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!


Grades are in, just this week! Summer has officially arrived for the academic.

It's arrived in Minnesota, as well -- we've got beautiful weather, sunny days, warm temperatures. I've been trying to get caught up on weeding and planting things in the garden, since I traveled a lot this month and the days of frost were pretty recent here. Today in the garden I noticed a bee, but only one. It's not surprising that it seems like there are fewer bees than usual out and about. Bee population collapses have been getting a lot more news: the population numbers aren't good, but we still love all the fruits and vegetables that bees help to pollinate.

We're still not sure exactly why bee numbers have declined so much, but it seems to be a complex interaction between parasitic mites that have invaded bee colonies and agricultural chemicals we use to suppress other insects.

What can you do? Look up information about what plants you can grow that help bee populations. In Minnesota, check out the U of MN's Bee Lab pages! Avoid certain types of pesticides and fungicides. Talk to your Lowe's or Home Depot about not selling plants and flowers treated with neonicotinoids, a pesticide that comes up through a plant and weakens bees who collect the pollen, or buy from a smaller distributor who doesn't use neo-nics.

On the math side, there's a lot of differential equations to model bee colonies and their populations! There are quantitative models of honey bee population dynamics and mathematical model of bee colony collapse disorder. There's an online simulation you can run. You can tweak the models yourself if you know enough math, and one honors thesis I found did just that.

So on the docket, coming soon, are some worksheets or activities that explore bee populations at a few different mathematical levels. As always, I want students to have entry points into this interesting problem from a wide range of mathematical starting points!

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....

Snowflakes link

Check out a fun article on the shape of snowflakes! Entering finals week soon here.... White Bear Lake is not forgotten....

White Bear Lake water level and precipitation

Ok, my previous graph was ugly & slightly wrong so I'm going to put up an updated version. The units are better and so it should be a better visualization. Blue is for water falling from the sky and green is for lake level.

White Bear Lake Precipitation And Level

White Bear Lake Precipitation And Level

You can see much more easily here how lake level and annual precipitation tracked pretty closely for many years, but since about 2003 they have decoupled. The USGS has a report discussing the analysis that went into showing this scientifically, but you can see it here too!

Worksheet soon... digesting Thanksgiving leftovers still...