Number · KS3

Negative Numbers

Numbers below zero follow two simple pictures: a walk along the number line for + and −, and a sign rule for × and ÷. Master both and the minus sign stops being scary.

National curriculum KS3: order positive and negative integers; use the four operations applied to negative numbers

1 · The number line

Negative numbers live to the left of zero. Adding moves you right. Subtracting moves you left. That is the whole game. Slide the controls and watch the walk.

Cold gets colder. Think temperature: −3°C and it drops 5 more degrees → −8°C. It rises 5 degrees → 2°C. If the number line feels abstract, a thermometer is the same picture stood on its end.

2 · Subtracting a negative

Two signs together is where marks are lost. 5 − (−3) says "take away a debt of 3". Removing a debt makes you better off, so it is the same as adding 3. Whenever two signs sit next to each other, merge them first.

Same signs → +
5 − (−3) = 5 + 3 = 8
4 + (+2) = 4 + 2 = 6
Different signs → −
5 + (−3) = 5 − 3 = 2
4 − (+2) = 4 − 2 = 2

3 · Multiplying and dividing

For × and ÷, work out the number part, then fix the sign: same signs give a positive, different signs give a negative. Try every combination.

Why does negative × negative make positive? −2 × 3 means "lose £2, three times": −£6. Now −2 × (−3) means the opposite of losing £2 three times — so you gain £6. Reversing a loss is a gain.

4 · Where negatives live

Temperature

−4°C overnight, 6°C by lunch. The rise is 6 − (−4) = 10 degrees.

Money

A bank balance of −£20 is £20 owed. Paying in £50 leaves −20 + 50 = £30.

Lifts and levels

Floor −2 is two levels below the ground floor. From floor 3 down to −2 is 5 floors.

Sea level

A diver at −15 m and a drone at 30 m are 45 m apart.

Worked examples

The temperature is −6°C. It rises 9 degrees. What is it now?
Start at −6. Rising means move right 9.
−6 + 9 = 3°C
Work out 7 − (−5).
Two signs together: − and − merge into +.
7 − (−5) = 7 + 5 = 12
Work out −3 × (−8) and −24 ÷ 6.
Numbers first, then signs: same signs → positive, different signs → negative.
−3 × (−8): 3 × 8 = 24, signs the same → 24
−24 ÷ 6: 24 ÷ 6 = 4, signs different → −4

Quick check

What you've learned

Adding moves right, subtracting moves left. Two signs together merge: same signs make +, different signs make −. For × and ÷, do the number part first, then set the sign: same signs give a positive answer, different signs give a negative one. When stuck, draw the number line or think in temperatures.