Category Archives: Precalculus

Nitrates never end

Nitrates, nitrates…

Well, the semester is coming to a close again. Two more weeks of class and then finals. My corn has grown pretty tall and survived a squirrel attack, and next weekend I should be able to plant it outside: April 30 is our average last frost date in my area. It's been a nice week weather-wise, though, and so the plants are outside in my thrown-together "cold frame." I'm using soil with built-in compost for the young plants since corn is a heavy feeder, and once I move them to the ground I'll put them into areas where I've pulled up the hairy vetch (which is supposed to fix nitrogen) and added more compost. We will consider manure, but for the moment it seems like we might make it with the contents of our own compost pile this summer. The benefit of working without commercial fertilizer is that we don't have to think very hard about whether we're contributing to the nitrates in the Mississippi river -- even considering that urban areas contribute less than 10% of Minnesota's nitrate runoff.

The Star Tribune (the Minneapolis paper) had an article this weekend about nitrates in well water in the rest of Minnesota. It's costing taxpayers a lot of money -- up to $3,300 per household in some areas to install nitrate-removal equipment and ensure that people can drink the water safely! In fact, nitrates made the news because in Randall, MN, people can't drink the water safely. Nitrate levels are too high and dangerous to human health, especially babies and the elderly. The majority of nitrates in Minnesota water come from agricultural fertilizer runoff, and dealing with the runoff has become a hot local political topic. It's not just a problem in the Gulf of Mexico when it's affecting our local drinking water and lakes.

Nitrates in drinking water would be a great topic for a social studies debate. Farmers need nitrates to grow their corn at a level that provides enough for food products, corn syrup, compostable cups, and ethanol. As taxpayers we subsidize some of these uses, and as taxpayers we pay for the removal of nitrates from the water. Paying for removal of nitrates is hugely expensive: besides the initial up-front cost of the equipment, it looks like it costs 15 to 35 cents per 1000 gallons of water to do the processing and maintenance, while low-nitrate water costs only 5 to 10 cents per 1000 gallons TOTAL. That's a huge percentage increase in water costs! (The figures are from the linked Minnesota Department of Agriculture file.)

Interestingly, I couldn't find nice data online about nitrates in water in Minnesota, so I returned to the tried and true USGS data from the Raccoon River near Des Moines, Iowa. The past month's data provides enough for a numerical integration and estimation worksheet:

Numerical Integration: Nitrates

It is a pretty basic worksheet -- provides enough information for a social-studies debate starting point, but not much more. It would be very cool to make some more sophisticated optimization problems for nitrate use, to help students (and perhaps the public) think about the tradeoffs involved in agriculture and environment.

Amazon version of book! and revisiting the Raccoon River

The EarthCalc book is now on Amazon, too! Amazon is a touch inconvenient because I couldn't include the worksheets and solutions as an automatic download, but it's a big platform that reaches a lot of people. At Leanpub you get the worksheets and solutions as well as epub/pdf/mobi formats for the book...

Raccoon River: A while ago I wrote some posts about the Raccoon River in Iowa, and the flow of nitrates into the river. The posts talked about how fertilizing our big corn and soybean fields can lead to problems with nitrate runoff, especially in spring when the snow melts and washes leftover fertilizer into rivers and lakes. The associated worksheets were about increasing/decreasing functions, average rate of change, interpretation of graphs, etc.

It's spring again! Nitrates are a topic with continuing relevance even though the worksheet data is from 2013. Recently, the city of Des Moines voted to sue three Iowa counties for not managing nitrate and nitrite runoff: according to the linked National Public Radio report, removing the nitrates in 2013 cost the city $900,000! The New York Times (coincidentally?) recently featured an article about no-till farming, which reduces fertilizer runoff.

So, how much agriculture is practiced in your state? How big an impact does fertilizer runoff have on your ecosystem? Consider asking your students to report on whether their families fertilize their lawns, and find out what your community is doing to deal with runoff into lakes and streams!


A few updates:

  • The book is just about ready. I've been updating it at Leanpub, and it's your last chance to grab it before it goes on sale at Amazon and elsewhere!
  • Job changes: as of January I've started a full-time position at the University of Minnesota, working with MCFAM. There I'm involved with teaching, research, and online education. What does this mean for you? While I'm more busy in some ways, there has been a freeing of psychic energy and that may manifest in more posts 🙂
  • Last, I've gotta expand beyond calculus. There may be some probability or linear algebra mixed in here!

My spring goal is to post regularly (max once a week, though), and to this end I'll start with some links on bees again. More research keeps coming out about possible causes in the decline of the bee populations world-wide, and there are many indications that a lot of factors have come together to contribute to the bee trouble.

