## Sunday, March 13, 2011

### How, and how not, to explain graphing trig functions

Graphing trig functions should be relatively easy for students who have already mastered general function transformations with quadratics and exponentials.  Nevertheless, there are so many steps involved that I think an emphasis on procedure and practice is justified in this case. With that in mind, I aimed to build on students' prior understanding of transformations while giving them an outline for how to graph trigonometric functions of the form f(x) = A*sin(B(x-C))+D.
I completely butchered the first lesson on graphing trig function and promised my students (all patience and humor to the end of that horrid, horrid hour) to make amends by presenting the procedure crystal clear the following lesson.

What I learned from the failed lesson:

• Save your geogebra files before moving into the classroom, as the computer can and will have a fit and shut down unexpectedly (bye bye 30 minutes of work) when you plug in the projector.
• When creating a nice mnemonic for how to graph these functions (I came up with BC AD - Before Christ, Anno Domini, get it?) go ahead and check first that this is actually a reasonable way to graph these functions. BCAD isn't, in the sense that it doesn't allow you to place points each step of the way.
• If something isn't working, stop doing it. I actually persisted for 40 minutes giving students one retarded way after another to graph trig functions, when I should have assigned some other work and taken a few minutes to think things through. It's a testimony to the loveliness of my students that they persevered and, when at last I came to my senses and called the whole thing quits, laughed with me (and, admittedly, at me).

In my defense, when I immediately after the lesson googled how to graph such functions... NOTHING came up. Most websites and videos explain A, B, and D - but skip the C. I sat down with my colleagues and we  came up with the following method:

• D gives principal axis, so make a line there.
• A gives amplitude, so make lines at D+A and D-A. Now we know the range of the function.
• C is the phase shift. For sine, put a point at (C, D). For cosine, put it at (C, D+A).
• B is the frequency, and 2*pi/B is the period. Figure out how long one period is and place a point at (C+period, D) for sine or (C+period, D+A) for cosine.

Now draw a full period of the function between the points you've made.

Mnemonic? I'll ask the kids to make one up. All my ideas involve dog ate cat bones and dingo ate chubby baby.

I'm afraid the lesson itself won't be any more exciting than demonstration and practice. But I think sometimes (especially after the havoc I put them through last time) demonstration and practice is what the kids want most of all. For homework, I give them a modeling activity involving the movement of the sun in three arctic cities.

1. Sine curves are all ugly. What you really need is a mnemonic that tells them how to

oh well, it's the best I could do.

2. I miss the kids back there. Last week I tried a new, non-rigorous way of explaining why the chain rule works. It wasn't as intuitive when I tried to explain it as it was when I prepared it and some of my students actually got kind of hostile.

3. Alex - that one is great, thanx!

Johan - Aw, so what did you do when that happened?

4. Update: this explanation seems to have done the trick. In the end I went with a modification of Alex' suggestion: Draw A Curve Brilliantly. :)

5. Kept a stiff upper lip, then cried my eyes out when I got home.

6. Johan, I know, that's just like you.

7. Thanks for this! I was looking for a way to make graphing these easier on my seniors :)

8. Thanks for the advice. Newbie Math Teacher in New York.