I occasionally use these videos in math circle, and think they're awesome! The videos are fun and silly and have a lot of great mathematical content. They can also work in other environments, and I know a bunch of students who watch them at home (so at least one of your students will likely have already seen the video you're about to show). I'll tell you about a few that I've used:

Making hexaflexagons is one of the best hands on math circle activities, and works for pretty much all ages (I've done it with middle schoolers). I used this video as an introduction and used this video later on in class (pausing at appropriate points for students to ask questions, make their own, etc).

There's a whole playlist called

*Pi and Anti-Pi*, which I used on Pi Day. Some of the videos are good for middle school, but others mention radians and trigonometry and therefore would be more appropriate for high schoolers. I think that students enjoy hearing about the Pi vs Tau argument, and it makes them think about the importance of definitions and that the way they're taught is not the only way.
Infinity is a great and mind-blowing topic for math circles, and Vi Hart has a playlist called The Infinite Series. I haven't used these videos myself, but I think it's a great topic for high schoolers.

I also really like the story of Wind and Mr. Ug, which is about living on a Mobius strip. I used it in conjunction with Math Improv: Fruit by the Foot for an exploration of what happens when you make

*n*half twists in a paper strip before taping the ends together. I used this with middle schoolers.
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