Is Pi Always 3.14...?

One of the greatest experiments students can do to celebrate pi day is measuring the circumference of a circle and dividing by the measured diameter to approximate pi. This is a cool exercise not only because it is an engaging experiment, but also because it gets students to think about and observe an essential characteristic of circles, namely that every circle is similar to every other, and C/d is a constant. They might even start to wonder, why is it that pi is approximately 3.14 and not some other value? This is a wonderful mystery that made me think: Is there a way to make pi something else?

Well, let’s first review where pi comes from. Imagine you get a puppy and name him Pie-Dough. Because Fido was already taken. You’ve got a nice big yard for him to play in, but he’s always getting into things he shouldn’t, so sometimes you put him on a 20 meter leash that is tied to a tree in the middle of the yard. Of course, he is always running around at the end of the leash. Here’s what that looks like:

Of course, Pie-Dough is happily running in a circle. The circumference of that circle divided by the radius (the leash) is 2pi. No surprises there. But things are about to get pretty fantastic. Imagine that you move to the North Pole, and you put Pie-Dough on a super lightweight leash that is, get this, 500 kilometers long! And Pie-Dough has crazy puppy energy, and still runs in a circle at the end of the leash. But I have a question for you: Is the leash pulled straight? Check out the image below, with the leash in blue and the path of Pie-Dough as a black circle:

Because the leash is so long, Pie-Dough’s leash is actually curved along the surface of the earth! He actually doesn’t get to run in as big of a circle as he could if the leash were straight. Can you see why? So now, if you divide the circumference of his circle by the leash length, will you still get 2pi?

Nope, you won’t. Because his “radius” is curved. The Circumference divided by the “radius” will be less than 2pi. This is even clearer in the diagram below, where Pie-Dough’s leash extends all the way to the equator (he can swim oceans, no problem). Can you figure out C/r in this case?

So you might be thinking, well yeah, the ratio is different because I’ve distorted the “radius.” That’s true. But another way to think about it is that I’ve changed the space we are working in from a flat surface to a curved surface. We stopped looking at the world from Euclid’s perspective, where the ratio C/r = 2pi, to a non-Euclidean perspective, where C/r < 2pi. Our universe changed from a flat plane to the surface of a sphere! In fact, the actual universe on large scales is curved, so there is good reason to look at it from a non-Euclidean perspective. The universe can be curved like the spheres we saw above, or it can be curved in such a way that C/r > 2pi. What does that look like?

So in the diagram above, imagine Pie-Dough’s yard is the funky red saddle shape. The sphere there is just to roughly show you Pie-Dough’s possible paths, if he could fly off the red saddle. But let’s say he can’t fly off the surface and he just runs around on the curved path where the blue sphere and the red saddle intersect. The points A and B show some examples of where Pie-Dough could be, with the leash shown as a dotted line from the center. Actually the leash should also be curving, stuck on the saddle, but this was the best approximation I could make using Geogebra. If you know of a better visualization let me know!

As you can see, Pie-Dough can run quite a distance along that topsy-turvy path, a lot more than he could when he was stuck on a flat surface. In this case, because the universe he’s running in is curved like this saddle, the “circumference” is huge compared to the “radius,” giving us a ratio of C/r > 2pi.

So there are two things I think are cool about this. First off, at large scales, the actual universe can be curved as shown above. Secondly, if you have a baseball, you can explain to someone how the circumference divided by the radius can be less than or greater than 2pi! It's all about thinking of it from the perspective of Pie-Dough--when he runs on his path in his curved universe, for him, the yard is flat, the radius is straight, and the path is a circle. But we know that because his universe is actually curved, the geometry get's weird, specifically, it gets non-Euclidean. Pie-Dough, we're not in Kansas anymore!

Oh, by the way, here’s another way to approximate pi quite accurately. It’s a freebie so check it out!

Finally, check out these other great Pi-Day posts by inspiring teachers! Happy 3.14! 

An InLinkz Link-up

Making Mastodons Real

For many years, in addition to teaching I ran an outdoor education program at a school in urban southern California. Most of my students lived far-removed from the natural world and outdoor education was literally a breath of fresh air for them. Usually we took students to Joshua Tree National Park, a land of giant granite boulders sculpted by the wind and glowing gold in the setting sun. One evening we drove into the park with a new group of middle schoolers, their faces pressed to the window in awe of the hulking stones looming like mastodons. One kid, who had studied the basics of geology in his science class, nevertheless innocently exclaimed, “Woah, who put these here?”, thinking that the fabricators of Disneyland had been hard at work in Joshua Tree. This climbing trip made those stones real for the students, bridging the gap between the abstraction of nature and the reality of rocks.

Climbing in Joshua Tree

Predicting The Great Pumpkin

Thanks to Doc Running who has organized a Halloween blog hop, you are in for a treat from several great math teachers! I've got a big story for you that I hope you'll share with students, and links to more great resources. Read on!

I always smile when I think about the scene in Big where giant Tom Hanks is coincidentally eating a tiny ear of corn. Partly I smile because it’s great comedy—he approaches the task with a seriousness reminiscent of Chaplin eating his shoe.

But I also like this scene because science! Corn (more precisely, maize) was domesticated

The Shepherd's Calculation

From the balcony of my new apartment I can see the Mediterranean Sea. This is what it looks like:

I live in Spain, where I’ve taught math at international schools for the past four years. I taught and surfed in southern California for many sunny seasons, and last week I discovered that you can surf in the Mediterranean too! Can you tell which photo below depicts surfing in California and which surfing on the Mediterranean?

7 Ways to Drop the Mic in Math Class

I like to think I’m an entertaining story-teller. Even the driest of mathematical procedures (quadratic formula derivation, anyone?) can come to life when I’m in performance mode. But over the years I’ve lectured less and less, giving students more autonomy to follow their own paths of inquiry. Students work in small groups on real world scenarios, experimenting, drawing conclusions and solving complex problems while I facilitate, motivate and occasionally lecture. Here a few tips on how to drop the mic and let students pick it up in math class.

1) Ask Curious Questions

How much taller is a human compared to a carpenter ant? How much faster can a sailboat go if you double its length? How many trees per person are there in the world?  If you flip two coins, are you more likely to get two heads or one head and one tail? If you double the radius of a pizza, how much more food do you get? What function is the best model for a car accelerating from a stop light, and why? Can you figure out the percentage of green m and m’s in the world from one bag?

A Snapshot of Interactive, Hands-On Math

What they're working on.

An Introduction To Courage To Core

Introduction
My first year of math teaching was in 1995 at an innovative private high school in San Francisco named Lick-Wilmerding. Students worked in small groups to explore algebra and geometry, reporting their findings in documents written as if they were lab reports. Students would receive direct instruction when necessary, but frequently, questions were explored and answered by members of the groups themselves. I adapted hesitantly to this new structure, as I myself had been taught by teachers and professors using lecture combined with extensive homework practice. Over my subsequent years of teaching in the states I explored a range of methods tailored to the various cultures of the schools at which I taught. This included straight lecture and homework, a mixture of lecture and independent class work, and collaborative group work supported by a textbook. Over the last 8 years of teaching at La Jolla Country Day in San Diego and two international schools in Spain, the American School of Valencia and the American School of Madrid, I’ve developed Courage To Core which allows my ninth and tenth grade math students to explore mathematical concepts as if exploring scientific concepts in laboratories.