an exciting fractal.
Leave students alone as much as possible. Allow students time to think and struggle a bit. It will help
them to come to a clearer understanding of the concepts in the lesson.
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Basic Assessment:
Walk around and help students who are really struggling. Be sure to allow students some time to work
before you jump in as it is always better for students to try to solve problems on their own.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• First, does the fractal work?
• Is their work complete?
• Does it show four levels?
• Make notes and share corrections/feedback with students.
5.8 Right Triangle Trigonometry
The Pythagorean Theorem
Activity-The TV Dilemma
The objectives of this activity are for students to employ problem solving skills and the Pythagorean
Theorem in a real world context.
Michael's father wants to buy as big a TV as he can fit in the 5 ft wide, 4 ft high wall space he has in their
basement. Most TV's are made with dimensions in a 16:9 (length to height) ratio, as described in the You
Know What? in this section of the text.
Students will need calculators, writing utensils and paper for this problem.
Determine the diagonal length of the largest TV that will fit in their basement. Round to the nearest
(appropriate) inch. Be careful that if you round up, your TV will still "fit" in the allotted space.
Be prepared to share an explanation of your thought process and work performed to attain your answer.
Have students share their work with each other and provide feedback.
Basic Assessment:
Students may need some guidance getting started with this one because "step by step" directions are not
provided. This is a chance for students to really think about a situational problem and come up with their
own method to solve it.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did students choose an appropriate method to solve the problem?
• Does a TV with their determined diagonal length fit in the allotted space?
• Were students able to describe their process and work to their peers?
• Did students round their final answer appropriately?
Here is a solution. You may allow for other answers to be acceptable as long as they do not exceed this
value, are close to this value and students can explain why they choose their specific length. For example,
students may start by saying that the horizontal length is 59" to be certain that it fits inside 5 ft. Use
your discretion here to decide what is sensible.
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Since the horizontal length cannot exceed 5 ft (60"), I chose to let the horizontal length of the TV equal
60". Using the ratio 16:9 for the length to height of the TV, I solved to find the height of 33.75". Now,
using Pythagorean Theorem, I can find the diagonal length of 68". I rounded down to 68" (the actual value
was 68.8") because I've already set the horizontal length equal to the maximum space. If I were to round
up to 69", then the horizontal length would exceed the allotted space. So, the largest possible TV to fit in
the allotted space would have a diagonal length of 68".
Converse of the Pythagorean Theorem
Activity-Figure Those Triangles!
The objective of this activity is to understand and use the converse of the Pythagorean Theorem to identify
right, acute and obtuse angles from side measures. Students will also use the Pythagorean Triples and
their multiples in this activity.
This is a game that has the students use number cards to figure out whether triangles are right triangles,
acute triangles or obtuse triangles.
To prepare for this game, use index cards and write one number on each card. Number the cards 2-25.
Each group will need a set of cards to play the game.
Divide the students into groups of three to four students.
Students are going to use the number cards to create as many different types of triangles as they can in
each of three rounds. At the end of each round, count how many correct combinations each team has made
and award points.
Round 1: Right triangles. Students are given a short period of time to use the number cards to create as
many different combinations of numbers that will equal right triangles. For example, using 3-4-5, the
students will have a right triangle because 3 2 + 4 2 = 5 2 . Students work together to write out as many as
possible using the numbers 2 - 25. Each card may only be used once.
Round 2: Acute triangles. Remind students that a 2 -\-b 2 > c 2 .
Round 3: Obtuse triangles. Remind students that a 2 + b 2 < c 2 .
Finally have students share their combinations.
Winning teams can be determined based on the number of points accumulated in each round.
Basic Assessment:
You should walk around the room while students play the game and offer assistance as needed. At the end
of each round have each group share their combinations and check each other's work as you check it too.
You may wish to assign a grade for the work students do in this game. Perhaps you expect each group
to come up with at least 3 combinations in each round and they may earn bonus points for additional
combinations. You should determine what the minimum number of combinations expected in each round
is based on the ability of your class. It should be a number that most (or even all) groups can achieve so
that they don't get frustrated.
Using Similar Right Triangles
Activity-The Model Train Problem
The objective of this activity is to practice identifying the similar triangles formed when an altitude is
drawn from the right angle to the hypotenuse of the right triangle. Students will also practice finding the
geometric mean in order to solve for specified lengths in these similar triangles.
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You are trying to build a support for a mountain range on your model train platform. You want one side of
the mountain to have a greater slope than the other side. You supports underneath the mountain surface
look like this:
B
A D 2 in C
Provide students with a copy of this diagram to reference and write on as they complete the problem.
In the diagram, B represents the peak of the mountain range which should be 6 in above the platform.
The supports, AB and Z?C, meet at a right angle at B. D represents a point directly below point B on the
platform (i.e. BD and AC are perpendicular). Label this information in your diagram.
Now, determine the following:
How far from point D is point A?
How long is support AB1
How long is support BC1
Basic Assessment:
Circulate and provide assistance while students are working on this problem. Students are likely to struggle
with setting up the appropriate proportions and/or geometric means. Encourage the students to draw the
three triangles separately and oriented the same way to make it easier for them to identify corresponding
sides. Students may also recognize that they can use the Pythagorean Theorem instead of proportions for
some parts of this problem you can decide whether or not to allow this.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Measure of AD correct? 18 in
• Measure of AB correct? approx. 18.97 in
• Measure of BC correct? approx. 6.32 in
• Are answers supported by clear and correct working?
