### Why do this problem?

This problem gives a great engaging context in which to
practise calculus. It is very useful for getting across important
ideas concerning integration and differentiation as operations,
which will be useful at university in both science and maths
courses.

### Possible approach

Initially play as a straightforward game to try to hit the
targets.

Students could then try to create new targets for their
friends to try to hit using the same initial functions.

You could then give the students the task of creating their
own game of Calculus Countdown.

Once students have hit a target they will need to communicate
their answers clearly. How might they write down an answer in a
clear and unambiguous way that can easily be interpreted by someone
else?

### Key questions

What happens when you integrate or differentiate the starting
functions?

What would you have to integrate or differentiate in one step
to hit the targets? Does this help?

### Possible extension

Interesting side questions which might emerge are: What
targets are possible? Can you prove that certain targets (such as
$\sqrt{x}$ or $3$) are impossible to hit? Clearly, proving the
impossibility of a target will require some clear thinking; be sure
that students concentrate on clearly writing down their
arguments.

### Possible support

Focus on the first 3 targets.