The Euler Characteristic (Steven Gubkin above) is potentially a good option - it feeds into, for example, how to make a football, and why there are pentagons in the domes at the Eden Project: but that is more than 10 minutes.

Another option is talking about the number line - and showing you can cover the rationals in intervals with intervals of arbitrarily small total length: related to the infinities question above. But I sometimes use this to show that the number line (which is used from primary school upwards) requires some careful thinking - and if I have time I explain how this feeds into that mathematics of continuity and change (a mention of Zeno's paradoxes gets in too).

If you want a 'why do maths' question such stories as the making of nuclear bombs/reactors (making sure that the bombs explode, but the reactors don't), or sending men to the moon and getting them back could provide a narrative (not just how much fuel, but how long does it take, therefore how much food etc) could provide motivation.

But if you are looking for material in relation to schools, the Mathematical Association has a good list of resources, which might also give some ideas.