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SPEER'S 

ARITHMETICS 



PRIMARY... 

f OR TEACHERS 



GIMN AND COMPANY 



jih-^TL., 



"L-fin n, 



REESE LIBRARY j 

OF THK 

UNIVERSITY OF CALIFORNIA. 



Vr v u 



Deceive J 



Accessions No. 



. OIKS No. 



PRIMAKY 

AKITHMETIC 



FIRST YEAR 



FOB THE USE OF TEACHERS 



BY 

WILLIAM W. SPEEK 

ASSISTANT SUPERINTENDENT OF SCHOOLS, CHICAGO 




BOSTON, U.S.A., AND LONDON 
GINN & COMPANY, PUBLISHERS 



1897 



L (6 1534- 

S 7 



COPYRIGHT, 1896 

BY WILLIAM W. SPEER 



ALL RIGHTS RESERVED 




PREFACE. 



THIS book is one of a series soon to be issued. The 
point of view from which it is written is indicated in 
the introduction. 

The essence of the theory of teaching arithmetic can 
be expressed in a few sentences. The fundamental thing 
is to induce judgments of relative magnitude. The pres- 
entation regards the fact that it is the relation of things 
that makes them what they are. The one of mathematics 
is not an individual, separated from all else, but the union 
of two like impressions : the relation of two equal magni- 
tudes. A child does not perceive this one until he sees the 
equality of two magnitudes. He will not become sensitive 
to relations of equality by handling equal units with the 
attention directed to something else, as the color, the tex- 
ture, or the how many ; nor by one or two experiences in 
comparing magnitudes. 

To aid the learner in seeing a 1 as the relation of two 
equal units, a 2 as the relation of a unit to another one half 
as large, a % as the relation of a unit to another twice as 
large, we must induce the repeated acts of comparing which 
bring these relations vividly before the mind. With this 
purpose the child is not required to build out of parts a 
whole which he has never seen, nor expected to discover 
a relation in the absence of one of the related terms. He 
does not begin with elements. He is not prevented from 



IV PREFACE. 

seeing things as they are by pushing elements into the 
foreground. The mind grasps something vaguely as a 
whole, moves from this to the parts, and gradually advances 
to a clearer and fuller idea of the whole. Whether the 
object of study be a flower, a picture, a cubic foot, or a six, 
the process of learning is the same. If we promote prog- 
ress in the discovery of relations of magnitude, we will 
make it possible for the compared wholes to be pictured in 
their full extent, thus affording opportunity for comparing, 
for activity in judging. There is no such opportunity 
when a child who has no idea of a thing constructs it 
mechanically from given parts. Creation, in any subject, 
requires a basis in elementary ideas. 

It is not to be forgotten that there is a wide difference 
between seeing that the relation of two particular things is 
8, and realizing 8 as a relation, realizing it in such a way 
that it can be freely used without misapplying it. 

There is no real progress unless the mind is gradually 
gaining power to think of things not present to sense, and 
to think of a relation apart from a particular thing. But 
there is no way to promote this progress except by securing 
continued activity of sense and mind. The child grows 
into the idea 8, slowly and unconsciously but surely, under 
right conditions. A cube does not become known by count- 
ing surfaces, edges, etc., again and again, but by observing 
other forms and many different cubes. Through repeated 
acts of dissociating and relating, what is particular sinks 
out of sight and the common trait stands out. This 
principle is of general application. 

There should be constant calls for reperception, for 

judging and verifying. Only by multiplying experiences 

in the concrete, by noting the same relation in many dif- 

\ ferent things and in many different conditions, does the 

child come to know a relation as it is. 



PREFACE. Y 

The slow development of the power to form perfectly 
quantitative judgments is considered. Hence the earlier 
work makes no demand for close analysis. It provides for 
a gradual advance toward exactness. The exercises are 
only suggestive. The condition of the child determines 
what he should do. But in any case, the work in the 
beginning should be so simple that it can be done easily ; 
it should look to the free action of both body and mind. 

The child interested in finding colors and forms wishes 
to move about, to touch and handle things. Out of school 
he combines thinking and acting. Why should he not do 
so in the school ? Interest will lead the child to control 
himself, but repression from without induces dullness, 
indifference, and antagonism. Force a child to preserve a 
regulation attitude, to keep his nerves tense, and you destroy 
the foundation of healthful mental activity. In the transi- 
tion from home to school life, careful provision should be 
made for the whole child to express himself. 

Attention is asked to the remarks upon over-direction, 
premature questioning, demands for analysis beyond the 
inclination and power of the pupil and for outward forms 
which are not the genuine expression of the child. 

Great importance is attached to that order of work which 
puts things before the pupil and leaves him free to see and 
to tell all he can before interfering with his action by ques- 
tioning or direction. Questions have their uses. They 
serve to arouse attention, to aid in testing the pupil's view, 
and may lead to the correct use of new forms of expression. 
But there are effects of questioning which are too often 
overlooked. Questions do for the child what he should do 
for himself ; they conceal his attitude toward the work 
and prevent your seeing what he would do unaided. They 
call attention to details for which the mind may not be 
prepared, and present a partial, fragmentary view. The 



VI PREFACE. 

questioning may be logical, but the learner connects only 
that which he himself relates. Questions cause the teacher 
to suppose that the child grasps what is not appreciable by 
him, and so prevent the adaptation of the work. To 
attempt to force through questions what you see in a 
poem, picture, or problem, instead of leaving the pupil to 
discover what he is prepared to see, is to ignore the true 
basis of advance, to disregard the law that the mind passes 
from vague ideas to those fuller and more exact, only 
through its own acts of analysis and synthesis. Free 
work reveals the pupil and makes it possible to meet his 
needs. 

This view furnishes no excuse for random, desultory 
work. The teacher must carefully select the means, whether 
the ideas into which he wishes to lead the child are mathe- 
matical, biological, or historical. 

In conclusion it is urged that any success is dangerous 
which lessens the susceptibility of the mind to new impres- 
sions. We may be so successful in training the child to 
reproduce as to destroy his power to produce. Progress is 
impossible without growing power to do unconsciously what 
was at first done consciously ; but accuracy is not to be 
desired at the expense of growth. The purpose of auto- 
matic action in education is not to restrict, but to set force 
free. When the work of the school is mechanical it 
weakens the relating power, the power to act in new cir- 
cumstances, and thus lowers the child in the scale of 
being. 

As insight into the subject and contact with the child 
enable us to open right channels for free action, there 
will be little occasion for drills. The fresh, vigorous effort 
of involuntary attention carries the child forward with sur- 
prising rapidity. Out of se(/*-activity comes the self-control 
which gives strength to persist. 




THEOET OF AEITHMETIO. 



DSTTKODUCTIOK 

THE following quotations may be found sugges- 
tive of working ideals. The teacher who enters 
into their spirit will feel the need of knowing both 
the child and the subject. She will see that 
attention is a condition of thinking, and interest 
a condition of attention ; that the mind is one 
and indivisible, and must be so treated if we would 
strengthen it. The mental as well as the physical 
nature is under law. When our teaching is in 
accord with this law, we shall find the forces of 
nature working for us ; the child will become 
strong with the strength of nature. 

Apprehension by the senses supplies, directly or 
indirectly, the material of all human knowledge ; or, 
at least, the stimulus necessary to develop every inborn 
faculty of the mind. Helmholtz. 

The products of the senses, especially those of sight, 
hearing, and touch, form the basis of all the higher 
thought processes. Hence the importance of developing 
accurate sense concepts. . . . The purpose of objective 



2 THEORY OF ARITHMETIC. 

thinking is to enable the mind to think without the help 
of objects. Thomas M. Balliet. 

The understanding must begin by saturating itself 
with facts and realities. . . . Besides, we only under- 
stand that which is already within us. To understand 
is to possess the thing understood, first by sympathy 
and then by intelligence. Instead of first dismember- 
ing and dissecting the object to be conceived, we should 
begin by laying hold of it in its ensemble. The pro- 
cedure is the same, whether we study a watch or a 
plant, a work of art or a character. Amiel. 

The action of the mind in the acquisition of knowl- 
edge of any sort is synthetic-analytic ; that is, uniting 
and separating. These are the two sides, or aspects, of 
the one process. . . . There is no such thing as a syn- 
thetic activity that is not accompanied by the analytic ; 
and there is no analytic activity that is not accompanied 
by the synthetic. Children cannot be taught to perform 
these knowing acts. It is the nature of the mind to so 
act when it acts at all. George P. Brown. 

Our children will attain to a far more fundamental 
insight into language, if we, when teaching them, con- 
nect the words more with the actual perception of the 
thing and the object. . . . Our language would then 
again become a true language of life, that is, born of 
life and producing life. FroebeL 

Voluntary attention is a habit, an imitation of natural 
attention, which is its starting-point and its basis. . . . 



INTRODUCTION. 3 

Attention creates nothing ; and if the brain is barren, 
if the associations are meagre, it functions in vain. 
Eibot. 

How, indeed, can there be a response within to the 
impression from without when there is nothing within 
that is in relation of congenial vibration with that 
which is without? Inattention in such case is insus- 
ceptibility ; and if this be complete, then to demand 
attention is very much like demanding of the eye that 
it should attend to sound-waves, and of the ear that it 
should attend to light-waves. Dr. Maudsley. 

Activity bears fruit in habit, and the kind of activity 
determines the quality of the habit. Alex JEJ. Frye. 

If a teacher is full of his subject, and can induce 
enthusiasm in his pupils ; if his facts are concrete and 
naturally connected, the amount of material that an 
average child can assimilate without injury is as aston- 
ishing as is the little that will fag him if it is a trifle 
above or below or remote from him, or taught dully or 
incoherently. 6r. Stanley Hall. 

Is it not evident, that if the child is at any epoch of 
his long period of helplessness inured into any habit 
or fixed form of activity belonging to a lower stage of 
development, the tendency will be to arrest growth 
at that standpoint and make it difficult or next to 
impossible to continue the growth of the child ? Wil- 
liam T. Harris. 



4 THEORY OF ARITHMETIC. 

We must make practice in thinking, or, in other 
words, the strengthening of reasoning power, the 
constant object of all teaching from infancy to 
adult age, no matter what may be the subject of 
instruction. . . . Effective training of the reasoning 
powers cannot be secured simply by choosing this 
subject or that for study. The method of study and 
the aim in studying are the all-important things. 
Charles W. Eliot. 

Intellectual evolution is, under all its aspects, a 
progress in representativeness of thought. Herbert 
Spencer. 

Consciousness implies perpetual discrimination, or 
the recognition of likenesses and differences, and this 
is impossible unless impressions persist long enough to 
be compared with one another. . . . Impressions persist 
long enough to be compared together, and accordingly 
there is reason. John Fiske. 

Thinking is discerning relations ; but we discern the 
relations of things. In order to discern relations we 
must compare ; hence, our powers to think are our 
comparative powers. These are our faculties to discern 
relations. Dr. M' Cosh. 

Thought consists in the establishment of relations. 
There can be no relation established, and therefore no 
thought framed when one of the related terms is absent 
from consciousness. Herbert Spencer. 



INTRODUCTION. 5 

Intelligence is virtually a correct classification. 
Dr. Maudsley. 

The thing is its relations, and although analytically 
we may separate them, attending now to this relation, 
now to that, we must never imagine the separation to 
be real. Gr. H. Lewes. 

All knowledge results from the establishment of 
relations between phenomena. J. B. Stallo. 

Every act of judgment is an attempt to reduce to 
unity two cognitions. Sir William Thomson. 

The primary element of all thought is a judgment 
which arises from a comparison. Francis Bow en. 

There is no enlargement of the mind unless there 
be a comparison of ideas one with another. Cardinal 
Newman. 

The extent or magnitude of a quantity is, therefore, 
purely relative, and hence we can form no idea of it 
except by the aid of comparison. Davies. 

Of absolute magnitude we can frame no conception. 
All magnitudes as known to us are thought of as equal 
to, greater than, or less than, certain other magnitudes. 
Herbert Spencer. 

Those who accept the above can hardly agree 
with the prevailing practices in the teaching of 
arithmetic. 



6 THEORY OF ARITHMETIC. 

It is hoped that the following brief presenta- 
tion of mathematics as the science of relative 
magnitude will aid teachers in bringing mathe- 
matical teaching into accord with educational 
principles. 



MATHEMATICS, DEFINITE KELATIONS. 

Mental advance from the vague to the definite. - 

Teaching which meets the needs of the developing 
mind must be successful. No other can be. 

The marvelous progress of the child during the 
first five or six years of its life is largely due to 
free action and spontaneous attention ; to the 
absence of demands unfavorable to growth. 

We recognize the incapacity of the infant. We 
watch and minister to its growth by creating an 
environment fitted for calling forth its activities. 
So should we acquaint ourselves with the mental 
state of the child, as shown in his work, his play, 
his questions ; in what he hears and sees ; in what 
he does and in what he tries to do ; in what he 
says and in what he does not say. From the 
basis of his experience and power our training 
should proceed. 

Complex conceptions cannot be imposed upon a 
mind incapable of receiving them; neither can 
simple truths. Nothing is self-evident save to 
him who sees it. The child no more knows that 
things equal to the same thing are equal to each 
other, until he sees it to be so, than he knows 
that yellow and blue make green. He sees only 
that which he has the power to see. 



8 THEORY OF ARITHMETIC. 

The change from the helplessness of the babe 
to the power of the child of six is a constant 
miracle; but its powers are still relatively feeble. 

Between the capacity for vague perceptions 
and for framing definite mathematical ideas there 
are many intervening stages. 1 The natural ap- 
proach to each higher thought-product is through 
the lower one, which is its necessary antecedent. 

The perception of equality is the basis of 
mathematical reasoning, a condition of definite 
thinking. But a child sees things as longer or 
shorter, larger or smaller, before he is able to see 
their perfect equality or exact degree of inequality. 2 
Until, without effort, he makes such discrimina- 
tions as are expressed by the terms long, short, 
large, small, etc., he is not ready to make the 
discriminations expressed by twice, three times, 
i or i. 

Analysis dependent upon representative power. 

Exact quantitative relations cannot be estab- 
lished without analyzing. Analysis fixes the at- 

1 " In early life the cerebral organization is incomplete. The 
period necessary for completion varies with the race and with the 
individual." Prof. Tyndall. 

2 " The conception of exact likeness," remarks Mr. John Fiske, 
" is a highly abstract conception, which can only be framed after 
the comparison of numerous represented cases in which degree of 
likeness is the common trait that is thought about." Cosmic 
Philosophy, vol. ii. p. 316. 




MATHEMATICS, DEFI 

tention in turn upon each part rather than upon the 
relation of the compared wholes. When the pupil 
enters upon the process of exact comparison he 
should be able to hold each term of the comparison 
so firmly that the necessary intrusion of a common 
measure will not efface either of them. Other- 
wise, the operations intended to throw into relief 
the precise relation of the magnitudes interpose 
as a cloud to render the relation invisible. 

Place a measure in the hands of a pupil and set 
him to marking off spaces on this and that and 
counting them before he is ready for such work, 
before anything has been done to induce the habit 
of looking from one magnitude to another, and 
you absorb him in a mechanical process which 
turns the thought from the relational element 
with which mathematics deals. He may write, 
" The door is 8 feet high," when he has simply 
counted 8 spaces. But he has made no mathe- 
matical comparison, observed no relation, done 
little which tends to develop power to think. 
If we ask him to find exact relations before he 
has sufficient representative power to bring each 
term of the comparison into consciousness and 
approximate its relations unaided, the probability 
is that the relation of the magnitudes as wholes 
will not be seen at all. 

Premature attempts to initiate the pupil into 
the ideas of mathematics will bewilder him with 



10 THEORY OF ARITHMETIC. 

the mechanism of the subject and create a condi- 
tion unfavorable to the perception of mathemati- 
cal or any other truth. 

"Not only is it true/' says Herbert Spencer, 
"that in the course of civilization qualitative 
reasoning precedes quantitative reasoning; not 
only is it true that in the growth of the individual 
mind the progress must be through the qualitative 
to the quantitative, but it is also true that every 
act of quantitative reasoning is qualitative in its 
initial stages" 

Unity of subject. The teacher must be clear 
as to what characterizes a science. Otherwise the 
essential may be lost sight of in the subordinate, 
and the energy of the pupil wasted in the effort to 
unite what should never have been separated. 

A living apprehension of the fact. that mathe- 
matics deals with definite relations of magnitude 
suggests the mode of beginning the study. It 
suggests the need of creating definite ideas ; it 
forbids presenting things as isolated, independent, 





absolute in themselves. It does away with arti- 
ficial distinctions between a fraction and an in- 
teger, by presenting each as a relation. Thus 2 is 



MATHEMATICS, DEFINITE RELATIONS. 11 

the relation of a unit to another half as large ; 
and one half is the relation of a unit to another 
twice as large. 

A relation the result of a comparison. To be 

conscious of a relation means more than to be 7 
conscious of the terms between which it exists. 
We may think of the taste of an orange or of a 
pear without connecting them in any way, but if 
we are considering their relative sweetness we 
must bring together in thought the taste of each ; 
a comparison must take place before we can assert 
that one is sweeter than the other. So we may 
think of a certain line as one yard, of another as 
six inches, without ability to assert their relative 
magnitude. We may go further and note the fact 
that in one yard there are six six -inches, and still 
remain without any appreciation of their relative 
magnitude. Before we can assert this, the intel- 
lectual act which brings the shorter line before 
the mind as equal to one sixth of the longer must 
take place. 

We cannot meet the demands of mathematics 
by observing things simply as distinct and sepa- 
rate. If relations are to come into consciousness, 
the comparing which brings them there must take 
place. An example may make this more clear. 
Suppose the magnitudes a, &, c, and rf, to be before 
the child. He notes likenesses and differences in 



12 THEORY OF ARITHMETIC. 

them just as he does in colors, leaves, fruit, or 
anything to which he attends. Noting d and a he 
sees that d is greater than a, that a is less than d. 







He has made comparisons and established rela- 
tions, but not exact relations. These relations he 
expresses by the indefinite words greater and less. 
If, by measuring, he effects an exact comparison 
of d and a, he needs language for stating that the 
relation of d to a is 3 ; the relation of a to d is J. 
He may call a | and d 1, or d 3 and a 1 ; or he 
may call d 12 and a 4, but their relation remains 
unchanged. 

The thing is its relations. Comparing c with 
a considered as 1, we call c 2. Comparing c with 
d, c becomes f, yet the magnitude c has not 
changed. The a which we dealt with as ^ when 
thought of in relation to d, as ^ in relation to c, 
we call 1 when compared with &, or with any 
other equal magnitude. 



MATHEMATICS, DEFINITE RELATIONS. 13 

Just as the child learned to know a line as long 
in comparison with another, short in comparison 
with a third ; to call a day warm or cold accord- 
ing to that with which it is compared, so he should 
learn to know a magnitude as 2 when seen in rela- 
tion to a magnitude equal to its ^ ; to see the same 
magnitude as 3 or 5 or ^ when compared with 
other magnitudes. 

Means of comparing. Effecting an exact com- 
parison requires analysis and synthesis,' just as 
every act does which results in a judgment. 



In order to discover the relation of 4 to 6, we 
may separate the 4 and the 6 each into 2's. By 
the analysis (subtraction or division) we find 3 
2's in the 6 and 2- 2's in the 4. Since 2 is ^ of 
3- 2's, we infer that 2 is of 6. (Why?) In 
order to make such an inference we must see that 
3* 2's equal 6, synthesis (addition or multiplica- 
tion). 

From successive relations of equality we pass to 
the final act of relating, which brings 2- 2's, or 4, 
before the mind as equal to f of 6. The final 
thought is not of the 4 nor of the 6, nor of the 
relation of the measuring unit to either ; but of 
the relation of the 4 to the 6. In no case have 
we established the relation sought until the com- 



14 



THEORY OF ARITHMETIC. 



pared wholes are brought into consciousness in 
that relation. 

Again, suppose we wish a child to discover the 
relation which exists between 7 and the sum of 
3 and 4. 



Mental acts must take place showing him in 
the 7 two magnitudes, one equal to 4, the other 
equal to 3, and bringing the 7 again before the 
mind as one quantity. But we must pass be- 
yond these steps and bring the seven and the 
sum of 3 and 4 before the mind in the relation 
of equality. A judgment of relative magnitude 
must be formed which unites the compared 
terms. 

Each of these judgments, like every other judg- 
ment, is the product of analysis and synthesis 
of separating and uniting ; of subtracting and 
adding ; of dividing and multiplying. There is 
no real synthesis without analysis, no addition 



MATHEMATICS, DEFINITE RELATIONS. 



15 



without subtraction, no multiplication without 
division. 

Conditions of comparing. 
In comparing there must be 
ideas to compare. In present- 
ing the magnitude 7, as well as 
3 and 4, we are merely meet- 
ing that condition of thinking 
" which requires that, in estab- 
lishing a relation, each of the 
compared terms must be pre- 
sent in consciousness. Through 
this comparison the pupil learns to know 7 in one 
relation ; through other comparisons he will enter 
into fuller knowledge of it. 

Meaning of a word depends upon experience. 

When the need of a name arises, give it. The 
principle is the same whether dealing with the 
qualitative or the quantitative. We do not leave 
the child without the name water because he does 
not know the elements of water. We tell him a 
certain object is a chair long before he has a com- 
plete idea of it. As his surroundings produce 
activity, he gradually comes to know special 
features of the chair, and to distinguish arm- 
chairs, rocking-chairs, etc. 

The name alone can avail nothing ; l but when 

1 Language attains definiteness for the individual only as it is 
associated with definite ideas. The square is a definite figure ; 



16 THEOBY OF AEITHMETIC. 

it will be serviceable in focusing the attention, in 
aiding the child to retain his grasp of a thing, and 
thus in facilitating his investigations, it should be 
freely given. 

