Integer– An integer is a whole number that can be either greater than zero (positive) or less than zero (negative). Zero is neither positive nor negative. They count discrete quantities.

Two integers that are the same distance from zero in opposite directions are called opposites. Every integer on the number line has an absolute value which is its distance from zero.

The number of units a number is from zero on the number line. The absolute value of a number is always a positive number (or zero). We specify the absolute value of a number *n* by writing *n* in between two vertical bars: |*n*|.

Zero Pair- Created when a negative and a positive are combined. Technically, there are an infinite number of zero pairs. If students understand zero pairs, then they would just cancel out the pairs and would be left with one negative counter.

Addition and Subtraction of integers:

When adding integers of the same sign, we add their absolute values, and give the result the same sign.

Examples:

2 + 5 = 7

(-7) + (-2) = -(7 + 2) = -9

(-80) + (-34) = -(80 + 34) = -114

When adding integers of the opposite signs, we take their absolute values, subtract the smaller from the larger, and give the result the sign of the integer with the larger absolute value.

Example:

8 + (-17) = ?

The absolute values of 8 and -17 are 8 and 17.

Subtracting the smaller from the larger gives 17 – 8 = 9, and since the larger absolute value was 17, we give the result the same sign as -17, so 8 + (-17) = -9.

Subtracting Integers:

Subtracting an integer is the same as adding its opposite.

Examples:

In the following examples, we convert the subtracted integer to its opposite, and add the two integers.

7 – 4 = 7 + (-4) = 3

12 – (-5) = 12 + (5) = 17

-8 – 7 = -8 + (-7) = -15

-22 – (-40) = -22 + (40) = 18

Note that the result of subtracting two integers could be positive or negative.

Multiplying Integers: Multiplication is essentially thought of as repeated addition or adding many groups of the same amount. We can apply that idea to integers.

For example, (3) x (-4) would be three groups of negative 4.

To multiply a pair of integers if both numbers have the same sign, their product is the product of their absolute values (their product is positive). If the numbers have opposite signs, their product is the *opposite* of the product of their absolute values (their product is negative). If one or both of the integers is 0, the product is 0.

Examples:

In the product below, both numbers are positive, so we just take their product.

4 × 3 = 12

In the product below, both numbers are negative, so we take the product of their absolute values.

(-4) × (-5) = |-4| × |-5| = 4 × 5 = 20

In the product of (-7) × 6, the first number is negative and the second is positive, so we take the product of their absolute values, which is |-7| × |6| = 7 × 6 = 42, and give this result a negative sign: -42, so (-7) × 6 = -42.

In the product of 12 × (-2), the first number is positive and the second is negative, so we take the product of their absolute values, which is |12| × |-2| = 12 × 2 = 24, and give this result a negative sign: -24, so 12 × (-2) = -24.

Dividing Integers: Keep in mind the differences between partitive and measurement division when thinking about a situation and how to model it.

To divide a pair of integers if both integers have the same sign, divide the absolute value of the first integer by the absolute value of the second integer.

To divide a pair of integers if both integers have different signs, divide the absolute value of the first integer by the absolute value of the second integer, and give this result a negative sign.

Examples:

In the division below, both numbers are positive, so we just divide as usual.

4 ÷ 2 = 2.

In the division below, both numbers are negative, so we divide the absolute value of the first by the absolute value of the second.

(-24) ÷ (-3) = |-24| ÷ |-3| = 24 ÷ 3 = 8.

In the division (-100) ÷ 25, both number have different signs, so we divide the absolute value of the first number by the absolute value of the second, which is |-100| ÷ |25| = 100 ÷ 25 = 4, and give this result a negative sign: -4, so (-100) ÷ 25 = -4.

In the division 98 ÷ (-7), both number have different signs, so we divide the absolute value of the first number by the absolute value of the second, which is |98| ÷ |-7| = 98 ÷ 7 = 14, and give this result a negative sign: -14, so 98 ÷ (-7) = -14.

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Mindy Hess and Mallory Welty