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Dynamic Mathematics with GeoGebra
In this article we introduce the free educational mathematics software GeoGebra. This open source tool extends concepts of dynamic geometry to the fields of algebra and calculus.
Loci
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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
- 2.3. Design Heuristics
- 3. Mirascope Construction Using GeoGebra
- 4. Using the Simulation: Explorations and Exercises
- 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
Bu, Lingguo, "Modeling the Mirascope Using Dynamic Technology," Loci (November 2010), DOI: 10.4169/loci003595
Discuss this article
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thread #1:
Will other shapes?
Since the publication of article, I have received several emails about the mirascope and its GeoGebra-based explorations. Among them are (1) can we use a sphere? (2)how could you use it in the dark (place a LED inside)?, (3) what is the point of having students make their own? I welcome your ideas and suggestions. Thanks for your interest.
thread #2:
The role of models for transferring understanding
I have a mirascope on my desk in my office. Based on the interactions I see between office visitors and the mirascope, it is the single most intriguing object in my office (and yes, this does check my ego). I put it on my desk because of a conversation I overheard one day while sitting in a coffee shop. There were two male university students next to me, and they were highly intrigued by a question that one of them had posed. He asked "Why does our brain confulse left and right when we see a reflection in a mirror, but we don't get confused with the up and down direction in the reflections?" These two people generated several possible explanations (I suppose I might call them hypotheses), and all of their reasons were based on incoherent bits of knowledge about psychology and neuroscience rather than notions about optics or even a passing reference to a physical model. They eventually encountered their reflections in the spoons on their table, dared to ask each other how their reflection could be upside down, and thus they were completely baffled (ending the conversation with a shrug and apparent feeling of helplessness). I had a laptop with GeoGebra on my table. I was dying to introduce them to a model that might help them tie these incoherent ideas together, but I chose to finish my work so that I could return home to play with my one year old daughter. If I had known of this article, I would have shown it to the two university students and let them disentangle their ideas. Lingguo Bu has very elegantly taken these ideas to the next level with a beautiful GeoGebra-based model of the basic optics involved in the mirascope, showing a complex integration of basic ideas that result in a fascinating phenomenon. Does the author or my fellow readers know of other GeoGebra models of basic ideas and related ideas that may be even more complex? I would love to put them together into a coherent unit related to optics and geometry.