METHODS OF TEACHING ARITHMETIC IN PRIMARY

SCHOOLS.1

BY LARKIN DUNTON, LL.D.,

Head Master of the Boston Normal School.

XIV.

DIVISION.

HERE are two kinds of division, namely, separating a num

THERE

ber 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 separation 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 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 processes 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 children, 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 process; 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.

DIVIDING BY TWO.

First Exercise.

Let the children add 2 successively to 2, 4, etc., so as to form the numbers 2, 4, 6, 8, 10, etc., to 20.

Question thus: How many are 2×2? 3×2? 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 succeeding 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.

A. O O O

B. 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, 16, 18, 20.

Second Exercise.

Two children have 15 apples, how many has each?

Of 14 apples each has 7 apples;

In the same way treat 3, 5, 7, 9, 11, 13, 15, 17, and 19.

Third Exercise.

Draw on the board two rows of circles with 10 circles in each row. This will show that half of 2 rows is 1 row; half of 2 tens

is 1 ten; half of 20 is 10.

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.

Fourth Exercise.

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, 86, 88.

Fifth Exercise.

Two persons have 3 dimes; how can they be divided? Each takes 1 dime, equal to 10 cents. They then exchange the other dime for 10 cents, and each takes 5 cents; so that each has 10 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.

Sixth Exercise.

Show on the numeral frame 3 rows of 10 balls each, and 1 row of 6 balls. What is half of them?

Half of 2 tens is 1 ten, and the other ten balls added to the 6 make 16 balls. Half of 16 is 8; so half of 36 is 1 ten and 8, or 18.

Or this: Divide 3 dimes and 6 cents equally between two persons. Let each take 1 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.

So treat all numbers which have even tens and odd units: 21, 23, 25, 27, 29; 41, 43, 45, 47, 49; 61, 63, etc.; 81, 83, etc.

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.

THE STUDY OF SYNONYMS.

ADVERTISE AND PUBLISH.

1. Both are derived from the Latin.

2. Both mean to make known.

3. ADVERTISE denotes the means, and PUBLISH the end. TO ADVERTISE is to direct public attention to any matter by means of a printed circular. To PUBLISH is to make known either by oral or printed communication. Every man that ADVERTISES his own excellence should write with some consciousness of a character which dares to call the attention of the public."

4. “

"The criticisms which I have hitherto PUBLISHED have been Imade with an intention rather to discover beauties and excellences in the writers of my own time, than to PUBLISH any of their faults and imperfections."

PALE, PALLID, WAN.

1. PALE and PALLID are derived from the Latin.

WAN is derived from the Anglo-Saxon.

2. Their common meaning is, deficient in ruddiness and freshness of color.

3. The absence of color, where color is a requisite quality, constitutes PALENESS.

PALLIDNESS is an excess of PALENESS, and WANNESS is an unusual degree of PALLIDNESS.

PALENESS in the countenance may be temporary and may be produced by fear or any sudden emotion.

PALLIDNESS and WANNESs are permanent. Protracted sickness, hunger, and fatigue bring on PALLIDNESS, and when these calamities are greatly heightened, they may produce WANNESS.

4. "Now morn, her lamp, PALE glimmering on the sight, Scattered before her sun reluctant night."

"Her spirits faint, her cheeks assume a PALLID tint."
"And with them comes a third with regal pomp,

But faded splendor WAN."