Meadows or Malls?
We were given the task to split up Durango’s land so that all the citizens were satisfied. We were told that there were 3 sections of land, Dalton Ranch, The Old Fort, and Boston Mine. These sections of land needed to be split between recreational and developmental, not equally but it had to satisfy the people. We assigned each section and type their own set of letters. These are called variables. The variables are a letter given to a number that can and most likely will change. Variables are very important for carrying on our problem and creating constraints.
Parcel
Improvement costs per acre for recreation
Improvement costs per acre for development
Dalton Ranch
$50
$500
The Old Fort
$200
$2,000
The Boston Mine
$100
$1,000
We were told that there were 550 acres for Durango to use in total and at least 300 of those had to go to development. At most 200 acres of the army base and mining land could go for recreation. The amount of army base land used for recreation and the amount of Dalton ranch land used for development together had to total exactly 100 acres. These sentences are called constraints. We can turn them into just numbers and mathematical notations to see what exactly restrictions and limitations our problem has. We write our constraints because it quantifies the range of values that'll fit the situation. Constraints give us a way to relate the variables to each other and they Help us define what the feasible region is/ will be.
After finding the corner points and feasible region using the constraints and matrices we checked the corner points with our constraints. The corner points have to follow the constraint. For example if you have a constraint the says X has to be less than 7 and Y had to be more than 4 (X<7, Y>4) but you got a corner point (9,3) that does not match your constraints so you know it is incorrect.
Using all of that information we came to the conclusion that the best possible solution would be 225 acres goes to ranch recreation, 75 acres to ranch development, 25 acres to army recreation, 75 acres to army development, 0 acres to mining recreation, and 150 acres to mining development which would come out to about $353,740.
Reflection
Throughout my junior year I have noticed that I don’t enjoy working collaboratively as much as I used to but this didn’t necessarily affect my skills to do so. Growing up I alway loved working with my peers but through my junior year I progressively started liking working on my own more. Although I don’t enjoy it as much as before I am still very able to work with group members and I still always ask what I can do to help out the team.
Covid did not directly affect me during this unit but it did help me grow as a student in the previous years. This year we were in person everyday and I was able to get the help I needed when I needed it but last year was different. We did lots of online school and I wasn’t able to talk to my teachers anytime I had a question. That experience has taught me to take every chance I get to ask questions and not wait, I just learned to not like time for granted. This did affect my success in Precalculus as a whole in a good way, I advocated for myself more and I got lots of confusion cleared up but it did not affect this project specifically.
I thought this unit content was very difficult. I would do the work to the best of my abilities, ask questions, and pay attention but the next day I would be just as confused as I was on the first. It felt like I was taking 1 step forward, 2 steps back. What I learned in this unit does not feel applicable to my life. The situation does, I might be looking to sell or buy land one day but I can’t imagine when I would ever use the math. The only thing I'm curious about this unit is who, how and why did someone come up with these ways of solving.
- RR is the number of acres of ranch land to be used for recreation
- RD is the number of acres of ranch land to be used for development
- AR is the number of acres of army land to be used for recreation
- AD is the number of acres of army land to be used for development
- MR is the number of acres of mining land to be used for recreation
- MD is the number of acres of mining land to be used for development
Parcel
Improvement costs per acre for recreation
Improvement costs per acre for development
Dalton Ranch
$50
$500
The Old Fort
$200
$2,000
The Boston Mine
$100
$1,000
We were told that there were 550 acres for Durango to use in total and at least 300 of those had to go to development. At most 200 acres of the army base and mining land could go for recreation. The amount of army base land used for recreation and the amount of Dalton ranch land used for development together had to total exactly 100 acres. These sentences are called constraints. We can turn them into just numbers and mathematical notations to see what exactly restrictions and limitations our problem has. We write our constraints because it quantifies the range of values that'll fit the situation. Constraints give us a way to relate the variables to each other and they Help us define what the feasible region is/ will be.
- RR + RD = 300
- AR + AD = 100
- MR + MD = 150
- RD + AD + MD ≥ 300
- AR + MR ≤ 200
- AR + RD = 100
- RR ≥ 0
- AR ≥ 0
- MR ≥ 0
- RD ≥ 0
- AD ≥ 0
- MD ≥ 0
After finding the corner points and feasible region using the constraints and matrices we checked the corner points with our constraints. The corner points have to follow the constraint. For example if you have a constraint the says X has to be less than 7 and Y had to be more than 4 (X<7, Y>4) but you got a corner point (9,3) that does not match your constraints so you know it is incorrect.
Using all of that information we came to the conclusion that the best possible solution would be 225 acres goes to ranch recreation, 75 acres to ranch development, 25 acres to army recreation, 75 acres to army development, 0 acres to mining recreation, and 150 acres to mining development which would come out to about $353,740.
Reflection
Throughout my junior year I have noticed that I don’t enjoy working collaboratively as much as I used to but this didn’t necessarily affect my skills to do so. Growing up I alway loved working with my peers but through my junior year I progressively started liking working on my own more. Although I don’t enjoy it as much as before I am still very able to work with group members and I still always ask what I can do to help out the team.
