# The Chaos Game

Towards the end of every school year comes testing week. It can be tough to juggle sometimes. In my school you could be administering the test, holding class as normal, or covering students not taking the test their teacher may be administering that day. Testing blocks also consume a good portion of the day and can leave you with planning difficulties. For instance, I saw my afternoon class periods somewhere around 10 times more than the others due to testing.

No matter how long you've been in the business, it never hurts to have ideas kicking around for days such as this. One I had been holding onto for a while is The Chaos Game. If you look around hard enough, there are some sites with embedded applets that will allow you to simulate thousands of dice rolls. These can be useful as a debrief. I first stumbled on this via Frank's180 Photos website.

Rules are simple. Create an equilateral triangle, assign values 1-2, 3-4, 5-6 to the corners and create a random starting point. Each roll of the die moves your point halfway to the given corner. Do this enough and you get a Sierpinski triangle.

It takes a bit of explaining to get the mechanics down, but after that it's smooth sailing. I dusted off some transparencies, borrowed some dice, and let them roll for about 10 minutes. I gathered up all the transparencies and found the ones where it was clear the student was detailed in their proceedings. When you overlay about 10 of them, you get a nice first order Sierpinski triangle. The kids are baffled as to how this happened.

I should mention I give them this activity with zero hints as to what we're trying to do. If anything, regardless of the lesson, that has been my most successful strategy. Can I cleverly hide the real objective? Can I get them to see the result without having to hand it to them?

So once I reveal the traingle out come the questions. How the heck did this happen? It was totally random. We all had diferent starting points. It's a great discussion about a branch of mathematics that doesn't get much love at the high school level. A few struggle with the idea that there could be order amid such chaos. Then if you get lucky, you can wander down a fractal rabbit hole.

A couple tips: a well-drawn triangle is key, ruler accuracy is key, and small dots are key. Otherwise it will just be a mish mash of points (with some falling in the center which shouldn't happen). I conducted this activity with two Algebra II classes (~60 students), so there were enough good results to succeed.

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