Thought and expression are inseparable. Words 
without ideas are dead ; images without words 
are elusive. The most effective method of mas- 
tering the means of expression in mathematics, 
or in any other subject, is the exercise of the ' 
mind upon realities, in mathematics, the real- 
ities are the relations of magnitude. 

Using divided magnitudes obscures wholes, weak- 
ens sense of coexistence. By presenting divided 
magnitudes (see n below), we destroy the wholes 
we wish compared, and call upon the child for 
a synthesis for which he is not prepared. The 
problem does not require him to make a com- 
parison of the magnitudes, but merely to count the 
how many. We force upon the attention isolated 
units and operations for which the mind has no 
need, and which, by being thus pushed into the 

but the child may handle many squares and repeat the definition 
of a square many times without any feeling of its definiteness. If 
we taught the child to say that the sum of 3 and 4 = 7, without 
his mentally seeing it to be so, we should be presenting symbols 
without significance. To refuse to give the name 7 to the magni- 
tude in this particular relation, because the learner is not fully 
conscious of the meaning of the term, is as if we refused to allow 
a child to talk of a star because his idea of it is not that of the 
astronomer. 



MATHEMATICS, DEFINITE RELATIONS. 17 

foreground, tend only to , intellectual chaos. A 
synthesis not accompanied by analysis must be 
artificial. There can be no real synthesis without 
analysis. 




In observing n (the divided magnitude) does 
the child consider the relative magnitude of the 
units or the how many? In comparing c (the 
undivided magnitude) with d, what receives the 
primary attention, the how many or the relative 
size ? 

The child grasps a dollar or a dozen as a unit, 
untroubled by its composition. So it should grasp 
a 12, a 17, a 100, a J, or a 7. So it will if you 
bring them into consciousness as wholes. 1 If you 
wish a pupil to note the relation between the 
length and the width of a desk ; or between a 12 
and a 3 ; a 1,200 and a 300 ; a 68 and a 17; or 

1 " Now, the fact is, that all objects of apprehension, including 
all data of sense, are in themselves, i.e. within the act of apprehen- 
sion, essentially continuous. They become discrete only by being 
subjected, arbitrarily or necessarily, to several acts of apprehension, 
and by thus being severed into parts, or coordinated with other 
objects similarly apprehended into wholes." J. B. Stallo, 



18 THEORY OF ARITHMETIC. 

a |- and a |, what are, in each case, the wholes to 
which you wish him to attend ? 

If, instead of bringing the terms of the compari- 
son before the mind as related wholes, we require 
the learner to begin by constructing them from 
the parts, 1 we destroy for him the continuity of 
the magnitudes. Consciousness is occupied with 
a succession of separate units, and but a vague 
sense of the relations of the given magnitudes is 
awakened. 

Undivided magnitudes; use induces analysis and 
synthesis. It must not be supposed that the 
mere use of undivided magnitudes will insure 
the perception of mathematical relations; 2 but it 
fosters such perception. It is a condition of pres- 
entation in accord with the familiar fact that the 

1 " Where the parts of an object have already been discerned, 
and each made the subject of a special discriminative act, we can 
with difficulty feel the object again in its pristine unity ; and so 
prominent may our consciousness of its composition be, that we 
may hardly believe that it ever could have appeared undivided. 
But this is an erroneous view, the undeniable fact being that any 
number of impressions, from any number of sensory sources, falling 
simultaneously on a mind WHICH HAS NOT YET EXPERIENCED THEM 
SEPARATELY, will fuse into a single undivided object for that mind." 
Wm. James. 

2 The material provided for mental nutrition is most important. 
But there is danger of relying too exclusively upon special methods 
and intrinsic values. Undue reliance upon any means or subject 
may blind us to the fact that the educational process is not going 
on at all. 



MATHEMATICS, DEFINITE RELATIONS. 19 

mind moves from the whole to the part and back 
again to the whole; that it analyzes through a 
desire for more intimate knowledge, in order that 
it may reach a better synthesis. We should present 
as wholes the magnitudes whose relations we wish 
established, and leave the way open for those suc- 
cessive acts of analysis and synthesis by which 
such relations are established. 

Freeing the mind from the concrete. Noting the 
same relation between many different magnitudes 
tends to free the mind from the concrete and the 
particular, and to make the relations the objects 
of thought. 

Thus the pupil sees magnitudes differing greatly 




in size, but discovers that 2 is the relation not 
only of c to rf, but of x to y, of a to &, of o to 
m, and of e to n\ he notes the unlikeness of the 
separate pairs, the likeness of their relations; he 



20 THEORY OF ARITHMETIC. 

is asked for inference after inference which turns 
attention to the ratio of the units. 1 Gradually 
he learns to know magnitudes in the only 
way that they can be known, in relation. 
The simple ratios of mathematics become real to 
him. 

Giving varying names to the units, as a 12 and 
a 6 ? a | and a J, a 100 and a 50, aids in separat- 
ing accidental from essential relations, and in 
preventing the error of mistaking the relative for 
the absolute. 

Through many, very many experiences, fitted 
for developing the power, he becomes able to dis- 
sociate the relation from the thing, and to deal 
with the 2, the 3, the , the |, etc., as uniform 
relations upon which far-reaching inferences may 
be based. 2 

1 " The higher processes of mind in mathematics lie at the very 
foundation of the subject." Sylvester. 

2 " The peculiarity of abstract conceptions is that the matter of 
thought is no longer any one object, or any one action, but a trait 
common to many ; and it is, therefore, only when a number of 
distinct objects or relations possessing some common trait can be 
represented in consciousness, that there becomes possible that 
comparison which results in the abstraction of the common trait 
as the object of thought." John Fiske. 

" The development of ideas is the slow, gradual result of contin- 
uous judgment." Francis Bowen. 

" What is associated now with one thing and now with another 
tends to become dissociated from either, and to grow into an 
object of abstract contemplation by the mind." Wm. James. 



MATHEMATICS, DEFINITE RELATIONS. 21 

Inference must succeed perception. The import- 
ance of bringing simple basic ratios definitely into 
consciousness is better understood when we look 
beyond them. 1 

The development of mathematics within the 
mind, and the development of the mind by means 
of mathematics, are alike impossible without that 
thinking, relating, reasoning, by which the mind 
"produces from what it receives." From the 
beginning we must address the mind and not one 
function ; give opportunity for inference to suc- 
ceed perception. By unduly crowding the sensing 
and recording of ratios we may so handicap the 
mind that it cannot move. As law or principle 
serves the man of science, so each simple truth 
should serve the child in lighting the way to 
other truths. 

By means of perceived relations we must pass 
to the inferring of relations. For example, it is 
not enough for the child to see the relation of d to 
a and of a to d. From these perceptions he must 
infer other relations. Rightly taught, such infer- 
ences as that the weight of d equals 3 times the 

1 " And if we neglect to educe the fundamental conception 
on which all his ulterior knowledge must depend we not only 
sow the seed of endless obscurity and perplexity during all his 
future advance in this science, but we also weaken his reasoning 
habits . . . and thus make our mathematical discipline produce, 
not a wholesome and invigorating but a deleterious and pervert- 
ing effect upon the mind." Whewell. 



22 THEORY OF ARITHMETIC. 

weight of a ; that 3 times as many inch cubes can 
be cut from d as from a ; that d will yield 3 times 
as much ashes as a; that the cost of d equals 3 
times the cost of a ; that the weight of a equals ^ 
the weight of d\ that a will yield \. the amount of 





ashes that d will yield ; that the cost of a equals \ 
the cost of d, will follow naturally and readily upon 
the perception of the ratio of d to a and of a to 
d. They will never follow without the generating 
conditions, and the generating conditions are per- 
ceptions of exact relations. Upon these equations, 
made known by the activity of the mind upon the 
magnitudes themselves, all mathematical deduc- 
tion depends. 

We frequently hear it said, " Is it not a proof 
that the child sees the conditions when he says that 
d will cost 3 x if a cost x ? " Or, if this is not 
enough, he can tell you that, " Because d is 3 
times a, etc." Experience shows, however, that 
many pupils who can do all this will tell you 
a little later that it will take 3 boys 3 times as 
long to do a piece of work as it will one boy ; 



MATHEMATICS, DEFINITE RELATIONS. 23 

that the weight of a 2 -inch- cube is twice that of 
an inch cube. As they advance, their seeming 
inaptitude for mathematics becomes more marked. 
Why is this ? Because a wrong direction was 
given to the mind in the beginning; because 
mechanical processes for securing results were 
substituted for those experiences which create 
ideas of equality and of exact ratio ; because 
using objects merely to teach children to count 
and to manipulate numbers, 1 instead of presenting 
them in such a way as to attract attention to 
their relative magnitude, leaves the mind without 
any basis for the deductions which are demanded. 
Out of number nothing comes save number. If 
we ask for conclusions concerning quantity we 
must see that the mind possesses a basis 2 for those 
conclusions. The indispensable groundwork of 
reasoning is the definite mental representation of 
the relation upon which an inference rests; and 
mathematical inferences rest upon ratios. 

Clear imaging ; clear thinking ; correct conclusion. 

The material upon which the mind can act 
from time to time depends upon its growing 

1 " It would indicate a radically false idea of number to wish 
to employ it in establishing the elementary foundations of any 
science whatever ; for on what would the reasoning in such an 
operation repose ? " Comte. 

2 " The attempt to found the science of quantity upon the 
science of number I believe to be radically wrong and educa- 
tionally mischievous." Win. K. Clifford. 




24 THEORY OF ARITHMETIC. 

power to represent in- thought the conditions upon 
which conclusions follow. 

Pupils accept the statement that it will take 
twice as long to paint a 2-inch square as a 1-inch 
square because they do not represent the squares 
mentally. 

If the pupil has been trained so that it is his 
habit to make the necessary mental representa- 
tions he will see for himself that if x is the num- 
ber of yards of carpet 1J yards wide required for 
a floor, 2 x yards f of a yard wide will be needed. 
No wordy explanation will be required. Yet pupils 
fail constantly in such simple exercises. They 
cannot make comparisons, because they have in 
their minds no images of the things they are to 
compare. They cannot deduce from symbols the 
relations of reals. 1 Asking pupils to reason about 
things which they do not see mentally, is asking 
the impossible and can only lead to confusion and 
discouragement. The power of representative 
thought, of imaging, underlies all intellectual 
progress, and we cannot prepare the mind for 
abstract thought without developing this power. 

Mathematics deals with realities. However 
divergent may be the lines of mathematical 

1 " How accurate soever the logical process may be, if our first 
principles be rashly assumed, or if our terms be indefinite or 
ambiguous, there is no absurdity so great that we may not be 
brought to adopt it." Dugald Stewart. 



MATHEMATICS, DEFINITE KELATIONS. 25 

thought, their beginnings are sensible intuitions, 
that is, the ideas of magnitude must be based 
on perceptions; and however long the line, its 
extension is in all cases by means of successive 
acts of comparing and inferring. 

Sylvester finds that " The study of mathematics 
is unceasingly calling forth the faculties of obser- 
vation and comparison ; that it has frequent 
recourse to experimental trial and verification; 
and that it affords a boundless scope for the 
highest efforts of imagination. ... I might go 
on," he says, " piling instance upon instance to 
show the paramount importance of the faculty of 
observation to the process of mathematical dis- 
covery." 

" Mathematics," says Mr. Lewes, " is a science 
of observation, dealing with reals, precisely as all 
other sciences deal with reals. It would be easy 
to show that its method is the same." The reals 
are the relations of magnitude. 

The order of truth changes ; the mental action 
which embraces it remains the same. We note 
the likenesses of two leaves or the exact likenesses 
of two magnitudes ; in each case we have a basis 
for inference obtained by comparing. When we 
turn to exact likenesses, we enter the domain of 
mathematics. 

Objects unfitted to awaken mathematical ideas. 
Were we concerned simply with the number of 



26 THEORY OF ARITHMETIC. 

things, beans, shoe-pegs, shells, leaves, pebbles, 
chairs, or the legs of frogs might serve as well 
as anything. But mere numerical equality will 
not serve as a basis for mathematical reasoning ; 
-exact results cannot be founded upon it. 

Dealing with units, without regard to their 
equality or inequality ; considering them only as 
distinct things; and reaching results true only 
numerically, has been called the indefinite calculus; 
but the indefinite calculus furnishes no basis for 
mathematical reasoning. If arithmetic is made 
merely a means of teaching number, and opera- 
tions with number, it should receive but brief 
time in the common-school course. Very little of 
it will suffice for the ordinary vocations of life. 
The cases in which mere numerical relations are 
considered are so simple as scarcely to stir the 
mind. 1 

A superficial knowledge of mathematics may 
lead to the belief that this subject can be taught 
incidentally, and that exercises akin to counting 
the petals of a flower or the legs of a grasshopper 
are mathematical. Such work ignores the funda- 
mental idea out of which quantitative reasoning 

1 In regard to the how many, to work which does not deal 
with definite relations, Comte said, " This will never be more 
than a point, so to speak, in comparison with the establishment 
of relations of magnitude of which mathematical science essen- 
tially consists. ... In this point of view, arithmetic would 
disappear as a distinct section in the whole body of mathematics." 



MATHEMATICS, DEFINITE RELATIONS. 27 

grows the equality of magnitudes. 1 It leaves 
the pupil unaware of that relativity which is the 
essence of mathematical science. Numerical state- 
ments are frequently required in the study of 
natural history, but to repeat these as a drill upon 
numbers will scarcely lend charm to these studies, 
and certainly will not result in mathematical 
knowledge. 

Vague ideas of the unlikeness of a rhomboid, 
a square, and a trapezium may be gained by count- 
ing them, and so may vague ideas of the relations 
of magnitude. If definite ideas of color, form, or 
weight come from counting and learning tables, 
then definite ideas of quantitative relations may 
come in the same way. 

Turning from the numbering of things to their 
mathematical comparison, we see at once why 
plants and animals are not well adapted for our 
purpose. In them, that which is material is 
obscured by that which is irrelevant. 2 It is diffi- 
cult for the undeveloped mind to view these objects 

1 " Equations constitute the true starting point of arithmetic." 
Comte. 

" The fundamental ideas underlying all mathematics is that of 
equality." Herbert Spencer. 

2 " The visible figures by which principles are illustrated 
should, so far as possible, have no accessories. They should be 
magnitudes pure and simple, so that the thought of the pupil may 
not be distracted, and that he may know what feature of the 
thing represented he is to pay attention to." Committee of 
Ten." 



28 THEORY OF ARITHMETIC. 

in their mathematical aspect. Their differences 
in magnitude are not easily appreciated by the 
senses. Their exact measurement is not easy. 
They lend themselves to accurate imaging far less 
readily than simple magnitudes, and do not result 
in those mental states which would be created 
were mathematical relations brought conspicu- 
ously and impressively into the pupils' experiences, 
That mathematics enters into other sciences is 
understood. The fruitfulness of physics for the 
teacher of mathematics is apparent. Advancing 
science is constantly making more clear the inter- 
dependencies of the various sciences. Each aids 
in the development of the others. 1 But it does 

1 " Although each science throws its light on every other, 
owing to the interdependence of phenomena and the community 
of consciousness, yet . . . phenomena are independent not less 
than interdependent. Mathematics cannot receive laws from 
chemistry, nor physics from biology ; the phenomena studied in 
each are special." Lewes. 

" This unification of all the modes of existence by no means 
obliterates the distinction of modes, nor the necessity of under- 
standing the special characters of each. ... If we recognize the 
one in the many, we do not thereby refuse to admit the many in 
the one" Lewes. 

" Sciences are the result of mental abstraction, being the 
logical record of this or that aspect of the whole subject-matter of 
knowledge. As they all belong to one and the same circle of 
objects, they are one and all connected together ; as they are but 
aspects of things, they are severally incomplete in their relation 
to the things themselves, though complete in their own idea and 
for their own respective purposes ; on both accounts they at once 
need and subserve one another." Cardinal Newman. 



MATHEMATICS, DEFINITE RELATIONS. 29 

not follow that different .classes of ideas will be 
equally excited by the same objects. 

The result of trying to call forth mathematical 
ideas by means of phenomena whose exact meas- 
urement is beyond the power of the pupil, is very 
similar to the result when no pretence is made of 
founding deduction upon perception. Why should 
it not be ? In neither case do mathematical rela- 
tions come definitely into consciousness. 

What objects will excite definite ideas ? Things 
whose exact relations can be most readily seen ; 
things which can be most accurately imaged and 
exactly compared ; things which tend most to 
excite definite intuitions and to result in definite- 
ness of mind,, should be given precedence in 
elementary instruction in mathematics. 

Comte observes, " The only comparisons capable 
of being made directly, and which could not be 
reduced to any others more easy to effect, are 
the simple comparisons of right lines." This is 
apparent to whoever gives thought to the 
matter. 1 

1 " On tracing them back to their origins, we find that the units 
of time, force, value, velocity, etc., which figures may indiscrimi- 
nately represent, were at first measured by equal units of space. 
The equality of time becomes known either by means of the equal 
spaces traversed by an index, or the descent of equal quantities 
* (space-fulls) of sand or water. Equal units of weight were 
obtained through the aid of a lever having equal arms (scales). 



30 THEORY OF ARITHMETIC. 

Since the measurement of all magnitudes is 
reducible to measurements of linear extension, and 
since the comparison of linear units alone reveals 
that perfect equality upon which the science of 
mathematics is built, since by such comparisons 
and only by them do we obtain the original 
materials of mathematical thought, since these 
experiences alone give rise to those abstract con- 
ceptions which enable us to use numbers intelli- 
gently, it follows that definite magnitudes should 
furnish the objective stimulus in laying a basis for 
mathematical knowledge. Out of ratios estab- 
lished by comparing right lines the ratios of 
surfaces and solids are inferred, and also the 
quantitative relations of units of value, force, and, 
in short, of all other magnitudes. 

The problems of statics and dynamics are primarily soluble, only 
by putting lengths of lines to represent amounts of forces. Mer- 
cantile values are expressed in units which were at first, and 
indeed are still, definite weights of metal ; and are, therefore, in 
common with units of weight, referable to units of linear exten- 
sion. Temperature is measured by the equal lengths marked 
alongside a mercurial column. Thus, abstract as they have now 
become, the units of calculation, applied to whatever species 
of magnitude, do really stand for equal units of linear extension, 
and the idea of coextension underlies every process of mathemat- 
ical analysis. Similarly with coexistence. Numerical symbols 
are purely representative ; and hence may be regarded as having 
nothing but a fictitious existence." Spencer, Principles of Psy- 
chology, vol. ii. p. 38. 

" Whenever I went far enough I touched a geometrical 
bottom." Prof. Sylvester, Address British Association, 1869. 



MATHEMATICS, DEFINITE RELATIONS. 31 

Means of passing beyond the range of percep- 
tion. It is the definite relations of magnitudes 
established by means of solids, surfaces, and lines, 
that enable us to conceive or interpret the rela- 
tions of quantities which cannot be brought within 
the range of perception. The ratios which we 
actually see are few, but out of these grows the 
science of mathematics. 

These primary relations, then, should be so 
repeatedly felt, so ingrained, that they will become 
elements in the mental life. This is possible only 
by confronting the pupil again and again with the 
conditions which force upon him the methods and 
ideas of mathematics. He should become so iden- 
tified with the kind of relations dealt with, that 
the abstract terms in which he afterwards reasons 
will be truly representative. Otherwise, he will 
restrict and misapply them. It is the certainty 
of the seen that makes us rationally certain of the 
unseen. 

The basis of drills the perception of relations. 
It is well understood that the use of language 
must become automatic if the mind is to move 
freely in the discovery of laws and principles. 

How is this needful familiarity with the means 
of making quantitative comparisons to be provided 
for? Not, certainly, by treating the means as 
though it were the end ; not by forcing premature 



32 THEORY OF ARITHMETIC. 

drill upon tables and routine work in combining 
and separating symbols. This is to ignore mathe- 
matics, to ignore natural sequences, both within 
and without the mind. Its tendency is to prevent 
energy from rising to that higher kind of power of 
which an intelligent being is capable. 

The drills should harmonize with the dominant 
idea of the subject and meet the conditions which 
favor retention without interfering with growth. 




In his observing and comparing, the pupil has 
dealt with the ratios 2, 3, 4, etc. He has seen 
that the ratio of 4 to 2 equals the ratio of 6 to 3, 
of 8 to 4, of 10 to 5, of to , etc. We bring 
these equal ratios together in the same table and 
associate them in his mind. Making the common 
thing, the ratio, prominent, unifies the work and 
relieves the memory. Grouping like ratios in the 
drills is analogous to the grouping required in 
solving problems. Thus, the pupil sees that the 
relation of the cost of 6 acres to the cost of 2 
acres is equal to the relation of their areas. From 
one truth he passes to another, and brings the 
differing ideas into unity. The drills should em- 
phasize this sense of likeness in the midst of 



MATHEMATICS, DEFINITE RELATIONS. 33 

difference without interfering with the flexibility 
of the mind. 

Drill work should be a means of increasing men- 
tal power by training the eye to quickness and 
accuracy, and the mind to attend closely and image 
vividly. 

In every exercise the first thing to secure is a 
clear mental picture. When the pictures are dis- 
tinct, work for rapidity. What is to be recog- 
nized at sight should be taken in through the eye. 1 
The visual image will be dimmed and blurred, and 

1 " A common error, into which beginners are apt to fall, is to 
try to combine, and therefore to confuse, the two methods of 
remembering, by sight and by sound." Dr. M. Granville. 

" When a child first sees a thing, it takes it in by the eye ; 
when it first hears a thing, it takes it in by the ear ; in each case 
the whole mind is concentrated on the sensation, which, as Dr. 
Carpenter says, * is the natural state of the infant.' But as soon 
as education begins, all this is changed, and the mind, instead of 
being concentrated upon one thing, is distracted by several." 
Kay. 

" We must attend to the formation of the original impression 
. . . and recall it in its entirety afterwards." Kay. 

