The problem states that the school had installed 550 lockers at school during summer vacation. The lockers are numbered from 1 to 550. When the students return from summer vacation, they decide to celebrate the lockers by working off some energy.
•The first student goes to all of the lockers and opens every locker.
•The second student then goes to all of the lockers and opens every other locker.
•The third student changes the state of every third locker (if it is closed, they open it, if it is open they close it)
•The fourth student changes the state of every fourth locker.
•The fifth student changes the state of every fifth locker.
•The sixth changes every sixth locker, and so on.
For me, the translation of this in my brain was like stating that each person is going to change the number of lockers open and closed until the 550th student goes, and that the point of this is to find out what the pattern of this means, and there was a nagging feeling telling me that the students were supposed to represent a mathematical term, but not how quite yet. What I fist did was write out the first 30 lockers and making them open because that was what the first student did. I then closed every other locker starting from two, and continued this open-close process until I reached the 15th locker. After I did each number I wrote it down because I knew that as soon as I was done changing the terms that needed to be changed, that would be there final position (1=open, 2=closed, 3=closed, 4=open, 5=closed, etc). With this process I was able to answer the first question that asked, 'After the first ten students go marching, which of these lockers are still open?', which the answer would be 1, 4, and 9. Numbers 2-4 on the other hand would be quite difficult. However the solution became clear when Ms. X (mathematics teacher) showed us a video of what one student did to figure out the problems. He showed us that the numbers left, after 50 lockers went through the process, were all perfect squares. The reason being is that perfect squares have an odd amount of factors, thus it started from opened and ended at open.
The second problem inquires, 'Which lockers will be open and which will be closed when they finish all 550 lockers? Why?', the answer to this is similar to what I previously explained before, that all the numbers that are perfect squares will be open because perfect squares have an odd number of factors. Examples are; 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, and 529. I stopped at 529 because there would only be 550 lockers and 24*24= 576, exceeding the amount of lockers by 26.
The third problem asks, 'What if the students were not in order when they went marching to open and close lockers. Would the same lockers be open in the end? Why or why not?'. This made me conclude that the lockers that were open in order will still be opened when not in order. This is due to the fact that the students act like factors for each number, the number representing the locker. Even if you had them going backwards or whatever way you would want to do it, it would be the same because the students wouldn't touch the lockers that don't have their number as a factor. To further understand the concept, if you were the sixth student, you would still only touch the numbers that six can go into easily (without any remaining factors making it go into decimals or fractions), as would be the same for 458.
The fourth question wants us to further investigate, 'Which lockers were only touched by two students? Were any lockers touched by only three students? Which number locker was touched by the most students?'. For the first part of this question I instantly recognized the loose term of students as what they were representing, factors. So the first thought coming into my mind after the previous realization, is that it was talking about prime numbers. So the answer would be prime number lockers were only touched by two students. To continue with the next part of this question, I immediately knew it would be perfect squares, however not all perfect squares because perfect squares just need an odd number of factors, not just three. Some examples to back up my conclusion is; 4 (1*4, 2*2), 9 (1*9, 3*3), 25 (1*25, 5*5), and 49 (1*49, 7*7). So overall, yes, there were lockers touched by only three students. The last part of the question asks simply requires a number, a number that has the most factors/ touched by the most students. I had no idea how to go about this question, truthfully, because the only option I was thinking of was getting all the even numbers between 2-550 and just start from there. However, I had no need to do this considering I had help from a third party who knew a more complicated, yet simpler method of finding the answer. The method was calledFundamental Theorem of Arithmetic, which simplified all numbers into a multiplication expression filled with primes, when multiplied together the product equals the beginning number. So the answer for this would be 512, in the Fundamental Theorem of Arithmetic it would be 2x2x2x2x2x2x2x2x2.
I had incorporated the Habits of Mind in this open-ended problem by using evidence. I did this by explaining my thought process for the whole thing, starting with how I just wrote out each locker and ending with how I learned from another student and added to it. I continued to use evidence by giving examples on most of the answers I had, so the reader would have a better understanding of what I was trying to do, and how they can test it out (by finding the factors of numbers given, or trying it out). The use of data was merged with the whole process, because if I didn't use the data in the question, I would be unable to do any work. This is just to further prove that I have used evidenceduring the whole process of this problem.
