Category Archives: Calculus

The Related Rates of an Automatic Pizza Saucer








Building a Better Review for the AP Calc Exam




The learning structures I used for review:



I loved













Any review structures you used that worked well?

Volume in Calculus: Conceptualizing before Formalizing










An Anchor Problem for Riemann Sums





duxford(isn’t this building beautiful??)








Whiteboarding Mode: Simultaneous Show and Tell




Draw a function such that…

  • \lim_{x \to -2}=3
  • f(-2) does not exist
  • \lim_{x \to 1} does not exist
  • and f(1)=-3.




The Dead Puppy Theorem and Its Corollaries






Teaching Through Concrete Examples: The Intermediate Value Theorem

I’ve been getting pretty into cognitive science lately. I realize some of it is useless, and a lot of the rest of it is made up of kind of common sense things once you really think about it, but regardless, I have found it so helpful to put scientific names and research to intuitions I have in the classroom. One of the ideas that I have really liked (from Daniel T Willingham’s Why Don’t Students Like School?) is that we learn everything by connecting it to things we already know, and much of what we already know is concrete. Thus, the more you can teach through concrete examples, the more likely students are to learn the material.

EXAMPLE: Speed, iPhone prices and the Intermediate Value Theorem

This year, while teaching the Intermediate Value Theorem in AP Calculus, I did not start with the theorem itself, as I always find that language so intimidating for what is actually a simple idea. Instead I started with this:

I showed them a video of a speedometer that cuts out for about 10 seconds in the middle (ah, you’re dizzy and you pass out for a second at the wheel!). Before the cut out spot, the car was going 60 mph, and after it was going 100 mph. I then asked the to tell me:

  1. What was a speed that you are 100% sure that you must have gone in the time in between? Why?
  2. What was a speed that you could have gone in the time between, but you aren’t 100% sure? Why?

We talked about this for a few minutes, letting the students argue a bit about their thoughts and came to an agreement as a class. Then I put up a new picture that showed the original iPhone prices at some intervals. It started at $599, a few months later was $399, and then two years later was $99. Then I asked very similar questions:

  1. What was a price that you are 100% sure that the iPhone must have had in the time in between? Why?
  2. What was a price that the iPhone might have had in the time between, but you aren’t 100% sure? Why?

Again, I let them argue for a bit and discuss. After we had settled on answers, I asked what was different about the situation, keeping in mind that we had already discussed continuity in the class, but I had never mentioned this in this situation. Students said wonderful things like “To get from one price to another, the iPhone doesn’t have to pass through the other prices” and “Prices can can jump whereas speeds can’t” and I let them continue to do that until one student finally realized “Speed is continuous, whereas price is not!”

Prepped with the ideas of theorem, we took the speed situation and translated it into a mathematical theorem before looking at the actual Intermediate Value Theorem. It took about 10-15 minutes of class, which was well worth having a strong conceptual understanding of the theorem. Students still struggled mightily with proving anything with the theorem (as they have in proving anything mathematically both this year and in previous years – any advice there?) but the conceptual development of the idea was not only quicker, but I think stickier.

Isn’t that better than starting with this?

Let THEM Figure Out the Power Rule

I have been reading and enjoying (though not fully buying everything in) Daniel T Willingham’s book Why Students Don’t Like School: A Cognitive Scientist Answers Questions About How the Mind Works and What It Means for the Classroom. One of the ideas that I think is really useful in planning instruction is that humans are wired to enjoy learning – some scientists believe that the brain releases a little bit of dopamine every time we solve a problem. We actually physically get pleasure from solving problems.

As an example, check out these two word picture puzzles (figure out the common expression indicated by the words and their placement):

Which of the two puzzles did you enjoy more? If you’re anything like me, or most human beings, you didn’t really enjoy the one that had the answer right above it. Even if you didn’t figure out the other one, you probably at least thought about it more than the other one (though Willingham points out that the physical response only occurs when a person solves a problem). How often do we give the answers to the riddles first in math instruction?

Here is a rule and here are examples of every type of problem you will have to do with it, now do problems like those even though you kind of already know the answer.

An example of posing math as a riddle instead:

It took a few days for students to learn the power rule this year, as opposed to me just writing f(x)=x^{n} so f'(x)=nx^{n-1}, which takes about 10 seconds (if you talk while you are doing it and write very, very slowly, and have to erase something in the middle because you forgot what you were doing). Despite the time needed, I felt that the cognitive payoffs with the progression I used were great, and students really internalized the idea because THEY FIGURED IT OUT THEMSELVES. Figuring out the Power Rule is something that is totally in their reach, and I would have been robbing them of some learning pleasure had I just given them the power rule at the beginning.

PHASE 1: What is a derivative? We started just by drawing tangent lines to f(x)=x^2 at a bunch of points, estimating the slope and then making a table of values. I chose this function specifically because with the derivative of f'(x)=2x, it’s easy to see the pattern for the slopes in the numbers without graphing them (saving one level of abstraction). Yay, the slope at any point is just twice the x-value!

Then we did this two more times, once on a small sheet of paper for \sin x, and then once, in groups, on a huge sheet of butcher paper for \frac{1}{3}x^{3}. This was laborious and took a ton of time in class, but by the end I felt like students really understood well the idea of a derivative. More importantly, were ITCHING for an easier way to find it. They had all these great ideas that they were proposing, so it was easy to funnel their energy into the next phase.

PHASE 2: Finding the Rules. Then, I introduced the derivative tracer, a GeoGebra applet that does in seconds what they did in 15 minutes. I gave them a sheet of functions (below) for them to find the derivative of using the derivative tracer (kind of like collecting data in a typical canned high school science lab) and asked them to make conclusions about the derivatives that they found.

Though it was interesting to talk through with them the idea of a constant function’s derivative being 0, and a linear function’s derivative being a constant, the highlight of the lesson was seeing students figure out the power rule. When students got to that section, they seemed really proud that they could see the pattern. I had numerous students raise their hand to call me over to ask me if their idea worked, and then were so excited that it did that they immediately gave me a high-five. Students raised their hand so that I would come give them a high-five… in math class. I know the power rule is kind of easy, but I felt like they were so much more invested in the quest of learning mathematics because they figured something out themselves. Further, instead of trying to get ideas from my math notation, they had the ideas first and then I formalized it with math notation (though many students could do this no problem for themselves).

Long story short: The excitement in the room while the students were discovering something mathematical was palpable, even though that thing had been discovered many many many times before, including by their classmates sitting a few seats down. There was no “real world” motivation in this progression, no gimmicks – just the pure pleasure of mathematical discovery. So, to add to my ever lengthening list general goals for the year: I hope to avoid at all costs robbing students of the pleasure of figuring something out for themselves.

Here is my derivatives “lab” using the GeoGebra derivative tracer. Note that I’m not quite as adventurous as some and still want some structure in the classroom while “discovery” is happening. This is part of my controlling personality – tell me if you think this is too guided given my goals.

By the way, the answer to the other word picture puzzle is “mathematical induction.”

Calculus Standards 2011-2012: Feedback Requested


Some notes:



Calculus Standards 2011-2012

Project Implementation Reflections and Questions




Any ideas would be greatly appreciated!

(Also, below is my rubric for grading these projects)