J. A. (James Alexander) McLellan.

The public school arithmetic : based on McLellan & Dewey's Psychology of number online

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'IN MEMORIAM

FLOR1AN CAJOR1




THE PUBLIC SCHOOL ARITHMETIC



Q_
*3r




THE



PUBLIC SCHOOL ARITHMETIC



BASED ON McLELLAN AND DEWEY'S

"PSYCHOLOGY OF NUMBER"



BY

J. A. McLELLAN, A.M., LL.D.

PRESIDENT OF THE ONTARIO NORMAL, COLLEGE ; AUTHOR (WITH DR. DEWEY)

OF "THE PSYCHOLOGY OF NUMBER," "APPLIED PSYCHOLOGY,"

"THE TEACHER'S HANDBOOK OF ALGEBRA," ETC.

AND

A. F. AMES, A.B.

HONOR GRADUATE IN MATHEMATICS ; FORMERLY MATHEMATICAL, MASTEB

ST. THOMAS COLLEGIATE INSTITUTE, ETC. ; SUPERINTENDENT

OF SCHOOLS, RIVERSIDE, ILL.



f0rk
THE MACMILLAN COMPANY

LONDON: MACMILLAN & CO., LTD.
1900

All rights reserved



COPYRIGHT, 1897,
BY THE MACMILLAN COMPANY.



Set up and electrotyped June, 1897. Reprinted September,
1897; January, July, 1898; July, 1900.



Nor fa not)
J. S. Cushing & Co. Berwick & Smith
Norwood Mass. U.S.A.



PREFACE



THE present reaction against Arithmetic as a school study is
perhaps due partly to irrational teaching and partly to the grow-
ing tendency to make things interesting to the learner by making
them easy. This reaction, whatever its cause, is to be deplored.
If irrational methods of teaching have led to waste of time and
unsatisfactory results, it is surely the part of wisdom to work,
not for a removal of the subject from the list of regular studies,
but for improvement in methods of teaching it. In view of the
present cry " for things, not abstractions," - as if things apart
from abstractions, the mind's action upon things, Avere the only
reality, - - there is urgent need of giving Arithmetic its rightful
place and having it taught on sound psychological principles.
When so taught, no subject can be substituted for this "Logic of
the Public Schools." There has been much nonsense uttered by
those who are clamorous for " Content '' studies as opposed to
" Form " studies. How is it possible to separate Form and
Content, and regard the one as good in itself and the other
as, at best, a necessary evil ?

" In the case of Number, form represents the measured adjust-
ment of means to an end, the rhythmical balancing of parts in a
whole, and therefore the mastery of form represents directness,
accuracy, and economy of perception, the power to discriminate
the relevant from the irrelevant, and ability to mass and converge
relevant material upon a destined end ; represents, in short, pre-
cisely what we understand by good sense, by good judgment, the



vi PREFACE

power to put two and two together. When taught as this sort of
form, Arithmetic affords, in its own place, an unrivalled means of
mental discipline." *

To help both teacher and pupil to make the most of this
" unrivalled means of mental discipline" is the purpose of "The
Public School Arithmetic." The treatment of the subject is in
line with what is believed to be the true idea of number and
numerical operations as developed in McLellan and Dewey's
"Psychology of Number," a work which, to use the words of
one of our greatest educators, " has caused more stir among
teachers, and elicited a greater number of favorable opinions
than any other pedagogical book of the day." Indeed, "The
Public School Arithmetic' 1 has been prepared in response to
requests from teachers and educators of every grade, for a text-
book on Arithmetic based on the principles set forth in "The
Psychology of Number." Some of the special features of the
book may be noted :

1. The treatment of the subject is in strict line with the idea
of number as measurement a process, that is, by which the mind
makes a vague whole of quantity definite ; the mental sequence
being the undefined whole, the units of measure, the defined
whole. The book from the beginning stimulates and promotes
this spontaneous action of the mind. For rational method in
arithmetic, i.e. method in harmony with psychical development,
a true conception of number is a prime necessity ; a false idea of
number hinders the normal action of the mind, dulls perception
and reason, cultivates habits of inaccurate and disconnected atten-
tion; in a word, makes the subject all but worthless as a means
of mental discipline.

