The warmest November

I've been playing with a lot of temperature data lately: you can see a short Python analysis of temperatures over at my personal blog. At work (at the University of Minnesota) I'm working with some masters' students on research into the finance of weather derivatives and catastrophe bonds, so I've been thinking a lot about temperature, El Niño, snow in Siberia, etc...! I've also been thinking about how to teach probability and statistics.

So, this post is about using the normal distribution and spreadsheets to deal with real data! November has been very warm, even though I picked the coldest days to go winter camping. How warm is our November here in Minneapolis-St. Paul?

Here is a Google spreadsheet with the minimum and maximum temperatures for every November 8 since 1970:

Google spreadsheet: max, min temps, with precipitation and graphs, Nov. 8

Included are histograms of max and min temperatures, and a scatter plot of max and min against each other. Continue reading

How are formulas created?

I saw this question on Reddit: "How are formulas created?"  (Turns out Reddit has a lot of people asking math questions!) The answer provided by commenters is "mathematical modeling" -- done in science courses! This answer makes me sad! Mathematicians make plenty of models, too! When you think about it, though, how many high schoolers get to see that? In fact, how many sophomores in college have ever seen math at work making models?

Well, here are a few examples if you're interested!

  • Stefan's equation for sea ice thickness: these two posts talk about modeling sea ice thickness with a differential equation but don't ask you to use data to create a model
  • Modeling tides on the California coast, with more here: these two posts give worksheets on creating your own model of the tidal patterns at Point Reyes Seashore using actual NOAA data
  • Lynx! for cute fuzzy animals with sharp teeth! These two posts have students develop their own trig model for lynx populations, see how bad that model is, and then use a logarithm composed with the trig function to get a model that better fits the sharp population peaks.

Sometimes I feel like teachers who make room for this material are swimming upstream since so many of our high school math curricula don't provide the time for experimental, living mathematics... but every now and then I meet someone who really makes it work. And maybe with modeling as one of the high school Common Core standards there will be some official space for this in high school classes! It is so sad that students can go through 12 years of school and never really see mathematical model-making at work.

Trapped by boring fake word problems in your math textbook?
Get intros to real-life issues in the natural world and see math at work! Geek out about butterflies and aquifers while upgrading your analytical skills :)
* = required field

powered by MailChimp!

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.

Spring fling: hairy vetch

It's spring in Minnesota, which found me flinging hairy vetch seeds from a flowerpot into the back yard.

Dirt where hopefully hairy vetch will fixate. Strawberries already started....

Dirt where hopefully hairy vetch will fixate. Strawberries already started....

Let's back up a moment. My garden fever has been ramping up; those of you who live in the great frozen north might understand the hunger to see green things. It's why all the undergrads were in shorts last week (that and we set a record of 84 degrees!). For the past few years my husband & I have tried to start seeds in egg containers, and every year something terrible happens: they blow off the porch, or they all drown in a big rainstorm, or squirrels eat them. No more. This year I marched into Eggplant Urban Farm Supply a few blocks away and made a stand. I spent what felt like an exorbitant amount of money, but it'll be less than a dollar a plant even if we have a few failures. I got a seed-starting tray and a seed heater for our cold house and some hairy vetch and inoculant for nitrogen fixation!

Nitrogen what?

Continue reading

A first draft of a monarch matrix worksheet

Well, let's get a first pass at a monarch worksheet posted. It's definitely about matrix models and graphs, and there are a few things you need to talk about with students prior to putting it to work:

South and north: Monarchs overwintering Mexico have hit pause on their reproductive lives. Really, it's called reproductive diapause! They get to live 6-9 months down in the highlands of central Mexico, living on the oyamel fir trees in the mountains.

MountainsOfMexicoThis is not a beach vacation: the monarchs cover the trees high up in the forest. I had the good fortune to visit Cerro Pelon butterfly reserve last January with Joel Moreno of Joel's Butterfly B&B, and these are pictures I took from that trip.

