There’s been a lot of laughing / crying about cubic models in the last few days, but I thought talking about modeling could make a nice lesson for my son who is reviewing calculus. Then I saw a great tweet from Carl Bergstrom that made me want to give it a go.

First we talked about cubic polynomials in general and what we can learn about these curves from calculus:

Next we talked about fitting a cubic polynomial to data. We have talked a bit about fitting curves before – my son mentioned this project on fitting temperature data which used some amazing work from John Shonder on looking at temperature changes in each US county for the last 100 years:

In this video I asked him for his ideas about why fitting data with a cubic curve might lead to problems:

Next we moved on to looking at the cubic model that was published and focused a bit on the fascinating tweet below from Carl Bergstrom. I think Bergstrom’s tweet is a great lesson for calculus students because he’s noticing that the “cubic fit” can’t be a 3rd degree polynomial because the 2nd derivative isn’t right:

So if it's a third degree polynomial fit, how they get a + / – / + pattern of second derivatives?

Magic marker?

(Not being sarcastic; the dotted part of the pink line might just be hand-drawn)

Finally, we looked at a few cubic and log-cubic fits to corona virus deaths in the US. I showed him that forcing a cubic fit to the data ended up with some strange results. Finally, I asked him why those strange results might be coming from the cubic fit (which is a pretty hard question for a 10th grader):

This was a nice project – and an especially nice one to show a calculus example directly related to current events.