•The first student goes to all of the lockers and opens every locker.
•The second student then goes to all of the lockers and opens every other locker.
•The third student changes the state of every third locker (if it is closed, they open it, if it is open they close it)
•The fourth student changes the state of every fourth locker.
•The fifth student changes the state of every fifth locker.
•The sixth changes every sixth locker, and so on.
For me, the translation of this in my brain was like stating that each person is going to change the number of lockers open and closed until the 550th student goes, and that the point of this is to find out what the pattern of this means, and there was a nagging feeling telling me that the students were supposed to represent a mathematical term, but not how quite yet. What I fist did was write out the first 30 lockers and making them open because that was what the first student did. I then closed every other locker starting from two, and continued this open-close process until I reached the 15th locker. After I did each number I wrote it down because I knew that as soon as I was done changing the terms that needed to be changed, that would be there final position (1=open, 2=closed, 3=closed, 4=open, 5=closed, etc). With this process I was able to answer the first question that asked, 'After the first ten students go marching, which of these lockers are still open?', which the answer would be 1, 4, and 9. Numbers 2-4 on the other hand would be quite difficult. However the solution became clear when Ms. X (mathematics teacher) showed us a video of what one student did to figure out the problems. He showed us that the numbers left, after 50 lockers went through the process, were all perfect squares. The reason being is that perfect squares have an odd amount of factors, thus it started from opened and ended at open.
The second problem inquires, 'Which lockers will be open and which will be closed when they finish all 550 lockers? Why?', the answer to this is similar to what I previously explained before, that all the numbers that are perfect squares will be open because perfect squares have an odd number of factors. Examples are; 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, and 529. I stopped at 529 because there would only be 550 lockers and 24*24= 576, exceeding the amount of lockers by 26.
The third problem asks, 'What if the students were not in order when they went marching to open and close lockers. Would the same lockers be open in the end? Why or why not?'. This made me conclude that the lockers that were open in order will still be opened when not in order. This is due to the fact that the students act like factors for each number, the number representing the locker. Even if you had them going backwards or whatever way you would want to do it, it would be the same because the students wouldn't touch the lockers that don't have their number as a factor. To further understand the concept, if you were the sixth student, you would still only touch the numbers that six can go into easily (without any remaining factors making it go into decimals or fractions), as would be the same for 458.
The fourth question wants us to further investigate, 'Which lockers were only touched by two students? Were any lockers touched by only three students? Which number locker was touched by the most students?'. For the first part of this question I instantly recognized the loose term of students as what they were representing, factors. So the first thought coming into my mind after the previous realization, is that it was talking about prime numbers. So the answer would be prime number lockers were only touched by two students. To continue with the next part of this question, I immediately knew it would be perfect squares, however not all perfect squares because perfect squares just need an odd number of factors, not just three. Some examples to back up my conclusion is; 4 (1*4, 2*2), 9 (1*9, 3*3), 25 (1*25, 5*5), and 49 (1*49, 7*7). So overall, yes, there were lockers touched by only three students. The last part of the question asks simply requires a number, a number that has the most factors/ touched by the most students. I had no idea how to go about this question, truthfully, because the only option I was thinking of was getting all the even numbers between 2-550 and just start from there. However, I had no need to do this considering I had help from a third party who knew a more complicated, yet simpler method of finding the answer. The method was calledFundamental Theorem of Arithmetic, which simplified all numbers into a multiplication expression filled with primes, when multiplied together the product equals the beginning number. So the answer for this would be 512, in the Fundamental Theorem of Arithmetic it would be 2x2x2x2x2x2x2x2x2.
I had incorporated the Habits of Mind in this open-ended problem by using evidence. I did this by explaining my thought process for the whole thing, starting with how I just wrote out each locker and ending with how I learned from another student and added to it. I continued to use evidence by giving examples on most of the answers I had, so the reader would have a better understanding of what I was trying to do, and how they can test it out (by finding the factors of numbers given, or trying it out). The use of data was merged with the whole process, because if I didn't use the data in the question, I would be unable to do any work. This is just to further prove that I have used evidenceduring the whole process of this problem.