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ENTARY

AND BROOKS

IN MEMORIAM

FLOR1AN CAJORI

EATIONAL

ELEMENTARY ARITHMETIC

GEORGE W. MYERS, Ph.D.

PROFESSOR OF THE TEACHING OF MATHEMATICS AND ASTRONOMY

SCHOOL OF EDUCATION, THE UNIVERSITY OF CHICAGO

AND

SARAH C. BROOKS

PRINCIPAL OF THE TEACHERS' TRAINING SCHOOL, BALTIMORE, MARYLAND

CHICAGO

SCOTT, FORESMAN AND COMPANY

1906

COPYRIGHT 1898, 1899, 1905

BY SCOTT, FORESMAN AND COMPANY

BOBT. O. X<AW CO.. nUiraVKB AND H1NDHUH, OHIOA8O.

PREFACE.

Many and substantial gains have been made in

GENERAL. the last decade in both theory and practice in elemen-

tary education, and arithmetic has gained largely from

this general advance. We have learned much of late as to practical

ways of securing in the teaching of the elements of mathematical science

the larger and the more significant educational aims strengthening

the judgment and the will, the power to think and to do.

The writer of an arithmetic who cuts himself off from these ad-

vances and satisfies himself with mere "figuring/' or the mere meeting

of an existing "consensus," cheapens his effort by denying the higher

purposes of education and renounces his opportunity to help even

the makers of programs by pointing the direction whence improve-

ment must come. Without neglecting either of these two important

requirements, the authors of this book have striven to make the arith-

metic work more thoughtful, and even the drill purposeful and

educative.

There is no school subject in which foreshortened

PLACE FOR views and distorted perspective work more harm than

FORMAL in elementary mathematics. Children as well as adults,

STUDY. learn new ideas by meeting them first in simple forms,

intermingled with familiar ideas arid fairly well-under-

stood uses of the new ideas.

After a little, the new idea makes itself felt as something new.

This is the time to differentiate it for formal study, to learn what it

really is. This is the stage for the study of process and for drill

enough to fix it and to make its use easy and facile.

The learner then desires to experience the added power the

mastery of the process has given him, and this calls for the applica-

tion stage. The treatment of new ideas, processes and topics in this

book is accordingly arranged on this three-fold plan of (1) its in'

iii

it PREFACE

formal use, (2) its formal study, and (3) its application. Examples

of this plan may be seen in the teaching of the tables.

The arrangement of number work for the grades

ORDER OF must be in accordance with the natural unfolding of

DEVELOP- the child's mind. Too often this important fact is

MENT. lost sight of in the logic of the subject itself. Strictly

speaking there can be no contradiction between the

demands of the child's mental development and the logical require-

ments of the subject. It is nly wLen 1 gic is construed to mean the

procedure of adult mind that the .emends ~f logic become mischievous

in the elementary school. Rightl^ understooa, logic means the nat-

ural procedure of the learning mind in mastering a subject. The

recent work of experimental psychologists has proved conclusively

that one law of human development is that what is best for the learner

at a given stage of his development is also best for him ultimately.

In the language of biological science, what is best for the tadpole

while he is a tadpole is also best for him when he is a frog.

This doctrine, now generally accepted by all students of educa-

tion, has done much toward the general accrediting of childhood at

its true worth and has given flat and final denial of the right to quarrel

with the child because he is not something else, by attempts of the

teacher to force upon him the logic of the adult. This modern doc-

trine is also accepted by this book and it is believed it has unified

the interests of logic with those of the child by making its logic the

logic of the learner at the stage he has reached.

Two things must be carefully provided for in an

MOVEMENT arithmetic for children. There must be continuous

AND UNITY. and progressive movement through the subject and

there must be Organic unity of parts. Too often

sequence of processes and grading f exercises have wrought havoc

with the essential integrity of arithmetic as a whole. The child looks

through a narrow slit at the passing column, but fails to see the proces-

sion as a whole or even any important part of it. Divers attempts

merely to make arithmetic easy, to make play of what must be work

and work of what ought to be play, to tell number stories when there

PREFACE v

is no story, which have gotten into recent texts, are open to the charge

of having lost the unities of the subject in a confusion of irrelevant

not to say remote associations.

Other writers, in the struggle for scientific unification and pro-

portion of parts, have secured a sort of unity and balance at a great

cost of practicability. The latter is a danger that leans toward

"virtue's side," but the danger is real and serious and must be avoided.

It is largely avoided by teaching nothing not even a partial notion

which is sometimes necessary in such a way at one stage that it

must be unlearned by the pupil at a later stage. How to secure both

economic movement and essential unity has been the most earnest

struggle of the authors of the RATIONAL ARITHMETICS. The task has

not been easy and the teaching public will say how well they have

succeeded.

The ideas of number and of the numerical proc- I

MATERIAL. esses must be derived from the concrete. Form and

number are the two main developments of quantity^

The process of numbering in its varied aspects is very closely paral- j

leled in the physical world by the process of measuring in its varied !

applications. This does not imply that numbering and measuring

are one and the same process, or set of processes. What it does imply

is that numbering is the mental side of the same problem of adjust-

ment of activity that has its physical expression in measurement. It

means that measurement is the most direct and certain route to cor-

rect notions of number, for one who has not yet acquired them.

That the theory of number has no necessary connection with

measurement is witnessed by the fact that perfectly sound and ade-

quate theories, constructed altogether on the basis of counting, are

familiar to all students of advanced mathematics. The continuum

of quantity can be constructed with perfect logical rigor out of the

elementary number discreta. But this mode of evolution of the num-

ber system and processes is neither the more economical nor the more

fruitful mode for the immature learner. The physical acts of the

measuring processes run so closely parallel to the mental acts of the

numbering processes, the one-to-one correspondence of steps in the

vi PREFACE

processes is so complete and so natural, that the child glides succes-

sively and with perfect ease and certainty into and through correct

ideas of the unit, of the assemblage, of times, of parting and dividing,

of fractioning, of ratio and proportion, and of valuation, to the idea

of continuous positive and negative magnitude and number. The chief

advantage to the child of making measurement fundamental to number

is that motor, tactual, auditory, and visual sensations serve powerfully

to re-enforce and to sharpen number impressions at every turn The

whole being thus engages in the struggle for mastery of the difficulties

of number and number process "The whole boy goes to school."

