Four Operations with Negative Numbers: KS3 Maths Rules

How do you add and subtract with negative numbers?

Think of a number line running from left (negative) to right (positive):

• Adding moves you to the right (up).
• Subtracting moves you to the left (down).

Worked examples:

What happens when you subtract a negative number?

Subtracting a negative is the same as adding its positive opposite. Two negatives next to each other in a subtraction become a positive.

"Minus a minus = plus"

Worked examples:

• 7 − (−3) = 7 + 3 = 10
• −4 − (−9) = −4 + 9 = 5
• −6 − (−6) = −6 + 6 = 0

Why? Removing a debt is the same as gaining money. If you owe £6 (−6) and that debt is cancelled (subtract −6), you are £6 better off: 0 − (−6) = +6.

What are the sign rules for multiplication?

When multiplying two numbers, only the signs matter for deciding the sign of the answer.

Memory shortcut: Same signs → positive. Different signs → negative.

Worked examples:

1. (−3) × (−7) = 21 (both negative → positive)
2. 6 × (−4) = −24 (different signs → negative)
3. (−2) × (−5) × (−1) = 10 × (−1) = −10 (multiply left to right; final sign is negative because three negatives overall is odd)

For three or more factors, count the negatives: an even number of negative signs gives a positive result; an odd number of negative signs gives a negative result.

What are the sign rules for division?

The sign rules for division are identical to those for multiplication.

Worked examples:

1. (−36) ÷ (−9) = 4 (same signs → positive)
2. 45 ÷ (−5) = −9 (different signs → negative)
3. (−18) ÷ 6 = −3 (different signs → negative)

What are the most common mistakes?

1. Confusing addition with multiplication rules. "Minus a minus is a plus" applies to subtraction (removing a negative) and to multiplying/dividing. It does not mean (−3) + (−5) = +8 — adding two negatives gives a more negative result: (−3) + (−5) = −8.

2. Forgetting to apply the sign separately from the magnitude. Work out the size of the answer first (ignore signs), then apply the sign rule. For example: (−7) × (−8) → size = 56, signs: (−)(−) = +, answer = 56.

3. Losing track of signs in multi-step calculations. Write each sign explicitly rather than carrying it in your head.

Frequently asked questions

Why does a negative times a negative give a positive?

One explanation: multiplication by −1 flips a number to the opposite side of zero on the number line. Flipping once gives a negative; flipping again (multiplying by another −1) brings you back to positive. So (−1) × (−1) = +1, and this generalises to all negative pairs.

Does the order of numbers matter when multiplying negatives?

No. Multiplication is commutative: (−4) × 5 = 5 × (−4) = −20. The sign rule only cares about how many negative factors there are, not their position.

How do I handle negative numbers in BIDMAS?

Treat negative numbers as exactly that — negative values. Apply BIDMAS as normal, but be careful with signs at each step. For example: −3² means −(3²) = −9, not (−3)² = 9. If you want to square −3, use brackets: (−3)².

Where do negative numbers appear in other KS3 topics?

Negative numbers appear throughout KS3 maths: temperatures and sea level in real-life contexts, coordinates in all four quadrants (negative x and y values), substituting negative values into formulae, straight-line graphs with negative gradients, and sequences that decrease below zero.

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