Planning Platforms Problem
Problem Statement:
For this problem, we were given a paper that described to us a problem from which we had to try and solve. However, the only thing known from this problem are the variables for which we have to take account for. The point of this problem was to try and stimulate our brain to form a formula without using numbers and that could be used for the situation below. The variables included to find the first formula would be the number of platforms needed, the increase of the height for each platform, and the height of the first platform. The second formula that we would need to find would need the height of the last platform and the area of a single side of the platform for decoration. The problem that we were given is also available below.
For this problem, we were given a paper that described to us a problem from which we had to try and solve. However, the only thing known from this problem are the variables for which we have to take account for. The point of this problem was to try and stimulate our brain to form a formula without using numbers and that could be used for the situation below. The variables included to find the first formula would be the number of platforms needed, the increase of the height for each platform, and the height of the first platform. The second formula that we would need to find would need the height of the last platform and the area of a single side of the platform for decoration. The problem that we were given is also available below.
Process
When first given this problem, I was confused with what we were trying to do. From the other problems that we have been given, the point of the problem was to try and find the answer, a single answer, where there could be no other options for this to be true. So when we tried to solve this problem, we were befuddled. We were supposed to find a general formula for this, no numbers were technically needed to solve this. However, our group found that it was easier showing our work and using examples to solve it out. To begin, we drew an example of what the platforms could look like, but it didn't help to make the situation clearer. But, it did help us to find a starting point on how to find a formula for the first part. Since the height of the platforms was constantly increasing by a fixed amount, the point-slope form, y=mx+b, could be applicable to this, so we tried to find out how the variables we had to use could be applicable to the point -slope form variables. We were able to conclude that the y would be the height of the tallest platform, m would be the number of platforms, x would be the height increase, and b would be the height of the shortest (or first platform). However, the answer we received was always off the height of the first platform so we took away the addition to the problem and we got our version of the answer. When we were sharing our work for this portion, we found out that our formula was only applicable to the measurement of feet, so if we tried the problem in inches, we would receive the incorrect answer. Another group came up with a concrete formula that could be applied to both feet and inches.
With this information, we began the second portion of this problem, where we find the formula to find the area of a side of a platform. We tried many different formulas, most including the height of the first platform and the increase of the platforms. An example of one of the formulas we came up with was h+d(n)+(h+d), which is translated as the height of the first platform added to the distance of the platforms added to the distance of the platforms multiplied by the number of platforms added to the height of the first platform. When we tried to see if it worked for different examples, it didn't pass the test, so we tried again. After doing this process for quite some time, I looked back at the paper that was given to us and noticed that it said, she needs to know how high the tallest platform will be, and with that in mind we tried to include the height of the tallest platform when doing the problem. We tried to add the height of the tallest and shortest platforms and multiply it be the number of platforms, but it seemed to be doubled the amount we needed, so we multiplied the number of platforms by .5 to create an equation that could work. To make sure that this worked correctly, we tested the formula out many times, and it worked perfectly. We then shared our work with other groups that may have needed help.
When first given this problem, I was confused with what we were trying to do. From the other problems that we have been given, the point of the problem was to try and find the answer, a single answer, where there could be no other options for this to be true. So when we tried to solve this problem, we were befuddled. We were supposed to find a general formula for this, no numbers were technically needed to solve this. However, our group found that it was easier showing our work and using examples to solve it out. To begin, we drew an example of what the platforms could look like, but it didn't help to make the situation clearer. But, it did help us to find a starting point on how to find a formula for the first part. Since the height of the platforms was constantly increasing by a fixed amount, the point-slope form, y=mx+b, could be applicable to this, so we tried to find out how the variables we had to use could be applicable to the point -slope form variables. We were able to conclude that the y would be the height of the tallest platform, m would be the number of platforms, x would be the height increase, and b would be the height of the shortest (or first platform). However, the answer we received was always off the height of the first platform so we took away the addition to the problem and we got our version of the answer. When we were sharing our work for this portion, we found out that our formula was only applicable to the measurement of feet, so if we tried the problem in inches, we would receive the incorrect answer. Another group came up with a concrete formula that could be applied to both feet and inches.
