Analyzing Exponential and Logarithmic Graphs

As I was looking ahead in my unit of exponentials and logs in Algebra II, I opened up a lesson plan whose first page read, "Note to self: This lesson sucked. Kids were totally bored."

Must have written that a year ago and forgotten...until now.

This is a topic that I teach in PreCalc also, so I was motivated to change this boring lesson.  But worse than being boring, my lesson honestly did not have kids exploring interesting mathematics.

What I really wanted was for kids to understand the inverse relationship between exponential and logarithmic functions before we talked about solving equations.  I wanted them to start to understand what happens graphically before we explored the analytic implications.

So, I made this matching activity.  I really broke it down for my Algebra II kids, but I think PreCalc students (or advanced Algebra II students) could dive right into it with little to no instruction on the teacher's part.  I limited the transformations of the graphs to shifts only, but, for more advanced students, it could be nice to show reflections also (though I might stay away from stretches/shrinks...).

I had my Algebra II students work ONLY with the exponential graphs first.  They shared a deck of cards with a partner, but each student was to fill in his/her own chart. Once they were done with that side, I had them figure out which log graph was the correct inverse for each exponential graph.  Lastly, I had them analyze the log graphs.

The activity is designed so that students can see the similarities/differences of exponential and log functions, beyond just "x's and y's switch."  Ok, so they switch...what does that mean?  If I have an exponential graph that shifted to the right 2 units, which direction will its inverse graph shift?  Why?

I think this was a considerably more interesting way to get kids more comfortable with log graphs.  And they were definitely noticing the types of patterns I was hoping they'd notice.  The nice thing is that since a lot of the patterns are obvious, kids can quickly check their own work for errors once you've had a discussion as a class about all the similarities that should occur in their charts.

Chart to record results (and key):

Exp Log Match Table

Deck of cards (6 exponential functions and their corresponding logarithmic inverses)--thanks as always, Desmos!:

Exp Log Match Graphs

A couple of the matches

"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?