Chain Rule--getting better

It's been over a year since I last taught calculus and pleaded for help with explaining the chain rule.  It was a lot harder to teach than I thought it'd be.  Usually I can predict where students are going to stumble, but not this time. Thankfully, the incredible online math teacher community came to my rescue.

When I posted last year, Sue and Bowman both suggested that for the first few examples I give, I only change the "outside" function and keep the "inside" function exactly the same.  Totally brilliant (and probably totally obvious to most other teachers).

And then when I cried out for more help on Twitter, Sam suggested I use something like this to pique curiosity.  I had actually tried and failed with this method when I taught Business Calc, so his encouragement was all I needed to resolve to try again.

This year the lesson was as follows:

• As a class:  Practice decomposing functions (i.e., identifying the inner and outer functions)
• As a class:  Differentiate y=(3x^2+x)^2 by expanding; compare our result to y'=2(3x^2+x)
• In groups of 3-4:  Try the same task but with a different given function; record results on the board:

• As a class:  Generalize chain rule
• As a class:  Practice the chain rule with multiple outer functions but same inside functions
• As a class:  Go over some potential places that could be stumbling blocks
• In groups/on their own:  Practice, practice, practice (i.e., group work and homework)

This worked so much better than last time.  Here are the cards I gave the students when they got into groups.  I color coded them for myself (different colors represented different levels of difficulty) so that I could differentiate a bit.

And here are the notes from my presentation:

2.4a Chain Rule.pdf

As a final note, I want to express my sincere gratitude for and love of this math community we have via blogs and Twitter.  Thank you to all the teachers--like Sue, Bowman, and Sam--who make me a better teacher.  Even though I've never met you, I so covet your advice, encouragement, and camaraderie.  You have my deepest respect.

Improvements Graphing Piecewise Functions

My PreCalc kids did better graphing piecewise functions this year than in the past, so there's a chance I actually improved at teaching this topic.  Just a couple notes (more for myself, so I don't forget this next year):

1. Draw a vertical, dotted "wall" at the possible point of discontinuity.
2. Determine which piece(s) of your function will have a closed circle at your wall and which one(s) will have an open circle.
3. Determine which function you'll use for all your x's to the left of the wall and which function you'll use for the right.
4. Graph the top function (use transformations); erase everything to the left or the right of your wall, depending on your decision from Step 3.
5. Repeat Step 4 for the bottom function.  Erase the oppose piece this time.

The key, for me, is "the wall."  I've used this concept before in analyzing limits in calculus graphically, but I don't know why it didn't dawn on me to use the same concept here until recently.  It worked like a charm--hardly any students drew the nonsensical, non-function relations that I've seen in the past.  Also, hopefully this gives us a leg up when we get to limits next semester.  Fingers crossed!

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.