Pythagoras smiled :)

Posted on November 9, 2011


Yesterday was the day for figuring out right triangle sides. The fearless instructor wrote “Pythagorean Theorem” on the board. One of the students doing very well drily declared that this was one of those things that teachers always explained, but nobody ever really understood. The teacher said that she wasn’t a betting woman, but she wagered that they would.
First she drew a right triangle and reminded ‘em what right triangle meant… which of course a lot of teachers don’t deign to do. After all, they learned all that in high school. She labeled its sides 3, 4 and 5 and asked them how they’d figure out the perimeter.
Notice, she’s asking them how, not what the answer is. That might not matter for the two people who’d have called out the answer – but for the five people who didn’t remember what perimeter does, who got to hear “add up the sides,” it does. These are the ones who would be thinking, “okay, whatever. It’s not on the test. I can only going to worry about what’s on the test – anything else would be too much. I”m not good at math, you know.”
Then she asks about the area, and **just like last time** is duly informed that the area is 12, because the formula is L x W. What would the area of the rectangle be? Oh… then somebody calls out the formula. Since this is review, we don’t spend time going back and nailing it down, just as we didn’t go back and nail it down last time. And you know what? Therefore, it’s a safe bet that any time it comes up in the future, most of these guys will think the area of a triangle is the base times the height. I also do believe that, had we let them play with real squares and area, they would have learned the difference.
Then she gives ‘em the Pythagorean Theorem. The square of this guy plus the square of that guy = the square of the other guy, with references to the Scarecrow in the Wizard of Oz.
She shows them a2 + b2 = c2 – and says squared… some of ‘em are still going to just say “a-two” etc, but we can at least model things well. She figures out the 3-4-5 and asks ‘em if it proved true. She draws attention to what the hypotenuse is, and which guys are the legs, and that the hypotenuse is always the longest one.
I toss in that yes, the answer makes sense. That the straight distance is going to be shorter than the angled-out distance, but that the distance can’t be more than the two of ‘em added together, because the 7 would b the line laid out flat. Of course, in my games, you would **see** that, not just hear me saying it.

Then comes the part that’s fundamentally different from the way our young man has probably heard it before. A2 + b2 = c2 and you label the sides, a b and c, and …
Perhaps they’ll start with a “concrete” example, as Salman Khan does. He’s got a sailboat, and a mast, and a rope from the top of the mast to the deck making a right triangle. He tells you that if you know two sides of a right triangle, “you *know* * that you can figure out the third … of course, if this is new to you, you don’t know. And he doesn’t realize that those little comments can add to the “everybody else already knows this and I’m the moron who doesn’t and who doesn’t belong here” feeling. I suppose it’s connected to the way that when good teachers introduce something, they start with what you know. I remember that “grounded” feeling when a teacher said “you know that…” and I *did* know it, and we built on it, and it was kind of exciting. However, if it is the *new* stuff a teacher is saying that about, wouldn’t that seem to indicate that the teacher in question is using “you know” as a filler phrase instead of actually accessing that prior information?
Salman Khan explains, as our teacher did, that the longer side is the one opposite the 90 degree angle and that that’s how you know where the long side is, and at least he doesn’t just call it a hypotenuse and hope you understand it.
Then, “part two,” is where he abandons all pretense of dealing with the concrete beyond the fact that I Can See The Triangle. Forget the mast and the boat; it’s just a triangle. We’re given two sides – the long one and a short one. We ascertain, with a solid reminder, that the longest one is the one opposite the right triangle. He emphasizes how important it is that we *don’t* add.
However, he then proceeds to copy the formula, with the addition sign, which to most people means you’re going to add something, and then do the algebra. If I’ve been doing missing addend problems as long as s/he has, I’m fine… but for many of the students? It’s all symbols.
That’s not how it was done yesterday.
It came back to what we call “understanding parts and wholes.” If you know two parts and you want to find the big piece, you add them.
If you know what the whole thing is, and what one of the parts would be, you would subtract to figure out the missing part.

I hate to break it to all the math teachers in the world, but to most people on the planet, that makes a whole lot more sense than “you want to get your variable by itself, and to do so, you need to subtract the same thing from both sides of the equation.”

Because you *know* the big side, and you’re looking for the little side, you square the two that you know, subtract the little guy from the big guy, and take the square root of the answer (since you don’t *care* what c^2 is; you want to know what C is).
(Now, in my animated version, the “squaring” would explode in two dimensions and you would see it get big… and then square rooting would shrink it back down. And since you can program a computer to do that, it would show nice grids so you could kinda see how yea, 10 by 10 is a hundred… 11 by 11 is 121…)

Today’s novel chapter: GPA, perhaps?