This week, Monday afternoon, I received a tweet from a former Key Curriculum Press colleague
so I did. This post is motivated partly by the fact that I shared the graph on Twitter, and I feel like the graph on its own feels empty. It's missing the conversation that we had around it, so it felt a bit like sharing my bullet-point notes for a story, rather than the story itself. So here's the story.@klockmath Can you recreate the famous pi-petaled rose for tomorrow using @Desmos ?— Vishakha Parvate (@VishakhaParvate) March 13, 2017
That evening at dinner, I decided to show my son. He loves anytime I offer to share computer work with him. (By sheer coincidence, we ate vegan pot pie for dinner. :) I started by taking out his circular turtle box and a string, and we marked diameter lengths on the string. I asked him how many of those lengths he thought we would need to go all the way around the box.
He replied, 200. He's not yet 4, so his number sense sometimes gets confused with his desire to exaggerate. Well we wrapped the string around the turtle and saw that it was a bit more than 3 diameter lengths (marked in red).
Then I showed him the Desmos graph, and we talked a bit about pi but then got distracted by other things. (He demanded to play Marbleslides.) This conversation put a bug in my head, and by morning I spontaneously decided that I would do this activity with my class the next day, pushing other topics to the side. So, on my way out the door in the morning, I grabbed a ball of twine from the utility closet and quickly grabbed a few portable circles (empty food tins and lotion tubs) and as many sharpies as I could find. I worked furiously on BART to make the graph presentable and come up with a plan for the class. As spontaneous as it was, I feel like it went really well. So here's what happened.
I started by drawing a circle on the whiteboard and talking about some of the vocabulary. I labeled a diameter and a radius, and we talked about the definition of a circle and also circumference/perimeter. I asked for a relationship between radius and diameter. In both classes (I teach one morning section and one afternoon section), we noted that the diameter is twice the radius, and the radius is half the diameter. So I wrote those sentences up on the board, and we wrote the equations that go along with them. In both classes, the first suggested equation was r2=d, so I put 5 m for the radius in my whiteboard circle and asked what the diameter was. Without hesitation, they all said 10 m. Then we checked the equation for these values, saw that it didn't make sense, and made the adjustment. I asked the question about how many diameters it would take go around the circle. I asked for guesses. I asked for too low and too high guesses. We didn't spend too much time on that, but I think I'd like to spend at least one minute more in future iterations of this activity.
Here I asked people to get out some circles. More than half the students had a drink of some sort in front of them, and I had 7 circles of my own that I was able to pass out for students to either use or trace & pass. Then, as I distributed markers and lengths of string, I asked people to shout out their estimates and asked for shows of hands for how many people also got that estimate. Estimates ranged from 2 to 4.5, with the majority in the 2.5 to 3.5 range. They all marked their strings with diameter lengths, and then wrapped the strings around their circles. (Incidentally, this also offered an opportunity to discuss the arbitrariness of units of measure. You can create your own ruler!) Very quickly, I had a student shout out, "I'd like to revise my estimate!" Everyone got 3 and a bit. We reflected on the fact that everyone had a different sized circle, yet we all got the same number of diameters to go around the circumference. My first class seemed largely unfazed by this, as they reasoned that when you scale a circle up or down, the diameter gets scaled proportionally. My second class, by contrast, were surprise at the constant multiple. But, as we talked about it and brought up the proportional scaling, they eased back in their seats and gave a contented nod of agreement. I also gave one of my favorite arguments, which is to hold up two circles of different size and say, "You can think of this as a smaller circle, or you can think of it as exactly the same size circle, but farther away." And then you'll see that doing the string wrapping experiment doesn't change for the different size circles. This style of argument makes me giggle because it always reminds me of Steven Wright's quote (which I can't find, so I'll mis-quote: I was out taking a walk, and a UFO landed in front of me, and three one-inch-tall guys got out of it. I said, "Are you really one inch tall?" and they said, "No, we're really very far away.").
So, moving from that to technology, I pulled up the Desmos graph. I started by introducing the model, noting that the orange point is moving outward along the radius at the same rate that the green point is moving along the circle, so that by the time the orange point reaches the outer edge of the circle, the green point has moved one radius length along the circumference.
I made sure everyone was on board with that much, and then we turned on the tracing of the orange point. I also switched to tracing one full diameter length (rather than a radius length), during which time the orange point went from the center to the outside of the circle, and back to the center.
We took a moment to remember that the arc being traced out by the green point along the circumference of the circle is one diameter length, and we noted that it does indeed look like about a third of the circle. Now we had a visual for the diameter length, which is the orange petal-looking thing. Now we're ready to see how many petals it takes to go around. Here I show a snippet of what we did, utilizing the power of the technology to try to get a little more precision than "3 and a bit."
Now, in each of the classes, I had shushed one or two students who started saying stuff about pi, because I didn't want everyone to lose the discovery in our process. But at this point I did ask, "What is this number that we're seeing?" And people shouted out pi, and I put π up on the calculator. Everyone has heard of pi, but from my experience, most students' understanding of it is flawed in some fundamental ways. One of the ways is that, I don't believe I've ever had a student who can tell me what pi is. If I ask a class what pi is, most often I hear variations of "3.14" and "it's a number that goes on forever." Never have I heard a student say, it's how many diameters it takes to go around the circumference of a circle. Including yesterday. But because of the work we did, we were able to talk about the equation C = πd, not as a formula for how you calculate the circumference of a circle, but as the definition of π. This notion blew several students' minds, but they got it. We now had some basis for talking about the actual value of pi, so we discussed the decimal representation, and the fact that some people like to memorize digits of pi, and some people like to celebrate pi day. Then I went back to the graph, and I started animating.
and I stopped the animation, at a time when I knew the many of students would be anticipating the "finishing" of the flower. I wanted them to be conscious of this anticipation. One student had to look away, she was so bothered by the unfinished flower. But we did finish it, and it looked like it finished on exactly 22 diameters. Here we also counted the number of revolutions we had made, by counting the rings. Seven revolutions. So I did the calculation.
They went oooooh. So I put up the value of pi again, right under the value of 22/7, and we saw that they were close but not the same. You can even zoom in on the rings to see that 22 diameters over-shoots 7 revolutions by a smidge.
I mentioned that some people use 22/7 as a rational approximation of pi. One student asked why anyone would want a rational approximation of pi. We talked a little about rationality, and how pi is not rational, and how so many people really really want all numbers to be rational. One student wondered why we care about pi at all. For this, because of the conversation we had, it did not take any convincing for her to agree that pi must show up any time you have circles around, so then I mentioned gears and machinery, and anything that has a linkage with a fixed length rod that pivots around one of its ends. In 10 seconds flat, she was able to go from questioning why anyone cares about this number to comfortably saying, oh yeah, of course. Well, that is, 10 seconds after the hour that we spent getting to that point.
Now, in full context, here is the Desmos graph.
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