Course Reflection
I came into this class not likening math and being very confused by any math in general. I cam out of this class feeling the exact same way if not way more confused. Due to being very lost and confused all the time my skills of communicating my thinking in a clear and accessible way as well as reflecting and synthesizing have became stronger. At the beginning of the yea I struggled with communicating where exactly I was confused and it helped my teacher and peers support me better. I also became very good at reflecting where I can improve and what the next steps I can take to be better next time are. I grew as an osprey through perseverance. A lot of days it was really hard to want to come to class because it felt like I was the only one who didn't understand what I was being taught. I had to persevere through discouragement to advocate for myself and my learning. It helped me become more independent and confident in my own abilities. I now know, for a fact, I hate math. But I also now know that sometimes you have to do things you don't like very much to succeed in life and that it's ok to ask for help.
Meadows or Malls?In this past unit, Meadows and Malls, we had to learn a lot of different things ranging from linear programming, to using corner points and matrices to find a given value. In the final unit problem we had to figure out the best way to distribute land so both the recreationally interested folk and developmentally interested people were satisfied. We were given three different pieces of land and we were told that different portions of the land had to be split up by recreational land and develop land. Naturally, the people of Durango disagreed on the way the land should be split up. In the chart below you can see the allocated land given and the cost it takes to build recreation or development land.
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POW 4Durango’s annual Fourth of July concert is going to be crazier than ever this year. The planner, Kevin, wants each of the baton twirlers to be standing on their own platform as shown below. Each pillar must be taller than the last one, with each height increment the same. What formula can we use to calculate the initial height and every pillar after’s height? What about the very last pillar?
Next, Camilla, the head decorator, wants to put a banner on the front of every pillar. Luckily, the rolls of fabric she will be using are all the same width as the pillars. She just needs to know the total height of each pillar so she can buy the right length of fabric. The Platform Display: Durango is getting ready for the big Fourth of July band concert that precedes the fireworks. The concert is always a major event, but this year the band leader, Kevin, plans to make it bigger and better than ever. Kevin wants to have each of the baton twirlers standing on an individual platform, as shown here to the right. The baton twirlers will toss batons up and down to one another. Kevin wants the difference in height from one platform to the next to be the same in each case. Kevin’s Decisions: Kevin has several decisions to make.
Camilla is in charge of building and decorating the structure. She needs a permit from the city to build the structure, so she needs to know how high the tallest platform will be. She also plans to hang a colorful strip of material on the front of each platform. Each strip will reach from the top of the platform to the ground. The width of the material is the same as the width of each platform, so she needs only one strip per platform. She needs to know the total length of material that she should buy. Camilla is going crazy because she can’t do her job until Kevin makes his decisions. Your Task: You are Camilla’s assistant, and she has asked you to be ready to give her the information she needs as soon as Kevin makes up his mind. Your task in this POW is to create two formulas that will allow you to do this instantly. One formula should tell you the height of the tallest platform. The other should tell you the total length of material that Camilla will need. Your formulas should give these results in terms of the number of platforms, the height of the first platform, and the difference in height between adjacent platforms. Kevin's Problem: If Px represents the height of the tallest pillar, what is Px ? The starting height of the first pillar is represented by “a,” and the increasing height of every pillar after that is represented by “m.” Variables:
The reason {b+m( n-1)=Pn} works for all the pillars is because regardless of which pillars’ height you are trying to solve, it will always be the height plus the product of m and the identity minus 1. (Ex, pillar 5 will only add 4m to b because b is the starting height.) Camilles’ Problem: Now that we figured out a formula for the heights of each pillar, we need to figure out the easiest and least complicated way to add all of them together in order to know how much fabric Camille should purchase. Variables:
If P1= b+m(1-1), P2=b+m(2-1), & P3 =b+m(3-1), then (b=m(1-1)) + (b+m(2-1))+ . . . +(b+m(x-m))= F (F = the total amount of fabric Camille needs). After figuring this out I realized it would be easier to simplify this equation into a summation notation. Summation Notation, AKA Sigma Notation, was originated by mathematicians who needed to describe long additions with a pattern that's more simple, as opposed to writing out equations such as {12+34+9…+7+10}. It should look like this: In this scenario, “i” on the bottom represents the starting value or initial height. “n” represents the final value, and “ai” represents the formula for each term. This means our notation should look something like this: Summation notation can get complicated because it's open form so “n” becomes a big number. So the final solution we should come to should be the closed form: (2b+m(2n-1))/2=F Solution: After all this, we can finally figure out the answer to Kevin’s issue. If the height of the initial pillar is “b”, the given pillar (pillar 1, pillar 2, etc.) is “n”, and the increasing height is “m”, then the height of the tallest pillar has to be equal b +m( n-1)=Pn . In simpler terms, the nth pillar is equal to the first pillars’ height, plus the added height times n minus 1. Now, for Camille. In her struggles to measure her lengths of fabric, we were able to find a solution. If the starting height is “b”, the increasing height is “m”, the given pillar is “n”, and total length of fabric is “f”, then the total length of fabric needed must equal (2b+m(2n-1))/2=f. In simpler terms, the total length of fabric is equivalent to 2 times the starting height, plus the added height, multiplied by double the given pillar, minus one, over 2. Reflection: I wasn’t too excited about this POW because it was just more math work to do. It was also challenging having a sub but she explained it in a good way that honestly helped me understand pretty well. It was pretty easy for the most part because it pretty much only consisted of variables. That just makes it a lot more simple and can be solved geometrically. I was definitely struggling where to start but our sub and my peers were able to offer a lot of help in my understanding of this problem. She seemed like she definitely knew what she was doing and it made me feel better about Julian not being here to help. After finishing this POW I have realized that I am slightly more capable in some aspects of math than I thought and I'm starting to realize I can reach out for help from more people than I thought. After the sub helped me I was still confused about the Summation Notation but after asking my peers, Skye and Harper for help they really helped me see what I was doing wrong and it made me more confident in my ability to advocate for myself. I hope in the future I can help myself by advocating and figuring out the right questions to ask and how to ask them. |