Introduction
To begin this project we started off by doing a worksheet called "Distance, Velocity, Acceleration". This worksheet was our introduction to quadratics. For this worksheet we learned about velocity, distance and, acceleration by looking at things like the speed and the distance traveled in moving objects such as cars, and rockets. The central question is about a rocket being launched for a firework display (victory celebration). For the next few worksheets that we were given we learned about parabolas, equations and vertex for parabolas (worksheets 2-7). The specific skills these worksheets taught us were converting equations to vertex form, practice with parabolas, practice with vertex form parabolas, graphing equations, and finding equations of parabolas using algebra and the graphing website Desmos to check our work. The next few worksheets we worked on we learned about parabolas con-caving up and down, distributing areas, changing vertex form to standard form, and more practice with parabolas (worksheets 8-14). The remainder of the worksheets (15-25) helped us learn skills such as, working with volume and area, writing equations in standard form without parenthesis, using The Pythagorean theorem, finding vertices and intercepts, and circling back to learning about another rocket. The goal of this project was to find the connection between algebra and geometry by learning all of these concepts, finding similarities and using skills from previous worksheets/ lessons to move on to the next. Something about this project that was really important for my success was using the habits of a mathematician. The habits that I used the most and that stuck out to me the most were stay organized and be systematic. These were significant to me because using them and keeping them in mind helped me not fall behind in my work and they helped me complete this project successfully.
Exploring the Vertex Form of the Quadratic Equation |
Other Forms of the Quadratic Equation |
Using this series of worksheets and Desmos we explored how the a,h and k affect the location and the shape of the parabola.
y=a(x-h)^2+k. The advantage of writing the quadratic equation in vertex form is that for me personally it is easy to graph, which is a concept I understood pretty fast. We learned that the a value in the equation determines weather the parabola concave's up or down and is determines how wide or narrow the parabola will be. The h value in the equation is the x coordinate of the vertex on the parabola. The k value in the equation is the y coordinate of the vertex on the parabola. The worksheets that helped me the most with these concepts were worksheets 8, 9 and 10 (see below). The habit of a mathematician that I used the most while using the quadratic equation in vertex form was collaborate and listen. I used this skill a lot in these worksheets because at first I was really struggling with them. After getting help and learning the different ways my peers did it these concepts, they really started to make sense which was a great feeling. Solving problems with quadratic equations. Kinematics:
Kinematics was introduced to us in the beginning of this project with a worksheet called "Victory Celebration" (see first picture below). This problem states that High Tech High is having a celebration because the sports teams had a good year. For the celebration they decided to have a fireworks show. These fireworks will be launched from a tower, this tower is 160 feet tall. The mechanism that will be launching the fireworks should initially launch them at 92 feet per second. Our task was to sketch the situation, describe how you might use the height formula to help answer the students questions, write a clear statement of the questions the students want to answer. using whatever methods you choose try to get some answers (or partial answers) to the students’ questions. This was the central problem of this project and it was our introduction to quadratics. This worksheet showed us the similarities between kinematics and quadratics, and how you can use skills from learning quadratics to apply to kinematic problems like this one. Geometry: An example of us using geometry in this project was worksheet #3 this worksheet is called "A Corral Variation" (see second picture below). This problem states that Johnson raises cattle and he needs to build a cattle pen for his cattle. He wants to be efficient so he figures that he will just use an existing fence he has as one of the walls of the pen. He needs to build the other 3 sides of his pen with the 500 feet of fencing he bought. We were asked to answer 5 questions about this problem: 1. What is the area of the cattle pen shown in the diagram? 2. Choose three other possibilities for the dimensions of the rectangular pen. Find the area of each pen. Keep in mind that dairyman Johnson can use as much or as little of the existing border fence as he likes. Also remember that he wants to use a total of 500 feet of fencing for the other three sides. 3. Suppose the pen extends x feet away from the border fence. For example, in the diagram, x would be 100. Find an expression for the area of the pen in terms of x. 4. Try to determine the value of x that will maximize this area. 5. Did you notice anything that this activity has in common with the path of a rocket? This worksheet helped us understand the similarities between geometry and quadratics, and how applying our skills from quadratics to geometry can benefit us. Economics: An example of an area of this project that we learned how to apply the quadratic equation to economics was worksheet #23 called Profiting From Widgets. Using the prior knowledge learned and the practice we had with the quadratic equation we were able to find the maximum profit you could make from selling widgets. Solving this problem was pretty simple using the equation the worksheet gave us which was 1000-5d in order to find the amount of widgets you would sell and the profit you would make from selling that amount of widgets. Me and my friend Alena worked on this problem together and we used the method conjecture and test. The d value in the equation stands for dollars (price of each widget sold). We plugged in different numbers until we found the maximum profit that you could make by selling 500 widgets for $100 each in order to make a $50,000 dollar profit. For me this worksheet was really fun and I really liked the method I used to solve it. This problem helped me apply quadratics to something that is a real world situation like economics (see last picture below). |
The standard form of the quadratic equation is ax^2+bx+c=y. The advantages of writing it in this form is that it is the most organized and it gives you the y-intercept, value c. Even though this is the most organized equation I still had trouble learning how to graph it. I think that standard form was definitely a difficult equation to understand at first but after a lot of practice using worksheets and getting some help from Desmos I began to catch on. The factored form of the quadratic equation is a(x-f)(x-r)=y. The advantage of writing the equation in this form is that it gives you the x-intercepts. This equation was still hard to understand at first, I think everything it when you first see it, but I think that I caught on to this concept faster then the standard form of this equation. When I was learning these forms of the quadratic equation the habit of a mathematician I used the most was be confident patient and persistent. I used this habit because these equations were difficult for me when I first learned them but by not giving up and being persistent I was able to understand them and explain them to my peers. Examples of these equations are below.
