SUGGESTIONS TO TEACHERS.

The eight lessons that immediately follow have been arranged and

introduced at the beginning of this book for the purpose of giving

pupils a thorough drill in the use of the digits from 2 to 9 inclusive.

Though for convenience of presentation the limit of the processes

is confined to 12 times —, the teacher may extend the operations of

addition and subtraction at will. The limit of 100, however, is all

that is desirable with numbers less than 9.

The treatment of each number includes the four processes of

addition, subtraction, multiplication and division, and recognizes the

fact that they are intimately and naturally connected, and that one

operation may be said to include all the others, for all the operations

are only the comparison of numbers, one with another. They are,

in fact, different methods of reading the same general relation of

numbers, thus: 4+4=8 is read 4 and 4 equals 8; 2×4=8 is read

2 times 4 equals 8; 4-2-2 is read 4 less 2 equals 2; and 4÷2=2 is

read 2 is contained in 4, twice, or 2 can be taken twice from 4, or 2 is

one-half of 4.

The consideration of these processes does not contemplate the use

of the terms add, subtract, multiply or divide, as these terms will be
more profitably discussed hereafter, but it is designed to make pupils
familiar first with the processes themselves, and with the use of
terms easily understood. It is assumed that pupils know how to
count, and are familiar with the forms, names and values of the
significant figures. While the teacher will be greatly assisted in this
instruction by the use of lines or dots upon the board, or by a
numeral frame, or small objects, such as grains of corn, beans, etc.,
it must not be forgotten that the memory should be made an import-
ant factor in the mastery of these relations. Constant appeals must
be made to the pupil's power to gather and retain through frequent
repetition.

Immediately succeeding the treatment of the abstract number, a
variety of exercises, both abstract and denominate, is given as models
for the teacher. These exercises may be variously diversified and
extended, according to the skill of the teacher and the necessities
of the pupils. The lessons should be used as follows: as for in-
stance, Lesson I; (1) reads 2, 4, 6, 8, etc., 24; (2) 1, 3, 5, etc., 23;
(3) 2, 4, 6, etc., 24; (4) 24, 22, etc., 2; (5) 23, 21, etc., 1; (6) once, twice,
3 times, etc., 12 times.

ADDITION TABLE.

1+1 1+2 1+3 1+4 1+5 1+6 1+7 1+8 1+9
2+12+2 2+3 2+4 2+5 2+6 2+7 2+8
3+3 | 3+4 | 3+5|3+6 | 3+7
4+2 4+3 4+4 4+5 4+6
5+1 | 5+2 | 5+3|5+4|5+5

3 3+1 3+2

2

2+9

4

3+8 3+9

5

4+1

4+7 4+8 4+9

5+6 5+75+8 5+9

5+85+9

9

9+1

10

8+3 8+4 8+5 8+68+7 8+8 8+9

9+2 9+39+4 9+5 9+6

9+7 9+89+9

11 12 13 14 15 16 17 18

MULTIPLICATION TABLE.

1's 2's 3's 4's 5's 6's 7's 8's 9's 10's 11's 12's

The above table is to be used as follows:

1. ADDITION. Let the pupil add to each number in a given column a number announced by the teacher, the pupil stating the successive sums only. Thus, the teacher will say, "Column A, add 5;" the pupil selected answers as rapidly as possible, "6, 8, 10, 12, 14," etc.

2. SUBTRACTION. Let the pupil subtract from a given number each number in a given column. Thus, the teacher announces "Column B, subtract from 17." The pupil subtracts each number, mentally, and states the result only, "8, 12, 14, 16,” etc. 3. MULTIPLICATION. The teacher selects a column and a multiplier. The pupil announces the product, using each number as a multiplicand. Thus, "C by 5," results in 60, 5, 15, 45, etc. 4. DIVISION. The teacher selects a column and announces a dividend; the pupil states the quotient, using each number as a divisor. It is not necessary that the dividend selected be a multiple of any of the divisors. Thus, the teacher may announce "D, dividend 20;" the pupil responds 24, 5, 20, 24, 6, etc.

In all of these exercises accuracy should be insisted upon from the beginning; rapidity should not be expected at first, but will result from constant practice.

(6)

MODEL

ELEMENTARY ARITHMETIC.

INTRODUCTION.

LESSON I.

DEVELOPMENT OF NUMBERS BY 2's.

1. 2+2+2+2+2+2+2+2+2+2+2+2=?
2. 1+2+2+2+2+2+2+2+2+2+2+2=?
3. 1x2=? 2x2=? 3x2=? 4x2=? 5×2=?
6x2=? 7x2=? 8x2=? 9x2=? 10×2=?
11x2=? 12x2=?

4. 24-2-2-2-2-2-2-2-2-2-2-2=?

5.

23-2-2-2-2-2-2-2-2-2-2-2-?

6. 2÷2 ? 4÷2=? 6÷2=? 8+2=? 10÷2=? 12÷2 ? 14÷2 ? 16÷2=? 18÷2=? 20÷2=? 22÷2 ? 24÷2=?

7. 4 is 2 more than what number? 2 less than what number?

8. 15 is 2 more than what number? 2 less than what number?

9. 12 is twice what number? 22 is twice what number? 10. What number should be doubled to obtain 24? What number is contained in 18 twice?

11. What number is 2 more than 18? 2 less than 16? 12. What number should be added to 12 to obtain 14? 2 can be taken from 6 how many times?

13. 6 is one half what number? 8 is one half what number?

14. 11 is one half what number? 18 is twice what number?

15. 12 is double what number? Of what number is 7 one half?

16. 9 is one half what number?

numbers equal 10?

What five equal

17. 13 is equal to what six equal numbers and one more? 8 times 2 and 1 more equals what number?

18. 11 times 2 and 1 more equals what number?

19. A boy had 16 cents, and gave 2 cents to his sister. How many cents had he left?

20. Alfred had 22 marbles, and found 2 more. How many did he then have?

21. If one yard of cloth costs 6 dollars, how much will 2 yards cost?

22. John has 9 chestnuts, and his sister has twice as many. How many has his sister?

23. If one top costs 2 cents, how many tops may be bought for 16 cents?

24. How many two-cent stamps may be bought for 24 cents?

25. A girl paid 22 cents for a book, and one half as much for a slate. How much did she pay for the slate? 26. Paid 9 cents for a lead pencil, and twice as much and 1 cent more for some paper. How much was paid for

the paper?