Inverse relationship (fact families)
Theory
Fact families show how addition and subtraction are inverse operations that undo each other. This Year 3 maths page explains part-part-whole, fact-family triangles, writing the four number sentences from three numbers, finding a missing part, and checking a subtraction by adding. Curriculum: VC2M3A01 / AC9M3A01.
Addition and subtraction are inverse operations. That means they undo each other: if you add a number then take the same number away, you end up back where you started.
Every group of three numbers that go together is made from two parts and one whole. Joining the two parts makes the whole. Taking one part from the whole leaves the other part. This is called part-part-whole.
A fact family is the set of number sentences you can write from the same three numbers. From the parts \(7\) and \(8\) and the whole \(15\) you can write two additions and two subtractions: \(7 + 8 = 15\), \(8 + 7 = 15\), \(15 - 7 = 8\) and \(15 - 8 = 7\).
The same three numbers, two parts and a whole, always make these four facts. Here the parts are \(7\) and \(8\) and the whole is \(15\).
How to use a fact family
- Find the parts and the whole. The whole is the biggest number; the other two numbers are the parts.
- Decide add or subtract. If the whole is missing, add the two parts. If a part is missing, take the known part from the whole.
- Check with the inverse. Undo your answer: add the parts to check a subtraction, or subtract to check an addition.
| \(3 + 8\) | \(=\) | \(11\) |
| \(8 + 3\) | \(=\) | \(11\) |
| \(11 - 3\) | \(=\) | \(8\) |
| \(11 - 8\) | \(=\) | \(3\) |
Two additions join the parts; two subtractions take a part from the whole.
| \(\text{parts}\) | \(:\) | \(40 \text{ and } 25\) |
| \(\text{whole}\) | \(:\) | \(65\) |
| \(65 - 25\) | \(=\) | \(40\) |
Taking one part from the whole leaves the other part, so the answer is \(40\).
| \(80 - 53\) | \(=\) | \(27\) |
| \(\text{check: } 53 + 27\) | \(=\) | \(80\) |
A part is missing, so subtract. There are \(27\) empty spaces.
| \(36 + 38\) | \(=\) | \(74\) |
Adding the answer back on gives \(74\), the number he started with, so \(74 - 38 = 36\) is correct.
Common pitfalls
Frequently asked questions
What is a fact family?
A fact family is a set of number sentences that use the same three numbers. From two parts and a whole you can write two additions and two subtractions, such as \(7 + 8 = 15\), \(8 + 7 = 15\), \(15 - 7 = 8\) and \(15 - 8 = 7\).
What does inverse operation mean?
Inverse operations undo each other. Addition and subtraction are inverse operations, so subtracting undoes adding. If you add \(8\) to reach a number, subtracting \(8\) takes you back to where you started.
How do you check a subtraction using addition?
Add the answer back on to the number you took away. If the total is the number you started with, the subtraction is correct. To check \(83 - 46 = 37\), work out \(37 + 46\), which gives \(83\).
How do you use a fact family triangle?
Put the whole at the top corner and the two parts at the bottom corners. Add the two parts to make the whole, or take one part from the whole to find the other. The three numbers give you four facts.
Should you add or subtract to find a missing part?
If a part is missing, subtract the known part from the whole. If the whole is missing, add the two parts together. The whole is always the biggest number.
Why do addition and subtraction help each other?
They are inverse operations that use the same part-part-whole numbers. Knowing one fact means you know the other three, so a hard subtraction can be solved by thinking of an addition you already know.