# A Busted Lesson + Polygraph

I brought Desmos into my class this week to try and help my students visualize and analyze exponential functions. The idea was to give them a bunch of situations — starting values, growth/decay rates, etc, and have them graph and answer questions about them. One thing I wanted them to take away from this lesson was the process of resizing the window — saying hey, this function starts at 400 and gets bigger, so I’m going to have to see more of the y-axis. Of course, if my students graph a pretty typical exponential function, like below, and the just zoom out, they end up with a bit of an awkward view:

When in most situations it’s a much better idea to visualize the function like this:

I had kids warm up with some pretty simple graphing and analysis, then they tried some problems where they’d need to do some resizing. I put together a little walk-through with some screenshots and arrows. Some kids figured it out, but most were confused. I brought the class together, modeled a problem, then let them try another. They were still really struggling — and all of their thinking energy was going into the window, and not into the problem-solving I was looking for with exponential functions. We stuck with it until the end of class for my first group, but I knew that I hadn’t been successful. My later classes, I scaffolded things more, but still saw them struggle, and kids were losing interest fast. I decided they were lacking some significant domain and range background knowledge, and I need to take a step back before we pursue this again.

So I have 20 minutes left and we need something to do. I decided to go with the Desmos lesson Polygraph: Hexagons. I’m planning on doing the Polygraph lesson for parabolas sometime soon, and it seemed as good a time as any to test out the lesson and do some mathematical thinking rather than scrap the class.

Quick summary: Polygraph let’s kids play Guess Who against each other, with math. One kid picks an object, in this game a hexagon, from this selection:

Then, a second student asks yes or no questions to try and narrow down the options. Looks something like this:

I set it up with very little introduction, and let them go.

Things I learned:

1. Students really like this. It’s fun to play against your peers, and the interface and game makes it engaging.
2. Some students needed more direct instruction on the exact structure of the game — one early game looked like this:

3. Most of my students didn’t arrive naturally on the idea of a “good question” — a question that will eliminate as many shapes as possible. Instead, there were lots of questions along these lines:

That aren’t specific, and will in most cases only eliminate one or two hexagons.
4. It’s pretty tempting to get silly when students get frustrated. No fault of theirs, they just kept losing, and their questions got lazy and looked something like this:

or this:

In summary: I’m really glad I did this in a low-stakes way before committing to it for parabolas. First, I think on the logic side of things, this lesson is harder than parabolas — parabolas have a few very distinctive, easy to name features like opening up or down that students can latch onto. Second, I think it’s fine to have kids start pretty cold, but I at least need to make sure they understand the structure of the game — yes or no questions to narrow down the possible choices. Then, I’ll be ready with a list of vocabulary I’m looking for students to develop, and stop the class to point out high-quality questions their peers asked. It would be awesome to pick out a few early pairs who win (a large majority of students failed to guess correctly with the hexagons) and highlight their questions. Then, keep shouting out students who use that vocabulary. That, after all, is the reason that Desmos built this lesson — to make mathematical vocabulary more meaningful, and my teaching in this lesson was not focused enough on that goal.

I’m excited to give it another go. We’ll see how it goes!