The habit of a mathematician that I used during this part of the project is called decribe and articulate. This skill was important for me to use because by visualizing the parabolas that these equations made I was able to understand them better. Converting between forms Learning how to convert between all of these forms of the quadratic equation was a long but interesting experience. Actually understanding these concepts and being able to explain how to convert them took a lot of practice but felt very rewarding and helpful in the end. I think that the most important habit of a mathematician that should be used while doing these conversions would be, being systematic. I think this habit is important because all of these conversions have a processes, system, or a list of steps you need to use in order to convert correctly. If your not being systematic you won't get the correct answer.
Vertex to standard: Converting from vertex form to standard form was one of the first conversions we learned, and actually was one of the easier conversions to learn how to do for me. To convert vertex form to standard form you start out by foiling the equation which means taking (x-h) and doubling it because it is squared then foiling it in the correct order which is first, outer, inner, last (FOIL). After you foil the equation you distribute the a value. Your last step would be to combine like terms and you will get your equation in standard form. y=ax^2+bx+c Standard to vertex: The easiest way to convert vertex form to standard form to me is completing the square. When I learned how to convert standard to vertex this way I understood it much faster then converting other forms. The first step to completing the square is to draw an area diagram and fill it out according to your equation after this step you should have an equation that looks like this a(x^2+bx+d-d)+c. Then you change where your parenthesis are placed so your equation looks like this a(x^2+bx+d)-d+c. after this you take what you got when you completed the square and put it as your x and h value and combine like terms. Your final equation in vertex form would be y=a(x-h)^2+k. Factored to standard: Converting factored form to standard form was one of the concepts a struggled with. Honestly converting this form is still pretty challenging to me and I still struggle with it. Even though this concept was really hard for me to understand there aren't very many steps you have to do to convert. The first thing you need to do is distribute using the foil method I explained in converting vertex to standard. Your next step would be to combine like terms and distribute the a value. After these steps your equation should be in standard form ax^2+bx+c=y. Standard to factored: This conversion was difficult for me but again most of them were. I understood this faster then converting standard to factored but I still had some trouble with it. The first step to converting standard form to factored form is doing the reverse of distributing which using an area diagram. After plugging in your numbers and solving the area diagram you simplify your equation and end up with the factored form a(x-f)(x-r)=y. The habit of a mathematician that helped me while learning these conversions was looking for patterns. This habit was important to learning this skill because looking for patterns and seeing if certain conversions had similar steps really helped me understand this concept better. |
Reflection
This was a very long project that had its ups and downs when it came to comprehension of concepts , time management, and learning things I struggled with in a short amount of time. I'll begin by talking about comprehension of concepts. To be completely honest I struggled with most of the concepts we were taught. Something good that came out of all this struggle was that I learned that asking my peers to explain there way of thinking towards a problem was really beneficial to me. I learned that the way a teacher explains things isn't always enough to fully understand a concept. For me personally seeing different perspectives and getting help from my peers was crucial to my learning. Next, time management. Time management has never been a strongpoint for me, but in this project you did't really have a choice to procrastinate. Dr. Drew gave us a new worksheet almost everyday, if not every two days. This was helpful in a lot of ways because it helped me stay motivated to get my work done on time so I could move on to the next worksheet. Now there were a few times in this project were I missed a worksheet and became kind of behind but I found myself on task while working for the majority of this project. Even though getting these worksheets so often helped me a lot with time manegement, it made me feel rushed to learn things I was really struggling with in a short amount of time. I felt like right when I was about to understand a problem the new worksheet would be given to us. This is understandable though because everyone works and learns at there own pace. Even though it was taking me kind of a long time to understand these topics I still managed to learn and finish the concepts/ worksheets on time by using the habits of a mathematican called generalize and starting small. I used the habit generalize to help me undrstand all concepts and the habit start small to help me not get so overwhelmed if I was given a large problem with multiple steps.