3/13 Pi Day - 1
This is my fourth experience with Pi Day or “Pi Day - 1” as I called it since we meet on Tuesdays.
See:
In a nutshell, because there’s pie to eat, the kids always have fun. But I was reminded of another perspective today from @evelyn_lamb
“Pi Day bothers me not just because it celebrates the the ratio of a circle’s circumference to its diameter, or the number 3.14159 … It’s also about the misplaced focus. What do we see on Pi Day? Circles, the Greek letter π, and digits. Oh, the digits! Scads of them! The digits of π are endemic in math gear in general, but of course they make a special showing on Pi Day. You can buy everything from T-shirts and dresses to laptop cases and watches emblazoned with the digits of π.”
I’m pretty much in total agreement with above. I’ve gently ranted in the past about pi digit memorization contests and other such trivialities. But as her article continues, there was a man behind the holiday, Larry Shaw the recently deceased director of the San Francisco Exploratorium. I think his vision was more than just eating pie but it was also an incredibly whimsical gesture which is why I believe its had as much cultural resonance.
So I take the day partly in that spirit of whimsy and also with the mission to always ground it in circle geometry in some way and as said at the start, the kids always have fun celebrating. Mathematics doesn’t have enough moments like this especially in school.
This year I decided to go back to the basics. I had initially toyed with talking about the unit circle and the derivation of radians versus degrees but on reflection I found so much material that I couldn’t fit that in. Instead, I started with a survey of student definitions of pi (while they were eating). This was surprisingly solid. The phrase “ratio of circumference to diameter” came up almost immediately. I then took a poll of how many kids had already done activities in class where they measured circular objects of some sort and divided them by their measured diameters to find pi approximations. Again, almost everyone had done so often several years ago in Elementary School.
So with everyone convinced already pi existed and it had a value it was time for some deeper questions. The first one I posed was “Is measuring a single object a good way to prove pi’s existence?” We chatted a bit about accuracy and sample sizes as well as whether from a mathematical perspective we can ever prove something from samples. My favorite version of this is
“What if only ordinary people sized circles have a ratio around pi and if we could measure microscopic or macroscopic versions we’d find something different?”
One of the kids then suggested approximating the circumference of a circle with polygons so we then did that on the board for the hexagon version. I cold called in this case which I usually don’t do to get a student to sum up the perimeter of the hexagon arriving at pi is approximately 3.
From there we took a quick digression to also do the area of a circle visual proof where you cut the circle up and form a rough rectangle that is pi*r by r in size. Again I had the kids fill in and compute the area.
Finally I noted that we don’t actually compute pi to a billion digits using geometry and asked if anyone knew of other ways to get it. This was a new idea for the room and a good setup for the 2 videos I chose for the day.
The first was this amusing (there were a lot of genuine laughs while watching) video of Matt Parker computing pi by hand using the alternating series 1 - 1/3 + 1/5 - 1/7 ….
But of course this doesn’t really explain why this works only that it appears to do so. So I also picked the very ambitious following one by 3blue1brown:
Its about as approachable as its going to get with this amount of background knowledge but still a stretch. I stopped several times to ask questions about some of the background concepts. There are several potential stumbling blocks here:
- law of inverse squares
- Inverse pythagorean theorem
- The general abstraction model used
- The number line can be thought of as a curve.
The last one was the one I chose to focus on the most and I framed it as a thought experiment “What if the number line isn’t really a line at all but a curve, we’re just at a small portion of it and just like with a curve if you magnify enough it appears to be straight.” My hope is that if nothing else stuck that idea was interesting and thought provoking (hello Calculus in the future) My informal survey is that most kids found it interesting but I may have had one where this pushed too far. So I am planning to do a little preamble next week “Its ok to give me feedback if you found anything too confusing and I also sometimes want you to focus on the big ideas in moments like this even if the details aren’t accessible yet”
P.O.T.W:
I gave out the last problem from MathCounts this year now that it was released: https://drive.google.com/open?id=1mvYa9rWcU04MMcykZixWhdg8ehHJIPHoniVSEBdQrPA
Its actually a fairly awkward merge of quadratic inequalities and dice counting problem but I wanted to provide a capstone to the kids experience there and dig into how to solve it.
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