Subtraction that makes sense
Maths · Key Stage 1 · The School
Two meanings, one symbol
The − sign says two different stories. TAKE AWAY: I had 8 sweets, I ate 3, how many left? DIFFERENCE: Sara has 8, Ali has 3, how many more does Sara have? Both are 8 − 3 = 5. Knowing which story you are in tells you how to picture it, and picturing it correctly is most of the work.
Count back or count on
For 12 − 3, counting BACK is quick: 11, 10, 9. For 12 − 9, counting back nine times is slow and error-prone — count ON from 9 instead: 10, 11, 12, that is 3 steps, so the answer is 3. Choosing the shorter journey is not cheating; it is what good mathematicians always do.
Addition checks subtraction
Subtraction and addition undo each other. If 15 − 7 = 8, then 8 + 7 must be 15. Always check that way — it catches almost every slip. These pairs are called fact families: 8, 7 and 15 make 8+7=15, 7+8=15, 15−7=8, 15−8=7. Learn one family and you get four facts free.
TAKE AWAY: 12 sweets, you eat 9, how many left? Count back from 12 — eleven, ten, nine, eight, seven, six, five, four, three. Three sweets left. DIFFERENCE: Sam has 12 stickers, Ali has 9, how many MORE does Sam have? Nobody eats anything. Count ON from 9 to 12 — ten, eleven, twelve, that is three. Both are 12 − 9 = 3. Same subtraction, two very different stories, and counting on is the easier route for the second.
Two questions that both say "take away"
Subtraction answers two different questions and it helps to know which you have. "I had 12 sweets and ate 5" is REMOVAL — something has gone. "Ali has 12 and Sara has 5, how many more does Ali have?" is DIFFERENCE — nothing has gone anywhere, you are comparing two amounts. Both are 12 − 5, and children who have only met removal get stuck on the second because nothing is being taken away. Drawing them differently helps: crossing out for removal, two bars side by side for difference.