ii- i ii i.
UNIVERSITY OF CALIFORNIA.
PRIMARY SCHOOLS *
LARKIN DUNTON, LL.D.
HEAD-MASTER OF THE BOSTON NORMAL SCHOOL
" OF THE
SILVER, BURDETT & CO., PUBLISHERS
NEW YORK . . . BOSTON . . . CHICAGO
BY LARKIN DUNTON-
TYPOGRAPHY BY J. S. GUSHING & Co., BOSTON.
PRESSWORK BY BERWICK: & SMITH, BOSTON.
IT has long seemed to the author that there was needed
in this country a manual for teachers, which should set
forth, in systematic order and fully, the process of procedure
on the part of the teacher in developing in the pupils ideas
of numbers and their relations, ideas of numerical processes,
and ideas of the signs by which both are represented. The
present work is an attempt to supply that need in the case
of teachers of primary schools.
For illustrations and method of treatment the author is
mainly indebted to the work of A. Bohme, formerly a nor-
mal school teacher in Berlin, Anleitung zum Unterricht im
Rechnen. But Bohme is not responsible for any of the
statements made in this work ; because such omissions,
additions, and other changes have been made as seemed
desirable in order to adapt the work to the needs of Ameri-
can schools. While portions of this work are free transla-
tions from Bohme, others are independent discussions.
The scope of the work and the arrangement of topics
can be seen by a glance at the table of contents.
While frequent reference is made to Arithmetic Charts
by the same author, the methods here described are not
dependent upon any particular kind of apparatus, but are
equally applicable, whatever may be the apparatus in use.
If, by the introduction of this manual, and of his Arith-
metic Charts, the author shall have made the teaching of
arithmetic easier for the teacher, and more educational for
the pupil, his purpose in preparing them will have 1
BOSTON, Sept. 15, 1888.
NUMBERS FROM ONE TO TEN 7
Apparatus for illustration 8
Use of apparatus in counting 1 1
Slate exercises in counting 14
Number pictures 17
Arithmetic charts 22
Separating numbers into two parts 24
Learning the use of figures 37
Written exercises in addition 38
Written exercises in subtraction 40
Combined addition and subtraction 42
NUMBERS FROM ONE TO TWENTY 44
Separating numbers into two parts 47
Teaching numbers from 1 1 to 20 53
NUMBERS FROM ONE TO ONE HUNDRED 61
Reading and writing numbers 65
Addition and subtraction . 78
Teaching the multiplication table 83
Applying the table to written work 90
Constructing the table 94
Preparation for division 97
Dividing by two 101
Dividing by three TIO
Dividing by other numbers to ten 113
Practice work 1 14
NUMBERS FROM ONE TO ONE THOUSAND 117
Counting and writing 117
Addition 1 19 .
Written and mental arithmetic 137
HIGHER NUMBERS 141
Of THE \
NUMBERS FROM ONE TO TEN.
INSTRUCTION in arithmetic should begin with count-
ing, since this is the foundation of all arithmetical
When children enter school, they can usually count
a little, that is, they can distinguish a few numbers
of similar things by the appropriate words. But it
frequently happens that the children know words
which stand for numbers, merely as a succession of
sounds, without knowing what number of things one
word or another signifies. The teacher should be
careful not to mistake this mechanical knowledge of
words for a real knowledge of numbers, or for the
ability to count. In order to give the child a definite
idea of the meaning of the number words which are
already partly known to him, we should direct his
ARITHMETIC IN PRIMARY SCHOOLS.
attention to the objects around him, and teach him
the words which express their numbers. . Through
conversation we may lead him to observe that there
are in the room one door and another door, and thus
to comprehend the expression that one door and one
door are two doors. When the meaning of the word
two is thus made clear, it may be applied to different
objects in the room, as to two hands, two feet, two
eyes, two ears, two windows, etc.
In this way we may lead the child to the compre-
hension and application of the words one, two, three,
four, etc., to ten.
We should thus use concrete objects in making the
child acquainted with numbers and the naming of
numbers. Since clear and correct ideas are based
upon perception alone, it is necessary to connect the
exercises in number in the lower grades continually
with concrete objects. In order to reach the proper
result in the shortest time, it is desirable to have
some simple apparatus, by means of which all the
necessary observations can readily be made.
2. APPARATUS FOR ILLUSTRATION.
Festal ozzi furnished such an apparatus in his table
of units. This consisted of a table containing ten
rows of rectangles with ten rectangles in each row.
In each rectangle in the first row was one line, in
each rectangle of the second row were two lines, and
APPARATUS FOR ILLUSTRATION. 9
so on to the tenth row, in each rectangle of which
were ten lines.
Other machines for illustrating number-teaching
have been devised, involving the same principle, and
recognizing fully the necessity of observation, which
have been improvements upon Pestalozzi's. One of
these consists of a black wooden board, about twenty
inches long and twenty inches wide, in which are
bored a hundred holes, in ten rows of ten each, so
arranged that there are ten horizontal and ten ver-
tical rows. In addition to this board are one hun-
dred buttons of white wood or bone, with stems that
can be stuck in the holes. This has several advan-
tages over the fixed table of Pestalozzi :
i. It allows each exercise to be observed alone;
2. The children can see the numbers produced ;
3. They can themselves perform the exercises ;
4. The things to be observed are objects, and con-
sequently much better than any signs ; 5. The
number pictures (to be explained hereafter) can be
formed. Besides, it is very handy, and the exercises
to be shown can easily be observed by the children
who sit in the farther part of the room.
Where such an apparatus is wanting, the teacher
may use for the same purpose wooden pegs, bits of
pasteboard, cubes, and the like. Mothers and nurses
might do much to prepare the children for instruc-
tion and education in numbers, would they take occa-
sion to have them count and compare, while at play
10 ARITHMETIC IN PRIMARY SCHOOLS.
with their little plates, soldiers, building blocks, etc.
It is possible, even while the sole purpose is play, so
to direct them as to do much to prepare them for
instruction and for the development of their powers.
Another piece of apparatus is the numeral frame,
one form of which is much used in this country.
The best kind is composed of a wooden frame about
four feet long and two feet wide, in which, running
horizontally from end to end, are fastened ten brass
or steel rods ; on each of these rods are ten easily
moved wooden balls about an inch and a half in
diameter. The whole is supported at a convenient
height by means of upright standards attached to
bars running crosswise at the bottom. It is well to
have the balls painted different colors, say three red
ones at the left on each wire, then three yellow ones,
and four green ones at the right ; or, two and two,
black, red, yellow, green, and white. One-half the
frame should be covered with a board, so as to con-
ceal all the balls that are not used in any example.
This apparatus has some special advantages. It can
be made to present many examples very readily, the
balls can be seen across the room, and, by means of
the different colors, the number of balls in sight on
any wire can be readily determined, even by the
The two pieces of apparatus just described may be
united in one by boring holes for the buttons in the
board used for a screen for the balls, as shown in the
USE OF APPARATUS IN COUNTING.
cut. This board may be made to perform still an-
other office by ruling vertical and horizontal lines
across it through the rows of holes, namely, the form-
ing of number pictures, to be explained hereafter.
There are other ingeniously constructed machines
for teaching the first steps in number ; but generally
the simplest form is the best. Perhaps for general
use the numeral frame that I have just described is
the most desirable.
3. USE OF APPARATUS IN COUNTING.
However necessary real observation may be for the
first steps in arithmetical instruction, and even in
exercises introduced later, yet, ultimately, observa-
tion must be replaced by ideas ; ideas in the mind
12 ARITHMETIC IN PRIMARY SCHOOLS.
must take the place of objects without. The child
learns to walk, at first, with help, then independently ;
he learns to dispense with assistance gradually. So
it is with the use of observation in arithmetical
It has already been explained how a child may be
made to gain clear ideas of the fundamental numbers,
that is, the numbers from one to ten. But the work
thus begun may be completed and the ideas impressed
upon the memory by means of the numeral frame,
in the following manner ; or, indeed, the very first
instruction may be so given :
Move out one ball on the upper wire, and ask,
" How many balls is that ? " The answer should be
given in a complete sentence, thus, "That is one
ball." Move out two balls on the second wire, and
to the question, " How many balls are there ? " should
the answer follow, "There are two balls." But if
the child does not know the word two, the sentence
must be given first by the teacher.
In the same way show the class all the numbers of
balls from one to ten, and teach the sentences, in
connection with the observation, " There is one ball,"
"There are two balls," and so on to, "There are ten
balls." Let these sentences be clearly and distinctly
pronounced, now by individuals, and now by the
class in concert, while the teacher points successively
to the different groups of balls. Finally, let the
children use the pointer.
USE OF APPARATUS IN COUNTING. 13
When these sentences have been learned as the
expressions of the several facts, let them be abbre
viated, thus :
1. One ball, two balls, etc., to ten balls.
2. One, two, three, etc., to ten.
Next, teach the following sentences in order, in
connection with the use of the balls :
1. After 'one comes two, after two comes three,
etc., to ten.
2. One and one are two, two and one are three,
etc., to ten.
Although the last two sentences are not exactly
counting, yet they are in place here, for they result
immediately from the facts learned in counting.
Up to this point we have had the children speak
the sentences in the order of the numbers, so as to
accustom them to the counting of objects, and to the
use of number words in their natural order ; because
counting is the foundation of all arithmetical opera-
tions. Now, however, exercises may be given out of
their order, so that the pupils may be led to count
any number of objects, as fingers, windows, etc., or
to tell their number.
When the children have become proficient in desig-
nating any number of objects up to ten by the appro-
priate word, in telling what number comes after each
number, and how many each number becomes when
increased by one, they, may then be taught to count
14 ARITHMETIC IN PRIMARY SCHOOLS.
backwards from ten to one. The proper order of
the steps may be indicated as follows :
1. There are ten balls, there are nine balls, etc., to
2. Ten balls, nine balls, etc., to one.
3. Ten, nine, eight, etc., to one.
4. Before ten comes nine, before nine comes eight,
etc., to one.
5. One from ten leaves nine, one from nine leaves
eight, etc., to one.
6. Ten less one is nine, nine less one is eight, etc.,
The method is the same as in counting from one
to ten. The numbers and their relations are to be
suggested by the observation of the balls, as they
are presented by the teacher.
4. SLATE EXERCISES IN COUNTING.
Many classes of beginners are so constituted that
the pupils are not all of the same degree of advance-
ment, so that they cannot properly be taught all
together. When this is the case, we have little need
to consider the question of slate exercises ; for these
are, at this stage of the work, merely makeshifts ;
and, however closely they may be connected with
the objects to be observed, the real teaching of the
numbers cannot be dispensed with. But when the
SLATE EXERCISES IN COUNTING. 15
class contains two or more divisions, then the teacher
must provide suitable exercises to occupy the rest of
the children while he is engaged in teaching one
division. For this purpose he must make use of the
blackboard. Of course, exercises in writing furnish
abundant means for occupation, especially where
reading and writing are taught together ; but exer-
cises should be devised which satisfy the aim of arith-
metical instruction. All exercises designed simply
to keep the children busy are an abomination. When
slate exercises are introduced as a means of teaching
counting, they should be closely connected with the
objects to be observed by the children; so that the
work done by them will be a means of fixing in their
minds the ideas gained by the observation of the
The children should not at this stage be made
acquainted with figures, for they are not yet able to
represent in their minds, by means of figures, what
the figures signify, because figures are purely arbi-
trary signs, and not pictures of numbers, in which
the children can again find the units which they
signify. Written exercises in number must, there-
fore, at first, consist of representations of numbers
To prepare the children for these exercises, the
teacher should write upon the board the following, or
similar groups of marks, while the children observe
and count. The board upon which these groups are
ARITHMETIC IN PRIMARY SCHOOLS.
written should be ruled in squares of convenient size,
so that each line, star, etc., will occupy one square.
o o o
o o o o
o o o o o
o o o o o o
o o o o o o o
o o o o o o o
o o o o o o
o o o o o
o o o o
o o o
These groups may be formed of other figures, at
the pleasure of the teacher ; for example,
DO + X T I.
The upper row of groups represents counting for-
wards, and each row of marks is one more than the
row above ; while the lower row of groups represents
counting backwards, and each row of marks is one
less than the row above.
The different lines, figures, etc., which the teacher
can increase at his pleasure, serve both for a change,
and for practice in writing and drawing. The slates
should be ruled in squares corresponding with the
NUMBER PICTURES. \J
ruling of the board ; which may be done by scratch-
ing the slates lightly. This marking of the slates
assists in establishing the habit of doing all work on
the slate in an orderly manner. It is of much use a
little later in fixing the habit of writing figures of
uniform size, and in vertical and horizontal lines.
When the children have seen the groups formed,
have counted the marks, compared and named them,
so that their numbers are all well known, they may
be required to copy them ; and the closer they are
required to follow the copy, the better, not only for
the training of the eye and hand, but also for the
arithmetic itself ; for, as exactness in the order light-
ens the work of computing the numbers, so it makes
clearer the knowledge of the relation of one number
to another. Moreover, exactness here is of great
moral value, inasmuch as it trains to habits of order
Chart I. will now be found useful for review in
counting and for copying.
5. NUMBER PICTURES.
Through the exercises already explained, the chil-
dren may gain ideas of all the fundamental numbers,
that is, all the numbers from one to ten, in their
unity. The eye, however, is not in condition to see
a large number of units lying side by side, or one
above another, and grasp the units as a number.
ARITHMETIC IN PRIMARY SCHOOLS.
The children, for the most part, have merely the
power to count, a power which is to be regarded
only as the foundation. If a child is to obtain a total
impression of a number at once, the number must be
in the form of a definite picture, in which he discov-
ers the number at a glance, and grasps it immedi-
ately in all its units. Such number pictures are pre-
sented at the bottom of Chart I.
FOR REVIEW IN COUNTING.
NUMBER PICTURES. 19
These rectangles with the enclosed dots should be
put upon the board, one after another, and when they
have been observed and the dots counted, they should
be carefully copied on the slates, in order to impress
them upon the eye and memory.
In regard to one, two, and three, there is nothing
of importance to be said.
In four we see two points above and two points
below ; or two points at the right and two points at
the left. Attention may be directed to the form of
the picture by questions : " What do you see in
four?" "Two points above and two points below."
"What else?" "Two points at the right and two
points at the left." But it will be taken for granted
generally in this book, that the teacher knows how
to develop the points of a lesson by proper questions,
when they are suggested.
As the children copy, the arrangement of the
dots makes clear to them the thoughts expressed
by the following sentences, which may be devel-
oped by the proper questioning on the part of the
1. From four we can make two twos.
2. Two and two are four.
3. Four less two is two.
4. Two times two are four.
5. The half of four is two.
6. There are two twos in four.
20 ARITHMETIC IN PRIMARY SCHOOLS.
In order to give further practice in the use of these
sentences, refer to objects in which the number four
appears ; e.g., a wagon has four wheels, two before
and two behind ; the cat, dog, mouse, etc., have each
four feet ; the table has four legs, etc.
Occasionally should practical problems be given :
George has two cents, and gets two more ; how many
has he now ? And so of the other relations of the
numbers. But the use of these problems should not
be carried too far, otherwise the arithmetical instruc-
tion lacks brevity and definiteness. I shall not intro-
duce these problems often, because the live teacher
can easily invent enough to fit the work upon the
numbers, as they are studied, one after another; or,
better yet, can find some good books of problems.
The number Jive may be produced from four by
putting a dot in the midst of the four.
From jfa^ we can make four and one ; it follows that
a. 4 and I are 5. c. 5 less I is 4.
b. i and 4 are 5. d. 5 less 4 is i.
Six consists of two threes ; it follows that
a. 3 and 3 are 6. c. 2 times 3 are 6.
b. 6 less 3 is 3. d. Half of 6 is 3.
e. There are 2 threes in 6.
f. 2 and 2 and 2 are 6. k. The third of 6 is 2.
g. 3 times 2 are 6. i. There are 3 twos in 6.
j. 2 and 2 are 4 ; 4 and 2 are 6.
NUMBER PICTURES. 21
Seven consists of six and one, the one being in the
middle ; hence,
a. 6 and i are 7. c. 7 less i is 6.
b. i and 6 are 7. d. 7 less 6 is i.
Eight consists of two fours ; therefore
a. 4 and 4 are 8. c. 2 times 4 are 8.
b. 8 less 4 is 4. ^/. Half of 8 is 4.
e. There are 2 fours in 8.
It is further obvious from the picture, that 2 and 2
and 2 and 2 are 8 ; or 4 times 2 are 8 ; the fourth of
8 is 2 ; and there are 4 twos in 8.
If the children are old enough and advanced enough
to make it easy for them to comprehend, the follow-
ing facts may be taught :
I of 8 is 2, If i apple costs 2 cents,
f of 8 are 4, 2 apples cost 4 cents,
f of 8 are 6, 3 apples cost 6 cents,
| of 8 are 8. 4 apples cost 8 cents.
\ of 8 away, 6 is left, If i orange costs 8 cents,
| of 8 away, 4 is left, | orange costs 4 cents,
f of 8 away, 2 is left, \ orange costs 2 cents,
| of 8 away, o is left. f orange cost 6 cents.
Nine consists of three threes ; therefore
a. 3 and 3 and 3 are 9. d. 9 less 6 is 3.
b. 3 times 3 are 9. e. 6 and 3 are 9.
c. 9 less 3 is 6. f. A third of 9 is 3, etc.
22 ARITHMETIC IN PRIMARY SCHOOLS.
Ten consists of two fives ; therefore,
a. 5 and 5 are 10. c. 10 less 5 is 5.
b. 2 times 5 are 10. d. Half of 10 is 5.
e. There are 2 fives in 10.
While the teacher is instructing the children in
these numbers, he must be careful, both in the oral
and written work, to make them able to name the
number pictures as soon as they are seen, and also to
construct them on the numeral frame, or draw them
on the board.
An excellent practice in comparing numbers grows
out of forming one number from another. For exam-
ple, put the number picture for five on the board,
then ask, What must be done in order to make a
seven ? Must something be added, or taken away ?
How many must be added ? Where must the dots
be put ? Again, How can seven be made from nine ?
etc. In all such cases, one number picture is to be
changed to another by either addition or subtraction
of the proper dots. Exercises of this kind are very
useful ; they exercise the children in the comparison
of numbers, and prepare them for the division of
numbers, which is about to be explained in detail.
6. ARITHMETIC CHARTS.
In addition to the apparatus for developing ideas
of number, which has been already described, a few
ARITHMETIC CHARTS. 23
arithmetic charts will be found very helpful for
reviews at every stage of elementary instruction in
arithmetic. The author of this book has arranged a
series of thirteen such charts, which are published
by Silver, Burdett & Co. Miniature copies of them
will be printed in this book, as they are needed for
illustration ; and they will be referred to simply by
their numbers. Where they are not furnished to
schools, teachers can put them on the blackboard, or
on large sheets of paper, and thus save themselves
A word in regard to the use of the charts. The
children should see each number and each exercise
produced ; that is, each illustration of a number, or
of a numerical operation, should be made by the
teacher, either upon the numeral frame, or upon the
board, or with objects, just when it is needed to
make the truth clear to the class. Hence all ready-
made charts, or other illustrations, are to be used
later, after this preliminary, but fundamental work
has been done. They are to serve as a means for
review and practice in what has already been made
clear to the understanding.
The charts representing matter for observation are
to be read forwards, backwards, vertically, and hori-
zontally. The special use to be made of the differ-
ent charts will be explained as they are introduced.
In general, they are designed to lighten the labor
of the teacher, while making the instruction more
thorough and systematic.
24 ARITHMETIC IN PRIMARY SCHOOLS.
7. SEPARATING NUMBERS INTO Two. PARTS.
Upon a pupil's facility in the use of numbers below
ten depends his progress in mastering numbers
above ten. The greater his facility in the use of
small numbers, if it is founded upon clear under-
standing, the surer and more rapid will be his prog-
ress in larger numbers. In order to attain this
facility depending upon understanding, we must have
the numbers regarded from as many sides as possi-
ble ; this comes from the division, or separation, of
the numbers into their component parts. From this
division we obtain results for all the different funda-
mental operations in arithmetic, which are the more
easily committed to memory, because they are all
grounded upon a single result, namely, that of divi-
sion. The results, however, which are obtained from
this division, must, by no means, be learned by heart,
as one commits to memory vocabularies or verses ;
they must become things of the memory through an
unlimited amount of reckoning, through practice.
