Locker Problem - Response

A school has 1000 lockers and 1000 students. On the first morning:

*The first student walks along and opens every locker.

*The second student walks along and closes every second locker.

*The third student walks along and changes the position of every third locker door.

And so on.

Once all 1000 students have completed this process, which lockers are open?

a) Briefly describe how you solved it?

In order to solve this problem, I created a chart on which I wrote "student #" and "locker #", and simulated a student running through the hallway and closing lockers over 10 trials and 10 lockers. After realizing that I couldn't draw any proper conclusion from this, I decided to simulate this over 20 trials and 20 lockers. I still didn't see the pattern, but upon recording the number of times each locker had been opened and closed, I realized that every so often, a locker had been opened/closed an odd number of times. In fact, these lockers remained open, and I found that their corresponding locker numbers were perfect squares. I found that the largest perfect square less than 1000 is 961, or 31^2. Hence, of the 1000 lockers, 31 lockers remained open.

b) Where do you think a student might get stuck?

I feel a student would get stuck right at the start of the problem, and have a hard time wrapping their head around what the question was asking. I know that I originally didn't read the question correctly, and it took me a second to understand that the number of the student corresponded to the lockers the student was closing. It was also very easy, when doing a test trial, to make mistakes of correctly marking the lockers which were open and which remained closed. It was also overwhelming, in general, that there were 1000 lockers in the problem; I was worried that I wouldn't find the pattern and be stuck for much longer creating more trials or abandoning the trial version altogether. I think they would likely get frustrated if they made mistakes in their logical deduction, because there would be no pattern emerging from the tiles. 

c) How might you assist that stuck student?

In this example, I really like the idea of manipulating algebra tiles; I might suggest using black for closed lockers, and white for open lockers. Because the problem is also initially intimidating, I would have the students work in pairs or even groups of three, especially because it is easy to make a mistake in preliminary reasoning. 

While algebra tiles work for kinesthaetic learners, it would make more sense for others to simply write down symbols to represent closed and lockers (e.g. X for closed and O for open), and this is the method I preferred. I found that it was much easier to catch my mistakes by writing down the trials than it would have been to simulate them with algebra tiles. This is especially true if there is no opportunity to work in a group, because a person may even forget the number of the trial they're finishing, and may feel an overwhelming urge to re-start the problem. Finally, if a student does not recognize perfect squares, then they may not identify that the lockers remaining open end up having a perfect square as their number; working in partners would help amend this. Finally, I would have the student write down the number of times a locker is being opened/closed in a table. It was only then that I noticed the pattern which existed in this activity.

d) What extensions could you offer students ready for a challenge?

In order to provide more of a challenge for any students who were willing, I would ask about what would happen if the students, after the 1000th student, repeated the process and ran down the hallway in the opposite direction. I could also have the student prove that the order of the students running through the hallway and which multiple the students represent doesn't matter, and that the same 1000 lockers would remain open after 1000 trials.