LIBRARY
OF THE
UNIVERSITY OF CALIFORNIA.
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Accession 90255 Class
Teachers' Manual
to Walsh's Primary
Arithmetic*
MATHEMATICS FOR COMMON SCHOOLS
A
MANUAL FOE TEACHEES
INCLUDING
DEFINITIONS, PEINCIPLES, AND EULES
AND SOLUTIONS OF THE MOEE
DIFFICULT PEOBLEMS
BY
JOHN H. WALSH
ASSOCIATE SUPERINTENDENT OF PUBLIC INSTRUCTION
BROOKLYN, N.Y.
PRIMARY ARITHMETIC
D. C. HEATH & CO., PUBLISHEES .
1900
_ A3 /
M/3
COPYRIGHT, 189%
BY JOHN H. WALSH.
J. S. Gushing & Co. — Berwick & Smith.
Boston, Mass., U.S.A.
CONTENTS
(PRIMARY AND ELEMENTARY ARITHMETIC MANUAL.)
•>• PAGB
INTRODUCTORY 1
Plan and scope of the work — Grammar school algebra — Con-
structive geometry.
II
GENERAL HINTS 5
Division of the work — Additions and omissions — Oral and
written work — Use of books — Conduct of the recitation — Drills
and sight work — Definitions, principles, and rules — Language
— Analyses — Objective illustrations — Approximate answers —
Indicating operations — Paper vs. slates.
Ill
EARLY ARITHMETIC TEACHING 13
Counting — Primary arithmetic — The Grube method — Slate
problems.
IV
NOTES ON CHAPTER ONE . . . . .* 17
V
NOTES ON CHAPTER Two . t 22
VI
NOTES ON CHAPTER THREE . .... • • . . 27
iii
IV CONTENTS
VII
NOTES ON CHAPTER FOUB 32
VIII
NOTES ON CHAPTER FIVE 41
ANSWERS . 1
MANUAL FOR TEACHEES
INTRODUCTORY
Plan and Scope of the Work, — In addition to the subjects
generally included in the ordinary text-books in arithmetic,
Mathematics for Common /Schools contains such simple work
in algebraic equations and constructive geometry as can be
studied to advantage by pupils of the elementary schools.
The arithmetical portion is divided into thirteen chapters,
each of which, except the first, contains the work of a term of
five months. The following extracts from the table of contents
will show the arrangement of topics :
FIRST AND SECOND YEARS
Chapter I, — Numbers of Three Figures. Addition and Sub-
traction.
THIRD YEAR
Chapters II, and III, — Numbers of Five Figures. Multipli-
ers and Divisors of One Figure. Addition and Subtraction of
Halves, of Fourths, of Thirds. Multiplication by Mixed Num-
bers. Pint, Quart, and Gallon ; Ounce and Pound. Roman
Notation.
1
2 MANUAL FOR TEACHERS
FOURTH YEAR
Chapters IV, and V, — Numbers of Six Figures. Multipliers and
Divisors of Two or More Figures. Addition and Subtraction of
Easy Fractions. Multiplication by Mixed Numbers. Simple
Denominate Numbers. Roman Notation.
FIFTH YEAR
Chapters VI, and VII. — Fractions. Decimals of Three Places.
Bills. Denominate Numbers. Simple Measurements.
SIXTH YEAR
Chapters VIII, and IX, — Decimals. Bills. Denominate Num-
bers. Surfaces and Volumes. Percentage and Interest.
SEVENTH YEAR
Chapters XI, and XII, — Percentage and Interest. Commercial
and Bank Discount. Cause and Effect. Partnership. Bonds
and Stocks. Exchange. Longitude and Time. Surfaces and
Volumes.
EIGHTH YEAR
Chapters XIII, and XIV, — Partial Payments. Equation of
Payments. Annual Interest. Metric System. Evolution and
Involution. Surfaces and Volumes.
INTRODUCTORY 3
While all of the above topics are generally included in an
eight years' course, it may be considered advisable to omit some
of them, and to take up, instead, during the seventh and eighth
years, the constructive geometry work of Chapter XVI. Among
the topics that may be dropped without injury to the pupil are
Bonds and Stocks, Exchange, Partial Payments, and Equation
of Payments.
Grammar School Algebra, — Chapter X., consisting of a dozen
pages, is devoted to the subject of easy equations of one unknown
quantity, as a preliminary to the employment of the equation in
so much of the subsequent work in arithmetic as is rendered
more simple by this mode of treatment. To teachers desirous
of dispensing with rules, sample solutions of type examples, etc.,
the algebraic method of solving the so-called " problems " in per-
centage, interest, discount, etc., is strongly recommended.
In Chapter XV., intended chiefly for schools having a nine
years' course, the algebraic work is extended to cover simple
equations containing two or more unknown quantities, and pure
and affected quadratic equations of one unknown quantity.
No attempt has been made in these two chapters to treat
algebra as a science ; the aim has been to make grammar-school
pupils acquainted, to some slight extent, with the great instru-
ment of mathematical investigation, — the equation.
Constructive Geometry, — Progressive teachers will appreciate the
importance of supplementing the concrete geometrical instruction
now given in the drawing and mensuration work. Chapter XVI.
contains a series of problems in construction so arranged as to
enable pupils to obtain for themselves a working knowledge of
all the most important facts of geometry. Applications of the
facts thus ascertained, are made to the mensuration of surfaces
and volumes, the calculation of heights and distances, etc. No
attempt is made to anticipate the work of the high-school by
teaching geometry as a science.
4 MANUAL FOR TEACHERS
While the construction problems are brought together into a
single chapter at the end of the book, it is not intended that
instruction in geometry should be delayed until the preceding
work is completed. Chapter XVI. should be commenced not later
than the seventh year, and should be continued throughout the
remainder of the grammar-school course. For the earlier years,
suitable exercises in the mensuration of the surfaces of triangles
and quadrilaterals, and of the volumes of right parallelopipedons
have been incorporated with the arithmetic work.
II
GENERAL HINTS
Division of the Work, — The five chapters constituting Part I.
of Mathematics for Common Schools should be completed by the
end of the fourth school year. The remaining eight arithmetic
chapters constitute half-yearly divisions for the second four years
of school. Chapter I., with the additional oral work needed in
the case of young pupils, will occupy about two years ; the re-
maining four chapters should not take more than half a year each.
When the Grube system is used, and the work of the first two
years is exclusively oral, it will be possible, by omitting much of
the easier portions of the first two chapters, to cover, during the
third year, the ground contained in Chapters I., II., and III.
Additions and Omissions, — The teacher should freely supple-
ment the work of the text-book when she finds it necessary to do
so ; and she should not hesitate to leave a topic that her pupils
fully understand, even though they may not have worked all the
examples given in connection therewith. A very large number
of exercises is necessary for such pupils as can devote a half-year
to the study of the matter furnished in each chapter. In the
case of pupils of greater maturity, it will be possible to make
more rapid progress by passing to the next topic as soon as the
previous work is fairly well understood.
•
Oral and Written Work, — The heading "Slate Problems" is
merely a general direction, and it should be disregarded by the
teacher when the pupils are able to do the work " mentally."
The use of the pencil should be demanded only so far as it may
5
6 MANUAL FOR TEACHERS
be required. It is a pedagogical mistake to insist that all of the
pupils of a class should set down a number of figures that are
not needed by the brighter ones. As an occasional exercise, it
may be advisable to have scholars give all the work required to
solve a problem, and to make a written explanation of each step
in the solution ; but it should be the teacher's aim to have the
majority of the examples done with as great rapidity as is con-
sistent with absolute correctness. It will be found that, as a
rule, the quickest workers are the most accurate.
Many of the slate problems can be treated by some classes as
" sight " examples, each pupil reading the question for himself
from the book, and writing the answer at a given signal without
putting down any of the work.
Use of Books. — It is generally recommended that books be
placed in pupils' hands as early as the third school year. Since
many children are unable at this stage to read with sufficient
intelligence to understand the terms of a problem, this work
should be done under the teacher's direction, the latter reading
the questions while the pupils follow from their books. In later
years, the problems should be solved by the pupils from the
books with practically no assistance whatever from the teacher.
Conduct of the Eecitation, — Many thoughtful educators consider
it advisable to divide an arithmetic class into two sections, for
some purposes, even where its members are nearly equal in
attainments. The members of one division of such a class may
work examples from their books while the others write the
answers to oral problems given by the teacher, etc.
Where a class is thus taught in two divisions, the members of
each should sit in alternate rows, extending from the front
of the room to the rear. Seated in this way, a pupil is doing a
different kind of work from those on the right and the left, and
he would not have the temptation of a neighbor's slate to lead
him to compare answers.
GENERAL H
As an economy of time, explanations of new subjects might be
given to the whole class; but much of the arithmetic work
should be done in "sections," one of which is under the im-
mediate direction of the teacher, the other being employed
in "seat" work. In the case of pupils of the more advanced
classes, "seat" work should consist largely of "problems" solved
without assistance. Especial pains have been taken to so grade
the problems as to have none beyond the capacity of the average
pupil that is willing to try to understand its terms. It is not
necessary that all the members of a division should work the
same problems at a given time, nor the same number of prob-
lems, nor that a new topic should be postponed until all of the
previous problems have been solved.
Whenever it is possible, all of the members of the division
working under the teacher's immediate direction should take
part in all the work done. In mental arithmetic, for instance,
while only a few may be called upon for explanations, all of the
pupils should write the answers to each question. The same is
true of much of the sight work, the approximations, some of the
special drills, etc.
Drills and Sight Work. — To secure reasonable rapidity, it is
necessary to have regular systematic drills. They should be
employed daily, if possible, in the earlier years, but should never
last longer than five or ten minutes. Various kinds are sug-
gested, such as sight addition drills, in Arts. 3, 11, 24, 26, etc. ;
subtraction, in Arts. 19, 50, 53, etc. ; multiplication, in Arts. 71,
109, etc. ; division, in Arts. 199, 202, etc. ; counting by 2's, 3's,
etc., in Art. 61 ; carrying, in Art. 53, etc. For the young pupil,
those are the most valuable in which the figures are in his sight,
and in the position they occupy in an example ; see Arts. 3, 34,
164, etc.
Many teachers prepare cards, each of which contains one of
the combinations taught in their respective grades. Showing
one of these cards, the teacher requires an immediate answer
8 MANUAL FOR TEACHERS
from a pupil. If his reply is correct, a new card is shown to
the next pupil, and so on. Other teachers write a number of
combinations on the blackboard, and point to them at random,
requiring prompt answers. When drills remain on the board
for any considerable time, some children learn to know the
results of a combination by its location on the board, so that
frequent changes in the arrangement of the drills are, therefore,
advisable. The drills in Arts. Ill, 112, and 115 furnish a great
deal of work with the occasional change of a single figure.
For the higher classes, each chapter contains appropriate
drills, which are subsequently used in oral problems. It happens
only too frequently that as children go forward in school they
lose much of the readiness ia oral and written work they
possessed in the lower grades, owing to the neglect of their
teachers to continue to require quick, accurate review work in
the operations previously taught. These special drills follow
the plan of the combinations of the earlier chapters, but gradu-
ally grow more difficult. They should first be used as sight
exercises, either from the books or from the blackboard.
To secure valuable results from drill exercises, the utmost
possible promptness in answers should be insisted upon.
Definitions, Principles, and Eulee, — Young children should not
memorize rules or definitions. They should learn to add by
adding, after being first shown by the teacher how to perform
the operation.' Those not previously taught by the Grube
method should be given no reason for " carrying." In teaching
such children to write numbers of two or three figures, there is
nothing gained by discussing the local value of the digits. Dur-
ing the earlier years, instruction in the art of arithmetic should
be given with the least possible amount of science. While prin-
ciples may be incidentally brought to the view of the children
at times, there should be no cross-examination thereon. It may
be shown, for instance, that subtraction is the reverse of addition,
and that multiplication is a short method of combining equal
GENERAL HINTS 9
numbers, etc. ; but care should be taken in the case of pupils
below about the fifth school year not to dwell long on this side
of the instruction. By that time, pupils should be able to add,
subtract, multiply, and divide whole numbers ; to add and sab-
tract simple mixed numbers, and to use a mixed number as a
multiplier or a multiplicand ; to solve easy problems, with small
numbers, involving the foregoing operations and others contain-
ing the more commonly used denominate units. Whether or not
they can explain the principles underlying the operations is of
next to no importance, if they can do the work with reasonable
accuracy and rapidity.
When decimal fractions are taken up, the principles of Arabic
notation should be developed ; and about the same time, or some-
what later, the principles upon which are founded the operations
in the fundamental processes, can be briefly discussed.
Definitions should in all cases be made by the pupils, their
mistakes being brought but by the teacher through appropriate
questions, criticisms, etc. Systematic work under this head
should be deferred until at least the seventh year.
The use of unnecessary rules in the higher grades is to be
deprecated. When, for instance, a pupil understands that per
cent means hundredths, that seven per cent means seven hun-
dredths, it should not be necessary to tell him that 7 per cent of
143 is obtained by multiplying 143 by .07. It should be a fair
assumption that his previous work in the multiplication of
common and of' decimal fractions has enabled him to see that
7 per cent of 143 is yfo of 143 or 143 X .07, without information
other than the meaning of the term " per cent."
When a pupil is able to calculate that 15 % of 120 is 18, he
should be allowed to try to work out for himself, without a rule,
the solution of this problem : 18 is what per cent of 120 ? or of
this: 18 is 15% of what number? These questions should
present no more difficulty in the seventh year than the following
examples in the fifth : (a) Find the cost of ^ ton of hay at $12
per ton. (b) When hay is worth $12 per ton, what part of a
10 MANUAL FOR TEACHERS
ton can be bought for $ 1.80 ? (c) If ^ ton of hay costs $1.80,
what is the value of a ton ?
When, however, it becomes necessary to assist pupils in the
solution of problems of this class, it is more profitable to furnish
them with a general method by the use of the equation, than
with any special plan suited only to the type under immediate
discussion.
In the supplement to the Manual will be found the usual defini-
tions, principles, and rules, for the teacher to use in such a way
as her experience shows to be best for her pupils. The rules
given are based somewhat on the older methods, rather than on
those recommended by the author. He would prefer to omit
entirely those relating to percentage, interest, and the like as
being unnecessary, but that they are called for by many success-
ful teachers, who prefer to continue the use of methods which
they have found to produce satisfactory results.
Language. — While the use of correct language should be
insisted upon in all lessons, children should not be required in
arithmetic to give all answers in " complete sentences." Espe-
cially in the drills, it is important that the results be expressed
in the fewest possible words.
Analyses, — Sparing use of analyses is recommended for begin-
ners. If a pupil solves a problem correctly, the natural inference
should be that his method is correct, even if he be unable to state
it in words. When a pupil gives the analysis of a problem, he
should be permitted to express himself in his own way. Set
forms should not be used under any circumstances.
Objective Illustrations, — The chief reason for the use of objects
in the study of arithmetic is to enable pupils to work without
them. While counters, weights and measures, diagrams, or the
like are necessary at the beginning of some topics, it is important
to discontinue their use as soon as the scholar is able to proceed
without their aid.
GENERAL HINTS 11
Approximate Answers, — An important drill is furnished in
the "approximations." (See Arts. 521, 669, 719, etc.) Pupils
should be required in much of their written work to estimate
the result before beginning to solve a problem with the pencil.
Besides preventing an absurd answer, this practice will also have
the effect of causing a pupil to see what processes are necessary.
In too many instances, work is commenced upon a problem before
the conditions are grasped by the youthful scholar ; which will
be less likely to occur in the case of one who has carefully
" estimated " the answer. The pupil will frequently find, also,
that he can obtain the correct result without using his pencil
at all.
Indicating Operations, — It is a good practice to require pupils
to indicate by signs all of the processes necessary to the solution
of a problem, before performing any of the operations. This fre-
quently enables a scholar to shorten his work by cancellation, etc.
In the case of problems whose solution requires tedious processes,
some teachers do not require their pupils to do more than to
indicate the operations. It is to be feared that much of the lack
of facility in adding, multiplying, etc., found in the pupils of
the higher classes is due to this desire to make work pleasant.
Instead of becoming more expert in the fundamental operations,
scholars in their eighth year frequently add, subtract, multiply,
and divide more slowly and less accurately than in their fourth
year of school.
