B. Number · Operations: Exponents, Factors, Fractions, Decimals, Integers, Percents and Proportions

Subtracting Integers

B2.4 A1About 45 minutes

Learning goals

I can use integer chips, adding zero pairs when needed, to subtract integers.
I can use a number line to find the difference between two integers.
I can explain why subtracting an integer is the same as adding its opposite.
I can solve problems about temperature change and elevation with integers.

Words to know

difference
The answer to a subtraction. On a number line, it is how far and in which direction you go from the second number to the first.

opposite
The integer the same distance from zero on the other side. The opposite of −4 is +4.

zero pair
(+1) and (−1) together. Adding a zero pair does not change a value.

adding the opposite
Subtracting an integer gives the same answer as adding its opposite: 5 − (−3) = 5 + (+3).

Remember

In the last lesson you added integers with chips and a number line, and you learned that a zero pair (+1) + (−1) equals 0. Subtracting integers uses the same tools, with one clever trick.

Let's start with a warm-up in the weather

One winter morning the temperature was −8°C. By the afternoon it was +5°C. How many degrees did it warm up?

On a thermometer, count from −8 up to 0: that's 8 degrees. Then from 0 up to +5: 5 more. Total: 13 degrees.

The change is final − start: (+5) − (−8) = +13. Subtracting a negative gave a bigger number!

From −8 to +5 is a jump of 13 to the right, so (+5) − (−8) = +13.

Model 1: Take away chips (add zero pairs if you need to)

Subtracting means taking away. Show the first number with chips, then take away the second number.

(+5) − (+2): show 5 positives, take away 2 positives. +3 is left.
(−6) − (−2): show 6 negatives, take away 2 negatives. −4 is left.

What if you don't have the chips you need to take away? Add zero pairs. They don't change the value, but they give you the chips you need.

(+2) − (+5): show 2 positives. You need to take away 5 positives, so add 3 zero pairs. Now there are 5 positives and 3 negatives (still worth +2). Take away the 5 positives. −3 is left.

(+2) − (+5): start with two + chips, add 3 zero pairs (the dashed box), then take away five + chips (crossed out). Three − chips are left, so (+2) − (+5) = −3.

Model 2: The number line

To find a − b, start at b and see how far and which way you move to reach a. Right is positive, left is negative.

(+5) − (−8): from −8 to +5 is 13 to the right → +13.
(−3) − (+4): from +4 to −3 is 7 to the left → −7.

The pattern: subtract = add the opposite

Subtraction	Answer	Adding the opposite
5 − (+2)	3	5 + (−2) = 3
5 − (+1)	4	5 + (−1) = 4
5 − 0	5	5 + 0 = 5
5 − (−1)	6	5 + (+1) = 6
5 − (−2)	7	5 + (+2) = 7

As the number being subtracted goes down by 1, the answer goes up by 1. Subtracting a negative is the same as adding a positive.

Worked example

Example 1: Subtract a negative

Find (+4) − (−3).

Chips: show 4 positives. There are no negatives to take away, so add 3 zero pairs (now 7 positives and 3 negatives, still +4).
Take away the 3 negatives. 7 positives are left.
Adding the opposite: (+4) + (+3) = +7. Same answer. ✓

(+4) − (−3) = +7

Worked example

Example 2: Negative minus positive

Find (−7) − (+3).

Add the opposite: (−7) + (−3).
Both negative: 7 + 3 = 10, so −10.
Number line check: from +3 to −7 is 10 to the left → −10. ✓

(−7) − (+3) = −10

Worked example

Example 3: Difference in elevation

A loon dives to −6 m (6 m below the surface of a lake). An eagle flies at +35 m. How far apart are they?

Difference = higher − lower = (+35) − (−6).
Add the opposite: (+35) + (+6) = +41.

They are 41 m apart.

Check with addition

Subtraction and addition undo each other. If (−2) − (−6) = +4, then (+4) + (−6) should give back −2. It does! ✓ Use this check whenever a sign feels surprising.

Hands-on activity

Winter temperature tracker

You need: a weather app, website or newspaper with a 7-day forecast (or the table below), paper and a pencil.

Write down the high and low temperature for each day of the week.
For each day, find how much colder the low is than the high: high − low.
Find the change from one day's high to the next day's high: new − old. Is it positive (warmer) or negative (colder)?
Model one of them with chips (use two kinds of buttons, coins or pasta) and one on a drawn number line.

If it isn't winter, use these: Mon high −2, low −11; Tue high +1, low −6; Wed high −5, low −14.

What to look for: Mon: (−2) − (−11) = 9 degrees. Tue: (+1) − (−6) = 7 degrees. Wed: (−5) − (−14) = 9 degrees. Monday to Tuesday highs: (+1) − (−2) = +3 (warmer). Tuesday to Wednesday highs: (−5) − (+1) = −6 (colder).

Math talk: "Why is the difference always positive when you subtract the low from the high?"

Just for fun

Why is subtracting a negative like cheering up a grumpy friend?

Take away the "not happy" and what's left is more happy!

Your turn

(+8) − (+3) = ?

(+3) − (+8) = ?

(−6) − (−2) = ?

(−2) − (−6) = ?

(+5) − (−4) = ?

(−7) − (+3) = ?

The temperature goes from −15°C to −4°C. What is the change?

More practice: warm-up

(+6) − (+6) = ?

(+4) − (−1) = ?

(−3) − (+2) = ?

(−5) − (−5) = ?

0 − (+3) = ?

More practice: on your own

(−12) − (+8) = ?

(+9) − (−11) = ?

(−4) − (−10) = ?

(+3) − (+15) = ?

A diver is at −18 m. A gull is flying at +25 m. How far apart are they?

Challenge

(−6) + (+10) − (−3) − (+8) = ?

Write a subtraction using two negative integers that has an answer of −7.

The day's high was −2°C and the overnight low was −17°C. Write a subtraction for the change from the high to the low. What does the sign tell you?

Check your understanding

Quiz

What is (+3) − (−5)?

What is (−4) − (+6)?

Which is the same as (−9) − (−2)?

To take away 4 positive chips from (+1), what should you do first?

The temperature changes from −3°C to +6°C. What is the change?

What is (−8) − (−8)?

For parents and teachers

Big idea: The Ontario curriculum (B2.4) asks Grade 7 students to use objects, diagrams and equations to represent, describe and solve situations involving addition and subtraction of integers. Subtraction is the trickier one. Students should model it with chips (adding zero pairs when needed) and number lines until they discover that subtracting an integer is the same as adding its opposite.

Common difficulties:

Thinking subtraction always makes the answer smaller. (+4) − (−3) = +7.
Changing the wrong sign when adding the opposite. Only the operation and the integer after it change.
Order in temperature change: change is always final − start.

How to help: Use winter forecasts to find daily ranges and day-to-day changes. Ask your child to check each subtraction with addition. This lesson builds A1 skills: reasoning from patterns, representing with models, and checking with inverse operations.