Skip to main content

Full text of "Mathematics Grade 10 Teachers' Guide - Siyavula WebBooks"

See other formats

Mathematics Grade 10 Teachers' Guide 

Siyavula WebBooks 


Bridget Nash 

Mathematics Grade 10 Teachers' Guide 

Siyavula WebBooks 


Bridget Nash 


< http://cnx.Org/content/colll341/l.l/ > 


Rice University, Houston, Texas 

This selection and arrangement of content as a collection is copyrighted by Bridget Nash. It is licensed under the 

Creative Commons Attribution 3.0 license (http://creativecommons.Org/licenses/by/3.0/). 

Collection structure revised: August 10, 2011 

PDF generated: August 10, 2011 

For copyright and attribution information for the modules contained in this collection, see p. 47. 

Table of Contents 

1 Overview 1 

2 Assessment and Assessment Support 9 

3 Chapter Contexts 25 

4 On the Web, Everyone can be a Scientist 29 

5 FullMarks User Guide 33 

6 Rich Media 37 

7 Maths Blog Posts 43 

8 Solutions 45 

Index 46 

Attributions 47 


Chapter 1 

Overview 1 

1.1 Overview 

1.1.1 Curriculum Overview 

Before 1994 there existed a number of education departments and subsequent curriculum according to the 
segregation that was so evident during the apartheid years. As a result, the curriculum itself became one of 
the political icons of freedom or suppression. Since then the government and political leaders have sought 
to try and develop one curriculum that is aligned with our national agenda of democratic freedom and 
equality for all, in fore-grounding the knowledge, skills and values our country believes our learners need to 
acquire and apply, in order to participate meaningfully in society as citizens of a free country. The National 
Curriculum Statement (NCS) of Grades R 12 (DoE, 2011) therefore serves the purposes of: 

• equipping learners, irrespective of their socio-economic background, race, gender, physical ability or 
intellectual ability, with the knowledge, skills and values necessary for self- fulfilment, and meaningful 
participation in society as citizens of a free country; 

• providing access to higher education; 

• facilitating the transition of learners from education institutions to the workplace; and 

• providing employers with a sufficient profile of a learner's competencies. 

Although elevated to the status of political icon, the curriculum remains a tool that requires the skill 
of an educator in interpreting and operationalising this tool within the classroom. The curriculum itself 
cannot accomplish the purposes outlined above without the community of curriculum specialists, material 
developers, educators and assessors contributing to and supporting the process, of the intended curriculum 
becoming the implemented curriculum. A curriculum can succeed or fail, depending on its implementation, 
despite its intended principles or potential on paper. It is therefore important that stakeholders of the 
curriculum are familiar with and aligned to the following principles that the NCS is based on: 



Social Transformation 

Redressing imbalances of the past. Providing equal 
opportunities for all. 

continued on next page 

1 This content is available online at <http://cnx.Org/content/m40349/l.l/>. 



Active and Critical Learning 

Encouraging an active and critical approach to 
learning. Avoiding excessive rote and uncritical 
learning of given truths. 

High Knowledge and Skills 

Learners achieve minimum standards of knowledge 
and skills specified for each grade in each subject. 


Content and context shows progression from simple 
to complex. 

Social and Environmental Justice and Human 

These practices as defined in the Constitution are 
infused into the teaching and learning of each of the 

Valuing Indigenous Knowledge Systems 

Acknowledging the rich history and heritage of this 

Credibility, Quality and Efficiency 

Providing an education that is globally comparable 
in quality. 

Table 1.1 

This guide is intended to add value and insight to the existing National Curriculum for Grade 10 Math- 
ematics, in line with its purposes and principles. It is hoped that this will assist you as the educator in 
optimising the implementation of the intended curriculum. 

1.1.2 Curriculum Requirements and Objectives 

The main objectives of the curriculum relate to the learners that emerge from our educational system. While 
educators are the most important stakeholders in the implementation of the intended curriculum, the quality 
of learner coming through this curriculum will be evidence of the actual attained curriculum from what was 
intended and then implemented. 

These purposes and principles aim to produce learners that are able to: 

• identify and solve problems and make decisions using critical and creative thinking; 

• work effectively as individuals and with others as members of a team; 

• organise and manage themselves and their activities responsibly and effectively; 

• collect, analyse, organise and critically evaluate information; 

• communicate effectively using visual, symbolic and/or language skills in various modes; 

• use science and technology effectively and critically showing responsibility towards the environment 
and the health of others; and 

• demonstrate an understanding of the world as a set of related systems by recognising that problem 
solving contexts do not exist in isolation. 

The above points can be summarised as an independent learner who can think critically and analytically, 
while also being able to work effectively with members of a team and identify and solve problems through 
effective decision making. This is also the outcome of what educational research terms the "reformed" 
approach rather than the "traditional" approach many educators are more accustomed to. Traditional 
practices have their role and cannot be totally abandoned in favour of only reform practices. However, 
in order to produce more independent and mathematical thinkers, the reform ideology needs to be more 
embraced by educators within their instructional behaviour. Here is a table that can guide you to identify 
your dominant instructional practice and try to assist you in adjusting it (if necessary) to be more balanced 
and in line with the reform approach being suggested by the NCS. 

Traditional Versus Reform Practices 


Traditional - values content, correctness of learn- 
ers' responses and mathematical validity of meth- 
ods. Reform - values finding patterns, making 
connections, communicating mathematically and 

Teaching Methods 

Traditional - expository, transmission, lots of 
drill and practice, step by step mastery of algo- 
rithms. Reform - hands-on guided discovery meth- 
ods, exploration, modelling. High level reasoning 
processes are central. 

Grouping Learners 

Traditional - dominantly same grouping ap- 
proaches. Reform - dominantly mixed grouping 
and abilities. 

Table 1.2 

The subject of mathematics, by the nature of the discipline, provides ample opportunities to meet the 
reformed objectives. In doing so, the definition of mathematics needs to be understood and embraced by 
educators involved in the teaching and the learning of the subject. In research it has been well documented 
that, as educators, our conceptions of what mathematics is, has an influence on our approach to the teaching 
and learning of the subject. 

Three possible views of mathematics can be presented. The instrumentalist view of mathematics assumes 
the stance that mathematics is an accumulation of facts, rules and skills that need to be used as a means 
to an end, without there necessarily being any relation between these components. The Platonist view 
of mathematics sees the subject as a static but unified body of certain knowledge, in which mathematics 
is discovered rather than created. The problem solving view of mathematics is a dynamic, continually 
expanding and evolving field of human creation and invention that is in itself a cultural product. Thus 
mathematics is viewed as a process of enquiry, not a finished product. The results remain constantly open to 
revision. It is suggested that a hierarchical order exists within these three views, placing the instrumentalist 
view at the lowest level and the problem solving view at the highest. 

1.1.3 According to the NCS: 

Mathematics is the study of quantity, structure, space and change. Mathematicians seek out patterns, 
formulate new conjectures, and establish axiomatic systems by rigorous deduction from appropriately chosen 
axioms and definitions. Mathematics is a distinctly human activity practised by all cultures, for thousands 
of years. Mathematical problem solving enables us to understand the world (physical, social and economic) 
around us, and, most of all, to teach us to think creatively. 

This corresponds well to the problem solving view of mathematics and may challenge some of our instru- 
mentalist or Platonistic views of mathematics as a static body of knowledge of accumulated facts, rules and 
skills to be learnt and applied. The NCS is trying to discourage such an approach and encourage mathemat- 
ics educators to dynamically and creatively involve their learners as mathematicians engaged in a process of 
study, understanding, reasoning, problem solving and communicating mathematically. 

Below is a check list that can guide you in actively designing your lessons in an attempt to embrace the 
definition of mathematics from the NCS and move towards a problem solving conception of the subject. 
Adopting such an approach to the teaching and learning of mathematics will in turn contribute to the 
intended curriculum being properly implemented and attained through the quality of learners coming out of 
the education system. 




Learners engage in solving contextual problems re- 
lated to their lives that require them to interpret 
a problem and then find a suitable mathematical 

Learners are asked to work out which bus service 
is the cheapest given the fares they charge and the 
distance they want to travel. 

Learners engage in solving problems of a purely 
mathematical nature, which require higher order 
thinking and application of knowledge (non-routine 

Learners are required to draw a graph; they have 
not yet been given a specific technique on how to 
draw (for example a parabola), but have learnt to 
use the table method to draw straight-line graphs. 

Learners are given opportunities to negotiate mean- 

Learners discuss their understanding of concepts 
and strategies for solving problems with each other 
and the educator. 

Learners are shown and required to represent situa- 
tions in various but equivalent ways (mathematical 
modelling) . 

