Key ideas
- Integers are whole numbers, positive, negative or zero. On a number line, numbers get smaller as you go left: −8 is smaller than −2.
- Algebra tiles help with adding and subtracting:
+1 is a yellow tile
−1 is a red tile
A yellow and a red make a zero pair: +1 + (−1) = 0
- Subtracting a negative is the same as adding: 5 − (−2) = 5 + 2. Adding a negative is the same as subtracting: 5 + (−2) = 5 − 2.
- Multiplying and dividing: same signs give a positive, different signs give a negative.
| Signs | Answer | Example |
|---|---|---|
| + and + | positive | 3 × 4 = 12 |
| − and − | positive | −3 × −4 = 12 |
| + and − | negative | 3 × −4 = −12 |
| − and + | negative | −12 ÷ 4 = −3 |
Ordering integers
Put these numbers in order, smallest first: −5, 3, −1, 0, −8(1 mark)
Place them on a number line.
In plain EnglishThe further left a number is, the smaller it is. Negative numbers are to the left of zero, and −8 is further left than −5.
Read from left to right.−8, −5, −1, 0, 3✓1
In plain EnglishThe biggest-looking digit (8) is actually the smallest number here, because it is the most negative.
⚠ Common mistake. Thinking −8 is bigger than −5 because 8 is bigger than 5. Think of temperature: −8°C is colder than −5°C.
Adding integers
Work out −3 + 5(2 marks)
Make each number with tiles.−3: three red tiles +5: five yellow tiles
In plain EnglishRed tiles are −1 and yellow tiles are +1. Put them all together, because we are adding.
Cancel the zero pairs.3 zero pairs = 0✓1
In plain EnglishMatch each red tile with a yellow tile. Each pair is worth 0, so take them away.
Count what's left.2 yellow tiles left, so −3 + 5 = 2✓2
In plain EnglishTwo yellow tiles are left over. That's +2.
Work out −2 + (−4)(1 mark)
Make each number with tiles.two red + four red
In plain EnglishBoth numbers are negative, so all the tiles are red.
Count them.six red tiles, so −2 + (−4) = −6✓1
In plain EnglishThere are no yellow tiles, so there are no zero pairs. Six red tiles make −6.
Subtracting integers
Work out 2 − 5(2 marks)
Start with 2.two yellow tiles
In plain EnglishSubtracting means taking tiles away. We need to take away 5 yellow tiles, but there are only 2.
Add zero pairs until you can take away.add 3 zero pairs: 5 yellow and 3 red✓1
In plain EnglishAdding zero pairs doesn't change the value (each pair is 0). Now there are 5 yellow tiles.
Take away 5 yellow tiles.3 red tiles left, so 2 − 5 = −3✓2
In plain EnglishThree red tiles are left: −3. On a number line, start at 2 and go back 5.
Work out 3 − (−2)(2 marks)
Start with 3, then add zero pairs.3 yellow + 2 zero pairs✓1
In plain EnglishWe need to take away 2 red tiles, but there aren't any. Add 2 zero pairs so there are 2 red tiles to remove.
Take away the 2 red tiles.5 yellow tiles left, so 3 − (−2) = 5✓2
In plain EnglishTaking away the red tiles leaves 5 yellow ones. Subtracting a negative is the same as adding: 3 − (−2) = 3 + 2.
Adding and subtracting
Work out −4 + 7 − (−2) − 6(2 marks)
Tidy up the double signs.−4 + 7 + 2 − 6✓1
In plain EnglishTwo signs next to each other: the same signs make +, different signs make −. So − (−2) becomes + 2.
Work left to right.= 3 + 2 − 6= 5 − 6= −1✓2
In plain English−4 + 7 = 3, then 3 + 2 = 5, then 5 − 6 = −1.
Multiplying integers
Work out (a) −4 × 3 (b) −5 × −6(2 marks)
(a) Multiply the numbers, then do the sign.4 × 3 = 12; different signs → −12✓1
In plain EnglishIgnore the signs and multiply: 12. One negative and one positive make a negative.
(b) Same again.5 × 6 = 30; same signs → 30✓2
In plain EnglishTwo negatives are the same sign, so the answer is positive.
Same signs → positive. Different signs → negative.
Dividing integers
Work out (a) −20 ÷ 4 (b) −18 ÷ −3(2 marks)
(a) Divide, then do the sign.20 ÷ 4 = 5; different signs → −5✓1
In plain EnglishDivision uses exactly the same sign rules as multiplication.
(b) Same signs.18 ÷ 3 = 6; same signs → 6✓2
In plain EnglishNegative ÷ negative is positive.
Multiplying and dividing
Work out −2 × 6 ÷ (−3)(2 marks)
Work left to right.−2 × 6 = −12✓1
In plain EnglishMultiplication and division are equal, so go left to right. Different signs give a negative.
Then divide.−12 ÷ (−3) = 4✓2
In plain EnglishSame signs (both negative), so the answer is positive.
Order of operations with integers
Work out (5 − 8) × (−4) + 2(3 marks)
Brackets first.(5 − 8) = −3✓1
In plain EnglishStart at 5 and go back 8: −3.
Then multiply.−3 × (−4) = 12✓2
In plain EnglishSame signs, so positive.
Then add.12 + 2 = 14✓3
In plain EnglishBODMAS works exactly the same with negative numbers.
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Your turn
Non-calculator. Type negative numbers with a minus sign, like -5. Stuck? Tap Need a hint?