Category Archives: math circle

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.

circle-billiards trapped pentagon

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?

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!

fields medalist poster final with blurb


Math Circle – Halving Fun

I recently led a recreational session at (MC)^2—that’s the Mercer County Math Circle. I called the session “Halving Fun”. I don’t remember exactly how I hit upon halving—maybe because of a partition puzzle I’ve been contemplating—but I knew I was excited to share how much great mathematics can come from such a seemingly simple concept. Here’s the session description:

Join us as we “halve” some fun while creating some art and solving some puzzles that involve breaking shapes in half. What’s the most beautiful way you could cut a square into two equal pieces? Is it always possible to cut a funky shape in half through a specified point? Whether on grids or with free-form shapes, halving tasks provide many fruitful mathematical opportunities. Come explore and these questions and more!

I kicked off the session by showing off some examples of halving art, and giving a little spiel about each.

This slideshow requires JavaScript.

Here’s the slide deck if it’s useful to you. I also shared the Yoshimoto Friends video that Paul and I made.

Then I set everyone loose to either make some art or try out some of the problem sheets I’d brought. I also had some physical “halving” puzzles and other models for attendees to play with.

Here are the problem sheets:

  • It’s Half – I’ve gotten a lot of mileage out of geometric fraction problems like these. On this sheet, you already know that each picture represents a half. The goal is to make clear just why this is the case.
  • Half Frames – These are two problems that anyone can work on, but that have rich connections to algebra and geometry.
  • Halving – I always share this sheet with my Geometry classes, ever since I found it in Girls’ Angle Bulletin a number of years ago. It’s also a great jumping-off point for students to make problems of their own!
  • Half Tetra – This is a template for a 3D puzzle to cut out and fold up. Two identical pieces combine to make a regular tetrahedron. Cardstock recommended. Via Futility Closet.
Can you say why the hexagon is half shaded?

Can you say why the hexagon is half shaded?

Attendees made some lovely Truchet-inspired drawings, and one attendee especially dug into the half frames problem—coming up with an interesting conjecture in the process. :) A middle school math teacher who attended even planned to share some of the problems with her students. Maybe you’ll find fit to do so, too. Let me know if you do!

Math Circle – Celtic Knots

On Saturday at (MC)^2, Harini and I led a session on Celtic knots. Many of the ideas for it were taken from these great articles on the NRICH website. Here’s the description we sent out ahead of time:

It’s St. Patrick’s Day this week! We’ll look at some ornate Celtic knots, observe and explain some patterns, and make some knot drawings of our own. We’ll also discuss the difference between knots and links, ways to identify each, and some other basic topological ideas. Come shamrock some great math with us!

Here is the warm-up sheet we gave to people on Saturday as everyone arrived and got settled.

How many bands are each of these designs made up of?

How many bands are each of these designs made up of?

We started with some introductory remarks and showing a few examples of Celtic knots:


Here are all of the examples we showed.

Then we played Vi Hart’s video “Snakes + Graphs”:

examples curveAfterwards we focused in on the first doodle game Vi shares. We drew and talked through a simple example on the board, then game folks time to try some of their own or to construct the over and unders on this example curve.

Next we shared a second, much more restrictive method of drawing knots. It produces images like the A and B weavings pictured above. We showed Alison’s video on this NRICH page twice and let folks follow along with this sheet (or work on whatever they liked).

Then I shared a third method for drawing knots that I learned through these lecture notes. Here was my first try, which I made in preparation for the session:


Here’s the handout of dot paper that we passed out for people to try on. You’ll need to tilt the page at a 45° angle.

We showed this photo of a project Harini did at a Bridges math art workshop a couple of summers ago, as inspiration for them to take next steps on their own at home.


Much of the 45 minutes of the session was giving attendees time to try out these methods on their own and giving them support and answering their questions as they arose. As a result, it was a much quieter math circle than what I’ve been used to, but still a lot of fun. People seemed pretty happy, and lots of great knot drawings got made.