Learners represent data using a graph, a table and 
a formula to represent the same data. 

Learners individually do mathematical investiga- 
tions in class, guided by the educator where nec- 

Each learner is given a paper containing the math- 
ematical problem (for instance to find the number 
of prime numbers less than 50) that needs to be in- 
vestigated and the solution needs to be written up. 
Learners work independently. 

Learners work together as a group/team to investi- 
gate or solve a mathematical problem. 

A group is given the task of working together to 
solve a problem that requires them investigating 
patterns and working through data to make con- 
jectures and find a formula for the pattern. 

Learners do drill and practice exercises to consoli- 
date the learning of concepts and to master various 

Completing an exercise requiring routine proce- 

Learners are given opportunities to see the inter- 
relatedness of the mathematics and to see how the 
different outcomes are related and connected. 

While learners work through geometry problems, 
they are encouraged to make use of algebra. 

Learners are required to pose problems for their ed- 
ucator and peer learners. 

Learners are asked to make up an algebraic word 
problem (for which they also know the solution) for 
the person sitting next to them to solve. 

Table 1.3 

1.1.4 Outcomes 

Summary of topics, outcomes and their relevance: 

1. Functions - 

- linear, quadratic, exponential, rational 

continued on next page 




Relationships between variables 
in terms of graphical, verbal and 
symbolic representations of func- 
tions (tables, graphs, words and 
formulae) . 

Functions form a core part of 
learners' mathematical 
understanding and reasoning 
processes in algebra. This is also 
an excellent opportunity for 
contextual mathematical 
modelling questions. 


Generating graphs and generalis- 
ing effects of parameters of ver- 
tical shifts and stretches and re- 
flections about the x-axis. 


Problem solving and graph work 
involving prescribed functions. 

Table 1.4 

2. Number Patterns, 

Sequences and Series 




Number patterns with constant 

Much of mathematics revolves 
around the identification of pat- 

Table 1.5 

3. Finance, Growth and Decay 




Use simple and compound 
growth formulae. 

The mathematics of finance is 
very relevant to daily and 
long-term financial decisions 
learners will need to make in 
terms of investing, taking loans, 


Implications of fluctuating ex- 
change rates. 

Table 1.6 

saving and understanding 
exchange rates and their 
influence more globally. 

4. Algebra 




Identifying and converting forms 
of rational numbers. Working 
with simple surds that are not 

Algebra provides the basis for 
mathematics learners to move 
from numerical calculations to 
generalising operations, 

simplifying expressions, solving 
equatKSHlftaHffiflih^ffiBg and 

inequalities in solving contextual 



Working with laws of inte- 
gral exponents. Establish between 
which two integers a simple surd 
lies. Appropriately rounding off 
real numbers. 


Manipulating and simplifying al- 
gebraic expressions (including 
multiplication and factorisation). 


Solving linear, quadratic and ex- 
ponential equations. Solving lin- 
ear inequalities in one and two 
variables algebraically and graph- 

Table 1.7 

5. Differential Calculus 




Investigate average rate of change 

The central aspect of rate of 

between two independent values 

change to differential calculus is a 

of a function. 

basis to further understanding of 
limits, gradients and calculations 
and formulae necessary for work 
in engineering fields, e.g. design- 
ing roads, bridges etc. 

Table 1.8 

6. Probability 




Compare relative frequency and 

This topic is helpful in developing 

theoretical probability.Use Venn 

good logical reasoning in learn- 

diagrams to solve probability 

ers and for educating them in 

problems. Mutually exclusive and 

terms of real-life issues such as 

complementary events. Identity 

gambling and the possible pitfalls 

for any two events A and B. 


Table 1.9 

7. Euclidean Geometry and Measurement 




Investigate, form and try to prove 
conjectures about properties of 
special triangles, quadrilaterals 
and other polygons. Disprove 
false conjectures using counter- 
examples. Investigate alternative 
definitions of various polygons. 

The thinking processes and 
mathematical skills of proving 
conjectures and identifying false 
conjectures is more the relevance 
here than the actual content 
studied. The surface area and 
volume content studied in 
real-life contexts of designing 
kitchens, tiling and painting 
rooms, designing packages, etc. 
is relevant to the current and 


Solve problems involving surface 
area and volumes of solids and 
combinations thereof. 

Table 1.10 

future lives of learners. 

8. Trigonometry 




Definitions of trig func- 
tions. Derive values for special 
angles. Take note of names for 
reciprocal functions. 

Trigonometry has several uses 
within society, including within 
navigation, music, geographical 
locations and building design 
and construction. 


Solve problems in 2 dimensions. 


Extend definition of basic trig 
functions to all four quadrants 
and know graphs of these func- 


Investigate and know the effects 
of a and q on the graphs of basic 
trig functions. 

Table 1.11 

9. Analytical Geometry 




Represent geometric figures on a 

This section provides a further 

Cartesian coordinate system. For 

application point for learners' al- 

any two points, derive and apply 

gebraic and trigonometric inter- 

formula for calculating distance, 

action with the Cartesian plane. 

gradient of line segment and co- 

Artists and design and layout in- 

ordinates of mid-point. 

dustries often draw on the con- 
tent and thought processes of this 
mathematical topic. 

Table 1.12 


10. Statistics 




Collect, organise and interpret 
univariate numerical data to de- 
termine mean, median, mode, 
percentiles, quartiles, deciles, in- 
terquartile and semi-interquartile 

Citizens are daily confronted 
with interpreting data presented 
from the media. Often this data 
may be biased or misrepresented 
within a certain context. In any 
type of research, data collection 
and handling is a core feature. 
This topic also educates learners 
to become more socially and 


Identify possible sources of bias 
and errors in measurements. 

Table 1.13 

politically educated with regards 
to the media. 

Mathematics educators also need to ensure that the following important specific aims and general prin- 
ciples are applied in mathematics activities across all grades: 

• Calculators should only be used to perform standard numerical computations and verify calculations 
done by hand. 

• Real-life problems should be incorporated into all sections to keep mathematical modelling as an 
important focal point of the curriculum. 

• Investigations give learners the opportunity to develop their ability to be more methodical, to generalise 
and to make and justify and/or prove conjectures. 

• Appropriate approximation and rounding skills should be taught and continuously included and en- 
couraged in activities. 

• The history of mathematics should be incorporated into projects and tasks where possible, to illustrate 
the human aspect and developing nature of mathematics. 

• Contextual problems should include issues relating to health, social, economic, cultural, scientific, 
political and environmental issues where possible. 

• Conceptual understanding of when and why should also feature in problem types. 

• Mixed ability teaching requires educators to challenge able learners and provide remedial support where 

• Misconceptions exposed by assessment need to be dealt with and rectified by questions designed by 

• Problem solving and cognitive development should be central to all mathematics teaching and learning 
so that learners can apply the knowledge effectively. 

Chapter 2 

Assessment and Assessment Support 1 

2.1 Assessment 

"Educator assessment is part of everyday teaching and learning in the classroom. Educators discuss with 
learners, guide their work, ask and answer questions, observe, help, encourage and challenge. In addition, 
they mark and review written and other kinds of work. Through these activities they are continually finding 
out about their learners' capabilities and achievements. This knowledge then informs plans for future work. 
It is this continuous process that makes up educator assessment. It should not be seen as a separate activity 
necessarily requiring the use of extra tasks or tests." 

As the quote above suggests, assessment should be incorporated as part of the classroom practice, rather 
than as a separate activity. Research during the past ten years indicates that learners get a sense of what 
they do and do not know, what they might do about this and how they feel about it, from frequent and regu- 
lar classroom assessment and educator feedback. The educator's perceptions of and approach to assessment 
(both formal and informal assessment) can have an influence on the classroom culture that is created with 
regard to the learners' expectations of and performance in assessment tasks. Literature on classroom assess- 
ment distinguishes between two different purposes of assessment; assessment of learning and assessment for 

Assessment of learning tends to be a more formal assessment and assesses how much learners have learnt 
or understood at a particular point in the annual teaching plan. The NCS provides comprehensive guidelines 
on the types of and amount of formal assessment that needs to take place within the teaching year to make up 
the school-based assessment mark. The school-based assessment mark contributes 25% of the final percentage 
of a learner's promotion mark, while the end-of-year examination constitutes the other 75% of the annual 
promotion mark. Learners are expected to have 7 formal assessment tasks for their school-based assessment 
mark. The number of tasks and their weighting in the Grade 10 Mathematics curriculum is summarised 


Weight (%) 

Term 1 






continued on next page 

1 This content is available online at <http://cnx.Org/content/m40351/l.l/>. 




Term 2 

Assignment /TestExamin£,tikfti 


Term 3 



Term 4 



School-Based Assessment Mark 


School-Based Assessment Mark (as a % of Promo- 
tion Mark) 


End-of-Year Examination 


Promotion Mark 


Table 2.1 

The following provides a brief explanation of each of the assessment tasks included in the assessment 
programme above. 

