Question

Interesting Math Topics (for High School Students)?

Answer 1

I feel like the best place to find what you're looking for is math magazines, they talk about interesting stuff in a cool fun way.

Answer 2

This just came to my mind, the fact that there are general methods to solve cubic and quartics. This will definitely require a presentation, seeing the actual expressions involved in these methods will make everyone understand why they are never used.

Answer 3

Proof of Poincaré conjecture

Answer 4

Explain Kelly betting (Fortune's Formula, by William Poundstone).

Explain the math behind some casino games: craps, for example. Lotteries. Figure out the house edge.

Or give rules of thumb for amortization calculations. Very practical knowledge they can show off to their parents.