Free Think Friday: Locker Lockdown
Problem Statement
In this problem, our group was tasked with finding which lockers would be open after waves of students would come through and open or close their respective locker. The first person would open all of the lockers, the second person would close every other locker, and the third person would close or open every third locker, and so on and so forth. The question stated was, "After the 552th student open or closed their respective locker, which lockers would be left open?" To aid us with this problem, a hint and first question was, "What would the lockers look like after the tenth student?" This would help us find the pattern that would be the key to solving this problem.
Process Description
I began by looking for a pattern in the equations with the lockers and how they were opened. I couldn't find anything substantial with what I found though, so I made a 10 by 10 grid that recorded which the locker number and wave of students would have a corresponding square and therefore we could keep up with when lockers opened and closed. I looked for which numbers were found more in the "closed" section and which were more common in the "open section, but we ended up not having to use this information. When we finished filling out the grid, we found that only the 1st, 4th, and 9th locker would be open at the end of the tenth wave. We went to expand this to confirm out suspicion of the answer, and confirmed it before we even finished the 25 by 25 grid.
We got a lot of hints before we were able to figure out the problem, and got confirmation for what the answers we weren't sure of, which helped a lot when it came to finding the answer. We knew from the hints that it had to do with the factors, and we had help correcting our mistakes with the 10 by 10 grid that wasn't filled out correctly the first time which was confusing us.
We also all took many different approaches; Katie started with a full number chart and put a bunch of stickies on to keep track of the pattern. Chris made a google excel sheet and gridded out his own version. Sarah started with trying to find equations that could maybe solve the problem or explain it.
We got a lot of hints before we were able to figure out the problem, and got confirmation for what the answers we weren't sure of, which helped a lot when it came to finding the answer. We knew from the hints that it had to do with the factors, and we had help correcting our mistakes with the 10 by 10 grid that wasn't filled out correctly the first time which was confusing us.
We also all took many different approaches; Katie started with a full number chart and put a bunch of stickies on to keep track of the pattern. Chris made a google excel sheet and gridded out his own version. Sarah started with trying to find equations that could maybe solve the problem or explain it.
Solution
The solution to this Free Think Friday problem was "Perfect Squares." The only lockers that would be open were those numbers that were perfect squares, such as 1, 4, 9, 16, 25, etc. This is because of the number of factors they have, that being odd. The rest of the numbers had even amounts of factors and thus would be closed when the last person would come by. The first factor meant it would be open, the second factor meant it would be closed, and the third factor meant that it would be opened again, and so on.
Self Assessment/Reflection
For this problem, I think I contributed just as fully, if not more than my group mates, so definitely a 9/10 or 10/10. I worked hard to finish up the problem for us and I noted everything we were finding in my notebook. It was me that started the 10 by 10 grid and found the answer for the first problem, and from there I got help by collaboration from my group mates who were still trying to figure out the problem with their own methods.
I learned about the factoring thing, about how perfect squares had an odd amount of factors. I think our group was on the right track with the gridding, and the hints just helped us fix our errors and get to the answer faster. Either way, I think we would have come to the answer eventually, just a little slower.
I think with this problem, we definitely had to "attend to precision" and "make sense of problems and persevere through them." I chose both of these because both of the mathematical habits were critical to our group and this problem. A huge problem we had at first was that we had errors with our data because we weren't being careful enough with our grids and methods, so we had a lot of "excess junk," which was keeping us from finding the golden nuggets of really important data. With a little help, we were able to fix it, but then we had to persevere and really think about the patterns and hints to find an answer.
Comparing and contrasting ourselves to other groups, we definitely were a bit slower to find the answer, but that didn't stop us from coming to the same conclusion as the others. We didn't have an explanation as to WHY the answer was what we got, but we were getting there (We knew that the perfect squares were only opened because the last person that came by opened them, rather than closed them unlike the other numbers), however we hadn't quite realized it was because of the numbers of the factors until another group came and explained it. I wouldn't say that we worked any less harder than any other group, we just were collectively a little slower because we all started out solo and came together with a bunch of different ideas that needed to be sorted out. I also took the liberty of leading the group, and I made mistakes in my own data that confused us and set us on a different path until we fixed our mistake with the hints we were given. We did finish the problem though, and I feel like we worked relatively well together.
