Inverse operations & fact families
Theory
Multiplication and division undo each other, so one trio of numbers makes a whole fact family of related facts. Use an inverse to find a missing number.
An inverse operation reverses another operation.
Multiplication and division undo each other; addition and subtraction undo each other.
A fact family is the set of related facts built from one trio of numbers.
From \(3\), \(4\) and \(12\): \(3 \times 4 = 12\), \(4 \times 3 = 12\), \(12 \div 3 = 4\) and \(12 \div 4 = 3\). Knowing one fact gives the others, and lets you find a missing number with the inverse.
One trio, four facts
| Trio | Products | Divisions that undo them |
|---|---|---|
| \(3,\ 4,\ 12\) | \(3 \times 4 = 12\) \(4 \times 3 = 12\) | \(12 \div 3 = 4\) \(12 \div 4 = 3\) |
| \(7,\ 9,\ 63\) | \(7 \times 9 = 63\) \(9 \times 7 = 63\) | \(63 \div 7 = 9\) \(63 \div 9 = 7\) |
Addition families work the same way
| Trio | Totals | Subtractions that undo them |
|---|---|---|
| \(6,\ 9,\ 15\) | \(6 + 9 = 15\) \(9 + 6 = 15\) | \(15 - 6 = 9\) \(15 - 9 = 6\) |
Build a fact family
- Multiply the two smaller numbers to get the product: this is the top of the triangle.
- Swap the factors to write the second multiplication.
- Divide the product by each factor to write the two division facts.
Find a missing number
- Find the operation done to the unknown.
- Apply the inverse to the number you know.
- Check by putting the answer back into the original sentence.
Two products, then the two divisions that undo them.
| \(4 \times 9\) | \(=\) | \(36\) |
| \(9 \times 4\) | \(=\) | \(36\) |
| \(36 \div 4\) | \(=\) | \(9\) |
| \(36 \div 9\) | \(=\) | \(4\) |
Divide to undo the \(\times 8\). Check: \(9 \times 8 = 72\).
| \(72 \div 8\) | \(=\) | \(9\) |
Sharing into equal groups is a division. Check: \(8 \times 6 = 48\).
| \(48 \div 6\) | \(=\) | \(8\) |
She fills \(8\) boxes.
Subtract to undo the \(+7\). Family: \(8 + 7 = 15\), \(15 - 8 = 7\).
| \(15 - 7\) | \(=\) | \(8\) |
Common pitfalls
Frequently asked questions
What is a fact family?
It is the set of related facts built from one trio of numbers, such as \(3 \times 4 = 12\), \(4 \times 3 = 12\), \(12 \div 3 = 4\) and \(12 \div 4 = 3\).
How many facts are in a multiplication fact family?
Usually four: two multiplications and the two divisions that undo them. If the two factors are the same, like \(5\) and \(5\), there are fewer.
How do I turn one multiplication into a division fact?
Divide the product by either factor. From \(7 \times 8 = 56\) you get \(56 \div 7 = 8\) and \(56 \div 8 = 7\).
How do I find a missing number with an inverse?
Do the opposite of the operation done to the unknown. For \(\square \times 8 = 72\), work out \(72 \div 8 = 9\).
Do addition and subtraction make fact families too?
Yes. The trio \(6\), \(9\), \(15\) gives \(6 + 9 = 15\), \(9 + 6 = 15\), \(15 - 6 = 9\) and \(15 - 9 = 6\).