Tag Archives: short projects

Lynx: the chain rule and a better model

As promised, a return to lynx. In my previous post about lynx I posted a worksheet modeling lynx populations with a cosine function, and mentioned that this is not the best model. Look at the derivative to see how bad it is -- the green and red lines ought to be matching up:


Graphing the log of the lynx data gives a transformed graph that is much more sinusoidal! The better model for the lynx data, then, is exp(something sinusoidal). Look at the graph below to compare Model 2 and its derivative to the data. The green and yellow curves are much more alike:


This worksheet guides students to developing this model after having them evaluate the previous sinusoidal model via technology.

The worksheet I'll include below is meant for a day when you have computer lab time with students. I know that this does not include everyone... but if you can head down to the lab for such an activity, there is a lot students can learn!

This worksheet applies knowledge of:

  • the chain rule, on compositions of trigonometric and exponential functions
  • numerical approximation of the derivative
  • shapes of graphs.

Along the way students must evaluate models and create one of their own.

As the instructor, you'll have to decide what software you want to use for this activity. I have had success using Excel, asking every student to email me their work on the way out of the lab, and these days you can use Google Drive if your institution uses Gmail. If you and your students are already quite familiar with R you could also use that. Beware of differences between Mac Excel and Windows Excel, especially in graphing -- work through the activity yourself on whatever platform students will use.

Chain Rule: Lynx

Lynx Pelt Data Spreadsheet

Incorporating short projects

So far I've provided worksheets for group work in class, as many instructors are not able to modify the curriculum or grading system for the first-year calculus courses they teach. Worksheets work well in this situation because you can just slip them into a class discussion when you've covered the basic lessons, or give them out to students who are more advanced. On the other hand, even in somewhat inflexible courses sometimes you have the freedom to give students a take-home mini-project. Many of the topics I've covered so far could be the seed for a mini-project.

When I say mini-project, I'm talking about a project (somewhat open-ended) that is less than two pages. Less than two pages ensures mini-ness and makes it easier to grade.

An example of a mini-project would be:

  • Find your own population to model (subject to instructor approval) or use the lynx population in the Yukon between 1821 and 1934. State clearly in words what population you are modeling, what type of equation (trigonometric, exponential, linear) you are using, and why.
  • Write an equation that models the population using appropriate mathematical symbols.
  • Create a graph with the data and your model clearly indicated.
  •  Discuss the strengths and weaknesses of your model.

One possible rubric, then, grades students on writing, modeling, and graphically presenting data:

Sample grading rubric for mini-projects

In other classes I've used a rubric that grades on clarity, conciseness, correctness, and completeness, especially useful for writing-intensive exercises. Peer grading is another option for situations in which instructor grading will not work, and has the benefit of exposing students first-hand to the difficulty of reading the mathematical work of others!

(A gratuitous sample peer-review rubric from another class -- not calc!)

I find that a very concrete rubric for mini-projects helps students structure their exploration: they know they need a topic that they can (1) model with an equation and (2) draw a graph for. These mini-projects are to reward exploration, which is something many students taking calculus as a terminal class really fear!

I love mini-projects. They're a good way to allow some creativity in a rigid structure -- replace a quiz! They are low-stakes: when students send you the panicked email about how long it needs to be, you say, "Less than two pages -- with pictures." Often students can come up with really cool topics from other classes or other life experiences, so with a solid rubric they're actually fun to read and not so hard to grade.