Paper vs. Slates, — To the use of slates may be traced very much
of the poor work now done in arithmetic. A child that finds the
sum of two or more numbers by drawing on his slate the number
of strokes represented by each, and then counting the total, will
have to adopt some other method if his work is done on material
that does not permit the easy obliteration of the tell-tale marks.
When the teacher has an opportunity to see the number of
attempts made by some of her pupils to obtain the correct quo-
12 MANUAL FOR TEACHERS
tient figures in a long division example, she may realize the
importance of such drills as will enable them to arrive more
readily at the correct result.
The unnecessary work now done by many pupils will be very
much lessened if they find themselves compelled to dispense with
the "rubbing out" they have an opportunity to indulge in when
slates are employed. The additional expense caused by the
introduction of paper will almost inevitably lead to better results
in arithmetic. The arrangement of the work will be looked
after ; pupils will not be required, nor will they be permitted, to
waste material in writing out the operations that can be per-
formed mentally ; the least common denominator will be deter-
mined by inspection ; problems will be shortened by the greater
use of cancellation, etc., etc. Better writing of figures and neater
arrangement of problems will be likely to accompany the use of
material that will be kept by the teacher for the inspection of
the school authorities. The endless writing of tables and the
long, tedious examples now given to keep troublesome pupils
from bothering a teacher that wishes to write up her records,
will, to some extent, be discontinued when slates are nr longer
-jsed.
w -
III
EARLY ARITHMETIC TEACHING
Counting, — While the majority of children are able, upon enter-
ing school, to repeat the names of the first ten or more numbers,
they are not always able to count things. The first duty of the
teacher is to secure correct notions of the first nine numbers, and
this can best be done by the employment of objects, such as beans,
splints, shoe-pegs, blocks, etc. A numeral frame is very useful
for this purpose.
In counting, it is very important to have the child understand
that the second splint is not two splints. This may be made clear
to a child by having him put on his desk one bean, then near it
two beans, three beans in another place, etc. After the pupil
can count understandingly to nine, he should be taught the
figures. The notation and numeration of numbers of two or
more figures will be discussed in later chapters.
Primary Arithmetic, — After children have learned to count
readily, experts disagree as to the best method of procedure.
Many excellent teachers believe that work should be commenced
at once upon numeration and notation, followed by the funda-
mental operations in the usual order. Some of the advocates of
this method favor the completion of each topic before proceeding
to the next ; that is, numeration and notation are taught at least
to billions ; then addition is taken up, beginning with small num-
bers and gradually increasing to examples containing numbers of
eight or nine figures. Subtraction, multiplication, and division
are each studied to this extent before the next is commenced.
The more intelligent advocates of teaching operations at the
IS
14 MANUAL FOR TEACHERS
outset, recognize the fact that it is neither necessary nor advisable
to defer the addition of small numbers until children are able to
write those of three or more periods, nor to postpone finding the
sum of ^ and ^ until after the properties of numbers have been
studied in the fifth school year. Their plan is to follow such
simple examples in the addition of small numbers as involve no
carrying, by corresponding ones in subtraction. More difficult
examples in both of these operations come next, followed by sim-
ple ones in multiplication and division. Easy work in fractions
is introduced at an early stage, and problems involving the more
common denominate units are brought in from time to time.
The Grnbe Method, — A growing number of educators believe
that early arithmetical instruction should be based upon the
study of numbers, rather than upon that of processes, — that
the former should be the prominent feature of the early instruc-
tion, and the latter incidental, at least for the first two years.
This method, called after its inventor, Grube, requires the
teaching of all of the processes in the case of each number before
proceeding to the next. Thus, when the number 4 is studied,
the pupil measures it by all numbers smaller than itself. Using
4 beans, he measures by 1, by arranging them as follows :
0000. In this way he sees that 1 + 1 + 1 + 1 = 4; that
there are 4 ones in 4, or 1 X 4 = 4; that 4 — 1 — 1 — 1 = 1; that
4^1 = 4.
Measuring by 2, 00 00, he sees that 2 + 2 = 4, 2x2 = 4,
4-2=2, 4-^2 = 2.
Measuring by 3, 000 0, he sees that 3+1 = 4, 1 + 3 = 4;
4-3=1, 4-1 = 3; that (1 X 3) + 1 = 4, and that 4 -*- 3 = 1
and 1 over.
The pupil then answers questions given by the teacher, first
using the counters and afterwards without them : —
Four is how many more than 3? Than 1? Than 2? Three
is how many less than 4 ? Two is how many less ? One is how
many less ?
EARLY ARITHMETIC TEACHING 15
How many ones in 4 ? How many twos ? Threes ? One-half
of 4 is what ? Two is % of what number?
Problems containing the foregoing combinations are then given
in great variety by the teacher until all of the facts about the
number 4 in its relation with the smaller numbers are fully
mastered.
In teaching any number, no larger number must appear in
any way whatever. During the study of 4, it is not permissible
to ask 4 twos, or that 4 is 1 less than what, etc., etc.
The work proceeds slowly and thoroughly, at least a year
being devoted to the numbers below 10. The second year is
given to the numbers from 10 to 20, and the third year to those
from 20 to 100. This is probably as far as the method is carried
in this country.
In the greater number of the schools using this method,
systematic instruction in the fundamental processes is commenced
by the beginning of the third year ; while in some, the Grube
method is used for oral work, and the teaching of slate addition
is carried on at the same time, even during the first year.
Slate Problems, — When, instead of receiving oral instruction
for some time, children are taught processes from the outset, it
frequently happens that many of them show little ability in
solving problems. While some attention should be given in the
early years to this side of arithmetic, it should not be permitted
to retard too much the advancement of pupils. Many of them
have to leave school soon, and they should be taught as rapidly
as is consistent with real progress to perform accurately the
ordinary operations in whole numbers, simple fractions, and
decimals. Being familiar with these tools, greater maturity
will, of itself, show which is to be used in such questions as are
likely to come up in ordinary avocations.
The teacher should exercise much care to give only such
problems as can readily be understood by the pupil, and which
do not contain too many conditions or numbers that bewilder
16 MANUAL FOR TEACHERS
the learner. While a beginner will have no difficulty in deter-
mining whether to add or subtract in a mental problem suited
to his capacity, the same kind of problem with larger figures
will give him much difficulty. For this reason, the earlier slate
problems should be the merest trifle beyond his ability to solve
mentally. In his attempt to work them out in his head, he will
determine whether addition or subtraction is needed, etc.
Problems in all grades should be "miscellaneous," and pupils
should be allowed as far as possible to determine for themselves
what operation is necessary to solve any given one.
IV
NOTES ON CHAPTER ONE
THE hints given as to the work of this chapter are intended
chiefly for the guidance of teachers of young children that are
beginning slate work in the fundamental processes without much
preliminary oral instruction. Pupils that have been taught for
two years by the Grube method should not be required to spend
unnecessary time on the simpler portions of the work.
Art. 4. — In teaching notation of numbers of two figures to
young children that have not been previously taught by the
Grube method, it is not advisable to lay stress on the local value
of the tens' figure. Show them how to read and write 10, 11,
12, etc., to 20; then 30, 40, 50, etc., to 90. After this, there is
but little difficulty.
7. By working an example for the pupils, teach them to place
under each column its sum. As their tendency is to begin work-
ing at the left, be careful to see that they always commence to
add at the right.
9. The problems will present no difficulty, as they involve
only addition.
11. These sight exercises may first be employed as drills to
teach children to use in blackboard addition as few words as
possible. The first figure should not be named, — only the sum
of the first and the second, then this total added to the third.
In subsequent drills upon these combinations, each pupil should,
in turn, give the sum of any set indicated by the teacher. The
work should be done rapidly to be of value.
17
18 MANUAL FOR TEACHERS
13. The making of original problems by the pupils should be
a feature of every grade.
15 and 16. Subtraction is here introduced by the " building-
up " method. Pupils find it easier to ascertain the. difference
between two numbers by going forward from the smaller to the
larger, than by " taking away " one from the other.
17 consists of sight exercises in the form of addition, leading
to the subtraction exercises in Art. 19.
21. While in adding, the use of the word and is considered
unnecessaiy ; in subtracting, it is used just before the figure that
is to be written.
For some advantages obtained by employing the " building-
up " method, see Art. 384, where it is used to obtain in one oper-
ation the difference between 1000 and 643 + 287 -f 25. In Art.
385, it is used to find a remainder in long division without writ-
ing the product of the divisor by the quotient.
23. Here begins the real problem work, as the pupil has now
to determine for the first time in slate examples whether the
result is to be reached by addition or subtraction. When the
pupils are able to solve one of these problems without using
the pencil, it should be repeated, but with such a change in one
of the numbers as will render necessary the use of the slate.
For the 10 cents in the first example, for instance, 14 cents or 24
cents may be substituted.
As many pupils attend rather to the numbers in a problem
than to its terms, some may subtract when they should add,
especially as this seems the natural operation when only two
numbers are involved. It is important that they should be led
to see that the size of the numbers does not change the nature of
the example, and that they can easily determine whether addi-
tion or subtraction is required, by considering what operation
NOTES ON CHAPTER ONE 19
they would employ in a similar example containing very small
figures.
It is not advisable as a regular thing to follow an oral problem
by a written one of exactly the same nature, as this tends to
make children inattentive to the terms of the latter when they
already know from the oral problem what operation is required.
28. It is inadvisable to waste time in endeavoring to make
clear to very young children the reason for " carrying."
37. Teachers should require pupils to write the proper sign
before working an example, as this tends to make them listen
more carefully in order to determine whether addition or sub-
traction is involved. In some problems that are too simple to
need the use of the pencil, changes may be made in the numbers
employed ; great care, however, should be taken not to use num-
bers so large as to confuse the pupils.
38. Have children uriderstand that when a number contains
the word " hundred," it should consist of three figures. Do not
explain.
54. These exercises are intended to lead up to the subtraction
with " borrowing " in the next article. Perhaps the following
would be a better arrangement :
? ? ? ? ? ?
+ 29 +37 +17 +86 + 75 + 90
41 50 25 90 100 150
As children are generally taught to begin with the bottom
figure in addition, they will naturally say in the first example,
9 and 2 are 11, writing the 2 in its place, etc.
55. Subtraction with " borrowing " is generally taught in one
of three ways. The "building-up" method given in the text
is the most readily taken hold of by young pupils.
20 MANUAL FOR TEACHERS
** By the second method, the child is instructed that
whenever he increases by ten any figure of the minuend,
g9 he must add 1 to the next figure of the subtrahend.
Seeing that he cannot take 9 from 1, he says 9 from 11
leaves 2 ; 1 (to carry) and 2 are 3, 3 from 4 leaves 1.
While this method is just as logical as the next, it is not so
easily " explained," and, for this reason, is not so much favored
by many teachers of the present day.
The third method consists of diminishing the next left- sii
hand figure of the minuend after " borrowing." Where
the minuend contains ciphers, this method is particularly —
confusing to beginners, especially where they are forbidden, as
should be the case, to write the changes that are made in the
figures of the minuend.
Except in the addition of long columns, children should be
required from the beginning of slate work to abstain from count-
ing, writing " carrying " figures, and the like. The guide fig-
ures introduced into the foregoing explanations of methods of
subtracting should not be used by pupils.
61. As a change from sight work, and to increase the pupils'
readiness in the solution of mental examples, these drills are
useful. Not requiring any preliminary writing on the board,
they can be taken up at any time the class is unoccupied for a
few minutes — waiting for the signal to go home, for example.
The pupils all stand ; the teacher announces the number to
be added, 2 for instance, and begins by saying 1 herself. The
first pupil says 3, then sits ; the next, 5 ; and so on. After 39,
or some other convenient number, is reached, the teacher begins
by saying 2, and the pupils, in order, give 4, 6, 8, etc., to 40.
The intelligent teacher will be careful to suit these drills to the
capacity of her pupils. She will not weary beginners by spend-
ing too much time on the more difficult drills with 7, 8, and 9 ;
nor will she waste the time of older scholars by dwelling on the
addition by twos.
NOTES ON CHAPTER ONE 21
The same kind of work may be employed as subtraction drills.
Subtract by twos :
40, 38, 36, etc.
39, 37, 35, etc.
By threes :
40, 37, 34, etc.
39, 36, 33, etc.
38, 35, 32, etc.
By fours :
40, 36, 32, etc.
39, 35, 31, etc.
38, 34, 30, etc.
37, 33, 29, etc.
V
NOTES ON CHAPTER TWO
74. Slate multiplication is commenced as soon as the table of
2 times is learned. The first examples contain no carrying.
76. Division tables should not be memorized.
81. Do not permit children to prefix an unnecessary cipher
in the quotient of 100^-2; that is, do not have the answer
written 050.
84. Many scholars think that when a slate problem contains a
very small number and a large one, they must either multiply
or divide. Examples 1—4 are given with simple numbers to
show them that the nature of the operation depends entirely
upon the conditions of a problem. While pupils should not be
required to use a pencil to solve a problem that can be solved
mentally, it would help the class to have these four examples
worked on the board as an indication that in the subsequent
examples there may be needed any one of the four operations
learned thus far, and to serve as a model in their arrangement
of the other problems.
While many teachers require the pupils to write the denomina-
tion of each addend, of the subtrahend and the minuend, of the
multiplicand, and of the dividend, it is scarcely necessary. In
later life it is not done ; and confusion is sometimes produced in
the minds of young scholars by attempting to make them under-
stand why, for example, 60 pints divided by 2 will sometimes
give a quotient of 30 pints, and at other times, as in the 6th
UHIT'EKSITT
NOTES ON CHAPTES^gSfKfcALlFQSS^ 23
problem, an apparent quotient of 30 quarts. It will be found
more satisfactory, even if less scientific, to have the denomination
written only with the result.
Although no formal instruction in finding halves and thirds of
numbers has as yet been given, the average pupil will be able to
solve problems 10, 11, and 14.
85. Lay no stress on the local value of the figures. Practice
will enable the children to read and write correctly numbers
of four figures. Teach the pupil to write the comma when the
word " thousand " is said and after the number of thousands, the
comma to be followed always by three figures.
97. Children should be led to see that 12 X 2 is the same as
12 + 12 ; so that when they come to 15 X 2, they will have no
difficulty in deducing the rule for writing 0 and carrying 1 when
they multiply the 5 by 2.
98. Give the pupils time to find for themselves the quotient
of 30-^2. If it becomes necessary to show some of them how to
work the example, do not elaborate the meaning of the 1 (ten)
remainder when the tens' figure, 3, is divided by 2. An experi-
enced mathematician, in dividing 9752 by 2, does not say 2 into
9 thousand 4 thousand times with a remainder of 1 thousand,
2 into 1700 8 hundred -times with a remainder of 1 hundred, etc.
In dividing 30 by 2, children should not be permitted to write
the first remainder, 1, before the 0, to indicate that 2 is to be
divided into 10 for the second quotient figure. Children learn
to work just as well without these unnecessary scaffolds.
104. While these drill exercises introduce a multiplier greater
than 2, they contain no combinations, except 3x3, other than
those found in the preceding work. After working these exam-
ples, the pupils will have learned that twice 9 is equal to 9 twos, —
that when he knows the table of 2's, he knows a portion at least
of the table of 3's, 4's, etc., to 9's.
24 MANUAL FOR TEACHERS
111. When the teacher places the pointer on a number in one
of the two outer spaces of the first circle, the pupil promptly
gives the result obtained by adding to it the number contained
in the inmost space. When this last number has been combined
with all the others, it is replaced by a different number.
112. These drills are useful to impress upon a child the fact
that when he knows, for instance, that 6 and 5 are 11, he should
also know that 6 and 15 are 21, that 6 and 25 are 31, etc. They
may also be employed as subtraction drills.
115. Division drills are necessary to enable pupils to acquire
facility in obtaining quotients and remainders. When pupils
are dividing by 2, the numbers from, say, 9 to 19 are written on
the board with 2 underneath.
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19
When the pointer is placed at the 9, the pupil answers 4 and
1; when placed at 14, he answers 7; at 17, 8 and 1; etc.
Other divisors may be employed, but care should be taken
not to have any quotient figure but 1 or 2 at this time, as pupils
have not yet learned the table of 3's. Thus, when 6 is used as
a divisor, the teacher should not use a dividend greater than 17.
When the three-times table is known, numbers from 12 to 29
may be written.
