In the Advanced Math & Logic elective I co-taught this spring, we talked about a lot of math concepts that often never seem to fit into the rigid state standards for Algebra, Geometry and beyond. Yet after finding so many great resources and ideas, I think any of these activities could and should be done in your regular high school math classrooms.
Some of them will be good at getting students interested when things seem boring, some would make great extra credit or challenge problems for your brightest students, and most of them have a place in your classroom no matter what.
This first part is all about topology.
Four Color Theorem
One of the first concepts we explored was this simple theorem. First, we did this simple Map Coloring activity. Then, I challenged students to try to see if they could follow the Four Color Theorem by filling in this blank U.S. Map. These activities could help hook students into a unit about proofs.
DIY Topology Puzzles
These are puzzles where you take various shapes and combine them into one larger shape (similar to tangrams). There's a lot of these puzzles in Professor Stewart's Cabinet of Mathematical Curiosities by Ian Stewart, the brilliant book that helped inspire these ideas. I think something like this would be a great way to kick off the year in Geometry, or a challenging project for a student who needs it.
Torus and Klein Bottle Games
There's six different types of online, Java-based games utilizing these concepts here, but the one that we played in class was the maze where you direct a mouse to cheese. The kids really enjoyed these. If you can find printable versions of this, in a book or otherwise, please let me know (after months of searching, I still can't find the site I was able to print from when I used this in class).
Seven classic topology problems
The Konigsburg Bridge problem (there's more good stuff about that one here) is included here among others. These should be printed out for student use.
Click here for Part 2, which features resources for other discrete math topics.