About This Chapter
Standard: Derive the equation of a parabola given a focus and directrix. (CCSS.Math.Content.HSG-GPE.A.2)
Standard: Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. (CCSS.Math.Content.HSG-GPE.A.3)
About This Chapter
This standard focuses on deriving the equations of various conic sections, including parabolas, ellipses and hyperbolas. In addition to deriving equations, students who understand the standard will be able to find the focus, directrix and definition of various conic shapes.
The lessons in this standard show students how to:
- Find the focus and directrix of a parabola
- Uncover the focus and definition of an ellipse and a hyperbola
- Discover the equation of a parabola from the focus and directrix
- Derive the equation of an ellipse or a hyperbola from the foci
Individuals with mastery of this standard will know how to find the focus and directrix of a parabola, as well as the focus and definition of an ellipse and a hyperbola. They will also be able to find the equation of a parabola, ellipse and hyperbola from the focus.
How to Use These Lessons in Your Classroom
Below are tips for utilizing these lessons in the classroom to complement instruction in the CCSS.Math.Content.HSG-GPE.A.2 and CCSS.Math.Content.HSG-GPE.A.3 standards:
Working with Parabolas Lessons
Assign the chapters The Focus and Directrix of a Parabola and Finding the Equation of a Parabola from the Focus and Directrix for homework. Have students break into small groups and switch off between finding the focus and directrix from a parabola equation and its opposite operation of using the parabola equation to discover the focus and directrix.
Using Ellipses and Hyperbolas
Split students into two groups: one that covers ellipses and another group that covers hyperbolas. Have each group answer questions requiring them to derive the equation and find the focus and definition of their respective conic section. After they've answered a few questions, tell the students to switch.
Conic Section Practice Lessons
Have students watch the Practice with Conic Sections video lesson and answer all example questions. Go over these practice quiz answers in class.
1. The Focus and Directrix of a Parabola
In this lesson, we will review what a parabola is, then we will look at the formal definition of a parabola, introducing the focus and directrix of a parabola. We will look at some examples to help solidify our understanding of these concepts.
2. Finding the Equation of a Parabola from the Focus and Directrix
A parabola is the familiar shape seen in many physical applications, like the path taken by a ball thrown upwards. This lesson explores equations for the parabola and shows how they may be obtained from two quantities: the focus and the directrix.
3. Foci and the Definitions of Ellipses and Hyperbolas
In this lesson, we'll look at the definition of an ellipse and a hyperbola. We'll use the foci of each of these to define them technically and formally, and we'll look at some examples to make the definitions more understandable.
4. Derive the Equation of an Ellipse from the Foci
In this lesson, you're going to learn the definition of an ellipse and foci, the standard forms of the equation for an ellipse, and how to find such an equation when given the foci.
5. Derive the Equation of a Hyperbola from the Foci
This lesson will go over what a hyperbola is and walk through the steps of finding the equation of a hyperbola given just the foci and vertex. After learning the process, we will look at an example of finding a hyperbola equation given this information.
6. Practice with the Conic Sections
Conic sections are shapes created by cutting through a 3D cone. In this lesson, learn how to identify each conic section from its graph and characteristic equation.
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