Operations with Negative Numbers

Starting at −3 on a number line, adding −4 means moving 4 more places to the left, landing on −7. Both numbers are negative, so combining two debts gives a bigger debt — if you owe £3 and then owe £4 more, you now owe £7 in total.

Some students believe “two negatives make a positive” and incorrectly get +7. That rule applies to multiplication and division only — not to addition. Adding two negative numbers always gives a more negative result.

Subtracting a negative is the same as adding its positive equivalent. Think of removing a debt: if your balance is £5 and a £3 debt is cancelled, your balance rises to £8. So \(5 – (-3) = 5 + 3 = 8\).

On a number line, subtracting −3 means moving 3 places to the right from 5, arriving at 8. This is why “minus a minus” always gives a larger result than the starting number.

Look at the pattern: \(-2 \times 3 = -6\), \(-2 \times 2 = -4\), \(-2 \times 1 = -2\), \(-2 \times 0 = 0\). Each time the second number decreases by 1, the answer increases by 2. Continuing: \(-2 \times (-1) = 2\), \(-2 \times (-2) = 4\), \(-2 \times (-3) = 6\).

Alternatively, think of it as reversing a debt: if you cancel 3 lots of a £2 debt, you’re £6 better off. Removing negatives makes things positive. The pattern argument and the context argument both point to the same conclusion: negative × negative = positive.

Division asks “what do I multiply by to get here?” We need a number that satisfies \((-4) \times ? = -12\). Since \((-4) \times 3 = -12\), the answer is 3. The sign rule for division is the same as for multiplication: negative ÷ negative = positive.

Students often expect a negative answer because both numbers are negative. But dividing two negatives produces a positive result, just like multiplying them. Think of it as splitting a £12 debt equally among 4 people who all cancel their debts — each person comes away £3 better off.

Example: \((-3) \times (-2) = 6\)

Another: \(-4 + 7 = 3\)

Creative: \((-10) \div (-2) = 5\) — using division rather than the addition and multiplication in the other examples. Or \(0 – (-1) = 1\) — subtracting a negative from zero.

Trap: \((-5) + (-3)\) — a student who believes “two negatives make a positive” might think this gives +8. But adding two negatives gives a negative: \((-5) + (-3) = -8\).

Example: \(5 – (-3) = 8\) — the answer 8 is greater than both 5 and −3.

Another: \(2 – (-6) = 8\) — greater than both 2 and −6.

Creative: \((-1) – (-100) = 99\) — a huge answer from two negative numbers. Or \(0 – (-0.001) = 0.001\), greater than both 0 and −0.001.

Trap: \(10 – 3 = 7\) — the answer 7 is greater than 3, but not greater than 10. For the answer to exceed both numbers, you need to subtract a negative.

Example: 3 and −5. Sum = −2 (negative). Product = −15 (negative).

Another: 1 and −4. Sum = −3. Product = −4.

Creative: 100 and −101. Sum = −1. Product = −10100. The sum is only just negative, but the product is hugely negative.

Trap: −2 and −3. Sum = −5 (negative ✓). But product = (−2) × (−3) = +6 (positive ✗). A student might think “both negative means everything is negative” — but negative × negative = positive.

Example: \(a = -2\). Then \(a^2 = 4 > -2\) ✓ and \(a^3 = -8 < 4\) ✓.

Another: \(a = -1\). Then \(a^2 = 1 > -1\) ✓ and \(a^3 = -1 < 1\) ✓.

Creative: \(a = -0.5\). Then \(a^2 = 0.25 > -0.5\) ✓ and \(a^3 = -0.125 < 0.25\) ✓. In fact, any negative number works — the square is always positive (greater than the original), and the cube is always negative (less than the positive square).

Trap: \(a = 2\). Then \(a^2 = 4 > 2\) ✓ — but \(a^3 = 8 > 4\), so the second condition fails. Students might check only the first condition and assume the second follows automatically.

Every negative number sits to the left of zero on a number line, and every positive number sits to the right. Since left means smaller, any negative number is always less than any positive number — even −0.001 is less than 0.001.

Students sometimes doubt this when the negative number “looks bigger,” e.g. they may feel −100 should be bigger than 2 because 100 is bigger than 2. But magnitude and value are different things: −100 has a large magnitude but a very small value. A useful check: “would you rather owe £100 or have £2?”

True when subtracting a positive number: \(10 – 3 = 7\), and 7 < 10. This matches the intuition that “subtraction makes things smaller.”

But false when subtracting a negative number: \(10 – (-3) = 13\), and 13 > 10. It also fails when subtracting zero: \(10 – 0 = 10\), which is equal, not less. The presence of negative numbers (and zero) breaks the primary-school rule that subtraction always reduces a value. This is one of the most common assumptions students carry from KS2 into KS3.

\((-3)^2 = (-3) \times (-3) = 9\), which is positive. \((-0.1)^2 = 0.01\), also positive. Squaring any negative number always gives a positive result, because negative × negative = positive.

Students often confuse \((-3)^2\) with \(-3^2\). The first means “negative three, all squared” = 9. The second means “the negative of three squared” = −9. The brackets make all the difference.

If \(a = 5\), then \(-a = -5\) (negative ✓). But if \(a = -3\), then \(-a = -(-3) = 3\) (positive ✗). And if \(a = 0\), then \(-a = 0\), which is neither positive nor negative.

The symbol \(-a\) doesn’t mean “a negative number” — it means “the opposite of \(a\)”, which could be positive, negative, or zero depending on the value of \(a\). This is one of the most persistent misconceptions in algebra.

The student is applying the “two negatives make a positive” rule to addition, where it doesn’t belong. This rule only applies to multiplication and division. When adding two negative numbers, the result is always negative.

The correct answer is \((-6) + (-2) = -8\). Think of combining two debts: owing £6 and then owing £2 more means you owe £8 in total — a bigger debt, not a positive balance.

The answer is correct — \((-5) \times (-4)\) does equal 20 — but the reasoning is dangerously wrong. The student says the negatives “cancel out”, which suggests they are simply ignoring the signs rather than applying the sign rules for multiplication.

This strategy would give the wrong answer for \((-5) \times 4\), where the signs don’t “cancel” but the student’s approach would still discard the negative and get 20 instead of the correct answer −20. The correct reasoning is: negative × negative = positive (a specific rule), then multiply the values. Getting the right answer doesn’t mean the method is sound.

The student has collapsed the double negative incorrectly, treating \(7 – (-5)\) as simply \(7 – 5\). They saw the subtraction sign and the negative sign and merged them into a single subtraction, losing one of the two operations.

Subtracting a negative means the operation changes to addition: \(7 – (-5) = 7 + 5 = 12\), not 2. A useful check: subtracting a negative should always give a result larger than the starting number, but the student’s answer of 2 is smaller than 7.

The student ordered the negative numbers by the size of the digit, as if they were positive. They treated the ordering as 1, 2, 7 and placed them in that order. But with negative numbers, a larger digit means further from zero — which means a smaller value.

−7 is the smallest (furthest left on a number line), not −1. The correct order from smallest to largest is −7, −2, −1, 3. A number line is the best tool for checking: the further left a number sits, the smaller it is.