# Introducing Inflection with Infection

One of the problems I have with math instruction that goes CONCEPT-PRACTICE-APPLICATION is that you miss out on some great opportunities to teach difficult concepts using application. When I introduced concavity this year, I did a great 10 minute activity that paid huge dividends when discussing what can become the somewhat tricky conceptual math of the second derivative and inflection points.

We got back from break this past Sunday, so I started just by asking what everyone had done for break. I had a secret agenda for this though, because from their answers I chose the person who went to the most exotic or most random place and chose them as Patient Zero for an infectious disease that was going to infect our classroom. I assigned everyone a number (I used a deck of cards, but I have done this before with the random number generator on the calculator too). Then, while Patient Zero stood on one side of the classroom everyone got as far away from him or her by standing on the other side of the classroom, so as not to catch their exotic disease. Then Patient Zero picked a card and whoever’s card got picked got infected with the disease and came over to the sick side of the room. I stood at the board and recorded data for how many total people had been infected up to each round. Then, all cards were replaced and all sick people picked someone to inflect. Then, again. Then again and again until the whole classroom was infected.

The disease spread exponentially at first, but once people started picking others that were already sick and the supply of healthy people dwindled, the spread of the disease slowed down, a nice beautiful logistic curve. Here is the data we collected:

After we hand-graphed the points together, I had them then write bullet point stories for why the graph looked like it did. Most students were quickly able to notice that the infection spread quicker and quicker at first and then started slowing down for the reasons I noted above. Many even picked out the inflection point (but not by name) by saying that this was the point the infection was spreading fastest. (The data weren’t as pretty in my other class, but that’s okay! The main point still worked, and it’s nice to see that models aren’t perfect).

After that, we were ready for Calculus. This was a perfect thing to do after a 3 week break, because I then had them tell me everything they could in terms of Calculus to fit with the stories they wrote. We had a very lively review of everything I had wanted to review grounded in this conversation about infection, and the conversation really primed us to talk about inflection points in a deep and meaningful way. The rest of the lesson was like cutting through sponge cake (is that a saying people say?) The idea of an inflection point possibly occurring when the second derivative is zero made far more sense than if I had tried to state this fact and then show why, or make more abstract graphical arguments.

My goal this year has been to motivate well all the math occurring in the classroom, both with application and pure mathematical ideas, and I think this is a good example of success! More to come on other applications of concavity.

Posted on January 15, 2012, in Calculus, Teaching. Bookmark the permalink. 12 Comments.

1. Donny Dickson - Bowman's Brother

THAT IS SO COOL…. I hope you gave a good name for the disease too. Definitely stealing that if I ever get to teach inflection points.

2. Great example of a good, accessible launch problem. The Escape the Textbook group calls these “anchor problems,” with the idea being that you can keep referring back to the problem to help students latch onto the new learning by remembering this solid, relate-able context. But so far I’ve had a lot of difficulty finding good problems that actually do “anchor” my units. This one seems like a good example. (Though I don’t teach calculus this year…maybe next year)

• ooh, i like that terminology. it helps to have words to describe stuff like this. thanks!
(also, this activity is great for logistic/exponential stuff in pre-cal of course!)

3. I do this for logistic growth in both precal and calculus, too. I just go ahead and tell them it’s a zombie disease. I also assign them numbers on the way in to class and use the ti84 calculator as a random number generator. They can just hit “enter” on the projected calculator to infect the next person.

We begin class with predictions about populations in different scenarios (a dozen mosquitoes in the USA, 5 billion people in earth, 400,000 frogs on a tiny island, 1 wolf in a forest, etc.) to correspond to different initial populations to guess how the logistic model may change with different initial conditions. They go over their predictions in groups and then as a whole class. Then we do the zombie infection to see how we can look at populations of infected and match that to different scenarios. (Can begin with only 1 zombie or 15 or whatever.)

I always think it would be interesting to give both sides a calculator (or cards as you use) and have both infecting and curing going on to see how that might change things, but we usually run out of time for the lesson and the math is more detailed than we could get into anyways.

4. You’re a genius. Love it love it love it.

5. suevanhattum

Someone else posted a while back about a werewolf attack, and using dice to simulate it. I’m doing that today, to achieve the sorts of things you mention in this post.

(I tried the cards last semester, and it took way too long. I might not have understood how you did it.)

If it goes well today, maybe I’ll post about it. (Thanks for all the inspiration.)

• Was it Shawn? I’d love to hear about it. You’re right though that the cards take a really long time. I did it with the random number generator this year and it was much quicker.

• It did go well on Wednesday, but the Werewolf attack gives exponential decay, not a logistic curve. (As soon as I saw that, I felt so silly that I hadn’t realized ahead of time.) It was my first day getting them thinking about exponential growth (etc), so it was a good inclusion anyway.

And then we made up a game with the dice in 1 minute that we completed in 5, which did achieve logistic. I said I had a horrid new disease and that I had just breathed on the 5 students nearest me. They rolled dice, and whoever got sick, the 5 nearest them (sometimes including the already at-risk) had to roll. We didn’t get to do it out with a table of values and a graph, but they could see the slow-fast-slow growth pattern, as it swept across the room. Cool!

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