## Functions and Derivatives

06 Oct

My calculus students had the following question this week:

A function f(x) exists such that $f''(x)=(x-2)^2(x+1)$. How many points of inflection does f(x) have? Their initial response looked like this:

One of the students had graphed y”. Using TI-Nspire Navigator, I made her the Live Presenter so that we could all look at her graph while we discussed the correct answer.

My students are entering into the Mathematical Practice of “reasoning abstractly and quantitatively”…They were using this information to make a y” number line to organize what they know about y” and where points of inflection might be. While we were discussing this, I decide to ask a different question. I froze my projector so that the students were all looking at the graph of the function. And I told them that the question was the same…how many points of inflection…but the given information was different. We were now given the graph of f’.

Everyone was able to translate the graph of f’ into information about f”. Then I decided I had one more question about the graph. The question was the same…how many points of inflection…but this time, we were given the graph of f.

A response like this certainly doesn’t happen every day in class. But I kind of like it when it does. I was sure not to reveal the answer when I showed students the results (if I had previously marked the answer, then in the Review Workspace, I would uncheck the box that says Show Correct Answer). Since I made up this Quick Poll “on the fly”, I didn’t previously have the correct answer marked.

I gave my students the opportunity to enter into the Mathematical Practice of “construct viable arguments and critique the reasoning of others”. The students had to talk to someone in the class who had answered differently and convince the other person why they were right. While they were talking to each other, I sent the poll again.

Everyone was successful…and I didn’t have to say anything. They taught each other. And they now have a much better understanding of what a point of inflection is. And how to use the derivative and the second derivative to determine how many points of inflection the original function has.

I would have never thought to make students convince each other that their answer was correct. I would have become the expert in the class and told the students (over half of whom had answered incorrectly) the correct answer. But a few years ago, I heard Jeff McCalla ( $T^3$ Instructor, and co-author of TI-Nspire for Dummies) share this idea at a session at a TI Regional Conference. I think he had learned the idea from Jill Gough ( $T^3$ Instructor and blogger).

After class, I actually had a student tell me that the class period had been productive; he understand a lot more than before he came to class.

And so the journey continues….