Since it's spring, you might consider planting bee and butterfly-friendly plants when you have a chance! It's not too hard to upgrade the insect-friendliness of your yard. In Minnesota, check out the U of MN resources on bee-friendliness. Other places will have different suggested plants. And look out for future Monday posts on EarthCalculus, starting with bees and butterflies!


Basic Bees

Alright. We'll start with a basics of differential equations worksheet -- short and straightforward, and unbearably simple in the end. That's ok. One of the things I love about teaching DEs is that you can start so simply and get to such complex ideas in a pretty step-by-step manner. This worksheet is just a step one: a system of linear differential equations and some questions about equilibrium.

Even with this simple set-up there are so many questions that could be asked that I don't explore! What happens if the recruitment rate from transformation from hive bee to forager bee increases? What happens if the death rate of forager bees increases or decreases? How does laying rate affect equilibrium solutions? These are good questions for a classroom discussion to extend the worksheet.

Then the next step is to mercilessly criticize the models and make them better. There is a lot to criticize about these models: they're hopelessly simplistic, and students can figure that out by looking carefully at the bee life cycle information you provide them. Students love ripping apart these models and putting them back together better, and hopefully you can use those discussions to set an important tone:

Critiquing math models is a fun and respectful conversation that is not about who's talking, but about what is being said.

There is so much conversation in our modern world that involves criticizing something because of who said it, rather than looking at the idea itself. Just look at the top politics stories in the New York Times. Yes, we teach math, but we can also teach how to have constructive conversations about matters of fact and substance.

So! Here's the reading referenced in the worksheet:

The normal worker bee life-cycle is relatively well-understood, and basically goes as follows: the queen lays eggs which are tended by worker bees in the hive. These eggs develop into adults over about three weeks. Once the bees become adults, they work in the hive tending to new eggs or doing hive maintenance tasks (hive-cleaning and construction!). Survival rates in a healthy hive are very high, though not perfect. Once bees are old enough (18-21 days after emerging from their honeycomb) they become forager bees. The rate at which bees switch from hive living to foraging does depend on the ratio of forager bees to the size of the hive -- if there are not enough foragers, hive bees switch more quickly so that enough foragers can be bringing back pollen to feed the entire group. Forager bees are exposed to many more dangers and die at a much higher rate. There's also another mysterious mechanism that can happen if there are too many foragers and not enough hive bees: some forager bees can be convinced to come back and be hive bees.

(Information from Khoury-Myerscough-Barron, this site about bee life-cycle with pictures, and the honors thesis. Several of these reference Winston's book.)

Basic Bees: DEs

Fun fact: all worker bees (and the queen) are female. Male bees are called drones. They mate with the queen but don't seem to do much useful work, and are all expelled from the hive in winter and die! Ok, maybe that part isn't fun...

Other fun links: Sherlock Holmes kept bees and (fictionally of course) wrote a book, "A Practical Handbook of Bee Culture, with some Observations upon the Segregation of the Queen." This love of bees is also important in Laurie King's Mary Russell series, in which Russell becomes Holmes' apprentice.

And if you don't need those basic bees... I'll soon put up a more sophisticated model!

Average change: nitrates again

Spring still keeps happening slowly in Minnesota, and snow is still melting up north. Iowa and other big agricultural states to our south have experienced all their snow melt, though, and are beginning the farming season.

Here's a worksheet that's pretty well inappropriate to the academic year -- no one is doing rate of change right now in calculus or precalculus! But it's written, so I may as well share. Again, it's about nitrate and nitrite runoff into the Raccoon River in Iowa. Spring is a good time to fertilize soil, but the runoff that's happening during the first snow melt is actually all from fertilizer applied last summer.

Average Change: Nitrates in the Raccoon River

The semester is coming to a close. Just a few more weeks of class and it's over. I'm working, as slowly as the spring, on some new worksheets about White Bear Lake water levels and about bee ecology. I learned today that Michelle Obama keeps bees near the White House garden. They're fascinating creatures!

More later... just a few final exams to go 🙂

Runoff in the Raccoon River

Did you know that raccoons wash their food when near a body of water? I love the image. I don't even know if raccoons live near the Raccoon River in Iowa anymore. But I do know where to find real-time nitrate and nitrite monitoring data for the Raccoon River.

Nitrate is NO3 and nitrite is NO2. They both occur naturally in soil and are also vitally important components of fertilizer. Fertilizer, of course, is necessary for the high-yield agriculture practiced in US states like Iowa. The difficulty is that nitrate and nitrite are highly water soluble. They're only useful to a plant if they're available to the plant at the right time in its growth cycle. If the soil is too dry for it to sink in and get to the roots of the plants or the plant doesn't grow due to bad weather, then excess fertilizer is left on the ground and runs off in the rain.