Covid did not directly affect me during this unit but it did help me grow as a student in the previous years. This year we were in person everyday and I was able to get the help I needed when I needed it but last year was different. We did lots of online school and I wasn’t able to talk to my teachers anytime I had a question. That experience has taught me to take every chance I get to ask questions and not wait, I just learned to not like time for granted. This did affect my success in Precalculus as a whole in a good way, I advocated for myself more and I got lots of confusion cleared up but it did not affect this project specifically.
I thought this unit content was very difficult. I would do the work to the best of my abilities, ask questions, and pay attention but the next day I would be just as confused as I was on the first. It felt like I was taking 1 step forward, 2 steps back. What I learned in this unit does not feel applicable to my life. The situation does, I might be looking to sell or buy land one day but I can’t imagine when I would ever use the math. The only thing I'm curious about this unit is who, how and why did someone come up with these ways of solving.
Planning the Platforms
Durango is hosting an annual 4th of July party. There will be baton twirlers on platforms passing batons up and down to each other. The event planner, Kevin, wants the difference from one platform to the next to be the same between all platforms. Kevin has lots of decisions to make.
My task was to create two formulas that I can give to Camila so she can figure out the numbers she needs as soon as Kevin makes up his mind. One formula needs to tell the height of the tallest platform and the second needs to tell you the length of material Camila needs to buy. The formulas need to give these results in terms of the number of platforms, the height of the first platform, and the difference in height between adjacent platforms.
To find out the first formula, the height of the tallest platform, I used the variables P, B, N, M, X.
Like all the other POWs I’ve done, this was very difficult. The math itself was very hard, especially with substitute teachers he two days we had class time for it but I did get help from other students. The write up wasn’t the hardest part but it was difficult trying to explain complex math.
- He needs to decide on the number of platforms. Kevin isn’t sure how many of his baton twirlers will be good enough to perform by the Fourth of July.
- He needs to decide on the height of the first platform. This will depend on how tall the baton twirler on the first platform is, and Kevin hasn’t decided who the first baton twirler will be.
- He needs to decide on the difference in height from one platform to the next. Kevin doesn’t know yet how high the twirlers will be able to toss their batons
My task was to create two formulas that I can give to Camila so she can figure out the numbers she needs as soon as Kevin makes up his mind. One formula needs to tell the height of the tallest platform and the second needs to tell you the length of material Camila needs to buy. The formulas need to give these results in terms of the number of platforms, the height of the first platform, and the difference in height between adjacent platforms.
To find out the first formula, the height of the tallest platform, I used the variables P, B, N, M, X.
- P = the height of the tallest platform
- B = the height of the initial platform
- N = the placement of the platform (1st, 2nd, 3rd …)
- M = the height that is added
- X = the total number of pillars
Like all the other POWs I’ve done, this was very difficult. The math itself was very hard, especially with substitute teachers he two days we had class time for it but I did get help from other students. The write up wasn’t the hardest part but it was difficult trying to explain complex math.
Precalculus: A Year in Review
I grew as a mathematician in many ways. I saw the most growth in communicating thinking in a clear and accessible way and recognizing and resolving errors. I have never been the best as trying to explain my thinking. If I have a question about something I don't fully understand it is hard for me to ask the question in a clear way. It is hard for me to ask the question in a way that someone can understand and answer, through this year I got better at that. Through being confused and asking a lot of questions, Julian taught me how to organizing my thinking into an askable question. We did lots of classwork in the form of worksheets this year. A day or 2 after the worksheet was due Julian would usually go over it as a class to clear up any confusion. Watching him go through problems I previously did, I learned how to recognize and resolve my errors.
During this year in pre calc I also improved on my advocacy and perspective. Being in a very welcoming learning environment such as Julian's classroom I learned to advocate for myself and my learning. I got better at asking questions when I needed to and take responsibility when I didn't finish and assignment or pay attention. All of the classwork we did in this class was based off of a real world problem like the ones you see above. We learned math through situations that different characters encountered in their "real" life. Being taught math this way helped me have a wider perspective on it. I never saw the importance in more advanced math because "when would I actually use this". Learning through real life situations helped me see a different perspective in which this kind of math is useful.
Through out this year, especially in math, I learned I need to become more motivated, whether I am interested or not. It was very easy for me to leave class when I didn't want to be there and completely miss out on lots of learning. All of my absences and confusion came from me simply not wanting to be there. This led me to struggle in class and it made my grade decline. I learned that I need to be there and be invested even when I don't want to be because the consequences will affect me, even if it looks like it in the moment or not.
During this year in pre calc I also improved on my advocacy and perspective. Being in a very welcoming learning environment such as Julian's classroom I learned to advocate for myself and my learning. I got better at asking questions when I needed to and take responsibility when I didn't finish and assignment or pay attention. All of the classwork we did in this class was based off of a real world problem like the ones you see above. We learned math through situations that different characters encountered in their "real" life. Being taught math this way helped me have a wider perspective on it. I never saw the importance in more advanced math because "when would I actually use this". Learning through real life situations helped me see a different perspective in which this kind of math is useful.
Through out this year, especially in math, I learned I need to become more motivated, whether I am interested or not. It was very easy for me to leave class when I didn't want to be there and completely miss out on lots of learning. All of my absences and confusion came from me simply not wanting to be there. This led me to struggle in class and it made my grade decline. I learned that I need to be there and be invested even when I don't want to be because the consequences will affect me, even if it looks like it in the moment or not.