4. This sign, -, named minus, which means less, is used in place of the word less.

Thus, instead of writing, 5 less 2 are 3, we may write, 52 3, and read, 5 minus 2 equals 3.

5. This sign, x, is called the sign of multiplication, and is used in place of the words multiplied by, or times. Thus, instead of writing, 4 multiplied by 3, or 3 times 4 are 12, we may write, 4 x 3 = 12, and read, 3 times 4 are 12.

When we read the word times in place of the sign x, we should read from right to left. Thus, 3 x 2 is read, 2 times 3, or 2 threes; $4 x 3 is read, 3 times 4 dollars, etc.

6. This sign,, is called the sign of division, and is used in place of the words divided by, or in.

Thus, instead of writing, 12 divided by 4, or 4 in 12, 3 times, we may write, 12 ÷ 4 = 3, and read, 12 divided by 4 equals 3.

This sign is often used by putting the two numbers in the place of the dots, which is a more compact and convenient form. Thus, 1243 is better written,

Both forms should be familiar to the eye.
Read the following:

12

= 3.

4

7. This sign, $, is called the dollar sign, and is used in place of the word dollars, and is always written before the number. Thus, instead of writing 5 dollars, we may write $5, and read 5 dollars.

25 feet

8 feet x 6 48 feet 25 feet÷5-5 feet

= 5

5 feet

LESSON V.

As soon as pupils have learned the signs of operations, and their meaning, they should be taught to write in mathematical language combinations similar to the following, and to read the same without hesitation.

Thus, $2 $2 $4; $8-$3

$5.

Read, $2 and $2 are $4, or $2 plus $2 equals $4; $8 less $3 are $5, or $8 minus $3 equals $5.

3 cents x4 = 12 cents; $12 ÷ 2, or

$12
2

= $6.

Read, 4 times 3 cents are 12 cents; $12 divided by 2, or

The fractional form of expressing division should be made perfectly familiar; and halves, thirds, fourths, fifths, and so on to tenths, should be counted, and illustrated in connection with all the "Table Exercises."

The symbols for these parts are as follows:

LESSON VI.

DEFINITIONS.

1. Counting two or more numbers of the same kind together is called adding, or addition.

The number obtained by adding is named the sum. Thus,

9 is the sum of 5 and 4, or 5+4.

Read the following, and give the sum:

4 + 3 =

2 + 6 =

9 men + 2 men =

2. Taking one number from another of the same kind is called subtracting, or subtraction.

- 4.

The number obtained by subtracting is named the difference. Thus, 5 is the difference between 9 and 4, or 9 Read the following, and give the difference:

3. Taking one of two numbers as many times as there are units in the other is called multiplying or multiplication. The number obtained by multiplying is named the product. Thus, 20 is the product of 4 times 5, or 5 × 4.

Read the following and give the product :

5×2 =

9×2=

4 quarts × 2 =

4. Finding how many times one number is contained in another of the same kind, or finding one of the equal parts of a number is called dividing, or division.

The number obtained by dividing is named the quotient. Thus, 4 is the quotient of 20 divided by 5, or 20÷÷5, or 2o. Read the following and give the quotient :

LESSON VII.

COUNTING.

1. Count 2. Count 10 by 2's.

Ans. 2, 4, 6, 8, 10.

2. Count 16 by 2's; 20 by 2's; 30 by 2's.

3. Count by 2's back from 10 to 0.

4. Count by 2's back from 12 to 0;

Ans. 10, 8, 6, 4, 2, 0. from 20 to 0.

5. Count 5 by 2's, and how many over; 9; 11; 13.

6. Count the lines on this page by 2's.

7. Count the words in each of the first ten lines by 2's.

8. Count 2 by 's.

Ans. 1, 1, 1, 2.

9. Give the sum of each group of two numbers at sight, reading rapidly from right to left, and reverse, 5 times.