Multiply and divide integers in context

Ontario Grade 8 Mathematics

Number

Use signs, equal groups, and real situations to make sense of integer calculations

An integer is a whole number, its opposite, or zero. Examples include

-4

,

0

, and

7

. In a situation, a positive number might mean a gain or rise. A negative number might mean a loss or drop. The context tells you what the signs mean. For example, if an increase is positive, a change of

-3

points means a loss of

3

points. This lesson uses equal groups and equal sharing to understand multiplication and division with integers.

What you will learn

Explain what positive and negative integers mean in a situation.
Multiply and divide integers using sign rules.
Explain what an integer product or quotient means in context.

Start with equal groups

Imagine that a player loses

3

points in each of four rounds. You can model the situation with four equal groups. Each group is a change of

-3

points. The total change is the sum of those four equal changes.

Multiplication is a quick way to find a total made from equal groups. Here, the number of groups is positive, and the change in each group is negative. The total is a loss, so it is negative.

On a number line, positive changes move right and negative changes move left. Starting at zero, four leftward moves of

3

points end at

-12

points. This picture connects the calculation to the meaning of the signs.

4\times(-3)=-12

Use multiplication to find a total made from equal groups.
Read the context to decide what positive and negative mean.
A negative change moves left on a number line when positive change is shown to the right.

Multiply and divide using sign rules

For multiplication and division, first work with the number parts, without their signs. Then use the signs to decide whether the answer is positive or negative.

When the two signs match, the product or quotient is positive. When the signs differ, the product or quotient is negative. For example, two negative numbers have matching signs, so their product or quotient is positive.

These rules work for signed amounts in different situations. The context does not change the calculation. It helps you explain what the answer means. A factor of zero makes a product of zero. Division by zero is not defined, so the divisor cannot be zero.

(+)(+)=+\quad(-)(-)=+\quad(+)(-)=-\quad(-)(+)=-

Matching signs give a positive product or quotient.
Different signs give a negative product or quotient.
Multiply or divide the number parts, then give the answer its sign.

Make division fit the question

Division can answer two kinds of questions about equal groups. It can find the amount in each group when the total and the number of groups are known. It can also find the number of groups when the total and the amount in each group are known.

For example, a total change of

-18

degrees across six equal intervals can be divided by six to find the change in one interval. The quotient is a signed change. In another question, a total change of

-18

points made from changes of

-3

points can be divided by

-3

to find how many changes occurred. That quotient is a count of groups.

Always ask what the quotient represents. It might be a signed amount per group, such as points per round, or a positive number of groups. Include the unit when the situation gives one.

\text{total}

\divamount in each group=number of groups

Divide the total by the number of groups to find the amount in each group.
Divide the total by the amount in each group to find the number of groups.
Explain the quotient using the question's units and meaning.

Check the answer in context

After calculating, check whether the sign and size make sense. Repeated losses should produce a negative total. Sharing a negative total change among a positive number of intervals gives a negative change in each interval.

For a group-count question, check by multiplying the number of groups by the amount in each group. The result should match the stated total. This check connects division back to equal groups.

Practise naming what each number represents before calculating. This makes it easier to choose multiplication or division and to give a clear final answer.

Check the sign against the direction of the change.
Use multiplication to check a division answer about groups.
State the result in words, with units when they are given.

Worked example

Repeated point losses

A player loses

3

points in each of four rounds. What is the total change in points?

Four equal point losses

Start at 0 and move left by 3 points four times. The endpoint is −12 points.

Identify the equal groups
There are four rounds, and each round changes the score by $-3$ points. Multiply the number of rounds by the change in each round to find the total change.
$4\times(-3)$
Apply the sign rule
The signs differ, so the product is negative. The number parts make $12$ , and the negative sign shows that the total is a loss.
$4\times(-3)=-12$
State the result
The total change is $-12$ points. In words, the player loses $12$ points overall.

Answer: The total change is

-12

points, a loss of

12

points.

