Which Shape Covers More: Orchestrating a Sixth Grade Math Debate

First, we must mourn for the loss of the Way We Used to Teach.

I love my work. If I let you scroll through the photos on my phone, you’d find hundreds — thousands? — of photos of kids at my school engaging in math. (Mixed in with photos of, you know, my actual children!) I miss so much about teaching in a classroom that didn’t have to be online or governed by social distancing measures.

But here we are. This year, I’m teaching sixth grade math in a school that is scheduled to be fully remote. The days are loonnnnng. I am grateful for my students and colleagues.

Throughout the year, I will continue to mourn for the way things used to be, but I will also celebrate the ways things are working.

One way things are working: I have unprecedented access to student work.

Setting the Stage

Right now, my primary tech tools are zoom and Desmos. I use other things, too — Canvas is our district LMS, and I incorporate jamboards and google slides, etc. — but I do not know what I would do without Desmos. It allows me to focus on student work and student thinking.

The other day, I asked my students to determine which shape covers more of the plane: blue rhombuses, red trapezoids, or green triangles. This task comes from IM 6-8 Math (Grade 6, Unit 1, Lesson 2), and it was adapted for Desmos by Cindy Whitehead (@finding_EMU).

I anticipated that some students would say that green triangles covered more of the plane, because there are more green triangles. I anticipated that others would say that red trapezoids covered more of the plane, because they take up more space. (1 red trapezoid takes up the same amount of space as 3 green triangles, so there would need to be three time as many green triangles as red trapezoids in order for the areas to be equivalent. Some students might even use some understanding about ratios (either intuitively or through instructional experiences) to see that each figure is the same hexagonal pattern repeated 8 times, which means that we can use what we know about the small hexagon and then scale up.

This visual is available for students to experiment with (fading the shapes in and out, and increasing the visibility of the orange frame) on a geogebra applet. I kept the applet open in a tab so that I could share it with students. I also kept a tab open with a link to the Math Learning Center web app for pattern blocks, in case students wanted to experiment with equivalence in that space.

Monitoring Students at Work

I assigned students to breakout rooms, and set them off to work. As they worked, even though I could not hear or see them in the breakout room, I could watch what developed on the screen via my dashboard. (click on images to enlarge)

I also received live updates of which shape students covered more. I did not anticipate we would have consensus, but the level of disagreement excited me.

(As exciting as it would be to have some of these famous mathematicians join us for sixth grade discussions, these are not my students actual names. Desmos offers a great anonymizing feature, which I use when I share this data with the class, too.)

With this much disagreement, I imagine we were ripe for a Mathematical Debate.


After bringing students back to the main meeting room, I presented the data. “What do you notice?”

“Is the answer actually d: they all cover the same amount?” Hanima asked.

I smiled. “No, I was not trying to trick you. I will say that I’m excited that people have made cases for each of the three shapes, which means we can have a — drum roll, please — math debate.”

This was only our fifth day of class, so I imagine students did not know what to expect. I continued: “I will put you into breakout rooms with people that disagreed with you. Your job: convince your classmates that you’re right. Or change your mind, so that you now agree with someone else.”

Grouping Students for Debate

In a typical classroom, I might have had students debate with their table groups. Or I might have had students go with pre-formed partnerships. Or I might have had students move into three different corners of the room to show their thinking — one corner for trapezoids, one for rhombuses, and one for triangles — and then broken students up into groups from there. Or some other strategy.

In the virtual space, I assign breakout rooms. Sometimes, I let zoom generate these randomly, but sometimes I create the breakout rooms manually so that I can be more deliberately about groups. So:

Quickly, I assigned the students arguing for red into 4 different breakout rooms. Then I assigned a student arguing blue into each room, and a student arguing green. Students who had not selected one of the three options were randomly assigned. The whole process only took about 30 seconds. It did mean that students watched me vamping. I stared intensely at my screen, working as quickly as I could.

