Fact families for addition, subtraction, multiplication and division

Quick answer

A fact family is a set of facts made from the same three numbers. From 5, 8 and 13 you get 5 + 8 = 13, 8 + 5 = 13, 13 − 8 = 5 and 13 − 5 = 8. Multiplication works the same way (3 × 4 = 12, 12 ÷ 4 = 3). Kids learn addition families in 1st grade and multiplication families in 3rd.

Fact families are one of the most useful ideas in early math, and one of the quickest to teach. They show children that addition and subtraction (and later multiplication and division) aren't separate sets of facts to memorize. They are the same facts, read in different directions.

What is a fact family?

A fact family is a set of facts made from the same three numbers. For addition and subtraction, two numbers are the parts and the biggest number is the whole. With 5, 8 and 13, the parts are 5 and 8 and the whole is 13.

Every family follows the same pattern: the two addition facts start with the parts, and the two subtraction facts start with the whole. That's the one rule worth repeating: in subtraction, the whole comes first.

When kids learn fact families

Here is how the Common Core standards, used by most states, build the idea. Your state or school may differ.

Addition and subtraction fact families

Start with objects. Put 4 red counters and 6 blue counters in a row. How many in all? 10. Swap the colors' order: 6 + 4 is still 10. Now cover the blue ones: 10 − 6 = 4. Cover the red ones: 10 − 4 = 6.

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• 4 + 6 = 10
• 6 + 4 = 10
• 10 − 6 = 4
• 10 − 4 = 6

The big payoff is subtraction. Common Core asks 1st graders to see subtraction as a missing-addend problem: to find 10 − 8, ask what number makes 10 when added to 8 (1.OA.B.4). So for 14 − 6, a child who knows 6 + 8 = 14 can answer 8 straight away, with no counting back. Our guide on how to teach subtraction shows how this “think addition” strategy fits with the others.

Multiplication and division fact families

In 3rd grade the same idea returns. Now the three numbers are two factors and a product. An array shows the whole family at once: 3 rows of 4 is 12, and 4 columns of 3 is also 12. Split the 12 dots into 3 equal rows and each row has 4; split them into 4 columns and each has 3.

• 3 × 4 = 12
• 4 × 3 = 12
• 12 ÷ 4 = 3
• 12 ÷ 3 = 4

Division becomes “think multiplication.” Common Core's own example: find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8 (3.OA.B.6). A child who knows 4 × 8 = 32 knows the answer is 4. That's why times tables come first; see our guides on teaching multiplication and teaching division.

Families with only two facts

When both parts are the same number, swapping them doesn't create a new fact. So doubles make families of just two: 6 + 6 = 12 and 12 − 6 = 6. Square numbers do the same in multiplication: 5 × 5 = 25 and 25 ÷ 5 = 5.

Zero is a special case for multiplication. 0 × 5 = 0 and 5 × 0 = 0, and 0 ÷ 5 = 0, but there's no fourth fact because you can't divide by 0.

How to make and use fact family triangle cards

Triangle flash cards are the classic fact family tool, and you can make a set in ten minutes.

Step 1: Make the cards

Cut triangles from index cards. Write the whole (the sum, or the product for ×/÷) in the top corner, and mark it with a dot or a circle. Write the two parts in the bottom corners. Start with one small group, such as the families that make 10.

Step 2: Cover one corner

Hold a card with your thumb over one number. Your child names the hidden number and says the fact: cover 13 and they say “5 + 8 = 13”; cover the 8 and they say “13 − 5 = 8.” Covering a bottom corner is the subtraction (or division) practice.

Step 3: Sort and review

Make two piles: “I know it” and “still thinking.” Come back to the second pile a day or two later. Quick self-quizzing and spacing practice out over days are both recommended by the U.S. Institute of Education Sciences in its guide to organizing instruction and study.

Fact family houses and games

• Fact family houses. Draw a house with a triangle roof and four windows. Write the three numbers in the roof; your child writes one fact in each window. Doubles houses only need two windows, a fun pattern to spot.
• Domino families. Pick a domino. The two halves are the parts and the total is the whole. Say or write all the facts. A double domino, like 4 and 4, makes a two-fact family.
• Dice families. Roll two dice, add them and write the family. For ×/÷, multiply the dice instead.
• Fact family sort. Write facts from three or four families on sticky notes, mix them up and sort them into their families.