Facility in division will come only by practice, and it may be
necessary for the teacher to supplement the examples of the
book by others of her own.
118. Do not fail to keep up practice in addition and sub-
traction.
119. Subtraction examples in which the subtrahend is given
before the minuend should occasionally be used.
NOTES ON CHAPTER TWO 25
121. Do not worry a pupil by attempting to explain, through
problem 9, the difference between division and partition. Let him
write ^ — without taking advantage of the opportunity
25 pounds
to show him that he should have an abstract quotient when the
divisor and dividend are both concrete.
140. The analysis of problem 3 should not be required. A
pupil that obtains from problem 2 the knowledge that 18 five-
dollar bills amount to $90 will probably get the correct answer
to the next problem, even though he may have to use 5 as a
multiplier instead of the 18 that the more common form of the
analysis would require. The other form should not be presented
at this stage.
143. Children should be permitted to determine for themselves
the method of obtaining the half of 36. It may require a little
longer time than to show them, but the time will not be wasted.
147. Roman notation is not of much importance. Most chil-
dren learn sufficient about it from the numbers affixed to their
reading lessons.
157. Teachers should not endeavor to show by drawings that
a quart measure is twice as large as a pint. If a pint measure is
represented by a rectangle, each side of the rectangle indicating
the quart should be only about 1£ times that of the former in
order to preserve the correct ratio, and children are not mathe-
maticians enough to understand that where one of two similar
solids has its corresponding dimensions 1-J- times those of the
other, the volume of the former is double that of the latter.
Use the measures themselves, borrowing them, if necessary, from
a neighboring store.
159. A few problems involving more than one operation are
here introduced. Avoid, if possible, giving help; and do not
26 MANUAL FOR TEACHERS
require the scholars to perform unnecessary work, or to follow
the same mode of solution or arrangement. In solving the first,
some may write on their slates only two numbers, viz. 15 and 35.
Others may set down 15, 15, and 20, etc. Do not teach yet
how to multiply by a mixed number.
VI
NOTES ON CHAPTER THREE
While the teaching of formal definitions should find no place
in the arithmetical instruction of the earlier years, the teacher
should not hesitate to employ such technical terms as are called
for by the work of the grade. Pupils gradually learn to under-
stand what is meant by multiplier, quotient, remainder, etc.,
even where no attempt is made to explain the signification of the
words. They will also become able to use each correctly, even if
they cannot state its exact meaning in language that will satisfy
a critical mathematician.
164. Sight exercises in division should be extended to cover
dividends that are not multiples of the divisor. The slate exam-
ples in division supplied thus far have no remainders, as children
find it more agreeable in the earlier stages of this work to have
the answer a whole number. The partial dividends, however,
do not always exactly contain the divisor, hence the need of
such drills as will enable the pupil to determine rapidly the
quotient figure and the remainder. Until Art. 176 is reached,
this remainder need not be given by the pupils in the form of a
fraction. See Art. 115.
168. In making " original problems," the pupil should strive
to be original. No problem should be accepted as satisfactory
that is substantially the same as one already furnished by
another pupil. If, for example, the following is given to illus-
trate 12 X 5 : " What will be the cost of 5 yards of ribbon at 12
cents a yard? " the teacher should not be satisfied with — " How
27
28 MANUAL FOR TEACHERS
much will be paid for 5 pounds of cheese at the rate of 12 cents
per pound?"
174. While the problems are gradually becoming more diffi-
cult, some of them can be done by bright pupils without using
the pencil. In these cases, require that only the answers should
be written. See previous notes on problem work. (Arts. 23,
84, and 159.)
178. Children should be permitted to follow their own 26
plan of finding the product of 26 by 1^-. Some may do +13
the work by simply placing 13 under 26. The regular 39
method should not be taught until, perhaps, the 25th
example, as the previous ones can be done by the children with-
out assistance. At this point, however, the systematic
124 way of multiplying by a mixed number may be pre-
2| sented, which should be followed in such subsequent
31 examples as are not so simple as to make this amount
248 of writing unnecessary, as is the case in the 26th.
279 In finding £ of 124, the pupil should not be permitted
to write the multiplicand, 124, in some other part of his
slate, and 4 as a divisor in front of it. No other writing of
figures should be allowed than is given above. A little practice
will enable scholars to perform this division and other similar
operations, without always bringing into close contact the num-
bers to be handled.
In some European countries, the multiplier is
760 X 1-J- placed at the right of the multiplicand, instead of
152 being written underneath. An example like the
912 26th would be worked in that case without writing
760 a second time. To small children, how-
ever, it would be confusing to be required to learn two 760
methods of working examples so nearly alike ; hence the 1^
advisability of uniformly following the plan originally 152
given, of first finding the fractional part, and then multi- 760
plying by the whole number. 912
NOTES ON CHAPTER THREE 29
180. The arrangement of work should begin to receive
some attention. In solving the second problem, some 2)70
children will find the cost of -^ pound of tea on one por- 35
tion of the slate, and then write this amount, 35^, on -J-25
another part, with 25^ underneath. They should be 60^
led to see how to avoid doing unnecessary work.
186. Some short examples in the addition and the subtraction
of horizontal columns are given, to accustom children to handle
numbers that are not arranged for work in the usual way.
The addition example could be used to explain the reason
for " carrying," but the explanation should be deferred for the
present.
191. Examples in division should occasionally be presented
to pupils in the form used in the second column. When children
recognize *f- as an example in division, they need no rule for
the reduction of an improper fraction to a whole or to a mixed
number.
197. Do not furnish the pupils with a method of solving the
9th example that is suited to a sixth year class in denominate
numbers. Leave them to their own resources as much as pos-
sible.
202. More drill examples are needed than are furnished in
the book.
203. To secure good work in division, much practice must
be given. Many more examples than are here supplied may
be needed by some classes.
213. While it is convenient to write the subtrahend under
the minuend, pupils should gradually accustom themselves to
perform the fundamental operations with numbers in other than
the usual positions.
30 MANUAL FOR TEACHERS
215. Children should be encouraged to avoid unnecessary
7U19 wr^ing. They should be led to see that after finding
Ty on one part of the slate that -f of 119 is 17, they should
X 5 not place this number in another place in order to multi-
"etcT ply by 5.
220-223. These drills are intended to lead up to the use
of larger numbers in the oral work of the pupils.
224. It is not advisable to begin formal instructions in frac-
tions at this stage of school life. There is no need of defining
the word "fraction" for the present. Eveiy member of the
class will be able to tell what is the sum of £ and -£, especially
if the question is put in the form of a problem.
It will be necessary, perhaps, to explain that 4 X % is another
way of expressing -J- of 4 ; that ^-X 10 means 10 halves. 1-^-J-
will also require translation into the form, " How many halves
in 1 ? " Pupils may be led to see this by being asked to indicate
by signs and figures the example, " How many twos are there
in eighteen?" The drills in the use of fractional divisors need
not be made prominent for the present.
230. Accustom children to writing the decimal point in the
product, as soon as it is reached in multiplying. Reasons should
not be dwelt upon.
231. The above applies to placing the decimal point in the
quotient.
238. Unless pupils have been carefully trained to give only
reasonable answers to slate problems, there will be some who
will obtain 171 as the sum of 13 J and 4J. They will first write
1 as the equivalent of ^ + ^-; and to this they will prefix 17,
obtained by adding 13 and 4. The special training in number
received by pupils taught by the Grube method prevents to
a great extent the absurd mistakes found in the answers
NOTES ON CHAPTER THREE 31
given JDV pupils, even of high-school classes, to simple problems.
When the early arithmetical instruction is largely given to work
in the fundamental processes, the teacher should make liberal
use of oral problems, to give the requisite knowledge of number
that will enable a pupil to know when his answer is very much
out of the way. Systematic instruction in finding " approxi-
mate " results is supplied in later chapters.
239. These examples are intended to lead up to finding the
difference between a whole number and a mixed number.
240. Pupils will find little difficulty in working out these
examples if they are left to themselves.
241. When the addition and the subtraction of mixed num-
bers containing halves are readily performed, the teacher will find
comparatively little trouble with the work under Arts. 241-245.
Encourage pupils to make diagrams; or, if necessary, to divide
circles into quarters, and to use these parts in performing the
required operations with the fractions.
To find, for instance, the sum of £ + £, it may be advisable
to permit some scholars to arrange the six quarter-circles in such
a way as to make a whole circle and a half-circle.
246. As children are more accustomed to dealing with halves
and quarters than with thirds, a little more illustrative work
may be needed in Arts. 246-250, than was required in the pre-
vious work in the addition and the subtraction of mixed num-
bers.
VII
NOTES ON CHAPTER FOUR
253-258. In the last chapter, pupils were required to add
only fractions containing the same denominator ; in this chapter,
an addition or a subtraction example may contain fractions
whose denominators are different. For the present, however,
it will not be necessary to call attention to the need of reducing
fractions to a common denominator. The average scholar can
solve these examples without assistance, if he has been able to
work out those found in Chapter III.
259. While these problems are becoming more difficult, they
are still well within the powers of a pupil that is really anxious to
solve them. When, however, they are found to be beyond the
capacity of many members of the class, the teacher may first
use them as " sight " problems, with some slight changes in the
figures.
If, for instance, after a pupil that reads the first from his book
declares that he is unable to obtain the answer mentally, the
teacher may give it as follows :
A sailor has 10 yards of cloth. He uses 4 yards for a coat
and 2 yards for a vest. How many yards has he left ?
In the second, 1|- pounds may be substituted for 1^ pounds;
in the third, 3 packages instead of 4; 20 dozen in the fourth,
instead of 3£ dozen.
Slate work on these problems should not be permitted until
so many have been solved in this way that the pupil has had time
to forget what operations have been used in each. This will
32
NOTES ON CHAPTER FOUR 33
require him to study the conditions of the different problems,
instead of relying upon his memory.
266. When the formal analysis of oral problems is made
a feature of the work, it is important that the statements be
not so long as to be tedious.
In the first, for example, the following would be sufficient,
after the pupil has stated the problem:
" If 8 ounces of tea cost 40 cents, 1 ounce will cost 5 cents,
and 5 ounces will cost 25 cents."
While the customary order has been followed in the systematic
treatment of the various topics, pupils are called upon in the
earlier chapters of Mathematics for Common /Schools to solve
many problems that are frequently deferred in other books to a
later stage of their arithmetical instruction. While scholars
readily solve this class of problems, they are not always able
to state in technical language the reasons for the various proc-
esses employed in obtaining the answers. A child who sees that
division is used to ascertain the number of ten-cent pies that can
be purchased for forty cents, cannot be made to understand thus
early in his school life that the same process is used to find what
part of such a pie can be bought for five cents. A correct state-
ment by the pupil of his method of reaching the result, should
usually be accepted as satisfactory. Even in the more simple
questions, set forms of analysis should be carefully avoided.
268. To prevent misunderstanding, parentheses have been
employed even when not required by arithmetical usage. The
quantities within the parentheses must be added, multiplied,
etc., before being operated upon by the quantity outside. The
third example becomes 30 X 3 ; the fourth, 80 -4- 4 ; the fifth,
\ of 80; the eighth, 70 -f- 7, etc.
269. These may be used as slate examples, if they are found
too difficult for " sight " work.
34 MANUAL FOR TEACHERS
271. Some of these questions may not require the use of a
pencil ; Nos. 6, 7, 8, 11, and 19, for instance.
272. The answers to the first ten examples should be given at
sight.
273. Use 49 to 57, inclusive, as " sight " examples ; also as
many as possible of those in the next section.
274. When the divisor ends in one or more
ciphers, the latter are set off by a vertical bar,
and also a corresponding number of figures from
the right of the dividend. To keep the pupil from omitting
these figures from the remainder, it is advisable to require him
to write the partial remainder as above, before he
8 1 0)434 [1 begins to divide. Then, using 8 as a divisor, he
54|^ writes the quotient figures in their places, and
completes the partial remainder by prefixing 2 to
the 1 that was originally brought down.
It being the usual practice in abstract examples 8|0)434!0
in division to refrain from reducing the fractional 54ff
part of the quotient to lowest terms, the above
method may be used in examples where both the
80)4340 divisor and the dividend terminate in a cipher.
54f Some teachers prefer, however, in this case, to can-
cel the cipher in each, and to^ give the quotient
of 4340 -*- 80 as 54f .
277. Employ in " sight " work.
278. The foot-rule and the yardstick should be used by the
children. They should ascertain, for instance, the length of
their slates in inches, the length of the blackboard in yards or
in feet, the height of the blackboard in feet, the dimensions
of the room, etc.
NOTES ON CHAPTER FOUR 35
280. It will be sufficient to accustom pupils to placing the
product by the tens' figure one place to the left without giving
the reason therefor. Neatness in the arrangement of the work,
and the careful writing of figures, will prevent some mistakes.
282. In short division, the scholar has been taught to place
the first figure of the quotient under the last figure
of its partial dividend, and to write under each — ^
succeeding figure of the dividend its corresponding
quotient figure. When his work is neatly arranged, he seldom
omits ciphers, nor does he often obtain two quotient figures from
one partial dividend.
To obtain the benefit of this experience, the pupil should
be taught in long division to write the quotient
over the dividend. By doing this, he will not be
tempted, as are some beginners that place the
quotient at the right, to give 23-2lr as the answer
to the above example ; nor will he be likely to
think that 252 contains 21, 111 times. This last
result is obtained by assuming that the second
partial dividend, 42, contains the divisor 1 time, with a re-
mainder of 21. This latter is then made a partial dividend,
with the above result.
285. While the pupil may write 16 as a multiplier in the
5th problem, he should be required to multiply by 80, ^
in order to shorten the work. The multiplication by 30 -,„
should be performed, also, without rewriting the numbers —
so as to place 30 under 16.
286-290. The special drills will be found of great value in
giving pupils a knowledge of numbers ; and many oral problems
employing these and similar combinations should be made by
the teacher. Oral problems containing large numbers should,
as a rule, require but one operation for their solution.
36 MANUAL FOR TEACHERS
In the oral addition of numbers of two figures, the pupil
should not commence, as in slate work, with the units' figures.
The special drills of the last chapter should have taught him to
think immediately of 80 when he sees 40 + 40, 60 + 20, 50 + 30,
etc. The next step in this work should contain such combina-
tions as 47 + 40, 63 + 20, 54 + 30, etc. In adding 54 and 30,
the pupil should be taught to first see the eighty, then the four.
The sum of 27 and 32 (the third step) should be obtained by
joining 27 and 30 to make 57, and adding 2 to this result to
obtain 59. If the pupil begins with the units, 7 and 2, he is
likely to forget the tens' figures. When the addition work
is readily performed, the pupil finds little trouble with the rest.
There being no carrying, he will readily obtain the product of
32 by 3, and the others given in Art. 288, especially if he begins
the multiplication at the tens' figure. After he becomes expert
in adding and multiplying, he will experience no difficulty in
subtracting and dividing.
294. The teacher should not encourage unnecessary work,
by permitting children to write the sum of 12J + 6J as 18| =
18 + 1 = 19. If, however, it be deemed advisable in the 4th
example, for instance, that the fractions should be expressed
with the same denominator, care must be taken to prevent
pupils fr°m making such mistakes as using the sign of
equality between ^ and J- in such a way as to represent
— i-l-4 that 50J is the equivalent of J-. A vertical line drawn
5 ¥ I ¥ between the two sets of fractions will serve to separate
the original example from the auxiliary portion. (See Art. 310.)
304. As some children merely look for the figures of a prob-
lem without paying attention to its terms, an occasional one
is given in which some or all of the numbers are expressed
in words.
306. In making out a bill, it is convenient to be able to write
the cost of 196 Ib. at 4^ per Ib. without using another sheet of
NOTES ON CHAPTER FOUR 37
paper and placing the 4 under 196. In working these examples,
the pupil is expected to write only one figure of the product at a
time. It is not intended that all of these twenty-five examples
should be done before proceeding with the subsequent work.
A few of them should be used from time to time throughout
the term.
307. The last sentence applies also to these examples. A few
of the easier ones should first be given. After more practice
in long division, the more difficult ones may be taken up.
310. Whenever it becomes necessary, in the 8
opinion of the teacher, to permit the rewriting 49J 7
of the fractions with a common denominator, 20|- 4
she should, as soon as possible, have her pupils 70f ^ = If
write the common denominator only once, as
above. When the common denominator is written
49|- J under each numerator, it is likely to be confusing to
20^- f -children, not to speak of the danger of its being added
-^ in occasionally with the numerators.
