Math Circle – Billiards

A new year of the Mercer County Math Circle will be gearing up soon. I’m also teaching a class once a week this year at PLC called “Math to Love”. It’s meant both for folks who like math and want more, as well as for those who don’t like math and so need something to love all the more. Hopefully I can attract some people to attend.

For our first Math to Love class, I made a lesson that I’m also hoping to share at MC^2 soon. It’s about billiards.

There is sooooooo much cool math that involves billiards.

The reason I picked billiards to feature at this particular moment is because two of this year’s Fields Medalists study billiards: Maryam Mirzakhani and Artur Avila. To find out more about these amazing mathematicians, see our recent Math Munch post.

Here are some ideas for sharing billiards with kids that I used, or will use, or that someone (you!) could use.

A couple of cool facts to mention, illustrate, or expand on:

• the short videos about Maryam and Artur, linked to above.
• the reflection property of the ellipse (billiards video), and potentially the other conics
• the fact that it’s unknown whether every triangle has a closed billiards loop. (Richard Evans Schwarz has made substantial progress on the problem.)
• Using Cinderella software, you could share this applet that can show a variety of billiard phenomena.

A couple of tasks:

•  this investigation of billiards in rectangles that I made. It’s something I first learned from Avery Pickford years ago. There are all kinds of great questions that arise through this investigation about topics like scaling, common factors, odds and evens, division with remainders, and more. (Similarly, at Illuminations.)
• some version of the “bouncing icon” WCYDWT that Dan Meyer shares in these posts.

Which corner pocket will the billiards ball land in?

One more way to bring this year’s Field Medalists into your classroom? Print out a copy of this poster that I created.

I had a lot of fun sharing these billiards ideas with my “Math to Love” class and I’m looking forward to doing it again at the math circle. Hope you found something here to enjoy and to use!