Preparation
Dr Evans' activity was based on one on the website www.FractalFoundation.org.
She printed several copies of triangles from that website, in two versions. The easy version, for the younger children, had lines filled into the triangles. The more difficult version, for the older children, had just the outline of the triangle. Both are available on that website, along with lots of great pictures that can supplement the lesson and illustrate what's going on.
Dr Evans also brought crayons, markers, and pencils for coloring triangles, and scissors for cutting them out.
In addition, she brought examples of fractals: two ferns she picked up at flower shop, and a book with pictures of mountains, snowflakes, etc.
Beginning
Dr Evans welcomed the children to Math Circles. She requested that the younger children sit on one side of the room, and the older ones on the other side, so they were pretty well sorted by ability at the beginning.
She asked students who could remember the Math Circles rules, and we went over them one more time:
Rules:
1. Make mistakes
2. Ask questions
3. Have fun
4. Help others have fun.
At this point, Dr Evans explained what a fractal is. A fractal is an object that is "self-similar": it contains a pattern that repeats again and again. She brought in several examples.
First example: A fern. Note that a fern has a stem with many stems growing out of it. Each of those stems also has many stems growing out of it. At the next level down, the veins in the leaves radiate out of the stem just as in the two previous levels, and so on. She passed around the two ferns she had brought.
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| A student showing the ferns, with Dr Evans in the background. Note also the fractal pyramid! |
Activity
After the discussion, Dr Evans handed out the papers she had copied with triangles on them. Those who had a blank triangle would be drawing a fractal on their paper. Those who already had a fractal inside their triangle would be coloring.
The picture above shows one student's first two levels in creating the fractal. First, an upside down triangle is drawn by connecting midpoints of the original large triangle. This leaves three upright triangles. Find the midpoints of each of these, and connect them. This leaves nine upright triangles. Find midpoints of these and connect them. Students can keep drawing triangles as long as they would like.
... And many of the children did draw triangles for a long time....
After the triangles were finished, the students were instructed to cut them out.
About 30 minutes into the lesson, Dr Evans took a group of those who were finished out into the hallway where there was a wide space to build a fractal out of the triangles that the students had colored and decorated. She asked some of the older ones to help her form a new, larger fractal. (Those who were still coloring stayed in the classroom -- remember how we recommended having extra adults for kids this age? This is one place where that really helped.)
Assembling the fractal in the hallway is tricky. Many students just want to stack triangles on top of each other. You can point out to them that they are making a mistake! Which is a good thing -- they're following the rules. But then they need to figure out how to fix the mistake.
Make sure they are building the triangles in a repeating pattern. The upside down triangles that were drawn in the steps above will be carpet when you spread the triangles on the floor. The smaller white triangles (with fractals drawn on them), will fill in the fractal.
In the above photo, the children have placed a few more triangle. In our class, it was really just two or three of the older children who got excited about placing the triangles on the floor. A lot of the others were very happy to continue coloring for a long time.
When the fractal was all done, the children lined up in a row on the bench behind their creation, and parents took pictures.
By then, 50 minutes had passed, so we reconvened in the classroom for cookies.
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