" Nothing needs more to be insisted on than that vivid and 
complete impressions are all-essential." Herbert Spencer. 

" There can be no doubt as to the utility of the visualising 
faculty when it is duly subordinated to the higher intellectual 
operation. A visual image is the most perfect form of mental 
representation wherever the shape, position, and relations of 
objects in space are concerned." F. Galton. 

" The more completely the mental energy can be brought into 
one focus, and all distracting objects excluded, the more powerful 
will be the volitional effort." Dr. Carpenter. 



34 THEORY OF ARITHMETIC. 

hence imperfectly remembered, if we attempt to 
call the ear into action at the same time that we 
address the eye. 

The way not to succeed in memorizing the tables 
is to repeat so many different impressions in the 
same exercise that none of them are distinct ; to 
confuse eye and ear training ; to make the work 
so difficult that it cannot be done easily and 

" It is a matter of common remark that the permanence of the 
impression which anything leaves on the memory is proportioned 
to the degree of attention which was originally given it." D. 
Stewart. 

" Most persons find that the first image they have acquired of 
any scene is apt to hold its place tenaciously. " F. Galton. 

" The habit of hasty and inexact observation is the foundation 
of the habit of remembering wrongly." Dr. Maudsley. 

" No ideas can long be retained in the memory which are not 
deeply fixed by repetition." Joseph Payne. 

" The leading principle is to learn very little at a time, not in 
a loose, careless way, but perfectly." P. Prendergast. 

"A few such items must be memorized and reviewed daily, 
adding a small increment to the list as soon as it has become 
perfectly mastered." W. T. Harris. 

" We usually attempt to master too much at once, and hence 
the impressions formed in the mind lack clearness and distinct- 
ness." Kay. 

" All improvement in the art of teaching depends on the atten- 
tion that we give to the various circumstances that facilitate 
acquirement or lessen the number of repetitions for a given 
effect." Prof. Bain. 

" It is not enough that impressions be received ; they must be 
fixed, organically registered, conserved ; they must produce per- 
manent modifications in the brain. . . . This result can depend 
only on nutrition." Th. Ribot. 



MATHEMATICS, DEFINITE RELATIONS. 35 

quickly ; to drill once or twice a month ; and to 
prolong the exercise until the power of attention 
is exhausted. 

The way to succeed is to develop vivid mental 
pictures, and to fix these pictures by bringing 
them again and again before the mind. 

Briefly summarized, we may say : Reasoning 
in arithmetic establishes equality of relations ; 
reasoning in any subject, equality or likeness of 
relations. 

We know magnitudes only in relation ; and the 
purpose of mathematical science is to establish 
definite relations between magnitudes. The funda- 
mental operation is comparison. Out of the rela- 
tions established by comparison grow inferences. 

Only through the activity of the mind in observ- 
ing and comparing can those equations be formed 
which are the groundwork of reasoning, the basis 
of advance from relations seen to relations which 
lie beyond the range of perception. 1 

That quantity is a ratio between terms which 
are themselves relative ; that mathematics is not 

lu The domain of the senses, in nature, is almost infinitely 
small in comparison with the vast region accessible to thought 
which lies beyond them. . . . By means of data furnished in the 
narrow world of the senses, we make ourselves at home in other 
and wider worlds, which are traversed by the intellect alone. . . . 
We never could have measured the waves of light, nor even 
imagined them to exist, had we not previously exercised ourselves 
among the waves of sound." Prof. Tyndall. 



36 THEORY OF ARITHMETIC. 

concerned with things as separate and absolute ; 
that it deals only with relations, are truths which 
have often been pointed out, but which the work 
of the school shows to be felt by few. 

In the light of these ideas, those arbitrary 
divisions, so fatal to the continuous unfolding of 
thought, are seen to belong to our language and 
our schemes of study, rather than to the subject. 

Make definite relations the basis, and the integer 
and the fraction are each seen as a ratio ; geo- 
metry, arithmetic, and algebra merge insensibly 
into one another. With definite relations as the 
center, it becomes clear that if we would teach 
mathematics, and not the mere mechanism of the 
subject, we must look to the development of the 
representative and comparative powers. Only thus 
can we lift arithmetic from a matter of memory, 
routine, and formula to its rightful place as a 
means of enlarging the mind. 




PEIMAET ARITHMETIC. 



FIRST STEPS. SENSE TRAINING. 




Finding solids. Place spheres, cubes, cylinders, and 
other forms of various sizes in different parts of the room 
where the children can find them. 

Show a sphere to the pupils. Ask : 

1. What is this? 

Find other balls or spheres. 
Find a larger sphere than this. Find smaller 
ones. 

2. Name objects like a sphere. Example ; An 
orange is like a sphere. 



38 PRIMARY ARITHMETIC. 

3. What is the largest sphere that you have, 
seen? 

What is one of the smallest spheres that you 
have seen? 

4. To-morrow tell me the names of spheres 
that you see when going from school and at home. 

Ask, to-morrow, for the names of the objects and where 
they were seen. 

5. What is the largest sphere you found? 
What is the smallest? 

Review and work in a similar way with other solids. 

"He should at first gain familiarity through the senses 
with simple geometrical figures and forms, plane and solid ; 
should handle, draw, measure, and model them ; and should 
gradually learn some of their simpler properties and rela- 
tions." Committee of Ten. 

Children recognize objects similar in form, color, etc., 
before they desire or have the ability to express what they 
see. 

Until a child can readily select a form he is not ready 
to make a statement of what he has found. Let the 
approach to telling be through doing ; through the activity 
of the pupil in discriminating and relating. 

The teacher, and such pupils as are able, should use the 
proper terms, so that pupils who have not heard the terms 
may learn to apply them. Children can discover like- 
nesses and differences relations but not the terms in 
which they are expressed. They should learn the terms 
unconsciously by living in an atmosphere where they are 
used. Since we think most easily in the names we have 
first and most familiarly associated with a thing, the right 



PRIMARY ARITHMETIC. 39 

term should be used from, the beginning. Providing fitly for 
expressing is an important means of arousing self-activity. 
The different exercises are to be continued from day to 
day, as the growing interest and powers of the child 
suggest, and until there is skill in performing and ease in 
expressing. The teacher should know the condition of the 
pupil's mind. His expression is the index to his mental 
state. Avoid anything which will tend to substitute 
mechanical expression for real expression. Any form 
which is not the outgrowth of what is within, which is not 
the genuine product of free activity, will mislead the 
teacher and weaken the child. 

"Forms which grow round a substance will be true, 
good ; forms which are consciously put round a substance, 
bad. I invite you to reflect on this." Carlyle. 

Finding colors. Tests in color should be given before 
the more formal work suggested below. For example : 
Group cards of the same color and threads of worsted. 1 

Provide ribbons, worsted, cards, etc., of different colors, 
to be found by pupils when looking for a particular color. 

Pin or paste squares of standard red and orange where 
they can be seen. Pin the red above the orange. 

1. Find things in the room of the same color 
as the red square. What things can you recall 
that are red ? 

1 These exercises are not to teach color, but are to train pupils 
to visualize, to attend, to compare, and to secure greater freedom 
in expressing through noting different relations. All pupils need 
such work before beginning the usual studies of the primary 
school. They lack needful elementary ideas, which must be 
obtained through the senses. The range of the perceptions needs 
to be widened. 



40 PRIMARY ARITHMETIC. 

2. Look at the orange square. Find the same 
color elsewhere in the room. Recall objects that 
have this color. 

3. Close the eyes, and picture or image the 
red square. Now the orange square. 

4. Which square is above? Which below? 
Name the two colors. 

5. To-morrow bring something that is red and 
something that is orange. Also tell the names of 
orange or red objects that you see in going to and 
from school. 

Pin or paste a square of yellow below the orange. 

1. Look at the yellow. Find the same color in 
the room. Recall objects having this color. 

2. Look at the red, then the orange, then the 
yellow. Close the eyes and picture the colors 
one after another in the same order. 

Cover the squares. 

3. Which color is at the top ? At the bottom? 
In the middle ? 

4. Name the three, beginning at the top. Name 
from the bottom. 

5. Which color is third from the top ? Second 
from the top ? Third from the bottom ? 

6. To-morrow bring something that is yellow 
and tell me the names of things tfiat you have 
seen that are yellow. 

Add a square of green. 



PRIMARY ARITHMETIC. 41 

1. Find green. Recall objects that are green. 

2. Try to see the green square with the eyes 
closed. 

3. Look at the four colors. 

4. Think of the four, one after another, with 
the eyes closed. 

Cover the squares. 

5. Think the colors slowly from the top down. 
From the bottom up. 

6. Name the colors from the top down. From 
the bottom up. Which is second from the top ? 
Third from the bottom ? Second from the bottom? 

7. Which color do you like best ? 

Add a square of blue and work in the same manner with 
the five as with the four. 

Add a square of purple. 

Work for a few minutes each day until the colors can 
easily be seen mentally in the order given. 

Show a standard color. Have pupils find tints and 
shades of this color, and tell whether they are lighter or 
darker than the standard. 

Have pupils bring things that are shades or tints of 
standard colors. 

Using colored crayon or water-colors, have pupils com- 
bine primary colors and tell whether the result is darker 
or lighter than the standard secondary color. Example : 
Mix red and yellow. Is the result darker or lighter than 
the standard orange ? 

Why is it one of the first duties of the schools to test 
the senses and to devise means for their development ? 



42 PRIMARY ARITHMETIC. 

Handling solids. Cover the eyes. 




Have a pupil handle a solid. Take it away. 

Uncover the eyes. Pupil finds a solid like the one 
handled. 

Cover the eyes. 

Give a pupil a solid. Take it away. Give him another. 

Are the solids alike ? 

Which is the larger ? Which is the heavier ? 

Eepeat the exercise from day to day. 

Judgment and memory should be carefully cultivated 
through the sense of touch as well as through the sense 
of sight. Touch and motion give ideas of form, distance, 
direction, and situation of bodies. " All handicrafts, and 
after them the higher processes of production, have grown 
out of that manual dexterity in which the elaboration of 
the motor faculty terminates." 

Similar solids. Have a pupil select a solid and think 
of some object like it. Have other pupils guess the name 
of the object. 

Ex. : I am thinking of something like a sphere. 

Is it an orange ? 

No, it is not an orange. 

Is it a ball of yarn ? 

It is not. 



PRIMARY ARITHMETIC. 43 

Relative magnitudes. Place a number of solids on the 
table. 

1. Find the largest solid. Find the smallest 
solid. 

2. Find solids that are larger than other solids. 
Ex. : This solid is larger than that one. 

Find solids that are smaller. 

3. Name objects in the room larger than other 
objects. 

Ex. : That eraser is larger than this piece of 
chalk. 

Name objects less than other objects. 

4. Give names of objects at home that are 
smaller than other objects. 

Ex. : A cup is smaller than a bowl. 

5. Recall objects that are larger than other 
objects. 

Ex. : An orange is larger than a .peach. Some 
beetles are larger than bees. 

6. What animals are larger than other animals? 

7. Recall objects that are smaller than other 
objects. 

Ex. : A base ball is smaller than a croquet 
ball. 

8. Find the largest pupil in the class. The 
smallest. 

9. To-morrow tell me the names of objects that 
are larger than other objects and the names of 
others that are smaller. 



44 PRIMARY ARITHMETIC. 

1. Find things that are higher than other things 
in the room. 

Ex. : The door is higher than that table. 

2. Find the tallest pupil. The shortest. 
Compare heights of pupils. 

Ex. : Mary is taller than Harry. 
Compare the heights of other objects. 

3. Recall objects that are longer than other 
objects. 

4. What leaves are longer than they are wide ? 
What leaves are wider than they are long ? 

5. To-morrow tell me the names of other leaves 
that are longer than they are wide. 

Cutting. Let the pupils at first cut and draw what 
they choose. After a number of daily exercises, when they 
have gained some command of the muscles, let them try to 
cut in outline objects which you place before them or 
which they have seen. Let the work be simple. 

The drawing and cutting should be done freely, without 
the restraint of definiteness. If you ask more than the 
pupil can easily represent, the strained, unnatural tension 
interferes with free muscular action. In the slow and 
painful effort to represent perfectly, the mind is absorbed 
in the parts and is prevented from seeing the whole. 
A premature demand for definite action is a fundamental 
error, in that it separates thought from expression. 

" The imperative demand for finish is ruinous because 
it refuses better things than finish." Euskin. 

"Of course one cannot understand a child's picture- 
speech at once, any more than one can his other utterances. 
We must study and learn it." H. Courthope Bowen. 



PRIMARY ARITHMETIC. 45 

Building. Have pupils build prisms equal to other 
prisms. 

Teacher shows a prism and the pupils build. 

Hold the attention to the relative size. This is the 
mathematical element. 

Avoid the analysis of solids until the habit of recogniz- 
ing them as wholes is formed. Do not ask for number of 
surfaces, lines, corners, etc. Such questions, if introduced 
prematurely, tend to destroy self-activity, to interfere with 
judgments of relative size and with the power to see 
relations. 

" Analysis is dangerous if it overrules the synthetic 
faculty. Decomposition becomes deadly when it surpasses 
in strength the combining and constructive energies of life, 
and the separate action of the powers of the soul tends to 
mere disintegration and destruction as soon as it becomes 
impossible to bring them to bear as one undivided force." 
Amiel. 

Ear training. Have pupils listen and tell what they 
hear. 

Have pupils note sounds when various objects are struck. 

Pupils close eyes. Teacher strike one of the objects. 
Pupils tell which was struck. 

Teacher strike two or more objects. 

Pupils tell by the sound the order in which they were 
struck. 

Train pupils to recognize one another by their voices 
and by the sounds made in walking. 

Pupils close eyes and listen. 

Drop a ball or marble two feet, then three. 

Pupils tell which time it fell the farther. 

" There are two ways, and can be only two, of seeking 
and finding truth. . . . These two ways both begin from 



46 PRIMARY ARITHMETIC. 

sense and particulars ; but their discrepancy is immense. 
The one merely skims over experience and particulars in 
a cursory transit ; the other deals with them in a due and 
orderly manner." Bacon. 

" It appears to me that by far the most extraordinary 
parts of Bacon's works are those in which, with extreme 
earnestness, he insists upon a graduated and successive 
induction as opposed to a hasty transit from special facts 
to the highest generalizations. " - Whewell. 




Touch and sight training. Pupils handle -solids : 

1. Find one of the largest surfaces of each 
solid. 

Ex. : This is one of the largest surfaces of this 
solid. 

2. Find one of the smallest surfaces. 

3. Find surfaces that are larger than other sur- 
faces. 

Ex. : This surface is larger than that one. 

4. Find surfaces that are smaller than other 
surfaces. 

5. Compare the size of other surfaces in the 
room. 



PRIMARY ARITHMETIC. 47 

6. Find the largest surface or one of the largest 
surfaces in the room. 

7. Close the eyes, handle solids, and find largest 
and smallest surfaces. 

8. Cover the eyes; handle and tell names of 
blocks and of other objects. 

The exercises for mental training are only suggestive of 
many others which teachers should devise. Be sure that 
the exercises are suited to the learner's mind, and to his 
physical condition. 

Visualizing. Place on the table three objects, for 
example : A box, a book, and an ink-bottle. 




1. What can you tell about the box? About 
the book? About the ink-bottle? Which is the 
heaviest? Which is the lightest? Which is the 
largest ? 

2. Look at the three objects carefully, one after 
another. 

3. Close your eyes and picture one after an- 
other. 

Cover the objects. 

4. Think the objects from right to left. From 
left to right. 



48 PRIMARY ARITHMETIC. 

5. Name the objects from right to left. From 
left to right. 

6. Which is the third from the right? The 
second from the left? 

" Our bookish and wordy education tends to repress this 
valuable gift of nature, visualizing. A faculty that is 
of importance in all technical and artistic occupations, that 
gives accuracy to our perceptions and justness to our gen- 
eralizations, is starved by lazy disuse, instead of being 
cultivated judiciously in such a way as will, on the whole, 
bring the best return. I believe that a serious study of 
the best method of developing and utilising this faculty 
without prejudice to the practice of abstract thought in 
symbols is one of the many pressing desiderata in the 
yet unformed science of education."- Francis Galton. 

When the position of every object in the group can 
easily be given from memory, place another object at the 
left or right. Add not more than one object in an exercise 
unless the work is very easy for the pupils. 

When a row of five is pictured and readily named in 
any order, begin with another group of five. Each day 
review the groups learned, so as to keep them vividly in 
the mind. 

Questions or directions similar to the following will test 
whether the groups are distinctly seen : 

Picture each group from the right. Name objects in 
each from the right. 

In the third group, what is the second object from the left ? 

What is the middle object in each group ? What is the 
largest object in each group ? 

When four or five groups can be distinctly imaged, 
this exercise might give place to some other. 



PRIMARY ARITHMETIC. 49 

Finding circles. Show pupils the base of a cup, a 
cylinder, or a cone, and tell them that it is a circle. 

Conduct the exercises so that the doing will call forth 
variety of expression in telling what is done. 

The correct use of the pronouns, verbs, etc., will thus 
be secured without waste of the pupils' energy. What the 
pupils see and do should lead to statements similar to the 
following : 

That circle is larger than this one. I have found 
a circle that is larger than that one. Helen has found 
a circle larger than that one. He has found a circle 
smaller than this one. They have found circles larger 
than this one. 

1. Find circles. 

2. Find circles that are larger than others. 
Find circles that are smaller. 

3. Find the largest circle in the room. 

4. Find one of the smallest. 

5. Find circles in going to and from school and 
at home, and tell me to-morrow where you saw 
them. 

Finding forms of the same general shape as those taken 
as types is of the highest importance. Unless this is done 
pupils are not learning to pass from the particular to the 
general. They are not taught to see many things through 
the one, and the impression they gain is that the particular 
forms observed are the only forms of this kind. Unless 
that which the pupil observes aids him in interpreting 
something else, it is of no value to him. Teaching is 
leading pupils to discover the unity of things. 



50 



PRTMAKY ARITHMETIC. 



Finding rectangles. Show pupils rectangles (faces of 
solids), and tell them that such faces are rectangles. 

1. Find other rectangles in the room. 
Ex. : This blackboard is a rectangle. 

2. Find larger and smaller rectangles than 
this one. 

3. Find square rectangles. Find oblong rect- 
angles. 

Finding triangles. Show the pupils the base of a tri- 
angular prism or pyramid. 

The base of this solid is a triangle. 

1. Find triangles in the room. 

2. Find triangles that are larger and smaller 
than other triangles. 

Finding edges or lines. Place solids where they can be 
handled. 




1. Show edges of different solids. 

Show one of the longest 1 edges of the largest solid. 

1 The form of the solid will, of course, determine the adjective 
to use. Every lesson should help to familiarize the child with 
correct forms of speech. 



PRIMARY ARITHMETIC. 51 

2. Look for the longest edges in each of the 
solids. 

3. Show the longer edges of other objects in 
the room. 

Ex. : This and that are the longer edges of the 
blackboard. 

4. Show the shorter edges of different objects. 

5. Find edges of different solids and tell whether 
they are longer or shorter than other edges. 

Ex. : This edge of this solid is shorter than that 
edge of that one. 

6. Find edges of objects in the room and tell 
whether they are longer or shorter than other 
edges. 

Ex. : This edge of the table is longer than that 
edge of the desk. 

7. Make sentences like this : This edge is longer 
than that one and shorter than this one. 

"Vision and manipulation, these, in their countless 
indirect and transfigured forms, are the two cooperating 
factors in all intellectual progress." John Fiske. 

Relative length. Scatter sticks of different lengths 
on a table. 

Use one as a standard. Pupils select longer and shorter, 
and state what they have selected. 

After pupil selects a stick and expresses his opinion, 
let him compare the sticks by placing them together. 
This will aid him in forming his next judgment. 

Select sticks that are a little longer or a little shorter. 
This exercise will demand finer discrimination than an 



52 PRIMARY ARITHMETIC. 

exercise where there is no restriction as to comparative 
lengths. 

Direction and position. Pupils and teacher point : 

1. Teacher: That is the ceiling. This is the 
floor. That is the back wall. This is the front 
wall. This is the right wall. That is the left 
wall. This is the north wall. That is the south 
wall. This is the east wall. That is the west 
wall. 

2. A pupil points and teacher tells to what he 
is pointing. A pupil points and the pupils tell to 
what he is pointing. 

3. Tell the position of objects in the room. 
Ex. : There is a picture of a little girl on the 

north wall. There are three windows in the west 
wall. 

Place groups of solids on three or four desks in different 
parts of the room, thus : 






1. Tell the position of each. 
Ex. : The cylinder is at the left at the back. 
The cube is at the right in front. 



PRIMARY ARITHMETIC. 53 

2. Without looking tell where the objects are. 

Tell where different pupils sit. 

Ex. : Mary sits on the 'second seat in the fourth row 
from the right. 

Place a number of objects on a table. 

Let pupils look not longer than ten seconds. Cover the 
objects. Have pupils tell what they saw. Practise until 
pupils learn to recognize objects quickly. 

Have a pupil from another class walk through the room. 
Ask pupils to tell what they observed. 

Such exercises as the following, if not carried to the 
point of fatigue, cultivate alertness of mind, concentration, 
and power to respond quickly to calls for action. 

Teacher occupy a pupil's seat, give directions slowly, 
then place hand where she wishes the pupils to place theirs. 

1. Place hand on the front of your desk. On 
the back. In the middle. At the middle of the 
right edge. At the middle of the left edge. On 
the right corner in front. On the left corner at 
the back. On the left corner in front. On the 
right corner at the back. 

2. Pupil place hand and teacher or other pupil 
tell where it is. 

3. Pupil place an object in different positions 
on the desk. Pupils tell where it is. 

Give each pupil a cube. Teacher use rectangular solid 
and follow her own directions. 

4. Place finger on upper base. On the lower 
base. On the right face. On the left face. On 
the front face. On the back face. 



