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Friday, January 22, 2010

Marginal Improvements

Is something that would flow like this any better? I don't know. It's better for ME, but I don't know how many kids are going to come along for the ride.

And now I'm enamored with Sue's comment on the pyramid-slide post:

Yeah, we've gotta build this crazy thing. Pharoah's given us this diagram, and wants it 432 Pharoah-feet from the center to the edge. You know it's gonna be coming in at a 53 degree angle, the way we're gonna build it. He wants us to tell him how tall it's gonna be - I think he wants to see whether it will beat the last pyramid we built for him.

1. I figured you were about to whip out some trigonometry, but then at :40 I get totally confused and I have no idea what you are up to.

2. I have to know how you added the angle tick mark for the 100 degree angle. I've been looking for a way to indicate congruent angles in a notebook presentation for a while. Thanks.

3. I think it's meant to tie what kids already know (or are supposed to know) about similar triangles to the present circumstance. The length of the chord scales up and down exactly in proportion to the radius of the circle, so the sine (which is a ratio) is independent of which circle you start with.

Of course, it's difficult to tell for certain without a mellifluous narration, so I'm just guessing here.

4. David - It's a total pain, but this is how I do it -

(For some reason I'm in "make a video and put it on youtube" mode right now for explaining things.)

5. Right - you get similar right triangles with the same central angle, so the side ratios don't depend on the radius of the circle.

I've been told (by someone, sometime...but I don't have a reference...and when we are talking about stuff that's that old, it's likely impossible to know for sure) that this problem - find the length of a chord given the radius and central angle - was the motivation behind the invention of trigonometry.

6. The Indian mathematician Aryabhata studied the chord-length function, and the half-chord-length function (ardha-jiva, shortened to jiva). Arabs transliterated this into jiba, still referring to the chord.

However, they wrote it (as they tended to do) in Arabic, which only writes consonants and uses diacritics sparingly for vowels. Leaving them off, it's hard to tell jiba from jaib, and that's what the European translators thought they wrote. Jaib, of course, means "bay" (like the Chesapeake Bay), and so they wrote sinus in Latin.

Of course, we also see this word come up in lunar features (almost all of them are ironically named after bodies of water) like Sinus Iridum (the "Bay of Rainbows"), and in anatomical features shaped like sacs or inlets, and so we might think that it has something to do with the shape of the sine curve. Maybe that's what the translators thought the Arabs had been thinking, but they just didn't know how to read!

7. Thanks John.

that this problem - find the length of a chord given the radius and central angle - was the motivation behind the invention of trigonometry.

Yes, it was astronomical: what is the distance between stars X and Y?

8. I don't really get that. If the stars are endpoints of a chord, where is the center of the circle? Not the Earth unless the stars were equidistant from Earth. Were they assuming all the stars were on a sphere with Earth in the center, or something?

9. This is maybe a nonsequitur? But in the vein of searching for a more natural way to introduce right triangle trig's central problem:

The awesome Jason Cushner, who at some point was Colorado Teacher of the Year or something like that, and now teaches at a private school in VT, used to (and perhaps still does) begin right triangle trig by taking his whole class outside and asking them how tall is a tree, or a flagpole, or a building, or something like that; and giving them tape measures, string, washers, protractors, straws and tape.

Plusses:
1) It's easy to measure distances along the ground because you don't have to climb anything;
2) Angles of inclination are trickier but can be done using the straw, protractor, string, washer and tape, again without climbing anything; thus
3) The need for a mathematical way to get from an angle and a side to another side is built into the situation naturally; and
4) The kids will never forget that day. This lesson was described to me 2 years ago by an adult former student of Jason's who'd been in his class in the 90s.

10. Thanks, Kate. I'm glad to know my lazy ways can stand up to the scrutiny of people who work much harder. ;^)

Before I teach trig or calc again, I really want to learn more of the history.

11. Yes, exactly! Beyond the spheres of all the planets lay the "sphere of fixed stars". They were "fixed" because they didn't move independently of each other the way the sun, moon, and planets (planetes: "wanderers") did.

Beyond that was the Primum Mobile: the "first mover", which sets all the celestial spheres in motion. And past that was the Empyrean: "the habitation of God and all the elect".

12. Ben - Done that! But not in January. :)

I've never done it like: figure it out, here are some tools you could use, go. I typed up a whole guided investigation deal. Which, you know what, kids learn &^%#-all from guided investigations. Except how to blindly follow directions and not know why. Consider this me swearing off them forever.

13. Ooh, brave woman! I want to hear how this one (tree, protractor, etc) goes with no guided investigation sheet. I would be so scared!

And I really need to do this. Why am I so scared of a bit of primitive engineering? I think some folks call the device an inclinometer?

14. The reference I used called it a "hypsometer." But inclinometer sounds right to me too.

15. So I did kind of a little history lesson thing with the sphere of stars. Motivating something by showing how it had to be invented to solve an actual problem is a thing, right? It went over pretty well, except in 4th period which I had to drag along kicking and screaming (because LOUD SIGH, everything is SO STUPID. SCOWL.) but that is pretty much every day with them.

Hi! I will have to approve this before it shows up. Cuz yo those spammers are crafty like ice is cold.