TEACHERS' MANUAL
BENJ.flSANBORN&CO
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THE
STONE ARITHMETIC
TEACHERS' MANUAL
GRADES ONE TO SIX
BY
JOHN C. STONE
7
BTATS NORMAL SCHOOL, MONTOLAIR, NEW JERSBT, AUTHOR OF THE
TEACHING OF ARITHMETIC, HOW TO TEACH PRIMARY NUMBER,
A CHILD'S BOOK OF NUMBER, AND JUNIOR HIGH SCHOOL
MATHEMATICS, AND CO-AUTHOR OF THE STONE-MILLIS
SERIES OF ELEMENTARY AND SECONDARY
MATHEMATICS
BENJ. H. SANBORN & CO.
CHICAGO NEW YORK BOSTON
1928
s i :).i
C. ■"
CoPTKIGHT, 1925
bt benj. h. sanbokn & ca
PREFACE
This book is more than a Manual. In preparing it, the
author has aimed to give teachers the results of his experi¬
ences of twenty-five years in teaching children and in pre¬
paring teachers. The latest and best principles of pedagogy
and psychology of number have been given in the simplest
and briefest possible manner. No methods have been given
that have not been fully tried out with numerous classes
and found to be more efficient than those discarded. Many
of the methods are printed here for the first time. Among
these are : A consistent use of the addition method of sub¬
traction throughout the text, and particularly the method
used in mixed numbers ; the division of fractions ; the in¬
troduction to decimals; the "second method" of long divi¬
sion (p. 115); the addition of fractions; the development
of measurements ; etc.
But all methods that differ from other texts have been
found to be much more efficient than those discarded. They
are used for one or more of several reasons : Because they
require the least possible change in the "transfer of skill"
from an already acquired skill ; because they prevent a con¬
fusion of skills ; because they require fewer mental responses
and are thus simpler ; or because they can be used with less
danger of error.
To aid the teacher in planning her work so as to cover the
year's work easily, the work of the textbook is divided into
ten units for each year. Of course, the abilities of children
vary and some will find the load more difficult than others,
iii
iv
PREFACE
but any normal child can easily finish the work outlined.
The text is a basal book. Much better results will be ob¬
tained by using it as directed in this Manual than by using
it to supplement other texts or by supplementing it with
other texts.
Although the text has sufficient drills and tests, the Manual
gives tests at the ends of grades three, four, and five so that
the teacher may have tests that she is sure have not been
prepared beforehand by the pupil. The tests of the third
and fourth year contain all of the primary facts and should
be used freely throughout the course.
Since the Manual is really a text on the pedagogy of arith¬
metic and the psychology of number, the teacher should
study the entire book very carefully — not merely the grade
that she is to teach — for every chapter has valuable help for
teachers of any grade.
John C. Stone
June, 1925.
CONTENTS
Chapter I. The New Trend in the Teaching of Arithmetic
Topics: Economy and efficiency (1) ; The modern curriculum
(2) ; Economy in learning (3) ; Errors made in the doctrine of
habit^formation (4) ; Why the quotient should be written below
(5) ; Avoiding a confusion of habits (7) ; The recurrence of pri¬
mary facts in drills (8) ; Making drill exercises (9) ; Primary skills
needed in finished processes (10) ; Tests and remedial drills (12) ;
Motivation (12) ; Habituation vs. rationalization (14) ; Learn¬
ing definitions (15) ; Reforms in the nature of problems (16) ;
Developing arithmetical reasoning (17).
Chapter II. Economy in Teaching; Fundamental Laws of
Drill
Topics: Cases I to XVI (20-32) ; Summary of principles (32).
Chapter III. Habit Formation and Diagnostic Testing . .
Topics : The 100 primary facts of addition (35) ; Abilities
needed in written addition (36) ; Diagnostic testing (40) ; Three
types of tests (41) ; How to tell the "time limit" in giving diag¬
nostic and inventory tests (42) ; Teaching subtraction (43) ;
Carrying in subtraction (44) ; Testing work in subtraction (45) ;
Multiplication (45) ; The primary facts (45) ; A second ability
needed in multiplication (47) ; Including 10,11, and 12 in the pri¬
mary facts (48) ; Zeros in the multiplicand (48) ; Two- and three-
figured multipliers (48) ; Zeros in the multiplier (49) ; Diagnosing
errors in multiplication (51) ; Habit formation in division (53) ;
The primary facts of division (54) ; A new habit to be established
(54) ; Drills needed for long division (55).
Chapter IV. Planning the Daily Lesson . . . .
Topics: The development lesson (57); The drill lesson (60);
The length of drill period (61) ; The apphcation lesson, or problems
(62) ; Using the textbook (63) ; Drilling on needed facts (65).
vi
CONTENTS
Chapter V. The Work of the First Two Years ... 66
Topics : The work of the first year (66) ; The work of the second
year : the objective (67) ; The use of a textbook (68) ; Three
essential phases of the work (68) ; Presenting the addition facts
(69) ; Drills in addition (69) ; How the pupil uses the drill cards
(70) ; A game with the cards (71) ; Flash-card drill (71) ; Diag¬
nostic testing (72) ; Developing the subtraction facts (73) ; Drill¬
ing upon subtraction (74) ; Problems using subtraction (74) ; Add¬
ing three or more one-figured numbers (75) ; Adding zero (75) ;
Adding two-figured numbers (75) ; Adding by endings (76) ; The
year's attaimnents (76) ; Tests (77-80) ; A Child's Book of Number
divided into ten units of work (80-87).
Chapter VI. The Third Year 88
Topics: The course of study, 3 B(88) ; 3 A(90) ; How to cover
the work outlined (90) ; Diagnostic testing and drilling (92) ; The
written work (93) ; The problems (94) ; Conclusion (96) ; A game
that is also a diagnostic test (96) ; Dividing the third year's work
into monthly units (97-106) ; Third-grade achievement tests (106-
109).
Chapter VII. The Work of the Fourth Year . . . 110
Topics: The course of study, 4 B (110) ; 4 A (112) ; How to
accomplish the work of the year (114) ; Long division (114) ; Two
methods of long division (115) ; The problems (115) ; Diagnosing
a solution that has a wrong answer (116) ; The work of the year
divided into ten units (117-127) ; Fourth-grade achievement tests
(127-129) ; Oral-pjoblem test (130) ; Written-problem test (130).
Chapter VIII. The Work of the Fifth Year . . . 132
Topics : The aims of the fifth year (132) ; Standards of skill
(132) ; How to get skill (133) ; Checking work (134) ; Developing
power to solve problems (134) ; Common fractions (135) ; Adding
and subtracting fractions (136) ; Subtracting mixed numbers
(137) ; Multiplying and dividing fractions (138) ; Drills, races,
and tests in fractions (140) ; Decimal fractions (140) ; The prob¬
lems of measurement (141) ; Measuring rectangles and prisms
(141) ; The work of the fifth year divided into ten units (142-151) ;
Tests on fundamental processes (151-154).
CONTENTS
vii
Chapteb IX. The Work of the Sixth Year .... 155
Topics : The aims of the sixth year (155) ; Developing skill
in computation (155) ; Reviewing fractions (156) ; Completing
and applying decimals (156) ; Large munbers and graphs (157) ;
The meaning and use of per cent (157) ; The uses of per cent in
business (158) ; Practical measures (158) ; The final review and
tests (158) ; The work of the year divided into ten units (159-
167) ; Standards of skill (167) ; Control of the processes (168) ;
Tests in problem-solving (168).
THE STONE ARITHMETIC
TEACHERS' MANUAL
GRADES ONE TO SIX
CHAPTER I
THE NEW TREND IN THE TEACHING OF ARITHMETIC
This chapter is given to show teachers some of the newer
phases of the teaching of arithmetic in order that they may
more fully appreciate the text and use it more wisely.
Economy and Efficiency
Economy and efficiency is the slogan in the industrial,
commercial, and educational worlds today. Just as trained
engineers are constantly at work on these problems in
machinery and in business — seeking to eliminate waste
or to turn it into by-products — so in the educational
world, by constant study and research, we are finding ways
to get better results with a more economical use of time.
Experts are continually studying the question of what is
worth while in arithmetic and what is not; how one skill
or habit may grow into another with the simplest possible
transfer of habit ; how much drill work is needed ; how it
should be distributed ; how to meet individual differences ;
and how to test results and apply remedial drill. They are
studying the fundamental processes to find the primary
skills used in the finished processes, and how to develop
1
2
TEACHERS' MANUAL
these skills separately in such a way that the transfer from
one skill to a related one is brought about with the least
possible effort. Careful research is being made to determine
when arithmetic should be begun ; in what grades the various
topics should be taught ; the social uses of arithmetic, in order
to ehminate as far as possible any waste of time and effort ;
and how to develop ' ' arithmetical thinking. ' ' Every page of
the text is organized to give the greatest possible skill with
the least possible effort, and the text should be carefully
followed according to the outline in later chapters.
The Modeen Cubriculum
There was a time when the three R's constituted the whole
of the curriculum. Today with our modern and enriched
courses of study, arithmetic is entitled to but about an
eighth or a tenth of the school day. When it was given a
much larger proportion of time, it was given quite largely
for "mental gymnastics." That is, in the belief that it gave
a valuable general mental discipline — that it trained the
reasoning powers. While the social, industrial, and com¬
mercial needs of the subject have always been recognized,
until recent years our courses of study went far beyond
current needs and retained much of the obsolete in the belief
current fifty years ago and even twenty-five years ago, as
expressed by Fitch, that "Arithmetic is conspicuously one
of those subjects of school instruction the value of which
extends beyond itself."
Today our courses of study and modern textbooks are
built upon the almost universally accepted principle that
In a course in arithmetic we should teach the child that which
he will need and can learn, and waste no time in teaching him
that which he will never need, or in trying to teach him that
which he cannot learn.
TREND IN THE TEACHING OF ARITHMETIC 3
While this is accepted as a principle, it is not always
carried out in practice. Many of our most recent text¬
books give much that will never be used. Perhaps this is
most noticeable in the work in fractions. About the only
fractions ever added or subtracted are those that result from
expressing a number of units of measure in terms of a larger
unit, as ounces in terms of pounds, inches in terms of feet,
etc. These fractions are halves, thirds, fourths, sixths,
eighths, ninths, twelfths, and sixteenths. To these natural
fractions may be added those arbitrary divisions of inches
into halves, fourths, eighths, and sixteenths. So why should
a child waste his time in adding and subtracting fractions
never used in the problems met in actual life situations!
And yet we find such fractions in almost all of our most
recent textbooks.
Economy in Learning
Economy in learning is a matter of proper habit formation.
The transfer from one habit to a related one must be done with
the least possible effort. Thus, to recognize the relations of
the two inverse processes (subtraction and division) to the
two direct processes (addition and multiplication) results
in a very great saving of time, as has been proved by many
recent studies. Thus, if a child knows that 2 and 5 are 7,
when asked, " Five pencils and how many more make
seven?" he will answer, "Two," without knowing that there
is a process called subtraction. Or when a child knows
7 "t" 3 = 10, he will give you the missing number in ? -p 3
= 10 or 7 -h ? = 10.
The only new thing for him to learn is that as a subtrac-
10
tion fact, the notation _7 asks the same question as
7 -f ? = 10. Namely, " Seven and what make ten ? "
4
TEACHERS' MANUAL
Likewise, to teach a child that " 3 ' gus-sin-tu ' 12 four
times " is to teach him a new and meaningless fact ; while,
if rightly taught, he knows the division facts as soon as he
knows the multiplication facts. Thus, when a child knows
that 3 X 4 = 12, he can tell you the missing number in
3 X ? = 12 or in ? X 4 = 12. That is, to the question,
" Three times what is twelve? " he will respond, " Four,"
without having ever heard of the process of division. Hence
it only remains to show him that the notation 3)1^ is the
question, " Three times what is twelve? "
These are illustrations of what is meant by sa3dng that the
transfer from one habit or skill to a related one should be
made in the easiest possible manner. These are the meth¬
ods of the text, and the teacher should not confuse the child
by using the methods and language of the older texts.
Erboks Made in the Doctrine op Habit Formation
While all agree with the fundamental principles of habit
formation and seek to establish as few unrelated habits as
possible, the doctrine has been carried to a ridiculous con¬
clusion by some recent amateur textbook-makers and by
some teachers. Among the most foolish of these is the
recent form of short division, viz. :
45
3)135'
It is such unbusinesslike forms as this, advocated by some
" reformer," that cause the business world to make sneering
remarks about " schoolroom " or " schoolmarm " methods.
There is but one argument by the advocates of this form
of short division. That argument is that since the pupil
puts the quotient above in long division, he will become con¬
fused in long division if he puts it below in short division.
TREND IN THE TEACHING OF ARITHMETIC 5
That may seem plausible to one who has not thought the
matter through, and who has never thought to make a test
to see if such a confusion ever arises. In five minutes one
may make a little test that will prove that no confusion
arises in the transfer from below to above, or from above to
below. Suppose that you belong to that small group of
teachers who have been misled into using the form shown
above. Give your pupils a dozen exercises in short division
and time them. Now work three or four exercises on the
blackboard, placing the quotient below, and give them the
same number of exercises of equal weight and ask the class to
write the quotients below. You will find no loss in speed or
accuracy. That will shatter the only argument for placing
the quotient above. I have tried this test many times.
Why the Quotient Should Be Written Below
There are so many arguments for putting the quotient
below in short division that it is almost inconceivable that
men who have made noted contributions to education in
other lines should advocate the other method.
Only a few examples are needed to show how impractical
such a method is.
Suppose that you wish to find the average of
several numbers. You must either put the
quotient below or rewrite the sum, a wasteful
use of time that the business world will never
practice.
Suppose that you wish to subtract the quo¬
tient, as in discounting a bill 25%, you must
either write the quotient below or rewrite both
dividend and quotient.
42
36
48
3)126
42
4)$600
150
$450
6
TEACHERS' MANUAL
$700 Suppose that you wish to find a fractional part
3 of a number, you must either write the quotient
4) $2100 below or rewrite the product.
$525
Suppose that you wish to make two or more
4)480 continued divisions, you should have to adopt
5)120 the habit of beginning at the bottom of a page
24 and working up, instead of working down a
page as we do in reading and writing.
Not only is the method impractical, as shown above, and
one that business men will never adopt, but it actually
develops a habit that leads to error in long division. The
author has observed that children taught this method place
the first quotient figure in long division in the wrong posi¬
tion and that it ^akes much time to break the habit. Thus,
they place the first figure as follows :
4 5 3
32JÎ389 46)2398 832)29763
This is due to the fact that the child uses the 3, 4, or 8
to get the " trial quotient " (as he should), and from the
" transfer of habit " from short division, he puts the quotient
figure over the right-hand figure of the dividend that was
used to get the "trial quotient." The above quotients should
be written :
4 5 3
32JT389 46)2398 832)29763
The purpose of placing the quotient above in long divi¬
sion, instead of at the right of the dividend as was done
until recent years, is to simplify the pointing off in division
of decimals. Hence if the quotient figures are not rightly
placed, errors result in pointing off quotients.
TREND IN THE TEACHING OF ARITHMETIC 7
Avoiding a Confusion of Habits
Some "reformers" who would put the quotient above in
short division to avoid a confusion of habits, still teach the
forms :
f-f=f ÎXf=^
in which there is a very real danger of a "confusion of
habits." All extensive tests and surveys have found a very
large number of such mistakes as :
f + f=7 fXf = f and f + i = ^
2
Such confusions as these are easily avoided. Fractions
to be added or subtracted should be written in columns as in
adding or subtracting whole numbers, and the obhque line
should be used in writing the fraction. Thus :
3
Vs
Vé
8
%
H
M
5
H
3
%
H
H
8
Vs
%
5
K
H
H
16
iVs
IH
Another great danger lies in a practice seen in many
schools and in many textbooks of indicating the processes
of addition and subtraction. Thus :
38 98 85 92
+ 24 + 56 - 24 - 36
In algebra, if these are all addition exercises, the answers
are 62, 154, 61, and 56. If they are subtraction, the answers
are 14, 42, 109, and 128.
It is certainly reasonable to expect that a pupil used to
such forms for eight years in arithmetic will be confused
by them when he sees them the ninth year in algebra.
8
TEACHERS' MANUAL
Such forms are not used in the text. While such forms
are found in recent texts, they are severely criticized by
all modern psychologists.
The Recurrence of Primary Facts in Drills
No one knows how many times a fact must recur in the
mind of a child before it becomes fixed. In fact, the number
of times it must recur depends upon the individual child,
the attention he gives to it, his interest in it, and his effort
to learn it. The more interest he takes in it, and the greater
the attempt to remember it, the sooner it will become fixed.
Recalling a fact does not fix it, hut recalling it with an effort
to remember it is what fixes it.
Perhaps if all facts were equally interesting and if all
should receive the same attention, it would take about as
many " exposures " to fix one as another. For that reason,
teachers should be sure that one fact of a group should be
recalled about as often as another when the facts are first
presented. For example, after the " 3-times " facts have
been presented, all facts should recur about equally in the
first drills that follow. But some of this group of facts are
used out of school more often than others, hence become
fixed much sooner than others. For this reason later drills
should bring in, those facts that are less used out of school,
and hence less permanently fixed, more often than the others.
If a child had a real out-of-school use of 7 X 9 more often
than of 2 X 5, it would become fixed more quickly than 2X5.
The fact is that he uses 2X5 hundreds of times to 7 X 9
once. Hence the latter requires much more drill in school.
In giving supplementary diill, be sure that the above prin¬
ciple is followed.
In the older textbooks there was but little if any consid¬
eration given to the recurrence of facts. In some cases.
TREND IN THE TEACHING OF ARITHMETIC 9
the easy facts recurred much more frequently than the more
difficult ones. That, of course, is a serious condemnation
of such drills. It means that pupils have wasted much time
in " overdrill " on facts that were permanently fixed to the
neglect of facts not yet made permanent in the child's mind.
Making Drill Exercises
In making a set of drill exercises it is very easy to make
any multiplication or subtraction fact recur at your will.
And by^taking the desired divisor and quotient and from
them making the dividend it is easy to control the recur¬
rence of the division facts. But in making the addition
drills the task is a much more difficult one. Thus, in adding
a column of numbers, use is made of the 100 primary facts
and of the related ones called " adding by endings." Add¬
ing 9 and 4 gives a number ending in 3. If this sum is added
to 8, it gives a number ending in 1. This necessitates using
one of the " easier facts." That is, 1 must be added to
some number. When counting the recurrence of facts
in an addition drill, count them by adding in both direc¬
tions, for a child should always check his work by adding
in both directions, and thus he gets combinations in one
that he does not in the other.
Later drills need not be as carefully made, for after a cer¬
tain period of recurrence errors are not so much the result
of not knowing the facts as from a break in the attention.
Our best accountants occasionally make errors not due to
lack of a "permanent bond," but to a lapse of attention.
It is known to all teachers that a child that can add five or
six single columns of four numbers each without an error
cannot add the same numbers all put in one column of
twenty or twenty-four numbers. Hence after facts are
fairly permanent the drills are to develop the attention
10
TEACHERS' MANUAL
span. Even in the later drills, however, all facts should
recur within frequent intervals, but it is not as important
that the more difficult ones should recur the most frequently
as it was when these primary facts were becoming fixed,
for it is the attention span that is now being developed.
Primary Skills Needed in Finished Processes
One of the recent advances made in developing skill in
computation is the recognition of the various skills needed
in each process. Not many years ago the child was drilled
in the " forty-five addition facts " and then expected to add
long columns of two- and three-figured numbers. Today
we recognize that there are at least ten skills and habits to
be built up separately before we may expect skill in written
addition. These skills and habits are represented by the
following types :
(A)
(B)
(O
(D)
(E)
(F)
(G)
(H)
(1)
(J)
3
2
13
5
16
9
34
48
3
38
5
3
5
6
8
5
6
46
4
7
8
8
21
7
74
4 48
That is, the pupil must (A) have an automatic control of
the 100 combinations of two one-figured numbers, including
zero ; (B) use these facts in adding three or four one-figured
numbers ; (C) add a two-figured and a one-figured number
whose sum is in the same decade; (D) use these skills in
adding three or four one-figured numbers ; (E) add a two-
figured and a one-figured number whose sum is in the next
decade ; (F) use this skill in adding three or four one-figured
numbers; {G) add two-figured numbers without carrying;
(H) add two-figured numbers with carrying; (/) build up
TREND IN THE TEACHING OF ARITHMETIC 11
the attention span ; and (J) use any of these skills in adding
numbers of two or more figures.
In subtraction there are but two skills to train : The 100
primary facts which come directly from the 100 addition
facts, and the habit of carrying 1 whenever a fact with 10
or more up to 18 occurs in the primary fact used. Hence
skill in subtraction is much more easily obtained than skill
in addition and much less drill is needed in this process.
In multiplication the following skills are needed, rep¬
resented by the following types :
(^)
(B)
(Q
(D)
(E)
3X7
4X6-1-3
78
28
365
5
47
208
That is, the child must (A) have an automatic control of
the primary facts ; (B) he able to think a product and add
a one-figured number (this is a very easy transfer from
" adding by endings" developed in addition) ; (C) get the
proper habit of carrying after he thinks the product ; (D)
get the habit of properly placing the partial products and
adding ; and (E) know where a partial product is placed
when there is a zero in the multiplier.
Unless there is conscious training on the part of the
teacher, there will be a confusion of the carrying with the
habit developed in addition. One of the most common
errors in the early work in multiplication is :
27
35
63
75
4
4
8
4
168
200
644
360
That is, the child carries before he multiplies, because in
addition he added the number carried to the first addend.
Division is the most complex of the processes and also
12
TEACHERS' MANUAL
requires the development of more primary skills before
final skill may be expected. The skills are represented by
the following types :
(A) (B) (C) (D) {E) (F)
3)9 2)^ 2)9 2)^ 20)^ 20)70
(G) (H) (I) (J)
21)945 30)^ 30)78 38)T728
This subject is given a more complete discussion in Chap¬
ter III of the Manual.
The textbook carefully organizes the drills so as to build
up proper habits. The teacher should understand the pur¬
pose of each drill and see that the fundamental skills are
properly developed.
Tests and Remedial Drills
If a teacher is to avoid overdrill or underdrill, discover
individual differences, and make the best use of her time,
there must be very frequent achievement tests. When an
individual has failed to make a satisfactory score, the reason
must be found by a set of diagnostic tests covering the
primary skills required in the achievement expected. If
primary facts are not known, inventory tests must be
given to find jußt what ones of a series of facts need fur¬
ther drill. It is only in this way that a teacher can accom¬
plish the maximum in a minimum amount of time. Such
inventory hsts are provided for in the text.
Motivation
Good teachers have always recognized the fact that the
more pleasurable we make learning, the greater the desire
to learn, and the greater attention the child pays to learn¬
ing, the more quickly he learns. To call this fundamental
TREND IN THE TEACHING OF ARITHMETIC 13
principle to the attention of teachers, educators coined the
word " motivation " several years ago to emphasize the
need of bringing about this pleasure, attention, and desire.
WhUe the principle itself is sound, many teachers, not seeing
its full meaning, have mistakenly felt that if the children
were led to enjoy the class period, the lesson had been " mo¬
tivated," and have not stopped to ask whether the. desired
end had been accomplished or not.
In many classrooms the number period is used in playing
some game in which the individual child makes but few
responses and these not to combinations that he needs.
Thus, the class may "play store" and a pupil maybe inter¬
ested in but a single purchase made by himself. Or it may
be a game in which the interest is in the game and not in a
desire to know the number facts. When games are used, they
must be of a nature that does not detract from interest in
number; they must include all facts of the series being
drilled upon ; they must require a response from each mem¬
ber of the class ; and they must offer an opportunity for a
large number of responses during the drill period. The
highest type of motivation is in number itself through num¬
ber races, timed drills, and class contests. A contest game
is given in Chapter VI that will hold attention and be a diag¬
nostic test at the same time.
Perhaps the most wasteful type of so-called motivation is
that of the dramatization of adult activities, particularly
as a type of drill lesson. There may be some excuse for an
occasional use of dramatization of certain activities in the
lower grades, but this type of recitation should be used very
sparingly. But there can certainly be no excuse in the
upper grades for dramatizing the banking business, insur¬
ance, the stock exchange, tax-collection, etc., as is done in
some schoolrooms and in some recent texts.
14
TEACHERS' MANUAL
Habituation vs. Rationalization
In the past there has been much time wasted in an
attempted rationalization that did not rationalize. Most
written processes were begun with abstract numbers and an
attempt was made to show the " why " of each step of the
process, as " carrying " in addition, through the decimal-
place-value-system of our notation, objectified by the use of
splints bound up in bundles of tens and hundreds. And
" explanations " were given in the textbooks, usually in
small type, for the teacher, but not within the comprehen¬
sion of the pupil.
Today each new process begins with a simple concrete
problem within the child's understanding. He is told in a
simple way what to think. Thus :
1. James spent 38fi for a knife and for a top. How much
did he spend for both?
Think, "8jé and 5fé are 13jé." Then think, "13jé makes 1
dime and 3fé."
— Write 3 jé under cents, and add the 1 dime to 3 dimes and
1 dime. Think, "1, 3, and 1 are 5." Write 5.
2. Mary wants a doll that costs 95jé. She has 67jé. How much
more money does she need ?
95 Since 5 is less than 7, think, "7 and 8 are 15." Write
^ 8. Since 8 and 7 are 15, carry 1 to 6, and think, "7 and 2
28 are 9." Write 2.
3. Frank sold 36 papers each day for 3 days. Find how many
he sold.
36 Think, "3X6 = 18." Write 8, carry 1.
3 Think, "3X3 = 9." Then add the 1 to carry to 9
108 and write 10.
4. John paid 96jé for 2 httle rabbits. How much was that for
each?
TREND IN THE TEACHING OF ARITHMETIC 15
Think, "9-5-2 = 4 and Inot divided." Write 4 under
^ 9. Now think, "1 of 9 not divided and the 6 to the
right of 9 makes 16 to divide."
Think, "16 2 = 8." Write 8 under 6.
Of course, through problems and objective illustrations,
the meaning of the processes must be rationalized, but the
written processes themselves need no further rationalization
than that shown above. In fact the child can understand
no more than this. This is the method followed in the text.
It is a waste of time to attempt more than given in the text.
Later in the course, when the child is able to follow a
rationalization, those subjects of infrequent recurrence may
be remembered better if rationalized. Rationalization may
lead to a safer use of them, it may save a confusion of facts
with other facts, and it may lead to greater interest in the
topic. Hence you will find a more complete rationalization
of the work given in the Intermediate and Advanced (Fifth
Year to Eighth Year) books than in the Primary (Third Year
and Fourth Year) book.
Learning Definitions
In the past much stress was placed upon definitions. To¬
day the modern textbook gives very few definitions. When
a child learns to add, subtract, multiply, or divide, he is told
what the process is called, but it is not really defined.
A child can subtract as well without knowing the names
" subtrahend," " minuend," and " difference," as he can by
knowing them.
He can multiply as well without knowing the names
" multiplicand," " multiplier," and " product " ; and so on
for all the processes.
After he has learned how to use the processes he is told
16
TEACHERS' MANUAL
the names of the terms so that he may use them in case he
needs to refer to them. The same is true of the terms used
in fractions and throughout the course.
The name or definition, when given at all, is given after
the child knows its meaning and use, not before as in the
older type of textbooks.
Reforms in the Nature of Problems
There has been a gradual improvement in the nature of
problems in recent years. Most writers have made a fairly
successful attempt to make problems that meet actual life
conditions. It is strange, however, that most of them are
afraid to introduce problems in the lower grades that con¬
cern dollars and cents, written $3.25. Instead of that,
when they want to introduce a problem including three-
figured numbers about money, they use $325, an amount
that the child cannot image, in place of $3.25, which he can.
Some of our most popular textbooks leave such numbers
until the fifth grade and give a chapter styled " United
States Money " just preceding decimals. This is a custom
handed down from Pike and other early writers who wrote
at a time when much of the money in circulation was that
from Europe brought over by the colonists and when our
own system wa^ new. To argue that the child in the third
grade will be confused in using the period to separate dollars
and cents is to argue that he will be confused in using it at
the close of a sentence or to dot an i. He gets it and uses it
as he does any other symbol used in writing words or num¬
bers. It is merely a matter of notation, not something to
explain.
Not only does the introduction of dollars and cents in the
very first written work allow a type of problem concrete to
the child and within his needs and interests, but it paves
TREND IN THE TEACHING OF ARITHMETIC 17
the way for all work with decimals, except two problems
— multiplying and dividing by a decimal.
While, as stated above, there is a marked improvement
in the nature of problems, many authors, in order to work
in a large variety of fractions never used in life, give prob¬
lems including :
I lb. f yd. f bu. etc.
Developing Abithmetical Reasoning
Fornierly, except for the " Miscellaneous " lists at the
end of a chapter or of the book, problems were classified as
to process and worked by a stereotyped form. Today most
writers follow each new process with a few classified prob¬
lems to show the meaning and use of the new process, after
which all problems are unclassified as to process. And yet,
all observers know that there is a sad lack of power among
the graduates of our schools to analyze a new situation, and
reason out what processes to apply. Without picturing
relationships or the meaning of results obtained, many of
our graduates juggle with all the numbers given in the prob¬
lem and thoughtlessly accept whatever that juggling brings
forth.
I recently gave the following problem to a number of high
school graduates :
A man paid $15,000 for a house and rented it for $125 per month.
The yearly rent was what per cent of the cost of the house ?
One solution was :
$15,000 X $125 = $1,875,000, the yearly rent.
$15,000 -Í- $1,875,000 = .008%, the yearly rate.
I recently saw a class in high school solve the following
problem given by their teacher.
18
TEACHEES' MANUAL
A dealer imported 448,000 lb. of coal for which he paid $12 per
ton. Find the import duty at $4 per ton. (Use 2240 lb. per ton.)
Many of the solutions were :
448,000 2240 = 200.
200 X $12 = $4800.
$4800 X $4 = $19,200.
Such errors as these raise a very serious question with
every thoughtful teacher. Pupils who make such errors are
either subnormal intellectually or have had improper teach¬
ing, or our textbooks have not offered proper material.
Barring the first of these, which is beyond our control,
let us consider the other two. Many believe that the solu¬
tions found in many schools are too formal and stereotyped,
and that the problems in most texts are too nearly of fixed
types which the child learns to solve through memory alone,
just as he learns the facts of computation.
For a number of years educators have done careful re¬
search work to find the " social needs " of arithmetic among
the people in the common walks of life. Writers have at¬
tempted to use only problems that meet the most common
social needs. These can be reduced to a few standard
types that the pupil can solve through memory. These
the pupil with low intelligence can learn to solve.
But there is another need, fully as valuable as the utili¬
tarian need, and that need is power to analyze a new situa¬
tion and discover the solution. Hence in every school
course there should be some attention paid to problems that
seldom occur in the mere " social needs," problems of a
type never met before by the pupil, problems of which he
has no stereotyped forms of solution, but problems in which
he must see the quantitative relationships and from them
reason out what to do. It is only through such problems
TREND IN THE TEACHING OF ARITHMETIC 19
that the intellectually alert pupil will get the training and
development of which he is capable.
The Stone text for intermediate grades (grades V-VI) and
advanced grades (VII-VIII) gives many problems of very
infrequent recurrence in actual life in order to develop
greater power to see quantitative relationships. These
problems should be used very carefully in an attempt to
develop " arithmetical reasoning."
The teacher should not expect all pupils to get all prob-
blems given in the text. If pupils of low intelhgence could
get all of the problems, there would be nothing that would
stimulate the alert and intellectual pupil. A textbook must
give some problems that only the most capable children can
solve.
CHAPTER n
ECONOMY IN TEACHING: FUNDAMENTAL LAWS
OF DRILL
Economical teaching and learning demand that proper
habits be built up in such a way that they relate to past
habits, and lead to new ones with the least possible loss in the
" transfer of habits " ; and that all habits function when
used in the final written work. Chapter III discusses the
habits needed in the four fundamental processes with whole
numbers.
This chapter deals with some of the fundamental laws of
learning through the " case method." These are typical
cases that have come under the observation of the author in
schools in many parts of the country.
CASE I. A teacher in the third grade was drilling with flash
cards with the numbers written 3 + 4 = , 2 + 5 = , etc. She
complained that while the pupils knew their number facts, they
could not do written addition.
This teacher did not realize that skill in giving the sums
from figures written in this form could not without very
great loss be transferred to figures written in columns. It is
in the column form that the child always uses them in writ¬
ten work, and this is the form that should be used in all drill
work.
Moreover, this teacher did not realize that there are many
skills to be developed before she could have expected skill
in written work. This is discussed in Chapter III.
20
ECONOMY IN TEACHING
21
CASE II. A teacher was drilling her class in 2A from cards
with the numbers in column form on one side and with dots to
represent the numbers on the other side :
3
5
When a child could not give the sum, she said, "Now think.
Think hard." After a few wrong guesses, the dots were shown
and the child was told to count them. This teacher complained
that the class did not seem to be able to remember the facts.
This teacher did not differentiate between the develop¬
ment of a fact and drill upon it. A few of the facts should
have been developed from concrete objects (these dots were
not the most vital kinds of concrete objects) to show the
child the meaning of addition ; but drill is to fix the facts
so they may be given automatically. When a pupil gave
a wrong answer or hesitated, the answer should have been
shown. She should have had the drills on one side of the
card and the complete fact on the other. Thus :
3
5
3
5
8
No guessing or counting should have been allowed.
CASE III. For seat-work in the second grade, a teacher wrote
the addition drills on the blackboard and had the class copy them
and write the sums. The papers were collected and corrected and
22
TEACHERS' MANUAL
handed back the next day. Many of the sums were wrong and the
class was making but little progress.
The teacher could not have devised a worse method of
procedure. A wrong sum written makes as lasting an impres¬
sion as a right one. If she felt that it was necessary to have
seat-work in order to keep the pupils out of mischief, she
could have had them copy the complete fact from the black¬
board. Thus, she should have written :
3 2 3 5 6
5 7 3 2 2,
----- and so on.
8 9 6 7 8
In copying them the child would have got an eye pic¬
ture of the fact, had practice in making the figures, and been
kept " busy." Or, better yet, she could have had seat-
work that would have been a valuable drill and a diag¬
nostic test at the same time. Thus, as each new fact was
presented, each child could have made a set of cards like the
second set in Case II, but much smaller, say 2" by 3". As
the child looked at the drill side of each card (all displayed
before him on his desk), he could have written the sum and
then turned the card over to check his work. If the answer
was wrong, he should have thrown that card out for further
study. He thus knows in a few minutes just what facts he
does not know and what ones he should study. The motive
should be that of playing a game. He should strive to see
how soon he can give all. Then he has " won " the game.
A child should spend his time on what he needs to know and
does not know, rather than upon what he already knows, or
upon what some other pupil does not know.
CASE IV. A teacher in the third grade used a large circle with
the nine digits (no zero) around the circumference and a number
in the center to be added. She would ask a child to see if he could
ECONOMY IN TEACHING
23
ride around the "merry-go-round" without falling off. She said
it was to "motivate" the drill as the children liked to do it.
This was a very poor form of drill. The children will
never have to add numbers in this order. Moreover, the
motive was a very artificial one. Of course, the more pleas¬
ant we can make the learning of a fact, the quicker it will be
learned. The desire to play, to win, and to excel are fun¬
damental instincts that can be used to make learning more
pleasant. But there are many better games than this, and
games fihat lead to better results. This will be discussed
when the work is taken up by the year in later chapters.
CASE V. A teacher in a fourth grade was spending a great
amount of time in drilling on adding five three-figured numbers
because she was working for a set standard in such exercises. The
pupils were very slow and inaccurate and seemed to be making but
little if any progress. She was giving no oral or sight work to build
up primary skills, for she "supposed" that this had been done in
former grades.
A diagnostic test showed that there were many of the
"100 primary combinations " that the pupils did not know,
but they found the sums by counting ; and that when adding
to a two-figured number as 16 -1- 8, 15 -1- 3, etc., the chil¬
dren always counted. The teacher thought that the pupils
would resent drilling upon the primary combinations and
" adding by endings " (16 -f 8, 25 -f- 3, etc.) as " baby
work." But they were shown that this was necessary
before they could add larger numbers rapidly and accurately ;
they helped the teacher fix a standard to be attained (the
number per minute) and took great interest in reaching the
standard. This was a use of the game spirit referred to
under Case IV.
There should be very frequent diagnostic tests given to
24
TEACHERS' MANUAL
see if needed abilities or habits have been established. When
not, they must be established before work needing them is
continued. And the pupil should always know the purpose
of any drill and enter heartily into developing any required
standard of skill.
CASE VT. A teacher who had always taught and used the
"taking-away" method in subtraction was required by a new
superintendent to teach the " addition" method. She did very well
in teaching the primary facts and in written work without carry¬
ing; but when she came to "carrying," she failed, for ^e taught
it as follows :
526 "6 less 8 you cannot take. Borrow 1 from 2 and add it to 6.
148 That makes 16. 8 and 8 are 16. Write 8."
" 1 less 4 you cannot take. Borrow 1 from 5 and add it to
1. That makes 11. 4 and 7 are 11. Write 4."
She was using language unfamiliar and meaningless, and
confusing the old method with the new. No wonder that
she could not teach it ! She should have shown that when
1 ß
seeing 6 over 8 we must think of the g fact ; and that in all
such cases 1 must be added to the next lower number (in
this case, 7). So the child should have developed the follow¬
ing habits :
526 Think, "8 and 8 are 16." Write 8; carry 1.
148 Think, "5 and 7 are 12." Write 7 ; carry 1.
378 Think, "2 and 3 are 5." Write 3.
This is much more easily taught than the " taking-away "
method, but the teacher must not confuse it in thought or in
language with the other method.
CASE VII. A teacher used a circle with the numbers 1, 2, 3,
and so on, to 12, written in the same order as those on a clock face
for a drill device in multiplication. The multiplier (in this case, 3)
ECONOMY IN TEACHING
25
was written in the center. The children were first asked to " say
the table " and were then to be rewarded by being allowed to see
how many hours they could run without stopping. If they missed
at 5, they ran 5 hours ; if at 8, 8 hours ; if they did not miss any,
they ran for 12 hours without stopping. This was her way of
"motivating the work."
It is hard to conceive of a more foolish or more wasteful
procedure. The pupils learned their " tables " in order.
In a brief test, it was found that when a fact, as 3 X 6, was
asked for, the pupils recited silently the table until they
came to 3 X 6 before giving the answer. Moreover, three
of the facts, 3 X 10, 3 X 11, and 3 X 12, or 25% of the
drill, were not primary facts needed in written work; and
3X0, which is needed, was not included. And finally, the
motive itself was a very weak and silly one that could not
have made a very strong appeal to a red-blooded child in
the third grade.
CASE Vlll. A teacher in the third grade used a ring-toss game
for most drill work in multiphcation. Twelve cup hooks were
screwed into a drawing board and one of the twelve numbers from
1 to 12 was written below each hook. The children tossed can
rubbers at the hooks. When a hook was ringed, its number was
multiplied by the number being studied. Thus, if the"3-times"
table was being studied, the number of the hook ringed was multi¬
plied by 3. Each pupil in a class of 36 or 40 took his turn in tossing
a ring. There was no teamwork and a pupil was interested only
in his own score.
The time spent in this form of drill was practically wasted.
The class was so large that each child got but 2 or 3 trials ;
very few hooks were ringed ; some children did not get a
chance to make a single combination during the drill period ;
10, 11, and 12, not needed, were as apt to be ringed as any.
26
TEACHERS' MANUAL
When such a device is used, it must be one in which some
score is made each trial ; teams must be organized so that
every pupil is interested in every score made ; and it must
be so arranged that all combinations are sure to occur
sometime during the game.
Such devices may, on account of the increased interest,
be justified at times ; and they may lead to the child's use
of the game out of school, and thus he may get drill that he
would not otherwise get. But any device by which the
combinations arise by chance should not become a major
part of a drill lesson. The teacher should control the com¬
binations that arise so as to drill the child upon what he
needs rather than trust to chance. See the games sug¬
gested under the work of the third year.
CASE IX. A test given a fourth-grade class that had spent a
semester in multiplying by two- and three-figured multiphers
showed that a large per cent of the class made the following error :
346
203
1038
6920
7958
It was found that the teacher had taught the class to fill out
the partial products with zeros. Thus, she taught :
48
24
192
960
1152
In attempting the rationalization of the process, she had
not only failed to rationalize it, but had built up a wrong
ECONOMY IN TEACHING
27
habit. It is habituation, not rationalization, with which we
are concerned in teaching the written forms of the processes.
The zero in the second partial product should not have
been written.
CASE X. In a test given to high school graduates on the exer¬
cise 20,304 X 53,241, over 50% failed, making the following error :
53241
20304
212964
1597230
1064820
122667264
Many of the students who made this mistake had high
grades in their mathematics. So a reason for the error
was sought. It was found that they had been taught to
find 203 X 487 as follows *
487
203
1461
9740
98861
In this case the answer was right, but the habit formed
was to bring down a zero when a zero occurred in the mul¬
tiplier and continue multiplying by the number to the left,
indenting each new partial product one place.
The zero should not have been written in the second
partial product (second example), but the " habit " formed
should have been to write the right-hand figure of each
partial product directly under the digit used as multiplier.
We must be sure that the child does not form habits that
will lead to errors in special cases.
28
TEACHERS' MANUAL
CASE XI. A teacher in a second grade used cards for drill in
subtraction, written :
II
CO
1
00
7 - 4 =
II
!
OS
In reciting, the children were required to say, "8 minus 3 equals 5,"
and so on. Progress was very slow. When a child could not give
an answer, he went to the blackboard, made a number of lines, and
erased part of them to find the answer.
There were several things wrong with this procedure:
(I) The pupils should not have seen the figures written in
this order. They should have been written in column form
as he is to see them in written work. (2) The child should have
given nothing but the answer. Repeating the phrase in no
way helped the memory. (3) The language " 8 minus 3 "
was new and meaningless to him. The new fact was thus
unrelated to the corresponding addition fact which he knew.
So there was no connection here between " the new and the
old." The question should have been, " 3 and what make
8? " when developing the first notion of subtraction.
(4) The child should not have found the fact in the way
he did. Thus found it became a new fact unrelated to the
addition fact already known. The drill cards should have
been :
8
3
3
?
8
When the child could not give an answer quickly when
seeing the drill side of the card, the other side should have
been shown. This connects the new with the old. That
ECONOMY IN TEACHING
29
is, it uses all that can possibly be used of the " transfer of
habit " from addition to subtraction.
CASE XII. In an application lesson in multiplication, the
teacher proceeded as follows: "An orange costs 5^. How can I
find the cost of 3?" The reply was, "Times 5ji by 3."
The teacher had evidently confused the two ways of
writing and reading multiplication, or she had attempted
to teach the children both forms and they were confused.
As the «children were not corrected, evidently they had been
taught to use this form. There are two ways of writing and
reading multiplication. 3 X 5^ is read, " 3 times 5^" ; and
5ji X 3 is read, " 5^ multiplied by 3." The first form is
preferable, for it means more to the child. One form only
should be taught. The teacher should be as careful to use
and require as good English in an arithmetic lesson as in an
English lesson.
