Introduction
Throughout this project we learned many different things beginning with projectile motion, to areas and volume, and The Pythagorean Theorem, to help us enhance our understanding of quadratic functions and methods for solving quadratic equations, my favorite subject in math. While doing this project we work on our ability to work with algebraic symbols and relating algebraic representations to situations. We learn that rewriting quadratic expressions in special ways, either in factored form or in vertex form, gives awareness to the graph of the function. The goal was to understand the connection between algebra and geometry.
Exploring Vertex Form of the Quadratic Equation
To begin this project we used a series of handouts and desmos to understand parabolas and how they affect the location and shape of the parabola. We learned the formula y=a(x-h)^2+k, and this formula moved the graph in many different ways depending on where we put a number or what number it was. To begin if I change the a to a number higher than 1 than it will make my parabola move in and make it skinner and if I make that number negative it will flip the parabola upside down and also make it skinner after its flipped. This is shown below:
Throughout this project we learned many different things beginning with projectile motion, to areas and volume, and The Pythagorean Theorem, to help us enhance our understanding of quadratic functions and methods for solving quadratic equations, my favorite subject in math. While doing this project we work on our ability to work with algebraic symbols and relating algebraic representations to situations. We learn that rewriting quadratic expressions in special ways, either in factored form or in vertex form, gives awareness to the graph of the function. The goal was to understand the connection between algebra and geometry.
Exploring Vertex Form of the Quadratic Equation
To begin this project we used a series of handouts and desmos to understand parabolas and how they affect the location and shape of the parabola. We learned the formula y=a(x-h)^2+k, and this formula moved the graph in many different ways depending on where we put a number or what number it was. To begin if I change the a to a number higher than 1 than it will make my parabola move in and make it skinner and if I make that number negative it will flip the parabola upside down and also make it skinner after its flipped. This is shown below:
Other Forms of the Quadratic Equation
Standard form of the quadratic equation is y=ax^2+bx+c. Factored form of the equation is y=a(x-p)(x-q). These functions are both shown in the examples below.
Standard form of the quadratic equation is y=ax^2+bx+c. Factored form of the equation is y=a(x-p)(x-q). These functions are both shown in the examples below.
Converting between Forms
This pictures below are factored form to standard form, standard form to vertex form, and vertex form to standard form. These examples are shown below with random numbers I chose.
This pictures below are factored form to standard form, standard form to vertex form, and vertex form to standard form. These examples are shown below with random numbers I chose.
Solving problems with Quadratic Equations
There are 3 types of real world problems that I can solve using quadratic equations. The first type of real world equation that I could solve would be Kinematics. The next real world work we did was geometry. The final real world work I learned was Economics. I really enjoyed this part of math because I love the fact that it shows how math can be used in the real world.
There are 3 types of real world problems that I can solve using quadratic equations. The first type of real world equation that I could solve would be Kinematics. The next real world work we did was geometry. The final real world work I learned was Economics. I really enjoyed this part of math because I love the fact that it shows how math can be used in the real world.
Habits of a mathematician
The habits of a mathematician are 11 habits that provide us with helpful learning habits and ways to learn math and learn effectively. The habits are listed below:
Look for patterns- Throughout this whole project I have had to look for patterns in every equation I get and every parabola and I feel very comfortable with this habit because it is something I will always have to use.
Start small- Starting small is definitely something I have gotten used to because I have never been able to just look at a problem and know what the answer is so I started small and figured everything out and worked to a point where I got the work.
Be systematic- I'm not the most systematic person, I'm always looking for a faster way out of it regardless of starting small I always have tried to just take big steps to get the work done quick but I have gotten better with taking my time and making small changes to see a difference.
Take apart and put back together- This is a very important habit of a mathematician because it helps so much with getting the work done and it helps my understanding of the math work we are doing because essentially I am just looking for something in the problem to break it down so it is easier to understand.
Conjecture and test- I don'y fully understand the conjecture and test habit of a mathematician and I feel like I have never used it.
