First day of class
Graphics class: drew a diagram of a pinhole camera, with an image appearing upside-down on the back wall. Replaced the box with an eyeball. Asked "why is the image upside-down? Why not flipped left-right, or something else? Why should the direction of up-or-down be privileged? Light rays aren't significantly affected by gravity (except near a black hole, which we aren't)."
Did some more thought experiments until one student said "it's not upside-down, it's rotated." So we discussed rotation, reflection, scaling, etc. and I promised that all that stuff they learned in Linear Algebra would actually make sense and be useful this semester.
Computer Architecture class: discussed units, and why E = mc^2 makes sense but E = mc^3 couldn't possibly have been right. Word problem: I walk at 2 steps/second, each step covering 2.5 feet. I jog at 3 steps/second, each step covering 4 feet. If I alternate jogging 75 steps with walking 75 steps, how fast am I moving on average?
Solved the problem two different plausible ways, getting different plausible answers. Discussed which one was wrong, and why. What happens if I lengthen my jogging stride to 5 feet? Obviously this is a 25% increase in jogging speed, but what's the percentage increase in average speed? What if instead of alternating every 75 steps, I alternated every 30 seconds, or every 100 yards? Good discussion.
Data structures class: let's explore a vague problem. In the game of SubDivvy, the players agree on a positive integer to start with, and on each turn, the current player chooses a proper divisor of the current number, subtracts it from the current number, and that becomes the new "current number." Tell me everything you can about the behavior of this game.
Then commenced a series of student-led discussion. Debated definition of "proper divisor"; agreed to exclude the current number itself (or the game end in one turn), and negatives (or the game goes forever). Not sure whether to exclude 1. Defined loss of game as "there is no proper divisor," which means the number is 1 (if we allow 1 as a proper divisor) or prime (if we don't). Proved that current number only decreases, never reaches 0, and never goes negative. Proved that in the allow-1 version of the game, game always ends after at most n-1 turns; still working on minimum number of turns. Proved that (still in the allow-1 version) 1 is a losing configuration, 2 is winning, 3 is losing, and 4 can be won or lost depending on what you choose as the first move -- in other words, strategy does matter in this game (there are games in which it doesn't). Wait: we've been assuming there are 2 players, which I never said. What if there were 3, or 4...? At the bell, left students with the challenge of characterizing winning and losing game states. Good discussion.
If only every day of classes went this way....