BFFs: f, f', and f''

In AP Calculus, we're currently working on applications of the derivative.  As I studied past AP Calc exams this summer, it was clear to me that students need a very firm understanding of the relationships between f, f', and f'' in order to be successful on the exam.  I've been gently guiding my students in this pursuit the entire semester (in fact, that's how they discovered derivatives of trig functions), but now we're diving in head first.  I know that this is not an easy concept to master.  Very few students "get" it right away (I didn't either at their age).  But, to me, that's what makes it super fun to teach.  Or try to teach.

So, here's what we have been doing in Calc AB to help students solidify these three relationships:

• Introduction to f, f', and f'' by matching their graphs in groups of 3-4 students.  The matching activity is very similar to this one.
• Students conceptualized what it means for the first derivative to be positive, but the second derivative to be negative (for example) by filling out these charts:

How Are f, f', And f'' Related

• Students described concisely in words through this chart:

• My still all-time favorite, Inflection via Infection
• Daily Warm Up where students have to answer about ten questions like:

1. If f is increasing then f' ______________.
2. If f has a point of inflection then f' _____________.
3. If f'' is negative then f ______________.
4. If f'' is negative then f' ______________.

And then the finale:  a nine-question clicker quiz similar to the questions above.  The students who scored less than a 50% on this quick assessment are being called into lunch next week to get further help (this was totally my colleague's idea...genius!).  What I loved about the clicker quiz was that I could post the results as soon as the kids were done and then we could talk about the questions that gave them the most trouble.

For the kids who are coming in for extra help, we have created a packet where they will be given a function and then instructed to graph the function and its first two derivatives.  Then they'll answer questions like "Where is f concave up?"  "Where is f'' positive?"  "Where is f' increasing?"  And, hopefully, they'll see that the answers to all three questions are the same.

It seems my students do fairly well when they are asked questions about what the first and second derivatives tell you about the original function.  However, they have a hard time telling you what the second derivative tells you about the first derivative.  They don't seem to make the connection that that's the same thing as asking what does the first derivative tell you about the original function (which, like I said, they can do just fine!).  For example, on the quiz, the first two questions were:

1. If f is increasing, then f' is ____________.
2. If f' is increasing, then f'' is ___________.

They did beautifully on the first question; horribly on the second.  When I asked them, "Do you see how the two questions are the same?  In each case, you've only derived once."  I got a few "Ah-ha!"'s, but I think several are still struggling to see the connection.  So, that led me to create this chart:

No words.  All symbols.  And I purposely did not call any of the functions f.  My hope is, if they can understand this flow chart, they will now be able to answer questions like #2 above.  We shall see how it goes.

What other things do you do to help students with these ever-important relationships?

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.

Understanding the Derivative via Strogatz

If you've never read Steven Strogatz's book The Joy of x, you should put it on your reading list.  Strogatz, in my opinion, is able to sell and teach the development of mathematics to a general audience--which is no easy task.  He's a brilliant teacher in this book and can be appreciated by both "math people" and "non-math people," educators and non-educators alike.

I have a class set of his books, and I got to put them to use for the first time this week.  I had my calculus students read the beginning of the chapter entitled "Change We Can Believe In."  Strogatz does such a great job explaining the value of a derivative in this chapter.  I gave my students an anticipation guide and explained the value of anticipating where an author is going with the material...before you read the actual material.  I think this is especially true in mathematics:  it took me a looooong time as a student to realize math textbooks could be used for more than just the problem sets.  But, when I did start to fully appreciate math texts for their entire content, I was invested in the material because I would make predictions about the proofs before reading.  If I could get through the proof without the help of the author, wohoo! (rare, but wohoo nonetheless).  If not, I had invested enough time and energy into the problem that, by golly, I was going to figure it now.  Which meant I needed to READ.

I digress.  This wasn't supposed to be a post on the value of this literacy strategy.  But there you have it anyway.

Here's the AG I gave the kids.  They did argue through a few of the statements, which is exactly what I'd hoped for.

The Joy of X-Change We Can Belive in (AG)

Students asked when they would get to read from the book again and where they could their own copy of the book...so I count this as a success.