Poliedros e Teorema de Euler
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Data
2013-04-13
Autores
Mialich, Flávia Renata [UNESP]
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Editor
Universidade Estadual Paulista (Unesp)
Resumo
Este trabalho tem como tema central Poliedros e o Teorema de Euler. Foi feita uma breve análise da definição de Poliedros e apresentadas algumas considerações históricas a respeito dos Poliedros e o Teorema de Euler. Foram abordadas duas versões/demonstrações do Teorema de Euler, a primeira para poliedros convexos, e a segunda, conhecida como de Teorema de Euler segundo Cauchy (que engloba certos poliedros não convexos, que são homeomorfos à esfera). Ainda, como consequência do Teorema de Euler, foi demonstrado o teorema da existência de apenas cinco poliedros regulares, conhecidos como Poliedros de Platão. Analisou-se também o conteúdo/ensino de Poliedros em certos documentos oficiais (PCN, Currículo do Estado de SP, Matrizes do SARESP e ENEM). Por último foi elaborada uma proposta de atividades explorando poliedros, o Teorema de Euler e os conteúdos de área, volume e planificação, bem como análise e resolução de algumas questões do SARESP e ENEM (relativas a tais conteúdos), utilizando, para melhor compreensão e visualização, o software matemático Poly, a fim de construir uma aprendizagem mais significativa para os alunos. Com o desenvolvimento do trabalho foi possível compreender melhor a definição de poliedros, o Teorema de Euler e refletir um pouco sobre o desenvolvimento das pesquisas Matemáticas a partir de alguns aspectos históricos. Através da análise do conteúdo, em certos documentos oficiais, pode-se verificar que o assunto/tema tratado no trabalho faz parte desses documentos e têm sido cobrados em avaliações, mas, em geral, de forma bastante simples. Observamos também, que em algumas questões analisadas os enunciados não estavam muito claros, o que pode confundir os alunos
This work has as its central theme Polyhedra and Euler's Theorem. We made a brief analysis of the definition of Polyhedra and some historical considerations about Polyhedra and Euler's Theorem. We considered two versions/proofs of Euler's Theorem, the first for convex polyhedra, and the second, known as Euler's Theorem according to Cauchy (which includes certain nonconvex polyhedra that are homeomorphic to the sphere). Also, as a consequence of Euler's theorem, it was demonstrated the theorem of existence of only five regular polyhedra, known as Plato’s Polyhedra. We also analyzed the content/teaching of Polyhedra in certain official documents (PCN, SP State Curriculum, SARESP matrices and ENEM). Finally we presented a proposal of activities exploring polyhedra, Euler's Theorem, the contents area, volume and planning, as well as an analysis and resolution of some questions from SARESP and ENEM (for such contents), by using, for better understanding and visualization, the Poly mathematical software in order to build a more meaningful learning for students. With the development of this work we got a better understand of the definition of polyhedra, the Euler's Theorem and we reflected a little on research and development of mathematics from some historical aspects. By analysing the content in certain official documents, it can be seen that the subject/topics covered in this work are parts of these documents and have been rated in tests, but generally in a quite simple form. We also observed that in some questions discussed, the statements were not very clear, which can confuse the students
This work has as its central theme Polyhedra and Euler's Theorem. We made a brief analysis of the definition of Polyhedra and some historical considerations about Polyhedra and Euler's Theorem. We considered two versions/proofs of Euler's Theorem, the first for convex polyhedra, and the second, known as Euler's Theorem according to Cauchy (which includes certain nonconvex polyhedra that are homeomorphic to the sphere). Also, as a consequence of Euler's theorem, it was demonstrated the theorem of existence of only five regular polyhedra, known as Plato’s Polyhedra. We also analyzed the content/teaching of Polyhedra in certain official documents (PCN, SP State Curriculum, SARESP matrices and ENEM). Finally we presented a proposal of activities exploring polyhedra, Euler's Theorem, the contents area, volume and planning, as well as an analysis and resolution of some questions from SARESP and ENEM (for such contents), by using, for better understanding and visualization, the Poly mathematical software in order to build a more meaningful learning for students. With the development of this work we got a better understand of the definition of polyhedra, the Euler's Theorem and we reflected a little on research and development of mathematics from some historical aspects. By analysing the content in certain official documents, it can be seen that the subject/topics covered in this work are parts of these documents and have been rated in tests, but generally in a quite simple form. We also observed that in some questions discussed, the statements were not very clear, which can confuse the students
Descrição
Palavras-chave
Poliedros, Euler, Teorema de, Matemática - Estudo e ensino, Polyhedra
Como citar
MIALICH, Flávia Renata. Poliedros e Teorema de Euler. 2013. 79 f. Dissertação (mestrado) - Universidade Estadual Paulista Julio de Mesquita Filho. Instituto de Biociências, Letras e Ciências Exatas, 2013.