Check: Four equal losses of

3

points make a total loss of

12

points.

Worked example

Find the change in each interval

A temperature reading changes by a total of

-12

degrees over four equal intervals. The change is the same in every interval. What is the change per interval?

Total temperature change

The total change goes from 0 to −12 degrees. Four equal interval changes are each −3 degrees.

Choose division
The total change is shared equally among four intervals. Divide the signed total change by the number of intervals.
$(-12)\div4$
Calculate the quotient
The signs differ, so the quotient is negative. The number parts give $12\div4=3$ .
$(-12)\div4=-3$
Explain the unit
The quotient is the change in one interval. The negative sign means the reading dropped by $3$ degrees in each interval.

Answer: The change is

-3

degrees per interval.

Check: Four changes of

-3

degrees add to a total change of

-12

degrees.

Worked example

Find the number of equal changes

A record shows a total change of

-12

points. Each equal change was

-3

points. How many equal changes were there?

Four changes of −3 points

Four equal leftward changes of 3 points reach −12 points.

Interpret the quotient
Divide the total change by the change in each group. The quotient will tell how many groups of $-3$ points make $-12$ points.
$(-12)\div(-3)$
Apply the sign rule
The signs match, so the quotient is positive. The number parts give $12\div3=4$ .
$(-12)\div(-3)=4$
Check in context
Four equal changes of $-3$ points make a total change of $-12$ points. A positive answer makes sense because it counts the groups.
$4\times(-3)=-12$

Answer: There were 4 equal changes.

Check: Four is a positive count, and four groups of

-3

points total

-12

points.

Common mistakes and how to avoid them

Assuming a negative sign disappears whenever numbers are multiplied or divided.

Correction: Compare the signs. Matching signs give a positive answer; different signs give a negative answer.

Giving a positive total for several repeated losses.

Correction: A positive number of groups of negative changes still gives a negative total.

Reporting a number without saying what it means.

Correction: State the result in context, such as a loss of 12 points or a change of −3 degrees per interval.

Dividing by the number of groups when the question asks how many groups there were.

Correction: For a group count, divide the total by the amount in each group. Check by multiplying the count by the amount in each group.

Lesson summary

Use context to decide what positive and negative integers mean.
For multiplication and division, matching signs give a positive answer and different signs give a negative answer.
Calculate the number parts, apply the sign, and explain the result with its unit.
Use division to find an equal share or a number of groups, depending on the question.
Check the answer by relating it back to the situation.

Check your understanding

Question 1

A hiker's recorded elevation change is

-12

metres over four equal sections. What is the change per section?

$-3$ metres per section
$3$ metres per section
$-8$ metres per section
$-48$ metres per section

Show answer and explanation

-3

metres per section

Divide the total change by the four sections. The signs differ, so the quotient is negative, and

12\div4=3

.

Question 2

What is

(-6)\times(-2)

?

$-12$
$12$
$-8$
$8$

Show answer and explanation

12

The signs match, so the product is positive. The number parts multiply to

12

.

Question 3

A total change of

-12

points is made from equal changes of

-3

points. How many equal changes were there?

$-4$ changes
$4$ changes
$9$ changes
$-9$ changes

Show answer and explanation

4

changes

Divide

-12

by

-3

. Matching signs give a positive quotient, and

12\div3=4

.

Key terms

Integer: A whole number, its opposite, or zero.
Equal groups: Groups that each contain the same amount.
Quotient: The answer to a division calculation.
Total change: The amount and direction of a change over the whole situation.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation B2.7. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

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B2.7 · Multiply and divide integers in context

Ontario Grade 8 Mathematics

Use signs, equal groups, and real situations to make sense of integer calculations

What you will learn

Start with equal groups

Multiply and divide using sign rules

Make division fit the question

Check the answer in context

Repeated point losses

Find the change in each interval

Find the number of equal changes

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 8 Math

About this lesson and its review