Without abating energy on the symbolical phases and processes, this

book seeks through joining, dividing, and measuring lines, surfaces,

and solids, weights, etc., to fill symbols with meaning and processes

with purpose.

The appeal to the concrete involves the use of certain materials,

and any book based on this method of instruction must provide

these within its pages. This the RATIONAL ELEMENTARY ARITHME-

TIC does in great variety; e.g., lines, squares, and surfaces, cubes,

and solids, pp. 1, 2, 4, 12, 20, 60, 100, 198, 204, etc., the construction

of figures for linear and square measure on pp. 2, 4, 173, and 184,

the visual presentation of fractions on pp. 158, 159, 161, 252, and 256.

By means of these the child is also led to make use of other material

which will aid him in grasping the principles involved. >He learns to

seize quantitative questions by the handle.

There are numerous lists of problems involving

USE OF real measurement, and incidentally also counting at its

MATERIAL. best. These lists are carefully graded and the teacher

is urgently recommended at all times to have pupils

solve all they can orally. The pencil and paper should be used only

when the difficulties of the problem make it too hard for the pupil

orally. Different pupils will show very different degrees of aptitude

for rapid oral work No plan of isolating the oral from the written

work can suit the varying needs of different pupils, and every pupil

has a right to the best sort of training of which he is capable. The

problems of life are handled in this way and the pupil should early

PREFACE vii

form the habit of using his head as much as possible and his pencil

only as an aid to his head.

It is also recommended that teachers follow the

CHOOSING . . . .. 11 i

practice of having pupils work rapidly through many

of the lists of problems, indicating the processes called

AND I** i i

tor and giving and recording estimates of about what

FORMING * t 7 f\ .

the answers must be, before any figuring is done.

Then the problems should be worked through and the

correct results compared with the estimates. This work

is of high value as training of judgment and as aiding the pupil to know

when, as well as how, to add, subtract, multiply, or divide.

One of the gains that accrue from basing number

THE TABLES, teaching on measurement is that by associating parts

of the multiplication table with certain facts of the

denominate number tables, both sets of facts may be learned at once,

and more easily than either may be learned alone. For example, the

2's may be based on the fact, 2 pt. = l qt.; the 3's on 3 ft.= l yd.;

the 4's on 4 gi. = l pt.; 4 pk.= l bu.; 4 qt. = l gal.; the 5's on 5c~

1 nickel, 5 nickels =1 quarter, on the divisions of the clock-face, the

number of school days in a week, etc. In this way the most common

denominate number facts are learned and at the same time they fur-

nish a concrete background and purpose for learning the multiplica-

tion table. The space idea is utilized in building the tables, so that

the products of the multiplication tables are abo seen to be mensu-

ration facts. This incidental use of form is of no mean value. It

is indeed one great reason why we need a multiplication table.

The problems from beginning to end deal with

PROBLEMS. realities, and appeal strongly to the child's environ-

ment and experience. This gives to the work a genuine

interest which cannot otherwise be secured, and cultivates the powers

of observation and of inference, and the ability to apply numbers to

the situations of every-day life and experience, and at the same time

strengthens the ability to see and make problems on one's own account

from the raw materials of quantitative situations. Every new idea

comes to the child in the form of a problem, and through the agency

viii PREFACE

of real problems he can be most easily made to feel the genuineness

and usefulness of arithmetical knowledge. The RATIONAL ELEMEN-

TARY ARITHMETIC works out the elementary ideas of arithmetic

through the agency of real problems.

The work of the first 5 pages includes a review and

THE PLAN summary of the 2's and 3's, which pupils will have

OF THE learned before the third grade. Then follow the intro-

BOOK. " duction of halves and fourths through a study and use

PART i. of the foot-rule, then the teaching of the tables, begin-

ning with the 4's, through the use of those standards of

measure which call for the several factors to be taught and tabulated,

and largely by aid of the mensurational facts involving these factors,

see pp. 11, 12, 19, 20, 49, 50, etc. The tables are developed through

the use of the 4, 5, 6, etc., as multiplicand, because this is the most

natural way to teach them, but in the final form for reference the 4, 5,

6, etc., are put into the relation of multiplier. In consequence of the

two-fold aspect of every number, the unit and the multiplicity (the

times), the products are most easily built up and grasped by keeping

the more difficult element, the multiplicity, as small as possible. One

6, two 6's, etc., are much easier than 6 ones, 6 twos, etc. Thus the

6's , for example, are first developed in the form

1X0= G

2X6=12

3X6=18

etc.

But these products are, speaking strictly, not the 6's at all, since the

6's are those products in which 6 enters as a multiplier. So that for

final reference the table is written in the form of The Sixes, thus

6X1= 6

6X2=12 (Seep. 61)

6X3=18

etc.

PREFACE ix

The same system is followed in the rest of the tables, up to and includ-

ing the 10's in Part I. Many uses of the 11 's and 12's are given in

Part I, but the final treatment and formulation of the ll's and 12's are

reserved for Part II. Ample drills, short and frequent, are given to

fix and make flexible the tabular facts taught.

Part II includes the final treatment of the ll's and

PART n. 12's, easy scale-drawing, further use of denominate

number tables, the teaching of the four fundamental

operations, many unified lists of problems, promiscuous as to process

but unified as to some central and interesting thought. See pp. 104,

105, 121, 132, 133, 146, 167, 168, etc., and considerable easy work

calling for simple fractions. The demands of form are met through

many useful problems in the mensuration of simple figures.

The last part of the book carries forward through

PART m. the fifth ffrade the application of the fundamental proc-

esses to an extended range of practical problems

calling often for more than one process , the development of the prod-

ucts by 12, 15, 16, and other factors of frequent occurrence in

business and the industries; it includes the mensuration of boxes,

scale-drawing, a fairly complete treatment of common and decimal

fractions, denominate numbers, together with percentage and interest.

