The Quintessence of Quadratics
Introduction:
Launching into a new project, we began our journey into our quadratic unit by revisiting kinematic equations and valuables such as acceleration and distance. We used the online program "desmos" as a tool to visualize the shape, size, and location of the parabolas, which came in handy as we moved on to rewriting quadratic expressions. The objectives of the project were simple: to not only understand quadratic functions but to understand their representations and how to use our set of skills to solve quadratic equations. Through the following weeks we worked with topics that ranged from Pythagorean theorem to projectile motion. The main focus of the unit was around a problem that revolved around a rocket being launched for a firework show.
Exploring vertex form and quadratic equations:
When it came to understanding variables such as A,K, and H, Desmos let students explore quadratics in a more visual way. When we were given handouts where we had to figure out how these variables related to the parabola. Paired with the online program, we discovered that when you piece together these values you create a distance equation using geometry and graphing. From there we were able to see the relations like how adding the A value to an equation like Y=x^2, we get Y=Ax^2. Just adding another value effects the shape of the parabola since a higher sum of A creates a more narrow curve and a wider curve is caused by a lower sum of A. Like finishing a puzzle piece, the next variable we added was K which is equal to the y-coordinate of the parabola's vertex when put into our equation, and our equation became a little more complex: Y=Ax^2+K. As for H, it is similar to K since it serves the same purpose for the x-coordinate. Finally our quadratic equation in vertex form was complete: Y=A(x-H)^2+K
Other forms of the quadratic equation:
Now that we had a grasp on vertex form, we slowly expanded our knowledge by working through a series of problems alongside our peers that strengthened our skills and introduced us to standard form and factored form. Standard form is when the A, B, and C in the equation can be any value. Due to this, it is easy to identify y- intercept of the correlating parabola. Standard form has a similar set up to vertex form and essentially you replace the A,K, and H variable with A, B, and C: Y=Ax^2+Bx+C. With factored form, you can think of it as an extension of standard form, and makes it easy to see the x-intercepts of the parabola: Y=A(x-B)(x-C).
Converting between forms:
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Area diagrams:
An area diagram can be useful for people to solve quadratic equations since it provides a visual to those who learn by being able to write things out and label the different squares. You can see specifically where each variable should go and makes the step by step process easier and keeps your work organized. When you fill in the square and complete it, it can break down the problem for you before you move on to writing it out.
An area diagram can be useful for people to solve quadratic equations since it provides a visual to those who learn by being able to write things out and label the different squares. You can see specifically where each variable should go and makes the step by step process easier and keeps your work organized. When you fill in the square and complete it, it can break down the problem for you before you move on to writing it out.
Solving problems with quadratic equations:
There are many real world connections that can be made from quadratic equation problems. Specifically, there are three that come to mind from our unit. Over the course of the project, as a class we covered Kinematics, geometry, and economics in the world of mathematics. one problem that is fitting for kinematics would be our handout titled "Another rocket" where the students had the task of solving different values for a rocket that is being shot over a school for a firework display. For Geometry, a handout that comes to mind was "Leslie's flowers", where we were tasked with helping a gardener plan the layout of her garden box, given that she used triangular structures. And for Economics, I would look at the handout titled "widgets" where we were tasked with solving for predicated sales as well as the profit they'd gain if their price was valued at an "x" amount for a company who sold widget gadgets.
Solving a problem:
On of the most memorable problems we covered in this project was the handout titled " How much can they drink?". In this problem we were tasked with helping a farmer find the maximum volume of a water trough as well as if the trough was increased by 120 inches. The handout was laid out as shown below:
On of the most memorable problems we covered in this project was the handout titled " How much can they drink?". In this problem we were tasked with helping a farmer find the maximum volume of a water trough as well as if the trough was increased by 120 inches. The handout was laid out as shown below:
In order to solve this problem, a series of steps were taken:
Question one states that the students must find the volume of the trough given that it's 30'' by 80'' by 5'' as well as how much water it can hold. When multiplied together(LxWxH), we get the volume of 12,000in^3.
The next question asks us to find the volume of two other possible troughs the farmer could make, so I filled in the variables with 80'' by 10'' by 20'' and got 16,000in^3 for the volume and used a 40'' by 60'' by 20'' rectangle to get a volume of 4,800in^3.
