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.
Awwwwwww, the same goes to you, so we're even!ReplyDelete
Rebecka, you are a sweetie. But I don't deserve your gracious thanks. I may have offered advice, but not the advice you took here. I've never done that (keeping the inside the same).ReplyDelete
I always get them to describe the function using the word "something". y=six(3x) becomes y=sin(something). Then the derivative is y'=cos(the same something)*(derivative of the something).
It wasn't good enough. Half my class forgot their chain rule on y=sin(3x)/x^2. They were busy using the quotient rule, and didn't even see the layers on that onion!
You totally deserve credit! It's the conversations that are so helpful. As they evolve, so does my teaching.Delete
But, there are some topics that, no matter how great they're taught, the best teacher is lots of exposure to them and lots of practice. I think that goes for the chain rule. Yes, I do feel like I explained it better this time around, but, ultimately, the kids need to practice and figure out their own potential stumbling blocks.
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