Month: November 2008

Sick again!

I was sick a couple weeks ago, before parent night. I recovered. I got a flu shot last week. I thought I was going to be in good shape.

I jinxed it. Now I’m sick again. I went to bed with a sore throat. I woke up sick. I hate teaching when I’m sick; all day I’m going to dream about lying in bed.

Rationalizing the Denominator, and Comment Writing

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 f(x)=\sqrt{x} 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: \frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{5}}. 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 \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{5}}. Why is it that students NEVER understand fractions?

Water Pistols and Children!

A while ago, I posted about some interesting problems posed in the Technology Review magazine.

  • Jerry Grossman has equipped n children with loaded water pistols and has them standing in an open field with no three of them in a straight line, such that the distances between pairs of them are distinct. At a given signal, each child shoots the closest other child with water. Show that if n is any even number, then it is possible (but not necessarily the case) that every child gets wet. Show that if n is odd, then necessarily at least one child stays dry.
  • Each of logicians A, B, and C wears a hat with a positive integer on it. The number on one hat is the sum of the numbers on the other two. The logicians take turns making statements, as follows:
    A: “I don’t know my number.”
    B: “My number is 15.”
    What numbers are on the hats of A and C?

I submitted my solutions (click here to read my submitted solutions) and lo and behold, one of the solutions got published in the latest Technology Review (click here)!