### Why do this problem?

This problem provides a great opportunity to introduce the concept
of representing operations on unknown numbers algebraically.
It leads to work on inverse operations and solving simple linear
equations.

### Possible approach

Either work through a few
examples using the interactivity, or, even better, read out the
instructions:

Think of a number

Add 4

Double

Subtract 7

Go round the class asking
learners what they finished with and then surprise them by
revealing their starting number!

As we don't want the
class to end up thinking maths is a strange magical mystery, it's
time to explain what's going on. One way to begin thinking about
this is to collect together some outputs and their corresponding
inputs, for example:

Finishing Number |
Starting Number |

11 |
5 |

17 |
8 |

3 |
1 |

23 |
11 |

14 |
6.5 |

This is not merely an
exercise in pattern spotting - once learners have figured out HOW
the machine is working out the starting numbers, they need to
understand WHY this works.

It would be good to
introduce a variety of methods for explaining what's going
on:

- Represent the instructions as a function machine, and
explore what happens when the functions are inverted.
- Represent the chosen number as $x$ and write an expression for
the final number $y$ in terms of $x$. Then rearrange to get $x$ in
terms of $y$.
- Discuss simplification to make the functions easier to invert
(so representing it as $y=2x+1$ rather than $y=2(x+4)-7$.

Now challenge the class
to come up with their own examples of "Think of a number"
instructions, which they can invert to deduce someone's starting
number from their final number. Encourage them to record their
working using function machines and algebra. Set challenges such as
having to include particular operations, at least 6 different
operations, and a set of instructions that's really quick to
invert.

At the end of the lesson, a selection of students can read out
their instructions for the class to try, and then work out starting
numbers from the final numbers.

### Key questions

How can I use function
machines and algebra to explain the "Think of a number"
puzzle?

How can I use function
machines and algebra to create a "Think of a number" puzzle?

### Possible extension

### Possible support

Start introducing algebraic notation gently through the problem

Your Number
Is...