Derive the equation of a parabola given a focus and directrix.
INTERPRETATION OF OBJECTIVE - G.GPE.A.2
This objective needs to interpretation - it is very straight forward - obviously we will extend our look at the conic sections by analyzing the parabolic curve. Again we will use the distance formula, some of our geometric relationships and the definition of a parabola to derive its equation. We then will look at parabolic curve properties.
(1) The student will derive the equation of a parabola.
(2) The student will be able to graph a parabola.
(3) The student will be able to determine the characteristics of a parabola based on its equation.
(4) The student will be able to determine the equation of a parabola using certain characteristics.
THE BIG IDEA
The study of conic sections provides us a nice bridge between geometry and algebra. The coordinate plane provides us the opportunity to analyze geometric shapes in new ways as we derive equations, look at characteristics and graph them.
TRAPS & PITFALLS
The big issue here is that we begin to get a little algebra heavy... not that it is difficult but by this point the algebraic operations and techniques used here are definitely rusty and so it isn't necessarily the parabola that trips them up... it is the ability to transform an equation into an alternate form... the ability to solve for a variable in different location of the equation... etc... In geometry we keep it simple to analyze parabolas with the origin as the vertex and I think this is wise.
We connect the parabola to the circle in terms of similarities and differences. We see how the definition leads us to set up an algebraic relationship to derive the general equation. We also see how equations can come in different forms and that each of those forms has their own strengths and weaknesses.
We are preparing students to study all kinds of relationships through coordinate analysis. While we cover circles here we will look at parabolas next and then later they will look at the rest of the conic sections.
MY REFLECTIONS (over line l)
The first time around I moved quickly thinking that the algebra skills were already in place but ultimately I found that they weren't and this really hurt a lot of students. The second time around I slowed down and did more algebra review to prepare them. This came with much more success. I also felt it important to really visualize the definition of a parabola and how it works before going to the algebra. This was also met with much more acceptance. They seemed to know what to do with the algebra better because they had a visual picture of the definition.