An interesting time to look at nitrate/nitrite runoff is as the snow is melting in the fields of Iowa. Fertilizer hasn't been applied for a season, so all that is left is the runoff from last year's application. It's often not raining yet, so the only water for runoff comes from snowmelt. Daily temperature fluctuations rule the amount of runoff each day for a few days.

So here's a graph-reading worksheet: it's not calculus, and is perhaps more focused on high-school or junior-high skills, but these are always worth a reminder....

Precalc: Nitrates in the Raccoon River


It's been a while. First set of midterms written, given, and graded. Spring may (?) be coming to Minnesota. Ski trips taken. Spring break coming.

I've been working on this nitrate run-off project for a while. Learning about the problem of nitrite and nitrate runoff from agriculture -- mostly from fertilizers -- has been a non-linear process! I've heard reports on the radio about the problems nitrate runoff causes, not only for drinking water in towns in Minnesota and Iowa, but in the dead zone it is causing in the Gulf of Mexico. I read some papers as well, looking for data and ideas tractable for calculus worksheets. Finally I found some real-time data tracking nitrate levels in Iowa rivers, including the Racoon River. Since Iowa is so heavily agricultural, nitrate levels are a significant problem for drinking water treatment plants.

This is the perfect time of year to look at nitrate levels because the spring thaw is either here now or coming soon. Over the winter, farmers did not fertilize -- that would be silly! -- but the spring thaw means a lot of water from snowmelt and precipitation washing over fields and into rivers and streams, bringing with it the leftover nitrate from last year's fertilization. There are some interesting things to see in the data: when temperatures are hovering around freezing, the daily freeze-thaw cycle can often be seen in the nitrate levels measured by the monitors.

It's not easy to model daily nitrate runoff because it depends so much on daily temperature, precipitation, level of snowmelt, and other factors that can change quickly. On the other hand, we can look at data over a period of time and use calculus to understand some of the factors involved. Now that I've wrestled the time and date formatting of the real-time data into compliance using the R programming language, I can make you some beautiful graphs and present some numerical integration worksheets estimating nitrate runoff as well as some graph interpretation worksheets asking students to come up with physical explanations for the data they see presented.

Coming soon....

White Bear Lake: graphing

This worksheet is again about White Bear Lake and its shrinkage. This time, though, it's all about visualizing data. Data about surface elevation and surface area from the Minnesota DNR is again presented in the worksheet, and students are asked to graph it two ways. Which way is better for presenting the information?

I want to provoke an argument here! Getting students to argue in a constructive and respectful way is a great tool for pushing critical thinking and reasoning, as well as practicing language skills. Here, students need to think about the meaning of the information they're trying to present and then argue for which graphical representation is most effective. If I were teaching this in eighth grade I'd get students to write a news article about White Bear Lake, using mathematics and their own research!

White Bear Lake: Graphing surface area

White Bear Lake: rate of change

The worksheet up today is a short one-pager about rate of change. It's technically not using any calculus, but it asks students to compare rates of changes and draw conclusions about the shape of White Bear Lake.

Using some data from a 1998 Minnesota DNR report about surface area and elevation, the worksheet asks students to compute average rate of change of surface acreage for elevations 903 and 913 feet above sea level and then for elevations 925 and 926.5 feet above sea level. Change of acreage is really dramatically different for these elevation ranges: you can really see how shallow the shoreline of White Bear Lake is, and how much effect simply losing one foot of water surface elevation has. No wonder the current drop is so noticeable!

White Bear Lake: Rate of Change

I have some more ideas for White Bear Lake worksheets, so we'll see what happens. I'm thinking about

  • modeling surface area vs elevation using a spreadsheet
  • numerical integration to calculate volume of the lake
  • graphing: what's the best way to convey mathematical information?

White Bear Lake take one

White Bear Lake has been in the news a lot recently: Minnesota Public Radio is doing a big project about our dropping groundwater levels, for instance. And now that the holidays are over (Happy New Year all!) I've got the first pass worksheet for exploring White Bear Lake mathematically. No calculus yet, just an exploration of the area and volume of the lake. On average, White Bear Lake is pretty shallow. I think that's one reason the drop is so noticeable, especially on the western shore of the lake. In the worksheet students can find out just how shallow WBL is and also deal with big numbers.

As a teacher, I think this is an appropriate worksheet for a class that's dealing with big numbers (scientific notation would be great here!) and units.

  • Scientific notation!
  • Changing units.
  • Average depth from knowledge volume and area.

Precalc: Volume And Average Depth

If you've got suggestions for improving the worksheet, let me know! And check out my email subscription list on the side -- I will email you about the EarthCalculus book I'm putting together right now....