Construção e medida de volume dos poliedros regulares convexos com o Cabri 3D: uma possível transposição didática

• Ano de defesa: 2016
• Autor(a) principal: Santos, Amarildo Aparecido dos
• Orientador(a): Silva, Maria José Ferreira da
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Pontifícia Universidade Católica de São Paulo
• Programa de Pós-Graduação: Programa de Estudos Pós-Graduados em Educação Matemática
• Departamento: Faculdade de Ciências Exatas e Tecnologia
• País: Brasil
• Palavras-chave em Português: Poliedros regulares convexos; Construção geométrica; Medida de volume
• Palavras-chave em Inglês: Regular convex polyhedra; Construction; Volume measure
• Área do conhecimento CNPq: CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA
• Link de acesso: https://tede2.pucsp.br/handle/handle/19665

This research aims to explore the construction of regular convex polyhedra in Cabri-3D as a possible didactic transposition on the construction of polyhedral surfaces developed by Euclid (300 BC), transformed to a current language, and verifying if this construction presents the necessary relations for the development of formulas for the volume measure calculation of these polyhedra. We oriented ourselves by the following question of research: Does the construction of regular polyhedra by Euclid method propitiate the measure calculation of their volumes as well as the composition and decomposition of dodecahedron and the icosahedron? The theorical referencial is based on the notion of Didatic Transposition and the Ecological Problematic of Yves Chevallard and the Registry of Durval’s Semiotics Representation specifically on the sequential, perceptive, operative and discursive apprehensions, in addition to the four ways of seeing (to look) the pictures in function of the roles that they perform in the activities of geometry: the botanical, the topographer geometer, the constructor and inventor-woodworker. The research is of qualitative nature of documental type because it is based on the reading, analysis and interpretation of Book XIII, the Elements of Euclid, that approach the construction of the regular tetrahedron, regular hexahedron, regular octahedron, regular dodecahedron and the regular icosahedron. The procedures are developed in three parts. On the first part we explored the constructions proposed by Euclid and we adapted them so that they could be constructed with the tools in the environment of Cabri-3D dynamic representation, and we noted that every regular convex polyedra can be constructed in this environment. On the second part we explored the constructions accomplished and search relations and measures that allow to deduce formulas for the volume measure calculation of these polyhedra, in function of the measure of their edges, as much as in function of spheres diameter measure that circumscribe them. The dynamism of the software favoured the visualization of these relations and measures. On the third part we searched the necessary conditions to determine if the pentagon based pyramid can or cannot be part of a regular dodecahedron, as well as the conditions for a tetrahedron can or cannot compound a regular icosahedron. From the determination of these conditions we could propose the construction of these two polyhedron by composition in Cabri-3D and deduct a formula for the volume measure calculation by the volume measure of one of the pyramids that compose it. Thus, we believe that our question was answered and the hypothesis raised during the research were validated, conducting ourselves to develop, futurely, a sequel for its education