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.
[If you want, my .doc for the worksheet above is here… and the next worksheet with problems to work on is here in .doc form too.]
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?
It is really wonderful reading the blog of somebody who gets it. I keep trying to engage my colleagues in math Ed discussions and its frustrating, I’m sure for them as we’ll as me… Thank you.
I’d love to hand out rubic cubes the first day of trig in precalc even before starting to play with the identities and encourage them to just manipulate them, however they wished. Never mention why, just say I’m trying something. In my mind, I can’t help but think 2-3 days of twisting and turning something, kinesthetically, might just prime their minds for the manipulation that solving trig identities would require…
Just a thought.
I commented about having the kids compete for shortest solution on one of your previous posts. Before I did that pure calculus class (with the other teacher that I previously mentioned), I only taut the Algebra 1 class, so even the kids looked at me as “not the per calculus teacher”. 1won one of the shortest solution contests on about day 3. Even the 5+year precalculus, calculus teacher couldnt best it. And I did it using algebra1 level math, creatively applied. The kids wanted my help more after that. Shortest correct solution got to take a lap of glory, high fives around the groups and I got cheering for mine. It was fun, exhilarating and something I’m glad the students got to enjoy too. Math classes should be fun.
A thought for the strange mathematical puzzles, The Moscow Puzzles by Boris Kordemsky (darn spellcheck). Though I do have a great dice demonstration which is wonderful in an algebra 1 class.
I think the “big idea” you are looking for must include CCSS Standard A-SSE.1: “Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.” So, the big idea is not just that expressions can be equivalent even if they look different. It’s also that you can learn to see big chunks of expressions that have meaning in and of themselves, and sometimes a really long expression is just the sum/difference/product/quotient, etc of identifiable chunks.
Is this what you wanted?
Click to access 2009HandoutE1.pdf
This is really beautiful. It’s how my book proved some of the basic trig formulas (I went a different route this year for these things, but I like the graphical approach here). Thank you!
Working on some long term planning for my first time around precalculus. I’m so glad to read you’re struggling with the why of these as well. I personally enjoy these kind of “math puzzles” and applying a bunch of identities to show that expressions are equal, so I hope to transfer some of that enthusiasm to my students even though it doesn’t seem very “useful.” I will have to make a note to come back to this closer to time to teach it! I hope you don’t mind if I use much of your worksheet?
Welcome to blogging! Yes, of course, feel free!
Hi Sam, I actually printed the worksheet and started filling it in, thinking I knew what was going to happen in section 1, but it didn’t – are some of the blanks supposed to end up containing the same expressions? I feel like I need to get them to label the side lengths and then fill in sin(90-α) etc for that to happen – am I totally on the wrong track? It would not be the first time….Section two I love the wickedly weird graph you get for all 4 functions, and I can’t wait to hear their reactions! I find that just punching them with the brackets in the right places is a great exercise, and is likely to lead to more careful interpretation/expression-writing later. Especially the sin²x, since there’s no way to input the square symbol so it will appear on the calculator screen the way we write it. Another way that the tech can enhance the comprehension!
Hihi! For section 1, yes, it’s about the cofunctions. So sin(a)=COsine(90-a), etc. So all the blanks will look the same: blah(a)=COblah(90-a). And that triangle makes it clear why (since the fxns are reciprocal functions). It only explains it in the 1st quadrant, but it’s good enough for me.
Ah, so what you want them to write in for sin α isn’t a ratio, it’s cosine(90-α) right away, and next to cos α they write sin(90 – α)? Just trying to get a feel for how you wanted them to happen upon those co-function relationships. And i agree about the beauty, and first quadrant’s good enough, we get 4 for the price of one that way!