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.

Why is it important for High School students to learn Math?

Why Math?

I’m sure all of us are familiar with the age-old quips: “Why do we need to learn math?” or better “When am I going to use this in my life?” In elementary school, teachers argue Math is important because it provides you with the tools in order to tell time, to make change, to measure distances/areas/volumes, and in cooking. While this is a solid argument, it isn’t enough. You can’t tell a 15-year-old that they need to learn math so that they can tell time, make change, or to cook: By that point, most are rather articulate in those skills. In fact, this clarifies the question. It should be “Why is it important for students to learn math?” but rather “Why is it important for students to learn math beyond arithmetic?”

Some teachers will argue it develops students’ critical thinking and problem-solving skills. While there’s no arguing that it can, at the same time one might still argue that a student can develop those same skills using a less symbolic form of thinking. English, Social Studies, and most of the humanities require some form of critical thinking, usually taking the form of research papers and projects. At the same time, Shop projects also rely on students’ problem-solving abilities, and these also seem terribly useful in students’ day-to-day lives. Modern education, in fact, is centered on the fostering of critical thinking and problem-solving skills in its youth. So this begs the question: If all of education can develop this form of thinking, is it really necessary for students to take math beyond arithmetic?

So where does that leave math? Some might argue that math is needed to do science. For students who intend to pursue careers in science, this might be enough. However, for those students who see science as yet another kind of math —i.e., one of those useless things that they teach you in school— arguing that math is useful in science is hardly a solid argument.

Those who know of Gardener’s theories of multiple intelligences might argue that while some students might not understand mathematics well, other will understand it better. For that reason, students should take mathematics. Even if this so, the opposite makes a stronger argument: If some students understand math better, let them take math; all others can be exempt.

Thus, this returns us to same problem: “Why is it important for students to take math beyond arithmetic?”

The reason is simple: It is a Literacy that humans have developed over the course of the past 5,000 years that enables humans with the ability to model and predict real life with great precision. Naturally, math is a useful tool to have if you go in one of the hard sciences, or economics, or computer science. It is of equal value to those who pursue degrees in the softer sciences: For them, statistical analysis will be a mainstay of their education, if not of their professional careers. Learning the mathematics that prepares students in performing these analyses will free them time to focus on learning theories and ideologies of greater importance to their studies.

But what of the students who intend to go into construction? What of those students who go into sales? Or of those who decide to run their own businesses? What about the students who drop out of school, who end-up working as janitors, or end-up in vice selling drugs, or running with a gang, or worse? Do they need math beyond arithmetic?

Even these students need math, and here’s why: construction, sales, hourly-jobs, even vice runs on the basic principles of business. And for any business to remain competitive, it requires the most up-to-date tools of the trade. In specific, there needs to be an understanding of how to maintain supply, what fuels the demand, how to balance budgets, and how to predict the market. To accomplish each need easily and successfully, at least some proficiency in algebra, basic geometry, data analysis, statistics, and even calculus is required. Any student that finds himself in a business with opportunities for advancement will soon discover those who understand the math of business, are more likely to succeed than those who do not.