Introduction:
In the very beginning of this project we derived the distance formula. But to get their we had to prove the velocity formula and the acceleration formula. Proving velocity was simple since it is a linear equation. d/t = v, and d = v(t). Next we decided to prove the slope of the line which was the change in time/ the change in distance. Through that we proved that velocity is the slope of a distance vs. time graph. Also the d = v(t) is the same formula as finding the area of a square a = l(w). The next portion we covered constant acceleration and how it affects the time graphs. We compared constant acceleration vs. time, to velocity vs. time. Constant acceleration had no slope to the line because every set amount of seconds it moved a set amount of units. But for velocity it had a slope. Another thing that we learned was you can find areas using these lines. You can find the area of what is directly below the velocity curve. With this you can find areas of shapes where the slope intersect through the y axis at a point other than 0. The equation that you use to find these kinds of ares is d=do+vo(t)+1/2at.^2. All of this lead up to the victory celebration problem which was ou first quadratic equation of the year!
Exploring vertex form and quadratic equations:
After the victory celebration parabola problem we were introduced to the quadratic forms. We were given handouts in class that had various problems about vertex form. We learned how to identify all the parts of the vertex form and their direct correlations to the chart. To start of we used the simplest quadratic equation in the form of y = a(x-h)^2+k, this was y = x^2. We then advanced the equation by adding an a variable to it. This then transformed the equation into y = ax^2 a effects the parabola’s curve by making it concave up or down based off it being positive or negative. If a<1 the concave will be wider but if a>1 it becomes narrower. We next added h which is y = ax^2+k k can be directly coordinated to the x point of the parabola's vertex. After this we added h which makes the equation y = a(x-k)^2+h h is directly coordinated to the y point of the parabola's vertex
Other forms of the quadratic equation:
Vertex form is extremely useful since you can take all of the values and see how they directly affect the parabola. But they’re other forms that can be just as useful and that can convert into Vertex form through math. One of the other forms is called Standard Form which looks like ax^2+bx+c a,b,c can take the form of any number. Through this for you can directly see where the y point is of the vertex of the resulting parabola. Another form that is useful is called Factor Form. Factor Form is y = a(x-q)(x-p) This is just an expanded form of Standard Form where a has the same effect as it does in Vertex Form p and q are correlated to the x intercepts of the parabola. This equation is useful because it allows you to find the x intercepts of any parabola.
Converting between forms:
When going from standard to factored you are going to un-factor the equation. This means you are going to take apart the x^2 into two x’s. Next you're our going to take the 36 and find the square root of it. After this you are left with (x+6)(x+6). Lastly you carry down the 0.
When going from vertex to standard you take your equation y = (x-4)^2+0 and you factor it out. You our then left with (x-4)(x-4)+0. From here you foil the numbers to un-factor them. After this you are left with y = x^2-8x+16.
|
When going from factored to standard you our given (x+6)(x+6) you multiply both of the 6’s to get 36 and carry down the zero. You are then left with two x’s which can be simplified into x^2
When going from standard to vertex form you factor out x^2+36+0 you are then left with (x+6)(x+6)+0. After this you simplify the equation into (x+6)^2+0.
|
Area diagrams:
Area diagrams are a way to provide a visual way to learn quadratics. They can also help you to keep your transferring between vertex and standard organized. Through an area diagram it is easy to find where each variable should go and can help you label out your quadratic equation if you find it difficult.
Solving problems with quadratic equations:
There are some real world implications that quadratics have. Such as Economics,Geometry, and Kinematics. Throughout the year we solved problems that had to do with all three of these topics. One of the problems that I had felt most confident in solving was a geometry problem called leslie's flowers. In this problem quadratics was especially incorporated to help find leslie a missing length of her triangular flower bed. A problem that had incorporated economics was the widgets problem where we had to solve a money problem for a company that was producing the ever grossing widget toys. We had to find the maximum cost per widget that would yield in the most pay out for the company. Last but not least a problem that incorporated kinematics was our first quadratic problem that we had done in this project. That was the victory celebration problem where we had to calculate the maximum height of a rocket being launched. It’s time spent in the air, and how far the rocket would go from its starting position.
Problem:
The problem that I had chosen was the one that I had felt most confident in and that was the Leslies Flowers problem. This quadratic problem was implicated into geometry. In the problem you are trying to find the area of her triangular flower bed with the information given in the problem. You are also solving for an unknown variable.
