Tag Archives: calculus

A new Calculus Activity Builder activity

Let me bring you up to date, in case you have not been following along.

I am on leave from my community college teaching this year, and am working at Desmos remotely from St Paul.

A large chunk of my time involves working on the pedagogy side of Activity Builder, which we released this summer.

Activity Builder lets you build a classroom activity using one of three basic screen types: graph, question, and text with image.

From time to time, I’ll take the opportunity to turn something I’ve done in the classroom before Activity Builder and make an online version. I did that yesterday. (Here is a link if you want to play along as a student—I recommend doing that!)

It’s a simple little calculus activity on the surface. You see a function that is graphed on the coordinate plane, except that parts of the graph are obscured by large black circles.


There are four such graphs, and I ask the same three questions of each one.

  1. Behind which circle(s) must there be roots for this function?
  2. Behind which circles might there be roots?
  3. Behind which circles is it impossible for there to be roots?

After each round of questions, you have the opportunity to move the circles aside to see for yourself whether there are roots.

This is a little routine I developed as a Calculus teacher to spur conversation, and it contrasts with a standard textbook approach, which asserts the importance of three conditions for knowing there are roots:

  • continuity on the interval in question, and
  • a sign change between the interval’s endpoints

In that spirit, you are told in this activity that the first three functions are continuous. You are not told that the last one is.

In a classroom setting, I’ll discuss these examples once students have worked through them. In that discussion, I want to get students to verbalize the following things:

  1. There are sometimes roots where you don’t expect them (Screen 8).
  2. There are sometimes not roots where it looks like there really ought to be.
  3. If the function starts negative and becomes positive, it has a root.
  4. And vice versa. (Screen 4)
  5. AS LONG AS THAT FUNCTION IS CONTINUOUS!!!!! (Screen 16 for crying out loud)

Only after that am I ready to state the Intermediate Value Theorem.

This activity illustrates a curricular principle I sketched out recently, which is that lessons build on students’ experience, and help them to structure that experience mathematically.

This activity creates an experience for students, and then it’s my job to help students structure that in a formal way—through statement of and exploration of the Intermediate Value Theorem.

I’m not a big fan of providing structure for things students haven’t experienced. Typically they see no need for it, and struggle to incorporate these structures into their view of the world. Also, students end up lacking meaningful mental images for representing and triggering the formal structures.

This is theme that plays out in all of my work, by the way. Math On-A-Stick, Oreos, Talking Math with Your Kids….all are predicated on Experience first, structure later.


Brief thoughts on being ready for calculus

A smart friend (whose permission I have not asked) read an article of mine that will be published in Mathematics Teaching in the Middle School sometime soon. The article is based on my NCTM talk last spring, titled “They’ll Need It for Calculus”.

This friend asked by email:

For clarification: are you arguing that the sorts of problems that you point to will help students better understand calculus, or that these sorts of problems will help students do better in their calculus classes?

I was pretty sure that you were making the first argument, but not the second.

My reply, which I stand by, is this:

That these two things are different from each other is a pretty damning critique of the whole affair, is it not?

You know what will help them do well in their calculus classes? Memorizing about 20 of these:


The Fundamental Theorem of Calculus

I have taught Griffin (9 years old) the Fundamental Theorem of Calculus.

That is…

\frac{d}{dx} \int_{a}^{x} f(t)dt=f(x)

Details and discussion coming soon.

In the meantime, see Kristin’s related post.

Full disclosure. Griffin was paid a sum of $0.25 for his performance.

A little gift from Desmos

Last summer, the super-smart, super-creative team at Desmos (in partnership with Dan Meyer, who may or may not be one of the Desmos elves) released a lovely lesson titled “Penny Circle“. It’s great stuff and you should play around with it if you haven’t already.

The structure of that activity, the graphic design, the idea that a teacher dashboard can give rich and interesting information about student thinking (not just red/yellow/green based on answers to multiple choice questions)—all of it lovely.

And—in my usual style—I had a few smaller critiques.

What sometimes happens when smart, creative people hear constructive critiques is they invite the authors of the critique to contribute.

Sometimes this is referred to as Put your money where your mouth is. So late last fall, I was invited to do this very thing.

