MEADOWS and Malls Pre calc project
Write up and reflection
Meadows or Malls:, Writeup, and Reflection
During this project we started off by refreshing our knowledge by looking at two variable problems and studying in which the problems intersect. When looking at multiple variables and problems, two or more of them will intersect at a corner point. At this corner point all of the constraints will be met and the problem will be satisfied. These corner points will lay upon the edges of the feasible region. A feasible region is an area on a graph where all of the requirements or constraints will be met. There are infinite points that lay on the feasible region, but the ones that lay on the corner points will be the best solutions, in terms of the final goal. When we have more than two variables we will have a graph that has more than two dimensions. For example, a problem with three variables will be three dimensional in which the third variable will be represented by the third dimension, as we know it is as the z plane. After practicing we applied this knowledge to four dimensions and more, since the unit problem had many dimensions, or variables. After playing around with the fourth dimension and trying to solve a four variable problem we decided it would be an easier option to use algebra instead of graphing.
After we explored corner points and feasible regions, we solved the inequalities by using the substitution method instead of graphing. We would line up the inequalities below and above each other so that the x y and z variables were on top of each other. This makes solving the problems without a graph possible. This method provided us with the point in which all the variables intercept and the values of each variable that would be most ‘profitable’.
Since our unit problem had way too many variables to solve on top of each other using the substitution method, we needed to find a new way to solve the problem without doing it manually. Julian taught us about Matrices. A matrix is a way to organize values in a way that is easier to solve than a table. After playing around with small matricules such as ones that were 2 x 2 values wide and ones that were 5 x 5 values wide, we felt ready to apply the matrices to our unit problem. We added each value of the constraint into a matrix. Each row represented one constraint and one row represented one variable. We plugged in the numbers, and counted how many times the variable appeared in each constraint. Writing either a 1 or a 0 in each slot of the matrix. Then after doing that we made a new matrix with the solution to each constraint. This matrix would be used to find the inverse of itself. The inverse will tell us what number to multiply matrix B ( one with the solutions) to get matrix A ( matrix with 1s or 0s). These new numbers will tell us what numbers will satisfy each combination of constraints. After doing so we only needed to do some simple math in order to find which would be most profitable. We tested each solution and took out each one that didn’t satisfy every constraint. That left us with only three options. We applied basic math to find the cost of each three solutions.
I believe that COVID has been the reason I’ve improved so drastically in my communication skills. I also feel way more observant of people in real life situations. I can analyze body language and tone of voice better now that I’ve spent so much time focusing on everything except for their face. This has helped me to complete the project better because I feel more empathy for my peers and group mates and want to complete things more thoroughly. I also feel more connected to my community.
During this project we started off by refreshing our knowledge by looking at two variable problems and studying in which the problems intersect. When looking at multiple variables and problems, two or more of them will intersect at a corner point. At this corner point all of the constraints will be met and the problem will be satisfied. These corner points will lay upon the edges of the feasible region. A feasible region is an area on a graph where all of the requirements or constraints will be met. There are infinite points that lay on the feasible region, but the ones that lay on the corner points will be the best solutions, in terms of the final goal. When we have more than two variables we will have a graph that has more than two dimensions. For example, a problem with three variables will be three dimensional in which the third variable will be represented by the third dimension, as we know it is as the z plane. After practicing we applied this knowledge to four dimensions and more, since the unit problem had many dimensions, or variables. After playing around with the fourth dimension and trying to solve a four variable problem we decided it would be an easier option to use algebra instead of graphing.
After we explored corner points and feasible regions, we solved the inequalities by using the substitution method instead of graphing. We would line up the inequalities below and above each other so that the x y and z variables were on top of each other. This makes solving the problems without a graph possible. This method provided us with the point in which all the variables intercept and the values of each variable that would be most ‘profitable’.
Since our unit problem had way too many variables to solve on top of each other using the substitution method, we needed to find a new way to solve the problem without doing it manually. Julian taught us about Matrices. A matrix is a way to organize values in a way that is easier to solve than a table. After playing around with small matricules such as ones that were 2 x 2 values wide and ones that were 5 x 5 values wide, we felt ready to apply the matrices to our unit problem. We added each value of the constraint into a matrix. Each row represented one constraint and one row represented one variable. We plugged in the numbers, and counted how many times the variable appeared in each constraint. Writing either a 1 or a 0 in each slot of the matrix. Then after doing that we made a new matrix with the solution to each constraint. This matrix would be used to find the inverse of itself. The inverse will tell us what number to multiply matrix B ( one with the solutions) to get matrix A ( matrix with 1s or 0s). These new numbers will tell us what numbers will satisfy each combination of constraints. After doing so we only needed to do some simple math in order to find which would be most profitable. We tested each solution and took out each one that didn’t satisfy every constraint. That left us with only three options. We applied basic math to find the cost of each three solutions.
- What growth have you noticed so far during your junior year in your collaborative skills, and how did these changes affect your ability to work with your group members?
- How has your experience with COVID during this unit helped you to grow as a student and mathematician, and how did it affect your ability to be successful in Precalculus and in completing this project?
I believe that COVID has been the reason I’ve improved so drastically in my communication skills. I also feel way more observant of people in real life situations. I can analyze body language and tone of voice better now that I’ve spent so much time focusing on everything except for their face. This has helped me to complete the project better because I feel more empathy for my peers and group mates and want to complete things more thoroughly. I also feel more connected to my community.