Breakout Rooms

I jumped between breakout rooms. In every room, I was greeted by a student’s passionate argument. In one room, sixth grader Lucy was sharing their screen to illustrate their argument better. (I took a video recording of this breakout room — however, I vowed that the recording would only be for me, so, sadly, I can’t post it here.)

Lucy’s screen

“The triangle is like 1. It’s the smallest shape, like a unit. Then three of those fit into the trapezoid. I labeled them 1, 2, 3. Then 2 fit into the diamond or rhombus or whatever,” Lucy explained. I could see Ibrahim, who was only visible from the eyes up, nodding along.

Lucy shared their image of the first pattern.

Lucy continued. “Then I… um, very meticulously… counted each triangle for each color,” they said with a laugh. “So I went through, and I was like, ‘okay, let’s do all the blues. Okay, 1, 2, 3… whatever! And I counted all of them. Once I figured out all the numbers, red covered the most.”

“Does anyone else want to share?” Lucy invited. (A natural teacher!) Feng and Jun looked way from their cameras. After a pause, Ibrahim began.

“I noticed that there are hexagons within the shape. I traced one, and tried to see out of one of them which one would take up the most space. I’m still debating on that. I kind of think that red wins over now, because… if you put red in the middle, it would equal a hexagon.” Ibrahim circled the three blue rhombuses that made up that same hexagon in the middle. “I don’t know. I just think red is bigger.”

There were still a few students that seemed quiet or unsure. They listened to their classmates, and I noticed that a few quietly changed their responses on their Desmos slide. Red went from 5 students to 8. Jun, who was in a breakout room with Ibrahim and Lucy, was one of them.

“I thought green was more,” Jun explained in her breakout room. “It’s kind of split up into cubes, and in each cube there are 7 green pieces, and 4 blue pieces, and 3 red pieces. There are 8 cubes, so… yeah.” She paused — a long pause. “There are more green but now I think the red covers more of the big shape.”

Polling the Class

While listening to students in breakout rooms in my first class, I set up a poll on zoom. My district recently upgraded my account to Zoom Pro, and I’m determined to take advantage of some of these features.

I forgot to take screenshots of the polls, and apparently Zoom does not save the results. But I swear: students were dramatically persuaded by their classmates! In most classes >90% of the students now believe the red covers the most area.

In one class, a full 100% of students changed their mind. In that class, they had been evenly split with people arguing red and green, with no one arguing blue. My hypothesis is that this means that there was a fundamental disagreement about whether the question called for the greatest quantity or the greatest area, and once students resolved that ambiguity, they arrived at the same answer.

It’s hard for me to know why other students changed their minds. I could have asked them to send me a private chat — since I did not have any other documents prepped — but we were already perilously close to the end of our class time.

Synthesizing Ideas

I asked 2 students in each section to share. Then, I introduced the word AREA.

original c/o Morgan Stipe

I told students that area is measuring how much space a surface covers. We would build on this idea the following day. For this particular lesson, I removed the definition from the word wall card (shown above). Students were still building their informal definitions, and I did not want to disrupt that process. I revealed the full definition the the following day.


Working in the breakout rooms has allowed for students to get to know one another better. Students at my school are coming from 8 different K-8 schools across the district, which means that they are getting to know new classmates in an exclusively virtual space. Some breakout rooms are silent. Cameras off. Others are chatty, with students working in parallel. Some are even lively, with students working collaboratively. We’re only a few days in.

Originally, this task was intended to be the first of two activities in Desmos. However, I let the breakout rooms go long. In general, students indicated that the amount of time in the breakout room was fine — not too long or too short — but it’s hard to know when I can’t easily take the ‘temperature’ of the class, like I can in a bustling in person classroom.

I think lessons are taking longer in the virtual world. I am new enough to synchronous middle school teaching that I cannot sense how much this may change as I improve my fluency with the technology and we develop stronger class community and routines.

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