Students Play with Dots: Math Circle Followup

I talked about my experience in a teacher math circle in a recent post and spent some of that time describing how we played with the initial prompt: How many squares are in a 4×4 dot pattern. What are the areas? You can read that post to get a run down on what I did and where the group went with it.  I wanted to present the same task to my students and see what they did. 

I had a plan for Friday morning which had students revisit some work from the day before and provide justifications to each other on the rules they had written for a series of pattern questions. I knew they all had had a rough go at it the day before and I wanted a bit more time to look into original responses and plan a better way to debrief and move forward so at the last minute I decided it was the perfect time to pitch the dot pattern prompt from the Math Circle. It was Friday. They could play and hopefully I’d inspired at least one to go home and think over the weekend. Here is what went down:

I projected the 4 by 4 dot pattern on the board and and asked students how many squares they could find and what the area of those squares were. They worked alone or in pairs with either graph paper on the table white boards.  Right away students started to dig in and after a bit of chatter, 14 became the it number to arrive at (with areas 1, 4, and 9 units squared). High fives all around, we were done!

Except… one student chimed in: “Mmmm… do diamonds count?”

Me: What do you mean? What is a diamond?

Std: This! (holds up some sticky notes at an angle)

All Stds: Shoot!

Me: ….?

Stds: Looks like we have more work to do.

They all quickly jumped back in to find more squares. I heard a few comments like ‘My brain hurts!” and “There is so much going on here!” but I wandered around a took note of different approaches. By this point, they’d self grouped into larger teams of about 3-4 students around a central paper or white board. (This seemed to fall in line with my experience. I played on my own until it looked more exciting to join up with others).

Part way through the group work, I asked by students to call out sizes, number of squares, and areas that they found to begin tracking on the class board. Then I asked students to come up and add in squares they had more trouble describing. As can be seen in the pictures, originally a group gave me a 4×4. I left it up until the group itself had trouble drawing it and then realized it was actually a 3×3.

Areas and Size of the diamond squared proved tricky for some students. I didn’t give away Pythagorean theorem (Thanks, Dan!) And instead encouraged them to visually determine the area. The 4 1/2 squares came out and they found an area of 2. I then asked them to tell me what that meant for a side length (√2) . While I was having that conversation with one group, another leaned over and said they’d used Pythagorean’s Theorem and asked if that was okay. I had them go up to the board and explain why it worked and then more groups began using it, especially for the 5 u² squares.

The students were excited. They’d done some great math, worked really well together. I didn’t have any cell phones or bathroom break requests. We could have stopped here. But, we end the fun. I clicked a button, and suddenly by 4 by 4 projected dot pattern was now a 5 by 5.

Me: “Hmm, maybe it was supposed to be a 5 by 5. Does that change your answer?”

Immediately students began debating  on how to count new squares.


I let them ride out the period in small groups exploring the larger pattern and them told them we’d revisit the question on Monday.

Monday: I had collected all the student work and made tables on the board with Size, #, Area for the 4×4 and 5×5 that had been found by students and left a blank 3×3 table and grid as well. Students came in and as they did, I asked them to draw up an example of one of the types of squares until the board was full and the 3×3 was completed as well.

I then asked the to ‘Notice and Wonder.’ We were really wrapped up in some great ideas, so I only have the aftermath board photo.  The two notices that seemed to peek the most interest are in the upper right hand corner of the photo (“Nice” was defined as teh non-tilted squares). But you can see lots more of the student thinking all over. A favorite of mine was the list of areas with the 13 * . I asked what the asterisk meant and a student added the ideas in the squiggly Areas Possible” section and said: “I predict that a 13 u² should happen, but we didn’t find it in the 5×5. I think we’d need a bigger grid, but that we’d still find it.” I told them a bit about Math Teacher Circle and how we all played with this task and found avenues to explore as well.


Those three ideas became the basis on my next move. We did have to get back into the regularly scheduled unit, so I told them they could choose one of those 3 questions (or anything else they noticed/wondered) and do some more exploring on their own and I’d give them some points on the MET. We have a Math Exploratory Tracker which means I ask students to do mathy things outside of the normal math class. Students keep track of how often they do mathy things. Brain teasers, puzzles, blocks, reading about math people….anything that sparks their interests. Points are just colored boxes. But they like to see their own colored box lines grow.  (The idea is similar to the reading log they keep for ELA class). I’ll update when I start getting their ideas in.

At first I wished I’d let them play more in class. We probably could have waited one more day to get back to our unit. But, on the other hand, we’d never reach “the end” since there isn’t really one, so I like the idea that they have something that they are invested in that still has room to explore. As students, they haven’t been asked to “explore” or “play” with math that often, so guiding them through the process a bit worked well to give them the tools to find something on their own and since we spent time and the questions were their own, hopefully they are more excited to continue to explore outside.


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