Fact family in math: what it means and how to teach it at home

A fact family is a group of addition and subtraction facts that use the same three numbers. The numbers 3, 4 and 7 make four facts: 3 + 4 = 7, 4 + 3 = 7, 7 - 3 = 4 and 7 - 4 = 3. Once a child sees how the facts connect, one fact helps them find the other three.

What is a fact family, with real objects?

Put 7 raisins on a plate, 3 on the left and 4 on the right. You can tell four true stories about them.

1. 3 raisins and 4 raisins make 7 raisins: 3 + 4 = 7.
2. 4 raisins and 3 raisins make 7 raisins: 4 + 3 = 7.
3. Eat the 3 on the left, and 4 are left: 7 - 3 = 4.
4. Eat the 4 on the right, and 3 are left: 7 - 4 = 3.

All four stories use the same raisins: the whole plate and its two piles. Use counters, buttons or cereal instead if raisins are not handy, and keep small items away from children under three.

“Fact family” is a classroom word; the US Common Core standards describe the idea without using the term. The Grade 1 standards ask children to use “the relationship between addition and subtraction,” and give the example that when you know 8 + 4 = 12, you know 12 - 8 = 4. Your child’s teacher may use “related facts” or “fact family” for the same thing.

What is a fact triangle?

A fact triangle is a triangle with the whole at the top and the two parts at the bottom corners. Cover one corner with a finger and ask for the hidden number.

1. Cover the 7: “3 and 4 together make what?” The answer is 7, so this is addition.
2. Cover the 3: “7 take away 4 leaves what?” The answer is 3, so this is subtraction.
3. Cover the 4: “7 take away 3 leaves what?” The answer is 4.
4. Write the four equations next to the triangle once the answers come easily.

You can draw a triangle on paper in a minute. Make a few with different numbers, such as 2, 6 and 8, or 5, 4 and 9, and keep them in an envelope for car rides and waiting rooms. Always put the largest number, the whole, at the top. Then covering the top corner always means adding, and covering a bottom corner always means taking away.

Why does a fact family help with subtraction?

If subtraction feels harder than addition to your child, a fact family gives them a way in: to solve 12 - 8, think “what do I add to 8 to make 12?” The 4 in 8 + 4 = 12 is the answer.

The Grade 1 standards state this directly. They ask children to “understand subtraction as an unknown-addend problem,” with the example of finding 10 - 8 by “finding the number that makes 10 when added to 8.” A child who knows 8 + 2 = 10 already knows 10 - 8 = 2, so they do not need a separate fact to learn. Our guide on how to teach subtraction covers “think addition” alongside other subtraction methods.

Here is a small routine. Say an addition fact, then ask for the two subtractions.

1. You say: “6 + 3 = 9.”
2. You ask: “So what is 9 - 3?” Let your child answer, and then ask how they know.
3. You ask: “And 9 - 6?”

If your child counts back, that is fine. Keep the triangle nearby and point to the corner so they see that the answer is in the family.

What about doubles and zero?

A double gives a family with only two facts. A family with zero has the usual four, but the facts can look odd at first.

• Doubles: 5, 5 and 10 give 5 + 5 = 10 and 10 - 5 = 5. There are only two different facts, because swapping the order changes nothing.
• Zero: 0, 6 and 6 give 0 + 6 = 6, 6 + 0 = 6, 6 - 0 = 6 and 6 - 6 = 0. Try these with objects: a plate with 6 raisins and an empty plate show why adding zero changes nothing.

Order matters for subtraction but not for addition. The Grade 1 standards give “if 8 + 3 = 11 is known, then 3 + 8 = 11 is also known” as an example of this. You can show it with the raisins: swapping the piles changes where they sit, not how many there are. You cannot swap the numbers in a subtraction, since 7 - 3 and 3 - 7 are different. For now, stick to the larger number first.

How are fact families connected to number bonds?

A number bond is a pair of parts that makes a whole. Each bond is the start of a fact family. The bond 3 and 7 for 10 gives 3 + 7 = 10, 7 + 3 = 10, 10 - 3 = 7 and 10 - 7 = 3. Our guide to number bonds to 10 shows seven ways to practice the bonds themselves, and the facts follow once the bonds are familiar. If addition itself is still shaky, start with how to teach addition, then come back to the family. A free ten frame shows each bond as two colors of counters.

When do schools expect children to know them?

In the US, the Grade 1 standards ask children to add and subtract within 20 and to be fluent within 10. The Grade 2 standards ask children to “fluently add and subtract within 20 using mental strategies” and to “know from memory all sums of two one-digit numbers” by the end of the year. Learning facts in families is one way to get there, because each fact a child knows leads to the others in its family. Our first grade curriculum page lists the adding and subtracting topics that sit around them.

In England, the national curriculum asks Year 1 children to “represent and use number bonds and related subtraction facts within 20,” and Year 2 children to “recognise and use the inverse relationship between addition and subtraction.” Both lines describe the idea behind a fact family.

Later, the same idea comes back with multiplication and division. The Grade 3 standards describe division as an “unknown-factor problem,” so 3 x 4 = 12, 4 x 3 = 12, 12 / 3 = 4 and 12 / 4 = 3 form a family too.

Which games help with fact families?

Both of these free games ask children to see how numbers fit together:

• Make Ten Match: find two cards that make 10. Each pair is the start of a fact family.
• Number Bonds: join the tiles that add to a target, so children split a whole into parts again and again.

After a round, pick one pair your child found and ask for the other three facts in its family. If your child finds these hard after many weeks of calm practice, talk to their teacher.