Math POW: Growing Tiles
Problem Statement
The Growing Tiles problem is one that shows a pattern in which you start with one square, then three squares were added on the top, left, and right sides, one on each side, of the original square. It then continued, the total number of tiles, to increase by three and each side increase by one. My job is to figure out a formula that can be used to find out the total number of tiles in each case, and figure out how many tiles there would be in the 100 case.
The Growing Tiles problem is one that shows a pattern in which you start with one square, then three squares were added on the top, left, and right sides, one on each side, of the original square. It then continued, the total number of tiles, to increase by three and each side increase by one. My job is to figure out a formula that can be used to find out the total number of tiles in each case, and figure out how many tiles there would be in the 100 case.
Process
The first thing I did when trying to solve this problem was to figure out how each case increased by, from the case # and the number of tiles added into each figure. I first noticed that each figure increased by three so I tried to go off from there since my other observation, multiply four to receive the number of tiles, only applied to case 1 and 2. I then multiplied the case # by three and had observed that the number multiplied by three and subtracted by two equaled the number of tiles in the figure corresponding with the case. I tested this equation with all the cases in the figure above and found out the equation that can be used to find the number of tiles for each case. I then configured the formula into a slope-intercept formula, where y equaled the total number of tiles and x equaled the case #, y=3x-2. When I showed the teacher he said that x could equal another part of the diagram, which turned out to be the vertical ascent of the tiles from the middle point.
The first thing I did when trying to solve this problem was to figure out how each case increased by, from the case # and the number of tiles added into each figure. I first noticed that each figure increased by three so I tried to go off from there since my other observation, multiply four to receive the number of tiles, only applied to case 1 and 2. I then multiplied the case # by three and had observed that the number multiplied by three and subtracted by two equaled the number of tiles in the figure corresponding with the case. I tested this equation with all the cases in the figure above and found out the equation that can be used to find the number of tiles for each case. I then configured the formula into a slope-intercept formula, where y equaled the total number of tiles and x equaled the case #, y=3x-2. When I showed the teacher he said that x could equal another part of the diagram, which turned out to be the vertical ascent of the tiles from the middle point.
We were then supposed to discuss the problem as a group, comparing ideas, and showing our process. A table mate had also found the same formula, but through a different process, one in which she created a table and started the process from y=x, y=2x, y=3x, until she found the formula she had found which applied to all cases. Although, that wasn't the only formula that the table came up with, another table mate came up the idea to figure out the total number of tiles. In the idea, one just needs to add one less than the case number three times and adds one more to figure out the y value. We then tried to show how our equation worked, the first one since it made more sense, by representing a graph that the teacher aided our thinking with, as shown above.
We then shared our ideas with the class, discovering even more equations that could've been used, the main one being y=x(n-1)+c. This was an important formula because it can be converted to figure out different portions any figure that has a similar shape to it. However, even though the formula was explained to us, I do not know how to apply it in this situation, so I reverted back to the original formula I had.
We then shared our ideas with the class, discovering even more equations that could've been used, the main one being y=x(n-1)+c. This was an important formula because it can be converted to figure out different portions any figure that has a similar shape to it. However, even though the formula was explained to us, I do not know how to apply it in this situation, so I reverted back to the original formula I had.
Solution
The solution that I got was y=3x-2, to explain this in the diagram we write out each variable in the equation. The coefficient three is there because the shape increases three ways, the top, right, and left side. The whole thing is subtracted by two because of several reasons, one would be due to that three subtracted by one would be two. Which is to say that if the first case had three tiles instead of one, they whole figure would just be multiplied by three. Another reason to explain this is similar to what was previously stated except how it would be later on in the process. To show this, look on the picture on the right, to see that I had drawn two imaginary squares to represent the figure being incomplete, and that the subtract two would be unnecessary if there were two more squares. It also works with any of the cases, as I will show with the fourth case.
