The Introduction
For our last block in math we began to learn about the quintessences of quadratics, in order to prepare ourselves for seventh grade math. The objective was to better understand quadratic equations and how to solve them. Throughout this whole project we formed a better understanding of how to work with and solve algebraic equations. The main question that we were asked in this block was What is a Quadratic Equation? I am going to explain to you how I got to the point of answering this question. It all started with our teacher arranging us in a new seating chart. I was lucky enough to have people that I work really well with and help me when I am struggling. We started with being introduced to a problem that involved us trying to find the height of a rocket, the time it takes to reach the climax of its journey, and the time it takes to journey back to the ground. We used the kinematics equation and obtained the distance formula. h(t)=d0+v0t+1/2at^2. We used the information we already had from this problem and plugged it in. Making our new equation h(1)160+92t-16^2. This equation had now opened up a whole part of math that we had never looked at. We stopped working on this problem and we started learning how to solve this quadratic equation.
Exploring the Vertex Form of the Quadratic Equation
As we began to look at parabolas, we learned that one way it’s equation can be expressed is in vertex form. This was the first forms of many of the quadric equations that we started learning in this block. The form looks like this y = a(x-h)2+k. To better understand how the a, h and k in this form affect the location and shape of the parabola we used multiple handouts and online graphic calculator called Desmos to help with the process. The first equation that is used in the part one of the Parabolas and Equations packet is y+ax^2. This equation was used to understand how a affects the width of the parabola. With the graphing calculator I was able to understand the a larger a makes the parabola smaller than a smaller a. There was also an effect on how the parabola took shape on the graph. If it is +a then then the parabola concaves up, but if it is -a then the parabola concaves down. For part two of the Parabolas and Equations packet we learned how k affects the highest and lowest part of the parabola. We also learned that the Y coordinate is equal to k because of it’s positioning on the vertex. For part three, the final part we learned about h is similar to k in a sense that it controls the X coordinate on the vertex.
We moved onto a new paper, Vertex form for Parabola to compile all of the information that we had learned from the three papers that we had just went over. I can personally say that this is where I started to get confused by the work that we were doing so I looked to my partners for help. They explained to me that we were using the knowledge we now have of a,h and k to finding the equation of a parabola. This information we could now use to find the vertex in the rocket problem. This was a moment where all the stress and confusion that a lot of us went through started to make sense. I truly feel that I am not the type of learner that needs to be put through loops for me to understand what the problem is asking and I wish that these three problems were given to us before we started the project.
We moved onto a new paper, Vertex form for Parabola to compile all of the information that we had learned from the three papers that we had just went over. I can personally say that this is where I started to get confused by the work that we were doing so I looked to my partners for help. They explained to me that we were using the knowledge we now have of a,h and k to finding the equation of a parabola. This information we could now use to find the vertex in the rocket problem. This was a moment where all the stress and confusion that a lot of us went through started to make sense. I truly feel that I am not the type of learner that needs to be put through loops for me to understand what the problem is asking and I wish that these three problems were given to us before we started the project.
Other Forms of the Quadratic Equation
(Graph #1)The standard form of a quadratic equation is ax^2+bx+c=y. This is the form gives us the y intercept through the value of c. It is much more simple then the other forms, causing it to be used more widely.(Graph #2) The factored form of a quadratic equation is (x-p)(x-q)=y. This is the form gives us the x intercept.
Converting between Forms
Solving Problems with Quadratic Equations
Kinematics (projectile motion): As a volleyball player I can use Kinematics to understand if the time that the volleyball stay in the air is shortened, then the reaction time for my opponents is shortened. Kinematics is also useful for when I am trying to determine the trajectory of the ball which helps me better my accuracy for games and practice. Geometry (triangle problems and rectangle area problems): I now have the understanding of how to create symmetrical shapes for when I work on future projects in class. I have the understands of how to measure triangles and find the area of a rectangle. Economics (maximizing revenue/profit or minimizing expenses/losses): As I continue to grow and become part of this world, I will have a better understand of how to control revenues and expenses. I can use this in my everyday life. If I am interested in becoming a manager of profit, I will have this knowledge in my back pocket.
One of the problems that we worked on in this block was the Is it Homer? The problem problem is "When a baseball is thrown or hit, the air path is almost a perfect parabola. When Mighty Casey hits it, it hits a maximum height of 80 feet and is 200 feet away from the home base. The center fence is 380 feet from the home plate and is 15 feet tall. But does Mighty Casey's ball clear the fence?" We start with plugging the already known factors into the vertex form. y=a(x-h)^2+k now becomes y=a(x-200)^2+80. You then follow the orders of operation to get y=15.2 as your answer. We have solved that it does clear.
