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:

Grade 10

Functions and Relationships

The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.

The student demonstrates algebraic thinking.

Grade 9

Functions and Relationships

The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.

The student demonstrates algebraic thinking.

Algebra

Reasoning with Equations and Inequalities

Understand solving equations as a process of reasoning and explain the reasoning

Represent and solve equations and inequalities graphically

Seeing Structure in Expressions

Write expressions in equivalent forms to solve problems

Functions

Building Functions

Build new functions from existing functions

Interpreting Functions

Interpret functions that arise in applications in terms of the context

Analyze functions using different representations

Linear, Quadratic, and Exponential Models

Construct and compare linear, quadratic, and exponential models and solve problems

Geometry

Circles

Understand and apply theorems about circles

Expressing Geometric Properties with Equations

Translate between the geometric description and the equation for a conic section

Integrated Mathematics

Geometry and Measurement

Competency Goal 2: The learner will use properties of geometric figures to solve problems.

Pre-Calculus

Geometry and Measurement

Competency Goal 2: The learner will use relations and functions to solve problems.

Secondary

Mathematical Analysis

MA.08 The student will investigate and identify the characteristics of conic section equations in (h, k) and standard forms. The techniques of translation and rotation of axes in the coordinate plane will be used to graph conic sections.

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

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.