Common mistakes (and how to help)

• Starting subtraction with a part. A child writes 5 − 13 = 8. Remind them the whole always comes first in subtraction and division, and point to the top of the triangle.
• Adding the parts in a subtraction fact. 13 − 5 = 18. Act it out: start with 13 counters and take 5 away. Can there be more than you started with?
• Mixing in a stranger. Writing 13 − 3 = 10 in the 5, 8, 13 family. Every fact must use exactly the three family numbers.
• Assuming order doesn't matter in division. 12 ÷ 4 = 3, but 4 ÷ 12 is not 3. Turnarounds only work for + and ×.

Practice with missing numbers

Each problem hides one number from a family. Ask your child which fact they used to find it.

Fact family practice (with answers)

1. 7 + ? = 12
2. 12 − 7 = ?
3. ? + 6 = 15
4. 15 − ? = 9
5. 9 + 4 = 13, so 13 − 4 = ?
6. 14 − ? = 7
7. Write the fact family for 2, 9 and 11
8. 3 × ? = 15
9. 15 ÷ 3 = ?
10. ? × 6 = 24
11. 24 ÷ 6 = ?
12. 7 × 8 = 56, so 56 ÷ 8 = ?

Show answers

For more practice, print our subtraction worksheets and use the related addition facts to check answers. To see where fact families fit in the year, visit our 1st grade and 3rd grade skills checklists.

Frequently asked questions

What is a fact family in math?

A fact family is a group of related math facts that all use the same three numbers. For addition and subtraction, two numbers are parts and the biggest is the whole: 3 + 7 = 10, 7 + 3 = 10, 10 − 7 = 3 and 10 − 3 = 7. For multiplication and division, two numbers are factors and the biggest is the product.

What grade learns fact families?

Addition and subtraction fact families are usually taught in 1st grade, when Common Core asks children to understand subtraction as an unknown-addend problem (1.OA.B.4) and find missing numbers in equations (1.OA.D.8). Multiplication and division fact families come in 3rd grade, with division as an unknown-factor problem (3.OA.B.6). Your state or school may differ.

How do you explain fact families to a 1st grader?

Use real objects. Put 5 red blocks and 3 blue blocks together and ask how many in all (8). Turn the line around: 3 + 5 is still 8. Then take the blue blocks away: 8 − 3 = 5. Take the red ones away instead: 8 − 5 = 3. Same three numbers, four facts, one family.

What are fact families for multiplication and division?

They are sets of facts built from two factors and their product. For 4, 6 and 24: 4 × 6 = 24, 6 × 4 = 24, 24 ÷ 6 = 4 and 24 ÷ 4 = 6. An array of 4 rows of 6 dots shows all four facts at once. Knowing the multiplication fact gives you both division facts for free.

Why do some fact families have only two facts?

When the two parts (or factors) are the same number, swapping them gives the same fact. So the family for 6, 6 and 12 is just 6 + 6 = 12 and 12 − 6 = 6. Square numbers do the same thing in multiplication: 5 × 5 = 25 and 25 ÷ 5 = 5.

How do fact families help with subtraction?

They let children solve subtraction with addition facts they already know. Instead of counting back for 14 − 6, a child thinks “6 plus what makes 14?” and remembers 6 + 8 = 14, so the answer is 8. Common Core describes this as understanding subtraction as an unknown-addend problem.

Sources

• Common Core State Standards: Grade 1, Operations & Algebraic Thinking — CCSSI
• Common Core State Standards: Grade 2, Operations & Algebraic Thinking — CCSSI
• Common Core State Standards: Grade 3, Operations & Algebraic Thinking — CCSSI
• Organizing Instruction and Study to Improve Student Learning (practice guide) — Institute of Education Sciences, What Works Clearinghouse

Written by The Mathing Team for parents and teachers. Grade expectations follow the Common Core State Standards for Mathematics; your state or school may sequence skills a little differently.