I'm sure most senior math teachers would agree that a lot of the difficulty kids have with trig identities has to do with the algebra involved, and not the trig. But it also comes from the fact that they often treat the identity as if it were an equation, and immediately start moving things from side to side or cross-multiplying, which is what their autopilot does as soon as it detects that = sign.
The subtlety that they're missing, and that I wanted to get across at the outset, is that when they solve an equation, they are already assuming it's true. It's the logical equivalent of saying "it's true because it's true." But identities are to be proven - and proving something is true is a lot trickier than assuming it's true - just ask a lawyer.
So while the rest of this week will be devoted to reinforcing their algebra skills, today I wanted to introduce some basic logical ideas, without actually saying them out loud. Instead I used my subliminal messaging powers, which will appear here in red italic text, which is why I have called this Subliminal Text Messaging!
I did this today with the whole class at first, no notes, no recorded lesson. The part you see below took about 15 minutes, after which, they worked in groups of 2-3. I said pretty much these actual words, but their answers are of course composites.
Me: True or false?
(x + 3)(x - 3) = x² - 9
All of them, immediately: True! (Identities are about algebra that you already know)
Me: Convince me.
Them: Well if you foil you get x² - 9. (a volunteer did this on the board): (Work on the LHS only)
Me: So what? What does that have to do with anything? (Wait a minute - what was the question again?)
Them: Well...it's the same as the other side.
Me: So what?
Them: Well since it came to the same thing as up here (point to x² - 9 on RHS) then we were right, it was true. (Are you saying that if LHS = something, and RHS = that same something, then LHS = RHS?)
Me: Assuming of course that your "foiling" was correct.
Them: Yes. Oh. Was it? (Just messing with their minds. :) And that we convince by using things we already know to be true.)
Me: It was, no worries. While you were doing this "foiling", did you need to look at the x² - 9?
Them: No. (This is different from solving an equation - you're not doing something to both sides here, you're looking at one side only, then comparing it to the other.)
Me: What about this - true or false?
(x + 2)(x - 8) + 6x = (x + 4)(x - 4)
Me: What's the matter? Why isn't anyone answering me?
Them: We're working on it.... (Students' likely subliminal message: Geez Miss, take a pill.)
Me: Oh this one isn't quite so obvious, eh? How come? (What's the difference between this one and the last one?)
Them: Because there's more steps.
Me: Well how about this: Susie you simplify the LHS, Johnnie, you do the RHS, and we'll see what happens:
(Two different people = the two sides are being done completely independently of each other - again, this ain't no equation being solved)
Susie: My side comes to x² - 16
Johnnie: My side comes to x² - 16
Them: It was true!
Me: How does that mean it was true? (Even when both sides got algebra-ed, if LHS = something, and RHS = that same something, then LHS = RHS? Sure about that?)
Them: Both sides came to the same thing, so they must have been equal. Like "this equals that". (Students' likely subliminal message: Isn't that just common sense?!?)
Me: Susie and Johnny, while you were doing your side, did you have to look at the other side in order to proceed?
Susie and Johnny: Nope. But I did at the end. (The only reason to check the other side is to see if it's the same)
Me: Great! Now how about this:
Me: Ah but I didn't ask you this time if it was true or false......in fact, simply by using the word "Prove", I'm already telling you that it's.....
Them: ...that it's true? ....(it's not about deciding true or false, it's about convincing by using other things we already know to be true, like algebra, trigonometry, and common sense....)
Me: Right! But now explain to me why you thought it was true.....
We then did the above really simple example together, then off they went in their groups to do harder ones. I caught a few egregious algebra crimes and nipped them in the bud, and gave some groups harder ones to sink their teeth into, so it was a good opportunity for differentiation.
I plan to have them submit three proved identities on their blogs later this week. If I got my message across, I'll see proofs, rather than autopilot solving. More later!
If you have had success in helping your students with trig identities, or if you have your own subliminal messages to share, please do!
(This was also posted at The Flipped Learning Journal.)