Introducing Differentials with Stock Predictions
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…
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Posted on January 17, 2012, in Calculus, Teaching. Bookmark the permalink. 9 Comments.
Bowman in Arabia 
I like this context. I usually use a “treasure hunt” that comes with a magical compass like on Pirates of the Caribbean. So, there’s some perfect path we’re supposed to walk on to get to the treasure and the compass always points the right heading when we’re on that path. So, what we do is check the compass, go a few steps in that direction, then check the compass again and adjust our course.
We can be more accurate if we check the compass more often, but that’s more tedious. We can be less accurate, but quicker if we take bigger steps or check less often. It just depends on how close we think we are and how treacherous the off-roading may be.
Nice. I like that both of these convey the idea of accuracy vs ease of calculation. Thanks for the idea
I don’t like this context for calculus, though it is a great one for statistical modeling. The problem is that the long-term trends bear almost no relationship to the short term noise. If you take the limit as delta-t goes to zero, the results are complete junk. This is *not* what I’d want to teach calculus students. For calculus, you want smooth systems, not noisy ones.
Agree that this would be bad for something like the limit definition of the derivative, but to me the benefits of giving the kids the model of making a prediction based on current y value and current rate of change outweighs the fact that local linearity is nonsense here. I mixed it with more abstract activities that got at the local linearity idea better for smooth curves, I just wanted the stock example to make the differential approximation equation more intuitive with a more concrete way to see it. So, disagree, but I do appreciate the comment because I hadn’t really considered that being an issue here.
I write slopey but with a negative slope.
You’re welcome.
Just saw this today and it was a shock to see myself up on the first line! I’m glad that you found the term useful. I’m curious though, how you are anticipating referring back to this problem in future lessons. This is the difficulty that I have with anchor problems–I think they are great ways to start off a unit, but I have a hard time seeing how to build the unit around them in a way that really compels students to reference the initial launch activity. It’s something that I’d like to get better at, because I see it as a powerful opportunity. So any thoughts you have on how to do this would be much appreciated.
well, it was only a 2.5 day unit, so this didn’t take me too far, but it still helped solve especially some of the AP style word problems that go along with differentials… delta y could be referred to as the actual change in stock price, dy was easy to refer to as our predicted change in stock price, calculating the volume of a spherical shell as opposed to a sphere that has a radius of 1.01 or something was easy to couch in terms of the stocks, connecting the graphical idea with concavity to determine if our predictions were high or low helped to have the image of the stock graph in our head, and the biggest way we referred to it was by using the phrase “how is our stock changing right now?” to help them remember that just about the most important part of doing these predictions is to figure out how the function is currently changing. so admittedly not as good as the previous problem (because you’re right, it was mostly me referencing instead of the students), but still got some mileage out of it.
I’ve been looking for interesting ways to introduce differentials, so I can’t wait to try this. Thanks!
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