54 PRIMARY ARITHMETIC. 

5. Pupils place finger and teacher tell where it 
is placed. 

6. Pupils place finger and tell where they have 
placed it. 

Place solids where they can be observed. 




"We overlook phenomena whose existence would be 
patent to us all, had we only grown up to hear it familiarly 
recognized in speech." William James. 

1. Tell the names of as many as you can. 

2. What is the name of the first at the left? 
Give name if none of the pupils know it. Of the 
second ? Of the third ? Of the first, second, and 
third ? Of the fourth ? Of the first, second, third, 
fourth? Of the fifth ? Of the five ? 

3. Look at the solids. Then think of them 
without looking. 

Cover the solids. 

4. Give names in order from left to right. 
From right to left. 

5. Tell position. 

Ex. : The square prism is the second solid from 
the left. 



PRIMARY ARITHMETIC. 55 

Building. Give pupils a number of cubic inches. 

1. Build a prism equal to this one (show prism 
only for an instant). 

Build a prism equal to this one. 

Build a cube equal to this one. 

Give other similar exercises from day to day. 

Cutting. 1. Cut a slip. Cut a longer slip. 

2. Cut a slip. Cut a shorter slip. 

Give each pupil a square two inches long. 

3. Cut larger squares than the square two 
inches long. 

What did I ask you to cut ? 

4. Cut smaller squares than the square two 
inches long. 

What did I ask you to cut ? 

5. Cut a square that is neither larger nor 
smaller than the square two inches long. 

Give other exercises. 

" Almost invariably children show a strong tendency to 
cut out things in paper, to make, to build, a propensity 
which, if duly encouraged and directed, will not only pre- 
pare the way for scientific conceptions, but will develop 
those powers *of manipulation in which most people are 
most deficient." Herbert Spencer. 

Drawing. 1. Draw a square. Draw a smaller 
square. 

2. Draw a large square, a small square, and 
one larger than the small square and smaller 
than the large square. 



56 PRIMARY ARITHMETIC. 

3. Draw two equal squares. 

4. Draw a line. Draw a longer line. 

5. Draw a line. Draw a shorter line. 

6. Draw a line. Draw another neither longer 
nor shorter than this line. Draw other equal lines. 

Do not push demands in advance of the child's growing 
power to do. 

Through the child's attempts to do that which it wishes, 
comes the fitting of the muscles for more definite and more 
complex movements. Above all things let the earlier 
movements be pleasurable, that an impulse to renewed exer- 
tion may be given. The desire to create is the truest 
stimulus to that action which gives muscular control. Our 
exactions may make the doing so disagreeable as to destroy 
the desire to produce. 

Relative magnitude. Place solids where they can be 
handled. 

1. Find solids that are a little larger than other 
solids. 

2. Find solids that are a little smaller. 

3. Find objects that are a little larger or a 
little smaller than other objects. 

Ex. : That desk is a little larger than this. 

4. Find surfaces of the solids that are a little 
larger or a little smaller than other surfaces. 

5. Find edges of the solids that are a little 
longer or a little shorter than other edges. 

6. Find edges of other objects that are a little 
longer and those that are a little shorter than 
other edges. 



PRIMARY ARITHMETIC. 57 

Cutting. 1. Cut a slip of paper. Cut another 
a little longer. Another a little shorter. Measure. 
Practise. 

2. Cut a square. Cut another a little larger. 
Another a little smaller. Measure. Practise. 

Drawing. 1. Draw a line. Draw another a 
little longer. Another a little shorter. Measure. 
Practise. 

2. Draw a square. Draw another a little longer. 
Another a little smaller. Measure. Practise. 

Cutting. 1. Cut a slip of paper. Try to cut 
another equal in length to the first. Look at 
them. Which is the longer ? Place them together 
to see if they are equal. Practise cutting and 
comparing. 

Give each pupil paper and an oblong rectangle. 

2. Cut a rectangle as large as, or equal to, the 
rectangle I have given you. What are you to 
cut ? Is the rectangle you cut as long as the 
rectangle I gave you ? Is it as wide ? Does the 
one you cut exactly cover the one I gave you ? 
Are the two rectangles equal ? Practise trying to 
cut a rectangle exactly the same size as or equal 
to the one I gave you. 

Equality. " The intuition underlying all quantitative 
reasoning is that of the equality of two magnitudes." 
Herbert Spencer. 

1. Find solids and other objects that are equal. 



58 PRIMARY ARITHMETIC. 

2. Find solids in which the surfaces are all 
equal. 

3. Find solids that have surfaces of only two 
sizes. 

4. Find solids that have surfaces of three sizes. 

5. Find solids in which the edges are all equal. 

6. Find solids that have edges of two different 
lengths. 

7. Find solids that have edges of three different 
lengths. 

8. Find a solid that has four equal surfaces. 
How many other equal surfaces has it? 

9. Find a solid that has two equal large sur- 
faces. 

10. Find a solid that has two equal small sur- 
faces. 

11. Find a solid that has four equal long edges. 

12. Show me an edge of one solid equal to an 
edge of another. 

13. Show me two edges of a solid which, if put 
together, will equal one edge of another. 

14. Find objects in the room that are equal, or 
of the same size. 

Ex. : Those two windows are equal. Those two 
erasers are equal. 

Give each pupil a square. 

1. Cut a square equal to the one I have given 
you. Compare. Is the square you have cut equal 



PBIMAKY AK1THMETIC. 59 

to the one I gave you ? Practise cutting and com- 
paring. 

Give each pupil a triangle. 

2. Cut a triangle equal to the one I have given 
you? Compare. Are they equal? Which is the 
larger ? 

1. Draw a line. Draw another equal to the 
first. Measure. Are the lines equal ? 

Give each pupil a square. 

2. Draw a square equal to the one I have given 
you. Do the squares look exactly alike ? Meas- 
ure. Are they equal ? 

3. Draw a triangle. Draw an equal triangle. 
Do the triangles look exactly alike? Are they 
equal ? 

1. Show me equal surfaces in the room. Equal 
edges. 

2. Show me the equal long edges of the black- 
board. How many equal long edges has the 
blackboard? How many short? Show me the 
two equal long edges and the two equal short 
edges of other surfaces. 

3. Show me the two largest surfaces of this 
box. 

4. A chalk-box has surfaces of how many sizes? 
Show a real brick or a paper model. 

5. How many equal large surfaces has a brick ? 



60 PEIMARY ARITHMETIC. 

How many equal small surfaces? How many 
other equal surfaces ? 

6. Show me a surface in one solid equal to a 
surface in another. 

7. Show me two surfaces which, if put together, 
will equal one surface that you see. 

8. Show me one of the longest edges of this 
box. One of the shortest. One of the other 
edges. 

9. How many equal long edges has the bc5x? 
How many equal short edges ? How many other 
equal edges ? 

10. How many rows of desks do you see ? 

11. Show me two equal rows. 

Pupil observe objects. Cover his eyes. Let another 
pupil substitute an object for one of those observed. 
Uncover eyes. Pupil tell what was taken away and what 
was put in its place. 

Secure sets of squares and of other rectangles of differ- 
ent dimensions. Scatter sets over the table. 1 

Train pupils to select those that are equal. 

Ex. : That square rectangle equals this one, or that 
oblong rectangle equals this one, or James found a square 
equal to this one. 

Secure variety of statement. 

Cutting. 1. Look at a cube 2 in. long and cut 
a square equal to one of its surfaces, or look at a 

1 Length of squares, 2 in., 2| in., 3 in., 3 in., 4 in. Dimen- 
sions of oblong rectangles, 1 X 2, 2x2, 3x2, 4x2, 5x2, 
and others 1 X 3, 2 x 3, 3 X 3, 4 X 3, 5 X 3, 6 X 3. 



PRIMARY ARITHMETIC. 61 

square rectangle 2 in. long and cut an equal one. 
What did I ask you to cut ? 

Let pupils criticise their own work. Do not tell them 
that the square rectangle they cut is too large or too small ; 
let them compare and tell you. The work will be good, 
no matter how crude or imperfect, if it is the best the 
pupil can do. Growth is possible only from the basis of 
genuine, natural expression. 

2. Practise cutting and comparing. 

3. Cut a square rectangle two inches long with- 
out observing model. 

4. Cut a rectangle whose length and width are 
the same. Measure. Are they equal ? What is 
the name of this figure ? Practise. 

To-morrow, have pupils cut the square rectangle again. 

Have them tell what they cut, in order to learn to asso- 
ciate the language with the thing. 

Give pupils square rectangles four inches long and train 
them to cut, first when observing, then from memory. 

Give pupils rectangles 4 in. by 2 in., and tell them to 
cut rectangles 4 in. by 2 in. 

5. What did I tell you to cut ? After cutting, 
compare and measure. 

6. What are the names of the three forms that 
you have cut ? 

7. What is the width of the square 2 in. long ? 
Of the square 4 in. long ? 

Why are a child's ideas necessarily crude rather than 
complete ? What, then, should be true of his outward 
representations ? 



OF THB 

UNIVERSITY 



62 PRIMARY ARITHMETIC. 

Why is it impossible to secure perfect forms from 
young children without interfering with mental and moral 
development ? 

" We shall not begin with a pedantic and tiresome insist- 
ence on accuracy (which is not a characteristic of the 
young mind), but endeavor steadily to lead up to it to 
grow it producing at the same time an ever-increasing 
appreciation of its value." H. Courthope Bowen. 

As before urged, let the work be done freely. Unnatural 
restraint in expressing results in lack of feeling. It 
lessens desire to see and to do. The use of things in 
which mathematical relations are conspicuous furnishes 
no excuse for disregarding the truth that progress in the 
power to represent either within or without is ever from 
the less to the more definite. The child is not troubled 
by a complexity or a definiteness which it does not see. 
Teaching in harmony with nature will permit the child to 
see freely and express freely. 

Exercise in judging will gradually increase the power 
of definite thinking ; and exercise in doing the power of 
definite action. 

Drawing. Draw 6-in. squares on different parts of the 
blackboard. 

Pupils observe and try to draw equal squares. Meas- 
ure, and try again. 

Let one pupil draw and others estimate whether the 
square is larger, smaller, or equal to the 6-in. square. 

Have pupils measure after drawing, so that they may 
see mistakes and make more accurate estimates. v 

Draw lines a foot long. Pupils observe the lines and 
try to draw equal lines. 

Let one pupil draw and others estimate whether the 
lines are longer, shorter, or equal. 



PRIMARY ARITHMETIC. 63 

Pupils find edges of objects that they think are a foot 
long. 

Without pupils observing you, draw lines a foot long, 
a little more than a foot long, and others a little less than 
a foot long. 

Arrange obliquely, horizontally, and vertically. Letter 
A, B, (7, etc. 

Pupils select different lines. Ex. : The line C is less 
than a foot long. Other pupils tell whether they agree 
or not. 

Have pupils find edges in the room a little more or a 
little less than a foot long. 

Without pupils observing you, draw a line 2 ft. 
long. 

Have pupils estimate the length. Let them measure. 

Without pupils observing you, draw lines on the board 
less than 2 ft., more than 2 ft., and 2 ft. Letter. 

Have pupils estimate the lengths. Ex. : I think the 
line B is more than 2 ft. long. Measure. 

Have pupils find edges in room a little more or a little 
less than 2 ft. long. 

Draw a 6-in. line on the board. Do not separate into 
inches. Draw a foot. Pupils look at both lines. How 
many 6-in. lines in the foot ? 

Draw a 4-in. line. Pupils observe and draw. Observe 
the foot and the 4-in, line. How many 4-in. lines in a 
foot? 

Place the solids where they can be handled. Pupils 
estimate the length of edges. Measure. 

Have pupils show edges of solids that they think are 
4 in. long. 

Have pupils tell how long, wide, and high they think 
each solid is. Ex. : I think this solid is 4 in. long, 2 in. 
wide, and 1 in. high, or it is 4 in. by 2 in. by 1 in. 



64 PBIMARY ARITHMETIC. 

" If the judgment made be original, then the standpoint 
of the one making the judgment is disclosed." William 
T. Harris. 

Building. If a direction is not understood, the teacher 
should explain by doing a thing similar to that she wishes 
done. Thus, if she says build a unit equal to f of this 
one, and the pupils do not understand, she should build a 
unit equal to f of it. Then the pupils should build units 
equal to f of other units. 

1. Using cubes, make a prism equal to this one. 

2. Using cubes, make a prism two times as 
large as this one. 

Continue to build prisms two times as large as those 
selected until this can be done easily. 

3. Build a block equal to f of this one. 

4. Build one equal to f of this one. Of this 
one. 

5. Build a block equal to ^ of this one. ^ of 
this one. 

"Doing, or rather, expressive doing, reveals to the 
teacher the nature of his pupil's knowledge ; exhibits to 
the pupil new connections and suggests others still ; de- 
velops skill or effectiveness in doing as mere exercise of 
information seldom does, or does but feebly ; and trains 
the muscles, the nerves, and the organs of sense to be 
willing, obedient, effective servants of the mind." H. 
Courthope Bowen. 

Cutting. Give pupils paper rectangles of different 
sizes. 



PRIMARY ARITHMETIC. 65 

1. Cut a rectangle into two equal parts. After 
cutting, place the parts together to see if they are 
equal. Practise cutting and comparing the two 
parts. 

2. Cut rectangles into three equal parts. Com- 
pare the parts. Are they all equal ? Practise. 

Drawing. 1. Draw a line. Place a point in 
the middle of the line. Measure to see if the 
parts are equal. Try again. Measure. Is one of 
the parts longer than the other ? Are the parts 
equal ? What is meant by equal ? Show me one 
of the two equal parts. Show me the other. 

2. Draw a line. Separate it into two equal 
parts. Measure. Are the parts equal ? Separate 
the line into four equal parts. Show me one of 
the four equal parts. Show me three of the four 
equal parts. Show me the four equal parts. 

3. Draw a line. Separate it into three equal 
parts. Measure. Are the parts equal ? 

4. Show me where the line should be drawn to 
separate the blackboard into two equal parts. 
Point to the two equal parts of the board. 

5. Can you see the two equal parts of the floor? 
Of the top of your desk? Show me two equal 
parts of other things in the room. 

Give each pupil a square. 

6. Measure the edges of the square. What is 
true of the edges of the square? Find other 
squares in the room. 



66 PRIMARY ARITHMETIC. 

7. Draw a square. Measure. Are the edges 
equal ? How many equal edges has a square ? 
Practise trying to draw squares. 

8. Draw an oblong rectangle. Measure the 
two long edges. Are they equal? Measure the 
two short edges. Are they equal ? Practise try- 
ing to draw oblong rectangles. 

Equality. Place solids where they can be handled. 

1. Show a part of that solid equal to this one. 

2. Show a part of one solid equal to another. 

3. Show a part of that rectangle equal to this 
one. 

4. Show other parts that are equal. 

5. What part of that solid equals this one ? 
(Give the name of the part if none of the pupils 
know it.) 

6. Show the part of that rectangle equal to 
this one. 

7. What is the name of the part of that rect- 
angle equal to this one ? 

Building. Give pupils cubes. Show a unit. 

1. Build a unit equal to this one. 

2. Separate the unit into two equal parts. 

3. This is J of the unit. 

Show the other half. Hold up the f . 
Put the halves together. Put one half on the 
top of the other. 



PRIMARY ARITHMETIC. 67 

Show a larger unit. 

4. Build a unit equal to ^ of this one. 

5. Build another unit equal to f of ito 

6. Build another unit equal to f of it. 

Relative Magnitude. 1. Draw a line. Sepa- 
rate it into two equal parts. This is ^ of the 
line. Show me the other half. Show me the f 
of the line. 

2. Show me J of the top of your desk. Show 
me of the blackboard. Show me % of this solid. 
Show me ^ of that solid. Show me f of that 
solid. 

3. Draw a line. Draw another as long as J of 
the first. Measure. 

4. Draw a line. Draw another two times as 
long. Show me the part of the second line that 
is as long as the first. What part of the second 
line equals the first? The first line is as long as 
what part of the second ? The first line equals 
what part of the second ? 

5. Cut a slip of paper. Cut another slip J as 
long. Measure. Cut a slip of paper. Cut an- 
other equal to J of the first. What did I ask you 
to do? 

6. Cut a rectangle. Cut another two times as 
large. Show me the second rectangle you cut. 
What part of the second rectangle is as large as 
the first ? 



68 PRIMARY ARITHMETIC. 

7. Use sticks and lay lines two times as long as 
other lines. 

8. Use sticks and make rectangles two times as 
large as other rectangles. 

Have pupils handle solids and tell into how many equal 
smaller solids a larger solid can be cut. 

Avoid the frequent use of any particular solid, surface, 
or line, in making comparisons. To use an inch cube, a 
two-inch cube, a foot, or a yard in the elementary work 
oftener than other units are used interferes with free 
mental action. 

Place on the table various solids, cardboard rectangles, 
both square and oblong, and other objects. Let each 
pupil take one object. 

Teacher : John, what have you ? 
I have a sphere. 

Other pupils tell what they have. Pupils tell what 
other pupils have. Ex. : William has a red square. 

Teacher: Who has the largest solid? Who have solids 
that are alike ? 

Place objects upon other objects and tell what was done. 

Ex. : I put a cone upon a cube. Mary placed a cone 
upon a cube. 

Place two objects together and tell what you did. 

Ex. : I put a square and an oblong rectangle upon the 
table. 

Tell what are in a group of three objects. 

Ex. : A knife, a pen, and a pencil are in that group. 
I have a sphere, a prism, and a cylinder. 



PRIMARY ARITHMETIC. 



Relative magnitude. 1. Tell all you can about 
A and B. 




2. B is as large as how many As ? 

3. What part of B is as large as At A equals 
what part of B ? 

4. B equals how many times A ? 

Place pairs of solids having the ratio two where they 
can be handled by the class. 

5. Observe solids and make sentences like this : 
This solid can be cut into two solids each as large 
as that one. 

6 . Have pupils discover all the relations they can. 

The things between which the relation ^-, -J-, f , J, 2, 3, 4, 
etc., is seen, should vary. Keep in view the fact that the 
thing is its relations. (See page 19.) That which the pupil 
sees as ^ when related to a unit twice its size he should 
see as -J- or 2 according to that with which it is compared. 
He will do so if there is a proper presentation. At first 
his perceptions of these relations will be dim. They will 
gradually develop according to his experience. 

" There must be accumulation of experiences, more 
numerous, more varied, more heterogeneous there must 
be a correlative gradual increase of organized faculty." 
Herbert Spencer. 



70 



PRIMARY ARITHMETIC. 



"The formation of an idea is anorganic evolution which 
is gradually completed, in consequence of successive expe- 
riences of a like kind." Dr. Maudsley. 



c 








a 




















Draw the units on the blackboard, making C 6 in. long. 

1. Tell all you can about these units or rect- 
angles. How many of these rectangles are square? 
How many oblong ? 

2. Find the units that are equal. 

3. The different units can be cut into what? 
Ex. : The unit Tcan be cut into two M's. 

4. Make sentences like this : One half of B 
equals C. 

5. Find units equal to ^ of other units. 

6. Find units two times as large as other 
units. 



PRIMARY ARITHMETIC. 



71 



7. Draw the units again to a different scale and 
continue the work. 





Draw the units on the blackboard, making B 6 in. long. 

1. Find out all you can about these units. 

2. Find the equal units. 

3. Find units equal to ^ of other units. 
Ex. : The unit / equals ^ of 0. 

4. Make sentences like this : One half of M 
equals A. 

5. The different units can be made into what 
units ? 

Ex. : The unit M can be made into four (7's. 

6. Find units two times as large as other units. 
Ex. : The unit M equals two times A. 

The number of repetitions needed will depend greatly 
upon the manner of presentation. But no art, no mode of 
work can alter the fact that time is required, that ideas 
are the result of an organizing process. 

Building. Show a prism 3 by 1 by 1. 
1. Build a unit equal to this one. 



72 PRIMARY ARITHMETIC. 

2. Separate the unit into three equal parts. 

3. Show me the three equal parts. 

4. Hold up two of the three equal parts. 

5. Show me one of the three equal parts. 
Show a unit 6 by 1 by 1. 

1. Build a unit equal to this one. 

2. Separate the unit into three equal parts. 

3. Show me the three equal parts. 

4. Show me one of the three equal parts. 

5. Show me two of the three equal parts. 
Show a unit 3 by 2 by 1. 

1. Build a unit equal to one of the three equal 
parts. 

2. Build another equal to two of the three 
equal parts. 

3. Build another equal to the three equal parts. 

Cutting. Give each pupil several rectangles of differ- 
ent sizes. 

1. Cut a rectangle into three equal parts. 
What did I tell you to do ? Place the three parts 
together. Are the three parts equal ? Practise 
cutting and comparing. 

Drawing. 1. Draw a line. Separate it into 
three equal parts. Measure. Is one of the parts 
shorter than one of the others? 

2. Draw lines of different lengths and practise 
trying to divide them into three equal parts. 



PRIMARY ARITHMETIC. 73 

3. Draw rectangles of different sizes and prao 
tise trying to separate them into three equal parts* 

4. Show me where lines should be drawn to 
separate the blackboard into three equal parts. 
Move your hand over each of the three equal 
parts of the blackboard. 

Select different solids. 

5. Show me where each should be cut to sepa- 
rate it into three equal parts. 

6. Find a solid that can be made into three 
parts, each as large as this solid. 

Ex.: That solid can be made into three solids 
each as large as this one. 

Give each pupil a piece of paper on which there is 
drawn a line equal to D. 



1. Draw a line equal to D. 

2. Draw a line two times as long as D. 

3. Draw a line three times as long as D. 

4. Name the lines D, A, B. 

5. A is how many times as long as D ? 

6. B is how many times as long as Z)? 

7. Show me ^ of A. B is how many times as 
long as i of A ? 

8. Show me ^ of A. Draw a line three times 
as long as J of A. 

9. Draw a line equal to the sum of D and A. 



74 PRIMARY ARITHMETIC. 

The sum of D and A equals what line ? 

