Homework 15
Portfolio Selection
Homework 15
In problems such as: homework 8 picturing pictures, homework 10 you are what you eat, rock n rap and homework 13 the big U, there are three main steps that a student must go through in order to solve such a problem. First, the student must explore the problem. This could mean anything. The student could write down the known information, create equations, guess and check or create an algorithm. These types of problems have a lot of depth to them and at a first glance the student may not be aware of everything necessary to solve the problem. This first step is to help the student feel comfortable with the problem and to possible discover parts or details of it that they didn’t see at first. The second step is to begin to work the problem out. In this step, the student will follow any leading thoughts or hypothesis formed from the exploration phase. This step includes solving equations, graphing them, and then finding a feasible region. All of the listed problems were based in linear programing. This second step is where the students will actually be computing and dealing with this subject specifically, as well as learning the basic rules and process of a linear programing problem. After all work is completed in this phase the student should arrive at an answer that they think is correct for the given problem. Finally, the last step is to prove that their answer that they arrived at in the working phase. Just arriving at the supposed answer is not enough. This last phase requires the students to mathematically justify why their answer is correct, allowing for a much deeper understanding of linear programing.
The following three activities helped me to formulate these three steps to solving such linear equation problems as well as gave me an opportunity to practice using the three phases.
1. Profitable Pictures
While working through the problem profitable pictures, I came to realize the importance of the last phase of proving that my answer was correct. When I worked out the problem, I arrived at an answer to maximize the amount of watercolor and pastel profit, but then upon discussion with my peers, I noticed that my proof was not mathematically sound. I was challenged to come up with a better defense of my solution and it drove me to work harder to try to justify my answer. Because I was so driven, I ended up coming up with a proof that I never thought I was capable of. This helped me realize how important of a step this is for students learning linear programing. The students will have a much deeper understanding of the concepts and details of the problem if they are asked to prove their answers because it will, in a sense, force them to look deeper into the details of the problem. The students will be exploring properties of linear programing that with a much higher cognitive demand.
2. Homework 8: Picturing Pictures
This problem will help students understand what they should be doing during the exploration phase. The three questions asked in the problem first ask the student to create equations based off the constraints, graph them, and then plot points for the profit. This sets a great example for students to think about when working through any type of problem similar to this is linear programing. I especially enjoyed this problem because it was able to walk the students through the proper way to explore, and at the same time allow for student self-discovery by keeping the questions intentionally brief.
3. Rock ‘n’ Rap
This linear programing problem will help students learn to think outside the box when proving their statements or thinking of ways to extend a given problem. The last question tells the students to suppose the constraints for profit were reversed, and to then see if this change would affect the amount of rock and rap CD’s to make. This problem helped me understand how to take what I know, and then move forward and to dive deeper into a problem. It prompted me to think in a different way about the same problem. My mind went straight to the graph and how the change would affect the lines that I graphed. Seeing how the lines previously change or do not change, will help strengthen the student’s understanding a linear programing graph. The question also asked for an explanation of the answer which will help students with being able to mathematically prove their answers.
In problems such as: homework 8 picturing pictures, homework 10 you are what you eat, rock n rap and homework 13 the big U, there are three main steps that a student must go through in order to solve such a problem. First, the student must explore the problem. This could mean anything. The student could write down the known information, create equations, guess and check or create an algorithm. These types of problems have a lot of depth to them and at a first glance the student may not be aware of everything necessary to solve the problem. This first step is to help the student feel comfortable with the problem and to possible discover parts or details of it that they didn’t see at first. The second step is to begin to work the problem out. In this step, the student will follow any leading thoughts or hypothesis formed from the exploration phase. This step includes solving equations, graphing them, and then finding a feasible region. All of the listed problems were based in linear programing. This second step is where the students will actually be computing and dealing with this subject specifically, as well as learning the basic rules and process of a linear programing problem. After all work is completed in this phase the student should arrive at an answer that they think is correct for the given problem. Finally, the last step is to prove that their answer that they arrived at in the working phase. Just arriving at the supposed answer is not enough. This last phase requires the students to mathematically justify why their answer is correct, allowing for a much deeper understanding of linear programing.
The following three activities helped me to formulate these three steps to solving such linear equation problems as well as gave me an opportunity to practice using the three phases.
1. Profitable Pictures
While working through the problem profitable pictures, I came to realize the importance of the last phase of proving that my answer was correct. When I worked out the problem, I arrived at an answer to maximize the amount of watercolor and pastel profit, but then upon discussion with my peers, I noticed that my proof was not mathematically sound. I was challenged to come up with a better defense of my solution and it drove me to work harder to try to justify my answer. Because I was so driven, I ended up coming up with a proof that I never thought I was capable of. This helped me realize how important of a step this is for students learning linear programing. The students will have a much deeper understanding of the concepts and details of the problem if they are asked to prove their answers because it will, in a sense, force them to look deeper into the details of the problem. The students will be exploring properties of linear programing that with a much higher cognitive demand.
2. Homework 8: Picturing Pictures
This problem will help students understand what they should be doing during the exploration phase. The three questions asked in the problem first ask the student to create equations based off the constraints, graph them, and then plot points for the profit. This sets a great example for students to think about when working through any type of problem similar to this is linear programing. I especially enjoyed this problem because it was able to walk the students through the proper way to explore, and at the same time allow for student self-discovery by keeping the questions intentionally brief.
3. Rock ‘n’ Rap
This linear programing problem will help students learn to think outside the box when proving their statements or thinking of ways to extend a given problem. The last question tells the students to suppose the constraints for profit were reversed, and to then see if this change would affect the amount of rock and rap CD’s to make. This problem helped me understand how to take what I know, and then move forward and to dive deeper into a problem. It prompted me to think in a different way about the same problem. My mind went straight to the graph and how the change would affect the lines that I graphed. Seeing how the lines previously change or do not change, will help strengthen the student’s understanding a linear programing graph. The question also asked for an explanation of the answer which will help students with being able to mathematically prove their answers.