Gabriel Neves, Jul 9, 2015

Book created as support for the G3 presentation. Describes the experience of giving a mini course in a university to a public composed for experienced teachers and researchers in mathematics, as well as the results of it, which is a 3D representation for a function of complex variables.

As the result of a mini course on the basic aspects of GeoGebra ministered at the Federal University of Rio de Janeiro (UFRJ), for teachers of the Institute of Mathematics, the present work describes the experience as well as the results of it. Widely inserted and explored in basic school, it is interesting to note the perspective of researchers from different fields of mathematics over the software. With a respectable range of mathematical software at disposal, many of them are in a comfortable position, what may retract their curiosity about new possibilities of resources and tools. With surprising results, the course revealed enormous potential of these professionals to develop knowledge and resources through GeoGebra. Given the domain of the basic functions of the software, brilliant constructions were created and great possibilities were glimpsed. As a suggestion made for one of the professors, a construction on complex variables functions was thought to illustrate some of the contents of a regular superior course in mathematics, and that's the main subject of this presentation. As a 2D model was first easily made, the a idea became to, somehow, go from the plane to the 3D space, making use of its possibilities in creating a better view of the function.

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This was the first model created. It was quite simple to be made, almost immediately after one of the lectures, with the ideas and collaboration of professor Felipe Acker. It takes, in the same plane, one circle, of which we are able to control the length of the radius, a point on this circle, which is animated in order to make it variate through all the complex numbers with a given absolute value, and, finally, the image of this point through a polynomial function. GeoGebra can make the translation between point in the plane and complex number, what is good for us, and we make use of it. As for the of the construction, the geometrical place of the image, as we variate the point on the circle, is built.

• 1. Função Polinomial de Variáveis Complexas - 2D

The 2D graphic, mentioned in chapter 2, was created, originally, with the intention to be evolved into a 3D representation. However, there are some limitations that GeoGebra, in its 3D mode, presents that have to be dealt with. Locus, for instance, was something that brought great difficult for the construction. Later on, the persistence of this idea motivated the construction that is subject of this chapter. Once the locus isn't so complete in the GeoGebra 3D mode, the use of trail took its place. In the same space, a cone is built to represent the growth of the radius of the circle created in the 2D construction. A list of planes is plotted and in each one is created the image of a point ranging on a circle (the intersection of each plane with the cone) in the same plane. So, it's basically, a systematic repetition of the 2D construction on the 3D space, some sort of the reorganization of the domain.The result is a beautiful graphic, representing the growth of the absolute values of the images, as we increase the absolute value of the domain.

• 1. Esquema Visual 3D de Função com Variáveis Complexas

As result of the mini course success, a publication in the form of teaching material is under development. It will be structured into five lectures, which were held during the mini-course, addressing from Euclidean geometry to the differential and integral calculus. There is, still, the possibility of a brand new mini course, more advanced, to be offered and, similarly, the production of educational materials, with topics ranging from programming on GeoGebra to complex variable functions. Much more important, however, are the products of this mini course. These two constructions, especially the second one, are the most memorable exemples, but much were learnt, much were built, making the whole event a really special experience.

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