312. See Art. 268.
316. Where the multiplier ends with ciphers, some
teachers think that time is saved by omitting the
ciphers from the partial products. The ciphers at the
right of the multiplier are written beyond the multipli-
cand, and are brought down at the end of the work. 98800
Other teachers prefer to place the numbers as is gen-
erally done in multiplication, writing a cipher under each one
in the multiplicand as its partial product, and writing
76 the partial products by 3 and 1 under these figures,
1300 respectively. This method will be found to give more
22800 satisfactory results later on, when pupils have such
76 multipliers as 20£, 300J, etc., in which a fraction fol-
98800 lows the ciphers.
38 MANUAL FOR TEACHERS
319. See Art. 310. When an addition example consists of
more than two mixed numbers with fractions of different denom-
inators, it may be advisable to permit young
children to write out the successive opera-
tions in the manner here indicated.
7*
Ji
22J
Many of the fifty examples on this page
should be used as "sight" work from the
= If = 1% blackboard, the pupils writing only the
results. Nos. 1-6, 8-9, 13-16, 23-24, 26,
31-37, 41-43, 49-50 can be treated in this way after they have
been worked out on the slate, if not in the first instance.
321. Until children obtain some knowledge of numbers, their
progress in long division is very slow. In dividing 918 by 17,
for instance, a pupil that is not properly instructed will some-
times take 1 as the first figure of the quotient. When, after
subtracting, he obtains a remainder of 74, he may realize that
he is wrong without being able to determine just how far astray
he is. In this case he tries 2 as the quotient figure, ascertaining
the product of 17x2 in a corner of his slate, and then trans-
ferring the 34 to its proper position under the first two figures
of the dividend. Another subtraction follows, with a resulting
remainder, again, perhaps, recognized as too great ; and so on.
The object of these drills is to enable the scholar to reach at
once a close approximation to the correct quotient figure. Their
use may be commenced in some such way as the following :
The teacher writes on the blackboard a convenient number of
those found among the first twenty, arranging them as shown
below, with the divisor preceding the dividend. Under these
she places the corresponding ones from the second and third
sets, respectively.
20)160 60)360 90)450 50)300 30)270
19)160 59)360 89)450 49)300 29)270
21)160 61)360 91)450 51)300 31)270
NOTES ON CHAPTER FOUR 39
Placing the pointer on those in the first row, successively, she
receives the quotients promptly. She then asks for the quotient
of the first in the third row, 21)160. If the pupil announces
8 as the result, he should be required to give mentally the prod-
uct of 21 X 8, which he will find to be too great. He is thus
led to see that the quotient is 7, with a remainder. The other
quotients in this row are then elicited. After a pupil discovers
that 21 is not contained 8 times in 160, that 61 is not con-
tained 6 times in 360, etc., he may be introduced to the second
row. A little questioning will enable him to perceive that if
160 -5- 20 = 8, the quotient of 160 •+• 19 must be at least 8, with
a remainder; that 360-^59 gives a quotient of 6, with something
over, etc. Regular practice with this particular set of drills
will rob division by 19, 29, 39, etc., of some of its terrors to
slow pupils, as they will be led to use 2, 3, 4, etc., as " trial
divisors" instead of 1, 2, 3, etc., whereby they will be able to
obtain their answers in a reasonable time.
After the children have become able to announce at once the
quotients of all the drills in the first three sets, and other similar
ones supplied by the teacher, they may take up the remaining
ones by degrees. When there is a remainder, the pupils should
not be required to calculate it.
324. The quotient of 2,800-^200 may be made more obvious
if the dividend is read 28 hundred, instead of two thousand eight
hundred.
328. See Art. 274, as to writing the partial remainder before
beginning to divide.
341. Do not give reasons for the location of the partial prod-
ucts. There is plenty of time for the science of arithmetic later
on in the course.
343. Although the divisors contain three or four figures, these
examples should not prove so difficult as many of those already
40 MANUAL FOR TEACHERS
worked. A pupil that has learned from the pre-
i_ vious drills that 800 -5- 200-= 4, will be able to see
201)8643 that 201 is contained 4 times in the first three fig-
ures of 8,643. The teacher should be careful to
see that the first quotient figure is written in its
etc. ,
proper place.
No. 36 may cause some hesitation until the pupil perceives that
he has to divide 81 hundred and something by 9 hundred and
something. No. 37 will become simple if handled in the same
way. In No. 47, 98 hundred divided by 12 hundred will give
the clue to the quotient ; in Nos. 48 and 50, nine thousand and
two thousand should be used for this purpose.
344. With such a multiplier as 209, some teachers
write a series of ciphers to denote the product by 0.
The method given in the text-book is the one gener-
ally followed in later school life, and is just as easily
taught to beginners as the above.
346. Where the multiplicands are small, as in nearly all of
these examples, the product by the fraction should be
determined " mentally" and written in its place. A pupil 3
should not be encouraged to waste time by indicating on — E
another part of his slate that 64 is to be divided
^— by 8, and that this product is to be multiplied by
3, and doing all this work to reach a result that
— - can be readily obtained without any writing whatever.
In Nos. 78 and 88, such pupils as need to use the pencil
in multiplying by the fraction should be permitted to do so.
The teaching of the common method of multiplying by a mixed
number is taken up at the beginning of the next chapter.
VIII
NOTES ON CHAPTER FIVE
347. The denominators of fractional multipliers have hereto-
fore been factors of the multiplicands, and the latter have been,
as a rule, small numbers. With the introduction of larger num-
bers and the occasional use of multiplicands that are not multiples
of the denominators of the fractions in the multiplier, it becomes
necessary to furnish pupils with a general method of dealing with
this class of examples. (See Arithmetic, Art. 347.)
348. In multiplying 27 by 13^, some pupils may
be tempted to follow the rule, and to multiply 27 by -.^
the numerator 1. In the first few examples this may q\ 07^
be permitted, but the scholars should soon be taught — Q-
to discontinue the practice, and to divide the multipli- ,
cand without rewriting it. (See Art. 178.)
350. In adding 56 and 17, the pupil should first combine 56
and 10 to make 66, and then add 7. (See Art. 286.)
351. Children taught subtraction by the " building-up " method
will ascertain how many must be added to 19 to make 66, by
saying 19 and 40 are 59, and 7 are 66 ; or 19 and 7 are 26, and
40 are 66. While the second plan is easier in some respects, it
gives the 40 and the 7 of the result in the reverse order, which
makes it necessary for the pupils to transpose them. In this
respect, the first plan is more satisfactory.
41
42 MANUAL FOR TEACHERS
When the other method of subtraction is practiced in slate
work, 66 is first diminished by 10 and then by 9. To find the
difference between 94 and 76, the pupil takes 70 from 94, leaving
24, and from this remainder takes 6.
352. In multiplying 24 by 4 the pupil begins at the tens.
Four times 20 are 80, to which is added 4x4, making 96.
353. While nearly the whole class will learn to give answers
mentally to the previous combinations, it may be necessary to
use the division drills as " sight " work chiefly.
359. See Art. 319.
362. Oral problems involving several operations, or those of
an unfamiliar type, should be solved from the book as "sight"
work, and should be followed later on by similar questions an-
swered without seeing the numbers. No. 5 is of the second kind ;
and it might be well to place it on the board, writing " 2 thirds '*
and " 1 third " to express the parts, instead of employing the
fractional form or that given in the book. In No. 7, the quotient
of 60-^40 will be expressed by 1J, instead of the 1|^- obtained by
writing the remainder over the divisor. No. 5 should not be
made an excuse for teaching a method of obtaining the cost of
the whole when that of a part is given.
These examples are introduced to give variety to the work, to
lay a foundation for subsequent systematic treatment of problems
of this kind, and to give a pupil an opportunity to use his think-
ing powers. The way to deprive them of value is to " explain "
how they should be done, or to require from the scholars too
much analysis.
363. If the school does not own these measures, the teacher
should endeavor to secure the loan of a quart, a peck, and a
bushel, for a few hours, at least. Sawdust could be used to show
pupils that the peck contains eight quarts, etc.
NOTES ON CHAPTER FIVE 43
364. While many of the problems of this article resemble the
previous oral problems, it may be advisable to solve a number of
them as " sight " work, changing the numbers when necessary.
The first may be read " How many 200-lb. barrels can be filled
from 6,000 Ib. ? " In the second and third, the fractions may be
omitted. The cost of the calico and of the ribbon in No. 4 may
be made 10 $. Nos. 5 and 6 need no change, perhaps.
370-372. Do not waste time by endeavoring to use these ex-
amples to explain " carrying " or the local value of digits.
374. The answers should be written directly from the book.
Do not permit scholars to copy the examples on their slates.
377. First, perform operations on the quantities enclosed within
the parentheses.
384. Very little preliminary explanation will be needed-. Place
f \ *> (a) on the blackboard, and ask a /^\ ?
125 pupil to write the missing number 125
632 in its place, one figure at a time, 632
999 beginning with the units' figure. 1000
Have another pupil work (b) in the same way. Nos. 1 to 5 may
be used as a class exercise, each pupil writing only the answer on
his paper, the examples being placed on the board.
385. In many German schools, children are not permitted in
long division to write the partial products. Examples 6-23 are
given to train pupils to omit these products when the quotient
contains but one figure. After a few of them are worked on the
board, the answers to the others may be written by all the pupils,
as suggested in the preceding article. In writing the answers,
the pupils should first set down the quotient figure, then the
divisor as the denominator of a fraction, and lastly the remainder
as a numerator. (See Art. 563, p. 55.)
44 MANUAL FOR TEACHERS
386-388. These examples should be placed on the board, and
the pupils should write the results one figure at a time.
397-401. See Art. 321.
405-406. See Arts. 306 and 307.
407. Prove the correctness of the grand total by comparing
the total of the 6th column with that of the llth row.
412. Permit the pupils to use their own method of working
these examples, and avoid giving unnecessary assistance.
413-414. Example 1 should be omitted where pupils do not
receive marks that are thus averaged. No. 2 may also be omitted
if the word "average" is not understood by the pupils.
424-426. See Arts. 286-290, page 34.
429. In Examples 1, 2, 5, 9, etc., it will hardly be necessary
to inform the pupils that 1 is not considered a factor of a number.
SUPPLEMENT
DEFINITIONS, PRINCIPLES, AND RULES
A Unit is a single thing.
A Number is a unit or a collection of units.
The Unit of a Number is one of that number.
Like Numbers are those that express units of the same kind.
Unlike Numbers are those that express units of different kinds.
A Concrete Number is one in which the unit is named.
An Abstract Number is one in which the unit is not named.
Notation is expressing numbers by characters.
Arabic Notation is expressing numbers by figures.
Eoman Notation is expressing numbers by letters.
Numeration is reading numbers expressed by characters.
The Place of a Figure is its position in a number.
A figure standing alone, or in the first place at the right of other
figures, expresses ones, or units of the first order.
A figure in the second place expresses tens, or units of the
second order.
A figure in the third place expresses hundreds, or units of the
third order ; and so on.
A Period is a group of three orders of units, counting from right
to left.
RULE FOR NOTATION. — Begin at the left, and write the hun-
dreds, tens, and units of each period in succession, filling vacant
places and periods with ciphers.
i
11 SUPPLEMENT
RULE FOR NUMERATION. — Beginning at the right, separate the
number into periods.
Beginning at the left, read the numbers in each period, giving
the name of each period except the last.
ADDITION
Addition is finding a number equal to two or more given num-
bers.
Addends are the numbers added.
The Snm, or Amount, is the number obtained by addition.
PRINCIPLE. — Only like numbers, and units of the same order
can be added.
RULE. — Write the numbers so that units of the same order shall
le in the same column.
Beginning at the right, add each column separately, and write
Ihe sum, if less than ten, under the column added.
When the sum of any column exceeds nine, write the units only,
and add the ten or tens to the next column.
Write the entire sum of the last column.
SUBTRACTION
Subtraction is finding the difference between two numbers.
The Subtrahend is the number subtracted.
The Minnend is the number from which the subtrahend is taken.
The Eemainder, or Difference, is the number left after subtracting
one number from another.
PRINCIPLES. — Only like numbers and units of the same order
can be subtracted.
The sum of the difference and the subtrahend must equal the
minuend.
RULES. — I. Write the subtrahend under the minuend, placing
units of the same order in the same column.
DEFINITIONS; PRINCIPLES, AND RULES ill
Beginning at the right, find the number that must be added to
the first figure of the subtrahend to produce the figure in the corre-
sponding order of the minuend, and write it below. Proceed in
this way until the difference is found.
If any figure in the subtrahend is greater than the corresponding
figure in the minuend, find the number that must be added to the
former to produce the latter increased by ten ; then add one to the
next order of the subtrahend and proceed as before.
II. Beginning at the units'1 column, subtract each figure of the
subtrahend from the corresponding figure of the minuend and
write the remainder below.
If any figure of the subtrahend is greater than the corresponding
figure in the minuend, add ten to the latter and subtract; then,
(a) add one to the next order of the subtrahend and proceed as
before ; or, (b) subtract one from the next order of the minuend
and proceed as before.
MULTIPLICATION
x.
Multiplication is taking one number as many times as there are
units in another number.
The Multiplicand is the number taken or multiplied.
The Multiplier is the number that shows how many times the
multiplicand is taken.
The Product is the result obtained by multiplication.
PRINCIPLES. — The multiplier must be an abstract number.
The multiplicand and the product are like numbers.
The product is the same in whatever order the numbers are
multiplied.
RULE. — Write the multiplier under the multiplicand, placing
units of the same order in the same column.
Beginning at the right, multiply the multiplicand by the number
of units in each order of the multiplier in succession. Write the
IV SUPPLEMENT
figure of the lowest order in each partial product under the figure
of the multiplier that produces it. Add the partial products.
To multiply by 10, 100, 1000, etc,
RULE. — Annex as many ciphers to the multiplicand as there
are ciphers in the multiplier.
DIVISION
Division is finding how many times one number is contained in
another, or finding one of the equal parts of a number.
The Dividend is the number divided.
The Divisor is the number contained in the dividend.
The Quotient is the result obtained by division.
PRINCIPLES. — When the divisor and the dividend are like num-
bers, the quotient is an abstract number.
When the divisor is an abstract number, the dividend and the
quotient are like numbei'S.
The product of the divisor and the quotient, plus the remainder ',
if any, is equal to the dividend.
RULE. — Write the divisor at the left of the dividend with a line
between them.
Find how many times the divisor is contained in the fewest fig-
ures on the left of the dividend, and write the result over the last
figure of the partial dividend. Multiply the divisor by this quotient
figure, and write the product under the figures divided. Subtract
the product from the partial dividend used, and to the remainder
annex the next figure of the dividend for a new dividend.
Divide as before until all the figures of the dividend have been
used.
If any partial dividend will not contain the divisor, write a
cipher in the quotient, and annex the next figure of the dividend.
If there is a remainder after the last division, write it after the
quotient with the divisor underneath.
DEFINITIONS, PRINCIPLES, AND RULES V
FACTORING
An Exact Divisor of a number is a number that will divide it
without a remainder.
An Odd Number is one that cannot be exactly divided by two.
An Even Number is one that can be exactly divided by two.
The Factors of a number are the numbers that multiplied to-
gether produce that number.
A Prime Number is a number that has no factors.
A Composite Number is a number that has factors.
A Prime Factor is a prime number used as a factor.
A Composite Factor is a composite number used as a factor.
Factoring is separating a number into its factors.
To find the Prime Factors of a Number.
RULE. — Divide the number by any prime factor. Divide the
quotient, if composite, in like manner; and so continue until a
prime quotient is found. The several divisors and the last quotient
will be the prime factors.
CANCELLATION
Cancellation is rejecting equal factors from dividend and divisor.
PRINCIPLE. — Dividing dividend and divisor by the same num~
ber does not affect the quotient.
GREATEST COMMON DIVISOR
A Common Factor (divisor or measure) is a number that is a
factor of each of two or more numbers.
A Common Prime Factor is a prime number that is a factor of
each of two or more numbers.
The Greatest Common Factor (divisor or measure) is the largest
number that is a factor of each of two or more numbers.
Numbers are prime to each other when they have no common
factor.
