Simetrias para o ensino de equações e funções na Educação básica

Abstract

Symmetries are widely found in nature, but do not have many written works on their properties and applications. In regard to school textbooks, more specifically those that deal with mathematics, treat them superficially. In the pedagogical practice textbooks with a deep approach to symmetries and their applications are rarely found. While reading those textbooks used in primary education and secondary education we have found symmetries applied frequently in the past four years, but the same does not happen with respect to a more intrinsic application in particular topics in the teaching of mathematics or geometry. Most textbooks are restricted to axial, translational and rotational symmetries applied to geometric figures, and seldom make a parallel to any interdisciplinary aspect, for example, to show that the majority of living beings exhibit bilateral symmetry. However, in the general sense, they do not make an analytical study of symmetries. One of the most important work on symmetries was written by the German physicist Kepler while attempting to explain the symmetries of the most coveted symmetrical object at that time: snowake of six points. In this dissertation, we describe a work that has been done over ten years in public and private schools, and that has had positive results on mathematics teaching. We begin with an approach to mathematical education, the ways in which symmetries arise in nature, in architecture, and visual arts. The importance of symmetries for the classical, relativistic and quantum physics. Then, we present an educational program which takes into consideration one logical-deductive sequence of teaching of mathematics whose scope is the resolution of problems whose solutions need the resolution of equations of first and second degrees, as well as systems of equations. The first-degree equations are solved using axial symmetry in which we consider the equal sign is the axis of symmetry that keeps the balance of equation. Second-degree equations are solved by the method of complete on squares keeping unchanged the axial symmetry of the equation, thus avoiding the use of formula of Bhaskara. Systems of equations are solved using the principle of translational symmetry, where one isolates the same term unknown in two or more equations of the system. This work culminates with an analytical study of the behavior of linear, quadratic, exponential, logarithmic, and trigonometric functions, considering applications of the operations of symmetry. The curve reections are held (axially symmetrical) representing the functions in relation to the coordinate axes, as well as compared to any straight line and even to the bisectrix of the odd quadrants of the orthogonal Cartesian plane, obtaining their several inverse functions; furthermore, we also analyzed the rotational symmetry of each curve representing the functions relative to the origin of the orthogonal Cartesian plane. From our teaching experience, we have noticed that students have had a great openness to interdisciplinary works and through this learning mechanism have developed a great ability to algebraic and geometric perception and a greater enthusiasm for the study of other sciences.