052e. Multiplying and Dividing Integers

Learning Intentions

To know that the sign of a product or quotient of two negative numbers is positive
To know that the sign of a product or quotient of a negative and positive number is negative
multiply and divide integers

Pre-requisite Summary

Understand that integers can be positive, negative or zero
Be able to compare integers on a number line
Recall multiplication and division facts with positive whole numbers
Know that multiplication and division are inverse operations
Understand that the sign of a number affects whether it is above or below zero
Be able to separate the sign of a calculation from the size of the numbers involved

Worked Examples

Worked Example 1

Determine the sign of each product or quotient:
a) $(- 3) \times (- 4)$
b) $(- 12) \div (- 3)$
c) $(- 5) \times 2$

Worked Example 2

Multiply integers:
a) $(- 6) \times 4$
b) $(- 7) \times (- 3)$
c) $5 \times (- 8)$

Worked Example 3

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

Worked Example 4

Use sign rules and known number facts:
a) $(- 9) \times (- 2)$
b) $(- 48) \div 8$
c) $(- 54) \div (- 9)$

Worked Example 5

Evaluate expressions involving multiplication and division of integers:
a) $(- 3) \times 5 \div (- 1)$
b) $24 \div (- 6) \times 2$
c) $(- 4) \times (- 3) \div 6$

Worked Example 6

Explain the sign of each answer, then evaluate:
a) Why is $(- 2) \times (- 7)$ positive?
b) Why is $(- 18) \div 3$ negative?
c) Why is $(- 20) \div (- 4)$ positive?

Problems

Problem 1

Determine the sign of each product or quotient:
a) $(- 4) \times (- 5)$
b) $(- 15) \div (- 3)$
c) $(- 6) \times 3$

Problem 2

Multiply integers:
a) $(- 8) \times 2$
b) $(- 5) \times (- 6)$
c) $7 \times (- 4)$

Problem 3

Divide integers:
a) $(- 28) \div 7$
b) $(- 42) \div (- 6)$
c) $36 \div (- 9)$

Problem 4

Use sign rules and known number facts:
a) $(- 7) \times (- 4)$
b) $(- 56) \div 8$
c) $(- 63) \div (- 7)$

Problem 5

Evaluate expressions involving multiplication and division of integers:
a) $(- 2) \times 6 \div (- 3)$
b) $30 \div (- 5) \times 4$
c) $(- 5) \times (- 2) \div 5$

Problem 6

Explain the sign of each answer, then evaluate:
a) Why is $(- 3) \times (- 8)$ positive?
b) Why is $(- 21) \div 7$ negative?
c) Why is $(- 32) \div (- 8)$ positive?

Exercises

Understanding and Fluency

State whether each answer will be positive or negative:
a) $(- 2) \times (- 9)$
b) $(- 4) \times 7$
c) $18 \div (- 3)$
State whether each answer will be positive or negative:
a) $(- 24) \div (- 6)$
b) $5 \times (- 11)$
c) $(- 36) \div 9$
Multiply the integers:
a) $(- 3) \times 8$
b) $(- 6) \times (- 4)$
c) $7 \times (- 5)$
Multiply the integers:
a) $(- 9) \times 3$
b) $(- 8) \times (- 7)$
c) $(- 12) \times 2$
Divide the integers:
a) $(- 20) \div 4$
b) $(- 45) \div (- 9)$
c) $63 \div (- 7)$
Divide the integers:
a) $(- 54) \div 6$
b) $(- 72) \div (- 8)$
c) $56 \div (- 8)$
Evaluate:
a) $(- 4) \times 5 \div (- 2)$
b) $36 \div (- 6) \times (- 3)$
c) $(- 7) \times (- 2) \div 7$
Evaluate:
a) $(- 3) \times (- 9) \div 3$
b) $48 \div (- 8) \times 2$
c) $(- 10) \times 4 \div 5$

Reasoning

Explain why the product of two negative integers is positive.
A student says $(- 6) \times 4 = 24$ because $6 \times 4 = 24$ . Explain the mistake.
Explain why $(- 30) \div (- 5)$ is positive.
A student says $(- 18) \div 3 = 6$ because “division makes numbers smaller”. Explain why this is incorrect.

Problem-solving

A submarine changes depth by $(- 6)$ metres each minute for $4$ minutes. Represent this with multiplication and find the total change in depth.
A bank account loses $(8) each day for $5$ days. Represent this with integers and find the total change.
A temperature changes by $(- 3)^{\circ} C$ per hour for $6$ hours. Find the total temperature change.
A lift moves down $4$ floors at a time for $3$ moves. Represent this with multiplication and find the total change.
A total loss of $(42) is shared equally over $(- 6)$ days in a model where the daily change is negative. Find the daily amount.
A game score changes by $(- 5)$ points each round for $8$ rounds. What is the total score change?

Potential Misunderstandings

Students may think two negative numbers always give a negative answer
Students may confuse the sign rule for multiplication and division with the sign rule for addition
Students may ignore the sign and calculate only the whole-number part
Students may think a negative divided by a negative is negative
Students may not separate the sign from the magnitude when calculating
Students may apply a memorised rule without checking whether the operation is multiplication or division
Students may think division by a negative changes only the divisor and not the sign of the quotient
Students may make arithmetic errors with multiplication facts after choosing the correct sign

Next: 053e. Order of Operations with Integers