10. If we call D I, what ought we to call A ? 
What ought we to call B ? 

11. The sum of A and D equals what? The 
sum of 1 and 2 equals what ? 

Relative magnitude. Give each pupil a square inch 
and an oblong 2 in. by 1 in. and another 3 in. by 1 in. 




1. What is the length of the square rectangle ? 
How long is the largest rectangle ? What is the 
length of the other rectangle ? 

2. Show me the rectangle 2 in. by 1 in. The 
rectangle 3 in. by 1 in. Point to each rectangle 
and describe it. 

Ex. : This is a square rectangle 1 in. long. 

3. Call the largest rectangle B 9 the smallest 0, 
and the other N. Show me 0. Show me B. 
Show me N. 

4. JVis as large as how many O's ? What part 
of N equals ? N equals how many times ? 
equals what part of JV? 

5. B is as large as how many O's ? B equals 
how many times ? Show me of N. B is how 
many times as large as J of JV? 

6. If we call , what is Nt What is 

7. Cut rectangles equal to 0, JV, and B. 



PRIMARY ARITHMETIC. 



75 



1. Place and ^together and make one rect- 
angle of the two. How long is the rectangle you 






N 


B 



have made ? How wide is it ? It is as large as 
what rectangle ? It equals what rectangle ? 

2. Place 0, JV, and B together, making one rect- 
angle of the three. How long is the rectangle ? 
This rectangle could be cut into how many B's ? 
Into how many N's ? 

3. Show me ^ of the rectangle. B is what 
part of the rectangle ? If you put two rectangles 
together, the new rectangle is called the sum. 
The sum of and N is what part of the rect- 
angle ? 

4. If we call 1, what ought we to call Nt 
What ought we to call. 5? Show me the unit 3. 
The unit 2. The unit 1. 

Use different magnitudes, and change their arrangement 
very often. If this is not done the objective representa- 
tions become the thing, and the relation, which is the 
essence of the subject, is not brought into consciousness 
at all. We prevent the perception of truth when our presen- 
tation limits the relation to particular things. (See preface.) 



1. Tell all you can about 
the units 1, 2, and 3. 

2. The unit 2 is how many 
times as large as the unit 1 ? 



76 PRIMARY ARITHMETIC. 

What part of 2 is as large as 1 ? The unit 3 is 
how many times as large as the unit 1 ? Show 
me J of the unit 2. The unit 3 is how many 
times as large as half of the unit 2 ? The unit 3 
is as large as how many halves of 2 ? The unit 
3 equals how many 1's? 

1. Place the units 1 and 2 together. The sum 



1 


2 


3 



of 1 and 2 equals what ? The unit 3 is how much 
greater than the unit 1 ? 

Ans. : The unit 3 is 2 greater than the unit 1, 
The unit 3 is how much greater than 2 ? How 
much less is the unit 2 than the unit 3 ? The 
unit 1 is how much less than the unit 3 ? The 
unit 3 is as large as the sum of what two units ? 
Two and what equal 3 ? One and what equal 3 ? 

2. Make one rectangle of 1, 2, and 3. The 
sum of 1 and 2 is what part of the rectangle ? 
The unit 3 is what part of the rectangle ? 

3. Show me the two equal units that make the 
rectangle. Show me the three equal units in the 
rectangle. What three unequal units do you see 
in the rectangle ? Separate the rectangle into two 
unequal units. What are the names of the two 
unequal units in the rectangle ? 

4. The rectangle equals how many 3's ? How 
many 2's ? 



PRIMARY ARITHMETIC. 77 

5. If the 1 is worth a nickel, what is the 2 
worth ? 

6. If you pay a nickel for the 1, how many 
nickels ought you to pay for the 3 ? 

7. If 2 cost a dime, 1 will cost what part of a 
dime ? 

8. The cost of the 1 equals what part of the 
cost of the 2 ? 

9. The cost of 3 equals how many times the cost 
of 1? 

10. Show me the part of 3 that will cost as 
much as 2. 

11. If an apple costs 3/ ? how many 3-/ will two 
apples cost? 

12. How many times as long will it take to 
walk two blocks as to walk one block ? 

13. What part of the time that it takes to walk 
two blocks will it take to walk one block ? 

14. If three tops cost 6/, what part of 6/ will 
two tops cost? 

Draw the three rectangles on the blackboard to the 
scale of 1 foot to the inch. 

1. If the length of the square rectangle is 1, 
what is the length of each of the others ? What 
is the height of each ? 

2. What is the number of feet in the length of 
each rectangle ? 

3. Show me the rectangle 1 ft. by 1 ft. The 
rectangle 1 ft. by 2 ft. The rectangle 1 ft. by 3 ft. 



78 PRIMARY ARITHMETIC. 

4. Show me the upper edge of the middle rect- 
angle. The lower edge. The right edge. The 
left edge. How many edges has each rectangle ? 
Show me the entire edge or the perimeter of each. 

5. How many feet in the perimeter of the 
square foot ? How many feet in the perimeter 
of the middle rectangle ? In the perimeter of the 
largest rectangle ? Letter the rectangles 0, A, 
and B. 

6. Have pupils tell all they can about the rela- 
tions -of the rectangles 0, A, and B. 

7. Name the rectangles 1, 2, and 3. Have 
pupils tell all they can about 1, 2, and 3. See 
questions on 1, 2, and 3 in preceding lesson. 

Cutting 1. Cut a rectangle, making its length 

and width equal. If we call the length of the 
rectangle 1, what ought we to call its width? 
Practise cutting rectangles whose edges are*l by 1. 

2. Cut a rectangle, making its length 2 and its 
width 1. Measure. The length of the rectangle 
is how many times its width ? If the width of 
this rectangle is 1, what is its length ? Cut rect- 
angles making the dimensions 1 by 3. Measure. 
Practise. 

Drawing. 1. Try to draw rectangles on the 
blackboard 1 by 1. Measure. 

2. Draw rectangles on the blackboard 1 by 2. 
Measure. 



PRIMARY ARITHMETIC. 



79 



3. Draw rectangles whose edges will be repre- 
sented by 1 and 2. 

4. The length is how many times as great as 
the height ? 

Drawing should prolong attention. For the teacher, 
drawing should be an index of what the child can see 
and do. 

Cutting. 1. This rectangle is 1. Cut a 1, a 2, 
and a 3. 

2. The 2 you have cut equals how 
many times the 1 ? 

3. The 3 you have cut equals how 
many times the 1 ? 

4. If you put the 1, 2, and 3 together, 

the sum will make how many 3's ? How many 
2's ? How many 1's ? 

Drawing. 1. This is 3. Draw a 3, a 2, a 1. 

2. Have a pupil draw a unit 
on the blackboard and name it 1, 
2, or 3. Have other pupils draw 
the other two units. Use lines 
and rectangles. 

Relative sizes. Place the cube 1 
in. long, the solid 1 in. by 1 in. by 2 in., and the solid 1 
in. by 1 in. by 3 in. where they can be seen. Name them 
C, D, and A. 

1. What is the name of the largest unit? The 
name of the smallest ? Of the other unit ? 




80 PBIMAKY ARITHMETIC. 

2. Look at the units C, D, and A, and tell all 
you can about them. 

3. D equals how many C"s? A equals how 
many C 's ? D is how many times as large as C ? 
A is how many times as large as C ? A equals 
how many times C ? 

4. Show | of D. What part of D equals (7? 
A equals how many times -| of D ? 

5. Put C and D together. The sum of C and 
D equals what unit ? The sum of C and D equals 
how many C's ? Put C, D, and A together. 
How many A's in the sum ? How many jD's ? 
How many C's ? 

6. If we call C 1, what ought we to call Z)? 
What ought we to call A ? Show me the 1. The 
2. The 3. 

7. Look at the units 1, 2, and 3, and tell all 
you can about them. 

8. The unit 2 is as large as how many 1's ? 
The 3 is as large as how many 1's? 

9. What part of 2 is as large as 1 ? Show me 
the part of 3 that is as large as 1 ? Show me the 
part of 3 that is as large as 2. 

10. 3 is how much greater than 1 ? 3 is how 
much greater than 2 ? 1 is how much less than 
3 ? 2 is how much less than 3 ? Put 1 and 2 
together. The sum of 1 and 2 equals what unit ? l 

1 The child understands spoken language before he uses it ; he 
acquires it unconsciously. Let him have the same opportunities 



PRIMARY ARITHMETIC. 81 

Unite the units 1, 2, and 3.. The sum equals how 
many 3's ? How many 2's ? 

11. If you put (7, -D, and A together, how high 
a post will they make ? What is -| of the height 
of the post ? Two inches equals what part of the 
height of the post ? The top of the post is what 
kind of a rectangle ? 

1. Cover the eyes of different pupils and place 
solids in their hands. Let pupils tell relative 
sizes of solids and surfaces and the relative lengths 
of edges. 

2. Find units that you can call 1, 2, and 3. 

3. Show a unit that is two times as large as 
this one. Show different units that equal two 
times other units. 

Ex. : This unit equals two times that unit. 

4. Show different units that equal three times 
other units. 

Ex. : This unit equals three times that one. 

5. Tell things like this : This is a 2, for it is two 
times as large as that unit. 

6. Tell things like this: That is a 1 ? for this 
unit equals ^ of it. 

in learning to associate ideas with sight-forms. From day to day 
place upon the blackboard the expression for the relations dis- 
covered. At first do not ask attention to them. When the child 
wishes to use them, a great step toward the power to express will 
have been taken. Let that which you write mean something to 
the child, as that which he hears does ; let it symbolize his thought. 



82 PRIMARY ARITHMETIC. 

7. Find solids whose surfaces represent 1 and 
2. How many of the surfaces may we call 1 ? 
How many 2 ? 

8. Find surfaces of different solids whose rela- 
tions are 1, 2, and 3. 

9. Find edges that we may call 1 and 2. Tell 
how many 1's and how many 2's you find in the 
edges of the solid. 

Relations of quart and pint. Show pupils the pint 
and quart measures. Have them find the number of pints 
equal to a quart by measuring. 




1. After measuring, tell all you can about the 
quart and the pint. 

This free work is far more valuable than that induced 
by questions. Both the weak and the strong have oppor- 
tunities to show their power, while the exercise tends to 
develop self-activity ; that is, it fosters a desire to discover 
when not acting under the stimulus of questions. 

Too much questioning interferes with the natural action 
of the mind in relating and unifying. It isolates ideas. 
It prevents the teacher from seeing the real state of the 
pupil's mind. What is wanted is a questioning attitude, 
a curiosity which will sustain interest and strengthen 
attention. 



PRIMARY ARITHMETIC. 83 

2. What is sold by the pint and by the quart ? 

3. A quart is how many times as large as a 
pint? 

4. What part of a quart is as large, or as much, 
as a pint ? 

5. A quart is how much more than a pint ? 

6. A pint is how much less than a quart ? 

7. A quart and a pint equal how many pints ? 

8. Show me 1| quarts. What have you shown 
me? 

9. 1J quarts equal how many pints ? 

10. If we call a pint 1, what ought we to call 
a quart ? Why ? 

11. If we call a quart 2, what ought we to call 
the sum of a quart and a pint ? 

12. If a quart is 1, what is a pint? 

Fill the quart and pint measures with water, and let 
each pupil lift the two measures. 

1. Which is the heavier, the quart of water 
or the pint ? 

2. The quart of water is how many times as 
much as the pint ? 

3. What part of the quart weighs as much as 
the pint ? 

4. The weight of a pint equals what part of 
the weight of a quart ? 

5. The weight of a quart equals the weight of 
how many pints ? 



84 PRIMARY ARITHMETIC. 

6. A pint of water weighs a pound; how much 
does a quart of water weigh ? 

7. What part of a quart of water weighs a 
pound ? 

8. The sum of a quart and a pint of water 
weighs how many pounds ? 

9. Compare the weight of different solids with 
the weight of a pint of water. 

Ex. : This solid weighs less than a pound, or 
this solid weighs a little more or a little less than 
a pound. 

10. If a pint of milk costs 3/, what ought a 
quart to cost ? 

11. In a quart there are how many pints ? In 
3 quarts there are how many 2-pints ? 

12. How much milk should be put into a quart 
measure to make it half full ? 

Relation of the foot and six inches. Have pupils try 
to draw lines of the same relative length as the foot and 
the 6-in. on paper and on the black- 
board. After the practice in drawing 
and in telling what they can about the 
relations, draw the foot and the 6-in. lines on the board. 

1. Tell all you can about these lines. 

2. What is the length of the longer line? 
What is the length of the shorter line ? 

3. Into how many 6-in. can a foot be separated ? 

4. A foot is how much longer than 6 inches ? 

5. 6 in. and how many inches equal a foot ? 



PRIMARY ARITHMETIC. 85 

6. 6 in. are how much shorter than a foot? 

7. Show me the part of a foot that equals 6 in. 

8. What part of a foot equals 6 in.? 

9. A foot is how many times as long as 6 in. ? 

10. 6 in. equals what part of a foot ? 

11. How many 6-in. in a foot ? In 2 ft. ? 

12. Two 6-in. equal what? 

13. 6 in. and 1 ft. are how many 6-in. ? 

14. How many 6-in. in 1^ ft. ? 

15. If we call 6 in. 1, what ought we to call a 
foot? 

16. If a foot is 1, what is 6 in. ? 

17. If we call 6 in. 1 ? what ought we to call 
2ft.? 

18. Why ought we to call 2 ft. 4, if we call 
6 in. 1 ? 

19. Review without observing the foot and the 
6-in. line. 

Relative length. Give each pupil an equilateral tri- 
angle having a 2-in. base. 

1. Cut an equilateral triangle as large as this 
one. Measure the edges. Are 

they equal ? Practise cutting and 
measuring. 

2. Draw an equilateral triangle. 
Measure the edges. Are they 
equal? Practise drawing and measuring. 

3. Try to draw a line equal to the sum of two 




86 PRIMARY ARITHMETIC. 

edges of the triangle. Is the line you have drawn 
two times as long as one of the edges of the tri- 
angle ? 

4. Draw a line equal to the sum of the edges of 
the triangle. Is the line you have drawn three 
times as long as one of the edges of the triangle ? 
Measure. Try again. 

5. Show me the perimeter of the triangle. 
How many 2-in. in the perimeter of the triangle ? 

6. Let one pupil try to draw an equilateral tri- 
angle on the board. Other pupils criticise. 

7. Tell all you can about this equilateral tri- 
angle. 

Relation of the yard and the foot. Draw a line a 



yard long on the blackboard. Draw another a foot long. 
Give the names of each. 

1. What is the name of the longer line? Of 
the shorter line ? Show me the yard. 

2. Tell all you can about the yard and the foot. 

3. How many feet do you think there are in a 
yard ? Measure. A yard is how much longer than 
a foot ? A foot is how much shorter than a yard ? 

4. A yard equals how r many times a foot ? 

5. Into how many equal parts must you sepa- 
rate a yard to make each part a foot long ? 



PRIMARY ARITHMETIC. 87 

6. A yard of ribbon contains how many feet ? 

7. Have pupils try to place points a foot apart 
on the blackboard. Pupils in class tell whether 
they are more or less than a foot apart. Measure. 
Practise. 

Estimate the number of yards in different 
lengths, heights, edges. 

Ex. : The height of that door is more than 
2 yds. but less than 3. 

8. How many feet in a yard ? How many 3-ft. 
in 2 yds.? In 4 yds.? 

Problems. 1. If I have 2 apples in my pocket 
and ^ as many in my hand, how many have I in 
my hand? , 

2. If I pay 4/ for a yard of ribbon, how much 
must I pay for ^ yd. ? 

3. If 1 ft. of molding costs 2/, how many 2-/ 
will 1 yd. cost ? 

4. If 1 ft. of molding costs 17/, how many 
17-/ will 1 yd. cost? 

5. If ^ barrel of flour lasts 1 month, how long 
will 1 barrel last ? 

6. I use 1 yd. of ribbon for a hat and f of a 
yard for a collar ; how many feet do I use ? 

7. I had 4 horses and sold J of them ; how 
many did I sell ? 

8. Mary had a quart of berries and sold a pint. 
What part of her berries did she sell ? 



88 PRIMARY ARITHMETIC. 

Relative magnitude. Show pupils 1, 2, 3. Call them 
t, t, 1- 

1. What are the names of these units ? 




2. What is the name of the largest ? Of the 
smallest ? Of the next to the largest ? 

3. Put and f together. The sum of and f 
equals what ? 

4. What must be added to the unit f to make 
the unit 1 ? 

5. The unit 1 is how much larger than the unit 

t? 

6. You can separate the unit 1 into how many 
thirds ? 

7. What part of f is as large as J ? 

8. What part of 1 equals the | ? 

9. What part of 1 equals the f ? 

10. The unit 1 is how many times as large as 
the*? 

11. i equals what part of f ? Of 1 ? 

12. Show f of the top of this table. Show f of 
it. 



PRIMARY ARITHMETIC. 89 

Select other solids Laving the same relative size, and 
call them -J-, f , 1. Pupils compare. Tell all they can. 





(Hi. 



1. Show f, f, and | of different objects in the 
room. 

2. Practise making units of cubes equal to f 
of other units. 

3. Practise making units equal to f of other 
units. 

4. Practise making units equal to ^ of others. 

Give each pupil a square inch, a rectangle 2 in. by 1 in., 
and one 3 in. by 1 in. Call them , f , 1. 

1. Tell all you can about the units ^ f, 
and 1. 

2. What part of f equals the J ? 

3. How many ^ in the 1 ? 

4. What part of the 1 equals the ? 

5. The unit 1 is how many times as large as 
the unit ? 

6. Show me ^ of the f. The unit 1 is how 
many times as large as ^ of the f ? 

Draw the units on the blackboard to the scale of 1 ft. to 
the inch. 



90 



PRIMARY ARITHMETIC. 



1. If the largest unit is 1, what is the name of 
each of the others ? 

2. Tell all you can about the relations of these 
units. 

Ask questions similar to those above. 

3. If the ^ is worth 5/ ? what is |- worth ? 

4. If the | is worth 3/ ? how many 3-/ is the 1 
worth ? 

Draw the figures of the diagram on the blackboard; 




making A 6 in. long. After pupils have studied and com- 
pared them, draw to some other scale. 

1. Tell all you can about the relation of these 
units. 



PRIMARY ARIT 




2. If A is 1, how many 1's in the diagram ? 
Can you find five other figures as large as A ? 

3. If A is 1, how many 2's do you see ? How 
many 3's ? 

4. If B is 1, how many 1's do you see ? How 
many 2's ? How many 3's ? 

5. If B is 1, how many of the figures are 
halves ? 

6. If Gr is 1, how many 1's in the diagram? 
How many 2's ? How many 3's ? 

7. If G is 1, what is A ? If G is 1, how many 
of the figures are thirds ? How many represent 
| ? How many f ? The figure If equals how 
many thirds? 

8. If A is a 6-in. square, each of the others 
equals how many 6-in. squares? 

9. Make sentences like this: The sum of A and 
B equals G. 

Draw a yard, a foot, and 6 in. on the blackboard. 



1. Tell all that you can about the relations of 
these lines. 

2. The yard equals how many feet ? The yard 
is how many times as long as the foot ? 

3. The foot is how many times as long as the 



92 PRIMARY ARITHMETIC. 

6-in. ? How many 6-in. in the foot ? In the 
yard? 

Problems. 1. The cost of 1 ft. of paving equals 
what part of the cost of 1 yd. ? 

2. 1 yd. will cost how many times as much as 
i of a yd.? 

3. 1 yd. will cost how many times as much as 
1 ft.? 

4. The cost of 2 ft. of molding equals what 
part of the cost of 1 yd. ? 

5. James has 3 marbles and John has f as 
many ; how many has John ? 

6. If a quart of milk costs 8/, what part of 8/ 
will a pint cost ? 

7. If a cup of sugar is used in making a cake, 
how many cups will be needed in making a cake 
3 times as large ? 

8. If the smaller cake is enough for 1 lunch, 
the larger is enough for how many lunches ? 

9. If 3 yds. of tape cost 24/, what part of 24/ 
will 2 yds. cost ? 

10. This line represents the cost 

of 1 yd. of cloth ; draw a line to represent the cost 
of | of a yd. 

11. This line represents the cost of 6 in. 
of ribbon ; draw a line to represent the cost of 
1 ft. Of 1 yd. 

12. If $2 is the cost of ^ of a ton of coal, what 



PKIMARY ARITHMETIC. 93 

is the cost of 1 ton of coal ? Show relative cost 
by drawing two rectangles. 

13. This line represents the cost of 2 ft. ; 

draw a line to represent the cost of 1 yd. 

14. 2 ft. of cord cost 6/. The cost of 1 yd. 
equals how many halves of 6/ ? 

Draw a square foot on the blackboard. 

1. Show the perimeter of the 
square foot. What have you 
shown ? How many feet in the 
perimeter of the square foot ? 

2. How many 6-in. lines in 
one edge of the square foot ? In 
the perimeter of the square foot? 

Ratios of length. Draw a foot on the blackboard. 
Draw a 4-in. line. Pupils practise drawing and meas- 
uring these lines. 

1. Tell all you can about these lines. 
Give the pupils the names of these lines. 

2. What is the name of the longer line ? What 
is the name of the shorter line ? 

3. Into how many 4-in. can a foot be divided ? 

4. 4 in. and how many 4-in. equal 1 ft. ? 

5. 2* 4-in. and how many inches equal 1 ft. ? 

6. Show the part of a foot that equals 4 in. 

7. What part of a foot equals 4 in. ? 

8. What part of a foot equals 6 in.? 



94 PRIMARY ARITHMETIC. 

9. 4 in. equal what part of a foot ? 