CASE XIII. A teacher in a third-grade class drilled in division
by having children recite the "tables." The children began,
"3 goes into 3 one time" ; " 3 goes into 6 two times " ; "3 goes
into 9 three times " ; and so on, to "3 goes into 36 twelve times."
The tables were written 6 -í- 3 = 2.
A worse procedure cannot be imagined. (1) Pupils should
not learn the facts by " tables." (2) The language was new
and meaningless. What mental picture could such language
give the child? (3) There was no connection between divi¬
sion and multiplication ; and thus the facts had to be mem¬
orized independently of former habits. (4) The written
form should have been 3)6, as the pupil is to see it in written
work. (5) 30 -Í- 3 ; 33 3 ; and 36 3 are not primary
facts.
When the " 3-times " facts are thoroughly known, the
30
TEACHERS' MANUAL
teacher should have drilled upon 3 X ? = 12 from sight
and dictation, saying, " 3 times what are 12? " and so on.
When this is well done, she should have shown that 3)12 is
another and shorter way of asking the question, " 3 times
what are 12 ? " In this way, the division facts are known
at once from the multiplication facts. The drill cards
should have been :
3)12
When a child hesitates, he should be shown the reverse
side of the card. He should give answers only. When he
has occasion to read 3) 12, he should say, " 12 divided by 3 is 4."
4
CASE XIV. A teacher in the third grade used the long-division
form in written division from the first, when having one-figured
divisors, in the thought that she was developing a habit that must
be used when the child comes to two-figured divisors.
She was " penny wise and pound foolish." She was build¬
ing up a very wasteful habit that will some day have to be
changed (and habits such as this are hard to break), to save
developing a new habit in long division that really is de¬
veloped with little if any difläculty.
CASE XV. A teacher in long division'had the pupils make out
a table for each exercise before beginning the division. Thus, for
972 -j- 36, the children made a complete table 1 X 36 ; 2 X 36 ;
and so on, to 9 X 36.
This procedure in no way developed the new ability needed
for skill in long division, that of estimating the quotient
figure. The ability to estimate the quotient figure is difiB-
cult to develop. There are many grades of difläculty, and
3 X ? = 12
ECONOMY IN TEACHING
31
the process of long division is the most complex and difiScuIt
of all the processes. It requires the development of a series
of abilities and a careful gradation of exercises as to diffi¬
culty. See the author's How to Teach Primary Number.
The needed abilities are carefully built up in the text.
CASE XVI. A teacher began long division with the "teen
family." Thus, 432 16; 490 -i-15; etc. The class made
many guesses and trials before finding the right quotient figures.
She had selected the most difi&cult type of two-figured
divisors. The ability to estimate closely the quotient
figure is the new ability to be developed. This comes from
the ability developed in short division, followed by the
ability to give such quotients as 80 20 ; 60 30 ;
85 20 ; 96 30 ; 92 -f- 20 ; and so on. This leads to
divisors nearly a multiple of ten. So the easiest divisors are
21, 31, 41, 51, and so on. The first exercises should be those
in which the " trial quotient " is the " true quotient." Such
exercises are followed later by those in which the " trial
quotient " is 1 larger than the " true quotient."
Vbky Easy Very Hard
1. 31)1395 1. 47)3572
2. 21)1344 2. 56)4872
3. 41)1476 3. 67)2546
In the " very easy " list the estimate in (1) comes from
3)13, known in short division, and is the "true quotient."
Such exercises are used until the steps in the procedure are
developed.
In the " very hard " group the very best of the common
procedures is to get the " trial quotient " in (1) from 4)35,
32
TEACHERS' MANUAL
and then to multiply mentally 47 by 8 (the trial quotient)
and find that the product is more than 357, and hence write
7 instead of 8 as the " true quotient." If this method is
preferred, there must be much drill in giving the product of a
two-figured number by a one-figured number without a
pencil. A better method is shown in this Manual under the
work of the fourth year, and in the textbook. It is given
briefly below.
This drill can be avoided by drill upon 40)357, and so on,
until the child can give the quotient and remainder at sight-
Then in 47)357 he thinks, " 8 and 37 remaining," as he looks
at 4 and 35. Then he thinks that 37 is less than 8X7.
The author has used this method with splendid success.
This, as any method, depends upon building up the proper
sequence of habits. In fact, all that any " method " re¬
quires is the building up of the needed habits that function
in the final finished process. The habits to be built up de¬
pend upon the final method desired. See Chapter VII.
SuMMAET OF PkINCIPLES
1. Always drill with a purpose. Know what the child
needs to know and see that he knows it before proceeding
to a new fact or process in which this knowledge is needed.
2. Suit the amount of drill to the difficulty of the task.
Facts that recur frequently in out-of-school activities need
less drill in order to fix them than those that do not. The
sum of " 2 and 3 " recurs hundreds of times to " 7 and 9 "
once. So the latter needs much more schoolroom drill than
the former does.
3. Make frequent inventory tests to find what the child
knows and what he does not know. Then drill him upon
what he does not know rather than upon what he does know.
4. Do not develop habits that will have to be broken
ECONOMY IN TEACHING
33
later, that lead to slow and inaccurate work, or that lead
to errors in special cases.
5. Do not develop schoolroom habits not used in life.
Thus, do not write the quotient above the dividend in short
division. It is but a " schoolroom habit " that must be
broken later.
6. Before teaching any written process, be sure to ana¬
lyze it and see the primary abihties needed. Then drill
upon exercises that will develop each ability until it be¬
comes a fixed habit.
7. Do your best to make the work attractive. Without
interest and attention there can be but little if any prog¬
ress.
8. When using games to motivate drill work, see that the
interest in number is greater than in the game itself. The
games must not be too highly organized ; all combinations
of the series must come up ; all pupils must be interested in
all combinations that arise; the slow pupil must not be
eliminated ; and the games should be of a nature that will
be used out of school.
9. The best type of motivation or interest comes from a
desire to excel a former record. A daily graph of progress
is the best way to stimulate interest.
10. A pupil should always know the purpose of the drill
lesson. Thus, in the upper primary grades, he must see the
necessity of trying to give the ICQ combinations in addition
in a fixed time, in order to do larger exercises in a fixed time ;
or that he must do a certain number of exercises in " adding
by endings " in a fixed time, in order to add a certain num¬
ber of columns in a fixed time.
But, in the final analysis, success in teaching any subject
depends upon the teacher. Teaching is not an easy-going
occupation. It is a very exacting and scientific profession.
34
TEACHERS' MANUAL
depending not only upon a careful study of psychology and
pedagogy, but upon our inborn ability to teach. Teachers
like poets are " born not made." A good teacher may over¬
come the handicap of a poor textbook. But a poor teacher
will fail with the best.
CHAPTER in
HABIT FORMATION AND DIAGNOSTIC TESTING
The child learns most or all of the 100 primary facts of
addition and subtraction during the second year, but when
he begins the work of the third year it should not be assumed
that he fcnows them ; that is, that he has an automatic con¬
trol of them so that he can give them quickly and
accurately.
Success in developing skill in any process in arithmetic
demands that all habits needed in the finished process be
built up in a way that each habit contributes to the next.
The 100 Peimaky Facts of Addition
{The page numbers refer to Stone's Primary Arithmetic and Third-Year
and Fourth-Year Arithmetic)
Before further progress can be made in addition, all pri¬
mary facts must be known. The pupil must not " guess,"
count, or hesitate in giving them. Thus, on page 2, are
given " adding 2." The sum is written so as to give an
eye picture of the fact. These facts should be read by
the pupil. That is, he should say, " 7 and 2 are 9 " ;
and when dictated or read by the teacher, he should respond,
" nine," to " 7 and 2." The " dictation drill " is as im¬
portant as the " eye drill," for, in using the facts in adding,
he must think and not see the number (a former sum)
to which a number is added. Thus, on page 3, the pupil
is applying the facts learned on page 2 to adding three num¬
bers. But in the first column he sees 6 and 1 and thinks 7.
35
36
TEACHERS' MANUAL
But without seeing 7, he must now think 9 when he sees 2.
It should be evident, then, that dictation drill is necessary.
On page 4, new facts are given to be learned in the same
way and then applied to adding three numbers. On page
6, exercises are given that use the facts already known in
adding three two-figured numbers. This is to develop the
habit of beginning with the right-hand column to add.
The plan throughout the text is to build up the necessary
habits needed to get skill in computation. The teacher
should observe carefully the purpose of each drill exercise
in the text as explained above for the first seven pages.
Abilities Needed in Written Addition
By analyzing the skills or habits used in the final written
addition it wül be seen that there are at least 10 habits to
be built up before skill in adding 6 three-figured numbers
can be expected. They are the following types :
(A)
(B)
(O
(D)
(E)
(F)
(G)
(H)
(1)
(J)
(K)
3
4
21
38
17
8
17
8
7
5
365
5
2
M
26
_2
6
_5
9
6
7
426
3
5
5
9
6
978
3
8
4
342
685
376
1. By (A) is meant the automatic control of the 100
primary facts of addition. This means all possible com¬
binations with two one-figured numbers, including zero,
written in reverse orders. Thus :
2 5 0 3 8 1
5 2 3 0 1 8 etc.
HABIT FORMATION AND DIAGNOSTIC TESTING 37
This one habit listed as (A) is really two habits. For the
pupil must be able to give the facts as well from dictation as
from seeing the figures.
2. By (B) is meant skill in using the 100 facts in adding
three or four numbers that include these facts only. Thus,
4 6
the example used in (B) uses two of the facts ^ and g.
Notice that this differs from (F), which uses one of (A)
8 14 8
g and one of (E) while (H) uses one of (A) ^ and one of
(G)^.
3. By (C) is meant using the skills developed in (A) and
(B) in getting the new habit of beginning to add with the
right-hand column. In reading, writing, and writing num¬
bers, the pupil has worked from left to right ; so it takes
some practice to get the habit of beginning at the right to
add.
4. By (D) is meant the new habit of recording the right-
hand digit of a sum and carrying the other to the next
column. Many of the errors in addition are due to the lack
of this habit, so (D) needs careful drill. That the three
needed habits may be carefully developed before this new
one is begun, this new habit of carrying is delayed untU
page 28 in the textbook. Why we carry needs no further
explanation than that given in the text. It is not " ration¬
alization," but " habituation " that we are working for.
5. By (E) is meant the ability to add a two-figured
number and a one-figured number where the sum remains
in the same decade. That is, where the sum of the ones
of the two-figured number and the one-figured number does
not exceed 9. We cannot assume that the " transfer of
38
TEACHERS' MANUAL
7 17
habit " from „ to „ comes without drill. With those pupils
with native ability the transfer is very easy, but with the less
intellectual the transfer is but slight and much drill is
needed. The adding of 15 and 3, 24 and 2, and all like com¬
binations, must be a single " response " and not a double
one. That is, a pupil must not " think " the sum of the
" ones," and then " think " the " tens," and then the whole
sum ; but he must, from drill and from associating the sum
with the {A) facts, see the sum at once. That is, from the
" key fact " he must announce the sum automatically,
giving such exercises about as quickly as he gives the 100
primary facts. Thus :
5 15 25 35
from 4 he must recognize _4 4 or 4
9 19 29 39
This may be called " adding by endings." That is, the
sums end like the primary facts.
There should be more " dictation drill " than " sight
drill " in developing this ability, for in using this skill in
adding columns, the pupil does not see the two-figured
number.
6. By (F) is meant applying (E) to adding three or more
one-figured numbers. It differs from (E) in that in these
drills the pupil either saw or heard both numbers. But in
(F) he must transfer this ability to " thinking " the two-
figured number and seeing the one-figured number. These
should be pure sight drills. There is but little if any value
in using these as dictation drills.
7. While (G) is adding a two-figured and a one-figured
number, it is a new habit ; for in ((?) the sum is a number in
HABIT FORMATION AND DIAGNOSTIC TESTING 39
the next decade ending like the " key fact " from the pri¬
mary facts whose sums are 10 or more. That is :
8 18 28 38
from _7 the pupil must give _2_ ®tc.
15 25 35 45
The pupil must do this as a single act and not by adding
the " ones " first. The drills must seek to get all possible
" transfer of habit " from associating these from the " key
facts " from which they come ; that is, by observing the
" endings." This is sometimes called " adding by endings."
The drills should be both sight and dictation drills.
8. By ( H) is meant applying (G) to adding three or more
numbers. This habit requires development, for the child
neither sees nor hears the two-figured number. That is
why most of the drill in type {G) must be oral.
9. By (7) and (/) is meant the use of all former habits
in longer columns. This is to cultivate the power of holding
the attention upon a thing for a longer time. In (5), {F),
and (i7), the pupil had to hold his attention to one thing
without a rest until he made two combinations. This so-
called attention span differs in different pupils. But in all
pupils there is a limit to which the attention can be held
upon a single thing without a break. So drills of the type
of (7) and (J) are necessary to build up the power to fix
the attention for a span without its breaking down. The
text limits such drills for the third grade to four two- or
three-figured numbers and to five one-figured nixmbers, so
that the attention span is limited to four combinations
without a pause.
10. For future work with longer columns there are no
new abilities or habits needed. But the pupil needs much
drill upon the 10 types of drill given here throughout all
40
TEACHERS' MANUAL
grades, in order that the " recall " of any needed fact be¬
comes a matter of automatic habit, so that the attention
span may be increased without fatigue.
The textbook takes up these types in order. This dis¬
cussion is to show the teacher their need in order that they
will not be neglected.
Diagnostic Testing
The secret of economical and successful teaching lies in
the axiomatic fact that
A teacher's time should he spent in teaching a child that
which he should know and does not already know, instead of
teaching him what he already knows or does not need to know.
As was shown in the preceding section, there are at least
ten skills required in simple double-column addition of six
numbers. The teacher must know just what ones of these
skills or habits are needed by each individual pupil. And the
pupil himseh must know what he needs and why he needs
it, and he must work with a purpose. Most drill work in
many schools is largely a sort of treadmill, purposeless
work, boresome to both pupil and teacher. In an attempt
to stimulate an artificial interest, to relieve the boresome-
ness, foohsh and silly games that lead nowhere are often
resorted to. The author recently saw a " drill lesson " in
multiplication in a fourth grade in a demonstration school
of a teachers' training school that well represents this waste
of time. The pupils were " playing store." Tables were
filled with empty boxes, cartons, cans, etc. The pupils
had brought prices of eggs, milk, butter, fruit, etc. These
were written on the blackboard in full view of all. The
prices were such as 20^, 35^5, etc. All numbers ended in
0 or 5. Three of the nine digits did not occur in the prices.
After a large part of the " drill period " had been spent in
HABIT FORMATION AND DIAGNOSTIC TESTING 41
getting ready, electing a clerk, etc., the pupils were told
to go up and buy any niunber of any one article they wished
and to pay the clerk for it. No child bought more than
two articles. So (excepting the clerk) no child got " drill "
in more than two of the 100 multiplication facts, and these
were from the " 2-times " facts, one of which was 2X5
or 2 X 0. As this had been a common practice for many
weeks, the pupils no longer needed the " 2-times " facts
and so the efficiency of this drill period was zero. And even
if the pupils needed such drill, the two facts that they had
during the 30-minute drill period might not have been the
ones needed. The chance is that they were not. I can
conceive of no possible situation in which such a lesson could
have been marked " 1% efficient."
1. The teacher must know what each child needs in a cer¬
tain grade.
2. By diagnostic testing, she must find what he knows and
what he does not know.
3. She must drill him upon what he does not know rather
than upon what he knows.
4. And she must remember that it is not merely recalling
facts, but recalling them with an effort to remember them that
fixes them.
Three Types of Tests
A pupil is expected to achieve a certain skill in a certain
grade. For example, he may be supposed to do about 7
or 8 addition exercises of 6 three-figured numbers in 5 min¬
utes at the end of the fifth grade. To give him such a group
of exercises, containing all of the 100 primary combinations,
is to give him an achievement test. Now in case he falls
far short of the required standard, the reason must be
found. To locate the trouble, a diagnostic test is given.
42
TEACHERS' MANUAL
This test selects a few exercises requiring the fundamental
skills needed. There should be several exercises of each
group so as to eliminate accidental errors. Since such a
test does not test in all the facts of each group, but seeks
to locate any lacking ability, another type of test is needed.
For example, the diagnostic test wül not show lack of skiU
in all of the primary facts. Then must follow inventory
tests. These are tests on all facts of a series. In addition,
they should include all of the 100 primary combinations ;
all the " adding-by-ending " group that have a sum in the
same decade ; and all of the group whose sums are in the
next decade. This will locate the facts that are not known
and must be followed by a remedial drill.
Following this must come drill in adding single columns
in order to build up the attention span.
The same plan is followed in each process.
How to Tell the Time Limit in Giving
Diagnostic and Inventoby Tests
Suppose that you are working for skill to add 5 " three-
by-six " exercises in four minutes, which would be an ex¬
cellent skill to expect at the end of the fourth grade. There
are 17 combinations in each exercise, or 85 combinations
in all. That means an average of over 21 combinations per
minute. Such an exercise is a complex of the several skills
as noted in a preceding section. Manifestly, on account
of the attention needed, and on account of the complex of
the skills used, we cannot expect as quick a response as
when responding to the exercises in the primary drills. It
is reasonable, then, in such exercises as ^ or that we
should expect a rate of from 28 to 30 combinations per
minute before getting 85 complex combinations in 4 minutes.
HABIT FORMATION AND DIAGNOSTIC TESTING 43
Also we should expect at least 13 exercises per minute such as
5 8
3 or 7, requiring two responses each, and 8 such exercises
7 6
5
3
as g, requiring three responses each, before expecting 21
6
combinations in more complex situations.
Thefihild should understand all this, know what he is work¬
ing for, the purpose of all drills, and of the time limit. This
should be the motive for drill, and there should he no need of
other devices except upon very rare occasions.
Teaching Subtraction
Economy in teaching and in learning demands that each
new habit, when possible, be made to grow out of some for¬
mer habit. Subtraction is but the inverse of addition. The
transfer of habit from addition to subtraction is very
easy if properly presented. The first notion of subtraction
should be such as " 2 and what make 5? " not " 2 from 5
leaves what ?" It takes drill, of course, to make the re¬
sponse automatic when seeing the written form. Until the
g
pupil gets an automatic control of the written form, as _,
O
he thinks " 5 and 3 are 8," and announces "3."
Through problems he must then see the three uses of sub¬
traction and use the proper language for each. This has
been carefully done in the text.
By analyzing the habits needed in written subtraction
" without carrying," it is seen that no new habit beyond the
table is needed. That is, the pupil needs that part of the
44
TEACHERS' MANUAL
table that includes one-figured numbers only. He has the
habit of beginning at the right, learned in addition.
In order to link closely subtraction with addition, using
the habits formed there, this first phase of subtraction is
taken up in the text on pages 7-15 before the addition facts
are finished.
Caebying in Subtbaction
Carrying in subtraction comes up in connection with
those primary subtraction facts of a two-figured and a one-
figured number as :
10 10 11 12 12 13 14 15
^_5_6^_3_8_9_6 etc.
In subtracting the pupil must associate ? with ^2»
äO O O
1 .,,11 2 ~12 , ~ ~
„ with „ with and so on.
6 _6 ^ _3
There are two fundamental habits, then, to be developed
in order to get skill in written subtraction. They are :
1. Thinking in the second group of the subtraction facts
when the digit is less than the one below it; and
2. Carrying 1 whenever these facts are used.
Subtraction, ^^hen, requires but two habits and a short
attention span, and hence it requires much less drill than
addition. To attempt to teach subtraction, however, be¬
fore the child has a perfect control of the corresponding
addition facts is a waste of time. One of the most serious
errors made in teaching is that of trying to teach a thing
before a child is prepared for it.
In developing the process, follow the method of the text.
If local problems are more vital and concrete than those in the
text, substitute them, but follow the method of the textbook.
HABIT FORMATION AND DIAGNOSTIC TESTING 45
Testing Work in Subtraction
In an achievement test in written work, all of the primary
facts should be included. Some subtractions should re¬
quire carrying and others not. If the result is not satis¬
factory, the reason is not hard to find. Give an inventory
test of all the 100 primary facts. If any of these are not
known, drill until they are. The only other trouble is that
the child fails to carry 1 when he should, or carries 1 when
he should not. Drill to fix the proper habits in these two
cases.'
Multiplication
As in the other processes, the final written process is a
complex of several primary abilities or habits. These are :
In simple multiplication —
1. The 100 primary facts.
2. Adding a one-figured number to a two-figured number.
In compound multiplication —
3. In addition to 1 and 2, placing the partial products in
their proper place.
The Primary Facts
The 100 primary facts include all possible combinations
of two one-figured numbers including zero, written in re¬
verse order. Thus :
0X0 = 0 3X0 = 0 0X3 = 0
3X5 = 15 5X3 = 15 etc.
In multiplication, the child sees the figures. Hence the
major part of the drill is sight drill. The form of the drill
differs from that of addition. In written addition the figures
appear in column form and hence the drills should be in this
46
TEACHERS' MANUAL
form. But in multiplication most products that must be
34Ô 6
recalled are not in column form. Thus, in g, the ^ is the
only pair in column form. Likewise the multiplication
needed in division is not in colunrn form. So the pupil must
become accustomed to giving the products when seeing the
figures written in any form.
Perhaps the forms 3 X 5, 4 X 6, 0 X 3, etc., are as good
for regular drill forms as any. These are better forms than
5 6 3
g ^ 0 these are not forms seen in written work while
they are forms used in addition and subtraction, and hence
may be confused with these processes. It is very probable
that errors made in multiplication in various surveys were
7 0 5
due to this form of drill. Children wrote 0 4 1^ etc.,
7 4 6
confusing them, no doubt, with addition. Had the tests
been written 0 X 7, 4 X 0, 1X5, the results might have
been different. Taking everything into consideration, it
seems better to use other than column forms for drills in
multiplication. Even the device of using a circle, as seen
in so many books and schoolrooms, is free from objection
in multiplication, but it is not a good form to use for the
other processes.
The flash card with the " drill " on one side and the
" facts " on the other is perhaps the best form for the early
drills :
3X7
3 X 7 = 21
HABIT FORMATION AND DIAGNOSTIC TESTING 47
or, they may include :
3X7
7X3
3 X 7 = 21
7 X 3 = 21
When the pupil cannot give the fact instantly, show him
the answer by turning the card. The common practice of
saying, " Now think. Think hard," heard in so many
classrooms, leads to wrong habits — to guessing or counting.
The facts should not be learned and recited as " tables."
They are given in the text as tables for reference and study,
and to show the child that he has a unit of work to
accomplish; but the drill upon them should be from a
miscellaneous arrangement.
A Second Ability Needed in Multiplication
Aside from the primary facts, in written work the child
must " think " a product and add some one-figured number
from 1 to 8 to it. In doing this, he sees neither the product
nor the number to add. This differs slightly from " adding
by endings " developed in addition, for there he saw the one-
figured number to be added. Since the " transfer of habit "
is so closely allied with that of addition, it is perhaps more
economical to get the needed habits from written multipli¬
cation itself than from other forms of drill. So written
multiplication is taken up in the text as soon as the facts are
learned.
In this way, the pupil develops the needed ability to add
a one-figured number to a product very gradually. Thus,
in multiplying by 2 he will never have to add more than 1 ;
in multiplying by 3, he will never have to add any number
but I or 2 ; and so on.
48
TEACHEES' MANUAL
Care must be used to see that he does not " carry " before
multiplying. The author saw a class in the third grade all
make the following errors :
36
28
47
38
3
2
3
2
128
66
Ï8Ï
86
The trouble is seen to be that they transferred the habit
of carrying in addition and thus added the number to be
carried before multiplying
Including 10, 11, and 12 in the Facts
That most teachers and most textbooks still include 10,
11, and 12 in their drills upon the primary facts shows how
purposeless is most drül, and how little thought is given to
building up needed habits for written work. It may be well
for a child to know 3 X 12 = 36, or that 3 X 25 = 75, etc.
But the primary facts needed in written work are merely the
100 combinations from 0 X 0 to 9 X 9, for these are the only
combinations ever needed in written work.
Zekos in the Multiplicand
Even children well-trained in the " zero facts," as 0 X 0,
3X0, 8X0, etc., often fail to " transfer the habit " to
written work ; hence such exercises as ^^2 ^^3» so on,
must be watched carefully when first presented.
Two- and Three-figured Multipliers
Before a child uses two-figured multipliers he should mul¬
tiply by 20, 30, and so on, to 90. This requires but one new
HABIT FORMATION AND DIAGNOSTIC TESTING 49
habit in addition to the habit already fixed — that of an¬
nexing a zero to the product. The form should be :
32 36 42 56
20 30 40 20 and so on.
In these he should first find 2 X 32, 3 X 36, 4 X 42, and
2 X 56, a habit already established. Then he should annex
a zero to the product, which is the only new habit. The
text develops this method.
Multiplying by a two-figured number is but a combina¬
tion of habits already formed ; (1) Multiplying by a one-
figured number ; (2) multiplying by 20, 30, 40, and so on ;
and (3) addition. It is a waste of time to take up this phase
of multiplication until these fundamental habits have been
well established.
In the written work, the form should be :
48 48
24 and not 24
192 192
96 960
n52 1152
as seen in some schools and in some textbooks.
Zeros in the Multiplier
Such an exercise as :
326
204
presents a new difficulty of which the teacher should be
conscious. It should be developed from
326 , 326
4 200
50
TEACHERS' MANUAL
just as 24 X 36 was previously developed from ^ 20*
The child should form the habit of writing the partial prod¬
ucts as follows ;
326
204
1304
652
66504
placing the first product of each multiplication imder the
multiplier used.
The habit of writing the products as shown below — a
form very commonly used — leads to errors in special cases :
326
204
1304
6520
66504
By using this form, instead of getting the proper habit of
placing the first product below the multiplier from which it
was obtained, he gets the " stair-step " habit of indenting
each new partial product. The author saw nearly all of a
class of high school graduates make the following error due
to the false habit shown above :
32456
20403
97368
1298240
649120
77991768
HABIT FORMATION AND DIAGNOSTIC TESTING 51
Diagnosing Errors in Multiplication
The author recently examined some test papers of a class
in 3B. The following were typical errors :
(^)
(B)
(C)
(D)
37
37
37
308
6
6
6
4
182
229
422
1502
In (A) the pupil did not have the habit of carrying ; in
(B) he did not know the " tables " ; in (C) he carried before
he multiplied ; in (D) he carried to 4 X 3 instead of to 4 X 0.
Each group needed " different treatment." Group (A)
needed to be drilled upon carrying ; group (B) needed more
drill upon certain primary facts; group (C) confused the
carrying habit with that of addition and had to be taught
how to carry in multiplication and be drilled until it becomes
a habit ; group (D) merely needs to be shown how to carry
to 0.
The author saw the papers of a test given at the end of a
" development lesson " in multiplication by two-figured
numbers. The test was given to see if the pupils could
write the work down in the proper form before giving them
home- or seat-work for practice.
Four pupils failed, making the following four types of
errors :
(^) (B)
32 41 32 41
^ _32 24 32
68 122 68 122
64 123
132 245
52 TEACHERS' MANUAL
(C) (D)
32
41
32
41
24
32
24
32
128
82
64
123
64
123
128
82
192
205
1344
943
Pupil (A) confused the work with the " habit " in addi¬
tion, that is, of adding each column separately. Hence he
found the product of the two numbers in each column.
This child evidently ranked very low in native intelligence
and had not the mental ability to get anything out of the
" development lesson."
Pupil (B) made the same mistake in the first partial prod¬
uct, but used the tens' digit properly in getting the second
partial product. However, he wrote it in the wrong place.
He did not get much from the development.
Pupil (C) had the ability to multiply by one-figured mul¬
tipliers, but failed to follow the development and generalize
as to where to place the products.
Pupil (D) multiplied by the tens' digit first, but wrote the
product as if it were from the ones' digit.
These children all need different treatment. Both (A)
and (B) evideutly lack both the primary habits needed for
this work and the native ability to profit by what the rest
of the class can and should do.
Both (C) and (D) evidently have the needed habits, but,
ranking somewhat lower than the average in native intelli¬
gence, need a little extra help outside of the regular drill period.
Pupils who had perfect papers should not be held back
and bored by having to listen to the teaching of these
four pupils. They should be allowed to go on and do all
the work of which they are capable, while these four are
HABIT FORMATION AND DIAGNOSTIC TESTING 53
being taught. Nothing bores a child like being re-taught
what he already knows. And we must remember that the
bright pupil is entitled to as much of our time as the dull
child is. In fact, we may be using time that belongs to the
bright pupil in attempting to teach a dull child that which
he cannot learn. In many schools the bright pupils often
become loafers, or are so bored by being held back for dull
students that they lose interest in the subject. Strive to
keep each student up to his native capacity.
•
Habit Formation in Division
A careful analysis of habits needed in both long and short
division will show the following general types :
(A) (B) (C) (D) (E) (F) (G)
2)8 2)^ 3)7 2)^ 20)^ 20)TO 30)95
(H) (7) (J) (K)
20)75 21)0^ 36)1692 18)810
Type (A) means the primary facts; (B) uses them in
written division without remainders; (C) is the division
with a remainder ; (D) is written division with a remainder ;
(E) is getting the quotient from 2^ or from the abilities
developed in (A) ; (F) is to recognize from 2)7_ that the
result is " 3 and 10 remaining " ; (G) is to recognize from 3)9
that the result is " 3 and 5 remaining"; (77) is to recognize
from 2)7 that the result is " 3 and 15 remaining " ; (7) is
to develop the steps in long division with divisors so that
2)9 gives the " trial quotient " 4, which is the " true quo¬
tient " ; (J) requires the ability to get the " true quotient "
from the " trial quotient " ; (K) requires a new skill, for 1)8
does not give a " trial quotient " that helps in the least in
getting the " true quotient."
54
TEACHERS' MANUAL
All these abilities have been carefully developed in the
textbook.
The Peimaby Facts of Division
The " transfer of habit " from multiplication to division
is simple and easy if proper language is used. If a
pupil knows a group of multiplication facts so well that he
can give the " missing number " when the product and one
of the numbers are known, he needs only the notation or
form of writing the division fact. Thus, if he can give the
products of 2 X 5, 3 X 4, 2 X 7, 3 X 6, 2 X 9, 3 X 8, and
so on, so weU that he can give the " missing number " in
2X? = 10 3X? = 12 2X? = 14
3X? = 18 2X? = 18 etc.
he is ready for the form :
2)10 3)12 2)14 3)18 2)18 etc.
The thought should be " 2 times 5 is 10," as he gives or
writes 5 ; "3 times 4 is 12," as he gives or writes 4 ; "2
times 7 is 14," as he writes 7 ; and so on.
In drill, the pupil should give the answer only. That is,
when looking at 2)10 he should say, " Five," not " 10 divided
by 2 is five."
Avoid meaningless words like " goes into," often pro¬
nounced " gus-sin-tu."
The drill in the primary facts must be both sight and
dictation drills.
A New Habit to Be Established
Analyzing the habits needed in dividing the following :
5)3795
759
HABIT FORMATION AND DIAGNOSTIC TESTING 55
you will see that the pupil made direct use of but one of the
primary facts. That was the last division of 5)45. And
in this he did not see the 45. He first saw 5)^, which is
not a primary fact. Next he had to " think " 5)29, not
a primary fact. Hence the pupil must get the habit of
calling the " quotient and remainder " for all numbers from
the divisor up to one less than ten times the divisor. Thus,
for dividing by 5 he needs 5)5, 5)6, 5)7, and so on, to 5)49.
For 8 he needs 8)8, 8)9, 8)10, and so on, to 8)79.
To give 5)^ as " 7 and 2 remaining," the pupil not only
thinks of the primary fact 5)^, but must think, " 35 and
2 are 37." You will notice, then, that in doing short divi¬
sion the pupil must not only know the primary facts, but he
must be able to subtract certain two-figured numbers with¬
out a pencil. This new habit or ability needed should be
developed through drill in the form needed and not from
drill in subtracting two two-figured numbers at sight. That
is, he should have much drill in giving the quotients and
remainders from 8^, 8)^, 8)76, etc., not from such drills
as 58 67 76
M, 72, as advised by some writers and given in
some textbooks.
The drills to form the proper habits are given after each
"table" in the text. They should be used daily until the
pupils answer with skill.
Drills Needed for Long Division
To give 20^, the pupil must see that the answer comes
from thinking 2)6. To give 20)70, he must look at 2 and 7,
but think, " 3 and 10 remaining " instead of " 3 and 1 re¬
maining," when it was 2^, instead of 20)70. To give
20)87, he looks at 2 and 8 and thinks 4 and sees the 7 as the
56
TEACHERS' MANUAL
remainder. To get 20)95, he sees 1 remaining from 2)9 and
then thinks 15 as the true remainder.
These abilities are all necessary in order that the " trial "
and " true " quotients can be found quickly in long division.
Drills to develop these are given in the text.
CHAPTER IV
PLANNING THE DAILY LESSON
There are three distinct types of work in arithmetic —
the development of a new fact or process to show its mean¬
ing and use; drill upon it to give the child an automatic
control of it ; and its use in life situations to lead to the
habit ^of using arithmetic in daily life and to develop the
reasoning power. The teacher must know which type of
lesson she is teaching, have a definite aim in teaching it, and
see that her aim has been accomplished.
The Development Lesson
A " development lesson " means, in general, that a new
process is built up from other established habits. So in
developing a new process, the teacher must know what old
habits are needed in the new and that the child has these
habits. For example, to develop the rule for multiplying
by a two-figured multiplier, two habits must have been
formed before developing the rule for this new process,
viz. : (1) Multiplying by a one-figured munber ; and (2) mul¬
tiplying by 20, 30, and so on.
Hence the teacher begins with these two problems and
finally combines them into one. (See page 210 ; 58.)
Taking a problem not given in the text, the steps are as
follows ;
1. One week Frank sold 9 rabbits at $.85 each. How much did
he get for them?
The children write : $ .85
9
$7.65
57
58
TEACHERS' MANUAL
2. The next week he sold 20 at $.85 each. How much did he
get for them?
The children write : $ .85
20
$17.00
3. How many rabbits did he sell in the two weeks? (Answer,
29.)
4. How much did he get for the 29 ?
The children write : $ 7.65
17.00
$24.65
5. Had he sold 29 all at one time at $.85 each, how much would
he have received? (Answer, $24.65.)
6. When you know how much he got for one and the number he
sold, how do you find out how much he got for all? (Answer,
multiply the cost of one by the number sold.)
7. To find out how much he got for 29 rabbits at $.85 each,
what is necessary? (Answer, multiply $.85 by 29.)
8. The teacher says, "This is the way we write it," as she
writes :
$.85
29
9. The teacher says, "We cannot multiply by 29 all at once,
so let's multiply by 9, just as if the 2 was not there." The children
write ;
$.85
^
$7.65
10. The teacher says, " See if this is what you got for the answer
to the first problem you had." They see that it is.
11. She says, "Since this is what he got for 9 rabbits, we must
still find what he got for how many more?" (Answer, 20 more.)
12. How do you multiply by 20? (Answer, multiply by 2 and
annex a zero to the product.)
PLANNING THE DAILY LESSON 59
13. See if you can multiply by 20 in this problem by not using
the 9.
The children write :
$ .85
29
$7.65
17.00
14. Here it may take a suggestion as to where to write the
product of 2 X $.85. If so, suggest that since they must annex a
zero, it should be written in the second place from the right.
15. Compare your answer with the one you got in problem 2.
(Answer, they are the same.)
16. In problem 4, how did you find what he got for 29 rabbits
when you knew how much he got for 9 and for 20 ? (Answer, add.)
17. Then what can you now do to get the answer?
The children write :
$ .85
29
$7.65
17.00
$24.65
The children now see that this is shorter than solving three
problems separately. It still remains to show that the zero
need not be written down as it does not affect the sum.
Neither is the first product pointed off. So have this form
written down :
$ .85
29
765
170
$24.65
Point out that the first product when multiplied by the
tens' digit is written in tens' place, or directly below it.
60
TEACHERS' MANUAL
Perhaps one more problem should be taken up in this
way. Then test results by giving three or four exercises to
see if the class has got the procedure. If so, follow with
drills to fix the right habit of procedure.
This is the most formal development that will occur in the
first four years. It is doubtful if this is of any great value.
However, it takes but a few minutes and the more intellec¬
tual members of the class may profit by it.
There is much less real " development work " now done
than formerly. The presentation in the text gives the child
all the " development " which he is capable of understand¬
ing. This, however, is not a return to " rote teaching."
In presenting any new fact or process, it is presented through
some concrete vital situation so that the child sees its mean¬
ing and the meaning of the notation.
The presentation is so simple in the text that a child should
be able to discover a new process for himself. Teach the
children to depend upon themselves by having them read
the developments in the text. If there are children who
cannot get the process from the text, let those who can ex¬
plain it to the others. Do not do for children what they
can do for themselves.
The lesson developed above was not divided into the
"five formal steps" of a development lesson. If interested
in a more formal discussion, see the author's The Teaching
of Arithmetic, Chapter XI.
The Deill Lesson
The drill lesson is the one in which most of the time is
devoted during the first four school years. And it is per¬
haps the most important. In planning it, both teacher and
pupil must know the purpose of it. Her aim may be more
far reaching than theirs, but both must work with a purpose
PLANNING THE DAILY LESSON 61
and realize at the end that the results have been accom¬
plished. The old copybook slogan that " practice makes
perfect " is only partly true. The full truth is " practice
makes perfect only when the results of practice are satis¬
fying to the one practicing."
One learns most easily when the learning is made pleasant.
The teacher's problem, then, is to make the work interesting
— to motivate it. Hence she must know what children
like and what bores them. They may, however, be satisfied
with her pleasant manner — a pleasing personality covers a
multitude of pedagogical blunders — a game, or other de¬
vice, or things that are not conducive to progress in arith¬
metic. I was recently invited to visit a second-grade class
to see how interesting the teacher made number work.
The teacher was attractive — pretty, bright, and alert.
The children were playing a game. They liked it and they
loved the teacher. You could see that they were sorry to
see the period close. They had more fun than if they had
been out at play. But in thirty minutes the pupils did not
give more than five or six addition combinations each, and
these combinations arose by chance, and may not have
been the ones needed. The children were interested, but
not in number. You must have interest, but be sure
that the interest is in number and not something irrelevant
to number.
It is not merely recalling a fact that fixes it, hut it is recalling
it with an attempt or a desire to remember it that fixes it.
The Length of Drill Period
No definite law can be laid down as to the proper length
of a drill period. We must " drill with attention." If the
drill is continued too long, fatigue may result and attention
lag. But the children are more likely to be bored by the
62
TEACHERS' MANUAL
mode of procedure than fatigued through concentrated
effort. It's the old story of
When I'm doing chores and errands,
I get so tired I think I'll drop ;
But when I'm playing hounds and hares,
I never want to stop.
In general, it is better to have but four or five minutes of
undivided attention in drills that do not have rest periods,
as giving the 100 primary facts, and then have a moment of
rest and change to another form of drill.
A longer period may be used when there are frequent
breaks or brief rests. Thus, in adding three-by-five exer¬
cises, the attention span is really 14 combinations with a
slight rest in going to the next exercise. From six to eight
minutes of continued drill in such exercises is not too much.
The time element, however, varies with the nature of the
drill and with the nature of the procedure. Fifteen or
twenty minutes without a change is not too much in drilling
through games. The time varies also with the age of the
child, and with his native abihty.
The Application Lesson, or Problems
In teaching a child to apply his knowledge of the processes
to problems, two things must be kept in mind : (1) He must
know the meaning of the processes; and (2) the problem
must picture conditions within his experiences.
The aim in giving a problem is not primarily to get drill in
the processes themselves (yet the numbers used should be
so chosen as to give valuable drill), but to develop the power
to interpret the problem and reason what to do. For that
reason, "problems without numbers " may be used to advan¬
tage. Thus :
PLANNING THE DAILY LESSON 63
1. Nell knows the cost of a yard of ribbon and wishes to know
the cost of several yards. How can she find it? (Answer, multi¬
ply-)
2. Mary knows how much money she had when she went shop¬
ping and how much she had when she returned home. How can
she find how much she spent? (Answer, subtract.)
3. Frank knows how much he gave for a whole flock of hens.
How can he find how much each hen cost him? (Answer, divide.)
4. Walter knows how much he earned each week for several
weeks. How can he find how much he earned during the whole
time? • (Answer, add.)
In the same way, problems containing numbers may be
used without actually doing the computation. Thus, the
class may read silently :
Walter earned $1.85 one week and $2.10 the next. He spent
$1.35 for a pair of skates. How much had he left?
Ask, " How should the problem be solved? " The answer
should be, " Add what he earned and then subtract what he
spent from the sum."
Using the Textbook
A textbook in arithmetic cannot be used as a reader or
history by assigning a certain number of pages each day.
The book is arranged topically and psychologically and
should be followed in the sense that each topic and sub-topic
must be understood before going on to the next. But there
should be very frequent reviews of back work. Part of the
class period may be spent in new work — reading a new
development or solving problems — and part of it in review¬
ing former work as fundamental drills.
The book is written for children and to children. The
boxed developments in italics are just what the teacher would
64
TEACHERS' MANUAL
do if the work was developed orally. Since one of the func¬
tions of the school is to fit pupils to educate themselves
through the printed page, these developments should be
read silently by the pupil either in class or as an out-of-class
assignment and he should get the how to do the work by him¬
self if possible. Let those that can get it explain it to the
class. There will be some work in the lower grades where
the teacher may have to go over the new work with the class,
developing it as it is done in the book. But in so far as
possible, train the pupil to interpret the printed page instead
of depending upon the teacher.
In solving the problems, the purpose is to train pupils to
interpret the conditions from the printed page and discover
what to do. So have the problem read silently and then call
upon some pupil to state the problem in his own words and
tell what process or processes to apply. When the aim is the
understanding of the problem and what processes to apply,
the computation need not be done during the class period,
but this may be left for seat-work.
Always be on the alert for real local problems and for
problems arising in school projects.
The drills should be done in two ways : The pupils should
turn to a drill exercise, work for a certain number of minutes,
then check theic work and announce their results. Results
should be kept and the same drill used later, and results
compared. The pupils are thus spurred on to do their best
and to try to beat former records. At other times the exer¬
cises should be dictated by the teacher and the numbers
copied and the work done. Always work for the greatest
possible concentration on the part of the pupil. When drills
are assigned for seat- or home-work, encourage pupils to keep
the time spent on an exercise so as to encourage their best
work instead of dawdling over it. Work always for accu-
PLANNING THE DAILY LESSON 65
racy and speed, but do not overlook the checking of results.
This must become a habit.
Drilling on Needed Facts
After a series of number facts have been taught, it will be
found that some of these facts become more quickly fixed
than others. This is due partly to the fact that the pupil
uses some of them more often than others. As soon as a
teacher discovers that certain facts are fairly well fixed,
more attention should be given to those that are not. Thus,
if the 100 primary facts have been taught, an " inventory
test " should be given to find what ones are well fixed and
what ones are not.
