**The Challenge: **Today’s math art challenge is to play the chaos game, and was inspired while I was perusing the excellent Power in Numbers by Talithia Williams. Her chapter on Fern Hunt indicated one of her research interests as Chaos Theory – something that always grabs kids attention, and leads to one of the more fascinating probability related math-art creations.*Materials Needed: *randomizer (die, coin, etc.), paper, pencil, possibly graphing software. *Math concepts* *you could explore with this challenge: *fractals, probability, randomness

Dr Hunt’s main focus was in “dynamical systems and applied probability”. Nevertheless, the connection to chaos is enough to open the door for a probability lesson where you can teach about Dr. Hunt and her work before leading into the following activity.

- Draw a triangle.
- Number each corner such that rolling a die indicates them each equally. (I numbered mine with {1,2} {3,4} and {5,6} on each corner so each vertex had a 1/3 probability of being selected with a fair 6-sided die roll.)
- Place a point in the triangle randomly.
- Randomly select a vertex (I rolled a die for this) and using a ruler, mark a point exactly halfway from your original point to that vertex.
- Repeat…. MANY times.

Ideally, this is an excellent activity to do on transparency paper in a class, and then overlay everyone’s images. The end result is rather surprising. Here is what I was able to do on my own (with about 2 hours of podcasts).

Simply because it takes a long time for the image to resolve itself if you’re doing this by hand, this may be a good one to try to code (not a current skill set of mine). This numberphile video does a nice job explaining and unpacking the chaos game, but if you start it where I have tagged it below (3:10) you can see the form taking shape:

Questions you might ask yourself or students participating in this:

- Can you predict the shape that will form?
- Were you surprised by what appeared?
- What is surprising and what may
*not*be surprising about the final image?

- What is surprising and what may
- What about the “rules” for creating this imposed structure here?
- How might you alter the rules to get different results? (The wikipedia page actually explores some neat answers to that.)
- What do you imagine are actual applications for “applied probability”?
- What do you notice or wonder about the accuracy of by hand (mine above or yours created) and computer generated images? Is there a boon to one over the other? When may “by hand” be more useful? What about “computer generated”?

*Depending on how you use this activity, you may engage with different mathematical standards. I’ve listed possible connected math content above. Here are a few suggestions for how you might integrate the **8 mathematical practices**. Feel free to add your own suggestions in the comments! *

2.) Reason abstractly and quantitatively. *How is “chaos” apparent here or not?* *How many trials do you need to make in order to see the pattern appear? *

6.) Attend to precision. *How is the effectiveness of the result affected by your precision in measuring? *