Yesterday (August 14, 2014) I saw this post from Dan Meyer, which is part of what I’m sure will be an interesting series:
Developing The Question: Good Work!
Something made this post stick in my mind for the rest of the day, though even now I have not been able to put my finger on precisely what it was. After the kids went to bed last night I decided to use one of the problems in the post as a starting point for a little math project this morning. My hope was that the combination of thinking through and then going through a little talk would help focus my thoughts on the original post. It failed in that regard, but talking through some of the ideas that were motivated by Meyer’s post was pretty fun nonetheless.
We started with a slightly modified version of one of the problems from the MathArguments180 blog:
Three Questions about three circles
I made the problem with four circles and a square rather than three circles and a triangle so that my younger son wouldn’t have to worry about finding the area of a triangle. Stepping through the three questions got the project off to a great start:
One of the things about this problem (and all three problems referenced in Meyer’s post) that I thought was interesting was the wide variety of possible extensions. One of my favorites goes back to multiplication. Chopping up a rectangle into smaller rectangles is an neat way to show a geometric interpretation of multiplication. Perhaps a little more surprising is that you can use the exact same idea to show why a negative number multiplied by a negative number is a positive number:
The idea from the negative times a negative piece of the last video has some other fun geometric extensions. By coincidence, yesterday my younger son and I were talking through a neat problem about a cube with some holes in it. He actually built a model of the object out of snap cubes to help him understand the problem and we used that model today. Yesterday’s question was about the number of 1×1 square pieces on the surface, so to mix it up a little today I asked about the number of cubes in the model. My goal here was to show how the “taking away more than you had” idea from the last video gives us a new way to study this cube:
For the last bit of the talk we went back to the whiteboard. I wanted to show an example from geometry that would look similar to the first problem, but might give the boys a chance to understand the “take away more than you have” example in a slightly different setting. They saw the problem in a little bit different way, though. That’s fine, obviously, so I ended up just showing them the “take away more than you have” approach at the end and showing that both approaches lead you to the same answer.
One last bit that I wanted to talk through that I didn’t have time to get to was something from an old 3D printing project we did. At the end of this post – Learning from 3D Printing – are some pictures showing a solution to the “Prince Rupert Problem.” Learning how to create the red shape in Mathematica was a really fun project whose main idea involved breaking up a space into pieces you already could understand. Fortunately Mathematica has built in capability to add and subtract shapes in 3 dimensions.
Anyway, I did really enjoy the 3 questions idea from Matharguments180 blog. It was also fun to talk through some of the various fun and surprising extensions of the ideas from the first problem. I may update this post if I am ever able to put my finger on why Dan Meyer’s post stuck in my mind, but for now the result of that stickiness will just have to be a fun little math project with the kids this morning.