Relate multiplication & division (fact families)
Theory
Fact families show how multiplication and division are linked. Because they are inverse operations, three numbers make four related facts: two multiplication turnaround facts and two division facts. This Year 3 maths page uses arrays and fact-family triangles to build these facts and to find missing numbers. Curriculum: VC2M3A03 / AC9M3A03.
Multiplication joins equal groups to make a total. Division splits a total back into equal groups. They undo each other, so we call them inverse operations.
A fact family is a group of facts that use the same three numbers. The two small numbers are the factors and the big number is the product (the total). The numbers \(3\), \(8\) and \(24\) make four facts: \(3 \times 8 = 24\), \(8 \times 3 = 24\), \(24 \div 3 = 8\) and \(24 \div 8 = 3\).
The two multiplication facts are a turnaround pair: the factors swap places but the answer stays the same. Each division fact starts from the product and divides by one factor to find the other.
These are the four facts for the family \(3\), \(8\) and \(24\). Every fact family follows the same pattern: two multiplication facts and two division facts.
How to write a fact family
- Find the three numbers. From an array, count the rows and the number in each row; the total is the product. From a triangle, the top is the product and the corners are the factors.
- Write the two times facts. Multiply the two factors, then turn them around to get the second fact.
- Write the two divide facts. Start from the product and divide by one factor, then by the other factor.
Missing numbers and checking
To find a missing factor, divide: \(4 \times \square = 20\) gives \(\square = 20 \div 4\). To find a missing number that is being divided, multiply: \(\square \div 3 = 7\) gives \(\square = 3 \times 7 = 21\). To check any division, multiply the answer by the number you divided by and see if you get back the total.
| \(3\text{ rows of }4\) | \(=\) | \(3 \times 4 = 12\) |
| \(\text{turnaround}\) | \(=\) | \(4 \times 3 = 12\) |
The facts are \(3 \times 4 = 12\) and \(4 \times 3 = 12\).
| \(3 \times 10\) | \(=\) | \(30\) |
| \(10 \times 3\) | \(=\) | \(30\) |
| \(30 \div 3\) | \(=\) | \(10\) |
| \(30 \div 10\) | \(=\) | \(3\) |
Two times facts and two divide facts make the family.
| \(\square\) | \(=\) | \(20 \div 4\) |
| \(\square\) | \(=\) | \(5\) |
Division undoes multiplication. Check: \(4 \times 5 = 20\).
| \(18 \div 3\) | \(=\) | \(6\) |
Sharing equally is division. Check with the family: \(3 \times 6 = 18\).
Common pitfalls
Frequently asked questions
What is a fact family in maths?
A fact family is a group of facts that use the same three numbers. Two factors and their product make four facts: two multiplication facts and two division facts. For \(3\), \(8\) and \(24\) they are \(3 \times 8 = 24\), \(8 \times 3 = 24\), \(24 \div 3 = 8\) and \(24 \div 8 = 3\).
How are multiplication and division related?
They are inverse operations, which means they undo each other. Multiplication joins equal groups to make a total, and division splits a total back into equal groups. So if \(3 \times 8 = 24\), then \(24 \div 8 = 3\).
How do you write a fact family from an array?
Count the rows and the number in each row. An array of \(3\) rows of \(8\) gives \(3 \times 8 = 24\) and the turnaround \(8 \times 3 = 24\). The total \(24 \div 3 = 8\) and \(24 \div 8 = 3\).
What is a turnaround fact?
A turnaround fact swaps the two factors in a multiplication. \(3 \times 8\) and \(8 \times 3\) both equal \(24\). The order of the factors does not change the answer.
Why do some fact families have only two facts?
When the two factors are the same, like \(5 \times 5 = 25\), the turnaround is the same fact and both division facts are the same. So a double makes just one multiplication fact and one division fact, \(25 \div 5 = 5\).
How do you find a missing number in multiplication?
Use division, because it is the inverse of multiplication. To solve \(4 \times \square = 20\), work out \(20 \div 4 = 5\). Check by multiplying: \(4 \times 5 = 20\). If the number being divided is missing, multiply instead: \(\square \div 3 = 7\) means \(3 \times 7 = 21\).