It is sufficient, at first, that a child, if he is to unite,
for example, five and three into a single number, adds
first one to five, then another, and still another, even
if he represents the process to his senses by means
of marks, fingers, etc. ; yet continued practice must
bring him to the point where the union of three and
five in eight is a simple conception, a thing of the
memory. If the child constantly perceives the three
SEPARATING NUMBERS INTO TWO PARTS. 2$
units in three, he will, in time, be able to unite three
to five at once. We shall be able to bring him to
this state of mind the more easily if we show him
that eight consists of a five and a three. When,
however, we have shown him this, he will be able,
from the single observation, to understand the fol-
lowing four sentences :
a. Three and five make eight.
b. Five and three make eight.
c. Three from eight leaves five.
d. Five from eight leaves three.
If now, we use these sentences as the expressions
for the truths which constantly appear before the
eyes of the child, the results will finally become
impressed upon his memory. This result will, of
course, be reached in the case of some children
quicker than with others.
Out of the above division we obtain two results in
addition and two in subtraction. Another example
will show that in a single division all the four funda-
mental operations of arithmetic may be illustrated.
Out of eight we can make two fours. It follows that
a. 4 and 4 are 8. c. 8 less 4 is 4.
b. 2 times 4 are 8. d. The half of 8 is 4.
e. There are 2 fours in 8.
In the first stages of arithmetical work, where the
numbers are small, and the results to be gained
ARITHMETIC IN PRIMARY SCHOOLS.
through division are correspondingly few, the direct
observation may result in clear mental pictures, or
ideas. Here the connection between the various
ground operations of arithmetic is so obvious from
the observation, that it seems unnecessary to sepa-
rate the treatment of the different operations. Hence,
in this and the following stage, that is, in the treat-
ment of numbers from one to ten, and from ten to
twenty, the four fundamental operations may be
united. All the different results are obtained because
every number below ten is divided into every two parts
of which it is composed, in the way shown in Chart II.
UNITING AND SEPARATING NUMBERS FROM Two TO TEN.
I will now give an explanation of Charts I. and II.
and their use.
The upper part of Chart I. is for review work in
counting. The dots may be counted under the direc-
SEPARATING NUMBERS INTO TWO PARTS. 2/
tion of the teacher ; and then they may be copied on
the slates. During the work of copying, the children
should always count the dots as they make them.
The lower part of Chart I. contains the number
pictures from one to ten. These pictures are designed
to furnish a means of impressing the ideas of the
fundamental numbers, that is, the numbers from
one to ten, upon the mind in such a way that they
may reappear in the imagination of the - pupil when-
These pictures should not be used as the sole
means of developing ideas of numbers, but rather as
a means of thorough review and impression. Figures
should not be taught in connection with these pict-
ures. The whole attention of the <pupil should be
given to the numbers and their production ; mere
figures will be prominent enough in his work by and
by, however much pains may be taken to avoid it.
Chart II. is to show the parts of numbers to ten.
Each number picture is to be formed at first by
the teacher on the numeral frame, the blackboard, or
other apparatus ; so that the attention of the children
can be directed to only one number. All the special
facts in regard to the composition of the number, and
the relation of its parts, are to be developed by proper
questions, and by pointing to the parts to be seen.
As fast as the number pictures have been treated
in this way they may be copied from the chart, and
thus much labor on the part of the teacher may be
ARITHMETIC IN PRIMARY SCHOOLS.
saved. After, for example, the number two has been
treated as indicated below, the first rectangle may be
copied by the children. By this means the truths
will be further impressed upon the mind.. The
teacher, however, should be sure that the children
connect the proper name of the number, and the
names of the parts, with what they write ; so that
numbers and names will become thoroughly asso-
ciated in their minds.
THE NUMBER TWO.
This is the first rectangle on Chart II. It shows
that the number tzvo can be divided into two units ;
hence the truth of the following sentences :
a. i and I are 2. c. 2 less I is I.
b. 2 times i are 2.
d. 1 of 2 is I.
e. i in 2 two times.
THE NUMBER THREE.
From three we can make a two and a one ; it
a. 2 and i are 3. c. 3 less i is 2.
b. i and 2 are 3. d. 3 less 2 is i.
SEPARATING NUMBERS INTO TWO PARTS. 29
THE NUMBER FOUR.
Four may be divided into : A. Three and one.
B. Two and two. It follows that
A. a. 3 + i = 4-
c. 4 1=
a. 2 + 2=4.
.4 2 = 2.
c. 2 X 2 = 4.
d. \ of 4 = 2.
e. 2 in 4 = 2 times.
THE NUMBER FIVE.
This may be separated into : A. Four and one.
B. Three and two. It follows that
A. 0. 4 + i = 5.
b. 1+4 = 5-
2 = 5.
^.2 + 3=5.
THE NUMBER SIX.
) ARITHMETIC IN PRIMARY SCHOOLS.
Six may be divided as follows :
A. Five and one.
a. $ + 1=6. c. 6 i = 5.
.1 + 5=6. dl 6 5 = i.
B. Three and three,
tf . 3 + 3 = 6. ^.3x2=6.
.6-3 = 3. *. Jof6 = 3.
^.2x3=6. /. j- of 6 = 2.
C. Four and two.
a. 4 + 2 = 6. d. 6 4 = 2.
.2+4 = 6. <?. 3 x 2 = 6.
c. 6 2 = 4. f. J of 6 = 2.
THE NUMBER SEVEN.
Seven may be divided into :
A. Six and one.
#. 6 + i = 7. ^.71=6.
b. 1+6 = 7. d. 7 6= i.
B. Five and two.
#. 5 + 2 = 7. ^.7 2 = 5.
.2 + 5=7. d. 7-5=2.
C. Four and three.
a. 4+3 = 7- c. 7-3=4-
^ 3 + 4 = 7- ^.7-4 = 3.
SEPARATING NUMBERS INTO TWO PARTS. 31
THE NUMBER EIGHT.
Eight may be divided into :
A. Seven and one.
#.7+1=8. c. 8 -- i = 7.
A i +7 = 8. d. 8-7= i.
B. Four and four.
a. 4 + 4 = 8. c. 2 X 4 = 8. ^.4X2
.8-4 = 4. d. of8=4. / Jof8
C. Five and three.
tf. 5 + 3 = 8. <:. 8 - 3 = 5.
b. 3 + 5 = 8. rf. 8 - 5 = 3.
D. Six and two.
#. 6 + 2 = 8. .8 2=6.
.2 + 6 = 8. rf. 8 6 = 2.
THE NUMBER NINE.
32 ARITHMETIC IN PRIMARY SCHOOLS.
Nine may be divided into :
A. Eight and one.
#. 8 + i = 9. <:. 9 i = 8.
.1+8=9. d.g8 = i.
B. Five and four.
C. Six and three.
#.6 + 3=9. d. 9 6 = 3.
*. 9 - 3 = 6. / \ of 9 = 3.
D. Seven and two.
a. 7 + 2=9.
b. 2 + 7 = 9.
c. 9 - 2 = 7.
d. 9-7 = 2.
THE NUMBER TEN.
SEPARATING NUMBERS INTO TWO PARTS. 33
Ten may be divided into :
A. Nine and one.
a. 9 + i = 10. c. io i = 9.
A 1+9=10. rf. io 9=1.
B. Five and five.
a. 5 + 5 = 10. c. 2 X 5 = 10.
b. 105= 5. d. \ of 10= 5.
C. Six and four.
a. 6 + 4 = 10. .10 4 = 6.
.4 + 6=10. ^.10 6=4.
D. Eight and two.
a. 8 + 2 = 10. d. io 8= 2.
b. 2 + 8 = io. e. 5 X 2 = io.
c. io 2= 8. / of 10= 2.
E. Seven and three.
a. 7 + 3 = io. c. 10-3 = 7.
.3 + 7 = 10. ^.107 = 3.
FURTHER USE OF CHART II.
For further instruction in regard to the use that
can be made of Chart II., I will explain some addi-
tional work upon the number eight. Arrange eight
balls on the numeral frame, as shown below.
34 ARITHMETIC IN PRIMARY SCHOOLS.
There are four balls at the left and four balls at
the right ; or, there are four balls above and four
balls below. So, 4 and 4 are 8. Four balls are to
be seen two times ; therefore, 2 times 4 balls are
equal to 8 balls. Two balls appear always under two
other balls ; hence 4X2 8.
If we take four balls from the eight balls, then
four balls remain : hence 84 4. This removal
may be indicated by covering part of the balls.
The line across the rectangle divides the dots into
two equal parts. This may be shown on the frame
by holding a pointer between the two fours. It fol-
lows that the half of 8 is four. Two balls appear
four times ; therefore the fourth part, or a fourth, of
eight is two.
These considerations prepare for the following
questions : How many are 4 + 4? 2X4? 4X2?
8 4 ? \ of 8 ? \ of 8 ? What number must one put
with 4 to make eight ? How many more is 8 than
4 ? How many less than 8 is 4 ? How many times
4 is 8 ? How many times 2 is 8 ? Of what number
is 4 the half ? Of what number is 2 the fourth ?
What part of 8 is 4 ? What part of 8 is 2 ? How
many is 8 less 2X2? How many is 8 less 3X2?
How many times 2 is 8 less 4 ? etc.
These exercises with pure numbers are the proper
preparation for such simple practical examples as
these : Charles has 4 cents, and Fred has 4 cents ;
how many have they together ? Charles has 8 cents,
SEPARATING NUMBERS INTO TWO PARTS. 3$
and gives 4 of them to Fred ; how many has Charles
left ? Charles got 4 cents yesterday, and 4 more
to-day ; how many times 4 cents has he ? How
many cents in all ? Charles had 8 cents, and lost
half of them ; how many has he now ? Charles and
Fred together had 8 apples, and divided them so that
each had an equal number ; how many did each then
have ? Each of 4 children had 2 pears ; how many
had they all together ? Four children divide 8 apples
equally among them ; how many does each receive ?
Give a boy 8 pencils, and let him give one each to
4 other boys, and then one more to each of them.
What is \ of 8 ? One apple costs 2 cents ; how
many cents do 2 apples cost ? 3 apples ? 4 ? A yard
of ribbon costs 8 cents ; how much does half a yard
cost ? A fourth ? Three-fourths ? etc., etc.
In giving practicable problems it is often necessary
to mention coins, measures, and weights. These
should not only be well known, but they should often
be shown to the children. The teacher should limit
his problems to those coins, weights, and measures
that are accessible to the children in their ordinary
intercourse. The copper, nickel, and smaller silver
coins are all the coins that should be mentioned in
these early problems ; the measures should be limited
to the inch, foot, yard, pint, and quart ; and the
ounce and pound weights are enough. It is well to
have all the measures involved in the problems given,
constantly before the eyes of the children, so that
36 ARITHMETIC IN PRIMARY SCHOOLS.
they will be impressed upon the memory. As the
work in numbers progresses, these illustrations may
be enlarged. Their application will be indicated as
The number pictures which are studied with the
children during the lesson should be copied upon the
slates as written work. The teacher can at first
make them upon the board, and subsequently have
them copied from Chart II. When this has been
repeated sufficiently, they may be written from
The chart will also serve a good purpose in con-
ducting reviews. What is represented in the chart
may be expressed in words. In addition, the verbal
expressions would run thus :
One and one are two. Four and one are five.
Two and one are three. Three and two are five.
Three and one are four. Five and one are six.
Two and two are four. Etc., etc.
This order from left to right on the chart is to be
interchanged with the movement from right to left,
from top to bottom, from bottom to top, and with
exercises out of order.
By regarding each picture as a number, and cover-
ing first the dots at the right and then those at the
left, numerous exercises in subtraction may be formed.
By means of these exercises all the facts of the ad-
dition, subtraction, multiplication, and division tables
LEARNING THE USE OF FIGURES.
may be learned, where the sum, minuend, product,
or dividend does not exceed ten. Since these
results are of the greatest use in all arithmetical
operations, they must be firmly fixed in the mem-
ory. This is to be done, however, by observing and
stating the facts as shown on the chart, by copying
the number pictures, and by written representations
in figures, not by learning the statements, as such, by
8. LEARNING THE USE OF FIGURES.
The ground already covered is sufficient for a
fourth of a year, and, under some conditions, for a
longer time. Hitherto the children have learned
only from observation ; now, however, they may
without danger pass from things to signs, from num-
bers to figures. This transition may be made by
means of the following chart :
USE OF FIGURES.
ARITHMETIC IN PRIMARY SCHOOLS.
Through diligent copying, pointing, and reciting,
the children will impress these forms upon the mind
so that they can be made without a copy.
The different rectangles of Chart II. may then be
copied, and with them the corresponding figures may
be copied in similar rectangles, as follows :
The rest of the chart may be treated in the same
9. WRITTEN EXERCISES IN ADDITION.
In order still further to represent in figures the
results gained through the preceding instruction, we
make use again of Chart II., at first for the purpose
of addition. The children must now learn the sign
of addition (+, plus, or and) and also the sign of
equality (=, is, or are). Then, by writing the figures
for the dots seen in the different rectangles, they can
form these series of numbers on their slates :
1 + i = 2, or i + i = 2.
2 + i = 3, or i + 2 = 3.
3 + i = 4, or i + 3 = 4.
3 + 2 = 5, or 2 + 3 = 5.
5 + i = 6, or i + 5 = 6.
3 + 3 = 6, or 3 + 3 = 6.
WRITTEN EXERCISES IN ADDITION. 39
2 + 2= 4, or 2 + 2= 4. 4 + 2 ^= 6, or 2 + 4 = 6.
4 + i = 5, or i + 4 5. etc. etc.
At first the work may be confined to a few of the
number pictures, but it should be extended gradu-
ally till the whole chart can be represented in fig-
In order to teach regularity and order in the
arrangement of the figures, it is worth while to have
the slates ruled upon one side in squares of about
three-eighths of an inch ; the other side may be ruled
in lines for writing. After the first year the squares
may be omitted, but at first they are very helpful.
A portion of the blackboard should be ruled in the
When Chart II. can be readily interpreted in this
way by figures, the reverse process should be intro-
duced. The children should be required to trans-
late figures into numbers. The teacher will write
upon the board, for example, 3+2 = 5, and the
children will copy the same, and then add the cor-
responding number pictures, thus :
For a review, and for drill in this work, the upper
part of Chart IV. furnishes a convenient means.
ARITHMETIC IN PRIMARY SCHOOLS.
WRITTEN REPRESENTATION OF CHART II.
1 + 1=2
3 + 2
5 + 2
6 + 2
9 + i
5 + i
4 + 3
5 + 5
3 + i-
3 + 3
7 + i
6 + 4
2 + 2 =
4 + 2
4 + 4
6 + 3
6 + 1
5 + 3
7 + 2
7 + 3
2 1 = 1
5'- I =
1C. WRITTEN EXERCISES IN SUBTRACTION.
For the first practice in written subtraction, Chart
II. may be used. The children must learn the mean-
ing of the sign of subtraction ( , less) and use this
in representing the results of their observation. By
observing all the dots in the rectangles, and then
covering first those in the right and then those in
the left, the following results will be reached :
A. 2 1 = 1, 4 2 = 2, B. 2 1 = 1, 4 2 = 2,
3~i = 2, 5-1=4, 3-2-1, 5-4=1,
4-i = 3 5-2 = 3, 4-3 = i 5-3 = 2,
etc. etc. etc. etc.
WRITTEN EXERCISES IN SUBTRACTION.
When Chart II. can be observed, and the corre-
sponding figures readily written, the process should
be reversed. The children should produce the num-
bers when the figures are shown. The work on the
pupils' slates may assume this form.
The lower part of Chart IV. furnishes a convenient
means of drill in the interpretation of figures denot-
As a final review of this kind of work, Chart V.
will be useful, inasmuch as it requires the pupil to
interpret the signs + and , as well as to indicate
FOR REVIEW OF CHART II.
2 + 1
2 + 6
4 + 4
4 + 5
2 + 8
5 + 3
8 + 1
6 + 3
i + i
3 + 7
2 + 2
i + 7
4 + 6
i + 5
3 + i
5 + 2
2 + 7
1 + 2
5 + i
5 + 5
3 + 5
7 + 3
4 + 2
3 + 6
1 + 8
4 + 3
7 + 2
42 ARITHMETIC IN PRIMARY SCHOOLS.
11. COMBINED ADDITION AND SUBTRACTION.
Up to this point the written exercises have been
connected immediately with the observation of the
chart. Nothing more has been required of the chil-
dren than the translating of the number pictures into
figures, and figures into number pictures. Now, in
order to free the written work from the necessity
of observation ; to replace immediate knowledge of
objects with ideas of objects, the results of the addi-
tions and subtractions may be united in the same
written exercises, so that the one may furnish the
clew to the other, thus :
1+2 = 3. 3 2 = 1.
The whole of Chart V. may be treated in this way.
While exercises in multiplication and division have
not been hitherto excluded, they are not numerous
enough in this stage to make it worth while to intro-
duce them into the written work as special topics.
Before proceeding to explain the treatment of num-
bers in the following stage of the work, I will remark
that it is of the utmost importance that the work in
numbers from one to ten should be thoroughly mas-
tered. Naming any number up to, and including,
ten, and also one part of the number, should instantly
suggest to the child the other part. The two parts
COMBINED ADDITION AND SUBTRACTION. 43
of each number should be so associated with each
other and with the number that one part cannot be
thought of as such without the idea of the other part
being at once called to mind. Haste here is not to
be desired. The results must be lastingly fixed, and
this can only be accomplished by much patient, atten-
tive, earnest effort.
I have suggested a progressive use of a few kinds
of apparatus, but I would by no means limit the
teacher to these. Variety of illustration is desirable ;
but it is also desirable to have some means of making
the children do such work as will cause the desired
results, which will not be a constant drain upon the
teacher's power. Hence the free use of the charts
for review is recommended.
44 ARITHMETIC IN PRIMARY SCHOOLS.
NUMBERS FROM ONE TO TWENTY.
12. COUNTING TO TWENTY.
As in teaching numbers from one to ten we began
with counting, so we do in teaching numbers from
ten to twenty. Put ten balls on the upper wire of
the numeral frame. Let the children find how many
twos there are in ten, how many fives, how many tens,
and how many ones. Then tell. them that ten ones
are called a ten, and that one is called a unit. Count
out ten units on the upper wire, and call the result
one ten. Put one ball out on the second wire ; then,
pointing first to the ten and next to the one, say :
"One ten and one unit make eleven units." Add
another ball, and then, pointing as before, say : " One
ten and two units are twelve units." And so pro-
ceed to the sentence : " One ten and nine units make
Add another ball, and there appear two rows of
ten each, thus :
The truth which the pupils gain from observing
COUNTING TO TWENTY. 45
these balls is expressed : Two rows are two tens, or
twenty units. These dots should then be copied by
the children on their slates from a copy made on the
board by the teacher.
For further practice let the above sentences be
repeated as the balls are shown, from ten to twenty ;
and then let the counting from one to twenty be
practised, introducing the following changes :
1. One, two, three, four, five, and so on to twenty.
2. After one comes two, after two comes three,
and so on to twenty.
3. One and one are two, two and one are three,
and so on.
4. One, three, five, seven, etc.
5. Two, four, six, eight, etc.
6. One, four, seven, ten, etc.
7. Two, five, eight, eleven, etc.
8. Three, six, nine, twelve, etc.
9. Twenty, nineteen, eighteen, etc.
10. Before twenty comes nineteen, etc.
11. Twenty lees one is nineteen, etc.
12. Twenty, eighteen, sixteen, etc.
13. Nineteen, seventeen, fifteen, etc.
14. Twenty, seventeen, fourteen, etc.
When the children can surely and readily perform
these exercises ; can unite a ten and a fundamental
number that is, a number from one to ten; can
change any number from eleven to twenty into tens
46 ARITHMETIC IN PRIMARY SCHOOLS.
and units ; and when they can, further, construct any
number on their slates by arranging the proper dots
in tens and units ; and can name any number shown
them by balls or marks, then and not till then, may
they be allowed to pass on to the representation in fig-
ures of numbers from eleven to twenty. Till they
have reached this ability, they may be kept practising
upon the written work connected with numbers from
one to ten. This will constitute a valuable review.
Written work should never precede corresponding
oral work ; for the written work, at this stage, is
simply designed to impress upon the mind what the
oral work has already made clear to the understand-
ing. It is useful for review, but should keep a few
steps behind the oral work, whenever the children
.are introduced to a new topic. Written work de-
mands more self-independence ; but in classes com-
posed of several divisions the pupils must be thrown
more upon their own resources. On account of
the weaker children, therefore, the written exercises
should be deferred till a perfect understanding is
gained and a certain degree of facility is reached. It
is well to bear this remark in mind constantly.
The written representation of numbers from eleven
to twenty is not difficult for children to comprehend.
The figure standing for the ten is put at the left, that
representing the units at the right, therefore :
I ten and I unit = 1 1 units.
I ten and 2 units = 12 units.
