Rough week

I'm not sure why, but it's been a rough week for teaching, particularly for my first class. On Monday we had a quiz which they didn't do so well on. I ended that class with an attempt at explaining the formula for the "Magic Number" in baseball standings, but I mostly got confusion from the students, particularly since the Yankees and the Red Sox were tied. ("If the Yankees have w_1 wins and the Red Sox have w_2 wins, how many games do the Red Sox have to win to take the division if the Yankees win all their games?" "It doesn't matter, they're tied." "Yes I know, but suppose they weren't tied." "It doesn't matter, they're tied." etc.) After grading the quizzes, I decided on Wednesday to go into a more thorough explanation of what "simplify" means, and when one is allowed to "cancel" factors from numerator and denominator. That turned out to be a lot harder than it seemed, particularly the latter part. I wanted to explain that you can only cancel things from numerator and denominator only if both are only made up of factors, like a*b, not a+b; but (a+b)(c+d) is okay, etc etc. Just a mess.

Then today I started into the word problem section of the book. It started out okay with some general problems, but then I started describing uniform motion and mixture problems, and I was probably trying to cover too much too quickly, because it was just really bad. I didn't explain that, in the uniform motion problem, x is the distance from some origin which you can pick arbitrarily. After the mixture bit, someone asked, "Can't we just use the formula in the book, which is much easier than all these steps?" Well, yes; I would like you to understand what the formula means, but.... Oh well.