Many thanks to Dan Meyer for the upgrades to my supposedly-educational crapola radar. (Don't you love how the textbook labels this "Egyptology" for us? That adds so much awesomeness to this problem. It really enhances the pixellated image.)

Now I don't really want to spend 30 minutes rehashing right triangle trig, just remind them about it before I peg a vertex to the origin and one side to the

*x*-axis and start spinning the hypotenuse around. As of now I'm at the filtering-signal-from-noise stage of devising something better.

Wow, that is

ReplyDeletetextbook, um, textbook. I too credit Mr. Meyer for helping me spot such way-too-"helpful" hooey much more easily."Egyptology." I love it.

I am not yet convinced that a textbook should never overlay a structure on a picture (at least until we get animated text books).

ReplyDeleteHowever, this is a particularly confusing diagram, and it cannot actually be useful, because, since the top of the pyramid is farther away from us than the bottom and the image has undergone perspective projection, either:

1) The 53 degree measure is the angle of inclination of the edge of the pyramid, in which case the drawn triangle is on the INSIDE of the pyramid but doesn't indicate it,

2) The 53 degree measure is the angle of inclination of the PROJECTION of the edge of the pyramid onto the photograph, in which case the upper vertex of the triangle isn't at the same height as the top of the pyramid anyway, or

3) The 53 degree measure is the angle between edges of the pyramid, so that the triangle is flush with the side of the pyramid, and the height of that triangle is not the height of the #($*& pyramid!

In conclusion, the fact that a diagram is drawn here is not this problem's worst feature. The worst feature is that the diagram is totally ambiguous and, in 2 out of 3 cases, incorrect! The one case that is correct (case 1) would be harder to measure (climb to the top with a measuring tape connected to the corner) than just measuring the height directly (climb to the top with an altimeter). But maybe you only have a measuring tape, I guess. A 700-foot measuring tape. And someone is threatening to kill you if you don't tell them how tall the pyramid is.

And I'll just mention the fact that google already knows how tall the freaking pyramid is anyway!

Riley thank you for adding a whole other dimension to my "How the hell is it EASIER to find that angle or the distance along the base to a point precisely under the apex than it is to measure the height?" objection.

ReplyDeleteSo if we're trying to rehabilitate this problem (my usual strategy, since in the past I never put in the kinds of prep time you amazing folks do), here's what I might say:

ReplyDeleteYeah, we've gotta build this crazy thing. Paroah'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.

Of course it would be so much better to have some real history - How did the Egyptians use trig? Did they develop it while building pyramids? Don't those textbook authors make lots of money? Can't they take a bit of time to look up the history, and make it real?

Sue, I think your revision is an improvement, but it has a glaring problem to me: the 53 degree angle. Where does that come from, and who cares?

ReplyDeleteIf you're just trying to get kids to remember trig properties, I'd say just use a triangle and SOH CAH TOA. It's possible to interest kids with abstract math, and this can be a good lesson.

If you want them to use trig to solve a problem, have them make a pyramid that's 10 cm tall and 10 cm wide out of paper. The paper will tell them if they're right or not: correct answers lead to good seams and correct heights, and wrong answers lead to everything messed up. Now they're using trig for something they care about: making something nice. Give each group a different size and compare notes at the end.

I figured their building methods provided some limitations, like they don't want it any steeper, but they can't make a smaller angle, or it would collapse. I figured they knew that from the last pyramid they had built. ;^)

ReplyDeleteThat sounds like a great problem, Riley! Have you done it with students? How'd it go? Can I, fumble fingers that I am, do it well enough to model it?

That pyramid photo's not actually pixellated. The Moslems took the smooth surface stones to build mosques in Alexandria.

ReplyDelete*ducks and runs*

Back to your regularly scheduled program:

The dihedral angle is 52o, so that clever Egyptian needs to create a height for the outer face and then ...

As to Sue's comment below - I have an old wooden toiletpaper holder that came from the woodshop. It's a certain width and height - I have them build the pyramid that JUST fits underneath it. I require them to make the height calculations before they can try it out for size.

ReplyDelete