Slave to the Formula
Today in the math lab, there were two precalculus students doing some problem that had two parts, that could be solved the same way. The first part looked like a problem that could be solved using some deterministic algorithm that would take a while but would get the answer. I asked her to stop what she was doing and just think about it for a second, and she discovered the answer fairly quickly.
“What would have happened if I had used the formula?” “Well, you would have gotten the same answer. But it would have taken you seventeen times longer, and then you would look at your result say, ‘Oh, I could have done this the easy way,’ and then you will feel sad.”
Later on, I had the same student complete a different problem and it ended up as something of the form “cosine of X equals Y”. I asked her if the number Y sense in the context of the problem and after going back and doublechecking her work, she couldn’t find any error. Now, that’s because there wasn’t any. She was totally right.
I then said, what “Would happen if Y was a number that didn’t make sense?” Her response was, “well, if Y was bigger than 2 then when you try to take the inverse cosine you’ll get an error.”
I explained that this is what I call a Noisy Error. Something like cos(X) = 2, or Sqrt(X) = -5, or e^X = 0, that sort of thing is a Noisy Error because when you go to solve it, you literally can’t. The error you made is very noisy in that it makes itself noticed very soon in an obvious manner.
Other errors are “silent errors.” In the above problem if Y was a number between 0 and 1. If the Y had been between -1 and 0, the formula would fail silently with no warning - that is, it would give you a perfectly valid answer, but to the wrong question. Those are the errors you should check for. It’s not just “will this give me an error on my calculator” but “does the answer make any sense?” If the answer doesn’t make sense in the context of the problem, then the most likely cause is an arithmetic error somewhere earlier.
Another student had the same issue and did get a negative value for Y. After doing a bit of rereading, we found that he accidentally used “3300″ instead of “3500″ in the formula, because he had copied it down incorrectly. His answer was valid but it answered a different question than was asked.
Moral of the story - don’t be a slave to the formula. Please ask yourselves if the values you have make sense - just because the formula spit them out doesn’t mean they’re right, and just because the formula spits out the answer doesn’t mean you had to do all the the extra work.