Question three proposes that the farmer bended the folded pieces up to have a width of X inches, and we are tasked with finding a formula for the volume of the trough. To do this, I drew a diagram outlining the sides that represented X and wrote an equation using what I knew: V=(40 -2x) *x*80. The (40-2x) represents the width after the two sides are bent up since we don't know the value of X yet. Next I simplified my equation by multiplying the 80 and the X which brought me to V=80x(40-2x). After distributing the 80x I boiled it down to 3,200x-160x^2, which meant that the equation was now in standard form. Now that I rewrote the equation to say
V= -160x^2 + 3,200x, I entered the equation into the desmos graphing tool I got a parabola and moved on to the next question.
Question one states that the students must find the volume of the trough given that it's 30'' by 80'' by 5'' as well as how much water it can hold. When multiplied together(LxWxH), we get the volume of 12,000in^3.
The next question asks us to find the volume of two other possible troughs the farmer could make, so I filled in the variables with 80'' by 10'' by 20'' and got 16,000in^3 for the volume and used a 40'' by 60'' by 20'' rectangle to get a volume of 4,800in^3.
Question three proposes that the farmer bended the folded pieces up to have a width of X inches, and we are tasked with finding a formula for the volume of the trough. To do this, I drew a diagram outlining the sides that represented X and wrote an equation using what I knew: V=(40 -2x) *x*80. The (40-2x) represents the width after the two sides are bent up since we don't know the value of X yet. Next I simplified my equation by multiplying the 80 and the X which brought me to V=80x(40-2x). After distributing the 80x I boiled it down to 3,200x-160x^2, which meant that the equation was now in standard form. Now that I rewrote the equation to say
V= -160x^2 + 3,200x, I entered the equation into the desmos graphing tool I got a parabola and moved on to the next question.
Question four states that we find the value of X that will be the maximum volume. I set up this question by dividing 3,200 by -160^2 which equals -3,200 over 320 which is equivalent to 10 over 1. Next, I took my equation V=-16,000+ 32,000= 16,000 which gave me my side length (10) and my maximum volume (16,000). It could also be written as V=(10, 16000).
Question five wanted the student to find question four would change if the trough length was 120'' instead of 80''. I took my already existing formula V=80x(40-2x) and plugged in my new numbers which gave me the expression V=120x(40-2x), which equals 4,800x-240x^2. using the same method as question three, I got V=240x^2=4,800x.Next I divided 4,800 by -240^2 which equals -4800 over 480, which was also equivalent to 10. This means that there would be no changes to question four if the trough length was 120".
Question five wanted the student to find question four would change if the trough length was 120'' instead of 80''. I took my already existing formula V=80x(40-2x) and plugged in my new numbers which gave me the expression V=120x(40-2x), which equals 4,800x-240x^2. using the same method as question three, I got V=240x^2=4,800x.Next I divided 4,800 by -240^2 which equals -4800 over 480, which was also equivalent to 10. This means that there would be no changes to question four if the trough length was 120".
Reflection:
The quintessence of quadratics was by far the project to challenge me the most, whether that was finding ways to grasp the math concepts or ways to connect it to my own life. If a packet wasn't clicking for me or if I wasn't understanding the material, I never gave up and figured I wouldn't be able to solve for it. I looked to my peers to explain the problems if needed and worked with different groups throughout the weeks to see different ways of solving and ways to approach the problems. Looking back on myself in the beginning of the year, this isn't something I would do, and I've realized my growth in areas such as asking for help and being able to admit I need help. This project has proven to be one that made me think about next years courses and beyond. I recognize that this unit was designed at the end of the year to lay down the base work of eleventh grade maths, and understanding the importance of that I knew this wasn't a project that you could just fly through, but rather one that I needed to work on and invest more time than usual into, since my understanding of the concepts wouldn't just affect my current self, but it would either set me up for early success or failure regarding next years classes. Since we have weekly SAT warm ups in class, I was able to see first hand that quadratic equations don't stop in tenth grade and are a very real part in advanced maths. As always, the habits of a mathematician connect to this project very much, and like most projects they have helped me along the way and aided me in growing as a mathematician throughout this unit.
