On a recent blog post, Dan Meyer professed his love for me. He did it in his own way, through his sweet dulcet tones, declaring me a reality TV host and a Vegas lounge act . LOVE!
He was lauding a worksheet… well, a single part of a worksheet… I had created. You see, I’m teaching Precalculus for the first time this year, and so I have the pleasure of having these thoughts on a daily basis:
What the heck are we teaching this for? IS THERE A REASON WE HAVE KIDS LEARN [fill in the blank]? WHAT’S THE BIG IDEA UNDERNEATH ALL OF THIS STUFF?
[Btdubs, I love teaching a new class because these are the best questions EVAR to keep me interested and to keep my brain whirring!]
And I went through those questions when teaching trig identities. And so I concluded the idea of identities is that two expressions that look different are truly equal… and they all derive from a simple set of ratios from a triangle in a unit circle. Equivalent expressions. When things are the same, when things are different…
So my thought was to make graphing central to trig identities. For the first couple days, every time kids were asked to show an identity was true, they were asked to first actually graph both sides of the equal sign to show they truly are equivalent. (And half the time, they weren’t!)
To introduce this, I made this worksheet (skip to Section 2… clearly I had to polish some stuff off beforehand):
Dan asked, I blogged.
To be honest, I still have some thoughts about trig identities that I need to sort out. I am still not totally satisfied with my “big idea.” I still have the “so what” banging around in my brain when thinking about equivalent expressions. I have come to the conclusion that the notion of “proving trig identities are true” is not really a good way to talk about proof. There’s also the really interesting discussion which I only slightly touched upon in class: “Are and equivalent expressions?” I have something pulling me in that direction too, saying that must be part of the “big idea” but haven’t quite been able to incorporate.
If I were asked right now, gun to my head to answer, I think I suppose I’d argue that “big idea” that a teacher can get out of trig identities are teaching trial and error, the development of mathematical intuition (and the articulation of that intuition), and the idea of failure and trying over (productive frustration). Because I think if these trig identities are approached like strange mathematical puzzles, they can teach some very concrete problem solving strategies. (To be clear, I did not approach them like strange mathematical puzzles this year.) Now the question is: how do you design a unit that gets at these mathematical outcomes? And how do you assess if a student has achieved those? (Or is truly being able to verify the identity the fundamental thing we want to assess?) 
 Except I got my teaching contact for next year, and I’ll be making more than the tops of those professions combined. YEAH TEACHING! #rollinginthedough
 Different ideas I remembered from a conversation on Twitter… Teachers have contests where they see how many different ways a student/group/class can verify an identity. And another idea was having students make charts where they have an initial expression, and they draw arrows with all the possible possibilities of where to go next, and so forth, until you have a spider web… What’s nice about that is that even if students don’t get to the answer, they have morphed the original expression into a number of equivalent and weird expressions, and maybe something can be done with that? I also wonder if having kids make their own challenges (for me, for each other) would be fun? Like they come up with a challenge, and I cull the best of the best, and I give that to the kids as a take home thing? Finally, I know someone out there mentioned doing trig identities all geometrically, with the unit circle, triangles, and labeling things… I mean, how elegant is the proof that ? So elegant! So coming up with equivalent expressions using the unit circle would be amazing for me. Anyone out there have this already done?