Sunday, March 16, 2014

Beware the Pies of March!

Above: Daniel made a pumpkin pie and then used edible markers (and geometry) to work out a nine-piece solution on it. I was so proud!

[I originally did this activity with my elementary math group right before Thanksgiving. Thanks to all the parents who helped out! People brought in protractors, scissors, plates, forks, paper towels...and lots of pies.

I took some pictures at the time, but I’m reconstructing the “proof” drawings. I have fictionalized the dialogue and changed kids’ names, although their suggested solutions and serious questions are based on what actually happened. I use three colors to represent the kids, but assume an actual class of about 15.

I didn't get this finished for Thanksgiving or Christmas, but it seems fitting to post it now in honor of Pi Day and the Ides of March. (Knives are involved in both. Plus, if you keep reading, you might spot some subtle treachery. Not least because I officially converted to Tau last year. Beware the Pies of March!)

If only we had done geometry and proofs like this when I was in school…--GMHB]

Psst! Hey, kids!
Wanna drive your adults crazy this Thanksgiving?
Great! First, make sure that an odd number of people want pieces of the apple pie.
But what if an even number of people want pieces?
Claim you don’t want one.
But I really do.
Can you get your little brother to say he wants a piece even though he doesn’t?
You’ll come up with something. Be creative. Claim your stuffed tiger will bite off Uncle Murray’s fingers unless she gets a slice. (At my house, that would even be true.)
Well…I don’t think Scheherezade (my tiger) would ever actually go through with it, but I’ve never been brave enough to find out. Anyway. Just be sure to insist on an odd number of pieces.
Have you ever tried to cut a pie into five or seven pieces?
I’ve never tried to cut a pie at all. My mom has this thing about knives.
*Sigh.* Okay. Let’s try this. Do you see this circle?
Pretend it’s apple pie.
I just remembered. I don’t like apple pie.
How about pecan pie?
I’m allergic to pecans.
Okay, we’ll pretend it’s pumpkin pie.
I love pumpkins! And the Great Pumpkin is coming soon--
Proselytize after class, Linus. Now, how would you cut it into four pieces?
That’s easy.

Great. And it only took you two “straight” lines.
Maybe next time I should use a ruler.
Yeah. Now, how would you cut it into five pieces?
Oooh! Lemme do it!
Okay, Alejandro. Come up and show us--

I mean, five equal pieces?
Um. Um. Wait, I know! A circle has 360 degrees. So I need to divide 360 by 5 which is…um, 72!
Nice calculator.
So all you need to do is cut the pie into five pieces with 72-degree angles.
Wait. How do we measure 72 degrees?
Well, kiddies, I’m glad you asked me that question. I happen to have here…[drumroll]…a protractor.
What’s a proctractor?
It helps us measure angles. Now, who has an idea for what to do next?
Oooh! Oooh! Me!
Okay, Griselda. Please demonstrate for the others.
Well, I’d start at the center of the circle, and then use this starting at zero, and then go over to 72…
Let me interrupt you for a second Griselda. You did a fine job measuring the degrees, but how do you know that’s the center of the circle?

Well, it looks about right…
That’s true, but what if you’re off? What if I said this dot over here were the center of the circle?
Yeah, that’s crazy, but look at what happens:

Now, those are all 72 degree angles, but does anyone think that looks fair?
*More giggling*
You won’t think it’s funny if you discover you’ve been served an unfair amount of Aunt Bertha’s chocolate pie! Think of the injustice!!!
But how can we find the exact center?
Good question. Any ideas? Okay, Algernon. Come on up.
Well, I’d take this ruler and measure the top to the bottom. It’s 7 and 1/2 inches, so I’d mark in the middle. And then I’d do it the other way, just to be sure…

That’s pretty good. Of course, if one of those measurements is off by even a quarter of an inch, you still have a problem.

So I’ll do three or four measurements.

Not bad. It’s better than just guessing, but there's still some variation. Besides, we’re out to drive the adults crazy, right?
Okay, so you’re going to insist that eyeballing, guessing, or even estimating with a ruler just isn’t good enough. And if the adults say you’re being too picky, what do you say?
“Pi’quality is a fundamental right…”
Very good, kids. You’ll go far. So, when you challenge the grown-ups, they’ll probably say “Do you have a better solution?” That’s a delicate moment in the negotiations, and you need to be prepared. As it happens, there is a better solution! I’ll walk you through it.
Pick a number between—
Infinity squared!
—We’ll go with 5, thanks. Now, I’m going to measure a 5 inch line that goes across the circle.
That doesn’t go through the center!
Yep, I know. It’s okay. A line like this is called a chord. If it went all the way through the center of the circle, it would be called a diameter.
What’s a chord?
Chords are when I play two notes at the same time!
Well, a chord is a musical term, that’s true. It’s also a geometry term. It has multiple meanings. Now, we’re out to drive your adults nuts, right?
Okay, so I encourage you to go home and ask your parents what a chord is.
Anyway, I’m going to draw three of these randomly around this circle. You could do it with just two, but three is even safer. And exactly halfway, at 2.5 inches, I’ll put a dot.
Next, I’m going to measure a right angle using this piece of paper, and then trace along the edge. This new line will be perpendicular, or at a 90 degree angle, to the first chord.

Then I’ll do it again for the other chords…
Miss Gail?
Yes, Helga?
Do all the chords have to be the same length?
Uh…wow, what a good question!
You did say that children were better at thinking imaginatively and asking questions.
She also said we’re better at learning new languages, though I don’t know why that matters in this case.
Yeah! And I’m better at “Angry Birds” than my dad!
Right. Well, in theory, it should work with different chord lengths. I can’t think of any reason why it wouldn’t work. It just never occurred to me to test it. Why don’t you all go home and play with that idea on your own?
Can’t we test it now? Please?
Fine. Let me try to scrounge up another color…
Okay, so let’s try a 2 inch, a 3 inch, and a 4 inch chord and see what happens.

Yup! Still in the center.  Does everybody agree this is the exact center of the circle?
I think it’s 2 nanometers off.

Don’t get  picky, Franz, or I’ll make you do percent error problems next time.
But you said we were supposed to drive adults crazy!
I meant other adults!
Okay, so now that we have a center that is within acceptable parameters—
It’s close enough.
I still think it’s 2 nanometers off.

[Quellingly] I’m ignoring unnecessary interruptions.

Now, it’s your turn. Here are some circles drawn on pieces of paper, and here are some protractors and rulers. You already have pencils. Work together in groups of 2-4 people.  Draw on the paper to show me how you would cut your circle into an ODD number of equal pieces. You can “buy” a piece of real, homemade pumpkin pie--
Did you make it yourself?
I’m sure we all hope so. Other families have also brought in various fruit pies and a chocolate pie. Anyway, you can have a real piece of pie once I have approved your solution.
Just remember: doing this on paper is for practice. In a few days, you can do this for real.

Below: Pictures from the actual Thanksgiving math meeting.


1 comment:

Jon said...

You should post more of your math group lessons. This was really cool.