An inventory test means one covering all the primary
facts that are supposed to be known, or that have been pre¬
sented, to see just how many of them have been so thoroughly
fixed that the child recalls them automatically.
CHAPTER V
THE WORK OF THE FIRST TWO YEARS
It has been shown by scientific studies that time spent in
formal number work in the first grade can be used to better
advantage in reading and in other school activities. And
most modern courses omit formal number work in the first
school year.
There are, however, many situations in the school and
home, and in play, that call for some knowledge of number.
But all these needs are confined to counting, to reading num¬
bers, and to certain vocabulary. But counting, reading
number, and the child's general idea of number are built up
in connection with his other school activities.
The Work of the First Year
1. vocabulary. There is a certain vocabulary that has
to do with size, number, and position that is gained through
regular home and school experiences. They include such
notions as : Right, left ; in front, behind ; above, below ;
long,''short ; large, small ; and such comparisons as : Long,
longer; short, shorter; high, higher; wide, wider; small,
smaller ; also : Taller, tallest ; wider, widest ; longer, longest ;
nearer, nearest ; etc.
In his vocabulary the child also uses such terms as : Inch,
foot,yard; pint,quart; pound; hour,minute; squares,rec¬
tangles, circles ; and there may be other units of measure in
certain localities, as : Bushel, peck, acre, mile, etc., that
should be included in his vocabulary.
66
THE WORK OF THE FIRST TWO YEARS 67
2. counting. The child likes to count and needs to
count many objects that enter into his interests, needs, and
activities. Perhaps this need and interest does not extend
beyond 100.
a. Rhythmic or rote counting. In order to get the sequence
or order of the names, the child should first learn to repeat
the names in proper order without having his attention called
to what they mean. They should be grouped as follows :
^ Counting by I's to 10.
Counting by lO's to 100.
Counting by I's from 10 to 20.
Counting by I's to 100.
h. Rational counting. By rational counting is meant the
applying of the terms learned to counting things. Thus, a
child may count the children present, the number of boys in
the class, the number of girls in the class, the number of times
a ball is bounced, the books, pictures, windows, walls, chairs,
etc.
3. reading numbers. The need of reading numbers
is largely confined to reading the number of a page in his
book, house numbers, the numbers on a calendar, room
numbers, and the numbers on a clock face.
4. writing numbers. The child should learn to make
the figures neatly and to write the nine digits, and if need
arises, write any number up to 100 or 200 from dictation.
The Work of the Second Year
The Objective
The chief objective of the second year is an automatic
control of the primary facts of addition and the correspond-
68
TEACHERS' MANUAL
ing subtraction facts. The child gets some skill during this
year in using these facts in adding three and four one-figured
numbers, and some skill in adding a two-figured and one-
figured number to prepare for further work of adding col¬
umns. He should enjoy his work with numbers, desire to
know more about them, and see that they are useful to him
in many ways.
The Use of a Textbook
{The page numbers refer to Stone's A Child's Book of Number)
While the primary facts may be taught without having a
book in the hands of the pupil, and proper habits may be
established by following the outline of this Manual, the child
will get more enjoyment from his work and learn to read a
problem and interpret the situation, and much of the time
of the teacher will be saved, by having A Child's Book of
Number in the hands of each child.
The book furnishes splendid material for silent reading,
and through its use the pupil will be much better able to
read and interpret the problems that must be met in the
following years.
Three Essential Phases of the Work
There are three important phases of the work that must
receive proper attention: (1) The first presentation of a
fact ; (2) the drill to develop an automatic control of it ;
and (3) its use in simple everyday life situations.
The importance of the first presentation is twofold :
Through a concrete presentation of the fact, the child sees its
meaning, use, and how to find it ; and he sees the meaning
of the notation.
The drill work is to give an automatic response to the
THE WORK OF THE FIRST TWO YEARS 69
primary combinations when the figures are seen or when
the names are heard.
The applications to life situations are to motivate learn¬
ing, and to lead to the habit of using number whenever it
is needed.
Pbesenting the Addition Facts
Enough of the facts should be developed with concrete
objects to show the child the meaning of addition and how
to find* the facts for himself. At the same time that the
facts are being developed, the child should be shown the
notation ; that is, how the figures say them. The child
that knows how to count can see at once that adding 1 is to
call the next number in the series of counting.
Showing the child a number of objects, add one more and
ask, " How many? " until he senses that he counts on one
more. Let him take page 3 of A Child's Book of Number.
Have the class read the page silently, filling the blanks.
Then ask some pupil to read it, placing the proper word
where the blank is.
Then the class should read silently, "What the figures say,"
at the bottom of the page, until anyone can stand and say ;
" One and one is two," " two and one is three," and so on.
In drill, the child should give the sum only, but this exercise
is to see that the child can interpret the meaning of the
symbols and the notation.
Drills in Addition
Throughout the text the complete fact is given on the
" development " page, as on pages 3, 4, 7, and so on. Have
each pupil copy these facts, when first developed, in black
crayon on cards of tag stock about 2" by 3". Then on the
70
TEACHERS' MANUAL
reverse side copy the figures above the line only. Thus, on
one side will be :
2 7 13
3 2 6 3 and so on.
5 9 7 6
On the reverse side will be :
2 7 13
3 2 6 3
This is done by the pupils as seat-work.
How the Pupil Uses the Cards
Have the pupil lay his cards on his desk, showing the
numbers and answers, and study them ; that is, thinking
what each one says. Then have him lay them with reverse
side up and as he "thinks " the sum, turn the card to see if
he thought right, having him throw out any card he missed,
for further study. Or instead of merely " thinking " the
sum, he may write it.
To use the cards in class period, the cards are laid on the
desk with the answer side down. The teacher says, " Show
me the numbers that make ," asking for any sum
learned. Thus, if she asks for 7, each pupil picks up the
cards that make 7, showing the teacher the side :
6 5 4 1 2 3
1 2 3 6 5 4
As he does this he sees the side :
6 5 4 1 2 3
1 2 3 6 5 4, and thus dis-
7 7 7 7 7 7
covers any error that may be made. In this way each pupil
is participating in the recitation all of the time, and any error
is immediately corrected.
THE WORK OF THE FIRST TWO YEARS 71
If a pupil makes an error, that card is thrown out for fur¬
ther study. There is thus a daily diagnosis of each pupil's
difficulties, and each pupil is working on the facts that he
needs instead of drilling on those that he does not need.
A Game with the Cards
For seat-work and home-work, teach the children to play
a game as follows : Lay all cards on the desk or table with
reverse side up and quickly think or write the sum of any
card anc^ then turn the card over. If he " thinks " wrong,
that card is laid to one side for study. When no card is
missed, the child wins the game. Have pupils report how
many games they won and how many they lost during their
seat-work or at home.
Be sure to impress upon pupils that when they cannot
give the sum instantly, the card must be thrown out for
further study. To hesitate is as bad as to fail, for the child
is either guessing or counting. We must be sure that no bad
habits are being formed.
Flash-card Drill
Besides the drills suggested with the pupils' cards, there
should be other forms. The teacher should make " flash
cards " about 4" by 1" on a good grade of tag stock. These
are quickly made by using a fine brush and india ink, or with
black crayon. But the figures must be large and distinct
so as to be easily seen in all parts of the room. In using
them, each child must make every combination. To secure
the attention of all, the class may be divided into two teams
and see which team wins the most points.
To use the cards, the " drill " is on one side and the com¬
plete fact on the other, just as the pupils' cards were made.
Wheal a pupil of one team makes an error or hesitates, the
72
TEACHERS' MANUAL
teacher says, "Next," and anyone gives the answer. The
teacher then shows the side of the card with the answer.
If the team whose member is reciting is the first to give the
answer, it keeps the other team from scoring. But if the
opposite team gives the answer first, it scores a point for that
team.
Suppose that you are drilling upon the sixteen facts on
7 2
page 11 (or thirty-two when written as 2 and rjy it is often
desirable to have one pupil give all of them while the other
pupils watch for an error or a hesitation. In such cases,
anyone giving all facts wins a point for his team, in addition
to the point won in the way shown above.
Diagnostic Testing
Each pupil should work with an aim by working at what
he needs rather than aimlessly studying or drilling upon
facts of which he has a control. Hence, besides the daily
examination that each pupil is making of his own needs by
using his cards in the way that has been shown, there should
be frequent diagnostic or inventory tests given as follows :
Make mimeographed sheets of all facts that should be
known, or of the group upon which you wish to test the
class.
Make another sheet with the same facts, and in the same
order, with answers. Allow only the amount of time that
the pupil needs to record the answers, so as to allow no time
for " guessing " or counting. Instruct the pupils to leave
any answer that they cannot recall at once so as to finish
all that they do know. Then give each pupil a sheet with
answers. On this sheet he makes a line under all that were
missed or not written. This sheet is kept by him for study.
Any child giving all of the facts is given a star or some mark
THE WORK OF THE FIRST TWO YEARS 73
of recognition. In this way, each pupil knows his own
needs and is working with an aim upon what he needs. Thus,
very rapid progress can be made.
Developing the Subtraction Facts
The subtraction facts are only the addition facts asked in
another form, and written in another notation. The drills
on pages 11 and 15 lead to them and the games on pages
16 and 11 lead to the use and notation. An outline form of
development is as follows :
1. Write upon the blackboard the addition facts that are
thoroughly known and have the sums given. As they are
given write them.
2. Now have the class close their eyes; with a small
card cover one of the addends. The class then opens eyes
and gives the number covered. If they cannot, they do not
have a good control of addition and are not ready for sub¬
traction.
3. Next erase one of the addends in each exercise and
have the class give the missing number. If this can be done,
you are ready for the next step.
4. Now place a question mark in place of the missing
number, thus :
8
?
?
7
5
?
?
3
4
?
?
6
9
5
6
9
6
8
Have the pupils see that this asks the question, "8 and
what are 9? " " 3 and what are 5? " and so on. Have them
ask the question and answer it until all can do it quickly.
5. The final step is to show the notation. That is, show
that the questions asked above are asked as follows, by writ-
74
TEACHERS' MANUAL
ing each exercise below the corresponding form shown above.
Thus:
9 5 6 9 6 8 6
8 3 4 7 5 6 3
Have the pupils read these as, "8 and what are 9?"
" 3 and what are 5? " and so on, answering each question.
Do not use the new language of " minus," " less," " take
from," or " subtract " yet. Merely tell the class that when
the question is " 8 and what are 9? " and so on, it is called
" subtraction."
Drilling upon Subtraction
At first the child will have to think what number below the
line with the one above it makes the top number, since this
is a new form of the picture that he knows. But drill must
continue until the answers can be given as quickly as those
in addition. Cards may be made by the pupils as was done
in addition. Thus, on one side is the drill and on the other
the complete fact. This is not so essential as in the addition
facts, for if the addition facts are well known, the pupils
get an automatic control of the subtraction facts very
quickly. In a single lesson, the author has had pupils give
all subtraction facts corresponding to the sixteen addition
facts on page ll, almost as quickly as they gave the addition
facts.
Problems Using Subtraction
The first problems in subtraction must be those that the
child answers from addition. Thus, " John wants to buy a
top costing 8 cents. He has only 6ji. How much more
money does he need ? " " Mary is going to make 5 paper
dolls. She has made 3. How many more has she to make ? "
and so on. See the problems on page 34.
THE WORK OF THE FIRST TWO YEARS 75
The next problem is to find what is left. Thus, "John
had 8 cents. He spent 5jé for an ice-cream cone. How
much money had he left?" Do not have him think in a
new form of "5 from 8 leave what?" but think, "5 and
what make 8?"
The next problem is that of differences. Thus, " J ames has
8 marbles and John has 6. How many more has James? "
Encourage pupils to ask such questions as those on pages
40 and 41.
Adding Thkee oe More One-figured Numbers
As soon as a pupil knows a group of the primary facts of
addition he should use them in adding three or more num¬
bers. Thus, on pages 20 and 21 the child uses the facts
learned. The only difference between adding such col¬
umns and giving the primary number facts is that he has to
" think " two sums before announcing a result — one from
two figures that he sees, and one from thinking one and
seeing the other.
Adding Zero
Unless properly presented and drilled upon as one of the
primary facts, adding zero presents a real difficulty. To
give it meaning, the facts that include zero must be intro¬
duced through scoring games, as on pages 28 and 29.
Adding Two-figured Numbers
To give interest to the drill in adding columns and to
develop the habit of beginning at the right to add, the addi¬
tion of three or four two-figured numbers without carrying
may be taken up on pages 46 and 47. Do not attempt any
more explanation than that given here. If it seems wise
and time permits, addition with carrying may be taken up
76
TEACHERS' MANUAL
late in the year, as on pages 82 and 83. But time should
not be taken for this if it is needed for drill upon the work
already outlined.
Adding by Endings
Before a child can add three or more numbers any of
which, before the last, is more than 9 he must have practice
in " adding by endings." Thus, in the following :
iA)
(B)
(0
(D)
(E)
(F)
3
2
4
7
5
8
5
6
5
8
6
7
4
5
9
4
8
6
(A), (B), and (C) need only the primary facts, while (D)
needs
15 11 15
4, (E) needs 8, and (F) needs 6.
To prepare for the addition of such columns, such drills
as those on pages 85, 86, 90, 94, and 95 must be given by the
pupil about as quickly as one of the 100 primary facts.
The Year's Attainments
There are no established standards of skill in the work of
the second year. But teachers wish to take stock of the
year's work and summarize results. The results of this
year may be classified under three heads : knowledge, skill,
and attitude.
As to knowledge, the pupil should have a complete control
of the 100 primary facts in both addition and subtraction.
And he should know the meaning of addition and subtrac¬
tion and how to use them in simple problems.
As to skill, he should be able to give the 100 addition facts
or the 100 subtraction facts in about 3 minutes. (See
Groups I, II, III, and IV in the following section.)
THE WORK OF THE FIRST TWO YEARS 77
He should be able to use the addition facts in adding three
or four one-figured numbers. Ten or twelve exercises per
minute with three one-figured numbers, or six or eight with
four one-figured numbers is a high degree of attainment.
(See Tests I and II in the following section.)
Also, he should have developed habits of order and neat¬
ness, and an increased power of attention. Adding columns
of three and four numbers was to develop the attention
span.
As to nttitude, he should take pride in his knowledge of
number ; enjoy working with number ; desire to know more
about number ; and see that number is useful to him in
many ways.
Primary Facts of Addition and Subtraction
Group I
The 55 addition facts whose sums are less than 10.
0 1
2
2
3
1
4
3
4
5
2
0 3
2
0
1
1
0
3
1
0
1
1 3
5
4
6
7
0
7
6
8
2
0 0
1
2
0
1
2
0
1
0
3
9 0
1
8
0
3
0
1
0
7
2
0 1
2
1
5
2
3
4
6
2
4
4 0
1
0
1
0
5
2
0
1
2
4 4
5
7
6
8
2
6
9
7
5
1 3
5
2
3
6
5
2
4
3
4
8 4
4
6
5
3
3
7
3
6
5
78
TEACHERS' MANUAL
Group n
The 55 subtraction facts corresponding to Group I.
0
4
4
2
4
2
4
6
5
5
3
0
3
2
0
1
1
0
3
1
0
1
1
3
6
6
6
8
2
7
7
8
5
0
0
1
2
0
1
2
0
1
0
3
9
1
3
9
5
5
3
5
6
9
6
0
1
2
1
5
2
3
4
6
2
4
8
4
6
7
7
8
7
8
9
8
7
4
4
5
7
6
8
2
6
9
7
5
9
7
9
8
8
9
8
9
7
9
9
8
4
4
6
5
3
3
7
3
6
5
Group m
The 45 addition facts whose sums are 10 and more.
5
9
6
7
8
6
8
2
9
5
2
6
3
8
4
4
8
1^
1
7
7
5
2
8
9
4
3
9
4
7
7
9
2
3
6
8
9
8
3
6
7
9
5
8
7
9
6
7
5
8
7
9
3
5
6
5
3
8
9
6
9
7
8
7
6
9
5
8
9
4
6
9
7
6
9
4
9
4
8
5
4
9
8
5
7
6
8
7
8
9
THE WORK OF THE FIRST TWO YEARS 79
Group IV
The 45 subtraction facts corresponding to Group III.
10
11
12
10
16
10
12
10
10
_2
_6
_8
_4
ji
_8
ID
11
14
12
11
10
12
10
11
_9
_4
_7
_7
_9
2
_3
_6
_8
18
14
10
11
15
16
14
11
12
_9
_6
_5
_8
_7_
_9
_5
13
11
12
13
17
15
13
13
17
_7
_6
_9
_5
_8
_9
_6
_9
16
14
14
11
15
12
15
13
13
9
8
5
7
6
8
7
8
9
Test I
(Add down)
This will test the child's ability to use the addition facts
of Group I. The test includes all but a few of the zero facts.
After these are copied the sums should be found in 2i or
3 minutes.
y may be used throughout the
term
for drills.
0
1
2
4
1
1
3
4 0
1
7
1
1
2
4
0
2 6
5
0
3
0
5
1
4
0 3
3
5
2
0
2
1
2
2 2
1
3
4
7
5
1
0
3 5
3
1
1
2
1
6
7
2 0
1
3
0
5
2
4
3
1 1
0
3
3
4
2
0
2
6 3
8
2
6
0
4
3
4
3 5
80
TEACHERS' MANUAL
Test n
This test includes all the facts of Groups I and III except
the zeros. It may be used very frequently for drills and
tests. The sums should be found in about 4 or 5 minutes.
7
1
2
1
5
3
1
4
1
2
0
2
8
1
5
5
2
3
1
9
7
5
4
4
9
6
9
1
2
6
1
2
4
1
5
6
1
3
2
4
1
4
1
1
3
8
6
2
5
8
8
9
8
3
7
3
8
7
2
3
4
3
1
1
1
1
2
1
1
5
3
2
3
8
2
6
9
6
4
7
7
2
6
1
2
2
4
2
1
5
5
1
7
4
7
3
6
6
2
5
4
6
5
8
3
5
9
6
3
4
5
3
2
3
2
3
5
2
1
3
6
3
4
6
4
4
9
8
7
9
7
8
9
7
7
"A Child's Book of Number"
Divided into Ten Units
It is highly recommended that the text be placed in the
hands of each pupil. It makes the work more interesting,
it is valuable silent reading, it helps the child to interpret
a number situation from the printed page, and it will thus
prepare for more efficient work in the next grade.
THE WORK OF THE FIRST TWO YEARS 81
If this cannot be done, there should be a copy of the text
in the hands of the teacher to guide her in developing the
work and to furnish proper drills.
Read the " Suggestions to Teachers," pages 133-137 of
the text, before beginning the year's work.
First Month
Cover the work on pages 1-19 inclusive. This includes
only adding 1 or 2 to numbers whose sums do not exceed 10.
1. Ttie pupils, no doubt, can count and know the names
of the figures, but they will enjoy reading pages 1 and 2.
2. Pages 3 and 4 develop the meaning of adding 1, and
give the notation of the number facts learned.
3. Have the children read these pages silently and think
what should fill the blanks. Then let them read those
pages aloud, filling the blanks.
4. Have each pupil make on cards about 2" X 3" a set
of the nine facts learned. On one side have :
1
2
3
4
5
6
7
8
9
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
8
9
10
On the reverse side of the cards have :
123456789
111111111
Later make nine more cards with the addends written
in reverse order. Have them used as described on page 70
of this manual.
5. Children will enjoy reading the little rhymes on pages
5 and 6 and will likely make others of their own. These are
given to help make number work more enjoyable and to give
training in getting a number situation from the printed page.
82
TEACHERS' MANUAL
6. Pages 7 and 8 develop adding 2 and give the notation
of the facts. Make cards and use them as suggested for add¬
ing 1.
7. Always use the reading matter for silent reading.
Children enjoy the rhymes.
8. On such pages as page 11, the sums are written so
that the child may have them for study. Then by covering
the sums and writing the answers, he has them as a check
before wrong sums are fixed in his mind.
9. Page 15 is the first subtraction. See that it grows out
of addition as on this page. This is called the " addition
method."
10. Pages 16 and 17 are games that prepare for subtrac¬
tion by the addition method.
11. Page 19 suggests an outdoor game. It is not advis¬
able to use such games during the class period. Those on
pages 16 and 17 are much better classroom games. But if
we can encourage children to use number in their out-of-
school games, they will thus get much valuable drill.
Let the children use this page as a lesson in silent read¬
ing and see who can tell how to play the game.
Second Month
Cover the wori; on pages 20-33 inclusive. The new work
is adding 3, adding 0, and learning to use the primary facts
already known in adding three numbers.
1. The work on pages 20 and 21 is to develop the ability
to use the facts learned in adding three numbers. The
numbers must be added from the top down in order to keep
within the facts that the child knows. Use these two pages
until all pupils can add each group of three about as fast
as they can give two of the primary facts. Also return to
these pages frequently throughout the year.
THE WORK OF THE FIRST TWO YEARS 83
2. Use page 22 for a reading lesson.
3. Use page 23 until the facts are given as quickly as the
primary facts.
4. Pages 24 and 25 develop adding 3. Use the facts on
cards as suggested for the former facts.
5. Pages 28 and 30 develop the meaning of adding 0. It
is only through a scoring game that adding 0 seems sensible
to a child. To ask " 3 apples and 0 apples are how many? "
seems foolish to a child.
6. Pages 30 and 31 suggest a good schoolroom game that
children like very much.
7. Page 33 suggests a good out-of-school game. Have
pupils read the game and then tell how to play it. Such a
page as this is very valuable work in silent reading.
Third Month
Cover the work on pages 34-48 inclusive. This includes
the four remaining facts of the group whose sums do not
exceed 10. They are :
4 4 4 5
4 5 6 5
1. Use pages 34 and 36 for reading lessons. Use page 35
very frequently during the month.
2. Page 37 includes the primary facts that are known, and
is to build up the ability to use them in adding three num¬
bers. Use this page until a sum can be given in about the
time it takes to make two primary combinations.
3. Pupils should now be able to give page 39 without
hesitating. If not, they must be drilled until they can.
They now have cards of all of these facts and by playing the
game suggested in these outlines they can find those facts
that need more drill.
84
TEACHERS' MANUAL
4. Encourage children to make problems like those sug¬
gested on pages 40 and 41.
5. Use page 42 very frequently. Children should find
the forty sums in 4 or 5 minutes.
6. Drill on the subtraction exercises on page 44 very
frequently.
7. Have pupils read page 46 silently and see who can find
how to add two-figured numbers.
8. Drill on page 48 until pupils can add the 24 exercises
in about 5 minutes. In all, there are 96 combinations or
mental responses to be made.
Fourth Month
Cover the work on pages 49-59 inclusive. This gives the
twenty primary facts whose sums exceed 10.
This group is not found through counting. Finding the
first group by counting was to show the meaning of addition.
1. If the device to aid in adding 9, given on page 52, is a
help, use it. If not, fix the facts through driU. The same
suggestion applies to page 54.
2. The game on page 53 may be used in or out of school.
3. Use pages 58 and 59 frequently during the remainder
of the year to get greater speed and accuracy.
Fifth Month
Cover the work on pages 60-72 inclusive. These are to
develop greater skill in using the facts learned.
1. The game on page 60 is a home game at which two
can play. Have it read silently and see who can tell how to
play it. Children like this game very much. It gives prac¬
tice in adding three numbers.
2. The game on pages 62 and 63 is an outdoor game. Use
THE WORK OF THE FIRST TWO YEARS 85
it as a reading lesson as suggested for other games. Make
the same use of the game on page 66.
3. The game on pages 68 and 69 is a splendid home game.
Encourage children to make it for use at home.
Sixth Month
Cover the work on pages 73-86 inclusive and continue
daily practice on the primary facts of addition and subtrac¬
tion. The new work is halves and fourths, a few units of
measure, and carrying in written addition.
1. The text suggests the method of teaching the material
on pages 73-79. Use objects both in teaching halves and
fourths and in teaching the units of measure.
2. Have pupils read pages 82 and 83 silently and see who
can learn from them how to carry in addition. If they need
help, use no more difficult explanation than that given in
the text.
3. Have pupils add the exercises on page 84 to see if the
sums are right. Then have the exercises copied without
the sums, then have them added and the sums compared
with those in the text.
4. Pages 85 and 86 give twenty of the " adding-by-end-
ings " facts. These are essential to column addition. So
practice until the sums can be given about as quickly as
the primary facts.
Seventh Month
Cover the work on pages 87-97 inclusive and continue
daily practice on the primary facts of addition and subtrac¬
tion.
1. The new work is written subtraction without carry¬
ing. Teach the " addition " method as shown on page 87.
This is much easier than the " taking-away " method used
86
TEACHERS' MANUAL
by many schools. Do not introduce definitions or technical
terms. The text gives all that the child can understand.
2. Practice until the pupils can write the answers to the
30 exercises on page 88 in 4 minutes.
3. Page 90 gives a group of " adding-by-endings " drills
whose sums are in the next decade. Be sure that pupils see
this. Drill until all facts are given as quickly as the pri¬
mary facts.
4. Pages 94 and 95 belong to the same general group.
Practice as suggested above.
5. Use pages 96 and 97 for a reading lesson and encourage
pupils to make up problems and solve them. Besides hav¬
ing a very great mathematical value in teaching pupils to see
the quantitative situation, they make valuable reading and
language lessons.
Eighth Month
Cover the work on pages 98-104 inclusive and give some
daily drill on the primary facts, and upon all the drills in
" adding by endings."
1. Spend much time on pages 98, 99,101, and 104. They
employ all facts learned and develop skill in using them.
Spend 10 minutes a day on such drills and the rest of the
time on the primary facts or on the " adding-by-endings "
facts.
2. Children who have made the game on page 103 for
home play have had much pleasure with it and have got
much valuable practice in number work. It should be used
but rarely for classroom drill.
Ninth Month
Cover the work on pages 105-116 inclusive and do much
reviewing of former work.
THE WORK OF THE FIRST TWO YEARS 87
1. The children know the " 2-times " facts from their
primary facts of addition. The new way of writing them is
all that has to be taught.
2. Observe that the " table of 5's " instead of the " 5-
times table " is taught. This is because the children know
how to count by 5's from counting nickels.
Tenth Month
Cover the work on pages 117-132 inclusive. The new
work is the " 3-times table." If pupils have not devel¬
oped the desired skill in addition and subtraction, it will be
better to omit this work and spend the time on former drills.
CHAPTER VI
THE WORK OF THE THIRD YEAR
This is a period of " habit formation " in the child's life.
Most of the year's work is devoted to the building up of those
habits that lay the foundation of future work in numerical
computation.
The problems this year are those vital to the child's inter¬
ests and needs. They are given to meet his needs ; to inter¬
est him in the study of arithmetic ; and to develop the habit
of using arithmetic in the everyday affairs of life.
In the second grade, oral work, or work without a pencil,
predominated. The oral work still predominates, but in this
grade the child has use of larger numbers and leams to com¬
pute with a pencil.
The Coukse op Stxjdy
3B
I. reading and writing numbers less than 10,000
II. addition and subtraction
1. Ability to give or write all the 100 primary facts of
addition and of subtraction without hesitation or error, at
the rate of at least 30 to 35 per minute.
2. Ability to add by endings, as :
3 13 23 8 18 28
5 _5 _5 and 7 _7 J
giving about 25 to 30 sums per minute.
88
THE WORK OF THE THIRD YEAR
89
3. Adding single columns of three, four, and five numbers
at the rate of 12 or 14 exercises per minute, when three num¬
bers are used ; about 8 or 10 when four are used ; and about
6 or 7 when five are used.
4. Adding two-figured numbers up to four numbers, at
the rate of about four " two-by-three " exercises per minute ;
and about five " two-by-four " exercises in two minutes.
5. Subtracting two- and three-figured numbers at the
rate of about five two-figured numbers per minute and about
seven three-figured numbers in two minutes.
6. Using addition and subtraction in problems including
the addition and subtraction of numbers representing dollars
and cents, as $1.75.
III. MULTIPLICATION AND DIVISION
1. Fix the " 2-times," " 3-times," and " 4-times " facts
from 0 to 9 inclusive and the corresponding division facts,
so they can be given at the rate of about 30 per minute.
2. Written multiplication and division by 2, 3, and 4.
Strive to get about three such products as 3 X 487, 4 X 319,
and so on, per minute ; and about the same number of quo¬
tients, such as 3)576, 4)732, and so on, per minute.
3. Use multiplication and division in problems including
dollars and cents, such as 3 X $2.35 and 3)$5.16.
4. Teach objectively the meaning of i, i, and i.
5. Teach that J of a number is found by dividing it by 2 ;
that i is found by dividing by 3 ; and that i is found by
dividing by 4.
Teach no tables of measure, but common units of measure
needed in other activities are taught when use requires them.
90
TEACHERS' MANUAL
. 3A
I. addition and subtraction
1. Frequent drills on the 100 primary facts, and adding
by endings.
2. Work for greater skill in written work. That is, get
greater speed and accuracy than the standards set for 3 B.
3. Use in problems of dollars and cents.
II. multiplication and division
1. Review facts taught in 3 B.
2. Complete facts to 9 X 9 and 9)81.
3. Drill on division combinations with remainders as
5)23, 6)35, and so on, to 9)89.
4. Written multiplication and division by one-figured
numbers.
5. Teach the use of such multipliers as 1^, 2^, to 9^.
6. Teach the meaning of i, i, to ^ as dividing by 5, 6, to 9.
How to Cover the Work Outlined
The work outlined for the third year is about what is
expected in our best schools. Many courses of study outline
much more, but do less. The common complaint is that
pupils are not »satisfactorily covering the work expected of
them. It is better to set up a reasonable objective and reach
it than to attempt more and do nothing thoroughly. Suc¬
cess in teaching arithmetic depends upon doing thoroughly
that which needs to be done, and not wasting time on that
which is unimportant. Too often work is attempted before
fundamental skills upon which it depends have been fully
estabhshed.
1. Do intensive drill work. Do not use devices that
require but few combinations from each pupil. During a
THE WORK OF THE THIRD YEAR 91
10-minute drill period each child should make from 150 to
200 responses, responding to the same fact many times.
2. Devote a part of each period to sight drill, part to
dictation drill, and part to written drill.
3. Work with a time limit, have each pupil keep a record
of his progress, and have as his chief motive a desire to reach
a fixed standard and to excel former accomplishments.
4. Avoid the use of games and other devices that detract
from the number work ; that consume any large part of the
time in the setting up ; or that are apt to bring up that which
is not needed more often than that which is needed. The
game at the end of this outline is excellent.
5. Make the work so attractive that the pupils want to
learn that which they should learn. The game spirit can be
used in many ways without actual wasteful games where the
main interest is away from number itself.
6. Spend the most time on the most difficult combina¬
tions. Teach what a pupil does not know and do not waste
time in drilling upon what he does know.
7. Give frequent diagnostic tests to find what facts need
drill and what ones do not.
8. Do not take time that belongs to the whole class to try
to teach a child that which he cannot learn. Except in
schools where the pupils have been graded and grouped
according to intelligence, there are pupils who lack the inborn
intelligence to profit by teaching that is fitted to normal
children. These children deserve our sympathy, but they
are not entitled to the time that belongs to those who need
our help and can profit by it. To hold a class back for a
weakling means that the average and superior children are
learning habits of idleness and of loafing instead of habits of
study and concentration. Besides that, they are liable to
lose interest in the subject.
92
TEACHERS' MANUAL
Diagnostic Testing and Drilling
Each child must know what he needs to drill upon, and the
teacher must know each child's needs. And yet she hasn't
the time to mark the papers and tests of 30 or 40 pupils
daily. But there are devices by which the pupils may mark
their own tests, or discover their own weaknesses. One of
the simplest ways is as follows :
Have each pupil make and keep a little drill card about
2 " by 3 " of each fact learned in any of the processes. On
one side is the drill. On the other is the fact. Thus :
for addition
6
9
6
_9
15
for subtraction
for multiplication
for division
12
5
5X7
5)^
5
?
12
5 X ? = 35
5 X 7 = 35
Let the pupils work in pairs, one showing the cards and the
other answering. When a pupil gives a wrong answer or
hesitates, the card is thrown out for further study.
THE WORK OF THE THIRD YEAR 93
Another good way is to have mimeographed sheets made,
one sheet having the test, and the other the test, in the same
order, with answers. Thus, one sheet would have :
3
6
7
9
8
6
5
9
3
4
8
5
The other would
have :
3
6
7
9
8
6
5
9
3
4
8
5
8
15
10
13
16
11
The children work in pairs. One child writes the answers.
The other follows with the answer sheet. When a wrong
answer is given or a child hesitates, a cross is made below
that exercise on the answer sheet, which is then given to the
child making the answers, for further study.
To hesitate is as bad as to miss, for it means that the child is
guessing or counting.
Keep the pupils enthusiastic about discovering and cor¬
recting their own weaknesses and striving to obtain a given
standard.
The Written Work
It is a waste of time to attempt written work that needs
certain habits until these habits are first established. Watch
for any error in the " transfer of habit " from the primary
habit to its use in the written work. Thus, the child may
know 7 + 0 = 7, or 7 X 0 = 0, and yet he may fail to use
the fact properly in written work. Watch for a confusion of
habits, such as confusing 7-1-0 = 7 with 7X0 = 0, and
5 -f 1 = 6 with 5X1=5. To avoid such errors it is best
to use the drill form :
94
TEACHERS' MANUAL
7 5
0 1
7 6
7X0 = 0
and 5X1=5
in drill upon primary facts.
If drill work is " made up " to supplement that of the text,
see that it includes the difficult facts that need more drill,
instead of facts which the child does not need. Impromptu
drill made up offhand without careful thought may be a
complete waste of time. There should be a definite purpose
in each driU exercise. The exercises should be made up to
include the facts that need drill. Make frequent use of the
achievement tests at the end of this chapter.
Use the same drills of the text over and over until required
skill is obtained. From time to time revert to former drills,
even to those of former grades. A textbook in arithmetic
is not taken page by page and then left, as readers or other
textbooks are.
Insist upon all written work being checked before the
answer is given.
Have the problems read silently. Then have a pupil tell
what process to apply. Thus the problem may be :
Frank earned $3.75 one week and spent $1.80 of it. How much
had he left?
After it has been read silently, call upon a pupil to tell
what is to be donej He should reply, " Subtract what he
spent from what he earned." There must be a silent analy¬
sis of the problem, of course, in order to know what to do,
but do not require written analysis. The pupil should write
only :
The Problems
THE WORK OF THE THIRD YEAR
95
$3.75
1.80
$1.95
Do not waste time in having the pupil say or write,
" Given : —" and " Required : —" a practice quite common
in many schools.
The fact that a pupil knows what to do and does it is
sufficient proof that he has analyzed the problem, knows
what was given, and what was required. To write it out or to
say it is a waste of time.
If the problem is, " John earned $1.85 each week for 5
weeks. How much did he earn in all ? " have a pupil say,
after reading it, " Multiply the amount earned in one week
by 5." And then have him write :
$1.85
5
$9.25
Do not allow the following wasteful " analysis " so often
seen :
1. Amount earned in 1 week = $1.85.
2. Amount earned in 5 weeks = 5 X $1.85.
This does not clarify his thinking and it consumes valu¬
able time. The fact that the pupil can tell what to do shows
that he was " thinking clearly."
Use the forms given in the text.
Be on the alert for real problems that arise in projects or in
other school or out-of-school activities. The teacher should
see that pupils develop the habit of using arithmetic in all
quantitative situations of life in which it is needed. But do
not make up unreal problems about real situations merely to
furnish an opportunity to use a process.
96
TEACHERS' MANUAL
A teacher should know the aim of a course in arithmetic
and the aim of each particular lesson, and should continually
ask herself, in the light of this, if she is making the best possi¬
ble use of her time. So much of the work in many schools
seems to be a mere matter of " marking time." School-
teaching should be taken as seriously and as scientifically
as any other profession. Time-killers are not teachers. A
prominent superintendent once remarked to the author that
so many of his teachers could travel all day around a three-
cent piece and never realize that they were getting no¬
where. See that each day accomplishes something definite.
Conclusion
The work of the third year, as outlined here, and as laid
out in the text, can he and should be well done in any school
by all normal children. Unless it is, the children will not be
able to do the work of the succeeding grades. But it cannot
be done if the time is wasted on the subnormal child who
cannot learn, or in unwise or wasteful methods of procedure.
Follow the methods and drills of the text, and keep every
normal child interested in the work and in his own progress.
Lack of enthusiasm on the part of the teacher and the pupils
is the greatest cause of poor work. Arithmetic should be
one of the most enjoyable periods of the day. Children
naturally like number work, but a teacher's indiflterence
toward the subject, or her lack of a knowledge of how to
teach it, may result in its becoming very boresome.
A Game That Is Also a Diagnostic Test
Select a definite group of facts and write them on the
blackboard. Divide the class into two teams and have a
contest. Select a pupil from one team to stand and give
the answers to the entire group. If he gives all, he wins a
THE WORK OF THE THIRD YEAR
97
point for his team. But if he misses any fact, anyone may
give the answer. If the opposite team gives the right answer
before his own team does, it scores a point for their team.
If his own team corrects the error before the other team
does, it saves the other team from scoring. This keeps the
attention of everyone on the drill and is very interesting.
A child should keep a little notebook of any facts missed.
He thus knows what to study.
Alternate, calling first upon a pupil from one team and
then from the other so as to allow an equal chance of scoring.
When a pupil hesitates, the teacher should say, "Class."
Then anyone may answer.
^To avoid disputes as to which team gave an answer first,
the opposite team only may correct errors or answer when
the teacher says, " Class."
Each pupil reciting is thus being tested to see what ones
of a group of facts need further drill.
The Third Year's Work Divided into Monthly Units
{The pages refer to Stone's Primary Arithmetic and Third-Year Arithmetic)
In order to help teachers do the year's work well and to
meet or surpass existing standards, the work of the textbook
is here divided into ten units, that being as many months as
we have in the school year.
First Month
Cover very thoroughly the work on pages 1-19 inclusive.
The pupils have had all these facts during the second year,
but they need a thorough review to make up sldll lost during
the summer, and to extend former skill.
1. In the addition andsubtraction " tables " for study, given
98
TEACHERS' MANUAL
after the development of each new series of facts, as on pages
2,4, 10,16, 17, and 19, the complete fact is given so that the
child may get a complete " eye picture." That is, from see¬
ing 3, he gets a picture so that 3 at once looks wrong to
him. After a careful study of these " tables," have the chil¬
dren cover the answers with a sheet of paper and write them.
Then lower the sheet so as to see the real answers and mark
their own papers and study any fact that was missed. Also
make cards of each group of facts as suggested on preceding
pages and use as suggested.
2. The groups of three and four numbers following each
group of " tables," as on pages 3, 4, 16, 17, and 19, make use
of the facts presented when " adding down." These require
a new skill, for in adding them the child does not see both
figures when making the second combination, but must
" think " one of them. Moreover, these exercises require
greater attention, for two or three mental combinations must
be made before announcing the sum.
3. Use the exercises in the text over and over until pupils
give them quickly, for they contain all of the facts.
4. See that each pupil discovers the facts that he finds
difiicult, and masters them. Arouse interest in mastering
the facts, not in playing a game in which the chief interest is
in the game. Contests to see who can give all the facts or
to see if one can beat a former record are good, but the
games where the combinations occur by chance are wasteful
forms of drill, however interesting they may be.
5. The only new habit to be mastered in " adding without
carrying," pages 6 and 7, is the habit of beginning with the
right-hand column.
6. Make subtraction grow out of addition, as on pages 7
4
4
7
9
Library
THE WORK OF THE THIRD YEAR 99
and 8. Do not use the terms " minus," " less," " from,"
etc. After the facts are fixed, show the other two uses of
subtraction as given on pages 8 and 9.
7. In written subtraction without carrying, there are no
new habits to establish. The child knows the " tables "
and has learned to begin at the right in addition.
Second Month
Cover the work on pages 20-34 inclusive. The new work
is adding zeros, carrying in addition, and adding by endings.
1. Introduce the meaning of adding zero through a scoring
game. To ask, " Five boys and zero boys are how many? "
is foolish and meaningless. But any child knows that if he
makes a score of 5 and his next is 0, his score is still 5.
2. Page 22 gives the ICQ primary facts including zero.
Use this page until pupils can give the 100 facts in 2 or 3
minutes. Find what facts each individual pupil does not
know and have him drill upon these rather than upon those
he knows.
3. The game on page 23 is merely a suggestion of an out¬
door game to encourage the use of the number facts out of
school. Use games very sparingly during the drill period.
There is too much danger of not getting the needed combina¬
tions. The teacher must know what facts a child needs and
hence must keep the control of them in her hands rather than
let them arise by chance. '
4. Drill on page 24 throughout the month until it can be
done at the rate of 2 or 3 exercises per minute. Strive to do
the 28 exercises in 10 minutes or less.
5. Teach the reading of dollars and cents and use adver¬
tisements from daily papers for practice in reading such
numbers.
6. In teaching carrying, pages 28 and 29, do not make any
100
TEACHERS' MANUAL
further attempt to teach the " why " than given in the text.
It is " how," not " why," that is important. Most children
should be able to read the text and learn how to add without
the aid of the teacher. Let those who can do so, and then
let them show those who cannot understand the text,
7. Drill on page 30 a few minutes daily until the habit of
carrying becomes automatic.
8. In teaching "adding by endings," be sure that the
children see that the ending is like the key fact from
the 100 primary facts. The " adding-by-endings " facts
on pages 31-34 give a sum in the same decade as the
two-figured number. The exercises consisting of single
columns of three figures, which follow the drills in " adding by
endings," are to develop skill in using the facts. Use them
very frequently.
Third Month
Cover the work on pages 35-50 inclusive. The new work
is zeros in subtraction, no carrying ; carrying in subtraction ;
and the " 2-times " and " 3-times tables."
1. Frequently review the addition exercises of the pre¬
ceding months for a few minutes.
2. Zeros in subtraction cause trouble unless carefully pre¬
sented. There fire just the three types that are given on
page 37. See that these three types are understood.
3. The subtraction facts on page 38 are those used when
there is carrying. See that the child has an automatic con¬
trol of them before taking up the written work.
4. Develop the written process as on page 39. The
important thing is " habituation," not " rationalization."
That is, when the child sees a smaller number above a larger
one, he must know what number fact of page 38 is to be
used, and that he must carry 1, as shown in the center of
THE WORK OF THE THIRD YEAR 101
page 39. Let all who can get the development from the
text rather than from the teacher.
5. Before finding the results on pages 39 and 40, have
pupils run through the exercises, beginning with exercise 3,
page 39, and say : " Think 12, carry 1 " ; " think 11, carry
1 " ; " think 10, carry 1 " ; etc.
6. Do not neglect problem work. The problems show
the child the need and use of each new process. Hence the
problems on pages 41 and 43 should receive proper attention.
7. Use the drill on page 42 very frequently, until proper
habits are fixed. Interest pupils to work daily for greater
skill. Try to do the whole page in 10 minutes or less without
an error. You can easily get skill beyond existing standards
by properly using this text ; but the class period must be
used in concentrated effort, not wasted in foolish games
and dramatization, as done in many schools. Always work
with a definite aim.
