"It Slices, It Dices"

I've written about how much I love Steven Strogatz's book The Joy of x and how I have a class set of his marvelous book and how I've used anticipation guides to get my students to read and debate selected chapters from this book.

So, I won't do that again.  But here's another anticipation guide for the first part of his chapter "It Slices, It Dices" [aka an introduction to the integral].

The Joy of X-It Slices It Dices (AG)

If you don't have his book handy, you can read the chapter in the New York Times Opinion Pages here.

Just a little review...

In Algebra II, I tried a different kind of review as we were preparing for final.  It worked well, so I thought I'd share:

1. Type up/select some review problems and number them as you go, just like you would a review guide or practice test (all my questions were multiple choice, but free response would work, too).  I'd write a few more problems than there are students so each kid will get 1-2 total.
2. Print off the problems and cut them into strips.
3. Pass out the strips of paper (more advanced students got harder problems).  Also, as students finish before others, you can give the fast workers another problem since you made some extras (mwahahaha).
4. As students finish, have them record their answer(s).  They can use their own paper or something like this for ease in assessing on the teacher's part.  Check students' work for accuracy as they finish.  If the answer is correct, they get a piece of tape.
5. Once everyone is finished, students put their name on the strip(s) of paper they received and are told to place their problem anywhere in the room.  The only two restrictions I gave were (1) each piece of paper had to be put in a place where even a person of my height could see it and (2) don't hang anything from the Smart Board.
6. After the problems are hung, the kids work each problem.  If they have a question on a problem, they are to consult the person whose name is written on that piece of paper.

The students worked all hour and I think I answered like two questions the whole time.  I even had one girl say, "Mrs. Peterson, can you...wait!  Never mind, I'm supposed to ask...[so and so]."

Hoorah!

I printed off the problems on a colored sheet of paper, just to make things more exciting, I guess.  But that turned out to be good because the kids asked if we could do this review again the next day, so I printed off more problems on a different color for the following class period.  My classroom looked like a hot mess for a couple days, but it was definitely worth it.

Do you have any other ways you love to review that put the onus on the students?

Scaffolding with Hypothesis/Conclusion Tasks

A couple years ago, Kate wrote about a good activity for discovering the Intermediate Value Theorem.  In a nutshell, students are given the theorem and asked to state the hypothesis and conclusion.  Then they are instructed to create four graphs:

• One where both the hypothesis and the conclusion are true
• One where the hypothesis is false but the conclusion is true
• One where both the hypothesis and the conclusion are false
• One where the hypothesis is true but the conclusion is false (impossible)

I did this activity with my calc students the second week of school.  I really do love it...but, looking back, I think I was asking too much of them too quickly.  I tried the same approach at least two other times to introduce different theorems (differentiability implies continuity and Rolle's Theorem) and the kids did get slightly better at these tasks, but I could still sense more frustration than I wanted.  Some frustration is good, but not so much that they feel defeated before we've ever done any true problems.

So, in order to talk about another important theorem, the Extreme Value Theorem, I used the same idea but with some more scaffolding.  As their warm-up, students were instructed to read from their texts what EVT says and then write the hypothesis (f is continuous on a closed interval) and conclusion (f attains both a min and a max on that interval).

I wrote both of these on the top of the board, too, for reference, and then underneath showed them these graphs:

I asked them to find someone around them and, together, decide for each graph whether (1) the hypothesis was satisfied and (2) the conclusion was met.  You know those moments in class where even you, as the teacher, are taken aback by the enthusiasm of your students?  This was one of those moments.  The kids were at once having rich mathematical discussions and teaching each other.  Maybe it was because, for once, I wasn't asking them to come up with these examples on their own.  But I'm going to chose to believe it was because the task was just the perfect mixture of difficulty and attainability.

After a few minutes of letting my students discuss, we went over the correct answers, putting an X over H or C if it was not met and circling the letter if it was met.  I asked them which case we never had (circle on H, X over C) and we discussed why such a case is impossible to draw.

What I love about this is that students are forced to use appropriate vocabulary.  I always want to send the message that I respect their intelligence and never want to dumb-down material.  I think that was met here.

Were my kids using the highest level of critical thinking--creating--in this task?  No, they weren't.  But, they were understanding, applying, analyzing, and evaluating.  Every single kid was.  And that's a trade-off I'm absolutely willing to make.  Next year, I will probably introduce most theorems in this manner.  Maybe I can build up to students creating their own examples.  But, I think that's an unrealistic expectation of my students during their first semester ever of calculus. 

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.