Monthly Archives: July 2013

Drugs in waterways: derivative mix

We return to naproxen (sold under the brand name Aleve). Naproxen is my "drug of choice" for these worksheets because it apparently occurs in a lot of our waterways and its decay is pretty well understood. Last time we discussed naproxen in particular, we looked at a function that gives the rate of photolysis for naproxen at a depth , the rate at which the substance breaks down in the presence of sunlight. There are a few different ways that substances like naproxen, ciprofloxacin (an antibiotic), cocaine, or bisphenol-A get taken out of waterways: breakdown in sunlight, breakdown by organic processes, or sedimentation. Naproxen breaks down easily in sunlight but it doesn't like to be filtered by sand or settle out into sediment even when the water is treated with ferric sulfate to make coagulation happen.

The linked abstract is for a paper about a pilot-scale drinking water purification plant, looking at how water from the River Vantaa could be used for drinking water if the groundwater source for Helsinki, Finland, were to be rendered unusable. Remember that groundwater usage is increasing enormously across the world, and so our nice clean aquifers are overtaxed in many locations. We should not waste so much water (agriculture and lawns, folks!) but will also need to learn a lot about how surface water can be purified so that we can drink it again.

The worksheet below has a mix of derivative and rate of change questions. It asks about some derivatives that require the chain rule (quotient rule and exponential function rule combined) and it also asks students, at the end, to switch variables and look at how the rate of photolysis changes as turbidity changes. After every heavy rain a lot of sediment enters a river and then settles out over time. Development and construction can also change turbidity substantially: digging up a lot of trees and plants to expose dirt allows a lot of that dirt to run off. Agriculture also has its role, as during the planting season fields can be vulnerable to erosion and run-off.

Chain Rule: Photolysis of Naproxen

If you're in a position to work with a science teacher or run experiments yourself, I found a fun page on experiments with turbidity appropriate to junior to senior high school students (and what college student wouldn't mind playing with mud, really?). This could make a cool big brother/big sister activity: high school seniors do the math and the freshmen or junior high students do some experiments on turbidity. In addition, there's a World Water Monitoring project and day (September 18) that you could join.

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

Stories and Graphing

I've been spending the last few days trying to edit some math papers I'm working on and learn more about polygon spaces. (What's the space of quadrilaterals in with side lengths (1,2,1.3,1.8) and one vertex at the origin, for instance? If you guessed that it looks like , you'd be right!)  It's always interesting to talk with students about what mathematicians do. They expect we do more with numbers, and when I talk about what I am working on they're surprised at all the funny little pictures I draw and all the writing we all do.

This activity is a good one for discussion. You could either print this or use a projector to put it up in front of the class for a group discussion. Group discussions can be a great way to get a class comfortable with each other and facilitate the verbal discussion of mathematics, which is another skill that must be learned and practiced. I had a hard time learning to "talk math" despite my good reading and essay-writing skills, and even now I need to gather my thoughts to produce coherent mathematical sentences in a way that I don't need to when talking about politics, for instance. I've concluded that I think about mathematics nonverbally and that the translation process simply takes some time, even now that I've had a lot more practice than your average freshman.

When I run a class discussion like this, I try to do a few things:

  • Give students a minute or two to think in silence and write down a tentative answer.
  • Start cold-calling students -- "Pahoua, what do you think about the first one? Jordan, what about you?" This requires a level of trust that is built over time.
  • If students are right, I don't necessarily say so and move on -- I keep asking other students, and ask other students if they think the first student is right or not. I don't want students practicing the Clever Hans strategy. (Clever Hans is a horse who counted by tapping his hoof, but he just watched his trainer until the trainer looked happy and then stopped.)
  • If students are wrong and I can identify the assumption that led to the mistake, pointing out that assumption can often help. "Danielle, are you assuming that the graph must be increasing?" Sometimes I ask other students for their thoughts. "It looks like Jason disagrees. Why?"
  • If a student is visibly hesitant or reluctant to answer, ask for a question or the beginning of the thought instead. "Alexi, what do you think about part c? ...   What strategy are you thinking of trying?" or "What do you notice first about the graph?" or "What question are you asking yourself right now?" These often prompt good discussions.
  • I do try to use students names consistently as I call on them, and
  • I try to keep track of who I call on and rotate through the whole class. There's plenty of research that shows that even the most enlightened and conscious instructors call more on boys than girls, etc., so bringing a class list and discreetly checking names off is an easy way of making sure you involve everyone in discussions. Telling students that you track who you ask is sometimes useful, too, because then they know that everyone will be asked and they should be prepared! Students do think that it's fair to do this, especially if you offer the outs suggested above if they are stuck.

So, with no further ado, here is a worksheet on matching graphs and stories, using the language of derivatives:

Graphing Stories and Stories of Graphs

Copyright details: The graphs on page one are:

(A) Water levels in Lake Superior measured at Marquette, MI, using data from as suggested by commenter TAO. 

(B) Well water depth data from observation well 27010, with data from

(C) Data on lynx populations using the dataset included in the R programming package. The data is from the Hudson Bay company's logs.

(D) A graph from, since I can't seem to find raw data for lemmings. Used under the fair use policy for copyrighted documents and documents produced by the federal government.