Modeling the Mirascope Using Dynamic Technology

Abstract

From a problem solving perspective, this article analyzes the physical and mathematical properties of the mirascope and further seeks to model the mirascope using new dynamic learning technology. Open-ended explorations will be discussed using the mathematical mirascope, showcasing the integration of algebra and geometry and the affordances of technological tools. Several intermediate dynamic worksheets are provided as scaffolding to serve the diverse needs of readers.

Contents

• 1. Introduction
• 2. Preliminaries and Heuristics
  • 2.1. The Plane Mirror
  • 2.2. The Parabolic Mirror
    • 2.2.1. How light is reflected by a parabolic mirror
    • 2.2.2. How to find the focus of a parabola
  • 2.3. Design Heuristics
• 3. Mirascope Construction Using GeoGebra
  • 3.1. Define Two Parabolas Algebraically
  • 3.2. Modeling Light Reflections Within the Mirascope
• 4. Using the Simulation: Explorations and Exercises
  • 4.1. Is the Whole Image Visible?
  • 4.2. Are More Light Rays Necessary to Locate the Image of a Point?
  • 4.3. Exercises
  • 4.4. Managing Complexity
• 5. Conclusion
  • Acknowledgements
  • References
• Additional Figures

This article uses the open-source mathematics learning environment GeoGebra as a primary platform. Java runtime is required to access the dynamic materials. Although GeoGebra installation is not required, readers are advised to have GeoGebra running through either WebStart or local installation in order to experiment with the ideas proposed in the article. It is also worth noting that right-clicking in GeoGebra provides users with most tools related to a mathematical object such as tracing and styles.

Dynamic Materials

(Each of the links below opens a Geogebra activity in a new window.)

• How we see things in a plane mirror
• How we might see things in a parabolic mirror
• How to find the focus of a parabola geometrically defined
• How to find the focus of a parabola algebraically defined
• Initial setup for the mirascope
• A mirascope with one light ray reflected out
• A complete mirascope with image of a point identified
• A complete mirascope for further exercises
• A complete mirascope with focus identified
• A complete mirascope with foci and vertices aligned
• The complete construction protocol