It seems like in undergrad, the line sounds kind of like this: “Just pick a nice open-ended question and have the students discuss it.” It sounds really good, too.

Except sometimes the students aren’t in the mood to talk. Or they would rather talk about a different part of the problem than you intended. Or their skill set isn’t strong enough in the right areas to engage the discussion. Or the loudest voice in the room shuts the conversation down. Or… something else happens. It’s a fact of teaching. Open-ended questions don’t always lead to discussions. And even if they did, class discussions aren’t always the yellow-brick road lead to the magical land of learning.

But sometimes they are. Today, it was. And today, it certainly wasn’t from the open-ended question that I was expecting. We are in the early stages of our unit on similarity. I had given a handout that included this picture…

… and I had asked them to pair up the corresponding parts.

The first thing that happened is that half-ish of the students determined that in figures that are connected by a scale factor (we hadn’t defined “similar” yet), each angle in one image has a congruent match in the other image. This is a nice observation. They didn’t flat out say that they had made that assumption, but they behaved as though they did and I suspect it is because angles are easier to measure on a larger image. So, being high school students and wanted to save a bit of time, they measured the angles in the bigger quadrilateral and then simply filled in the matching angles in the other.

Here’s were the fun begins. You see, each quadrilateral has two acute angles. Those angles have measures that are NOT that different (depending on the person wielding the protractor, maybe only 10 degrees difference). And it worked perfectly that about half of the class paired up one set of angles and the other half disagreed. And they both cared that they were right. It was the perfect storm!

Not wanting to remeasure, they ran through all sorts of different explanations to how they were right, which led us to eventually try labeling side lengths to try to sides to identify included angles for the sake of matching up corresponding sides. But, that line of thought wasn’t clear to everyone, which offered growth potential there as well.

By the time we came to an agreement on where the angle measures go, the class pretty much agreed that:

1. Matching corresponding parts in similar polygons is not nearly as easy as it is in congruent polygons.

2. Similar polygons have congruent corresponding angles.

3. The longest side in the big shape will correspond to the longest side in the small shape. The second longest side in the big, will correspond to the second longest side in the small. The third… and so on.

4. Corresponding angles will be “located” in the same spot relative to the side lengths. For example, the angle included by the longest side and the second longest side in the big polygon will correspond to the angle included by the longest side and the second longest side in the small polygon. (This was a tricky idea for a few of them, but they were trying to get it.)

5. Not knowing which polygon is a “pre-image” (so to speak) means that we have to be prepared to discuss two different scale factors, which are reciprocals of each other. (To be fair, this is a point that has come up prior to this class discussion, but it settled in for a few of them today.)

I’d say that’s a pretty good set of statements for a class discussion I never saw coming.

I like the orientation of the quadrilaterals. The diagram is a great example for students to think about how they can show the two triangles are similar using transformations. A dilation is not enough. We would need a set of rigid motions along with a dilation to show that one figure can be obtained from the other. Thanks for sharing this!