Add and subtract integers in context

Ontario Grade 8 Mathematics

Number

Use meaning, direction, and a number line to solve situations with positive and negative numbers.

Integers are whole numbers, their opposites, and zero. Positive integers are greater than zero. Negative integers are less than zero. For example, a temperature of

-3^\circ\mathrm{C}

is three degrees below zero. In this lesson, you will use integers to describe changes and positions. You will add or subtract them and explain what your answer means. A number line is a line with numbers placed in order. Moving right means the value increases; moving left means it decreases.

What you will learn

Recognize positive and negative integers in everyday situations.
Use a number line to model adding or subtracting integers.
Choose an operation that matches the meaning of a situation.
Explain whether an answer makes sense in its context.

Connect the situation to an integer

A positive integer can represent a gain, an increase, or a position above a starting level. A negative integer can represent a loss, a decrease, or a position below a starting level. Zero means there is no change or that the position is at the reference point. The reference point is the value chosen as zero, such as sea level or zero degrees.

Before calculating, decide what the integers describe. The sign is part of the value:

-4

is not the same as

4

. For example, a change of

-4

means a decrease of four units. A change of

4

means an increase of four units.

On a number line, adding a positive integer moves right. Adding a negative integer moves left. The number you start at matters: begin there, then move the number of spaces shown by the integer being added.

a+b

Positive values are to the right of zero; negative values are to the left.
Adding describes combining a starting value and a change.
A negative change moves left on the number line.

Use subtraction to describe a difference

Subtraction can answer how far apart two integer values are, or what change takes one value to another. To find the change from an initial value to a final value, subtract the initial value from the final value. This order matters.

For example, if a temperature changes from

-4^\circ\mathrm{C}

to

3^\circ\mathrm{C}

, the change is final temperature minus initial temperature. The number line shows a move from

-4

to

3

, which is seven spaces to the right. So the change is a rise of

7^\circ\mathrm{C}

.

A useful way to think about subtraction is: start at the first number and ask how far, and in which direction, you must move to reach the second number. When subtracting a negative integer, the result can be greater than the starting value. Check the direction in the situation instead of relying only on a memorized rule.

\text{change}=\text{final value}-\text{initial value}

For a change, use final value minus initial value.
A rightward move is an increase; a leftward move is a decrease.
Keep the context and the order of the values clear.

Choose a model and check the meaning

A number line is useful when a situation has a starting value and a change. Mark the starting value, then move left or right according to the change. Read the ending value as the answer.

You can also write a short equation to keep track of the situation. Use the starting value first. Add a positive change for an increase, or add a negative change for a decrease. For a question asking how much a value changed between two points, subtract the initial value from the final value.

After calculating, state the answer with its meaning and units. A result of

-3

in a temperature situation means three degrees below zero, not a decrease of three unless the question is asking for a change. Ask whether the answer fits the direction and size of the situation.

\text{ending value}=\text{starting value}+\text{change}

Model the starting value and the change before calculating.
Use units such as degrees when the context provides them.
Check that the sign of the answer matches the situation.

Worked example

A temperature rises

At sunrise, the temperature is

-3^\circ\mathrm{C}

. Later, it rises by

8^\circ\mathrm{C}

. What is the later temperature?

Temperature rise

Start at $-3$ and move right eight units to $5$.

Identify the start and change
The starting temperature is $-3^\circ\mathrm{C}$ . A rise is a positive change of $8^\circ\mathrm{C}$ .
$-3+8$
Move on the number line
Start at $-3$ and move eight spaces to the right because the temperature rises. You pass zero and land at $5$ .
$-3+8=5$
State the result
The later temperature is $5^\circ\mathrm{C}$ . It is above zero, which fits a rise of eight degrees from $-3^\circ\mathrm{C}$ .

Answer: The later temperature is

5^\circ\mathrm{C}

.

Check: The result is eight degrees greater than

-3^\circ\mathrm{C}

.

Worked example

A position moves downward

A lift is at level

4

relative to the ground floor, which is level

0

. It moves down seven levels. What level does it reach?