I have been working with Team Desmos and Dan Meyer on Function Carnival. Today we release it to the world. Click through for some awesome graphing fun!


It was a ton of fun to make. I was delighted to have the opportunity to offer my sharp eye for pedagogy and task design, and to argue over the finer details of these with creative and talented folks.

Go play with it.

Then let us know what we got right and what we got wrong (comments, twitter, About/Contact page).

Because I just might get the chance to work on the next cool thing they’re gonna build.

Ready for Calculus

I’m working on a paper that will be submitted for publication. It mostly records and expands on my NCTM presentation in Denver a while back.


Along the way, I decided to a bit of looking into the claims I have made about the standard view of what it means. I did it the lazy way. Google search on “Ready for Calculus”. 

See for yourself. (Click on the images to be taken to original sources on the web.)

Exhibit A

Screen shot 2013-06-26 at 4.11.13 PM

exhibit B

Screen shot 2013-06-26 at 4.12.32 PM

Exhibit C

Screen shot 2013-06-26 at 4.15.57 PM


Exhibit D

I did find one that had a different nature

The most important precalculus concept is the notion of a functional relationship between two variable quantities. This relationship may take many forms: linear functions, power functions, exponential functions, logarithmic functions, trig functions, polynomial functions, rational functions… Functions from these basic families may be combined, transformed, and inverted to produce still more functional forms. Functions also appear in various representations: formulas, graphs, data sets… You will have to be familiar with the basic families of functions, and all of their representations, in order to succeed in your study of calculus. The concept of function underlies everything that calculus considers.

This was nice, and if you’re at all interested in this topic, you should go read the essay in its entirety. (Don’t worry, it’s short.)


And now let’s all imagine how a community college developmental math program that took Exhibit D more seriously than Exhibits A—C would be different from the present-day state of affairs.


The goods [#NCTMDenver]

Good turn out for my session Saturday morning (EIGHT O’CLOCK!).

Thanks to Ashli Black (@Mythagon) for the shot of title screen.

I’ll get some more details up here sometime soon. In the meantime, here’s the handout (.pdf). And here’s the slide deck (.zip, and which—to be honest—was just a photo album on the iPad; the simplicity of this was liberating).

Here are Alison Krasnow’s notes from the session.

road.to.calculusOne last thing…this is the absolute best form of session feedback, as far as I am concerned—getting to read someone else’s notes on the session speaks volumes about what participants experienced (in contrast sometimes to what I think we did).

The slides:

UPDATE: This talk has been adapted to a paper submitted to Mathematics Teaching in the Middle School. I’ll keep you posted on its progress.

Polar coordinates

More lazy email(ish) excerpts. This one from a conversation about polar coordinates, and my Calculus II students’ struggles with them last semester.

I don’t fully understand the usefulness of polar coordinates as traditionally presented. I feel like a great deal of the focus is on the pretty pictures we get when trig functions interact with polar coordinates. Cardioids and all that.

What do we gain from graphing this in PreCalc?

When I taught Calculus 2 this past semester, it was clear that my students were struggling to sort out differences between cartesian coordinates and polar coordinates. They knew how to convert between polar and cartesian coordinates, but they didn’t seem to know why one would do that, nor did they seem to see polar coordinates as a self-contained system. Polar coordinates were always (in their minds) in relation to cartesian coordinates.

My students struggled to think about an angle as an independent variable that could change (and correspondingly a radius as a dependent variable that could change).

They couldn’t view a function defined in polar coordinates as a dynamic relationship. They could identify points one at a time. They could make their graphing calculators display polar graphs. But they couldn’t think about the process of tracing out a polar graph. This seemed to be true even for students who could talk dynamically about cartesian graphs (increasing, decreasing, approaching an asymptote-this was terminology my students could apply to cartesian graphs, but not to polar ones).

The precise conceptual nature of the relationship between polar coordinates and cartesian parametric equations is unclear to me. My students saw some relationships that they couldn’t quite articulate. I’m interested in exploring this territory a bit.

For the record, I was flying blind through this material last semester. It was the first time I had taught polar coordinates in any serious way. I have never taught precalculus. I had never taught Calculus 2 before. So I kept bumping into obstacles that I hadn’t imagined would be there.