8. The child knows the "2-times" facts from the addition
facts, but not the new way of writing them. In the written
work on pages 44 and 45, the new habit is that of carrying.
Be sure that he carries after thinking the product, not before.
If the child does not see this, he is apt to carry before he
multiplies, confusing it with the habit formed in addition of
carrying to the first addend.
9. Have the pupils find the sums of three like addends as
on page 46 and make their own " 3-times table." This
is to show them the meaning of the "tables" and that multi¬
plication is to save addition when the addends are alike.
Fourth Month
Cover the work on pages 51-62 inclusive, and frequently
review the most difficult work of former months. Not only
seek to keep all skills developed, but try to increase them.
102
TEACHERS' MANUAL
So, constantly revert to former drill pages for a few
minutes.
1. The Tiew work of this month is the " 4-times
table " ; the division facts arising from all multiplication
facts learned ; and written division by 2, 3, and 4.
2. Page 51 is important. Be sme that pupils understand
that interchanging the two factors does not affect the prod¬
uct. Have pupils find the new facts of the " 4-times
table" for themselves, and make their own "tables."
3. The review on page 54 is to prepare for division.
When a child can give the missing number of 2 X ? = 14, he
knows the division fact. He should see that 2)^ asks the
same question, viz. : " Two times what is 14? "
When the child first sees the forms :
2)8 2)Í2 3)1^ 3)^ 4)^ 4)16 etc.
he should think, " 2 times what is 8? " "2 times what is
12? " " 3 times what is 15? " and announce the answer.
Never use the expression " goes into." It means nothing.
4. The " tables " at the top of page 55 are for reference.
In drilling, use the forms :
2)2 2)6 3)9 4)12 2)18 3)1^ etc.
5. In developing written division, use the method on
pages 58 and 59. No further explanation is needed or can be
understood.
6. Use drill page 60 xmtil pupils can give the quotients
and remainders quickly. Practice until the act becomes
practically a single mental response. That is, practice until
the child does not have to think a product and a difference.
Thus, in 3)17, he should not have to think " 3 X 5 = 15 ;
15 and 2 are 17," but should practice until he responds " 5 and
2 remaining " about as quickly as he responds to 3)15.
THE WORK OF THE THIRD YEAR 103
Fifth Month
Cover the work on pages 63-76 inclusive. The last
month of the semester reviews the skills already developed
rather than introducing new work. There is a new notation
and a new idea introduced — the meaning of half, third, and
fourth — but no new number facts or skills are required.
1. Concentrate on developing greater skill in written addi¬
tion and subtraction. Review the elementary facts of former
months if necessary. Strive to attain the standards given
in the tests on pages 73-76.
2. Introduce the idea of halves, thirds, and fourths
objectively, as on page 68. The new idea is to show that ^
of a number is found by dividing by 2 ; that ^ of a number is
found by dividing by 3 ; and that i of a number is found by
dividing by 4.
3. Throughout the semester you should anticipate the
tests and standards given on pages 73-76, and throughout the
year give frequent tests to see that pupils are gradually
working toward these standards. Of course, the brighter
pupils can easily surpass them, while the duller ones may not
reach them ; but all should approximate them if they are to
reach those of the next semester.
Sixth Month
Cover the work on pages 77-92 inclusive. No new work is
introduced except counting change and measuring tempera¬
ture. The aim of the month is largely to develop greater
skill in using the facts and processes learned and to develop
greater power to use the processes in problems.
1. "A zero in the multiplicand " will give trouble unless
consciously approached as in the text.
2. In teaching a pupil to solve a problem, be sure that he
104
TEACHERS' MANUAL
understands what the problem is and what to do, before
beginning the computation. You will notice that the prob¬
lems are unclassified as to process. The pupil must under¬
stand the problem and think what process to apply. Do not
require any oral or written analysis. The child has made
the proper analysis when he has correctly discovered what
process to apply. Use the forms of solution of the text.
Seventh Month
This month covers the work on pages 93-110 inclusive.
The main purpose is the development of greater skill in using
the facts and processes already learned. Aside from a few
common measures, the new work is " adding by endings "
where the sum is a number in the next decade. Have the
pupils see that the sum ends like the primary fact to which it
is related, but that it is a number in the next decade.
1. Spend much time on the drills on pages 93-95. Unless
the child develops skill with these, he cannot add such exer¬
cises as those on page 96 with skill.
2. The " race " idea on page 96 is a good way to motivate
drill work. Here the child is using the game spirit, but his
attention is on niimber instead of away from it, as often hap¬
pens in other types of games.
3. On page *98 are given the names of the terms. No
other definition should be given. In fact, it is not very
important that these names be learned at this time. How
to perform the processes and to use them in problems is the
important thing.
4. Give much drill on page 99. That is, use it very fre¬
quently.
5. In teaching the measures on pages 101-105, use the
actual units of measure. Otherwise the work is of no value
and may as well be omitted.
THE WORK OF THE THIRD YEAR 105
6. Strive throughout the month to develop greater skill
in computation. Give frequent reviews of the fundamental
drills of former months. Use the tests of pages 73-76 to see
if pupils are growing in skill.
Eighth Month
Cover the work of pages 111-124 inclusive. The new
work is the "5-times" and "6-times tables " and the corre¬
sponding division facts. Note that there are but nine new
facts in these two "tables, " hence use these new facts more
often than the others. With the notion of division already
developed, the division facts are known when the multiplica¬
tion facts are. That is, 5)35 merely asks, " 5 times what is
35?"
1. Before skill in written division can be expected, there
must be a splendid control of such exercises as those on pages
114 and 121. So spend much time on these two pages.
2. Keep up and extend all former skills by frequent re¬
views. You are working for the standards on pages 149-152
at the end of the year. So begin now to see that pupils are
nearing them. Both teacher and pupils must work each
day with a definite aim. That is, toward a definite goal.
3. Do not neglect the problems. Use all in the text and
as many local ones as you can find that are real and interest¬
ing.
Ninth Month
Cover the work on pages 125-141 inclusive. The new
work is the remaining multiplication facts and the corre¬
sponding division facts. Observe that in the last three
" tables" there are but six new facts.
1. Be sure this month that all pupils have an automatic
106
TEACHERS' MANUAL
control of all the primary facts of multiplication and division.
Review all former " tables." The " races" on pages 125-128
are for this purpose. Use them very frequently during the
month.
2. Make thorough use of the problems included in these
pages. The number facts and processes are of no value
unless the pupil knows how to use them.
3. Give much time to the drills in division on pages 132,
136, and 140. Keep the final standards for the year in mind.
Get the pupil to want to attain them. It is only when a
pupil is drilling with a definite aim that he accomplishes any¬
thing. Work without a purpose is valueless.
Tenth Month
Cover the work on pages 142-152 inclusive. There is no
new work given this month. Pages 142-147 furnish an
apphcation through problems of the facts and processes
learned during the year. Hence much of the month may be
spent in drills from former months to build up the standards
for the year from pages 148-152.
As laid out by months for the teacher, she should be able
to cover the work easily and reach standards never before
attained if her work has been done in a way to eliminate
all waste of tfine. But to attain proper standards, both
teacher and pupils must work daily with a definite goal in
mind.
Third-grade Achievement Tests
Test I
This test gives all of the 100 primary facts except a few
of the zeros. The entire test requires 132 combinations and
should be done in about 6 or 7 minutes.
«
THE WORK OF THE THIRD YEAR 107
Add dovm :
1.
2.
3.
4.
5.
6.
674
819
617
639
868
565
973
937
966
908
489
514
819
172
891
390
267
970
910
427
950
856
506
768
3376
2355
3424
2793
2130
2817
7.
8.
9.
10.
11.
12.
469
340
854
379
726
799
*956
247
287
142
156
305
845
359
586
325
489
649
729
678
374
653
429
897
2999
1624
2101
1499
1800
2650
Test II
This test contains all of the subtraction facts (except 0 — 0)
where there is no carrying. It should be done in 1§ or 2
minutes.
Subtract :
1.
2.
3.
4.
5.
6.
494
645
768
924
876
954
304
301
348
801
710
452
190
344
420
123
166
502
7.
8.
9.
10.
11.
12.
987
866
975
719
953
892
625
416
273
218
703
312
362
450
702
501
250
580
13.
14.
15.
16.
17.
18.
915
733
678
682
938
875
304
410
501
261
520
562
611
323
177
421
418
313
108
TEACHERS' MANUAL
4
Test in
This test includes all of the 45 subtraction facts that
require carrying. Some of the more difficult ones occur
twice.
The 24 exercises should be done in about 8 minutes.
Subtract :
1.
2.
3.
4.
5.
6.
9370
8291
7305
8057
7635
8013
2627
3819
4652
1584
1794
2373
6743
4472
2653
6473
5841
5640
7.
8.
9.
10.
11.
12.
9077
8730
7528
9160
8944
6128
3849
1679
3855
5803
5849
4590
5228
7051
3673
3357
3095
1538
13.
14.
15.
16.
17.
18.
8717
6254
4916
8046
8135
9265
6048
1718
2839
3171
5047
7575
2669
4536
2077
4875
3088
1690
19.
20.
21.
22.
23.
24.
7568
8534
5320
9211
8476
7381
5659
2616
4331
6213
7689
5786
1909
5918
989
2998
787
1595
Test IV
This test includes all of the multiplication facts except
0 X 0, 1 X 0, and 1 X 1, but not in reverse order. They
should be done in about 6 minutes.
THE WORK OF THE THIRD YEAR 109
Multiply:
1.
2.
3.
4.
5.
6.
416
782
619
720
385
748
2
2
3
3
3
4
832
1564
1857
2160
1155
2992
7.
8.
9.
10.
11.
12.
605
319
702
615
908
716
4
4
5
5
6
6
2420
1276
3510
3075
5448
4296
13.
14.
15.
16.
17.
18.
917
809
908
519
590
291
7
7
8
8
9
9
6419
5663
7264
4152
5310
2619
Test V
This test contains all quotient figures except zeros. It
should be done in about 6 minutes.
1.
2.
3.
4.
5.
6.
2)590
2)772
2)834
3)2148
3)1776
3)2502
295
386
417
716
592
834
7.
8.
9.
10.
11.
12.
4)872
4)3024
4)1756
5)1455
5)1930
5)2735
218
756
439
291
386
547
13.
14.
15.
16.
17.
18.
6)2916
6)3438
6)1314
7)1995
7)1379
7)2548
486
573
219
285
197
364
19.
20.
21.
22.
23.
24.
8)2272
8)3160
8)1408
9)1647
9)2484
9)4455
284
395
176
183
276
495
CHAPTER Vn
THE WORK OF THE FOURTH YEAR
The work of the fourth school year contains less really
new work than that of any other school year. Long multi¬
plication and long division is about the only new work.
There is but little that is new in multiplication, but the
process of long division is a difficult process to master. It
requires the mastery of a long series of related habits. Prep¬
aration for it requires much drill in multiplication and in
short division, as well as other drills given in the text.
But the major part of the year is not devoted to these two
new processes. All processes and facts taught in former
grades must be performed with greater accuracy and greater
speed. The child who does not know thoroughly all the
fundamental processes with whole numbers, and who has
not formed proper habits in computation at the end of this
year, will have trouble in arithmetic throughout the remain¬
ing grades.
The Course op Study
4B
I. beading and writing numbers to 5,000,000
II. addition and subtraction
1. Drill for greater speed and accuracy. Make frequent
use of the achievement tests given in the third grade of this
Manual.
110
THE WORK OF THE FOURTH YEAR 111
2. Continue drill upon the more difficult combinations of
the 100 primary facts. Work for a speed of 50 primary
combinations per minute.
3. Continue drill upon "adding by endings," working
for at least 35 or 40 combinations per minute.
4. Continue drill in adding single columns up to six one-
figured numbers. Work for at least 20 or 25 combina¬
tions per minute.
5. Drill in adding two or more two- and three-figured
numbers up to adding six three-figured numbers. Work
for at least 20 combinations per minute. For example,
there are 17 combinations to be made in a three-by-six exer¬
cise. Six of these in five minutes would be 102 combina¬
tions in 5 minutes.
6. Drill on the 100 primary facts of subtraction, working
for 40 or 50 combinations per minute.
7. Drill upon written subtraction up to four-figured
numbers, working for at least 15 or 20 subtractions per
minute.
Note. — The reason for seeming fewer combinations in subtraction than in
addition is from the fact that when there is carrying, there are two facts to
3106
"think" in each subtraction. Thus, in 1587 there are seven combinations:
7 and 9 ; 1 and 8 and 1 ; 1 and 5 and 5 ; 1 and 1 and 1.
III. MULTIPLICATION AND DIVISION
1. Continue drill upon the primary facts until about
40 or 50 per minute can be written or given.
2. Drill in written multiplication until reaching about
4 or 5 exercises per minute with a one-figured multipher
and a three-figured multiplicand. Make frequent use of the
achievement tests of the third grade of this Manual.
3. Develop and drill in multiplying by 10, 20, and so on,
to 90.
112
TEACHERS' MANUAL
4. Develop and drill upon two-figured multipliers. Work
for two or three exercises per minute with two figures in
both multiplier and multiplicand.
5. Drill in short division with one-figured quotients and
a remainder, as 4)27, 6)38, 9)74, and so on, from 2)3 to 9)89,
working for 15 or 20 exercises per minute.
6. Drill in short division with quotients up to three figures,
working for six- or eight-quotient figures per minute.
IV. THE TABLES OF COMMON MEASUEES
Teach the "tables" of length, dry measure, liquid measure,
weight, and time.
V. FHACTIONS
1. Teach objectively the meaning and notation of frac¬
tions.
2. Use fractions in expressing the relation of one unit of
measure to another and in interpreting a fractional unit in
terms of lower units.
3. Use fractions in finding the cost of a part of a pound,
yard, etc.
4 A
I. BEADING ;^ND WRITING NUMBERS UP TO NINE DIGITS
II. ADDITION AND SUBTRACTION
1. Continue drill upon exercises given in the preceding
grades, working for greater speed and accuracy.
2. Continue drill in written work with larger .numbers,
and, in addition, with longer columns to develop the atten¬
tion span. Work for greater speed and accuracy.
3. Make frequent use of the achievement tests at the
end of this chapter of the Manual.
THE WORK OF THE FOURTH YEAR 113
III. MULTIPLICATION AND DIVISION
1. Continue all forms of drill given in former grades,
working for greater accuracy.
2. Use three-figured multipliers and multiplicands con¬
taining zero, as 507, 309, etc.
3. Develop and drill upon division by 10, 20, 30, and so
on, to 90, using short-division forms to prepare for long
division.
4. Develop long division and use easy divisors, as 21, 31,
41, and so on, continuing with these until the pupils get the
form. *
5. Give much drill in estimating the quotient figures.
6. Give drill in long division with harder divisors, as 35,
46, 24, and so on.
7. Drill in dividing by 13, 14, and so on, to 19. Give
much drill in finding products of such numbers by 2, 3, and
so on, to 9, without a pencil.
8. Express remainders in long division as decimals to
hundredths.
IV. EMPHASIZE THE PEOBLEM-SOLVING SIDE OF THE
SUBJECT DUEING THIS SEMESTEE
V. FEACTIONS
1. Review the meaning and notation of a fraction and
use it in finding parts of a number, as in 4 B.
2. Teach adding and subtracting halves and fourths,
using them in problems.
VI. MEASUEING SUEFACES
Teach the measurement of small rectangular areas and
use in problems.
Note. — The last two topics may be omitted if time is needed to reach
the desired standards set up for whole numbers.
114
TEACHERS' MANUAL
How to Accomplish the Work op the Year
Read the suggestions that follow the outline of work for
the third year. To accomplish the work requires frequent
diagnostic tests and carefully planned and well-motivated
drills to overcome defects. If pupils come to this grade
without the preparation that is assumed, intensive work
must be done until they " catch up." This can soon be done
with children with average intelligence. If pupils have
been promoted to this grade who are below the standard
because they are below standard in native intelligence, it
will be impossible to bring them up to the standards expected
of normal children. While such children are entitled to our
sympathy, they are not entitled to time that belongs to
others. They should be put in special classes. The legiti-
rñate standards of our schools cannot be lowered to the level
of abilities of the subnormal.
Long Division
Long division is the most complex and the most difficult
of the processes to teach. Moreover, it has the fewest real
applications that are vital to the child at this age. For that
reason, interest must be kept up through a sense of growing
skill and pride in skillful work. The drills in dividing by
10 (page 248 of the Primary Arithmetic, page 96 of the
Fourth-Year Arithmetic), in dividing by 20, 30, and so on
(pages 250-256 of the Primary Arithmetic, pages 98-104 of
the Fourth-Year Arithmetic), are to prepare for long division.
The first exercises (pages 256-261 of the Primary Arith¬
metic, pages 104-109 of the Fourth-Year Arithmetic) are
those in which the "trial " quotient is the " true " quotient.
With these the child will get the various steps in the written
work.
THE WORK OF THE FOURTH YEAR 115
The second step (page 262 of the Primary Arithmetic,
page 110 of the Fourth-Year Arithmetic) introduces exer¬
cises in which the " trial " quotient is one larger than the
"true" quotient. There are two methods of getting the
true quotient. They are illustrated below.
First Method
The " trial" quotient comes from 11 2 and is 5.
46 Multiply 24 by 5 mentally and see that the product
24)1104, is larger than 110 and, hence, write 4 as the quo-
96 tient.
144 To get the second quotient figure, the " trial" quo-
144 tient is 7 from 14 -j- 2. Mentally see that 7 X 24
is larger than 144 and write 6. This method re¬
quires ability to multiply a two-figured number by a one-
figured number without a pencil. If it is the method you
choose, give much drill in announcing such products with¬
out a pencil.
Second Method
From the drills in dividing by 20 the child sees
46 110 and the 2 (of 24) and thinks, " 5 (the 'trial'
24)1104 quotient) and 10 remaining." He then thinks,
96 " 10 is less than 5 X 4," and writes down 4.
144 Next he thinks, " 7 and 4 remaining." Then he
144 thinks, " 4 is less than 7 X 4," and writes 6. The
author has got very skillful work by this
method. It takes less mental work, and is more nearly
based upon skills developed in former work. Use the
method that you think you can make the more efficient.
The Problems
The problem-solving side of arithmetic receives greater
and greater attention as the facts and processes become
116
TEACHERS' MANUAL
matters of habit. The problems are to develop " power to
think " in numerical situations ; and to develop the habit
of using arithmetic in life situations of vital need or interest.
Making problems, problems without number, and incom¬
plete problems, all add to this development.
Do not waste time in " written analysis " or in labeling
the steps. Have the pupil read the problem silently, think
what to do, and do it. Use the forms used in the text.
The following form, seen in schools and in textbooks, is a
great waste of valuable time :
1. Cost of 3 books = $3.75.
2. Cost of 1 book = I of $3.75 = $1.25.
3. Cost of 5 books = 5 X $1.25 = $6.25.
It was the " thought " that led to what to do that was
valuable, not the writing out of the thought.
The solution should have been :
3)$3.75
$1.25
5
$6.25
Diagnosing a Solution That Has a Wrong Answer
Frequent tests should be given and wrong results carefully
analyzed. The trouble may be easily remedied, or it may
be beyond our help, as shown below.
1. A wrong result may be due to an error in computation.
This should never happen. A pupil should be taught to check
all computations and never consider the work finished until
he knows that the computation is correct.
2. The trouble may be due to a careless reading of the
problem. A problem should not be given to a pupil who
lacks experiences through which the situation described
THE WORK OF THE FOURTH YEAR 117
may be made concrete and vivid if carefully read. Then
insist upon a careful silent reading of the problem until an
accurate mental picture is formed and the child knows what
is to be found.
3. The error may be due to applying the wrong process.
If so, see that the meaning of the processes is understood.
Each process should be developed concretely when first
taken up, and then followed up by problems of one step
until the child fully understands its meaning and use. But
in future work, if it is found that a child does not know what
• '
a process means, it must be taught him again. He must
not make a " guess " at what process to apply from the
looks of the figures. This often happens.
4. The trouble may be due to lack of native ability to
reason out what to do. Such children can never reason out
new situations, but must memorize a few needed type forms.
If the teacher can teach such children a few type forms of
problems of frequent recurrence, so that through memory
they recognize what to do in a few of life's most conamon
needs of arithmetic, she has done her duty and has done all
that she can do. But she must not neglect those mentally
alert children capable of more valuable training.
The Work of the Fourth Year Divided into Ten Units
(TÄe re/er to Stone's Primary Arithmetic and Fourth-Year Arithmetic)
First Month
Cover the work on pages 153-168 (1-15) inclusive. No new
facts or processes occur. But the work of the third grade is
reviewed and extended to more difficult applications to
further develop the reasoning powers, and to longer drill
exercises to further develop skill and to develop the atten¬
tion span.
118
TEACHERS' MANUAL
1. Pages 153-155 (1-3) give concrete and interesting
problems that include the four processes learned. Have
pupils read these problems silently, then stand and give the
substance of the problem and tell what process to apply.
These are to develop " arithmetical reasoning," not to fur¬
nish drill in computation.
2. Page 156 (4) is given for those pupils who may have
had their grade work from some other text in which this
notation was not taught ; and also for those who may have
forgotten it dming the summer. It should not require
much time.
3. Pages 157-161 (5-9) are a review of work using num¬
bers that mean money. See that there is no mistake made
in using the period. (Do not call it a decimal point as is
so often done.)
4. Pages 162-164 (10-12) are to develop greater skill in
written addition. Pages 163 (11) and 164 (12) should be
used very frequently during the month.
5. Pages 165-168 (13-16) are to develop greater skill in
subtraction. Observe that the exercises on page 168 (16)
contain larger numbers than those previously used. This
is to develop the power to hold the attention for a longer
span.
Second Month
Cover the work of pages 169-189 (17-37) inclusive. This
does not include any new work except multiplying by 11
and 12, which may be omitted if the time is needed for other
work. The purpose of the work of the month is to review
multiplication and division and to develop greater skill in
computation, and greater power to solve problems.
1. Practice on page 169 (17) until the desired skill is at¬
tained. The last drill exercise is to develop a skill needed
in written multiplication.
THE WORK OF THE FOURTH YEAR 119
2. Page 170 (18) reviews the most difficult facts of multi¬
plication, so use it frequently. While no number smaller
than 6 is employed as multiplier, all digits are used in the
multipUcand.
3. Pages 171 (19) and 172 (20) teach an important fact.
See that pupils understand thoroughly that in such a prob¬
lem as, " Find the cost of 24 oranges at 4f5 each," that while
24 is the real multiplier, 4 is used as the actual multiplier to
save work. In such exercises, use abstract numbers only.
24
Do not use the form 4^ seen in so many classrooms.
96 )é
4. Teach carefully the problems on pages 172 (20), 173
(21), 175 (23), 181 (29), 184 (32), 185 (33), and 186 (34).
Problem-solving gradually assumes a more important place
as the processes become fixed. Use no oral or written forms
of analysis. Have the problem read sUently and studied
carefully ; then have a pupil tell what to do. Being able to
tell what to do shows that he has really analyzed the situa¬
tion. Otherwise he would not know what to do. Insist
upon neat work, but do not have it written out in steps.
5. Give much attention to the drills in division on pages
174 (22), 176 (24), 177 (25), 179 (27), and 180 (28). With¬
out skill in such drills you cannot expect skill in written
division.
6. There is not much value in giving the work on " mul-
tipl3dng by 11 and 12," given on pages 186-189 (34-37).
It is given in the text because it is still required in most
courses of study. It is well to know the "tables," but
written work with 11 and 12 as " one-figured " multipliers
is seldom used, although it saves some time. Hence this
work may be omitted and the time spent in getting a
greater mastery of the more important work.
120
TEACHERS' MANUAL
Third Month
Cover the work on pages 190-204 (38-52) inclusive. The
work of this month is mostly new, but there should be fre¬
quent short reviews of preceding work.
1. In teaching pages 190 (38) and 191 (39), make much
use of objects and concrete illustrations. The aim is to show
the meaning of a fraction and of the notation of a fraction.
Thus, the child must see that " 5 of the 6 equal parts " of
anything is called " five-sixths " of it, and written f. And
likewise when he sees the notation f, he must interpret it as
" 3 of the 8 equal parts " of the thing considered.
2. Teach the child to read the divisions on his ruler and
to see the relation of halves, fourths, and eighths.
3. To find f of 40, the child should see that this means
" 3 of the 5 equal parts of 40 " ; and hence requires dividing
by 5 and multiplying by 3. However, show, as on page
195 (43), that in using such fractions we multiply first and
then divide, the upper term being the multiplier and the
lower term the divisor.
4. Pages 196 (44) and 197 (45) are to show how to ex¬
press remainders as fractions. Use concrete problems, as
in the text, to show this ; and then drill until the pupil uses
it correctly.
5. In teaching pages 198-201 (46-49), have the measures
present. Page 202 (50) is to show how to change fractions to
smaller terms, for when expressing the remainder as a frac¬
tion, the fraction should always be in smallest terms. Thus,
36-5-8 should be 4^, not 4|.
6. Use many local problems like those on pages 203 (51)
and 204 (52). Teach this month's work slowly and care¬
fully, making every possible local use of it in problems that
arise in and out of school.
THE WORK OF THE FOURTH YEAR 121
Foubth Month
Cover the work on pages 205-213 (53-61) inclusive.
This covers fewer pages than the work of some of the pre¬
ceding months, but the work is new — multiplying by a
two-figured number.
1. Drill on the primary facts of multiplication on page
205 (53), until the pupils all have an automatic control
of them. That is, until they can give any of them without
hesitation.
2. *rhe drill at the bottom of page 205 (53) should be used
daily until pupils can give any product without hesitation.
This drill prepares for the multiplication that must be done
later when dividing by numbers from 13 to 19 inclusive.
3. In multiplying by 10, be sure that pupils do not get
the idea that annexing a zero to $2.45 multiplies it by 10.
4. Observe the way the exercises on page 207 (55) are
written. This is not the way of most texts.
34
To write brings in but one new habit, that of annexing
zero to the product of which he knows how to find.
The steps should be : (1) Multiply by the 2, as has been
done in former work ; then (2) annex a zero to the product.
5. In the development on pages 210 (58) and 211 (59),
have the pupils study the work carefully and get the
method of multiplying by a two-figured number for them¬
selves. Let the brighter ones explain it to those who cannot
get it alone.
6. The drill exercises on pages 211 (59) and 212 (60)
should be used over and over until pupils do not hesitate in
placing the partial products where they belong.
7. Use the problems on page 213 (61) and have pupils
122
TEACHERS' MANUAL
bring in real problems in which the multiplier is a two-
figured niunber.
8. The work of this month will allow considerable time
to review the term's work and prepare for the tests on pages
222-228 (70-76). It would be well to have pupils try these
tests occasionally, to see how nearly they approach the
standards. In this way the pupils are working toward a
definite goah-
Fifth Month
Cover pages 214-228 (62-76) inclusive. This month
gives no new work except Roman numerals, but is a review
of preceding work for the final tests of the semester.
1. The review on pages 214^220 (62-68) is through
problems and thus gives training in problem-solving.
2. If there are any pupils who are weak in any of the
phases of computation, the fundamental drills of the year
should be used by them throughout the month.
3. If pupils have solved the problems given in the text
throughout the term in a thoughtful way, as suggested in
this outline, they should be able to make a score of 80%
or more on the test on page 228 (76). Many fourth-grade
pupils can get all of this test. Give the pupils all the time
needed and insist that the computation be carefully checked
before the answers are turned in. If pupils are found weak
in problem-solving, stress this phase of the work more and
more in the future. Ability to compute is of no value unless
pupils can tell what process to apply in solving a problem.
4. Use pages 202-207 (50-55) for frequent drills as well
as for final tests. That is, pupils should use them over and
over during the month and keep a record of the scores they
make each time. Let each pupil know what is expected of
him so that he will practice with a purpose.
THE WORK OF THE FOURTH YEAR 123
Sixth Month
Cover the work on pages 229-245 (77-93) inclusive.
The chief aim of the month is to develop the power to solve
a problem, and to develop greater skill in computation.
1. The game on page 229 (77) is to suggest problem-
making by the pupils. This phase of the work — the use of
arithmetic in solving problems that occur in life — is gradu¬
ally stressed more and more as pupils get a better grasp of
the computation side of the subject.
2. Page 230 (78) is a review of " expressing remainders
as fractions." Drill on this page until pupils can give " the
cost of 1 " quickly and accurately. The answers must be
given in fractions. Thus, at 3 for lOji one costs 3ij¡5. This
does not mean that you could pay 3^^ if but one is bought,
but it is the actual cost of one if you buy 3 for lOjé.
3. Use pages 231 (79) and 232 (80) very frequently dur¬
ing the month. They are to develop greater skill. They
review all the primary facts of addition. Have pupils keep
a record of the score made each time they use them.
4. Pages 233 (81) and 234 (82) are very important in
developing the power to analyze a situation and discover
what to do. Return to these pages frequently until pupils
can answer any of the questions.
5. Use pages 235 (83) and 236 (84) very frequently dur¬
ing the month.
6. Pages 237-242 (85-90) inclusive are to develop power
to solve a problem. Do not give help, and also show the
class that in getting help from others they are not learning
to solve problems by themselves as they will have to do out
in life. But teach the pupils how to " attack a problem."
To do this, have a problem read silently. Then have a
pupil tell what the problem is. Thus, to the first problem,
124
TEACHERS' MANUAL
a pupil will respond, " A boy knows the price of two coasters
and wants to know the difference in cost." Another pupil
will tell what process to apply.
Do not waste time, however, in useless analysis. It may
be unnecessary to spend so much time on the first one.
When two processes are required, a more carefiil analysis
is needed.
7. Page 242 (90) introduces a new feature of multiplica¬
tion. The only new thing is the proper placing of the second
partial product. Use form A and not form B seen in many
schools.
A B
346 346
208 208
3068 3068
692 6920
72,268 72,268
8. Page 244 (92) introduces a new form of checking divi¬
sion. Show the pupils that this is a better way than going
over the work a second time. When they wish to be abso¬
lutely sure, it is well to use both methods — this method,
and going over the division a second time.
Seventh Month
Cover the work on pages 245-261 (93-109) inclusive.
The work of the month is to develop greater power in
problem-solving and to build up the skills needed for long
division.
1. A pupil should now be able to solve all of the problems
of the test on pages 246 (94) and 247 (95). Use this page
several times during the month.
2. From page 248 (96) the aim is to build up the skills
needed in long division. Teach each page very thoroughly.
THE WORK OF THE FOURTH YEAR 125
3. The first long-division exercises, pages 256-258 (104-
106), are to develop the mode of procedure; that is, the
habit of following the four steps in proper order. The exer¬
cises are so selected that the "trial" quotient is the "real"
quotient. Use the method of presentation given in the
text. That is, in the exercise 21)483, the pupil looks at 2
and 4 and thinks 2. This he writes over 8. Be sure that
the child gets the mode of procedure before introducing more
difficult exercises ; that is, those in which the " trial " quo¬
tient is not the " real " quotient.
4. Such drills as those on pages 258 (106) and 259 (107)
are very essential if skill is to be developed in long division.
Also give much drill on the exercises on page 260 (108).
5. Page 261 (109) still contains exercises in which the
"trial" quotient is the "real" quotient. Use it until
pupils can divide such exercises with great skill.
Eighth Month
Cover the work on pages 262-277 (110-125) inclusive.
The most important work of this month is to teach the
pupil to tell whether the " trial " quotient is the " real "
quotient or not, before writing it down. This is the skill
that is hardest to develop in long division.
1. The second method on page 262 (110) is by far the
best method to use. It makes use of skills already developed
and saves much mental work. It is a new method not yet
in general use. Thus, in exercise 3, page 263 (111), the pupil
looks at 3 and 29 and thinks, " 9 and 2 remaining." Then
he thinks, " The 2 remaining and the 8 following make 28."
He then thinks that 28 is less than 9X6, and thus dis¬
covers that the " trial " quotient (9) is too large and writes 8.
In exercise 4, he looks at 3 and 22 and thinks, " 7 and 1
126
TEACHERS' MANUAL
remaining." Then using the 1 with the 6, he thinks, " 16
is less than 7 X 6," and writes 6 in the quotient.
In exercise 7, he looks at 6 and 33 and thinks, " 5 and 3
remaining." Then he compares 39 with 5X4 and sees
that it is greater, so in this case the " trial " quotient is the
" real " quotient, and so he writes 5.
In exercise 10, the " trial " quotient is 6 and 22 remaining.
But 22 is less than 6X6, so the " real " quotient is 5,
By a little practice the pupil can easily tell the real quo¬
tient without much effort.
2. Teach very carefully the problems on pages 265-27C
(113-118) and on pages 274-277 (122-125). They are tc
develop greater power in " arithmetical reasoning," with¬
out which the power to compute is of but little value.
Ninth Month
Cover the work on pages 278-293 (126-141) inclusive,
Division by numbers from 12 to 19 inclusive is more diffi¬
cult than the work in long division that has been given, foi
to find the " real " quotient requires more mental multi¬
plication. The method is shown on page 278 (126).
1. In order to build up this new skill needed in long divi¬
sion by numbers from 12 to 19 inclusive, give much oral
drill on the exercises on page 279 (127).
2. Expressing the remainders in long division, given or
pages 281-284 (129-132), is a new feature introduced foi
the first time in this text. The older type of texts eithei
left them as " remainders " or wrote the remainder aboví
the divisor and thus formed a type of fraction meaningless
to the child and never used in the real problems of life,
Teach carefully these pages.
3. Note carefully the work on page 283 (131). Mösl
pupils make the mistake of leaving the last decimal founc
THE WORK OF THE FOURTH YEAR 127
in the quotient. In life we use the " nearest " number as
shown here.
4. Pages 286-293 (134-141) give a little simple work in
fractions. This is not vital as a preparation for the work
of the fifth year, for it is not assumed in the work of that
year that the child has had any work in fractions. This
work, then, is optional. If time is needed to get required
skill in work with whole numbers, it had better be spent in
this way.
Tenth Month
Cover the work on pages 294-306 (142-154) inclusive.
The only new work is measuring the surface of a rectangle.
1. With square inches or square feet, have actual areas
measured until the child sees the meaning of measuring areas.
2. In computing areas, use abstract numbers and avoid
letting the child get the erroneous notion that " inches
times inches give square inches," and " feet times feet give
square feet," an error found in many schools.
3. Make much use of the tests on pages 300-306 (148-
154) throughout the month. By following the text and the
course of study, as laid out in this Manual, pupils should be
far superior to most of the pupils of the country as a whole
and able to surpass existing standards.
Fourth-geade Achievement Tests
Note. — For shorter tests, use those of the third grade. There should be
an increase of 10% or 15% in skill.
Test I
This test contains all the addition facts except a few of
the zero facts and the columns are long enough to test
the attention span. The 12 should be done in about 5
minutes.
128
TEACHERS' MANUAL
Add down:
1.
2.
3.
4.
6.
6.
7.
8.
9.
10.
11.
12.
3
6
8
2
7
3
2
6
9
2
9
5
5
5
4
4
9
9
6
1
8
7
3
9
7
7
5
6
2
2
8
7
3
6
8
7
8
2
6
3
1
1
7
7
6
5
6
2
6
4
4
9
2
3
2
8
8
1
5
8
9
9
2
4
6
6
1
5
8
9
1
5
3
3
1
9
4
2
8
3
1
8
6
9
4
9
9
1
3
3
6
8
7
3
9
7
2
4
7
5
5
4
9
7
5
5
6
6
5
6
4
1
3
8
7
9
6
7
5
4
62
65
60
44
42
41
66
61
61
63
68
62
Test n
This test contains all but a few of the zero facts. The
5 exercises should be done in about 5 minutes.
Add down:
1.
2.
3.
4.
6.
368
247
392
469
475
836
739
359
957
457
341
262
958
271
881
752
118
694
142
617
475
396
523
495
928
246
873
738
968
493
971
658
967
555
971
819
471
643
894
598
4808
3764
6274
4761
6420
Test III
This test uses all the facts of multiplication except the
I's and O's, some of them two or three times. The 16 should
be done in about 7 minutes.
THE WORK OF THE FOURTH YEAR 129
Multiply:
1. 2.
3.
4.
5. 6.
7.
8.
72 93
76
95
64 86
69
47
28 29
27
52
49 58
76
79
2016 2697
2052
4940
3136 4988
5244
3713
9. 10.
11.
12.
13. 14.
15.
16.
84 92
63
87
56 95
67
92
38 48
39
63
47 38
48
79
3192 4416
2457
5481
2632 3610
3216
7268
Test IV
This test contains all primary facts except a few of tl
I's and O's. The 12 should be done in about 8 or 9 minute
1.
2.
3.
4.
5.
6.
916
729
938
318
807
624
38
26
48
74
57
67
34,808 18,954
45,024
23,532 45,999
41,808
7.
8.
9.
10.
11.
12.
690
629
915
863
928
294
46
95
75
35
38
54
31,740 59,755
68,625
30,205 35,264
15,876
Test V
The 8 should be done in about 10 minutes.
Give quotients and remainders:
1. 2. 3.
309^
46)14245
5.
758^
85)64499
790^
54)42679
6.
876^
94)82420
587^
63)37040
782^
4.
659^
67)52447
74)48830
8.
689^
56)38637
130
TEACHERS' MANUAL
Test VI
Problem Test
Do not use a time limit. Pupils should reason out what
to do, not " guess " at it.
Give answers vñthout a pencil:
1. John had 50^, earned 20ji, and spent 30ji. How
much had he left ? Ans. 40)í
2. Frank earned 60)i and spent i of it. How much had
he left? Ans. 40j5
3. Walter earned 30¡í an hour for 4 hours and spent ^
of it. How much did he spend? Ans. 60fi
4. At the rate of 3 apples for lOfi, how many can you
buy for 50jé? Ans. 16
5. When oranges are selling at 4 for 15)¿, how much will
a dozen cost? Ans. 45ff
6. After spending 30^5, John had 60^5 left. How much
had he at first? Ans. 90^
7. If Frank can ride 24 miles in 3 hours, how far can he
ride in 2 hours at the same rate ? Ans. 16 mi.
8. When oranges are GGji a dozen, how much should
8 oranges cost? Ans. 40jí
9. A picnic lunch for 6 children cost $2.50. How much
would it cost 12 children at this rate? Ans. $5.00
10. At 80^ a pound, how much should f of a pound of
candy cost? Ans. 60jí
Test VII
Written Problem Test
Children will score all the way from 40% to 100% on this
test. 50% should be a passing grade.
1. Frank earned $2.75 a week for 7 weeks and spent
$5.80 of it. How much did he save? Ans. $13.45
THE WORK OF THE FOURTH YEAR 131
2. Walter wants a bicycle that costs $28. He has
$16.45. In 7 weeks he can earn enough more to buy it.
How much is he earning per week? Ans. $1.65
3. John had $3.85 and earned $1.65 more. If he spends
$2.70 for a pair of skates, how much will he have left?
Ans. $2.80
4. Twenty-four children went on a picnic party. Their
fare cost them $8.40 and their lunch cost $9.60. What did
it cost each if they shared the expense equally? Ans.
5. .A boy bought 5 dozen apples at 40 ¡¡5 per dozen and
sold them at 5 jé each. How much did he make ? Ans. $1.00
6. To help buy a picture costing $36.50, some children
gave an entertainment. They sold 80 tickets at 35^ each.
How much more money will be needed to buy the picture?
Ans. $8.50
7. Ned paid 3jé each for papers that he sold for bi each.
How many must he sell to make $1.50? Ans. 75
8. Walter sold $22.50 worth of vegetables from his gar¬
den. The total expenses for seeds and plants were $3.60.
He worked 42 hours. How much did he make per hour?
Ans. 45jé
9. Frank earned $3.45 per week for 6 weeks and saved
$16.80 of it. He spent an average of how much per week?
Ans. 65f5
10. At a " fair " to raise money for a phonograph the
children at the candy booth made $3.60 by selling candy at
40jé per pound that cost them 25per pound. How many
pounds did they sell? Ans. 24 lb.
CHAPTER Vm
THE WORK OF THE FIFTH YEAR
{The pages refer to Stone's Intermediate Arithmetic and Fifth-Year Arithmetic)
The aims of the fifth year are :
1. To develop greater accuracy and speed with whole
numbers.
2. To develop power to reason more accurately in the
solution of problems.
3. To develop skill in computing with fractions and power
to use them in problems.
4. To use those decimals that have their parallel in United
States money.
5. To solve problems that involve areas of rectangles and
the volume of rectangular prisms.
Standards of Skill
The following standards should be attained by the end
of the fifth year :
I. addition
1. Twenty-five exercises of six one-figured numbers in
five minutes or less.
2. Twelve exercises of four three-figured numbers each
in five minutes or less.
II. subtraction
Fourteen exercises of five-figured numbers with two or
three carryings in five minutes or less.
132
THE WORK OF THE FIFTH YEAR 133
III. multiplication
1. Twelve exercises of two-figured multipliers and mul¬
tiplicands in five minutes or less.
2. Six exercises of a four-figured number by a two-fig¬
ured number in five minutes or less.
IV. division
Five exercises of a five-figured number by a two-figured
number with a three-figured quotient and a remainder, in
five minutes or less.
How to Get Skill
Skill in anything comes through much practice with atten¬
tion and with an aim. Merely recalling a fact does not fix
it, but recalling it with an attempt to remember it is what
fixes it. Hence let the pupil see what he is to attain, and
see that he works with an aim.
For example, to add twenty-five exercises of six one-figured
numbers in five minutes he must make 125 combinations, or
25 per minute. To add an exercise of four three-figured
numbers he must make 11 combinations if there is carrying
from each column. So in each drill period have the pupil
compute how many combinations per minute he made, and
have him keep a record of it. It is well to keep these records
in the form of a graph on squared paper.
There must be frequent diagnostic tests to find individual
weaknesses. The child must know where his weakness is
and strive to overcome it.
Drills upon the primary facts must continue with a def¬
inite aim as to skill, especially upon the more difficult
ones.
Nothing is gained through " dawdling " through a drill.
A short period with the maximum attention is better than
134
TEACHERS' MANUAL
a long period with but partial attention. Drill without a
definite purpose is a waste of time.
Checking Work
The pupil should develop the habit of checking all work
before it is considered finished. While he should work for
100% accuracy in the first computation, even accountants
never attain this and always check their work.
Developing Power to Solve Problems
The solution of problems becomes a greater factor in this
grade than in former grades. To develop this power, the
pupil should read the problem carefully, picture the situa¬
tion, and tell what to do. There is nothing gained through
the so-called " written analysis " or the labeling of terms.
The forms in this text should be those required of the pupil.
The " incomplete problems," the " problems without
number," and the frequent " tests " in problem-solving are
all to develop this power. They should be carefully and
thoughtfully used. They should not be worked for a pupil.
The only way in which he can develop power is to work them
for himself.
