Deangelo Concepcion
December 9, 2012
Math 3
POW #4: The Staircase Problem
Problem Statement
This problem is about legos. The main task is to find how many legos are required to build a certain level staircase. For example, it takes ten (10) legos to build a four-level staircase. There were also three other sub-tasks. The first one is to find how many legos are needed to make a five-level, six-level, and ten-level staircase. The second sub-task is to find how you would determine the number of legos needed for a staircase with 100 levels. The third sub-task is how many levels could be built with 10,000 and how many would be left over.
Process
1. For the first sub-task, we started with the five level staircase. In order to figure it out, we created our own five-story staircase by drawing out squares on a piece of paper, a square representing one lego. After drawing each staircase, we counted how many legos that were in it. It had a total of 15 legos in it, meaning that 15 legos were required to build a five-level staircase, We used the same method to find the amount of legos in a six-level, and a ten-level staircase. The six level had a total of 21 legos in it while the ten-level had 55 legos.
2. In the second sub-task, we looked at ways that you would be able to determine the number of legos needed for a one-hundred level staircase. We first started out by looking for patterns in the staircases. We looked at how many legos each staircase increases by. We noticed that each staircase was the amount of legos in the previous staircase plus the number of levels in a staircase. For example, for a five-level staircase, the number of legos in the four-level staircase, which is 10, plus the number of levels, which is five, totals out to 15 legos, which is exactly the amount of legos in that staircase. After finding this, we looked at better ways to calculate the legos. We noticed that the number of legos in each staircase is actually the levels before that all the way down to zero plus itself added up to get the answer. An example of this is that, for a four-level staircase, it is 4+3+2+1+0, or level four plus level three plus level two plus level one plus level zero. This all equals out to 10. We figured that this was the most basic way to solve this problem. Using it, we found that the number of legos in a one-hundred level staircase is 4984
Solution
For the first sub-task, the amount of legos required to build a five-level staircase is 15, the amount of legos required to build a six-level staircase is 21, and the amount of legos needed to build a ten-level staircase. For the second sub-task the most basic way we could determine the amount of legos in a one-hundred level staircase was by adding the levels prior to that one all the way down to zero to itself
Reflection
In this problem, I learned about looking for patterns. The main pattern that I saw in this was that each level of a lego staircase was an addition to the amount of legos in the level before it. The increase was varied on the height the level was at. I used a lot of evidence to support much of my answers. I showed it in much of my drawings of the lego staircases. This POW required a lot of this kind of evidence in order to solve all of the problems.
December 9, 2012
Math 3
POW #4: The Staircase Problem
Problem Statement
This problem is about legos. The main task is to find how many legos are required to build a certain level staircase. For example, it takes ten (10) legos to build a four-level staircase. There were also three other sub-tasks. The first one is to find how many legos are needed to make a five-level, six-level, and ten-level staircase. The second sub-task is to find how you would determine the number of legos needed for a staircase with 100 levels. The third sub-task is how many levels could be built with 10,000 and how many would be left over.
Process
1. For the first sub-task, we started with the five level staircase. In order to figure it out, we created our own five-story staircase by drawing out squares on a piece of paper, a square representing one lego. After drawing each staircase, we counted how many legos that were in it. It had a total of 15 legos in it, meaning that 15 legos were required to build a five-level staircase, We used the same method to find the amount of legos in a six-level, and a ten-level staircase. The six level had a total of 21 legos in it while the ten-level had 55 legos.
2. In the second sub-task, we looked at ways that you would be able to determine the number of legos needed for a one-hundred level staircase. We first started out by looking for patterns in the staircases. We looked at how many legos each staircase increases by. We noticed that each staircase was the amount of legos in the previous staircase plus the number of levels in a staircase. For example, for a five-level staircase, the number of legos in the four-level staircase, which is 10, plus the number of levels, which is five, totals out to 15 legos, which is exactly the amount of legos in that staircase. After finding this, we looked at better ways to calculate the legos. We noticed that the number of legos in each staircase is actually the levels before that all the way down to zero plus itself added up to get the answer. An example of this is that, for a four-level staircase, it is 4+3+2+1+0, or level four plus level three plus level two plus level one plus level zero. This all equals out to 10. We figured that this was the most basic way to solve this problem. Using it, we found that the number of legos in a one-hundred level staircase is 4984
Solution
For the first sub-task, the amount of legos required to build a five-level staircase is 15, the amount of legos required to build a six-level staircase is 21, and the amount of legos needed to build a ten-level staircase. For the second sub-task the most basic way we could determine the amount of legos in a one-hundred level staircase was by adding the levels prior to that one all the way down to zero to itself
Reflection
In this problem, I learned about looking for patterns. The main pattern that I saw in this was that each level of a lego staircase was an addition to the amount of legos in the level before it. The increase was varied on the height the level was at. I used a lot of evidence to support much of my answers. I showed it in much of my drawings of the lego staircases. This POW required a lot of this kind of evidence in order to solve all of the problems.