# College Math Teaching

## April 1, 2010

### Portability of skills and optimization problems

Filed under: calculus, student learning — collegemathteaching @ 3:25 pm

Consider the following optimization problem: you have a long wall and you wish to put a rectangular plot next to the wall; the wall forms a barrier for one of the 4 sides:

Say that the fencing for the sides perpendicular to the wall costs 3 dollars per foot and the side parallel to the wall costs 10 dollars per foot.

Naturally, there are two questions you could ask:
1. Given that the area must be, say, 1000 square feet, what is the cheapest fence that you can build or
2. Given that you have only 420 dollars to spend, how big of an area can you enclose?

Neither of these questions is difficult, but then I am teaching this to a class which has a median math ACT of 22 (off semester for this course). This fact is relevant for what follows.

I did problem number 2 first:

Objective: Maximize $A = xy$
Constraint: $C= 6x + 10y = 420$
(of course, $x > 0, y > 0$)

Then we proceed as follows: use the constraint to eliminate a variable:
$420 - 6x = 10y$ which gives $42 - (6/10)x = y$
Now substitute into the objective: $A = x(42 - (6/10)x)$
$A =42x -(6/10)x^2$
Differentiate the objective and set equal to zero:
$42-(12/10)x =0$ which has solution $x =35$ and $y = 21$.

No biggie, right? Well, pay attention to the step in which we solved for $y$.

Now when we did the dual problem, I had the students “help me out”.

Objective: minimize $C= 6x + 10y$
Constraint $xy = 1000$

So what did one student do? You guessed it: $y = 1000 -x$
Yes, this student had passed their algebra class.

What gives? Why didn’t the student simply say $y = 1000/x$?

After seeing a few more answers like this it dawned on me: on the first problem they saw me subtract, so they figured that subtraction was the step you used to go from the constraint to the objective!

Had I just asked them “solve for $y$ if $xy=1000$ without having the optimization problem to do, they could have done it.

In short, their algebra skills are not yet “portable”; they don’t see the “constraint to the one variable objective function” as merely an application of the algebra that they already “know”.

Sometimes it is tough to see where a bad student gets lost, given that most of us who teach calculus were strong students when we first took the course.