Subtraction
Subtraction answers two questions that turn out to be the same one: how many are left when you take some away, and how far apart two numbers are.
The subtraction tool
Taking away, and the gap between two numbers
12 0 12
Tap a counter to take it away.
How subtraction works
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Decide which question you are answering
Taking away starts with a group and removes some of it: twelve counters, four taken away, how many are left? Finding the difference compares two amounts and measures the space between them: how much more is 12 than 7? Both are subtraction, and both give the same number.
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Count back one step at a time
To find 12 − 3, count down three steps: 11, 10, 9. Count the steps, not the starting number — 12 is where you stand, not the first step back. Put a finger down on each step so you can check you took exactly three.
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Switch to counting on when it is faster
Counting back seven times from 12 is slow and easy to lose track of. Ask instead: what do I add to 7 to reach 12? From 7 to 10 is 3, from 10 to 12 is 2, so 5. The difference is 5, so 12 − 7 = 5. This is why knowing the sums to twenty makes subtraction easy.
Worked examples
12 − 3
Count back three steps from 12: 11, 10, 9, so the answer is 9. The common slip is to count 12 as the first step and land on 10 — you take three steps, not four.
12 − 7
Seven steps back is a long walk. Count on instead: 7 to 10 is 3, 10 to 12 is 2, and 3 + 2 = 5. So 12 − 7 = 5, and 7 + 5 = 12 checks it.
The fact family of 8, 9 and 17
Three numbers give four sentences: 8 + 9 = 17, 9 + 8 = 17, 17 − 9 = 8, 17 − 8 = 9. Learn the family and every question about those numbers becomes one you already know.
Where it goes wrong
- Counting the starting number as a step For 12 − 3, a child counts 12, 11, 10 and answers 10 — one step too few, because 12 is where the count starts rather than a step back from it. Count the jumps, not the numbers: three jumps from 12 land on 9.
- Counting back when the fact is already known Counting back in ones works, but it is slow, and slow counting is where errors creep in. Once the sums to twenty are secure, 12 − 7 should be answered by thinking 7 + 5 = 12, not by counting seven fingers backwards.
- Losing the ten Working on the ones and forgetting the ten underneath: 12 − 3 answered as 7, because 10 − 3 was calculated and the extra 2 in 12 was dropped. Read the whole number, then subtract — the frame and the number line on this page both show the ten staying put.
Questions
How do I teach counting back without the off-by-one mistake?
Make the steps physical and separate from the number you start on. Put a counter on 12 on a number line and move it three jumps to the left, counting the jumps rather than the squares it lands on. Saying "twelve is where I stand, then one, two, three" gives the mouth and the hand the same job, and the mistake largely disappears.
Can subtraction be done in either order, like addition?
No, and it is worth saying plainly: order matters. 12 − 7 = 5, but 7 − 12 is a number below zero, which is a different idea you will meet much later. Subtraction always answers "how many remain" or "how far apart", and neither question changes if you swap the numbers around.
What is the difference between take-away and difference?
They are the same operation seen two ways. Taking away starts with a group and removes some, leaving 5 from 12. Finding the difference measures the gap between 7 and 12, which is also 5. The number line on this page shows both at once, and children who can see the two pictures stop treating subtraction as a rule to memorize.
What is a fact family, and why does it help?
A fact family is the set of four sentences that use the same three numbers, such as 7 + 5 = 12, 5 + 7 = 12, 12 − 7 = 5, and 12 − 5 = 7. Learning subtraction this way means every subtraction you meet is one you may already know from addition — and the number of facts to remember halves.
When is a child ready to move on from subtraction?
When sums within twenty come back instantly and can be used backwards without counting. If 8 + 5 = 13 is automatic, then 13 − 5 is close to automatic too. A child who needs fingers for facts within ten is still building those facts, and that is worth finishing before multiplication starts asking for the same facts in a new shape.