A while ago I saw a challenge in a book. The challenge was to take pentagons made up of five triangles and find all the different ways in which a person can color them with five different colors. What counts is the relationship between the colors, so red-blue-green-yellow-black is the same as blue-green-yellow-black-red or even black-yellow-green-blue-red. The challenge sounded fun so I printed off a bunch of little pentagons on some cardpaper and started coloring. My kids joined in.

The next step of the challenge was to take the little cardpaper pentagons and arrange them so that they could be taped together into a dodecahedron so that the edges of the tiles touch identically colored edges. This is possible, though I discovered that I had to color my tiles on both sides as some of my original 12 tiles were the mirror images of what I needed.
After creating an amazing little dodecahedron I felt the need to repeat the activity but laying the tiles out as a flat net instead of taping them together. Try it! It is actually incredibly hard to do, but easier if you cheat and fold the dodecahedron up as you go along and then untape it and glue it flat. The really neat thing is the pentagons that end up only attached to just one other pentagon, and how before you glue them you can sort of roll them along and see how they would attach to the other pentagons if the net were folded up.
My next projects I got the kids to help me with even more and these I managed to get pictures of. One was to glue some of the small pentagons to a piece of black paper to see what patterns we could create. The result was the above where it almost looks like every star could have five circles surrounding it and every circle could be part of surrounding three stars…. but not quite, because at some point the circles start to join together.
I wanted to see what the shape would look like if I continued it larger, so I started trying to duplicate it on the computer. I was using a open office draw, and copying and pasting pentagons, and since I am just new to using that particular draw program I couldn’t figure out how to specify that I want something rotated to a specific degree. Instead I was trying to turn them by hand and at first, I was viewing the pentagons surrounding the circle as though they were pointing in ten different directions, each of which I had to carefully tilt. Then I realized it is just as possible to see the same shape as a combination of five pentagons pointing up and five pointing down. The picture below was created to show that different way of looking at the same shapes:
As I worked on the shapes and tried to share them as best I could with my children, I found myself reflecting on the very idea of what education is. I kept wanting to justify, in some way, “teaching” them this stuff. Spacial reasoning is important for math, isn’t it? For engineering? Then I found myself comparing a picture to a story. There are so many stories I want my children to know, so that they can recognize the stories when they are referenced to, and because the stories share some way of looking at the world and addressing the tough questions of life. Could a geometric figure have similar relevance? Is it a mathematical story explaining the relationship between angles and lengths?

The next creation was inspired by a quilting pattern in a quilting book. Partway through making it one of the kids stole the scissors for his own project and I took to just folding and ripping the paper. I was kind of rushed and the kids were helping so its not great art, but the math in it is fun. Each color makes up just about 1/4 of the overall design, so for each color you need a square the quarter the size of the picture. That square is cut into quarters. Each quarter is cut in half (to make a triangle an 1/8th the size of the original square obviously, or 1/16th the size of the final image). One of those pieces is glued into a corner of the quarter, and the next piece is cut in half again. Half is glued on, the other half is cut in half again… repeat, repeat, repeat until the pieces get too small to work with.

Edited to add: I just stumbled across another great resource. The Toymaker has lots of paper toy patterns. Many of them just involve printing, cutting and gluing but some of them can be colored. I printed out her dodecahedyron pattern and got the kids to draw snowmen in ten of the pentagons. Then we cut, folded up and hung it on the Christmas tree. Five of the snowmen had to point outward, away from a central pentagon, and the other five (the top ones) had to point in towards the central pentagon.

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