Polygraph: Parabolas and Productive Struggle

Taught the Desmos lesson Polygraph: Parabolas yesterday. If you haven’t seen it, go check it out. Ok, if you’re too lazy to check it out, students join the lesson on your class’s device of choice, and play a game of Guess Who with these parabolas:

One student picks a parabola, and the other student asks yes or no questions about it and tries to eliminate parabolas until they discover which graph their partner picked.

Is this really worth doing with computers?
It absolutely is. Our Chromebooks are kindof a pain to get kids onto, getting everyone’s login right and et cetera, but the Desmos lesson imposes some important structure as opposed to the in-person version — it forces students to play honestly, it only allows answers in the form of yes or no, and it connects students so that they can play 8 games against 8 different opponents, seamlessly. Plus it’s fun.

Productive struggle….or just struggle?
There’s a ton of opportunity for productive struggle here. But I think it’s also important to avoid unnecessary struggle and focus on what kids should be struggling with. I tried this lesson with hexagons a few weeks ago at the end of a different lesson didn’t go well. The biggest thing students struggled was asking yes or no questions. This time, I really carefully framed what questions students would be asking, and made sure students understood what that meant before we got into the lesson — having students ask “does it open up or down” when the other person can only say yes, no, or I don’t know is a distraction from the potential for learning here. Also, you should mention to your students that the grid a “guesser” sees is not the same grid the “picker” sees. So asking “is it in the first row”, “is it in the second row” is not a good strategy. Students figure this out, but it’s not worth the time wasted trying it.

Speaking of those questions…
So I launched the lesson with a little bit of explanation, and a prompt for students to think of questions that would be useful to ask. Here are my students responses
Some students who struggle with math:

And students who are a bit stronger:

They clearly got the idea that they were asking yes or no questions. Awesome, let’s put that confusion aside and talk about quadratics — because the range of questions here is pretty limited. We nailed down the vocabulary — clarifying that when students asked if a quadratic was positive, what they were asking was whether it opened up. Brought in vertex,axis of symmetry, intercepts, quadrants. Pointed out that “is it wide” is a bit subjective (I wish I’d actually had students debate which ones were wide; this question kept popping up). Put this all on the board for them to reference. Now students  were armed with some good vocabulary, and could get started.

I actually did that whole part wrong
Ok I don’t have any huge regrets, but my students have done a bunch of work with quadratics. They’re not experts, but they’ve seen much of the vocabulary before. The right way, in my opinion, to do this lesson is to introduce quadratics with Polygraph, let students struggle with the language, and then introduce the vocabulary. Desmos says as much in their lesson outline, and I will definitely use it as an introduction next year. That said, it was really useful formative assessment and practice for features of quadratics.

So what did students actually do?
So there was plenty of silliness. Students wrote things like this:

But as the lesson went on, students developed the vocabulary. I think it’s important to remember that the logical skills to work through this lesson do not come naturally to every 14 year old. That said, students really wanted to win, and encouraged each other, or gave each other a hard time if they weren’t taking it seriously. As the lesson went on, there were a lot more rounds that looked like this:

Ok so they really like asking whether it’s wide. And the questions still aren’t perfect. But this is pretty significant growth from where students were. I think the biggest evidence of learning for me is that classes usually started with about a 25% success rate, but by the end they were winning around 75% of games. That’s a big difference. And most important, students were constructing arguments, critiquing the reasoning of others, laughing and having fun during math class, and really, really engaged. That’s a big win.

How to manage this?
Teaching other Desmos lessons, I’ve let the teacher view sit up front on the projector — in Function Carnival, kids can reference other kids’ graphs, or in Water Line they can see other the objects appearing in the cupboard (go try those lessons if you haven’t already!). In this lesson, I floated around the room a bunch, but I mostly perched in the back, watching the teacher view and keeping an eye out for key learning moments or issues coming up. It was kindof nice to avoid the temptation to write a goofy name to show off for friends or to ask a question just to get laughs.

Anyway, each class I brought everyone together partway through to anonymously highlight a few rounds and ask students what they thought of the questions being asked and how they could be improved. I think this was a key part in their evaluation and analysis of quadratics. That said, I think the most important thinking and reasoning came between students. Hearing students badger each other (mostly playfully) when they made a mistake, or push each other to ask better questions so they can win, or yell across the room that they meant to push yes and not no. Students were engaged, and real argument and reasoning has to come from real engagement. It was just fun to be in the room and to see how much they enjoyed it. My most important teacher moves came when I saw students losing over and over again.

That’s a screenshot from the teacher view with names cut out — pretty easy to tell which students need me to look over their shoulder and give them some extra coaching.

The teacher view is one of the best parts of the lesson. I can see at a glance how each student is doing, as well as a scrolling view of the current games being played. I can also zoom in on an indivudual game, or a student. It’s not too hard to use at a glance, and I was able to respond to much of the information immediately.

A bit of an afterthought
In between rounds, students had to answer a question. There were four total, and they were great, something like:

I wish I had framed this for students, and emphasized it as a meaningful part of the lesson, rather than an afterthought. I got lots of lazy answers because students wanted to get back to the game, and I hadn’t messaged to them that it was important. These questions promote some great thinking from all students, and could serve as a much more authentic source of formative assessment if they had taken the questions more seriously.

In sum:
This lesson is awesome. There was almost no prep, kids loved it, there was some great learning, and I will definitely revisit Polygraph later this year. I think there are some challenges in getting students to ask the right questions, and creating meaning in the vocabulary rather than sticking with “is it wide” and “does it open up” over and over again. I would also love to see a way to track the class winning percentage over time as a low-stakes way to encourage students to take it seriously. But all in all, I’m a huge fan of getting students excited about and talking about quadratic functions in class, and I can’t wait to see what Desmos puts out next.

3 thoughts on “Polygraph: Parabolas and Productive Struggle”

1. Pingback: parabolas! | sonata mathematique

2. Lisa

I appreciate that you tried to get students to think about good questions ahead of time by showing them parabolas on paper. If I had computers for my students to use and was planning on doing the polygraph activity, I think I would want to do something similar to help direct their questions. I think I would include these questions also:
“In between rounds, students had to answer a question. There were four total, and they were great, something like…”
In fact, I think I would start with those questions first and then move to the groups of four. Maybe that would help them think of better questions to ask.

3. Ben

I’m super late to the Polygraph Party… However, I just did my first one (Polynomials) and having read this beforehand was really beneficial, especially the idea of doing a brief introduction example to orient them to the question type. Thanks!