With this information, we began the second portion of this problem, where we find the formula to find the area of a side of a platform. We tried many different formulas, most including the height of the first platform and the increase of the platforms. An example of one of the formulas we came up with was h+d(n)+(h+d), which is translated as the height of the first platform added to the distance of the platforms added to the distance of the platforms multiplied by the number of platforms added to the height of the first platform. When we tried to see if it worked for different examples, it didn't pass the test, so we tried again. After doing this process for quite some time, I looked back at the paper that was given to us and noticed that it said, she needs to know how high the tallest platform will be, and with that in mind we tried to include the height of the tallest platform when doing the problem. We tried to add the height of the tallest and shortest platforms and multiply it be the number of platforms, but it seemed to be doubled the amount we needed, so we multiplied the number of platforms by .5 to create an equation that could work. To make sure that this worked correctly, we tested the formula out many times, and it worked perfectly. We then shared our work with other groups that may have needed help.
Solution
For the first solution, we chose the formula we found as a class because the one we found as a groupone that we reviewed as a team which was the n(d)-d+h=t, where n represents the number of platforms, d represents the constant increase of the platforms, h represents the height of the first platform, and t represents the height of the tallest platform. Another way to rewrite this formula would be d(n-1)+h=t because when you multiply n and d together you would have nd-d+h=t so it is capable for the formula to be undistributed so equal d(n-1)+h=t. Even though we reviewed this problem, I am still uncertain on how this equation works to give us the height of the tallest platform. For the second solution, we got (t+h)(.5n) where t represents the height of the tallest platform, h represents the height of the shortest, and n represents the number of platforms. This formula works because the problem stated that for this problem we would need to know the tallest platform, so we incorporated the information we already knew and applied it to the formula. We checked to see if it applied to different examples, and since it passed for five different examples we knew that this would work.
For the first solution, we chose the formula we found as a class because the one we found as a groupone that we reviewed as a team which was the n(d)-d+h=t, where n represents the number of platforms, d represents the constant increase of the platforms, h represents the height of the first platform, and t represents the height of the tallest platform. Another way to rewrite this formula would be d(n-1)+h=t because when you multiply n and d together you would have nd-d+h=t so it is capable for the formula to be undistributed so equal d(n-1)+h=t. Even though we reviewed this problem, I am still uncertain on how this equation works to give us the height of the tallest platform. For the second solution, we got (t+h)(.5n) where t represents the height of the tallest platform, h represents the height of the shortest, and n represents the number of platforms. This formula works because the problem stated that for this problem we would need to know the tallest platform, so we incorporated the information we already knew and applied it to the formula. We checked to see if it applied to different examples, and since it passed for five different examples we knew that this would work.
Reflection
In this problem, I feel that I was systematic and that I collaborated and listened well because of the way that I approached I did the problem. This was because I made small changes to formulas that I thought would work until it seem to applied to the cases I had, and I continued to do this in a manner where I only change one variable at a time. I used collaboration and my listening skills because when I didn't understand what was going on, we reviewed it as a class and what I understood from the discussion, I used to further proceed in the problem. I think that we had a hard time cooperating as a group because we were divided because we went to different areas to work on the problem. So that whenever one of us thought we got the correct answer, the other didn't understand how they got to that conclusion. Next time, I think that we should stay together and work on the problem together and help each other out so we all understand how we are doing what we are, and how we got to that step.
In this problem, I feel that I was systematic and that I collaborated and listened well because of the way that I approached I did the problem. This was because I made small changes to formulas that I thought would work until it seem to applied to the cases I had, and I continued to do this in a manner where I only change one variable at a time. I used collaboration and my listening skills because when I didn't understand what was going on, we reviewed it as a class and what I understood from the discussion, I used to further proceed in the problem. I think that we had a hard time cooperating as a group because we were divided because we went to different areas to work on the problem. So that whenever one of us thought we got the correct answer, the other didn't understand how they got to that conclusion. Next time, I think that we should stay together and work on the problem together and help each other out so we all understand how we are doing what we are, and how we got to that step.