About Quadratic Equations
In this unit of quadratics, we learned a lot about quadratic equations, functions, and graphs, where they come from, how to find them, and how to manipulate them. We started out with the basics (standard, factored, and vertex form) then slowly worked our way up to the more complex things like equivalent expressions, completing the square, and the Quadratic Formula using many tools and techniques such as online graphers (Desmos) and visual patterns!
Work
The most challenging and most interesting thing we did this unit was deriving the quadratic formula out of standard form. Before we did the activity as a class, I was given the problem to solve on my own without the puzzle that we did where the steps were given to me. I was simply told to do it. I got very excited and started to get to work immediately but, instead of being simple, easy, and fun like I thought it’d be, it proved to be very difficult. It pushed my understanding and view of quadratics very far! I started the puzzle on my own doing every type of manipulation I could think of which turned out to be very good in the beginning but then started taking me in circles until I admitted I was stuck. I first looked to my friend that was also given the puzzle for help but without much success. Then I looked to the teacher. She showed me that everything I had been doing, though interesting and intellectual, was too much. All I needed to was complete the square which is exactly what the teacher had been teaching the class as I blindly worked on the problem. I ignored what she was doing because I believed I knew it all already which was partly true but I should have paid more attention to it because the answer was right there in front of me. :)
Questions
Right now, as I reflect on the unit, I have two mathematical questions and two reflection questions. The mathematical questions are: Is there another way of deriving the quadratic formula from standard form without using completing the square? and, Is there a way of deriving it from, let’s say, vertex form or something else? The two reflection questions are as follows: What more is there to algebra that I have yet to discover? and, Will it be similar to what I have learned so far? All these questions I am hoping will be answered for me in the near future! :)
Self-Assessment and Reflection
Before this unit, I thought I had a pretty basic, but good, understanding of quadratic equations. I knew all the formulas, the functions, the forms, and the vocabulary so I started this unit thinking it would be a bore. I couldn’t have been more wrong. It turns out that review was very good indeed (since I had forgotten a lot) and I learned a lot of new things! Not only that but I discovered things about myself and math that I really needed! I realized that math isn’t so bad, especially algebra. I discovered that I actually really, really like algebra. Puzzle solving, manipulation, thinking outside the box, those are all things that make up algebra and what I really like! Before, I learned everything step by step and never got to think for myself about how to solve a problem or, more importantly, why the problem is solved that way. Since I already had the basic knowledge of quadratics, I was able to go deeper into my understanding of that knowledge and push my thinking! This means that I used habits of a mathematician #__ (persists in solving problem) and #__ (asks questions to push thinking) the most! Actually, more like a mix between the two. Like, “persists to push thinking” or something because I really thought deeply about each problem and tried to gain the very best possible understanding of it even if it meant continuing to work on it waay after class which i did quite a few times!