VI SUPPLEMENT
The greatest common divisor of two or more numbers is the
product of their common prime factors.
PRINCIPLES. — A common divisor of two numbers is a divisor
of their sum, and also of their difference.
A divisor of a number is a divisor of every multiple of that
numb e?' ; and a common divisor of two or more numbers in a
divisor of any of their multiples.
To find the Common I rime Factors of Two or More Numbers.
RULE. — Divide th( numbers by any common prime factors,
and the quotients in like manner, until they harp, no common
factor ; the several divisors are the common prime factors.
To find the Greatest Common Divisor of Numbers that are Easily
Factored.
RULE. — Separate the numbers into their prime factors ; the
product of those that are common is the greatest common divisor.
To find the Greatest Common Divisor of Numbers that are not
Easily Factored.
RULE. — Divide the greater number by the less ; then divide
the last divisor by the last remainder, continuing until there is no
remainder. The last divisor is the greatest common divisor.
2f there are more than two numbers, find the greatest common
divisor of two of them; then of that divisor and another of the
numbers until all of the numbers have been used. The last divisor
is the greatest common divisor.
LEAST COMMON MULTIPLE
A Multiple of a number is a number that exactly contains that
number.
A Common Multiple of two or more numbers is a number that
is a multiple of each of them.
The Least Common Multiple of two or more numbers is the
smallest number that is a common multiple of them.
DEFINITIONS, PRINCIPLES, AND RULES Vll
PRINCIPLES. — A multiple of a number contains all the prime
factors of that number.
A common multiple of two or more numbers contains each of
the prime factors of those numbers.
The Least Common Multiple of two or more numbers contains
only the prime factors of each of the numbers.
To find the Least Common Multiple of Two or More Numbers.
RULE. — Divide by any prime number that is an exact divisor of
two or 'more of the numbers, and write the quotients and undivided
numbers below. Divide these numbers in like manner, continuing
until no two of the remaining numbers have a common factor.
The product of the divisors and remaining numbers is the least
common multiple.
FRACTIONS
A Fraction is one or more of the equal parts of anything.
The Unit of a Traction is the number or thing that is divided
into equal parts.
A Fractional Unit is one of the equal parts into which the num-
ber or thing is divided.
The Terms of a Fraction are its numerator and its denominator.
The Denominator of a fraction shows into how many parts the
unit i-s divided.
The Numerator of a fraction shows how many of the parts are
taken.
A fraction indicates division ; the numerator being the divi-
dend and the denominator the divisor.
The Value of a Fraction is the quotient of the numerator divided
by the denominator.
Fractions are divided into two classes — Common and Decimal,
A Common Fraction is one in which the unit is divided into any
number of equal parts.
A common fraction is expressed by writing the numerator above
the denominator with a dividing line between.
SUPPLEMENT
Common fractions consist of three principal classes — Simple,
Compound, and Complex,
A Simple Traction is one whose terms are whole numbers.
A Proper Fraction is a simple fraction whose numerator is less
than its denominator.
An Improper Fraction is a simple fraction whose numerator
equals or exceeds its denominator.
A Compound Fraction is a fraction of a fraction.
A Complex Fraction is one having a fraction in its numerator, or
in its denominator, or in both.
A Mixed Number is a whole number and a fraction written
together.
The Eeciprocal of a Number is one divided by that number.
The Reciprocal of a Fraction is one divided by the fraction, or
the fraction inverted.
PRINCIPLES. — Multiplying the numerator or dividing the de-
nominator multiplies the fraction.
Dividing the numerator or multiplying the denominator divides
the fraction.
Multiplying or dividing both terms of a fraction by the same
number does not alter the value of the fraction.
Reduction of fractions is changing their terms without altering
their value.
To reduce a Fraction to Higher Terms,
RULE. — Multiply both numerator and denominator by the same
number.
To reduce a Fraction to its Lowest Terms,
RULE. — Divide both terms of the fraction by their greatest
common divisor.
A fraction is in its lowest terms when the numerator and the
denominator are prime to each other.
DEFINITIONS, PRINCIPLES,
To reduce a Mixed Number to an Improper Traction,
RULE. — Multiply the whole number by the denominator ; to the
product add the numerator ; and place the sum over the denom-
inator.
To reduce an Improper Traction to a Whole or to a Mixed Number,
RULE. — Divide the numerator by the denominator.
A Common Denominator is a denominator common to two or
more fractions.
The Least Common Denominator is the smallest denominator
'common to two or more fractions.
To reduce Tractions to their Least Common Denominator.
RULE. — Find the least common multiple of all the denomi-
nators for the least common denominator. Divide this multiple by
the denominator of each fraction, and multiply the numerator by
the quotient.
ADDITION 'OF FRACTIONS
PRINCIPLE. — Only like fractions can be added.
RULE. — Reduce the fractions, if necessary, to a common denom-
inator, and over it write the sum of the numerators.
If there are mixed numbers, add the fractions and the whole
numbers separately, and unite the results.
SUBTRACTION OF FRACTIONS
PRINCIPLE. — Only like fractions can be subtracted.
RULE. — Reduce the fractions, if necessary, to a common denom-
inator, and over it write the difference between the numerators.
If there are mixed numbers subtract the fractions and the whole
numbers separately, and unite the results.
MULTIPLICATION OF FRACTIONS
RULE. — Reduce whole and mixed numbers to improper frac-
tions ; cancel the factors common to numerators and denomina-
tors, and write the product of the remaining factors in the numer-
ators over the product of the remaining factors in the denominators.
SUPPLEMENT
DIVISION OF FRACTIONS
RULES. — I. Reduce whole and mixed numbers to improper
fractions. Reduce the fractions to a common denominator. Divide
the numerator of the dividend by the numerator of the divisor.
II. Invert the divisor and proceed as in multiplication of frac-
tions.
To reduce a Complex Fraction to a Simple One.
RULES. — I. Multiply the numerator of the complex fraction
by its denominator inverted.
II. Multiply both terms by the least common multiple of the
denominators.
DECIMALS
A Decimal Fraction is one in which the unit is divided into
tenths, luindredths, thousandths, etc.
A Decimal is a decimal traction whose denomination is indi-
cated by the number of places at the right of the decimal point.
The Decimal Point is the mark used to locate units.
A Mixed Decimal is a whole number and a decimal written
together.
A Complex Decimal is a decimal with a common fraction
written at its right.
To write Decimals.
RULE. — Write the numerator ; and from the right, point off as
many decimal places as there are ciphers in the denominator,
prefixing ciphers, if necessary, io make the required number.
To read Decimals.
RULE. — Read the numerator, and give the name of the right-
hand order.
PRINCIPLES. — Prefixing ciphers to a decimal diminishes its
value.
DEFINITIONS, PRINCIPLES, AND RULES XL
Removing ciphers from the left of a decimal increases its value.
Annexing ciphers to a decimal or removing ciphers from its
right does not alter its value.
To reduce a Decimal to a Common Fraction.
RULE. — Write the figures of the decimal for the numerator, and
1, with as many ciphers as there are places in the decimal, for the
denominator, and reduce the fraction to its lowest terms.
To reduce a Common Fraction to a Decimal,
RULE. — Annex decimal ciphers to the numerator, and divide it
by the denominator.
To reduce Decimals to a Common Denominator,
RULE. — Make their decimal places equal by annexing ciphers.
ADDITION AND SUBTRACTION OF DECIMALS
Decimals are added and subtracted the same as whole numbers.
MULTIPLICATION OF DECIMALS
RULE. — Multiply as in whole numbers, and from the right of
the product, point off as many decimal places as there are decimal
places in both factors.
DIVISION OF DECIMALS
RULE. — Make the divisor a whole number by removing the
decimal point, and make a corresponding change in the dividend.
Divide as in whole numbers, and place the decimal point in the
quotient under (or over) the new decimal point in the dividend.
ACCOUNTS AND BILLS
A Debtor is a person who owes another.
A Creditor is a person to whom a debt is due.
Xii SUPPLEMENT
An Account is a record of debits and credits between persons
doing business.
The Balance of an account is the difference between the debit
and credit sides.
A Bill is a written statement of an account.
An Invoice is a written statement of items, sent with merchan-
dise.
A Keceipt is a written acknowledgment of the payment of
part or all of a debt.
A bill is receipted when the words, " Received Payment," are
written at the bottom, signed by the creditor, or by some person
duly authorized.
DENOMINATE NUMBERS
A Measure is a standard established by law or custom, by
which distance, capacity, surface, time, or weight is determined.
A Denominate Unit is a unit of measure.
A • Denominate Number is a denominate unit or a collection of
denominate units.
A Simple Denominate Number consists of denominate units of
one kind.
A Compound Denominate Number consists of denominate units of
two or more kinds.
A Denominate Fraction is a fraction of a denominate number.
A denominate fraction may be either common or decimal,
Eeduction of denominate numbers is changing them from one
denomination to another without altering their value.
Eeduction Descending is changing a denominate number to one
of a lower denomination.
RULE. — Multiply the highest denomination by the number re-
quired to reduce it to the next lower denomination, and to the prod-
uct add the units of that lower denomination, if any. Proceed
in this manner until the required denomination is reached.
DEFINITIONS, PRINCIPLES, AND RULES xiii
Keduction Ascending is changing a denominate number to one of
a higher denomination.
RULE. — Divide the given denomination successively by the
numbers that will reduce it to the required denomination. To this
quotient annex the several remainders.
To find the Time between Dates,
RULE. — When the time is less than one year, find the exact
number of days ; if greater than one year, find the time by com'
pound subtraction, taking 30 days to the month.
PERCENTAGE
Per Cent means hundredths.
Percentage is computing by hundredths.
The elements involved in percentage are the Base, Bate, Per-
centage, Amount, and Difference.
The Base is the number of which a number of hundredths is
taken.
The Eate indicates the number of hundredths to be taken.
The Percentage is one or more hundredths of the base.
The Amount is the base increased by 'the percentage.
The Difference is the base diminished by the percentage.
To find the Percentage when the Base and Eate are Given,
RULE. — Multiply the base by the rate expressed as hundredths.
To find the Eate when the Percentage and Base are Given,
RULE. — Divide the percentage by the base.
To find the Base when the Percentage and Eate are Given,
RULE. — Divide the percentage by the. rate expressed as hun-
dredths.
To find the Base when the Amount and Eate are Given.
RULE. — Divide the amount by 1 + the rate expressed as hun-
dredths.
XIV SUPPLEMENT
To find the Base when the Difference and Rate are Given,
RULE. — Divide the difference by I — the rate expressed as hun-
dredths.
PROFIT AND LOSS
Profit or Loss is the difference between the buying and selling
prices.
In Profit and Loss,
The buying price, or cost, is the base.
The rate per cent profit or loss is the rate.
The profit or loss is the percentage.
The selling price is the amount or difference, according as it
is more or less than the buying price.
COMMERCIAL DISCOUNT
Commercial Discount is a percentage deducted from the list
price of goods, the face of a bill, etc.
The Met Price of goods is the sum received for them.
In Commercial Discount,
The list price, or "| .
The face of the bill j 1S the base'
The rate per cent discount is the rate.
The discount is the percentage.
The list price diminished by the discount is the difference.
In successive discounts, the first discount is made from the list
price or the face of the bill ; the second discount, from the list
price or face of the bill diminished by the first discount ; and so
on.
COMMISSION
Commission is a percentage allowed an agent for his services.
A Commission Agent is one who transacts business on com-
mission.
> ig
)
DEFINITIONS, PRINCIPLES, AND RULES XV
A Consignment is the merchandise forwarded to a commission
agent.
The Consignor is the person who sends the merchandise.
The Consignee is the person to whom the merchandise is sent.
The Net Proceeds is the sum remaining after all charges have
been deducted.
In buying, the commission is a percentage of the buying price;
in selling, a percentage of the selling price; in collecting, a per-
centage of the sum collected; hence:
The sam invested, or
The sum collected
The rate per cent commission is the rate.
The commission is the percentage.
The sum -invested increased by the commission is the amount.
The sum collected diminished by the commission is the differ-
ence.
INSURANCE
Insurance is a contract of indemnity.
Insurance is of three kinds — Tire, Marine, and Life,
Fire Insurance is indemnity against loss of property by fire.
Marine Insurance is indemnity against loss of property by the
casualities of navigation.
Life Insurance is indemnity against loss of life.
The Insurance Policy is the contract setting forth the liability of
the insurer.
The Policy Pace is the amount of insurance.
The Premium is the price paid for insurance.
The Insurer, or Underwriter, is the company issuing the policy.
The Insured is the person for whose benefit the policy is issued.
In Insurance,
The policy face is the base.
The rate per cent premium is the rate.
The premium is the percentaae.
XVl SUPPLEMENT
TAXES
A Tax is a sum of money levied on persons or property foi
public purposes.
A Personal, or Poll Tax, is a tax on the person.
A Property Tax is a tax of a certain per cent on the assessed
value of property.
Property may be either personal or real.
Personal Property consists of such things as are movable.
Eeal Property is that which is fixed, or immovable.
In Taxes,
The assessed value is the base.
The rate of taxation is the rate.
The tax is the percentage,
DUTIES
Duties are taxes on imported goods.
Duties are either Specific or Ad Valorem.
A Specific Duty is a tax on goods without regard to cost.
An Ad Valorem duty is a tax of a certain per cent on the cost
of goods.
In Ad Valorem Duties,
The cost of the goods is the base.
The rate per cent duty is the rate.
The ad valorem duty is the percentage.
INTEREST
Interest is the sum paid for the use of money.
The Principal is the sum loaned.
The Amount is the sum of the principal and interest.
The Bate of Interest is the rate per cent for one year.
The Legal Bate is the rate fixed by law.
"Usury is interest at a higher rate than that fixed by law.
Simple Interest is interest on the principal only.
DEFINITIONS, PRINCIPLES, AND RULES
To find the Interest when the Principal, Time, and Eate are Given.
RULE. — Multiply the principal by the rate expressed as hun-
dredths, and this product by the time expressed in years.
To find the Time when the Principal, Interest, and Kate are Given.
RULE. — Divide the given interest by the interest for one year.
To find the Eate when the Principal, Interest, and Time are Given.
RULE. — Divide the given interest by the interest at one per
cent.
To find the Principal when the Interest, Eate, and Time are Given,
RULE. — Divide the given interest by the interest on $ 1.
To find the Principal when the Amount and Time and Eate are
Given,
RULE. — Divide the given amount by the amount of $1.
LNTEKEST BY ALIQUOT PARTS.
To find the Interest for Tears, Months, and Days.
RULE. — Find the interest for one year and take this as many
times as there are years.
Take the greatest number of the given months that equals an
aliquot part of a year and find the interest for this time. Take
aliquot parts of this for the remaining months.
In the same manner find the interest for the days.
The sum of these interests will be the interest required.
To find the Intarest when the Time is Less than a Tear,
RULE. — Find the interest for the time in months or days that
will gain one per cent of the principal.
Find by aliquot parts, as in the first rule, the interest for the
remaining time.
The sum of these interests will be the interest required.
XViii SUPPLEMENT
INTEREST BY Six PER CENT METHOD.
To find the Interest at 6%.
RULE. — For Years : Multiply the principal by the rate ex-
pressed as hundredth^, and that product by the number of years.
For Months : Move the decimal point two places to the left, and
'multiply by one- half the number of months.
For Days : Move the decimal point three places to the left, and
multiply by one-sixth the number of days.
To find the interest at any other rate per cent, divide the in-
terest at 6% by 6, and multiply the quotient by the given rate.
To find Exact Interest.
RULE. — Multiply the principal by the rate expressed as hun-
dredths, and that product by the time expressed in years of 365
days.
ANNUAL INTEREST
Annnal Interest is, interest payable annually. If not paid when
due, annual interest draws simple interest.
To find the Amount Due on a Note with Annual Interest, when the
Interest has not been Paid Annually,
RULE. — Fmd the interest on the principal for the entire time,
and on each annual interest for the time it remained unpaid.
The sum of the principal and all the intei'est is the amount due.
COMPOUND INTEREST
Compound Interest is interest on the principal and on the un-
paid interest, which is added to the principal at regular inter-
vals. The interest may be compounded annually, semi-annually,
quarterly, etc., according to agreement.
To find Compound Interest.
RULE. — Ftnd fhe amount of the given principal for the first
period. Considering this as a new principal, find the amount of
DEFINITIONS, PRINCIPLES, AND RULES xix
it foi the next period, continuing in this manner for the given
time.
