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.

More thoughts on the Chain Rule

I posted about how teaching the Chain Rule was a lot harder than I thought it would be.  I'm still convinced this is at least partially due my students' lack of understanding/recognizing a composite function.  For example, just the other day we needed to simplify the expression



And a student (one of my top students, I might add) suggested we "divide out an ln."

Hold up.

Let's ignore the fact that dividing by any number other than 1 would change the expression.

Dividing by ln?  So...somehow there's not a connection that ln is meaningless without an argument.  "Dividing by ln" is akin to "dividing by √ " or "dividing by cos."  An empty square root or an empty cosine doesn't have any kind of value, and really doesn't mean a thing.

I was further disturbed when I gave my Business Calculus students a function like



and was told that in order to find the derivative, we should use the Product Rule.

Wha...?

Do students view ln as some sort of constant?  Like e?  It seems maybe so if the function above is thought of as a product and if we can indeed "divide out ln."

I decided we needed to revisit the Chain Rule.

I started by showing a slide that had a composite function at the top and four expressions beneath it such as:

I asked my students to tell me why we needed to use the Chain Rule and then to identify the derivative of the outside function (holding the inside) and the derivative of the inside function.  The next slide highlighted the former in red and the latter in blue:

We did several of these.  Then we concluded with other types of functions to test if they knew when to use what rules.

I think I will start with this type of presentation the next time I teach the Chain Rule.  Giving the students a limited amount of options to start out with seemed to worked fairly nicely.

There are still definitely some issues.  But I now have a better understanding of what needs to be emphasized  in terms of composite functions.  I will try to make them more densely populated in my algebra and pre-calculus classes from now on.

If you're interested, here's the slideshow we worked through:

Test 3 Review

The Chain Rule was harder than I thought

Last week I taught the Chain Rule to both my Calculus I and my Business Calc class.  Let me rephrase that.  Last week I was supposed to teach the Chain Rule to both my classes.  I'm pretty sure I didn't quite get there.  At least not yet.  I was really looking forward to teaching it, because it shows up everywhere.  Also, I never remember thinking the Chain Rule was a particularly hard concept.  But maybe I'm romanticizing my beginning calculus experience.

This is my first go at teaching either class, so I'm sticking pretty close to what the books say.  I figure the authors are the experts on both the subject and the audience, so it's a good starting point.  Both books teach the Chain Rule quite differently, so I was excited to try both and compare and contrast.

Unfortunately, one thing I found across the board was that many of my students don't have a firm grasp on composition functions. Sure, they can compute fog, but ask them to go the other way--to decompose a function--and all of a sudden at least half of them look at you like you've asked them to please go swim across the Atlantic Ocean.  It was a frustrating moment as a teacher because I couldn't find a way to explain decompositions without using the typical vague words like "inside" and "outside" functions. I tried saying that the "inside" function is what has parenthesis around it, or the expression you could put parenthesis around without changing anything.  Yeah...that works for functions like

f(x)=sin(3x−5)

And

f(x)=4x2+2−−−−−−√

But when we got to

f(x)=e5x−1

they told me the inside function was e.

Not e to some power. Just e.

FAIL.

On me, not my students.

Note to self: learn how to teach the decomposition of functions.

The Chain Rule via Leibniz notation did go a bit better. We talked about how if a company that produces video games wants to know how much it is making per minute, it could take how much it makes per game sold and multiply that by how many games it sells per minute:

Similarly, if y=f(u) changes 1/2 as fast as u, and u=g(x) changes 3 times as fast as x, then we can conclude that y changes 1/2 times 3, or 1.5, times as fast as x:

y=(3x2−4)2

Or...

Another thing I tried that I stole from the Business Calc book was beginning with a "guess" for the derivative of a function such as

f(x)=(x3+5)2

For the "guess" for f'(x) we applied the Power Rule to the "inside" function and got

f′(x)=2(x3+5)

Then we found the actual derivative for f(x) by expanding it and using the Sum Rule.  We found that the derivative was the same as our guess but multiplied by 3x^2!

I thought this would be a great "AH-HA!" moment.  Alas.  It was not.

I just got a bunch of, "So, what was the guess for?"  "What's the final answer?"  "How would you enter that into MyMathLab?"

Sigh.

I think the "guess" thing really could have been powerful.  I just need to ponder how to present it better.

So, that's the Chain Rule.  Some things worked.  Some didn't.  Most didn't.  But this is one of the amazing parts of teaching mathematics--learning how others learn math.

y′=2(3x2−4)