The problems in the text should be used in two ways:
Some should be assigned for out-of-class study; others
should be used for the first time in class. When the aim
of the recitation is to develop power to solve a problem,
have the pupils turn to some unassigned page. Do not take
the problems in order, but ask the class to read a certain
problem silently. After they have had time to read it, ask
a pupil to stand and give, without the book, the problem
and tell how to solve it. He need not be required to give
the exact numbers, but he may say, " The problem gives the
amount earned per week, the amoimt spent, and the num-
THE WORK OF THE FIFTH YEAR 135
ber of weeks, and requires the total amount saved. So sub¬
tract the amount spent each week from the amount earned,
and multiply the difference by the number of weeks."
In using the tests, it is recommended that they be given,
unassigned for study, during the class period. Problem-
solving depends upon judgment and reasoning, which vary
with native ability. Hence pupils will vary greatly in their
standing in such a test owing to native ability. It is not
unusual for pupils to vary from 30 to 40 points up to 100.
This does not mean to imply that native ability cannot be
trained, but that through proper training the alert and in¬
telligent child will develop much more rapidly than one less
intelligent. Hence in work requiring reasoning, there will
be a much greater variation than in work done by rote.
These tests should meet the needs of all. They contain
enough difficult problems to interest the strongest pupil and
enough easy ones to give some work for the dullest. The
same tests may be given many times during the year, but
not so frequently that the pupil will solve them through
memory.
The first chapter is a very important one both in drill and
in problem-solving and should be used frequently throughout
the year.
Common Fractions
The teaching of any process depends first upon the mean¬
ing of the notation. The addition facts with whole numbers
would have no significance if the meaning of the figures were
not first known. Likewise, before work in fractions can
have any meaning, the notation of a fraction must be
understood.
As on page 47, divide an object and have the pupil describe
a part of it without fractions, as, " Three of the five equal
136
TEACHERS' MANUAL
parts," etc., and then show him the short way of saying and
writing it — " three-fifths " and " f
Use fractions in expressing the relation of a part to a whole
and of one unit to another, and so on, until you are sure
that such symbols as | mean " five of the eight equal parts
of," and so on.
In developing the meaning and notation of a fraction,
use concrete illustrations and bring out, as on pages 49 and
50, that fractions are added, subtracted, multiplied, and
divided as whole numbers are; and thus make the pupil
feel at home with them. Do not bring in definitions until
they are introduced in the text. They do not aid in the
understanding, and they may cause confusion.
Adding and Subtracting Fractions
When fractions are written in line form to add, as,
" f + T>" pupils are apt to confuse addition with multiplica¬
tion later, and add both numerators and denominators. The
author has seen pupils in the upper grades, and even in high
school, do this. To avoid this, all fractions to be added or
subtracted should be written in column form. This is not
only to avoid such confusion, but to connect the work with
the habit established in adding whole numbers. Follow the
notation and development of the text.
To add halves and fourths (pages 58-60), the pupil must
learn the relation of halves and fourths objectively and
remember them. These should be drilled upon, just as the
primary facts with whole numbers were, until they are
recalled automatically. Likewise, the relations on page 62
and throughout the text must be treated in the same way.
Work for a high degree of skill. Motivate the drills by
" races," as on page 64.
The fractions that one has to add or subtract in the pr/?lv
THE WORK OF THE FIFTH YEAR 137
lems that arise in life are halves, thirds, fourths, sixths,
eights, ninths, twelfths, and sixteenths. But only certain
ones of these occur in the same problem. The work in this
text is as difficult as any found in real problems ; so strive
for skill with these fractions rather than waste time in add¬
ing those never met in life.
After a pupil has learned the relations of the fractions
that he needs, he should, through them, see inductively that
if one term of a fraction is multiplied by any number, the
other liprm must be multiplied by the same number in order
to give a fraction of the same value. Then if the pupil ever
needs to change some fraction to some one whose relation
is not known, as f to 15ths, he will reason that since 15 is
5X3, the numerator must be 5 X 2.
SUBTEACTING MiXED NUMBEKS
Observe the three types of problems on pages 65, 66, and
67, and the easy development of each by the addition
method. This is not the method of other texts, but much
simpler and more efficient. So the teacher should be thor¬
oughly acquainted with it before teaching it, in order not to
confuse it with former methods. Observe the two steps in
the type of problem on page 67. Do not allow the pupil
to put down more work than is given in the boxed solutions
of the text. Have him " think " as in the directions of the
text and you will be surprised at the ease with which he sub¬
tracts. Do much drill work to develop skill. " Races,"
as on page 70, motivate the work.
At the end of the first semester, pupils should pass the
four tests on pages 77, 78, and 79 with a perfect score. No
" time " is given, for it is knowledge and not speed that is
being tested.
138
TEACHERS' MANUAL
Multiplication and Division of Fractions
To rationalize the work with fractions the order of de¬
velopment must be :
1. Multiplying a fraction by a whole number.
2. Dividing a fraction by a whole number.
3. Multiplying a fraction by a fraction.
4. Dividing a fraction by a fraction.
Multiplying by a whole number
This is developed from the meaning of multiplication and
the addition of fractions. Follow the development on page
89. The pupil knows from the meaning of multiplication
that 3 X $5 may be found by writing $5 three times and
adding. In like manner, to find 3 X f he may write |
three times and add it. Thus he sees that the numerator
only is multiplied when multipl3ang a fraction by a whole
number.
Dividing by a whole number
Develop the rule concretely as on page 96. In addition
to the development of the text, if further demonstration is
needed, suppose that f of a pie (using a picture) is to be
divided equally between two boys. Just as if they are
apples, each boy will get ^ of a piece and J remains to be
divided into two equal parts, so each boy would get i and
i or |. Thus, f ^ 2 = |. Use several such problems.
Then inductively it is seen that you may divide a fraction
by a whole number by dividing the numerator or by multi¬
plying the denominator.
Multiplying a fraction by a fraction
The method of development is shown on page 100. The
development depends upon three things already established :
THE WORK OF THE FIFTH YEAR 139
(1) The meaning of finding a fractional part of anything,
as Î X 15 means multiplying by 3 and dividing by 4 ; that
is, it is 3 X 15 -Í- 4; (2) how to multiply a fraction by a
whole number ; and (3) how to divide a fraction by a whole
number.
Just as f X 15 = 3 X 15 4,
So Î X f = 3 X I -Î- 4, or f -7- 4 or
Work out several answers from the meaning of the process
and have the pupils see inductively that work is saved by
multiplying the numerators and the denominators.
In all development work, have those pupils who are able
to do so get the development from the text rather than from
the teacher. Give help only when it is necessary.
Dividing a fraction by a fraction
The methods of rationalizing the division of fractions
used in the past, although logical, were beyond the com¬
prehension of children. Therefore they served no purpose.
They neither helped the child remember how, nor helped
him in any way to develop the how if forgotten. The
method used in the text is easily understood by any child
in the fifth grade. It depends upon just two facts that the
child already knows : (1) When the divisor is 1 the quotient
is the same as the dividend ; and (2) when both dividend and
divisor are multiplied by the same number the quotient is
not affected.
Let the problem be :
2\3
sH
The child can see that the divisor (|) multiplied by |
gives 1. Multiplying both divisor and dividend by |, the
problem becomes :
m
9
¥
140
TEACHERS' MANUAL
Hence, to divide by a fraction, multiply by its reciprocal.
This is a universal law applicable to all numbers. Thus :
24 ^ 2 = i X 24
16 -r- i = 2 X 16
75 ^ 8 = .125 X 75
64 -i- 12^ = .08 X 64, and so on.
The Drills, "Races," and Tests in Fractions
The drills throughout the text bring up over and over
those fractions that the child will use in life so that he may
get the same automatic control of them that he has of whole
numbers. They should be used very frequently until de¬
sired skill is attained.
The " races " are drills, but they are diagnostic tests as
well. For example, each " race " on page 64 is a step in
advance of the one preceding it as to difiBculty. The " race "
in subtraction on page 70 is a test of all the child has had in
subtraction. In the same way, each "race" is both a
motivated drill lesson and a diagnostic test.
The tests are graded to show the weakness in any phase
of the work. Thus, on page 87, Test I has like denomina¬
tors; Test II has related denominators; and Test III has
unrelated denominators. The same is true of the subtrac¬
tion tests on page 88. If the tests show a weakness in any
child, that child should be given remedial drill and the topic
should be redeveloped if necessary.
Decimal Fractions
Decimal fractions are new ideas in name only to the child.
The first use of a decimal is in expressing the quotient in an
inexact division. This was taken up in the fourth grade.
THE WORK OF THE FIFTH YEAR 141
Follow the development of the idea and use of a decimal on
pages 112-115,
Adding and subtracting decimals are exactly like adding
and subtracting dollars and cents and need no new rides
or development. Likewise, multiplying and dividing by
whole numbers introduces nothing new. In this way, the
work in decimals in the fifth grade is kept easy, simple, and
natural.
The Problems of Measurement
•
Pages 127-129 take up the addition, subtraction, mul¬
tiplication, and division of what was formerly called Com¬
pound numbers. The term, however, is not used in this
text. This is not an important topic and may be omitted.
The more common way is the " second way " given under
each process. Such work is still found in other books, in
certain standard tests, and in most courses of study.
Measuring Rectangles and Prisms
Notice that the method of development differs from that
of other texts. For generations, teachers and textbooks
have taught that the number of square units along one
dimension is the same as the number of linear units in this
dimension, and that there are as many such rows as there
are units in the other dimension. So to find the area of a
rectangle 8 ft. by 15 ft., they have taught the pupil to write
this, 8 X 15 sq. ft. = 120 sq. ft. While this seems simple and
concrete it does not seem to mean much to the pupil and you
still hear " feet by feet gives square feet," just as " a: by a; is
Hence this text abandons the former method and, after
developing the rule inductively (page 133), gives the rule
and formula and then uses abstract numbers (page 133) in
finding the area. It is urged that all work in measurement
142
TEACHERS' MANUAL
be done with abstract numbers as on pages 133 and 134.
The same plan should be used in working with prisms and
all future problems in measurement.
The Work of the Fifth Year Divided into Ten Units
To aid the teacher in covering the work of the year the
text is here divided into ten units. See that each unit is
properly completed in the time given. The work of the
year can then be accomplished easily.
First Month
The first three months should be devoted to the first 46
pages. The work of this month is pages 1-15 inclusive.
The text is so arranged as to give variety between problem-
solving and computing.
1. Cover the first six pages slowly and completely. They
are to develop power to analyze a situation and discover the
solution. Do not give direct help, but, if necessary, show
pupils how to attack a new problem. To show a pupil how
to solve a problem does not train his reasoning powers. It
merely helps him to solve a like problem. Impress upon
pupils that the problem must first be read carefully and a
picture of the situation clearly formed. Then they are to
discover what processes are to be used and the order in
which they are to be applied.
2. Page 5 is a valuable type of work to bring out a care¬
ful reading and the analysis of a situationi
3. Page 6 is another valuable type of work to develop
the power to solve a problem.,
4. Always be on the alert to find good local problems,
but never bring in unreal problems merely to offer an excuse
for bringing in a process of computation. Be sure, too,
that the problems are vital and interesting.
THE WORK OF THE FIFTH YEAR 143
5. Pages 8 and 9 are to impress upon pupils the need of
checking all work. Insist that all computation be 100%
accurate before it is considered finished.
6. Test II, page 11, is to build up the attention span.
Pupils who can make a high score on Test I will usually make
a much lower one on Test II, owing to the fact that it
requires a longer span of attention. Use these drills a few
minutes daily.
7. The drills on pages 12-15 are to develop greater skills
in four of the primary abilities needed in written addition,
viz. : (1) Primary facts ; (2) adding by endings ; (3) add¬
ing by endings with sums in the next decade; (4) using
these skills in adding single columns, to build up the power
to hold the attention for a longer span. Make very fre¬
quent use of these four drills.
8. Do not take the work page by page, but divide each
lesson up into drill and problem-solving. Never let the
work become boresome. Use contests, time limits, and a
desire to beat past records.
Second Month
Cover the work on pages 16-31 inclusive. As in the first
month, the aim is greater skill in computation and greater
power to solve a problem.
1. Page 16 is really an achievement test. If pupils lack
the desired skill, page 17 should be drilled upon until the re¬
quired standard is attained.
2. Use " test in solving problems," pages 18 and 19, to
discover how pupils rank in ability to reason. Use it as a
class exercise without previous assignment for study.
3. The problems, pages 19-22, are very valuable in de¬
veloping " arithmetical reasoning." Encourage pupils to
solve all they can without help. Impress upon them that
144
TEACHERS' MANUAL
to get help will not help them when they meet a new type
of problem. Try to work up a spirit of determination to do
all work without help. Get a pupil to take great pride in
doing the work himself. Have him see that this is the only
way to become a strong student. He must take pride in
his work and in his own ability if he is to succeed.
4. See that all pupils understand the meaning of mul¬
tiplication (pages 23 and 24) and the names of the terms.
5. Tests I and II, page 24, are achievement tests. If it
is found the pupils lack skill, use pages 25-28 very frequently.
The drill sets contain all possible combinations.
6. Teach thoroughly pages 30 and 31. This was given
in grade four, but pupils need much more practice in using it.
Third Month
Cover the work on pages 32-46. This month is devoted
chiefly to building up those abilities needed in long division.
Not much skill could be expected in the fourth year. For
that reason, long division is stressed here.
1. Have pupils see clearly the two uses of division as given
on pages 32 and 33.
2. The oral drills on pages 37, 38, and 39 are very im¬
portant and should be used daily.
3. The drill on page 39 is to develop the power to give the
" real " quotient from the " trial " quotient, which is the only
difficult thing in long division. This drill should be used
5 or 10 minutes daily.
' 4. Use pages 44-46 frequently throughout the term.
Fourth Month
Cover the work on pages 47-64 and give frequent reviews
of pages 44-46.
This is new work and should be done very carefully and
THE WORK OF THE FIFTH YEAR 145
slowly. See that pupils understand every phase of the work.
Otherwise there can be no progress.
1. Pages 47-50 are to develop the meaning and the nota¬
tion of a fraction. Use concrete illustrations. Three
eighths should mean " three of the eight equal parts."
2. Finding a fractional part of a number (pages 52-53)
is one of the most common uses of a fraction. No new skills
are required, but the child must see that the upper term is
always a multiplier and that the lower term is always a
divisor. This comes from the meaning of a fraction.
3. Now that the child knows the meaning of a fraction,
he should express remainders in division as fractions (pages
53 and 54). Be sure that this is fully understood.
4. Page 55 suggests a type of local problems that pupils
can make and solve. Always show the need of each new
process by use of some local or personal problem. The prob¬
lems of the text are of this nature, but problems are even
more vital when made from some local situation.
5. Begin addition of fractions with mixed numbers with
like fractions, as in the text. This gives a more real type
of problem and uses fractions in column form as in whole
numbers. The oblique line is used to make the numerators
stand out so as to be easily seen.
6. Never write fractions in line form, as i + i = f, for
the pupil is likely to confuse addition with multiplication
later and add both numerator and denominator. The writ¬
ing of fractions to be added, in column form with oblique
lines, is a distinct feature of this book, for the author has
found that children taught by the older method make such
mistakes as | 4-1 = f.
7. Drill on such relations as i = f = I, 4 = I, f = f,
etc., until the pupil knows them as he knows the primary
facts of addition and multiplication. If this is done, he
146
TEACHERS' MANUAL
does not have to compute the change to a common denom¬
inator and write down the new fractions, but merely
" thinks " them into the forms wanted. Use the forms of
the text, making the changes mentally.
8. Observe that the four " races " on page 64 are really
diagnostic tests. That is, each race requires a different
ability or skill. Use this page until you get great skill in
adding such fractions. Develop a spirit of contest so as to
get interest and attention.
Fifth Month
Cover the work on pages 65-80 inclusive and continue to
drill on pages 44-46. The new work is that of subtraction.
1. Observe that there are three distinct types of exer¬
cises. Page 65 gives the first type. Use the method given
here ; that is, the " addition " method. This method is
much easier than the older methods, especially when applied
to the next two types of subtraction.
2. The second type of subtraction is given on page 66.
Observe that the pupil has to use much less work than in the
method taught in older books. You, no doubt, have always
used the older methods, for this method is a new one pre¬
sented here for the first time. So be sure that you do not use
a " mixture " with other methods in presenting it. The
author has seen children use this method with skill within
two minutes of the first presentation.
3. The third type, given on page 67, is the one that re¬
quires much more work by older methods than by the method
given here. Observe the two steps to such problems given
on page 68. If former abilities have been properly devel¬
oped, this third type of subtraction gives but little trouble.
4. The " practice in subtraction," pages 68 and 69, are
diagnostic tests. Test I is on the first type of subtraction ;
THE WORK OF THE FIFTH YEAR 147
Test II is on the second type ; and Test III is on the third
type. Use these drills until pupils have a good control of
each type.
5. The " race " on page 70 takes in all the skills developed
and should be used over and over many times.
6. Have pupils memorize the relations at the bottom of
page 74. If this is done, fractions need not be rewritten as
in problem 1, page 75.
7. If the month's work has been well done, pupils should
make, almost perfect scores on the tests on pages 77-80.
Sixth Month
Cover the work on pages 81-95 inclusive.
1. Be sure that the general review of former work on
pages 81-84 is thoroughly done before beginning new work.
2. The work on pages 85 and 86 is important. One of the
most important uses of fractions is that of expressing the
relation of one number to another. Have the pupils see
clearly that the number compared is the numerator and the
number with which it is compared is the denominator.
3. The tests on pages 87 and 88 will reveal any weak¬
ness in adding or subtracting fractions. If any weakness is
found, be sure that it is overcome before proceeding with the
new work.
4. Observe that when multiplying a fraction by a whole
number, the rule is developed from the meaning of multipli¬
cation. That is, 3 X I means f + f + î, just as 3X4
means 4-1-4 + 4.
5. The problems are to show the use and need of fractions
and to develop power to reason. Bring in local problems
when possible.
6. The " race " on page 95 brings in all abilities now
developed. It should be used over and over until it can be
148
TEACHERS' MANUAL
done with great skill. It is really a review of all former
work in fractions, and at the same time a diagnostic test.
If any skill is lacking, apply remedial drills to build it up.
Seventh Month
Cover the work on pages 96-111 inclusive. The new work
is division of fractions.
1. Develop the division of a fraction by a whole number
objectively. This is the " partition " use of division.
2. Notice on page 98 how to divide a large mixed number
by a whole number. This is much more economical than
the method generally taught in other texts.
3. Multiplying a fraction by a fraction, pages 100 and
101, depends upon three abilities already established. They
are ; (1) Multiplying a fraction by a whole number ; (2) di¬
viding a fraction by a whole number; and (3) finding a
fractional part of any number.
4. Make much use of the " race " on page 103. It is a
review of the abilities developed in multiplication and divi¬
sion already taught. Be sure this is well done before taking
up the final case of division by a fraction. It is really a
test. If weaknesses are discovered, apply remedial drill.
5. In teaching a child to divide a fraction by a fraction,
be sure that he knows two fundamental principles : (1) That
whenever the divisor is 1, the quotient is like the dividend ;
and (2) that multiplying both dividend and divisor by the
same number does not affect the quotient.
Thus, 1H> for, from the meaning of division, this means
f
" 1 times f is
In the problem if both dividend and divisor are mul¬
tiplied by f, the problem becomes 1^; so the quotient is
That is.
THE WORK OF THE FIFTH YEAR 149
The quotient was found by multiplying the dividend by the
reciprocal of the divisor.
This method of development is much more simple than
any former method, and the process is much more easily
remembered when developed in this way.
6. Use the exercises on page 108 over and over until the
method is fixed.
7. Make much use of the drills and " races " on pages 110
and 111, and return to them frequently throughout the re¬
mainder of the year.
Eighth Month
Cover the work on pages 112-129 inclusive. The work of
this month is new, but the former drills and races in whole
numbers and fractions should be used occasionally to retain
and extend former skills.
1. The first use of a decimal is to express quotients in
uneven divisions. See that the pupils appreciate the mean¬
ing and use of decimals. Observe that the development
grows out of what the child has done in working with num¬
bers written as dollars and cents. That is why the child
is first taught to read hundredths and then tenths. It makes
the " transfer of habit " very simple and easy. In fact,
all of the work in decimals given in this month has its par¬
allel in work with dollars and cents. For that reason, it
involves no new difläculty. The treatment in the text is
unique and the simplest possible treatment. Hence do not
attempt to teach more than given here, or by methods given
in other texts. A pupil is confused by too many methods.
The presentation of the text can easily be read and under¬
stood by the pupils. Do not give any help where pupils
can help themselves.
2. The work in practical measurement on pages 123-126
150
TEACHERS' MANUAL
inclusive is very important and should be done thoroughly
and carefully.
3. Pages 127-129 inclusive show how to add, subtract,
multiply, and divide compound numbers, as such. This
may be of some interest, but it is of little value. It is given
in the text because most courses of study require it. The
usual way in life, when using such numbers, is to express
them as mixed numbers as in the " second way " given in
the text. The " first way " may be omitted, but the " sec¬
ond way " is important. But if you are fitting for some of
the so-called " standard tests," the " first way " must be
taught.
Ninth Month
Cover the work on pages 130-147 inclusive.
1. Be sure that the pupil fully understands the meaning
of measuring a surface. Use the method on page 131, giv¬
ing many other exercises.
2. In teaching the pupils to " compute " the area of a
rectangle, follow the inductive method of page 132. Ob¬
serve that in finding an area only abstract numbers are used.
This is to avoid giving a child a wrong notion that he
is " multiplying feet by feet " or " inches by inches." More¬
over, to use g perfectly correct form of 3 X 4 sq. ft. in find¬
ing the area of a rectangle 3 ft. wide and 4 ft. long, leads to
confusion when the child comes to measure surfaces of other
forms. Thus, in finding areas of triangles, trapezoids, or
circles he is at a loss to know what factor to select as multi¬
plicand (and make it a concrete number) and what ones to
use as abstract numbers. It is useless to impose any such
confusion upon him. The only way to avoid it is to use
nothing but abstract numbers, as is done in the text.
3. Teach the child from the very first how to interpret
THE WORK OF THE FIFTH YEAR 151
and use a formula. Have him see that it is only a short
way to express a long sentence, and hence that it saves time,
and pictures to the eye a simple fact easily remembered.
4. As in all cases, use each new fact or rule in some local
situation. The problems on page 135 may suggest other
local problems.
5. Be sure that pupils do not confuse area with perimeter.
In many tests given high school graduates I have found such
a confusion.
6. Use the suggestions concerning areas in teaching vol¬
umes on pages 140 and 141.
7. The problems of the month are to fix the rules and
facts, to show their uses in real problems of life, and to
develop " arithmetical reasoning," so give help sparingly.
Tenth Month
Cover the work on pages 148-160 inclusive. As in the
former years, the last month is devoted to reviews and to
the extension of former work instead of to new work.
1. The work in decimals on pages 148-156 inclusive does
not include anything new, except reading thousandths and
ten thousandths. It is a review of the four processes taught
in a former month.
2. The tests on pages 157-160 inclusive should have been
anticipated during the year and the children should have
worked toward them.
Tests on Fundamental Pbocesses
No time limit is given in these four tests, for they are
inventory tests to show whether the pupil has control of all
types of computation that have been taught. Hence a
pupil should be given all the time he needs. If the tests
152 TEACHERS' MANUAL
show any weakness in any of the different types, remedial
drill should be given.
Test I
Addition
1.
2.
3. 4.
5.
6. 7.
8.
24
37
Vs K
%
% SH
5%
_5
_9
K K
K
H 8H
7%
9.
10.
11.
12.
13.
14.
7
58
$3.78
5.8
26M
19%
9
65
5.67
16.53
57%
43%
6
9
2.89
7.08
84%
8%
8
86
7.58
45.9
9%
7%
15.
16. 12.5 + 4.85 + .9 + 36.5
3 lb. 8 oz.
17. 365 + 8.46 + 39.5 + .87
6 lb. 12 oz.
18. 3^
+ 61 + 71 +
7
7 lb. 14 oz.
19. 5f
+ 7i + 9| + 8^
20, 368 + 554 + 775 + 826 + 644 + 992 + 499 + 274
Answers:
1.
29
5.
1 5
9. 30
13. 178
17. 413.835
2.
46
6.
1Ä
10. 218
14. 79|
18. 18|
3.
1
7.
Hi
11. $19.92
15. 18 lb. 2 oz.
19. 31i
4.
3
1
8.
13|
12. 75.31
16. 54.75
20. 4932
Test n
Subtraction
1.
2.
3.
4. 5.
6.
7.
89
71
201
% %
%
$3.50
54
36
139
% %
%
1.48
THE WORK OF THE FIFTH YEAR 153
8. 9. 10. 11. 12. 13.
9.36 35006 48.3 3.68 16^ 90
1.37 12898 7.58 1.9 9^ 36%
14. 15. 16. 17. 3i - If
28% 7 lb. 12 oz. 5 lb. 8 oz. 18. 36.9 - 19.3
19% 5 lb. 8 oz. 1 lb. 12 oz. 19. 54.3 - 3.48
20. 75.16 - 24.9
Answers:
1. 35 5. i 9. 22,108 13. 53| 17. If
2. 35 6. ^ 10. 40.72 14. 8f 18. 17.6
3. 62 7. $2.02 11. 1.78 15. 2 lb. 4 oz. 19. 50.82
4. f 8. 7.99 12. 7i 16. 3 lb. 12 oz. 20. 50.26
Test in
Multiplication
1.
2.
3.
4.
5.
6.
385
507
486
520
$7.69
5.96
29
86
907
760
40
29
7. f X 265 8. I X 345 9. 3i X 176 10. f X f
11. i X 12. 3 X 98i 13. 2i X 16f 14. f X 75i
15. 45 X 26.4 16. 208 X 709 17. 3i X $8.75
18. 5| X 18.6 19. 200 X 7.95 20. 5 X 6Ib. 7oz.
Answers:
1. 11,165 5. $307.60 9. 668^ 13. 41| 17. $30.62J
2. 43,602 6. 172.84 10. ^ 14. 56f 18. 106.95
3. 440,802 7. 159 11. f 15. 1188 19. 1590
4. 395,200 8. 301|^ 12. 295| 16. 147,472 20. 32 lb. 3 oz.
154
TEACHERS' MANUAL
Test IV
Division
Give quotients and remainders:
1. 36)15820 2. 54)42703
Carry quotients to nearest hundredths:
3. 85)68668
4. 37)2092
5.
43)1817
Divide:
7. i ^2
11.
H 3
15.
3
T
_ 1
Î
19.
S. i ^2
12.
2i ^ 4
16.
5
S
_ 3
T
20.
9. f s- 3
13.
175i ^ 4
17.
Ii
3
10. 1^3
14.
e . 2
T • T
18.
2i-
li
Answers:
1. 439 ; 16 rem.
6. 42.26
9.
i
13.
43f
2. 790 ; 43 rem.
6. 9.38
10.
2
V
14.
3
3. 807 ; 73 rem.
7. i
11.
i
15.
li
4. 56.54
8. Î
12.
A
16.
li
6. 63)590.7
1 3
20. 3)10 lb. 14 oz.
17. 4
18. 1|
19. ^
20. 3 lb. 10 oz.
CHAPTER IX
THE WORK OF THE SIXTH YEAR
{The pages refer to Stone's Intermediate Arithmetic and Sixth-Year Arithmetic)
The aims of the sixth year are :
1. To develop greater skill in accuracy and speed in com¬
puting with whole numbers, extended to larger numbers.
2. To teach short methods of computation.
3. To review and extend the use of fractions.
4. To review and complete the work in decimals begun
in the fifth year.
5. To develop more fully the use of ratio in arithmetic.
6. To introduce the use of the graph in picturing number
relations.
7. To introduce per cent as a method of expressing ratio
and to show its general uses.
8. To teach common business terms and problems.
9. To review measures, and to extend the measurement
of areas to triangles, parallelograms, and trapezoids.
10. To emphasize the problem-solving side of arithmetic.
Developing Skill in Computation
To develop skill in computation requires constant practice
with a purpose. Chapters III and IX of the text show the
pupil the standards he is to attain. For suggestions as to
drills and tests, see the preceding chapters in this Manual.
The pupil must see the need of drill and take an interest in
drilling in order to attain a fixed goal.
155
156
TEACHERS' MANUAL
Reviewing Fractions
Fractions are reviewed to give the pupil greater skill in
computing with them, to extend their meaning and use, and
to develop greater power in using them in problems. Their
use in expressing a ratio (pages 164-172; 4-12) should
receive careful attention. The tests and drills are important
and should be used frequently throughout the year.
Completing and Applying Decimals
All but two problems in decimals have their parallel in
work in United States money and have been taken up in the
fifth grade. The two new problems are multiplying by a
decimal and dividing by a decimal.
Notice the easy and simple way in which decimals are
treated in the text. The pupil can have no trouble whatever
until the two new problems are met. And these can give but
little trouble as taken up in this text. You will notice that
the new rules are not developed through " an extension of our
decimal notation to the right of one's place," or by changing
the decimals to common fractions — practices still used by
most textbooks and teachers. On page 205 (45) the pupil
takes up three problems and knows the answer is right from
three different types of reasoning. From these he sees induc¬
tively the truth of the rule at the bottom of the page. Do not
make the " explanation " complex by using any of the older
methods. The method here given is simpler and sufficient.
Likewise, the second problem, dividing by a decimal
(pages 207 and 208 ; 47 and 48), is developed in a simple way
that any pupil in the sixth grade can understand.
Develop the habit of proceeding in division according to
the four steps on page 208 (48), and in this order.
The pupil should not feel that he has completed decimals
THE WORK OF THE SIXTH YEAR 157
until he can pass a 100% test on the tests on pages 213-216
(53-56).
Large Numbers and Graphs
Chapter IV is important. In general reading and in his
other studies, the pupil meets large numbers and graphs, and
to interpret rightly what he reads, he must be able to under¬
stand the relationships that they represent.
The pupil must be able to express the approximate ratio of
large numbers in fractions in order to interpret much that he
reads. Likewise, he must be able to interpret and construct
simple graphs.
The Meaning and Use op Per Cent
Long experience with high school graduates and in reading
thousands of examination papers by candidates for teacher's
license to teach, has shown the author that, in general, pupils
leave school with a wrong notion of per cent. The topic is
too often taught by new forms and rules and the pupil sees
no connection with other work. Per cent is only a new nota¬
tion and expression for the ratio of two numbers. The only
new thing is the sign and the name. To get this notion of it
clearly before the pupil, this text spends the first seventeen
pages of Chapter V on the one problem of finding what per
cent one number is of another. This reverses the order of
other texts. That is, the first problem in most of the other
texts is that of finding a per cent of a number. At the end
of these seventeen pages the pupil should have the proper
attitude toward per cent, and see that it is but another way
of expressing the " ratio relation " of one number to another.
Moreover, the average pupil leaves school with the habit
of dividing the smaller of the two numbers by the larger.
Over and over in the text he is shown that " the number
158
TEACHERS' MANUAL
compared is the dividend and the number with which it is
compared is the divisor." Do not take up the second
problem (finding a per cent of a number) until the pupil fully
understands the meaning of per cent as the relation of one
number to another. Constantly emphasize the ratio idea.
Unless the pupil makes 100% in the work on pages 262-264
(102-104), he does not understand the work of the chapter.
The Uses op Per Cent in Business
In teaching Chapter VI, make every possible use of local
situations. Discount sales are common in every town;
most children have sold something on commission or have
known those who have; they know that merchants must
make a profit on what they sell ; they have seen the monthly
accounts or bills that come into the home ; they have perhaps
kept their own accounts; they have heard the subject of
borrowing or loaning money discussed in the home; they
have seen a bank and have some notion of its use.
Practical Measures
Observe that abstract numbers are used in computing
areas and see the reason for this given in the fifth grade.
In developing the rules, cut forms from cardboard rather
than use pictuj-es drawn upon the blackboard.
Bring in local problems as suggested in the text. Seek to
make the subject practical and interesting.
The Final Review and Tests
The teacher should be familiar with the tests given in Chap¬
ter IX and use the standards given as goals toward which to
work. During the entire year the pupils should know of
these standards and work to attain them. It would be
well, then, to use the tests frequently during the year and
THE WORK OF THE SIXTH YEAR 159
have the pupils keep a record of their achievements. If
there is unsatisfactory progress, the teacher should seek the
cause. It may be that some primary ability has not been
sufficiently developed.
If pupils make satisfactory standards in the tests of
Chapter IX, schools will cease to be criticized by the business
world for failing properly to train in arithmetic.
In addition to this, frequently use the tests for the fifth
year given in this Manual. Pupils should make 100% on
those Jour tests.
The Work of the Sixth Year Divided into
Ten Units
First Month
Cover the work on pages 161-176 (1-16) inclusive. The
work of the month stresses the meaning and use of ratio.
Two numbers are always compared either by subtraction or
division. That is, in comparing the cost of a $25 bicycle
with a $30 bicycle we say either, " It is $5 less," or, " It is f
as much." The quotient of any two like numbers is their
ratio. In the review, then, of fractions and decimals see
that the meaning and use of ratio stand out.
1. Solve the problems on pages 166-169 (6-9) by the
" ratio " method as illustrated in the text, instead of by the
" unitary-analysis " method sometimes used in such prob¬
lems. This gives a training needed later, and helps to
develop the power to see and express relationships.
2. Pages 170 (10) and 171(11) give valuable drill in inter¬
preting fractional relations. They are not a type of problem
that arises often in the practical affairs of life, and but few
such problems are given in the text. They are given here to
develop power to see and interpret numerical relationships.
160
TEACHERS' MANUAL
3. Pages 174-176 (14-16) are to introduce the new idea
of per cent as only another way to express a ratio, and to
leave the proper notion that it is but an expression of the
ratio of two numbers.
Second Month
Cover the work on pages 177-195 (17-35) inclusive. This
is a final review of the processes with fractions.
1. Be sure that every point is understood. The three
tests on pages 193 (33), 194 (34), and 195 (35) cover all
fundamental abilities needed. Give these tests early in the
month to find if there are any weaknesses to be overcome.
If so, apply remedial drill found in the work of former
grades.
2. Drill daily for speed and accuracy. Have all problems
carefully solved, and encourage pupils to bring in real prol>
lems met in other activities.
Thikd Month
Cover the work on pages 196-210 (36-50) inclusive. This
is a review of all decimal work of the fifth year and teaches
the two new problems, namely, multiplying and dividing by
a decimal.
1. Encourage pupils to solve all problems on pages 196-
203 (36-43) without help. A pupil may be helped in seeing
how to " attack " a problem, but to solve it for him is of little
if any value.
2. Teach carefully the new principle on page 204 (44).
3. The rule for pointing off a product is developed on
page 205 (45). Notice that by three different types of rea¬
soning the truth of the rule at the bottom of the page is
justified.
THE WORK OF THE SIXTH YEAR 161
The rule is easy to develop, to use, and to remember.
Hence it should be easily fixed by the exercises on page
206 (46).
Make it very clear, however, that any product is wrong
unless the decimal point is in the right place.
4. The rule for dividing by a decimal is developed on
pages 207 (47) and 208 (48). This is based upon two facts
already known : (1) When dividing by a whole number, the
decimal point is directly above the decimal point in the
dividend, when each quotient figure is placed directly above
the right-hand figure of the partial dividend that is being
used ; and (2) multiplying both dividend and divisor by the
same number does not affect the quotient.
5. The note at the bottom of page 208 (48) gives a way in
which the rule may be developed, and should be known by
the pupils in order that they may use it as a check in pointing
off a quotient.
6. Develop the habit in division of taking the four steps
in the order in which they are given in the exercise worked
on page 208 (48) of the text.
7. Drill on exercises like those on page 209 (49) until the
pointing off of products and quotients becomes automatic.
Fottrth Month
Cover the work on pages 211-230 (51-70) inclusive. The
work of the month is devoted mainly to increasing skill in
abilities already developed.
1. See that in all future work pupils use what is taught on
page 211 (51). This saves much time in using such divisors.
2. Pupils should now make 100% on such exercises as
those on pages 212 (52) and 213 (53). These are really
diagnostic tests to show any weakness that may exist.
162
TEACHERS' MANUAL
3. Pupils should now make 100% in the two tests on
pages 213 (53) and 214 (54).
4. You cannot expect 100% of any but superior students
on the " test in using decimals," on pages 215 (55) and 216
(56), for this is a test of power to reason, not merely a test of
facts learned.
5. Pages 217-224 (57-64) are given to test and drill the
child in the use of whole numbers. If the pupils have
attained the required standards, these pages may be omitted.
6. See that each pupil understands and can use the short
methods given on pages 225 (65) and 226 (66). Only the
most important aliquot parts of 100 are given.
7. Stress the saving of time by using common fractions
for tte decimals .5, .25, and .125, page 226 (66).
Fifth Month
Cover the work on pages 231-240 (71-80). This is
mostly new work. This being the last month of the semes¬
ter, some time should be devoted to any work of the preced¬
ing months that needs particular attention. Use former
drills as needed.
1. Observe how work is saved when divisors or multipliers
end in zeros, pages 231 (71) and 232 (72), and insist upon
pupils using this method in all future work.
2. Teach carefully how to express the ratios of large
numbers. Also show how to tell which of a series of approx¬
imate ratios is nearest the true ratio.
3. Be sure that pupils can interpret and construct the
two kinds of graphs given in this chapter. Find local data
to use when possible.
Use both squared paper and paper ruled by the pupils, in
constructing graphs.
THE WORK OF THE SIXTH YEAR 163
Sixth Month
Cover the work on pages 241-252 (81-92) inclusive.
1. This month is devoted to but one thing — getting the
pupil to see that per cent is but a particular name for a ratio
expressed in hundredths.
This differs from the work of other texts. Most texts
first teach how to find a per cent of a number. The change in
presentation is due to the author's experience with high
school graduates who have not learned by the older methods
what per cent means. Before a pupil can use per cent intel¬
ligently, he must see that it is merely the ratio of one quan¬
tity to another and that the name and sign is used for hun¬
dredths.
2. Be sure that pages 241 (81) and 242 (82) are thoroughly
understood before taking up the following work.
3. Use page 243 (83) as oral work until you are sure that
it is thoroughly understood.
4. Pages 244 (84) and 245 (85) are to give a clear eye pic¬
ture of relations expressed by per cent.
5. Teach pages 246-249 (86-89) very thoroughly.
6. " Relations larger than one," pages 249 (89) and 250
(90), are brought in here to prevent the erroneous notion
that the larger of the two numbers is the divisor, and that
the answer is less than 100%.
Impress constantly that the number being compared is
the dividend and the number with which it is compared is
the divisor. Unless the pupils understand this, their knowl¬
edge of the notation of per cent is of no value to them. And
yet many high school graduates do not know this essential
fact of per cent. That is why it is stressed so in this text.
7. Page 251 (91) is very important. See that pupils can
give the common fraction that is approximate in value to
certain per cents.
164
TEACHERS' MANUAL
8. " Increases and decreases expressed in per cent,"
pages 251 (91) and 252 (92), take in one new step — sub¬
traction, before the division.
Use the forms of solution given in the text.
Seventh Month
Cover the work on pages 253-264 (93-104) inclusive.
Pages 253-257 (93-97) still present the same problem that
was given in the sixth month, but in other forms, to fix the
meaning and use of per cent as a ratio.
1. If the work on pages 241-257 (81-97) inclusive is well
done, the pupil has a thorough foundation for all future work
with per cent.
2. At the bottom of page 257 (97) a new problem is pre¬
sented — finding a per cent of a number. Have the pupils
see that this is only multiplication by a decimal after the per
cent is expressed as a decimal. But he must correctly
express the per cent as a decimal. Hence much drill is
needed like the drill on page 259 (99).
3. Show pupils that there are a few per cents (those on
page 260 ; 100) in which the multiplication is shorter when
the per cent is expressed as a common fraction instead of as
a decimal. But never express such a per cent as 80% as f,
for then there must be a multiplication by 4 and a division
by 5. To multiply by .8 instead of i takes but half the time.
4. After the review on page 262 (102) and the drills on
page 263 (103), a pupil should score 100% in the test on
pages 263 (103) and 264 (104). If not, discover the reason
and re-teach any point that is not understood. Unless a
pupil has a thorough knowledge of the work in Chapter V of
the text, he will be greatly hampered in all future uses of per
cent.
THE WORK OF THE SIXTH YEAR 165
Eighth Month
Cover work on pages 265-281 (105-121) inclusive. No
new problem of per cent is given. The text teaches but two
problems : (1) Finding what per cent one number is of an¬
other ; and (2) finding a per cent of a number. The aim of
Chapter VI is to teach the meaning and use of certain busi¬
ness terms and problems.
1. The use of " discount " on pages 265 (105) and 266
(106) is the common use of it as applied to any reduction of
a forrtier price, and has nothing to do with the so-called
" trade or commercial discount." The problems are kept
within the child's interests by dealing with special sales that
the child sees advertised in the daily papers, and through
other uses with which he is familiar. Encourage pupils to
solve the problems without help. Show them how to attack
a problem, but do not solve it for them.
2. Most children know of people who get a commission
for selling. If they do, encourage them to make up real
problems.
3. In finding interest for a fraction of a year, use the
method of the text. This method depends upon the mean¬
ing of interest and hence is easily understood and remem¬
bered. The various special methods given in most texts
are soon forgotten. In attempting to use them, many mis¬
takes are made. Finding interest recurs so infrequently
that special methods are soon forgotten. The method given
in the text will not be forgotten, for it depends upon the
meaning of interest.
Ninth Month
Cover the work on pages 282-308 (122-148) inclusive.
This assignment contains more pages than are usually given
166
TEACHERS' MANUAL
for one month, but much of the work is a review of former
work.
1. Chapter VII has been given before, but is given again
here for a final review of common " tables."
2. The reduction on pages 285 (125) and 286 (126) comes
from the meaning and use of fundamental processes and
should give no trouble if taught as in the text.
3. The four fundamental processes with compound num¬
bers may be omitted and the exercises and problems solved
as fractions, unless you are fitting for some standard test
that uses them. In life, fractions are used in all such prob¬
lems. That is, the forms " as fractions " given in the text
are used.
4. In teaching Chapter VIII, follow the method of the
text. Most of the work is a review of former work.
See that pupils fully understand the meaning and use of
the formulas.
5. In all computation, use abstract numbers as in the text.
That will avoid such erroneous statements as 3 ft. X 4 ft.
= 12 sq. ft. 3 ft. X 6 sq. ft. = 18 cu. ft., as seen in many
schools.
Tenth Month
Cover the jtests and drills on pages 309-322 (149-162)
inclusive.
It is not intended that the month be devoted wholly to
these tests. But they should be given early in the month to
see what, if any, remedial work needs to be done. They are
really achievement tests. If required standards have not
been reached, diagnostic tests should locate the weakness
and remedial drills found throughout the text should be
used.
If the work of the textbook has been carefully done, as
THE WORK OF THE SIXTH YEAR 167
suggested in this outline, the criticism of the work of our
schools will soon cease.