Monarchs in the trees at Cerro Pelon

Monarchs in the trees at Cerro Pelon

Continue reading

Matrix models and DEs: an example of which when

Last week I left you with the question of matrix models for populations versus differential equations. Matrix models are discrete -- they jump from time 1 to time 2 to time 3 -- and differential equations give a continuous description relating rates of change to the quantities in the system. Bees and butterflies are both pollinators, both pretty, both summer insects (up here in the north) -- why would we use different models for the two?

First off, if you're familiar with solving systems of differential equations, you might remember that matrix methods are pretty useful in that endeavor!

  • Matrices allow you to solve systems of linear differential equations.
  • Euler's method basically reduces differential equations to difference equations/matrix methods.

Differential equations can be really hard (or currently impossible) to solve. Matrix models are computationally advantageous and let us deal with small populations really concretely. If we can chunk up the life stages of a population, as with the turkeys in last week's post, we can do some pretty slick matrix modeling.

I think it's the structure of the lifecycles and lifestyles of bees vs butterflies that drives the choice. Let's think about this: bees live in hives, the same one for a long time. We can think of a hive as a population whose health we want to model. There are different classes of bees in the beehive, but they all live in the hive at the same time. Butterflies live as individuals rather than in hives or herds, so we can't look at any population smaller than a regional one. Moreover, the migration of monarch butterflies is a really big deal. The winter monarchs -- the ones who fly to Mexico -- have very different lives than summer monarchs. They live a lot longer and in different places. It's almost as if there are two kinds of butterflies separated in time. The time and space dimensions for modeling these populations, then, are pretty different.

So, that's one set of reasons for using different modeling techniques for these different populations. Can you think of others?

Here's a fun fact, though: you can use discrete methods for some bee modeling. In fact, the Fibonacci sequence comes up in bee math! I was too busy this weekend pondering the game theory of pricing books on Amazon (suddenly relevant) to complete the desired insect life worksheet, but I found some really cool resources while reading:

Looks like I'm getting drawn toward longer projects here, like the bees and the butterflies... we'll see what happens!

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!

Bees and butterflies: differential equations vs matrix models

If you look back at the Basic Bees and DEs post that went up a while ago, you'll see some baby differential equations. You can write DEs as

rate of change = increase - decrease

and get some pretty cool models for populations, for instance. (My favorite is looking at predator-prey interactions: write two differential equations, one for foxes and one for rabbits, for instance. Foxes eat rabbits, so the populations depend on each other. What happens as one increases and the other decreases? Check out a puma version here.)

However, differential equations can be really hard to solve. Sometimes it's nicer to take a discrete rather than continuous approach: use a matrix model! In a matrix model, you divide time up into discrete steps: months or years or stages of life. Then you multiply a population vector that gives population at step n by a matrix that tells you how each population changes. That gives you a new vector that gives population at step n+1.

Here's a non-insect example: wild turkeys. We can classify wild turkeys as poults (ages 0-1), yearlings (ages 1-2), and adults (ages 2+). Every year turkeys get a year older, as we all do! Only yearlings and adults can reproduce. Then you can do some research to find how the population structure works:

  • The number of poults each year depends on the reproduction of yearlings and adults. So P(n+1) = F2*Y(n)+F3*A(n): number of poults at time n+1 is a reproductive constant times number of yearlings at time n and a constant times number of adults at time n.
  • The number of yearlings at time n+1 is given by how many poults survive! Y(n+1) = Q1*P(n). Q1 is less than one.
  • The number of adults at time n+1 is given by how many yearlings survive plus how many adults at time n survive. So that's A(n+1) = Q2*Y(n) + Q3*A(n). Here Q2 and Q3 are also less than 1 (no magical birth of old birds).

It seems like the literature on bees all uses DEs, while the literature on monarch butterfly populations uses mainly matrix models. This might be because of monarchs' special lifecycle: most monarch live, mate, and die up north, in Canada, the eastern US, or the midwest of the US, but some make the long trip to central Mexico to overwinter there. (There's a smaller population that has the same pattern, but with the Rocky Mountains and California replacing the North and Mexico.) The overwintering monarchs live a much longer lifespan and really have a totally different life than the summer monarchs.

I'm working on a worksheet for monarch modeling with a matrix. In the meantime, you can find educational links at Education World and Monarch Watch. Spring is the time to start thinking about butterfly activities, as the monarch migration north starts in April!


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!