For example, the first part should look like this:
2 _6_
6 ~ AD
AD = 18
The second two parts could be solved using proportions or the Pythagorean Theorem.
Special Right Triangles
Activity- Triangle Tiling
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The objective of this activity is to get students thinking about and using the ratios of the sides in isosceles
right triangles and 30-60-90 right triangles. Students will also see how these triangles appear in familiar
polygons.
For this activity, you will need to cut out a bunch of squares and equilateral triangles of different sizes and
colors. There should be enough for each student to get one of each figure to work with.
Tell students that they need to measure their square and figure out the length of the diagonal of each
square. Then they must show that the ratio of the diagonal to the length of a side is approximately ^ : 1.
The measurements and the calculations should be shown on the back of the square.
Next, students will find the altitude of their equilateral triangle. They may do this a number of different
ways (draw the angle bisector, the perpendicular bisector of one side, etc). Students should measure the
sides of the triangle and verify that the altitude bisects the base of the triangle. Next, tell students that
they need to figure out the length of the altitude of each triangle. Then they must show that the ratio
of the altitude to the length of half of the base is approximately V3 : 1. The measurements and the
calculations should be shown on the back of the equilateral triangle.
Once all calculations have been made and ratios have been verified, students will create a tiling design on
a wall in the classroom with their figures.
Have students elect a design team (2 students who will map out the design). You can have them cut some
of the squares and triangles into the special right triangles and keep some of them squares and equilateral
triangles for more variety in the design. You may also wish to allow students to cut out additional figures
(squares, equilateral triangles, isosceles right triangles and 30-60-90 right triangles) in appropriate colors
to complete their design idea. You may wish to have students make their design on the floor first where
they can change things before they attach the final design to the wall.
Basic Assessment:
You will need to check each student's work and calculations before they begin making the tiling design.
Award points for correct measurement and calculations.
Tangent, Sine and Cosine
Activity-Ladder Tangents
The objective of this activity is to have students identify the different parts of right triangles and use an
appropriate trigonometric ratio to solve a real world problem.
Students will use what they have learned about trigonometric ratios to solve the following problem.
A building is 25 ft tall. Victor and his roofing company are going to replace the roof on the building. Victor
requires that the angle that the ladder makes with the ground does not exceed 70° for safety reasons. Use
this information and the diagram to answer the following questions.
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Label the hypotenuse of the right triangle.
Which trig function can be used to determine the minimum length of a ladder needed to reach the top of
the building?
Set up an equation using this ratio to find the length of the ladder.
How far from the bases of the building should Victor place the base of the ladder?
At what angle will the ladder meet the building?
Basic Assessment:
Walk around as the students are working. Notice which students are having difficulty and offer support.
Allow students time to share their work when finished.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did students correctly identify the hypotenuse of the right triangle in the diagram?
• Students should use the sine ratio to find the length of the ladder as shown:
sm7U = —
x
25
x
sin 70°
The ladder needs to be at least 26.6 ft long.
• Students should use either the tangent or cosine ratios or the Pythagorean Theorem to find the
distance from the building that the ladder should be placed. This distance is approx 9.09 ft.
• The angle at which the ladder meets the building is 20°.
Inverse Trigonometric Ratios
Activity- Write Your Own Word Problem
The objective of this activity is to practice using the inverse trigonometric functions and gain experience
writing and interpreting word problems. Students usually struggle the most with translating word problems
into mathematical equations. Having students write their own problems helps them to better understand
the terminology used in writing these problems.
Students will work in pairs to write two trigonometric word problems. One problem will require that a
trigonometric function be used in order to solve for the length of a side in a right triangle. The second
problem will require the use of one of the inverse trigonometric functions to find an acute angle in a right
triangle. You may wish to discuss the following with students before hand:
• Angle of elevation
• Angle of depression
• What minimal information is required to find a side in a right triangle using trigonometry?
• What minimal information is required to find an acute angle in a right triangle using inverse trigono-
metric functions?
Students should write out the word problems on poster paper and include a diagram of the situation. Each
pair should have a worked out solution to their problem on notebook paper.
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Have student pairs put up their problems around the room. You may wish to check their work to make
sure it is sensible first, or let other groups determine this as they attempt to solve it in the next step.
Now, each pair will circulate around the room and solve the problems of each of the other pairs. (If you
have a large class, you may wish to have students work in groups of three or only solve some of the other
groups' problems to save time).
Have students share their solutions so that the other groups can check their work. Provide feedback and
assistance as needed.
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did each pair create and solve two problems correctly?
• Did each pair correctly solve the other pairs' problems?
• Did pairs provide constructive feedback to other students whose problems had design flaws?
5.9 Circles
Parts of Circles and Tangent Lines
Activity-Memory Game
The objective of this activity is to have students review the definitions in this section by creating a memory
game then playing the game.
Students will create pairs of cards-one with a term and one with a picture illustrating the term.
Example:
Each group should create cards for each of the following terms:
Radius
Chord
Diameter
Tangent Line
Secant Line
Congruent tangent segments
Common Internal Tangent
Common External Tangent
Internally Tangent Circles
Externally Tangent Circles
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Concentric Circles
Congruent Circles
Once students have made their cards, they can play the memory game.
Basic Assessment:
This activity is mainly designed to help students "study" the terms. You may wish to award completion
points for groups who have correctly created the cards for the game.