Looking for patterns: When being introduced to a new problem I always kept this is mind as a way to quickly understand what the problem was asking as well as how to solve. For example, if we were given a problem where we needed to solve for a rocket (which was a common theme throughout this unit) I could look back at my prior work to see if the question was similar or if I had already solved part of it.
starting small: When it came to challenging problems as well as converting from form to form, starting small always helped me in learning step by step how to solve the problem.
being systematic: when one way of solving the problem wasn't working or if I got lost in my work, I always reminded myself to go back each step at a time and look for the tiny mistakes I might have made. getting a second pair of eyes to go over my work also helped.
taking apart and putting back together: In many of the problems were the questions were broken into different parts and A,B,C's, I always tried to take the work I'd done so far and then break it down so I could see what I was doing and what I had to solve for.
conjecturing and testing: If I was struggling in a specific worksheet and I knew that I had an error somewhere in my work, I would simply try something new and if that didn't work, I would try again until I was a little but closer to the answer than I was the first time.
staying organized: I'm naturally an organized person, and I believe that staying organized in math helped me catch my mistakes before I made more mistakes based on that one as well as keep track of all my work and be able to go back and correct my handouts while the class went over corrections.
describing and articulating: when I didn't understand something and needed help from my peers, getting the correct answer wasn't my top priority but rather understanding WHY it was the correct answer and figuring out how I get to that answer. this meant that when I did ask for help I was very specific as to what I was struggling with and what I needed help on.
seeking why and proving: When I did get a concept, I would really value all the practice problems we completed since I could strengthened my skills and perfect what I was still lacking.
being confident, persistent, and patient: Even when I was struggling with a problem I always reminded myself that I had the capability of solving it and simply needed to look at the problem a different way or come back to it with a fresh pair of eyes. In the end this helped me stay focused on the end goal as well as kept me from being discouraged.
collaborating and listening: If I was working in a group, it was a team effort in the sense that everyone was contributing to the group and speaking up if they needed help or for someone to double check their work which I thought was really great since it showcased what it meant to have a math community at HTH.
generalizing: To grasp the problems I kept track of all the terms and definitions from my class notes which I referred to so I could understand the variables and the set up of the problems better.
starting small: When it came to challenging problems as well as converting from form to form, starting small always helped me in learning step by step how to solve the problem.
being systematic: when one way of solving the problem wasn't working or if I got lost in my work, I always reminded myself to go back each step at a time and look for the tiny mistakes I might have made. getting a second pair of eyes to go over my work also helped.
taking apart and putting back together: In many of the problems were the questions were broken into different parts and A,B,C's, I always tried to take the work I'd done so far and then break it down so I could see what I was doing and what I had to solve for.
conjecturing and testing: If I was struggling in a specific worksheet and I knew that I had an error somewhere in my work, I would simply try something new and if that didn't work, I would try again until I was a little but closer to the answer than I was the first time.
staying organized: I'm naturally an organized person, and I believe that staying organized in math helped me catch my mistakes before I made more mistakes based on that one as well as keep track of all my work and be able to go back and correct my handouts while the class went over corrections.
describing and articulating: when I didn't understand something and needed help from my peers, getting the correct answer wasn't my top priority but rather understanding WHY it was the correct answer and figuring out how I get to that answer. this meant that when I did ask for help I was very specific as to what I was struggling with and what I needed help on.
seeking why and proving: When I did get a concept, I would really value all the practice problems we completed since I could strengthened my skills and perfect what I was still lacking.
being confident, persistent, and patient: Even when I was struggling with a problem I always reminded myself that I had the capability of solving it and simply needed to look at the problem a different way or come back to it with a fresh pair of eyes. In the end this helped me stay focused on the end goal as well as kept me from being discouraged.
collaborating and listening: If I was working in a group, it was a team effort in the sense that everyone was contributing to the group and speaking up if they needed help or for someone to double check their work which I thought was really great since it showcased what it meant to have a math community at HTH.
generalizing: To grasp the problems I kept track of all the terms and definitions from my class notes which I referred to so I could understand the variables and the set up of the problems better.