Negative numbers trip up KS3 students because the rules — "minus times minus is plus" — feel arbitrary. They're not. Once you have the right mental model, the rules become obvious and you stop slipping. This guide gives the model and the four operations, with the slip-prone moves named.
The mental model: position on a number line
Imagine a number line with 0 in the middle. Positive numbers to the right, negative numbers to the left.
... -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 ...
↑
origin
Two things matter:
- Position = the value (how far from 0)
- Direction = positive (right) or negative (left)
Every operation becomes a movement on this line.
Adding and subtracting
Adding a positive = move right
3 + 4 = 7 → start at 3, move 4 right → land at 7.
Adding a negative = move left
3 + (-4) = -1 → start at 3, move 4 left → land at -1.
Subtracting a positive = move left
3 - 4 = -1 → start at 3, move 4 left → land at -1.
Subtracting a negative = move right (the double negative)
3 - (-4) = 7 → start at 3, subtracting -4 means moving in the opposite of left = move 4 right → land at 7.
The shortcut: subtracting a negative cancels — turn it into addition. 3 - (-4) = 3 + 4 = 7.
Multiplying and dividing
The rule, with the model:
Same signs → positive. Different signs → negative.
Why? Think of multiplication as repeated movement:
3 × 2 = 6: start at 0, move 2 right, do it 3 times → land at 6 ✓-3 × 2 = -6: start at 0, move 2 right, but reverse direction (negative count) → land at -6 ✓-3 × -2 = +6: start at 0, reverse direction (negative count) AND move left (negative number) → both reversed = forward = land at +6 ✓
For division, exactly the same rules.
| Operation | Result sign |
|---|---|
| pos × pos | + |
| pos × neg | − |
| neg × pos | − |
| neg × neg | + |
| (Division is identical.) |
Worked examples
Example 1: -7 + 12 = 5
Start at -7, move 12 right, end at 5.
Example 2: -3 - 5 = -8
Start at -3, move 5 left, end at -8.
Example 3: 4 - (-6) = 10
Double negative cancels: 4 + 6 = 10.
Example 4: -2 × -8 = 16
Same signs → positive. 2 × 8 = 16.
Example 5: -15 ÷ 3 = -5
Different signs → negative. 15 ÷ 3 = 5.
Example 6: -20 ÷ -4 = 5
Same signs → positive. 20 ÷ 4 = 5.
The four slip-prone moves
Slip 1: Forgetting a leading negative
❌ -3 + 5 = -8 (subtracted instead of added)
✅ -3 + 5 = 2
Fix: read the sign of the first number carefully — -3 starts on the left of 0.
Slip 2: Subtract-a-negative not cancelled
❌ 5 - (-2) = 3
✅ 5 - (-2) = 5 + 2 = 7
Fix: rewrite as + before computing.
Slip 3: Sign error in multiplication chain
❌ -2 × 3 × -4 = -24 (lost track of signs)
✅ (-2) × 3 = -6; then (-6) × (-4) = +24
Fix: do one multiplication at a time and track the sign.
Slip 4: Dropping the sign on the answer
❌ Final answer 5 when it should be -5
✅ Write the sign every time, even on simplified steps.
In equations
x - 5 = -3 → add 5 to both sides → x = -3 + 5 = 2.
-2x = 10 → divide both sides by -2 → x = 10 ÷ -2 = -5.
This is where careless sign-handling costs marks. See solving linear equations.
In algebra: -3(x - 2)
-3 × x = -3x
-3 × -2 = +6
Result: -3x + 6
The second term flips to positive because -3 × -2 = +6. See expanding brackets.
How Professor Pi teaches this
Pi's hint ladder for -3 × -4:
- L1: "What do we know about same-sign multiplication?"
- L2: "Same signs give a positive answer. What is 3 × 4 (ignoring the signs)?"
- L3: "Right, 12. Now apply the sign — same signs → positive."
- L4: worked twin
-2 × -5 = +10, your turn.
FAQ
Why is a negative times a negative positive?
Multiplication is repeated movement. Negative times negative reverses both the direction and the count — two reversals cancel back to forward.
How do I subtract a negative number?
Subtracting a negative is adding a positive. 5 − (−3) = 5 + 3 = 8.
What's the difference between a minus sign and a negative sign?
Position. Minus sits between two numbers (5 − 3). Negative sits before one (−3). The maths is the same; the distinction helps avoid sign-flip slips.
Related reading
Pedagogy from Professor Pi at aitutors.me. Updated 20 May 2026.