SEPARATING NUMBERS INTO TWO PARTS. 47
I ten and 3 units = 13 units, and so on to nineteen.
The following series may be explained and copied :
10+1 = 11. 10+ 8 = 18. 14+1 = 15.
10 + 2 = 12. 10+ 9=19. IS + I = I6.
10+3 = 13. 10+10 = 20. 16+1=17.
10 + 4=14. 10+ 1 = 11. 17+1 = 18.
10+5 = 15. 11+ 1=12. 18+1 = 19.
10 + 6=16. 12+ 1 = 13. 19+1=20.
10 + 7 = 17. J 3 + i = T 4-
13. SEPARATING NUMBERS INTO Two PARTS.
In general the same course is to be followed in the
division, or separation, of numbers from eleven to
UNITING AND SEPARATING NUMBERS FROM ELEVEN TO TWENTY.
48 ARITHMETIC IN PRIMARY SCHOOLS.
twenty as was recommended in regard to numbers
from one to ten. The results needed in all the fun-
damental rules may be obtained by observing the
separation of the several numbers into the various
pairs of which they are composed, as shown on the
The treatment of numbers at this stage of the
work is almost the same, in general, as in the case of
numbers below ten. The division of each number
between eleven and twenty is to be indicated by the
arrangement of the balls on the numeral frame by
the teacher. From each division two results in addi-
tion and two in subtraction are to be obtained ; and,
in the case of numbers composed of factors, at least
one result for division and one for multiplication.
No division is to be made, however, that will make
one part of the divided number greater than ten.
The division of the numbers here meant is simply
the separation of the numbers into two parts.
The divisions are to be carefully shown, one after
another, the number-pictures made by the children,
and the truth stated orally, with frequent repetition,
before the chart is called into use. The chart is for
review only : first, by reciting the facts as shown by
the arrangement of dots ; second, by copying the
number-pictures on the slates ; and, third, by repre-
senting the dots by figures.
I will first indicate the results to be reached, and
then make suggestions as to the manner of doing the
SEPARATING NUMBERS INTO TWO PARTS. 49
THE NUMBER ELEVEN.
Eleven can be divided into :
a. Ten and one ; hence,
10+ i = n, ii i IO,
1 + io ii, ii 10= I.
b. Six and five ; hence,
6+5 11, 11 5=6,
5+6 n, 116 5.
c. Nine and two ; hence,
9 + 2 ii, 11 29,
2 + 911, 119 2.
d. Eight and three ; hence,
8 + 3-1 1, 11-3-8,
3 + 8-n, 11-8-3.
e. Seven and four ; hence,
7 + 4-1 1, 11-4-7,
4 + 7=n> 11-7=4.
THE NUMBER TWELVE.
Twelve can be divided into :
a. Ten and two ;
10+ 2 12, 12 2 IO,
2 + IO 12, 12 IO 2.
b. Six and six ;
6 + 612, 6 in 12 two times,
1266, 6 X 2 12,
2 X 6 12, \ Of 12 2,
\ of 1 2 6, 2 in 1 2 six times.
5O ARITHMETIC IN PRIMARY SCHOOLS.
c. Eight and four ;
8 + 4= 12, 12 8 = 4,
4+8-12, 3 X 4-12,
12-4= 8, \ of 12= 4,
4 in 12 three times.
d. Nine and three ;
9 + 3 = 12, 12-9-3,
3 + 9-12, 4x3-12,
12-3-9, i of 12- 3,
3 in 12 four times.
*. Seven and five ;
7 + 5-12, 12- 5 -7,
5 + 7- 12, 12-7-5.
THE NUMBER THIRTEEN.
Thirteen can be divided into :
a. Ten and three ;
10+ 3 - 13, 13 3 = 10,
3 + 10-13, 13-10- 3.
b. Eight and five ;
8 + 5 =13, i3-5=8>
5 + 8 -13, 13-8-5.
c. Nine and four ;
9 + 4 = 13, 13-4-9,
4 + 9 = I 3> 13-9=4-
d. Seven and six ;
7 + 6 =13, 13-6 = 7,
6 + 7 =13, 13 7 =-6.
SEPARATING NUMBERS INTO TWO PARTS. 5 1
THE NUMBER FOURTEEN.
Fourteen can be divided into :
a. Ten and four ;
10 + 4, 4+io, 14 4, 1410.
b. Seven and seven ;
7 + 7, 14 7, 2x7, 7 in 14, i of 14.
c. Eight and six ;
8 + 6, 14-8, \ of 14,
6 + 8, 7X2, 2 in 14.
d. Nine and five ;
9+S> 5 + 9> H- 5> I4-9-
THE NUMBER FIFTEEN.
Fifteen can be divided into :
a. Ten and five ;
10+ 5, 15- 10, iof 15,
5 + 10, 3 X 5> 5 in 15.
b. Nine and six ;
9 + 6, 15-9, iofis,
6 + 9. 5 x 3> 3 in 15.
c. Eight and seven ;
8 + 7, 7 + 8, 15-7. IS-8.
52 ARITHMETIC IN PRIMARY SCHOOLS.
THE NUMBER SIXTEEN.
Sixteen can be divided into :
a. Ten and six ;
10 + 6, 6+10, 16 6, 16 10.
b. Eight and eight ;
8 + 8, 8X2, 4X4, 4 in 16.
16 8, \ of 16, 2 in 16,
2X8, | of 16, i of 16,
. Nine and seven ;
9 + 7, 7 + 9> 16-7, 16-9.
THE NUMBER SEVENTEEN.
Seventeen can be divided into :
a. Ten and seven ;
10 + 7, 7+io, I7~7> 17-10.
b. Nine and eight ;
9 + 8, 8+9, 17-8, 17- 9.
THE NUMBER EIGHTEEN.
Eighteen can be divided into :
a. Ten and eight ;
10+ 8, 18 10, 9 X 2, 9 in 18,
8+10, 18 8, |ofi8, 2 in 18.
b. Nine and nine ;
9 + 9, 9 in 1 8, 6 X 3,
18-9, 3 X 6, iofiS,
2x9, \ of 18, 6 in 18.
\ of 18, 6 in 18,
NUMBERS FROM ELEVEN TO TWENTY. $3
THE NUMBER NINETEEN.
Nineteen can be divided into :
Ten and nine ;
10 + 9, 9+io, 19 9, 19 10.
THE NUMBER TWENTY.
Twenty can be divided into :
Ten and ten ;
10 + 10, 10 in 20, 5 X 4,
20 10, 4 X S, i of 20,
2 X 10, \ of 20, 4 in 20.
\ of 20, 5 in 20,
14. TEACHING NUMBERS FROM ELEVEN TO
It has already been remarked that it is of the
highest importance for the pupils to know every two
parts of which each number from one to ten consists.
As an indication of the way the work in developing a
knowledge of numbers from eleven to twenty should
be managed, I will show by a few examples how to
utilize this knowledge of the parts of the fundamental
If 5 is to be added to 8, let 2 be added first, so as
to make 10. If this 2 be taken from the 5, 3 remains ;
and this 3 added to 10 makes thirteen ; therefore, 5
added to 8 makes 13. In general, first add enough
54 ARITHMETIC IN PRIMARY SCHOOLS.
to make 10 ; then add the rest of the number to be
If 5 is to be subtracted from 13, first subtract 3, so
that the remainder will be 10; then from the 10 take
away the rest of the 5, namely, 2, and the remainder
The relations, or truths, shown by the number
pictures for 8 and 5, may be indicated as follows :
8 + 5 may be resolved into 8 + 2 10;
10 + 3-13.
5 + 8 may be resolved into 5 + 5 = 10;
10 + 3-13.
13 5 may be resolved into 13 3 10;
10 2 8.
13 8 may be resolved into 13 3 10.
The numbers 9 and 7 may be treated thus :
9 + 7 may be changed into 9 + i 10 ;
10 + 6 16.
7 + 9 may be changed into 7 + 3 10 ;
10 + 6 16.
16 7 may be changed into 16 6 10 ;
10 i 9.
16 9 may be changed into 16 6 10 ;
These processes and results are first to be shown
by means of the balls on the numeral frame, then by
the number pictures, which are first to be made by
NUMBERS FROM ELEVEN TO TWENTY. 55
the teacher on the board and afterwards copied by
the children on the slates. As fast as the number
pictures have been treated in this way, Chart VI. may
be used as a means of review.
The chart is to be read from left to right, right to
left, top to bottom, and bottom to top. If the child
hesitates in this reading, the teacher should lead him
to see the divisions of the numbers to be added or
subtracted, such that the results first obtained will
always be 10. By this means the pupil will learn to
think to the desired result without counting. When
this reading of the chart can be gone through with
rapidly and correctly, the chart may be copied picture
by picture, thus :
10+ i = u, 11 10= i,
I + IO = II, II - I = IO.
Next should follow the reverse of this process,
namely, writing the corresponding number pictures
when the figures are given. On Chart VII. are the
figures corresponding to the number pictures on Chart
VI. Let the pupils copy these figures, and at first
produce the corresponding pictures ; but later the
results may be written immediately in figures, or
ARITHMETIC IN PRIMARY SCHOOLS.
WRITTEN REPRESENTATION OF CHART VI.
9 + 4
9 + 6
9 + 8
10 + 8
9 + 9
7 + 4
10 + 9
IO + 2
IO + IO
I 5~ 5
I 5~ 7
14 - 6
20 - 10
Charts VIII. and IX. are designed to assist in the
final review of the addition and subtraction of num-
bers from one to twenty. This work completes the
learning of the tables of addition and subtraction,
which was begun in sections 9 and 10, and hence
should be made very thorough. These charts should
be copied by the children. Sometimes the corre-
sponding pictures should be constructed, and some-
times the results should be written at once in figures.
The drill should be partly oral ; at one time the pupil
reading from the chart and giving the result ; at an-
other, the teacher should read. The work may be
varied by letting the reading and reciting both be
done by pupils.
FOR REVIEW OF CHART VI.
3 + 10
10 + I
2 + IO
7 + 10
9 + 4
IO + 2
I + 10
8 + 10
8 + 6
6 + 10
FOR REVIEW OF CHART VI.
7 + 4
2 + 9
5 + 9
5 + 8
5 + 6
5 + 1
12 - 10
9 + 10
13 - I0
58 ARITHMETIC IN PRIMARY SCHOOLS.
If you exercise the children upon numbers from ten
to twenty, as was recommended in regard to numbers
from one to ten, namely, by having the results of the
additions and subtractions reached in all cases through
the performing of the necessary operations upon the
objects themselves, and then by having the results
fixed in the memory through the repetition of the
processes by which they were reached, and not by
the saying of the sentences which express the results,
you will have laid a most thorough foundation for the
following stage ; that is, the treatment of numbers
from twenty to one hundred. But not every child
possesses a sufficiently strong memory for numbers.
It would be tiresome to dwell on this stage of the
work till every child was perfect in all the operations
practised. This perfection is to be reached in the
next stage of the work, where the exercises are simi-
lar, where they are more varied, and where, on account
of their greater variety, they are less fatiguing.
It always makes a difference whether a child is
taught alone, or with many others, as in school. In
the one case, the work may be graduated to the
individual ; but in school, if one attempts to make
the weakest perfect, the brightest, and even those of
moderate talent, are kept back too much. In school,
neither the brightest nor the dullest, but the average,
must determine the progress of the class. All must
always be made to comprehend the work in hand, at
least so far as is necessary for understanding what is
TEACHING NUMBERS. 59
to follow ; but readiness in doing may often be
secured through the reviews necessarily practised in
what follows. In this case addition and subtraction
of numbers above twenty will make imperfections
here disappear, if the same processes are continued.
Special attention ought to be given to the numbers
12, 15, 1 6, 1 8, and 20, because they afford an oppor-
tunity to prepare the children for multiplication and
division. The following suggestions are offered :
\ year = 6 months ; f year 12 months.
1 ft A (t ' % (f Q (f
"3 ~~ 4 >
i " =3 " ; I " = 6
* " =2 " ; | " - 4 "
TV " = I " J T 2 2 " = ^
If I apple costs 2 cents, what cost 2, 3, 4, 5, 6 apples ?
1 3 " 2, 3, 4 apples?
I" 4 " 2, 3 apples ?
6 12 " i, 2, 3, 4, 5 apples?
4 " 12 " " i, 2, 3 apples ?
3 " 12 " " i, 2 apples ?
2 " 12 " " i apple?
The written work on numbers from ten to twenty
is to follow the illustrations of Chart VI. , as previ-
It is recommended that the addition and subtrac-
tion should be limited to the fundamental numbers,
because additions in the second ten are grounded
upon those in the first ten.
60 ARITHMETIC IN PRIMARY SCHOOLS.
If the child knows that :
1 + 3 = 4, he knows that 11+3 = 14.
2 + 6-8, " " 12 + 618.
4 + 3-7, " " 14 + 3-17.
It is only necessary to call his attention to these
NUMBERS FROM ONE TO ONE HUNDRED.
FOR the purpose of extending the ideas of numbers
to one hundred use should be made of the large
numeral frame with 100 balls. First move out two
rows of balls on the frame. These, as the children
already know, contain 2 tens, or 20 units.
Add to these another row, and we have now
3 tens, or 30 units ;
so may be shown 4 tens, or 40 units,
5 tens, or 50 units,
6 tens, or 60 units,
7 tens, or 70 units,
8 tens, or 80 units,
9 tens, or 90 units,
10 tens, or 100 units.
The statements of the truths thus exhibited may
be practised by
a. Naming the numbers in order, forwards and
b. Questioning on the numbers out of their order ;
c. Pointing and having the children name ;
62 ARITHMETIC IN PRIMARY SCHOOLS.
d. Naming and having the children point ;
e. Forming the series in order and having them
10 + 10 = 20 ; 20 + 10 = 30, etc.
100 10 = 90 ; 90 10 = 80, etc.
The expression of the numbers in figures should
be omitted at first, so that the ideas of the numbers
may not be confused with the figures. If written
work for the pupils is desired, enough may be found
in a review of the work on numbers from ten to
Now the teacher may go back again to one ten,
and have the numbers n, 12, 13, and so on to 20,
formed by the addition of one unit at a time, as was
recommended in the development of numbers from
ten to twenty. In the same way should the numbers
from 21 to 30, 31 to 40, 41 to 50, etc., to 100 be
formed. The numerical facts thus illustrated may
be expressed :
Twenty and one are twenty-one ;
Twenty and two are twenty-two ;
Twenty and three are twenty-three ;
and so on to 100; and also,
Two tens and one unit are twenty-one units ;
Two tens and two units are twenty-two units ;
Two tens and three units are twenty-three units,
and so on to 100.
At each new ten the teacher should stop and prac-
tise the children in the numbers already learned, by
questioning them on the numbers out of their order.
For example, point to 24, 27, 29, 22, or 30 balls, and
ask, How many balls ? Tell the pupil to point to
different numbers. Pointing to 25, ask, How many
tens and how many units are there? What is the
number called? How many is it more than 20? How
many less than 30 ? Ask :
What number comes after 25, 22, 29 ?
What number comes before 25, 22, 29?
Count forward from I to 30.
Count backward from 30 to I.
Count 2, 4, 6, 8, 10, and so on to 30.
Count i, 3, 5, 7, 9, and so on to 29.
Count 30, 28, 26, 24, and so on to o.
Count 29, 27, 25, 23, and so on to I.
With the introduction of each ten review from the
When the school is not furnished with a large
numeral frame, a chart like the following, Chart X.,
may be used as a means of giving the children an
intuitive knowledge of numbers from one to one
hundred, and of the decimal system of numbers.
Both the top and bottom parts of the chart may be
used for counting by tens. If a piece of stiff paste-
board or a ruler be cut in this form, it may be so
64 ARITHMETIC IN PRIMARY SCHOOLS,
held as to cover any number of units in any row ;
and so by moving it down the lower part of the chart,
and then across the chart, the formation of all num-
bers from one to one hundred may be shown to the
eye, the same as by the numeral frame.
COUNTING BY TENS.
COUNTING TO ONE HUNDRED.
Chart X. may be used profitably as a means of
reviewing numbers from one to one hundred, either
by counting by tens or counting by units ; or for
READING AND WRITING NUMBERS.
showing numbers to be named, or to be reduced to
tens and units ; or for showing any number of tens
and units that may be named ; as well as for various
16. READING AND WRITING NUMBERS.
Written work, that is, use of figures, should not be
introduced till the pupils are able to find any number
on the numeral frame or on the chart, to resolve any
number which may be shown on the chart or frame
into tens and units, or to unite any number of tens
and units into the number which they constitute. It
WRITTEN REPRESENTATION OF CHART X.
66 ARITHMETIC IN PRIMARY SCHOOLS.
is of the utmost importance at this stage of the work
that figures are not mistaken for numbers ; and to
secure this, much work should be done with objects
that can be numbered, before the pupils are intro-
duced to the use of figures, which are the mere signs
of the numbers themselves.
The preceding chart, Chart XL, corresponds to
Chart X. It is simply the written signs of the num-
bers which the children have just learned.
A word as to the use of this chart. The children
are to read the expressions of the different numbers
in figures, to find the expression for any number
which the teacher may name, to find the expression of
any number which the teacher may show on Chart X.,
and to show upon Chart X. the number correspond-
ing to any figures to which the teacher may point.
If the children understand into how many tens and
units any number may be separated, it will generally
be sufficient to tell them that the tens are written at
the left and the units at the right. Chart XI. may
now be copied by the pupils. The teacher may now
have the expressions for different numbers which he
finds on the chart read and then copied.
To impress the written expressions of the different
numbers from one to one hundred upon the minds
of the pupils, the following series may be constructed
and written by the children :
1 + 1-2; 2+1-3; 3 + i = 4>
and so on to 100.
In the two first courses, that is, in the study of
numbers from one to ten, and of numbers from one
to twenty, we have recommended the simultaneous
treatment of the four ground rules of addition, sub-
traction, multiplication, and division. At this point
they should be separated. A few words in explana-
tion of the reason will be added.
With the size of numbers the parts into which the
numbers can be separated multiply ; so that the point
is soon reached where the resulting facts can no
longer be impressed upon the memory, and, indeed,
where this is no longer necessary. As a rule, the
memory is to be burdened with those facts only
which constitute the foundation upon which future
progress depends ; for example, the addition and sub-
traction tables. In teaching these we were able to
ground all the written exercises upon the direct
observation of the charts and other objects ; but in
the treatment of higher numbers this is impossible.
But the ability of the pupils at this point has so
increased that they are able to solve a much larger
number of problems in the same time. We must be
able, therefore, especially where there are several
divisions to be occupied at the same time, to select
problems which will be easy to assign, which will
make the work of correction easy, and which will
afford much occupation for the pupils.
68 ARITHMETIC IN PRIMARY SCHOOLS.
More than this, arithmetic is partly an art, and in
art facility in doing presupposes practice. Facility,
however, can never be attained unless the same thing
is practised for a long time. If a piano player wishes
to make a movement absolutely his own, it is not
enough for him to practise it in its turn along with
twenty other movements ; he must repeat this move-
ment by itself over and over. So facility in a definite
numerical operation, be it addition or subtraction or
any other, is attained only through continued prac-
tice in this very operation. This practice, however,
must not be mere mechanical routine, but rather,
thoughtful practice. To secure this, careful work
must be done in the addition of units.
The instruction must begin with what is easiest,
and proceed gradually to the most difficult ; begin
with the addition of two, then add three, and so on
to nine. The exercises are to be given at first with
the help of apparatus for actual observation ; but
gradually the apparatus is to be dispensed with, and
the pupils are to be taught to reach the required
results by processes of thinking. The following may
serve as an example of the proper work in teaching
the addition of units to tens or to tens and units.
Suppose the number seven is to be added to one and
to the succeeding results; the steps would be as
I + 7 = 8, which is already known.
8 + 7 = 15, may be thought as 8 + 2=10,
and 10+ 5 = 15.
15 + 7 = 22, may be thought as 1 5 + 5 20,
and 20 + 2 =- 22.
22 + 7 29, may be thought as 2 + 7 9,
and 20 + 9 = 29.
2 9 + 7 = 36, niay be thought as 29 . + I = 30,
and 30 + 6 = 36.
36 + 7 = 43, may be thought as 36 + 4 = 40,
and 40 + 3 43.
43 -f- 7 = 50, may be thought as 3 + 7 = 10,
and 40+10 = 50.
SO + 7-S7.
57 + 7 = 64, may be thought as 57 + 3 = 60,
and 60 + 4 = 64.
64 + 7 71, may be thought as 64 + 6 70,
and 70+ i = 71.
71 + 7 = 78, may be thought as 7 + I = 8,
and 70+ 8_= 78.
