NCTM's Illuminations has some good pi-related lessons on a range of timelines. I'd like to build up to the Archimedean approximation in my classes some time, but usually I end up doing small activities and demonstrations on Pi Day. Here are a few.
- Buffon's Needle Experiment. Simulates dropping thousands of needles onto lined paper. The probability of a needle crossing a line is 2/pi, so we can estimate the value of pi through repeated trials.
- Multifactorials. Cool to show in conjunction with pi formulas from Ramanujan and others.
- Doomsday Algorithm. Not related to pi but fun to teach on Pi Day.
- What Pi sounds like. A cool new musical rendering of digits (there are a few other neat ones like this if you search).
- Pi10k. An experiment mapping notes to digits. Like the video about but not as good-sounding.
- Pi Rap Battle. Near and dear to me.