Conics and Reflection
This material is replicated on a number of sites
as part of the
SERC Pedagogic Service Project
Initial Publication Date: March 17, 2010
Summary
In this activity, students will investigate the reflective properties of the parabola, the ellipse, and the hyperbola. They will look at applications that relate to both the collection of incoming signals and signals that are emitted from a source and reflected by the conic. They will first make discoveries utilizing interactive animations and then will complete a worksheet where they will write down these observations.
CollapseLearning Goals
1. Develop and understanding of the reflective properties of the parabola, the ellipse, and the hyperbola
2. Figure out the use of the reflective properties in the context of lighting devices, receptors, and other technologies.
3. Understand that the conics sections are more than just a set of equations or geometric axioms.
2. Figure out the use of the reflective properties in the context of lighting devices, receptors, and other technologies.
3. Understand that the conics sections are more than just a set of equations or geometric axioms.
Context for Use
This activity is designed for students in a PreCalculus or College Algebra class. Students can either be college student or high school students. The students should already be familiar with the equations and shapes of the conic sections. Students will also need to use a computer that is equipped with the JAVA plug-in.
Description and Teaching Materials
The students start out by exploring the following three JAVA applets (links unavailable):
1. www.ies-math.com/math/java/conics/focus/focus.html
2. www.ies-math.com/math/java/conics/focus_ellipse/focus_ellipse.html
3. www.ies-math.com/math/java/conics/focus_hyper/focus_hyper.html
The best situation would be for the students to work in a computer lab environment where they explore the properties. Otherwise, the instructor can present the applets using an Internet enabled computer with a projector. During the exploration period the instructor can facilitate discussion that will lead the students towards more thoughtful answers to the worksheet questions.
After exploring with the applets, the students are given the worksheet. Students can either work on their own or in pairs or groups.
1. www.ies-math.com/math/java/conics/focus/focus.html
2. www.ies-math.com/math/java/conics/focus_ellipse/focus_ellipse.html
3. www.ies-math.com/math/java/conics/focus_hyper/focus_hyper.html
The best situation would be for the students to work in a computer lab environment where they explore the properties. Otherwise, the instructor can present the applets using an Internet enabled computer with a projector. During the exploration period the instructor can facilitate discussion that will lead the students towards more thoughtful answers to the worksheet questions.
After exploring with the applets, the students are given the worksheet. Students can either work on their own or in pairs or groups.
Teaching Notes and Tips
The instructor should first make sure that a new enough version of the JAVA plug-in is installed. Many students are not used to this type of mathematical inquiry and will need plenty of guidance. Students can first be encouraged to verbalize their answers to a partner while the partner writes down what is said.
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Assessment
The assessment is up to the instructor. The instructor can first assess via participation in the exploration part. Then the instructor can look at the worksheets that the students turn in to see whether the students understood what reflection is, how it differs depending on what conic is used as the reflector. The instructor can access the students understanding of light coming in vs. light being emitted and also the understanding of the application of reflection of each conic.
References and Resources
Manipula Math (Conics)
This site contains 9 JAVA applets that allow the student to interactively discover the reflective and other geometric properties of conic sections.
This site contains 9 JAVA applets that allow the student to interactively discover the reflective and other geometric properties of conic sections.