What Makes A Circle, A Circle?

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In the seventh grade curriculum, geometry studies are few and far between. Sometimes students will stumble upon a textbook question regarding angles, triangles, area, or perimeter, but there is little exposure in isolation of geometric properties in our grade level. (Of course, all mathematical concepts are connected, but I digress). In our school, 6th grade is the year for circles, so we were advised to insert a review or to backfill for any or all students who possibly did not receive exposure to the concept. This makes it tricky when some students have lots of experience and other students have none or simply no memory of it. And although this might sound like an unusual problem to someone without 25 or more different personalities in class, it is often par for the course for every teacher and student each day. Some concepts are completely mastered by some students, others are sort of remembered, a few have surpassed curriculum expectations through independent study, and some have no recollection, regardless of the topic at hand. The reasons are plentiful. We as teachers all do our best to expose students to topics that they will claim they never saw, even if we previously taught it to them in the same year! The question is: how do we as teachers prevent this scenario?

We are supposed to be able to differentiate for all levels and reach all students in every lesson. Although I try via various modalities and methods, I have yet to figure out how to accomplish this with fidelity each day. However, one method that I find differentiation is embedded, is in well-developed three act tasks. Today, I used Dan Meyer’s Best Circle task. Here is a link: http://threeacts.mrmeyer.com/bestcircle/.

I am sure there are lots of different ways to utilize this lesson, which I would love to hear about from you if this fact applies. All I can share is what I did with it.

In the first act, four different gentlemen draw a circle. I paused the video clip and asked students to vote on which of the four circles was the best circle. I wrote the names of the circle creators (Chris, Timon, Andrew, and Nathan) on the board. Immediately, Chris received most of the votes. Students let me know it looked the least like an oval. I did not agree or disagree with their comment, but simply repeated, “Ok, less like an oval than the rest, got it.” I then asked, “Is that a true defense?” Students shrugged and expressed that they thought it was as good a reason as they could collectively argue at that point in time. Other than that, students did not have much to sustain their reasoning for choosing Chris. Although they would find out that the majority of their initial guesses was the correct one, a good math teacher always asks, “but why?” A student excited about math wants to be able to answer that question. And boy, were they interested.

Next, I assigned a new task to the students by asking a question: What makes a circle, a circle? I gave them about 10 minutes to determine the answer to this question. They were permitted to use their Chromebooks, textbooks, and each other to come up with research and data to support their opinions.  Students were allowed to revise their original hypothesis based on any information they found.

Work options were offered. Some students worked independently, others partnered with one student, and some collaborated with a group of three or four peers. One student spotted compasses on my desk and asked if he could use it. He went up to the board where the frozen screen shot with the circles remained. He slowly used the compass on each circle. As students researched and revisited words like equidistant, radius, and diameter, I was asked if they could go to the board and use a ruler to take measurements of the circles. When one student witnessed another student trying it out, she commented, “Oh, that is a good idea.” She joined the student, and then another and before my eyes, students formed a small cohort within the class in front of the screen shot with the four circles. Students began debating each other what part of the circle they should measure and whether or not they could make determinations from their measurements. Some were discussing the midpoint, others radius, hemispheres, quadrants, circumference, etc. A few were cheering themselves for their initial guess and others were disappointed that new information changed their previous prediction. When the 10 minutes was up, I shared act 2 with the additional information provided by the lesson. Students were asked whether the area, circumference and coordinate points of the circle were offered for each circle would provide a clue to help solve the problem. This is where it came out that students recalled very little in the way of formulas and the inner workings of a circle from previous learning. I showed them the resolution (the third act), but I quickly realized we weren’t done with the task today. There is so much left to dissect, I did not even want to enter into the explanation Dan provided in the teacher guide. They weren’t ready.
I do have a bit of formal review to do with students, perhaps a day or two of lessons, but we will return to this three act lesson to see if students can in fact determine why the best circle was in fact the best circle. The quick review will come…ahem…full circle. Maybe, just maybe at this time in the next school year, students will still remember some details about what makes a circle a circle.