Stay organized- I think that this is the most important habit because it corresponds with everything outside of my math work also. Staying focused while doing my math work is a huge thing that so I can understand what I'm doing and not just get confused.
Describe and articulate- I use this habit a lot too, because I am a very visual learner and this habit is basically saying to draw out what I can and approach it as something visual and not just numbers.
Seek why and prove- I don't use this habit of a mathematician usually at all, because once I get my answer I like to be done with my work and not have to come back to it.
Be confident patient and persistent- I am very persistent with my math usually I get annoyed sometimes and I'll wanna give up but once I've started my work I don't want to stop because I want the answer.
Collaborate and listen- I prefer to collaborate in math because I feel like when I work with other people It gets my brain going and I feel like I can get work done more efficiently.
Generalize- I do feel that I look to understand my math but only because I need too because I'm in 10th grade and I have SAT's next year.
The habits of a mathematician are 11 habits that provide us with helpful learning habits and ways to learn math and learn effectively. The habits are listed below:
Look for patterns- Throughout this whole project I have had to look for patterns in every equation I get and every parabola and I feel very comfortable with this habit because it is something I will always have to use.
Start small- Starting small is definitely something I have gotten used to because I have never been able to just look at a problem and know what the answer is so I started small and figured everything out and worked to a point where I got the work.
Be systematic- I'm not the most systematic person, I'm always looking for a faster way out of it regardless of starting small I always have tried to just take big steps to get the work done quick but I have gotten better with taking my time and making small changes to see a difference.
Take apart and put back together- This is a very important habit of a mathematician because it helps so much with getting the work done and it helps my understanding of the math work we are doing because essentially I am just looking for something in the problem to break it down so it is easier to understand.
Conjecture and test- I don'y fully understand the conjecture and test habit of a mathematician and I feel like I have never used it.
Stay organized- I think that this is the most important habit because it corresponds with everything outside of my math work also. Staying focused while doing my math work is a huge thing that so I can understand what I'm doing and not just get confused.
Describe and articulate- I use this habit a lot too, because I am a very visual learner and this habit is basically saying to draw out what I can and approach it as something visual and not just numbers.
Seek why and prove- I don't use this habit of a mathematician usually at all, because once I get my answer I like to be done with my work and not have to come back to it.
Be confident patient and persistent- I am very persistent with my math usually I get annoyed sometimes and I'll wanna give up but once I've started my work I don't want to stop because I want the answer.
Collaborate and listen- I prefer to collaborate in math because I feel like when I work with other people It gets my brain going and I feel like I can get work done more efficiently.
Generalize- I do feel that I look to understand my math but only because I need too because I'm in 10th grade and I have SAT's next year.
Reflection
To reflect on this quadratics unit I personally enjoyed it. I had fun with this unit because i've done this work before and I love going over math I've already learned because I already know what I'm doing. I really enjoyed this unit because out of everything we've worked on I felt like this one related the most to the real world and thats my biggest problem with math, I feel like I'll never need it, but I understand how I can use some of these concepts in the real world and I loved that about this project. Throughout this last semester and this last year I have really learned to use the habits of a mathematician. I have learned to use all of the habits of a mathematician while approaching my math work and while working on my math work. To conclude I really enjoyed this quadratics work and I feel like I have a well enough understanding of quadratics to use it in a real world situation.
To reflect on this quadratics unit I personally enjoyed it. I had fun with this unit because i've done this work before and I love going over math I've already learned because I already know what I'm doing. I really enjoyed this unit because out of everything we've worked on I felt like this one related the most to the real world and thats my biggest problem with math, I feel like I'll never need it, but I understand how I can use some of these concepts in the real world and I loved that about this project. Throughout this last semester and this last year I have really learned to use the habits of a mathematician. I have learned to use all of the habits of a mathematician while approaching my math work and while working on my math work. To conclude I really enjoyed this quadratics work and I feel like I have a well enough understanding of quadratics to use it in a real world situation.