78 + 7 = 85, may be thought as 78 + 2 = 80,
and 80+ 5 = 85.
85 + 7 = 92, may be thought as 85 + 5 = 90,
and 90 + 2 = 92.
92 + 7 = 99, may be thought as 2 + 7 = 9,
and 90 + 9 = 99.
If the result falls within the given ten, only the
units are to be increased ; but if the result reaches
into the next ten, the given units are first to be
increased to ten and the remaining units added to
the next ten. Along with this exercise in the succes-
sive additions of seven, let the corresponding parts
70 ARITHMETIC IN PRIMARY SCHOOLS.
of the addition table be carefully practised. Write
upon the board the numbers i, 2, 5, 8, 3, 9, 7, 4, 6,
and 10, and have the number 7 added to each, until
the results are perfectly committed to memory. The
separating of the number to be added into two parts,
the adding of the first part to the units, and the
adding of the second part to the next ten will disap-
pear with continued practice ; the addition of 26 and
7, for example, will soon be reduced to a single opera-
tion, when the pupil is perfectly familiar with the
fact that 6 + 7 = 13.
Facility in addition grounded upon a thorough
memorizing of the addition table is the end for which
the teacher should strive; but he will not succeed in
having all pupils reach this facility in the time which
can properly be given to the first steps in addition.
If the teacher insists upon a perfect memorizing of
the addition table by the weakest pupils, under all
circumstances, before proceeding to the addition of
larger numbers, he does a wrong to the brightest,
because he holds them back upon the first stage of
addition so long that they become weary of the work.
This is a pedagogical sin, which avenges itself no
less upon the individuals than upon the class. The
reason for the unequal acquisition of facility lies in
the unequal talent of the pupils for impressing num-
bers upon the memory. Number memory is not the
same in all pupils.
The teacher may, however, console himself with
the reflection that clear understanding and definite
comprehension of the process is of more use to the
student than great facility in performing the process.
Every exercise must be brought within the compre-
hension of the pupil before he is allowed to enter
upon a new stage of work. The attainment of a
reasonable amount of facility is no less desirable in
arithmetic than in other branches of study ; and yet
arithmetic has this advantage, that the following
stages always take up the exercises of the preceding,
and thus furnish an opportunity to increase the pupil's
facility in preceding processes.
Even in private instruction it would not be advisa-
ble to keep a pupil of weak memory for numbers
upon the first steps so long as would be required in
order to reach the extreme of facility. To weary the
pupil, to destroy his desire for arithmetical knowl-
edge, is an injury which outweighs any facility in
One of the easiest ways of assigning examples for
practice at this stage of the work is to have series of
numbers formed, at first by the addition of the same
number, later by the addition of different numbers
alternately. Such series occupy profitably one divis-
ion while the teacher is busy with another. A few
written figures will indicate the desired lesson. A
glance at the slate containing the pupil's work shows
72 ARITHMETIC IN PRIMARY SCHOOLS.
whether it is correct or not. The following examples
will serve for explanation :
+ 7= 7,
i + 7= 8,
2 + 7= 9,
4 + 7= n.
7 + 7 = 14,
8 + 7 = i5,
9 + 7-16,
11+7= 1 8.
14 + 7-21,
15 + 7-22,
16 + 7-23,
18 + 7= 25.
21 + 7 = 28,
22 + 7 29,
23 + 7-30,
25 + 7- 32.
28 + 7 = 35,
29 + 7 = 36,
30 + 7 = 37,
32 + 7= 39-
35 + 7-42,
36 + 7-43,
37 + 7-44,
39 + 7- 46.
42 + 7-49,
43 + 7 = 50,
44 + 7-51,
46 + 7= 53-
49 + 7-56,
50 + 7-57,
53 + 7- 60.
56 + 7-63,
57 + 7-64,
58 + 7 = 65,
60 + 7 67.
63 + 7-70, 64 + 7-71, 65 + 7-72, 67 + 7- 74.
70 + 7 = 77, 7i+7 = 78, 72 + 7 = 79, 74 + 7= 81.
77 + 7-84, 78 + 7-85, 79 + 7-86, 81+7-88.
84+7-91, 85 + 7-92, 86 + 7-93, 88 + 7- 95-
9i+7=9 8 , 92 + 7 = 99, 93 + 7=ioo, 95 + 7-102.
This table represents the pupil's work. The prob-
lems a, by c, and d may be assigned by telling the
class to add 7 to o, i, 2, and 4 fourteen times ; or
simply by writing + 7, 1+7, etc.
Since the first numbers are o, i, 2, and 4, the
results in any horizontal line vary by i, 2, and 4, and
so do the final results. A glance at one or two
places in the vertical line and at the end will show
whether the work is all right. The final results will
be 14 X 7 plus i, 2, and 4. So any series may be
dictated, beginning with any number from i to 9,
and adding any number from i to 9, and all the
results known at a glance.
To give a greater variety, and at the same time
provide for reviews, two numbers may be added
alternately, for example :
2 + 4- 6.
9 + 4=13-
i3 + 3 = i6.
16 + 4--= 20.
20 + 3 23, etc.
Compare this with the series marked c above.
2 + 7- 9.
9 + 7 = 16.
16 + 7 = 23, etc.
and it is obvious that the series will end with 100.
A word in regard to written exercises in general.
While it is true that in the beginning of the study of
numbers figures are a positive hindrance, this is by
no means universally the case. Written exercises
are of the greatest importance, provided they are
properly connected with observation and oral work.
Practical life requires the use of written arithmetic
and therefore the school must prepare the pupils for
it. But the pedagogical reason is still stronger.
Children are not all alike in ability. It often happens
that in oral work the brightest pupils have too little
to do in proportion to their ability ; or that the weak-
ARITHMETIC IN PRIMARY SCHOOLS.
est are behindhand in the solution of the problems,
so that their real progress is hindered. Now written
exercises, especially such series as have just been
recommended, are adapted to all conditions of the
class. Each can do in a given time what he is able,
and all will do good work. The bright ones are not
kept back, and the weakest are not overdriven. If all
the work is not done by all the pupils, what is done
is good for all. Written work, then, is adapted to
all, while oral work is often adapted only to the
Moreover, exercises in written work for the class
FOR PRACTICE IN THE GROUND RULES.
allow the teacher time to give individual instruction
to the weak pupils.
The preceding chart, marked Chart XII., will be
found very useful in assigning work to be done out-
side the recitation hour/ as well as for exercises, both
oral and written, to be performed in the class. Let
the numbers from one to ten be added to each of the
numbers and we have one thousand examples in
Let the numbers 2, 3, 4, etc., to 10, be subtracted
from 100 and from the successive remainders, and
the exercises will be the reverse of those explained
under addition ; for example :
100 7 may be thought as 10 7 3,
and loo 7 = 93,
93 7 may be thought as 93 3 = 90,
and 90 4 = 86,
86 7 may be thought as 86 6 = 80,
and 80 i = 79,
79 7 may be thought as 9 7 = 2,
and 79 7 = 72,
72 7 may be thought as 72 2 = 70,
and 70 5 = 65 ;
and so on till the remainder is less than seven.
If the minuend consists of tens only, the subtra-
hend is to be taken from 10, and the remainder
added to the next ten below; for example : 1007
76 ARITHMETIC IN PRIMARY SCHOOLS.
becomes 10 7 3, and 90+3 = 93. If the minuend
consists of tens and units, and the units are more
than the subtrahend, the subtrahend is to be taken
from the units and the remainder added to the tens ;
for example : 797 is changed to 9 7 2 and
70 + 2 72. If the minuend consists of tens and
units, and the units are less than the subtrahend, the
units of the minuend are to be subtracted first, and
then from the tens are to be taken the difference
between the units already subtracted and the subtra-
hend ; for example : 93 7 is changed to 93 3 = 90,
and 90 4 86.
This shows us how important it was to teach the
separation of the numbers below 10 into two parts ;
and also reminds us of the propriety of a careful
review of the corresponding number before begin-
ning a new exercise in subtraction. For example,
before giving exercises in the subtracting of seven,
the reviews should cover the following ground :
7 = 6 + i or 1+6,
7-5 + 2 or 2 + 5,
and also 7 = 4+3 or 3 + 4;
10-7, 7-7, 13-7, ii -7> 9~7> iS-7,
12-7, 14 7> 8-7, 16-7, and 17 - 7.
In the addition of 9, the children will often reach
the result by adding 10 and subtracting i ; so in
subtracting 9, they will often reach the result by
subtracting 10 and adding i. Such practices should
not be allowed unless they are understood; which
will be the case if they are discovered by the chil-
dren. But the teacher should examine and, if neces-
It does not follow, however,* that a pupil should be
allowed to continue a practice in numerical computa-
tion simply because he has hit upon it and under-
stands it. It is usually better for the teacher to
train the pupils in the most expeditious methods of
performing operations ; and, as a rule, one method is
better than two. For example, if the pupil is always
required to reach results in the addition and subtrac-
tion of units in the way described above, whenever
he is obliged to go through a conscious process in
reaching the result, he will reach the point where
such operations are automatic much quicker than he
will if he is allowed to reach his results now by one
process and now by another.
Written exercises may be given which are easy to
correct, consisting of series of subtractions. The
series, 100 7, etc., will end with 2 ; for 7 in 100 =
14, and 2 remainder. The series 99 8 ends with 3 ;
since 99-^8 = 12, and 3 remainder. Thus exercises
may be set consisting of a dozen or more subtrac-
tions, so constructed that a glance at the final result
will show whether the work is correct.
Chart XII. may be made helpful in oral drill,
since i, 2, 3, etc., may be subtracted from each
number ; and thus the labor of the teacher may be
78 ARITHMETIC IN PRIMARY SCHOOLS.
19. CONNECTED ADDITION AND SUBTRACTION.
For the sake of variety, as well as for the purpose
of reviewing the work of addition, series of alternate
additions and subtractions may be given for written
work ; for example :
97-7 = 9>
90 + 4-94, 97 - 3 - 94 ;
87 + 4-91, 94-3=9i;
and so on. Since subtracting 7 and adding 4 reduces
a number 3, every second result must be the same as
if 3 were subtracted.
8-3- 5> 1+4-5-
5 + 7- 12,
12-3-9, 5+4 = 9> etc -
Since adding 7 and subtracting 3 increases a num-
ber 4, every second result must be the same as if 4
Work of this kind is easy to assign and easy to
examine. The above illustrations are designed merely
as suggestions of what may be done.
So far in our treatment of numbers consisting of
two places we have added and subtracted only units ;
but it would be a good preparation for work with
CONNE C TED ADDITION AND S UB TRA C TION. 79
numbers from i to 1,000 to add and subtract num-
bers larger than 10 at this stage. The following is
suggested as a good order of work :
a. Tens to tens.
20 + 30,
10 + 20, etc.
b. Tens to tens and units.
30 + 40 = 70 ; so
36 + 40 = 76.
c. Tens and units to tens.
30 + 25,
30 + 20 - 50,
50+ 5 = 5$.
a. Tens from tens
90 70, etc.
b. Tens from tens and units.
96 - 40,
90 40 = 50 ; so
96 40--= 56.
c. Tens and units from tens.
90 - 63,
90 - 60 =-- 30,
30- 3 = 27.
d. Tens and units to tens and units.
32 + 44 32 + 40 + 4,
68 + 28 = 68 + 20 + 8.
d. Tens and units from tens and units.
96 - 34 = 96 30 - 4,
These exercises, at least those marked a, b, and c,
should be readily performed by the pupils orally
before they are changed to written exercises. In
the written work, a union of addition and subtraction
may take place in the same series of exercises, as
shown below :
80 ARITHMETIC IN PRIMARY SCHOOLS.
1. IO+2O = 3O. 2. 10+30 = 40. 3. 100 30 = 70.
20 + 20 = 40. 20+30=50. 90 30 = 60.
4. IOO 2O=
IO + 5O
4 + 40
10 + 25
19. 100 25
22. 12 + 24
IO + 4O
10 + 46
20. 100 46
23. 7 + 36
Similar series of numbers may be given indefinitely
as the needs of the class require.
The following rules may be useful :
To add one number between 10 and 100 to an-
other, add first the tens and then the units ; for
example: 57 + 39; S7 + 30=87; 8^+9 = 96.
To subtract one number between 10 and 100 from
another, subtract first the tens and then the units ;
for example: 77~49; 77 4O = 37; 37 -9^28.
A teacher ought to be satisfied with the weak
9. IOO 5O
9 + 50
CONNECTED ADDITION AND SUBTRACTION. 8 1
pupils if they can solve problems in these ways, and
not try to teach them shorter processes. It is better
for a pupil to be certain in one way than to be uncer-
tain in several.
It was previously shown that there was great
advantage in being able to increase any fundamental
number to 10; there is a like advantage in being
able to increase any number below a hundred to a
hundred. It is well, therefore, to drill the pupils in
such exercises as these :
86 and how many are 100 ?
86 + 4 = 90; 90+10=100; hence 14.
67 and how many are 100?
67 + 3 = 7o; 70+30=100; hence 33.
48 and how many are 100?
48 + 2 = 50; 50+50=100; hence 52.
Chart XII. affords abundant matter for drill in
the addition of numbers below 100 ; for example, in
adding 48 to each number on the chart, there are
100 additions. But each other number below 100
may be added ; which makes 5,000 examples in addi-
tion. Or, by how many does 53 differ from each
number on Chart XII. ? In answering this, the
child performs 100 subtractions. But the same may
be asked of all the other numbers below 100; which
gives 5,000 examples in subtraction. Add 24 to each
number in the first five vertical columns ; in the first
four horizontal lines, etc. Remember that practice
82 ARITHMETIC IN PRIMARY SCHOOLS.
The multiplication table is the foundation of the
process of multiplication. It is the tools without
which neither multiplication nor division can be per-
formed. Hence the child must make it so completely
his own that it cannot be forgotten and that it will
always be present to him in the twinkling of an eye
when it is needed for use. It is in the power of the
teacher to render such help to the little ones as will
spare the tears which, without such help, will be
sure to flow when the demand is made upon them to
learn the multiplication table by heart.
If the teacher wishes happily to avoid these break-
ers he must be sure that two things always exist in
proper relation one to the other, intelligence and
practice. Intelligence, which is gained only through
direct observation, was formerly neglected ; but the
tendency at the present time is to neglect the prac-
tice. Instead of drilling the pupils thoroughly in the
multiplication table in school, by means of recitations
and questioning, and by means of connecting the
work at every step with the preceding lessons, many
teachers are satisfied with making the children under-
stand how it is formed, and leaving the memorizing
to be done as home lessons. But it is the special
task of the teacher to show the children how they
should learn. In order to point out to inexperienced
teachers what exercises they may introduce to advan-
TEACHING THE MULTIPLICATION TABLE. 83
tage while the pupils are committing the multiplica-
tion table to memory, it will be necessary to go
somewhat into details.
Since thorough drill requires a long time, it is
recommended to make a preparation for the learning
of the multiplication table while teaching addition
and subtraction. When the pupils have thoroughly
learned to add and subtract the number two, they
may be taught to multiply by two ; when they have
learned to add and subtract the number three, they
may learn the threes of the multiplication table, etc.
By this course sufficient time may be secured for
reviews, which here are indispensably necessary,
since upon them depends the impressing of numbers
upon the memory.
21. TEACHING THE MULTIPLICATION TABLE.
TWOS OF THE MULTIPLICATION TABLE.
In our treatment of numbers from i to 20 we have
already found once 2, 2 times 2, and so on to 10
times 2, and we will rejoice at whatever has remained
in the memory ; still it is necessary to develop the
Place two balls on the numeral frame, or two
points on the board beside each other, thus :
and ask, How many balls are there ? Then put two
more balls with them, thus :
84 ARITHMETIC IN PRIMARY SCHOOLS.
and ask, How many times two balls are there ? How
many are two times two balls ? How many are two
times two ? How many are two twos ?
Just so may the ideas of 3, 4, 5, 6, 7, 8, 9, and 10
times 2 be developed. In doing this the following
figure will be formed, and the following expressions
of the truths which it represents should be repeated
many times, both by individuals and in concert :
3 ' "
" 1 6.
. . 9
It will help if the children are led to find that, for
example, 3 times 2 units are just as many units as 2
times 3 units. Thus, in the following figure there
are 3 rows of 2 points each, and there are also 2 rows,
of 3 points each.
TEACHING THE MULTIPLICATION TABLE. 85
So it may be shown that
4 times 2 are 8, and 2 times 4 are 8.
5 " 2 " 10, " 2 " 5 " 10.
6 " 2 " 12, " 2 " 6 " 12.
7 " 2 " 14, " 2 " 7 " 14.
8 " 2 " 1 6, " 2 " 8 " 1 6.
9 " 2 " 1 8, " 2 " 9 " 1 8.
IO " 2 " 2O, " 2 " IO " 2O.
The results stated at the right are already known
as the sums of equal numbers. The one set of state-
ments assists the pupil in remembering the other,
yet the truths ought not to be confused. Practical
examples like the following will guard against such
A mother gave her son 4 apples yesterday and 4
to-day ; how many times did he receive 4 apples ?
How many are 2 times 4? How many times 4 is 8?
How many fours can be made of 8?
A woman gives her child 2 apples daily ; how
many times 2 apples does he receive in 4 days ?
How many are 4 times 2 ? How many times 2 is 8 ?
How many twos can be made of 8 ?
Let the children illustrate the multiplication table,
as shown above, with points on their slates, and affix
the results, thus :
If they are to study the multiplication table at
86 ARITHMETIC IN PRIMARY SCHOOLS.
home, let them first construct it ; otherwise it is apt
to have little meaning. The impressing of the facts
out of their order is to be effected mainly through
question and answer, and is proper work for the
schoolroom. The more varied the practice, however,
the more firmly the facts are impressed upon the
memory. Hence the reverse form of viewing the
facts is to be used. For example : How many times
is 2 in 12? How many twos in 12? How often is
2 contained in 12? How often can we take 2 from
12? etc. It is well to spend a week or two on the
number 2 ; and a portion of each lesson should be
given to practical applications ; for example :
1 whole = 2 halves, i apple costs 2 cts.
2 wholes 4 halves. 2 apples cost 4 cts.
3 = 6 " 3 " " 6 "
etc. etc. etc. etc.
THREES OF THE MULTIPLICATION TABLE.
Let the course of instruction be as fol-
a. Construct the table on the frame or
board, as in the margin.
b. Practise alone and in concert forwards.
c. Practise alone and in concert back-
d. Question out of the regular order.
e. Let the children make the same on
TEACHING THE MULTIPLICATION TABLE. 8/
The results, i, 2, 3, 4, 5, 6, and 10 times 3, the
children will easily retain ; for 3 X 2 has been already
learned in studying the twos ; 3 X 3, in studying the
number picture for 9 ; 4X3, in the treatment of 12 ;
5 X 3, in the study of 1 5 on Chart VI. ; 6 X 3, in the
study of 1 8 on Chart VI. ; and 10X3 = 3X10 = 3
tens. These results will now afford little difficulty ;
7, 8, and 9 times 3 will cause more. But 9X3
10x3-1x3; 7><3 = 7+7 + 7; 8x3-8 + 8 + 8;
and all these are to be taught from rows of points.
The following applications are suggested :
1 orange costs 3 cents,
2 oranges cost 6 cents,
3 oranges cost 9 cents, etc.
FOURS OF THE MULTIPLICATION TABLE.
The course of exercises is the same as in teaching
the twos and threes. More or less are already known
of i, 2, 3, 4, 5, and 10X4; so fix these numbers
first. Connect 9X4 with 10X4; 6X4 with the
known 5x4. Take special pains with 7X4 and
8X4. Apply as follows :
1 horse has 4 legs,
2 horses have 8 legs,
3 horses have 12 legs, etc.
FIVES OF THE MULTIPLICATION TABLE.
a. The pupil knows i, 2, 3, 4, 5, 10 X 5.
b. He learns 5 X 5 = 25 easily from the sound.
88 ARITHMETIC IN PRIMARY SCHOOLS.
C. 9 X 5 = 10 X 5 I X 5.
d. 6x5 5x5 and 5 = 30.
e. 8x5 = 2 times 4X5 = 2 times 20, or 4 times
2x5= 4 times 10.
f. 7 x 5 is to be connected with 6x5.
Always direct the practice so as to connect the
thing to be learned with what precedes ; first a, then
b y then a and b ; then c, then a y by and c ; then d,
then ay by c, d\ then e y then #, by c y dy e y etc. Apply
thus : i five-cent piece = 6 cents, etc.
SIXES OF THE MULTIPLICATION TABLE.
a. i, 2, 3, 4, 5, and 10 X 6 are known.
b. 6 X 6 is remembered by the sound.
c. 9 X 6 is to be connected with 10 X 6.
d. 7 X 6 is to be connected with 6x6.
