Interactivate: Conic Flyer Equations

Abstract

This lesson utilizes the geometric interpretations of the various conic sections to explain their equations.

Objectives

Upon completion of this lesson, students will:

understand the concept of conic sections
be able to use the distance formula to derive the equations of conic sections
understand how the equations of the conic sections relate to one another
understand the correlation between the geometric and algebraic definitions of conic sections

Standards Addressed:

Student Prerequisites

Geometry: Students must be able to:
- identify circles, parabolas, hyperbolas, and ellipses
- identify and describe cross-sections
Algebra: Students must be able to:
- work with two-dimensional graphs
- evaluate basic equation operations
Technology: Students must be able to:
- perform basic mouse manipulations such as point, click, and drag
- use a browser for experimenting with the activities

Teacher Preparation

Access to a browser
A copy of the Worksheet for each student

Key Terms

coordinate plane	A plane with a point selected as an origin, some length selected as a unit of distance, and two perpendicular lines that intersect at the origin, with positive and negative direction selected on each line. Traditionally, the lines are called x (drawn from left to right, with positive direction to the right of the origin) and y (drawn from bottom to top, with positive direction upward of the origin). Coordinates of a point are determined by the distance of this point from the lines, and the signs of the coordinates are determined by whether the point is in the positive or in the negative direction from the origin
cross section	A two-dimensional "slice" of a three dimensional object
function	A function f of a variable x is a rule that assigns to each number x in the function's domain a single number f(x). The word "single" in this definition is very important
graph of the function f	The set of all the points on the coordinate plane of the form (x, f(x)) with x in the domain of f

Lesson Outline

Focus and Review

Remind students of information they have learned in previous lessons that is relevant to the task at hand.

Ask the following opening questions:

What is the geometric definition of a circle, ellipse, hyperbola, and parabola?
How are all of these definitions related?
Is there a way to convert these geometric definitions into algebraic equations?

Objectives

Let the students know what it is they will be doing and learning today. Say something like this:

Today, we are going to learn how to derive the algebraic definitions of conic sections from their geometric definitions.
We are going to use computers to visualize these conic sections, but please do not turn your computers on or go to this website until I ask you to. I want to show you a little about this first.

Teacher Input

Introduce the derivation of conic sections by leading the class in a discussion.

If students have trouble with the concept, review the distance formula and the geometric definitions of a circle, ellipse, parabola, and hyperbola.

Guided Practice

Bring up the Conic Flyer activity on an overhead or LCD projector to demonstrate how the activity works.
Explain how manipulating the sliders at the bottom alters the layout of the graph by changing the equation.
Work through at least one example problem with the class:
- Start with an equation with values for h, k, and r already defined.
- Substitute into the equation values for x to find the corresponding values for y.
- Plot the points on graph paper and draw a conic section connecting them.
- Graph the same conic section using Conic Flyer to check your work.

Independent Practice

Have students work in pairs to sketch their predictions of the graphs of the equations on the worksheet.

When each group finishes estimating, have them calculate a few points on each graph to check their work.

When each group has estimated and plotted points for all graphs, allow them to check their work using Conic Flyer.

Closure

As a class, discuss the findings of the lesson.

Review the questions on the worksheet and compare results.
Ask students if they can now derive equations of conic sections from their geometric definitions.
Review the geometric functions of h, k, a, b, and r in the conic equations.

Alternate Outline

If students are already familiar with the equations for conic sections, this lesson can be rearranged in the following way:

Start by solving the equation of each conic section for y.
Demonstrate how these solved equations relate to the distance formula.
Discuss how the geometric interpretation of each conic section is equivalent to the algebraic interpretation.

Suggested Follow-Up

Students who understand the algebraic and geometric derivations of conic sections can explore other types of cross-sections with Cross-Section Flyer.

Interactivate

Conic Flyer Equations