Over the vast majority of the second semester, we have been mainly focused around the topic of quadratics. We received and worked through 25 handouts to help us better understand quadratics. We started off with distance, velocity, and acceleration practice problems. Handouts 4-6 were all focused on parabolas, and handout 7 introduced vertex form. Parabolas and vertex form led us to figure out where the vertex is just looking at the equation and and where it crosses the axis which was covered in hndouts 8 and 9.
We started fully to work with the quadratic equation when we got handout 12. We were taught to use an area diagram for factoring the given numbers into ax^2+bx+c. We practiced thin until handout 14. On handout 15 is when we worked on completing the square. To complete the square you divided b in the previously stated equation. From there you square that number to get c. The handouts after this were mainly to continue with practicing with the quadratic equation.
Exploring the Vertex Form of the Quadratic Equation
Going more in depth about how we used the vertex form of the quadratic equation, I will explain the most helpful of the handouts. "The Victory Celebration" handout was what kicked off the entire quadratics project. This is where we worked with figuring out what a, h, and k meant in vertex form. As well as how they affect the function. We also learned how to put a, h, k into vertex form and looks like this: y = a(x-h)^2 + k.
The variable known as a affects how wide the parabola is. When the number for a is larger, the parabola gets narrower. If a is positive, the parabola opens upwards. If a is negative, then the parabola opens downwards. Otherwise known as concave up or concave down.
Moving on, the h is what controls if the parabola is left or right of the y axis. For example, if we say h is -5, the vertex of the parabola would touch the x coordinate of 5. If it was +5, the vertex of the parabola would touch the x coordinate of -5. Finally, the variable k is what controls the height(whether it is above or below the x axis)of the parabola. For example, if we say k is -8, the vertex of the parabola would then touch the y coordinate of -8. If it was +8, the vertex of the parabola would touch the y coordinate of +8.
Other Forms of the Quadratic Equation
There are two other forms of the quadratic equation known as "standard form" and "factored form." The standard form of the quadratic equation looks like this: ax^2 + bx + c. This controls the width of the parabola. If it's bigger than 0, the parabola opens upwards. If it's less than 0, the parabola opens downwards. One example of standard form would be "-x^2 - 4x + 4."
The factored form of the quadratic equation is a (x-r)(x-s). The factored form shows the x coordinates of the parabola ( x = r and x = s ). Sometimes the factored form can't be solved. This is due to the fact that there aren't always x intercepts. An example of factored form is "y = 2 ( x - 4 ) ( x + 3 )."
Converting Between the Forms
There's four converting steps usded in the quadratic equation. These are the steps:
1: Vertex to Standard
2: Standard to Vertex
3: Factored to Standard
4: Standard to Factored
Solving Problems with Quadratic Equations
In this section, I personally have chose a handout I felt like I learned the most from and felt the most comfortable with. The handout I have chose is handout 14 - "Square It." In this handout, we have been shown how the quadratic function was written and then how to find the vertex without any computation. The paper also showed what standard form looks like as well as how to convert them. We had to use an area diagram to solve a series of expressions and turn them into standard form.
In question 1 a, I was instructed to turn (x + 3)^2 into standard form. Ipersonally like to and tend to use F.O.I.L more than the area diagrams. For some reasoon the diagrams just confuse me a bit more. Before you F.O.I.L, you have to factor out (x +3)^2 into (x + 3)(x + 3). After the F.O.I.L process is done, you end up with x^2 + 3x + 3x + 9. The next step is to combine like terms which is now x^2 + 6x + 9. B, C, and D are solved the exact same way.
Reflection
Overall, I think this was my favorite topic to study this year. It felt like it was easieer for me to understand and connect with. It was actually a lot of fun and for the most part, enjoyable. I did find myself using the Habit of a Mathmatician: Colaborate and Listen a lot during this project. In general I have a hard time undertanding topics in math since I have not yet found the best way for me to learn yet. I asked my peers for a lot ofhelp and they were always willing to help me through problems step by step if needed.
I also liked this project because it really got us all to use each Habbit of a Mathmatician. I already explained how I used Colaberate and Listen but here is how I used the rest throughout this project:
Be Confidence, Patient, and Persistent: Like any other project, you have to be patient and persistent as well as have confidence in order to reach your goal. In this project, I definetly had to take it slow and pe very patient if I truly wanted to understand the whole unit.
Start Small: If I was lost, I would take it one step at a time, because if I tried to tackle the entire problem at the same time, I would get super stressed and frustrated and most likely give up. This would result in me sitting in class for who knows how lon, not knowing what to do because I was too frustrated to seek help.
Seek Why and Prove: With any problem, there's alwasy a way to go back and check to see if your answer is correct. With this unit, I had to do that a lot if my answer didn't make any sense. If I realized my answer was wrong, I could go back through and see where I messed up and work through those mistakes.
Look for Patterns: There were certain patterns you had to look for in the different forms of the quadratic equations in order to know how to convert from one form to another.
Describe and Articulate: When working with others, I was able to understand some of the problems and help some people. I helped people when they weren't sure where to start and broke it down for them step by step based off what I was taught by my other peers.
Conjecture and Test: One example of this is when you can multiply two numbers and add those same numbers to get both b and c in a quadratic equation. (n * m = 6 and n + m = 5 (n = 3 and m = 2))
Take Apart and Put Back Together: This is very similar to Start Small in some ways. Taking on difficult problems in full can be really challenging if you're already havnig trouble undertanding the topic as is. Taking the problem apart has mad it easier for me to fully digest what we're doing in class. If I try to tackle a challenging problem head on, chances are I won't get very far. By taking it apart, I can slowly work through a problem and knw I will eventually reach some sort of answer.
Be Systematic: When solving a quadratic problem, there can be a lot of steps involved. You need to be methodical and work through the problem in the correct corresponding steps.
Stay Organized: This one is huge. I personally sruggle with organization most of the time but I somehow kept all of my papers together for the most part. Since there were 25 hand outs, you were kind of forced to stay on top of it. If you lost papersbefore they were do, you either had to fond them or redo them. I was almost late with some of my papers since I did lose them Thankfully I was able to find them before the due dates. Generalize: If you truly understand the criteria, you'd be able to create a generalized statement on how to solve the problem. Generalizing has helped me to geth through problems faster and keep my paper more organized.
Once again, I really enjoyed this project and haveing as much freedom as we did in choosing how we were to work through the handouts. A lot of trust was put into us in chosing people we felt we would work best with. If we chose people that we knew we weren't going to get work done with and were struggling to complete the assignments, that was on us. I really looved colaberating with my peers though. I love seeing how other people's minds work in processing problems and coming up with solutions. It's so facinating to me. I almost wish we had a bit more time to go more in depth about other appoaches.