The solution that I got was y=3x-2, to explain this in the diagram we write out each variable in the equation. The coefficient three is there because the shape increases three ways, the top, right, and left side. The whole thing is subtracted by two because of several reasons, one would be due to that three subtracted by one would be two. Which is to say that if the first case had three tiles instead of one, they whole figure would just be multiplied by three. Another reason to explain this is similar to what was previously stated except how it would be later on in the process. To show this, look on the picture on the right, to see that I had drawn two imaginary squares to represent the figure being incomplete, and that the subtract two would be unnecessary if there were two more squares. It also works with any of the cases, as I will show with the fourth case.
y=3x-2
x=4
y=3(4)-2
y=12-2
y=10
If you refer to the problem statement, one will see that there is a total of ten tiles in case #4, thus proving this equation true.
x=4
y=3(4)-2
y=12-2
y=10
If you refer to the problem statement, one will see that there is a total of ten tiles in case #4, thus proving this equation true.
Refection
Something that I learned from this problem is to try to show my work more and that cooperation is key. I learned to start to show my work more because I found out that since I did most of the math in my head, I didn't have much work written done and thus, not much work to show how I had gotten to my answer. I had looked at one of my table mates work and she had shown all her thoughts, all her ideas that didn't work, and it was easy to understand how she was able to conceive an answer from all her work. From now on, when working on math, I will do my best to show my whole process so that when others look at my work they are able to understand where my answer came from or where I had gone wrong. The other part, about cooperation is key, is something important I learned because I was able to see different views and different formulas that people came up with to solve the problem. This had opened my eyes, so that even though that people go through different processes and problems, they are still capable to come to the same conclusion.
The Habits of a Mathematician I had used for this open-ended problem would be Look for Patterns and Collaborate and Listen. I used Look for Patterns in this problem by how I first came up with the formula I had. If you look closer on the second picture, you will see little notes that I wrote near the cases I had wrote to figure out the relationship the figures had with one another. When I realized that when the case number, multiplied by three and subtracted by two equals the total number of tiles, I tested it on all the cases, and realized I had found a pattern. It had existed because the three in the relationships refers to the number of directions the cube goes, and the number that is subtracted by would be the number of directions it goes subtracted by the base number. To refer this to any figure, imagine that there was a fourth line of squares going down, the equation would be changed to y=4x-3.
I used Collaborate and Listen in this problem when we were discussing what we had our first share out, I tried to get everyone to share out and explain how they had gotten this formula. As well as, making sure everything we decided as a team, everyone was on board about our decision and that everyone had their ideas, as much as possible, incorporated in the problem. I especially showed this habit when we were creating our poster by making sure that everyone got involved, even in the littlest way. This is why I think I deserve 10/10.
Something that I learned from this problem is to try to show my work more and that cooperation is key. I learned to start to show my work more because I found out that since I did most of the math in my head, I didn't have much work written done and thus, not much work to show how I had gotten to my answer. I had looked at one of my table mates work and she had shown all her thoughts, all her ideas that didn't work, and it was easy to understand how she was able to conceive an answer from all her work. From now on, when working on math, I will do my best to show my whole process so that when others look at my work they are able to understand where my answer came from or where I had gone wrong. The other part, about cooperation is key, is something important I learned because I was able to see different views and different formulas that people came up with to solve the problem. This had opened my eyes, so that even though that people go through different processes and problems, they are still capable to come to the same conclusion.
The Habits of a Mathematician I had used for this open-ended problem would be Look for Patterns and Collaborate and Listen. I used Look for Patterns in this problem by how I first came up with the formula I had. If you look closer on the second picture, you will see little notes that I wrote near the cases I had wrote to figure out the relationship the figures had with one another. When I realized that when the case number, multiplied by three and subtracted by two equals the total number of tiles, I tested it on all the cases, and realized I had found a pattern. It had existed because the three in the relationships refers to the number of directions the cube goes, and the number that is subtracted by would be the number of directions it goes subtracted by the base number. To refer this to any figure, imagine that there was a fourth line of squares going down, the equation would be changed to y=4x-3.
I used Collaborate and Listen in this problem when we were discussing what we had our first share out, I tried to get everyone to share out and explain how they had gotten this formula. As well as, making sure everything we decided as a team, everyone was on board about our decision and that everyone had their ideas, as much as possible, incorporated in the problem. I especially showed this habit when we were creating our poster by making sure that everyone got involved, even in the littlest way. This is why I think I deserve 10/10.