One thing I love about math is that there are so many ways to see things. My 11 year old friend Zack came over and was very excited to show me something he learned at school. He got a ruler and paper and sat down and meticulously created a square, then found the midpoints of the sides to create another square and then found the midpoints of those sides to create another square and on and on. I, of course, completely jumped to the conclusion that I knew what he was doing even though he said, "It's not what you think."
Naturally I imagined a traditional symmetrical design that highlights the squares and midpoints. - like the one to the left. But then Zack began coloring his design starting with a tiny triangle in the center and working outward. I couldn't figure out what he was doing and when I came back about a half hour later his work completely surprised me and others who had been visiting at the time.
I
realized then that Zack and I could spend days exploring his drawing along with a myriad of other drawings that share the exact same lines. This is when I longed for a technology tool that would enable us to draw a perfect square and could easily find the midpoints of the sides and could connect those midpoints to create a subsequent square; then we could make copies, stretch it, turn it, color it in a variety of ways and really see what we could find out about it. It's these times when technology can take your mind off the measuring and drawing, and can give you a chance to explore the pure mathematics of the design. At times like these I'm glad to have a tool like
Geometer's Sketchpad, which I went ahead and played with to make the designs above.
But to begin, Zack and I don't need anything. I'll probably just ask Zack if there's more pink or more orange in his drawing and how he could prove it one way or the other. Then I imagine we'd start thinking about ways to color the original square such that there's the same amount of pink as orange. And that would just be the start of the conversation. At the point when he wants to start drawing another copy I'll probably show him Geometer's Sketchpad and see where it leads. Stay tuned! Zack will be back and we'll see what we come up with.
I've been playing with Zack's spiral design which I mentioned in the previous post. It's fun to have it in a tool like Geometer's Sketchpad where it's a real mathematical object and you can play with it and change it and color it in different ways. I was surprised to think somehow this design was not like Zack's because this had four spirals and it seemed Zack's had only two. However, re-visiting Zack's design I see he used each color twice: a simple difference but it seemed quite different to my brain. 
I keep thinking about this in terms of an assessment I saw at a conference where they gave students a square and asked them to shade it to show four equal parts, and to do this in several different ways. The four color version would be a good solution. Zack's two color version would be a good solution to shading the square to show two equal areas. Of course in each case there's a little empty square in the middle that would have to be divided as well.