Properties of Arcs
Activity-How Much Pie do I get?
The objective of this activity is for students to integrate what they know about arc measures and central
angles with ratios, proportions and algebra skills.
Veronica is going to divide up an apple pie that she made amongst her siblings. Each child gets a portion
of the pie based on his/her age. For example, if Veronica is 10 years old and her oldest brother, Frank, is
15 years old, then the ratio of the arc measure of her piece of pie to that of his will be 10:15 or 2:3.
Veronica has three other siblings as well as the older brother. They are Chad, 8, Sally, 7, and Mikey, 5.
Use what you know about circle arcs and proportions to determine what the central angle measure of each
child's slice of the pie should be.
Use a compass to make a circle and a protractor to create a template for the slicing. You should label each
slice of pie in the diagram with the corresponding child's name.
Be prepared to share your method and solution to the problem with the class.
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
Did students set up a correct equation based on the information given to find the angle measures?
Did students find the correct measures?
Is the diagram correctly drawn to scale?
Solution: Students should come up with the equation:
5x + 7x + 8x + IOjc + 15x = 360°
45;t = 360°
x = 8°
Now we can substitute to find that the following central angles for each of the children and make the
diagram below. The radius of the circle can be anything, but the arcs and central angles should be the
same.
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Properties of Chords
Activity-Find the Center
The objective of this activity is to integrate what students know about chords, radii and diameters of a
circle to locate the center of a circle given an arc.
Before beginning this activity it is a good idea to review Investigation 1-4 in the textbook on constructing
a Perpendicular Bisector of a segment.
Students have learned in this section that the perpendicular from a the center of a circle to a chord will
bisect the chord. Now they will use the converse: If a segment is a perpendicular bisector of a chord, then
it passes through the center of the circle.
A group of students find a treasure map that was supposedly made by a pirate. The treasure is located
on the circular path through the three rocks shown on the map. It is 100° counterclockwise from the rock
furthest to the West. Use construction tools and a protractor to complete the circular path and mark to
stop on the map where the treasure could be found. The rocks are denoted by points A, B and C. Each
student will need a copy of the map below.
N
t
Basic Assessment:
Teachers should make their own solution on the same size paper as what the students are using. Then
make an "overhead" slide of the solution so that it is easy to lay it on top of the students' work and check
for accuracy. The solution below shows all required markings and the location of the treasure. Students
may make a different pair of chords by connecting different pairs of rocks.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
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points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did students create two chords using the rocks as the endpoints?
• Did students use a perpendicular bisector construction to locate the center of the circle?
• Did students correctly construct the rest of the circle?
• Did students correctly locate the buried treasure on the map?
Solution:
X
This is where the treasure should be.
Inscribed Angles
Activity-Find the Inscribed Angles
The objective of this activity is to find the measures of inscribed angles and the arcs they intersect.
For this activity, students are going to work in small groups to write a problem. Each problem must have
a circle with a triangle or a quadrilateral inscribed in it.
Students are going to use a protractor to measure the angles of the inscribed angles.
Students are assigned the task of writing a word problem for another student to solve.
Students must include a diagram and have an answer key to check student work.
Allow time for students to write their problems.
If students are having difficulty, have them brainstorm things with circles and other things in them-like a
circus ring with a stand in it, an amphitheater, a circular pond with walkways crossing it, etc.
When students are finished, have them switch papers with another group and solve each others' problem.
Allow time for students to share when finished.
There is a learning curve for this and don't be surprised if some of the initial attempts at writing a problem
are missing essential components. If this is the case, have the students try again to rewrite the problem.
Be sure that diagrams are accurate-this will make a huge difference.
Basic Assessment:
Walk around and observe students as they work. If students are having difficulty, offer suggestions to help
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them think up a scenario that makes sense. Help students to understand how to solve the problems. Help
students to be sure that the problems that they are writing are complete. Offer suggestions/feedback as
needed.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did students come up with a viable scenario for their word problem?
• Did students create an accurate diagram?
• Did students correctly measure the inscribed angles?
• Did students complete a correct solution to the problem?
• Perhaps additional points can be awarded for creativity.
Angles of Chords, Secants, and Tangents
The objective of this activity is to have students visualize the different angles formed by chords, radii,
secants and tangents and how their measures relate to the intercepted arcs.
Students will need white paper, colored paper, a compass, a straightedge and a protractor for this activity.
Students will start by creating a large circle using a compass on a piece of paper. Now students will mark
off two points on the circle, A and B. Now students will complete the following steps, using different colors
for each angle.
• Construct a central angle that intercepts AB.
• Construct an inscribed angle that intercepts AB.
• Construct intersecting chords in the circle such that one endpoint of one chord is point A and one
endpoint of the other chord is point B.
• Construct secant segments with a common endpoint outside the circle and the other endpoints on A
and B.
• Construct a tangent line to the circle at point B. Make an angle with the vertex at this point of
tangency and the one side AB and the other side on the tangent line.
Now students should measure all angles created in their diagram and set up equations using the properties
they have learned in this chapter to verify their findings.
Basic Assessment:
You may find it helpful to do an example with the class before having them complete this task individu-
ally. Walk around as students work and offer assistance as needed. If students are having difficulty, remind
them to refer back to the text. Collect student work when finished, and assess student understanding based
on the accuracy of student work. Offer feedback and correction as needed.
