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.

 

Advertisement

New year, new job

If you’ve been following along (and honestly, I cannot imagine how anyone could possibly have time to do so!), you are under the impression that I’m on sabbatical leave this year.

There has been a change of plans.

I’ve taken an unpaid leave from my college and am spending the bulk of my professional time on curriculum development work at Desmos as a (nearly) full-time teaching faculty member.

The job actually involves almost no sitting on small children.

Our team is growing (Do you know any awesome designers? Send them our way, please!) We have a lot of great stuff in the pipeline. I’m delighted and grateful for the opportunity to work with this amazing team. It’s well known that I’ve been working with them on an extracurricular basis for some time now (>1.5 years), and this has made the transition super smooth.

I am especially fortunate to be able to set aside a portion of my professional life for ongoing projects that are outside the scope and focus of Desmos (although they are certainly consistent with the overall Des-mission of more and better mathematics for all learners!) I’ll spend a couple mornings a week in a kindergarten most of this school year, for example, and Which One Doesn’t Belong is still slated for a 2016 release from Stenhouse. (We still need to sort out Math On-A-Stick for next summer, but that’s a year away.)

New Desmos lesson(s)

You should seriously go check out Polygraph. Four versions of a delightful and challenging game:

1. Lines
2. Parabolas
3. Rational functions
4. Hexagons

The hexagons will be familiar to long-time readers of this blog.

I have run the parabolas version in College Algebra, and the hexagons version in my Ed Tech course. It was a huge hit both times—lots of conversation happened both electronically and out loud in the classroom. It’s a ton of fun.

I am especially pleased with the rational functions version. It makes for challenging work—even among the mathematically astute Team Desmos in recent trial runs.

Read the Desmos blog post on the matter if you like.

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.