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This is a project I've done for three years. It is not necessary that this be part of a unit on fractals, though it certainly could. It is part of the assessment package I use for my unit on fractals and chaos, which lasts about 3-4 weeks. The project itself is introduced two or three days into the unit, after the students have been given some exposure to fractals, and to the concepts of self-similarity, iteration and fractional dimension.We start off by seeing a number of fairly famous fractals that the students could easily create by hand:
In each of these fractals, emphasis is placed on self-similarity and iteration. Self-similarity can be described as "a piece looks like the whole thing, and a piece of the piece looks like the piece, etc." Iteration can be described as the repetition of a process over and over again, where the output of one iteration becomes the input for the next.
- Koch snowflake
- Sierpinski gasket
- Cantor dust
- Dragon curve
- "Purina dog chow"
It is at this point where I give the students the assignment for the project - they are to create an original hand-drawn fractal. It must show self-similarity and evidence of an iterative process. It must go to at least stage 3 or 4, and could certainly go further, depending on the complexity of the fractal. I must see each stage separately, from stage 0 to stage n (the final product), although stages 0 through n-1 may be in a rougher draft. The only other "instruction" I give (if you can call it that) is that "color often looks nice."
This is often a difficult project for the students, for a number of reasons. First, they are fairly unfamiliar with fractals, so the start of the process is a bit rocky. Also, they are given no guidelines other than those mentioned above. I find this to be a help rather than a hindrance, however, as it lets the students turn loose their creativity, and doesn't give them the "crutch" of trying to create what they think I want to see.
Paul Kelley
Anoka High School
Anoka, MN 55303
pkelley@informns.k12.mn.us
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