*?. 8 X 6 demands special work.
Application : i week has 6 working-days, etc.
SEVENS OF THE MULTIPLICATION TABLE.
a. i, 2, 3, 4, 5, 6, and 10 X 7 are known.
b. 7 X 7 is easy to remember from the sound.
c. 9 X 7 is to be connected with 10x7.
d. 8 X 7 is to be connected with 7x7.
Application : i week has 7 days, etc.
EIGHTS OF THE MULTIPLICATION TABLE.
a. i, 2, 3, 4, 5, 6, 7, 10 X 8 are known.
b. 8x8 is easy to learn from the sound.
TEACHING THE MULTIPLICATION TABLE. 89
c. 9X8 10X8 1X8.
Application : 8 boys sit in i row, etc.
NINES OF THE MULTIPLICATION TABLE.
a. i, 2, 3, 4, 5, 6, 7, 8, 10 x 9 are known.
b. 9 X 9 is learned from the sound, also connected
with 10 x 9.
Application : i yard costs 9 cents, etc.
TENS OF THE MULTIPLICATION TABLE.
The result is already known.
Application : i dime is worth 10 cents, etc.
If the children are made to observe, to recall, and
to connect the unknown with the known, in the way
just pointed out, they may soon be brought to under-
stand any part of the multiplication table. But the
teacher must discriminate sharply between under-
standing and knowing. Knowing presupposes con-
tinued practice and diligent repetition of what pre-
cedes ; hence the pupil should never pass to a new
sentence without reviewing what goes before. Let
the new sentence be a reward for what is already
learned, so that the children will be accustomed to
find the reward for learning in the act of learning.
Knowing the multiplication table implies readiness
for use. The child must remember only the result,
not the process of reaching it. Question must follow
answer instantly. In a word the multiplication table
90 ARITHMETIC IN PRIMARY SCHOOLS.
must be absolutely a thing of the memory. On
thought of the words seven times jive the thought of
the word thirty-five must instantly follow. Perfect
understanding comes through illustration ; perfect
memorizing, through diligent use and through fre-
quent repetition almost endlessly continued.
What we have already explained is only a prepa-
ration for learning the multiplication table. This
preparation was aimed at in the addition of numbers
from one to a hundred ; for example, when the pupil
was exercised in the successive additions of six,
ground was broken for learning the sixes of the table.
The results can be fixed in the mind only through
continuous application. All up to this point is only
a preparation for learning the table in its written
22. APPLYING THE TABLE TO WRITTEN WORK.
The written sign for multiplication is an inclined
cross, thus, X, and means time or times.
If we should write down the numbers from I to
10, and ask the pupils to use them successively as
multipliers of a given number, we should by this
means assist them to reach the results by the suc-
cessive additions of the number to itself. However
necessary this order may be in the development of
the table and for its thorough comprehension, still a
practical mastering of the same, a ready working
knowledge of it, demands its application out of this,
APPLYING THE TABLE TO WRITTEN WORK. 9!
In order to fix in the minds of pupils that they are
always working for results, and not merely for prac-
tice, write down the numbers from i to 10 in the
following order :
i, 2, 5, 8, 3, 7, 4, 9, 6, 10.
Now, partly for the purpose of introducing variety
into the work, and partly for the sake of review, con-
nect both addition and subtraction with exercises in
multiplication. The beginning of the work of mul-
tiplying with two may be as follows :
1X2+1=3. 8x2 1 = 15. 9X2+1= .
1X21= I. 3X2+1= . 9X21= .
2x2+1=5. 3x21= . 6x2 + 1= .
2x21=3. 7x2 + 1= . 6x21= .
5X2+1 = 11. 7X21= . 10X^2+1= .
5X2 I= : 9. 4X2 + 1= . 10X21 =
8X2+1 = 17. 4X21= .
Substitute the numbers 2, 3, etc., to 10, for i in
the above exercises, as the numbers to be added and
subtracted, and you have 200 examples in multiplica-
tion by 2. Now substitute the numbers 3, 4, 5, etc.,
to 10, in place of 2 in the above examples, as the
numbers to be multiplied, and you have 1,800 exam-
ples in multiplication. With one-half of these are
connected examples in addition, and with the other
half examples in subtraction.
In assigning work of this kind it is only necessary
for the teacher to write or dictate one or two exam-
ARITHMETIC IN PRIMARY SCHOOLS.
pies of a kind ; for the pupils can readily invent the
rest of the series up to 20 examples.
When the pupils are familiar with the multiplica-
tion of whole numbers, as indicated above, the mul-
tiplication of fractions may be introduced with profit.
If the treatment of fractions is to be easy and pleas-
urable, the pupils must be made entirely familiar
with fractions themselves, as well as with the mode
of expressing them ; and for this purpose the appli-
cation of the multiplication table furnishes an excel-
lent opportunity. We will begin with the represen-
tation and multiplication of halves.
That one whole is equal to two halves may be
illustrated by the actual division of a piece of paper,
APPLYING THE TABLE TO WRITTEN WORK. 93
an apple, etc., into two equal parts. Then the same
may be illustrated by dividing a line or a circle. As
a result of the treatment of lines and circles in this
way, the preceding work will appear on the board
and on the pupils' slates.
It is only necessary to tell the pupils that half is
written thus, T , and that the number of halves is
shown by the figure above the line. The method of
writing fractions needs much practice on the slates.
When the children have become familiar with the
meaning and representation of halves, they may per-
form the following series of examples :
i+l = f. 8-| = 9 + 1 =
1-1 = 1- 3 + 1= 9-1 =
+ l = f.
+ ^ =
+ ^ =
8 + 1- 4-^1 =
These exercises may be increased to almost any
extent by adding and subtracting more than 1.
A preparation for division may be made by orally
questioning the children in this way : How many
whole ones in --/- ? How many whole ones and
halves in -^-P
In a similar manner may thirds, fourths, etc., to
tenths, be treated.
ARITHMETIC IN PRIMARY SCHOOLS.
23. CONSTRUCTING THE TABLE.
The practice of beginning the work in multiplica-
tion by committing to memory a ready-made multi-
plication table cannot be too strongly condemned.
But if the pupil writes down the facts in tabular form
as fast as he learns them, he will construct for him-
self the following table, designated as Chart XIII.
This will not only serve to recall the facts, but will,
at the same time, be a means of teaching the facts
That five times four are twenty may be shown
thus : Count down the chart at the left to 4 ; there
are four rectangles ; at the right of these are four ;
CONSTRUCTING THE TABLE. 95
and so on to the row beginning with 5 in the upper
row. That is, 5 fours are 20. If now we count the
rectangles along the top row from i to 5, we find five
in the row ; below these is another row of five ; and
so on to the row beginning with 4 in the left-hand
column. That is, four fives are 20.
In this way we obtain an intuitive knowledge that
5 fours are 20 and that 4 fives are 20. In the same
way all the facts of the multiplication table may
be demonstrated. Such demonstration will lay the
foundation for the fact, to be learned by and by, that
the product is not affected by the order of the
This table is well fitted to teach the resolution of
numbers into their factors ; for if the children know
6 x 4 = 24,
24 - 6 X 4 ;
4 x 6 = 24,
24 - 4 X 6 ;
3 x 8 - 2 4 ,
24 = 3 X 8 ;
8 x 3 = 24,
24 = 8 X 3 ;
and every fact in multiplication should be followed
by the corresponding fact in factoring. This is an
excellent preparation for division. So also is the
changing of fractions to whole numbers ; for exam-
ple : f = 4 andi; y = 5 and f.
In order to prepare the pupils for the work of
multiplication when dealing with larger numbers,
they should here be taught to multiply numbers
consisting of tens and units. The following are
illustrations of the work :
96 ARITHMETIC IN PRIMARY SCHOOLS.
3 x 24 - .
3 x 20 = 60.
3>< 4 == i2.
4 x 18 =
4 x 10 = 40.
4 x 8-32.
8x10 = 80.
8 x 2 = 16.
3 x 24 = 72. 4 x 18 = 72. 8x12= 96.
The written work in multiplication at this stage is
limited. We can, however, use the following series
of numbers from Chart XII. :
a, b y c, d by 2 ; a, b by 5 ; a by 8 ;
a, b, c by 3 ; a, b by 6 ; by 9 ;
a, b by 4 ; # by 7 ; # by 10.
In what precedes we have shown how, through
objective illustrations, the products of numbers 2, 3,
etc., to 10, may be understood by children, and how
these products may be fixed in their minds by oral
and written exercises. These products form the
so-called multiplication table, by the help of which
many arithmetical operations, which might be per-
formed by the repeated addition of the same number,
may be materially shortened. While constructing
this table the pupil has found that the multiplication
of a number is finding the sum obtained by additions
of the same number. He has himself found the
product by the addition of the same number ; and
he can in the same way find it again, should it escape
his memory. But facility in computation requires
that these products be made things of the memory.
Remembering how to find a product is to be distin-
guished, from remembering the product itself. In a
; B N
PREPARATION FOR DIVISION. 97
subject like arithmetic, where the understanding is
constantly called into exercise, there must be no
halting of the memory. Hence we seek to make the
facts of the multiplication table so appropriated by
the mind that they will seem to be the necessary
qualities of the memory itself. This is to be accom-
plished through continuous practice in computation,
provision for which has been made in what precedes.
24. PREPARATION FOR DIVISION.
It is well so to treat the subject of multiplication
as to prepare the pupils for division. We have been
finding products when we knew the factors ; but the
process is to be reversed, and we are to find the
factors when the product is given ; or, we are to find
one factor when the product and the other factor is
given. To divide 24 objects, beans, sticks, etc., into
4 equal parts, put first one object in each of 4 differ-
ent places, then distribute 4 more in the same way,
then 4 more, and so on till the 24 are all distributed.
We now have 6 objects in each place. One of the
four parts, which together contain 24 objects, con-
tains 6 objects. It follows that 24 is 4 times 6, and
also that the fourth part, or \, of 24 is 6.
This finding of the second factor is accomplished
through successive subtractions of the same number ;
but facility in reckoning requires the pupil to be
able, given the product and one factor, to know the
98 ARITHMETIC IN PRIMARY SCHOOLS.
other instantly. He must be taught . to perform a
process the opposite to what is required in finding
the product from the factors. He must be able to
tell at once how large a certain part of a number
is, how often a certain number can be taken from
another, or how often a certain number is contained
Much may be done in connection with multiplica-
tion to prepare the student for such work. The
Illustrates these truths :
4 times 6 = 24 ; 6 times 4 = 24 ;
\ of 24=- 6; i of 24= 4;
4 in 24 = 6 times ; 6 in 244 times.
Therefore, to the usual questions, How many are
4 times 6 ? etc., add : From what number can 4 sixes
be taken ? 6 fours ? From what number can 4 be
taken 6 times ? Six 4 times ? In what number is 4
contained 6 times ? Six 4 times ? What is the fourth
part of 24 ? The sixth ? Of what number is 6 the
fourth part ? Four the sixth part ?
If 4 apples cost 24 cents, how much will i cost ?
If 6 cost 24 cents, what costs i ? Charles stands 24
soldiers in 4 rows ; how many stand in i row ? What
part of 24 is one row? 24 is how many times 6?
How many times 4 ? etc.
If such questions as these are asked in connection
with the development and application of the multipli-
cation table, a good preparation will be made for the
next stage of the work, namely, division.
There are two kinds of division, namely, separating
a number into equal parts, and finding how often one
number is contained in another. As an example of
the first kind, suppose 6 children have 48 cents, and
the question is, How many cents will each child
have, if the cents are equally divided among them ?
We reason that each child will have one-sixth of 48
cents, or 8 cents. Here is an actual division, a sepa-
ration of the 48 cents into 6 equal parts.
As an example of the second kind of division, let
the question be, Among how many children can 48
cents be divided if each child receives 6 cents ? We
reason thus : From 48 cents 6 cents apiece can be
given to as many children as the times that 6 cents
can be taken from 48 cents, or the times that 6 cents
are contained in 48 cents, namely, 8 times ; hence,
among 8 children. Here we have found how many
times 6 cents are contained in 48 cents.
In both of these examples the number 48 is divided
into 6 equal parts ; but while the answer to the first
100 ARITHMETIC IN PRIMARY SCHOOLS.
question is 8 cents, the answer to the. second is 8
children. From these examples it appears that the
solution should always correspond to the question.
A confusion of the ideas involved in these two pro-
cesses is a sign of a thoughtless solution. The
teacher should guard against this confusion from the
first, and never allow such solutions as the following :
1. If 48 cents are divided equally among 6 chil-
dren, each child will receive as many cents as 6 is
contained in 48. Six children are not contained in
48 cents. Here 6, that is, 6 cents, is contained in
48, that is, 48 cents, 8 times, and not 8 cents ; and
the comparison is really between the number of
cents and the number of times that 48 contains 6.
A better solution would be this : Each child would
receive one-sixth of 48 cents, or 8 cents.
2. Among how many children can 48 cents be
divided, if each child receives 6 cents ? One-sixth
part of 48 is 8 ; therefore, 8 children. But 48 was
48 cents, and not 48 children. A better solution
would be this : If each child receives 6 cents, 48
cents could be divided among as many children as
the times that 6 cents could be taken from 48 cents,
namely, 8 times : hence, among 8 children.
Both forms of division must be made clear to the
pupils through practical problems ; for both forms
are of equal use.
Division, as soon as it deals with numbers beyond
the multiplication table, is a very complicated pro-
DIVIDING BY TWO. IOI
cess ; hence it is necessary to be very patient in
teaching it, and to proceed very gradually from the
easier to the more difficult. If the first difficulties
are really overcome, much has been done to lighten
the subsequent work.
26. DIVIDING BY Two.
Let the children add 2 successively to 2, 4, etc., so
as to form the numbers 2, 4, 6, 8, lo, etc., to 20.
Question thus ? How many are 2X2? 3X2? etc.
Two in 2 how many times ? In 4 ? In 6 ? etc. How
many times can 2 be taken from 2 ? From 4 ? etc.
How many twos in 2 ? In 4 ? etc.
Give this question : Two children are to divide 12
cents equally ; how many will each child receive ?
Although the children are prepared, from what
they have already learned, to answer this and similar
questions, yet, partly to prepare them for the suc-
ceeding stage, and partly to show the teacher by an
example how to manage when the difficulties involved
appear in a new place, we will explain the process of
working. In this example the teacher may use the
cents themselves first, then marks upon the board.
The latter may be arranged as those below. Having
written A and B, place first a circle for a cent which
A is to take, then under it one for a cent which B
takes, and so on till the 12 are represented.
102 ARITHMETIC IN PRIMARY SCHOOLS.
A. o o o o o o
B. o o o o o o
Each has taken 6 cents. When a number is
divided into 2 equal parts, each part is a half. The
half of 12 cents is 6 cents ; the half of 12 is 6.
In the same way develop the idea of the half of 2,
4, 6, 8, 10, 12, 14, 1 6, 1 8, 20.
Two children have 15 apples, how many has each?
Of 14 apples each has 7 apples ;
" i apple " " \ apple.
" 15 apples " " 7| apples.
In the same way treat 3, 5, 7, 9, n, 13, 15, 17,
Draw on the board two rows of circles with 10
circles in each row. This will show that half of 2
rows is i row ; half of 2 tens is I ten ; half of 20
In the same way may the idea of half of 20, 40, 60,
80, and 100 be developed.
The numbers 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 40,
60, 80, and 100 can be divided immediately, that is,
without being separated into parts, because they
appear in the twos of the multiplication table, if we
regard 20, 40, etc., as 2 tens, 4 tens, etc. Numbers
which do not so appear must be separated.
DIVIDING BY TWO. IOJ
Two persons together have 4 ten-cent pieces and
8 cents ; how shall they divide them ?
Each person takes 2 dimes and 4 cents, equal to
24 cents ; so half of 48 is 24.
Or the teacher may write on the board 4 rows of
10 circles each, and 8 circles. Half of 4 rows is 2
rows ; half of 8 circles is 4 circles ; half of 4 tens is
2 tens ; half of 40 is 20 ; half of 8 is 4 ; therefore,
half of 48 is 24.
So may be developed the idea of half of those
numbers whose tens and units are even numbers
22, 24, 26, 28 ; 42, 44, 46, 48 ; 62, 64, 66, 68 ; 82, 84,
Two persons have 3 dimes ; how can they be
divided ? Each takes I dime, equal to 10 cents.
They then exchange the other dime for 10 cents, and
each takes 5 cents ; so that each has 1 5 cents.
Or the teacher may draw on the board 3 rows of
10 circles each. Half of 2 rows, or 20, is 10 ; and
half of the other row is 5 ; so that the half of 30 is 15.
Treat 50, 70, and 90 in the same way.
Show on the numeral frame 3 rows of 10 balls
each, and i row of 6 balls. What is half of them ?
Half of 2 tens is I ten, and the other ten balls
104 ARITHMETIC IN PRIMARY SCHOOLS.
added to the 6 make 16 balls. Half of 1 6 is 8 ; so
half of 36 is i ten and 8, or 18.
Or this : Divide 3 dimes and 6 cents equally be-
tween two persons. Let each take I dime ; exchange
the other dime for 10 cents, which added to 6 cents
make 16 cents. Let each take 8 cents, which with
the dime make 18 cents.
So treat 32, 34, 36, 38 ; 52, 54, etc. ; 72, 74, etc. ;
92, 94, etc.
What is half of 49 apples ?
Half of 40 apples is 20 apples ;
8 " 4 apples;
i " i apple.
" 49 apples is 24^- apples.
So treat all numbers which have even tens and
odd units: 21, 23, 25, 27, 29; 41, 43, 45, 47, 49; 61,
63, etc. ; 8 1, 83, etc.
What is half of 57?
Half of 40 is 20 ;
16 " 8;
Treat in the same way all numbers whose tens and
units are odd numbers: 33, 35, 37, 39; 53, 55, etc.;
73> 75> etc. ; 93, 95, etc.
DIVIDING BY TWO. 10$
We will show by an example of the last exercise
(exercise eight) what the full treatment of a problem
in division should be, as it has been developed in the
successive stages of work in division. Division re-
quires a series of conclusions, and in this fact lies
the difficulty which it presents to the children. There
is no cause for discouragement, however ; for if divis-
ion by 2 is thoroughly mastered, the remaining
numbers can be passed over much more rapidly. Do
not introduce the children to the formal, written
representation of the process till they have attained
considerable facility in explaining it. If they need
to be occupied with written work, there is material
enough in the review of what precedes, especially
in addition, subtraction, and multiplication. Proba-
bly it will take from four to six weeks to ground a
class thoroughly in division of numbers below 100
by 2. Division by the numbers from 3 to 10 will
scarcely require more time. The successive steps in
the solution of a question in the division of a number
by 2 when both the tens and units are odd numbers
may be brought out thus :
(1) Teacher. We will find the half of 75. Can we
divide 75 immediately ; that is, all at once ?
Scholar. We cannot divide 75 immediately.
(2) T. Why not?
S. Because 75 is not found in the twos of the
106 ARITHMETIC IN PRIMARY SCHOOLS.
(3) T. Can we divide 7 tens immediately ?
5. We cannot divide 7 tens immediately.
(4) T. Why not ?
vS. Because 7 is not found in the twos of the
(5) T. What is the next number below 7 that is
found there ?
5. Six is the next number.
(6) T. What is the half of 6 tens, or 60 ?
5. Half of 6 tens, or 60, is 3 tens, or 30.
(7) 7! How many of 75 remain to be divided when
we have divided 60 ?
5. Fifteen remain to be divided.
(8) T. Can we divide 15 immediately?
S. We cannot divide 1 5 immediately.
(9) T. Why not?
vS. Because 15 is not found in the twos of the
(10) T. What is the next number below 15 that is
found there ?
S. Fourteen is the next number.
(11) T. What is half of 14?
5. Half of 14 is 7.
(12) T. How many still remain to be divided?
5. One still remains to be divided.
(13) T. What is half of i?
5. One-half is half of i.
(14) T. How many did we at first obtain, when
we divided 60 ?
DIVIDING BY TWO. IO/
5. We at first obtained 30.
(15) T. Then how many ?
5. Then 7.
(i 6) T. Then?
(17) T. Add them all together.
(18) T. Give the entire solution.
5. What is half of 75 ?
Half of 60 = 30.
Half of 14= 7.