Find the difference between the last amount and the given
principal, which will be the compound interest.
PARTIAL PAYMENTS
Partial Payments are part payments of a note or debt. Each
payment is recorded on the back of the note or the. written
obligation.
UNITED STATES RULE. — Find the amount of the principal to
the time when the payment or the sum of two or more payments
equals or exceeds the interest.
From this amount deduct the payment or sum of payments.
Use the balance then due as a new principal, and proceed as
before.
MERCHANTS' RULE. — Find the amount of an interest-bearing
note at the time of settlement.
Find the amount of each credit from its time of payment to the
time of settlement ; subtract their sum from the amount of the
principal.
BANK DISCOUNT
Bank Discount is a percentage retained by a bank for advanc-
ing money on a note before it is due.
The Sum Discounted is the face of the note, or if interest-bear-
ing, the amount of the note at maturity.
The Term of Discount is the number of days from the day of
discount to the day of maturity.
The Bank Discount is the interest on the sum discounted for
the term of discount.
The Proceeds of a note is the sum. discounted less the bank dis-
count.
Problems in bank discount are calculated as problems in
interest.
XX SUPPLEMENT
In Bank Discount,
The sum discounted is the principal.
The rate of discount is the rate of interest.
The term of discount is the time.
The bank discount is the proceeds.
EXCHANGE
Exchange is making payments at a distance by means of drafts
or bills of exchange.
Domestic Exchange is exchange between places in the same
country.
Foreign Exchange is exchange between different countries.
Exchange is at par when a draft, or bill, sells for its face
value ; at a premium when it sells for more than its face value ;
at a discount when it sells for less.
The cost of a sight draft is the face of the draft increased by
the premium, or diminished by the discount.
The cost of a time draft is the face of the draft increased by
the premium, or diminished by the discount, and this result,
diminished by the bank discount.
To find the Cost of a Draft
KULE. — Find the cost of $ 1 of the draft; multiply this ly the
face of the draft.
To find the Face of a Draft.
RULE. — Divide the cost of the draft by the cost of $1 of the
draft.
EQUATION OF PAYMENTS
Equation of Payments is a method of ascertaining at what time
several debts due at different times may be settled by a single
payment.
The Equated Time of payment is the time when the several
debts may be equitably settled by one payment.
The Term of Credit is the time the debt has to run before it
becomes due.
DEFINITIONS, PRINCIPLES, AND RULES XXi
The Average Term of Credit is the time the debts due at different
times have to run, before they may be equitably settled by one
payment.
To find the Equated Time of Payment when the Terms of Credit
begin at the Same Date,
RULE. — Multiply each debt by its term of credit, and divide the
sum of the products by the sum of the debts. The quotient will be
the average term of credit.
Add the average term of credit to the date of the debts, and the
result will be the equated time of payment.
To find the Equated Time when the Terms of Credit begin at
Different Dates,
RULE. — Find the date at which each debt becomes due. Select
the earliest date as a standard.
Multiply each debt by the number of days between the standard
date and the date when the debt becomes due, and divide the sum
of the products by the sum of the debts. The quotient will be the
average term of credit from the standard date.
Add the average term of credit to the standard date, and the
result will be the equated time of payment.
RATIO
Ratio is the relation one number bears to another of the same
kind.
The Terms of the ratio are the numbers compared.
The Antecedent is the first term.
The Consequent is the second term.
The antecedent and consequent form a couplet.
PRINCIPLES. — See Fractions.
PROPORTION
A Proportion is formed by two equal ratios.
The Extremes of a proportion are the first and last terms.
The Means of a proportion are the second and third terms.
XX11 SUPPLEMENT
PRINCIPLES. — The product of the means is equal to the prod-
uct of the extremes.
Either mean equals the product of the extremes divided by the
other mean.
Either extreme equals the product of the means divided by the
other extreme.
RULE FOR PROPORTION. — Represent the required term by x.
Arrange the terms so that the required term and the similar
known term may form one couplet, the remaining terms the other.
If the required term is in the extremes, divide the product of the
means by the given extreme.
If the required term is in the means, divide the product of the
extremes by the given mean.
PARTNERSHIP
Partnership is an association of two or more persons for busi-
ness purposes.
The Partners are the persons associated.
The Capital is that which is invested in the business.
The Assets are the partnership property.
The Liabilities are the partnership debts.
To find the Profit, or Loss, of Each Partner when the Capital of
Each is Employed for the Same Period of Time,
RULE. — Find the part of the entire profit, or loss, that each
partner's capital is of the entire capital.
To find the Profit, or Loss, of Each Partner when the Capital of
Each is Employed for Different Periods of Time,
RULE. — Find each partner's capital for one month, by multi-
plying the amount he invests by the number of months it is
employed; then find the part of the entire profit, or loss, that each
partner's capital for one month is of the entire capital for one
month.
DEFINITIONS, PRINCIPLES, AND RULES
INVOLUTION
A Power of a number is the product obtained by using that
number a certain number of times as a factor.
The First Power of a number is the number itself.
The Second Power of a number, or the Square, is the product of
a number taken twice as a factor.
The Third Power of a number, or the Cubs, is the product of a
number taken three times as a factor.
An Exponent is a small figure written a little to the right of the
upper part of a number to indicate the power.
Involution is finding any power of a number.
To find the Power of a Number,
RULE. — Take the number as a factor, as many times as there
are units in the exponent.
EVOLUTION
A Eoot is one of the equal factors of a number.
The Square Eoot of a number is one of its two equal factors.
The Cube Eoot of a number is one of its three equal factors.
Evolution is finding any root of a number.
Evolution may be indicated in two ways : by the Radical
Sign, V", or by a fractional exponent.
The Index of a root is a small figure placed a little to the left
of the upper part of the radical sign, to indicate what root is to
be found. In expressing square root, the index is omitted.
In the fractional exponent, the numerator indicates the power
to which the number is to be raised; the denominator indicates
the root to be taken of the number thus raised.
To find the Square Eoot of a Number,
RULE. — Point off in periods of two figures, commencing at
units. Find the greatest square in the first period and place the
root in the quotient. Subtract this square from the first period,
and bring down the next period.
XXIV SUPPLEMENT
Multiply the quotient figure by two, and use it as a trial divisor.
Place the second figure in the quotient, and annex it also to the
trial divisor. Then multiply the figures in the trial divisor by the
second quotient figure, and subtract.
Bring down the next period, and proceed as before until the
square root is found.
To find the Square Boot of a Traction.
RULE. — Reduce the fraction to its simplest form, and find the
square root of each term separately.
To find the Onbe Eoot of a Number.
RULE. — Point off in periods of three figures each, beginning at
units.
Find the greatest cube in the first period and place the root in
the quotient. Subtract this cube from the first period, and bring
down the next period.
Multiply the square" of the first quotient figure by three and
annex two ciphers for a trial divisor. Place the second figure in
the quotient. Then, to the trial divisor add three times the prod-
uct of the first and second figures, also the square of the second.
Multiply this su?n by the second figure and subtract.
Bring down the next period, and proceed as before until the cube
root is found.
To find the Cube Eoot of a Traction.
RULE. — Reduce the fraction to its simplest form, and find the
cube root of each term separately.
STOCKS AND BONDS.
Capital Stock is the money or property employed by a corpora-
tion in its business.
A Share is one of the equal divisions of capital stock.
The Stockholders are the owners of the capital stock.
The Par Value of stock is the face value.
The Market Value of stock is the sum for which it may be sold.
DEFINITIONS, PRINCIPLES, AND RULES XXV
Stock is at a premium when the market value is above the
par value ; at a discount, when below par.
Bonds are interest-bearing notes issued by a government or a
corporation.
A Dividend is a percentage apportioned among the stockholders.
A Stock Broker is a person who deals in stocks.
Brokerage is a percentage allowed a stock broker for his services.
In Stocks and Bonds,
The par value is the base.
The rate per cent premium, or discount, is the rate.
The premium, ~)
discount, or > is the percentage.
dividend J
mi i j i • M f amount, or
The market value is the « 7 . ~
( difference.
NOTES, DRAFTS, AND CHECKS.
A Promissory Note is a written promise to pay a specified sum
on demand, or at a specified time.
The Pace of a note is the sum named in the note.
The Maker is the person who signs it.
The Payee is the person to whom the sum specified i$ to be
paid.
The Indorser is the person who signs his name on the back of
the note, thus becoming liable for its payment in case of default
of the maker.
An Interest-bearing Note is one payable with interest.
If the words " with interest " are omitted, interest cannot be
collected until after maturity.
A Demand Note is one payable when demand of payment is
made.
A Time Note is one payable at a specified time.
A Joint Note is one signed by two or more persons who jointly
promise to pay.
XXVI SUPPLEMENT
A Joint and Several Note is one signed by two or more persons
who jointly and severally promise to pay.
In a joint note, each person is liable for the whole amount,
but they must all be sued together. In the joint and several
note, each is liable for the whole amount, and may be sued
separately.
A Negotiable Note is one that may be transferred or sold. It
contains the words " or bearer," or " or order."
A Non-negotiable Note is one not payable to the bearer, nor to
the payee's order.
The Matnrity of a note is the day on which it legally falls due.
A Draft, or Bill of Exchange, is a written order directing the
payment of a specified sum of money.
The Face of a draft is the sum named in it.
The Drawer is the person who signs the draft.
The Drawee is the person ordered to pay the sum specified.
The Payee is the person to whom the sum specified is to be
paid.
A Sight Draft is one payable when presented.
A Time Draft is one payable at a specified time.
An Acceptance of a time draft is an agreement by the drawee
to pay the draft at maturity, which he signifies by writing across
the face of the draft the word " accepted " with the date and his
name.
A Check is an order on a bank or banker to pay a specified
sum of money.
PRIMAEY ARITHMETIC
ANSWERS.
Page 72.
32.
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ANSWERS.
37.
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ANSWERS.
175.
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30.
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28.
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31.
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20.
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29.
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65,865.
32.
4072.
21.
305.
30.
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36.
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25.
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34.
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37.
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31,185.
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$ 1230.75.
38.
17,080.
27.
935.
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36,837.
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$901.25.
39.
27,648.
28.
1050.
37.
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40.
32,664.
29.
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38.
39,924.
Page 88.
41.
41,336.
30.
3456.
39.
45,225.
1.
104.
42.
49,872.
40.
46,557.
2.
112.
43.
60,544.
Page 92.
41.
54,162.
3.
120.
44.
66,320.
1.
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42.
57,078.
4.
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72,536.
2.
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43.
69,165.
5.
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75,264.
3.
135.
44.
75,330.
6.
176.
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81,872.
4.
180.
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81,603.
7.
184.
48.
96,368.
5.
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8.
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6.
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9.
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7.
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10.
248.
8.
216.
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11.
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Page
89.
9.
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95,922.
12.
280.
1.
13.
10.
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51.
13.
13.
328.
2.
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52.
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14.
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3.
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53.
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15.
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4.
20.
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5.
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6.
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7.
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15.
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24.
576.
65.
46.
27.
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16.
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25.
657.
66.
61.
ANSWERS.
67.
73.
98.
112f.
19.
4329.
8.
2175.
68.
80.
lOOf.
20.
1221.
9.
3139.
69.
91.
99.
50f.
10.
1200.
70.
29.
45.
Page 99,
11.
4104.
71.
43.
100.
78f.
1.
$897.42.
12.
944.
72.
47.
69f.
2.
$ 740.87.
13.
819.
73.
56.
101.
114f.
3.
1 1226.38.
14.
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74.
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4.
$7700.88.
15.
775.
75.
74.
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5.
$86,322.53.
16.
2180.
76.
83.
41f.
6.
$168.64.
17.
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77.
92.
7.
$116.93.
18.
5376.
78.
101.
Page 95.
8.
$599.93.
19.
5238.
79.
203.
1.
73,188.
9.
$81.89.
20.
1342.
80.
305.
2.
92,345.
10.
$497.27.
81.
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3.
67,172.
11.
$386.08.
Page
108.
82.
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4.
98,789.
12.
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1.
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24,246.
13.
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6.
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3.
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7.
14,286.
Page 100.
4.
72.
86.
935.
8.
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14.
$20.95.
5.
100.
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9.
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6.
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7.
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89.
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Page
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$ 136.50.
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9.
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2.
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93.
3.
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11.
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$21.50.
Page 110.
92.
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7.
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Page 101.
180.
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1.
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13.
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2.
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5.
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6.
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17.
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6.
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18.
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7.
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ANSWERS.
7. 242.
27. 2343.
47. 64,988.
66. 672.
264.
2556.
70,896.
616.
8. 253.
28. 3564.
48. 50,237.
67. 828.
276.
3888.
54,804.
759.
9. 264.
29. 5016.
49. 34,045.
68. 170r6T.
288.
5472.
37,140.
156A-
10. 275.
30. 5577.
50. 32,989.
69. 213T2T.
300.
6084.
35,988.
195T5^.
11. 341.
31. 6765.
70. 333T7r.
372.
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Page 111.
305|£.
12. 352.
32. 7920.
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54,192.
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59. 108.
79. 1284.
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60. 120.
80. 1560.
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94,896.
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22. 682.
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61. 144.
81. 3300.
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99,240.
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23. 803.
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82. 4272.
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89,604.
330.
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46. 69,520.
65. 492.
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75,840.
451.
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ANSWERS.
86. 1109if
114. 810.
3. 850.
63. 1297\.
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115. 183.
4. 1080.
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Page 112.
10. 2880.
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119. 405.
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111. 243.
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113. 420.
2. 640.
62. 126ft.
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8
ANSWERS.
2. 48$.
2. 299.
43. 6656.
84. 91,696.
3. 75$.
3. 793.
44. 6630.
85. 93,005.
4. 88$.
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6. 83.
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47. 9620.
88. 99,078.
7. 62f.
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48. 9375.
89. 99,891.
8. 26J.
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50. 8964.
91. 91,350.
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51. 10,332.
92. 89,642.
11. 53$.
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52. 11,220.
93. 97,363.
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53. 10,028.
94. 96,768.
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54. 10,323.
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14. 93$.
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56. 9975.
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16. 96.
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98. 99,970.
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Page 120.
21. 3$.
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62. 22,104.
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31. 1496.
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76. 43,112.
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78. 69,160.
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20. 222.
Page 119.
41. 4464.
82. 90,300.
21. 11.
1. 182.
42. 5544.
83. 95,961.
22. 12.
ANSWERS.
9
23. 13.
4. 66f.
Page 128.
19. 60 miles.
24. 23.
5. 81f.
51. $916.61.
20. 150 days.
25. 31.
6. 98|.
52. $778.91.
26. 112.
7. 91f.
53. f 1780.53}.
1. 31.
27. 213.
8. 99f.
54. $3431.48*.
2. 31.
28. 313.
9. 99f.
55. $76.11.
3. 24.
29. 211.
10. 52f.
57. $657.66.
4. 31.
30. 122.
58. $4.17.
5. 42.
31. 311.
Page 127.
59. $76.50.
6. 23.
32. 213.
11. 42f.
61. $58.20.
7. 41.
33. 311.
12. 46f
62. $457.12.
8. 11.
34. 231.
13. 78.
63. $977.67.
9. 21.
35. 33.
14. 46.
64. $963.69.
10. 21.
36. 113.
15. 19.
65. $725.04.
11. 11.
37. 23.
16. 40.
66. $31.25.
12. 21.
38. 34.
17. 73.
67. $.62.
13. 11.
39. 31.
18. 95.
68. $108.06.
14. 21.
40. 34.
19. 89.
69. $37.12£.
15. 31.
41. 31.
20. 80. '
70. $.34.
16. 41.
42. 32.
21. 3J.
17. 23.
22. 2|.
Page 130.
18. 31.
Page 122.
27. 3£.
1. $4.14.
19. 111.
1. 200 feet.
28. 18J.
2. $22.50.
20. 111.
2. $20.70.
29. 35£.
3. $14.
21. 321.
3. 11 sheep.
30. 53£.
4. 21 cents.
22. 322.
4. 900 inches.
31. 88f
5. $252.
23. 300.
5. 480 ounces.
32. 7£.
6. $750.
24. 302^.
6. 96 cents.
33. 37f.
7. 5 yards.
7. 16 pages.
34. 15£.
