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Academy11+

11+ Maths · Topic explained

Function machines

19 came out of a '× 3, then + 4' machine. What went in?

The big idea

A function machine does steps to a number; to go backwards, undo each step in reverse order.

Picture it

Like getting undressed: the last thing you put on comes off first.

Key words

input
The number that goes in.
output
The number that comes out.
operation
A step: +, −, × or ÷ a number.
inverse
The step that undoes it: ÷ 3 undoes × 3.

Worked examples

  1. 1Warm-up

    6 goes into a '× 2, then + 5' machine. What comes out?

    • 6 × 2 = 12
    • 12 + 5 = 17
  2. 211+ standard

    A '× 3, then + 4' machine gives 19. What went in?

    • Undo in reverse: − 4 first, then ÷ 3
    • 19 − 4 = 15
    • 15 ÷ 3 = 5
  3. 311+ standard

    8 goes into a '× ?, then − 6' machine and 34 comes out. What is the missing number?

    • Undo − 6: 34 + 6 = 40
    • 8 × ? = 40, so ? = 5
  4. 411+ standard

    A machine does '× 2, then − 1'. Write the output when n goes in

    • n × 2 = 2n
    • 2n − 1
    • Output: 2n − 1

The common mistake

The mistake

To undo '× 3, then + 4', do ÷ 3 first, then − 4.

Instead

Undo in reverse order: − 4 first, then ÷ 3.

Why: The last step done is the first one to undo.

Where it comes up in the 11+

Forwards and backwards machines, missing steps, and machines written as algebra.

The video

Video coming soon

We're making a short video that explains function machines step by step. Until then, everything it covers is on this page.

Free printable

Function machines worksheet

Practice questions on this topic to print and try on paper, at home or on the way to school.

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Quick quiz

Have a go first, then tap to see the answer.

  1. 1. 9 goes into '− 4, then × 5'. What comes out?

    • 25
    • 41
    • 45
    • 29
    Show the answer

    25

    9 − 4 = 5, and 5 × 5 = 25.

  2. 2. A '× 4, then − 3' machine gives 33. What went in?

    • 7.5
    • 9
    • 129
    • 8
    Show the answer

    9

    33 + 3 = 36, and 36 ÷ 4 = 9.

  3. 3. 12 goes into '÷ 3, then + ?' and 11 comes out. What is the missing number?

    • 4
    • 7
    • 8
    • 1
    Show the answer

    7

    12 ÷ 3 = 4, and 4 + 7 = 11.