2. This true idea of number running through the whole work
establishes the unity of the whole. The unity of arithmetic is
the unity of the fundamental operations, and is the indispensable
condition of interest in the study. The book practically develops

* The Psychology of Number, preface, page xii. .



PREFACE v ii

and applies the true relation between addition and subtraction
(not "two" rules), between addition and multiplication (not
" identical ' operations), between multiplication and division
(psychologically not two much less four processes), and
between these operations and fractions. This leads to the true
idea of number, gives meaning to the primary operations, culti-
vates facility in their applications, and imparts to all operations
and processes a vitalizing interest.

3. The real meaning of the primary operations being practically
developed, fractions are divested of their traditional difficulty by
being placed in their true relation to " integers " ; requiring, in
fact, merely the conscious recognition of ideas which have from
the first been freely used. The ideas of multiplication and divi-
sion, which are implicit in all the fundamental operations, - in
the very idea of number become explicit in fractions. Every
" rule " in fractions has a meaning, and therefore has an interest
that leads to facile mastery.

4. The same remark applies to Greatest Common Measure,
Least Common Multiple, etc., the real significance of which is
rarely comprehended by the pupil. The clear conception of what
number is, gives clearness and interest to every principle and
process. This is true even of the "mechanical" operations.

5. As fractions are but an application of " The Simple Rules,"
especially multiplication and division (ratio), so percentage in all
its forms is but an application of fractions. The pupil is not
perplexed with numerous " rules and cases " ; he grasps the idea
of number as measurement, and all the " rules " referred to simply
require the use of the one underlying principle. In fact, the
pupil is constantly passing from ideas to principles and " rules,"
which he formulates for himself.

6. Great care has been taken in selecting and grading the
examples, many of which have been prepared specially for this
work. The typical solutions are clear and concise; there is no
perplexing verbiage under the name of " logical analysis."



v iii PREFACE

7. This treatment of arithmetic will prove a good preparation
for algebra. The main difficulty for the beginner in algebra is in
the symbols. He must learn to think in the language of algebra ;
a higher degree of abstraction is required. If, then, the symbols
and operations in arithmetic have little meaning for him, what is
his plight when he is confronted with the higher abstractions of
algebra ?

He passes from abstractions which have little meaning to
abstractions which have no meaning. The teacher is obliged to
lay a foundation which should have been laid in arithmetic. Too
often this foundation is never laid ; essential ideas are left vague,
and all advanced work shares in the vagiieness. For the highest
abstractions in mathematics are an evolution from the lowest.
On the other hand, to the student who lias grasped the true mean-
ing of number and arithmetical operations, the transition- -hardly
a transition is easy and sure. There is no unbuilding and re-
building to be done. Algebra becomes a thing of beauty and of
power, an effective factor in " unfolding the Laws of the Human
Intelligence."

8. The idea and method of the book has been tested in actual
work ; and if interest, not to say enthusiasm, on the part of both
teachers and pupils is any test, there is good reason for the
appearance of " The Public School Arithmetic."

9. This book will be followed in a short time by a Primary
book in which the measuring idea will be strictly carried out.
Meantime every method and every device given in the Primary
book is being tested in actual work.

We are indebted to J. C. Glashan's Arithmetic for some good
suggestions and problems.

It is recommended that starred chapters or parts of chapters be
omitted from the Grammar School Course.

JUNE, 1897.