2.1.1 Tests 

All mathematics educators are familiar with this form of formal assessment. Tests include a variety of 
items/questions covering the topics that have been taught prior to the test. The new NCS also stipulates 
that mathematics tests should include questions that cover the following four types of cognitive levels in the 
stipulated weightings: 

Cognitive Levels 


Weighting (%) 


Estimation and appropriate 
rounding of numbers. Proofs of 
prescribed theorems. Derivation 
of formulae. Straight re- 
call. Identification and direct 
use of formula on informa- 
tion sheet (no changing of the 
subject). Use of mathemati- 
cal facts. Appropriate use of 
mathematical vocabulary. 


continued on next page 


Routine Procedures 

Perform well known proce- 
dures. Simple applications and 
calculations. Derivation from 
given information. Identification 
and use (including changing 
the subject) of correct for- 
mula. Questions generally similar 
to those done in class. 


Complex Procedures 

Problems involve complex cal- 
culations and/or higher reason- 
ing. There is often not an obvious 
route to the solution. Problems 
need not be based on real world 
context. Could involve making 
significant connections between 
different representations. Require 
conceptual understanding. 


Problem Solving 

Unseen, non-routine problems 
(which are not necessarily dif- 
ficult). Higher order understand- 
ing and processes are often in- 
volved. Might require the ability 
to break the problem down into 
its constituent parts. 


Table 2.2 

The breakdown of the tests over the four terms is summarised from the NCS assessment programme as 

Term 1: One test of at least 50 marks, and one hour or two/three tests of at least 40 minutes each. 
Term 2: Either one test (of at least 50 marks) or an assignment. 
Term 3: Two tests, each of at least 50 marks and one hour. 
Term 4: One test of at least 50 marks . 

2.1.2 Projects / Investigations 

Investigations and projects consist of open-ended questions that initiate and expand thought processes. 
Acquiring and developing problem-solving skills are an essential part of doing investigations and projects. 
These tasks provide learners with the opportunity to investigate, gather information, tabulate results, make 
conjectures and justify or prove these conjectures. Examples of investigations and projects and possible 
marking rubrics are provided in the next section on assessment support. The NCS assessment programme 
indicates that only one project or investigation (of at least 50 marks) should be included per year. Although 
the project/investigation is scheduled in the assessment programme for the first term, it could also be done 
in the second term. 

2.1.3 Assignments 

The NCS includes the following tasks as good examples of assignments: 

• Open book test 

• Translation task 

• Error spotting and correction 



• Shorter investigation 

• Journal entry 

• Mind-map (also known as a metacog) 

• Olympiad (first round) 

• Mathematics tutorial on an entire topic 

• Mathematics tutorial on more complex/problem solving questions 

The NCS assessment programme requires one assignment in term 2 (of at least 50 marks) which could also 
be a combination of some of the suggested examples above. More information on these suggested examples 
of assignments and possible rubrics are provided in the following section on assessment support. 

2.1.4 Examinations 

Educators are also all familiar with this summative form of assessment that is usually completed twice a 
year: mid-year examinations and end-of-year examinations. These are similar to the tests but cover a wider 
range of topics completed prior to each examination. The NCS stipulates that each examination should 
also cover the four cognitive levels according to their recommended weightings as summarised in the section 
above on tests. The following table summarises the requirements and information from the NCS for the two 




Content and Mark 

Mid- Year Exam 

100 50 + 50 

One paper: 2 hours 


Two papers: each of 1 


Topics completed 

End-of-Year Exam 

100 + 

Paper 1: 2 hours 

Number patterns (±10) 
Algebraic expressions, 
equations and inequali- 
ties (±25) 
Functions (±35) 
Exponents (±10) 
Finance (±10) 
Probability (±10) 

continued on next page 



Paper 2: 2 hours 

Trigonometry (±45) 

Analytical geometry 


Euclidean geometry and 

measurement (±25) 

Statistics (±15) 

Table 2.3 

In the annual teaching plan summary of the NCS in Mathematics for Grade 10, the pace setter section 
provides a detailed model of the suggested topics to be covered each week of each term and the accompanying 
formal assessment. 

Assessment for learning tends to be more informal and focuses on using assessment in and of daily 
classroom activities that can include: 

• Marking homework 

• Baseline assessments 

• Diagnostic assessments 

• Group work 

• Class discussions 

• Oral presentations 

• Self-assessment 

• Peer-assessment 

These activities are expanded on in the next section on assessment support and suggested marking rubrics 
are provided. Where formal assessment tends to restrict the learner to written assessment tasks, the informal 
assessment is necessary to evaluate and encourage the progress of the learners in their verbal mathematical 
reasoning and communication skills. It also provides a less formal assessment environment that allows learners 
to openly and honestly assess themselves and each other, taking responsibility for their own learning, without 
the heavy weighting of the performance (or mark) component. The assessment for learning tasks should be 
included in the classroom activities at least once a week (as part of a lesson) to ensure that the educator is 
able to continuously evaluate the learners' understanding of the topics covered as well as the effectiveness, 
and identify any possible deficiencies in his or her own teaching of the topics. 

2,1,5 Assessment Support 

A selection of explanations, examples and suggested marking rubrics for the assessment of learning (formal) 
and the assessment for learning (informal) forms of assessment discussed in the preceding section are provided 
in this section. Baseline Assessment 

Baseline assessment is a means of establishing: 

• What prior knowledge a learner possesses 

• What the extent of knowledge is that they have regarding a specific learning area 

• The level they demonstrate regarding various skills and applications 

• The learner's level of understanding of various learning areas 

It is helpful to educators in order to assist them in taking learners from their individual point of departure 
to a more advanced level and to thus make progress. This also helps avoid large "gaps" developing in the 
learners' knowledge as the learner moves through the education system. Outcomes-based education is a more 
learner-centered approach than we are used to in South Africa, and therefore the emphasis should now be 
on the level of each individual learner rather than that of the whole class. 


The baseline assessments also act as a gauge to enable learners to take more responsibility for their own 
learning and to view their own progress. In the traditional assessment system, the weaker learners often drop 
from a 40% average in the first term to a 30% average in the fourth term due to an increase in workload, 
thus demonstrating no obvious progress. Baseline assessment, however, allows for an initial assigning of 
levels which can be improved upon as the learner progresses through a section of work and shows greater 
knowledge, understanding and skill in that area. Diagnostic Assessments 

These are used to specifically find out if any learning difficulties or problems exist within a section of work 
in order to provide the learner with appropriate additional help and guidance. The assessment helps the 
educator and the learner identify problem areas, misunderstandings, misconceptions and incorrect use and 
interpretation of notation. 

Some points to keep in mind: 

• Try not to test too many concepts within one diagnostic assessment. 

• Be selective in the type of questions you choose. 

• Diagnostic assessments need to be designed with a certain structure in mind. As an educator, you 
should decide exactly what outcomes you will be assessing and structure the content of the assessment 

• The assessment is marked differently to other tests in that the mark is not the focus but rather the 
type of mistakes the learner has made. 

An example of an understanding rubric for educators to record results is provided below: 

- indicates that the learner has not grasped the concept at all and that there appears to be a fundamental 
mathematical problem. 

1 - indicates that the learner has gained some idea of the content, but is not demonstrating an under- 
standing of the notation and concept. 

2 - indicates evidence of some understanding by the learner but further consolidation is still required. 

3 - indicates clear evidence that the learner has understood the concept and is using the notation correctly. 
An example of a diagnostic assessment to evaluate learners' proficiency in calculator skills is provided 

below. There is a component of self-assessment as well as a component on educator assessment and how to 
group the various questions to diagnose any gaps or problems with learners' calculator skills. 

