Distância Entre Curvas Algébricas: Uma Discussão para o Ensino Médio

Abstract

This research investigates the various methods used to teach ŞDistance between Closed CurvesŤ in High School, by the association between abstract mathematical concepts and the various objects which are taught at this school level. Hence, the introduction of basic topology concepts was proposed for the elementary and middle school grades. Distance between Closed Curves was the main focus was this study. The guiding question for this investigation was: how can we determine the distance between curves? For a didactic transposition, this project sought to promote a relationship between high school studentsŠ elementary mathematical knowledge and abstract notions of distance between curves on a plane. To further expand this analysis, other questions were asked to complement the Ąrst one. They were: what is distance?; and, how can students use the intuitive understanding of distance to place it between objects on a plane? Methodically, this study was based on bibliographic studies, on the elaboration of didactic sequences and on the use of technology to optimize and ensure studentsŠ successful learning of this subject even in elementary and middle school grades. The results show that there is a connection between plane algebraic curves, curve spacing, metric space, Haudorff distance, and distance between curves that may, if presented in an articulated manner, help students to associate the concept of distance between sets in metric space and the distance between closed plane curves. Therefore, although this research assumed the existence of ideal teaching conditions, such as accessibility to computer programs in the classroom, teaching these complex contents in High School is feasible, contributing to the development of diverse mathematical notions. Keywords: Plane Curves, Metric Space, Hausdorff Measures, Didatics, Mathematics (High School) - Study and teaching - Methodology.