10. A foot is how many times as long as 4 in. ? 
As 6 in. ? 

11. How many 4-in. in a foot ? 

12. Show me f of a foot. How many 4-in. in f 
of a foot ? 

13. If we call 4-in. 1, what should we call a 
foot? 

14. If a foot is 3, what is 4 in. ? 

15. If 4 in. is , a foot is how many thirds ? 

16. Show the part of a foot that is 2 times as 
long as 4 in. 

What part of a foot is 2 times as long as 4 in. ? 
What part of a foot equals 6 in. ? 
Keview without observing the lines. Have pupils prac- 
tise placing dots 1 ft. apart. Six inches apart. 

Relative size. Let pupils handle solids which repre- 
sent 1, 2, 3, and 4. Call them A, B, C, and D. 







1. What is the name of the largest solid ? Of 
the smallest? Of the next to the largest? Of 



PRIMARY ARITHMETIC. 95 

the next to the smallest ? Give the names in 
order, beginning with the smallest. What is the 
name of the unit that is three times as large 
as At 

2. Tell all you can about the units. 

Let it be the constant practice first to permit the pupils 
to see what they can. The questions of the book are to 
aid the teacher and not to enslave the pupil. Questions 
have their value, but when they force details upon a mind 
unprepared for them, when they destroy the significance 
of the whole, when they limit individual seeing, when they 
interfere with the relating, unifying action of the mind, 
they are intellectual poison. 

3. Into how many A's can you divide each 
unit? 

4. Each unit equals how many As ? D equals 
how many .Z?'s ? 

5. Place A and B together. The sum of A 
and B equals what unit ? 

6. Place A and C together. The sum of A and 
C equals what unit ? 

7. The sum of A and C equals how many 5's ? 

8. Place B and D together. How many (7's in 
the sum of B and D ? 

9. The sum of C and B equals how many A'a ? 
10. The unit B is how many times as large as 

A ? The unit C equals how many times A ? The 
unit D equals how many times B ? The unit D 
equals how many times A ? 



96 



PRIMARY ARITHMETIC. 



11. Show me f of .C. The unit D is how many 
times as large as f of C ? 

12. What part of D equals A ? What part of 
C equals A ? Show the part of D that is as 
large as A. 

13. Show me f of C. What part of C is as 
large as B ? 

14. If A is 2, B is how many 2's? C is how 
many 2's ? D is how many 2's ? 




1. Show me the part of D that is as large as B. 
What part of D equals 5? 

2. (7 is how many times as large as A ? Show 
me ^ of B. C is how many times as large as \ 
of B? 

3. Z) is how many times as large as A ? D is 
how many times as large as B ? Show me J of (7. 
D is how many times as large as ^ of (7? Z) 
equals how many thirds of C ? 

4. I of C equals what unit? } of C equals 
what part of B ? ^ of (7 equals J of what unit ? 



PRIMARY ARITHMETIC. 97 

5. Move your finger from the top to the bottom 
of A. Over of B. Over of C. Over of Z>. 
What is true of these four units ? What units are 
of the same size as A ? Show me again the four 
equal units. What are the names of the four 
equal units ? 

6. Move your finger over B. Over f of C. Over 
^ of D. Show me the three equal units again. 
What are the names of the three equal units ? 

7. | of C equals what unit? f of C equals 
what part of D ? 

8. If you cut D into 4 equal parts, or into 
fourths, how many of the fourths will make a 
unit as large as ' C ? f of D equals what unit ? 

Use other solids having the same relations as A, B, C, 
D. Give different names to the solids, and review. Then 
review without solids. 

1. If we call A 1, what ought we to call 5? 



2. If A is ^, what is each of the other units ? 

3. If B is f, what is each of the other units ? 

4. If A is 3, how many 3's in each of the other 
units ? 

5. If A is worth 5/, how many 5-/ are each of 
the other units worth? 

6. If A is a box which holds a quart, how 
many quarts will each of the other boxes hold ? 
How many pints will each box hold ? 



98 



PRIMARY ARITHMETIC. 



Cutting. 1. This is a 1. Cut a 1, a 2, a 3, a 4. 
You must make the 2 how many times as large as 
the 1 ? Have you made the 2 
equal to 2'1's? Measure. Have 
. you made the 3 equal to 3 times 
1, or 3 times as large as the 1 ? 
Measure. 

2. How large have you made 
the 4? 

1. This is a 2. Cut a 1, a 2, a 3, a 4. The 1 
you cut equals what part of the 2 ? 

2. The 3 you cut is how many 
times as large as ^ of 2 ? The 4 
you cut is how many times as 
large as the 2 ? 

Drawing. Let a pupil draw a unit on the blackboard, 
and others draw related units and tell what they have 
drawn. 

Relative size. Place solids having the relation of 
1, 2, 3, 4 where they can be handled. 

1. If the smallest unit is 1 ? what is the name 
of each of the other units? What is the name of 
the largest unit ? 

2. Tell all you can about the units. 

3. Tell the sums that you see. 
Ex.: The sum of 1 and 2 equals 3. 

4. Tell how much greater one unit is than an- 
other. Ex. : 4 is three greater than 1. 



PRIMARY ARITHMETIC. 



99 



5. 2 and what equal 4 ? 2 and 2 equal what? 
4 equals how many 2's ? 

6. Put 4 and 2 together. The sum of 4 and 2 
can be divided into how many 3's ? Into how 
many 2's ? 

7. The sum of 4 and 2 is how many times as 
large as 3 ? It is how many times as large as 2 ? 

Give each pupil a square rectangle 2 in. long, a rectangle 
4 in. by 2 in., a rectangle 6 in. by 2 in., and a rectangle 
8 in. by 2 in. 



1. Tell all you can about the relations of 1, 2, 
3,4. 

2. In each of the rectangles 2, 3 ? and 4, cover 
all except the part equal to 1, and tell what part 
is equal to the 1. 

3. Show all the parts that are 2 times as large 
as 1 and give the name of each. 

4. Look for units that are equal to ^ of other 
units. 

5. Estimate the dimensions of each of the rect- 
angles ; i.e. tell how long and wide you think they 
are. Measure. State the dimensions. Without 
observing the rectangles tell the dimensions of 
each. 



1. Place the rectangles 1 and 2 together, 
sum of 1 and 2 equals what unit ? 



The 



100 PRIMARY ARITHMETIC. 

2. Show the part of the unit 4 equal to the sum 
of 1 and 2. What part of 4 equals the sum of 1 
and 2? 

3. Place the rectangles together and make one 
rectangle of the four. Show the f of this rect- 
angle. The sum of what units makes ^ of the 
rectangle ? What units make the other half ? 
The sum of 2 and 3 equals the sum of what other 
units ? 

4. 2 equals what part of 4 ? How many 4's do 
you see in the rectangle ? Can you find 2^'4's in 
this rectangle? Show me the 2'4's. Show me 
the half of 4. Point to the 2^'4's. 

5. The large rectangle can be made into how 
many rectangles as large as 3 ? Show me the 
3'3's. What part of another 3 do you see ? 

6. Into how many 2's can the rectangle be 
made ? 

7. If we should call the square 2, what ought 
we to call each of the other rectangles ? If we 
should call the square ^, what number of halves 
would we see in each of the other rectangles ? If 
there are 4 sq. in. in the square rectangle, how 
many 4-sq.-in. in each of the other rectangles ? 

Draw the rectangles on the blackboard to the scale of 
3 in. to the inch. 

8. Tell all you can about the rectangles 1, 2, 3, 
and 4, drawn on the blackboard. 



PRIMARY ARITHMETIC. 101 

Problems^ 1. 4/ will buy how many times as 
many marbles as 2/ ? 

2. If you can buy a barrel of flour for $4, how 
much can you buy for $3 ? 

3. If 1 yd. of cloth costs $3, how much cloth 
can be bought for $4 ? 

4. If of a basket of fruit is worth 25/ ? how 
many 25-/ is the basket of fruit worth ? 

5. If 1 doz. eggs costs 15/, how many dozen 
can be bought for 4'15/ ? 

6. If |- ton of coal lasts 1 week, how long will 
1 ton last ? 

7. If 1 Ib. of butter lasts a family 1 week, what 
part of a week will f of a Ib. last ? 

Show pupils units that represent 1, 2, 3, 4. 




1. If we callJ. 1, what is (7? What is 
What is B ? 



If pupils cannot give the names (J, J, f , 1), tell them. 

2. Show me the 1. The f The f The f 

3. What is the name of the largest unit? Of 



102 PRIMARY ARITHMETIC. 

the smallest ? Of the next to the smallest ? Of 
the next to the largest ? 

4. What are the names of these units ? 

5. Pat % and | together. The sum of and 
equals what ? 

6. Put | and f together. The sum of and f 
equals what ? 

7. What part of the unit 1 is as large as the J ? 

8. What part of the J equals the | ? 

9. Show the part of the f that equals the % ? 
What part of the f equals the ^ ? 

10. | and what equal f ? 

11. 1 is how much more than ^ ? 

12. | and what equal 1 ? 

13. The unit 1 is how many times as large as 



14. The unit 1 is how many times as large as 
the unit % ? 

15. The unit ^ is how many times as large as 
the unit ? 

16. If the ^ weighs 5 oz. ? how many 5-oz. do 
each of the other units weigh ? 

Eeview without observing the units. 

Building. 1. Build a unit equal to |- of this 
one. 

2. Build another equal to J. 

3. Another equal to ^, Another equal to |. 
Show a different unit. 



PRIMARY ARITHMETIC. 103 

4. Build a unit equal to | of this one. 

5. Build the . Build the f Build the f . 

Cutting. Give each pupil a rectangle. Call it 1. 
Cut another 1. . Cut . Cut . Cut f . 

Relation of gallon and quart. Eeview lesson on quart 
and pint. Have pupils practise rilling the gallon measure. 




Empty it. Fill it full. Empty it. Fill it full. 
Empty it. Fill it f full. After measuring, have pupils 
tell all they can about the gallon and quart. 

1. What is sold by the gallon ? 

2. A gallon is how many times as much as a 
quart ? 

3. What part of a gallon equals 1 qt. ? 

4. If you should make 2 equal parts of a gallon, 
how many quarts would there be in each ? How 
many quarts in | gal. ? 

5. A gallon is how much more than a quart ? 



104 PRIMARY ARITHMETIC. 

6. How many quarts must be added to half a 
gallon to make a gallon ? 

7. 1 qt. equals what part of a gallon ? 

8. If we call a quart 1, what ought we to call 
a gallon ? If we call the gallon. 1, what ought we 
to call the quart ? 

9. A quart equals how many pints ? A gallon 
measure will hold how many quarts ? A gallon 
measure will hold how many 2-pts. ? 

10. In 3 gals, there are how many 4-qts. ? 

Relation of gallon, quart, and pint. 1. If a pint 
of water weighs 1 lb., how much does a gallon 
weigh ? 

2. If a quart of milk costs 6/ ? what part of 
6/ will a pint cost ? 

3. If a quart of milk costs 6/, how many 6-/ 
will a gallon cost ? 

.4. If 1 gal. of milk costs 24^ ? what part of 24/ 
will 1 qt. cost? 

5. If \ of a gal. of milk costs 6^, how many 6-/ 
will a gallon cost? How many 6-/ will |- of a gal. 
cost ? How many 6-/ will f of a gallon cost ? 

6. The cost of 3 quart boxes of berries is 25/. 
The cost of 4 boxes equals how many thirds of 
25/? 

Relations of magnitude. Place units representing 2, 
4, 6, 8 where they can be handled. Teach the names 2, 4, 
6, 8. If the children know the number relations of 2, 4, 
6, and 8, use letters instead of numbers. 



PRIMARY ARITHMETIC. 105 

1. What is the name of the smallest unit ? Of 
the largest ? What is the name of the smaller of 
the other two ? Of the larger ? Name the units 
in order, beginning with the smallest. Name 
them in order, beginning with the largest. 

2. Put two units together and tell what the 
sum equals. 

Ex. : 4 and 2 equals 6. 

3. Tell how much less one unit is than another. 

4. The sum of 6 and 2 equals what ? The sum 
of 4 and 4 equals what ? The sum of 4 and 2 
equals what ? 

5. 4 less 2 equals what ? 8 less 4 equals what? 
6 less 4 equals what ? 8 less 2 equals what ? 2 
and what equal 4 ? 4 and what equal 8 ? 4 and 
what equal 6 ? 2 and what equal 8 ? 

6. How many 2's in each unit ? 

7. Make sentences like this : 2 equals ^ of 4. 

8. Make sentences like this : 8 equals 4 times 2. 

9. Tell the part of 4, of 6, of 8 ? that is as large 
as 2. Tell the part of 6 and of 8 that is as large 
as 4. Tell the part of 8 that is as large as 6. 

10. 4 equals how many times 2 ? 4 equals 
what part of 6 ? Of 8 ? 

11. 6 equals how many times 2 ? It equals 
how many times ^ of 4 ? It equals how many 
times | of 8 ? 6 is 3 times as large as what unit ? 

12. 8 equals how many times 2 ? How many 
times ^ of 4 ? How many times ^ of 6 ? 



106 PRIMARY ARITHMETIC. 

13. 2 is how many times as large as 1 ? 4 is 
how many times as large as 2 ? 8 is how many 
times as large as 4 ? 

14. What unit is 2 times as large as 1 ? As 2 ? 

As 4? 

1. Find other sets of solids that may be called 
2, 4, 6, 8. Tell all the relations that you can. 

2. Find surfaces that may be called 2, 4, 6, 8. 
Tell the relations. 

3. Find edges that may be called 2, 4, 6, 8. 
Tell the relations. 

4. Make statements like this : If we call this a 
2, we should call this 4, for it is 2 times as large 
as the 2. 

5. Make statements like this : If this is 8, then 
this is 4, for it equals ^ of 8. 

6. Call the blackboard 8. Show the part that 
is as large as 4. As large as 2. As large as 6. 

7. Make statements like this : If we call the 
edge of this table 2, we must have an edge 2 times 
as long if we wish to call it 4. 

Cutting. 1. This is a 2. Cut a 2, a 4, a 6, 
and an 8. Measure to see if you 
have made each unit the right 
size. 

2. Try again. Cut a large 
rectangle and call it 2. Cut a 4, 
a 6, and an 8. Measure. 



PRIMARY ARITHMETIC. 107 

Drawing. 1. Draw a rectangle. Call it 2. 
Draw a 4, a 6, and an 8. 

Problems. 1. If you can clean the blackboard 
in 8 minutes, what part of the board can you clean 
in 4 minutes ? In 2 minutes? In 6 minutes? 

2. The money that you pay for 4 apples equals 
what part of the money that you pay for 6 apples? 
For 8 apples ? 

3. If 2 Ibs. of candy cost $1, how much will 
'8 Ibs. of candy cost? 

4. How many times as long will it take to walk 
8 miles as to walk 2 miles ? 

5. If it takes 2 hours to walk 8 miles, how long 
will it take to walk 4 miles ? 

6. 6 yds. of ribbon will cost how many times as 
much as 2 yds. ? The cost of 4 yds. equals what 
part of the cost of 6 yds. ? 

7. 1 yd. of ribbon will cost how many times as 
much as ^ yd. ? How many times as much as 
iyd.? 

8. A gallon measure holds how many times as 
much as a quart ? 

9. If a quart of molasses costs a dime, how 
many dimes will a gallon cost ? 

10. 6 baskets of apples cost 75/. What part 
of 75/ will 2 baskets cost? What part will 4 
baskets cost ? 

11. 8 hours equals how many thirds of 6 hours? 
The distance you can walk in 8 hours equals how 



108 PRIMARY ARITHMETIC. 

many thirds of the distance you can walk in 6 
hours ? 

Comparing surfaces Give each pupil a square 2 in. 

long, a square 4 in. long, and a rectangle 2 in. by 4 in., and 
one 2 in. by 6 in., or draw figures on the blackboard of the 
same relative size. 

1. Use the small square as a measure and tell 
what you can about the relations of the rect- 
angles. 

2. Teach the names A, B, C 9 and JD. If we call 
A 2, what ought we to call each of the others ? 

3. Call A 1 ; what is the name of each of the 
others ? 

4. Call A i ; what ought we to call each of the 
others? 

5. If A is ^, how many fourths in each of the 
others ? 

6. If C is 1, A equals what part of 1 ? B 
equals what part of 1 ? D is how many times as 
large as the third of 1 ? 

7. Call D 1-, B equals what part of another 1 ? 
A equals what part ? C equals what part ? 

8. If C is 3, what is .? What is A ? What 
isD? 

9. If you can make 3Ts of A, how many Ts 
can you make of B ? Of C ? Of D ? 

10. What is the length of A ? How many 2-in. 
in the perimeter of A ? Of B ? Of D ? 



PRIMARY ARITHMETIC. 109 

Ratios of length. 1. Practise drawing a foot. 
Practise placing points 1 ft. apart. Try to draw 
lines a foot long with eyes closed. 
Measure. With your eyes closed try 
to place points 1 ft. apart. Measure. 

2. Draw a foot. Draw a line equal 
to |- ft. To | ft. To | ft. Practise 
drawing and measuring groups of 
these lines. 

3. If we call the shortest line 1, what ought we 
to call each of the other lines ? 

4. If the shortest line is ^, what is each of the 
other lines ? 

Ans. : 1, f, 2. 

5. If the shortest line is , what is each of the 
other lines ? 

6. Call the next to the shortest line 1 ; what is 
the name of each of the other lines ? 

7. If the shortest line is , find the 1. If the 
longest line is 1, find |. 

8. If the longest line is ^ what part of ^ is 
each of the other lines ? 

9. Call the longest line 12 ; what part of 12 is 
each of the other lines ? 

10. Call the shortest line 3; how many 3's in 
each of the other lines ? 



110 



PRIMARY ARITHMETIC. 



Have pupils assign different values to the units and tell 
what the other units are. Ex. : Call A 1 ; what is each of 
the others ? 




Have pupils compare different units with the other 
units. Ex. : A equals 2 times B, f of (7, | of D, etc. 

Draw lines on the blackboard 1 ft., 9 in., 6 in., 3 in. 
long. Teach the names of the lines. 

1. What is the name of the long- 
est line "I Of the shortest ? Of the 
line that is -J ft. long ? Of the line 
that is f ft. long ? 

2. Name the lines in order, be- 
ginning with the shortest. Repeat, 

Name in order, beginning with the longest. 



ARITHMETIC. Ill 

3. Make sentences like this : The sum of 6 in. 
and 3 in. equals 9 in. 

4. The 3-in. line equals what part of each of 
the other lines ? 

5. The 6 -in. line equals how many times the 
3-in. line ? It equals what part of each of the 
other lines ? 

6. Compare the 6-in. line with each of the 
other lines again. 

7. The 9-in. line equals how many times the 
3-in. line ? It equals how many halves of the 6-in. 
line ? It equals what part of the foot ? 

8. The ft. equals how many times the 3-in. line ? 
It equals how many times the 6-in. line ? Show ^ 
of the 9-in. line. The ft. is as long as how many 
thirds of the 9-in. line ? It equals how many thirds 
of the 9-in. line ? 

9. A foot is how many times as long as 6 
inches ? A foot is how much longer than 6 inches ? 
6 inches equal what part of a foot ? 4 inches 
equal what part of a foot ? A foot equals how 
many 4-inches ? What part of a foot is as long 
as 8 inches ? 

10. What is J ft. ? What is % ft. ? What is 
f ft. ? What is i of 6 in. ? What is | of 9 in. ? 
What is f of 9 in. ? What is f of a ft. ? Picture 
the lines in your mind and tell all you can about 
them. 

Review without observing the lines. 



PRIMARY ARITHMETIC. 



Problems. 1. If it takes all your money to pay 
for a loaf of bread the size of B, what part of your 
money will it take to pay for a loaf the size of D ? 
Of C ? Could you pay for a loaf the size of A ? 
The money you have would pay for a loaf three 
times as large as what part of A ? 




v 2. | of B is worth a nickel ; B is worth how 
many nickels ? A is worth how many nickels ? 
For C you must pay how many times as much as 
for I of C ? As for \ of A ? The cost of D equals 
what part of the cost of each of the other units ? 

3. Call the blocks cakes. If C is enough for six 
people, A is enough for how many people ? D for 
what part of twelve people ? If D is enough for 
three people, B will supply how many ? 

4. One dollar will buy 8 First Readers. What 
part of one dollar will pay for 6 First Readers ? 
For 4 ? For 2 ? 

5. A gallon of oil will cost how many times as 
much as \ of a gallon ? 

6. The cost of 2 Ibs. of raisins equals what part 
of the cost of 8 Ibs. ? 



PRIMARY ARITHMETIC. 



113 



1 doz. 

oooooo 
o o o o o o 



1 doz. 
oooo 

0000 

oooo 



1. How many 6's in a doz. ? 

2. How many 4's ? 3's ? 2's ? 

Give each pupil a rectangle 3 in. by 4 in., another 2 in. 
by 4 in., and a third 1 in. by 4 in. 




1. If M is a doz., what part of a doz. is each of 
the other units ? 

2. Show ^ of a doz. f of a doz. f of a doz. 

3. A doz. is how many times as large as | of 
a doz. ? 

4. M equals how many halves of D ? 

5. A doz. equals how many halves of f of a doz. ? 
Call C 4 ; what is D ? What is Jf ? 

6. 4 equals what part of 8 ? Of a doz. ? 

7. 8 equals how many times 4 ? It equals what 
part of a doz. ? 

8. A doz. equals how many times 4 ? It equals 
how many halves of 8 ? Which is the more, 4 or 
JofS? 



114 PRIMARY ARITHMETIC. 

9. What two equal units in 8 ? What three 
equal units in a doz. ? 

Keview, using the rectangles. Eeview without them. 

Problems. 1. A doz. oranges will cost how 
many times as much as 6 oranges ? As 4 ? 