Half of i -
Half of 75 -
This example will be sufficient to show how com-
plicated is the process of division, and how very
nicely the work should be graded, so as to lead the
pupils to ask and answer by themselves all the neces-
sary questions. At first the teacher asks the ques-
tions ; but soon the brighter pupils may act as teach-
ers. They may take their places in turn before the
class, and question their fellow pupils. This must be
continued till all, even the dull ones, are able to do
the same. The brightest children may be set to
questioning single rows or small divisions. Gradu-
ally the pupils will begin to unite the several succes-
sive processes ; at first two, and finally all. It is the
same here as in other complex mental processes : at
first we perform the successive steps consciously ;
108 ARITHMETIC IN PRIMARY SCHOOLS.
and then, as they are repeated, we seem to omit
more or less of the intervening steps and to reach
the conclusion at once. To make this result possible,
however, it is absolutely necessary that each step
be not only expressed but understood. Hence the
importance of thorough illustration and also of well-
graded and abundant practice.
When the pupil has been through all the work in
division of numbers by 2 which has now been pointed
out, so that he is prepared independently to arrange
the conclusions in order, he may be put at written
work. The form of this may be the following :
\ of 75. 2 in 75.
\ of 60 = 30. 2 in 60 = 30 times.
\ of 14 = 7. 2 in 14 = 7 times.
\ of i = \. 2 in i = \ times.
of 75 = 371 2 in 75 = y\ times.
Such work as this is the best preparation for the
division of larger numbers ; but then the written
form must be the result of a thorough comprehen-
sion. It is of importance that every figure stands in
its proper place. For this purpose it is well to divide
the slates into little rectangles, and to have one
figure put in each. The written solution of a prob-
lem should be a picture of order. The orderly
arrangement of the work makes it possible to dis-
pense with many words ; it tends to mathematical
brevity and definiteness ; and it materially shortens
the teacher's work of correction.
DIVIDING BY TWO. 1 09
If the division of small mumbers is to be a prepa-
ration for the division of larger numbers, the forms
which represent the two processes should agree. The
arrangement of the written work, in case of large
numbers, is the following :
The separation of the dividend into parts, in the
form given above, is sufficiently like this to give no
trouble in later work ; but any other separation as
into 70, 4, and i, or into 72, 2, and I, because the
children happen to know the half of these numbers
would be wrong practice, because it wottld not be a
preparation for higher work ; for in the division of
large numbers the result is to be reached figure "by
figure, and hence the same should be true in the
division of small numbers. The experienced arith-
metician, especially when his work is wholly mental,
is bound by no rules. He often reaches the result
by a short cut. But it is never to be lost sight of
that it is the business of this stage of the work to
develop the power of arithmetical calculation. It
will be time enough later to teach shorthand pro-
Then, too, the importance of performing the work
I 10 ARITHMETIC IN PRIMARY SCHOOLS.
at one stage in such a way as to prepare the pupils
for subsequent stages is too often overlooked. It is
much easier to make children comprehend the pro-
cesses of division, as they stand related to one an-
other, when dealing with numbers completely within
their comprehension than in the treatment of incom-
prehensible numbers ; hence the importance of slow
and well-graded progress in division of small numbers
The young teacher should not be impatient if
much time is spent with the number 2 ; for if the
fundamental conceptions of division are here made
clear, subsequent progress will not only be more
rapid, but much freer from that confusion which
results from an attempt to teach a new principle in
connection with indistinctly formed ideas.
27. DIVIDING BY THREE.
First develop through illustrations as, for exam-
ple, balls on the numeral frame, sticks, buttons, or
marks of various kinds on the board the third part
of 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 60, 90; then the
third part of i and 2. The latter may be done by
dividing a piece of paper into 3 equal parts, and
naming each part \ ; then by treating a second piece
in the same way. A practical problem may be used
for the same purpose : three children have 2 apples
to divide among them ; what part does each receive ?
DIVIDING BY THREE. Ill
They cut one apple into 3 equal parts, and each
takes i part, or \ of an apple. They then treat the
second in the same way. Each child then has f of
an apple ; so \ of 2 apples is | of i apple. Or, lay
two equal circles of paper one upon the other, and
cut them into 3 equal parts. Each double part is \
of the 2 pieces, or f of i piece.
In the development of the process of dividing by 3
it will not be necessary to divide the work into stages
so carefully graded as in the treatment of division
by 2. I will, however, indicate the corresponding
stages, as they are always useful in the treatment of
dull pupils. They are the following :
a. 3, 6, 9, 12, 15, 1 8, 21, 24, 27, 30, 60, 90.
b. i, 4, 7, 10, 13, 1 6, 19, 22, 25, 28.
c. 2, 5, 8, n, 14, 17, 20, 23, 26, 29.
d- 33> 36, 39> 6 3> 66, 69, 93, 96, 99.
e. 42, 45. 48, Si. 54, 57> 72, 75, 78, 81, 84, 87.
f. The rest of the numbers below 100.
An example will show the agreement of the treat-
ment with that of dividing by 2.
Three boys are to divide 89 cents among them ;
they have 8 dimes and 9 cents. How many cents
does each receive ?
Teacher. How many dimes can each take ?
Scholar. Each can take 2 dimes.
T. What will be left ?
5. 2 dimes and 9 cents = 29 cents.
112 ARITHMETIC IN PRIMARY SCHOOLS.
T. Can they divide 29 cents at once ?
5. No, for 29 is not found in the threes of the
T. What is the next number below 29 that is
found there ?
*. 27 is the next number.
T. What is a third of 27 ?
5. 9 is a third of 27.
T. How many cents has each now ?
5. 20 + 9 = 29.
T. How many cents are still to be divided ?
5. 2 cents.
T. How much does each receive of 2 cents ?
5. | of a cent.
T. How many cents has each altogether?
5. 20 + +
The pupil must finally be brought to the point
where he can give the following solution in sub-
stance alone :
We cannot divide 89 immediately by 3, because it
is not found in the threes of the multiplication table.
The next number found there below 89 is 60. A
third of 60 = 20. Now 29 remains to be divided.
As 29 is not in the threes of the multiplication table,
we divide the next lower number, 27. A third of
27 is 9. We have now divided 87, and 2 remains to
be divided. A third of 2 = f. So that the third
part of 89 = 20 -I- 9 + f = 29^.
DIVIDING BY OTHER NUMBERS. 113
Written exercises in dividing by 3 are not to be
assigned to the pupils till they have gained a thorough
understanding of the matter and some facility in
solving problems. The written expression for the
example just given will appear as follows :
i of 89. 3 in 89.
\ of 60 = 20. 3 in 60 = 20 times.
\ of 27 ^ : 9. or, 3 in 27 = 9 times.
\ of 2 - f . 3 in 2 = f times.
-J- of 89 = 2gf. 3 in 89 = 29! times.
Which form is to be used will depend upon the
special statement of the question and its solution.
28. DIVIDING BY OTHER NUMBERS TO 10.
After the treatment of division by 2 and 3 it will
not be necessary to go into particulars in regard to
dividing by 4, 5, 6, 7, 8, 9, and 10. The same method
is to be followed with all these numbers. When
they have all been taught, the pupils may be re-
quired, by way of review, to divide a number by each
of the fundamental numbers in turn. Great facility
should be attained in the division of those numbers
which are of special importance in business ; as, 12,
24, 25, 30, 50, 60, 100.
It will be observed that division, as indicated
above, depends upon the separation of numbers into
tens and units, although the words tens and units
+^ OF THE
114 ARITHMETIC IN PRIMARY SCHOOLS.
have been for the most part avoided. .Let us divide
89 by three, using these terms :
89 consists of 8 tens and 9 units. A third of 8
tens is 2 tens, with a remainder of 2 tens, which are
equal to 20 units. Add 9 units and the sum is 29
units. A third of 29 units is 9 units, with a remain-
der of 2 units. A third part of 2 units is |. Hence,
\ of 89 = 2 tens + 9 units + f r = 29!.
This solution requires more statements than the
one given above ; and there is danger that some of
these may escape the memory. Frequently the pro-
cess is shorter if the number to be divided is sepa-
rated into the parts indicated in the first solution ;
as, 89 = 60 + 27 + 2. The separating of numbers in
this way is the most important part of division.
29. PRACTICE WORK.
In the preceding work on division it must appear
that there is a necessity for the most careful separa-
tion of the work into stages founded upon the degree
of difficulties to be overcome ; so that the work will
conform to the educational maxim : From the easier
to the more difficult. First come the numbers found
within the multiplication table ; then follow those
without this table, but divisible without a remainder ;
and finally, those numbers which lie beyond the
table, but are not divisible without a remainder.
Some of the work in the division of numbers below
PRACTICE WORK. 115
a hundred is no doubt more difficult than work with
numbers between 100 and 1,000; and yet these are
fundamental difficulties, and it is better to conquer
them in connection with small numbers.
In some cases, however, it may be well to limit the
division of numbers at this stage, and not apply it to
all numbers. The following is a good selection of
numbers for this purpose :
By 2 divide numbers from i to 20.
(t 3 (( (t (( J (( OQ
^ tt (t t< j U ^o
H (( (( (t -r (i gft
" 6 " " " i " 60.
(( tj It (( <( T ft <TQ
" 8 " " " i " 80.
" 9 " " " i " 90.
" 10 " " " i " 100.
Chart XII. will be found useful at this stage,
because it contains all numbers below 100, so arranged
that they can be readily assigned for practice in
division : . ,
By 2, series a, b.
" 3, " *, *, c.
" 4, " a, b, c, d.
" 5, " a, b, c, d, e.
" 6, " a,b,c,d y e,f.
" 7, " a,b,c,d,e>f,g.
" 9, " a, b, c, d, e, /, g, h, /.
" 10, " ^, ^, c, d, e,f, g, k, /, k.
Il6 ARITHMETIC IN PRIMARY SCHOOLS.
The number 100 should receive special considera-
tion at this time. One kind of exercises is the rind-
ing of every two parts of which 100 consists ; for
99 + i, 98 + 2, 97+ 3, 96 + 4, 95+ 5,
94+ 6, 93 + 7> 9 2 + 8 > 9 J +9> 9+ IO >
89 + 1 1, and so on to 50 + 50.
Another kind of exercises is the following : 100 is
i x 100, 2 x 50, 3 x 33 + i, 4 x 25,
5X 20, 6x16 + 4. 7x14 + 2, 8x12 + 4,
and so on to 50 X 2.
A third kind of work, which ought to be done
before passing on to the treatment of numbers to
1,000, is making change from a dollar for any smaller
sum ; for example : 33 cents from a dollar ; the
change may be 2 cents, a nickel, a dime, and half a
dollar, or 2 quarters.
NUMBERS FROM ONE TO A THOUSAND. II 7
NUMBERS FROM ONE TO A THOUSAND.
3O. COUNTING AND WRITING.
101 to 200.
FOR the same reason that it was thought best for
pupils to be made acquainted with numbers from
eleven to twenty before studying numbers from
twenty to one hundred, it is here recommended
that they be made somewhat familiar with numbers
from 101 to 200 before they are required to deal at
all with numbers from 201 to 1,000. The following
steps will bring the pupils to the desired result :
1. Counting from 101 to 200.
2. Counting from 200 to 101.
3. Writing numbers from 101 to 200.
4. Reading numbers written on the board by the
5. Writing the numbers from dictation.
6. Separating the numbers from 101 to 200, into
a. Hundreds, tens, and units, as, for example :
1341 hundreds, 3 tens, and 4 units.
150-=! " 5 " " o "
105 = 1 " o " " 5 "
200 = 2 " o " " o "
Il8 ARITHMETIC IN PRIMARY SCHOOLS.
b. Tens and units, as, for example :
134= 13 tens, and 4 units.
150=15 " " o "
105 = 10 " " 5 "
200 = 20 " " o "
If the work on numbers below one hundred has
been thoroughly done, there will be need of but little
objective teaching at this stage of the work. It
would be well for the teacher to be provided with a
few strings of one hundred buttons each, or a few
bunches of sticks, each bunch containing one hun-
dred ; so that he can illustrate his work objectively,
and afford an opportunity for the dull pupils to
handle the objects themselves. But it is by no
means necessary to present all the numbers at this
stage of the work in the form of objects in the hands
of every child. Too much objective teaching of num-
bers is only less stultifying than too little. It is now
time to appeal to the imagination and to the power of
201 to 1,000.
The pupils should now be taught to count by hun-
dreds to 1,000 ; then the numbers 201, etc., to 1,000
should be treated in the same way, and by the same
steps as were recommended in the case of numbers
from 101 to 200, including the separation of the
numbers into hundreds, tens, and units, as shown
above under a and b.
This work of counting, reading, writing, etc., should
be continued till the pupils have clear ideas of all
numbers below 1,000, know them as composed of
units, tens, and hundreds, and know how their com-
ponent parts are represented by figures. The test
of this last item of knowledge is the ability of the
pupil to select the groups of objects, and the single
objects, for which the different figures of any number
below 1,000 stand. When this test can be easily
borne, it is time to advance to the different funda-
mental operations, but not before.
The more thoroughly pupils are drilled in adding
numbers expressed by two figures, that is, numbers
below a hundred, the easier will be the work of learn-
ing the addition of numbers represented by three
figures. Hence it is well at this point to make a
thorough review of the addition of numbers below a
hundred. For this purpose Chart XII. will be found
very convenient. If all numbers from eleven to one
hundred are in turn added to each of the numbers
on the chart, the pupils will have 90 times 100, or
9,000 examples ; while the teacher will be spared the
labor of copying any of them on the board. The fol-
lowing suggestions are offered as to the proper stages
of the review here recommended.
Let the pupils add the numbers written below to
each of the numbers on Chart XII., proceeding from
left to right :
120 ARITHMETIC IN PRIMARY SCHOOLS.
First Stage, a 10, b 20, <: 30, d 40 - - 100.
Those exercises should, as far as possible, be per-
formed orally ; but, when necessary, the pupils may
be required to indicate the work on their slates. Sup-
pose it is required to add the number 47 to each of
the numbers on Chart XII., the work will appear on
the slate in the following order :
1+47- 48 76 + 47-123 43+47- 90
12+47- 59 84 + 47-131 52 + 47- 99
26 + 47- 73 96 + 47-143 70 + 47-117
34 + 47- 81 2 + 47- 49 73 + 47 = 120
47 + 47- 94 19 + 47= 66 87 + 47-134
53+47-100 22 + 47- 69 93+47-140
67+47 114 31+47 78 etc. etc. etc.
Or the pupils may be required to write the results
only ; then the results of the above additions would
assume this form :
a. b. c. d. e. f. g. h. i. k.
I. 48 59 73 81 94 100 114 123 131 143
m. 49 66 69 78 90 99 117 120 '134 140
n. etc., etc.
It will be noticed that the numbers to be added
first are those consisting of tens only, then those
consisting of tens and units. In adding the latter
class of numbers, the tens should be added first and
then the units ; for example : 95 + 47 ; 95 + 40 = 1 35 ;
I35 + 7 = I42; hence, 95+47-142.
The pupils should acquire considerable facility in
adding numbers of two places before proceeding to
the addition of larger numbers. When they are pre-
pared to advance to work with larger numbers, they
should be assigned examples in the following order :
a. Both numbers containing tens only ; as,
40 + 30- 70.
80 + 701 50.
b. One number containing tens and units, the
other tens only ; as,
45 + 30- 75.
48 + 60- 1 08.
c. Both numbers containing tens and units ; as,
43 + 24- 67.
86 + 75 161.
d. One number containing hundreds, tens, and
units, and the other only tens and units ; as,
238 + 46-284.
475 + 48 = 523.
e. Both numbers containing hundreds, tens, and
units ; as,
436 + 398 === 834-
The above is the order in adding two numbers
122 ARITHMETIC IN PRIMARY SCHOOLS.
only ; the addition of columns of numbers comes
later. This work is all to be done mentally before
any part of the result is written. In 'adding two
numbers, the following rule is universally to be fol-
The first number is not to be separated into parts ;
when the second consists of tens and units, the tens
are to be added first, then the units ; when it con-
sists of hundreds, tens, and units, the hundreds are
to be added first, then the tens, and lastly the units,
367 + 86; 367 + 80-447; 447 + 6-453. 378 +
285; 378 + 200 = 578; 578 + 80 = 658; 658+5-
Only a few examples like the last should be given ;;
and these mainly to the ablest pupils.
The pupils should have enough work like the
above, on pure numbers, to make them familiar
with the processes of addition ; and then the knowl-
edge and power thus gained should be applied to the
solution of simple practical problems. Indeed, some
practical problems should be given with almost every
Before beginning the subtraction of numbers
above a hundred, the subtraction of numbers below
a hundred should be thoroughly reviewed. When.
this has been done, a judicious use of Chart XII.
will save the teacher much time and labor.
In the first stage of this work the minuend should
not exceed 200. If, now, the pupil is taught to think
of each number on Chart XII. as 100 larger than it
is, and to use these increased numbers as minuends,
each number on the Chart may be used as a subtra-
hend, and thus the teacher will have, ready made,
100 x 100, or 10,000, examples in subtraction.
These examples should be assigned in the following
a. The subtrahend containing tens only ; as,
101 20 101 60
112 20 112 60
etc., etc. etc., etc.
b. The subtrahend containing tens and units ; as,
112 24 112 67
etc., etc. etc., etc.
Any number from 201 to 1,000 may be used as a
minuend, and each of the numbers on Chart XII. as
a subtrahend, and thus we have 80,000 more exam-
ples in subtraction without the trouble of inventing
them or writing them on the board.
These examples are all to be solved mentally.
When it is desirable to have the results written
down by the pupils, the written work may assume
this form :
124 ARITHMETIC IN PRIMARY SCHOOLS.
147- 1 = 146 147- 2-145
147 -12 = 135 147 19 = I2 8
147 26 121 147 22 = 125
etc., etc., etc. etc., etc., etc.
Or the work may be more briefly represented
a. b. c. d. e. f. g. h. i. k.
/. 146 135 121 113 100 94 80 71 63 51
m. 145 128 125 116 104 95 77 74 60 54
//. etc., etc.
The rule for subtracting numbers consisting of
units and tens, as previously given, is this : First
subtract the tens, then the units ; as,
132-47; 132-40-92; 92-7-85; so 132-47
The brightest pupils may be encouraged to find
new ways of reaching the result ; for example :
47-50-3; 132-50-82; 82 + 3-85.
47-42 + 5; 132-42-90; 90-5-85.
If the work here indicated on numbers below 200
is thoroughly done, no special difficulty will be found
in the subtraction of numbers between 200 and 1,000.
It would be well, however, to grade the work in the
following way :
SUB TRA CTION. 1 2 5
a. The minuend containing hundreds and tens;
the subtrahend containing tens only ; as,
940 80 = 860
770 80 690
etc., etc., etc.
b. The minuend containing hundreds, tens, and
units ; the subtrahend containing only tens ; as,
877-60 = 817
etc., etc., etc.
c. The minuend containing hundreds, tens, and
units ; the subtrahend containing tens and units ; as,
930 67 863
etc., etc., etc.
d. Both minuend and subtrahend containing hun-
dreds, tens, and units ; as,
In all cases the rule for the subtraction of a num-
ber consisting of units, tens, and hundreds is this :
Never separate the minuend into parts ; but subtract
the hundreds of the subtrahend first, then the tens,
and last the units ; for example :
126 ARITHMETIC IN PRIMARY SCHOOLS.
993-267; 993-200 = 793; 793-60 = 733; 733
Here the subtraction can sometimes be most easily
performed by adding enough to the subtrahend to-
make a sum equal to the minuend ; for example :
It is possible to construct drift exercises in addition
.and subtraction, by making the processes alternate,
which will at the same time require much work on
the part of the pupils and very little on the part of
the teacher. Suppose the exercise set for the class
to be this : Beginning with 746, alternately add 248
and subtract 273. The work would assume the fol-
lowing form on the slates :
746 + 248 - 994; 994-273 = 72i.
72 1 + 248 - 969 ; 969 - 273 - 696.
696 + 248 - 944 ; 944 - 273 - 67 1 .
671 +248919; 919 273 646.
The final result is just 100 less than 746, the num-
ber with which the work began. The reason is that
273, the number to be subtracted, is 25 more than
248, the number to be added, and consequently each
addition and subtraction diminishes the original num-
ber 25 : and four additions and subtractions diminish
it just 100.
The teacher can readily construct any amount of
drill work, such that the correction of slates will be as
easy as in the example given ; for he has only to sub-
stitute any other number for 476, and any other num-
'bers differing from each other by 25 for 248 and 273.
This kind of examples is well adapted to furnishing
every pupil, dull or bright, with all the work he can
do in a given time ; for the work in the problem
given above, is only to be continued in order to fur-
nish 29 examples in addition and as many in subtrac-
tion, which can be corrected at a glance.
While facility in addition and subtraction is of the
greatest importance in practical business, care must
be taken not to weary and discourage the pupils.
