- how we know sin(x+h) = sin x cos h + cos x sin h
- the squeeze theorem
- how we know (sin x) / x approaches 1 as x approaches 0
- how we know (cos x - 1) / x approaches 0 as x approaches 0
Thursday Morning Lecture
I love lecturing! I agree that it's not usually an effective way to teach. I keep that in mind when I lecture, and do everything I can to draw my students in. I started out by telling them that as they go on in math it becomes more and more about proof. I warned them it would take a lot of effort to work their way through the reasoning I presented, and asked them to please tell me whenever something wasn't making sense.
I used index cards to call on people, and asked them to provide some of the simpler steps. I asked for the whole class at once to call out the very simplest parts: "sin2 x + cos2 x = ... " I used different colors of marker to point out various triangles we were considering. I had the students do a few algebra steps individually at their desks before I wrote them on the board. Except for redrawing the diagram and doing those algebra steps, I told them there was no need to take notes, since I would be giving them a handout at the end.
I asked afterwards how many were able to stay with it. I think over half of them raised their hands.
As we worked our way through this diagram, the students were getting a much-needed (for most of them) review of trig.
I need to modify this so that it doesn't look like there's a straight line formed by the terminal sides of α + β and -β. Darker lines and bigger angle and point identifications would be nice too.
We used this to prove our trig identity, and used that to get as far as...
We are through two of the four pages of my handout. For homework, I told them to use their calculators to fill in a table of values, evaluating the two expressions in the limits for h = .1, .01, .001, etc. On Monday, we'll look at the squeeze theorem to help us find these two limits, which will finish off our proof. (Maybe I'll give the second handout at the beginning of class, and ask them at the end of class which worked better for them - getting the handout afterward like we did on Thursday, or getting it before.)
This proof is a good review of trig, a good way to see the power of the squeeze theorem, and a good way to think about limits from some new perspectives. (We have not done a unit on limits. I skip that chapter, and come back to it during our last unit, before considering integration.)
I was so excited after my 80-minute performance, I wanted to make a video of it. I don't know if it would be useful to anyone else, but a number of my students would have liked to watch it this weekend. A lot of what I do right in my lectures would be hard to reproduce on a video - I need the audience to get me pumped up, and I need to see a confused face to realize that I should say more. But one of my students agreed to be my audience, and another offered to join her. So I might find a way to do my video lesson with just a bit of student participation. If it works out well, maybe I can do a bunch. We'll see...
Just in case, I set up a youtube channel for Math Mama. (Thank goodness that name wasn't taken!)
An Alternate Proof
When I got home from work, I re-read my blog post from last year. One of the commenters had posted a very different proof for the derivative of sine and cosine. It's very short and very visual. I love it. One of my reactions was to worry that dragging my students through the longer proof was unnecessary. But I think our work will help the students better understand the sorts of proof that use lots of algebra, and will give them a great context for the squeeze theorem. I also think this alternate proof isn't quite complete. (At some angles, adding a bit to the angle increases the cosine instead of decreasing it. What then?) But it is so cool, I have to show it to my students. I think I'll wait a day or two, so they'll have done all they want with our conventional proof first.
What do you think? Would you use this alternate proof?