A child who knows that 3 + 4 = 7 already knows three more facts, without memorising anything new. They just haven't noticed yet. Fact families are how schools help children notice. This guide explains what a fact family is, how to write one, the tricky cases that catch children (and some websites) out, and a few five-minute ways to practise at home.
A fact family is a small group of related number facts that all use the same three numbers. Two of the numbers are the parts, and together they make the third number, the whole.
Take the numbers 3, 4 and 7. The parts are 3 and 4, and the whole is 7. Here is the family:
The two additions put the parts together to make the whole. The two subtractions start with the whole and take one part away, which leaves the other part. Same three numbers, four facts.
If your child has met number bonds to 10, they already know this idea. A number bond shows a whole split into two parts, such as 10 split into 6 and 4. A fact family writes that same picture out as number sentences. It is the part-part-whole idea seen from a different angle.
Children can write any family in three steps:
Let's try 6, 8 and 14. The whole is 14, and the parts are 6 and 8:
Many classrooms draw a family as a triangle. The whole goes in the top corner and the two parts go in the bottom corners. For the family above, 14 sits at the top with 6 and 8 underneath. Cover any one corner with a finger and you have a question: "What's hidden?" If the top is covered, your child adds the parts. If a bottom corner is covered, they subtract from the whole.
In addition, the order of the parts doesn't change the answer: 3 + 4 and 4 + 3 both make 7. Subtraction is different. 7 − 3 = 4, but 3 − 7 does not make 4. The National Curriculum in England asks Year 2 pupils to understand exactly this: addition can be done in any order, but subtraction can't. That is why every subtraction in a family starts with the whole.
You may read that every fact family has exactly four facts. That's true most of the time, but not always.
When the two parts are the same number, swapping them changes nothing. Take 6, 6 and 12:
Writing "6 + 6" a second time with the parts swapped gives the same fact again, so a doubles family has just two different facts. This is a good thing to talk about with your child, because it shows they understand why families usually have four.
Doubles are worth knowing well for another reason. They are anchors for nearby facts: if you know 6 + 6, you can quickly work out 6 + 7. Our near doubles strategy guide shows how.
Families with zero behave like any other family, because the two parts (0 and 9) are different. Take 0, 9 and 9:
Fact families aren't just a worksheet exercise. They change how children think about subtraction.
Many children find subtraction harder than addition. Fact families let them turn one into the other. For 15 − 9, ask: "9 plus what makes 15?" The answer is 6, because 9 + 6 = 15, so 15 − 9 = 6. The US Common Core standard 1.OA.B.4 describes this as understanding subtraction as an unknown-addend problem.
Children are often asked to find a missing number, such as 8 + ? = 13. Fact families make this easy: the whole is 13 and one part is 8, so the missing part is 13 − 8 = 5. Both curricula ask for this. In the US it is standard 1.OA.D.8. In England, Year 1 pupils solve problems like 7 = ? − 9. Here 7 and 9 are the parts, so the missing number is the whole: 16, because 16 − 9 = 7.
A family also gives children a built-in way to check their work. If they think 14 − 6 = 8, they can add back: 8 + 6 = 14, so the answer is right. England's Year 2 curriculum asks pupils to use this inverse relationship to check calculations.
One family covers four facts, so learning families instead of single facts cuts the work by a lot. Once these small facts are secure, children can move on to bigger numbers with the ideas in our speedy subtraction tips.
The same idea works with multiplication and division, usually from around Grade 3 or Year 3. Now the two smaller numbers are factors and the biggest number is the product. Take 4, 6 and 24:
Division then becomes a missing-factor question. To find 32 ÷ 8, ask: "8 times what makes 32?" Since 8 × 4 = 32, the answer is 4. Common Core standard 3.OA.B.6 uses this very example, and standard 3.OA.C.7 points out that knowing 8 × 5 = 40 tells you 40 ÷ 5 = 8.
The special cases look familiar too:
Multiplication families halve the number of facts a child needs to learn: once they know 4 × 6, they get 6 × 4 and two division facts for free. Our guide on how to learn times tables fast covers learning the tables themselves, and the times tables page is a handy place to practise them.
Cut a triangle from paper and write a family on it, with the whole at the top. Cover one corner with your thumb. Your child names the hidden number and then says one fact from the family.
Put 7 small toys on the table. While your child looks away, hide some under a cup. They count the toys still showing, work out how many are hidden, and say the subtraction. If 3 are showing, then 4 are hidden, because 7 − 3 = 4.
Give three numbers and ask, "Do these belong together? Why?" Mix real families with fakes. 5, 6 and 11 are a family, because 5 + 6 = 11. 5, 6 and 12 are not, because the parts make 11, not 12.
On the Practice page, do a short round of Sum (1 digit), then a round of Subtract (1 digit). After each subtraction, ask your child to say the matching addition out loud. For older children, choose Table (till 10) and have them say the two division facts that go with each answer. If they want a game to finish, try a round of Math Balloon Pop.
Writing out four sentences neatly isn't the same as understanding. Look for these signs:
5 + ? = 12, ? − 5 = 7 and 12 − ? = 5. The answers are 7, 12 and 7.A fact family is a group of related facts that use the same three numbers. Two numbers are the parts and the third is the whole. For 3, 4 and 7, the family is 3 + 4 = 7, 4 + 3 = 7, 7 − 4 = 3 and 7 − 3 = 4.
Usually four: two additions and two subtractions, or two multiplications and two divisions. Families where the two parts are the same number, such as 6, 6 and 12, have only two.
Because swapping two equal parts gives the same fact again. For 5, 5 and 10, there is only 5 + 5 = 10 and 10 − 5 = 5.
Addition and subtraction families usually come in Grades 1 and 2 in the US and Years 1 and 2 in England. Multiplication and division families follow in Grade 3 or Year 3, once children know some times tables.
They let children turn a subtraction into an addition they already know. For 15 − 9, they think "9 plus what makes 15?" and get 6, because 9 + 6 = 15. They can also check any subtraction by adding back.
It is the same idea with × and ÷. For 4, 6 and 24, the family is 4 × 6 = 24, 6 × 4 = 24, 24 ÷ 6 = 4 and 24 ÷ 4 = 6. Square numbers such as 7, 7 and 49 have only two facts.
Keep it short and playful. Use paper triangle cards with one corner covered, hide toys under a cup, or play family detective with real and fake families. A quick round of Sum (1 digit) and Subtract (1 digit) on the MathApp Practice page also works well.