I was looking for a way for Geometry students to wrestle with similarity ratios. Daily, I try to escape the "this is how you do this kind of problem" mindset that plagues them, that they prefer, that is a trap we keep falling into. I suspected that Avery's "rank these rectangles according to squareness" problem would be worthwhile. I was excited by what the kids did with it.
Here's the handout I gave out, that I basically copied from Avery:
We had to really unpack 3. "devise a measure of squareness" to get them beyond "it just looks closer to a square." They were able to understand what I meant, but only after I stated it several different ways.
- Come up with a single number for rectangle E, and a single number for rectangle D, that shows that E is more square than D.
- Find a way to do a calculation for each rectangle, so that you can compare all the results, to see which is the most square.
- etc, rephrased a dozen different ways.
Ideally I would like to find a concise way to phrase this for students, that would get them closer to understanding what we are looking for by themselves.
Eventually, lots of groups came up with either or both:
- shorter/longer, and closer to 1 is more square.
- longer - shorter, and closer to 0 is more square.
No groups on their own noticed that you could break the subtraction method by choosing uncooperative rectangles (and I don't know how to get them to do this organically.) But after we reported out our methods, and I said "I'll tell you that the subtraction method doesn't always work. See if you can find some examples of rectangles where the subtraction method gives you misleading results." they were interested in looking for ways to break it, and lots of groups found counterexamples.
I asked each group to summarize their method in a google doc...here is the raw output.
We had some really interesting methods, too!
- Draw a diagonal and measure the angles made by the diagonal and the sides. The closer the two angles are, the more square. (Way to draw a connection to the properties of special quadrilaterals! I pounced all over that like...something that pounces enthusiastically on something. So cool.)
- Attempt to circumscribe a circle around the rectangle. The closer you can get the vertices to all touching the circle, the more square it is. (We did not have time to delve into this today but MAN, THAT IS GOOD. I would really like to find a way to get them to all think deeply about that.)
- A somewhat crazy method involving comparing the average of the four side lengths to the length and width separately. I encouraged this group to write their method algebraically, but they ended up with an expression that always reduces to zero. However, it doesn't always reduce to zero when they do it with numbers. Would like to spend more time on this, too.
In a nutshell, I had a great time this morning. Philosophical side note: I have been engaging in these thought experiments recently, often, for some reason, about the idea "Could I be happy...?" (Could I be happy if I won the lottery and didn't have to work? Could I be happy working at Walmart? Could I be happy being a housewife? Could I be happy as a math teacher forever?) and what I have come up with so far is that I can be happy as long as I am spending a non-trivial amount of my time learning something fundamentally new - about myself, about other people, about the way things work... ("fundamentally" implying, well, fundamental...specifically not something stupid like how to navigate the menus in a new app or what Loft has on sale this week.) In order to be happy being a math teacher forever, I need more days like today, where I am getting kids excited about learning. As ultimately cheeseball as that sounds.