I've just been working with my 12-year-old daughter on Cantor's diagonal argument, and countable and uncountable sets.
Why? Because the maths department at her school is outrageously good, and set her the task of researching a mathematician, and understanding some of the maths they did - the real thing.
So what else could we have done - thinking that we know our multiplication tables and fractions, but aren't yet completely confident with equations which have letters for unknown numbers?
I did think of proving that there are infinitely many primes - we can follow an argument - other suggestions welcome.
And incidentally, tell your local high school to do this ...