Adding Integers: Why Minus Plus Minus Is Not Plus

Negative numbers break for one reason: children are taught rules about signs before they are given a picture of what a negative number is. Give them the picture and most of the rules become unnecessary.

Picture one: the number line. Numbers live on a line. Adding a positive means walking right; adding a negative means walking left. So −3 + 5 is "start at −3, walk 5 right" = 2. There is no rule to remember, just a direction.

Picture two: money and debt. +7 is seven dollars in your pocket; −7 is seven dollars you owe. −4 + 9 is "you owe 4, you earn 9" — you clear the debt and have 5 left. Children who resist the number line usually accept this one instantly.

The one genuinely confusing case: subtracting a negative. 6 − (−2) = 8. In the debt model: someone cancels a debt of 2, so you are 2 better off. Say it that way once and it stops being magic.

What to practise, in order. First, ordering: which is bigger, −8 or −3? (−3 — it is further right.) Second, adding a negative. Third, subtracting a positive that takes you below zero. Only then, subtracting a negative. Mixing all four on day one guarantees confusion.

Where children actually meet this. Temperature below zero, floors below ground in a lift, a submarine's depth, a bank balance. Every one of those is a number line stood on its end, and it is worth pointing at all four in real life rather than only on paper.

Next: ordered pairs (where negatives create the four quadrants), function machine games, math word problems and games for 5th graders.