Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- How far apart are −5 and 3?
Answer
8: zero is between them, so 5 + 3. - What is the opposite of −4?
Answer
4. - The temperature is −2 °C and rises by 6 degrees. What is it now? (Count up a thermometer.)
Answer
4 °C: 2 steps up to zero, 4 more above it.
Words used in this lesson
- zero pair
- a positive and a negative counter together, worth 0. For example: +1 and −1.
- add
- put together; on a number line, move by the number. For example: −4 + 9 = 5.
- subtract
- take away; the same as adding the opposite. For example: 3 − 8 = −5.
- rise / fall
- go up / go down. For example: a fall of 12 degrees is −12.
- change
- new value minus old value. For example: −9 − 3 = −12.
A pair that is worth nothing
Take one positive counter and one negative counter. Together they are worth +1 + (−1) = 0. This is a zero pair. You can put zero pairs in, or take them out, as often as you like, and the value never changes.
Try 5 − (−2): start with 5 positive counters. You have to take away 2 negatives, but there are none. So put in 2 zero pairs, which changes nothing, and now there are 2 negatives to take. Predict the answer, then press Next step and watch the value readout.
Start: 5 positive counters, worth 5.
Show me the steps
- 5 − (−2). Start with 5 positive counters: worth 5.
- There are no negatives to take away, so add 2 zero pairs. Now there are 7 positives and 2 negatives, still worth 5.
- Take away the 2 negatives. 7 positives are left, so 5 − (−2) = 7, the same as 5 + 2.
- 2 − 6. Start with 2 positives, add 4 zero pairs (6 positives and 4 negatives, worth 2), take away 6 positives: 4 negatives are left, so 2 − 6 = −4.
- −3 + 5. Put 5 positives with 3 negatives. 3 zero pairs cancel and 2 positives are left, so −3 + 5 = 2.
Adding a negative number is the same as subtracting its opposite: 7 + (−3) = 7 − 3 = 4.
Subtracting a number is the same as adding its opposite: 5 − (−2) = 5 + 2 = 7, and 3 − 8 = 3 + (−8) = −5. On a number line, adding a positive moves right; adding a negative moves left.
The same rule from a pattern
Read down the table. The number being subtracted goes down by 1 each row, and the answer goes up by 1. Carry on past zero and the pattern decides what subtracting a negative must give.
| Calculation | Answer |
|---|---|
| 5 − 2 | 3 |
| 5 − 1 | 4 |
| 5 − 0 | 5 |
| 5 − (−1) | 6 |
| 5 − (−2) | 7 |
Counters, the number line under them and the pattern all give 7, so the rule holds.
Watch the sign in front of the number, not only the operation. −3 − 7 starts at −3 and moves 7 further left, to −10. It is not 4, which is a move to the right, and not −4, which takes 3 from 7 and keeps the minus sign.
Worked examples
In context. At dawn it is −7 °C in a mountain village. By midday it has risen by 12 degrees. What is the midday temperature?
- A rise is adding: −7 + 12.
- Start at −7 and move 12 up: 7 steps to zero, 5 more. 5 °C.
Your go. The first step is given. In the evening the temperature falls from 3 °C to −9 °C. What is the change in temperature?
- Change = new − old = −9 − 3.
- Finish it, then check. Should the answer be positive or negative?
Check the answer
−9 − 3 = −9 + (−3) = −12. A fall of 12 degrees, so the change is −12 °C.
Check before you go on
1. Which calculation has the same answer as 4 − (−3)?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Work out −5 + 8.
3. Work out 4 − 11.
Level 2 · Building
4. Work out −6 − (−9).
5. Work out −3 − 7.
6. Tokyo is 9 hours ahead of Coordinated Universal Time (UTC), written +9. Mexico City is 6 hours behind UTC, written −6. How many hours ahead of Mexico City is Tokyo?
Level 3 · Stretch
7. Find the missing number: −8 − □ = 5.
8. At noon it is 4 °C. By midnight the temperature has fallen by 11 degrees, and by morning it has risen by 3 degrees. What is the morning temperature, in °C?
Something to think about
Without working any of them out, put these in order from smallest to largest: 7 − (−3), 7 + (−3), −7 − 3, −7 + 3. What did you look at to decide?
One way to think about it
−7 − 3 = −10, −7 + 3 = −4, 7 + (−3) = 4 and 7 − (−3) = 10. The starting number puts each pair on one side of zero, and the move (left 3 or right 3) decides the order inside each pair.
Exit ticket
9. What is −3 − (−5)?