Lift movement

Start at level $4$ and move left seven levels to level $-3$.

Represent the movement
The lift starts at level $4$ . Moving down seven levels is a change of $-7$ .
$4+(-7)$
Follow the change
Start at $4$ on the number line and move seven spaces left. The lift passes level $0$ and reaches level $-3$ .
$4+(-7)=-3$
Interpret the answer
The lift reaches level $-3$ , which is three levels below the ground floor. The negative sign shows its position relative to level zero.

Answer: The lift reaches level

-3

.

Check: Moving down seven levels from level

4

passes zero and ends three levels below it.

Worked example

Find the change in temperature

A temperature changes from

-4^\circ\mathrm{C}

to

3^\circ\mathrm{C}

. What is the change from the first reading to the second?

Temperature change

The change from $-4$ to $3$ is seven units to the right.

Choose the subtraction order
To find the change, subtract the initial reading from the final reading. The final reading is $3^\circ\mathrm{C}$ and the initial reading is $-4^\circ\mathrm{C}$ .
$3-(-4)$
Read the distance and direction
On a number line, moving from $-4$ to $3$ goes seven spaces to the right. That is an increase of seven degrees.
$3-(-4)=7$
Explain the change
The temperature increased by $7^\circ\mathrm{C}$ . The subtraction order matches the movement from the first reading to the second.

Answer: The temperature increased by

7^\circ\mathrm{C}

.

Check: The move from

-4

to

0

is four degrees, and from

0

to

3

is three more; together the increase is seven degrees.

Common mistakes and how to avoid them

Treating a negative change as an increase because addition is used.

Correction: The sign of the change tells its direction. Adding a negative change means moving left or decreasing.

Subtracting the initial value from the final value in the wrong order.

Correction: For change from an initial value to a final value, calculate final value minus initial value.

Leaving off the meaning or unit of the answer.

Correction: State what the integer represents, such as a level below zero or a rise of a certain number of degrees.

Lesson summary

Integers include positive whole numbers, negative whole numbers, and zero.
Adding a positive integer moves right on a number line; adding a negative integer moves left.
To find a change from an initial value to a final value, subtract initial from final.
Interpret the answer in the original situation and include its unit when one is given.

Check your understanding

Question 1

The temperature is

-5^\circ\mathrm{C}

and rises by

9^\circ\mathrm{C}

. What is the new temperature?

$4^\circ\mathrm{C}$
$-14^\circ\mathrm{C}$
$-4^\circ\mathrm{C}$
correctIndex уст

Show answer and explanation

4^\circ\mathrm{C}

A rise is a positive change, so

-5+9=4

. The new temperature is

4^\circ\mathrm{C}

.

Question 2

A hiker is at position

-2

relative to a marked starting point and moves down five units. What is the new position?

$3$
$-7$
$-3$
correctIndex уст

Show answer and explanation

-7

Moving down five units is adding

-5

. Thus

-2+(-5)=-7

.

Question 3

A temperature changes from

-6^\circ\mathrm{C}

to

2^\circ\mathrm{C}

. What is the change?

A decrease of $8^\circ\mathrm{C}$
An increase of $4^\circ\mathrm{C}$
An increase of $8^\circ\mathrm{C}$
correctIndex уст

Show answer and explanation

An increase of

8^\circ\mathrm{C}

Subtract the initial reading from the final reading:

2-(-6)=8

. The positive result means an increase.

Key terms

Integer: A positive whole number, a negative whole number, or zero.
Change: The amount and direction a value moves from its starting value.
Reference point: The value chosen as zero for describing positions or changes.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation B2.4. 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.

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B2.4 · Add and subtract integers in context

Ontario Grade 8 Mathematics

Use meaning, direction, and a number line to solve situations with positive and negative numbers.

What you will learn

Connect the situation to an integer

Use subtraction to describe a difference

Choose a model and check the meaning

A temperature rises

A position moves downward

Find the change in temperature

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