## Reasoning Analytically

'Burle Marx (Rio)'

Sarah Morris, 2011

Sarah Morris, 2011

**"**How can I make sense of and impact the world around me using math?

**"**

## Description of portfolio item

This quarter in Math B we are working on exponential growth and functions. The Four Plans is a problem that is about patterns and totals in exponential growth. The goals my teacher had in mind for this problem was to be able to calculate the total amount earned from various payment plans & explain why a payment plan is the best for each of the parties involved.

To sum it up, in the ancient kingdom of Montarek there was this peasant that saved the a king’s daughter, so the king arranged for the peasant to get a reward. The king said that the peasant could get anything that she wanted. The peasant, who happened to be a chess player, wanted to have the king place one ruba on the first square of her chessboard, two on the second, four on the third, eight of the fourth, and so on so that all 64 squares were covered. The peasant wanted amount of ruba’s to double on each new square.

The king told the queen about the reward he promised the peasant and the queen said that reward would cost more than the entire royal treasury. The king then came up with a different plan where the chessboard would only have 16 squares and he would triple the number of rubas on each new square. The queen also came up with her own plan, with a board of 12 squares, where the amount of rubas on each new square would quadruple. The royal financial advisors also came up with a fourth plan using a 64 square chessboard, which would start out with 25 rubas and keep increasing by 5 rubas every square.

Each person in every table group chose what plan they would be analyzing(figuring out how many rubas each plan would give to the peasant), and we would be putting it all together in the end to see which plan was the best for the king. I chose to complete the peasant’s plan, which was a terrible decision because it was to hardest to do.

The steps I had to complete my part of the four plans were:

1. I shared the Four Plans slides with my group so we could start working on the questions in it.

2. I made a table of the number of rubas on each square according to the peasant’s plan.

1. I shared the Four Plans slides with my group so we could start working on the questions in it.

2. I made a table of the number of rubas on each square according to the peasant’s plan.

3. I calculated how many rubas would be on the last square by laying out every square and the amount of rubas that would be on it (the previous number x 2) in google sheets all the way up to 64 squares. The amount of rubas that would be on the last square is 9.223372037+18 (the +18 is the scientific notation because the number would be so large). This was a very lengthy process because I had to physically type out every single square on the computer and I had to keep multiplying every number by 2 on the calculator until I has calculated all 64 squares.

4. Next, I calculated how much the king would have to pay the peasant in total by writing it all out in vertically in google sheets, highlighting it all, and finding out the grand total. This also took a long time because I had to copy and paste every number so that they were listed vertically. This was the only way I could get the sum of it all.

5. Then, I wrote an equation of how the number of rubas increased per square. I had a hard time with this, maybe because I was doing it so late at night, but nothing came to me so my equation ended up being r=2n -1, which makes no sense at all.

6. In addition to the questions above, there were some follow-up questions that asked about exact locations of which number is on which square so it was helpful that I wrote every number down on the google sheet because these questions were quick to answer.

7. The last part of this project was to create a graph with the values of the rubas from squares 1-10. The x-axis had the number of rubas and the y-axis had the square number. The graph showed the curve of how the number of rubas would increase.

In the future, I can use these skills to encourage myself to keep working on something because it will all pay off in the end. I can use the skill of persevering with logic in any area of life, which is why it is such a good skill to learn. I now know that you can’t just keep trying with no thought, you have to step back and use logic to figure something out.

6. In addition to the questions above, there were some follow-up questions that asked about exact locations of which number is on which square so it was helpful that I wrote every number down on the google sheet because these questions were quick to answer.

7. The last part of this project was to create a graph with the values of the rubas from squares 1-10. The x-axis had the number of rubas and the y-axis had the square number. The graph showed the curve of how the number of rubas would increase.

In the future, I can use these skills to encourage myself to keep working on something because it will all pay off in the end. I can use the skill of persevering with logic in any area of life, which is why it is such a good skill to learn. I now know that you can’t just keep trying with no thought, you have to step back and use logic to figure something out.

During this project the math skill that I was working on was persevering with logic and recognizing patterns. I know that I’ve learned this skill because there was a lot of work I had to do to find the answers to some questions and I just kept going because I knew I would get it. If I found something challenging, I would keep trying to figure it out, then, if I couldn’t, ask a teacher.

At the end of this project, I had accomplished a fully complete analyzation of the peasant’s plan. My group members had gone through the same process I did for their plans (queen, king, and financial advisor) and we compared data to see which plan would make the king give the least amount to the peasant. Unfortunately, one of my groupmates did not complete their questions, so we couldn’t really have a final answer.

At the end of this project, I had accomplished a fully complete analyzation of the peasant’s plan. My group members had gone through the same process I did for their plans (queen, king, and financial advisor) and we compared data to see which plan would make the king give the least amount to the peasant. Unfortunately, one of my groupmates did not complete their questions, so we couldn’t really have a final answer.

## How portfolio item represents SUSTAINABILITY skill

The sustainability skill that represents the Four Plans project is “Reasoning Analytically” with a succeeding because I broke the problem down into parts that I could answer. For example, before I did anything I calculated the amount of rubas on every single square, so answering the questions would be easier. This also helped with explaining my answers because I really understood the pattern of how the rubas increased. I based my judgements of where a certain number of rubas was or which square had the first scientific notation on the actual data that I put together on the google sheets.

I took the time to do the calculations and it shows in my answers to the problem. I faced the challenge of finding out every single amount of rubas on each square with reason because I figured out some shortcuts to help me along the way. For example, I learned that if I listed all the amounts of rubas vertically on google sheets, I could highlight it all and get the sum. This helped me because I didn’t have to types everything out in a calculator. I looked back on my data for help on the questions, so it was really worth the time and effort of doing it all. For example, one of the questions was ‘What is the first square on which the King will place at least one million rubas?’. This would have been hard for someone who didn’t do the calculations, but I could just look at my data and answer the question easily. The answer is Square 21.

This project helped deepen my understanding of how to “reason analytically” because I now realize that the easiest way to deal with a problem is to break it down into parts you can solve. It is important you do that so you can get a full picture of what you have to do in order to figure out the problem. Also, you can’t just guess, you have to look at evidence or data you have and make judgments from there.

## Iteration & Personal growth

I feel like I had some big iterations in the process of completing this project. I had a bit of trouble in the beginning with figuring out how to calculate all the answers, so I just went with typing everything into a calculator. This takes a very long time, and I did go through with it, but I realized that I could have just put everything into google sheets and let it calculate it for me. This saved me a lot of time in the end. Another iteration I had was some of the answers to the questions. When we first got this assignment, I was not that interested and put down some random answers for the questions on the slides. I ended up changing those answers because I figured this project was worth a big chunk of my math grade. I did all the calculations and got my answers.

Some growth that I went through in this process was really understanding what it meant to “persevere with logic”. The calculations we did took a super long time to complete and I just kept going because I knew that I would eventually get the answer and be done with the project. To persevere with logic is to not keep on trying on a problem, but keep on trying with a new idea to complete it each time. You have to think about how you are going to do something before you do it, and completing challenging problems requires a lot of trial and error.

In the future, I can use these skills to encourage myself to keep working on something because it will all pay off in the end. I can use the skill of persevering with logic in any area of life, which is why it is such a good skill to learn. I now know that you can’t just keep trying with no thought, you have to step back and use logic to figure something out.

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