Calculator worksheet - diagnostic skills assessment 

Question 1 


a) 242 + 63 

b) 2 - 36 x (114 + 25) 

c) V144 + 25 

d) ^729 

e) -312 + 6 + 879 - 321 + 18 901 

Question 2 

a) ! + ! = ___ 

b , )2 Ms = - 

c - 2 §3+t = - 
d) 4 - § x § = . 

e MlO 9/ • 5 

f)2x(|) 2 -(i) = 

1 9 4_ 

4 16 — 

Self- Assessment Rubric: Name: 






If X, write down sequence of keys pressed 
















Table 2.4 

Educator Assessment Rubric: 

Type of Skill 


Needs Practice 


Raising to a Power 

Finding a Root 

Calculations with Fractions 

Brackets and Order of Operations 

Estimation and Mental Control 

Table 2.5 

Guidelines for Calculator Skills Assessment: 

Type of Skill 



Raising to a Power 

Squaring and cubing 
Higher order powers 

la, 2f 

continued on next page 



Finding a Root 

Square and cube roots 

lc, 2g 

Higher order roots 


Calculations with Fractions 

Basic operations 

2a, 2d 

Mixed numbers 

2b, 2c 

Negative numbers 

le, 2c 

Squaring fractions 


Square rooting fractions 


Brackets and Order of Opera- 

Correct use of brackets or order 

lb, lc, 2e, 

2f, 2g 


of operations 

Estimation and Mental Control 



Table 2.6 

Suggested guideline to allocation of overall levels 
Level 1 

• Learner is able to do basic operations on calculator. 

• Learner is able to do simple calculations involving fractions. 

• Learner does not display sufficient mental estimation and control techniques. 

Level 2 

• Learner is able to do basic operations on calculator. 

• Learner is able to square and cube whole numbers as well as find square and cube roots of numbers. 

• Learner is able to do simple calculations involving fractions as well as correctly execute calculations 
involving mixed numbers. 

• Learner displays some degree of mental estimation awareness. 

Level 3 

Learner is able to do basic operations on calculator. 

Learner is able to square and cube rational numbers as well as find square and cube roots of numbers. 

Learner is also able to calculate higher order powers and roots. 

Learner is able to do simple calculations involving fractions as well as correctly execute calculations 

involving mixed numbers. 

Learner works correctly with negative numbers. 

Learner is able to use brackets in certain calculations but has still not fully understood the order of 

operations that the calculator has been programmed to execute, hence the need for brackets. 

Learner is able to identify possible errors and problems in their calculations but needs assistance solving 

the problem. 

Level 4 

Learner is able to do basic operations on calculator. 

Learner is able to square and cube rational numbers as well as find square and cube roots. 

Learner is also able to calculate higher order powers and roots. 

Learner is able to do simple calculations involving fractions as well as correctly execute calculations 

involving mixed numbers. 

Learner works correctly with negative numbers. 

Learner is able to work with brackets correctly and understands the need and use of brackets and the 

"= key" in certain calculations due to the nature of a scientific calculator. 

Learner is able to identify possible errors and problems in their calculations and to find solutions to 

these in order to arrive at a "more viable" answer. 

2,1,6 Other Short Diagnostic Tests 

These are short tests that assess small quantities of recall knowledge and application ability on a day-to-day 
basis. Such tests could include questions on one or a combination of the following: 

• Definitions 

• Theorems 

• Riders (geometry) 

• Formulae 

• Applications 

• Combination questions 


l.Points A (-5 ; -3), B (-1 ; 2) and C (9 ; -6) are the vertices of AABC. 

a) Calculate the gradients of AB and BC and hence show that angle ABC is equal to 90 . (5) 

b) State the distance formula and use it to calculate the lengths of the sides AB, BC and AC of AABC. 
(Leave your answers in surd form). (5) 


1. Write down the formal definition of an exponent as well as the exponent laws for integral exponents. 

2. Simplify: *jg£ x ^ (4) 

1. A jet leaves an airport and travels 578 km in a direction of 50 E of N. The pilot then changes direction 
and travels 321 km 10 °W of N. 

a) How far away from the airport is the jet? (To the nearest kilometre) (5) 

b) Determine the jet's bearing from the airport. (5) Exercises 

This entails any work from the textbook or other source that is given to the learner, by the educator, to 
complete either in class or at home. Educators should encourage learners not to copy each other's work and 
be vigilant when controlling this work. It is suggested that such work be marked/controlled by a check list 
(below) to speed up the process for the educator. 

The marks obtained by the learner for a specific piece of work need not be based on correct and/or 
incorrect answers but preferably on the following: 

• the effort of the learner to produce answers. 

• the quality of the corrections of work that was previously incorrect. 

• the ability of the learner to explain the content of some selected examples (whether in writing or orally) . 

The following rubric can be used to assess exercises done in class or as homework: 


Performance Indicators 

continued on next page 



Work Done 

2 All the work 

1 Partially completed 

No work done 

Work Neatly Done 

2 Work neatly done 

1 Some work not neatly 

Messy and muddled 

Corrections Done 

2 All corrections done 

1 At least half of the cor- 
rections done 

No corrections done 

Correct Mathematical 

2 Consistently 

1 Sometimes 


Understanding of Math- 
ematical Techniques 
and Processes 

2 Can explain concepts 
and processes precisely 

1 Explanations are am- 
biguous or not focused 

Explanations are con- 
fusing or irrelevant 

Table 2.7 Journal entries 

A journal entry is an attempt by a learner to express in the written word what is happening in Mathematics. 
It is important to be able to articulate a mathematical problem, and its solution in the written word. 
This can be done in a number of different ways: 

• Today in Maths we learnt 

• Write a letter to a friend, who has been sick, explaining what was done in class today. 

• Explain the thought process behind trying to solve a particular maths problem, e.g. sketch the graph 
of y = x 2 2x 2 + land explain how to sketch such a graph. 

• Give a solution to a problem, decide whether it is correct and if not, explain the possible difficulties 
experienced by the person who wrote the incorrect solution. 

A journal is an invaluable tool that enables the educator to identify any mathematical misconceptions of the 
learners. The marking of this kind of exercise can be seen as subjective but a marking rubric can simplify 
the task. 

The following rubric can be used to mark journal entries. The learners must be given the marking rubric 
before the task is done. 


Competent (2 

Still Developing (1 

Not Yet Developed 
(0 Marks) 

Completion in Time 

Correctness of the Ex- 

Correct and Relevant 
use of Mathematical 

continued on next page 


Is the Mathematics Cor- 

Has the Concept Been 
Interpreted Correctly? 

Table 2.8 Translations 

Translations assess the learner's ability to translate from words into mathematical notation or to give an 
explanation of mathematical concepts in words. Often when learners can use mathematical language and 
notation correctly, they demonstrate a greater understanding of the concepts. 

For example: 

Write the letter of the correct expression next to the matching number: 

x increased by 10 a) xy 

The product of x and y b) x 2 

The sum of a certain number and c) x 2 

double that number d) 29x 

Half of a certain number multiplied by itself e) jx2 

Two less than x f ) x + x + 2 

A certain number multiplied by itself g) x 2 

Two consecutive even numbers h) x 29 

x + x + x + to 29 terms i) x + 2x 

x.x.x.x.x to 29 factors j) x + 10 

A certain number divided by 2 Group Work 

One of the principles in the NCS is to produce learners who are able to work effectively within a group. 
Learners generally find this difficult to do. Learners need to be encouraged to work within small groups. 
Very often it is while learning under peer assistance that a better understanding of concepts and processes 
is reached. Clever learners usually battle with this sort of task, and yet it is important that they learn how 
to assist and communicate effectively with other learners. Mind Maps or Metacogs 

A metacog or "mind map" is a useful tool. It helps to associate ideas and make connections that would 
otherwise be too unrelated to be linked. A metacog can be used at the beginning or end of a section of work 
in order to give learners an overall perspective of the work covered, or as a way of recalling a section already 
completed. It must be emphasised that it is not a summary. Whichever way you use it, it is a way in which 
a learner is given the opportunity of doing research in a particular field and can show that he/she has an 
understanding of the required section. 

This is an open book form of assessment and learners may use any material they feel will assist them. 
It is suggested that this activity be practised, using other topics, before a test metacog is submitted for 
portfolio assessment purposes. 

On completion of the metacog, learners must be able to answer insightful questions on the metacog. This 
is what sets it apart from being just a summary of a section of work. Learners must refer to their metacog 
when answering the questions, but may not refer to any reference material. Below are some guidelines to 
give to learners to adhere to when constructing a metacog as well as two examples to help you get learners 
started. A marking rubric is also provided. This should be made available to learners before they start 



constructing their metacogs. On the next page is a model question for a metacog, accompanied by some 
sample questions that can be asked within the context of doing a metacog about analytical geometry. 
A basic metacog is drawn in the following way: 

• Write the title/topic of the subject in the centre of the page and draw a circle around it. 

• For the first main heading of the subject, draw a line out from the circle in any direction, and write 
the heading above or below the line. 

• For sub-headings of the main heading, draw lines out from the first line for each subheading and label 
each one. 

• For individual facts, draw lines out from the appropriate heading line. 

