Calculus: From Secant Lines to the Tangent

Our semester began on Monday. I'm teaching Calculus I (as always, because it's my favorite class), Statistics, and Algebra for Statistics. All three classes were a joy to teach this week, even though I was a bit underprepared because of the chaos in my personal life.

On Thursday I was working on wrapping up the exercise from Active Calculus that the students had been working on since Tuesday. I showed the velocity curve we'd been exploring on Desmos, and limited the domain to the appropriate times, 0 to 3 seconds (which I learned how to do with the face project I described in my previous blog post). I had a little trouble remembering how to make a secant line attached to one stable point and one moving point, but I got it. (And helped the students get it. This took some hard thinking for many of them.)

Then I had a wonderful surprise. When I pulled the moving point over the stable point, the line disappeared and Desmos said "x= undefined or undefined" (not sure where their stutter came from...). I gasped. I hadn't expected that, and it was a perfect way to start talking about this problem calculus has of needing two points to figure slope, but needing to use just one point from the function in order to have a tangent. I got to talk about Newton and Bishop Berkeley and fluxions and infinitely close. It was great fun for me. On Monday I'll find out how much the students got out of it.

Calculus: Reviewing Functions with Desmos

I like to dive into the calculus ideas in my calc I course, so I do not start with review. (I use a just-in-time approach, reviewing what we need when we need it.)

But I know that the students' understanding of functions is weak and needs to be brought to mind. So I was excited about having them outline their own faces in Desmos as a homework assignment, which I learned about at Twitter Math Camp from Deb Boden (@debboden).





Here are my instructions:

Desmos Graph of Yourself

1. Set up an account on desmos.com. (It’s free.)
2. Upload a selfie into desmos. (Click the + in the upper left corner of the desmos calculator screen to add your image. Photos with you facing front are easiest to use.)
3. Use various functions to outline features of your face. (At least: lines, arcs of circles and ellipses, parabolas, and trig functions. Try including exponential and log functions, hyperbolas, and cubics.)
4. When you’re done, you can hide your photo to display the icon you’ve created. You can also hide the axes by clicking on the wrench in the upper right corner.
5. Add a link to your completed desmos work on our class google doc: [Link removed, for student privacy. I didn't need google, actually. They could have submitted directly to Canvas. But we may need the google doc for our viewable collection.] (My icon is linked there. You can check it out to figure out how to do this.)
6. We’ll share these in class and see how many classmates everyone can recognize.

Every time I got stuck, I googled my question. For example, "Desmos function restrictions" helped me make short pieces of the curves I used. If you are still stuck, start a discussion item here.


Rubric
50% Required function types  (lines, arcs of circles and ellipses, parabolas, and trig functions) 10% each (Extras can bring this score up to 60%)
15%   Good Match with Photo
15% Visually Engaging
20% for using Desmos and Canvas (our "learning management system")

(I didn't post the rubric until after they turned in their work, but I will next time.)

31 out of 44 students turned it in. I am loving Canvas, which our college just started using. (We used d2l before and I hated it. Yes, I have strong feelings about things.) I took hours grading this, but once I get good at it, I think I could do this in about an hour. Canvas made it easy. And now I know that a third of my class is having trouble. So I know I need to intervene somehow. Good information to have.

Here are some of my favorites...