The work can easily be done if done systematically. But
it carmot be done by wasteful methods of teaching and of
drill. Neither can it be done by a purposeless use of the
text used here and there to supplement other work. The
text is a basal text and should be used consecutively. One
cause of poor results is that pupils do not complete the work
of the textbook, but are given a page here and a page there.
Only by completing the work of the text can satisfactory
results be attained.
Standaeds of Skill
I. addition of whole numbeks
1. About 30 or 35 single columns of six digits in 5 minutes.
2. About 18 or 20 single columns of ten digits in 5 min¬
utes.
3. About 18 exercises of four three-figured numbers in
5 minutes.
4. About 8 exercises of eight three-figured numbers in
5 minutes.
II. subtraction of whole numbers
1. About 20 or 25 four-figured numbers in 5 minutes with
two carryings.
2. About 14 or 16 six-figured numbers with three carry¬
ings in 5 minutes.
III. multiplication of whole numbers
1. About 8 or 10 exercises of a three-figured number by a
two-figured number in 5 minutes.
2. About 4 or 5 exercises of a four-figured number by a
three-figured number in 5 minutes.
168
TEACHERS' MANUAL
IV. division of whole numbeks
About 6 or 8 exercises with a two-figured divisor giving a
three-figured quotient and a remainder in 5 minutes.
Control of Processes
The pupil should make 100% in the four tests at the end
of the Manual for the fifth year. In addition to these he
should have complete control of the decimal point in prod¬
ucts and quotients, and be able to change any number to a
per cent, or a per cent to a decimal, fraction, or mixed number
as the case may require. These are all covered by the tests
given in the text.
Tests in Problem-solving
The tests on pages 316-322 (156-162) of the text will test
the power to solve problems. Pupils will vary greatly on
any one of these tests owing to native ability. Superior
students should be able to get all of the four tests in
" quick thinking," pages 316-319 (156-159). Fair students
should get at least 60% of them.
The three tests in " problem-solving," pages 319-322 (159-
162), give problems more like those met frequently, and
some of them will be solved from memory. A " fair "
student should average 60% or 70% on these, and a " supe¬
rior " student should be able to get all of them.
ANSWERS
ANSWERS
THE STONE ARITHMETIC
THIRD YEAR
Page 6.—2. 48(i. 3. 49(i. 4. 78. 5. 67. 6. 67. 7. 79. 8. 79.
9. 89. 10. 97. 11. 99. 12. 97. 13. 98. 14. 99. 15. 99. 16. 99.
17. 99.,
Pages 6-7.— 1. 67. 2. 69. 3. 77. 4. 78. 6. 89. 6. 89. 7. 89.
8. 99. 9. 79. 10. 68. 11. 79. 12. 89. 13. 89. 14. 79. 15. 79.
16. 89. 17. 79. 18. 87.
Pages 11-13. — 3. 32?!. 4. 43 mi. 5. 42 mi. 6. 73. 7. 33. 8. 42.
9. 42. 10. 26. 11. 22. 12. 62.
Pages 13-13. — 2. 44?!. 3. 14. 4. 43?!. 5. 41. 6. 35. 7. 25.
8. 32. 9. 52. 10. 22. 11. 35. 12. 51.
Pages 13-14. — 2. 14?!. 3. 12 lb. 4. 12. 5. 15. 6. 33?!. 7. 11 in.
8. 43. 9. 75. 10. 41. 11. 43. 12. 64. 13. 23. 14. 33. 15. 13.
16. 23. 17. 46?!. 18. 23 yr. 19. 32ji. 20. 13.
Page 15. — 1. 27. 2. 62. 3. 21. 4. 51. 5. 74. 6. 21. 7. 25.
8. 23. 9. 61. 10. 41. 11. 51. 12. 24. 13. 62. 14. 63.
15 . 52. 16. 53. 17. 52. 18. 22. 19. 42. 20. 32. 21. 23.
22 . 42. 23. 21. 24. 27. 25. 21. 26. 61. 27. 81. 28. 46.
29. 13?!. 30. 51?!. 31. 251b.
Page 21. —16. 88. 17. 87. 18. 89. 19. 95. 20. 97. 21. 97.
22. 67. 23. 98. 24. 77ji. 25. 78. 26. 87?!. 27. 75?!. 28. 91 da.
Page 34.— 1. 67. 2. 85. 3. 86. 4. 77. 5. 68. 6. 68. 7. 97.
8. 97. 9. 78. 10. 88. 11. 79. 12. 99. 13. 96. 14. 89. 15. 87.
16. 89. 17. 89. 18. 99. 19. 67. 20. 87. 21. 79. 22. 107.
23. 88. 24. 99. 25. 88. 26. 99. 27. 99. 28. 98.
Page 35.— 1. 75?!. 2. 23?!. 3. 46?!. 4. 21?!. 5. 98?!. 6. 17.
7. 23. 8. 98. 9. 16. 10. 49?!. 11. 86?!.
Pages 38-39. — 2. 43?!. 3. 63?!. 4. 72?!. 6. 70?!. 7. 80?!.
9. $6.83. 10. $5.80. 11. $10.73. 12. $17.60. 13. $5.43.
14. $8.83.
Page 30. — 1. 119. 2. 131. 3. 158. 4. 166. 5. 190. 6. 117.
7. 168. 8. 118. 9. 131. 10. 130. 11. 142. 12. 156. 13. 119.
14. 133. 15. 123. 16. 126. 17. 127. 18. 120. 19. 110.
20. 123. 21. 146. 22. 127. 23. 166. 24. 155. 25. 167.
26. 161. 27. 184. 28. 119.
1
2
ANSWERS
Page 34.— 1. 160. 2. 176. 3. 166. 4. 246. 6. 206. 6. 203.
7. 177. 8. 145. 9. 149. 10. 190. 11. 178. 12. 179. 13. 153.
14. 161. 15. 188. 16. 247. 17. 235. 18. 236. 19. 192.
20. 170. 21. 200. 22. 106. 23. 117. 24. 135. 25. 161.
26. 130. 27. 149.
Page 37. — 4. 40)é. 6. 40.
10. 50. 11. 43.
Page 39. — 3. 35. 4. 34.
10. 32. 11. 27. 12. 69.
6. 54. 7. 70.
6. 26.
8. 45. 9. 59.
Page 40. — 1. 54. 2. 34.
8. 36. 9. 35. 10. 58.
16. 66.
23. 13.
30. 29.
37. 69.
5. 33.
13. 16.
3. 35.
11. 38.
18. 27.
25. 46.
32. 16.
39. 18.
7. 47. 8. 29. 9. 28.
14. 15. 15. 44. 16. 33.
34. 5. 24. 6. 34.
12. 23. 13. 25.
7. 36.
14. 45.
21. 45.
28. 26.
35. 53.
42. 56.
15. 35. 16. 66. 17. 28. 18. 27. 19. 35. 20. 37.
22. 19. 23. 13. 24. 29. 25. 46. 26. 57. 27. 39.
29. 39. 30. 29. 31. 24. 32. 16. 33. 35. 34. 22.
36. 51. 37. 69. 38. 74. 39. 18. 40. 25. 41. 62.
43. 52jf.
Page 41. —1. 83?;. 2. 82?;. 3. 17?;. 4. 27i. 5. 40)i. 6. 17?;.
7. 59?;. 8. 33^. 9. 33?;. 10. 12ji. 11. 16?;. 12. 80?;.
6. 536.
13. 814.
19. 386.
25. 135.
31. 124.
Page 43. — 1. 336. 2. 336. 3. 263. 4. 564. 5. 144.
7. 671. 8. 512. 9. 183. 10. 432. 11. 373. 12. 433.
15. 343. 16. 522. 17. 442. 18. 414.
21. 169. 22. 383. 23. 244. 24. 396.
27. 236. 28. 218. 29. 161. 30. 252.
33. 164. 34. 448. 35. 417. 36. 253.
14. 332.
20. 573.
26. 127.
32. 143.
Page 43. — 1. 193 mi. 2. 29 mi. 3. $1.85. 4. $1.60. 5. 80?;.
6. $1.75. 7. 181 mi. 8. 260. 9. 164.
Page 45. —2. 78?;. 3. 90?;. 4. 72. 5. 68^ 6. 96. 7. 70.
8. 144. 9. 182. 10. 128. 11. 146. 12. 164. 13. 190. 14. 148.
15. 166. 16. 192. 17. 178. 18. 134. 19. 168. 20. $2.70.
21. $3.70. 22. $4.80.
Page 47. — 3. 222.
9. 225. 10. 1404.
15. 1455. 16. 2268.
Page 48. — 2. 72 mi.
7. $1.62. 8. $4.05.
Page 63.—2. $7.40.
8. $8.76. 9. $13.12.
14. 992.
4. 93. 6. 255. 6. 129. 7. 288. 8. 204.
11. 2763. 12. 2178. 13. 2943. 14. 2376.
3. $1.35.
9. $5.55.
4. $2.60.
10. 1116.
4. $1.17.
10. $2.55.
5. $9.84.
11. 1472.
6. $1.05. 6. $19.20.
6. $7.92.
12. 2188.
7. $6.68.
13. 2772.
Page 67. —1. 65?;. 2. $1.60. 3. $2.55. 4. $1.35. 6. $7.00.
6. $3.00.
ANSWERS
3
Page 61. —1. 48. 2. 54. 3. 58. 4. 67. 6. 78. 6. 96.
7. 63. 8. 89. 9. 75. 10. 65. 11. 74. 12. 94. 13. 42. 14. 45.
15. 35. 16. 54. 17. 48. 18. 57. 19. 56. 20. 52. 21. 63.
22. 36. 23. 66. 24. 59. 25. 32. 26. 34. 27. 39. 28. 40.
29. 42. 30. 43. 31. 52. 32. 54. 33. 49. 34. 36. 35. 37.
36. 54.
Page63. — 2. 45. 3. 58. 4. 54. 6. 65. 7. 45fí. 8. 85¿.
9. $1.35. 10. $1.35. 11. 75íí.
Page 63.—3. 53. 4. 87. 5. 55. 6. 64. 7. 73. 8. 72.
9. 74. 10. 69. 11. 76. 12. 78. 13. 58. 14. 66. 15. 75.
16. 76.
Page 64. — 8. 1369. 9. 1499. 10. 1899. 11. 1798. 12. 1800.
13. 2277. 14. 1279. 15. 1156. 16. 1305. 17. 1483. 18. 1689.
19. 1076.
Page 65. — 2. 15. 3. 34. 4. 26. 5. 16. 6. 25. 7. 17. 8. 26.
9. 27. 10. 27. 11. 24. 12. 48. 13. 47. 14. 36. 15. 35.
16. 39. 17. 48. 18. 58. 19. 42. 20. 14. 21. 43. 22. 46.
23. 59. 24. 29.
Page 66-67. — 1. $3.48. 2. $3.28. 3. $7.24. 4. $4.28. 5. $0.87-
6. $3.26. 7. $0.87. 8. $0.83.
Page 67.— 1. 358. 2. 348. 3. 558. 4. 542. 5. 652. 6. 463.
7. 432. 8. 465. 9. 454. 10. 435. 11. 471. 12. 313. 13. 578.
14. 578. 15. 585. 16. 477. 17. 473. 18. 579. 19. 517.
20. 372. 21. 453. 22. 437. 23. 345. 24. 156.
Page 73. — 5. 18. 6. $2.34. 7. 34. 8. 23. 9. 33. 10. 44.
11. 54. 12. 91. 13. 57. 14. 79. 15. 93. 16. 49. 17. 68.
Page 73. — Test B. 1. 30. 2. 26. 3. 27. 4. 28. 5. 30.
6. 25. 7. 27. 8. 27. 9. 28. 10. 25.
Teste. —1. 221. 2. 241. 3. 255. 4. 261. 5. 211. 6. 234-
7. 211. 8. 260.
Page 74. — Test B. 1. 14. 2. 22. 3. 25. 4. 46. 5. 38.
6. 25. 7. 29. 8. 19. 9. 45. 10. 27. 11. 17. 12. 34. 13. 25.
14. 16. 15. 38. 16. 48.
Teste.— 1. 216. 2. 318. 3. 428. 4. 327. 5. 524. 6. 119.
7. 307. 8. 319. 9. 528. 10. 318. 11. 512. 12 . 306. 13. 339.
14. 198. 15. 267. 16. 186. 17. 388. 18. 278.
Page 75. — Test B. 1. 153. 2. 129. 3. 128. 4. 105. 5. 148.
6. 78. 7. 90. 8. 111. 9. 104. 10. 117. 11. 112. 12. 74.
13. 147. 14. 76. 15. 81. 16. 152.
Test e. — 1. 384. 2. 2250. 3. 2476. 4. 1540. 5. 2448. 6. 2600.
7. 1290. 8. 1120. 9. 2880. 10. 760. 11. 870. 12. 940.
TestD. — 1. 1700. 2. 1914. 3. 2781. 4. 2992. 5. 1584.
6. 2502.
4
ANSWERS
Page 76. — Test B. 1. 24. 2. 28. 3. 36. 4. 47. 6. 43.
6. 32. 7. 23. 8. 24. 9. 28. 10. 26. 11. 18. 12. 16. 13. 14.
14. 18. 16. 24. 16. 16. 17. 23. 18. 19.
Teste.— 1. 192. 2. 155. 3. 241. 4. 178. 5. 284. 6. 146.
7. 183. 8. 143. 9. 234. 10. 244.
TestD. — 1. 89. 2. 145. 3. 188. 4. 158. 5. 241. G. 89.
7. 133. 8. 218. 9. 164. 10. 181.
Page 77. — 1. $3. 3. IH. 4. $1.20. 5. 30»!.
Page 78. — 1. $6.75. 2. $3.20. 3. $5.50. 4. $2.17. 6. $1.82.
6. $1.90. 7. $13.95. 8. $2.15. 9. $4.13. 10. $1.55. 11. $0.85.
Page 79. — 1. $10.58. 2. $9.24. 3. $10.35. 4. $9.29. 5. $8.91.
6. $9.52. 7. $9.24. 8. $9.38. 9. $9.00. 10. $9.48. 11. 35»!.
12. 36»!; 72i; 54»!; $1.62. 13. 45»!; 30»!; 60»!; $1.35.
Page 80.— 1. 1033. 2. 1384. 3. 993. 4. 969. 5. 1305.
6. 1110. 7. 1022. 8. 1130. 9. 1419. 10. 1167. 11. 1245.
12. 1398. 13. 606. 14. 881. 15. 1319. 16. 580. 17. 541.
18. 1005. 19. 1052. 20. 958. 21. 820. 22. 888. 23. 902.
24. 1180.
Page 81. — 1. 359. 2. 238. 3. 259. 4. 307. 5. 428. 6. 617.
7. 493. 8. 284. 9. 385. 10. 573. 11. 394. 12. 226. 13. 465.
14. 249. 15. 595. 16. 373. 17. 318. 18. 175. 19. 646.
20. 581. 21. 147. 22. 455. 23. 378. 24. 156.
Page 83. — 1. $1.52. 2. $1.80. 3. $1.08. 4. $3.40. 5. $3.00.
6. $2.28. 7. $1.48. 8. $7.40. 9. $5.34. 10. $2.60.
Page 87. — 5. 1040. 6. 1227. 7. 1521. 8. 915. 9. 2706.
10. 1803. 11. 2024. 12. 2832. 13. 2416. 14. 3612. 15. 3220.
16. 2280. 17. 2670. 18. 1620. 19. 2680. 20. 3920. 21. 1710.
22. 2760. 23. 1800.
Page 87. — 1. 714 mi. 2. 2240. 3. $11.40. 4. $8.15. 5. $1.50.
6. $2.65.
Pages 88-89. — 1. $1.15. 2. $3.22. 3. $1.75. 4. $4.14.
5. $1.93. 6. $3.80. 7. $1.80. 8. $2.84. 9. $4.22. 10. $0.58.
11. $1.90. 12. $0.85. 13. $11.55.
Page 90. — 1. 1417. 2. 778. 3. 923. 4. 1189. 5. 1452.
6. 1285. 7. 257. 8. 363. 9. 443. 10. 544. 11. 217. 12. 643.
13. $7.35. 14. $9.24. 15. $14.70. 16. $7.12. 17. $8.36.
18. $7.20. 19. 474. 20. 687. 21. 647. 22. 433. 23. 591.
24. 879. 25. 356. 26. 653. 27. 492. 28. 924. 29. 768.
30. 876.
Pages 91-92. — 4. 7°. 5. 12°. 8. 18°. 9. 40°. 10. 48°.
11. 103°. 12. 24°. 13. 13°. 14. 180°. 16. 6°. 16. 4°. 18. 18°.
ANSWERS
5
Page 96.— 1. 230. 2. 192. 3. 182. 4. 184. 6. 162. 6. 136.
7. 180. 8. 128. 9. 160. 10. 143. 11. 162. 12. 198. 13. 156.
14. 151. 15. 169. 16. 165.
1. 164. 2. 373. 3. 353. 4. 454. 5. 152. 6. 458. 7. 382.
8. 292. 9. 638. 10. 447. 11. 647. 12. 684. 13. 187. 14. 486.
Page 97. — 2. 2162. 3. 1353. 4. 4449. 5. 4764. 6. 2436.
7. 5535. 8. 3543. 9. 7764. 10. 3528. 11. 2606. 12. 1372.
13. 2520. 14. $6.65. 15. 182 mi. 16. 236 mi.
Page 99.— 1. 187. 2. 266. 3. 378. 4. 199. 5. 455. 6. 398.
7. 188. 8. 156. 9. 239. 10. 284. 11. 89. 12. 144. 13. 183.
14. 131. 15. 229. 16. 155. 17. 239. 18. 78.
Page 100. — 2. 144. 3. 212. 4. 195. 5. 312. 6. 192. 7. 332.
8. 201. 9. 348. 10. 276. 11. 236. 12. 268. 13. 87. 14. 152.
15. 184. 16. 228. 17. 116. 18. 344. 19. 380. 20. 714.
21. 1047. 22. 501. 23. 1638. 24. 807. 25. 1134. 26. $2.24.
27. $2.55. 28. $3.80.
Page 101. — 3. 72 ia. 4. 79 in. 5. 33 in. 6. 18 in. ; 12 in. ; 9 in.
Page 104. — 7. 7?!. 8. 20^. 9. $3.20. 10. 90f^. 11. 80^;.
12. 45{i. 13. 12fi. 14. 70^.
PagelOO. — 8. 84(é. 9. 54^. 10. $1.15. 11. 95^. 12. 63^.
13. 98)í. 14. $1.75. 15. $1.20.
Page 109. — 1. 1020. 2. 1506. 3. 2010. 4. 2724. 5. 2340.
6. 927. 7. 3612. 8. 3480. 9. 3560. 10. 2824. 11. 2036.
12. 1880.
Page 110. — 1. 74. 2. 36. 3. $33.30. 4. $3.70. 5. 45 qt.
Page 113. — 5. 1625. 6. 2315. 7. 3640. 8. 4680. 9. 4905.
10. 3540. 11. 3450.
PagellS. — 4. 49. 5. 40. 6. 68. 7. 328. 8. 469. 9. 278.
10. 537. 11. 842. 12. 735.
Pages 114^115. — 1. $1.40. 2. 47. 3. 39. 4. 57^. 5. $1.70.
6. 35 da. 7. 39. 8. 26. 9. $1.90. 10. 37. 11. $13.75. 12. 18.
Page 117. — 1. $12.90. 2. $4.68. 3. $3.24. 4. $12.90. 5. $4.30.
6. $4.74. 7. $4.55.
Page 118. — 1. $1.25. 2. $1.68. 3. $2.10. 4. $1.26. 5. $1.08.
6. $1.41. 7. $3.00. 8. $2.40.
PagellO. — 4. 150. 5. 204. 6. 522. 7. 576. 8. 234. 9. 288.
10. 3024. 11. 4248. 12. 1854. 13. 3642. 14. 5412. 15. 4854.
Page 131. — 1. 228. 2. 392. 3. 459. 4. 297. 5. 594. 6. 771.
7. 889. 8. 792. 9. 576. 10. 423. 11. 529. 12. 453. 13. 582.
14. 396. 15. 727. 16. 936.
6
ANSWERS
Page 133. — 1. 2U. 3. 81fí. 4. 78^. 6. $2.73. 7. $1.68.
8. $1.62. 9. $1.12. 10. $1.30. 11. $1.61. 12. $1.35. 13. $1.71.
14. $0.80.
Page 133. — 1. 1770. 2. 2256. 3. 2886. 4. 1404. 5. 5868.
7. 1758. 8. 2910. 9. 3822. 10. 2574. 11. 3468. 12. 38.
13. 29. 14. 46. 15. 67. 16. 59. 17. 423. 18. 576. 19. 629.
20. 538. 21. 741. 22. 1296. 23. 2082. 24. 2280.
Page 135. — Race I. 1. 235. 2. 202. 3. 261. 4. 213. 5. 213.
6. 194. 7. 249. 8. 221.
Racen.— 1. 1675. 2. 1385. 3. 1699. 4. 1214. 5. 1706.
6. 1873. 7. 1222. 8. 1245. 9. 1386. 10. 1925. 11. 1294.
12. 1306.
Page 136. — Race I. 1. 13. 2. 58. 3. 28. 4. 29. 5. 35.
6. 46. 7. 46. 8. 26. 9. 44. 10. 46. 11. 37. 12. 17. 13. 37.
14. 38. 15. 47. 16. 49.
Racen.— 1. 393. 2. 314. 3. 285. 4. 336. 5. 292. 6. 208.
7. 504. 8. 285. 9. 493. 10. 491. 11. 153. 12. 234.
Race m.— 1. 448. 2. 223. 3. 245. 4. 566. 5. 444. 6. 415.
7. 445. 8. 442. 9. 165. 10. 299. 11. 604. 12. 249.
Page 137. — Race I. 1. 256. 2. 84. 3. 195. 4. 94. 5. 260.
6. 504. 7. 117. 8. 104. 9. 285. 10. 272. 11. 156. 12. 235.
13. 384. 14. 430. 15. 465. 16. 504.
Racen.— 1. 2130. 2. 2156. 3. 852. 4. 1745. 5. 1116.
6. 1576. 7. 3222. 8. 2512. 9. 3645. 10. 4896. 11. 4620.
12. 1580.
Race m.— 1. 1188. 2. 1840. 3. 2811. 4. 4956. 5. 1544.
6. 3595. 7. 2388. 8. 1470. 9. 5622. 10. 2538. 11. 2164.
12. 3294.
Page 138. — Race I. 1. 39. 2. 143. 3. 64. 4. 89. 5. 143.
6. 79. 7. 46. 8. 184. 9. 75. 10. 97. 11. 162. 12. 93.
Racen.— 1. 406. 2. 308. 3. 203. 4. 407. 5. 605. 6. 309.
7. 77, 3 r. 8. 76, 4 r. 9. 143, 1 r. 10. 192, 4 r. 11. 127, 3 r.
12. 182, 3 r.
Pages 138-139. — 1. 3 ; 15(i. 2. 2nd ; 2. 3. John ; 50^ 4. 26.
5. $6.55. 6. $2.35. 7. $2.25. 8. $1.40. 9. $2.88. 10. $1.10.
11. $2.85. 12. $1.30. 13. 24 mi. 14. $1.15.
Pages 133-133. — 1. $5.25. 2. $3.36. 3. $2.52. 4. $6.65.
5. $5.60. 6. 7doz.; ISfi. 7. $5.95. 8. $4.65. 9. 75)é.
Pages 133-134. — 1. $2.17. 2. $1.05. 3. 17»!. 4. 65»!. 6. $5.88.
6. $4.06. 7. $4.83. 8. $1.95.
ANSWERS
7
Page 134. — 1. 2376. 2. 2368. 3. 2046. 4. 2115. 5. 1801.
6. 1906. 7. 754. 8. 579. 9. 136. 10. 336. 11. 408. 12. 552.
13. 322. 14. 264. 15. 461. 16. 504. 17. 207. 18. 414.
Page 136. — 1. 297. 2. 386. 3. 456. 4. 814. 5. 495. 6. 746.
7. 639. 8. 847. 9. 587. 10. 692. 11. 478. 12. 654.
Page 137. — 1. 38. 2. 56. 3. 72. 4. 74. 5. 94, 4 r. 6. 85.
7. 246. 8. 357. 9. 826. 10. 937. 11. 854. 12. 763. 13. 23 mi.
14. 316. 15. 45 qt. 16. 32. 17. 30 da. 18. 31.95. 19. 370.00.
20. 152 mi.
Page 138. — 1. 318.80. 2. 314.80. 3. 56;!. 4. 31.46. 5. 31.20.
6. 31.75. 7. 36.80. 8. 31.52. 9. 33.68. 10. 36.80.
Page.140. — 1. 92. 2. 57. 3. 28. 4. 65. 5. 39. 6. 48.
Page 141. — 1. 648. 2. 747. 3. 855. 4. 576. 5. 783. 6. 531.
7. 324. 8. 432. 9. 65. 10. 48. 11. 56. 12. 73. 13. 92.
14. 87. 15. 315. 16. 36.25.
Page 141 {Problems). — 1. 88. 2. 28 yr. 3. 107 lb. 4. 36.85.
5. 24 mi.
Pages 142-143. — 1. 320.25. 2. 339.15. 3. 332.85. 4. 310.75.
5. 37.65. 6. 30.65. 7. 30.34. 8. 38.45. 9. 34.30. 10. 35;!.
11. 39.15.
Pages 143-144. — 1. 39.80. 2. 31.96. 3. 317.55. 4. 32.35.
5. 31.93. 6. 315.80. 7. 3104. 8. 35.85. 9. 34.90. 10. 30.90.
Page 145. — 3. 35.00.
Page 146. — 1. 3170.
6. 30.55. 7. 30.35.
Page 147. — 1. 72 mi.
6. 31.28. 7. 32.80.
Page 148. — 1. 151.
6. 35.45. 7. 34.75.
4. 32.48. 5. 37.34.
2. 35.80. 3. 35.35.
2. 24 mi. 3. 84 mi.
8. 31.60. 9. 34.75.
2. 32. 3. 329.20.
8. 151. 9. 208 1b.
4. 33.75. 6. 35.65.
4. 21 mi. 6. 30.92.
4. 35.10. 5. 90.
10. 382.50.
Pagel49. — A. 1. 28. 2. 30. 3. 25. 4. 31. 5. 25. 6. 29.
7. 28. 8. 29. B. 1. 201. 2. 248. 3. 286. 4. 246. 6. 269.
6. 277. 7. 294. 8. 287. C. 1. 1493. 2. 2355. 3. 1950.
4. 1801. 5. 2843. 6. 2911. 7. 2812.
PagelSO. — A. 1. 69. 2. 44. 3. 38. 4. 48. 5. 16. 6. 58.
7. 26. 8. 17. 9. 29. 10. 26. 11. 22. 12. 38. 13. 18. 14. 36.
16. 45. 16. 34. B. 1. 218. 2. 373. 3. 447. 4. 565. 5. 365.
6. 508. 7. 448. 8. 124. 9. 565. 10. 490. 11. 470. 12. 206.
13. 308. 14. 169. C. 1. 7925. 2. 5134. 3. 2106. 4. 1818.
5. 6805. 6. 5328. 7. 2506. 8. 4408. 9. 4735. 10. 4406.
11. 6092. 12. 7219.
8
ANSWERS
PagelSl. — A. 1. 196. 2. 282. 3. 380. 4. 399. 6. 570.
6. 688. 7. 288. 8. 576. 9. 684. 10. 420. 11. 342. 12. 348.
13. 644. 14. 342. 16. 588. 16. 665. B. 1. 1456. 2. 2340.
3. 3520. 4. 5040. 6. 7256. 6. 7740. 7. 3780. 8. 3045.
9. 4410. 10. 4824. 11. 3330. 12. 3560. 13. 4248. 14. 8550.
C. 1. 2086. 2. 3402. 3. 7304. 4. 2916. 6. 2985. 6. 3483.
7. 3408. 8. 6048. 9. 5622. 10. 7000. 11. 4930. 12. 3144.
Page 153.— A. 1. 75, 4 r. 2. 142, 4 r. 3. 133, 5 r. 4. 96.
6. 152, 3 r. 6. 121, 7 r. 7. 99, 5 r. 8. 137, 5 r. 9. 151, 1 r.
10. 193, 2 r. 11. 128, 1 r. 12. 166, 2 r. B. 1. 1538, 3 r.
2. 995, 5 r. 3. 1389, 5 r. 4. 1628. 6. 973, 8 r. 6. 1323. 7. 1196.
8. 1496. 9. 1530, 1 r. 10. 1283, 3 r. C. 1. 907, 1 r. 2. 680, 3 r.
3. 805, 3 r. 4. 690, 4 r. 5. 809, 2 r. - - - -
8. 590, 5 r. 9. 906, 3 r. 10. 807, 4 r.
3. 894. 4. 799. 5. 839. 6. 982.
10. 398.
6. 780, 3 r. 7. 609; 2 r.
D. 1. 698. 2. 967.
7. 769. 8. 926. 9. 903.
ANSWERS
THE STONE ARITHMETIC
FOURTH YEAR
(The page references in parentheses apply to the six-book edition)
Pages 153-154 (1-2). —1. Yes. 2. 132 mi. 3. 7:30. 4. 5 hr.
6. 23 mi. 6. 1 hr. 15 min.
Page 154 (2). —1. S1.60. 2. S5.60. 3. $5.00. 4. $5.20. 5. $3.15.
6. $9.4?; $9.48.
Page 155 (3). — 1. 828 mi.; 138 mi. 2. $1.84; $11.04. 3. 45(!.
4. $27.75. 5. $33.30. 6. 96 mi. 7. $79.68. 8. $9.96.
Pages 157-158 (6-6).—2. $13.89. 3. $14.29. 4. $12.50. 5. $11.81.
6. $18.29. 7. $14.42. 8. $11.50. 9. $12.10. 10. $10.80.
11. $14.59. 12. $16.79. 13. $13.68. 14. $13.30. 15. $11.90.
16. $12.10. 18. $7.48. 19. $6.95. 20. $7.54. 21. $11.65.
22. $3.22. 23. $10.75. 24. $0.80. 26. $9.03. 26. $9.73.
27. $43.02.
Page 159 (7). —1. $30.80. 2. $43.68. 3. $75.36. 4. $41.85.
6. $51.03. 6. $32.76. 7. $42.63. 8. $26.28. 9. $49.98.
10. $23.68. 11. $13.72. 12. $1.28. 13. $1.75. 14. $1.59.
16. $1.58. 16. $2.35. 17. $4.72. 18. $2.95. 19. $4.79.
20. $6.77. 21. $3.64. 22. $5.24.
Page 163 (10). —4. 999. 6. 1093. 6. 1041. 7. 937. 8. 775.
9. 595.
Page 164 (12). —1. 3585. 2. 2883. 3. 3549. 4. 3657. 6. 3522.
6. 3580. 7. 2262. 8. 3639. 9. 2793. 10. 2456. 11. 2860.
12. 2190. 13. 2667. 14. 3567. 16. 2235. 16. 3405. 17. 3702.
18. 4215.
Page 165 (13).—2. 2437. 3. 3513. 4. 3549. 6. 1938. 6. 6814.
7. 3482. 8. 3412. 9. 4172. 10. 2175. 11. 2214. 12. 4773.
13. 2335. 14. 4171. 16. 1353. 16. 3844. 17. 5273. 18. 2211.
19. 3035.
Page 166 (14). —1. 3074. 2. 7779. 3. 3269. 4. 5419. 6. 6293.
6. 1835. 7. 5731. 8. 2871. 9. 3328. 10. 3571. 11. 1852.
12. 1304. 13. 1968. 14. 2154. 16. 1604. 16. 3609. 17. 3583.
18. 3711. 19. 5149. 20. 3184. 21. 3462. 22 . 5172. 23. 2681.
24. 2556. 26. 3328. 26. 4682. 27. 2515. 28. 1636. 29. 2217.
30. 2154.
10
ANSWERS
Page 168 (16). —Z)., 196,837. 2. J5., 14,234. 3.236,178. 4.351,617.
6. 492,808. 6. 216,211. 7. 205,184. 8. 524,206. 9. 517,909.
10. 247,151. 11. 314,268. 12. 683,162. 13. 151,109. 14. 547,702.
15. 318,122. 16. 317,237. 17. 464,155. 18. 225,318. 19. 262,561.
Page 170 (18).—2. 2046. 3. 1536. 4. 4950. 6. 2958. 6. 2322.
7. 4476. 8. 1351. 9. 1988. 10. 5355. 11. 1953. 12. 2506.
13. 1148. 14. 2224. 15. 2952. 16. 4328. 17. 5824. 18. 7560.
19. 4904. 20. 3834. 21. 7317. 22. 5373. 23. 4923. 24. 5742.
25. 8208.
Pages 172-173 (20-21). —1. $1.41. 2. 33. 3. 85^. 4. 40. 6. 95^.
6. 55fi. 7. $3.25. 8. $2.15. 9. $0.90. 10. $2.20. 11. $0.80.
Page 174 (22). —1. 1798. 2. 3657. 3. 8469. 4. 3968. 5. 4786.
6. 2544. 7. 1938. 8. 2877. 9. 2657, 1 r. 10. 2741. 11. 1833.
12. 1493. 13. 2389. 14. 1981. 15. 2279.
Page 175 (23). —1. $7.96. 2. $2.40. 3. $6.60. 4. 12. 5. 36 ; 27.
6. 18. 7. $1.37. 8. $17.00. 9. $1.84.
Page 177 (25). —1. 1527. 2. 1944. 3. 1597. 4. 1872. 6. 1747.
6. 1439. 7. 1327. 8. 1454. 9. 1591. 10. 882. 11. 528.
12. 827. 13. 948. 14. 756. 15. 894. 16. 1953. 17. 1419.
18. 789. 19. 956. 20. 496. 21. 789. 22. 645. 23. 786.
24. 924. 25. 729.
Page 178 (26). —1. 23. 2. 69ff. 3. $2.38. 4. $2.28. 5. 22 mi.
6. $1.58. 7. $19.50. 8. $2.45. 9. 570 mi.
Page 180 (28). —1. 219. 2 . 238. 3. 276. 4. 349. 5. 384. 6. 427.
7. 631. 8. 928. 9. 328. 10. 238. 11. 349. 12. 276. 13. 384.
14. 328. 15. 427. 16. 219. 17. 439, 1 r. 18. 631. 19. 427.
20. 851. 21. 369. 22. 478. 23. 613, 5 r. 24. 739. 25. 769.
Page 181 (29).— 1. $1.75. 2. $0.65. 3. $15.90. 4. $2.16. 5. 25.
6. $14.80. 7. $3.15. 8. 60fi less.
Pages 184^185 (32-33). —1. $2.25. 2. $1.44. 3. $1.36. 4. $5.05.
5. $2.15. 6. ^1.20. 7. 40.
Page 185 (33). —1. $1.20. 2. $4.86; $1.14. 3. Yes. 4. $3.12.
Page 186 (34). — 1. 34fi. 2. $1.28. 3. 96 qt. 4. 12 qt. 5. 84 qt.
6. 14 qt.
Pages 187-188 (35-36).-2. $1.60. 3. $1.50. 4. $3.20; $16.00.
5. 80>i. 6. 5ji; 4^. 7. 48 in. 8. 53 in.
Page 189 (37).—3. $780. 4. 3564. 5. 5115. 6. 3124. 7. 4169.
8. 3201. 9. 3223. 10. 2556. 11. 3888. 12. 7052. 13. 5700.
14. 3420. 15. 2304.
Page 194 (42). —1. 27^. 2. 50fi. 3. 20^.
ANSWERS
11
Page 195 (43). —3. $3.85. 4. $7.70.
Page 197 (45). —2. 182f 3. 256^. 4. 191f. 5. 254|. 6. 274|.
7. 19H. 8. 141i. 9. 184^. 10. 233i. 11. 164^. 12. 127l.
13. 147^. 14. 198|. 15. 165|. 16. 122^. 17. 143|. 18. 1544.
19. 162^. 20. 137^. 21. 127f. 22. 1214. 23. 95|. 24. 574.
25. 97i. T F Ï Ï
Pages 198-199 (46-47). —2. $1.35. 5. 8; $2.80. 6. 16 qt. 7. 2pk.;
16 qt. 8. 9. H-
Page 200 (48).—3. 24 oz.; 20 oz.; 22 oz. 4. 60«!. 5. $1.25. 6. 63«!.
7. 60?!. 8. 90?!.
Page 207 (55). —4. 480. 5. 1050. 6. 1840. 7. 1620. 8. 3250.
9. 1200.
Page 208 (56). —1. $28.00. 2. $22.50. 4. $3.00. 5. $49.40.
6. $14.00. 7. $3.60.
Page 209 (57).—4. 720. 5. 1440. 6. 1140. 7. 2280. 8. 2360.
9. 2790. 10. 2720. 11. 3350. 12. 1880. 13. 4200. 14. 4380.
15. 4800. 16. 800. 17. 1500. 18. 1200. 19. 4800. 20. 2100.
21. 5400.
Pages 211-212 (59-60). —2. 19,260. 3. 14,946. 4. 15,064. 5. 14,945.
6. 34,808. 7. 14,472. 8. 19,908. 9. 8046. 10. 30,828.
11. 20,878. 12. 12,672. 13. 27,262. 14. 17,391. 15. 12,190.
16. 63,684. 17. 60,528. 18. 29,792. 19. 61,455. 20. 11,248.
22. 22,792. 23. 23,923. 24. 19,404. 25. 50,132. 26. 67,776.
27. 44,283.
Page 313 (60). — 1. 4922. 2. 10,400. 3. 17,346. 4. 11,730.
5. 17,888. 6. 11,792. 7. 14,580. 8. 16,384. 9. 24,912.
10. 14,588. 11. 18,213. 12. 30,816. 13. 15,640. 14. 21,090.
15. 26,880. 16. 30,870. 17. 28,880. 18. 35,100. 19. 27,324.
20. 33,696. 21. 40,803. 22. 19,812. 23. 16,281. 24. 15,675.
Page 313 (61). —1. $6.72. 2. $38.70. 3. $13.70. 4. 3380. 5. $13.60.
6. $1820. 7. $8.10. 8. $174.60. 9. $148.75.
Page 314 (62). —4. 24?!. 5. 10?!. 6. 24?!. 7. 10?!. 8. 40?!.
9. 8 oz. 10. 4 oz. 11. 4 oz. ; 8 oz. ; 12 oz.
Page 315 (63).—7. 1320 ft. 8. 220 yd. 9. 42,240 ft. 10. 660 ft.
11. 2640.
Page 316 (64). —2. 60?!; 30?!. 3. 56?!; 7?!. 4. 80?!. 5. 3 gal. 6. 32.
7. 40 pt. 8. $4.00.
Page 317 (65). —1. 8qt.; 32 qt. 2. 8 pk. 3. 32 da. 4. $3.20.
5. $2.88. 6. $1.28. 7. 20?!. 8. 70?!. 9. $2.30. 10. 8 da.
Pt^e 318 (66). —1. 8oz. 2. 8 oz. 3. 80^. 4. 40^. 5. 15?!.
7. 6000 lb. 8. 1000 lb. ; 500 lb.
12
ANSWERS
Page 233 (71). —1. 1796. 2. 1995. 3. 1678. 4. 1895. 6. 1588.
6. 1691. 7. 2001. 8. 1946. 9. 1959. 10. 1904. 11. 2056.
12. 2160.
Pages 223-224 (71-72). — 1. 2619. 2. 2142.
5. 5154. 6. 5373. 7. 1724. S. 4151.
11. 3731. 12. 4221. 13. 3243. 14. 1413.
17. 5391. 18. 5754. 19. 1912. 20. 1523.
23. 2451. 24. 4131. 25. 3220. 26. 4510.
29. 3512. 30. 7011.
Page 224 (72). —1. 29,616. 2. 29,235. 3. 43,813. 4. 50,464.
6. 23,696. 6. 33,888. 7. 36,960. 8. 69,176. 9. 66,724.
10. 78,624. 11. 61,112. 12. 19,929. 13. 33,834. 14. 43,630.
15. 38,172.
1. 3552. 2. 3948. 3. 4725. 4. 3060. 5. 4018. 6. 2175.
7. 2535. 8. 4464. 9. 5460. 10. 2553.
Pages 225-226 (73-74). —II. 1. 346. 2. 481. 3. 589. 4. 674.
5. 837. 6. 387. 7. 418. 8. 436. 9. 451. 10. 859. 11. 364.
12. 481. 13. 598. 14. 647. 15. 738. III. 1. 485, 5 r.
2. 633, 2 r. 3. 675. 4. 633, 3 r. 5. 716, 4 r. 6. 637, 4 r.
7. 514, 2 r. 8. 787, 4 r. 9. 844, 4 r. 10. 987, 4 r. 11. 1383, 2 r.
12. 1228, 4 r. 13. 1212, 4 r. 14. 1633, 2 r. 15. 1342, 6 r.
IV. 1. 780. 2. 930. 3. 860. 4. 740. 5. 680. 6. 960.
7. 870. 8. 590. 9. 480. 10. 670. 11. 860. 12. 690.
13. 850. 14. 730. 15. 940.
Pages 226-227 (74-75).—I. 1. 1962. 2. 1746. 3. 3483. 4. 478.
5. 1662. 6. 1277. 7. 20,720. 8. 548. 9. 37,360. 10. 367.
11. 1. 1795. 2. 2725. 3. 23,680. 4. 397. 5. 3423. 6. 1413.
7. 5173. 8. 847. 9. 37,680. 10. 489. III. 1. 2090.
2. 3632. 3. 33,120. 4. 528. 5. 1841. 6. 5006. 7. 3003.
8. 608. 9. 32,280. 10. 736. IV. 1. 2005. 2. 2063.
3. 73,040. 4. 628. 5. 1885.
Page 228 (76). —1. 38>!. 2. $4.32. 3. $1.20. 4. 26 mi. 5. 93.
6. 203 mi. 7. $1.92. 8. $1.90. 9. 35. 10. $22.20.
Page 231 (79). —1. 1730. 2. 2705. 3. 2788. 4. 2683.
6. 3061.
Page 232 (80).— 1. 1612. 2. 2935. 3. 3374. 4. 2300.
6. 2382. 7. 2704. 8. 2201. 9. 2087. 10. 2474.
12. 2339. 13. 2100. 14. 2696. 15. 2207. 16. 2566.
18. 2700. 19. 2592. 20. 3187. 21. 3781.
Page 236 (84). —1. 7485. 2. 5232. 3. 4241. 4. 1415.
6. 3741. 7. 2374. 8. 2493. 9. 3518. 10. 2291.
12. 2757. 13. 1629. 14. 1417. 15. 3388. 16. 2963.
18. 3933. 19. 5150. 20. 3181. 21. 1349. 22. 2415.
3. 3613. 4. 4423.
9. 1414. 10. 1323.
15. 3074. 16. 1721.
21. 2267. 22. 2632.
27. 2413. 28. 4611.
5. 2492.
5. 2405.
11. 2404.
17. 2592.
5. 2812.
11. 1607.
17. 1309.
23. 1702.
ANSWERS
13
24. 3014. 25. 1577. 26. 4683. 27. 3606. 28. 2664. 29. 2189.
