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)

b)

c)

Worked Example 2

Multiply integers:

a)

b)

c)

Worked Example 3

Divide integers:

a)

b)

c)

Worked Example 4

Use sign rules and known number facts:

a)

b)

c)

Worked Example 5

Evaluate expressions involving multiplication and division of integers:

a)

b)

c)

Worked Example 6

Explain the sign of each answer, then evaluate:

a) Why is positive?

b) Why is negative?

c) Why is positive?

Problems

Problem 1

Determine the sign of each product or quotient:

a)

b)

c)

Problem 2

Multiply integers:

a)

b)

c)

Problem 3

Divide integers:

a)

b)

c)

Problem 4

Use sign rules and known number facts:

a)

b)

c)

Problem 5

Evaluate expressions involving multiplication and division of integers:

a)

b)

c)

Problem 6

Explain the sign of each answer, then evaluate:

a) Why is positive?

b) Why is negative?

c) Why is positive?

Exercises

Understanding and Fluency

Exercise 1.

State whether each answer will be positive or negative:

a)

b)

c)

Exercise 2.

State whether each answer will be positive or negative:

a)

b)

c)

Exercise 3.

Multiply the integers:

a)

b)

c)

Exercise 4.

Multiply the integers:

a)

b)

c)

Exercise 5.

Divide the integers:

a)

b)

c)

Exercise 6.

Divide the integers:

a)

b)

c)

Exercise 7.

Evaluate:

a)

b)

c)

Exercise 8.

Evaluate:

a)

b)

c)

Reasoning

Exercise 9.

Explain why the product of two negative integers is positive.

Exercise 10.

A student says because . Explain the mistake.

Exercise 11.

Explain why is positive.

Exercise 12.

A student says because “division makes numbers smaller”. Explain why this is incorrect.

Problem-solving

Exercise 13.

A submarine changes depth by metres each minute for minutes. Represent this with multiplication and Solve the total change in depth.

Exercise 14.

A bank account loses $(8) each day for days. Represent this with integers and find the total change.

Exercise 15.

A temperature changes by per hour for hours. Find the total temperature change.

Exercise 16.

A lift moves down floors at a time for moves. Represent this with multiplication and find the total change.

Exercise 17.

A total loss of $(42) is shared equally over days in a model where the daily change is negative. Find the daily amount.

Exercise 18.

A game score changes by points each round for 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