Fact Families

Three numbers that belong together make a little team called a fact family. Give me the team, and I can write four true number sentences from it — without doing any new work. The three numbers just take turns.

Take the team 3, 4 and 7. Two of them add up to the third, so:

3 + 4 = 7 \qquad 4 + 3 = 7

7 - 4 = 3 \qquad 7 - 3 = 4

Four facts, one family. You already met this idea when you learned that subtraction undoes addition — a fact family is that idea packed into a single, tidy picture.

The adding-and-subtracting family

The clearest picture of an add/subtract family is the part–whole bar: one long bar is the whole, split into two parts. Put the parts together to add; take one part off the whole to subtract. The same three numbers do all four jobs.

Press Refresh for a fresh family, and step through its four facts:

Notice the two adding facts are the same sum written both ways, and the two subtracting facts each start from the whole and peel off a part. The whole is always the biggest number in the family — it is what you build up to, and what you take away from.

The multiplying-and-dividing family

The very same trick works for multiplying and dividing. Arrange the dots in a rectangle — an array — and one picture again gives four facts. The rows and columns are the two factors; the total is their product.

A 3 \times 4 array has 12 dots, so:

3 \times 4 = 12 \qquad 4 \times 3 = 12

12 \div 4 = 3 \qquad 12 \div 3 = 4

Same shape of family, different operations: two multiplying facts (the array counted by rows or by columns) and two dividing facts (share the total into rows, or into columns). Here the product is the biggest number — the multiplying twin of "the whole is biggest".

This is why fact families make learning your times tables so much lighter. The moment you know 6 \times 7 = 42, you also know 7 \times 6 = 42, 42 \div 6 = 7 and 42 \div 7 = 6 — four facts memorised for the price of one. There are far fewer families to learn than there are separate facts.

The superpower: find the missing number

A fact family turns every "fill in the blank" into an easy question, because the family always tells you which fact to use. If a part is missing, subtract; if the whole is missing, add. If a factor is missing, divide; if the product is missing, multiply.

Worked examples

\square + 7 = 15. The whole 15 and a part 7 are known, so use the family's subtraction fact: 15 - 7 = 8. The blank is 8.
13 - \square = 5. The whole is 13; one part is 5; the missing part is 13 - 5 = 8.
\square \times 6 = 42. The product 42 and a factor 6 are known, so divide: 42 \div 6 = 7.
56 \div \square = 8. The family is 7, 8, 56, so the missing number is 7 — check it: 56 \div 7 = 8. ✓

The four facts share the same three numbers. If you write 7 - 3 = 5 you have wandered outside the family 3, 4, 7 — the numbers no longer match. Always check the three numbers are the ones you started with.
Don't mix the two families. A trio like 3, 4, 7 makes an add/subtract family (because 3 + 4 = 7); it does not also give 3 \times 4 = 7. Multiply/divide families come from arrays, where the biggest number is the product — for 3 and 4 that trio is 3, 4, 12.