A Review of Mr. Stadel’s Rolling Tires

The teachers in my grade level were instructed to insert a circle unit (just the basics folks) into the scope and sequence for the advanced classes this year. The idea of teaching circles made me happy because I loved teaching circles to my fifth graders; back when I taught at the elementary level. One activity based on the now defunct Growing with Mathematics program involved students’ self-discovery of the pi ratio. Although I used the activity as a brief introduction, I knew I needed to up my game for my seventh graders.

Upon my searching of all circle lessons (wow there are so many!), I stumbled across Andrew Stadel’s 3-Act Math plan for Rolling Tires. Essentially, Mr. Stadel is rolling two different sized tires towards a tower of toilet paper on a makeshift table. That is the premise:  toilet paper and tires. Naturally the students were captivated.

I had rehearsed my 3 act math questioning technique with a few Dan Meyer lessons already, so I felt prepared. I showed act one and asked the students to discuss the questions that came to mind. Oh my goodness, they couldn’t get their questions out fast enough. “Will the tire knock down all of the toilet paper? Will the tire hit the target? Is the tower of toilet paper sitting on a table or something else? How quickly are the tires rolling? What is the diameter of each tire? What is the circumference? How many times does each tire need to rotate to get to the target?”

Many students asked me to replay the video over and over again. I was instructed to pause it at certain points, and conversation exploded in the room. The conversations were all about math, geometry, and calculations in real life and it must be said, the students and I were happy.

I asked my student investigators, before showing act two, to consider what specific pieces of information they would want to be provided with in order to satisfy their own varying levels of curiosity. It was the moments following that question which proved Mr. Stadel created a genius lesson. Without saying another word, my math family began to debate each other about the merits of having the radius, diameter, or circumference of each circle. In the middle of the debate, one of the students (I will call her Jan) interjected, “Hey, it doesn’t matter which piece of information we have, if we have the radius, we can find the area, circumference, and diameter. We already know pi is a constant ratio.”

Boom. Just like that, that one student changed the course of the classroom conversation. I asked the other students to clarify Jan’s comments to each other as I walked around and listened in on their dialogue. As I checked in with the students, it became clear that they all understood what she meant and I had this lesson to thank for the reinforcement.

I then showed act 2, which presented the diameter of each tire, the distance of the tire track, and the distance from the tire to the target (if it actually hit the bullseye). The question Mr. Stadel presented, which many students had asked in their own inquiry, was how many rotations would it take for each tire to hit the target and would it actually hit the target?

The students worked furiously and shouted out numbers to each other regarding the circumference. One student started to calculate the area and her partner corrected her with a visual. She actually rolled her water bottle on its side and demonstrated why circumference (kind of like the perimeter) would be important information to answer the question.  She asked if I still had string left over from the other lesson and showed her how the length of the string would be helpful to see the distance each rotation covered on the ground.

When most students arrived at their own conclusions (most saw no reason that both tires would not hit the target, but questioned whether or not the force would be powerful enough from the smaller tire to knock all of the toilet paper over), I asked if they were ready to view act three and most responded yes. However, one student protested, “No, don’t show it yet, wait!!! I am not done!” This is a math teacher’s dream.

We all agreed to wait a few more minutes, and then I played act three. The students screamed when the big tire hit the target and complained endlessly when the small tire missed the target. Questions continued to abound.

“Why did he miss? Was his aim bad? Did it have something to do with the tire being smaller? Why did he highlight the central angle of the circle? Why is that significant?”

I don’t need to describe this lesson any further because it is clear that it had students going beyond any math textbook exercise and yet still provided so much actual understanding of a concept. As I continue to search for lessons that produce results like these I must give Mr. Stadel a huge and grateful shout out for this lesson. Mr. Stadel, I am officially a big fan!!!