These latter features will commend themselves to teachers who de-

sire to do as much as possible for the boys and girls who are unable

to go on with their school work beyond the fifth year.

Exercises and problems for practice are to be

DRILL. found frequently throughout the book and they every-

where stand in organic relations to what precedes

and follows. This book does not "teach by drill," but everywhere

puts the drill feature in such relations to interesting and useful work

as to make the drill both purposeful and interesting. Drill should

always be for the purpose of fixing facts that have already been

taught and for the sake of establishing right habits of work. Even

in the army the manual of arms and the manoeuvres are first care-

fully taught and the drill is kept up to make correct procedure

habitual and natural. Something like this is the correct mode in

x PREFACE

number teaching. Judicious and intelligent drill constitutes an im-

portant feature of this book. The drill-master must live up to his

reputation of being a master of drill.

A fairly well-rounded elementary treatment of

FRACTIONS, the essentials of common and decimal fractions, of

PERCENTAGE, percentage, and of interest is given at the close of

INTEREST. Part III. The attempt has been to give only a fairly

complete first view, rather than fullness and finality.

As a means of recapitulating and bringing out more clearly the con-

nections and meaning of the fundamentals than could be done in

the earlier grades, this will be recognized at once as a feature no

less valuable to the pupil who goes on to the more mature work of

the grammar school, which builds on these fundamentals, than to

the pupil who is forced to leave school at the end of the fifth year,

and is very considerably benefited by a knowledge of these practi-

cal subjects.

The pleasant task now remains to the authors to make acknowl-

edgment of their indebtedness to many superintendents, principals,

and teachers who have assisted in perfecting the book by suggestions

and corrections both in the manuscript and the proofs. To Mr. F.

W. Buchholz, Professor of Mathematics in the Chicago Normal

School, the authors are under especial obligations for the great pains

and efficient service so kindly bestowed upon the proofs. His insight

and broad experience as a mathematical teacher and supervisor have

wrought wholesale improvement in the form, no less than in the sub-

stance of this book.

Chicago, Sept,, 1905. THE AUTHORS.

CONTENTS.

PART I.

MEASURING LENGTH 3

SURFACE MEASURE-MULTIPLES 5

ADDITION AND SUBTRACTION REVIEWS 7

TABLE OP FOURS.

USES, LIQUID MEASURE 9

USES, DRY MEASURE 11

BUILDING TABLE. 13

APPLICATION, WEIGHT 15

TABLE OP FIVES.

USES, U. S. MONEY 17

USES, TIME 21

BUILDING TABLE . .... 23

REVIEW PROBLEMS 25

EXERCISES FOR PRACTICE 27

FUNDAMENTAL OPERATIONS.

ADDITION 29

SUBTRACTION 33

MULTIPLICATION 37

DIVISION 41

LAUNDRY BILLS 45

DRAWING TO A SCALE 47

TABLE OF SIXES

USES, LINEAR MEASURE 49

BUILDING TABLE 51

APPLICATION 58

BOX MAKING 55

SCHOOL GARDEN 57

TABLE OF SEVENS.

USES, TIME 59

BUILDING TABLE 61

APPLICATION 63

AREA OF TRIANGLES 65

TABLE OF EIGHTS.

USES, DRY MEASURE 67

BUILDING TABLE 69

APPLICATION 71

MEASURING SOLIDS 73

BUILDING SOLIDS 75

CONTENTS OP BOXES 77

xl

xii CONTENTS

REVIEW. 79

TABLE OF NINES.

USES, SQUARE MEASURE 81

BUILDING TABLE 83

DRILL ON NINES 85

APPLICATION 87

WEIGHT 89

TABLE OF TENS.

BUILDING TABLE 91

APPLICATION, U. S. MONEY 93

FUNDAMENTAL OPERATIONS IN U. S. MONEY ; 95

USES OF ELEVENS AND TWELVES. 97

COST OF MEALS 99

PART II.

MEASURING A FLOWER GARDEN 101

PURCHASES AND WAGES 103

PROBLEMS ON CANDY RULES 105

TABLE OF ELEVENS.

BUILDING TABLE AND DRILL 107

TABLE OF TWELVES.

USES, LINEAR MEASURE 109

BUILDING TABLE Ill

APPLICATION 113

FIVES, SIXES, SEVENS, AND TWELVES 115

WRITING AND READING NUMBERS 117

THREES, SIXES, NINES, ELEVENS, TWELVES 119

SCALE DRAWING OF TILING 121

TWOS, FOURS, AND EIGHTS 123

MEASURES OF WEIGHT 125

THE DOZEN 127

FUNDAMENTAL OPERATIONS.

EXERCISES IN ADDITION 129

PROBLEMS IN ADDITION 131

FURNISHING A HOME 133

PROBLEMS IN ADDITION 135

EXERCISES IN SUBTRACTION 137

PROBLEMS IN SUBTRACTION 139

DRILL IN FUNDAMENTAL OPERATIONS. 141

MULTIPLICATION 143

PROBLEMS IN MULTIPLICATION 145

DIVISION LONG. 149

PROBLEMS IN DIVISION. 151

EXERCISES IN DIVISION 153

PROBLEMS OF SALE 155

MISCELLANEOUS PROBLEMS 157

FRACTIONS 159

MIXED NUMBERS 161

PROBLEMS IN FRACTIONS 168

MISCELLANEOUS PROBLEMS 165

PROBLEMS IN TIME 107

BUYING GROCERIES.... 169

CONTENTS xiii

EXERCISES FOR PRACTICE 171

MEASURES OF LONG DISTANCES.. 173

MEASURES OF DISTANCE 175

TWELVE AND A HALF 179

TIME AND DISTANCE 181

REVIEW PROBLEMS 183

PART III.

