A story from the tutoring room, where I encountered a student taking first semester chemistry for science majors who was struggling with a simple density problem.

(Why they teach complicated algebra problems before they teach any chemistry is a mystery to me, but I no longer remember what pedagogy my chemistry classes followed.)

Given that our chemistry class has college algebra as a prerequisite, I think the problem itself is instructive.

After a significant amount of prompting on an easier problem, the student was able to get all of the given data for this problem into the appropriate chem-SI units of g and cm. This left something like the following problem, except written as a fraction:

2.705 = 276/(30.48 * 1219.2 * X)

Solve for X.

The student was,it appeared, utterly helpless when faced with all of those numbers.

My suggestion that he cross multiply the X and 2.705 met the sort of look I would expect if I had said it in Japanese. OK, clear fractions? Still no luck. I don't think he saw this as a fraction. Multiply both sides by X? Ah, progress. Now he could compute the right side and solve 2.705*X = Number. Just to check, I asked him about

5 = 7/(3*X)

No problem there, although slow as molasses doing it.

When I started telling this story to a chemistry colleague, she starting laughing so hard that she almost fell out of her chair before I was halfway through it. She regularly sees this at the start of every semester. Now I know why she says my students aren't like hers. Many (but not all) of the kids who make that kind of mistake are weeded out by pre-calc and trig before they get to me.

When I asked a couple of math colleagues about this problem, I learned that they include problems with "messy" numbers on pre-calc exams, but not in college algebra. They also said they suspect that not all pre-calc classes give messy application problems. So that is why I was not surprised that a student in Becky Hirta's Calculus Circus had

trouble with a graph where the answer was not obviously going to come out in simple integer steps.

**Footnote:** Two of my physics students messed up a problem essentially as follows:

2.66 = 7.79+3.47*X

2.66/7.79 = 7.79/7.79 + 3.47*X

0.3415 = 3.47*X

To save you the effort, 2.66/7.79 is about 0.3415. Bet you didn't know that 7.79/7.79 was zero!

Yep, that is how at least two of them "canceled" that number that was added on the right. Again, my math colleagues tell me that this sort of error is not uncommon among students entering calculus 1, and that they are more likely to make it with numbers rather than symbols. I have to wonder if they would have subtracted 7.79 if it had been at the end

2.66 = 3.47*X + 7.79

instead of at the beginning....

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