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## Wednesday, February 2, 2011

### We Have a Winner

Alex, on Files for Riley's Intro to Trig, asking the question backwards and displaying the kind of lesson-design genius that I wish came easily to me:
Challenge the class to figure out sin57 with just a ruler and protractor - no calculator. Hopefully some bright spark will put the equipment together with last night's homework, and draw a right-angled triangle with a diagonal of 1.

So, same challenge - cos23. This time, draw the triangle yourself, putting the 'angle' in the same place.

Finally, draw the unit circle in. Pick a third point on the circle, in the same quadrant. Draw an 'x' there.

Hopefully someone (perhaps someone who's done the homework) will now tell you that you should draw the triangle, and that the two straight sides will give you sin and cos of the angle.

1. Just for the lulz:

If all you need is to the nearest hundredth, you don't even need a ruler and protractor to find sin(57)!

sin(60) = sqrt(3)/2 ~ 0.866
I know sin(57) is going to be less, but I need to figure out how much.
Converting to radians, 3 degrees is slightly bigger than 0.05 (pi/60). And since cos(60) is 1/2, that works as the slope of the tangent line for sin at 60. So I subtract 0.025 from 0.866 to get 0.841.

Actual Value:
sin(57) ~ 0.839

2. I don't think that the 1, 2 punch of Riley's intro and this post's follow-up would work for me. Mostly, because I'm not sure that my students have protractors at home, and I certainly don't have protractors for them at school.

I'm also confused by Alex's suggestion (no disrespect, Alex). If we're working from scratch, that means we're not assuming that students even know what the words sine or cosine mean. So how can we ask them to find sin(57)? Isn't that just askfdjsakfd(57) to them?

That having been said, I'm trying to work out a different sequence of pre-trig lessons that involves learning how to graph a circle (which we haven't learned yet). I'm working from the thought that what's most important to lay down before trig is (a) that the points on a unit circle can be described using right triangles (b) that, as a consequence, there are clear patterns connecting radius with the same reference angles, (c) that you've got to watch out for the signs when relying on the reference angle trick and (d) to introduce students to the idea that there's a cycle.

The sequence that is emerging for me is something like:
1) Review pythagorean theorem
2) Use that to understand the equation describing circles
3) Apply that to find the equation describing the unit circle.
4) Tell them the x coordinate at the unit circle and ask them to find the y coordinate. They'll realize that they need to use pythag, and we'll wait for someone to figure out that there are two possibilities. Maybe I'll make a show of it (NO--I AM THE TEACHER AND I SAY THAT THERE'S ONLY ONE!!!!)
5) Then we'll start introducing angles, and I'll ask them to fine X_COORINDATE(40) and X_COORDINATE(-40) and stuff like that.
6) Then we'll spill the beans: new name, sin(*) and cos(*).

Does that sequence sound like it makes sense?

3. @MBP, I should have explained that my students are already familiar with sin, cos, and tan as the ratios of sides of a right triangle. They learn about it in Algebra 1, use it in Geometry, and we review it at the beginning of this unit. I hope that clears up the confusion.

But your way makes sense to me, too.

4. Oh, ha, got it. My bad for wasting all those words. I'm in NY too, so my kids also learned that stuff.

Still, I stopped assuming that my students remember anything from Algebra I or Geometry a month or two ago. I have no sense for what they retain, and so I try to start from scratch as much as possible. After we get through sine and cosine as x and y coordinates I'll offer an explanation of how they tie in to the old concepts, but I'm happy to take the "new" definitions as foundational and explain the "old" ones in terms of them.

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