6th Grade - Multiplying and Dividing Integers

Introduction

• Integers are a set of whole numbers (0, 1, 2, 3, ....) and their opposites. For example, 1, -1,  2, -2, 3, -3, and so on.
• 0 is an integer. It is also called the origin. 0 does not carry any sign. When any integer is multiplied by 0, the answer is always 0. e.g. $\left(-5\right) \times 0 = 0$
• Symbols for Multiplication:

1. $\mathbf{\times }$,  like the letter "x" . e.g. $\left(-3\right) \times \left(2\right) = -6$

2. $·$ , a dot, E.g. $\left(-3\right)·\left(2\right) = -6$

3. $\left(\right)\left(\right)$, no symbol between parentheses,  e.g. $\left(-4\right)\left(5\right) = -20$

• Symbols for Division:

1.   $÷$; e.g. $12 ÷ 3$

2.  A long division sign

3.  A fraction sign; e.g. $\frac{3}{5}$

Multiplying and Dividing Integers

• Rule 1: When multiplying or dividing two integers with the same sign, multiply/divide the integers and give the result a positive sign.

$\mathbf{(}\mathbf{+}\mathbf{)}\mathbf{ }\mathbf{\times }\mathbf{ }\mathbf{(}\mathbf{+}\mathbf{)}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{(}\mathbf{+}\mathbf{)}$

$\mathbf{(}\mathbf{-}\mathbf{)}\mathbf{ }\mathbf{\times }\mathbf{ }\mathbf{(}\mathbf{-}\mathbf{)}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{(}\mathbf{+}\mathbf{)}$

$\mathbf{(}\mathbf{+}\mathbf{)}\mathbf{ }\mathbf{÷}\mathbf{ }\mathbf{(}\mathbf{+}\mathbf{)}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{(}\mathbf{+}\mathbf{)}$

$\mathbf{(}\mathbf{-}\mathbf{)}\mathbf{ }\mathbf{÷}\mathbf{ }\mathbf{(}\mathbf{-}\mathbf{)}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{(}\mathbf{+}\mathbf{)}$

• Rule 2: When multiplying or dividing two integers with different signs, multiply/divide the integers and give the result a negative sign.

$\mathbf{(}\mathbf{+}\mathbf{)}\mathbf{ }\mathbf{\times }\mathbf{ }\mathbf{(}\mathbf{-}\mathbf{)}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{(}\mathbf{-}\mathbf{)}$

$\mathbf{(}\mathbf{-}\mathbf{)}\mathbf{ }\mathbf{\times }\mathbf{ }\mathbf{(}\mathbf{+}\mathbf{)}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{(}\mathbf{-}\mathbf{)}$

$\mathbf{(}\mathbf{+}\mathbf{)}\mathbf{ }\mathbf{÷}\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{-}\mathbf{)}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{(}\mathbf{-}\mathbf{)}$

$\mathbf{(}\mathbf{-}\mathbf{)}\mathbf{ }\mathbf{÷}\mathbf{ }\mathbf{(}\mathbf{+}\mathbf{)}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{(}\mathbf{-}\mathbf{)}$

• Follow the order of operations when multiplying and dividing more than two integers.

Solved Examples

Example 1: Multiply: $\left(-7\right)\left(-9\right)$

Solution: Follow the rules of multiplying integers with the same sign.

$-7 \times -9= 63$

 

Example 2: Find the product: $-3·4$

Solution: Follow the rule of multiplying integers with different signs.

$-3 · 4 = -12$

 

Example 3: Solve: $\left(4\right)\left(-8\right)$

Solution: Follow the rule of multiplying integers with different signs.

$= 4 \times -8$

$= -32$

 

Example 4: Divide: $-36 ÷ -9$

Solution: Follow the rule of dividing integers with the same sign.

$-36 ÷ -9 = 4$

 

Example 5: Solve:$\frac{36}{-6}$

Solution: Follow the rule of dividing integers with different signs.

$\frac{36}{-6} = -6$

Cheat Sheet

• Multiplying or dividing integers with the same sign:

Positive x Positive = Positive

Negative x Negative = Positive

Positive $÷$ Positive = Positive

Negative $÷$ Negative = Positive

• Multiplying or dividing integers with different signs:

Positive x Negative = Negative

Negative x Positive = Negative

Positive $÷$ Negative = Negative

Negative $÷$ Positive = Negative

Blunder Areas

• When multiplying or dividing integers, follow the rules of multiplying and dividing integers. Do not get confused with the rules of addition or subtraction of integers.
• Do not misinterpret parentheses. E.g. $-3(-2) \ne -3 -2$

$-3\left(-2\right) = 6$, follow the rule of multiplying integers with the same sign.

$-3-2 =-5$, follow the rule of subtraction of integers with the same sign.