Metacogs are one's own property. Once a person understands how to assemble the basic structure they can 
develop their own coding and conventions to take things further, for example to show linkages between facts. 
The following suggestions may assist educators and learners to enhance the effectiveness of their metacogs: 

• Use single words or simple phrases for information. Excess words just clutter the metacog and take 
extra time to write down. 

• Print words - joined up or indistinct writing can be more difficult to read and less attractive to look 

• Use colour to separate different ideas - this will help your mind separate ideas where it is necessary, 
and helps visualisation of the metacog for easy recall. Colour also helps to show organisation. 

• Use symbols and images where applicable. If a symbol means something to you, and conveys more 
information than words, use it. Pictures also help you to remember information. 

• Use shapes, circles and boundaries to connect information - these are additional tools to help show 
the grouping of information. 

Use the concept of analytical geometry as your topic and construct a mind map (or metacog) containing all 
the information (including terminology, definitions, formulae and examples) that you know about the topic 
of analytical geometry. 

Possible questions to ask the learner on completion of their metacog: 

• Briefly explain to me what the mathematics topic of analytical geometry entails. 

• Identify and explain the distance formula, the derivation and use thereof for me on your metacog. 

• How does the calculation of gradient in analytical geometry differ (or not) from the approach used to 
calculate gradient in working with functions? 

A suggested simple rubric for marking a metacog: 


Competent (2 

Still Developing (1 

Not Yet Developed 
(0 Marks) 

Completion in Time 

Main Headings 

continued on next page 


Correct Theory (Formu- 
lae, Definitions, Termi- 
nology etc.) 



Table 2.9 

10 marks for the questions, which are marked using the following scale: 

- no attempt or a totally incorrect attempt has been made 

1 - a correct attempt was made, but the learner did not get the correct answer 

2 - a correct attempt was made and the answer is correct Investigations 

Investigations consist of open-ended questions that initiate and expand thought processes. Acquiring and 
developing problem-solving skills are an essential part of doing investigations. 

It is suggested that 2-3 hours be allowed for this task. During the first 30 - 45 minutes learners could 
be encouraged to talk about the problem, clarify points of confusion, and discuss initial conjectures with 
others. The final written-up version should be done individually though and should be approximately four 

Assessing investigations may include feedback/ presentations from groups or individuals on the results 
keeping the following in mind: 

• following of a logical sequence in solving the problems 

• pre-knowledge required to solve the problem 

• correct usage of mathematical language and notation 

• purposefulness of solution 

• quality of the written and oral presentation 

Some examples of suggested marking rubrics are included on the next few pages, followed by a selection of 
topics for possible investigations. 

The following guidelines should be provided to learners before they begin an investigation: 

General Instructions Provided to Learners 

1. You may choose any one of the projects/investigations given (see model question on investigations) 

2. You should follow the instructions that accompany each task as these describe the way in which the 
final product must be presented. 

3. You may discuss the problem in groups to clarify issues, but each individual must write-up their own 

4. Copying from fellow learners will cause the task to be disqualified. 

5. Your educator is a resource to you, and though they will not provide you with answers / solutions, 
they may be approached for hints. 

The Presentation 

The investigation is to be handed in on the due date, indicated to you by your educator. It should have 
as a minimum: 

• A description of the problem. 

• A discussion of the way you set about dealing with the problem. 

• A description of the final result with an appropriate justification of its validity. 

• Some personal reflections that include mathematical or other lessons learnt, as well as the feelings 
experienced whilst engaging in the problem. 



• The written-up version should be attractively and neatly presented on about four A4 pages. 

• Whilst the use of technology is encouraged in the presentation, the mathematical content and processes 

must remain the major focus. 

Below are some examples of possible rubrics to use when marking investigations: 
Example 1: 

Level of Performance 



• Contains a complete response. 

• Clear, coherent, unambiguous and elegant ex- 

• Includes clear and simple diagrams where ap- 

• Shows understanding of the question's math- 
ematical ideas and processes. 

• Identifies all the important elements of the 

• Includes examples and counter examples. 

• Gives strong supporting arguments. 

• Goes beyond the requirements of the problem. 


• Contains a complete response. 

• Explanation less elegant, less complete. 

• Shows understanding of the question's math- 
ematical ideas and processes. 

• Identifies all the important elements of the 

• Does not go beyond the requirements of the 

continued on next page 



• Contains an incomplete response. 

• Explanation is not logical and clear. 

• Shows some understanding of the question's 
mathematical ideas and processes. 

• Identifies some of the important elements of 
the question. 

• Presents arguments, but incomplete. 

• Includes diagrams, but inappropriate or un- 


• Contains an incomplete response. 

• Omits significant parts or all of the question 
and response. 

• Contains major errors. 

• Uses inappropriate strategies. 

• No visible response or attempt 

Table 2.10 Orals 

An oral assessment involves the learner explaining to the class as a whole, a group or the educator his or her 
understanding of a concept, a problem or answering specific questions. The focus here is on the correct use 
of mathematical language by the learner and the conciseness and logical progression of their explanation as 
well as their communication skills. 

Orals can be done in a number of ways: 

• A learner explains the solution of a homework problem chosen by the educator. 

• The educator asks the learner a specific question or set of questions to ascertain that the learner 
understands, and assesses the learner on their explanation. 

• The educator observes a group of learners interacting and assesses the learners on their contributions 
and explanations within the group. 

• A group is given a mark as a whole, according to the answer given to a question by any member of a 

An example of a marking rubric for an oral: 

1 - the learner has understood the question and attempts to answer it 

2 - the learner uses correct mathematical language 

2 - the explanation of the learner follows a logical progression 

2 - the learner's explanation is concise and accurate 

2 - the learner shows an understanding of the concept being explained 

1 - the learner demonstrates good communication skills 

Maximum mark =10 

An example of a peer-assessment rubric for an oral: 

My name: 

Name of person I am assessing: 




Mark awarded 

Maximum Mark 

Correct Answer 


Clarity of Explanation 


Correctness of Explanation 


Evidence of Understanding 




Table 2.11 

Chapter 3 

Chapter Contexts 1 

3.1 Chapter Contexts 

3.1.1 Functions and Graphs 

Functions form a core part of learners' mathematical understanding and reasoning processes in algebra. This 
is also an excellent opportunity for contextual mathematical modelling questions. 

3.1.2 Number Patterns 

Much of mathematics revolves around the identification of patterns. 

3.1.3 Finance, Growth and Decay 

The mathematics of finance is very relevant to daily and long-term financial decisions learners will need to 
take in terms of investing, taking loans, saving and understanding exchange rates and their influence more 

3.1.4 Algebra 

Algebra provides the basis for mathematics learners to move from numerical calculations to generalising 
operations, simplifying expressions, solving equations and using graphs and inequalities in solving contextual 

3.1.5 Products and Factors 

Being able to multiply out and factorise are core skills in the process of simplifying expressions and solving 
equations in mathematics. 

3.1.6 Equations and Inequalities 

If learners are to later work competently with functions and the graphing and interpretation thereof, their 
understanding and skills in solving equations and inequalities will need to be developed. 

lr This content is available online at <http://cnx.Org/content/m40352/l.l/>. 



3.1.7 Estimating Surds 

Estimation is an extremely important component within mathematics. It allows learners to work with 
a calculator or present possible solutions while still being able to gauge how accurate and realistic their 
answers may be. This is relevant for other subjects too. For example, a learner working in biology may need 
to do a calculation to find the size of the average human kidney. An erroneous interpretation or calculation 
may result in an answer of 900 m. Without estimation skills, the learner may not query the possibility of such 
an answer and consider that it should rather be 9 cm. Estimating surds facilitates the further development 
of this skill of estimation. 

3.1.8 Exponentials 

Exponential notation is a central part of mathematics in numerical calculations as well as algebraic reasoning 
and simplification. It is also a necessary component for learners to understand and appreciate certain financial 
concepts such as compound interest and growth and decay. 

3.1.9 Irrational Numbers & Rounding Off 

Identifying irrational numbers and knowing their estimated position on a number line or graph is an im- 
portant part of any mathematical calculations and processes that move beyond the basic number system of 
whole numbers and integers. Rounding off irrational numbers (such as the value of n) when needed allows 
mathematics learners to work more efficiently with numbers that would otherwise be difficult to "pin down", 
use and comprehend. 

3.1.10 Rational Numbers 

Once learners have grasped the basic number system of whole numbers and integers, it is vital that their 
understanding of the numbers between integers is also expanded. This incorporates their dealing with frac- 
tions, decimals and surds which form a central part of most mathematical calculations in real-life contextual 

3.1.11 Differential Calculus: Average Gradient 

The central aspect of rate of change to differential calculus is a basis to further understanding of limits, 
gradients and calculations and formulae necessary for work in engineering fields, e.g. designing roads, bridges 

3.1.12 Probability 

This topic is helpful in developing good logical reasoning in learners and for educating them in terms of 
real-life issues such as gambling and the possible pitfalls thereof. 

