### Why do this problem?

This problem provides a great opportunity to introduce a very
useful way of representing operations on unknown
numbers visually and symbolically (algebraically).

### Possible approach

Ask learners to each think of a number and read out the
following instructions:

Look surprised, when all of them reveal they have ended up with the
same number.

"Did you all start with the same number then?"

"Perhaps none of you started with a fraction, or decimal, or a
negative number - shall we try it again with one of those?"

More surprise that they still end up with the same number.

Now show the interactivity, or alternatively draw the
representation on the right hand side of the screenshot above, and
ask learners to think about how the visual representation helps us
to understand what is happening. Give them a couple of minutes to
discuss in pairs, and then draw the class together to share their
insights.

Next, suggest that the visual representation might be too
time-consuming, particularly if we'd added and multiplied by larger
numbers. Introduce a parallel representation using algebra:

$$x$$ $$x+3$$ $$2(x+3) \text{ or }2x+6$$ $$2x+10$$ $$x+5$$
$$5$$

Now invite three members of the class up to the board to each use a
different representation simultaneously for some other "think of a
number" sets of instructions.

For example:

Think of a number

Multiply by 4

Add 2

Halve it

Add 7

Divide it by 2

Take away the number you first thought of.

This is what the board might look like when the three learners have
finished:

Finally, challenge the class to come up with their own examples of
"Think of a number" instructions, which will lead to an anticipated
solution. Encourage them to record their working using both the
visual and the symbolic representation.

At the end of the lesson, a selection of students can read out
their instructions for the class to try, and then reveal the
anticipated solution (written in advance) with a flourish.

### Key questions

How can I use the visual or symbolic representation to explain
the "Think of a number" puzzle?

How can I use the visual or symbolic representation to create
a "Think of a number" puzzle?

### Possible extension

### Possible support

Focus on the visual representations until learners are secure with
this, before introducing symbolic notation.