MKASURING SURFACES 187

APPLICATION OF SQUARE MEASURE 189

PROBLEMS OF DIVISION 191

PROBLEMS IN VALUES RATIO 193

PRODUCTS BY 14 195

TRIANGLES 197

AREAS 199

MISCELLANEOUS PROBLEMS 201

PROBLEMS IN TIME 203

SOLIDS AND CAPACITY 205

TABLE OF CUBIC MEASURE 209

PRODUCTS BY 15 211

APPLICATIONS OF LIQUID MEASURE. 213

PRODUCTS BY 16 215

APPLICATIONS OF DRY MEASURE. 217

PRODUCTS BY 16% 219

READING AND WRITING NUMBERS 221

EXERCISES IN ADDITION AND SUBTRACTION 223

PROBLEMS IN MULTIPLICATION , 225

PROBLEMS IN DIVISION 227

PRODUCTS BY 20 229

PROBLEMS IN WEIGHING .'.... 231

PROBLEMS IN TIME 233

MISCELLANEOUS PROBLEMS 235

COUNTING PAPER 237

MISCELLANEOUS PROBLEMS 239

DENOMINATE NUMBER EXERCISES 243

DENOMINATE NUMBER PROBLEMS 245

BILLS AND ACCOUNTS. 247

PROBLEMS IN FRACTIONS 251

EXERCISES IN FRACTIONS 253

PROBLEMS IN FRACTIONS 255

ADDING AND SUBTRACTING FRACTIONS 257

MULTIPLYING AND DIVIDING FRACTIONS 259

DECIMAL FRACTIONS 261

MULTIPLYING DECIMALS

PERCENTAGE 265

PROBLEMS IN INTEREST .'. 269

PRINCIPLES OF INTEREST 273

SCALE DRAWING OF MANTEL ^

TABLES OF WEIGHTS AND MEASURES 277

PART FIRST.

O

A boy made a cardboard house, one end the size of this drawing.

Without measuring write what you think are these distances:

1. The width. The greatest height. Length of the slant of

the roof. Height to the lowest point of the slant.

2. Width and height of the lower window and of the door.

3. The distance between the right side of the house and the

door; between the door and the lower window; between

the lower window and the left side of the house; be-

tween the sill of the lower window and the floor.

4. Measure each of these distances with your ruler and com-

pare them with what you thought they would be.

l

RATIONAL ELEMENTARY ARITHMETIC.

E

FG

1. Find the lines A, B, C, D, E, F, G.

2. Which is longer, A or B? C or D? D or E? E

or F? F or G?

3. How many lines like A will make one like C? B?

E? D? F? G? F and G?

4. Measure each line with your ruler.

5. How many inches long is A? B? C? D? E? F? G?

6. How many inches long are A and B together?

7. A, B, and C together are as long as which line?

8. F and G are together how many inches long?

9. What is the name for the measure which is as

long as F and G together?

10. How many inches are there in one foot?

11. What is a measure 3 feet long called?

12. Hold your hand a yard from the floor.

13. How many feet high are you? How many yards

high? How many inches high?

14. How many inches do you measure around your

chest?

15. Stretching your arms straight sidewise, how many

yards and inches is it from the tips of the rin-

gers of one hand to those of the other?

16. How many feet can you reach upward from the

floor when standing close to the wall?

MEASURING LENGTH. 3

1. How many inches long is the front edge of your desk?

2. How many inches high is your desk?

3. How many feet high is a window sill in your schoolroom?

4. How many feet wide is the window?

5. Your schoolroom is how many feet long? How many

yards long?

6. Your schoolroom is how many feet wide? How many

yards wide?

7. How long, in feet, is one end of your schoolroom? How

long are the two ends? How long is one side? How

long are the two sides? How many feet are there around

the walls of your schoolroom?

8. Measure a strip of the blackboard 3 yards long; 5 feet long;

15 inches long; 2 yards 1 foot long; 2 feet 10 inches long.

9. Step off or walk 5J yards.

10. How many inches in one foot? In 2 feet? In 3 feet?

TABLE

1 yard = - feet?

1 foot = inches?

11. How many feet in 36 inches? In 24 inches?

12. A blackboard 6 feet long is how many yards long?

13. A box that is 2 yards long is how many feet long?

14. A room 12 feet wide is how many yards wide?

15. 3 and 2 are 5 \

or 2 added to 3 equals 5 ( Thege ftll haye the game mea ning.

3 plus 2 equals 5 I

3 + 2 = 5

16. 3 + 4=? 6 + 2=? 4+2=? 6+1 = ? 5+2=?

8+1=? 3+5=? 5+4=? 7+2=? 4+3=?

17. Add:

6 pounds 8 dollars 6 feet 6 cents 7 yards 7 bushels

5 pounds 2 dollars j? feet 4^ cents 2_ yards 5_bushels

RATIONAL ELEMENTARY ARITHMETIC.

SURFACE MEASURE MULTIPLES.

1. Draw the line a one inch long. Draw

b, c, and d as in the figure A.

2. What kind of a figure have you made?

3. How long is a? bf c? df How long,

then, is each side of the square?

Then what kind of a square may it

be called?

4. How much space (surface) does A cover?

5. Find a surface (space) on page 4 which is equal to A.

6. On page 4, find a surface equal to two times A. What is

it marked? How long is it? How wide? What kind of

a figure is it? How many square inches are in it, or,

what is its area? Draw an oblong the same size as B.

7. Draw an oblong as long as A and B (page 4) together.

8. How long is the oblong (rectangle) you have drawn?

How wide? What is its area?

9. Find on page 4 a rectangle (oblong) the same size as the

one you have just drawn.

10 On page 4, A and C together form a rectangle how long?

How wide? What is its area?

11. Make a square equal to A and C together. What kind

of a square is it called?

12. 2 twos equal 4 \

or 2 multiplied by 2 = 4 I Thege ^ haye thft game meaning<

2 times 2 = 4 I

2X2 = 4

Which is the shortest way to make this statement?

13. 2X3=? 2X7=? 2X 9=? 2X11=? 3X 9=?

