## The Circumference of a Circle

23 Mar

Thanks to Andrew Stadel’s CMC-South session, we started our lesson this year with a focus on construct viable arguments and critique the reasoning of others.

Create an argument for comparing the height and circumference of the bottle.

Now find someone who answered like you did. Share your arguments. Make yours stronger. Practice applying your math flexibility. (Thanks to Andrew for this idea in particular. I’ve had students partner with others with opposing arguments on many occasions; I had not thought about the importance of partnering with others with the same argument to make your argument stronger. In the session I attended, we shared our argument with someone who answered like we did at least twice.)

Now find someone who didn’t answer like you did. Share your arguments. Critique each other’s reasoning. Have you been convinced to answer differently?

Uh-oh. There were apparently some pretty good convincers from the height < circumference argument. I thought fast about what to do next. I didn’t want to immediately call on someone right or wrong to share her argument with the class – I wasn’t ready for the individual/partner thinking to stop.

So without resolving the first solution, I showed another picture.

And sent another Quick Poll.

Now find someone who answered like you did. Share your arguments. Make yours stronger. Practice applying your math flexibility.

Now find someone who didn’t answer like you did. Share your arguments. Critique each other’s reasoning. Have you been convinced to answer differently?

I could go with a whole class discussion based on these results.

CK reflected on this task in a Math Practices journal: “My first instinct was to say, ‘yes, the height is greater than the circumference’, because just looking at the can gave me the impression that the circumference was not very much. Then I was told to prove my argument, so I drew a diagram. …” (I think it’s interesting that CK chose to reflect on SMP1, make sense of problems and persevere in solving them, for this task, even though I emphasized SMP3, construct viable arguments and critique the reasoning of others, in class. The practices complement each other so well.)

We went on to think a little more about pi, using some data that students had measured at home and submitted via a Google doc and some data through the automatic data capture feature of TI-Nspire.

Based on feedback from students, I think this will be the last year for our What is Pi? lesson in its current form. We are getting students in high school who have learned math with the standard: CCSS-M.7.G.B.4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. And so our students are now coming to us with some understanding of the formulas for the area and circumference of a circle, unlike before.

I’ve recently learned that several of my geometry students wish that we weren’t learning the geometry the way that we are. They like their previous math classes better because they didn’t have to always think about why.

We are trying to change the habits and practice of how students learn mathematics. Focusing on the Standards for Mathematical Practice has required me to think through and plan learning episodes differently than before. Focusing on the Standards for Mathematical Practice requires my students to interact in those learning episodes differently, even though some don’t prefer to. And so the journey continues …

### 4 responses to “The Circumference of a Circle”

1. March 23, 2015 at 4:37 pm

Hhaving a few student wishing they were doing it “the old way” is possibly an improvement on having a lot of kids wishing they didn’t have to do it at all. They’ll get over it !

• March 23, 2015 at 6:55 pm

Yes … Thank you!

2. March 23, 2015 at 6:25 pm

Hi Check your last two p-graphs typo repeat.
Hopefully your students noticed the tennis ball brand. :^)
I like the 3-ball problem better than what I also do: Hold up a spent toilet paper roll/tube. Mark on it A around H height S same. Now have them choose which is greater. Next cut the tube length-wise to form a flat piece of cardboard. Ask ‘What shape will it be to validate your answer?i.e. what are the dimensions of the quadrilateral?’ Then tape a tube and the net to the wall.
A variation it to get a paper towel tube and cut it carefully (so they cannot see the cut marks) to a height of π*D. This always boggles my mind.
I like the tennis ball better as per your students reflection.
Thanks.

• March 23, 2015 at 6:58 pm

Thanks for noticing the repeat. Somehow when I paste my text into the editor I always think the last paragraph or two is missing, but it’s obviously not! I like the tube idea … I’ve only ever used a sheet of paper rolled up to form a “cylinder” when we talk about surface area of cylinders. Thank you!