3.1.13 Euclidean Geometry and Measurement 

The thinking processes and mathematical skills of proving conjectures and identifying false conjectures is 
more the relevance here than the actual content studied. The surface area and volume content studied in 
real-life contexts of designing kitchens, tiling and painting rooms, designing packages, etc. is relevant to the 
current and future lives of learners. 

3.1.14 Trigonometry 

Trigonometry has several uses within society, including within navigation, music, geographical locations and 
building design and construction. 


3.1.15 Analytical Geometry 

This section provides a further application point for learners' algebraic and trigonometric interaction with the 
Cartesian plane. Artists and design and layout industries often draw on the content and thought processes 
of this mathematical topic. 

3.1.16 Statistics 

Citizens are daily confronted with interpreting data presented from the media. Often this data may be biased 
or misrepresented within a certain context. In any type of research, data collection and handling is a core 
feature. This topic also educates learners to become more socially and politically educated with regards to 
the media. 


Chapter 4 

On the Web, Everyone can be a Scientist 1 

4.1 On the Web, Everyone can be a Scientist 

Did you know that you can fold protein molecules, hunt for new planets around distant suns or simulate 
how malaria spreads in Africa, all from an ordinary PC or laptop connected to the Internet? And you don't 
need to be a certified scientist to do this. In fact some of the most talented contributors are teenagers. The 
reason this is possible is that scientists are learning how to turn simple scientific tasks into competitive online 

This is the story of how a simple idea of sharing scientific challenges on the Web turned into a global 
trend, called citizen cyberscience. And how you can be a scientist on the Web, too. 

4.1.1 Looking for Little Green Men 

A long time ago, in 1999, when the World Wide Web was barely ten years old and no one had heard of Google, 
Facebook or Twitter, a researcher at the University of California at Berkeley, David Anderson, launched an 
online project called SETI@home. SETI stands for Search for Extraterrestrial Intelligence. Looking for life 
in outer space. 

Although this sounds like science fiction, it is a real and quite reasonable scientific project. The idea is 
simple enough. If there are aliens out there on other planets, and they are as smart or even smarter than 
us, then they almost certainly have invented the radio already. So if we listen very carefully for radio signals 
from outer space, we may pick up the faint signals of intelligent life. 

Exactly what radio broadcasts aliens would produce is a matter of some debate. But the idea is that if 
they do, it would sound quite different from the normal hiss of background radio noise produced by stars 
and galaxies. So if you search long enough and hard enough, maybe you'll find a sign of life. 

It was clear to David and his colleagues that the search was going to require a lot of computers. More 
than scientists could afford. So he wrote a simple computer program which broke the problem down into 
smaller parts, sending bits of radio data collected by a giant radio-telescope to volunteers around the world. 
The volunteers agreed to download a programme onto their home computers that would sift through the bit 
of data they received, looking for signals of life, and send back a short summary of the result to a central 
server in California. 

The biggest surprise of this project was not that they discovered a message from outer space. In fact, 
after over a decade of searching, no sign of extraterrestrial life has been found, although there are still vast 
regions of space that have not been looked at. The biggest surprise was the number of people willing to help 
such an endeavour. Over a million people have downloaded the software, making the total computing power 
of SETI@home rival that of even the biggest supercomputers in the world. 

1 This content is available online at <http://cnx.Org/content/m40353/l.l/>. 



David was deeply impressed by the enthusiasm of people to help this project. And he realized that 
searching for aliens was probably not the only task that people would be willing to help with by using the 
spare time on their computers. So he set about building a software platform that would allow many other 
scientists to set up similar projects. You can read more about this platform, called BOINC, and the many 
different kinds of volunteer computing projects it supports today, at 2 . 

There's something for everyone, from searching for new prime numbers (PrimeGrid) to simulating the 
future of the Earth's climate ( One of the projects,, involved 
researchers from the University of Cape Town as well as from universities in Mali and Senegal. 

The other neat feature of BOINC is that it lets people who share a common interest in a scientific topic 
share their passion, and learn from each other. BOINC even supports teams - groups of people who put their 
computer power together, in a virtual way on the Web, to get a higher score than their rivals. So BOINC is 
a bit like Facebook and World of Warcraft combined - part social network, part online multiplayer game. 

Here's a thought: spend some time searching around BOINC for a project you'd like to participate 
in, or tell your class about. 

4.1,2 You are a Computer, too 

Before computers were machines, they were people. Vast rooms full of hundreds of government employees 
used to calculate the sort of mathematical tables that a laptop can produce nowadays in a fraction of a 
second. They used to do those calculations laboriously, by hand. And because it was easy to make mistakes, 
a lot of the effort was involved in double-checking the work done by others. 

Well, that was a long time ago. Since electronic computers emerged over 50 years ago, there has been 
no need to assemble large groups of humans to do boring, repetitive mathematical tasks. Silicon chips can 
solve those problems today far faster and more accurately. But there are still some mathematical problems 
where the human brain excels. 

Volunteer computing is a good name for what BOINC does: it enables volunteers to contribute computing 
power of their PCs and laptops. But in recent years, a new trend has emerged in citizen cyberscience that 
is best described as volunteer thinking. Here the computers are replaced by brains, connected via the 
Web through an interface called eyes. Because for some complex problems - especially those that involve 
recognizing complex patterns or three-dimensional objects - the human brain is still a lot quicker and more 
accurate than a computer. 

Volunteer thinking projects come in many shapes and sizes. For example, you can help to classify 
millions of images of distant galaxies (GalaxyZoo), or digitize hand- written information associated with 
museum archive data of various plant species (Herbaria@home). This is laborious work, which if left to 
experts would take years or decades to complete. But thanks to the Web, it's possible to distribute images 
so that hundreds of thousands of people can contribute to the search. 

Not only is there strength in numbers, there is accuracy, too. Because by using a technique called valida- 
tion - which does the same sort of double-checking that used to be done by humans making mathematical 
tables - it is possible to practically eliminate the effects of human error. This is true even though each 
volunteer may make quite a few mistakes. So projects like Planet Hunters have already helped astronomers 
pinpoint new planets circling distant stars. The game Foldlt invites people to compete in folding protein 
molecules via a simple mouse-driven interface. By finding the most likely way a protein will fold, volunteers 
can help understand illnesses like Alzheimer's disease, that depend on how proteins fold. 

Volunteer thinking is exciting. But perhaps even more ambitious is the emerging idea of volunteer 
sensing: using your laptop or even your mobile phone to collect data - sounds, images, text you type in - 
from any point on the planet, helping scientists to create global networks of sensors that can pick up the 
first signs of an outbreak of a new disease (EpiCollect), or the initial tremors associated with an earthquake 
(, or the noise levels around a new airport (NoiseTube). 

There are about a billion PCs and laptops on the planet, but already 5 billion mobile phones. The 
rapid advance of computing technology, where the power of a ten-year old PC can easily be packed into a 

2 http://boinc. 


smart phone today, means that citizen cyberscience has a bright future in mobile phones. And this means 
that more and more of the world's population can be part of citizen cyberscience projects. Today there are 
probably a few million participants in a few hundred citizen cyberscience initiatives. But there will soon be 
seven billion brains on the planet. That is a lot of potential citizen cyberscientists. 

You can explore much more about citizen cyberscience on the Web. There's a great list of all sorts of 
projects, with brief summaries of their objectives, at 3 . BBC Radio 4 
produced a short series on citizen science 4 and 
you can subscribe to a newsletter about the latest trends in this field at 5 . 
The Citizen Cyberscience Centre, 6 which is sponsored by the South African 
Shuttleworth Foundation, is promoting citizen cyberscience in Africa and other developing regions. 


4 http:// 





Chapter 5 

FullMarks User Guide 1 

5.1 FullMarks User Guide 

FullMarks can be accessed at: 2 . 

Siyavula offers an open online assessment bank called FullMarks, for the sharing and accessing of 
curriculum-aligned test and exam questions with answers. This site enables educators to quickly set tests and 
exam papers, by selecting items from the library and adding them to their test. Educators can then download 
their separate test and memo which is ready for printing. FullMarks further offers educators the option of 
capturing their learners' marks in order to view a selection of diagnostic reports on their performance. 

To begin, you need to a create a free account by clicking on "sign up now" on the landing page. There is 
one piece of administration you need to do to get started properly: when you log in for the first time, click 
on your name on the top right. It will take you to your personal settings. You need to select Shuttleworth 
Foundation as your metadata organisation to see the curriculum topics. 

