What to do when you don't know what to do.

Solving maths problems generally involves breaking complex problems down into smaller steps. Sometimes the steps are obvious, sometimes they aren't! What should you do when you don't know where to begin?

Something I've been telling my students for years: in an ideal world, for every exam question you encounter, you'll have a clear idea of how to solve it, the method you'll use for each step, in sequence, how each step leads to the next step, right up to the final answer.

With harder, longer problems, the answer won't always leap out at you like this. When students don't know where to start I always ask them:

• "What do you know?
• "What can you do with that?"

Like an archeologist brushing away the sand to see if there's something underneath, sometimes you just have to start the first step without knowing where it's going to lead. The hope is that successfully completing the first step reveals some more information or at the least, a hint about what to do next!

Take this question as a case in point: we know precious little about the two forms except that their surface area is equal. In many ways, this makes the "next right thing" (NRT) easy to do: logic suggests it must have something to do with the surface area.

If you know the formulae for the surface areas involved, you can write them out, if not you can look them up on the formula sheet (on this occasion, this would lead to the next NRT: that the formula for the surface area of a sphere must be halved to give the curved surface of a hemisphere)

The curved surface area of a cone depends on L, the 'slant height'. Since this isn't given in the question (even as an algebraic term) we then have our next NRT: how to calculate the slant height from the information we're given (a fairly simply case of using Pythagoras' theorem with the height and radius of the cone). Substituting that into the equation for the surface area of the cone then reduces the problem to rearranging and simplifying algebra.

(If you're confident with algebra, this is a matter of routine. If you're not, you can also apply the "next right thing" method: the question has asked us to find an expression for h in terms of x. We can look at the equation as written on the page and ask "if I want to get h by itself, what do I need to get rid of and how do I do that? What do I need to move first")

In the wider world, the problems confronting us are often ill-defined. Unlike GCSE or A Level Maths, there is rarely a method to follow, seldom a known outcome. Deciding what to first can be daunting and confusing. In such situations, you could do a lot worse than asking:

• "What do I know?
• "What can I do with that?"