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)
And here are the notes from my presentation:
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.