B. Number · Operations: Rational Numbers, Integers, Fractions and Proportional Reasoning

Adding and Subtracting Integers: Choosing Strategies

B2.4 A1About 45 minutes

Learning goals

I can add and subtract integers using a number line, zero pairs or "add the opposite".
I can add and subtract strings of three or more integers by grouping positives and negatives.
I can find a missing integer using the relationship between addition and subtraction.
I can solve problems with temperatures, elevations, money and scores.

Words to know

integer
A whole number, its opposite, or zero: ..., −3, −2, −1, 0, +1, +2, +3, ...

opposite
The number the same distance from zero on the other side. The opposite of −7 is +7.

zero pair
A positive and a negative of the same size, like (+1) and (−1). Together they make 0.

add the opposite
The rule that subtracting an integer is the same as adding its opposite: a − b = a + (−b).

Remember from Grade 7

In Grade 7 you added and subtracted integers with integer chips and number lines, and you discovered that subtracting is the same as adding the opposite: (−3) − (−10) = (−3) + (+10) = +7. In Grade 8 you will choose the best strategy for each question, work with longer strings of integers, and solve multi-step problems.

Let's start with a hockey stat

In a simple version of hockey's plus-minus stat, a player's score goes up by 1 (+1) when their team scores while they are on the ice, and down by 1 (−1) when the other team scores while they are on the ice. (The real stat leaves out some goals, like power-play goals.) Over five games, Jaya's game scores were +2, −3, +1, −1, +4.

Group the positives: 2 + 1 + 4 = +7. Group the negatives: (−3) + (−1) = −4. Total: (+7) + (−4) = +3.

Grouping is one strategy. Today you will practise choosing the strategy that is fastest and safest for each question.

Bars above 0 are positive games and bars below 0 are negative games. Grouping gives (+7) + (−4) = +3.

Strategy	Best for	Example
Number line	Two integers, small numbers, temperature changes	(−7) + (+10): start at −7, move 10 right → +3
Zero pairs	Seeing why answers make sense	(+5) + (−3): 3 zero pairs cancel, +2 is left
Add the opposite	Every subtraction	(+4) − (−11) = (+4) + (+11) = +15
Group positives and negatives	Strings of 3 or more integers	(+6) + (−9) + (+2) + (−4) = (+8) + (−13) = −5
Look for opposites	Strings with matching numbers	(+7) + (−3) + (−7) = (−3), because (+7) + (−7) = 0

Choose the strategy that fits the question.

(−7) + (+10) + (−5): start at −7, jump 10 right to +3, then 5 left to −2. The sum is −2.

Worked example

Example 1: A longer string

Evaluate (−12) + (+5) − (−9) + (−4).

Rewrite the subtraction as adding the opposite: (−12) + (+5) + (+9) + (−4).
Group the positives: (+5) + (+9) = +14.
Group the negatives: (−12) + (−4) = −16.
(+14) + (−16) = −2. There are more negatives, so the answer is negative.

−2

Worked example

Example 2: Elevation

A diver is at −18 m (18 m below the surface). She rises 7 m, then goes down 11 m. Where is she now?

Up is positive and down is negative: (−18) + (+7) + (−11).
Negatives: (−18) + (−11) = −29. Then (−29) + (+7) = −22.

−22 m (22 m below the surface)

Worked example

Example 3: The difference between two temperatures

One year, a town's highest temperature was +23°C and its lowest was −31°C. What is the difference?

Difference = higher − lower = (+23) − (−31).
Add the opposite: (+23) + (+31) = +54.
Check on a number line: from −31 up to 0 is 31, and from 0 up to +23 is 23. 31 + 23 = 54. ✓

54 degrees

Worked example

Example 4: A missing integer

Find the missing number: (−6) + ___ = +4.

Subtraction undoes addition, so ___ = (+4) − (−6).
(+4) + (+6) = +10.
Check: (−6) + (+10) = +4. ✓

+10

Sign check

Before you calculate, ask: are there more positives or more negatives? That tells you the sign of the answer. Then find the size. And always rewrite subtraction as adding the opposite before you group, so no sign gets lost.

Hands-on activity

Integer card rounds

You need: a deck of cards with the face cards removed (Ace = 1), paper, a pencil. In this game black cards are positive and red cards are negative.

Deal yourself 5 cards and write them as integers. Find the sum by grouping positives and negatives.
Now choose two of your cards to subtract instead of add. Try to make the result as close to 0 as possible.
Check each other's work with a number line or with zero pairs (a black and a red card of the same value cancel).
Play 5 rounds. Lowest total distance from zero wins.

Model answer: black 7, red 4, red 9, black 2, red 3 → (+7) + (−4) + (−9) + (+2) + (−3) = (+9) + (−16) = −7. Subtracting the red 4 instead: (+7) − (−4) + (−9) + (+2) + (−3) = +1, which is much closer to zero.

Just for fun

Why are +7 and −7 terrible at group projects?

Every time they team up, they end up with nothing.

Your turn

(−8) + (+13) = ?

(−6) + (−9) = ?

(+4) − (−11) = ?

(−3) − (+7) = ?

(−15) + (+6) + (−2) = ?

(+9) − (−4) + (−20) = ?

At 6 a.m. it is −14°C. At 2 p.m. it is +3°C. What is the change in temperature?

More practice: warm-up

(+5) + (−5) = ?

(−2) + (−3) = ?

(−7) − (−7) = ?

(+3) − (+8) = ?

(−10) + (+4) = ?

More practice: on your own

(−25) + (+40) + (−12) + (+8) = ?

(−6) − (−6) − (−6) = ?

A class fund has $120. It pays $85 for a bus and $60 for snacks, then raises $40 at a bake sale. What is the balance?

Find the missing number: ___ − (−8) = −3.

A golfer's scores compared with par over four days are −2, +1, −3, +2. What is her total compared with par?

Challenge

Two integers have a sum of −3 and a difference of 11. What are they?

Evaluate (−1) + (+2) + (−3) + (+4) + ... + (−19) + (+20).

A submarine is at −250 m and a plane is flying directly above it at +1 850 m. How far apart are they vertically?

Check your understanding

Quiz

What is (−9) + (+4)?

What is (+3) − (−6)?

What is (−4) − (+5)?

Which expression is equal to (−8) − (−3)?

The temperature goes from −6°C to −15°C. What is the change?

What is (−2) + (+7) + (−5)?

For parents and teachers

Big idea: Grade 7 introduced adding and subtracting integers. Ontario's Grade 8 expectation B2.4 asks students to add and subtract integers, using appropriate strategies, in various contexts. The focus now is on choosing efficient strategies and handling longer, real problems.

Common difficulties:

Changing the sign of the wrong number when adding the opposite. Only the number being subtracted changes, and the minus becomes a plus.
Finding a change as start − final instead of final − start.
Dropping a sign in long strings. Encourage writing brackets around every integer.

How to help: Track the overnight low and afternoon high from a weather app for a week and find each day's change. Use a thermometer picture as a number line. Ask "Is the answer above or below zero? How do you know?"