Mary O’Keeffe took the lead in developing this circle session on Pythagorean Fractal Auctions for the 2011 Circle on the Road workshop.

**Target audience:** middle school students who are familiar with the Pythagorean theorem

**Apprentices:** Andrew Ma, Shira Polster, Brandy Wiegers

**Abstract:** we will use a mathematical auction to encourage small groups of students to work together to explore Pythagorean relationships, the idea of self-similarity, and fractals.

**Series A:** 3-4-5 Pythagorean tree fractals

The base fractal in A1 is built around a triangle with dimensions 3-4-5, with squares constructed on each side of the triangle. In A2, we extend the original A1 figure by constructing two triangles similar to the original triangle on the sides of the two smaller squares, as shown. In A3, we continue this process of constructing triangles similar to the original triangle on the sides of the smaller squares, as shown. And so on … What happens to the area and perimeter as you do this?

**Series B:** Isosceles Pythagorean tree fractals

All the triangles in the B-series are isosceles right triangles. The biggest square in each figure has area 64. What happens to the area and perimeter of the fractals as you extend the series?

**Series C:** 4-12-16 square Pythagorean tree fractals

All the triangles in the C series are similar triangles. The three squares in the first figure, C1, have areas 4, 12, and 16. What happens to the area and perimeter as you extend this series?

After they have time to investigate these fractals, we will run an auction in which the groups can “bid” for the right to present their partial solutions to a particular fractal.

What is neat about this process is that it encourages students to present even partial solutions and to observe and build and learn from each other, because groups will get to observe the other presentations and then bid for the right to improve their solutions. There is a fun element of “gambling” built in–deciding how much to bid and when to bid. I have noticed that some shy groups are more cautious and reluctant to bid in the first round, but then they observe the other presented partial solutions and realize, “Hey! We can improve on that!” and bid in succeeding rounds.

Credit: All the fractals were created in Wolfram Alpha. It is easy for you to create more Pythagorean fractal trees using any right triangle you like as the “seed”. Just use this link:

http://www.wolframalpha.com/input/?i=pythagorean+tree

A session hand out is attached.

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Pythagorean Fractal Auctions Houston handout.pdf | 184.55 KB |