This unit was all about quadratic equations and how you could go about solving them. I was surprised by how many "forms" a quadratic could be in. We started this unit with a rocket. We had to figure out the rockets maximum height. We were measuring the comparison between height and time. Luckily, this was a relatively small rocket so the numbers were not to complex this was using the habit of a mathematician which is starting small. This was a kinematic equation. The kinematic equation was h(t)=-16t^2+92t+160. This was in y=ax^2+bx+c. From this, you can figure out that c is the constant which means it is the initial height, b is the velocity so it will change with time, and a is the gravitational pole the rocket will have to endeavor. We needed to figure out three questions... "How high will the rocket go?" "How long would it take to get there?" "How long will it take it to hit the ground?"
Exploring the Vertex Form of the Quadratic Equation:
The "vertex form" is one of the many forms of quadratic equations. Vertex form is written as y=a(x-h)^2+k. Figuring out h and k is how you figure out the parabola. This was the first form of quadratics we were introduced to. We used demos which is an online graphing calculator and looked for patterns in it. This was using the habit of a mathematician which is looking for patterns. This visual representation was very helpful to me. This is a very helpful way to figure out the vertex because otherwise you would have to figure out the two x intercepts which would take a lot of work. Sometimes, there are not even x intercepts to find and you are just wasting your time. Without this formula, it would require a lot of trial and error and plugging in numbers. Sometimes, it might make you want to pull your hair out when the x intercept value does not even exist.
The formula being used in the first worksheet pictured was y+ax^2. I did that worksheet after messing around with demos for a little bit and I had a clear visual of what patterns meant what. From simply looking at an equation, I was able to figure out if the parabola conceived up or down, and where it would cross the x intercept. With the equation being y=ax^2, it was obvious the first step was to figure out a. What surprised me was these quadratic equations also could tell me how wide the parabola was. The smaller a is = the wider the parabola is. It is direct cause and effect. In order for the parabola to concave down it needs to be negative.
Once we got that down, then we moved onto our next worksheet which is pictured above. We were now trying to figure out how to find k. This was adding a new labor to our quadratic equation because now it did not always start at (0,0). k determines where the parabola starts. it was the same equation but adding something on. The equation was now y = ax^2 + k. k is equal to the y coordinate on the access. For example, if k is 7, the point of the vertex is (0,7).
Our next worksheet (pictured above) was to figure out how h effects the parabola. Like k, the vertex of the parabola changes when you change your k. The quadratic equation for this was y = (x - h)^2. Opposite of k, h moved on the x axis. For example if k was one the coordinates of the vertex would be (1,0). Now that we knew how to figure out both the x and why coordinates for the vertex of a quadratic function, we officially knew how to find a parabola. We then had a worksheet putting all these individual skills together. This was confusing for me but I used the habit of a mathematician which is staying organized because I had to remember which formula yes with what. The worksheet looked scary but, I used the habit of a mathematician which is break apart and put back together and it made it a lot less overwhelming. The worksheet is pictured below:
Other Forms of Quadratic Equations:
Pictured above is the standard form. The standard form is useful for finding the y intercept of the parabola which we figured out is always c. The standard form is my favorite form because I think it reads the most simple. There are no parenthesis and it is not very complicated.
Pictured above is the factored form. It is called the factored form because you factor a out. I'm not crazy about the factored form, I think it is complicated and kind of extra. Despite Dr. Drew's discouragement typically when I see the factored form I FOIL (first, outside, inside last). In this factored equation p and x are the x intercepts.
Step by Step Examples:
An area diagram can help understand the distribution of multiplication over addition by giving a clearer visual to a visual learner. Pictured below is a picture that I found very helpful.
Completing the square is also a helpful tool for a visual learner. It allows them to stay organized and remember what pairs with what. Factoring a quadratic equation is also a way to simplify the process and stay organized. It is using the habit of a mathematician staying organzied.
Solving Problems with Quadratic Equations:
Now that I have learned all this math around quadratics, I can solve some real world problems just like the original rocket problem. I can figure out the coordination between velocity / time of any moving object or thing. I can also find out the height / time of any given thing. I could do a problem figuring out how fast a car goes, a snail goes, a person goes or a plane goes! Using kinematics, I can figure out anythings projectile motion. Using geometry, I can figure out triangle problems and rectangle area problems. Using Economics, I can figure out how people can get the most for their money. Below is an example of me solving real world problems with quadratics. I needed to figure out how to make the most profit from widgets.
Habits of a Mathematician : How did I use them in this project?
I collaborated with my peers while working through this worksheet. I had to listen to their ideas and decide if I agreed or disagreed.
I seemed why and proved my formulas correct and what h and k were.
I visualized my parabolas.
I stayed organized by keeping all the worksheet in order and not loosing them.
I experimented through conjectures through experimenting with formulas I did not know were right.
I looked for patterns in desmos.
I solved a simpler equation by doing the quadratic equations with all variables first to see what I was working with with ought having to worry about messing up calculations.
I generalized what a parabola would look like by the numbers I saw. I could assume about how wide it would be if I were looking at it on a graph.
I was persistant... I did not understand this whole unit 97% of the time but I asked for help and kept trying.
I was systematic because I had a plan and a goal and reached it in a timely and organized way.
Reflection:
Honestly, this unit was hard and scary. I think this is the hardest unit we had all year. I am still not confident on the concepts. I can't seem to remember what goes with what formula. I am very worried about my math SAT store because I am not very strong at math. I think I will get by in grade 11 math but still not be an amazing mathematician. I can also seem to get by without really understanding the concepts but that will not help me on the SAT. I think I need to reach out for help more in grade 11 so I do not feel so behind there is no catching up. It is confusing to me how to remember what quadratic equation goes with what formula but I think I could have understood it better if I took better notes. While the lessons were happening, I was paying attention but not jotting down notes and that screwed me over, I should have at least taken a picture of the board so I had something as a reference.