Here is a sample solution:
Central angle is 110°
Inscribed angle is \ (110°) = 55°
Angle formed by chords is \ (48° + 110°) = 79°
Angle formed by secants is \ (110° - 40°) = 35°
Inscribed angle formed by tangent and chord is \ (110°) = 55°
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Segments of Chords, Secants, and Tangents
Activity-Discover the Properties
The objective of this activity is to practice identifying similar triangles and to discover the formulas for
the lengths of segments associated with circles.
Give students a copy of the circles below.
The solid line segments represent the chords and secants in the diagram. The dashed lines represent the
third side of a triangle formed by these segments. Use the diagrams to answer the following questions:
1. Identify all congruent angles in each circle. What do these congruent angles tell us about the triangles?
2. Write a similarity statement for the triangles formed by the intersecting chords.
3. Use the corresponding sides of the triangles to write a proportion which shows the relationship
between AE,EC,BE, and ED.
4. Given that AE = 3, EC = 8 and BE = 4, find ED.
5. Write a similarity statement for the triangles formed by the secant segments in the second circle.
6. Use the corresponding sides of the triangle to write a proportion which shows the relationship between
XY,XV,XZ and XW.
7. Given that YV = 10, XY = 6 and XW = 12, find XZ.
Students may need some coaching with this one, but in completing this activity they will be reviewing a
number of concepts and seeing these properties in a slightly different manner.
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Were students able to correctly identify the congruent pairs of angles in each diagram?
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Were students able to conclude which triangles were similar?
Do their similarity statements have the correct corresponding sides?
Are their proportions correct?
Did students find the correct values for ED(6) and XZ(8)?
Solution:
First diagram: iDAE = iCBE, LADE = iBCE, IDEA = iCEB, AAED ~ ABEC, §§ - EC , v
Second diagram: LX = zX 9 lYWZ = iZVY, iXZV = iXYW, AXYW ~ AXZV, f£ = ff , ±§ = g
Extension: Writing and Graphing the Equations of Circles
Activity-Find the Center of the Circle
The objective of this lesson is to practice using the definition of a circle to write an equation of a circle
and to review properties of circles.
Students will need a piece of graph paper, a straightedge and a compass for this activity.
The points (-3, 6), (9, -2) and (9, 6) are the vertices of a right triangle. Plot this triangle on the coordinate
plane. Find the equation of the circle circumscribed about this triangle.
To do this, first identify where the center of the circle should be located. Think about what you know
about right triangles inscribed in a circle.
Next, find the radius of the circle. Use the distance formula to do this.
Write the equation of the circle. Use your compass to make the circle on the coordinate plane.
Basic Assessment:
Students may struggle with getting started on this problem. Once they realize that the hypotenuse of the
right triangle must be the circle's diameter, the rest will follow. The center of the circle is the midpoint of
the diameter and the radius is the distance from this point to any vertex of the triangle. You may wish to
remind students about what they learned earlier in the course about the center of the circle and how it is
the point where the perpendicular bisectors of the sides of the triangle intersect.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
Did students plot the points correctly and draw the right triangle?
Did students correctly identify the center of the circle?
Were students able to use the distance formula correctly to find the radius?
Do students have the correct equation of the circle?
Here is a completed graph and equation.
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^(9 -3) 2 + (-2 -2) 2 = V36 + 16
V52 = 2Vl3
Equation of the circle is:
(x-3) 2 + (y-2) 2 = 52
5.10 Perimeter and Area
Triangles and Parallelograms
Activity-Flooring Dilemma
The objective of this activity is to practice dividing up an irregular region into known polygonal units to
find total area. This activity also involves a real world problem solving element.
Ralph is working for a flooring company for the summer. He is assigned the task of figuring out the area
of a strangely shaped room at the library. Here is a diagram of the room that Ralph needs to measure.
Help Ralph find the total area of the floor that needs to be covered. Provide students with the diagram
below and the questions that follow.
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Fill in missing measurements as needed based on the given information in the diagram. You may assume
that angles that appear to be right angles are right angles and that segments that appear to be parallel
are parallel.
Next, divide your diagram into rectangles and triangles and find the areas of these parts.
Now, find the total area of the floor.
If the cost of the flooring material is $5.95 per square foot, how much will the material cost?
Basic Assessment:
Circulate and assist students as needed to complete this assignment. You may wish to have them work in
pairs.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
Did students divide up the area into triangles and rectangles?
Are the measurements accurate?
Are the areas of each figure correct?
Is their total area correct?
Did students calculate the correct cost?
Check all work and offer correction/feedback when necessary.
One possible solution is below:
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\36
\
120
12
A
96
6
^v^lO
24\.
6| 10 | 8
Based on this diagram the total floor area is
* Students may have trouble recognizing that the equilateral triangle on the top of the diagram can be split
into two 30-60-90 right triangles to find the height
Trapezoids, Rhombi, and Kites
Activity-It's a Kite
The objective of this activity is to understand the relationships between the areas of quadrilaterals, specif-
ically, kites, rectangles and rhombuses. Students will identify these special quadrilaterals and apply the
appropriate area formulas.
You need to design a kite to fly in a school competition. The parameters required that your kite fit within
a 2 ft by 3 ft rectangular box.
First, students should draw three a scaled down rectangles to represent the box. They may choose their
own scale (for example, perhaps 1 ft = 2 in on their paper).
