Navegando por Assunto "Geometria"

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Um breve estudo sobre a curvatura média e o teorema de Aleksandrov
(2022-06-08) Lira, Yasmin Alves Sobrinho; Gomes, Renato Teixeira; http://lattes.cnpq.br/0570606157057337; http://lattes.cnpq.br/4862014205090674
When looking for the eigenvalues of the differential of the normal Gauss map dN, naturally arise in its characteristic polynomial two functions that are invariant by change of base of this operator: the determinant of the matrix of the normal Gauss map, called Gaussian curvature and the trace of this application. As this linear map is self-adjoint,there is an orthonormal basis in which its matrix is written diagonally in terms of the principal curvatures, and its determinant and trace are given by det(dN) = (−k1)(−k2) and its dash by tr(dN) = −(k1+k2). The negative half of the H = k1 + k2/2 is the so-called mean curvature, which was introduced by French mathematician Sophie Germain when studying a problem related to membrane vibrations. At this time, a problem proposed by Lagrange, which later received the name of Plateau’s problem, a Belgian physicist who carried out several experiments and in-depth studies on soap films around 1850, was, roughly speaking, to determine a surface that has the smallest area among those which have the edge given by a prescribed Jordan curve. It can be shown that such a surface has zero mean curvature at its regular points. Such surfaces are called minimal and are named after Lagrange. In this work we will make a brief study on mean curvature and minimal surfaces,demonstrating some results and presenting some examples of such surfaces. Furthermore, we will demonstrate Aleksandrov’s theorem which under certain assumptions says that the only compact surface with constant mean curvature in R3 is the sphere. For this, we will demonstrate this result with a different “machinery” the one used by Aleksandrov. We will follow R. Reilly’s approach in his article “Mean Curvature, the Laplacian and Soap Bubbles” which makes use of more basic knowledge of differential and integral calculus and surface theory for its demonstration.
Construções de fractais com o GeoGebra e dimensão fractal
(2021-12-21) Souto, Rafael Almeida; Tanaka, Thiago Yukio; Didier, Maria Ângela Caldas; http://lattes.cnpq.br/9721552594807972; http://lattes.cnpq.br/3394446426392577; http://lattes.cnpq.br/5859671240920200
In this work, we will present the elements and concepts related to Fractal Geometry such as its definition, classification into types, properties, and some measurable characteristics such as area, perimeter, and dimension measurements. At first, we will focus on the characterization of the most classic fractals of the theory, such as the Sierpinski triangle, the Koch curve, the Cantor set, among others. We will show how to build these objects using the mathematical tool of geometric transformations and their matrix translations and the implementation of these concepts through GeoGebra dynamic geometry software, which allows us to build the vast majority of fractals that will be mentioned during the work. Finally, we will also present a study on the concept of fractal dimension, whose applications are vast in various areas such as Economics, Medicine, Biology, among others. More precisely, we will present two methods of obtaining a fractal dimension, the first using the Hausdorff-Besicovitch method and a second way using the box-counting method. We believe that this monograph can be used as the first guiding material for studies and research in the field of Fractal Geometry, mainly due to the richness of details, especially for those who are unaware or know little about the theory. In addition, by bringing construction methods with GeoGebra, we believe that the material also serves as a guide to guide the use of theory in the classroom for students of the Licentiate Degree in Mathematics, future teachers. Finally, for those who already have a basic knowledge of fractals, the dimension study serves as a basis for guiding the application of this object.
Ensino e aprendizagem dos polígonos regulares: uma abordagem lúdica sobre os movimentos de simetria no ensino fundamental
(2022-10-19) Amorim, Welisson de Almeida; Silva, Thiago Dias Oliveira; http://lattes.cnpq.br/7439995985621562; http://lattes.cnpq.br/3297151195837218
The teaching of geometry in elementary school is linked to the visualization of the geometric object and essential mathematical notions. When possible, one should co-relate physical objects with the theory in order to establish a better understanding for the students. Thus, this work uses this argument for its construction and addresses an intimate relationship with the use of physical objects and theory. This work focuses on presenting the teaching and learning process of regular polygons about their symmetry movements. Given in a systematic way, this process is given through the relationship with the theory of dihedral groups and the use of playful materials that permeate children’s daily lives. In particular, such materials are related to the character “Patrick Estrela" from the children’s series “SpongeBob” and a children’s board game called “Gometric board”. Regular polygons but also their characteristics. Speaking of which, a very important characteristic of regular polygons is their invariant form when rotation and reflection movements are applied. Hence, this work uses this property and the physical/playful objects already mentioned to establish a better assimilation in the teaching and learning about regular polygons When formulated, such relationships will be applied in a 6th grade elementary school class, explaining at the end all the results of the intervention.