Solving:
First we find out the area of the smaller triangle. The formula to find the area for a triangle is 1/2bh but for that we need to find an unknown side length, h. To find this length we need to do pythagoreans theorem which is a^2+b^2=c^2. Now we need to plug in the values that we have for the problem, x^2+h^2=13^2. But before we can find the answer to this problem we need to use quadratics to solve for the relationship between h^2 and the bigger triangle. To set up this quadratic equation we simply plug in the values we our given. h^2+(14-x)^2=15^2, this is also equal to h^2=(14-x)^2-15^2. Next we set up our other equation x^2+h^2=13^2 the same way, h^2=13^2-x^2. Now we can set both of these equations equal to h^2 which means that both of the triangles equations equal each other. Another way to think of these equations as equal to each other is because they both our needing to solve for h^2 which means each squared is the same value for both of the triangles. Therefore both of the equations to find h^2 must be equal to each other. Now that they are equal to each other we need to set up the equation. 13^2-x^2=15^2-(14-x)^2. Now we can solve for x. After solving for x we now know it is equal to 5. Now we know the bases for both triangles, this means we can solve for the heights or h. H is equal to 12. With this new information we can find out the area of the triangle which comes out to be 84^2
Reflection:
Overall this project had it's ups and downs. At some points it seemed like the math was almost impossible to do and others it seemed like I was solving it with ease. This unit of math was easily the most unique I have done in all of my schooling, just because of how universal quadratic equations are as a whole. I feel as if this changed my perspectives of 11th grade math because it is only going to get more challenging and this project shows how much I need to improve in my learning as a mathematician. After this project I feel way more confident in my abilities to solve quadratic equations and explain their implications to the outside world more confidently now. Throughout this project I have learned to stay more organized as a mathematician because it is so easy to get lost in your own work and to lose your own work in general. I feel as if this has prepared me for the SATs and college in a positive way, showing me how to be confident in my math skills, and to break down hard challenging problems and put them back together into 1 easy solvable equation. Overall I am very pleased with how this project has turned out and I can't wait for math in 11th grade and the SAT.
Habits of A Mathematician I Used:
Stay Organized:throughout this project I had to make sure that all of my math work was organized on paper because it was so easy to get lost in your own math. it also was important for me to keep all of my work organized so I could revisit if I couldn't remember a topic that we had gone over
Being Systematic:If my answers for any of my problems didn't make sense I would have to go back step by step searching for where I had made my mistake. Or when starting a problem I made sure to write down all of my steps that went towards the answer.
Conjecturing and Testing: If I couldn't get the answer to a problem the first time I would go over it again but in a new way weather that meant trying the problem in my head, or writing it into an area diagram so I could better understand it
Describing and Articulating: If i wa having an especially hard time with a problem I wasn't afraid to seek help from my peers not to just get the answer but to find out why that was the answer and where I had gone wrong in my own work so I would improve upon my own mistakes.
Starting Small: If a problem seemed to be too long or challenging I would break it down into manageable steps and solve them one by one. After that I would put all of my solutions together to solve for the final answer to the problem.
Taking Apart and Putting Back Together: A lot of the problems in this unit were broken down into multiple step problems for example a,b,c. I would look at all of the things I was solving for and put them together to see the bigger problem I was actually solving.
Looking For Patterns: When first looking at a problem it was crucial to look for any patterns. This could possibly decrease the time it takes to solve a problem and also make it way easier to solve for.
Generalizing: When generalizing our problems it became clear to what each individual part of the quadratic formula did to the parabola. Also it helped me identify what the values in the quadratic formulas were directly correlated to
Seeking Why and Proving: When solving a problem I didn't just expect my answer to be right, I looked past the answer and actually thought of how my answer was right and used that to strengthen my skills as a learner.
Collaborating and Listening: Communicating to my group on problems we hadn't learned about before and listening of how they thought to solve for them helped me improve as a learner by being able to take in multiple opinions and not rule them out. After all there are many ways to solve math problems.
Being Confident,Persistent, and Patient: Not giving up on a challenging problem was definitely hard for me. But having confidence in my skills in a mathematician played a big role in helping me. Also understanding that the answer wasn't going to immediately appear after moments of working on the problem. Last but not least not giving up after I was wrong multiple times and wanting to find the corrects steps to the right answer.
Being Systematic:If my answers for any of my problems didn't make sense I would have to go back step by step searching for where I had made my mistake. Or when starting a problem I made sure to write down all of my steps that went towards the answer.
Conjecturing and Testing: If I couldn't get the answer to a problem the first time I would go over it again but in a new way weather that meant trying the problem in my head, or writing it into an area diagram so I could better understand it
Describing and Articulating: If i wa having an especially hard time with a problem I wasn't afraid to seek help from my peers not to just get the answer but to find out why that was the answer and where I had gone wrong in my own work so I would improve upon my own mistakes.
Starting Small: If a problem seemed to be too long or challenging I would break it down into manageable steps and solve them one by one. After that I would put all of my solutions together to solve for the final answer to the problem.
Taking Apart and Putting Back Together: A lot of the problems in this unit were broken down into multiple step problems for example a,b,c. I would look at all of the things I was solving for and put them together to see the bigger problem I was actually solving.
Looking For Patterns: When first looking at a problem it was crucial to look for any patterns. This could possibly decrease the time it takes to solve a problem and also make it way easier to solve for.
Generalizing: When generalizing our problems it became clear to what each individual part of the quadratic formula did to the parabola. Also it helped me identify what the values in the quadratic formulas were directly correlated to
Seeking Why and Proving: When solving a problem I didn't just expect my answer to be right, I looked past the answer and actually thought of how my answer was right and used that to strengthen my skills as a learner.
Collaborating and Listening: Communicating to my group on problems we hadn't learned about before and listening of how they thought to solve for them helped me improve as a learner by being able to take in multiple opinions and not rule them out. After all there are many ways to solve math problems.
Being Confident,Persistent, and Patient: Not giving up on a challenging problem was definitely hard for me. But having confidence in my skills in a mathematician played a big role in helping me. Also understanding that the answer wasn't going to immediately appear after moments of working on the problem. Last but not least not giving up after I was wrong multiple times and wanting to find the corrects steps to the right answer.