How to Solve a Quadratic Equation: Do You Know?
Think back to when you were in high school, maybe even middle school. You can maybe recall your teacher telling you how to graph a parabola given an equation. You can probably recall (ax)^2+bx+c, and perhaps you can even recall the quadratic formula. But do you even remember how to find the equation of a quadratic function using the vertex form? Yeah, me either.
You may be thinking: "Yeah, so what? Why is the vertex form of an equation important? Can't I just use the form I already know? Why isn't that good enough to teach students?" I'm not saying that the standard form of a quadratic equation isn't important, and I completely agree that it is essential that students know how to use the standard form of an equation, since it will be extremely useful as they get to know more families of functions. However, I don't think that the vertex form of an equation really gets the special spotlight that it deserves, and here's why.
First of all, there's a really great article here that discusses the implications of students' lack of knowledge in relation to the vertex form of an equation. Using the example of Colin, a high school honors student, this article ends up showing that Colin relates vertex form f(x)=(x-h)^2+k to a translation of the standard form, f(x)=x(ax+b)+c, which is an incorrect assessment. Students need to know how to use the vertex form because they (1) need to adopt multiple ways of solving a problem and (2) must have an understanding of where the equation of a quadratic function comes from, and I think that the vertex form of an equation is much better in delivering this understanding.
As a second reason why the vertex form of an equation is important, allow me to use a real-life example. I am a tutor at the Math Center at my university, and in my brief time at this job, I have seen several students that have no memory of learning the vertex form of an equation, either because their high school teachers didn't deem it important and never taught it, or because their teachers were so quick to brush over it that they had no time to retain the information. There is a gap in information here, since university professors expect students to have knowledge of this form from their high school education, and barely cover it as well, leaving students with no real way of knowing how they are supposed to find the equation of a parabola simply given the vertex and a graph. So, the vertex form of an equation can be useful not only in high school situations, but in post-secondary education as well.
I actually never learned how to do the vertex form of an equation in high school, and never truly learned of its existence until this past summer. I wasn't sure why it was important, and only had to teach myself of what it was to accomplish my goal of creating a Selected Response Assessment for one of my education courses. I never even knew why it was important and why we should be teaching it until I saw in the article (above) how its lack of representation affected student understanding of quadratic functions. This leads me to my last point.
Many students are curious about the "why" surrounding mathematics. For example, "Why do we use the standard form of a quadratic equation? Where does it come from?" The vertex form of an equation can answer this question. If we look at the vertex of a parabola, then we can find the vertex (-b/2a, f(-b/2a)), and using the vertex form of an equation, find the formula y= ax^2+bx+c. This work is modeled below:
You may be thinking: "Yeah, so what? Why is the vertex form of an equation important? Can't I just use the form I already know? Why isn't that good enough to teach students?" I'm not saying that the standard form of a quadratic equation isn't important, and I completely agree that it is essential that students know how to use the standard form of an equation, since it will be extremely useful as they get to know more families of functions. However, I don't think that the vertex form of an equation really gets the special spotlight that it deserves, and here's why.
First of all, there's a really great article here that discusses the implications of students' lack of knowledge in relation to the vertex form of an equation. Using the example of Colin, a high school honors student, this article ends up showing that Colin relates vertex form f(x)=(x-h)^2+k to a translation of the standard form, f(x)=x(ax+b)+c, which is an incorrect assessment. Students need to know how to use the vertex form because they (1) need to adopt multiple ways of solving a problem and (2) must have an understanding of where the equation of a quadratic function comes from, and I think that the vertex form of an equation is much better in delivering this understanding.
As a second reason why the vertex form of an equation is important, allow me to use a real-life example. I am a tutor at the Math Center at my university, and in my brief time at this job, I have seen several students that have no memory of learning the vertex form of an equation, either because their high school teachers didn't deem it important and never taught it, or because their teachers were so quick to brush over it that they had no time to retain the information. There is a gap in information here, since university professors expect students to have knowledge of this form from their high school education, and barely cover it as well, leaving students with no real way of knowing how they are supposed to find the equation of a parabola simply given the vertex and a graph. So, the vertex form of an equation can be useful not only in high school situations, but in post-secondary education as well.
I actually never learned how to do the vertex form of an equation in high school, and never truly learned of its existence until this past summer. I wasn't sure why it was important, and only had to teach myself of what it was to accomplish my goal of creating a Selected Response Assessment for one of my education courses. I never even knew why it was important and why we should be teaching it until I saw in the article (above) how its lack of representation affected student understanding of quadratic functions. This leads me to my last point.
Many students are curious about the "why" surrounding mathematics. For example, "Why do we use the standard form of a quadratic equation? Where does it come from?" The vertex form of an equation can answer this question. If we look at the vertex of a parabola, then we can find the vertex (-b/2a, f(-b/2a)), and using the vertex form of an equation, find the formula y= ax^2+bx+c. This work is modeled below:
So, as a last point, the vertex form of an equation can be used to figure out the "why" of a quadratic function, and can help answer student questions.
In conclusion, the classroom can be much improved with the addition of the vertex form of an equation, and can help students in a variety of ways. I hope that I can implement this in my own classroom in the future and help students to succeed and understand more about quadratic functions.
I like how you connected your own exerience to what you've found as a tutor.
ReplyDeleteclear - maybe spaces between paragraphs for readability.
coherent - strong point
complete - go ahead and give the citation for the article. It's a little sketchy to link it bc of copyright. You guys have access through the U.
More importantly, I'd love to know more about "So, as a last point, the vertex form of an equation can be used to figure out the "why" of a quadratic function, and can help answer student questions. "
content +
consolidated +
5 Cs 4/5