Tag Archives: assessing models

Related Rates: Compost oh compost

I'm on a nitrogen kick, I guess. While researching the previous post on beets, which ought to have a follow-up when we get to integration (did you do the worksheet?!), I learned a bit about carbon-nitrogen ratios in compost. The ratio of carbon to nitrogen is important because compost gets hottest when this ratio is around 30:1. Empirical evidence supporting this is not too hard to find, but a mathematical model is hard to find! This is why you need to mix "browns" and "greens" in your home compost.

Again, nitrogen is a very important nutrient for vegetable and grain growth -- if you use up the nitrogen from your soil you'll have small flowers and no tomatoes. Nitrogen conservation in your compost is thus pretty important. But again, I had a hard time finding math equations for this -- unless you count all the cool papers that solved twelve nonlinear systems simultaneously.

Finally, after days of trolling through the library files and becoming much more aware of what a HUGE BUSINESS waste processing is, I found a paper with some polynomials. It's also got some great 3D graphs and some visualizations, in Figure 4, of how these functions depend on their independent variables. Might be food for another post. The authors of "Optimizing composting parameters for nitrogen conservation in composting" took an approach similar to the beet-research people: they did a bunch of experiments, measuring values for aeration of the compost heap, moisture content, particle size, and time after start, and ran a big backward computation to come up with polynomials in those variables (A, M, P, and t) that predicted the carbon/nitrogen ratio (among other things) pretty well. That is the polynomial that the worksheet below focuses on.

This worksheet covers

  • related rates, applying the product and chain rules a few times
  • physical reasoning: I ask students to examine the assumptions of the model, which I then violate for mathematical purposes!
  • and writing English sentences explaining a math result.

Teaching tips: students often freak out about all the symbols in here. Reassure them that many are constants in the problems they're asked to work out. Remind them that the derivative of a constant is zero, even if the constant is one they don't know!

I would like to have had some other "real-life scenarios" or more interpretation, so I'll think about what sorts of related-rates problems could be added to this.

Related Rates: Nitrogen

If you've got refinements or modifications, let me know!

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


Trigonometry is often a terrifying topic for students taking a first calculus class in college. America's trig teaching seems a bit haphazard, and we can't assume students have a good grasp of periodic functions.

This summer project is clearly not offering a full calc curriculum, but instead offering supplemental material. A good lesson on the unit circle is a first step in any college calc discussion of trig. After that, though, students may find practice with a variety of trig problems useful. You can look at particle motion and springs and all sorts of mechanical applications, or you can use a cute and fuzzy animal that also has sharp teeth and claws.... the lynx!

A lot of arctic populations including lynx, snowshoe hare, and lemmings have cyclic population fluctuations and there is a lot of research about why this is. If you've got students in ecology you can ask them to tell you about this in more detail: this is a great topic for a small project or extra credit assignment.

I love this family of examples: there is so much room for discussion of the real world, from foxes and rabbits in my own neighborhood to the wolves and moose of Isle Royale. The examples work so well from precalculus through multivariable, linear algebra, and differential equations. Students can easily experiment with changing parameters in their models, using Excel or more sophisticated programs. And it's all about the circle of life, one of the most compelling stories we as humans know!

Alright then. To the worksheet. I may be a bit dissatisfied with this one yet for reasons I'll outline below. The instructor should display a graph of the actual data, included below. Students then work through 

  • constructing a trigonometric function from the amplitude and period, 
  • considering the shape of the actual data as they construct the function, and
  • critiquing their model at the end.

As so often is true in modelling, our first instinct -- the sinusoidal function -- is not actually most accurate. If you've got access to the lynx data (you can also find it as a dataset in R, the stats program) you can check that the logarithm of the data is actually more sinusoidal! Look at the long troughs in the the non-log graph, along with the sharper peaks. With students you can use the idea of symmetry to discuss why a sinusoidal approximation is not the best for the non-log-transformed data: the graph does not have reflect-and-glide symmetry, unlike cosine and sine. I, though, honestly don't yet know a good mathematical explanation for the fact that the logarithm of the population data has a more sinusoidal shape. I'd love to hear one.

