A function machine takes a number in, follows a rule, and gives a number out. If you know the input and the rule, you can find the output. If you know the output, you can work backwards to find the input.
Why this matters: Working backwards is how you check a bill, share money fairly, or fill in a missing number in a puzzle. It builds the thinking you will use in algebra.
Finding the output (going forwards)
Put a number in, do the rule, read the answer out.
Here 5 goes in, the rule is "+ 3", so 5 + 3 = 8 comes out.
Finding the input (going backwards)
To go backwards, use the inverse (opposite) operation. The opposite of + is −, and the opposite of × is ÷.
| Rule | Inverse (go back) |
|---|---|
| + 3 | − 3 |
| − 4 | + 4 |
| × 2 | ÷ 2 |
A machine adds 3 and gives out 10. What went in?
Work backwards: 10 − 3 = 7. So 7 went in.
Work backwards: 10 − 3 = 7. So 7 went in.
Put these steps in order to work backwards:
- Start with the output number
- Do the inverse (opposite) operation
- The answer is the missing input
Missing numbers
A box □ stands for the missing number. To solve 7 + □ = 15, do the inverse: 15 − 7 = 8, so □ = 8. To solve □ − 4 = 10, do 10 + 4 = 14, so □ = 14.