Linear Programming

My general feeling with math is that as teachers we make the argument to students that math has applications in real life, but rarely do we provide them with situations in high school where this math could be used. In fact, it is my feeling that the years between Algebra I and Calc tend to be a wasteland of math barren of practical applications (but full of forced ones). Therefore, if I were to teach a class during that time, I would make it a priority to teach linear programming, probably after covering systems of linear inequalities. (Ironically, I did not learn of linear programming until sometime during undergrad, even though it’s a rather simple topic and has very real applications. I believe that it’s in some Algebra II textbooks, but I suspect that even then it’s rarely taught.)

I would introduce the topic using a rather standard problem like the following:

Dee works at a doughnut factory that specializes in making chocolate doughnuts and frosted doughnuts. Each tray of chocolate doughnuts takes 2 lb of dough, and each tray of frosted doughnuts takes 1.2 lb of dough. Dee has at most 250 lb of dough to use. Each tray of frosted doughnuts takes 0.4 lb of frosting. Dee has no more than 58 lb of frosting to use. Each tray of chocolate doughnuts takes 0.2 hours to bake. Each tray of frosted doughnuts takes 0.1 hours. Suppose Dee has no more than 15 hours of baking time. If Dee makes a profit of $8 for each tray of chocolate doughnuts and $10 for each tray of frosted doughnuts, how many doughnuts of each kind should she make to maximize her profit? How much money will she make?

After allowing my students to try to solve this problem (some might make the connection from before to graph inequalities, but probably wouldn’t know what to make the feasibility region or the how to calculate the maximum profits, let alone the amount of money Dee will make). I would then teach them about linear programming using the example as a foil; explaining ideas like the feasibility region, the constraints, and how to calculate the maximum (or minimum) for each subject.

For homework, I would then give them a series of word problems that would require them to apply what they learned using the concept of linear programming. I’d start them off with items that asked less than the problem presented in class, and then progress in difficulty to same level as the problem in class. As a challenge, I’d also provide questions that required the use of more than two variables; however, I would note to the students that these problems might require innovations of their own to be solved. I would also try to make the problems a little more relevant to their lives, possibly using a social justice math slant.