LESSON PLAN #1 of 1

 

Name: Robert Duncan

 

Title of lesson: Equation of a Parabola for Dobsonian Telescopes

 

Date of lesson: 3/29

 

Length of lesson:

           One 90 minute period

 

Description of the class:

                     Course Title: Algebra II

                     Grade level: 11

 

Source of the lesson: Original

 

TEKS addressed:

                       ¤111.33. Algebra II (One-Half to One Credit).

(a) Basic understandings.

            (1) Foundation concepts for high school mathematics.

            (2) Algebraic thinking and symbolic reasoning.

            (3) Functions, equations, and their relationship.

            (4) Relationship between algebra and geometry.

            (5) Tools for algebraic thinking.

(c) Algebra and geometry: knowledge and skills and performance descriptions.

            (1) The student connects algebraic and geometric representations of functions.

                        (A) The student identifies and sketches graphs of parent functions,                including linear (y = x), quadratic (y = x2), square root (y = … x), inverse                  (y = 1/x), exponential (y = ax), and logarithmic (y = logax) functions.

                        (B) The student extends parent functions with parameters such as m in y =               mx and describes parameter changes on the graph of parent functions.

(d) Quadratic and square root functions: knowledge and skills and performance descriptions.

            (1) The student understands that quadratic functions can be represented in                different ways and translates among their various representations.

                        (B) The student relates representations of quadratic functions, such as                       algebraic, tabular, graphical, and verbal descriptions.

                        (C) The student determines a quadratic function from its roots or a graph.

 

The Lesson:

I.               Overview

Students explore the definition of a parabola first graphically, then algebraically.  Students then relate the parabola with the mirrors they will be using.

 

                    II.  Performance or learner outcomes

Students will be able to:

            -Understand the definition of a parabola

            -Find the equation of a parabola with focus on the y-axis

            -Predict changes to the graph of a parabola given changes in the                         focus and directrix

            -Understand how a circle and parabola behave similarly near the                         vertex

            

III.             Resources, materials and supplies needed

Geometer's Sketchpad

Computers for Students

(preferably)LabPro or other projection monitor

 

 

IV. Supplementary materials, handouts.

            
Five-E Organization

 

Teacher Does                                              Student Does

Engage:

Intro to Sketchpad activity

 

       Questions

 

 

Students follow directions on intro page to gain familiarity/skill with sketchpad.

 

      Student Response

                                                                 Evaluate

Visual inspection while walking around the room

 

Explore:

Students are given the definition of a parabola and explore the graph of the parabola.

 

Teacher helps with sketchpad and clarifies directions as necessary.

 

       Questions

 

 

First activity: Students are given a point, a line, and a circle.  The screen displays the distance from the circle to the point and to the line.  Students move the circle until the two distances are equal and then put a point there.  Repeat until they notice the form the graph is taking.

 

Second Activity: Students are given a point which always stays the same distance between the focus and directrix.  The point leaves a trail as it is dragged, so that students can better see the form of a parabola.

 

Third Activity: Students are given a graph and told to change the location of the directrix and observe its effect on the shape of the graph and location of the focus.  The goal is for students to notice the relationship between the location of the focus and the equation of the directrix.

 

Fourth Activity: Students answer questions about the directrix and focus.  Using the distance formula, (and probably some help from the teacher) they will generate the formula for a parabola.

 

Fifth Activity: Using the equation of a parabola, students make conjectures about how changes in the graph will affect the formula, and vice versa.

 

     Evaluate

Have students answer numbered questions on their own paper, as well as through visual inspection and class discussion.

 

Explain:

Going from parabolas to parabolic mirrors.

If students have not done the reflection lesson yet, give a short explanation on how light reflects off of a parabolic mirror.

 

If necessary and if time permits, also explain why incoming light rays come in perpendicular.

       Questions

 

 

 

Sixth Activity: Given a parabolic mirror and incoming light rays, students drag the point of reflection around and see the effect on the reflected light rays.  (ctrl+b will clean up the traced lines)  When students conjecture about the location of the focus, the button on the screen will show its location.

 

 

 

      Student Response

     Evaluate

Class discussion

 

Extend / Elaborate:

Show how parabolic mirrors and circular mirrors behave similarly near the vertex

 

Explain that this is the idea the idea they will be using when finding the focal length of their mirrors later.

 

       Questions

 

 

Seventh Activity: Similar to Activity 6, but this time with a circular mirror.  Students again drag the point of reflection around and see when the reflected light rays converge.  Once they identify the location, students are told to drag point D to this location.  The page shows the distance between point D and the center of the circle, as well as point D and the vertex of the graph.  At the focal point, these distances will be equal.