Hence it is recommended that problems like those
given above be not introduced too often, nor contin-
ued too long.
Before beginning the multiplication of numbers
between 100 and 1,000, the multiplication table
should be thoroughly reviewed. Then should fol-
low the multiplication of numbers between 10 and
100, which will be also partly a review. The three
stages of the work will be the following :
a. MULTIPLICATION OF TENS.
9 X 20 = 9 x 2 tens = 1 8 tens =? 1 80.
9x70 = 9x7 " = 63 " =630.
128 ARITHMETIC IN PRIMARY SCHOOLS.
The pupils are to be familiar with the changing of
tens to units. Then this work of changing units to
tens and of changing tens to units may be soon
omitted, and the result reached at once ; as,
9 x 20 1 80 ; 9 x 70 = 630.
b. MULTIPLICATION OF TENS AND UNITS.
6x 30-= 180
6x 9- 54
The above example illustrates both the order of
procedure in purely mental reckoning, and also a
good order of arrangement when the results of men-
tal operations are to be recorded.
c. MULTIPLICATION OF HUNDREDS, TENS, AND UNITS.
2X478; 2X400 = 800; 2X70=140; 800+140 =
940 ; 2x8=16; 940 + 16 = 956.
The order of multiplication is first the hundreds,
then the tens, and lastly the units. In multiplying
in the head, always begin with the highest order, and
work towards the units. By this procedure we are
compelled to repeat the partial results more than by
the reverse process. Then, too, when results are to
be united, they should be united as quickly as possi-
ble, so as to cause the least draught upon the mem-
ory. This applies especially to the multiplication of
If the numbers from I to 10 are written on the
board in an irregular order, as i, 2, 5, 8, 3, 7, 4, 9,
6, 10, problems in multiplication can be easily set
for a class, and so can examples requiring either addi-
tion or subtraction to be combined with multiplica-
tion. The following will serve as suggestions both
for the invention of the problems and for the arrange-
ment of the work by the pupils :
i x 70 70
2 X 70= 140
5 x 36 180
8 x 70 =-- 560
3 x 70 210
4 x 70 = 280
9 x 70 = 630
6 x 70 420
6x 36 216
iox 70 700
10 x 36 - 360
i x 70 + 42
i x 36 + 27 =
i x 70 42 =
i x 36-27 =
2 X 70 + 42 =
2x36 + 27 =
2 X 70 - 42 =
2 X 36 - 27 =
5 x 70 + 42 =
5X36 + 27 =
In performing the first of such exercises, when the
results are to be written, the teacher should insist
upon having the work so arranged as to show all the
steps in the solution ; as,
130 ARITHMETIC IN PRIMARY SCHOOLS.
5 X36+ 27
S xso= 150
5 x 6 = 30
5 X27 = 180
180 + 27 207
Examples of the multiplication of hundreds, tens,
and units will be but few, if the product does not
Chart XIII, will furnish an abundance of examples
in multiplication. The teacher may require every
number on the chart to be multiplied by i, 2, etc., to
10. Whatever tends to lessen the labor of the teacher
without injuring the quality of his work is not to be
It is well, at this stage of the work, to extend the
pupils' knowledge of the multiplication table to 1 1
and 12. For ordinary students it is hardly worth
while to go beyond 12.
After the pupils have been well drilled in the mul-
tiplication of pure numbers, special attention should
be given to the solution of practical problems. It
will be necessary for the teacher to assist the pupils
in the study of these problems. He will also have
an excellent opportunity to impart much practical
information in regard to those matters to which the
problems refer, and to drill the pupils in the expres-
sion and application of the numerical ideas which
they have already acquired, and of the practical
MUL TIPLICA TION. 1 3 1
truths which he imparts. In the solution of such
problems the pupils necessarily receive continuous
drill in sustained trains of reasoning. The following
is an example :
If beans are sold at 12 cents a quart, or 80 cents a
peck, how much is saved by buying a peck all at
once rather than by the single quart ?
If one quart costs 12 cents, a peck, or 8 quarts,
would cost 8 X 12 cents, or 96 cents; and 96 cents
are 16 cents more than 80 cents ; therefore 16 cents
would be saved by buying a peck at a time.
The study of a problem of this kind gives the
teacher an opportunity to impart to the pupils some
elementary ideas upon wholesale and retail trade, as
well as upon domestic economy ; while the pupil is
exercised in the expression of the relation of num*
bers, and also in going through a train of connected
reasoning and its expression.
The development of the reasoning power in con-
nection with the learning of arithmetic is too often
undervalued. Arithmetic does not mean simply pro-
ducing one number from another by adding and sub-
tracting, multiplying and dividing ; but, rather, judg-
ing, thinking, reasoning. Operations with numbers
can be introduced only after conclusions are reached
through the power of thought. A practical arithmeti-
cian is not a man who has attained great skill simply
in uniting numbers to form new numbers, skill in
numerical operations ; but rather one who knows
132 ARITHMETIC IN PRIMARY SCHOOLS.
how, as well, to make the judgments .necessary to
be used in the solution of practical problems. If a
person wishes to make others understand what he
himself has clearly thought out, he must take pains
to set it out in clear words and sentences. If instruc-
tion in arithmetic is to result in something more than
mechanical skill in numerical operations, the pupil
must be practised, at every stage of the work, both in
performing the steps of the reasoning processes re-
quired in the solution of problems and also in the
brief, exact, and definite verbal expression of such
reasoning. However valuable mechanical skill in
performing operations upon numbers may be, it is of
only secondary importance. Of much more impor-
tance is the ability to perform the reasoning pro-
cesses which lead to the solution of practical problems.
This reasoning discovers the numerical operations
necessary for reaching the desired result ; and with-
out the reasoning the operations could not be per-
Before beginning the division of numbers between
I and 1,000, a careful review of the division of num-
bers from i to 100 should be made ; for the teacher
cannot too often remember that the new is always to
be united with the old.
DIVIDING BY TWO.
The first thing to be done here is to make clear to
the children, and then give them ample practice in
finding, the half
a. Of 2, 4, 6, 8, 10, 12, 14, 16, 18, units.
b. Of 20, 40, 60, 80, 100, 1 20, 140, 1 60, 1 80, tens.
c. Of 200, 400, 600, 800, ijOOO, hundreds.
This done, the following problem and solution will
show the proper treatment of numbers which should
Divide 573 by two.
The number 573 consists of 5 hundreds, 7 tens,
and 3 units. We must first divide four of the five
hundreds ; half of 4 hundreds is 2 hundreds = 200.
I hundred = 10 tens ; 10 tens and 7 tens make
17 tens. Half of 16 tens is 8 tens 80
One ten remains, equal to 10 units ; to which
add 3 units, and we have 13 units. Half of 12
Half of I = i
Half of 573 = 286%
In order to bring the pupils to the point of facility
in the strictly mental division of numbers from I to
1,000, it is necessary for them to use the greatest
brevity of thought and expression ; hence they should
soon be taught to separate the number to be divided
into divisible parts without the use of the words "hun-
134 ARITHMETIC IN PRIMARY SCHOOLS.
dreds," "tens," and "units." The written expression
of the work given above will then assume the fol-
lowing form :
*of 573 =
\ of 400 = 200
\ of 160= 80
J-of 12 = 6
1 of 573=286^
From the preceding we derive the following : First
divide the hundreds which are divisible without a
remainder; reduce the hundreds which cannot be
directly divided to tens, and to these add the tens ;
divide what of the tens can be divided without a
remainder ; reduce the rest of the tens to units, and
to these add the units, etc.
The following figures will show how thoroughly
the foregoing work prepares the pupils for the usual
written form of division :
After the detailed explanation in regard to division
of numbers from I to 100, it must be unnecessary to
go further into detail here. If the practice there
recommended is here reviewed, the pupils will now
generally find no difficulty in dividing numbers below
1,000 by 3, 4, 5, 6, 7, 8, 9, 10, or by 20, 30, 40, etc.,
to 100. It will sometimes happen, however, that chil-
dren will come from an unskilful teacher, or will be
generally so dull that it will be desirable, at this stage,
to introduce division first by 2, then by 3, etc., to 10,
and to drill on each number by itself. In such case,,
it is recommended that the divisible hundreds, tens,
and units be treated at first by themselves ; for exam-
ple, the division, by 3, of
a. 3, 6, 9, 12, 15, 18, units;
b. 30, 60, 90, 120, 150, 1 80, tens;
c. 300, 600, 900, hundreds.
So may the division by the other units be intro-
duced, and frequently with profit.
Practical problems are necessarily omitted in these
papers : but they are by no means to be omitted from
the pupils' work. The young teacher is earnestly
recommended to make use of books of problems, and
not to rely solely upon his power of invention. Such
books should be used a part of the time by the pupils
themselves. In this way the power of reading and of
interpreting the written page is developed. A part
of the time the problems should be read to the pupils
by the teacher. If, now, the teacher adopts the in-
136 ARITHMETIC IN PRIMARY SCHOOLS.
variable practice of reading a problem but once, the
pupils' power of attention will be greatly strength-
The material for mental work in all the fundamen-
tal rules can be indefinitely increased by the use of
Chart XIIL, as has been explained heretofore. The
pupils can use it as the basis of thousands of exam-
ples to be performed in school upon the slate, which
take the place of mental work without the slate.
More of such exercises in the fundamental rules will
now be suggested.
To 365 add each number on Chart XIIL That
gives 100 problems in addition. Now, instead of 365,
each number from 101 to 1,000 may be used; which
gives 900 X 100 = 90,000 additions.
The correctness of the results may be proved at
the end of the hour by letting the pupils change
slates and read the answers through.
Let each number on Chart XIIL be subtracted
from 365. Then, in place of 365, use each number
from 101 to 1,000, and we have, in all, 900 X 100 =
Multiply each number on the chart by 2, 3, 4, etc.,
to 12, and we have 1,100 multiplications. Write i, 2,
WRITTEN AND MENTAL ARITHMETIC. 137
3, 4, etc., to 12, before each number on the chart, and
multiply by 2, 3, 4, etc., to 12, and we have 121,000
Put the figure i before each number of the chart,
and divide the resulting number by 2, 3, 4, etc., to 12,
and there are 1,100 divisions. Replace the i by 3, 4,
5, etc., to 12, successively, and we have n x 1,100 =
The ingenious teacher will be able to save time and
labor in other ways by the use of this chart.
35. WRITTEN AND MENTAL ARITHMETIC.
Heretofore written and mental arithmetic have not
been separated. The form of the written exercises
has corresponded strictly to the course of thought in
the mental exercises. It is possible, however, to man-
age the written work in such a way as to save both
space and time. But, although this saving is impor-
tant, it is not to be gained at the expense of clear un-
derstanding. The mind of the learner needs to be
prepared beforehand for obtaining a clear insight into
the reasons for the shorter processes of written work,
which are of special advantage in dealing with larger
numbers ; and on this account their consideration has
been postponed to a later stage.
Sometimes the terms mental arithmetic and written
arithmetic are set over against each other, as though
138 ARITHMETIC IN PRIMARY SCHOOLS.
they stood for two distinct kinds of arithmetical work.
Such a division, however, is incorrect ; for all arith-
metical computation is made by the mind. The use
of the terms oral arithmetic and memory arithmetic
on the one hand, and of slate arithmetic and figuring
or ciphering on the other, is often faulty for the same
It is true that sometimes figures are used to assist
the memory in retaining the numbers under consid-
eration, and at others the work is done by the mind
without such help ; and perhaps no better terms have
been invented to indicate these two facts than the old
names of written and mental arithmetic. It is certain
that the use of no other terms would change the facts,
or make the two processes either more or less alike.
What problems belong to written arithmetic and
what ones to mental arithmetic depends upon the
ability of the pupils to hold in the memory more or
fewer, larger or smaller numbers. Then, too, a pupil
well drilled in mental arithmetic will often solve prob-
lems without the use of figures, when others would
require the aid of pen or pencil. There can no abso-
lute limit to either class of problems be drawn. In
general, however, it is sufficient if the problems of
ordinary business, which do not involve numbers
larger than a thousand, can be solved without the
aid of figures ; though, of course, problems may
sometimes be solved mentally which involve much
WRITTEN AND MENTAL ARITHMETIC. 139
It certainly is well for all practical business men to
be able to use readily numbers below 1,000, without
recourse to written figures. In order to secure this
ability, work in written arithmetic, with its own
proper methods, has been deferred to a later period
than usual. After practice in numbers below 1,000
has given the pupils skill in mental computation, and
a clear comprehension of its principles, insight into
the principles of written arithmetic will be gained
much more easily.
In order that the acquired facility in mental arith-
metic should be retained, it is absolutely necessary
that mental arithmetic, in the narrow sense of the
term, should be closely connected with written arith-
metic, whether instruction in the two kinds of work
is given in the same or in different hours.
Written arithmetic, as well as mental arithmetic,
should not be practised mechanically, so that the
operations are performed merely by rule. The short-
ened processes of written arithmetic should be devel-
oped out of the processes of mental arithmetic which
have already been explained. If this is done, the
scholar will come to know not only the processes and
rules, but their reasons. The pupil is never to work
by a rule, like a mathematician by his formula, till he
understands the reason for his procedure and for the
War is to be continually waged against all mechan-
ical management of mathematical instruction, and
140 ARITHMETIC IN PRIMARY SCHOOLS.
against all learning of facts without reasons ; yet, in
the four fundamental operations of arithmetic, the
pupil is to attain an ease and a rapidity of working
which closely resemble that of a perfect machine ; so
that all his mental power may be given to the reason-
ing processes which the solution of the problems
If the work previously suggested has been well
done, the pupils are now prepared to enter upon the
stage of written arithmetic proper, and readily to
understand its processes.
HIGHER NUMBERS. 141
NUMERATION properly means counting ; but here it
has an enlarged meaning. It signifies counting, form-
ing higher or complex units out of a definite collec-
tion of less complex, or simple units, and also repre-
senting these different units by means of figures.
In studying numbers from i to 10, i to 20, and i
to 1,000, the pupil has incidentally learned something
of the nature of the decimal system of numbers, and
of the method of representing numbers by the Arabic
system of notation ; but it is now time to make his
knowledge more definite and systematic, and to ex-
tend it still farther. For this purpose review the
grouping and writing of numbers.
Call attention to the fact that a single ball on the
numeral frame, a single dot on Chart X., a single
finger, etc., is represented by the figure i standing
alone ; two balls, two dots, two fingers, etc., by the
figure 2 ; three balls, three dots, three fingers, etc.,
by the figure 3 ; and so on to nine.
Next show that a group of ten balls, ten dots, etc.,
is not represented by another figure, but by the figure
142 ARITHMETIC IN PRIMARY SCHOOLS.
I standing in the second place from the right ; that
two groups of ten balls, ten dots, etc., each are repre-
sented by the figure 2 ; three such groups by the
figure 3 ; and so on to nine groups. Name these
groups tens, and make the use of the name familiar
by counting the rows of balls on the numeral frame,
and the rows of dots on Chart X., etc., thus : one ten,
two tens, three tens, etc.
Then explain that ten groups of ten each, or ten
tens, are called a hundred, and are represented by the
figure i standing in the third place ; that two hun-
dreds are represented by the figure 2 ; three hundreds
by the figure 3 ; and so on to nine hundreds.
Explain that ten hundreds are called a thousand,
and are represented by the figure i placed in the
fourth place ; that two thousands are represented by
the figure 2 ; three thousands by the figure 3 ; and
so on to nine thousands.
A good set of objects for illustrating the grouping
of numbers and the use of figures may be easily made
of large buttons. Single buttons are units ; strings
of ten each, tens ; bundles of ten strings each, hun-
dreds ; packages of ten hundreds each, thousands.
The figures may be written as the groups are shown.
Still another excellent apparatus for this purpose
consists of ten cubes an inch on a side ; nine sticks
an inch square and ten inches long, marked with lines
an inch apart, so as to represent inch cubes ; and nine
pieces of board ten inches square and an inch thick,
NUMERA TION. 1 43
marked off with lines an inch apart, so as to represent
100 inch cubes each. The cubes are the units ; the
sticks represent the tens ; the pieces of board stand
for the hundreds ; while all, laid up in the form of a
cube, represent a thousand small cubes.
Several kinds of apparatus are better than any one
kind, and good apparatus may be so used as to save
the teacher much labor. But, somehow, the writing
of numbers should be illustrated objectively, till the
pupils can readily write any numbers from i to 1,000,
when the objects, grouped as has just been indicated,
are shown them ; till they can find the objects and
groups of objects representing any written number
from i to 1,000; find the number of tens and units
in any number of single objects ; the number of units
in any number of objects grouped in tens, as strings
of buttons ; the number of units in any number of
tens and units ; the number of hundreds in any num-
ber of tens ; the number of tens in any number of
hundreds ; the number of tens in any number of hun-
dreds and tens ; the number of hundreds, of tens, and
of units in a thousand ; and the number of hundreds,
tens, and units, of tens and units, and of tens, in any
number of thousands, hundreds, tens, and units.
When all this can be readily done with the objects
themselves, the pupils should be drilled in changing
written numbers into equivalent numbers with differ-
ent groupings ; as, for example,
ARITHMETIC IN PRIMARY SCHOOLS.
1,328 i thousand, 3 hundreds, 2 tens, and 8 units.
1,328 13 " 2 " " 8 "
1,328- 132 " " 8 "
1,328- 1,328 "
Or the following :
-4,807 24 thousands, 8 hundreds, o tens, 7 units.
34,807- 248 " o " 7 "
24,807 2,480 " 7 "
24,807 24,807 "
It must be made perfectly clear to the pupils that,
a. Units are represented by the ist figure.
Tens " " " " 2d "
Hundreds " " " " 3d
Thousands " " " " 4th "
b. 10 units i ten ;
10 tens i hundred ;
10 hundreds thousands, etc. ;
so that always ten units of a lower order are equal to
one unit of the next higher order.
NUMERATION. 1 45
In this scheme are first written i, 10, 100, 1,000.
Then follows the explanation that
i ten 10 units,
i hundred = 10 tens 100 units.
i thousand = 10 hundreds = 100 tens = 1000 units.
Then should follow the writing and analysis into
thousands, hundreds, tens, and units of other num-
bers ; as, 1,328, 4,807, etc.
The writing of numbers from dictation, by the aid
of this scheme, should gradually give place to the
writing of dictated numbers without such aid.
If the writing and analysis of numbers below 1,000
is thoroughly mastered, it will be but little work for
the teacher to make clear to the pupil the extension
of the same principles to numbers above 1,000. For
this purpose the scheme given above may be extended
nine or ten places. These should be broken up into
groups of three places each ; which can readily be
done by double lines, as shown above between the
thousands and hundreds. The headings of the sec-
ond group, or period, would be, thousands, ten-thou-
sands, hundred-thousands ; and so of the millions,
When this work has been well done, the pupil
needs but two more suggestions :
a. To read any number, begin at the right and
divide it into periods of three figures each, except the
last, which may contain three, two, or one figure ; read
146 ARITHMETIC IN PRIMARY SCHOOLS.
each period as though it stood alone, adding the name
of the last place in the period, except in case of the
last period read; as, 24,341,101,268, to be read,
twenty-four billion, three hundred forty-one million,
one hundred and one thousand, two hundred sixty-
b. To write any number, begin with the highest
period, and fill each subsequent period, using zeros
when the period is wholly or partly omitted ; for
example, to write twenty-four millions and seven-
teen, put three zeros in the thousands period, and a
zero in place of the hundreds in the units period,
The directions for extending numeration so as to
cover all the higher numbers are put here for the
.sake of completeness, and for the use of the bright
pupils ; but it is well to introduce the writing of
large numbers gradually, as the pupils have occasion
to write them. The subject of numeration is here
dwelt upon so fully because it is so intimately con-
nected with the decimal system, and because the
decimal system is usually the weakest place in arith-
metical teaching in this country. When the decimal
system of numbers and the Arabic notation are thor-
oughly understood, arithmetic is half learned. Young
teachers are therefore earnestly advised to advance
.slowly and thoroughly through the subject.
In written addition it is more convenient to begin
with the lowest place, that is, with the units, and to
work towards the highest, thus reversing the process
of mental addition. The first examples should be the
addition of numbers below 1,000, because the pupils
are already familiar with these numbers.
