Al's Revenge

My good friend, Al Harmon, figured out the puzzle I posted, and decided to reply in kind with a problem of his own. I post it here for a challenge to all my readers...

" Robin went to a concert that began between 6 PM and 7 PM and ended between 9 PM and 10 PM. The positions of the clock hands at the end of the concert were switched from their positions at the beginning (hour hand where the minute hand was, and vice-versa). What were the exact starting and ending times of the concert?" I assume he means a regular 12 hour clock, not the kind I have on the image at top... which looks like a clock I saw in Florence, and will mention later.

While your working on that, you may need a good chuckle, so here is a little document I pulled out of the file cabinet of unknown origin. I think it reflects the frustration all teachers feel when kids seem determined not to use their brain. Every teacher has one that drives them especially crazy... I remember a long argument among some mathteachers over how much should be counted off if a kid, given the problem "simplify the fraction ¹⁶/₆₄" responds with

Two days ago I asked a student to find the square root of two... she pressed buttons on her calculator and said "One point four one four."... Close enough, so I ask for the square root of three... again the calculator and an answer... finally I ask what is the square root of four... she picks up the calculator, pushes the buttons, and then starts to turn red.... Ok... I set her up for that one... but if you want to send your own "pet peave student answer" I will collect and display after awhile... in the meantime, here is the page I found in the file cabinet today:

A TEN DAY SYLLABUS FOR PRECALCULUS
DAY 1: Teach them that (a+b)/c is (a/c) + (b/c)

DAY 2: Teach them that a/(b+c) is NOT (a/b) + (a/c)

DAY 3: Teach them that x / ln(x) is NOT "1 / ln"

DAY 4: Teach them that you can't solve (sin(kx)) = 1 by saying "x = 1/sin(k)"

DAY 5: Remind them that a/(b+c) is NOT (a/b) + (a/c)

DAY 6: Show them a movie of a student sitting in a field, writing "(a+b)^2 = a^2 + b ^2" and then getting HIT BY A TRAIN!

DAY 7: Remind them that a/(b+c) is NOT (a/b) + (a/c)

DAY 8: Teach them that if the domain of the a function f is the reals, the graph of y = f(x) is NOT a blank pair of axes, that perhaps they should adjust the "window"

DAY 9: Teach them that x/(y+z) is NOT (x/y) + (x/z)

DAY 10: Group work: Bring a trout to class. Have them solve sin(kx) = 1. If they get x = 1/sin(k), hit them with the trout. Make it a big trout.