Habits of a Mathematician
This year has felt incredibly long and honestly, pretty stressful. That being said, pre-calc was a difficult class for me. Not because I don’t know how to do the math well, but because it didn’t feel as important as my other classes. Since pre-calcs’ content was extremely easy for me, I never prioritized it as a grade.
After finishing the year off with a less than ideal grade in math, I realized how much self-discipline I lost throughout the pandemic. I lost my ability to persevere through hardship and boredom which I regret.
Although my perseverance lacked, I never forgot how to advocate for myself and I am really glad that I did. I told my teacher how I was feeling and why I wasn’t meeting certain expectations, and he helped me set up a plan to raise my grade.
On the other hand, I grew as an osprey and individual by learning how to collect more evidence. This past year I have made a point to journal and take a lot more notes than I normally do. I am really glad that I chose to do that because it made studying a whole lot easier and more efficient. Taking notes by hand also proved to aid me in retaining information better.
In addition, I feel that my refinement has greatly improved. My attention to detail has really transformed the quality of work that I turn in to every class. I am glad to see my refining skills develop because now I feel genuinely proud of my work and I also get better grades.
As far as absences and participation go, I believe that I have consistently remained exceptional, especially in participation. I have always enjoyed classwork in math as well as collaborating with my peers.
After finishing the year off with a less than ideal grade in math, I realized how much self-discipline I lost throughout the pandemic. I lost my ability to persevere through hardship and boredom which I regret.
Although my perseverance lacked, I never forgot how to advocate for myself and I am really glad that I did. I told my teacher how I was feeling and why I wasn’t meeting certain expectations, and he helped me set up a plan to raise my grade.
On the other hand, I grew as an osprey and individual by learning how to collect more evidence. This past year I have made a point to journal and take a lot more notes than I normally do. I am really glad that I chose to do that because it made studying a whole lot easier and more efficient. Taking notes by hand also proved to aid me in retaining information better.
In addition, I feel that my refinement has greatly improved. My attention to detail has really transformed the quality of work that I turn in to every class. I am glad to see my refining skills develop because now I feel genuinely proud of my work and I also get better grades.
As far as absences and participation go, I believe that I have consistently remained exceptional, especially in participation. I have always enjoyed classwork in math as well as collaborating with my peers.
POW 4: Planning the Platforms
Durango’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’s Dilemma 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. Kevins’ problem: What is Px ( if Px represents the height of the tallest pillar)? As you can see below the initial height of the first pillar is “a”, and every pillar after that increases by m. I decided to use these variables:
The reason {Pn= b+m( n-1)} works for every pillar is because no matter what pillars’ height you are calculating, it will always be the initial height plus the product of m and the identity minus 1. (E.g. the 5th pillar will only add 4m to b because b is the initial height.) Camilles’ Problem: Now that I have a formula for the heights of the pillars, I need to figure out the simplest and most efficient way to add them all together so I know how much fabric Camille should buy. Here are my variables:
P1+P2+P3 . . . +Px represents the total height of all the pillars, which is equivalent to the total length of fabric camille will need. 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 (if F = the total amount of fabric camille needs). After some collaboration with Victor we came up with the [closed form] F= (2b+m(2n-1))/2 After viewing Julian’s work I realized it would be easier to simplify this equation into summation notation. Summation Notation (also known as Sigma Notation) was created by mathematicians who wanted to describe long additions with a pattern more clearly, as opposed to writing out equations such as {1+2+3+4…+9+10}. It looks something like this: In this case, the “i” on the bottom represents the starting value or initial height. The variable“n” represents the final value, and “ai” represents the formula for each term. That means our notation should look something like this: Summation notation, although helpful, is an open form which means it’s tricky to use when “n” becomes a big number. So our final solution should just be the closed form; F= (2b+m(2n-1))/2 Solution: Kevin (a.k.a. me) came up with the perfect mathematical formula for his problem! If “b” = the height of the initial pillar, “n” = the pillar identity (first pillar, second pillar, third pillar, etc.), and “m” = the added height, then the height of the tallest pillar must equal Pn= b +m( n-1). If you dissect this equation, what it is saying in “real-world terms” is that the nth pillar is equal to the first pillars’ height, plus the added height times n minus 1. Camille (a.k.a me 2.0) also devised a strategy for measuring her lengths of fabric! If b = initial height, m = added height, n = the pillar identity, and F = total length of fabric, then the total length (F) of fabric she needs must equal F= (2b+m(2n-1))/2. In real world terms, the total length of fabric is equal to 2 times the initial height plus the added height times double the identity minus one, all over 2. I know that is confusing, but I literally can’t explain it any other way. It’s math, pretty simple. Reflection: I actually found myself pretty excited for this POW because it was such a fun problem to solve. The fact that it only consisted of variables was super fun because that meant it could be solved geometrically quite easily. I struggled in the beginning with knowing exactly where to start, but our sub (as well as Victor, the honorary T.A.) was verry helpful! She proved to be talented in demonstrating her knowledge while still allowing me to do my own authentic work. She even helped me with SAT practice which seemed to perfectly fit into this POW. After completing this POW I have realized that I am more capable in certain areas than I previously thought. For example, I always had the belief that I was utterly terrible at generating ideas. I realized that wasn’t true when we began coming up with our own variables. Victor made me realize I am perfectly capable of identifying what needs to be included in the problem and what doesn’t. I hope that I can continue to find the same enthusiasm and motivation for problems like this in the future, because it proved to be very eye-opening. Extension Question: How would one write a formula for pillars that increase as well as decrease linearly? |