MS Director Blog "Orchard Middle School Math Connects To The Real World Through Applied Science

Jamie Napier
Italicized text is by fifth grade science teacher David Oyama

We are going to play a game and in the game we will have different groups of animals. Some of you are going to be insects, some of you are going to be frogs, some are going to be snakes, and, finally, some of you are going to be hawks. We will have three groups of insects: grasshoppers, walking sticks and fireflies. We will have two types of frogs: wood frogs and green frogs. There will be one snake group called copperheads and one hawk group, the red tails. In order to survive you need to collect color beads. Here is how you collect a bead from one of the orange strings…
There is a college class called differential equations. Typically, universities will give it a nickname: Calculus-Four, D.E. or diffy-Q; the last one playing the double entendre of implying it is a difficult class. Similar to organic chemistry tending to separate future physicians from future non-physicians, differential equations tend to separate future engineers from future non-engineers. It is D.E. where air resistance stops getting ignored in physics problems and the math underpinning thermodynamics is also no longer relegated to statistical guessing. It is also in D.E. that one encounters the Lotka-Volterra predator prey model. A picture is below.

The beads represent different needs that each organism needs. Red beads represent food or nutrients, blue beads represent water, and green beads represent shelter. All organisms, in order to survive have to acquire these items. If you are an insect you need to collect one of each color beads in order to survive. If you are a frog you need to collect two water beads and two insects. If you are a snake you need three water, three shelter, and three frogs or insects. If you are a hawk you need four water, four shelter, and to successfully attack four other organisms. Hawks have the most work to do, but nothing attacks hawks. There are rules to the attack, in order to attack another organism . . .
In order to solve the Lotka-Volterra nonlinear differential equation one does not get an answer in the classic way. Your answer is actually a set of possible answer sets called eigenvectors. Each eigenvector is a set of numbers one can plug into the original set of equations and achieve a balance. If the eigenvector set of numbers is off, then chaos will come into the system and it may go anywhere, but usually it disintegrates into nothing or runs off into infinity. 
After a successful attack the attacking organism gets to claim any other organisms they tagged. This brings needed beads or nutrients into your group. After a group has lost a member it is immune to attack for two-minutes. Slowly the predator population will grow and the prey population will shrink over time. 
The idea of a steady state predator-prey model has been applied to biology, business competition, thermal variances in liquids. And sometimes we don’t want populational stability. When applying an antibiotic to an invasive bacterium that is making us sick, we desire the invasive species to undergo a population collapse as quickly as is safely possible. Yet ingesting a gallon of antibiotic is hardly desirable even if it guarantees rapid populational collapse. 
We are going to play the game again. This time the forest is smaller so you will not have as much space to move around. This represents shrinking natural habitats. Notice how many students start in each group and how many are in each group by the end?
When studying abstract higher math, it may defy practical application. Even some adults, having passed through a required study of advanced math do not see its connection to reality. The fact our fifth-grade science class is playing a game that reflects the Lotka-Volterra predator prey model and its associated quest for eigenvector solutions is sort of amazing. Jean Piaget generated volumes of prose dedicated to the importance of creating empirical experiences to help understand the abstract. It is in these small moments that Orchard students will be able to look back and recall a specific lived example rather than abstract mental construction when processing how to understand advanced math when the time comes.

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