Next, students should draw a different kite in each box that fits perfectly in the required dimensions.
Find the lengths of the sides of these kites. Students should recognize here that they need to draw the
diagonals, measure the lengths of the diagonals and use the Pythagorean Theorem. Also, remind students
that they need to convert their lengths back to feet based on their scale.
Next, students should find the area of each kite. Again, students should convert this answer to feet based
on their scale. Remind students that they are now comparing areas and should adjust their scale factor
accordingly. For example, if they are converting square inches to square feet they must divide by 144, not
12.
Do you notice anything about the areas of your kites?
What is the ratio of the areas of the kites to the area of the rectangle?
Can you explain this phenomenon?
Share your findings with the class.
Basic Assessment:
Circulate and assist students as needed to complete this assignment. You may wish to have them work in
pairs.
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Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did students create three correctly scaled rectangles and draw three different kites in the rectangles?
• Did students correctly find the side lengths? These answers will vary.
• Did students determine that the area of all of the kites is 3 ft 2 l
• Did students determine that the ratio of the areas of the kites to that of the rectangles is 1:2?
• Were students able to communicate that the diagonal lengths would be the same as the lengths of
the sides and since the formula for area of a kite is \d\d2^ the area of the kite will be half the area
of the rectangle?
Area of Similar Polygons
Activity-Paint It!
The objective of this activity is to have students understand the relationship between the scale factor of
similar polygons and their areas and to use this relationship to solve a problem.
A scale drawing of a barn is shown below. Students will find the total surface area of the barn in centimeters
and then convert this area to meters based on the given scale. Now, the roof needs to be painted white
and the faces of the barn need to be painted red. If one can of paint will cover 50 square meters, how many
cans of each color paint are required to complete the task?
1 cm = 0.5 m
17 cm
20 cm
45 cm
30 cm
Basic Assessment:
Circulate and assist students as needed to complete this assignment. You may wish to have them work in
pairs.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did students find the correct area in the diagram?
• Did students correctly convert this area to meters?
• Did students determine the correct number of cans of paint?
Solution: The total surface area of the barn is: 1530 cm 2 (roof)+240 cm 2 (triangles) + 1200 cm 2 (front and back) +
1800 cm 2 (sides) = 4770 cm 2 .
Convert this to meters-the length ratio would be 2 : 1 (2 cm = 1 m), so the area ratio is 4 : 1 (4 cm 2 = 1 m 2 ).
So, the total surface area in square meters is 1192.5m 2 .
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Now, we can separate the roof surface to get 382.5 m 2 , which will require 8 cans of white paint. The rest
of the barn is 810 m 2 , which will require 17 cans of red paint.
Circumference and Arc Length
Activity-The Fish Pond Border
The objective of this activity is to practice calculating circumference and arc length of a circle.
A local park has a very large fish pond. The park authority has decided to create bridges over the pond
in such a way that there will be three paths to a point in the center of the pond.
2x°
3x°
The circumference of the pond will be divided as shown in the diagram.
What is the arc measure of each section of shoreline?
If the diameter of the pond is 50 ft, how many feet of bridge will be constructed?
How long is each section of shoreline?
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did students find the correct arc measures?
• Did students find the correct total length of bridge?
• Did students find the correct lengths of shoreline?
Solution: The arc measures are 135°, 135° and 90°. There will be a total of 150 ft of bridge pathways. The
sections of the shoreline are: two sections of ^100n « 117.8 ft and one piece of ^1007r « 78.5 ft.
Area of Circles and Sectors
Activity-Circle Segments
The objective of this activity is to calculate the area of a circle, a sector and to use the area of a triangle
to find the area of a circle segment.
Use your knowledge of various area formulas to determine the area of the large square, the circle and the
small square in the diagram below.
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10
Next, shade one of the four pieces of the circle that extends beyond the small square.
Find the area of this shaded region.
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did students calculate the correct area of the large square? (100w 2 )
• Did students calculate the correct area of the circle? (25nu 2 )
• Did students calculate the correct area of the small square? (5 V2J = 50u 2
• Did students calculate the correct circle segment area? \(25n - 50) u 2
5.11 Surface Area and Volume
Exploring Solids
Activity-Solids
The objectives of this lesson are for students to identify and draw isometric, orthographic, cross-sectional
and net representations of polyhedra.
Part 1: Give students a geometric solid to work with. For this figure, students are going to create four
different things.
1. Create an orthographic projection of their solid.
2. Create a cross-section of the solid.
3. Create a net for the solid.
4. Use the net to create an actual model of the solid.
Part 2: Students are going to use the net that they created to design a mobile of solids.
For this, students can create the same solid in different sizes, or they can create nets for different solids.
Perhaps they can "trade" nets with their classmates two. Each student should create a minimum of 5
solids from nets for their mobile.
Encourage students to use color and creativity in their mobile.
Students can connect all of the created solids together with string and wooden dowels.
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Have materials on hand for students to complete their mobiles when finished.
You can create a great display of mobiles and hang them all around the classroom.
Basic Assessment:
Some students really struggle with the idea of a net and how it works. For many, looking at a two-
dimensional representation of a three-dimensional figure is daunting. They struggle with visualizing how
the parts can be folded to configure a three-dimensional solid. Teachers are encouraged to help students
in their creation of their nets to insure that they will work.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Are the nets created correctly?
• Did the student use different sized versions of the same solid or different solids?
• Is the mobile finished? Are there at least five solids?