Worksheet constructing periodic function for lynx trapping

Plot of lynx trapping over time

Plot of logarithm of lynx trapping over time

I know that someone other than me has looked at this blog in the past week! Question of the day: do you use cute fuzzy animals to increase interest and engagement in class?

Group work and a power function for atmospheric pressure

We instructors of calculus know that linear models aren't everything, even though linearization is in some sense the point of the differential calculus. Since the first week of many calculus courses begins with a precalc review including power functions, I'll just move smoothly along to a power function model for atmospheric pressure! (Don't worry: we'll get to lynx trapping in the Yukon for a trig review activity in a day or two -- not everything is about physics or the atmosphere.)

It's always important to remind students about the difference between power functions and exponential functions, not least because they've got different differentiation rules. One nice way to look at power functions and exponential functions is by looking at growth -- we know that  x^2 and  2^x grow at very different rates. But everyone does that... and I was having fun with atmospheric pressure! This worksheet has a very funky power -- 1/0.19... -- and might be a good way to acquaint students with the messiness of real-life models. I will return to this topic when we get to derivative and integrals, too, because this equation is actually fairly easy to derive.

The worksheet below tries to foreshadow the idea of the derivative fairly heavily. It asks about 

  • composition of functions,
  • intervals of increase and decrease, and
  • slope of the tangent line.

As usual, I try to incorporate a bit of writing and thinking about the meaning of a model as well. There's definitely room for discussion around these worksheets.

In addition, you might notice that there's a bit of tedious calculation at the beginning. Why would an enlightened modern instructor do that? I like to give these worksheets to students in groups. At the beginning of the semester I always give a student survey asking about past math experience, major, problems or gifts I should know about, outside interests, favorite dessert, favorite color.  In the first few weeks of class I use this to arrange student groups and ask them to figure out how I've grouped them (by major, dessert, color, last name...!). Giving them just a bit more tedious calculation than most people would enjoy gives me a chance to encourage and incentivize conversation within groups even more.

This week's worksheet: atmospheric pressure as a power function!

A better model: power function

There's also a natural place to discuss solving for the inverse function here, and I might add a worksheet about that soon too.

Do you use group work or worksheets with students? Why or why not? What kinds of constraints do you have to deal with in considering group work?

A linear beginning

I've taught calculus now at several different college and universities. Calc is a funny class these days: students who have done quite well in high school math now often enter a linear algebra or multivariable calculus class directly, so students in calculus come from a variety of backgrounds but often did not have a good high school math experience. Often students in college calculus classes took the AP calculus offered in high school with varying results. A surprisingly high number of students I've seen in college calc did not even take precalculus in high school. This is generally a recipe for disaster.

During the fall and spring semesters I start every calculus section with a conversation about what calculus is good for and what other classes a student could take to fulfill requirements. I like to push statistics and math for liberal arts majors courses: statistics is becoming crucial to survival and innovation in a number of fields, from finance to medicine, and math for liberal arts majors classes often expose a student to graph theory, voting theory, and other useful techniques for looking at problems we all encounter in life. There is so much beautiful mathematics! Don't get stuck on calculus as the only way forward!

After that, it's time to start with review. I phrase it as a warm-up: here are things you will need to dredge out of your brain and reacquaint yourself with to be successful in calculus. First is the point-slope equation for a line.

The worksheet attached explores the change in atmospheric pressure as we increase in altitude from San Francisco to Denver to the peak of Mount Everest. In it, students develop a linear approximation for pressure in kilopascals from two data points (no use of derivatives), and then examine the validity of their approximation. Use this to explore:

  • how to work through a word problem
  • the point-slope equation for a line, introducing the idea of slope as rate of change
  • critical thinking about models: comparing theoretical and actual results can point out weaknesses in a model!

When discussing this worksheet, remember that temperature, humidity, and weather patterns affect the pressure of our atmosphere as well. Some of the lowest pressure readings ever taken near the Earth's surface have been in the centers of hurricanes, for instance. In calculus we study "baby problems" so that we can eventually build up the techniques to model situations more accurately -- like scales before playing Beethoven.

Linear Functions -- Head in the Clouds Worksheet

How do you start out your calculus classes in the first few days? How much review do you do? Do you think all of your students are best served by calculus?