To solve the locker problem I first drew out 10 lockers because 10^3 is 1000, so using a sample size of 10 seemed reasonable. My process was to evaluate what the first 10 students did and then see if I noticed any patterns. The pattern I noticed was that all perfect square lockers (1,2,4,9,16,...) were closed while the rest were open. From this I concluded that non perfect square lockers will be open and perfect square lockers will be closed.
I think all the perfect square lockers are closed because they have an odd number of factors. For example, 1 has 1 factor, 4 has 3 factors (1,2,4), 9 has 3 factors (1,3,9), 16 has 5 factors (1,2,4,8,16), etc. Because they have odd number of factors, each perfect square locker will be touched an odd number of times and hence will be closed.
Hi Nandini. Good diagram and initial approach, where you noticed a pattern that pointed towards the square numbers. But this is not complete! You have not written anything about WHY the square numbers might have anything to do with which lockers are left open! Please revise this post with the addition of your explanation about why the square numbers are involved. And think about the 'why?' questions you will be asked by your students... it's not enough to say that it seems to be a certain kind of pattern (because sometimes the appearance of a pattern can be a bit of an illusion). You need to be able to understand the 'why' and to convince yourself and others of your understanding, and you'll want to ask your students to do the same!
ReplyDeleteThanks for the update explaining the 'why' of this question!
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