AJRi A l,

!' '"' ;

ENTARY

AND BROOKS

IN MEMORIAM

FLOR1AN CAJORI

EATIONAL

ELEMENTARY ARITHMETIC

GEORGE W. MYERS, Ph.D.

PROFESSOR OF THE TEACHING OF MATHEMATICS AND ASTRONOMY

SCHOOL OF EDUCATION, THE UNIVERSITY OF CHICAGO

AND

SARAH C. BROOKS

PRINCIPAL OF THE TEACHERS' TRAINING SCHOOL, BALTIMORE, MARYLAND

CHICAGO

SCOTT, FORESMAN AND COMPANY

1906

COPYRIGHT 1898, 1899, 1905

BY SCOTT, FORESMAN AND COMPANY

BOBT. O. X<AW CO.. nUiraVKB AND H1NDHUH, OHIOA8O.

PREFACE.

Many and substantial gains have been made in

GENERAL. the last decade in both theory and practice in elemen-

tary education, and arithmetic has gained largely from

this general advance. We have learned much of late as to practical

ways of securing in the teaching of the elements of mathematical science

the larger and the more significant educational aims strengthening

the judgment and the will, the power to think and to do.

The writer of an arithmetic who cuts himself off from these ad-

vances and satisfies himself with mere "figuring/' or the mere meeting

of an existing "consensus," cheapens his effort by denying the higher

purposes of education and renounces his opportunity to help even

the makers of programs by pointing the direction whence improve-

ment must come. Without neglecting either of these two important

requirements, the authors of this book have striven to make the arith-

metic work more thoughtful, and even the drill purposeful and

educative.

There is no school subject in which foreshortened

PLACE FOR views and distorted perspective work more harm than

FORMAL in elementary mathematics. Children as well as adults,

STUDY. learn new ideas by meeting them first in simple forms,

intermingled with familiar ideas arid fairly well-under-

stood uses of the new ideas.

After a little, the new idea makes itself felt as something new.

This is the time to differentiate it for formal study, to learn what it

really is. This is the stage for the study of process and for drill

enough to fix it and to make its use easy and facile.

The learner then desires to experience the added power the

mastery of the process has given him, and this calls for the applica-

tion stage. The treatment of new ideas, processes and topics in this

book is accordingly arranged on this three-fold plan of (1) its in'

iii

it PREFACE

formal use, (2) its formal study, and (3) its application. Examples

of this plan may be seen in the teaching of the tables.

The arrangement of number work for the grades

ORDER OF must be in accordance with the natural unfolding of

DEVELOP- the child's mind. Too often this important fact is

MENT. lost sight of in the logic of the subject itself. Strictly

speaking there can be no contradiction between the

demands of the child's mental development and the logical require-

ments of the subject. It is nly wLen 1 gic is construed to mean the

procedure of adult mind that the .emends ~f logic become mischievous

in the elementary school. Rightl^ understooa, logic means the nat-

ural procedure of the learning mind in mastering a subject. The

recent work of experimental psychologists has proved conclusively

that one law of human development is that what is best for the learner

at a given stage of his development is also best for him ultimately.

In the language of biological science, what is best for the tadpole

while he is a tadpole is also best for him when he is a frog.

This doctrine, now generally accepted by all students of educa-

tion, has done much toward the general accrediting of childhood at

its true worth and has given flat and final denial of the right to quarrel

with the child because he is not something else, by attempts of the

teacher to force upon him the logic of the adult. This modern doc-

trine is also accepted by this book and it is believed it has unified

the interests of logic with those of the child by making its logic the

logic of the learner at the stage he has reached.

Two things must be carefully provided for in an

MOVEMENT arithmetic for children. There must be continuous

AND UNITY. and progressive movement through the subject and

there must be Organic unity of parts. Too often

sequence of processes and grading f exercises have wrought havoc

with the essential integrity of arithmetic as a whole. The child looks

through a narrow slit at the passing column, but fails to see the proces-

sion as a whole or even any important part of it. Divers attempts

merely to make arithmetic easy, to make play of what must be work

and work of what ought to be play, to tell number stories when there

PREFACE v

is no story, which have gotten into recent texts, are open to the charge

of having lost the unities of the subject in a confusion of irrelevant

not to say remote associations.

Other writers, in the struggle for scientific unification and pro-

portion of parts, have secured a sort of unity and balance at a great

cost of practicability. The latter is a danger that leans toward

"virtue's side," but the danger is real and serious and must be avoided.

It is largely avoided by teaching nothing not even a partial notion

which is sometimes necessary in such a way at one stage that it

must be unlearned by the pupil at a later stage. How to secure both

economic movement and essential unity has been the most earnest

struggle of the authors of the RATIONAL ARITHMETICS. The task has

not been easy and the teaching public will say how well they have

succeeded.

The ideas of number and of the numerical proc- I

MATERIAL. esses must be derived from the concrete. Form and

number are the two main developments of quantity^

The process of numbering in its varied aspects is very closely paral- j

leled in the physical world by the process of measuring in its varied !

applications. This does not imply that numbering and measuring

are one and the same process, or set of processes. What it does imply

is that numbering is the mental side of the same problem of adjust-

ment of activity that has its physical expression in measurement. It

means that measurement is the most direct and certain route to cor-

rect notions of number, for one who has not yet acquired them.

That the theory of number has no necessary connection with

measurement is witnessed by the fact that perfectly sound and ade-

quate theories, constructed altogether on the basis of counting, are

familiar to all students of advanced mathematics. The continuum

of quantity can be constructed with perfect logical rigor out of the

elementary number discreta. But this mode of evolution of the num-

ber system and processes is neither the more economical nor the more

fruitful mode for the immature learner. The physical acts of the

measuring processes run so closely parallel to the mental acts of the

numbering processes, the one-to-one correspondence of steps in the

vi PREFACE

processes is so complete and so natural, that the child glides succes-

sively and with perfect ease and certainty into and through correct

ideas of the unit, of the assemblage, of times, of parting and dividing,

of fractioning, of ratio and proportion, and of valuation, to the idea

of continuous positive and negative magnitude and number. The chief

advantage to the child of making measurement fundamental to number

is that motor, tactual, auditory, and visual sensations serve powerfully

to re-enforce and to sharpen number impressions at every turn The

whole being thus engages in the struggle for mastery of the difficulties

of number and number process "The whole boy goes to school."