1 This content is available online at <http://cnx.Org/content/m40356/l.l/>. 
2 http:// 




5.1.1 What Can I do in FullMarks? 

Access and 
share questions 


Create tests \ / 
from questions/ *y 

class lists 

Create \ 
scoresheets ) 

/Analyse learners'^ 
V performance J 

Figure 5.1 

5.1.2 How do I do Each of These? Access and Share Questions 

Sharing questions: use either the online editor or OpenOffice template which can be downloaded from the 
website (Browse questions — > Contribute questions — » Import questions). Take your test/worksheet/other 
question source. Break it up into the smallest sized individual questions that make sense, and use the 
template style guide to style your page according to question/answer. Upload these questions or type them 
up in the online editor (Browse questions — > Contribute questions — > Add questions). Do not include overall 
question numbering but do include sub numbering if needed (e.g. la, 2c, etc.). Insert the mark and time 
allocation, tag questions according to grade, subject i.e. a description of the question, and then select the 
topics from the topic tree . Finalise questions so that they can be used in tests and accessed by other 
FullMarks users. 

Accessing questions: there are three ways to access questions in the database. Click on "Browse questions" 
— > click on the arrow to the left of the grade, which opens out the subjects — * keep clicking on the arrows 
to open the learning outcomes or, following the same process, instead of clicking on the arrow, click on the 
grade — > now you can browse the full database of questions for all the subjects in that grade or, from the 


landing page, click on "Browse questions" — > below the banner image click on "Find Questions" — > search by 
topics, author (if you know a contributor), text or keywords e.g. GrlO mathematics functions and graphs. Create Tests from Questions 

So, you have all these bits of tests (i.e. many questions!), but what you really want is the actual test. How 
do you do this? Well, you can simply click "add to test" on any question and then click on the "Tests" tab at 
the top right of the page, and follow the simple instructions. Alternatively you can create a test by starting 
with clicking on that same "Tests" tab, and add questions to your test that way. Once done, simply print 
off the PDF file of the questions and the file for the memo. Issue your test, collect them once complete, and 
mark them. Create Class Lists 

But now you are asking, how can I keep track of my classes? Is Johnny Brown in class A or B? Well, you 
can make a class list by clicking on the "Class lists" tab at the top right of the page, and either import a 
CSV file, or manually enter the relevant information for each class. Now you can issue tests to your classes, 
and have a class list for each class. And what about capturing their marks? Create Scoresheets 

For each test you can create a scoresheet. Select the "Tests" tab at the top right, click on "Marks" below 
the banner image, and select the test and follow the instructions to input their marks. You can then export 
these as a CSV file for use in spreadsheets. Analyse Learners' Performance 

And finally, you can print out reports of class performance. Click on "Reports" at the top right of the page, 
which opens various reports you can view. There are reports to see class performance, learner performance, 
class performance per topic, class performance per question, learner strengths and weaknesses, and learner 

So now you know how FullMarks works, we encourage you to make use of its simple functionality, and 
let it help you save time setting tests and analysing learner marks! 


Chapter 6 

Rich Media 

6.1 Rich Media 
6.1.1 General Resources 

Science education is about more than physics, chemistry and mathematics... It's about learning to think and 
to solve problems, which are valuable skills that can be applied through all spheres of life. Teaching these 
skills to our next generation is crucial in the current global environment where methodologies, technology 
and tools are rapidly evolving. Education should benefit from these fast moving developments. In our 
simplified model there are three layers to how technology can significantly influence your teaching and 
teaching environment. First Layer: Educator Collaboration 

There are many tools that help educators collaborate more effectively. We know that communities of practice 
are powerful tools for the refinement of methodology, content and knowledge as well as superb for providing 
support to educators. One of the challenges facing community formation is the time and space to have 
sufficient meetings to build real communities and exchanging practices, content and learnings effectively. 
Technology allows us to streamline this very effectively by transcending space and time. It is now possible 
to collaborate over large distances (transcending space) and when it is most appropriate for each individual 
(transcending time) by working virtually (email, mobile, online etc.). 

Our textbooks have been uploaded in their entirety to the Connexions website 
( 2 ), making them easily accessible and adaptable, as they are under an 
open licence, stored in an open format, based on an open standard, on an open-source platform, for free, 
where everyone can produce their own books. Our textbooks are released under an open copyright license - 
CC-BY. This Creative Commons By Attribution Licence allows others to legally distribute, remix, tweak, 
and build upon our work, even commercially, as long as they credit us for the original creation. With them 
being available on the Connexions website and due to the open copyright licence, learners and educators 
are able to download, copy, share and distribute our books legally at no cost. It also gives educators the 
freedom to edit, adapt, translate and contextualise them, to better suit their teaching needs. 

Connexions is a tool where individuals can share, but more importantly communities can form around 
the collaborative, online development of resources. Your community of educators can therefore: 

• form an online workgroup around the textbook; 

• make your own copy of the textbook; 

• edit sections of your own copy; 

1 This content is available online at <http://cnx.Org/content/m40357/l.l/>. 



• add your own content into the book or replace existing content with your content; 

• use other content that has been shared on the platform in your own book; 

• create your own notes / textbook / course material as a community. 

Educators often want to share assessment items as this helps reduce workload, increase variety and im- 
prove quality. Currently all the solutions to the exercises contained in the textbooks have been uploaded 
onto our free and open online assessment bank called FullMarks ( 3 ), with 
each exercise having a shortcode link to its solution on FullMarks. To access the solution simply go to 4 , enter the shortcode, and you will be redirected to the solution on FullMarks. 

FullMarks is similar to Connexions but is focused on the sharing of assessment items. FullMarks contains a 
selection of test and exam questions with solutions, openly shared by educators. Educators can further search 
and browse the database by subject and grade and add relevant items to a test. The website automatically 
generates a test or exam paper with the corresponding memorandum for download. 

By uploading all the end-of-chapter exercises and solutions to the open assessment bank, the larger 
community of educators in South Africa are provided with a wide selection of items to use in setting their 
tests and exams. More details about the use of FullMarks as a collaboration tool are included in the FullMarks 
section. Second Layer: Classroom Engagement 

In spite of the impressive array of rich media open educational resources available freely online, such as 
videos, simulations, exercises and presentations, only a small number of educators actively make use of 
them. Our investigations revealed that the overwhelming quantity, the predominant international context, 
and difficulty in correctly aligning them with the local curriculum level acts as deterrents. The opportunity 
here is that, if used correctly, they can make the classroom environment more engaging. 

Presentations can be a first step to bringing material to life in ways that are more compelling than are 
possible with just a blackboard and chalk. There are opportunities to: 

• create more graphical representations of the content; 

• control timing of presented content more effectively; 

• allow learners to relive the lesson later if constructed well; 

• supplement the slides with notes for later use; 

• embed key assessment items in advance to promote discussion; and 

• embed other rich media like videos. 

Videos have been shown to be potentially both engaging and effective. They provide opportunities to: 

• present an alternative explanation; 

• challenge misconceptions without challenging an individual in the class; and 

• show an environment or experiment that cannot be replicated in the class which could be far away, 
too expensive or too dangerous. 

Simulations are also very useful and can allow learners to: 

• have increased freedom to explore, rather than reproducing a fixed experiment or process; 

• explore expensive or dangerous environments more effectively; and 

• overcome implicit misconceptions. 

We realised the opportunity for embedding a selection of rich media resources such as presentations, simu- 
lations, videos and links into the online version of the FHSST books at the relevant sections. This will not 
only present them with a selection of locally relevant and curriculum aligned resources, but also position 
these resources within the appropriate grade and section. Links to these online resources are recorded in the 
print or PDF versions of the books, making them a tour-guide or credible pointer to the world of online rich 
media available. 

3 http:// 
4 http:// 

39 Third Layer: Beyond the Classroom 

The internet has provided many opportunities for self-learning and participation which were never before 
possible. There are huge stand-alone archives of videos like the Khan Academy which cover most Mathematics 
for Grades 1-12 and Science topics required in FET. These videos, if not used in class, provide opportunities 
for the learners to: 

• look up content themselves; 

• get ahead of class; 

• independently revise and consolidate their foundation; and 

• explore a subject to see if they find it interesting. 

There are also many opportunities for learners to participate in science projects online as real participants. 
Not just simulations or tutorials but real science so that: 

• learners gain an appreciation of how science is changing; 

• safely and easily explore subjects that they would never have encountered before university; 

• contribute to real science (real international cutting edge science programmes); 

• have the possibility of making real discoveries even from their school computer laboratory; and 

• find active role models in the world of science. 

In our book we've embedded opportunities to help educators and learners take advantage of all these re- 
sources, without becoming overwhelmed at all the content that is available online. 