• Did the student use color and creativity in his/her design?
• Some students struggle with making projects like this look nice. You may wish to assign bonus points
for particularly well done mobiles as a way to reward those who excel at artistic endeavors without
penalizing those who do not.
Surface Area of Prisms and Cylinders
Activity-Prism Match
The objectives of this activity are to practice measuring, finding the surface area of prisms and use deductive
reasoning skills to solve a problem.
Part 1: This activity requires some preparation. You will need a bunch of boxes. You could collect them
over time yourself or have each student in the class bring in a box for this activity.
Next, give a box to each student. Have them figure out the surface area of their box. Some students may
need help with this. You will want to label each box with a letter for the game. It is a good idea to make
sure that each surface area is correct before beginning the next part. Write each area on an index card. It
is advisable to label each index card with a number and keep a "key" with the correct matches and names
of the students who calculated each area. This way, if one of the surface area calculations is incorrect, you
can determine which figure it was supposed to be for and even who calculated it. Mix up the numbers and
letters, however, so that students know that box A does not go with measure 1, etc.
Part 2: Place all of the boxes around the classroom.
To complete the activity, each of the students is given a ruler/tape measure and an index card with a
surface area on it.
Students will need to take notes as they work-remind them to do this. Also encourage students to think
of strategies before beginning the task. Does a large box have a large surface area or a small surface area?
Then the students need to figure out which box has their correct surface area. This will take some time.
Don't rush the students, but encourage them to work together.
Once they have found their box, the students need to create a diagram of it. Label the edges of the figure
with the lengths they found and show the calculations of the surface area. This final page should be
submitted for a grade.
Allow time for the students to share their work when finished.
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Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did each student calculate a correct surface area in part 1?
• Did the students match up the correct prism with their measurements?
• Is the diagram labeled accurately?
• Is the surface area calculation clear and correct?
Surface Area of Pyramids and Cones
Activity-Create Your Own Pyramid
The objectives of this activity are to identify pyramids, find the surface area of a pyramid using a net
or formula and to find the volume of a pyramid. Students will also explore the relationship between the
surface areas and volumes of similar solids.
For this activity, the students are going to design their own pyramids.
Part 1: Students should decide what polygon to use for the base of their pyramid. Students should
consider how big they want their final pyramid to be (i.e. determine their scale factor) and draw scale
diagram of the pyramid.
Now, students should use their diagram to calculate the surface area and volume of their pyramid.
Part 2: Students will construct a net (in actual size) and cut it out to construct the pyramid.
Students should use the actual lengths of the edges to determine the surface area and volume of their
constructed pyramid.
Compare the surface areas and volumes of the diagram to the actual pyramid. DO you notice anything?
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
Check student work for accuracy.
The design should be drawn to scale-so the design should not be the same size as the model.
Listen during presentations for student understanding.
Can the student explain what they did and why they did it?
Are the surface area measurements/calculations accurate?
Are the volume measurements/calculations accurate?
Did the student recognize that the ratio of the areas is the scale factor squared?
Did the student recognize that the ratio of the volumes is the scale factor cubed?
Offer correction/feedback as needed.
Volume of Prisms and Cylinders
Activity-Maximum Volume
The objective of this activity is to have students practice using the formulas for surface area and volume
and explore the concept of a maximum volume for a given surface area.
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For this activity students will be determining the dimensions of the can or box with "maximum" volume
given the surface area.
Students should work in pairs or groups of three for this activity. The surface area is 294w 2 .
Half of the pairs/groups should be given the premise that they are making a can and that the surface area
of the can cannot exceed 294w 2 . The can is a cylinder.
Half of the pairs/groups should be given the premise that they are making a box and that the surface area
of the box cannot exceed 294w 2 . The box is a rectangular prism.
Students should experiment with several different dimensions in their pairs/groups and try to determine
what the dimensions are of the can or box of maximum volume.
Have pairs/groups share their results and discuss them as a class.
They may not actually come up with a perfect solution and that is okay. You can open a discussion about
how more advanced mathematical techniques (i.e. calculus) provide tools for finding the best solution
to this problem. You may wish to discuss the practical applications of such a problem (i.e. minimizing
packaging materials for a product to save costs and waste.)
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did students develop several possible figures within the surface area limitation?
• Were students able to discuss the results with the class?
• Were they able to generalize their results?
Volume of Pyramids and Cones
Activity-Find the Figure with the Greatest Volume
The objectives of this activity are to evaluate and compare volumes of a cone and pyramids.
First ask students to find the volumes of a cone and a square pyramid. The cone has a diameter of 6 in
and a height of 10 in. The square pyramid has a base with diagonal of length 6 in and a height of 10 in.
Which has the greater volume? Can you explain why?
What if the side lengths of the square base are 6 in? Which has the greater volume in this case?
Draw the bases of the three figures in a single diagram. (This should look like a square inscribed in a circle
inscribed in a larger square.)
Discuss the relationship between the volumes of the two square pyramids.
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did students find the correct volumes for the cone and the smaller square pyramid?
• Did students discover that the cone has a larger volume and explain why this is so?
• Did students find the correct volume of the larger square pyramid?
• Did students make a correct drawing?
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• Did students recognize the relationship between the two square pyramids?
Solution:
The small square pyramid has a volume of ^(-7=) (10) = 60 in 3
The cone has a volume of |(3) 2 tt(10) « 94.25 in 3
The cone has the larger volume because the area of its base is greater than that of the small square pyramid.