2. The cost of 9 oranges equals what part of 
the cost of a doz. ? 

3. How many 3's in a doz. ? How many 4's ? 
How many 6's ? 

4. If 6 pens cost a dime, how many dimes will 
a doz. pens cost ? 

5. One half doz. pears will cost how many 
times as much as ^ of a doz. ? 

6. A doz. eggs cost 15/. 4 eggs cost what 
part of 15/ ? 8 eggs cost what part of 15/ ? 

7. A doz. bananas will cost how many times as 
much as 4 bananas ? 

8. The cost of | of a doz. pencils equals what 
part of the cost of f of a doz. ? 

9. The cost of 8 buttons equals what part of the 
the cost of a doz. buttons ? 

10. One doz. is how many more than 8 ? 4 is 
how many less than one doz. ? 

11. 6 and what equal a doz. ? 8 and what equal 
a doz. ? 4 and what equal a doz. ? 3 and what 
equal a doz. ? 9 and what equal a doz. ? 

Relation of rectangles. Give each pupil a square rect- 
angle 3 in. long, a rectangle 3 in. by 4 in., a rectangle 
2 in. by 3 in., a rectangle 1 in. by 3 in. 



PRIMARY ARITHMETIC. 



115 



1. What are the names of the rectangles in the 
order of their size ? 





2. Tell all you can about the rectangles. 

3. If H represents a doz. ? what part of a doz. 
does each of the others represent ? 

4. The sum of B and D equals what unit ? It 
equals what part of a doz. ? 

5. B equals what part of each of the other units ? 

6. What is the relation of D to each of the 
other units ? Of C ? Of HI Of the doz. ? 

7. If B is 3, what is each of the other units ? 

8. How many 3's in 6 ? In 9? In 12, or a doz. ? 

9. What is the relation of 3 to each of the other 
units? Of 6? Of 9? Of 12? Of a doz. ? 

10. If H is worth 10/, what part of 10/ is each 
of the others worth ? 

11. If B cost 5/ ? what is the cost of each of the 
others ? 

12. If 3 cost 5/, how many 5/ will 6 cost? 
How many 5/ will 9 cost ? 1 doz. ? 

Eeview, using the rectangles. Eeview without them. 
Drill. 



116 PRIMARY ARITHMETIC. 

Ratios of time. 1. How long is it from Christ- 
mas to the next Christmas ? From one birthday 
to the next ? 

2. Draw two lines, one representing a yr., and 
the other J yr., or 6 mos. 

3. Tell all you can about the yr. and 6 mos. 



4. How many 6-mos. in a yr. ? What part of a 
yr. equals 6 mos. ? 1 yr. and 6 mos. equal how 
many 6-mos. ? 

5. How many 6-mos. in 1^ yrs. ? 6 mos. equal 
what part of 1^ yrs. ? 1 yr. equals what part of 
l^r yrs. ? How many 6-mos. in | of a yr. ? 

Draw three lines, one representing a year, one -J- of a 
year, and the other f of a year. 

1. If the shortest line represents 4 mos., how 
many 4-mos. does each of the other lines represent ? 
How many months does 

each of the other lines re- 
present ? How many thirds 
of a yr. does each of the 

lines represent ? 

2. Compare 4 mos. with 8 mos. ; with a yr. 

3. Compare 8 mos. with 4 mos. ; with a yr. 

4. Compare 1 yr. with 4 mos. ; with 8 mos. 



PRIMARY ARITHMETIC. 117 

Problems. 1. The money that Harry can earn 
in 6 mos. equals what part of the money that he 
can earn in a yr. ? In 8 mos. ? 

2. The number of months in 1 yr. equals how 
many times the number in J yr. ? In ^ yr. ? 

3. The number of days in 3 mos. equals what 
part of the number in 4 mos. ? In 6 mos. ? In 
9 mos.? In 1 yr. ? 

4. One yr. is how much longer than 8 mos. ? 

5. The time from New Year to New Year equals 
how many halves of 8 mos. ? 

Relation of dime and nickel. 1. If A represents 
a dime, what is the name of the piece of 
money represented by B ? 

2. How many nickels equal 1 dime ? 

3. A dime and a nickel equal how many 
nickels ? 

4. A nickel equals what part of a dime ? 

5. The candy you can buy for a nickel 
equals what part of the candy you can buy for a 
dime? 

6. A nickel equals how many cents ? 

7. A nickel and how many cents equal a dime ? 

8. A dime equals how many 5/ ? 

9. 5/ equals what part of a dime ? 

10. A dime and 5/ equals how many 5/ ? 

11. 1^- dimes equal how many 5/ ? 

12. 5/ equals what part of 1^ dimes ? 

13. A dime equals what part of 1 J dimes ? 



118 PRIMARY ARITHMETIC. 

Relative values. - - 1. If the shortest line 
represents 2/, what do each of the other lines 
represent ? 

2. Point to the different lines and tell what 
each represents. 

3. 2/ equals what part of 4/ ? Of 6/ ? Of 8/ ? 
Of a dime ? 

4. Compare 4/ with each of the other units. 
Compare 6/ with each. Compare 8/ with each. 
Compare a dime with each. 

Problems. 1. 2/ will buy an apple. 4/ 

will buy how many apples? 

. How many will 6/ buy ? A 

dime ? 

2. A boy sells papers for 2/ 
each. How many does he sell 
2 ft to receive a dime ? 

3. 2/ is ^ of Nellie's money. How much 
money has she ? 

4. John has 10/ and loses ^ of it ; how much 
does he lose ? How many 2/ has he left ? 

5. 4 peaches equal what part of 6 peaches ? 
Of 8? Of 10? 

6. f of a Ib. of cheese cost 7/. How many 7/ 
will | of a Ib. cost ? 

7. If f of a doz. pencils cost 6/ ? what is the 
cost of a doz. ? 



PRIMARY ARITHMETIC. 



119 



Ratios of solids. Place solids which represent 1, 2, 
3, 4, 5 where they can be handled. 




1. Learn the names A, B, C, D, and E. 

2. Tell all you can about these units. Tell all 
you can about these units without looking at them. 

3. Unite different units and tell what they equal. 
Ex. : The sum of A and B equals C. 

4. Make statements like this : E less A equals D. 

5. B and what equals C ? A and what equals 
C ? C and what equals D ? B and what equals D ? 

6. Look at B. How many As equal B ? What 
part of C equals B ? What part of D equals B ? 
What part of E equals B ? 

7. D equals two times what unit ? It is two* 
times as large as what part of C ? It is two times 
as large as what part of E ? 

8. What part of C is two times as large as A ? 
What part of D is two times as large as A ? 



120 PRIMARY ARITHMETIC. 

9. B equals how many times At It equals 
what part of each of the other units ? 

10. C equals how many times ^ of Dt How 
many times ^ of B ? It equals what part of each 
of the other units ? 

11. D equals how many times At D equals 
how many times Bt D is how many times as 
large as f of C t D equals two times what part 
of Et 

1. E equals D and what part of another Dt 
E equals C and what part of another C ? E equals 
how many B's ? Am. : E equals 2|- j?'s. E equals 
how many ^L's ? 

2. Call E 1 ; each of the other units is what 
part of another 1 ? 

3. Call D 1 ; what is each of the other units ? 

4. Call C 1 ; what is each of the other units ? 

5. Call A 1 ; each of the other units equals 
how many times 1 ? 

6. Call A 2 ; what is each of the other units ? 

7. Call A I ; what is each of the other units ? 
Ans. : B is , C is f , D is 1 and E is 1 \. 

8. Call A | ; what is each of the other units ? 

9. Call A \ ; what is each of the other units ? 
* 10. Call A 3 ; what is each of the other units ? 



PRIMAKY ARITHMETIC. 



121 



Drawing and cutting. 1. Draw a line. Divide 
it so that one part will represent the unit 2, and 
the other the unit 3. Measure. 

2. Draw a rectangle. Divide it so that one 
part will represent 2, and the other 3. Measure. 
Practise. 

3. Cut rectangles. Divide them so that they 
will represent the unit 2 and 3. Measure. Practise. 
Practise drawing, dividing, and measuring. 



Relative areas. 1. Cut the units 2, 4 ? 3, 5, 1, 2. 

2. Measure 1 each by 2, and tell how many 2's 
in each. 

Ex. : In 5 there are 2f 2's. In 1 there is of 2. 

3. Tell how much more one unit is than another. 
Ex. : 4 is three more than 1. 4 is two more than 2. 

4. Tell how much less one unit is than another. 

5. Unite units and tell what the sum equals. 
Ex. : The sum of 2 and 1 equals 3. 

1 First, make estimates with the eye. Afterward test judg- 
ments by using a measure. 




122 PRIMARY ARITHMETIC. 

6. What two units equal 4 ? What other units 
equal 4 ? 

7. What two units equal 5 ? What other units 
equal 5 ? What three units equal 5 ? 

8. If the unit A is 1, each of the other units 
equals how many halves ? 

9. Compare each unit with the unit 2. 

10. If the unit 2 is worth a dime, what is each 
of the other units worth ? 

11. Draw units, making each two times as large 
as the 2, 4, 3, etc. Measure to see if you have 
made the units two times as large. Write the 
names 2, 4 ? etc. 

12. Tell all you can about the units you have 
drawn. 

Eeview. 

"The starting point is, constantly and necessarily, the 
knowledge of the precise relations, i.e. of the equations, 
between the different magnitudes which are simultaneously 
considered." - Comte. 

1. The unit 3 is how much more than the 
unit 1 ? 1 is how much less than 3 ? 3 apples are 
how many more than 1 apple ? 

2. 4 is how much greater than 2 ? 2 and 2 
equal what ? 4 is how many times as large as 2 ? 
2 equals what part of 4 ? 

3. 5 is how much greater than 1 ? Than 3 ? 
Than 2 ? Than 4 ? What must be added to 3 to 
make 5? To 1 to make 5? To 2 to make 5? 



PRIMARY ARITHMETIC. 



123 



5 pens are how many more than 3 pens ? Than 
2 pens ? 

4. The sum of 3 and 2 equals what? Of 1 
and 4 ? Of 2 and 2 and 1 ? Of 1 and 2 and 2 ? 
Of 3 and 2 ? 

5. Henry paid 3/ for candy and 2/ for nuts ; 
how much did he pay for both ? 

6. Nellie spent 5/ for pears and 2/ for pins; 
how much more did she pay for the pears than for 
the pins ? 

Separating and combining. 1. How many 1's 
do you see in this diagram ? How many 2's ? 
How many 3's ? How many 4's ? 5's ? 



2. If d is 2, what is the name of the units 
under each letter ? 

3. Unite the two units under each letter and 
think the unit to which the sum is equal. 

Ex. : Look at the units under e and think Jf,. 

4. Look at diagram and name sums. 



124 PRIMARY ARITHMETIC. 

Ex. : Look at the two units under e and say . 

5. Draw units on the blackboard and have 
pupils practise thinking sums. 

6. After observing the units carefully, turn 
away from them and pronounce the sums under 
each letter. 

The expression for quantitative ideas should be acquired 
as the everyday vocabulary has been, by repeatedly 
bringing into consciousness the relations which the terms 
express. 

As the pupil advances, sight forms should suggest ideas, 
just as spoken words do. But as reading should be ap- 
proached through sense training, an interest in things, and 
the power to talk freely, so should the use of written 
forms in mathematics. At the proper time the teacher 
should find occasion to present the written expression 
freely and in such manner that the primary attention is 
still held to the relations discerned. Gradually the use of 
language in mathematics should become as automatic as 
the use of language in other subjects. 

The principles which govern practice in aiding a child' 
to think in symbols apply in mathematics as elsewhere. 
For example, when we wish to acquaint the child with the 
written symbols for his thought of the color of a black 
dog, we write, " The dog is black." So, when a pupil tells 

3 
you that 3 and 2 equal 5, write 2 so that his eye may take 

5 

in the expression as a whole. We should represent the 
complete, not the partial thought of the pupil, the 
equation, not a part of it. Fix the thought so firmly 
that finally one side of the equation will suggest the 
other. 



PRIMARY ARITHMETIC. 



125 



Ask the following questions, and write answers on the 
blackboard : 



3 and 1 equal what ? a b 

1 and 2 equal what ? 31 

2 and 2 equal what ? _? 1 
2 and 3 equal what ? 52 

3 



Answers, 
c 
2 
2 
4 



d 

2 
1 
3 



e 
1 
3 
4 



1 

Observe 1 : 
2 

2 



image ; write ; 
practise. 



Observe 2; image; write; 
4 practise. 



Observe 1 2 ; image; 

5 write. 

3 T 
Observe 2 1; image 

5 2 write. 

312 
Observe 212; image ; write ; practise. 

524 

Continue adding one combination at a time, until the 
pupils can image and write the five readily. 

Tell the combination under each letter, thus : 2 
is under a. 5 

What combination is under c ? Under 6? Under &? 

Show the combination at the left. The second 
from the right. Image and think each combina- 
tion with its sum, beginning at the top. 

Image each combination and pronounce the sum. 

1 " The habit of hasty and inexact observation is the founda- 
tion of the habit of remembering wrongly." Dr. Maudsley. 

" A few such items must be memorized and reviewed daily, 
adding a small increment to the list as soon as it has become per- 
fectly mastered." W. T. Harris, 



126 PRIMARY ARITHMETIC. 

Continue to work with these five combinations until 
they are indelibly fixed. 

31222 
Write on blackboard : 22213 

Think the sum of each. Pronounce the sum. 
Do not say 3 and 1 are 4, nor 3, 1, 4 ; but 

observe g and say 4. 

Name sums from right to left, without observ- 
ing the board. From left to right. What is the 
second sum from the right ? The third from the 
left? etc. 

Make columns of the combinations, omitting sums, thus : 

abed Have pupils look at each 

column carefully and image 

3211 the sum of each combination 

2213 of two figures. Picture, slow- 

ly at first, the combinations 

1321 under a: 5, 2, 4, 3, 4, then 

1221 more quickly, but not so 

quickly as to destroy the 

2213 visual image. 

2122 It will be easy to secure 

rapidity after the habit of 

2132 imaging has been established. 

1312 Image, 1 beginning at the 

bottom. 

1131 Image from right, thus : 

3 1 2 2 4, 2, 4, 5. 

1 " There can be no doubt as to the utility of the visualizing 
faculty where it is duly subordinated to the higher intellectual 



PRIMARY ARITHMETIC. 



127 



1. Measure each unit by 2. 1 By 3. By 4. 

2. If c is 2, what is the name of each of the 
other units ? 

3. On each unit that you draw or cut, write the 
name. 

4. Tell all you can about these units. 

f 



5. What two units are as large as 6 ? 

6. Into what two equal units can you separate 4? 
7* What two equal units in 6 ? What three 

equal units in 6 ? What two unequal units do you 
see in 6 ? 

operations. A visual image is the most perfect form of mental 
representation wherever the shape, position, and relations of 
objects in space are concerned." Francis Galton. 

"Addition, as De Morgan somewhere insisted, is far more 
swiftly done by the eye alone ; the tendency to use mental words 
should be withstood." Francis Galton. 

1 Do not permit counting. Wait until the pupil observes and 
becomes conscious of the relative size of the units. 



128 PRIMARY ARITHMETIC. 

8. The unit 7 is how much larger than the 
unit 4 ? Than the unit 3 ? 4 is how much 
less than 7 ? What must be taken out of 7 to 
leave 4 ? 

9. 4 and what equal 6 ? 2 and what equal 6 ? 

10. 4 and what equal 7 ? 3 and what equal 7 ? 

11. 6 and what equal 7 ? 2 and what equal 7 ? 

12. 5 and what equal 7 ? 

13. 7 cherries are how many more than 3 cher- 
ries ? Than 5 ? Than 2 ? Than 1 ? 

14. The sum of 3 and 3 equals what ? 6 equals 
how many times 3 ? 3 equals what part of 6 ? 
6 is how much greater than 3 ? 3 is how much 
less than 6 ? 

15. Cora paid 5/ for paper and 2/ for a pencil. 
How much did she pay for both? How much 
more did she pay for the paper than for the pencil ? 
How much more for both than for the paper ? 
The cost of the pencil equals what part of the cost 
of the paper ? 

16. If C is a rug containing 2 square feet, how 
many square feet in each of the other rugs ? 

17. Call C 1. What is the name of each of the 
other units ? 

18. If the width of E is 1 foot, what is its 
perimeter ? How many more feet in the perimeter 
of E than in the perimeter of C ? 

19. Call C J. What is the name of each of the 
other units ? 



PRIMARY ARITHMETIC. 

Draw units on the blackboard. 



129 



1. How many 1's do you see in this diagram? 
How many 2's ? How many 3's ? How many 4's ? 
How many 5's ? 

2. Practise looking at the diagram and thinking 
the sums. 

3. Look and name the sums. 

4. Think and name sums without looking at 
the blackboard. 

Ask the following questions, and write answers on the 
blackboard : 



3 and 3 equal what ? 

2 and 5 equal what ? 

4 and 1 equal what ? 
4 and 2 equal what ? 

3 and 4 equal what ? 
(See method of study on 

pages 125, 126.) 



4 
3 

7 



Answers. 
215 
442 
657 



3 
3 
6 



130 



PRIMARY ARITHMETIC. 



Use any set of solids having the relation of 1, 2, 3, 4, 5. 

1. If 2 is the name of the smallest unit, what 
is the name of each of the others ? 

2. Give the names beginning with the smallest 
unit. Give the names beginning with the largest 
unit. 

3. Tell all you can about these units. 



10 




4. Unite different units, and tell what the sum 
equals. 

5. Make sentences like this : 8 less 6 equals 2. 

6. 4 and what equals 6 ? 2 and what equals 6 ? 
4 and what equals 8 ? 6 and what equals 10 ? 

7. Tell what two equal units are found in each 
unit. Ex. : In the unit 6 there are 2'3's. 

8. How many 2's in each unit? Each unit 
equals how many 2's ? 

9. What is the relation of 2 to each of the 
other units ? 

Use another set of solids having the same relations. 
Name them 3, 6, 9, 12, 15. Work with these units as you 
did with the 2, 4, 6, 8, 10. 



PRIMARY ARITHMETIC. 131 

Show a solid. Give it a name, and ask pupils to find a 
related solid. 

Ex. : This is 9; find 3. This is 10 ; find 2. 

Drawing. Draw rectangles having the relations of 4, 
8, 12, 16, 20 on blackboard. Work with these units as you 
did with 2, 4, 6, 8, and 10. 

Draw a rectangle. Give it a name. Pupils draw re- 
lated rectangles. Ex. : This is a 12 ; draw a 4. 

Draw rectangles either larger or smaller than these 
abov^e, but having the same relations. Teach these 
relations through the language 5, 10, 15, etc. 

Cutting. This is a 1. Cut a 1, 2, 3, 4, 5. 
This is a 2. Cut a 2, 4, 6, 8, 10. 
This is a 3. Cut a 3, 6, 9, 12, 15. 
Give other exercises in cutting and drawing, 
which will fix the relative sizes of these units. 

Separating and combining. 1. Measure each 
unit by 2. 

Ex. : There are If 2's in 0. 

2. How many of the units contain an exact 
number of the 2's ? 

3. Measure each unit by 3. 

4. If A is 2, what is the name of each of the 
other units ? 

5. Tell all you can about the relations of these 
units. Measure each by 2. 

Ex. : There are If 2's in 3. 

6. What units united will make 8 ? What two 
equal units in 8 ? What four equal units in 8 ? 



132 



PRIMARY ARITHMETIC. 



What two unequal units in 8 ? What other un- 
equal units in 8 ? 

1. The unit 8 is how much larger than the 
unit 6 ? Than the unit 5 ? Than 3 ? Than 4 ? 

2. How many 4's in 8 ? 8 is how much more 
than 4 ? 8 equals how many times 4 ? 4 equals 



what part of 8 ? How many 2's in 8 ? In 6 ? 
6 equals what part of 8 ? 

3. Show me the. unit equal to -f of 8. Show 
me the unit equal to | of 6. 

4. What is the sum of 4 and 4 ? Of 6 and 2 ? 
Of 4 and 3 ? Of 5 and 3 ? Of 2 and 6 ? Of 3 
and 5? 

5. 4 and what equal 8 ? 6 and what equal 8 ? 
2 and what equal 5 ? 2 and what equal 8 ? 5 and 
what equal 8 ? 3 and what equal 6 ? 3 and what 
equal 5 ? 3 and what equal 8 ? 



PRIMARY ARITHMETIC. 



133 



6. A boy had 8 marbles and lost ^ of them. 
How many had he left ? 

7. A little girl had 8 dolls. She gave of 
them to some poor children. How many did she 
give away? 

Draw the units on the blackboard. 



Have pupils practise thinking units and suras under 
each letter. 

Have pupils think and name sums without looking at 
diagram. 

Ask the following questions and write answers on the 
blackboard : 

4 and 3 equal what ? Answers. 

5 and 2 equal what ? 432 2 3 

6 and 2 equal what ? 4 j> 6 5 4 
5 and 3 equal what ? 88877 
4 and 4 equal what ? 



134 



PRIMARY ARITHMETIC. 



4 


3 


2 


3 


4 


5 


5 


5 


3 


2 


3 


2 


5 


6 


4 


6 


4 


3 


2 


2 


4 


4 


5 


6 


3 


4 


2 


2 


4 


4 


5 


6 



(See method of study, 
pages 125, 126.) 



Draw units on the blackboard. 1 

1. Draw these units. Write the names : thus, 
1, 2 ; 2, 4 ; 3, 6 ; 4, 8 ; 5, 10. 

2. Tell all you can about these units. 




3. Make sentences like this : In 6 there are 
2'3's. 

4. The sum of 4 and 4 equals what? Make 
sentences like this : The sum of 4 and 4 equals 8. 

5. One half of 6 equals what ? Make sentences 
like this : 3 = f . (Read : 3 equals of 6.) 

6. The sum of 10 and 5 equals how many 5's? 

7. If a 10 and a 5 are put together, the sum 
equals how many 5's ? Make sentences like this : 
The sum of 10 and 5 equals 3'5's. 

