V = πr2h
But she didn't stop there. She asked herself and the class:
Why is that the formula?
She went on to describe how it's like taking a bunch of CDs and stacking them. We know we can find the area of a CD.
That's when things got interesting. I heard comments like these:
- We use the height to determine how many circles it takes to make up the cylinder.
- But it's just like a line [it has no height]. How do you get volume with that?
- If you have a thing of CDs and stack them...you're going from 2D to 3D.
- But area has no height. We're stacking things that have no height.
So we talked a little about finding the volume of a rectangular prism is just counting the number of cubic units it takes to fill the prism and how doing that with a cylinder is tough because its round. I liked the direction they were going with this by trying to talk about something larger by defining it by its smaller parts. So small, in fact, that the height is practically 0.
I ask, So how can you find the volume of a sheet of paper?
A student takes a sheet of paper puts a ruler up to it as if he's going to measure the height of one sheet of paper and one of his group members says:
Dude, why don't we just make a stack that's 1" tall and divide by the number of sheets it took to make it?
They all agreed that would work and I thought we were ready to move on when a boy asks, "So what's the volume for a sphere?"
They wrestle with this for a minute or two when one of them turns to me and says, "Mr. Cox, you know, huh?"
But you're not going to tell us are you?
A student opens his planner to the Math Formulas page and starts to tell us what the formula is when one says,
I don't care what the formula is. I want to know why that's the formula.
Next semester's gonna be fun.
What grade do you teach? And this is calculus!
Now we need to know why area of a circle is pi r^2. And once we get that down to adding a bunch of circumferences, then why is C = pi * d? Do we even remember where pi comes from? Who would make up a number like that unless it came from a weird definition.
These are 8th grade algebra students and I never thought of the area of a circle as being the sum of a bunch of circumferences. Any suggestions for putting them in a position to try to grind that out?
Well, it's the same concept, just taking it down a dimension. A bunch of concentric rings/strings to make the circle. And once you've gone that way, though, you have to be prepared for going the other...hyperspheres in 4D!
Maybe introduce with a picture of a tree stump?
Right, but how do we take 2*pi*r and end up with pi*r^2? It seems like the formula should be 2*pi*r^2. It doesn't seem as intuitive as multiplying the area of the circle by h for volume.
Oh. Yeah, that's a bit less intuitive, I guess. I'm not coming up with a good way to do it without calculus. =(
I am curious: Do your kids arrive in your class with this level of inquisitiveness, or do you single-handedly breed it?
Wait! I remembered this:
It uses area of a triangle. So, we'd have to know where that comes from, too (maybe you've already done that one, though).
I get them for two years. As 7th graders, most are good at school and a few are really inquisitive. We just keep on encouraging inquiry and by the time they get to 8th grade, the majority of the class has a solid respect for learning and one another. So to answer your question:
I think they come to me inquisitive, but some just don't remember how to do it because it's been trained out of them.
Cool, David. How inspiring. :) It's really interesting that you get to keep them for two years. I've taught some kids back-to-back before, but usually it's a fluke (ie. I happen to be shuffling my schedule after the year is done). It'd be awesome to be allowed to keep the same kids for two years.
By the way, I was thinking about the inquiry of 2*pi*r --> pi*r^2. You might be able to show it using Calculus concepts but presented in a fairly intuitive way. The inquiry part can be for kids to take a bunch of ropes (with width ~ 1cm) to form concentric rings in a circle, and then lay the ropes out side by side. What they should notice is that:
1. All the concentric rings have lengths (circumferences) that are increasing linearly with their distance away from the center of the circle.
2. All of the rings/ropes together covered the entire inside of the circle, so you must be able to figure out the circular area using those ropes (sum of length of each rope X width).
3. From there, you can somehow guide them to figuring out that to add the length of all the strings is the same as adding the shortest and the longest, times n/2, where n = number of concentric circle ropes.
(ie. the arithmetic series formula. For example, adding 1 + 2 + 3 + ... 100 is the same as adding 1 + 100, times 50 pairs. Since 1 + 100 = 2 + 99 = 3 + 98 = ... = 49 + 50 = 101, and there are 50 "pairs" of those numbers each with a pair sum = 101. This series formula is very intuitive, even though it's part of Precalc. My dad taught me this algorithm in like the 3rd or 4th grade, so I'm pretty sure your kids could understand it as middle-schoolers.)
4. This simplifies to (0 + 2*pi*r)(1)(r/2) = pi*r^2. Here, 0 is the theoretical length of the innermost (shortest) ring. 2*pi*r is the length of the outermost (longest) ring. 1 is the thickness of each rope. r/2 is how many ropes there are, divided by 2to get how many "pairs" you have. Tada!
The two year loop is one of the things that really drew me to take this position.
I really like your idea for the area of a circle. Looks like a trip to the hardware store is in order. Gonna need some rope.
There's the other way of getting the area of a circle from the circumference--take circles and cut them into smaller and smaller wedges (fourths, eighths and sixteenths work well). Rearrange the wedges top to tail to get shapes that are closer and closer to parallelograms. Then your calculus is just imagining what happens if you could cut into infinitely small wedges to do this with, and you conclude that your rearranged shape (in the limit) is a parallelogram (or a rectangle) with base = 1/2 circumference (because the circumference is distributed half on top and half on bottom) and height = radius.
It does sound like it's going to be a great semester!
Awesome! Check out http://www.youtube.com/watch?v=aLyQddyY8ik
Hey I saw http://www.youtube.com/watch?v=whYqhpc6S6g from one of my students and thought of your post and our suggestion of using ropes.
Their proof is simpler.
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