First, I had a sample of 10 lockers and 10 students and tried to observe some patterns. I drew the line that separated the final states and changing states. I also underlined numbers where the change of states happened.
Since there seemed to be no obvious patterns, I turned to collect information on the total number of open and closed lockers for each locker and each student. Unfortunately, I still could not see any special.
Then, I focused on the last row and surprisingly found that the only lockers that were closed are perfect square number lockers. However, this finding should be rationally supported before I made a conclusion.
Thus, I looked back to the sample of 10. I counted the numbers of changes in states for each locker and realized that they are the numbers of factors for each locker number.
Finally, I double-checked my solution with a sample of 20 that I made in a spreadsheet. It matched what I had got!
Reflection: Although I underlined numbers that change occurred, I did not get it in the first place since I forgot to include the initial states of lockers.
Nice use of the spreadsheet to help you record your results, Yiwei. Great approach to start with a smaller problem of 10 lockers. By checking just the first 20 lockers, are you convinced that lockers with numbers that are perfect squares would be closed (or opened in the original problem as you missed the initial state as you mentioned) for 1000 lockers? Why or why not?
ReplyDeleteHello Erica, thank you for your reply and question. I was convinced since by using the spreadsheet, I also collected the numbers of changes and confirmed the relationship between them and numbers of factors. I guess checking a larger amount would be better when explaining to students, but it would be also important to show them when to accept the answers they get.
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