8. 96 jars.
Page 132.
8. 96 packages
. 35. 21|.
9. 9 gallons.
25. 20&.
9. 25 gallons.
36. 27^.
10. 75 cents.
26. 40|f
10. 7 ounces.
37. 25f.
11. $1620.
27. 50&.
11. $50.
38. 36£.
12. $95.
28. 203.
12. $100.
39. 8}.
13. $80.
29. 202.
13. $175.
40. 39|.
30. 20%.
14. 15 miles.
41. 17f.
Page 131.
31. 202&.
42. 37f.
14. 68 days.
32. 101.
Page 126.
43. 15f
15. 36 bushels.
33. 101^.
1. 41|.
44. 45£.
16. $2.56.
34. 203.
2. 62f.
45. 47.
17. $240.
35. 304.
3. 57f
60. 4f
18. 260 feet.
36. 200^.
10
ANSWERS.
37. 304.
25. 39.
64. 15.
4. 48,300.
38. 430&.
26. If
65. 13.
5. 78,300.
39. 203.
27. lOf.
66. 200.
6. 98,400.
40. 431^.
28. lOf
67. 48.
7. 98,800.
41. 202.
29. lOf
68. 60.
8. 91,000.
42. 120ff
30. 10.
69. 32.
9. 72,000.
43. 221T8T-
31. 9f.
70. 60.
44. 123.
32. 20£.
Page 141.
45. 325.
33. 41f
1. $5.
10. 90,000.
46. 231^j.
34. 18J.
2. 150 stamps.
11. 88,800.
47. 101&.
35. 26f
3. 62 cows.
12. 84,150.
48. 34^.
36. 31f.
4. $16.
15. 95,000.
49. 122£f
37. 67f.
5. 28 pounds.
16. 77,400.
50. 103.
38. 19f
17. 83,700.
51. 20&.
39. If
Page 139.
18. 89,100.
40. 19f.
6. 144 pieces.
19. 93,000.
Page 137.
7. $12.80.
20. 67,200.
1. 47.
Page 138.
8. 40 boxes.
21. 95,370.
2. 84.
41. 185.
9. 49 inches.
22. 99,540.
3. 32.
42. 48.
10. 234 eggs.
23. 81,480.
4. 73.
43. 203.
11. $60.
24. 88,480.
5. 81f
44. 19.
12. 5 cents.
6. 68£.
45. 90.
13. 15 cents.
Page 143.
7. 38f
46. 29.
14. $2.94.
1. 7f.
8. 20f.
47. 70.
15. 67 cents.
2. 9f
9. 70f.
48. 4. .
16. 7 packages.
3. 16f
10. 83f
49. 428.
17. $3.
4. 19f
11. 6f
50. 17,376.
18. 750 pounds.
5. 20fr.
12. 18f.
51. 1000.
19. 15 cents.
6. 19f
13. 19|.
52. 5600.
20. 46 cents.
7. 48.
14. 17.
63. 5600.
21. 20 cents.
8. 37.
15. 50.
54. 78.
9. 22}.
16. 26.
55. 126.
Page 140.
10. 45f.
17. 31f
56. 168.
22. 30 cents.
11. 84f
18. 13f
57. 144.
23. 5 yards.
12. 89f
19. lOf
58. 144.
24. 196 pounds.
13. llf.
20. 22f.
59. 144.
25. 20 pieces.
14. 39f.
21. 2f
60. 84.
15. 48f
22. 13f
61. 10.
1. 70,800.
16. 7.
23. 21.
62. 4410.
2. 71,200.
17. 22f
24. 30$.
63. 15.
3. 67,000.
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19.
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30.
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20.
66.
61.
109.
31.
9j-
21.
72.
62.
112.
32.
'21*.
22.
88.
63.
107.
33.
H-
23.
123.
64.
97ff.
34.
22|.
24.
138.
65.
89ft.
35.
H-
25.
77.
66.
54.
36.
11*.
26.
133.
67.
97.
37.
15*.
27.
37.
68.
102.
38.
14*.
28.
77.
69.
92.
39.
16*.
29.
115.
70.
350.
40.
Wf
30.
97.
71.
232^.
41.
19f.
31.
85J|.
72.
108f$.
42.
28f.
32.
97H.
73.
174f|.
43.
38*.
33.
92.
74.
139^-.
44.
47*.
34.
66*|.
75.
136ft.
45.
50*.
35.
68.
76.
119ft.
46.
8*.
36.
95.
77.
131ff.
47.
28f.
37.
143.
78.
86**.
48.
21|.
38.
541.
79.
120ff.
49.
7f.
39.
328.
80.
206*}.
40.
216.
81.
304ft.
Page 145.
41.
304.
82.
206¥87.
1.
54.
42.
271.
83.
234*|.
2.
54.
43.
74.
84.
178ft.
3.
54.
44.
143.
85.
116**-
4.
54.
45.
206.
86.
120H-
5.
44.
46.
184.
87.
400ft.
6.
32.
47.
136.
88.
545ft.
7.
24.
48.
108.
89.
555ft.
8.
33.
49.
204.
90.
355.
ANSWERS. 11
91. 91.
92. 142.
93. 113.
94. 112ff.
95. 87f£.
96. 51ff.
97. 103$f.
98. 102f$.
99. 103ft.
Page 146.
1. 136.
2. 32.
3. 34.
4. 122.
6. 87.
6. 75.
7. 24.
8. 65.
9. 25.
10. 49.
Page 147.
1. 136/&.
2.
3.
4.
5.
6.
7.
8- 80^.
9-
10.
12
ANSWERS.
28. 27|.
16. 96,266.
55. 92,640.
29. 16f.
17. 94,520.
56. 85,181.
30. 53f.
18. 94,518.
57. 82,926.
31. 31.
19. 90,750.
58. 64,684.
32. 26$.
20. 93,396.
59. 76,020.
33. 21*.
21. 96,170.
60. 97,768.
34. 4$.
22. 97,908.
61. 90,752.
35. 44$.
23. 89,159.
62. 70,455.
36. 11$.
24. 87,472.
63. 98,049.
37. 8f.
25. 97,768.
64. 86,592.
38. 16|.
26. 95,918.
65. 98,245.
39. 24$.
27. 43.
66. 71,604.
40. 32f.
28. 78.
67. 99,770.
Page 149.
41. 40f.
29. 32.
68. 98,802.
1. 27f.
42. 48f.
30. 24.
69. 81,804.
2. 47$.
43. 56$.
31. 14.
70. 95,081.
3. 51f.
44. 64f
32. 13.
71. 98,245.
4. 38f.
45. 72f
33. 14.
72. 98,245.
6. 66f
46. 79£.
34. 12.
73. 92,486.
6. 99f.
47. 27f
35. 11.
74. 75,072.
7. 68£.
48. 17f
36. 9.
8. 94f.
49. 29^.
Page 153.
9. 95f.
50. 62f.
Page 152.
75. 88.
10. 85.
37. 8.
76. 105.
11. 99.
38. 13.
77. 99.
12. 23.
Page 151.
39. 15.
78. 98.
13. 44$.
1. 99,684.
40. 13.
79. 69.
14. 70f. '
2. 85,731.
41. 24.
80. 69.
15. 27*.
3. 95,772.
42. 23.
81. 168.
16. 42|.
4. 94,770.
43. 45.
82. 95.
17. 53$.
5. 94,095.
44. 75.
83. 90.
18. 98$.
6. 89,622.
45. 33.
84. 93.
19. 83f.
7. 96,882.
46. 22.
85. 186.
20. 35f.
8. 95,914.
47. 8.
86. 154.
21. 6$.
9. 99,507.
48. 4.
87. 232.
22. 23f.
10. 91,344.
49. 6.
88. 368.
23. 22$.
11. 86,592.
50. 4.
89. 297.
24. 31f.
12. 97,020.
51. 68,580.
90. 100.
25. 31$.
13. 93,832.
52. 96,621.
91. 102.
26. 12f
14. 79,328.
53. 96,859.
92. 205.
27. 10.
15. 91,464.
54. 96,740.
93. 255.
ANSWERS.
13
94. 320.
33. 42,372.
74. 155.
9. 2£ yards.
95. 456.
34. 107,028.
75. 138.
10. 108 quarters.
96. 675.
35. 96,444.
76. 123.
11. $72.50.
97. 880.
36. 92,376.
77. 109.
12. 3 cents.
98. 615.
37. 337£.
78. 406.
13. 86 feet.
38. 673f.
79. 308.
14. $7.80.
Page 154.
39. 1237£.
80. 203.
15. $2.40.
1. 360.
40. 1897$.
81. 170.
16. 95 cents.
2. 1125.
41. 1683$.
82. 146.
17. 99 cents.
3. 800.
42. 2880f.
83. 123.
18. 48 eggs; 144
4. 1200.
43. 1869f.
84. 105.
eggs.
5. 1770.
44. 4575f.
85. 104.
19. 40 bushels.
6. 2800.
45. 6286£.
86. 98.
20. $60.
7. 2331.
46. 21,441|.
87. 48.
21. 413 butter-
8. 16,044.
47. 40,138.
88. 33^fo.
flies.
9. 14,883.
48. 44,500f.
89. 24.
22. 99 cents.
10. 39,234.
49. 42,274f.
90. 19.
11. 22,243.
50. 99,682f.
91. 16.
Page 160.
12. 4400.
51. 65,166f.
92. 14.
23. 130 yards.
52. 4231.
93. 12.
24. 7f acres.
Page 155.
53. 3152.
94. 11.
25. $1.75.
13. 6578.
54. 2405.
QK QQ
14. 23,922.
55. 1600.
*/«-!• t/O.
96. 87.
1. 91,448.
15. 43,190.
56. 1623.
97. 75.
2. 86,400.
16. 49,260.
57. 1405.
98. 33.
3. 97,886.
17. 17,922.
58. 1234.
99. 23.
4. 90,288.
18. 61,479.
59. 1035.
100. 9.
5. 89,415.
19. 85,200.
60. 2305.
101. 4.
6. 88,971.
20. 82,810.
61. 2046.
102. 8.
7. 89,208.
21. 6888.
62. 1653.
8. 82,766.
22. 13,552.
63. 1408.
Page 158.
9. 99,696.
23. 39,528.
64. 1305.
1. $7.
10. 73,140.
24. 51,968.
65. 1060.
2. $2.82.
11. 82,602.
25. 14,610.
66. 1003.
3. 192 pints.
12. 99,960.
26. 25,280.
67. 3265.
4. 20 yards.
13. 97,633.
27. 50,904.
68. 907.
5. $2.40.
14. 96,348.
28. 65,400.
69. 807.
6. 40 cents.
15. 180.
29. 84,252.
70. 486.
16. 232.
30. 96,560.
71. 325.
Page 159.
17. 348.
31. 79,380.
72. 247.
7. $3.
18. 567.
32. 30,537.
73. 189.
8. $2.25.
19. 864.
14
ANSWERS.
20.
1120.
8.
99*.
6.
538ft.
47.
6.
21.
777.
9.
82*.
7.
713*f.
48.
3.
22.
945.
10.
108$.
8.
282ff
23.
1100.
11.
64*.
9.
1636$f.
Page 164.
24.
1343.
12.
91.
10.
1966$f
1.
31 barrels.
25.
496.
13.
95.
11.
811 A-
2.
6 yards.
26.
1454f.
14.
37$.
12.
787**.
3.
380 inches.
27.
96,000.
15.
26.
13.
478f*.
28.
99,712.
14.
222**,
Page 165.
29.
96,888.
Page
162.
15.
279ff.
4.
H yards.
30.
99,328.
16.
86.
16.
m**.
5.
25 cents; $L
31.
77,608.
17.
81$.
17.
182*.
6.
16 cents.
32.
99,450.
18.
49.
18.
432f#f.
7.
98 cents.
33.
99,902.
19.
83$.
19.
153$ff
8.
3i pounds.
34.
95,841.
20.
87$.
20.
86&V
9.
39 pints.
35.
61,845.
21.
64$.
21.
181^..
10.
145 sheep.
36.
99,102.
22.
28$.
22.
113|$f.
11.
$2.
37.
96,696.
23.
37$.
23.
104f$$.
12.
93 cents.
38.
92,976.
24.
35*.
24.
HI***-
13.
6 weeks.
39.
99,051.
25.
54$.
25.
70f$f
14.
35 gallons.
40.
93,345.
26.
69f.
26.
709jfj.
15.
10 Ib. 5 oz.
41.
96,744.
27.
30*.
27.
219ft.
16.
$2.
42.
88,920.
28.
81$.
28.
132&V
43.
99,601.
29.
1-
29.
42m.
Page 166.
44.
99,485.
30.
75$.
30.
182fff.
17.
41 pounds.
45.
2000.
31.
H
31.
157$f|.
18.
$ 3.60.
46.
2800.
32.
6$.
32.
49f|-f.
19.
150 days ; 3
47.
24,000.
33.
18$.
33.
25f£f.
days.
48.
24,500.
34.
43f.
34.
40ft.
20.
$2.05.
49.
99,000.
35.
12f.
35.
30rm?.
21.
$39.
50.
96,000.
36.
llf.
36.
SS^s^
22.
$225.
51.
81,081.
37.
24*.
37.
5<fV9oV
23.
$2.25.
38.
18*.
38.
12****.
24.
$ 8.22.
Page 161.
39.
27*.
39.
19*{|**.
25.
§40.
1.
129$.
40.
40f.
40.
14****.
2.
92*.
41.
23ii|. page 169.
3.
79|.
1.
2857^T.
42.
VtfoV
1.
603,275.
4.
97f.
2.
3134$f
43.
3*ffft
2.
678,456.
5.
69$.
3.
1225$f
44.
17/j2/j.
3.
759,795.
€.
27*.
4.
1622|f
45.
3T^ilT-
4.
641,426.
7.
47*.
5.
990/T.
46.
4JT17T-
5.
$2714.42.
ANSWERS.
15
6. $8502.43.
45. 963,976.
86. 3134ff.
125. 34.
7. $7269.80.
46. 887,112.
87. 31425V
126. 125.
8. $9885.02.
47. 629,405.
88. 3009¥\.
127. 5.
9. 300,424.
48. 890,765.
89. 3034ff.
128. 138.
10. 913,092.
49. 933,725.
90. 3050f£.
129. 150.
11. $220,119.
50. 2123.
91. 30717V
130. 78.
12. $1912.09.
51. 1203.
92. 2016Ty?.
13. $359,809.
52. 1303.
93. 1234if£.
Page 172.
14. 414,867.
53. 1203.
94. 1132f|f.
1. 760 yards.
15. $161,715.
54. 1031.
95. 1355¥522j.
2. 240hf.pt.
16. 173,929.
55. 2402.
96. 504.
3. $3.45.
17. $2952.51.
56. 3002.
97. 306.
4. $248.
18. 399,952.
57. 3030.
98. 203.
5. $210.
19. $1624.43.
58. 10,444.
99. 105.
6. 4| pounds.
59. 1060.
100. 109f$ff
7. 41$ bushels.
Page 170.
60. 1011.
8. $4.98.
20. 868,980.
61. 1012.
Page 171.
21. 895,048.
62. 1013.
101. 59$.
Page 173.
22. 954,048.
63. 1011.
102. 19$.
9. 1$ minutes.
23. 996,450.
64. 1101.
103. 9639.
10. 1440 matches.
24. 592,320.
65. 1102. »
104. 12,141.
25. 864,128.
66. 220.
105. 96.
1. 5f.
26. 970,485.
67. 303.
106. 96.
2. 12$.
27. 940,215.
68. 150.
107. 12$.
3. 15f.
28. 967,890.
69. 606.
108. 60|.
4. 20.
29. 954,087.
70. 222.
109. 61$.
5. 27$.
30. 906,205.
71. 300|ff|.
110. 300.
6. 42f.
31. 968,464.
72. 306|^|.
111. 300.
7. 38$.
32. 886,730.
73. 219.
112. 2.
8. 56$.
33. 864,565.
74. lOlffff.
113. 8.
9. 134$.
34. 941,408.
75. 154|§ff.
114. 78.
10. 134}.
35. 948,708.
76. 112fK-f.
115. 162.
11. 88$.
36. 972,930.
77. 112$fjff.
116. 231.
12. 50.
37. 761,472.
78. 861^V
117. 36.
13. 44$.
38. 955,320.
79. 833 Jȴ.