I learned about the factoring thing, about how perfect squares had an odd amount of factors. I think our group was on the right track with the gridding, and the hints just helped us fix our errors and get to the answer faster. Either way, I think we would have come to the answer eventually, just a little slower.
I think with this problem, we definitely had to "attend to precision" and "make sense of problems and persevere through them." I chose both of these because both of the mathematical habits were critical to our group and this problem. A huge problem we had at first was that we had errors with our data because we weren't being careful enough with our grids and methods, so we had a lot of "excess junk," which was keeping us from finding the golden nuggets of really important data. With a little help, we were able to fix it, but then we had to persevere and really think about the patterns and hints to find an answer.
Comparing and contrasting ourselves to other groups, we definitely were a bit slower to find the answer, but that didn't stop us from coming to the same conclusion as the others. We didn't have an explanation as to WHY the answer was what we got, but we were getting there (We knew that the perfect squares were only opened because the last person that came by opened them, rather than closed them unlike the other numbers), however we hadn't quite realized it was because of the numbers of the factors until another group came and explained it. I wouldn't say that we worked any less harder than any other group, we just were collectively a little slower because we all started out solo and came together with a bunch of different ideas that needed to be sorted out. I also took the liberty of leading the group, and I made mistakes in my own data that confused us and set us on a different path until we fixed our mistake with the hints we were given. We did finish the problem though, and I feel like we worked relatively well together.
Free Think Friday: Fibonacci Fun
Problem Statement
For this session of Free Think Friday, I chose to assess the Fibonacci Fun problem. There was four triangles, with each iteration having less and less black space. Our quest was to find how the black region changes as the pattern continues through the iterations, and then to find the area as each of the colored triangles are added. The triangles would have triangular smaller pieces replacing parts of the black triangles.
Process Description
We began by trying to find the how the triangle was divided up. I knew it had to do something with the Fibonacci sequence, so I researched how this could possibly help us in it figuring out the pattern and how it changes. I stumbled upon the “golden spiral” and from there the “golden triangle.” Meanwhile, my groupmates found that when the triangle was originally divided, the larger portion would be halved. From there, each of the “larger” black triangles would be halved. We estimated that the next iteration of the triangle would have all equally sized triangles. But, we weren’t really sure what to do with the data we collected. We tried looking at how the percentages of black were changed, but nothing really showed much of a regular pattern. We then tried finding what the Fibonacci sequence had to do with the problem.
Solution
The Fibonacci sequence is numbers that follow this equation: Fn = Fn-1 + Fn-2. At first we wondered if this was the answer, however, we found that it didn’t quite work when applied. This is how far we got with our research and figuring out the problem before presentation time came, so we never got to find the answer, however I have a feeling it has to do with something about the Fibonacci spiral and the way it is divided, except backwards. Either way, the solution we found was that eventually the black part will just keep getting halved, especially when it reaches the point when all the black space is equal and therefore will equally take away space and replace it with equal colored space. What these numbers are and what the pattern is I’ve yet to figure out.
Self-Assessment/Reflection
I learned a lot through my research of the Fibonacci sequence. I learned first what it is, and secondly was the golden spiral was, specifically how it worked. I learned that the Fibonacci sequence could be used with right triangles, but how to apply that to our problem evaded me. I think I at least deserve full credit, as I tried to find ways to help our group even when I didn't know what to do either, such as researching and helping writing out our work. I tried to make myself useful when I couldn’t necessarily understand the problem, and did work to try and help myself and my group understand it. I feel like I really used the mathematical practice “make sense of problems and persevere in solving them.” The problem still feels unfinished and I’m curious as to what the answers are. I tried to best make sense of the problem with my own skills I could remember, and when that wasn't coming up with anything, I persevered to try and find a clue.