Derivatives of Trig Functions

One of the things I find challenging to balance is convincing kids of mathematical truths without overwhelming them.  Sometimes, I know, there is a time and a place for a bit of hand-waving.  And, sometimes, I know, there is a time and a place for formal proofs.[1]  But I think most of the time the sweet spot is somewhere in between a formal proof and "this is how it is--just memorize these rules."

In search of that happy medium, I created decks of 12 cards (6 with the graphs of the basic trig functions {orange} and 6 with the graphs of their derivatives {blue}).  I had students match them up with a partner.

Matching a function to its derivative using only graphs is new for my kids, so I knew this would be a challenge if I didn't lead them quite a bit.  However, gathering data from a graph is so heavily tested on the AP exam that I figured it wouldn't hurt to start making some connections.

After they matched them up, I followed up with these questions:

Derivative of Trig Functions Follow Up Questions

Here are the cards I made, if you're interested (thanks, Desmos!).

6 basic trig functions (enough for 16 decks):

Trig Functions Matching Cards1

6 derivatives (enough for 16 decks):

Derivative of Trig Functions Matching Cards2

[1]  Although I'm beginning to think I show proofs more for myself than my kids.

They're going to be prepared for calc...so help me God

I've written before about how I feel like a concept we think our students get that they really don't get is the composition of certain functions, specifically trig functions and log functions.  I made a vow to myself to emphasize compositions with my Pre-Calc students a lot this year, so that when they do get to calculus, the Chain Rule and u-substitutions will be two of their best friends, as opposed to worst enemies.

I was reminded of this vow when I asked a student to read an exercise from the book out loud.  The exercise started like this:



And this is how she read it:

"Sin" [as in a transgression, not a trigonometric function] "times pi over two minus x."

I wanted to say, "When have you EVER heard anyone say it like that, girl?"  But, I remained calm.  I ignored the mispronunciation (we have bigger fish to fry here), and focused on the "times" part.

It seems like every time I have this conversation ("It's not 'f times x,' it's 'f of x,' guys."), I feel like the kids are just nodding to get me to shut up.  I can't blame them.  I did the same thing in grad school [way] more than once.  As long as I make the prof think I understand what he's saying, all will be well.

But, inputs.  They're kinda a big deal.  What worries me is that it seems like students often view inputs as some kind of multiplication as opposed to actual arguments, which makes sense as the notation is very similar (parenthesis for both).

I continued to notice this was a problem as we were verifying trig identities.  I don't know if the kids just got so into the proofs that they forgot a few fundamental things...like what sine and cosine are...or what was going through their heads exactly.  But, let me tell you, I saw crap like following slide all. the. time.  So, I made them figure it out:

I would not tell them what was wrong, but I did mention it was subtle.  When they finally started figuring it out, we talked about why we need all those theta's!  Our dear trig functions are meaningless without them!

It's a small step, but if it gets them to remember that these trig functions must have an angle at which they're to be evaluated, even if that angle is arbitrary, well, then, that's a good thing.

Trig Art Projects 2012

At our school, our Pre-Calc/Trig students create trig art projects every fall.  I know it's a pretty popular thing to do, but I gotta say, I'm pretty sure I got some of the most amazing trig art ever.

The math criteria:

• Use at least 3 trigonometric functions
• Each function must have at least 2 periods
• There must be at least 2 shifts (vertical or horizontal)

Once students figure out their functions, they graph each function on a transparency or a sheet of wax paper.  This acts as their blueprint.  From there, they copy their blueprint onto a quarter of a poster board and add their artistic genius.  Finally, they fill out a rubric, telling me things about each function, such as its domain and range (they must state the domain and range that they used for THEIR project; i.e., "all real numbers" does not cut it), amplitude, and any shifting.

Students working with Desmos in the Math Lab

I took a day to let them play on Desmos.  This was fantastic because they were able to see immediately what  their transformations did to each function, without having to graph each function fifty times by hand.  Also, I showed them how to restrict the domain of their functions, which taught them a teeny bit of programming.  I feel that using math to understand technology is one of the greatest things I can teach my students.  So a day in the Math Lab to play with trig functions was well worth it, in my opinion.


I let my students vote on the Most Creative Award.  Here's their winner from 1st hour:

And from 4th hour:

Each winner got a gift card to Target in the approximate amount of $3pi ($9.42).  And, no, when I asked for that amount, the cashier's face was not nearly as disconcerted as I was hoping.

But the kids laughed.  And that's what matters.

Review Game: TRIG BRAIN POWER!

To prepare for an upcoming test last week, my Pre-Calc/Trig students and I had a great time playing TRIG BRAIN POWER! (because I can't think of a better name, and yes, it must be in all caps).

It's quite simple.  On the SMART Board, my "home page" was a unit circle with six adorable colored brains sitting on the positive x-axis:

I split students up into six teams, one team for each colored brain.  I would show a review exercise, and when every single person on the team had a written response, the team raised their hands, which signaled me to come check their answer.  The first team with the correct answer got to move three spaces on the unit circle, the second team got to move two, and the third team got to move one.  After each exercise, I had the winning team explain their strategy, and if there were no questions we moved on to the next exercise.  The team that rotated through the greatest angle at the end of the hour won (we had to add 2pi rads a couple times!).

I essentially adapted a game called BrainSavvy that I found on SMART Exchange.  If you want the Notebook file for this particular edition of TRIG BRAIN POWER!, just shoot me an email.

I liked this because it involved minimal prep, the students had to work with each other, they worked hard, and they had fun.