CONTENTS

CHAPTER I

PAGB

DEFINITIONS 1

CHAPTER II
NUMERATION AND NOTATION 10

CHAPTER IH
ADDITION 20

CHAPTER IV
SUBTRACTION ........... 32

CHAPTER V
MULTIPLICATION 43

CHAPTER VI
DIVISION 59

CHAPTER VTI
COMPARISON OF NUMBERS 76

CHAPTER VIII
SQUARE ROOT 80

CHAPTER IX

GREATEST COMMON MEASURE AND LEAST COMMON MULTIPLE . 87

ix



x CONTENTS

CHAPTER X

PAGE

FRACTIONS 100

CHAPTER XI
DECIMALS 139

CHAPTER XII
COMPOUND QUANTITIES 157

CHAPTER XIII
PERCENTAGE 197

CHAPTER XIV
INTEREST 239

CHAPTER XV
RATIO AND PROPORTION 274

CHAPTER XVI
POWERS AND ROOTS .......... 288

CHAPTER XVII
MENSURATION 297

CHAPTER XVIII

METRIC SYSTEM 314

CHAPTER XIX
MISCELLANEOUS EXERCISE ... . 322



ARITHMETIC



CHAPTER I

DEFINITIONS

1. Arithmetic is the science of numbers and the art of com-
puting by them.

2. A Unit is a quantity regarded as a standard of reference
with which to compare or measure quantity of the same kind;
as $1, 1 five-dollar bill, 1 in., 1 lb., 1 two-ounce weight,
1 three-inch measure, 1 doz. eggs.

3. Number is the repetition of the unit to measure a given
quantity, i.e. number determines the hoiv much of the quan-
tity by determining hoiv many units make up the quantity.

4. The number by itself shows the relative value or the
Ratio of the quantity to the unit. Thus the ratio of the
quantity $6 to the unit $1 is the number 6; the ratio of
the quantity 4 (3 apples) to the unit 8 apples is the num-
ber 4. The number and the unit together show the absolute
magnitude of the quantity. If the quantity is measured by
the number 8 and the unit of weight 4 lb., then its absolute
magnitude is 8 times 4 lb. or 32 lb.

If I measure the length of a room with a two-foot measure
and find it to be 16 ft., the unit of length, 2 ft., has been



2 ARITHMETIC

repeated 8 times to measure the length of the room, and
the ratio of the length of the room to the unit of measure is
represented by the number 8.

If a box has been found to contain 30 doz. of eggs, the
unit, 1 doz. eggs, has been counted 30 times to measure the
quantity of eggs.

If I buy 3 qt. of milk, the milkman fills and empties the
unit of capacity, his one-quart measure, 3 times in order to
measure the quantity of milk.

5. In the above instances the wholes to be measured are :
the length of the room, the eggs in the box, and the milk
sold. The units of measure are 2 ft., 1 doz., and 1 qt. The
numbers telling how many of these units are needed are
8, 30, and 3.

The unit of measure may itself be, and in exact computa-
tion is, measured or compared with some other unit, for con-
venience called the primary unit ; that is, it may be a part or
a multiple of such unit.

6. $ 1, 1 in., 1 lb., are all primary units. A five-dollar
bill, a two-pound weight, a four-inch measure, are examples
of derived units, because the primary units, $ 1, 1 lb., 1 in.,
are repeated 5, 2, and 4 times respectively, to give the derived
units.

If a quantity of chestnuts be counted into groups of 4 or
5, the derived units, 4 or 5 chestnuts, and the number of
these groups or derived units measures the whole quantity
of chestnuts. Similarly, eggs counted by 4's and 6's, and
apples by 3's or 5's, are examples of the use of the derived
unit. The primary units are 1 egg and 1 apple, while 4 eggs,
6 eggs, 3 apples, and 5 apples are derived units.



DEFINITIONS 3

7. With reference to 1 ct., $ 1 is a derived unit, and 100
is the number expressing $1 in terms of the primary unit,
1 ct.

Similarly, 1 wk. is a derived unit, and 7 is the number
expressing 1 wk. in terms of the primary unit, 1 da.

In ^ ft., the primary unit of reference is 1 ft. The foot
is divided into 4 equal parts, one of which is the derived unit
of measure. The number 3 shows how many of these de-
rived units make up the given length.

Exercise 1

1. Name three units of length used to measure short distances,
and state the number of times each unit must be repeated to make
the next larger.

2. What unit of length is used in stating the distance between
two cities ?

3. Name instances in which 1 sec. is used as the unit of time.
1 min. 1 hr. 1 da. 1 mo. 1 yr. State how often each of these
units must be repeated to make the next larger.

