The progression during this class was that my students:
1. classified statements as Always true, Sometimes true, or Never true
2. filled in the blanks in a statement so that it was Always true
3. wrote their own statements that are Always true
Part 1: Classifying:
Here are the statements I wrote on the eboard, each of which they then labeled with A, S, or N, and after each of which we discussed why (that's the italics):
Θ = Θ + 2π
N - The notion that = means "is the exact same number as". Coterminal is another thing entirely.
sinΘ = sin(Θ + 2π)A - A lot of confidence about this being an A, due to the previous discussion. The seed was planted that finding an A is kind of a big deal.
cos Θ = 1S - I insisted on hearing some Θ's for which it was true.
cos Θ = sin Θ
S - Same as before, except this time, tell me all of the Θ's for which this is true
Here I had to pause and get them comfortable with locating angles like Θ + π, π - Θ, etc on the Unit Circle, so that they could visualize the next statements. I had them drawing random angles for Θ, then the corresponding Θ + π etc. Once they were ready, I asked them to classify this one:
cos (Θ) = cos (π - Θ)N - Lots of lovely arguing, many said well they're equal but opposite. How to say that algebraically ..... and let's convince ourselves with a few angles on the calculator (in degrees though!) ...now it was time to segue to part 2.
Part 2: filling in blanks to make an A:
sin Θ ______ sin (π - Θ)
cos Θ _______ cos (π + Θ)
sin Θ _______ sin (π + Θ)
This part was done in groups. I witnessed some fantastic discussion - which I was unable to copy and paste due to a tech glitchy thing, but there were drawings being done, there were angles being tested on the calculators, there was correct vocabulary being used....it was truly exciting to see the strategies they were using to decide and then convince. I didn't have to say much. I shut up really well.
Part 3: writing A statements from scratch:
This time all I said was "Tell me about the relationship between cos Θ and cos (-Θ)."
Back into their groups they went. (There was no need to say, oh by the way, I want the truth.) They figured out where -Θ was in relation to Θ, they looked at their x-coordinates, they wrote a statement, they tested it out on random angles, and then ALL groups proclaimed:
cos Θ = cos (-Θ).
We repeated this to get (in a lot less time btw):
sin Θ = -sin (-Θ)
and talked about whether or not we could also say that
-sin Θ = sin (-Θ)
Tell me about cos Θ and cos (Θ + π/2). And I want the truth. I can handle it.
They'll always be root 2 units apart in the cartesian plane. Does that count?ReplyDelete
Hey, if it's true, it counts! And cool, I never thought of that. What I was hoping they'd see, via sketching, was the relationship between cos (Θ + π/2) and sin Θ. So far no takers, might have been too much of a stretch.Delete