Attention should be called to the fact that 2 boys
and 3 slates make neither 5 boys nor 5 slates, etc. ;
by which the pupils will be led to see that only like
quantities can be added. This will show the reason
for adding units to units, tens to tens, hundreds to
hundreds, etc. ; and for writing numbers, when they
are to be added, so as to bring units under units, tens
under tens, etc., as a matter of convenience. These
explanations made, introduce an example, as the fol-
3 First add the column of units. The result
88 is 32 units ; which are equal to 3 tens and 2
9 units. The 2 units are written under the
45 units ; and the 3 tens are added to the column
13 of tens. The result is 23 tens, equal to 2 hun-
77 dreds and 3 tens. The whole sum is thus
232 found to be two hundreds, three tens, and two
units, or 232. At first the pupil may be allowed
to write the tens resulting from adding the units,
with a small figure over the column of tens, as the
148 ARITHMETIC IN PRIMARY SCHOOLS.
After the pupils have had considerable practice in
adding numbers below 1,000, the addition of numbers
above 1,000 may be introduced. The work and ex-
planation will appear as follows :
2 43 i The column of units give 1 5 units i
6742 ten and 5 units. Write the five units under
982 the line in the column of units, and add the
3752 ten to the column of tens. This gives 30
97602 tens = 3 hundreds and o tens. Write the
785 o tens under the tens, and add the 3 hun-
6742 dreds to the column of hundreds. There
116,605 results 46 hundreds - 4 thousands and 6
hundreds. Write the 6 hundreds under the
column of hundreds, etc.
The above examples are sufficiently long for this
stage of the work. Practical problems should be in-
troduced constantly ; but for this purpose it is better
to depend upon a good text-book.
Experience shows that it is very easy to make mis-
takes in the easiest of mathematical processes, that
is, in addition ; so that when certainty of results is
desired, it is well to perform the additions twice, once
beginning at the bottom of the columns and once at
the top. Or, if the columns are very long, they may
be divided into two parts, the parts added separately,
and then the partial results added. If the result thus
reached agrees with the result of adding the entire
columns at once, the result is probably right.
The word " subtraction" means taking away. In
arithmetic it signifies the process of taking one
number from another, or of finding how much larger
one number is than another, that is, how many more
units one number contains than another. The num-
ber which is to be diminished is called the minuend ;
the number which is to be taken away is called the
subtrahend ; the number which is left, or which shows
how many units the minuend is greater than the sub-
trahend, or how many units the subtrahend contains
less than the minuend, is called the remainder or dif-
These definitions should be developed from one or
two examples ; as 3 from 5, 4 from 6. It would not
be without profit to illustrate the terms by perform-
ing first the act of taking one number of objects
from another number ; as, for example, 4 boys from 6
boys ; and then the act of comparing one number with
another, to find the difference ; as, for example, com-
paring 4 boys with 6 boys, to find how many more
there were in one group than in the other. When-
ever numbers are to be seen in new relations, the
teacher cannot take too much pains to make sure
that the ideas of the numbers are clear and distinct.
Write the minuend under the subtrahend, so that,
as in addition, units of the same kind will stand under
150 ARITHMETIC IN PRIMARY SCHOOLS.
Problems should be given first in which each figure
in the subtrahend stands for a smaller number than
the corresponding figure in the minuend ; as,
976 5 units from 6 units = I unit ;
435 3 tens " 7 tens =4 tens;
== 541 4 hundreds " 9 hundreds = 5 hundreds.
Therefore there remain 5 hundreds, 4 tens, and i
When the pupils have been made familiar with
such examples, problems should be introduced in
which one or more figures in the subtrahend stand
for larger numbers than the corresponding figures in
the minuend. The two following examples with their
explanations will make the principles plain upon which
they are to be solved.
7 units cannot be taken from 5 units ; so
495 we separate one of the nine tens into units,
-257 which gives 10 units; and these 10 units
= 238 added to the 5 units make 15 units. From
15 units take 7 units, and 8 units remain.
Then 8 tens, which were left, minus 5 tens, leave 3
tens; and 4 hundreds 2 hundreds = 2 hundreds.
The remainder, then, is 2 hundreds, 3 tens, and 8
units = 238.
The following problem presents an additional diffi-
99IO 2 units cannot be taken from o units.
1000 Since, now, there are no tens and no hun-
732 dreds, we change I thousand into 10 hun-
= 268 dreds ; we change i of these hundreds into
SUB TRA CTION. 1 5 1
10 tens, and i ten into 10 units. There remain
then in the minuend no thousands, but 9 hundreds,,
9 tens, and 10 units. Then, 2 units from 10 units =
8 units ; 3 tens from 9 tens = 6 tens ; and 7 hundreds,
from 9 hundreds - 2 hundreds. So the remainder is
2 hundreds, 6 tens, 8 units = 268.
Such examples as the last two may be readily
solved by the application of the principle, that if two
numbers are equally increased, the difference remains
the same, and by remembering that 10 units of any
order is equal to I unit of the next higher order :
As 6 units cannot be taken from 4 units,
274 add 10 units, making 14 units ; 6 units from
- 146 14 units leaves 8 units. Add i ten to the 4
128 tens, making 5 tens, which taken from 7 tens
leaves 2 tens ; and 2 hundreds i hundred =
i hundred. So that the remainder is i hundred, 2
tens, 8 units = 128. It will be perceived that we have
added 10 units to the minuend, and their equivalent,.
i ten, to the subtrahend ; and, consequently, have
not changed the difference. This method of explana-
tion and practice is believed to be easier of applica-
tion than the method first explained, and is therefore
Since the minuend the subtrahend = the remain-
a. The minuend the remainder = subtrahend.
b. The subtrahend + remainder = minuend.
152 ARITHMETIC IN PRIMARY SCHOOLS.
We can therefore prove the correctness of the work
in subtraction, either by subtracting the remainder
from the minuend, in which case the subtrahend is
obtained, or by adding the remainder to the subtra-
hend, in which case we obtain the minuend. The
latter proof is more practical than the former, and
should be occasionally used by the pupils.
All the principles involved in subtraction can be
learned by the use of small numbers ; so that it is
better to give the pupils much practice with these,
before introducing large numbers. In the use of
practical problems small numbers are much prefera-
ble ; since the imagination of pupils can then be
more easily appealed to when necessary.
The meaning of the terms used in multiplication
may be made clear to the pupils in the following way.
Let the teacher write, in a horizontal line on the
board, seven dots, and ask, " How many dots have I
made ? " Then let him make, under these, two rows
more, and ask, " How many rows of seven dots each
have I made ? How many times are seven dots re-
peated ? How many dots are there in all ? "
Then let the explanation follow, while the teacher
continually points to the single dots, the rows, or the
whole mass. This whole process is called multiplica-
tion. The number of dots in the first row, namely 7,
is the multiplicand. The number of rows, namely 3,
is the multiplier. The whole number of dots, namely
21, is the product. We have repeated 7 3 times, and
the result is 21.
The work on the board, as it has grown up under
the hand of the teacher, will appear thus :
When the pupils have followed several such illus-
trations, they will comprehend the following defini-
Multiplication is the process of finding how many
units result from repeating a number a given number
of times. The number to be repeated is called the
multiplicand. The multiplier is the number showing
how many repetitions are to be made. The product
is the number showing how many units result from
the repetitions. The multiplier and the multiplicand
are called the factors of the product. Thus, in the
above example, 7 is the multiplicand, 3' the multiplier,
21 the product, and 7 and 3 are the factors of 21.
It is not worth while at this stage to have these defi-
nitions committed to memory ; but the terms should
be thoroughly understood, so that they will bring up,
in the minds of the pupils, clear and distinct ideas.
154 ARITHMETIC IN PRIMARY SCHOOLS.
a. The examples first introduced for explanation
and practice should contain two or three places in
the multiplicand and only one in the multiplier. The
following will show the proper explanation :
3x7 units = 21 units = 2 tens and I unit.
247 Write the i unit under the column of units.
3 3X4 tens 12 tens, to which add the 2 tens
741 from the 21 units, and the sum is 14 tens= i
hundred and 4 tens. Write the 4 tens under
the tens. 3x2 hundreds = 6 hundreds, to which add
the i hundred from the 14 tens, and the sum is 7
hundreds, which is to be written under the hundreds.
The result is 7 hundreds, 4 tens, and i unit = 741.
b. Next follow examples with tens only in the mul-
tiplier. Here should come the explanation of the
process of multiplying a number by 10. In the treat-
ment of numbers from i to 1,000 it was shown that,
10 x 20 = 200, 10 x 89 = 890,
10 x 27 = 270, 10 x 93 = 930,
10 x 39 = 390, 10 x 72 = 720, etc. ;
in all which cases we obtained just as many tens as
there were units in the multiplicand. Now since tens
are indicated by a zero at their right, to multiply a.
number by 10 we have only to put a zero at the right,
thus setting the units' figure in place of the tens', the
tens' figure in place of the hundreds', etc.
Or the same may be shown by such examples as,
the following :
MUL TIPLICA TION. I 5 5
73 10X3 units = 30 units = 3 tens ;
X 10 10 X 7 tens = 70 tens =7 hundreds ;
730 and 7 hundreds and 3 tens 730.
Suppose, now, we wish to multiply a number by
40. We may first multiply by 4, and then by 10,
since 10x4 times a number is 40 times the number.
For example, 40 x 23.
X40 4x23=92; 10x92 = 920.
It follows that to multiply by tens we have only to
multiply by the number of tens and put a cipher after
c. The third class of problems should be those with
tens and units in the multiplier.
4 X 38 units ^152 units.
38 38 20 X 38 = 2 x 10 X 38 = 760 = 76 tens ;
X 24 X 24 and since no units can arise from the
152 152 multiplication of any number by tens,
760 76 we may omit the zero as in the second
912 912 example in the margin, and begin to
write the product under the tens.
Now, since 24 times a number is 4 times the num-
ber plus 20 times the number, we have only to add the
partial products in order to obtain the entire product.
d. Finally, examples should be introduced with
three or more figures in the multiplier ; as, 243 x
156 ARITHMETIC IN PRIMARY SCHOOLS.
This is equivalent to saying multiply 3,576 by 3, by
40, and by 200, and add the results.
3 X 3576= 10728, to be written units un-
3576 der units, etc.
243 40 X 3576 = 143040 = 14304 tens, to be
10728 written tens under tens, etc.
14304 200 x 3576 = 2 x 100 x 3576 = 2 x 10 x 10
7152 X 3576 = 715200 = 7152 hundreds, to be
868,968 written hundreds under hundreds, etc.
If, now, we add 3 times, 40 times, and 200 times
the number together, we have 243 times the number.
If the pupil has observed that multiplying numbers
by 10, 100, 1,000, etc., simply sets the figures one,
two, three, etc., places towards the left, he at once
comprehends the reason for the rule : Write the first
figure of each partial product under the place of the
number with which you multiply. If you multiply
by units, write the first figure under units ;
" tens, " " " " " tens;
" hundreds, " " " " " hundreds;
e. If the pupil thoroughly comprehends the in-
struction above suggested, he will have little diffi-
culty with numbers containing ciphers,
The number 201 is to be repeated 3 times; then
400 (loox 4) times ; and then the partial products are
to be united.
The number is to be repeated, first, 80 (10 X 8)
times, then 30,000 (10,000 X 3) times, and the partial
Such examples as the last would be introduced at
this stage only for the benefit of the brightest pupils,
who, by such work, may be interested and benefited.
Perhaps the easiest proof of multiplication is to
make the multiplier a multiplicand and the multipli-
cand a multiplier, and multiply again. Later, the
product may be made a dividend and the multiplicand a
divisor, when the quotient should equal the multiplier.
Practical problems are to be introduced at each
stage of the work of multiplication, which is heyre
marked a, b, c, d y and e. For most of these, however,
the teacher should depend upon a good text-book ;
and this should be, a part of the time, in the hands of
The number to be divided is called the dividend ;
the number by which we divide, the divisor ; the
number which shows how many times the divisor is
158 ARITHMETIC IN PRIMARY SCHOOLS.
contained in the dividend, or what part of the divi-
dend the divisor is, the quotient ; the number left
when the division is not completed, the remainder.
For example, the process of finding how many times
7 is contained in 21, or what a seventh part of 21 is,
is the division of 21 by 7. Here 21 is the dividend,
7 is the divisor, and 3 the quotient. Had we at-
tempted to find how many times 7 is contained in 23,
we should have found that it was contained 3 times,
with a remainder of 2.
Whether we attempt to find how many times a
number is contained in a given number, or what the
corresponding part of the given number is, the pro-
cess is the same ; for example, take the above num-
bers, 21 and 7. The seventh part of 7 is i, the
.seventh part of 2 times 7 is 2 ; and, in general, the
seventh part of 21 is as many units as 7 is contained
times in 21, namely 3. So that it is not necessary to
consider the two kinds of division separately, although
the pupil should always be required to know and to
state what he is doing.
The degree of difficulty in division depends upon
the constitution of the divisor ; so that the divisor
determines the stages of the pupils' work in division.
They are the following :
a. The divisor is composed of units ;
b. " " " " " tens ;
c. " " " " " tens and units ;
d. " " contains 2, 3, etc., places.
DIVISION. 1 59
Since, however, it is easier to make the process
understood if the dividend is small, examples should
be chosen for the first work where that is the case.
DIVIDING BY UNITS.
We will first explain an example of division by 2.
a b c
2)759(379 2)759(300 of 759
6 . . x 2 600 \ " 600 == 300
15. 758 iS9( 70 " 140- 70
14 i 140 " 18= 9
19 759 ~i9(__9 i " i = i
18 ^8379 iof759=37 9
#. The number 759 consists of 7 hundreds, 5 tens,
and 9 units. We first divide 6 hundreds by 2, and
we have 3 hundreds. These 3 hundreds we write at
the right. We indicate that 2X3 hundreds, or 6
hundreds, have been divided by writing 6 under 7.
We subtract 6 hundreds from 7 hundreds, and i hun-
dred remains, to which we unite the 5 tens, and have
15 tens. We divide 14 tens by 2, and the result is 7
tens, which we set in the tens' place. We write 2x7
tens= 14 tens under the tens to show that they have
been divided. We subtract the 14 tens, and there re-
mains i ten, to which we add the 9 units, and we
have 19 units. We divide 18 units by 2, and obtain
9 units, which we write in the units' place in the quo-
1 60 ARITHMETIC IN PRIMARY SCHOOLS.
tient. We subtract 2x9 units, or 18 units, from the
19 units, and have a remainder of i. So that the re-
sult, or product, is 371, and I remainder.
Under b the division is indicated more fully. The
parts of the dividend which have been divided (600,
140, 1 8), as well as the parts of the quotient (300, 70,
and 9), are written out in full. The third form, c, is
the form with which the pupil is familiar in his men-
tal work. It is added here to make the new and
shortened form of division still clearer. The form b
is recommended only for the purpose of explaining
the reason of form c. The first form is the one to be
used in practical work.
Here, more than anywhere else, is it necessary for
the pupil to write the different figures in the proper
places, units under units, tens under tens, etc. To
assist the teacher in securing this, the division of the
board, and also the slate, into little rectangles, as
formerly advised, is very helpful.
Abundant practice in dividing by 2 should precede
the dividing by other units. A clear comprehension
of the reason for the different parts of the process, as
well as great facility in the operation, cannot be too
strenuously insisted upon, before the pupil is allowed
to go on to new work. Time spent here is more than
It is important that the pupil learn to determine,
as soon as he begins to divide, how many places there
must be in the quotient ; because this explains the
reason for putting a zero in the quotient, whenever
the, divisor is not contained in the number of units of
any order in the multiplicand. An example will make
Since I hundred-thousand can-
6)184549(30,758 not be divided by 6 and produce
1 8 a whole number, we divide 18 ten-
45 thousands by 6, and the result is 3
42 ten-thousands. This shows the pu-
34 pil that there must be 5 places in
30 the quotient, which the beginner
49 may indicate by 5 points. Since 4
48 thousands divided by 6 produce no
i thousands, a zero must be put in
the quotient in the thousands' place ;
else the quotient would not contain 5 places, and the
first figure, 3, would be read 3 thousands ; for 4 thou-
sands and 5 hundreds, or 45 hundreds, divided by 6,
give 7 hundreds. By such examples the pupil will
learn to put a zero in the quotient whenever the num-
ber shown by bringing down a figure of the dividend
is not divisible by the divisor.
The correctness of the work in division may be
tested by multiplying the divisor by the quotient, and
adding the remainder to the product. The sum should
equal the dividend.
After the pupils have had a good deal of practice
in dividing by numbers represented by one figure,
using the form given above, they may be allowed to
1 62 ARITHMETIC IN PRIMARY SCHOOLS.
divide by the same numbers, writing . simply the
divisor under the dividend, thus :
2)9 1 5 1 7 1 2 6 i
At first the remainder may be written in small fig-
ures over and a little at the left of the next place.
DIVIDING BY TENS.
It is well to make a distinct step of dividing by
numbers consisting of tens only, because it throws
light on the succeeding steps in division. It would
be profitable, at this point, to review the mental pro-
cesses of multiplying, and also of dividing, by 10, 20,
30, etc., to 100. This done, an example or two will
make this step understood.
The twentieth of 41 hundreds is 2
20)4165(208 hundreds, and a remainder of I hun-
40 dred ; to this I hundred, or 10 tens,
165 add 6 tens, and the sum is 16 tens ;
1 60 which is not divisible by 20, and so
5 there are no tens in the quotient, and
the tens' place must be filled with a
zero. Dividing 165 units by 20, and we have 8 units,
with a remainder of 5-
It will soon be obvious to the pupils that dividing
by 10 is accomplished by cutting off the unit figure,
and regarding it as representing tenths ; and so,
later, of dividing by 100, 1,000, etc.
DIVIDING BY NUMBERS OF TWO PLACES.
The difficulty of dividing by such numbers arises
from the fact that the pupils do not know the multi-
plication table for numbers so large ; and hence the
products of these numbers, that is, the divisors used,,
must be found. For this purpose it is often neces-
sary for the pupils at first to proceed by way of trial.
This trial consists in finding how often a convenient
number of about the same size as the divisor that
is, a number whose product by 2, 3, 4, etc., to 9, is
already known is contained in the number to be
divided, and then multiplying the divisor by this quo-
tient. For example, if I wish to find how many
times 53 is contained in 480, I first see how often 50
is contained in it. This I know by knowing the
products of the tens, that is, 20, 30, etc., by 2, 3, etc.
Since 50 is contained 9 times in 480, therefore it is
probable that S3 is contained 9 times; this is here
the fact, for 9 X 53 =477. Here the probability and
the truth agree ; but were the divisor 54, this would
not be the case ; for 9 x 54 486. In this case the
quotient must be diminished by i. The nearer the
convenient number, or trial divisor, is to the true
divisor, the greater the probability is that the trial
quotient will prove to be the true quotient. Hence
in dividing by 56, 57, 58, or 59, it would be better to
use 60, rather than 50, as a trial divisor, while 50
would be more likely than 60 to give us the true num-
ber, if we were dividing by 50, 51, 52, 53, or 54. In
1 64 ARITHMETIC IN PRIMARY SCHOOLS.
the former case it would often be necessary to in*
crease the trial quotient by I in order to obtain the
After much experience, and practice, the pupil can
make this trial in his mind without writing down any
of the work ; and this poVer the teacher should try
to develop in the pupil.
The following is a typical explanation : 7 hundreds
-5-23 gives no hundreds; 7 hundreds + 5
2 3)75 I (3 2 tens ^75 tens; 75 tens -5-23 = 3 tens, for
69 3x23 tens = 69 tens. There remain 6
6 1 tens, to which add I unit, and we have
46 6 1 units. 6 1 units 20 = 3; but 3x23
15 =69 ; so 23 is not contained in 61 units
3 times, but one less than 3 times, or 2
times ; 2 X 23 =46 ; and 61 46= 15, the remainder.
89899 l6 994
7 ten-millions are not divisible by 9254;
78 millions are not divisible by 9254 ;
786 hundred-thousands are not divisible by 9254;
7863 ten-thousands are not divisible by 9254;
78632 thousands -f- 9254 = 8000 ; 8000 X 9254 =
74032 thousands, which taken from 78632 thousands
leave 4600 thousands = 46000 hundreds, to which
add 5 hundreds, and we have 46005 hundreds ; 46005
hundreds -^92 54 = 400; 400 x 9254 = 37016 hundreds,
which subtracted from 46005 hundreds leaves 8989
hundreds 89890 tens, to which add 9 tens, and the
sum is 89899 tens; 89899 tens -f- 9254 = 90 ; 90 X
9254=83286 tens, which taken from 89899 tens
leave 6613 tens = 66130 units, to which add 4 units,
and we have 66134 units ; 66134 units -^9254 = 7 ; 7
X 9254= 64778, which from 66134 = 1356, remainder.
Of course examples of this length would be given
only to older and brighter pupils.
The pupil can now prove his multiplication by
dividing the product either by the multiplier or the
quotient. If the work is right, the quotient will be
the other factor.
UNIVERSITY OF CALIFORNIA LIBRARY
JUL 1 8 1962