What is a fact family?
A fact family is a group of three numbers and the math facts they make together. In fact family math, 3, 9 and 12 belong together because 3 + 9 = 12, so the family is 3 + 9 = 12, 9 + 3 = 12, 12 − 3 = 9 and 12 − 9 = 3: two addition facts and two subtraction facts. The same idea works with multiplication: 4, 5 and 20 give 4 × 5 = 20, 5 × 4 = 20, 20 ÷ 4 = 5 and 20 ÷ 5 = 4. The biggest number is always the whole; the two smaller ones are its parts.
Here are fact family examples children meet early on:
| Numbers | The facts |
|---|---|
| 2, 5, 7 | 2 + 5 = 7 · 5 + 2 = 7 · 7 − 2 = 5 · 7 − 5 = 2 |
| 6, 6, 12 (a double) | 6 + 6 = 12 · 12 − 6 = 6 (only two different facts) |
| 3, 8, 24 | 3 × 8 = 24 · 8 × 3 = 24 · 24 ÷ 3 = 8 · 24 ÷ 8 = 3 |
Fact family triangles and houses
Fact family triangles put the whole in the top corner and the two parts in the bottom corners, with the signs in the middle. Cover one corner with a finger and the other two tell you what is hidden: cover the top and you add (or multiply), cover a bottom corner and you subtract (or divide). That is why a triangle with one number missing is such a good warm-up. Fact family houses put the three numbers in the roof and give each of the four facts its own room, which helps children see that the family always has four members. Both are on this page for fact family addition and subtraction and for multiplication and division.
How to teach fact families
- Start from a part-part-whole picture. A number bond or a ten frame with two colors shows the same three numbers before any facts are written.
- Turn the addition around. 3 + 9 and 9 + 3 make the same whole, so one fact gives a second one for free.
- Read subtraction backwards. 12 − 9 asks "9 and what make 12?", which children already know from the addition.
- Do the same with multiplication. An array of 3 rows of 8 is 24, and the same array split into rows or columns gives 24 ÷ 3 and 24 ÷ 8. The multiplication chart shows each pair in one glance.
| Grade | Fact family skill | Common Core |
|---|---|---|
| 1st grade | Use turn-around facts; see subtraction as a missing addend | 1.OA.B.3, 1.OA.B.4 |
| 2nd grade | Add and subtract within 20 from memory | 2.OA.B.2 |
| 3rd grade | Division as a missing factor; multiply and divide within 100 | 3.OA.B.6, 3.OA.C.7 |
Fact family calculator, worksheets and blank template
Fact family calculator: type three numbers in any order, such as 15, 5 and 3 or 5 7 35, and the tool finds whether they add or multiply to the largest one and lists the facts. Type just two numbers and it treats them as the parts, using the Facts setting to choose addition or multiplication. If the numbers don't make a family, it tells you why. A double such as 6, 6, 12 correctly gives two facts instead of four, and 0 is refused for multiplication because you can't divide by 0.
Fact family worksheets: print 1 to 8 triangles or 1 to 4 houses per page, with four answer rows of boxes and the signs already in place. Choose numbers up to 10 or 20 for addition, or times tables up to 5, 10 or 12 for the fact family multiplication and division sheets. One missing hides one corner of each triangle so children find it first. New worksheet draws a new set and repeats a family only when the page has more spaces than there are families (times tables up to 5 have just six), and Answers fills in every box in red. The fact family blank template prints empty triangles or houses with the signs, for families children choose themselves.
How we tested this tool
We checked the tool by script in Google Chrome on Windows. The calculator was given numbers in different orders and forms (3, 9, 12; 12 3 9; 5 7 35; "15, 5 and 3"; 62 15 77; two numbers only) and returned the right family each time; doubles such as 6, 6, 12 gave two facts, 2, 2, 4 followed the Facts setting because it works both ways, 0 was refused for multiplication, and numbers that don't fit, such as 3, 4, 9, got an explanation instead of facts. We then generated 400 worksheets and blank templates (both fact types, triangles and houses, 1 to 8 per page, every number range, with and without a missing number and answers, on Letter and A4) holding 1,400 fact families: in every one the parts add or multiply to the whole, the numbers stay in range, each answer row matches its fact, every triangle, house and box stays inside its space, and no two numbers or labels overlap. The first house layout let the lower roof circles sit on the first answer row, so the rows now start below them. We printed sheets to PDF with Microsoft Edge (one US Letter portrait page) and tested the page at a 375 px phone width.
Frequently asked questions
What is a fact family in math?
Three numbers that belong together and the facts they make. For 3, 9 and 12 the fact family is 3 + 9 = 12, 9 + 3 = 12, 12 − 3 = 9 and 12 − 9 = 3. The largest number is the whole and the other two are its parts.
What are the 4 facts in a fact family?
Two facts that join the parts and two that take a part away. For addition that is two addition facts and two subtraction facts; for multiplication it is two multiplication facts and two division facts, such as 4 × 5 = 20, 5 × 4 = 20, 20 ÷ 4 = 5 and 20 ÷ 5 = 4.
Why does a double have only two facts?
Because turning the parts around gives the same fact. 6 + 6 = 12 is the same both ways, and 12 − 6 = 6 is the only subtraction, so the family has two different facts instead of four.
How do you find the missing number in a fact family triangle?
If the top number (the whole) is missing, add the two bottom numbers, or multiply them on a multiplication triangle. If a bottom number is missing, subtract the other bottom number from the top, or divide.
Can a fact family have 0 in it?
For addition, yes: 5, 0 and 5 give 5 + 0 = 5, 0 + 5 = 5, 5 − 0 = 5 and 5 − 5 = 0. For multiplication, no, because the division facts would need dividing by 0.
Is there an answer key for the worksheets?
Yes. Set Answers to Show and print again. Every box is filled in red, and on one-missing sheets the hidden corner is shown too, on the same triangles or houses as the sheet on screen.