100. Multiplying and Dividing Integers

Learning Intentions

To understand that the product or quotient of two integers will be positive if the two integers have the same sign
To understand that the product or quotient of two integers will be negative if the two integers have opposite signs
find the product and quotient of two or more integers

Pre-requisite Summary

Know that integers can be negative, zero or positive
Be able to identify the sign of an integer
Recall basic multiplication facts and related division facts
Understand that multiplication and division are inverse operations
Know that a product is the result of multiplication
Know that a quotient is the result of division
Be able to multiply and divide positive whole numbers accurately
Understand that sign rules apply before interpreting the final answer

Worked Examples

Worked Example 1

Determine whether each result is positive or negative:

a) $(- 4) \times (- 3)$
b) $(- 5) \times 2$
c) $18 \div (- 6)$

Worked Example 2

Calculate each product:

a) $(- 7) \times 3$
b) $(- 6) \times (- 8)$
c) $4 \times (- 9)$

Worked Example 3

Calculate each quotient:

a) $(- 24) \div 6$
b) $(- 35) \div (- 5)$
c) $42 \div (- 7)$

Worked Example 4

Find each value:

a) $(- 2) \times 3 \times 4$
b) $(- 5) \times (- 2) \times 6$
c) $(- 3) \times (- 2) \times (- 4)$

Worked Example 5

Find each value:

a) $48 \div (- 6) \div 2$
b) $(- 72) \div 8 \div (- 3)$
c) $(- 90) \div (- 5) \div 3$

Worked Example 6

Evaluate each expression:

a) $(- 4) \times (- 3) \div 2$
b) $36 \div (- 6) \times (- 2)$
c) $(- 5) \times 2 \times (- 3) \div 5$

Problems

Problem 1

Determine whether each result is positive or negative:

a) $(- 8) \times (- 2)$
b) $7 \times (- 4)$
c) $(- 54) \div 9$

Problem 2

Calculate each product:

a) $(- 9) \times 2$
b) $(- 4) \times (- 7)$
c) $6 \times (- 5)$

Problem 3

Calculate each quotient:

a) $(- 28) \div 7$
b) $(- 45) \div (- 9)$
c) $63 \div (- 9)$

Problem 4

Find each value:

a) $(- 3) \times 2 \times 5$
b) $(- 4) \times (- 3) \times 7$
c) $(- 2) \times (- 6) \times (- 5)$

Problem 5

Find each value:

a) $60 \div (- 5) \div 3$
b) $(- 84) \div 7 \div (- 2)$
c) $(- 96) \div (- 4) \div 6$

Problem 6

Evaluate each expression:

a) $(- 6) \times (- 2) \div 3$
b) $54 \div (- 9) \times (- 4)$
c) $(- 8) \times 3 \times (- 2) \div 4$

Exercises

Understanding and Fluency

Decide whether each answer will be positive or negative:
a) $(- 3) \times (- 7)$
b) $(- 9) \times 4$
c) $28 \div (- 4)$
d) $(- 32) \div (- 8)$
Calculate each product:
a) $(- 4) \times 6$
b) $(- 8) \times (- 5)$
c) $7 \times (- 9)$
d) $(- 3) \times (- 11)$
Calculate each quotient:
a) $(- 42) \div 6$
b) $(- 56) \div (- 7)$
c) $72 \div (- 8)$
d) $(- 81) \div 9$
Find each product of three integers:
a) $(- 2) \times 4 \times 5$
b) $(- 3) \times (- 6) \times 2$
c) $(- 4) \times (- 2) \times (- 3)$
Find each quotient of three integers:
a) $48 \div (- 6) \div 2$
b) $(- 64) \div 8 \div (- 2)$
c) $(- 90) \div (- 9) \div 5$
Evaluate each expression:
a) $(- 5) \times (- 4) \div 2$
b) $30 \div (- 5) \times 3$
c) $(- 6) \times 2 \times (- 3) \div 4$
Write the sign, then calculate:
a) $(- 7) \times (- 2)$
b) $(- 36) \div 6$
c) $(- 45) \div (- 5)$
d) $9 \times (- 8)$
Complete the patterns:
a) $3 \times 4 = 12, (- 3) \times 4 = \underset{―}{}, (- 3) \times (- 4) = \underset{―}{}$
b) $20 \div 5 = 4, (- 20) \div 5 = \underset{―}{}, (- 20) \div (- 5) = \underset{―}{}$

Reasoning

Explain why $(- 6) \times (- 4)$ is positive but $(- 6) \times 4$ is negative.
A student says that a negative number times a negative number must be negative because both numbers are negative. Explain the mistake.
Explain how you can predict the sign of $(- 2) \times (- 3) \times 5$ before calculating the exact answer.
Noah says that $(- 48) \div (- 6) = - 8$ . Is he correct? Explain your answer.
A student works out $(- 3) \times 7 \times (- 2)$ and says the answer must be negative because the first number is negative. Describe the error.

Problem-solving

A submarine changes depth by multiplying a scale factor of $- 3$ by a movement of $4$ . What is the result?
A game score changes by $- 6$ points in each round for $5$ rounds. What is the total change in score?
A temperature change of $- 24^{\circ} C$ is shared equally across $6$ hours. What is the change each hour?
A business makes a loss of $$ 8$ each day for $7$ days. Represent this with integers and find the total result.
A number machine multiplies by $- 2$ , then divides by $5$ . What is the result when the input is $- 25$ ?
A diver changes depth by multiplying $(- 4) \times (- 3)$ . What integer result does this give, and is the result above or below zero?

Potential Misunderstandings

Students may think two negative integers always give a negative result
Students may confuse the sign rules for addition and subtraction with the sign rules for multiplication and division
Students may look only at the first sign and ignore the second sign
Students may forget that two integers with the same sign give a positive product or quotient
Students may forget that two integers with opposite signs give a negative product or quotient
Students may apply the sign rule correctly but make an error with the multiplication or division facts
Students may not realise that the sign can be determined before calculating the numerical part
Students may lose track of the sign when working with three or more integers
Students may think division with negatives follows a different sign rule from multiplication, when the pattern is the same
Students may not use brackets clearly, causing confusion about which number is negative