The only difference between the two figures is the areas of the bases.
The larger square pyramid has a volume of ^(6) 2 (10) = 120 in 2
The area of the base of the larger square pyramid is twice that of the smaller square pyramid. Therefore
its volume will be twice as large too. The length of the sides in the two squares differ by a factor of V2,
when the area of the base is found, this factor is squared to get a factor of 2.
Surface Area and Volume of Spheres
Activity-Measuring Spheres
The objective of this activity is to practice finding the surface area and volume of spheres.
Bring in a bunch of different size balls. The students will need string, measuring tape, rulers, paper and
pencils to complete this assignment.
Students may complete this assignment in pairs or groups of three. Each group will select a ball to work
with.
Using their tools, students need to figure out the surface area and the volume of this first ball. Students
should summarize in a few sentences the process they used to determine the surface area and volume of
their ball.
Then they select another ball and figure out the surface area and the volume for the second ball.
Then students need to draw a diagram comparing the two (radii, surface areas and volumes) and write
about any conclusions that they can determine about the two balls.
Allow time for students to share their work when finished.
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
Did students find the correct surface area and volume for each sphere?
Is their description of their process clear and correct?
Were their findings about the two spheres correct? did they recognize that the spheres are similar
and that in fact all spheres are similar?
Did students recognize the correct relationships between the radii, surface area and volumes? They
should recognize here that the ratio of the surface areas is the ratio of the radii squared and the ratio
of the volumes is the ratio of the radii cubed.
Listen for student understanding during the presentations.
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Extension: Exploring Similar Solids
Activity-Cone Sizing
The objectives of the lesson are to calculate the volume of a composite solid and to compare volume and
surface area of similar solids.
The "Cherry on Top" ice cream shop wants to make a giant ice cream cone to put on top of its roof so that
drivers on the nearby interstate will see the shop. The larger-than-life cone needs to be 22 ft tall from the
bottom of the pointed cone to the top of the hemispherical ice cream scoop on top.
The sugar cone base of the ice cream cones that the shop sells have a height of 4 inches and radius of 1.5
inches. The ice cream scoop is spherical and fits perfectly on top of the cone with no overlap (in other
words, its radius is also 1.5 inches.)
What are the volume and surface area of the ice cream cone sold in the shop in terms of n and rounded to
the nearest hundredth?
What is the scale factor between the actual cone and the giant cone?
What are the dimensions of the giant cone?
What are the volume and surface area of the giant ice cream cone? Can you find this using a proportion
rather than the formulas? How?
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did students find the correct surface area and volume of the actual cone?
• Did students find the correct dimensions of the giant cone?
• Were students able to find the correct surface area and volume of the giant cone in the correct units?
• Did students use the scale factor to find the surface area and volume of the giant cone?
Solution:
The volume of the actual cone is ±(1.5) 2 tt(4) + \ (§)tt(1.5) 3 = ^tt « 16.49 in 3 .
The surface area of the actual cone is tt(1.5) Vl.5 2 + 4 2 + ±4tt(1.5) 2 = 10.9itt « 34.27 in 2
The scale factor between the small cone with height from tip of the cone to the top of the ice cream scoop
of 5.5 in to the giant cone with height of 22 ft is 1:48
Using the ratio of 1.5:4 as the ratio of the radius to the height of the cone, and the fact that the radius
plus the height of the giant cone is 22 (here units don't matter), we can set up the proportion:
1.5 x
4 22 -x
4x = 33 - 1.5*
5.5x = 33
x = 6
This means that the radius of the giant cone (and the hemispherical scoop of ice cream) is 6 ft. The height
of the cone is thus 16 ft.
We can use the scale factor to find the volume and surface area of the giant cone.
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The scale factor of the lengths is 1:48, so the ratio of areas is 1:2304. The surface of the giant cone is thus
78,958.08 in 2 . Let's convert this measure from square inches to square feet by dividing by 144 (as there
are 144 square inches in a square foot-students may need an explanation of this). The surface are of the
giant cone is 548.32 ft 2 .
The ratio of the volumes will be 1:110592. The volume of the giant cone will be 1,823,662.08 in 3 . Now
let's convert this measure to feet by dividing by 12 3 to get 1,055.36 ft 3 .
5.12 Rigid Transformations
Exploring Symmetry
Activity-Magazine Symmetry Hunt
The objective of this activity is to have students begin to recognize and classify different types of symmetry
in the world around them.
For this activity, students are going to hunt through magazines of example of symmetry. You will need to
prepare and bring a stack of magazines to class. Architecture and nature magazines will be very helpful.
They are going to create a collage of the pictures that they find. Students will need paper, glue, scissors,
a ruler and a pencil.
When the students select a given picture, they need to cut it out and attach it to the collage. Then they
draw in the lines of symmetry or indicate the rotational symmetry.
Students do this until their collage is complete.
Then have the students write a paragraph explaining their collage-any themes and the types of symmetry
found in the collage.
Allow time for students to share their work when finished.
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Observe students as they work and help them classify the symmetries as necessary.
• Read student descriptions of their collages for accuracy.
• Were students able to correctly identify all symmetries in each figure used in their collage?
• Did students develop a theme in their work? (This may or may not be a requirement for full points.)
Translations
Activity-Find the Pre-Image
The objective of this activity is to practice making and identifying translations and writing mapping
notations for these translations in the coordinate plane.