Geometria na OBMEP: uma análise de questões da primeira fase do nível 2 à luz da BNCC
(2023-09-20) Silva, Luiz Guilherme Machado e; Barboza, Eudes Mendes; http://lattes.cnpq.br/9426464458648172; http://lattes.cnpq.br/0988088024884324
The Brazilian Mathematical Olympiad for Public Schools (OBMEP) is applied nationally and reaches a large proportion of Brazilian students. Thinking about it, this work aims to carry out a data survey, in order to clarify which are the skills of the National Common Curricular Base (BNCC) of the thematic unit of Geometry that have more recurrence in OBMEP. For this, the questions dealing with Geometry at OBMEP, level 2, 1st phase, between the years 2017 and 2022 were analyzed.It is worth noting that in the years 2020 and 2021 there were no applications of the tests due to the Covid-19 pandemic. Thereby, this work is a facilitator material for teachers who seek to work Geometry in the classroom using Olympic questions and also for students who intend to participate in Mathematics Olympics.
Jogo baralho geométrico: possibilidades e limitações para o ensino de geometria no 2º ano do ensino fundamental
(2019-02-08) Berto, Maria Lucicleide da Silva; Teles, Rosinalda Aurora de Melo; http://lattes.cnpq.br/8888500885370084; http://lattes.cnpq.br/1111622301812100
This paper discusses the use of the game Geometric Deck as a didactic resource to teach geometry, especially the association between graphical representations of geometric figures with graphical representations of objects of the physical world and the use of correct nomenclature. We seek support in research in the field of Mathematics Education, especially in those who defend Geometry as one of the best ways for children to understand the shapes and the space around them. Also in some that discuss the use of games in Mathematics classes, as a pedagogical resource that assists in teaching and learning, as well as in curricular guidelines in force in our country. From this, we propose as a research question: how the use of the game Geometric Deck in the early years of elementary school can contribute significantly to the students to make associations between objects of geometry and objects of the physical world using the correct nomenclatures? The main objective of this Course Conclusion Work was to analyze possibilities and limitations of the use of the game Geometric Deck for the teaching of geometry in the initial years of elementary school. It has as its starting point an action research, developed in two stages in a municipal school, in a class of second year of elementary school. We use as data collection instruments, a pre-test and the application of said game. In addition, videotaped observations were made in order to better understand students' performance in the activities proposed with the game. The pre-test, composed of four multiple-choice questions, made it possible to delineate the research direction, pointing out the knowledge that the students already had in relation to the subject and what were the main difficulties, especially those related to the use of correct linguistic terms for each flat or spatial geometric figure, and the association of geometric figures with physical objects. The intervention with the game aimed at minimizing such difficulties, so that they could evolve in the construction of their own knowledge. As well as to verify the possibilities and limitations that the game offered, serving as subsidies to propose reformulations of the same for new applications. In the experience of the game with a small group of students, we found difficulties regarding the association of geometric figures with physical objects and the use of the correct names of the geometric figures persisted. After the necessary steps to understand the rules, we tried to identify the procedures that the players were creating to achieve the goal of the game, it was verified that there was a progress in the development of the students' knowledge. It was also possible to identify limitations of the game, such as the lack of clarity of some rules. Among the possibilities for exploring the potential of this game, we suggest the reflection on the plays in each round. Finally, we conclude that frequent use, or at least more than once, of this game could provide a number of other learning situations for students, as well as strengthening others already built.
Poliedros de Platão e Arquimedes
(2023-04-27) Silva, Sunny Matheus Gomes da; Souza, Cícero Monteiro de; http://lattes.cnpq.br/7540654793551489; http://lattes.cnpq.br/5982672936748488
This work is focused on polyhedra, as well as the development of the first studies made by philosophers and mathematicians until the most modern times, in order to understand geometry and its relationship with nature and the universe. From the first indications made by the Greeks to Euclid, who masterfully organized all these studies on Geometry, as well as the demonstrations of polyhedra. The intention of this work is to understand the entire construction of polyhedra, planning and the elaboration of drawings to form the polyhedra, as well as the calculation of their areas and volumes. A point to be considered is that this topic is of great importance for Elementary and High School, as students can visualize and describe the world through geometric figures, as well as polyhedra.
A técnica de simulação de Monte Carlo aplicada ao cálculo de áreas no ensino médio
(2025-01-31) Araújo, Roberta Elaine Domingos de; Barros, Kleber Napoleão Nunes de Oliveira; http://lattes.cnpq.br/1338915220161592; http://lattes.cnpq.br/6420618435459342
This study presents the application of the Monte Carlo integration technique for calculating areas of plane figures, aiming to enhance Mathematics education in high school. The methodology includes a practical and experimental approach, involving computational simulations using R software and engaging classroom activities, such as the use of tangible materials to estimate areas of geometric shapes. The theoretical framework encompasses concepts from geometry, probability, and statistics, providing a solid foundation for the method’s application. The results indicated that the use of the Monte Carlo technique improved students’ understanding of mathematical concepts and increased their interest in the subject. Additionally, it was observed that increasing the number of random samples enhances the precision of the estimates, validating the method’s effectiveness. It is concluded that integrating interactive practices and computational tools into teaching enables more meaningful and contextualised learning, making it a valuable strategy for various educational levels.