30. 3782. 31. 3243. 32. 5972. 33. 3804. 34. 1204. 35. 2543.
36. 5478.
Page 337 (85). —1. $2.77. 2. $0.85. 3. $1.92. 4. $1.54. 5. $1.03.
6. $1.21. 7. $1.52. 8. $0.52. 9. $3.16. 10. $8.14. 11. $0.84.
12. 45.
Page 338 (86). —1. 160. 3. 5ff. 4. 5j!. 5. $1.00. 6. $1.35.
Page 339 (87). —1. 3510. 2. 3552. 3. 3910. 4. 5358. 5. 4592.
6. 6734. 7. 5859. 8. 4756. 9. 2184. 10. 6264. 11. 7980.
12. 4256. 13. 4656. 14. 6278. 15. 7268. 16. 3216. 17. 3528.
18. 4988. 19. 2881. 20. 4218. 21. 5780.
Pages 339-340 (87-83). —1. $1.29. 2. $7.63. 3. $5.70. 4. 296.
5. 80. 6. 22. 7. 62fi. 8. $11.70. 9. $1.65. 10. $2.05.
11. $4.00. 12. 15)é.
Pages 341-343 (89-90). —1. $12.48. 2. $1.50. 3. $3.12. 4. $39.50.
5. $7.90. 6. $3.60.
Pages 343-343 (90-91).—2. 185,976. 3. 168,746. 4. 169,092.
5. 323,466. 6. 119,966. 7. 241,082. 8. 653,050. 9. 429,768.
10. 308,586. 11. 411,849. 12. 374,673. 13. 779,876. 14. 618,240.
15. 354,050. 16. 657,400. 17. 375,700. 18. 381,930. 19. 374,850.
20. 773,260. 21. 777,140. 22. 561,370. 23. 454,020. 24. 422,820.
26. 578,210. 26. 423,200. 27. 490,200. 28. 636,400. 29. 641,700.
30. 611,800. 31. 467,400.
Page 344 (92).— 6. 95. 7. 189. 8. 173. 9. 234. 10. 245.
11. 347. 12. 257. 13. 383. 14. 352. 15. 399. 16. 467.
17. 275. 18. 392. 19. 432. 20. 475.
Page 345 (93). —2. 74, 2 r. 3. 98, 1 r. 4. 93, 6 r. 5. 69, 7 r.
6. 172, 4 r. 7. 98, 2 r. 8. 134, 5 r. 9. 116, 2 r. 10. 224, 4 r.
11. 196, Ir. 12. 449, 3 r. 13. 237, 2 r. 14. 219, 3 r. 15. 345, 5 r.
16. 380, 6 r. 17. $2.30. 18. 38. 9. 42.
Pages 346-346 (93-94). —1. $1.30. 2. 50?;. 3. 16|i; $1.28. 4. 47?;;
$3.29. 5. $1.44. 6. $2.16. 7. 35. 8. $2.25.
Pages 346-347 (94-95). —1. 27?;. 2. $1.26. 3. 8 yd. 4. 6 mi.
5. 96 mi. 6. $38.25. 7. 6. 8. S. 9. 12. 10. 10 yr. 11. Mrs.
B's;3mo. 12. 25. 31. 8 yr. 14. 18. 16. 40?;. 16. 15.
17. 15?;.
Page 353 (100).—3. 47?;. 4. 18?;. 5. 18?;; 17?;; $3.40. 6. 18?;.
7. 19é. 8. 32?;. 9. 32?;. 10. 33?;. 11. 27?;. 12. 28?;. 13. 49^.
14. 21?;. 15. 43|í. 16. 54?;. 17. 73?;. 18. 32?;. 19. 84?;.
14
ANSWERS
Page 254 (102).—2. 243, 12 r. 3. 268, 33 r. 4. 281, 23 r. 5. 256, 7 r.
6. 344, 26 r. 7. 33, 28 r. 8. 254, 10 r. 9. 238, 10 r. 10. 22, 50 r.
11. 155, 50 r. 12. 293, 50 r. 13. 55, 11 r. 14. 44, 32r. 15. 35, 15 r.
16. 39, 57 r. 17. 67, 45 r. 18. 39, 37 r. 19. 44, 68 r. 20. 24, 6 r.
21. 42, 16 r.
Pages 255-256 (103-104). — 2. 38if!. 4. $1.37, H r. 6- $1-87, H r-
6. $0.65, \H r. 7. $0.44, 23>i r. 8. $0.52, 35^ r. 9. $0.32, 28ÍÍ r.
10. $2.17,14jifr. 11. $1.54,18)1; r. 12. $2.16, 35)é r. 13. $1.53, 34^ r.
14. $1.30, 45»; r. 16. $1.88, 15^ r. 16. $1.54, 25)i r. 17. $0.98, H r-
18. $0.58, 2H r. 19. $0.81, 10»; r.
Pages 257-258 (105-106). —4. 23. 5. 24. 6. 26. 7. 18. 8. 43. 9. 32.
10. 16. 11. 16. 12. 24. 13. 36. 14. 21, 2 r. 15. 32. 16. 46.
17. 18. 18. 63. 19. 34. 20. 32. 21. 74. 22. 42. 23. 26.
24. 36. 25. 84. 26. 35. 27. 21. 28. 63. 29. 31, 1 r. 30. 35.
Pages 259-260 (107-108). —1. 38 mi. 2. 16 mi. 3. $3.99. 4. 78 mi.
5. $9.85. 6. 35?;. 7. 21; $9.70.
Page 261 (109). —1. 63. 2. 74. 3. 56. 4. 84. 5. 95. 6. 65.
7. 57. 8. 62. 9. 86. 10. 96. 11. 73. 12. 76. 13. 92. 14. 48.
15. 71. 16. 84. 17. 58. 18. 96. 19. 53. 20. 83. 21. 43, 13 r.
22. 52, 13 r. 23. 73, 14 r. 24. 52, 8 r. 25. 64, 26 r. 26. 64, 6 r.
27. 94, 11 r. 28. 73, 8 r. 29. 85, 5 r. 30. 94, 7 r. 31. 63, 8 r.
32. 56, 4 r. 33. 68, 12 r. 34. 73, 2 r. 35. 58, 12 r.
Page 263 (III). —3. 83. 4. 63. 5. 74. 6. 96. 7. 53. 8. 57.
9. 54. 10. 57. 11. 46. 12. 63. 13. 38. 14. 53. 15. $.28.
16. $.27. 17. $.28. 18. $.39. 19. $.51. 20. $.28. 21. $.58.
22. $.48.
Pages 264r-265 (112-113). —1. 234. 2. 436. 3. 527. 4. 438. 5. 236.
6.9 248. 7. 423. 8. 534. 9. 342. 10. 267. 11. 537. 12. 265.
13. 286. 14. 374. 15. 536. 16. 628. 17. 384. 18. 267.
19. 548. 20. 634. 21. 793, 41 r. 22. 1843, 23 r. 23. 958, 60 r.
24. 325, 8 r. 25. 1055, 5 r. 26. 155, 70 r. 27. 543, 39 r.
28. 715, 28 r. 29. 1147, 21 r. 30. 266. 31. 1117, 6 r.
32. 2384, 32 r. 33. 1343, 6 r. 34. 293, 44 r. 35. 601, 10 r.
36. 1233, 68 r. 37. 381, 50 r. 38. 550, 41 r. 39. 1152, 55 r.
40. 1520, 23 r. 41. 1322, 29 r. 42. 1399, 17 r. 43. 1726, 46 r.
44. 537, 73 r.
Page 267 (115). —1. $2.35. 2. 35»;. 3. $2.00. 4. 144. 5. 16 ; 20.
6. $4.50. 7. 33,27»;; 3,28»;.
Pages 268-269 (116-117). — 1. Track, $1.80. 2. $4.20. 3. 85»; more.
4. 59^ more. 6. 95»;. 7. 60^. 8. $1.00. 9. $16.25. 10. 90^.
Page 269 (117). —1. 3181. 2. 2036. 3. 2289. 4. 2586. 5. 2977.
6. 2203.
ANSWERS
15
Pages 269-370 (117-118). —1. 25^. 2. 25. 3. $6.25. 4. $1.52. 6. $2.
6. 10. 7. 19. 8. 5. 9. 7. 10. 25(í. 11. 25 1b. 12. 64.
13. 46^
Page 271 (119). —1. 33. 2. 30. 3. 42. 4. 38. 6. 36. 6. 36.
7. 30. 8. 37. 9. 40. 10. 41. 11. 32. 12. 36. 13. 36. 14. 33.
16. 35. 16. 39. 17. 38. 18. 36. 19. 34. 20. 35.
1. 2015. 2. 2460. 3. 1793. 4. 2000. 5. 2774. 6. 2212.
7. 2349. 8. 2222. 9. 2316. 10. 2419.
Page 272 (120). —1. 45,695. 2. 9863. 3. 40,542. 4. 13,863.
5. 49,840. 6. 69,372. 7. 16,090. 8. 22,826. 9. 1973.
10. 13,161. 11. 34,590. 12. 27,487. 13. 69,082. 14. 75,376.
15. 16,798. 16. 22,381. 17. 37,445. 18. 35,780. 19. 55,150.
20. 65,678.
1. 263. 2. 316. 3. 444. 4. 517. 6. 408. 6. 551. 7. 143.
8. 593. 9. 544. 10. 253.
Page 273 (121). —1. 288. 2. 162. 3. 567. 4. 470. 6. 329. 6. 585.
7. 232. 8. 228. 9. 335. 10. 384. 11. 413. 12. 498. 13. 430.
14. 582. 15. 336 16. 354.
1. 3384. 2. 2726. 3. 2835. 4. 8342. 5. 4988. 6. 3525.
7. 5074. 8. 4324. 9. 7252. 10. 5655. 11. 1274. 12. 6300.
13. 6003. 14. 5264. 15. 6794. 16. 3808.
1. 52, 21 r. 2. 38, 20 r. 3. 45, 8 r. 4. 42, 34 r. 5. 61, 75 r.
6. 68, 49 r. 7. 52, 25 r. 8. 57, 11 r. 9. 35, 7 r. 10. 74, 78 r.
11. 55, 35 r. 12. 73, 26 r. 13. 48, 9 r. 14. 46, 21 r. 15. 69, 25 r.
Pages 374^375 (122-123). —1. 5. 2. 8. 3. 50. 4. 24. 5. 54.
6. 16. 7. 12. 8. 51. 9. 85. 10. 13. 11. 24. 12. 18.
Pages 375-376 (123-124). —1. 60ji. 2. 20 yd. 3. 19fi. 4. 60(i. 5. 80jf.
6. 40^. 7. 90^. 8. 90^. 9. $1.35. 10. 15;!. 11. 15;!. 12. 36^.
13. $1.20. 14. $2.52. 15. $3.00. 16. 72^. 17. 32;!.
18. Hhr.; 2^ hr.
Page 377 (125). —1. 15;!. 2. 8. 3. 6. 4. 40;!. 5. 8. 6. 20;!.
7. 4oz. 8. 30;!. 9. 28;!. 10. 6. 11. 16 da. 12. 75;!. 13. 16 yr.
Page 378 (126). —3. 46. 4. 57, 5 r.
Pages 379-380 (127-128). —1. $1.65. 2. 18 mi. 3. 35;!. 4. 45;!.
5. 45^. 6. 56;!.
Page 380 (128). —1. 38. 2. 48. 3. 48. 4. 56. 6. 67. 6. 57.
7. 76. 8. 72. 9. 68. 10. 93. 11. 86. 12. 84. 13. 296.
14. 486. 15. 392. 16. 529. 17. 387.
Page 381 (129).—2. 209. 3. 307. 4. 608. 5. 704. 6. 506. 7. 407.
8. 503. 9. 605. 10. 903. 11. 608. 13. 70. 14. 130, 11 r.
15. 260. 16. 340, 15 r. 17. 420.
16
ANSWERS
Page 283 (131).—4. 24.75. 6. 34.16. 6. 37.75. 7. 27.75. 8. 23.75.
9. 18.25 mí.
Page 284 (132).—2. 22.38. 3. 5.27. 4. 4.67. 6. 5.21. 6. 2.83.
7. 5.26. 8. 5.33. 9. 5.15. 10. 4.21. 11. 4.04. 12. 23.67 mi.
13. 32.36. 14. 3.29 qt.
Page 289 (137). —4. 6. 5. 6|. 6. 5. 7. 4^. 8. 6^. 9. 6J. 10. 6.
11. 7i. 12. 5|. 13. 5|.
Page 290 (138). —4. 2J. 6. 2^. 6. 2. 7. 2. 8. 3. 9. f 10.
11. If. 12. H. 13. i- 14. |. 15. i. 16. 2^. 17. 2^. 18. |.
19. 2|. 20. i. 21. i.
Pages 290-291 (138-139). — 1. H yd. 2. 3^ lb. 3. | Ib. 4. IJIb. 5. 3f
6. 54. 7. 4f. 8. 6i. 9. 4^. 10. 11. 11. 3^. 12. 6|. 13. 5J
14. 4|. 15. 5. 16. 12. 17. 3|. 18. 8^. 19. 8J.
Page 292 (140). —4. 3. 5. Sf. 6. 18|. 7. H. 8. 2. 9. H.
10. 1^. 11. 3. 12. 2i. 13. 5i. 14. 2. 15. 6|. 16. 3^.
17. 2J. 18. 7^. 19. 4J. 20. 3J. 21. 2f. 22. 5|. 23. 5J.
24. 5.
Page 293 (141). —1. 2^ lb. 2. 22^ lb. 3. 4f lb. 4. 61b. 5. 7 yd.
6. 3ilb. 7. 12ilb. 8. 3i hr. 9. $4.80.
Pages 298-299 (146-147). — 1. Ralph, 20 sq. ft. 2. Nell, 4 sq. ft.
3. Frank, 90 sq. ft. 4. Mary, 384 sq. in. 5. 7 by 9, 3 sq. ft. more. 6. $12.
Page 299 (147).—2. 720 sq. in. 3. 135 sq. ft. 4. 72 sq.m.; 36 sq. in.
5. 4^ sq. ft. 6. 52 sq. in. 7. 54 sq. in. 8. 20 sq. yd.
9. 24 sq. yd.
Pages 300-301 (148-149). —1. $15.30. 2. 80»!. 3. $62.50. 4. 18|i.
5. $1.45. 6. $15.60. 7. $8.00. 8. $1.75. 9. $4.50. 10. $6.96.
Page 301 (149). —1. 34. 2. 30. 3. 30. 4. 38. 5. 43. 6. 34.
7. 35. 8. 36. 9. 42. 10. 34. 11. 42. 12. 33. 13. 39. 14. 35.
15. 34. 16. 36. 17. 45. 18. 36. 19. 37. 20. 41.
Page 302 (150). —1. 54. 2. 58. 3. 55. 4. 61. 5. 57. 6. 58.
7. 51. 8. 56. 9. 56. 10. 58.
Pages 302-303 (150-151). —1. 1886. 2. 1616. 3. 2115. 4. 1851.
5. 2457. 6. 1661. 7. 2025. 8. 1613. 9. 2016. 10. 1681.
Page 303 (151). —1. 3435. 2. 3967. 3. 4957. 4. 4479. 5. 4609.
Pages 303-304 (151-152). —1. 266. 2. 337. 3. 475. 4. 446. 5. 323.
6. 442. 7. 184. 8. 455. 9. 194. 10. 473. 11. 422. 12. 416.
13. 322. 14. 673. 15. 251. 16. 364. 17. 736. 18. 727.
19. 559. 20. 635.
ANSWERS
17
Page 304 (162). —1. 14,778. 2. 33,164. 3. 26,180.
5. 44,394. 6. 32,660. 7. 13,013. 8. 61,137.
10. 47,898. 11. 68,688. 12. 34,376. 13. 21,768.
16. 55,872.
1. 4512. 2. 2592.
7. 6557. 8. 4183.
3. 3219. 4. 7200. 6. 4704.
9. 4485. 10. 5624. 11. 4872.
Page 305 (163).
1.
19,008
26,391
60,669
25,984
66,8ip
2.
25,758
42,048
25,668
51,224
32,752
3.
33,880
45,828
45,888
38,416
23,716
6.
2. 1366, 1 r. 3. 799, 5 r.
874. 7. 1261, 2 r. 8. 493, 4 r.
11. 1526, 4 r. 12. 930, 5 r.
16. 423, 6 r. 16. 595, 4 r.
83, 63 r. ; 64, 23 r. ; 74, 89 r. ;
2. 66, 5 r.; 64, 29 r. ; 75, 11 r.; 77, 56r.; 51, 45 r.
85, 17 r. ; 85, 20 r. ; 86, 25 r. ; 82, 10 r. ; 74, 8 r.
-1. 5. 2. 10. 3. 40{f.
9. 64fi. 10. 60{i.
1. 771, 2 r.
6. 949, 5 r.
10. 949, 2 r.
14. 848, 7 r.
1. 87, 50 r. ;
Page 306 (164).
7. 8. 8. 75jé.
4. 38,664.
9. 68,368.
14. 66,241.
6. 4648.
12. 3264.
4.
59,264
47,538
63,058
33,423
66,534
4. 1298, 2 r.
9. 763, 6 r.
990, 3 r.
13.
60, 64 r. ;
3.
96, 16 r.
92, 34 r.
4.
11.
2^li.
80>i.
6. 7H- 6- 65 mi.
12. $1.20.
ANSWERS
THE STONE ARITHMETIC
FIFTH YEAR
Pages 1-2.— 1. $47.25. 2. $4.25. 3. $15.25. 4. $3.65.
5. $31.46. 6. $9.16. 7. $15.60. 8. $1.75. 9. $10.80.
10. $1.86. 11. $17.92. 12. $4.06.
Pages 2-3.— 1. $.85. 2. $.17. 3. $.64. 4. $.06. 6. $8.71.
6. $.14. 7. $.24. 8. $2.40. 9. $65.70. 10. $4.00. 11. $2380.00.
12. 350 gal. 13. 19 mi.
Page 4.— 1. $36.80. 2. $19.95. 3. $7.60. 4. $3.50. 5. $.85.
6. $163.80. 7. $1430.00. 8. $75.00. 9. $21.75. 10. $.54.
11. $4.05. 12. $336.00.
Page 9. —1. 548. 2. 538. 3. 565. 4. 535. 5. 593. 6. 556.
7. 499.
Subtraction. — 1. 18,934. 2. 11,388. 3. 25,278. 4. 45,565.
5. 51,094.
MultipHcation. — 1. 278,388. 2. 472,507. 3. 438,738.
4. 722,160. 5. 701,795.
Pages lO-ll. — Test I. 1. 37. 2. 29. 3. 36. 4. 34. 5. 33.
6. 36. 7. 31. 8. 39. 9. 32. 10. 33. 11. 27. 12. 40. 13. 32.
14. 37. 15. 34. 16. 35. 17. 32. 18. 33. 19. 33. 20. 30.
Testn. —1. 63. 2. 57. 3. 70. 4. 69. 5. 65. 6. 62. 7. 59.
8. 59. 9. 58. 10. 58.
Testm. —1. 2958. 2. 4281. 3. 4243. 4. 3795. 5. 3134.
6. 4074.
Pages 14H5. ^ Drm IV. 1. 26. 2. 30. 3. 35. 4. 32. 6. 29.
6. 30. 7. 29. 8. 29. 9. 24. 10. 21. 11. 29. 12. 34. 13. 32.
14. 28. 15. 25. 16. 22. 17. 34. 18. 31. 19. 29. 20. 33.
21. 24. 22. 22. 23. 23. 24. 27. 25. 29. 26. 38. 27. 24.
28. 31. 29. 33. 30. 31. 31. 27. 32. 26. 33. 36. 34. 34.
35. 19.
Test in speed. — 1. 5241. 2. 5105. 3. 5419. 4. 5257.
5. 5567. 6. 5607.
Page 16. — Test I. 1. 7917. 2. 4336. 3. 5537. 4. 3439.
5. 3771. 6. 4619. 7. 2418. 8. 2414. 9. 3183. 10. 2238.
11. 2079. 12. 2683.
18
ANSWERS
19
Testn. —1. 587,085. 2. 452,489. 3. 645,314. 4. 512,267.
5. 491,913. 6. 359,117. 7. 381,835. 8. 554,219. 9. 304,742.
10. 181,959.
Page 18. — 1. 10,884. 2. 23,495. 3. 57,859. 4. 40,064.
5. 46,328. 6. 32,841. 7. 40,741. 8. 28,036. 9. 37,839.
10. 20,188. 11. 11,393. 12. 42,679. 13. 44,732. 14. 53,676.
15. 65,755.
Pages 18-19. — 1. 39 hr. 2. $35.75. 3. 591. 4. $2.60.
5. $129. 6. $8.80. 7. 173; $60. 8. 69. 9. 95. 10. 12 wk.
Page 23. — 1. 351 mi. 2. $80. 3. $2520. 4. 900 mi.
5. 624 da. 6. $8352. 7. 4745 ft. 8. 5440 bu. 9. $360.
Page*24. — Test I. 1. 3572. 2. 4845. 3. 3082. 4. 6882.
5. 5074. 6. 5976. 7. 3948. 8. 4464. 9. 5460. 10. 4484.
11. 6256. 12. 6264. 13. 6188. 14. 2736.
Testn. —1. 268,105. 2. 619,158. 3. 247,008. 4. 315,445.
5. 836,679. 6. 820,416.
Pages 26-27. — Set A. 1. 144,628. 2. 59,511. 3. 53,900.
4. 306,760. 6. 109,944. 6. 111,776. 7. 655,248. 8. 178,875.
9. 345,654. 10. 192,073. 11. 299,616. 12. 578,961.
SetB. —1. 221,673. 2. 49,426. 3. 229,652. 4. 158,125.
6. 493,884. 6. 393,386. 7. 552,144. 8. 734,373. 9. 394,442.
10. 244,489. 11. 203,808. 12. 384,237.
SetC. — 1. 95,961. 2. 34,864. 3. 61,388. 4. 61,780.
5. 474,336. 6. 137,606. 7. 128,784. 8. 178,713. 9. 80,892.
10. 208,201. 11. 211,656. 12. 292,464.
Pages 37-38. — Set A. 1. 210,285. 2. 229,464. 3. 259,224.
4. 508,523. 5. 182,358. 6. 307,172. 7. 833,424. 8. 514,512.
SetB. — 1. 431,596. 2. 205,672. 3. 187,717. 4. 796,188.
5. 685,092. 6. 90,930. 7. 394,108. 8. 274,620.
Page 38.— 1. 392,904. 2. 631,164. 3. 406,182. 4. 410,795.
5. 638,385. 6. 584,445. 7. 474,660. 8. 511,360. 9. 710,600.
10. 528,960. 11. 226,860. 12. 94,640.
Page 39.— 1. $10.24. 2. $3.24. 3. $49.30; $47.85. 4. $14.40;
$1.80. 5. $2.80. 6. $2.60. 7. $17.13. 8. $12.00.
Page 30. — 9. 388,000. 10. 185,000. 11. 288,000. 12. 315,000.
13. 588,000. 14. 558,000. 15. 224,000. 16. 483,000.
17. 456,000.
Pages 34-35. — 1. $1.86. 2. $.86. 3. $3.65. 4. $2.26. 5. 35.
6. $1.25. 7. $.09. 8. $.50.
20
ANSWERS
Page 36. —3. 55 ; 59 r. 4. 72; 60 r. 5. 77; 68 r.
6. 113; 49 r. 7. 41; 78 r. 8. 54; 46 r. 9. 65; 42 r.
10. 74; 49 r. 11. 1264; 42 r. 12. 923; 24 r. 13. 473;
38 r. 14. 1399; 17 r.
Page 37. —1. 82; 23 r. 2. 53; 20 r. 3. 52; 3 r.
4. 38; 19 r. 5. 76; 14 r. 6. 63; 8 r. 7. 57; 13 r.
8. 54; 3r. 9. 57; 8 r. 10. 38; 2 r. 11. 28; 4 r.
12. 27; 2 t. 13. 28; 11 r. 14. 46; 13 r. 15. 48; 48r.
16. 87; 9r. 17. 48; 7 r. 18. 64; 8 r. 19. 73; 13 r.
20. 57; 28 r.
Page 40.— 1. 923 ; 24 r. 2. 1264 ; 42 r. 3. 715; 28 r.
4. 1147; 21 r. 5. 1117; 6 r. 6. 1343; 6 r. 7. 381; 50r.
8. 389; 39 r. 9. 1520; 23 r. 10. 591; 18 r.
11. 1322; 29 r. 12. 1399; 17 r. 13. 1726; 46 r.
14. 1116; 35 r. 16. 1067; 35 r. 16. 1755; 26 r.
17. 1603 ; 27 r. 18. 984; 82 r.
Page 41. —12. 52; 85 r. 13. 72; 92 r. 14. 54; 464 r.
15. 89; 403 r. 16. 88; 441 r. 17. 93; 626 r.
18. 96; 645 r. 19. 94; 126 r. 20. 99; 275 r.
21. 71; 276 r. 22. 66; 627 r. 23. 60 ; 637 r.
25. 122; 285 r. 26. 58; 248 r. 27. 116; 48 r.
28. 90; 529 r. 29. 115; 328 r. 30. 103 ; 236 r.
31. 82; 2r. 32. 85; 565 r. 33. 19; 684 r.
34. 27; 2498 r. 35. 16; 4163 r. 36. 12; 5676 r.
Page 42. — 1. $1.25. 2. $1.25. 17. $48.52. 18. $.54.
Page 43.— 1. 738; 33 r. 2. 147; 76 r. 3. 2319; 1 r.
4. 2029; 17 r. 5. 596; 263 r. 6. 631; 239 r.
7. 913; 919 r. 8. 2321; 291 r. 9. 2253; 195 r.
10. 2013; 5r. 11. 395. 12. 164; 481 r. 13. 931; 321 r.
14. 647; 30 r. 15. 706; 50 r. 16. 643; 323 r.
Page 44.— 1. 2889. 2. 3347. 3. 3572. 4. 3473. 5. 2659.
6. 3019. 7. 3748. 8. 2767. 9. 3441. 10. 3081. 11. 3090.
12. 2937. 13. 3727. 14. 2881.
Pages 44-45. ^ Race I. 1. 2687. 2. 3436. 3. 565. 4. 1182.
6. 2822. 6. 3075. 7. 5472. 8. 1586. 9. 308. 10. 7622.
11. 3282. 12. 2209. 13. 2533. 14. 5862. 15. 5166. 16. 208.
17. 195. 18. 4545.
Racen.— 1. 7137. 2. 4531. 3. 2443. 4. 1671. 5. 6121.
6. 2483. 7. 1182. 8. 3092. 9. 3119. 10. 1252. 11. 5417.
12. 1466. 13. 2469. 14. 3061. 15. 5562. 16. 459. 17. 1870.
18. 6239.
Pages 45-46. — Race I. 1. 302,744. 2. 400,551. 3. 430,376.
4. 410,504. 5. 443,004. 6. 167,301. 7. 233,772. 8. 778,428.
9. 481,992. 10. 576,975. 11. 716,268. 12. 312,984.
ANSWERS
21
Racen.— 1. 443,136. 2. 666,060. 3. 544,166. 4. 626,188.
5. 795,430. 6. 733,728. 7. 416,745. 8. 524,520. 9. 808,821.
10. 377,024. 11. 532,779. 12. 548,864.
Race m.— 1. 490,852. 2. 384,422. 3. 499,284. 4. 766,671.
6. 799,848. 6. 565,364. 7. 414,333. 8. 724,722. 9. 457,954.
10. 215,412. 11. 259,502. 12. 428,016.
Page 46. — Race I. 1. 568. 2. 467. 3. 846. 4. 781. 5. 493.
6. 693. 7. 489. 8. 465. 9. 796. 10. 793. 11. 921.
12. 526.
Racen.— 1. 738. 2. 528. 3. 486. 4. 847. 6. 638.
6. 827. 7. 624. 8. 784. 9. 864. 10. 468. 11. 387.
12. 563.
Pages 63-53. — 2. 12 oz. 3. 10 oz. 4. 1 ft. 8 in. 5. 12 qt.
7. 222. 8. 154. 9. 484. 10. 699. 11. 861. 12. 720. 13. 730.
14. 710. 15. 744. 16. 1017. 17. 795. 18. 278. 19. 54.
20. $2.50.
Pages 54-55. — 1. 4|. 2. 8#. 3. 6|. 4. 7f. 5. 8|. 6. 3f.
7. 5|. 8. 4|. 9. 6|. 10. 7i. 11. 5|. 12. 4|. 13. 4^.
14. 3|. 15. 3^. 16. 2^. 17. 2^. 18. 8|. 19. 6f. 20. 8|.
21. 75. 24. $2.57. 25. $3.24. 26. $1.75. 27. $.70 less.
Pages 56-57. — 2. 9cs. 3. 5 pt. 4. 51b. 6. 6^ doz., $1.17.
7. 13 pt. 8. $4.05. 9. 13. 10. 12. 11. 21^. 12. 26|. 13. 20.
Pages 59-60. — 6. 12^ lb. 7. 10 1b.; $4.50. 8. $5.17. 9. 3 yd.;
$1.92. 10. 22^. 11. 27f. 12. 24|. 13. 24. 14. 23^.
Pages 60-61. — 7. $4.60. 8. 24i lb. ; $8.69. 9. 85 1b. 10. 19^.
11. 23f 12. 23|. 13. 25|. 14. 20|. 15. 15. 16. 19^.
17. 13. 18. 15. 19. 25|. 20. 19^. 21. 15.
Pages 63-63. — 7. 9. 8. llf 9. IH. 10. 9|. 11. 9J.
12. 6|. 14. 25Jft. 15. 5ihr.; $1.04.
Page 63. — 1. 19|. 2. 26. 3. 23. 4. 20|. 5. 16J. 6. 7.
7. 12|. 8. llf. 9. 16^. 10. llj. 11. llj. 12. 18. 13. lOJ.
14. 17i. 15. 13. 16. 174. 17. 10. 18. 12J.
Page 64. — Race I. 1. 16^^. 2. 22. 3. 22^. 4. 24J. 5. 19|.
6. 21f. 7. 26. 8. 20|.
Racen. —1. 19J. 2. 22^. 3. 18J. 4. 19^. 5. 21^. 6. 18J.
7. 14. 8. 22i.
Race ra. —1. 21^. 2. 22|. 3. 22|. 4. 15|. 5. 17|. 6. 23.
7. 12|. 8. 18J.
Race IV.— 1. 16. 2. 12|. 3. 20J. 4. 18f. 5. 17f 6. 171-
7. 9J. 8. 16|.
22
ANSWERS
Pages 65-66. — 5. 31- 6. 5^. 7. SJ. 8. 5J. 9. 2^. 10. 3^.
11. 8i. 12. 84. 13. 14f. 14. 7|. 16. 7^. 16. 13|. 17. 4^.
18. 22|. 19. 224. 20. 284. 21. 14f. 22. 24f.
Pages 66-67. — 5. 3| Ib. 6. I4 Ib. 8. 27f 9. 23f. 10. 25|.
11. 284. 12. 644. 13. 254. 14. 144. 15. 334. 16. 464.
17. 56|. 18. 584. 19- 1384- 20. 634. 21. 3064. 22. 223|.
Pages 67-68. — 4. 54. 6. 2|. 6. 44. 7. 44. 8. 4.
Pages 68-69. — I. 1. I44. 2. 94. 3. 224. 4. 24|. 6. 74.
6. 44. 7. 224. 8. 144. 9. 254. 10. 204. II. 1. 144.
2. 15|. 3. 354. 4. I64. 5. 25|. 6. 28f. 7. 264. 8. II4.
9. 25f. 10. 134. III. 1. 144. 2. 23f. 3. 334. 4. 224.
5. 64. 6. 174. 7. 28|. 8. 74. 9. 34|. 10. 27f.
Page 69.— 1. I4. 2. 1|. 3. 1|. 4. I4.
7. 14. 8. 1|. 9. 4. 10. 1. 11. 2. 12. 24.
15. 4. 16. 24. 17. 4. 18. 4. 19. f. 20. 4.
23. 4. 24. 4.
Page 70. — 1. 4. 2. 4. 3. f. 4. 4. 5. 4.
8. 4. 9. f. 10. 4. 11. 4. 12. |. 13. 4.
16. 4. 17. 4. 18. 4. 19. 4. 20. 4. 21. 4.
24. 4. 26. 4. 26. 4. 27. f. 28. 4. 29. f.
32. 14. 33. 14. 34. 14. 35. 1|. 36. If.
39. 14. 40. f. 41. 4. 42. 1|. 43. I4.
46. 4. 47. f. 48. f. 49. f. 50. |. 51. f.
54. 4. 55. f. 56. f. 67. 4. 68. f- 59. 44.
62. 14. 63. 4. 64. 14. 65. I4. 66. 4.
69. 44. 70. 14. 71. 4. 72. f.
Page 75. — 2. 1944. 3. 2544- 4. 25^. 5. 2344. 6. 25^-
7. 22/j. 8. 25^. 9. 21^- 10. 22^. 11. 244. 12. 19^.
13. 23^. 14. 19^. 15. 1144. 16. 17,^. 17. IIA. 18. 94.
19. 84. 20. 84.
Pages 76-77. — 1. 4,4. 2. f. 3. 44 Ib. 4. 14 yd. 5. 54.
6. 54. 7. 44. 8. 24. 9. 5A. 11. 214. 12. llf. 13. 204.
14. 31|. 15. 344. 16. 2|lb. 17. 1|. 18. 17|.
Pages 78-79. —.Test HI. 1. 3^. 2. 2^. 3. 24. 4. 244.
5. 244. 6. 241. 7. 2f. 8. 3A- 9. 4. 10. 4. 11. f.
12. A- 13. A- 14. A. 15. A- 16- A- 17- A- 18- 15334.
19. 1921A. 20. 16794. 21. 1715^- 22. 154|. 23. 21A-
Test IV. —1. 624. 2. IO84. 3. 65|. 4. 1734. 5. 74|. 6. 684.
7. 4144. 8. 1274. 9. 4934. 10. 3864. 11. 3II4. 12. 405|.
13. 277|. 14. 2364. 15. 273|. 16. 156|. 17. 3444. 18. 385|.
19. 7174. 20. 1734.
Page 80. —1. 40(i. 2. 5fi. 3. 40^. 4. 5 hr. 5. 30. 6. 40 mm.
7. OOjf. 8. 20jé. 9. 18mm. 10. Ohr. Ii. $1.10. 12. $1.60.
13. 20 hens. 14. $.75.
5. 14.
6.
14-
13. If.
14.
3
"ff*
21. f.
22.
i-
6. f.
7.
f*
14. 4-
15.
f*
22. f.
23.
f*
30. 4.
31.
n-
37. 14.
38.
ih
44. 4.
45.
7
■ff*
62. 4.
53.
1
7*
60. f.
61.
H-
67. 14.
68.
H*
ANSWERS
23
Page 83. —13. If. 14. 2. 16. U. 16. 1^. 17. If. 18. W.
19. 2^. 20. 2i. 21. 1^. 22. 53|. 23. 60?. 24. 48A-
26. 58|. 26. 67A- 27. 76^-
Pages 83-84. —12. 25^. 13. 36i. 14. 24f. 16. 19J. 16. 28A.
17. 22^. 18. 25^. 20. 3^. 21. 3^. 22. 2|. 23. 3f. 24. 34.
26. 2i. 27. 2i. 28. |. 29. i. 30. |. 31. 4. 32. 4.
33. 34. |. 36. f. 36. 37. 74. 38. 134. 39. 254.
40. 34f. 41. 81- 42. 24f. 43. 34Î. 44. 17|.
Pages 89-90.—2. 6| lb. 3. 16| cs. 6. 20 cs. sugar; 12 cs.
milk; 18 tbs. cocoa; 12 tbs. butter. 6. $1.86. 7. 14. 8. 34.
9. 3^. 10. 5^. 11. 6|. 12. 4|. 13. 4|. 14. 4^. 16. 64.
16. 5|. 17. 3J. 18. 2|. 19. If. 20. 3|. 21. 3f. 22. 7f.
23. $2p.30.
Pages 90-91. — 4. 10. 6. 6. 6. 6f. 7. 7f 8. 34. 9. 54.
10. 3|. 11. 4f. 12. 7f. 13. 5f. 14. 6. 16. 10. 16. 10.
17. 14. 18. 9. 19. 6f. 20. 4f. 21. 7f. 22. lOf. 23. 14|.
Pages 91-93. — 1.4 eggs ; 2f cs. brown sugar ; If cs. butter ;
If ts. vanilla ; 4f ts. cream of tartar ; 2f ts. soda ; f ts. salt ;
6f cs. flour. 2. 6 eggs ; 4 cs. brown sugar ; 2 cs. butter ; 2f ts. vanilla ;
6| ts. cream of tartar ; 3| ts. soda ; f ts. salt ; 9f cs. flour.
4. 4 cs. sugar ; 2 cs. flour ; 2f cs. butter ; 2 cs. broken walnuts ;
6 eggs ; 9 sq. chocolate. 6. 6f cs. sugar ; If cs. orange juice ;
3 cs. strawberry juice ; 4f cs. water ; If cs. lemon juice ; 3f cs. pine¬
apple juice. 6. 16f cs. sugar ; 3| cs. orange juice ; 7f cs. strawberry
juice; llfcs. water; 3f cs. lemon juice ; 8f cs. pineapple juice.
7. 4fyd. 8. 9 yd. 9. 22f yd. 10. 7 yd. 11. 14 qt. 12. $14.85.
13. No; $.07. 14. $.88.
Pages 94-95. — 1. $1.05. 2. $3.00. 3. $4.05. 4. $8.10.
6. If lb. 6. $2.07. 7. I yd. 8. 3 qt. 9. 16f yd. 10. 22f lb.
11. 3fyd. 12. fyd. 13. 5 in. 14. 4f yd. 16. 5f yd.
Page 95.— 1. 4f. 2. 3. If 4. f 6. f. 6. ü
7. 4. 8. 4. 9. If. 10. 2. 11. 6. 12. 2f. 13. 5f. 14.
-- ■- - 18. f. 19. If. 20. If. 21.
26. 7f. 26. 27. 28.
32. 5. 33. If. 34. 2f. 36.
39. 2f. 40. 7f. 41. 42.
46. A- 47. 4|. 48. 34. 49. K
63. If. 64. f. 66. Iff. 66.
60. |.
16.
16. |.
17. f.
22.
8.
23. If.
24. f.
29.
H-
30. If.
31. 3.
36.
5
■ff*
37. f.
38. 2f.
43.
lA.
44. If.
46. f.
60.
3f.
61. A-
62. f|.
67.
5i.
68. f.
59. f.
Pages 96-97. — 3. A- 4. A- 6- A- 6- f
9. A- 10. A- 11. f- 12-, A- 13. A- 14, „
16. A- 17. A- 18. «• 21. f. 22. A. 23. f.
25. A- 26. A- 27. A- 28. A. 29. A- 30. A
IS*
a I
32. f.
24
ANSWERS
7 *
16. 81f.
22. 161^%.
28. 159^.
33. 26|bu.
Pages 97-98. — 2. 3. f. 4. f. 5. f. 6. A
8. 9. 10. f. 11. 12. 13. f.
ir. 105Ä- 18. 97^. 19. 187H- 20. 242|. 21. 206^-
23. 197|. 24. 249Ä- 25. 149^- 26. 105^- 27. Höf.
29. 315A. 30. 143^. 31. 48i bu. 32. 6| mi.
34. 1| yd.
Pages 99-100. — 2. 12,^ Ib. 3. 3| Ib. 4. 98H bu. 5. 8fi Ib.
6. f bu., Ü bu. 7. 3f Ib. 8. | qt., yes. 9. 2^ Ib. 10. ^ Ib., 3M Ib.
12. A'-
19. y.
26. i.
34. 3.
41. f.
Page 103. — 1. 1 c. sugar ; J c. molasses ; 1 tbs. vinegar ; ^ c. butter ;
2 tbs. water. 2. 2 es. sugar ; ^ c. molasses ; 2 tbs. vinegar ; 1 c. butter ;
4 tbs. water. 3. f c. sugar ; c. molasses ; f tbs. vinegar ; f c. butter ;
1^ tbs. water. 4. | c. butter ; 2 es. sugar ; 2 es. brown sugar ;
^ 0. molasses ; 1 c. cream ; 4 squares chocolate. 5. f c. butter ;
1| cs. sugar ; cs. brown sugar ; |- c. molasses ; f c. cream ;
3 squares chocolate. 6. J c. butter ; 1 c. sugar ; 1 c. brown sugar ;
^ c. molasses ; ^ c. cream ; 2 squares chocolate. 7. 5 cs. sugar ;
2f cs. milk ; 7 squares chocolate ; 3 tbs. butter.
Pages 100-101. —
8. A
9. f.
10. f.
11. A.
13. ff.
14. A-
16.
T
TT-
16.
5
TT.
17.
s
TT.
18. f.
20. A-
21. A-
22.
1
I-
23.
2
T.
24.
5
'S-
26. f.
27. A-
29. If.
30.
If.
31.
1.
32.
1.
33. 3f.
36. 3f.
36. 4A.
37.
2i.
38.
3f.
39.
2.
40. 1|.
42. 3f.
43. 3f.
Page 103.
8. 6. 9.
15. h
22. 16^.
29. 17.
36. 15f.
43.
60.
67.
64. f.
71. 6|.
1. 1^.
10.
16. 2|.
23. f.
30. 2^.
37. 13f.
44. 19i.
61. ^
68. 2^.
66. 1.
72. 8f.
2. 2f. 3. f.
^ 11. 7§.
17. 2i.
24. f.
31.
62. 23J.
69. IJ.
66. 4.
18. 2|.
26. f.
32. 4J.
39. 3i.
46. 20.
63. 63.
60. 4^.
67. 2|.
4. f 6.
12. 11.
19.
26. If.
33. 5f.
40. 4f.
47. 30.
64. 78.
61.
68. 3^.
1. f. 2. f.
10. h
16. If.
11.
1 '
4. If.
12. If.
f. 6. f.
7.
3
Y'
13. llf.
14.
1
ITS'
20. f.
21.
8J-.
27. 2f.
28.
7f.
34. 2f.
36.
3
'S'
41. 3f.
42.
3
S'
48. f.
49.
3^-
66. 10.
66.
10.
62. Sf.
63.
2|.
69. 2.
70.
f-
f. 6. 6.
7.
4.
13. 16f.
14.
26.
Page 105.
8. f. 9. y
15. If.
Page 108. — 1. If. 2. If. 3. If. 4. If. 6. 2f.
7. 3. 8. A- 9- 1|- 10. lA- 11- If 12. If-
14. Iff. 15. A- 16. 2A. 17. |. 18. A- 19. 3f.
21. 8|. 22. 9ff. 23. 21f. 24. 19A- 26. 2f.
27. 5. 28. Iff. 29. 2^- 30. 5ff. 31. 8f. 32. 15f.
34. If. 36. 8f. 36. A- 37. lA- "
41. If. 42. If. 43. 4ff. 44. 7f.
48. 2A.
