Fact Families: Addition and Subtraction

What is a fact family?

A fact family is a small group of three numbers that belong together, just like people in a real family. From those three numbers you can build four number facts: two additions and two subtractions.

Take the numbers 3, 7 and 10. Here is their whole family:

• 3 + 7 = 10
• 7 + 3 = 10
• 10 − 3 = 7
• 10 − 7 = 3

The two big numbers are the parts (3 and 7), and the largest number is the whole (10). Parts join to make the whole; the whole breaks back into its parts.

Why addition and subtraction live together

Addition and subtraction are opposites. Subtraction undoes addition. If you add 7 to 3 and get 10, then taking 7 away from 10 brings you straight back to 3. Because they reverse each other, they belong in the same family.

This is the why behind the whole idea: the three numbers never change, you just arrange them in different ways. The biggest number always sits alone in a subtraction (it is the one you take from) and always sits at the end of an addition (it is the total).

If you would like to see the opposite-operations idea on its own, our lesson on checking with the inverse builds on this directly.

Reading a fact family table

Let's lay out a few families side by side.

Part	Part	Whole	The four facts
2	5	7	2+5=7, 5+2=7, 7−2=5, 7−5=2
4	6	10	4+6=10, 6+4=10, 10−4=6, 10−6=4
8	5	13	8+5=13, 5+8=13, 13−5=8, 13−8=5

Notice the pattern: the two parts can swap in the additions, and the whole goes first in both subtractions.

Worked example 1: building the family

The numbers are 6, 9 and 15. Write the whole family.

• Additions (parts first): 6 + 9 = 15 and 9 + 6 = 15
• Subtractions (whole first): 15 − 6 = 9 and 15 − 9 = 6

Four facts, all from three numbers.

Worked example 2: finding a missing number

A family is 8, ☐ and 12. What is the missing number?

The whole is 12 and one part is 8. Use a subtraction fact: 12 − 8 = 4. So the missing number is 4, and the full family is 8 + 4 = 12, 4 + 8 = 12, 12 − 8 = 4, 12 − 4 = 8.

Worked example 3: a family with a double

When two numbers are the same, the family is smaller. Take 5, 5 and 10:

• 5 + 5 = 10
• 10 − 5 = 5

Swapping the parts gives the same addition, and both subtractions are identical, so this family has only two different facts.

Why fact families matter

Fact families show that maths is connected, not a list of separate sums to memorise. Once you spot the family behind a fact, you can answer related questions instantly and check your own work by reversing an operation. This habit pays off later in algebra, where undoing operations is the key to solving equations.

Try it yourself

You will need three small hoops or drawn circles and some counters.

1. Put a pile of counters in two of the hoops, say 6 and 4.
2. Tip them both into the third hoop and count the total: 10.
3. Say all four facts aloud while pointing at the hoops.
4. Now start with the whole hoop and split it into two parts a new way.
5. Challenge: Write a family on a card but hide one number. Can a partner find it?

What's next?

Fact families open the door to faster, smarter arithmetic. Try number bonds to 20 to build more families, then practise both operations together in addition and subtraction.