1 In all similar exercises the teacher should draw the units on 
the board, making them of such size that the eyes of the pupils 
will not be unduly taxed in observing them. 



PRIMARY ARITHMETIC. 135 

8. Compare each of the upper units with the 
one below it. 

Ex.: 4 = 2 times 2. 

9. Compare each of the lower units with the 
one above it. 

Tell everything you can about the units without 
observing them. 

Practise thinking the sums of the two units. 

Ex. : Look at 1 and 2 and think 3 ; at 4 and 2 
and think 6. 

Think sums without observing diagram. 

Name sums without observing diagram. 

Problems. 1. If 3 peaches cost a nickel, how 
many nickels will 6 peaches cost ? 

2. If 6 peaches cost a dime, what part of a dime 
will 3 peaches cost ? 

3. 4 books cost a dollar; what is the cost of 
8 books ? 

4. 10 Ibs. of coffee cost how many times as 
much as 5 Ibs. ? 

5. The cost of 5 Ibs. of coffee equals what part 
of the cost of 10 Ibs. ? 

6. The weight of 3 Ibs. of sugar equals what 
part of the weight of 6 Ibs. ? 

7. The cost of a pt. of milk equals what part of 
the cost of a qt. ? 

8. If 2 apples cost 4/, what part of 4/ will 
1 apple cost ? 



136 



PRIMARY ARITHMETIC. 



9. If 2 baskets of apples cost 75/ ? what part of 
the 75/ will 1 basket cost ? 

10. If John can walk to school in ^ hour, how 
long will it take him to walk to school and home 
again ? 

Separating and combining. In this work, aim to secure 
the association of the three figures in the mental picture. 
After imaging, test the mental picture by having the 
pupils supply from memory the figures denoting the sums. 
Do this in all similar exercises. 

2 4 35 
2 4 3 _5 

4 8 6 10 

2 4 

Observe 2 ; image; write. Observe 4 ; image; write; 
4 8 practise. 

24 3 

Observe 2 4 ; image; Observe 3 ; image; write. 
4 8 write. 6 

243 

Observe 2 4 3 ; image ; write ; practise. 
486 



Observe 5 ; image ; write ; practise. 

10- 



35 



24 

Observe 24 3 _5; image; write. 
4 8 6 10 



PRIMARY ARITHMETIC. 137 

Practise until pupils can write the four combinations 
easily and quickly from memory. 

Have pupils practise thinking the sums. 

1. Image and pronounce the sums 4, 8, etc. 

2. What two equal units in 4 ? In 8 ? In 6 ? 
In 10? 

3. What is f ? (Read : What is of 4 ?) 

4. What is f ? What is -\f>- ? What is f ? 

2. 4, 3, 5. 

5. Image two of each of the above figures, with 

5 
the sum. Ex. : Image_5 . Practise. 

10 

Ask pupils the following questions, and write their 
answers on the blackboard. After having written answers 
for several days on the blackboard, without calling direct 
attention to them, see if some of the brighter pupils can- 
not read the answers. The child should learn the expres- 
sion without separating it from the thought. 

Questions. Answers . 

2 == what part of 4 ? 2 = f 

3 = what part of 6 ? 3 = f . 

4 = what part of 8 ? 4 = f . 
8 = what part of 10 ? 5 ~ 1 /-. 

2 = how many times 1 ? 2 = 2 times 1 . 

4 = how many times 2 ? 4 = 2 times 2. 

6 = how many times 3 ? 6 = 2 times 3. 

8 = how many times 4 ? 8 = 2 times 4. 

10 = how many times 5 ? 10 = 2 times 5. 



138 PRIMARY ARITHMETIC. 

Draw on the blackboard. 



1. Tell all you can about these units. 

2. Measure each by 3. By 2. 

3. How many of the units can be exactly 
measured by 3 ? 

4. The unit A is equal to what part of (7? 
Of 5? Of^? 

5. If A is 3, what is the name of each of the 
other units ? 

6. Into what equal units can you separate 9 ? 

7. Into what two equal units can you separate 6 ? 

8. What three equal units in 6 ? 

9. What two unequal units in 9 ? What other 
unequal units in 9 ? 



PRIMARY ARITHMETIC. 



139 



1. 9 is how much greater than 6? Than 7? 
Than 5 ? Than 4 ? 

2. 5 is how much less than 9 ? 3 is how much 
less than 9 ? 4 is how much less than 9 ? 

3. 3 and what equal 9 ? 3 and what equal 7 ? 
7 and what equal 9 ? 5 and what equal 7 ? 2 and 
what equal 7 ? 2 and what equal 5 ? 6 and what 
equal 9 ? 6 and what equal 8 ? 

4. How many 3's in 9 ? 3 equals what part 
of 9 ? 6 equals what part of 9 ? 9 equals how 
many times 3 ? 9 is how much more than 3 ? 

5. 9/ are how many more than 5/ ? A nickel 
and how many cents equal 9/ ? 

6. A lady can dress 3 dolls in a day. How 
many can she dress in 3 days ? 

7. A house has 5 rooms on the first floor and 
4 on the second. How many rooms on the two 
floors ? 

Ratios. Draw the units on the blackboard. 



12 




U 




16 




20 




18 



1. Draw the units and write the names. 

2. Tell all you can about these units. 



140 PRIMARY ARITHMETIC. 

3. Compare each of the upper units with the 
one below it. 

4. Compare each of the lower units with the 
one above it. 

5. Make sentences like this : The sum of 12 
and 6 equals 3'6's. 

1. The number of eggs in 1 doz. equals how 
many times the number in |- doz. ? 

2. The number of pts. in 12 qts. equals how 
many times the number in 6 qts. ? 

3. The number of pts. in 6 qts. equals what 
part of the number in 12 qts. ? 

4. The number of days in 7 wks. equals what 
part of the number in 14 wks. ? 

5. The number of cents in 16 nickels equals 
how many times the number in 8 nickels ? 

6. The cost of 8 bu. of potatoes equals what 
part of the cost of 16 bu. ? 

7. Eggs are 10/ a doz, How many doz. can 
be bought for 20/ ? 

8. 9 ft. equal 3 yds. 18 ft. equal how many 
3-yds. ? 

See work on previous similar table, pages 136, 137. 

6 8 7 10 9 

_6 _8 J_ 10 _9 

12 16 14 20 18 

1, What two equal units in 12 ? In 20 ? 

In 14 ? In 16 ? In 18 ? 



(UNIVERSITY 



PRIMARY ARITlm^^^UFOR^^^ 141 

2. What is -\ 2 - ? ( Read : What is of 12 ?) 

6 8 7 10 9 
Image two of each of the above figures, with 

9 

the sum. Ex. : Image_9 ; practise. 

18 

Separating and combining. Draw these units on the 
blackboard. 

a b d 



1. Point to each and tell its name. 

2 . What is the sum of the units under each letter ? 

3. Observe A and think 9. 

4. Practise imaging A 9 B, etc., and think sum. 

5. Draw units on blackboard and practise think- 
ing sums. 



142 



PRIMARY ARITHMETIC. 



Ask the following questions, and write answers on the 
blackboard. See method of study, pages 125, 126. 



5 and 3 


equal what? 


6 and 3 


equal what ? 4 


2133 


7 and 1 


equal what ? 5 


7765 


7 and 2 


equal what ? 9 


9898 


5 and 4 


equal what ? 




5 


1 


3 


1 7 5 


2 3 




4 


6 


6 


714 


7 6 




3 


3 


1 


522 


4 6 




6 


6 


7 


477 


5 1 


Ratios. Draw the units on the blackboard. 


3 


6 




9 




12 


15 
















2 


4 


6 


8 


10 


m 














2 


3 


4 


5 



1. Draw the units and write their names. 

2. Tell all you can about these units. 

3. In each set of three units, compare each with 
the other two. Ex. : 2 = |, |. 

4 = 2 times 2, *p. 
6 = 3 times 2, *p. 



PRIMARY ARITHMETIC. 143 

4. Tell everything you- can about these units 
without observing them. 

1. At I/ each, how many postal cards can you 
buy for 3/ ? 

2. At 2/ each, how much will 3 postage -stamps 
cost? 

3. If Mary buys 3 rolls at 2/ each, how much 
must she pay ? 

4. If each edge of a triangle is 2 ft. long, how 
many ft. in the perimeter of the triangle ? 

5. If a yd. of ribbon costs 3'10/, how many 10/ 
will f of a yd. cost ? 

6. If a lady pays 5/ for a ft. of picture framing, 
how much ought she to pay for a yd. ? 

7. 6 yds. of cloth will make how many times 
as many doll's dresses as 2 yds. ? 

8. The cost of the cloth to make 4 dresses equals 
what part of the cost to make 6 dresses ? 

9. What is the relation of the cost of 6 yds. to 
the cost of 4 yds. ? 

10. 9 books will cost how many times as much 
as 3 books ? 

11. The cost of 3 marbles equals what part of 
the cost of 6 marbles ? Of 9 marbles ? 

12. The number of cents that 12 roses cost equals 
how many times the number that 4 will cost ? 

13. There are 8 pts. in 4 qts.; how many 8-pts. 
in 12 qts. ? 



144 PRIMARY ARITHMETIC. 

14. The number of ft. in 12 yds. equals how 
many halves of the number in 8 yds. ? 

15. A string 15 ft. long is how many times as 
long as a string 5 ft. long ? 

16. A string 15 ft. long is how many times as 
long as'half of a string 10 ft. long ? 

2 43 5 
2435 
2 A _5 
6 12 9 15 

Practise until the combinations can be readily written 
from memory. Try to secure, in each combination, the 
mental seeing of the three figures and their sum. 

This mental habit greatly lessens the labor of learning 
tables. 

What three equal units in 6 ? In 12 ? In 9 ? 
In 15? 

What is -V 2 - ? (Read : What is of 12 ?) What 
is | ? What is f ? What is - 1 /- ? 



Image three of each with the sum. Example : 
5 
5 

Image_5 ; practise. 
15 



PRIMARY ARITHMETIC. 145 

Ask questions and write answers of pupils on black- 
board. 

Questions. Amwers. 

2 = what part of 4 ? Of 6 ? 2 = |, f . 

3 = what part of 6 ? Of 9 ? 3 = f , f . 

4 = what part of 8 ? Of 12 ? 4 = f , - 1 /-. 

5 = what part of 10 ? Of 15 ? 5 = -V -, J^. 

Questions. 

2 = how many times 1 ? What part of 3 ? 

4 = how many times 2 ? What part of 3 ? 

6 = how many times 3 ? What part of 9 ? 
8 = how many times 4 ? What part of 12 ? 

10 = how many times 5 ? What part of 15 ? 



Answers. 

2 = 2 times 1, ^a. 
4 = 2 times 2, - 2 g 6 -. 
6 = 2 times 3, 2 ^. 
8 = 2 times 4, 
10 = 2 times 5, 



Questions. 

3 = how many times 1 ? How many halves of 2 ? 

6 = how many times 2 ? How many halves of 4 ? 

9 = how many times 3 ? How many halves of 6 ? 

12 = how many times 4 ? How many halves of 8 ? 

15 = how many times 5 ? How many halves of 10 ? 



146 



PKIMAEY ARITHMETIC. 



Answers. 



3 = 3 times 1, 

6 = 3 times 2, * 

9 = 3 times 3, ^.. 

12 = 3 times 4, s^ 8 -. 

15 = 3 times 5, ^p. 

Ratios -- Draw units on the blackboard. 



(Read : f of 2.) 



4 




8 




12 




16 




20 



3 




6 




9 




12 




15 




1. Draw units and write their names ; 1, 2, 3, 4 ; 
2, 4, 6, 8 ; etc. 

2. Tell all you can about these units. 

3. In each set of four units compare each unit 
with the other three. 



PRIMARY ARITHMETIC. 147 

I O - T^} 7j j "5~~" 

6 = 2 times" 3, ^VV 2 -- 
9 = 3 times 3, *, a^a. 
12 = 4 times 3, 2 times 6, - 4 ^. 



1. If 4 tops cost 20/, what part of 20/ will 2 
tops cost ? One top ? 3 tops ? 

2. 3 hats cost $12 ; what is the cost of 1 hat ? 
Of 2 hats ? Of 4 hats ? 

3. 2 doz. buttons cost 3 dimes; what is the cost 
of 4 doz. ? Of 1 doz. ? Of 3 doz. ? 

4. 12 Ibs. of butter cost $2 ; what part of $2 
will 3 Ibs. cost? What will 6 Ibs. cost? What 
part of $2 will 9 Ibs. cost ? 

5. Call 6 | ; what is 12 ? What is 3 ? What 
is 9? 

6. 9 boxes of strawberries cost T5/; what part 
of 75y do 6 boxes cost ? 3 boxes ? 12 boxes ? 

7. 16 color boxes cost a certain sum ; what part 
of the sum will 4 cost ? 8 ? 12 ? 

8. What is the relation of 4 to 12 ? Of 8 to 12 ? 
Of 16 to 12 ? 

9. A doz. cost a dime ; what is the cost of 4 ? 
Of 8 ? Of 16 ? 

10. There are 20 things in a score ; 5 equals 
what part of a score? 10 equals what part? 
15 equals what part ? In 5 score there are how 
many 20's ? 

11. 5 is -J- of what unit ? 10 equals what part 



148 PRIMARY ARITHMETIC. 

of the unit ? 15 equals how many 4ths of the 
unit? 



Ratios. (d) 


2 


4 


3 


5 




2 


4 


3 


5 




2 


4 


3 


5 




2 


4 


3 


5 




8 


16 


12 


20 



1. Learn (d) as the other tables have been 
learned. 

2. Compare each unit with the other three ; 
thus : 2 = |, |, f . 

4 = 2'2, -V- (read, f of 6), f . 
6 = 3'2, -y-, 
8 = 4'2, 2'4, -V- 

3. What four equal units in 12 ? In 8 ? In 16 ? 
In 20? 

4. What is f ? -y-? -V 6 -? -Y-? 

5. What is |? -y-? -V-? -Y-? 

2435 

6. Image four of each of the above figures, 

5 
5 

with the sum. Ex. : Image 5 ; practise. 

_5 
20 



PRIMARY ARITHMETIC. 149 

Ask questions, and write pupils' answers on the black- 
board. 

Questions. 

2 = what part of 4 ? Of 6 ? Of 8 ? 

3 = what part of 6 ? Of 9 ? Of 12 ? 

4 = what part of 8 ? Of 12 ? Of 16 ? 

5 = what part of 10 ? Of 15 ? Of 20 ? 

Answers. 

468 
- 2? 3? ' 

3 = I, I, -- 

4 = f,-VW 6 - 

5 = --> -,- 

Questions. 

2 = how many times 1 ? What part of 3 ? Of 4 ? 
4 = how many times 2 ? What part of 6 ? Of 8 ? 
6 = how many times 3 ? What part of 9 ? Of 12 ? 
8 = how many times 4 ? What part of 12 ? Of 16 ? 
10 = how many times 5 ? What part of 15 ? Of 20 ? 

Answers. 

2 = 2 times 1, a^a, f . 
4 = 2 times 2, ^, f . 
6 = 2 times 3, ^-, *. 
8 = 2 times 4, *^-a, \-. 
10 = 2 times 5, i^-&, -^ 

1. What is V-? O f8? O f4? 

2. What is |? Of 12? Of 6 ? 



150 PRIMARY ARITHMETIC. 

3. What are ^ ? Of 9 ? Of 6 ? 

4. 5 equals what part of 10 ? Of 20 ? Of 15 ? 

5. What is the relation of 10 to 20? Of * 
to -Y- ? Of -V- to -V- ? 

6. 2'3's equal what part of 9 ? Of 12 ? 

7. 6 equals f of what ? 6 equals f of what ? 

8. 16 equals how many times 8 ? How many 
times f ? 

9. What is the relation of 12 to f ? 

Problems. 1. A boat sails 4 miles in J hr. ; 
how far does it sail in 1 hr. ? In 1J hrs. ? 

2. James is 5 yrs. old. His age equals ^ of his 
brother's ; how old is his brother ? 

3. $10 is f of my money, what is ^ ? 

4. If 1 apple costs 3/ ? how many apples can be 
bought for 9/ ? For 12/ ? 

5. 3 bonnets cost $9, how many bonnets can 
be bought for $6 ? 

6. If a family uses 12 loaves of bread in 1 week, 
what part of a week will 9 loaves last ? 

7. I paid ^ of my money for coal and the rest 
for flour ; what part of my money did I pay for 
the flour ? 

8. If 2 horses eat a bushel of oats in a day, 
how much do 3 horses eat ? 

9. If 3 girls sweep the floor in 10 min., what 
part of the floor will 2 girls sweep in the same 
time? 



PRIMARY ARITHMETIC. 151 

10. 3 oranges cost 5/, how many *5^ will 12 
oranges cost ? A doz. ? 

11. John has 6/, and his brother has 2 times 
as many ; how many has his brother ? The two 
boys have how many 6/ ? 

12. 3 pairs of shoes cost $9, how many $9 will 
6 pairs cost ? 

13. If it takes 15 boys one day to dig a ditch, 
what part of the ditch can 5 boys dig in 1 day ? 
How many days will it take the 5 boys to dig the 
other f of the ditch ? 

14. At $ J a bushel, how many bushels of apples 
can be bought for $3 ? 

15. Mary is 5 yrs. old. Jane is 4 times as old, 
How old is Jane ? 

16. Roy walks 2 blocks, while his sister walks 

1 block. Roy walks how many times as fast as 
his sister ? If Roy can walk to school in 4 min. ? 
it will take his sister how many 4-min. ? How 
many min.? If John walks 3 times as fast as Roy, 
John will walk how far, while Roy is walking 

2 blocks ? 

17. Draw a line and call it the distance Roy 
walks in 1 min. Draw another, showing how far 
his sister walks in the same time. Draw one 
showing how far John walks in the same time ? 

18. Caroline has 8 roses ; to how many little 
girls can she give 2 and yet keep 2 herself ? 

19. Nettie and Addie are in the middle of the 



152 PRIMARY ARITHMETIC. 

room. If *Addie walks 3 yds. north, and Nettie 
2 yds. south, how far apart will they be ? 

20. A boy had 20/, and lost 16/; what part of 
his money did he lose ? 

21. Mr. Jones lives 4 blocks east of the school- 
house, and Mr. Brown 3 blocks west; how far 
apart do they live ? 

22. Howard and Frank bought a box of marbles 
for 6/. Howard paid 4/, and Frank 2/ ; what 
part of the marbles ought each to have ? 

23. A boy sells papers at 2/ each; how many 
does he sell to receive 10/ ? To receive 8/ ? 

24. If he sells of 10 papers, he will receive 
how many cents ? If he sells f of 10 papers ? 

25. A lady gave to Carrie 6 apples and to 
Fannie f of 8 apples ; to which did she give the 
greater number ? 

26. 4 peaches equal what part of 6 peaches ? 
Of 8 peaches ? Of 10 peaches ? 

27. The cost of 6 peaches equals what part of 
the cost of 8 ? Of 10 ? 

28. The jelly that 4 peaches will make equals 
what part of the jelly that 6 peaches will make ? 
That 10 will make ? That 8 will make ? 

29. How many pts. equal 1 qt. ? A gallon equals 
how many qts. ? A year equals how many 6-mos. ? 
How many 4-mos. ? How many 3-mos. ? How 
many 6's in a dozen ? How many 4's ? How 
many 3's ? 



PRIMARY ARITHMETIC. 153 

30. A dime equals how many nickels ? 2 dimes 
equal how many nickels ? 

31. 1 yd. equals how many ft. ? 5 yds. equal 
how many 3-ft. ? How many yds. in 6 ft. ? In 
9 ft. ? In 12 ft. ? 

32. If a yd. of cord is worth 15/, what are 3 ft. 
of cord worth ? 

33. In a score there are how many 10's ? How 
many 5's ? 10 equals what part of a score ? 15 
equals what part of a score ? 

34. There are 7 days in a week ; how many 
7-day s in 9 weeks ? 14 days equal how many 
weeks ? 7 days equal what part of 3 weeks ? 

35. There are 5 school days in a week; how 
many school days in 3 weeks ? 

36. 9 tons of coal last a family 6 mos. ; how 
many 9-tons will last a yr. ? How many tons ? 

37. If 2 barrels of flour last 4 mos., how many 
2-barrels will last a yr. ? How many barrels ? 

38. If you pour a pint of milk into a qt. 
measure, it fills what part of it ? 

39. Mr. Robinson sells 2 pts. of milk for 6/; 
how much ought he to receive for a qt. ? 

40. 2 qts. of water fill what part of a gallon 
measure? 3 qts. of water fill what part of a 
gallon measure ? 

41. If you take a qt. of milk out of a gallon of 
milk, what part of a gallon remains ? 

42. From a piece of cloth 20 yds. long a 



154 PRIMARY ARITHMETIC. 

merchant cuts 5 yds. ; the smaller piece equals 
what part of the larger piece ? How many 5-yards 
in the larger piece ? How many yards ? 

43. A lady paid $8 for a dress. She paid $4 
more for a cloak than for the dress ; how much 
did she pay for the cloak ? How much for both ? 

44. There are 4 gills in 1 pt. How many gills 
in 3 pts. ? In pt. ? In 4 pts. ? 2 gills equal 
what part of a pint ? How many pints in 8 gills ? 
In 12 ? In 6 ? 6 gills equal how many halves of 
a pint ? In a qt. there are how many gills ? A qt, 
is how many times as much as a gill ? 

45. How many 6-in. in 1|- ft. ? How many 
inches ? 

46. How many half-dozen in 1 doz. and 6 ? 

47. What is the sum of a dime and 4/? Of 
2 dimes and a 5/ piece ? 

48. Two dimes equal how many nickels ? 

49. The candy that a nickel will buy equals 
what part of the candy that can be bought for 
2 dimes ? 







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