So, I don't love tricks? You know, the "don't worry about the why, kid, just remember this little song" variety. But we're in the weeds of finding areas and perimeters of composite shapes. Of the quarter-circle stuck on a rectangle stuck to a triangle variety. Yes I want them to have a intuitive grasp that the distance around a circle is a little more than three diameters, and yes I want them to see a circle deforming into a rectangle whose width is r and length is half a circumference. And yes we are mostly solving problems with tracks and whatnot.

Buuuut....I am not confident about how much of that they are going to be able to retrieve when they are sitting for their SAT's a year and a half from now. So, we are shamelessly using a trick, that a student heard somewhere else, for keeping circumference and area of a circle straight. "Chocolate pi is delicious, and Apple pis are, too." I'm storing it in the same dark place as "All Students Take Calculus," as in, it makes me feel a little dirty, but it seems to be a necessary evil.

Wow, I had never heard that mnemonic before. And I'm embarrassed to admit how long it took me to figure out what the heck it was supposed to mean, so I won't admit it.

ReplyDeleteI mostly agree with your feelings about this one. The definition of pi is something that needs to be memorized, there's no way to figure it out, so C = pi d really is just something to memorize. The understanding to be gained there is the reason that pi is constant for any size circle.

Then the area formula is pretty tricky. You can do lots of unfolding circles into approximate piles of triangles and things like that, and there is a deeper reason that the 1/2 in the triangle area formula also shows up here, but it's essentially conceptual calculus to get that far.

I don't think I ever tried this in a classroom, but maybe it would make sense to teach these as "circles are like squares, except you get pi instead of 4". In other words, the perimeter of the square is 4 times the distance straight across the middle (which is one thing that would make sense to call a "diameter" of the square), and the area is 4 times the square of the "radius" (half of the "diameter").

It took me a while to figure out the mnemonic, and I'd never be able to remember it. What stops the kids from misremembering it as "Apple pies are delicious, and chocolate pies are too" and getting the mnemonic backwards?

ReplyDeleteJoshua's suggestion of remembering the exponent (like squares, but with different multipliers) at least has the advantage of being mathematically relevant. Personally, I remember circumference, area, surface area, and volume formulas all in terms of radius, not diameter. (And it helps me a lot that surface area is the derivative of volume with respect to radius, but that probably wouldn't help your kids.)

Because chocolate pie is, in fact, delicious, I guess. We all just do the best we can with what makes sense to us and our kids. Personally, I find 2*pi*r vs pi*r^2 to invite way more confusion than pi*d.

ReplyDeleteTwinkle, twinkle little star, circumference equals 2*pi*r

ReplyDeleteI agree 2pir creates more confusion and issues. But, that is part of the reason I would prefer to use it. I have many 9th graders that could easily solve 10 = pid, but are flown for a loop by the three factors when solving 10 = 2 pi r. I don't think it is in their interest to avoid this issue.

ReplyDeleteI feel similarly about tricks, but this is the mnemonic I remember:

ReplyDeletePie are squares? No, pie are circles!

I think the point Joshua touched on about the ratio between circumference and diameter being constant and area being related to length^2 is quite important and also grokable via software exploration. I've never played with GeoGebra and the like, but I assume they allow you to scale up polygons and then compute perimeter, area, and "radius"? There might be physical objects you can use to demonstrate the principle for volume/mass as well.

ReplyDeleteYou could also tie this exploration in with actual physical units, too.

P.S. I assume congratulations might be in order for Joshua for making the US Sudoku team? (unless there are two mathy puzzle-solving Josh Zuckers?)

So, like I SAID, it's not like I disagree with "the ratio between circumference and diameter being constant and area being related to length^2 is quite important and also grokable." There are many wonderful ways to build understanding of the relationships between dimensions in a circle. I am a fan of all of them. I am just not convinced that something more tightly encapsulated and easily remembered long in the future is not necessary, for the future kid sitting in the SAT's eventuality.

ReplyDeleteI don't think using or teaching mnemonic devices is anything to be ashamed of. I was always capable of figuring out the quadratic formula, but it did take me a few moments of thought. My life changed after I started teaching and learned the formula can be sung to the tune of "Pop Goes the Weasel." Now I always sing it in my head when writing the quadratic formula, and I can't imagine going back!

ReplyDeleteI personally prefer "Awesome Students Take Calculus," but either way it's one of my favorites! :)

I totally agreed with you, Emily. Until Your last line. I hate the a.s.t.c. thing, and I don't understand why the textbooks include it. It's quick and easy (or can be) to visualize where your angle ends up and decide whether the x and y are positive or negative. Why throw a mnemonic in to complicate matters?

ReplyDeleteWhen I first started teaching math, I always had the quadratic formula in my notes if I was going to need it. I sure wouldn't want to derive that while teaching something else. But students always need help with visualizing, and anything that implies they should memorize that feels to me like a huge detriment.

Don't be scared! The area of a circle is pi r squared.

ReplyDelete(I was slightly disconcerted by a group of 14-15 year old boys chanting this in unison when I asked them how to find the area of a circle...)

ASTC - All Students Turn Crazy! (or All Santas Take Cookies, or All Shamrocks turn colors, etc). I've never heard of the pies, thanks for sharing. And chocolate pie does rock!

ReplyDeleteUsing similarity as your base tends to help a lot here.

ReplyDeleteThe area of a circle with radius 9 is 81 times as big as the area of a circle with radius 1, so the area needs to be 81 somethin somethins, and in general the area formula better have r^2 in it somewhere or math explodes.

I also like showing a hexagon inscribed in a circle. The distance around the hexagon is 6r, and the circle is pretty close to that but a little bit more. So the circumference of a circle is [abitover6] times r.

Similarity helps a lot with area and volume formulas, because students should at least know what kinds of units to expect; this helps them know that a formula like 4*pi*r^2 has to be a formula for some sort of area while 4/3*pi*r^3 has to be a formula for volume. Sorry this is probably too late to be of much use ;)

PS: Our books do briefly mention SOHCAHTOA as a way of memorizing that mess, but ASTC is dead to us. Cosine is the x-coordinate, sine is the y-coordinate, and since tangent is sine/cosine, tangent is the slope. ASTC and the related-angle formulas pretty much come out of that for free.

ReplyDelete- Bowen