Yesterday I wrote about some of the difficulties I ran into and poor decisions I made while teaching the fractals unit/project in my geometry class. [See part of the project here, and the grading scale/rubric here.] Today I want to focus on two awesome aspects of that project.
First, the students loved this online site where you can manipulate line segments to produce fractals like the Koch snowflake curve and the Koch quadric island curve (see third variant here). Part of it may be just because playing on computers is both more natural and fun to them than drawing by hand (Note: I didn’t get any “Mr. Yates, this is too much work!” ‘s during this section of the project).
But part of what made this exciting may also have been that this was the only section of the project where the students actively related fractals to real-world natural objects, like I had shown them during my lecture on the subject. In addition to the Koch snowflake and quadric island curves, I asked them to create: one fractal that looked like a cloud, one that looked like a tree, one that looked like an ocean wave, one that looked like a coastline, one that overlapped itself (compare some of my overlapping Koch-inspired creations here – can these still be called fractals?), and one that didn’t necessarily look like anything but was just so cool it had to be included anyway.
Even more amazing was the chance to collaborate with an art teacher! Ceilon Aspensen, an art teacher at my school, is both a great teacher and interested in math, and she has taught fractals to her students before. At the start of this project, I got together with her to discuss how we could collaborate and build a project with cross-curricular connections. We decided upon creating an enormous fractal using linoleum and printmaking techniques.
The students picked the Sierpinski Triangle (ST) as what they would build together. First they used their measurement skills to draw an equilateral triangle on the linoleum, and their knowledge of the iterative ST process of connecting midpoints and removing the central downward-facing triangle to create a second-iteration ST.
Ms. Aspensen then taught them how to carve out all the linoleum that would not be a part of the raised printing surface (think of a stamp). Then she taught how to ink the surface and make a print (without smudging). And we were ready to go to work making a large triangle!
At this point, the students measured the paper and we went through the powers of 2 to determine how many of our triangles could fit given the length of the paper / board it was hanging on. With the paper about 72″ long, we could make a 2^6 = 64″-base equilateral triangle with 16 of our prints across the bottom. We also, after an initial mistake on the next row up, looked over the interesting patterns in creating each subsequent row of Sierpinski’s Triangle (print, don’t print, print, don’t print, etc. for the second row from the bottom; other patterns further up).
So, here’s what’s done so far! A 2/3-done (or, if you count the large empty upside down triangle in the middle, 3/4-done) sixth-iteration Sierpinski Triangle. Not quite at this level of detail, but still quite nice.
Ms. Aspensen had her art club do some work on this along with / after my geometry students last week, and I believe they will finish it this week, traversing the art-math connection in the other direction. When it is finished, we hope to hang it prominently along the school library hallway! I’ll be sure to post another picture of the entire fractal then!