Loci (2008)

Mathematical Brooding over an Egg, André Heck

Dynamic geometry software like Cabri Geometry, Cinderella, and Geometer's Sketchpad are computer programs that turn out to be very suitable tools at secondary school level for investigations and experimentations in plane geometry. To a certain extent they can also be used for studying topics in analysis. What complicates the latter type of use of dynamic geometry software is that the descriptive geometry approach and the analytical geometry approach are, in most packages, not at an equal level. Analytical expressions for geometrical objects can be obtained by introducing a coordinate system. But conversely, coordinates and algebraic equations can not be used to define geometrical objects. This imbalance motivated Markus Hohenwarter (2007a) to develop GeoGebra as a dynamic mathematics program in which geometrical and algebraic representations are bi-directionally connected. Noticeable in the applications in both domains is that the information technology use is relatively close to the problems and figures that one finds in mathematics textbooks. The main difference is the dynamic nature of the computer program. This observation also holds in many classroom projects in which a realistic situation is investigated with a dynamic geometry package. An example is the German project Mathematik rund ums Ei [in German; Mathematics about Eggs], in which students investigated the shape of an egg, amongst other things, and estimated the volume and surface area of an egg. Students used the dynamic geometry package Cinderella only to construct egg-shaped curves; a real egg was quickly out of sight.

But this can be done differently in dynamic geometry packages that allow dynamic use of digital images. Cabri II Plus, Cinderella, Geometer's Sketchpad and GeoGebra are examples of such packages in which a digital image may serve as a background for a geometric construction and may even be dynamically connected with geometric objects in a construction. This offers various opportunities to create a strong connection between mathematics and the real world, for example in the form of a practical investigation task. In this article I illustrate the mathematical investigation of a real egg, using the free and multi-platform dynamic mathematics package GeoGebra. The following research question is discussed in particular: "What is the volume and surface area of a given hen's egg?" Here, algebra and geometry go hand in hand. In the last section of this article I briefly discuss opportunities that measurement and manipulation of digital images in a dynamic geometry package offer in education on mathematical modeling. I hope and expect to give an inspiring example the use of information technology in a practical investigation in which mathematics is linked with doing experiments with an object from daily life.