Topics appropriate to teach high school students

Knot theory is always fun because you can bring in physical examples to play with.

Graph theory is a good one too,there's a lot of interesting and accessible things to do.

I would probably avoid all the topics you mentioned if you are just working with average high schoolers, and not specifically a math camp or something
I'd like to back up your idea for basic group theory. It's was the first time that I felt compelled to study math outside of class and while it wasn't what started me on this path (that had already happened at that point) and it was my first real introduction to abstraction.
I'd recommend something that they can play around with themselves outside of class. While you might get them to a point where they can understand a certain topic, it's much better if you can make them think about it beyond the scope of your presentation.

Most introductory finite combinatorics is good here - the binomial theorem, catalan numbers, derangements, graph theory, etc. There's plenty of room to give motivations, fun stories, history, etc. in this field to make the subject more human, which I think is super important when teaching to younger students (i.e. up to and including college freshmen/sophomores).

Probability is also a good supplemental topic with lots of counterintuitive questions you can ask them which they'd have a shot at puzzling out. (E.g. Monty Hall.)
When I was in late high school and had an extension lecture like this, the thing that really stuck with me was seeing a group isomorphism. Specifically, I saw how the symmetry group of a nonsquare rectangle and the multiplicative group of integers modulo 8 were both the dihedral group of order 4 (I think? My group theory is a little rusty, but the former two were definitely isomorphic, even if they weren't D4). Blew little me's mind. In that vein, I like your idea of looking at the group law on elliptic curves, because even a hardened analyst like me thinks they're beautiful. I'd stay well away from set theory though. That's too deep for high schoolers, even interested ones imo.
What about comparing infinities? I think the results would seem interesting and counter intuitive but they should be able to follow along enough to actually understand it which could be pretty motivating for them.
A couple of your topics seem a bit optimistic. In my opinion, your goal shouldn't be to "advertise" a specific field but rather give them something that allows them to experience that wonderful feeling of discovering some mathematical truth. I would recommend making the process self-directed rather than sit-and-listen. Often questions are more illustrative than answers.

Some suggestions off the top of my head:
- Hilbert's hotel and Cantor's infinities.
- Length, area and volume. What are they? How to define them for lines, rectangles, boxes; circles and spheres; general regions? (You can pretty much introduce the Hausdorff measure here, but limits need to be taken for granted).
- Zeno's paradox. Basic concept of a limit.
- Symmetries. Basic concept of an isomorphism.
- It is possible to introduce some calculus of variations with only very basic calculus. The baristochrone problem is a good topic. Soap films and bubbles are always fun.
Look into recreational math. You can use puzzles, games, magic, juggling, to introduce numerous sophisticated, but accessible topics, like nondecimal bases, the geometric series, and modular arithmetic.
The exposure I got in high school to basic matrix theory (matrix multiplication, determinants) and vectors (dot products, cross products, etc) was an absolutely indispensable head start for me in college
I learned algebraic structures. Basic Group Theory. It was fun. And a great introduction to proofs.
As a high schooler(9th grade so I don’t know ur target audience), I find number theory cool and works synergistically to function(function=geometry + algebra) which really aided my study of polynominal quartic and cubic functions or any other functions. I think high schoolers are willing to learn set theory, so u shouldn’t worry about the motivation part.( I briefly studied set theory and first order logic for interest)

0 like 0 dislike