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1.05 Multiplication and division of integers

Lesson

Multiplying integers

We have looked at how to multiply whole numbers when they are positive. Now let's look at what happens when negative integers are included in questions. Remember the product is the answer when two numbers are multiplied together.

Exploration

The applet below allows you to select two integers to multiply by using the sliders to change the value and the check boxes to change the sign.

  • Blue tiles represent positive integers
  • Red tiles represent negative integers
  • The product is shown as an array 

Guiding questions

  1. What is the sign of the product of two positive integers?
  2. If you are looking at the product $3\times\left(-4\right)$3×(4), how many tiles are in $3$3 groups of $-4$4 tiles?
  3. What is the sign of the product of one positive and one negative integer?

  4. Is the product of $3\times\left(-4\right)$3×(4) the same as $-4\times3$4×3? Check this for more products.

  5. What is the sign of the product of two negative integers?

 

Conclusions

To get comfortable with multiplying integers, we can use tools like the applet above to help us identify the patterns. Unlike adding and subtracting integers where we can use the number line or counters, multiplication comes down to looking at the sign of factors.

Product of two integers

Preciously, we have seen that the product of two positive integers is a positive integer

The product or quotient of a positive integer and a negative integer is a negative integer.

The product of two negative integers is a positive integer.

If we are working with integers, we just need to determine what the sign will be and then we can multiply the absolute values of the integers as we already know how to do.

 

Worked examples

question 1

Evaluate: $-4\times5$4×5

Think: We know that $4\times5=20$4×5=20, so we just need to determine what the sign will be. We have the product of negative integer and a positive integer, so the product will be negative. 

Do:    

$-4\times5$4×5 $=$= $-20$20

 

question 2

Evaluate: $-7\times\left(-5\right)$7×(5)

Think: We know that $7\times5=35$7×5=35, so we just need to determine what the sign will be. We have the product of two negative integers, so the product will be positive. 

Do:    

$-7\times\left(-5\right)$7×(5) $=$= $35$35

 

Practice questions

Question 3

Evaluate: $8\times\left(-12\right)$8×(12)

Question 4

Evaluate: $-9\times\left(-11\right)$9×(11)

 

Dividing integers

The same principals that help us to multiply integers also apply to divide. Check that the rules for multiplication also work for division.

 

Quotient of two integers


We have seen that the quotient of two positive integers is a positive integer

The quotient of a positive integer and a negative integer is a negative integer.

The quotient of two negative integers is a positive integer.

 

Worked example

question 5

Evaluate: $48\div\left(-6\right)$48÷​(6)

Think: We know that $48\div6=8$48÷​6=8, so we just need to determine what the sign will be. We have the quotient of one positive and one negative integers, so the quotient will be negative. 

Do:    

$48\div\left(-6\right)$48÷​(6) $=$= $-8$8

 

Practice questions

Question 6

Evaluate: $21\div\left(-7\right)$21÷​(7)

Question 7

Evaluate: $\frac{-60}{10}$6010

Outcomes

6.6a

Add, subtract, multiply, and divide integers (on the state assessment, items measuring this objective are assessed without the use of a calculator)

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