6.1.2 Embedded Content 

Throughout the books you will see the following icons: 




Aside: Provides additional information about con- 
tent covered in the chapters, as well as for exten- 


An interesting fact: These highlight interesting 
information relevant to a particular section of the 

continued on next page 



Definition: This icon indicates a definition. 

Exercise: This indicates worked examples through- 
out the book. 


Tip: Helpful hints and tips appear throughout the 
book, highlighting important information, things to 
take note of, and areas where learners must exercise 

FullMarks: This icon indicates that shortcodes 
for FullMarks are present. Enter the shortcode into 23 , and you will be redi- 
rected to the solution on FullMarks, our free and 
open online assessment bank. FullMarks can be ac- 
cessed at: 24 

Presentation: This icon indicates that presen- 
tations are in the chapter. Enter the shortcode 
into 25 , and you will be 
redirected to the presentation shared on SlideShare 
by educators. SlideShare can be accessed at: 26 

continued on next page 


Simulation: This icon indicates that simula- 
tions are present. Enter the shortcode into 27 , and you will be redi- 
rected to the simulation online. An example is 
Phet Simulations. The website can be accessed at: 28 

Video: This icon indicates that videos 

are present. Enter the shortcode into 29 , and you will be 
redirected to the video online. An example is 
the Khan Academy videos. The website can be 
accessed at: 30 

URL: This icon indicates that shortcodes are 
present in the chapter and can be entered into 31 , where you will be redi- 
rected to the relevant website. 


Table 6.1 

23 http: 
24 http: 
25 http; 
26 http; 
27 http: 
28 http: 
29 http: 
30 http: 
31 http: 



Chapter 7 

Maths Blog Posts 1 

7.1 Blog Posts 

7.1.1 General Blogs 

Teachers Monthly - Education News and Resources 

• "We eat, breathe and live education! " 

• "Perhaps the most remarkable yet overlooked aspect of the South African teaching community is its 
enthusiastic, passionate spirit. Every day, thousands of talented, hard-working teachers gain new 
insight from their work and come up with brilliant, inventive and exciting ideas. Teacher's Monthly 
aims to bring teachers closer and help them share knowledge and resources. 

• Our aim is twofold . . . 

• To keep South African teachers updated and informed. 

• To give teachers the opportunity to express their views and cultivate their interests." 

• 2 

Head Thoughts - Personal Reflections of a School Headmaster 

• blog by Arthur Preston 

• "Arthur is currently the headmaster of a growing independent school in Worcester, in the Western 
Cape province of South Africa. His approach to primary education is progressive and is leading the 
school through an era of new development and change." 

• 3 

7.1.2 Maths Blogs 

CEO: Circumspect Education Officer 4 - Educating The Future 

• blog by Robyn Clark 

• "Mathematics teacher and inspirer." 

• 5 

dy/dan - Be less helpful 

1 This content is available online at <http://cnx.Org/content/m40358/l.l/>. 

2 http:// 






• blog by Dan Meyer 

• "I'm Dan Meyer. I taught high school math between 2004 and 2010 and I am currently studying at 
Stanford University on a doctoral fellowship. My specific interests include curriculum design (answering 
the question, "how we design the ideal learning experience for students?") and teacher education 
(answering the questions, "how do teachers learn?" and "how do we retain more teachers?" and "how 
do we teach teachers to teach?")." 

• 6 

Without Geometry, Life is Pointless - Musings on Math, Education, Teaching, and Research 

• blog by Avery 

• "I've been teaching some permutation (or is that combination?) of math and science to third through 
twelfth graders in private and public schools for 11 years. I'm also pursuing my EdD in education and 
will be both teaching and conducting research in my classroom this year." 

• 7 

Overthinking my teaching - The Mathematics I Encounter in Classrooms 

• blog by Christopher Danielson 

• "I think a lot about my math teaching. Perhaps too much. This is my outlet. I hope you find it 
interesting and that you'll let me know how it's going." 

• 8 

A Recursive Process - Math Teacher Seeking Patterns 

• blog by Dan 

• "I am a High School math teacher in upstate NY. I currently teach Geometry, Computer Programming 
(Alice and Java), and two half year courses: Applied and Consumer Math. This year brings a new 
21st century classroom (still not entirely sure what that entails) and a change over to standards based 
grades (#sbg)." 

• 9 

Think Thank Thunk — Dealing with the Fear of Being a Boring Teacher 


blog by Shawn Cornally 

"I am Mr. Cornally. I desperately want to be a good teacher. I teach Physics, Calculus, Programming, 

Geology, and Bioethics. Warning: I have problem with using colons. I proof read, albeit poorly." 10 

6 http://blog. 
7 http://mathteacherorstudent. 
8 http://christopherdanielson. 
10 http:// 

Chapter 8 
Solutions 1 

8.1 Solutions 

The solutions to the exercises can be found embedded throughout the textbooks on Connexions. 
The Maths textbook can be accessed at 2 

1 This content is available online at <http://cnx.Org/content/m40361/l.l/>. 

2 Siyavula textbooks: Grade 10 Maths [CAPS] <> 




Index of Keywords and Terms 

Keywords are listed by the section with that keyword (page numbers are in parentheses). Keywords 
do not necessarily appear in the text of the page. They are merely associated with that section. Ex. 
apples, § 1.1 (1) Terms are referenced by the page they appear on. Ex. apples, 1 

A Assessment, § 2(9) 

B Blog posts, § 7(43) 

C CAPS, § 8(45) 

Citizen cyberscience, 

F FullMarks, § 5(33) 


G Grade 10, § 1(1), § 8(45) 

Grade 10 Maths, § 2(9), § 3(25), § 4(29), 
§ 5(33), § 6(37), § 7(43) 

M Mathematics, § 1(1), § 8(45) 

R Rich media, § 6(37) 

S Siyavula, § 1(1), § 2(9), § 3(25), § 4(29), 
§ 5(33), § 6(37), § 7(43) 
Solutions, § 8(45) 

T Teachers' Guide, § 1(1), § 2(9), § 4(29), § 5(33) 
Teachers' Guides, § 3(25), § 6(37), § 7(43) 



Collection: Mathematics Grade 10 Teachers' Guide - Siyavula WebBooks 

Edited by: Bridget Nash 


License: http://creativecommons.Org/licenses/by/3.0/ 

Module: "TG Maths - Overview" 

Used here as: "Overview" 

By: Bridget Nash 


Pages: 1-8 

Copyright: Bridget Nash 


Module: "TG Maths - Assessment and Assessment Support" 

Used here as: "Assessment and Assessment Support" 

By: Bridget Nash 


Pages: 9-24 

Copyright: Bridget Nash 


Module: "TG Maths - Chapter Contexts" 

Used here as: "Chapter Contexts" 

By: Bridget Nash 


Pages: 25-27 

Copyright: Bridget Nash 


Module: "On the Web, Everyone can be a Scientist" 

By: Bridget Nash 


Pages: 29-31 

Copyright: Bridget Nash 


Module: "FullMarks User Guide" 

By: Bridget Nash 


Pages: 33-35 

Copyright: Bridget Nash 


Module: "Rich Media" 

By: Bridget Nash 


Pages: 37-41 

Copyright: Bridget Nash 



Module: "TG - Maths Blog Posts" 

Used here as: "Maths Blog Posts" 

By: Bridget Nash 


Pages: 43-44 

Copyright: Bridget Nash 

License: http://creativecommons.Org/licenses/by/3.0/ 

Module: "TG Maths - Solutions" 

Used here as: "Solutions" 

By: Bridget Nash 


Page: 45 

Copyright: Bridget Nash 


Mathematics Grade 10 Teachers' Guide - Siyavula WebBooks 

This collection is the Teachers' Guide for the Siyavula WebBook - Grade 10 Maths. 

About Connexions 

Since 1999, Connexions has been pioneering a global system where anyone can create course materials and 
make them fully accessible and easily reusable free of charge. We are a Web-based authoring, teaching and 
learning environment open to anyone interested in education, including students, teachers, professors and 
lifelong learners. We connect ideas and facilitate educational communities. 

Connexions's modular, interactive courses are in use worldwide by universities, community colleges, K-12 
schools, distance learners, and lifelong learners. Connexions materials are in many languages, including 
English, Spanish, Chinese, Japanese, Italian, Vietnamese, French, Portuguese, and Thai. Connexions is part 
of an exciting new information distribution system that allows for Print on Demand Books. Connexions 
has partnered with innovative on-demand publisher QOOP to accelerate the delivery of printed course 
materials and textbooks into classrooms worldwide at lower prices than traditional academic publishers.