Reflection
I truly feel that this was one of the blocks that I was really passionate about understanding. It was a rocky start, considering I was getting very distracted by other people who also didn't have interest in learning. I would leach onto other people to get the answers last minute. I remember the exact moment when that all stopped. I had just handed in a packet that was half done and not up to standard. My teacher said to the class that you can always copy off of someone's work, but you will never be able to explain to me what we are learning in class. You can get an A, but you will struggle for the rest of your school year. This really hit me considering it was something I already understood, but never took the time to really consider. I decided from that point on I was going to start working on my own, to push myself to ask questions and get help from people who were willing. I was lucky enough to be sat next to a person who was at the same learning level as me. Who was willing to ask questions and test theories. I finally felt happy in my standing place in math class. I worked hard to complete my assignments, yes there were times when I slipped up and didn’t give myself the time to finish an assignment, but I tried my hardest to complete everything that I needed.
This block has changed the way I look at math completely. Before I thought it was just the subject that I was going to struggle till the day I die, but I understand now that you are suppose to struggle. That it is okay to struggle. It is also okay to ask as many questions as you want and need. You have a right to question the colors of the walls or the size of paper if it means that it is helping you solve a problem. For the longest time I thought that it was wrong of me to ask questions. That if I did I was just stupid. I understand now that there is going to be much more challenging math next year and it is on me to be an advocate for my learning and my peers. I’ve have struggled for the longest time to get to this point of being comfortable in something I never thought was possible.
This block has changed the way I look at math completely. Before I thought it was just the subject that I was going to struggle till the day I die, but I understand now that you are suppose to struggle. That it is okay to struggle. It is also okay to ask as many questions as you want and need. You have a right to question the colors of the walls or the size of paper if it means that it is helping you solve a problem. For the longest time I thought that it was wrong of me to ask questions. That if I did I was just stupid. I understand now that there is going to be much more challenging math next year and it is on me to be an advocate for my learning and my peers. I’ve have struggled for the longest time to get to this point of being comfortable in something I never thought was possible.
Habits of a Mathematician
Look for Patterns:
One area that where I looked for patterns was when we first began to use graph to understand how a,h and k affected the parabola. It was very important to observe in the changes I made and the ones that spawned.
Start Small:
When I worked with my partners and we came across problem that was challenging we would start small and work our way to open up the problem. There were times when it was wrong, but we learn from our mistakes.
Be Systematic:
When we started to work with converted one form to another, it was important to be systematic with each of the operations.
Take Apart and Put Back Together:
When we we're working with completing the square we had to take apart the equation to add what we needed.
Conjecture and Test:
There was multiple moments when my partners and I would test something to see if there was any credibility and come up short. Once we did our own testing we would ask our teachers for help, knowing that we didn’t know how to do it.
Stay Organized:
Organization was a big part of this block, but I only made a big deal out of half way through. I lacked in this habit, but I can now understand how important it is in math.
Describe and Articulate:
When my partners and I would mess up we would have to explain to our teacher how we got to that point of confusion to return ourselves back to understanding.
Seek Why and Prove:
Every part of this block was followed by why it was and how we can prove it to ourselves. I feel that because I had this requirement apart of the project, I have a better understanding and ability to explain it.
Be Confident, Patient and Persistent:
I am not the type of person to ask teachers questions, but I decided that to grow and learn in this block I needed to take it upon myself to ask as many questions as possible.
Collaborate and Listen (with Modesty):
I worked with a big part of my classmates and learned that I work well with a lot of people that I never thought I would because of this project.
Generalize:
Throughout this whole block I was looking back at my note, because I generalized everything that we learned to help me understand the next steps.
One area that where I looked for patterns was when we first began to use graph to understand how a,h and k affected the parabola. It was very important to observe in the changes I made and the ones that spawned.
Start Small:
When I worked with my partners and we came across problem that was challenging we would start small and work our way to open up the problem. There were times when it was wrong, but we learn from our mistakes.
Be Systematic:
When we started to work with converted one form to another, it was important to be systematic with each of the operations.
Take Apart and Put Back Together:
When we we're working with completing the square we had to take apart the equation to add what we needed.
Conjecture and Test:
There was multiple moments when my partners and I would test something to see if there was any credibility and come up short. Once we did our own testing we would ask our teachers for help, knowing that we didn’t know how to do it.
Stay Organized:
Organization was a big part of this block, but I only made a big deal out of half way through. I lacked in this habit, but I can now understand how important it is in math.
Describe and Articulate:
When my partners and I would mess up we would have to explain to our teacher how we got to that point of confusion to return ourselves back to understanding.
Seek Why and Prove:
Every part of this block was followed by why it was and how we can prove it to ourselves. I feel that because I had this requirement apart of the project, I have a better understanding and ability to explain it.
Be Confident, Patient and Persistent:
I am not the type of person to ask teachers questions, but I decided that to grow and learn in this block I needed to take it upon myself to ask as many questions as possible.
Collaborate and Listen (with Modesty):
I worked with a big part of my classmates and learned that I work well with a lot of people that I never thought I would because of this project.
Generalize:
Throughout this whole block I was looking back at my note, because I generalized everything that we learned to help me understand the next steps.