118. 648.
14. 26f.
39. 969,855.
80. 903^.
119. 46$.
15. 36|.
40. 976,372.
81. 982$f.
120. 70f.
16. 9f.
41. 926,328.
82. 1313$|.
121. 12,126.
17. 46}.
42. 925,245.
83. 2196ff.
122. 187,440.
18. 147$.
43. 856,674.
84. 2218ff.
123. 68.
19. 37$.
44. 977,724.
85. 2279£|.
124. 975.
20. 73$.
16
ANSWERS.
21. 7J.
24. 969,600.
3. 39ft.
42. 70J.
22. 4.
25. 617,120.
4. 70ft.
43. 3S&.
23. 86f.
26. 434,420.
5. 81.
44. 7J.
24. 29£.
27. 47,196.
6. 9ft.
45. 9|.
25. 34f
28. 47,272.
7. 21ft.
46. 8A.
26. 59$.
29. 47,082.
8. 7ft.
47. 78J.
27. 61|.
30. 137,598.
9. 92ft.
48. 55TV
28. m.
31. 59,660.
10. 57f.
49. 47A-
29. 17f
32. 59,508.
50. 39ft.
30. 7i
33. 137,427.
Page 177.
31. 8|.
34. 78,150.
11- ft*
Page 179.
32. 18f
35. 209,664.
12. 2ft.
1. 3210.
33. 49|.
36. 844,662.
13. 4ft.
2. 4321.
34. 25£.
37. 979,016.
14. 4ft.
3. 765.
35. 74£.
38. 998,016.
15. 18ft.
4. 3450T$¥.
36. 23J.
39. 17,329f
16. 33ft.
5. 5403.
40. 58,8671.
17. 53ft.
6. 4506Tf*.
Page 174.
41. 760,249.
18. 76.
7. 6063.
1. 46,512.
42. 369,123.
19. 105.
8. 7006.
2. 144,536.
43. 17,367f
20. 48ft.
9. 6003.
3. 253,840.
44. 30,250.
21. 9ft.
10. 6005.
4. 132,435.
45. 850,950.
22. 61&.
11. 7001.
5. 306,130.
46. 95,482$.
23. 63&.
12. 5203.
6. 354,488.
47. 938,475.
24. 7ft.
13. 6715T6&.
7. 87,084.
48. 935,712.
25. 15&.
14. 5701.
8. 199,014.
49. 781,1371.
26. 32^.
15. 1020TVT.
9. 784,770.
50. 954,320.
27. 29ft.
16. 2034.
10. 934,164.
28. 29ft.
17. 3240.
11. 784,770.
29. 82.
18. 4003.
12. 934,164.
30. 82f
19. 5041T^.
13. 30,504.
31. 21J.
20. 4774TVf.
14. 54,756.
32. 31f
21. 1789^.
15. 37,260.
33. 41 f.
22. 1509=£fr.
16. 138,624.
34. 51f.
23. 1155ft£
17. 616,302.
35. 61f.
24. 263 H&f
18. 104,148.
36. 61f
25. 2347iff-
19. 805,460.
37. 71f
26. 298 If ft.
20. 93,912.
38. 84*.
27. 1435^.
21. 151,782.
Page 176.
39. 79TV
28. 4f>9?i4.
22. 548,730.
1. 20^.
40. 65^.
29. 1545fff.
23. 846,300.
2. 25ft.
41. 59f
30. 720^°T.
ANSWERS.
17
31.
2117^ft.
3.
45**-
44.
31f
Page 189.
32.
1707ff*.
4.
66ft.
45.
13*-
4.
$4.
33.
1607*H.
5.
79ft-
46.
32|.
5.
29 tons.
34.
1615***.
6.
75ft.
47.
13*.
6.
195 days.
35.
1191ft*.
7.
17ft.
48.
14*.
7.
13 cents.
36.
1053***.
8.
16ft.
49.
.41*.
8.
416 yards.
37.
1008***,
9.
49ft.
50.
20ft.
9.
$405.
38.
879f§£.
10.
38ft.
10.
537 pounds.
39.
990Hf
11.
18ft.
11.
35 plants.
40.
6oo?yT.
12.
42ft.
Page 186.
12.
341 passengers.
41.
608Tyft.
13.
65ft.
1.
117 ounces.
13.
$ 3000.
42.
461^5.
14.
29H-
2.
4 Ib. 5 oz.
14.
Lost $ 20.
43.
307fl&.
15.
75H-
3.
20 gal. 2 qt.
15.
1799.
44.
185|f£f.
16.
116*.
4.
59 quarts.
16.
8 years.
45.
153|£*|.
17.
50ft.
5.
23 qt. 1 pt.
46.
25**ff
18.
65 J.
6.
57 pints.
Page 190.
47.
7()2|9|.
19.
92*.
7.
75 pecks.
17.
$3.
48.
30ff|f.
20.
97ft.
8.
143 quarts.
18.
$420.
49.
32***f
21.
28*.
9.
12 pk. 1 qt.
19.
$225.
50.
283||**.
22.
59.
10.
21 bu. 3 pk.
20.
46 boys.
51.
2302%°^T.
23.
12*.
11.
1568 quarts.
21.
$2.
52.
251***f.
24.
99ft.
12.
180 inches.
22.
$216.
25.
96f.
13.
44 feet.
23.
10 cents.
Page 181.
26.
251
14.
159 inches.
24.
9 months.
1.
240 bushels.
27.
71*.
15.
9 ft. 11 in.
25.
200 eggs.
2.
58 cents.
28.
56*.
16.
23 yd. 1 ft.
3.
5 cows.
29.
25ft.
17.
44 pounds.
1.
835,539.
4.
90 cents.
30.
3ft-
18.
88 gallons.
2.
759,645.
5.
$5500.
31.
14*.
19.
65 quarts.
3.
888,732.
6.
226£ acres.
32.
59*.
20.
151 bushels.
4.
869,649.
7.
$5.88.
33.
38|.
21.
43 pecks.
5.
805,050.
8.
$90.
34.
31*.
22.
85 ft. 3 in.
6.
746,108.
9.
402.
35.
18i.
23.
60 bu. 3 pk.
7.
902,000.
10.
4 cows.
36.
31*.
24.
13 ft. 1 in.
8.
963,214.
11.
16 days.
37.
30*.
25.
67 gal. 2 qt.
9.
855,922.
12.
3 yards.
38.
22f.
10.
957,032.
13.
80 quarts.
39.
41*.
11.
704,175.
40.
37*.
Page 188.
12.
593,164.
Page 185.
41.
13*.
1.
80.
13.
986,592.
1.
18ft-
42.
3*.
2.
$1200; $240.
14.
962,304.
2.
24ft.
43.
27*.
3.
$1.
15.
943,114.
18
ANSWERS.
16. 831,875.
55. 1046ff.
96. 144||f.
17. 833,316.
56. 1033$f.
97. 821|f$.
18. 505,134.
57. 841f|.
98. 91TVFV
58. 6G9|f.
99. 241ff||.
Page 191.
59. 215ff.
100. 237|1§|.
19. 190|.
60. 223AV
101. 63^V
20. 500.
61. 260|||.
102. 181|||f.
21. 420.
62. 89f|f.
22. 1845.
63. 83$$|.
23. 987.
64. 40ff$.
Page 192.
24. 1071.
65. 78^7-
103. 97.
25. 1612.
66. 81^^.
104. 48.
26. 1G45.
67. 32f|f.
105. 32.
27. 2583.
68. 99$$$.
106. 24.
28. 3885.
69. 43$|f.
107. 18.
29. 4100.
*y f\ o£ A 4 9
iu. ^^^rlJoT'
108. 16.
30. 780,096.
n9^1 511
• .-OOy tJirT.
109. 14.
31. 991,782.
72. 281|ff|.
110. 12.
32. 943,260.
70 17Q2807
/O. 1 / <?J5§7.
111. 11.
33. 984,328.
74. 166flft.
112. 44.
34. 892,320.
75. 51325%V
113. 33.
35. 952,408.
76. 107£f££.
114. 22.
36. 933,450.
77. 207$|f.
37. 875,706.
78. SOffff.
38. 952,714.
*7Q QA6 342
/ «7. ^^§^^X»
Page 193.
39. 970,169.
80. 579$|.
1. 1*.
40. 954,530.
81. 2332ff
2. 4TV
41. 3519.
82. 767$f.
3. 7TV
42. 3616.
83. 628ff.
d Ifi1 !
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43. 6132.
84. 1398|f.
5. 37if.
44. 4557.
85. 10217f.
6. ISA.
45. 9568.
86. 1051ff.
7. 11 A-
46. 10,791.
87. 974f£.
8. 31 A-
47. 17,572.
88. lOS^ff
9- 8yj-
48. 39,333.
89. 27822jV
10. 71A-
49. 76,775.
90. 841H.
11. lsf$-
50. 97,460.
91. 184TVV
12. 26if.
51. 69,000.
92. 905^.
13 53$.
52. 2218|f.
93. 554fff.
14. 991.
53. 786i|.
94. 951 f|£.
15. 911.
54. 1618J$.
95. 285|||.
16. 4 7 A-
17. 77^.
18. 86H-
19. 54.
20. 63$.
21. 88^.
22. 93|.
23. 94f.
24. 93f
25. 81$J.
26. SI
27. 76H-
28. 88$.
29.
30.
31. 71$.
32. 3f.
33. 24f.
34. 18f.
35. 21|.
36.
37.
38.
39.
40.
ANSWERS.
19
6. 594,672.
36. 577,771.
Page 197.
7. 531,696.
37. 194,142.
69. 59$ff.
8. 178,654.
38. 806,922.
70. 18$f|.
9. 194,508.
39. 834,725.
71. Slfff
10. 177,045.
40. 696,822.
72. 123fff.
11. 329,141.
41. 92,501$.
73. 21$H.
12. 537,966.
42. 205,979.
74. 33^f.
43. 301,058.
75. 3223iff.
44. 293,336|.
76. 370j%V
Page 196.
45. 397,087.
77. 410iff
13. 503,036.
46. 564,389f
78. 908fff.
14. 354,585.
47. 378,670$.
79. 930$|f.
15. 348,087.
48. 489,303$.
80. 460|f|.
16. 781,529.
49. 571,693$.
81. 417fff.
17. 75,854.
50. 352,315$.
82. 263f£f.
18. 63,616f.
51. 3646$f
83. 255 J fff.
19. 56,818$.
52. 2376$f
84. 197iflf.
20. 80,647$.
53. 1002|f
85. 194ff$f.
21. 77,371f.
54. 1578ff
86. 116ff|f.
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55. 326£f
87. 54fff|.
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56. 711W-
88. 68^1-
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57. 361$f
89. 29fff£.
25. 434,661f.
58. 441 /T.
90. 76|fff.
26. 447,673.
59. 3297|$.
91. 135$$|f.
27. 488,748.
60. 977$|.
92. 228$$ff.
28. 551,5361.
61. 2159$|.
93. 29||f|.
29. 486,029|.
62. 311ff.
94. 119ffff
Page 195.
30. 881,001$.
63. 1157&-
95. 47f§§|.
1. 260,512.
31. 101,376.
64. 10737V
96. 97ff|f.
2. 110,124.
32. 162,582.
65. 1048$$.
97. 50f§ff.
3. 387,024.
33. 145,116.
66. 314ff.
98. 63?W?>
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34. 130,245.
67. 63|f|.
99. 72%Y7.
5. 505,580.
35. 84,632.
68. 172//J.
100. 5$ff$§.
Elementary Mathematics
AtWQOd's Complete Graded Arithmetic. Presents a carefully graded course, to
begin with the fourth year and continue through the eighth year. Part I, 30 cts.; Part
II, 65 cts.
Badlam's Aids tO Number. Teacher's edition — First series, Nos. i to 10, 40 cts.;
Second series, Nos. 10 to 20, 40 cts. Pupil's edition — First series, 25 cts.; Second
series, 25 cts.
Branson's Methods in Teaching Arithmetic. 15 cts.
Hanus's Geometry in the Grammar Schools. An essay, with outline of work for
the last three years of the grammar school. 25 cts.
Howland's Drill Cards. For middle grades in arithmetic. Each, 3 cts.; per hun-
dred, $2.40.
Hunt's Geometry for Grammar SchOOls. The definitions and elementary con-
cepts are to be taught concretely, by much measuring, and by the making of models
and diagrams by the pupils. 30 cts.
PierCC'S Review Number Cards. Two cards, for second and third year pupils.
Each, 3 cts.; per hundred, $2.40.
Safford's Mathematical Teaching. A monograph, with applications. 25 cts.
Sloane's Practical Lessons in Fractions. 25 cts. Set of six fraction cards, fot
pupils to cut. 10 cts.
Sutton and Kimbrough's Pupils' Series of Arithmetics. Lower Book, for
primary and intermediate grades, 35 cts. Higher Book, 65 cts.
The New Arithmetic. By 300 teachers. Little theory and much practice. An excel-
lent review book. 65 cts.
Walsh's Arithmetics. On the, "spiral advancement" plan, and perfectly graded.
Special features of this series are its division into half-yearly chapters instead of the
arrangement by topics; the great number and variety of the problems ; the use of the
equation in solution of arithmetical problems; and the introduction of the elements of
algebra and geometry. Its use shortens and enriches the course in common school
mathematics. In two series: —
Three Book Series — Elementary, 30 cts.; Intermediate, 35 cts.; Higher, 65 cts.
Two Book Series — Primary, 30 cts.; Grammar school 65 cts.
Walsh's Algebra and Geometry for Grammar Grades. Three chapters from
Walsh's Arithmetic printed separately. 15 cts.
White's TWO Years With Numbers. For second and third year classes. 35 cts.
White's Junior Arithmetic. For fourth and fifth years. 45 cts.
White's Senior Arithmetic. 65 cts.
For advanced -works see our list of books in Mathematics.
D.C. HEATH & CO., Publishers, Boston, New York, Chicago
Elementary Science.
Austin's Observation Blanks in Mineralogy. Detailed studies of 3S minerals.
Boards. 88 pages. 30 cts.
Bailey's Grammar School Physics. A series of inductive lessons in the elements
of the science. Illustrated. 60 cts.
Ballard'S The World Of Matter. A guide to the study of chemistry and mineralogy;
adapted to the general reader, for use as a text-book or as a guide to the teacher in giving
object-lessons. 264 pages. Illustrated. $1.00.
Clark's Practical Methods in MicrOSCOpy. Gives in detail descriptions of methods
that will lead the careful worker to successful results. 233 pages. Illustrated. $1.60.
Clarke's Astronomical Lantern. Intended to familiarize students with the constella-
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giving twenty-two constellations. $4 50.
Clarke's HOW tO find the Stars. Accompanies the above and helps to an acquaintance
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Guides for Science Teaching. Teachtrs' aids in the instruction of Natural History
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I. Hyatt's About Pebbles. a6 pages. Paper. 10 cts.
II. Goodale's A Few Common Plants. 61 pages. Paper. 20 cts.
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V. Hyatt's Corals and Echinoderms. Illustrated. 32 pages. Paper. 30 cts.
VI. Hyatt's Mollusca. Illustrated, f 5 pages. Paper. 30 cts.
VII. Hyat's Worms and Crustacea. Illustrated. 68 pages. Paper. 30 cts.
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XIII. Richard's First Lessons in Minerals. 50 pages. Paper. 10 cts.
XIV. Bowditch's Physiology. 58 pages. Paper. 20 cts.
XV. Clapp's 36 Observation Lessons in Minerals. 80 pages. Paper. 30 cts.
XVI. Phe MY 's Lessons in Chemistry. 20 cts.
Pupils' Note- Book to accompany No. 15. 10 cts.
Rice's Science Teaching in the School. With a course of instruction in science
for the lower grades. 46 p*g s. Paper. 25 cts.
Ricks's Natural History Object LeSSOnS. Supplies information on plants and
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332 pages. $1.50.
Ricks's Object Lessons and How to Give them.
Volume I. Gives lessons for primary grades. 200 pages. 90 cts.
Volume II. Gives lessons for grammar and intermediate grades. 212 pages. 90 cts.
Staler S First Book in Geology. For high school, or highest class in grammar school
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Shaler'S Teacher's Methods in Geology. An aid to the teacher of Geology.
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Smith's Studies in Nature. A combination of natural history lessons and language
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See also our list of books in Science.
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