The Space Between the Numbers commented on my last post that some people call launch problems like the infection one I did for concavity “Anchor Problems,” the idea being (and I quote from her) “you can keep referring back to the problem to help students latch onto the new learning by remembering this solid, relate-able context.” I think that this describes what I was trying to do perfectly, and I love having new language to talk about my teaching. So, thanks for that!

I wanted to share one more Anchor Problem that I am using in my AP Class tomorrow to introduce Differentials (which I used to what I thought great success in my non-AP class last year). I am relating the idea of making a prediction with differentials/a tangent line to making stock predictions. With charts like this:

… I’m going to have them predict the stocks price in the near and far future using the graph in any way they can. Most students last year figured out to draw a tangent line, use the slope of that tangent line to see how fast it is changing currently and then multiply that by the number of months to get the change in price, and then add that to the original price. It’s an intuitive idea that sounds way more complicated when you try to describe it. I added the “% confidence column” this year to try to get at the idea of a prediction being less and less accurate the further you are from known data.

Eventually, I want us to codify our process into a rough equation like this:

Which we can then use to look at the “equation” for using differentials:

(I think my face is crooked… I always write slopey)

This worked well last year because even when we were solving abstract problems that had nothing to do with stocks, I would ask questions like “well, how fast is your stock changing right now?” and “how many months in the future or past are we predicting?“ which served to connect the abstract to the intuitive situation. I also think it gave students a mental picture of what they are doing.

Hope it works this year too…

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.