4. What is the prime standard unit for money value? For
weight ? For area ? For length ? For time ? For volume ?

5. What unit of area is used to convey a definite idea of the
size of a farm ? Of a country ?

6. What unit is used to measure wood ?

7. With what unit of capacity is milk measured ? Kerosene ?

8. What unit is used to measure a quantity of strawberries?
Potatoes ? Why are these convenient units for the purpose ?

9. State at least three reasons why the bushel would be an
inconvenient unit to measure strawberries.

10. Early in the season strawberries are sold in pint boxes;
later, in quart boxes. Explain why different units of capacity
are chosen.



4 ARITHMETIC

11. Why is the pint box chosen as the unit to measure red
raspberries in preference to the quart ?

12. Name different quantities which are weighed and sold by
the Ib. By the oz. By the T.

13. Give instances in which the following units are used:
1 sheet, 1 quire, 1 doz.

14. In each of the following quantities name the units and
give the ratio of each quantity to its primary unit: 5 ft., 4 hr.,
6 sq. in., 7 qt., 365 da., 12 oz.

15. What is the quantity which contains the unit 4 times,
when the unit is 6 in. ? 9 hr. ? 8 yr. ?

16. If the unit is $4, and this unit is repeated 6 times, what
quantity will be produced ?

17. If the ratio of the size of a farm to the unit of area, 8 A.,
is equal to 6, what is the size of the farm ?

18. In the following examples, what are the quantities which
contain their respective units the given number of times ?

UNITS OF MEASUBE NUMBERS

$2 6

8 qt. 2

7 da. 3

5 hr. 4

1 doz. 16

4 in. 4

19. Name the primary units of measure, name in two ways
the derived units, and state the number of derived units which
measure these quantities : $ T ? Q-, ft., - yd., J- of a dime, -f- of a
wk., | of a da., f of a doz. eggs.

20. Name the coin which gives the derived unit of value in
each of the following : $ ft, $ , $ J, f f f, $ 10, $ 5, $ 20, f of a
nickel, A- dime, % of a quarter of a dollar, f of half a dollar, f of
half a dollar, ff of half a dollar,



DEFINITIONS 5

21. The following quantities contain their respective units
how often ?

QUANTITY UNIT

21 da. 7 da.

24 hr. 8 hr.

1 gal. 1 qt.

1 min. 1 sec.

10 dimes 2 dimes

$ 18 worth of hats $3 for 1 hat

30 ct. worth of rnilk 6 ct. a qt.

22. What is the unit of measure in reckoning population?
How many of these units give the population of the town in
which you live ?

23. What is the number of times each of the following units
must be repeated to make the next higher unit : 1 in., 1 ft.,
1 ct., 1 dime, 1 da., 1 hr., 1 qt. ?

24. How many times must the following units of measure be
repeated to make 3 ft. : 2 in., 3 in., 4 in., 6 in., 9 in., 12 in. ?

25. State how each of the following units may be derived from
the next higher: 1 ft., 1 in., 50 ct., 25 ct., 1 da., 1 min., 1 qt.,
and 1 pt.

26. A quantity of cherries is measured by using as the unit
as many cherries as will fill a dish holding 3 qt. ; 9 of these
dishes are filled. How many qt. are there in the whole
quantity ?

27. At 30 bu. to the A., how many bu, would there be on
10 A. ? What is the unit here ? What gives this particular
unit ? If 10 bu. to the A., how many A. to produce an equal
quantity ?

28. What coin is equal in value to 10 of the unit 1 ct. ? 100
of the unit 1 ct. ? 10 of the unit 1 dime ?

29. What coin is equal to 10 of the unit 1 nickel? 5 of the
unit 1 nickel ? 20 of the same unit ?



tf ARITHMETIC

30. How many ct. are there in 5 of the unit $1? 6 of the
unit $ 1 ? 10 of the unit $ 1 ?

31. What coin is equal in value to 500 of the unit 1 ct. ?
100 of the unit 1 nickel ?

32. What coin is equal to one-tenth of a five-dollar gold piece ?



8. In the preceding diagram we have 36 dots, signifying
36 units of any kind, arranged in 4 rows of 9 dots each, and at
the same time 9 rows of 4 dots each. Hence we think of 36
as equal to 4 times 9 or 9 times 4. Arrange the dots to show
that 36 is equal to 3 x 12 or 12 x 3, and also to 2 x 18 or 18 x 2.