For this activity, the students will be creating a game. Divide students onto teams of two.
The first step is for students to design translations on a coordinate grid. They will need to create a
pre-image and four different translations of this pre-image in the coordinate plane.
Then have them write the mapping notations for each translation on a separate index card.
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To help students keep their index cards organized have them put letters on their figures in the coordinate
plane and numbers on the index cards and record a "code" which matches the correct image with its
mapping notation. (There will not be a card for the pre-image)
Now, each team joins another team.
Student pairs will need to identify the pre-image created by the other team and correctly match each
translated figure with the correct mapping notation. This will require some guess and check work.
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
Are the mapping notations correct?
Were student pairs able to correctly deduce which figure must be the pre-image?
Were student pairs able to correctly match the mapping notations with the correct translated figure?
Offer correction/feedback when necessary.
Reflections
Activity-Reflection in the Coordinate Plane
The objective of this activity is to practice making a reflection in the coordinate plane and verify that it
is an isometry.
First, the students will need graph paper, rulers and colored pencils for this activity.
Tell students to begin by drawing a quadrilateral on a coordinate grid. They can use color if they wish to.
Then tell them that they will be drawing a reflection of this quadrilateral over the line y = x. Students
should draw this line on the coordinate plane and reflect the vertices of their quadrilateral over it.
Then the students draw in the new quadrilateral on the coordinate grid.
Now, students should calculate the lengths of the sides of their pre-image quadrilateral and their image
after the reflection. Students should verify that the lengths of the corresponding pairs of sides are the
same.
Finally, have students exchange papers and complete a peer check. Each person needs to check the other
person's work.
Allow time for the students to share their reflections in small groups when finished.
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
Were students able to create the correct translation of their quadrilateral?
Did students correctly calculate the lengths of the sides of both figures?
Did students find that the lengths of the corresponding pairs of sides are the same?
Did students thoroughly check each other's work and offer constructive feedback as necessary?
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Rotations
Activity-Rotating Triangles
The objective of this activity is to rotate a triangle using a compass, ruler and protractor and verify that
it is an isometry.
Students should start by constructing a triangle on white paper and marking a center for their rotation.
Next students will use their tools to rotate their triangle. They may choose how many degrees to rotate
their triangle. You may wish to encourage students to rotate their triangles far enough to prevent overlap
of the pre-image and image. For some students having overlapping figures will make the subsequent steps
more difficult.
For each vertex of the pre-image, students will need to create a segment connecting this point to their
center of rotation. Next, students will draw a ray x° away from this segment. Now students can use their
compass to mark a point equidistant from the center as the vertex on the pre-image on this new ray. This
is the image of the vertex.
Repeat the process above for the other two vertices and then connect the vertices to make the image
triangle. Be sure to correctly label the vertices of the image.
Now students must verify that the image is congruent to the pre-image to show that rotations are an
isometry. Students may do this using any of the triangle congruence theorems learned in this course.
Basic Assessment:
Students may need significant assistance with the rotation of the first vertex. You may wish to do an
example with them before beginning this activity.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Did students correctly/accurately rotate their triangle?
• Were students able to prove that their image is congruent to their pre-image?
• Did students correctly apply a triangle congruence theorem?
Composition of Transformations
Activity-Glide Reflections
The objective of this activity is to have students create a design using a glide reflection.
To begin this activity it may be helpful to refer back to the foot prints in the lesson. These are an example
of a glide reflection "design".
Students will need pencils, colored pencils or markers, scissors, cardstock or cardboard and large white
paper to complete this activity.
Students are going to create a glide reflection design of their own. They may choose to use a geometric
figure or other shape of their choosing in this project.
In order to insure the congruence of the images in the design, students should cut out a template of their
figure from cardstock or cardboard that they will trace.
Once students have created their template, they will need to draw their line of reflection on their paper.
Now students will trace their template, glide it (a particular distance that will be repeated) and reflect it
over the line multiple times to create their design.
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Student's final work should have at least six of the figures. Students should color in their design to make
it aesthetically pleasing. Students should turn in their template with their final work.
Basic Assessment:
Circulate and help students throughout this process. Some will have an easier time with this than others.
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Students' work should be accurate-all figures congruent and evenly spaced.
• You may wish to award extra points for outstanding work.
Extension: Tessellating Polygons
Activity-Designing Tessellations
The objective of this activity is to further develop student understanding of tessellation, whether or not a
particular regular polygon with tessellate and to design an original tessellation.
For this lesson, the students are going to design a tessellation that uses two different polygons. Students
first need to identify which polygons will tessellate and which won't.
Each student will need colored pencils, rulers, paper, cardstock, and large paper.
Explain to the students that they will be graded on their work for accuracy, neatness and aesthetic appeal.
Each student needs to create a "sample" pattern to demonstrate that the tessellation that they have
proposed does work.
Students need to have a peer check their work. Students need to have an instructor check their work.
When the pattern has been approved, students may begin work on their final design.
This process will help students to problem solve and eliminate redoing work.
Allow time for the students to present their work when finished.
Basic Assessment:
Teachers should create a rubric where each piece of the assignment is worth points. Exactly how many
points each component should be worth is left up to individual teachers. The following is a list of possible
components:
• Does the pattern tessellate?
• Did the student use two polygons?
• Is the student's work neat and accurate?
• You may wish to award extra points for exceptional work.
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