Tiling a Patio- Problem of the Week
Isabella Norton
October 26, 2016
Math
Problem Statement
This problem states that Dr. Drew is planning to construct a patio at his house that will be laid out using only square tiles. Those tiles are inexpensive and nice, however he got beneficial advice to get a minimal amount of a more expensive type because of their less maintenance/better quality. People walking from the garage door to the back door will always walk along a certain diagonal, to one patio corner and to the opposite, so the diagonal tiles will get wear-tear. Every tile that is attached or touching a segment of the diagonal line must be the expensive tile. Even if one square tile touches the segment, it can be a rectangular tile. This problem also includes a 4-by-6 courtyard situation, which has 8 expensive tiles. Question one asks how many of the special tiles does Dr.Drew need to order for the 63-by-90, 63 rows of tiles with 90 tiles in each row, tile patio. Then, question two asks if Dr.Drew were to try a patio with a different dimension, could we find a general formula, to determine how many special tiles he would need. (We are supposing the courtyard had r rows with c tiles in each row.)
Process Description and Solution
In my attempt to solve both questions or the problem, I had the approach of observing the 4-6 tile patio, finding patterns or similarities, and I found that since there are 8 shaded tiles and 24 tiles total, one third of the patios tiles are expensive tiles. I also found that the diagonal segment creates a linear equation and the slope is rise-1 by run-2. This created an observation that helped me conclude and make a conjecture. Utilizing my data and observations, I figured that on this diagonal segment going across the tile patio, no matter how many rows with tiles inside, the slope is always going to be rise-1 by run-2. Consequently, I then figured out how many tiles in total were in a 63-by-90 tile patio and that was 5,670 tiles. I applied the slope rise-1 by run-2 and then found that you would need 1,890 special tiles for Dr.Drew's 63-by-90 tile patio. For question number two, we needed to find a formula that worked for all potential sized patios. I concluded since r is rows and c is tiles, you could determine how many special tiles you needed by multiplying r and c, diving your answer by 3, and then find the one third of special tiles you need for all size patios. Therefore, the full equation is y equals r times c, then divided by 3. (Picture of work is located below)
Extensions and Further Exploration
My extension of this problem came from a curious thought. I wondered what the new formula, equation, or special tile count would be if I used an even number (5 by 5 or 10 by 10...etc) for rows with the same even number for tiles inside. I did a couple of sketches of different even tile amount patios and quickly saw a slope of rise- 1 and over-1. Therefore, I drew an 8 by 8 patio and saw 8 squares would be special tiles, if you have an even sized patio, and your equation would be y equals r times c, then divided by r. I got the exact results when I drew a 5-by-5 square, and my whole new approach is needed to determine how many special squares you would need for an even dimension patio. Now I have another formula to determine this, and this will adapt to this problem and benefit me because I have a better understanding on these two formulas and the slopes I discovered. (Picture of work is located below)
Reflection
Overall, this problem gave me a better understanding on making conjectures and testing, looking for different patterns, and starting small/ breaking the problem down. My strength was definitely finding similarities/patterns because I quickly found the slope for each patio, which was the first step to accomplishing the whole purpose of the problem. I also took pride in documenting my thoughts and work because on this POW, I wanted to make sure I kept all my thoughts together in one place, so I could again see the small similarities. As a mathematician, I believe the habit I used the most and benefited me the most was conjecture and test because I learned it brings you closer to a discovery, asking yourself questions that led me to the slope, which all related to the habit seek why and prove.
October 26, 2016
Math
Problem Statement
This problem states that Dr. Drew is planning to construct a patio at his house that will be laid out using only square tiles. Those tiles are inexpensive and nice, however he got beneficial advice to get a minimal amount of a more expensive type because of their less maintenance/better quality. People walking from the garage door to the back door will always walk along a certain diagonal, to one patio corner and to the opposite, so the diagonal tiles will get wear-tear. Every tile that is attached or touching a segment of the diagonal line must be the expensive tile. Even if one square tile touches the segment, it can be a rectangular tile. This problem also includes a 4-by-6 courtyard situation, which has 8 expensive tiles. Question one asks how many of the special tiles does Dr.Drew need to order for the 63-by-90, 63 rows of tiles with 90 tiles in each row, tile patio. Then, question two asks if Dr.Drew were to try a patio with a different dimension, could we find a general formula, to determine how many special tiles he would need. (We are supposing the courtyard had r rows with c tiles in each row.)
Process Description and Solution
In my attempt to solve both questions or the problem, I had the approach of observing the 4-6 tile patio, finding patterns or similarities, and I found that since there are 8 shaded tiles and 24 tiles total, one third of the patios tiles are expensive tiles. I also found that the diagonal segment creates a linear equation and the slope is rise-1 by run-2. This created an observation that helped me conclude and make a conjecture. Utilizing my data and observations, I figured that on this diagonal segment going across the tile patio, no matter how many rows with tiles inside, the slope is always going to be rise-1 by run-2. Consequently, I then figured out how many tiles in total were in a 63-by-90 tile patio and that was 5,670 tiles. I applied the slope rise-1 by run-2 and then found that you would need 1,890 special tiles for Dr.Drew's 63-by-90 tile patio. For question number two, we needed to find a formula that worked for all potential sized patios. I concluded since r is rows and c is tiles, you could determine how many special tiles you needed by multiplying r and c, diving your answer by 3, and then find the one third of special tiles you need for all size patios. Therefore, the full equation is y equals r times c, then divided by 3. (Picture of work is located below)
Extensions and Further Exploration
My extension of this problem came from a curious thought. I wondered what the new formula, equation, or special tile count would be if I used an even number (5 by 5 or 10 by 10...etc) for rows with the same even number for tiles inside. I did a couple of sketches of different even tile amount patios and quickly saw a slope of rise- 1 and over-1. Therefore, I drew an 8 by 8 patio and saw 8 squares would be special tiles, if you have an even sized patio, and your equation would be y equals r times c, then divided by r. I got the exact results when I drew a 5-by-5 square, and my whole new approach is needed to determine how many special squares you would need for an even dimension patio. Now I have another formula to determine this, and this will adapt to this problem and benefit me because I have a better understanding on these two formulas and the slopes I discovered. (Picture of work is located below)
Reflection
Overall, this problem gave me a better understanding on making conjectures and testing, looking for different patterns, and starting small/ breaking the problem down. My strength was definitely finding similarities/patterns because I quickly found the slope for each patio, which was the first step to accomplishing the whole purpose of the problem. I also took pride in documenting my thoughts and work because on this POW, I wanted to make sure I kept all my thoughts together in one place, so I could again see the small similarities. As a mathematician, I believe the habit I used the most and benefited me the most was conjecture and test because I learned it brings you closer to a discovery, asking yourself questions that led me to the slope, which all related to the habit seek why and prove.