4 and 9 are called factors of 36, and 36 is called the
product of 4 and 9.

This illustrates a law of great importance in Arithmetic.

Thus we think of 24 as equal to 2 x 12 or 12 x 2, 3 x 8 or
8 x 3, 4 x 6 or 6 x 4.



9. If in the above arrangement we think of each dot as rep-
resenting $1, then the diagram shows that $12 -=- $2 = 6.

What other measurement is shown by the same arrange-
ment ?



Exercise 2



1. Arrange 30 dots in rows in as many ways as you can, and
express the results as in the preceding paragraph.

2. Express the following numbers as products in two ways:
6 (i.e. 2 x 3 or 3 x 2), 8, 10, 15, 21, and 35.



DEFINITIONS 7

3. Express the following numbers as products in as many ways
as possible, and arrange the products in corresponding pairs : 12,
16, 20, 28, 42, and 60.

4. Give all the factors of 32, 40, and 48.

5. Place dots to show the measurement of $ 20 by a $ 5 unit.
What other measurement does it show ?

6. Place dots to show the measurement of $24 by a $2 unit;
a $ 3 unit ; a $ 4 unit. AVhat other measurements are shown ?

7. What is the price of 6 yd. of cheese-cloth at 8 ct. a yd.?
Explain each of these statements :

The whole price = 6 (8 ct.).
The whole price = 8 (6 ct.).

8. Show that 10 units of $ 5 each is equal to 5 units of $ 10
each.

9. If a line 3 ft. long is repeated 6 times to measure the
length of a room, how long is the room ? How often would a
line 6 ft. long have to be repeated to measure the room ?

10. How many apples will be required to make 7 rows with
9 apples in each row ? What is the unit of measurement ? What
other convenient unit might be used ? What would be the ratio
of the whole quantity to this unit ?

11. Draw a straight line 36 in. long, cut strips of paper
respectively 6 in., 7 in., 8 in., 9 in., 10 in., 11 in., and 12 in.
long. Measure along the line 3 times with each of these strips
of paper.

Use a yardstick divided into in. to measure your results, and
prove the following :

3 x 6 in. = 18 in. ; 3 x 9 in. = 27 in. ;

3 x 7 in. = 21 in. ; 3 x 10 in. = 30 in. ;

3 x 8 in. = 24 in. ; 3 x 11 in. = 33 in. ;

3 x 12 in. = 36 in.



g ARITHMETIC

12. Draw a line 36 in. long. Make a 3-in. measure. Measure
along the line respectively 6, 7, 8, 9, 10, 11, and 12 times, and
prove that :

6 X 3 in. = 18 in. ; 9x3 in. = 27 in. ;

7x3 in. = 21 in. ; 10 x 3 in. = 30 in. ;

8x3 in. = 24 in. ; 11 x 3 in. = 33 in. ;

12 x 3 in. = 36 in.

13. What quantities are measured by the following: 6x4
in., 4x6 in., 5 x $ 8, 8 x $5, 10 x 5 pears, 5 x 10 pears ?

14. What two units of length each longer than 4 in. can be
used to measure a line 35 in. long ? State in each case the ratio
of the length of the whole line to each unit.

15. What are the convenient units of money to pay a debt
of $35? |80?

16. What are convenient units to pay debts of 75^ ?
34^ ? 87^ ?

17. A fruit dealer sells apples at the rate of 3 for
What is the unit to measure his apples ? What is the unit of
value ? How many units are there in 24 apples ? What is their
selling price ?

18. Bananas are sold at the rate of 4 for 5^. What is the
measuring unit for the bananas ? What is the unit of value ?
How many units of value in 20 bananas ? 36 bananas ? What
is their value ?

19. Oranges are sold at 20^ a doz. What are the two meas-
uring units ?

20. If A can do a piece of work which is represented by 36
units in 9 da., how much will he do in 1 da. ?

21. If a piece of work is represented by 60 units and A can do
5 units in one da., in how many da. can he do the entire work ?

