Just noticed in my description I forgot to say that the second student opens every "other" locker, and so on.
Do you think she got a "D" because the teacher thought she could have done much better with her analysis?
On Aug 30, 2013, at 11:28 AM, Wayne Bishop <firstname.lastname@example.org> wrote:
> "The Locker Problem" has been around forever by lovers of POW (no, not what you're thinking, it's Problem Of the Week) and I am reminded of a colleague at a sister campus who had identical twin daughters who - instead of always wanting to be together - preferred the opposite. Up through 8th grade, he had been able to put them into separate classrooms but, there was only one "gifted" math class in 8th grade and since both qualified and this was the best mathematics teacher in the school, they were in the same class always with a POW. His policy, admirable I thought, was to answer specific, well-formulated questions but nothing more; nothing "leading". They were on their own working separately (as was their preference). There were only 30 lockers, not 1000, so "brute force" was possible, much as the pictured scissors at Dan's site, and that is what one daughter did - sketched a row of 30 closed doors, a row of 30 open doors after the first student had opened every door, a row where the even-numbered doors were closed as the 2nd student went by, down to the 30th row where only the 1st, 4th, 9th, 16th, and 25th doors remained open. Done. No "conjecture" or conclusion of any kind other than the row of doors. Grade? A. > > Now for the other student? No pictures at all, only the argument: It depends on the number of divisors (probably said "factors") of the student's number. For example, consider 12. It's divisors are 1, 2, 3, 4, 6, and 12. Since this is an even number of divisors, the door will be closed each time it is opened so will be closed at the end. Done. Grade D. Obviously, she hadn't taken care of every door. Now, admittedly, she did miss an important detail; since every divisor less than sqrt(n) has a "buddy" divisor greater than sqrt(n), the numbers of divisors of n is even except for the special case where n is a perfect square. Thus for 30 or 1000, the remaining open lockers correspond to the perfect squares less than or equal to the number of lockers. But still, this was the strongest math teacher in the school. Without following up the insightful solution with the nice little introduction to number theory it offers, they would've been far better off doing "cookbook" algebra. > > I didn't look at Dan's beyond the table of scissors. I hope he did get beyond that for 1000? > > Wayne > > At 08:07 PM 8/29/2013, Robert Hansen wrote: > >> http://blog.mrmeyer.com/?p=17848 >> >> The locker problem is as follows... >> >> 1. There are 1000 lockers, all shut and unlocked, and 1000 students. >> 2. The first student goes along and opens every locker. >> 3. The second student goes along and closes every locker starting with locker 2. >> 4. The 3rd student goes along and opens every locker starting with locker 3. >> 5. And so on... >> >> What is the state of the lockers after the 1000th student completes the task? >> >> This is a problem meant to be worked out in your HEAD. That is the whole point of this problem. To think (logically) and organize your thoughts. This is a great problem. It isn't overly difficult yet supports the use of some formality. To a real teacher this problem is a goal for their students to reach and they plan it into their curriculum with that intent. It is an opportunity for their students to arrive. >> >> But not Dan. For Dan this is an opportunity to enjoy himself with HIS hobby of activity making. And since Dan had decided to ignore all of that organization and structure we call "mathematics" he doesn't have to fret about his students' cognitive development anymore and thus has plenty of time for his hobby. Dan gives us a new twist on the phrase "busy work". Unlike the common use of that phrase in these discussions (work just meant to keep students busy) this is work just meant to keep the teacher busy. >> >> Bob Hansen >> >