Geometria fractal no ensino médio: potencialidades para o ensino da matemática

Abstract

Fractal Geometry is a recent field of mathematics that aims to study the geometry of nature, considered "formless" by Euclidean Geometry, which receives greater emphasis in the Basic Education curriculum. Characterized as qualitative and exploratory in nature, this research aimed to analyze how Fractal Geometry can be approached in High School and contribute to the teaching of contents already provided in the curriculum. From a bibliographic survey on the evolution of Fractal Geometry, its characteristics and aspects of the exact self-similarity fractals Cantor Set, Cantor Dust, Koch Curve and Island, Menger Sponge, and Sierpinski Triangle and Carpet, characteristics of these that relate to contents provided in the Mathematics and its Technologies curriculum were highlighted, and finally, suggestions of activities that can be developed in High School classes were elaborated. Among the skills expected for the High School stage in the area of mathematics, four skills from the National Common Curricular Base (BNCC) were identified that open up space for the study of Fractal Geometry. Furthermore, it was shown that the fractals of exact self-similarity highlighted in this study are constructed from figures and solids of Euclidean Geometry, which allows their study based on students' prior knowledge. In this way, it is possible to study the geometric relations of self-similarity fractals (length, perimeter, area and volume) based on knowledge that students have already constructed regarding Euclidean objects. The study of Fractal Geometry, when carried out in this way, provides an opportunity to review and reinforce content from the Elementary School stage and to develop a more complex mathematical thinking, which is the objective of the High School stage as established by the BNCC. From this perspective, Fractal Geometry tends to enhance the teaching of mathematics, sharpening students' mathematical thinking and making them aware of the scope of this area of knowledge.