A B o D E F

I i | i i i

22. AB is a line which represents any primary unit of measure,
and AC, AD, AE, and AF are derived units. What part is the



DEFINITIONS 9

primary unit AB of each of the derived units ? What is the
ratio of each derived unit to the primary unit ? If AF is the pri-
mary unit, what would AB be ? What is the ratio of the derived
unit AF to the derived unit AD ? Of AD to AF? Of AC to
AE? OfAEtoAC? Of AF to AC? Of AC to AF?

23. One field contains 7 units of area, and a second field con-
tains 9. What is the ratio of the area of the first field to the
second ? Of the second field to the first ? Illustrate by a drawing.

24. The distance from A to B is divided into 5 parts of 3
mi. each, and that from A to C into 6 parts of 3 mi. each. What
is the ratio of the distance AB to AC? Of AC to AB? Illus-
trate your answer by a diagram.

25. The money in my purse is measured by the number 8 and
the unit $5. I owe a debt measured by the number 6 and the
unit $ 5. What is the ratio of the debt to the money in my
purse ? What is the ratio of the money in my purse to the debt ?
How much shall I have left after paying my debt ?

26. If the amount of work required to dig a trench 800 yd.
long is represented by 40 units, what does 1 unit represent ?



CHAPTER II

NUMEKATION AND NOTATION

10. Numeration is counting, or the expression of number
in words.

The ordinary system of numeration is the Decimal System,
so called because it is based on the number ten.

11. The names of the first group of numbers in regular
succession are: one, two, three, four, five, six, seven, eight,
nine.

Other number-names are : ten, hundred, thousand, million,
billion, trillion, etc.

12. The number one applied to any unit denotes a quantity
which consists of a single unit of the kind named.

The number two applied to any unit denotes a quantity
which consists of one such unit and one unit more.

The number three applied to any unit denotes a quantity
which consists of two such units and one unit more.

And so on with the numbers four, five, six, seven, eight,
nine ; applied to any unit they denote quantities increasing
regularly by one such unit with each successive number.

13. The number next following nine is ten, which applied
to any unit denotes a quantity consisting of nine such units
and one unit more.

Counting now by ten units at a time, as before we counted

10



NUMERATION AND NOTATION 11

by single units, we get the numbers ten, twenty, thirty, forty,
. . ., ninety.

The names of the numbers between ten and twenty are, in
order : eleven, twelve, thirteen, fourteen, . . ., nineteen.

The names of the numbers between twenty and thirty,
thirty and forty, . . ., are formed by placing the names of
the numbers one, two, three, . . ., nine, in order after
twenty, thirty, . . ., ninety.

14. The number hundred applied to any unit denotes a
quantity which consists of ten ten-units.

Counting now by a hundred units at a time, as before we
counted by single units, we get the numbers one hundred,
two hundred, . . ., nine hundred.

The names of the numbers between one hundred and
two hundred, two hundred and three hundred, . . ., are
formed by placing the names of the numbers from one to
ninety-nine in regular succession after one hundred, two
hundred, . . ., nine hundred.

15. The number thousand applied to any unit denotes a
quantity which consists of ten hundred-units.

Counting now by a thousand units at a time, as before we
counted by single units, we get the numbers one thousand,
two thousand, . . ., nine thousand, ten thousand, eleven
thousand, twelve thousand, . . ., twenty thousand, . . .,
one hundred thousand, . . ., two hundred thousand, . . .,
nine hundred and ninety-nine thousand.

The names of the numbers between one thousand and two
thousand, two thousand and three thousand, . . ., are formed
by placing in order the names of the numbers from one to
nine hundred and ninety-nine, the numbers preceding a



12 ARITHMETIC

thousand, - - after one thousand, two thousand, . . ., nine
hundred and ninety-nine thousand.

16. The number million applied to any unit denotes a
quantity which consists of a thousand thousand-units.

The number billion applied to any unit denotes a quantity
which consists of a thousand million-units.

The number trillion applied to any unit denotes a quantity


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Online LibraryJ. A. (James Alexander) McLellanThe public school arithmetic : based on McLellan & Dewey's Psychology of number → online text (page 1 of 21)