Without abating energy on the symbolical phases and processes, this

book seeks through joining, dividing, and measuring lines, surfaces,

and solids, weights, etc., to fill symbols with meaning and processes

with purpose.

The appeal to the concrete involves the use of certain materials,

and any book based on this method of instruction must provide

these within its pages. This the RATIONAL ELEMENTARY ARITHME-

TIC does in great variety; e.g., lines, squares, and surfaces, cubes,

and solids, pp. 1, 2, 4, 12, 20, 60, 100, 198, 204, etc., the construction

of figures for linear and square measure on pp. 2, 4, 173, and 184,

the visual presentation of fractions on pp. 158, 159, 161, 252, and 256.

By means of these the child is also led to make use of other material

which will aid him in grasping the principles involved. >He learns to

seize quantitative questions by the handle.

There are numerous lists of problems involving

USE OF real measurement, and incidentally also counting at its

MATERIAL. best. These lists are carefully graded and the teacher

is urgently recommended at all times to have pupils

solve all they can orally. The pencil and paper should be used only

when the difficulties of the problem make it too hard for the pupil

orally. Different pupils will show very different degrees of aptitude

for rapid oral work No plan of isolating the oral from the written

work can suit the varying needs of different pupils, and every pupil

has a right to the best sort of training of which he is capable. The

problems of life are handled in this way and the pupil should early

PREFACE vii

form the habit of using his head as much as possible and his pencil

only as an aid to his head.

It is also recommended that teachers follow the

CHOOSING . . . .. 11 i

practice of having pupils work rapidly through many

of the lists of problems, indicating the processes called

AND I** i i

tor and giving and recording estimates of about what

FORMING * t 7 f\ .

the answers must be, before any figuring is done.

Then the problems should be worked through and the

correct results compared with the estimates. This work

is of high value as training of judgment and as aiding the pupil to know

when, as well as how, to add, subtract, multiply, or divide.

One of the gains that accrue from basing number

THE TABLES, teaching on measurement is that by associating parts

of the multiplication table with certain facts of the

denominate number tables, both sets of facts may be learned at once,

and more easily than either may be learned alone. For example, the

2's may be based on the fact, 2 pt. = l qt.; the 3's on 3 ft.= l yd.;

the 4's on 4 gi. = l pt.; 4 pk.= l bu.; 4 qt. = l gal.; the 5's on 5c~

1 nickel, 5 nickels =1 quarter, on the divisions of the clock-face, the

number of school days in a week, etc. In this way the most common

denominate number facts are learned and at the same time they fur-

nish a concrete background and purpose for learning the multiplica-

tion table. The space idea is utilized in building the tables, so that

the products of the multiplication tables are abo seen to be mensu-

ration facts. This incidental use of form is of no mean value. It

is indeed one great reason why we need a multiplication table.

The problems from beginning to end deal with

PROBLEMS. realities, and appeal strongly to the child's environ-

ment and experience. This gives to the work a genuine

interest which cannot otherwise be secured, and cultivates the powers

of observation and of inference, and the ability to apply numbers to

the situations of every-day life and experience, and at the same time

strengthens the ability to see and make problems on one's own account

from the raw materials of quantitative situations. Every new idea

comes to the child in the form of a problem, and through the agency

viii PREFACE

of real problems he can be most easily made to feel the genuineness

and usefulness of arithmetical knowledge. The RATIONAL ELEMEN-

TARY ARITHMETIC works out the elementary ideas of arithmetic

through the agency of real problems.

The work of the first 5 pages includes a review and

THE PLAN summary of the 2's and 3's, which pupils will have

OF THE learned before the third grade. Then follow the intro-

BOOK. " duction of halves and fourths through a study and use

PART i. of the foot-rule, then the teaching of the tables, begin-

ning with the 4's, through the use of those standards of

measure which call for the several factors to be taught and tabulated,

and largely by aid of the mensurational facts involving these factors,

see pp. 11, 12, 19, 20, 49, 50, etc. The tables are developed through

the use of the 4, 5, 6, etc., as multiplicand, because this is the most

natural way to teach them, but in the final form for reference the 4, 5,

6, etc., are put into the relation of multiplier. In consequence of the

two-fold aspect of every number, the unit and the multiplicity (the

times), the products are most easily built up and grasped by keeping

the more difficult element, the multiplicity, as small as possible. One

6, two 6's, etc., are much easier than 6 ones, 6 twos, etc. Thus the

6's , for example, are first developed in the form

1X0= G

2X6=12

3X6=18

etc.

But these products are, speaking strictly, not the 6's at all, since the

6's are those products in which 6 enters as a multiplier. So that for

final reference the table is written in the form of The Sixes, thus

6X1= 6

6X2=12 (Seep. 61)

6X3=18

etc.

PREFACE ix

The same system is followed in the rest of the tables, up to and includ-

ing the 10's in Part I. Many uses of the 11 's and 12's are given in

Part I, but the final treatment and formulation of the ll's and 12's are

reserved for Part II. Ample drills, short and frequent, are given to

fix and make flexible the tabular facts taught.

Part II includes the final treatment of the ll's and

PART n. 12's, easy scale-drawing, further use of denominate

number tables, the teaching of the four fundamental

operations, many unified lists of problems, promiscuous as to process

but unified as to some central and interesting thought. See pp. 104,

105, 121, 132, 133, 146, 167, 168, etc., and considerable easy work

calling for simple fractions. The demands of form are met through

many useful problems in the mensuration of simple figures.