38. t
46. U.
39. f.
46. 3f.
6. 4f
13. 2ff
20. If
26. 9|f
33. f
40. lA
47. f
ANSWERS
25
Pages lOS-110. — 1. 10. 2. Yes. 3. yd. less. 4. 12? A.
6. 3?wk. 6. 2flb. 7. 2AT. 8. $3.69. 9. $.22?. 10. $.08.
11. $.15. 12. 46-mch; $.53. 13. $9.08.
Page 110. — 1. 2f. 2. 2?. 3. 3?. 4. 6. 6. ?. 6. A.
7. 3?. 8. 5. 9. 9. 10. 5?. 11. |. 12. f. 13. 132.
14. 300. 16. 503?. 16. 329?. 17. 309?. 18. 368?. 19. 4?.
20. 4?. 21. 2??. 22. 6?. 23. 6^. 24. 5^- 25. ?. 28. A.
27. A- 28. ?. 29. Ä- 30. ??. 31. 81?. 32. 93??.
33. 102^. 34. 72??. 35. 181?. 36. 295?. 37. 1?. 38. 1?.
39. 1?. 40. ?. 41. 1?. 42. 1?. 43. 44. ??. 45. ?|.
46. lA. 47. ??. 48. 3A.
Page III.—Race I. 1. ?. 2. f 3. A. 4. f. 5. ?. 6. ?.
7 ?. 8. ? 9. A. 10. ?. 11. A- 12. A- 13. ? 14. |.
15. 16. ?. 17. A- 18. ?. 19. ?. 20. ?.
Racen. —1. ?. 2. ?. 3. A- 4. ?. 5. A. 6. ?. 7. A-
8 |. 9. I 10. ?. 11. A- 12. A- 13. A. 14. ?. 15. ?.
16. A- 17. ?. 18. ?. 19. 1. 20. 1.
Race I.— 1. 1?. 2. 1?. 3. 2?. 4. 1?. 5. ?. 6. 2?. 7. 1?.
8. f. 9. 1?. 10. lA. 11. lA- 12. lA- 13. lA- 14. 3A.
15. 1?. 16. A. 17. |. 18. 1?. 19. 1|. 20. 2?.
Race n. 1. A. 2. 1?. 3. 1?. 4. 1?. 5. A. 8. 1?.
7. 2A. 8. 3?. 9. 2??. 10. ?. 11. 3?. 12. ?. 13. 1?.
14. 1?. 15. 2?. 16. 1?. 17. 1?. 18. 3?. 19. 1?. 20. ?.
Page 113. — 5. 24.75 bu.
Pages 114r-116. — 6. 22.11 mi. 6. 158.34 mi. 7. 44.64 mi.
8. 38.82 bu. 9. Yes. 10. 119.48 1b. 11. 82.84 mi. 12. 135.36 mi.
13. 19.44 mi.
Pages 116-117. — 3. .645 lb. 4. .56. 5. .33. 6. .46. 7. .74.
8. .52. 9. .88. 10. .83. 11. .28. 12. .50. 13. .29. 14. .60.
15. .25. 16. .25. 17. .35. 18. .19.
Pages 117-118. — 1. $13.35. 2. 2.75.
7. 433.12. 8. 859.83. 9. 597.94.
12. 620.56. 13. 446.48. 14. 319.30.
17. 310.41. 18. 272.22. 19. 161.78.
I.78 bu.
Pages 119-130. — 1. $60.75. 3. 18.60 mi. 4. 46.25 mi.
5. 185.01. 6. 257.16. 7. 439.84. 8. 518.4. 9. 42.1. 10. 287.1.
II. 3286.05. 12. 41,336.4. 13. 567.12. 14. 3219.06. 15. 252
mi. 16. 3058 bu. 17. 180 mi.
Page 130. — 1. 1.3. 2. .2. 3. 1.05. 4. .35. 4. 2.4. 6. 1.85.
7. 1.35. 8. 3.2. 9. 4. 10. 1. 11. 4.5. 12. .45. 13. .66.
14. .54. 15. 3. 16. .30. 17. 2. 18. .75. 19. .25. 20. 1.4.
21. 9.5. 22. 3.75. 23. 18. 24. 4.35. 25. 3.52. 26. 4.7.
27. 8.23. 28. 1.05. 29. 15. 30. 1.25. 31. 2. 32. 1. 33. 7.8.
5. 396.58.
10. 309.53.
15. 178.09.
20. 1.43 mi.
6. 333.78.
11. 655.62.
16. 252.48.
21. First,
26
ANSWERS
34. 12. 36. 2.05. 36. 2.88. 37. 5.28. 38. 2.6. 39. 26.
40. 6.75. 41. 1. 42. .81. 43. 5.4. 44. .56. 46. 7.2.
46. 10.8. 47. 1.2. 48. .7.
Page 121. — 1. $4.75. 4. 18.5 mi. 6. 42.75 bu. 6. 2.75 lb.
7. 142.75 1b. 8. 1.81 T. 9. 20.67. 10. 36.06. 11. 28.31.
12. 48.68. 13. 30.90. 14. 9.20. 16. 13.63. 16. 4.98.
17. 3.00. 18. 5.84. 19. .65. 20. .69. 21. .80. 22. .74.
23. .78.
Page 122. — 1. 1.8 mi. 2. .25 mi. 3. .4 mi. 4. 4.8 mi.
6. .2. 6. This season, .12. 7. 1.6 bu. 8. John's, .05 A; .95 A.
9. Our team, .15. 10. 2.4 mi.
Page 126. — 1. $.84. 2. $3.54. 3. $2.24. 4. $2.34. 6. $2.88.
6. $2.25. 7. $.68. 8. $2.00.
Page 127. — 3. 19 ft. 2 in. 14. 15 ft. 2 in. 6. 18 ft. 2 in.
6. 17 ft. 8 in. 7. 5^1^ yd. 8. No. 9. 15|| yd.
Page 128. — 2. 1 yd. 24 in. 3. 2 ft. 7 in. 4. 2 ft. 6 in.
6. 9 ft. 3 in. 6. 12 ft. 6 in. 7. 18 ft. 6 in. 8. 9 ft. 6 in. 9. 2 ft. 6 in.
10. 6 ft. 9 in. 11. 6 ft. 10 in. 12. 3 yd. 30 in.
Page 129.—2. 16 ft. 6 in. 3. 14 ft. 8 in. 4. 28 ft. 6 in.
6. 22 ft. 8 in. 7. 2 ft. 9 in. 8. 2 ft. 7 in. 9. 2 yd. 2 ft. 6 in.
10. 1 yd. 2 ft.
Pages 133-134. — 10. 224 sq. ft. 12. 408^ sq. ft. 13. 351f sq. ft. ;
$1.76. 14. 2| yd. by 32 in. 16. 2| yd. by 54 in. 17. 41 pupils.
Pages 135-136. — 2. 1 in. by 1^ in. ; 1^ sq. in., 1350 sq. ft. 3. 2 in.
by 4 in. ; 7200 sq. ft. 4. 3200 sq. ft. 6. 60 ft. by 90 ft. 6. 45 ft.,
75 ft., 150 ft. 8. 300 mi. 10. 61^ mi. 11. 70 mi.
Pages 137-138. — 6. 432 sq. in. ; 720 sq. in. ; 72 sq. ft. ; 45 sq. ft. ;
1^ sq. yd. ; 72 sq. in. ; 36 sq. in. ; 3 sq. ft. ; 24 sq. in. ; 16 sq. in.
8. 5445 sq. ft. 9. Larger. 11. .41+ A. 12. 5.23 A.
Pages 138-139. — 2. 314 ft. 3. 118^ ft. 4. $.83. 6. .30 mi.
6. 286 yd. 7. $86.40. 8. 74 ft. 9. 122 ft. 8 in.
Pages 143-144. — 4. 864 cu. in. 6. 1296 cu. in. 6. 18 cu. ft.
7. 20lf cu. ft. 8. 64 cu. ft. 9. 96 cu. ft. 10. 80 cu. ft. 11. .63 cd.
12. $30. 13. 10 loads. 14. 8100 cu. ft.
Pages 144-146. — 3. 2| bu. 4. 3.625 pk. 6. ^ bu. 6. J bu.
7. $1.80. 8. $2.70. 10. 2 bu. 5 qt. 11. 10 pk. 1 qt. 12. 16 bu.
14. 2 bu. 2 pk. 16. 4 bu. 2 pk. 16. 1 pk. 6 qt. 17. 7 bu. 3 pk.
19. 8 bu. 1 pk. 20. 23 bu. 3 pk. 21. 15 pk. 4 qt. 22. 17 pk. 4 qt.
23. 19 bu. 1 pk. 24. 52 bu. 2 pk.
Pages 146-147. — 4. Yes. 6. Yes. 6. 225 gal. 7. .6 gal. less.
9. 375 gal.; 370 gal. 10. 54,552^ gal.
Pages 148-150. — 2. $.72. 3. $71.43. 4. $6.16. 6. 8.671b.
7. 19.76 mi. 16. 6.38. 17. 7.11. 18. 6.56. 19. 8.67. 20. 6.99.
ANSWERS
27
21. 13.42. 22. 13.04. 23. 7.10. 24. 8.97. 25. 6.86. 26. 109
27. 1.23. 28. 1.44. 29. 2.49. 30. 2.93.
Page 151. — 2. .938. 3. .905. 4. .762. 5. .528. 6. .813
7. .733. S. .906. 9. .694. 10. .531. 11. .679. 12. .682.
13. .895. 14. .43751b. 16. .444. 16. .4. 17. .351.
Page 153. — 11. 1.735. 12. 2.005. 13. 8.645. 14. 24.285.
16. 7.64. 16. 30.268. 17. 25.025. 18. 2.88. 19. 3.44. 20. 4.91.
21. .761. 22. .718. 23. 1.75. 24. 4.42. 25. 5.13. 26. .325.
27. .364. 28. .237. 29. .162. 30. .454. 31. .371. 32. .454.
33. 2.25. 34. .79. 35. 2.05. 36. 2.36. 37. .415. 38. .625.
39. .32. 40. .262. 41. .028. 42. .064. 43. .025. 44. .025.
Pages 154-155. — 1. 411.1 mi. 2. 128.3 mi. ; 21.4 mi.
3. 19.3 mi. per hr. 4. 24.9 mi. 6. 11:45 a.m. 7. 39 min.
8. 40i2 mi. 9. 17.5 mi. per hr. 10. 19.4 mi. per hr. 11. 6 hr. 3 min.
Pages 155-156. — 1. 31.2 mi. 2. 12.3 mi. 3. 31.2. 4. 16.3 mi.
7. 28.2 mi. 8. 19.6 mi.
Pages 156-157. — Test I. 1. 13.113. 2. 248.16. 3. 1771.
4. 2.13. 6. 40.49. 6. 24.775. 7. 790.5. 8. .536.
Test II.— 1. 58.15. 2. 111.65. 3. 160.835. 4. .231. 6. 5.745.
6. 10.206. 7. 236. 8. .467.
Test m.— 1. 404.78. 2. 15.34. 3. 785. 4. .403. 6. 1.018.
6. 5.574. 7. 151.68. 8. 16.009.
Page 157. — 1. 46. 2. 50. 3. 48. 4. 38. 5. 45. 6. 46.
7. 46. 8. 51. 9. 46. 10. 47. 11. 55. 12. 45. 13. 50.
14. 52. 15. 51. 16. 51.
Page 158. — 1. 4483. 2. 5157. 3. 4405. 4. 4621. 5. 4909.
6. 5021.
Subtraction. — 1. 6834. 2. 2751. 3. 4107. 4. 4429. 6. 1661.
6. 3479. 7. 4617. 8. 3862. 9. 5493. 10. 4087. 11. 3959.
12. 2479. 13. 1195. 14. 5823. 15. 3434. 16. 1732. 17. 5812.
18. 1637. 19. 4772. 20. 1738. 21. 3146. 22. 5687. 23. 4475.
24. 6065.
Page 159. — Test A. 1. 6921. 2. 7152. 3. 4716. 4. 6524.
5. 7605. 6. 6132. 7. 4320.
TestB. — 1. 29,300. 2. 38,344. 3. 59,934. 4. 58,566. 5. 60,588.
6. 66,332. 7. 59,744. 8. 76,941. 9. 38,465. 10. 58,037. 11. 45,828.
12. 71,544.
Page 159. — Division. A. 85 ; 38 rem. 55 ; 55 rem. 86 ; 33 rem.
74; 31 rem. 80; 1 rem. 69; 57 rem. 90; 17 rem. B. 77; 3 rem.
55; 56 rem. 81; 17 rem. 81; 28 rem. 68; 48 rem. 86; 67 rem.
81; 37 rem. C. 84; 33 rem. 86; 48 rem. 83; 41 rem. 80; 27 rem.
67; 17 rem. 74; 39 rem. 51; 11 rem.
Page 160. — 1. 2. 8J)ii. 3. 16 min. 4. 4 hr. 5. 5 da.
6. 2i. 7. 5. 8. 81.60. 9. 30. 10. IH- H- 4 wk. 12. 10 min.
13. 30.
ANSWEES
THE STONE ARITHMETIC
SIXTH YEAR
(The page references in parentheses apply to the six-book edition)
Pages 161-163 (1-2). —4. 451b. 5. 24^ lb. 6. 141b.
7. 16 1b. 8. 15001b. 9. $2.70. 10. 990 steps.
Pages 163-163 (2-3). —17. 18. 4. 19. 5J. 20. 3|. 21. 6|.
22. 3f. 23. 24. 6|. 25. 8^. 26. 4J. 27. 5f. 28. 6|.
Page 164 (4). —1. 53|. 2. 59f. 3. 65^. 4. 79f. 5. 85^-
6. 82^. 7. 83^. 8. 128f. 9. 97^. 10. 87f. 11. 95|-
12. 95f. 13. 86|. 14. 63^. 15. 97^. 16. 87|.
Pages 164-165 (4^5). —1. f. 2. |, |. 3. f. 4. |, |. 5. f, f.
6. |. 7. |. 8. 9. f. 10. f. 11. f. 12. |. 13. f.
14. |. 15. |. 16. f. 17. f. 18. f. 19. f. 20. f. 21. f.
22. f.
Pages 165-166 (5-6).—2. I, |. 3. f. 4. f, |. 5. J.
Pages 166-167 (6-7).—2. Ralph, $1.20; Donald, $1.60. 3. Helen, $.80;
Her sister, $1.20. 5. $455. 6. 8f T. 7. H hr. 8. 4^ hr.
9. $3.90.
Pages 168-169 (8-9).—2. 102| ft. 3. 300 ft. 4. 180 ft. 5. 48 ft.
Page 169 (9). —1. f, f. 2. 10 lb. 3. 301b. 4. 201b.
5. 25 lb. grass ; 10 lb. clover. 6. 10 lb. meal ; 20 lb. oats. 7. 4 T. pea
coal ; 12 T. egg coal.
Pages 170-171 (10-11). — 1. 15. 2. 53| mi. 3. $8. 4. 8. 5. 36 yr.
6. 4hr. 7. $25. 8. 20. 9. 15. 10. 6. 11. $45. 12. $36.
13. 6. 14. 15. 15. 4fi. 16. 30.
Pages 173-174 (12-14). —5. 1.30. 6. 2.23. 7. 2.15. 8. .63.
9. .63. 10. 1.48. 11. 1.26. 12. .86. 13. .54. 17. 75%.
18. 75%. 19. 96%. 20. 86%. 21. 93%.
Pages 175-176 (15-16).—2. 72^%. 3.12^%. 5. 5^%- 6.96%.
7. 96%, 4%. 8. 9.3f%. 9. 83^%. 10. 26f%. 11. 40%.
12. 60%, 40%. 13. 20%. 14. 24%. 15. 54%.
Pages 177-178 (17-18). —8. 3f 9. 2^- 10. 2. 11. 24. 12. 3.
13. 2Î. 14. 2^. 15. 2U- 16. 34. 17. 2^. 18. 2^.
19. 341- 20. f. 21. 4. 22. 4. 23. |. 24. A- 25- i-
26. A. 27. 4. 28. 4|. 29. Ä- 30. 44. 31. Ä-
28
ANSWERS
29
Page 178(18).-!. 1^- 2. 2. 3. l^. 4. 1|. 5. 1^. 6. 1^.
7. 1^. 8. 2^. 9. If. 10. If. 11. 1^. 12. If. 13. 2f.
14. 2^. 15. lA. 16. Iff.
Pages 179-180 (19-20).—4. 11,%. 5. 12f. 6. 16f. 7. 20. 8. 18f|.
9. OA- 10. 16,%. 11. 13,%. 12. 21f. 13. 16f. 14. 16|.
15. 21. 16. 16|. 17. 18^. 18. 18. 19. 20f. 20. 33f.
21. 27i. 22. 26f. 23. 30f. 24. 33. 25. 35f. 26. 28A.
27. 30f.
Pages 180-181 (20-21).—2. 77f. 3. 87f. 4. 82|. 5. 83f. 6. lOlA.
8. 87f. 9. 83|. 10. 201f. 11. 261f. 12. 150f. 14. 51|.
15. 79f. 16. 64f. 17. 116|. 18. 11% 19. 118|. 20. 61f.
21. 115f. 22. lOlf. 23. 128f.
Pag# 183 (22).—Test! 1. f. 2. |. 3. If. 4. f. 5. f. 6. f.
7. If. 8. f. 9. If. 10. f. 11. f. 12. f. 13. ,%. 14. 1^.
15. lA- 16. lA- 17. ff. 18. lA.
Testn. — 1. 7f. 2. 6f. 3. 7f. 4. 8f. 5. 9f. 6. 9|.
7. 12f. 8. 7f. 9. 10|. 10. llf. 11. 13|. 12. 8|. 13. 12f.
Testm. — 1. If. 2. If. 3. If. 4. |. 5. If. 6. 3|.
7. 2f. 8. 3f. 9. 3f. 10. 4f. 11. 2f. 12. 5f. 13. If.
14. If.
Page 183 (23). — 2. 15 loaves ; 5 lb. ham ; 2f lb. butter. 4. 34f lb.
5. 61f. 6. 59f. 7. 45f. 8. 69f. 9. 104. 10. 90f. 11. 96f.
12. 146f. 13. 121f. 14. 114f. 15. 149f. 16. 99f.
Page 184 (24).—2. $5.41. 3. $13.25. 4. $2.48. 5. $4.81.
6. $1.33. 7. 322. 8. 632f. 9. 433f. 10. 503f. 11. 451f.
12. 309f. 13. 298f. 14. 429f. 15. 306f. 16. 352f. 17. 317f.
18. 259|.
Pages 186-187 (26-27).—20. f. 21. f. 22. |. 23. f. 24. f. 25. f.
26. f. 27. f. 28. f. 29. f. 30. A- 31. f. 33. 8|.
34. 27ff. 35. 32ff. 36. 41f. 37. llff. 38. 13ff. 39. 14A-
40. 23ff. 41. 51f. 42. 43f. 43. 14ff. 44. 48ff. 45. lA ya-
46. 190fbu. 47. If lb. 48. 6| mi. 49. 4|f lb. 50. f yd.
51. 5f mi.
14. 6|.
Page 188 (28). — 3. f.
10. A* 11* A* ^3. f.
18. f.
30
ANSWERS
59^. 3.
9. 192X.
Page 191 (31).—2.
7. If lb. 8. 8f mi.
Page 192 (32). —2.
7. 207f. 8. 336Í.
13. 91ff.
Page 192 (32).— 1. 9625.
5. 24,5334. 6. 19,800.
10. 14,325. 11. 9975.
Page 193 (33).— 5. 11
^Ib. 3
9. 86f mi.
1724.
10.
34f bu. 4. 142| mi.
10. 5flb.
5. 2flb. 6. 8|lb.
4. 84|.
125A-
43A.
2> If» If-
^ • If) ; 12f, 30 ;
2. 19,200.
7. 7675. 8.
12. 19,166|.
15: 21f; 14|; 17-^.
5. 156M-
11. 218f.
3. 11,650.
11,950.
12 1 -
7ST* ^TS f
170f
6.
12.
121f.
91-j^.
4. 8475.
9. 25,500.
6. 28f; 47|; 23f; 14|;
Page 194 (34). — 5. 17f; 20ff; 21|; 22^; 24|.
14|; 33|.
7. 2f; 13f; 8ff.
8.
83 3 1- IRl 11 *
■ Til Tj T I ■1"H»
6. 8|; 20i; 14f;
70f, 40A,
3» 2, -i^,
Page 195 (36). —6. 16^, 27f, 22^, 27#, 22«- 6. 22f, 8f, 23f, 23|, 27f.
12f, lOf, 43f, 52f, f, f, 2f, If, 6^.
8.
TSf fy ^'^y
If) 2, 2f, If, 5, If
Ï) T) "T-
Page 196 (36). — 2. 2393.6 gal. 4. 39,831 gal.
48.59 bu. 2. 1.37 lb.
6. 28.44 mi.
Pages 197-198 (37-38). — 1
4. 61.38 mi. 5. 56.75 mi.
3. 187.69 bu.
6. .375 bu.
12. .875.
18. .644.
Page 199 (39).—3. .58 ft. 4. .68 bu. 5. .78 bu.
7. .375 A. 8. .59. 9. .54 mi. 10. 3.84 mi. 11. .75.
13. .937. 14. .708. 15. .594. 16. .694. 17. .881.
19. .853. 20. .469. 21. .484. 22. .845.
Page 200 (40). — 2. 133.817. 3. 139.827. 4. 186.046. 5. 133.871.
6. 324.132. 7. G. = 134.57 bu. ; W. = 138.43 bu.; Dif. = 3.86 bu.
9. 14.887. 10. 49.685. 11. 104.625. 12. 64.146. 13. 60.881.
Page 201 (41). — 1. 58.6 mi. ; 66 mi.
3. 28.2 mi.; 67.5 mi.
404.6 mi. 2. 1:06 p.m.
5. 125.7 mi.
8. About 8:30 a.m.
Page 202 (42). —
4. 20 mi. per hr.
7. About 4:30 p.m.
10. 28 mi. per hr.
Page 203 (43). — 1. About 20 mi. per hr.
3. About 16 mi. per hr. 4. 9.6 mi. 5.
7. 22.4 mi. per hr. 8. Past Hyde Manor.
Page 206 (46). —4. 39.12. 5. 25.088.
8. 44.626. 9. 55.693. 10. 67.536.
13. 17.136. 14. 2.66. 15. 7.774. 16.
18. 210.8 mi. 19. 320.4 mi. 20.
22. 13801b.
2. 67.3 mi. ; 109.4 mi.
3. About 4 p.m.
6. 18.5 mi. per hr.
9. 23 mi. per hr
2. About 20 mi. per hr.
15.8 mi. 6. 17 mi. per hr.
9. 5:30 p.m.
6. 55.62.
11. 360.36.
3.618 bu.
$365.79.
7. 29.412.
12. 12.225.
17. 15.375 mi.
21. 57.5 1b.
ANSWERS
31
Pages 207-309 (47-49). —3. 17.1 mi. per gal. 4. $508.77.
6. 12.48 mi. per hr. 6. 38.9 mi. per hr. 8. 87.67. 9. 169.09. 10. 476.94.
11. 175.53. 12. 24.03. 13. 20.07. 14. 9. 15. 10.91. 16. 12.87.
17. 34.99. 18. 9.56. 19. 9.45. 20. 18.02. 21. .50. 22. 3.11.
Page 210 (60). — 1. 2.94 mi.; Yes. 2. Better. 3. 294.3 mi.
4. 19.3 gal. 5. 17.5 mi.
Pages 211-212 (51-52). — 3. 2.407 Ib. 4. $2.33. 5. $42.93. G. $45.38.
7. 58.591. 8. 76.634. 9. 12.152. 10. 10.566. 11. 6.253.
12. 32.46. 13. 41.59. 14. 3.958. 15. 20.109. 16. 53.47.
17. 28.2623. 18. 9.207. 19. 14.4767. 20. 11.2779. 21. 6.1365.
Page 214 (54). —3. (o) 45.424; (b) 19.693. 4. (a) 1.516; (6) 30.64;
(c) 21.275. 5. (fl) 1382.4; (6) 8.15; (c) .2924; (d) .00266.
Pages*215-216 (55-56). —1. 604.01 gal. 2. 8.351b. 3. 26.8 mi. per hr.
4. 158.1 mi. 5. 288.02 bu. 6. 706.25 Ib. 7. 150 Ib. 8. 2.47.
9. .381b. 10. .9. 11. 1.84. 12. 2.8 bu. 13. 1811.435 mi.
Page 218 (58). —1. 25. 2. 33. 3. 46. 4. 36. 5. 46. 6. 49.
7. 55. 8. 44. 9. 46. 10. 47. 11. 26. 12. 31. 13. 48.
14. 49. 15. 51. 16. 49. 17. 37. 18. 25. 19. 39. 20. 58.
Page 219 (59). —1. 4592. 2. 4090. 3. 3610. 4. 4626. 5. 4026.
6. 4682. 7. 4416. 8. 5019. 9. 3210. 10. 4299. 11. 4641.
12. 3723. 13. 3902. 14. 3965. 15. 4298. 16. 4892. 17. 3776.
18. 4253.
Page 221 (61). —1. 2889. 2. 3267. 3. 4546. 4. 5764. 5. 6342.
6. 2187. 7. 1415. 8. 3125. 9. 4153. 10. 2604. 11. 3142.
12. 6073. 13. 3647. 14. 2694. 15. 5945. 16. 4296. 17. 4949.
18. 4905. 19. 1342. 20. 3435. 21. 3591. 22. 3945. 23. 4593.
24. 4596. 25. 2405. 26. 1061. 27. 6708. 28. 2605. 29. 5595.
30. 1182. 31. 5074. 32. 4434. 33. 5371. 34. 6645. 35. 1166.
36. 5639.
Page 222 (62). — 1. $87.75.
Page 224 (64). —1. 36,864. 2. 67,065. 3. 41,662. 4. 75,932.
5. 62,342. 6. 73,140. 7. 35,908. 8. 49,474. 9. 63,825.
10. 60,696. 11. 60,291. 12. 70,848. 13. 32,832. 14. 47,127.
15. 66,470. 16. 45,288. 17. 60,043. 18. 47,042. 19. 58,820.
20. 47,232. 21. 60,965. 22. 64,032. 23. 31,104. 24. 32,524.
25. 298,890. 26. 254,503. 27. 372,812. 28. 376,244. 29. 598,968.
30. 552,636. 31. 580,678. 32. 423,255. 33. 236,402. 34. 346,610.
35. 713,592. 36. 557,282. 37. 262,515. 38. 419,608. 39. 670,273.
40. 559,364. 41. 247,923. 42. 353,328.
Page 236 (65). —1. 4825. 2. 21,875. 3. 39,450. 4. 15,600.
5. 25,500. 6. 17,075. 7. 28,450. 8. 10,575. 9. 19,800.
10. 12,800. 11. 44,750. 12. 26,200. 13. 24,125. 14. 8,400.
15. 14,583J. 16. 64,800. 17. 63,000. 18. 3000. 19. 21,900.
20. 69,750. 21. 56,400.
32
ANSWERS
Page 226 (66). —3. 81. 4. 144.75. 5. 98.1. 6. 86.4. 7. 57.9.
8. 211.5. 9. 211.05. 10. 1389.75. 11. 43.75. 12. 194.4.
13. 235.2. 14. 2625. 16. 221.975. 16. 358.2. 17. 367.2.
18. 1050. 19. 6300. 20. 3200. 21. 1050. 22. 60. 23. 6400.
24. 80. 26. 600. 26. 56. 27. 300. 28. 1.5. 29. 24.
Page 227 (67). — 1. 894|. 2. 549f. 3. 319J. 4. 432|. 6. 572^.
6. 991|. 7. 961^. 8. 937|. 9. 928f. 10. 927^. 11. 960|.
12. 949f. 13. 699f. 14. 896J. 16. 761. 16. 936|. 17. 904f.
18. 827. 19. 192.63. 20. 142.56. 21. 149.63. 22. 107.95.
23. 52.46. 24. 91.15. 26. 91.74. 26. 81.61. 27. 82.21.
28. 74.80. 29. 64.97. 30. 66.21.
Page 228 (68). —1. $4.20. 2. $157.80; $15.17. 3. $13.40. 4. $8.22;
$1.90. 6. $42.50. 6. $5.43|. 7. $60.55. 8. 29.46 bu. 9. 1431b.
10. 18.99 mi.
Page 229 (69). —2. 7.7. 3. 13.84. 4. 1.92. 6. 20.16. 6. 45.6.
7. 4.95. 8. 5.88. 9. 3.6. 10. 1.5. 11. 1.6. 12. 15.68.
13. 3.2. 14. 3.04. 16. 13.76. 16. .64. 17. .9. 18. $1.40.
19. 14 yd. 20. $.48. 21. $.18. 22. $1.20.
Pages 231-232 (71-72).—2. 7.43 bu. 3. 29.3 bu. 4. 31.99 bu.
Pages 232-233 (72-73). — 20. 8,480,000,000 lb. ; 4,240,000 T.
21. 10,070,000 T. 22. 530,000,000 bu. 23. $3,500,000,000.
24. 153,440,000 barrels. 26. $4,212,000. 26. $112,320,000.
Pages 234-235 (74-76). —6. i. 6. J. 7. f
Pages 247-249 (87-89).—2. 22.2%. 6. 64.5%. 7. 12.2%. 8. 23.3%.
9. 29.4%. 10. 30.2%. 11. 50%. 12. 26.2%. 13. 35.9%.
14. 73.9%. 16. 88.6%. 16. 73%. 17. 49.8%. 18. 28.8%.
19. 8.4%.
Page 251 (91).—21. 27%, J. 22. 49%,^. 23. 75%, f. 24. 12%, i-
26. 27%, i. 26. 49%, i. 27. 25%, i- 28. 24%, J. 29. 49%,
Pages 251-252 (91-92).— 1. 33^%. 2. 75%. 3. 25%. 4. 15%,
6. 40%. 6. 14.4%. 7. 413.3%. 8. 111.4%. 9. 337.5%.
Page 255 (96),»—2. 77.7%; 22.3%. 3. 91.7%; 8.3%. 4. 63.6% worn
6. 64.9% won.
Page 255 (96).— 1. 92%, 8%. 2. 93%, 7%. 6. 90%. 6. 75%.
7. 83%.
Page 256 (96). — 1. 80%. 2. 86%. 3. 70%. 4. 95%. 6. 70% nearest
stock ; 90% middle of ear ; 60% silk end. 6. 41.6%. 7. 60%.
Page 257 (97). —1. 62.6%. 2. 33.9%. 3. 87.5%. 4. 25.8%.
6. 16.0%. 6. 11.3%.
Pages 258-259 (98-99).—2. $11.20. 3. $14.70. 4. $7.15. 5. $9.70.
6. $43.28. 7. $9.63. 8. $4.50.
ANSWERS
33
Page 260 (100). —1. 44.8 mi. 2. 155.7 bu. 3. 39.761b. 4. 70.2 ft.
6. 104.88 gal. 6. $94.50. 7. $57.40. 8. 53.12 bu. 9. $1187.50.
10. 14501b. 11. 2345 bu. 12. 706.88. 13. 55.68. 14. 44.45.
16. 96.60. 16. 105. 17. 204. 18. 702.6.
Page 261 (101). —1. $24.25. 2. $29.20 saved; $14.60 for clothes ;
$7.30 to sister ; $7.30 for recreation. 3. $32, $64. 4. $50 saved ;
$12.50 to Ralph ; $12.50 for recreation. 6. $56.70. 6. 147 eggs.
7. 1472 bu.
Page 264 (104).—7. $57.75. 8. 270 mi. 9. 150 bu. 10. $90.
11. 3.2 ft. 12. $1800. 13. $16.50. 14. 50 bu. 15. $28.
16. 125%. 17. 180 bu. 18. 150%. 19. 6. 20. 2.47. 21. 200%.
22. 3050. 23. 100%. 24. 200%. 25. 40%.
Pages 265-266 (105-106). —1. $1232.50. 3. $23. 4. $327.25. 5. $18.
6. $16, $112. 8. 224%. 9. 25%. 10. 10%. 11. $23.80. 12. $6.83;
$1.96; $2.45. 13. First, 2.1%. 14. $2.02.
Pages 267-268 (107-108). —1. $.50. 2. $2.90. 3. $45. 4. $62.40.
5. 40%. 6. $237.50. 7. $271.50. 8. $323.95. 9. $348.
10. $11.20; $100.80.
Pages 268-269 (108-109). —1. 20%. 2. 25%ofC.; 20% of S. P. 3. $5.
4. 12^% of C. ; 10% of S. P. 5. 37^% of C. 6. $1.40 per basket.
7. 33i%ofC.; 25% of S. P. 8. 33|%ofS. P.; 50% of C.
9. 21.15%of S. P.; $132. 10. $5.00; 11^%.
Page 270 (110). —4. 183.02%. 5. 31.42%. 6. 66f%. 7. 9.24%.
8. 50%. 9. $420 to producer ; $126 to trans, co. ; $84 to wholesaler ;
$210 to retailer. 10. $412.50 to handler ; $337.50 to farmer.
Pages 271-272 (111-112).—2. 9%, $29.25. 3. Cash and carry.
4. 12%, 14.4%. 5. $54. 6. $.15.
Page 273 (113).—2. $6.77. 3. $4.95. 4. $5.02. 5. $4.97.
Page 274 (114).—2. $1.05. 3. $1.25. 4. $2.10. 5. $.60. 6. $4.90.
Pages 275-276 (115-116). —1. $75.00. 2. $87.50. 3. $112.50. 4. $175.00.
5. $255.00. 6. $165.00. 7. $162.50. 8. $190.00. 9. $111.00.
10. $123.75. 11. $100.50. 12. $91.75. 13. $136.25. 14. $165.60.
15. $255.60. 16. $96.25; $48.13. 17. $50. 19. $62.50. 20. $31.25.
21. $54.75. 22. $50.63. 23. $83.40. 24. $65.25. 25. $97.50.
26. $105. 27. $84.38. 28. $58.50. 29. $38.75. 30. $67.50.
31. $88.69. 32. $126. 33. $178.75. 34. $129.38.
Pages 377-278(117-118). —6. $240, $120. 7. $875. 8. $677.50;
$5420. 9. $7000 at 6%, $45. 10. $127.50.
Pages 379-380 (119-120). —2. $54. 3. $50. 4. $969. 6. $12.75.
7. $23.13. 8. $21. 9. $42.35. 10. $16.25. 11. $5.40. 12. $21.
13. $19.80. 14. $12.60. 15. $8.71. 17. $15.83. 18. $7.50.
19. $10.00. 20. $8.44. 21. $5.94. 22. $10.75. 23. $12.80.
24. $12.50. 25. $13.20. 26. $8.25. 27. $15.19.
34
ANSWERS
Page 281 (121).—2. $6.25, $493.75. 3. $8, $392. 4. $8.33, $491.67.
5. $10.50, $689.50. 6. $4.50, $895.50. 7. $8.00, $792.
8. $3.75, $746.25. 9. $12.67, $937.33. 10. $4.88, $645.12.
11. $4.56, $715.44. 12. $11.52, $948.48. 13. $16.10, $1133.90.
14. $18.00, $1182.00. 15. $10.00, $1490.00. 16. $19.20, $1580.80.
17. $4.80, $1195.20; $1200.
Page 282 (122). —1. 125 in. 2. 10^ ft. 3. 44 m.; 3| ft.
Pages 285-286 (125-126). —2. 120 oz. 3. 69 in. 4. 21 qt. 5. 39 pk.
6. 13 pt. 7. 206 oz. 8. 51 pt. 9. 23 ft. 10. 234 in.
11. 720 sq. in. 12. 12,096 cu. in. 13. 108 sq. ft. 14. 243 cu. ft.
15. 2400sq. rd. 17. 27 in. 18. 7 pt. 19. 8 in. 20. 140 sq. rd.
21. 1296 cu. in. 22. 34 pk. 23. 18 cu. ft. 24. 24 qt. 25. 3H in.
26. 280 rd. 28. 5 lb. 7 oz. 29. 6 ft. 4 in. 30. 3 yd. 32 in.
31. 9 qt. 1 pt. 32. 2 lb. 13 oz. 33. 2 A. 30 sq. rd. 34. SJft,
35. 44 gal. 36. 3| lb. 37. 2^1 A. 38. 2^ sq. ft.
Page 287 (127). — 2. 18 ft. 2 in. 3. 23 ft. 3 in. 4. 19 lb. 13 oz.
5. 17 gal. 1 qt. 6. 10 yd. 1 in. 7. 18 lb. 8. 11 bu. 3 pk.
Page 288 (128). — 2. 4 ft. 9 in. 3. 3 ft. 9 in. 4. 5 lb. 12 oz.
6. 6 lb. 12 oz. 6. 1 lb. 8 oz. 7. 5 in.
Page 289 (129). — 2. 20 ft. 6 in. 3. 50 lb. 10 oz. 4. 49 gal. 2 qt.
5. 40 bu. 2 pk. 7. 7| yd. 8. 22 bu. 9. 46 ft.
Page 290 (130). — 2. 1 lb. 5 oz. 3. 2 ft. II4 in. 4. 9 lb. 4^ oz.
5. 4 ft. 54 in. 6. 5 bu. 3f pk. 7. 3 gal. 14 qt. 8. 5 lb. 6| oz.
9. 3 hr. 16 min. 10. 5 gal. If qt. 11. 4 lb. 7f oz. 13. 10 in.
14. 8 ft. 74 in. 15. 2 ft. 10 in. 16. 21 plants. 17. 31 plants.
18. 34 posts.
Page 293 (133). — 15. 2250 sq. ft. 16. Same. 18. Donald, 50 sq. ft.
Pages 293-295 (133-135). —1. 10.94 sq. ft. 2. 4.69 A. 3. .61 A.
5. 106.75 bu. ; Frank. 6. Donald, 189.19 bu. ; Ralph, 188.33 bu.
7. .75 A. 8. 1.725 A.
Pages 295-296 (135-136).— 1. $48.00. 4. $108.80. 6. $8.40.
7. $3.30. 8. 5.31 A. 9. $941.62. 10. $2612.50. 11. $.20.
Page 298 (133). —5. 360 sq. ft. 6. .675 A. 7. $648. 8. 3600 sq. ft.
Page 299 (139). — 6. 54 sq. in. 7. .3 A.
Page 300 (140). — 7. 2.34 A. 8. 213.3 bu.
Pages 301-302 (141-142). — 2. 40 ft. 3. 54 ft. 4. R., 45 ft. ; C., 30 ft.
5. 8 ft.; 9.6 ft. 6. 300 ft.; 360 ft. 7. 250 ft. 8. 84 ft. 9. 28 ft.
10. 36 ft. 11. $40. 12. 4,^ ft. 13. 20 ft. 14. 314 in.
15. 20 rd.
Page 304 (144). —3. 80 bd. ft. 4. $34.62. 6. $23.40. 7. $6.91.
ANSWERS
35
Page 306 (146). — 1. 96 cu. in.
4. 324 cu. in. 6. 325 cu. ft.
8. 1134 cu. in. 9. 100 cu. ft.
216 bu. 12. 3|bu. 14. .039 gal.
17. 336 cu. ft.; 2.62 cord. 18.
20. Ralph, 38 cu. ft.
Pages 307-308 (147-148).—1. 8.73 gal. 2. 29,920 gal.
4. .502 bu. 5. 16.67 bu. 6. 873.6 bu. 7. 17.6 T.
9. 12 cd.
2. 144 cu. in. 3. 200 cu. ft.
6. 1008 cu. ft. 7. 952 cu. ft.
10. 64 cu. in. 11. 270 cu. ft. ;
15. 155.84 gal. 16. 106.6 loads.
20.57 T. 19. 3 times as much.
232.7 gal.
8. 33 T.
Page 308 (148). — 2. 2.14 in.
Page 309 (149). —Test I. 1. 4733.
6. 5^9. 6. 5554. 7. 5327.
Test II.— 1. 5829. 2. 6423. 3.
6. 6306. 7. 5994.
3. 7.29 ft. 4. Very near.
2. 4577. 3. 4089. 4. 5348.
6071.
4. 6601.
5. 5628.
Page 310 (150).—Test I. 1. 17,876,615. 2. 8,708,077.
5. 45,996,482.
9. 17,930,281.
14,638,499.
5. 4,728,951.
9. 22,667,394.
14,827,812.
54,440,622.
36,377,047.
6. 25,368,965.
10. 32,088,752.
4. 24,793,693. 5. 45,996,482. , 6.
8. 20,287,886. 9. 17,930,281. 10.
12. 23,248,344.
Test n. — 1.
4. 24,896,212.
8. 52,764,713.
12. 36,268,777.
Page 311 (151)-a. — Test I. 1. 320,775.
4. 214,165. 5. 296,194. 6. 735,590.
Testn. — 1. 804,922. 2. 5.38,552.
5. 455,421. 6. 874,965. 7. 728,712.
■ Testni. —1. 5,838,890. 2. 6,854,220.
6. 7,680,820. 6. 6,957,960. 7. 5,218,815.
3. 4,516,378.
7. 44,513,869.
11. 7,391,902.
3.
7.
11.
8.
3.
21,312,647.
54,132,552.
37,565,659.
3. 577,728.
486,772.
4. 467,861.
4. 2,976,768.
Page 311 (161)-6. — Test I. 1. 790 ; 72 rem.
847 ; 53 rem.
923 ; 68 rem,
Test n. — 1. 529 ; 23 rem.
4.
8.
667;
72 rem.
13 rem.
4. 385 ; 59 rem. 5. 385 ; 66 rem.
8. 775 ; 55 rem. 9. 883 ; 78 rem.
Test in. — 1. 784; 11 rem. 2.
4. 620 ; 21 rem. 5. 482 ; 32 rem.
8. 494 ; 55 rem. 9. 397 ; 50 rem.
Page 313 (152). — Test I. 1. 280J.
5. 355. 6. 310t2J.
2. 422,.304.
7. 561,297. 8.
3. 618,696.
521,284.
4,519,166.
8. 3,935,975.
2. 797 ; 49 rem.
5. 685 ; 67 rem. 6. 696 ; 44 rem.
9. 516; 46 rem. 10. 771; 16 rem.
2. 489 ; 40 rem. 3. 396; 42 rem.
6. 811; 19 rem. 7. 751; 25 rem.
10. 820 ; 12 rem.
836 ; 24 rem. 3. 639 ; 58 rem.
6. 301; 37 rem. 7. 303; 17 rem.
10. 489 ; 55 rem.
2. 346Î. 3. 3114. 4. 256|.
Test n. — 1.
6. 191A.
2404
2. 285|. 3. 345A- 4. 328. 5. 17Ï4.
36
ANSWERS
3. 274. 4. 58^.
18^. 10. 37A-
16. 374. 16. 47|.
Page 312 (162). —1. 19J. 2. 2
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Page 315 (166). — Test I.
1. 308.632.
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ANSWERS
37
Test IV. —1. 26.95. 2. 153.60. 3. 284.7. 4. 12.6. 5. 10^.
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10. 40.
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10. 160 bu.
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