by Joye Walker
It was in the UCSMP Algebra 2 book and I encountered it during my first year of teaching. Here was the opening linear programming example.
Stuart Dent decided to investigate one of his typical meals, fried chicken and corn on the cob. He compiled the data in the following table
|Vitamin A||Potassium(mg)||Iron (mg)||Calories|
Stu let f= the number of pieces of chicken and e= the number of ears of corn. After deciding the minimum amounts of each needed from this meal he wrote the system:
100f+310e>=1000 (at least 1000 units of vitamin A)
151e>=200 (at least 200 mg potassium)
1.2f+e>=6 (at least 6 mg iron)
122f+70e>=600 (at least 600 Calories)
The point was to find the cheapest diet for a healthy life. One could argue that we are not talking about healthy foods here, but let’s not bog down with that. The objective function is C=0.90f+0.75e where a piece of chicken costs $0.90 and an ear of corn costs $0.75. Let’s also not bog down about whether those prices are reasonable, even back when UCSMP algebra was written, probably the early 1990s. The vertices of the feasible region, rounded to the nearest hundredth when necessary, are (0, 60/7), (3.76, 2.01), and (5.89, 1.32).
1. No one eats 5.89 pieces of chicken and 1.32 ears of corn. Instruction is needed (but not provided in the example) to help students find the lattice points nearest the vertices of the feasible region, but that are contained in the feasible region. Recall that this is the opening example of linear programming.
2. I sketched the feasible region on graph paper, taking great pains to use a ruler and be accurate. The inequalities were not pleasant to graph. I used the two-intercept method to graph each line, but when one of the boundaries is y=200/151, it took a bit of hand waving to make a good graph.
3. If a student is inclined to work through the examples in the book (I was always that student, and over the years of my career, I taught many such students), it is extremely tedious to get it graphed with such nasty coefficients in the inequalities, and it takes some pretty good precision to graph such that identified intersections are indeed vertices of the feasible region.
4. Opening examples should not contain nasty numbers. Students need to learn the concepts first. The tedious calculations can come later. We don’t start teaching students to multiply 1.96 times 6.7789. We start with multiplying 2 times 7 first. We get more sophisticated when they can handle it.
5. Students who are not good at following examples such as linear programming for the first time, are not going to stay with this one. They will tune out and not have a clue how to tackle their homework.
6. I completely rewrote that section of the textbook (my first year of teaching) and shared my lessons and problems with colleagues. I did use “real-world” problems in the sense that, for example, a farmer had a certain amount of acres to plant in corn and beans, and used information that gave reasonable vertices and objective functions. I completely avoided the nasty numbers because, while they can appear in higher level exposures to linear programming, they shouldn’t appear in the very first examples when students have to translate the given problem into a system of inequalities.
I absolutely refuse to ask my students to work problems that were unnecessarily cumbersome and where the numbers took away from the concepts being explored. I’m sure they were thrilled when I retired last year.