Infection Points: The Shape of a Graph

Everyone here knows that I think Bowman Dickson is the bee’s knees, the cat’s pajamas, ovaltine! Recently he posted about how he introduces inflections points in his calculus class… and just a couple days later, I was about to introduce how we use calculus to find out what a function looks like.

Usually, I introduce this in a really unengaging lecture-format. But he inspired me to … copy him. And so I did, extending some of his work, and I have had an amazing few days in calculus. So I thought I’d share it with you.

The Main Point of this Post: By creating the need for a word to talk about inflection points on graphs, we actually saw the math arise naturally. And through interrogating inflection points, we were able to articulate a general understanding of concavity. In other words… the activity we did motivated the need for more general mathematical concepts.

First, definitely read Bowman’s post. All I did was formalize it, and extend it in a few ways, by making a worksheet. I put my kids in pairs and I had them work on it (.docx):

What naturally will happen when students generate their graphs is they will get a logistic function. (Which has a beautiful inflection point! But they don’t know the word… they just see the graph.)

So here we are. The students have a graph, and they’ve been asked to explain their graph for (a) the layperson and (b) the mathematician. Most get some of it done with their partners, and then they take it home to finish individually.

The next day, at the start of class, I assign students to work in groups of 3 (with different people than their partners the previous day). They are asked to take a giant whiteboard and:

(Now I want to give credit where credit is due. I have really been struggling with using the giant whiteboards well, and having students present their work effectively and efficiently. My dear friend Susanna, when I told her about this activity, suggested the groups, the underlining of the mathy words, etc.)

This worked splendedly.

(click to enlarge)

And they had such great observations. Some groups picked up on that change where the function was increasing in one way to increasing a different way. Others talked about how the rate of change (of infected over time) was greatest. Others talked about how the function was “exponential” for the first thing, seemingly linear for the middle third, and “something else” for the last third.

Those gave rise to good short discussions, and we came up with the language for inflection points (which I call INFECTION POINTS!!! GET IT!?!) and concave up/down.

After they had a sense what those words meant, I had students work in partners on the following (.docx):

The point was to get students comfortable with the ideas before we delve into the heavy mathematical lifting. It was powerful. Especially the last page, which got students thinking about patterns, exceptions, and ways to generalize. Our big conclusions:

And with that, I’m too exhausted to type more. But that’s the general sense of what went on in an attempt to teach how to use calculus to analyze the shape of a function.



  1. Really cool lesson–I think it makes all the difference when students see that they can mathematically reason even before learning the formal mathematics. I noticed the link to was all to the same YouTube video instead of BD’s post:

    Also, what program did you use to make those fantastic graphs in your worksheets? I’d love to make worksheets that pretty.

  2. Thank you so much! I fixed it – bowman deserves totes a ton of credit!

    As for the graphs, I use a free program called Winplot (google it!). It has a slightly high learning curve, but once you get it, it’s totally worth it.

  3. AMAZING! I think inflection points tend to be harder to grasp (or certainly harder to visualize) than relative extrema. For this reason, students get frustrated–and rightfully so. A project like this puts so much more meaning behind the math lingo.

    Many thanks to both you and Bowman Dickson. I love this idea and plan on stealing it. :)

  4. I did steal it. I modified the handout slightly to reflect our school. I used it to introduce logistic functions to a precalc class. It worked beautifully- I knew it captured their attention when the English teacher targeted as the virus originator had three or four groups of kids visit his room to tell him about it. Only one unanticipated problem. Most of my kids had never used the random integer function on their calculators, so they all got the same data. Next time I’ll have them play with the function a bit before starting the worksheet. I did the whiteboards too with great success

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