The last part of the book carries forward through

PART m. the fifth ffrade the application of the fundamental proc-

esses to an extended range of practical problems

calling often for more than one process , the development of the prod-

ucts by 12, 15, 16, and other factors of frequent occurrence in

business and the industries; it includes the mensuration of boxes,

scale-drawing, a fairly complete treatment of common and decimal

fractions, denominate numbers, together with percentage and interest.

These latter features will commend themselves to teachers who de-

sire to do as much as possible for the boys and girls who are unable

to go on with their school work beyond the fifth year.

Exercises and problems for practice are to be

DRILL. found frequently throughout the book and they every-

where stand in organic relations to what precedes

and follows. This book does not "teach by drill," but everywhere

puts the drill feature in such relations to interesting and useful work

as to make the drill both purposeful and interesting. Drill should

always be for the purpose of fixing facts that have already been

taught and for the sake of establishing right habits of work. Even

in the army the manual of arms and the manoeuvres are first care-

fully taught and the drill is kept up to make correct procedure

habitual and natural. Something like this is the correct mode in

x PREFACE

number teaching. Judicious and intelligent drill constitutes an im-

portant feature of this book. The drill-master must live up to his

reputation of being a master of drill.

A fairly well-rounded elementary treatment of

FRACTIONS, the essentials of common and decimal fractions, of

PERCENTAGE, percentage, and of interest is given at the close of

INTEREST. Part III. The attempt has been to give only a fairly

complete first view, rather than fullness and finality.

As a means of recapitulating and bringing out more clearly the con-

nections and meaning of the fundamentals than could be done in

the earlier grades, this will be recognized at once as a feature no

less valuable to the pupil who goes on to the more mature work of

the grammar school, which builds on these fundamentals, than to

the pupil who is forced to leave school at the end of the fifth year,

and is very considerably benefited by a knowledge of these practi-

cal subjects.

The pleasant task now remains to the authors to make acknowl-

edgment of their indebtedness to many superintendents, principals,

and teachers who have assisted in perfecting the book by suggestions

and corrections both in the manuscript and the proofs. To Mr. F.

W. Buchholz, Professor of Mathematics in the Chicago Normal

School, the authors are under especial obligations for the great pains

and efficient service so kindly bestowed upon the proofs. His insight

and broad experience as a mathematical teacher and supervisor have

wrought wholesale improvement in the form, no less than in the sub-

stance of this book.

Chicago, Sept,, 1905. THE AUTHORS.

CONTENTS.

PART I.

MEASURING LENGTH 3

SURFACE MEASURE-MULTIPLES 5

ADDITION AND SUBTRACTION REVIEWS 7

TABLE OP FOURS.

USES, LIQUID MEASURE 9

USES, DRY MEASURE 11

BUILDING TABLE. 13

APPLICATION, WEIGHT 15

TABLE OP FIVES.

USES, U. S. MONEY 17

USES, TIME 21

BUILDING TABLE . .... 23

REVIEW PROBLEMS 25

EXERCISES FOR PRACTICE 27

FUNDAMENTAL OPERATIONS.

ADDITION 29

SUBTRACTION 33

MULTIPLICATION 37

DIVISION 41

LAUNDRY BILLS 45

DRAWING TO A SCALE 47

TABLE OF SIXES

USES, LINEAR MEASURE 49

BUILDING TABLE 51

APPLICATION 58

BOX MAKING 55

SCHOOL GARDEN 57

TABLE OF SEVENS.

USES, TIME 59

BUILDING TABLE 61

APPLICATION 63

AREA OF TRIANGLES 65

TABLE OF EIGHTS.

USES, DRY MEASURE 67

BUILDING TABLE 69

APPLICATION 71

MEASURING SOLIDS 73

BUILDING SOLIDS 75

CONTENTS OP BOXES 77

xl

xii CONTENTS

REVIEW. 79

TABLE OF NINES.

USES, SQUARE MEASURE 81

BUILDING TABLE 83

DRILL ON NINES 85

APPLICATION 87

WEIGHT 89

TABLE OF TENS.

BUILDING TABLE 91

APPLICATION, U. S. MONEY 93

FUNDAMENTAL OPERATIONS IN U. S. MONEY ; 95

USES OF ELEVENS AND TWELVES. 97

COST OF MEALS 99

PART II.

MEASURING A FLOWER GARDEN 101

PURCHASES AND WAGES 103

PROBLEMS ON CANDY RULES 105

TABLE OF ELEVENS.

BUILDING TABLE AND DRILL 107

TABLE OF TWELVES.

USES, LINEAR MEASURE 109

BUILDING TABLE Ill

APPLICATION 113

FIVES, SIXES, SEVENS, AND TWELVES 115

WRITING AND READING NUMBERS 117

THREES, SIXES, NINES, ELEVENS, TWELVES 119

SCALE DRAWING OF TILING 121

TWOS, FOURS, AND EIGHTS 123

MEASURES OF WEIGHT 125

THE DOZEN 127

FUNDAMENTAL OPERATIONS.

EXERCISES IN ADDITION 129

PROBLEMS IN ADDITION 131

FURNISHING A HOME 133

PROBLEMS IN ADDITION 135

EXERCISES IN SUBTRACTION 137

PROBLEMS IN SUBTRACTION 139

DRILL IN FUNDAMENTAL OPERATIONS. 141

MULTIPLICATION 143

PROBLEMS IN MULTIPLICATION 145

DIVISION LONG. 149

PROBLEMS IN DIVISION. 151

EXERCISES IN DIVISION 153

PROBLEMS OF SALE 155

MISCELLANEOUS PROBLEMS 157

FRACTIONS 159

MIXED NUMBERS 161

PROBLEMS IN FRACTIONS 168

MISCELLANEOUS PROBLEMS 165

PROBLEMS IN TIME 107

BUYING GROCERIES.... 169

CONTENTS xiii

EXERCISES FOR PRACTICE 171

MEASURES OF LONG DISTANCES.. 173

MEASURES OF DISTANCE 175

TWELVE AND A HALF 179

TIME AND DISTANCE 181

REVIEW PROBLEMS 183

PART III.

