In addition to the topology problems included in Part 1, my experience this spring showed me a lot of ways that we could approach many other higher level, discrete mathematics topics in high school math courses.
Cellular Automata and Conway's Game of Life
Using examples from Professor Stewart's book, this Wikipedia entry, and in this Math.com article, I made a simple graphic organizer which showed students what to make on Conway's site.
Before starting, review the “rules of life,” make sure to show them the interface, and push them to test out the shapes on the pull-down menu. Also, ask them to experiment with changing the speed and size. Finally, I had them try out a shape of their own, like their name or something like that. The results are always interesting.
(PS: this example of breeders is just really cool.)
I used this Discovery Education lesson plan as a basis for a lesson on chaos theory. To start, students play the “Chaos Game” with dice, a ruler and three different colored markers. As they follow the rules on the game handouts, they should start to see a Sierpinski Triangle start to take shape. They do need to measure very carefully, as the students who were eyeballing it were getting a much more crude shape that wasn't as easy to distinguish. Then, students played on the online Chaos Game, where they apply the rules of the earlier game in order to land inside the highlighted part of a Sierpinski Triangle.
I showed the students Foldit, the protein-folding game that's helping scientists solve difficult problems. We discussed the importance of knot theory, an area of math that was once thought of as pointless, and how playing the game could eventually help cure diseases. Here's a YouTube video introducing how the game works.
You would need to be able to install the game on the computers for students to use, which limits it's usefulness in the classroom (unless you have that kind of access). I think that if you sell this to your students, however, those with computers and Internet access at home will almost surely try it out.
We also spent a lot of time on logic games. There are a few good ones in Professor Stewart's book, but I used mostly offline resources similar to these logic puzzle books on Amazon.com.
I hope you see the possibilities of these as I do! If you have similar resources on other advanced math or logic topics, please share them in the comments.