We get tomorrow off of school for “Election Day.” Translated, that is the day teachers at my school write narrative comments for all their students discussing their first quarter grades. We’ll all be holed up in our apartments, trying to come up with various ways to say “this student is doing great,” “this student is doing okay,” and “this student is not doing well.” Luckily, I’m pretty fast at writing these, so I’m not concerned.

In other news, I gave my Algebra II students a quiz last week, and one of the skills covered was rationalizing the denominator where there are radicals involved. (Multiplying the top and bottom of the fraction by the conjugate.) My three musings:

(1) Why do we math teachers care so much about this? I know it’s a good skill to teach because sometimes it really does simplify expressions, but do we always want to insist that the denominator is rationalized? I always thought that it was a bit dumb — and no one really has been able to justify why teachers insist on it with such vehemence. Any ideas? [1]

(2) Ummm… in Calculus, we’re starting to work on the formal definition of the derivative and guess what? To find the derivative of using the formal definition, you have to rationalize the NUMERATOR. Harumph.

(3) For extra credit, for students who had some extra time after finishing their Algebra II quiz, I asked them if any of them could somehow rewrite the following without any radicals in the denominator: . Although no one got it, I loved watching them work on it. [2]

[1] My high school Algebra II teacher told us: “Why don’t we want radicals in the basement? BECAUSE THEY BUILD BOMBS!” I will never forget that. Love it. I totally use it. His legacy lives on.

[2] Even though I was horrified that some students’ initial step was to rewrite that as . Why is it that students NEVER understand fractions?