MKASURING SURFACES 187

APPLICATION OF SQUARE MEASURE 189

PROBLEMS OF DIVISION 191

PROBLEMS IN VALUES RATIO 193

PRODUCTS BY 14 195

TRIANGLES 197

AREAS 199

MISCELLANEOUS PROBLEMS 201

PROBLEMS IN TIME 203

SOLIDS AND CAPACITY 205

TABLE OF CUBIC MEASURE 209

PRODUCTS BY 15 211

APPLICATIONS OF LIQUID MEASURE. 213

PRODUCTS BY 16 215

APPLICATIONS OF DRY MEASURE. 217

PRODUCTS BY 16% 219

READING AND WRITING NUMBERS 221

EXERCISES IN ADDITION AND SUBTRACTION 223

PROBLEMS IN MULTIPLICATION , 225

PROBLEMS IN DIVISION 227

PRODUCTS BY 20 229

PROBLEMS IN WEIGHING .'.... 231

PROBLEMS IN TIME 233

MISCELLANEOUS PROBLEMS 235

COUNTING PAPER 237

MISCELLANEOUS PROBLEMS 239

DENOMINATE NUMBER EXERCISES 243

DENOMINATE NUMBER PROBLEMS 245

BILLS AND ACCOUNTS. 247

PROBLEMS IN FRACTIONS 251

EXERCISES IN FRACTIONS 253

PROBLEMS IN FRACTIONS 255

ADDING AND SUBTRACTING FRACTIONS 257

MULTIPLYING AND DIVIDING FRACTIONS 259

DECIMAL FRACTIONS 261

MULTIPLYING DECIMALS

PERCENTAGE 265

PROBLEMS IN INTEREST .'. 269

PRINCIPLES OF INTEREST 273

SCALE DRAWING OF MANTEL ^

TABLES OF WEIGHTS AND MEASURES 277

PART FIRST.

O

A boy made a cardboard house, one end the size of this drawing.

Without measuring write what you think are these distances:

1. The width. The greatest height. Length of the slant of

the roof. Height to the lowest point of the slant.

2. Width and height of the lower window and of the door.

3. The distance between the right side of the house and the

door; between the door and the lower window; between

the lower window and the left side of the house; be-

tween the sill of the lower window and the floor.

4. Measure each of these distances with your ruler and com-

pare them with what you thought they would be.

l

RATIONAL ELEMENTARY ARITHMETIC.

E

FG

1. Find the lines A, B, C, D, E, F, G.

2. Which is longer, A or B? C or D? D or E? E

or F? F or G?

3. How many lines like A will make one like C? B?

E? D? F? G? F and G?

4. Measure each line with your ruler.

5. How many inches long is A? B? C? D? E? F? G?

6. How many inches long are A and B together?

7. A, B, and C together are as long as which line?

8. F and G are together how many inches long?

9. What is the name for the measure which is as

long as F and G together?

10. How many inches are there in one foot?

11. What is a measure 3 feet long called?

12. Hold your hand a yard from the floor.

13. How many feet high are you? How many yards

high? How many inches high?

14. How many inches do you measure around your

chest?

15. Stretching your arms straight sidewise, how many

yards and inches is it from the tips of the rin-

gers of one hand to those of the other?

16. How many feet can you reach upward from the

floor when standing close to the wall?

MEASURING LENGTH. 3

1. How many inches long is the front edge of your desk?

2. How many inches high is your desk?

3. How many feet high is a window sill in your schoolroom?

4. How many feet wide is the window?

5. Your schoolroom is how many feet long? How many

yards long?

6. Your schoolroom is how many feet wide? How many

yards wide?

7. How long, in feet, is one end of your schoolroom? How

long are the two ends? How long is one side? How

long are the two sides? How many feet are there around

the walls of your schoolroom?

8. Measure a strip of the blackboard 3 yards long; 5 feet long;

15 inches long; 2 yards 1 foot long; 2 feet 10 inches long.

9. Step off or walk 5J yards.

10. How many inches in one foot? In 2 feet? In 3 feet?

TABLE

1 yard = - feet?

1 foot = inches?

11. How many feet in 36 inches? In 24 inches?

12. A blackboard 6 feet long is how many yards long?

13. A box that is 2 yards long is how many feet long?

14. A room 12 feet wide is how many yards wide?

15. 3 and 2 are 5 \

or 2 added to 3 equals 5 ( Thege ftll haye the game mea ning.

3 plus 2 equals 5 I

3 + 2 = 5

16. 3 + 4=? 6 + 2=? 4+2=? 6+1 = ? 5+2=?

8+1=? 3+5=? 5+4=? 7+2=? 4+3=?

17. Add:

6 pounds 8 dollars 6 feet 6 cents 7 yards 7 bushels

5 pounds 2 dollars j? feet 4^ cents 2_ yards 5_bushels

RATIONAL ELEMENTARY ARITHMETIC.

SURFACE MEASURE MULTIPLES.

1. Draw the line a one inch long. Draw

b, c, and d as in the figure A.

2. What kind of a figure have you made?

3. How long is a? bf c? df How long,

then, is each side of the square?

Then what kind of a square may it

be called?

4. How much space (surface) does A cover?

5. Find a surface (space) on page 4 which is equal to A.

6. On page 4, find a surface equal to two times A. What is

it marked? How long is it? How wide? What kind of

a figure is it? How many square inches are in it, or,

what is its area? Draw an oblong the same size as B.

7. Draw an oblong as long as A and B (page 4) together.

8. How long is the oblong (rectangle) you have drawn?

How wide? What is its area?

9. Find on page 4 a rectangle (oblong) the same size as the

one you have just drawn.

10 On page 4, A and C together form a rectangle how long?

How wide? What is its area?

11. Make a square equal to A and C together. What kind

of a square is it called?

12. 2 twos equal 4 \

or 2 multiplied by 2 = 4 I Thege ^ haye thft game meaning<

2 times 2 = 4 I

2X2 = 4

Which is the shortest way to make this statement?

13. 2X3=? 2X7=? 2X 9=? 2X11=? 3X 9=?