Adding and Subtracting Negative Numbers

Adding two negative numbers means combining two debts. If you owe £3 and then owe another £5, you now owe £8. On a number line, start at \( -3 \) and move 5 places to the left, landing on \( -8 \).

Students often claim “two negatives make a positive” and write \( -3 + (-5) = 8 \). But that rule applies to multiplication, not addition. When you add negative numbers, you move further left (further into the negatives), so the result is always more negative than either starting number. A quick check: if the answer were \( +8 \), adding two debts would somehow create a profit — that makes no sense.

Subtracting a negative is the same as adding its positive. Think of it as removing a debt: if your account stands at £5 and a £3 debt is cancelled, you gain £3, taking your balance to £8.

Visualising with Counters: To subtract \(-3\) from \(5\), you need to remove 3 negative counters. Since we only have positive ones, we add 3 “Zero Pairs” first. When we take away the 3 negatives, we are left with 8 positives.

Whether using a number line or debt, removing something negative always results in a larger value.

Start at \( -4 \) on the number line and move 7 places to the right. You pass zero and land on \( +3 \). Here, the positive number (7) has a greater magnitude than the negative (4), so the result crosses zero and lands positive.

A context helps: if the temperature is \( -4 \)°C and rises by 7°C, it ends at 3°C — clearly above zero, not below. This demonstrates that adding a positive always “moves right,” potentially out of the negatives.

Starting at \( -2 \) on the number line, subtracting 3 means moving 3 more places to the left: \( -2 \to -3 \to -4 \to -5 \). When you subtract a positive number from a negative number, the result is always further from zero in the negative direction.

Check with the reverse: \( -5 + 3 = -2 \). This proves that \( -5 \) is 3 units to the left of \( -2 \).

Example: \( -4 + (-6) = -10 \)

Another: \( -1 + (-9) = -10 \)

Creative: \( -2.5 + (-7.5) = -10 \) — using decimals. Or \( -0.01 + (-9.99) = -10 \).

Trap: \( 5 + (-15) = -10 \). While this sum is correct, the number 5 is positive. The question specifically requires two negative numbers.

Example: \( 3 \; – \; (-5) = 8 \)

Another: \( -2 \; – \; (-7) = 5 \)

Creative: \( -99 \; – \; (-100) = 1 \) — subtracting a large negative gives a small positive result.

Trap: \( -3 \; – \; (-2) = -1 \). Students often think subtracting any negative number makes the answer positive, but if the starting number is “more negative” than the amount added, the answer stays negative.

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

Another: \( -10 \; – \; (-3) = -7 \)

Creative: \( 0 \; – \; 7 = -7 \) or \( 100 + (-107) = -7 \).

Trap: \( 3 \; – \; (-4) = 7 \). A student might write this thinking “the negative sign makes the answer negative,” but the operation actually increases the starting value.

Insight: \( n \; – \; (-n) = 2n \). We need \( 2n > 10 \), so \( n > 5 \).

Example: \( n = 6 \): \( 6 \; – \; (-6) = 12 \)

Creative: \( n = 5.01 \): \( 5.01 \; – \; (-5.01) = 10.02 \).

Trap: \( n = -6 \). A student might think 6 being bigger than 5 is enough, but \( -6 \; – \; (-(-6)) = -6 \; – \; 6 = -12 \), which is not greater than 10.

Adding two negative numbers always moves you further left on the number line. If you owe money and then owe more money, your total debt must be larger. There is no pair of negative numbers whose sum is zero or positive.

It depends on the magnitude. If the negative number is further from zero (\( 3 + (-7) = -4 \)), the result is negative. If the positive number is further (\( 7 + (-3) = 4 \)), it’s positive. If they are equal (\( 5 + (-5) = 0 \)), it’s zero.

This depends on the starting number. If you start at 5 and add \( -2 \), you land on 3, which is closer to zero. If you start at \( -5 \) and add \( -2 \), you land on \( -7 \), which is further from zero. This challenges the idea that “adding negatives” always increases magnitude.

Subtracting a positive number makes the result smaller (\( 10 \; – \; 3 = 7 \)). But subtracting a negative number makes it larger (\( 10 \; – \; (-3) = 13 \)). This targets the primary-level misconception that subtraction always reduces value.

The student has applied a multiplication rule to an addition problem. The rule “a negative times a negative equals a positive” only applies to multiplication and division.

Correction: When adding two negatives, you are combining two “piles” of negative value. Starting at \( -3 \) and moving 4 more places into the negative side gives \( -7 \). If you owe £3 and add a debt of £4, you definitely don’t have £7 profit!

The answer is correct, but the reasoning is a “dangerous shortcut.” “Ignoring signs” fails when both numbers are negative.

Example of failure: If the student used this rule for \( -8 \; – \; (-3) \), they might write \( 8 + 3 = 11 \). But the correct answer is \( -5 \). The proper reasoning is that subtracting a negative is the same as adding a positive because removing a debt increases your value.

The student is treating the calculation as \( 5 \; – \; 3 \), then applying a sign. They are ignoring the direction of movement on the number line.

Correction: \( -5 \; – \; 3 \) means starting at \( -5 \) and moving 3 more units to the left. This lands on \( -8 \). The student’s answer of \( -2 \) would only be correct for \( -5 + 3 \).

The student correctly calculated the first part (\( 7 \)), but committed the “brackets are just for show” error. They ignored the negative sign inside the brackets.

Correction: \( 7 \; – \; (-4) \) is the same as \( 7 + 4 = 11 \). Subtracting negative 4 is a move to the right, increasing the value. The student’s error resulted in an answer that was 8 units too small.