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Orientation of a real vector space
Macek, Lukáš ; Halas, Zdeněk (advisor) ; Škorpilová, Martina (referee)
In this thesis, we focus on creating a visual understanding of orientation of a real vector space and its subsequent connection to the mathematical definiton. As a result, this thesis can be used as supplementary material in higher education or serve as in- spiration for teachers. First, we develop the idea behind the equivalence of two bases, then we examine its connection to permutations of vectors in ortonormal basis, moti- vating the definition of parity of permutation. We continue by observing the behavior of the equivalence during a transition to the opposite half-space, noting the connection to volumes, and based on that, we motivate the concept of determinants. Next, we delve into the method of computing determinants, providing a complete derivation. Finally, we demonstrate how the determinant of a transition matrix between two bases relates to their equivalence and we define the orientation of vector space. 1
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Loci of points in non-Euclidean metrics
Skálová, Zuzana ; Moravcová, Vlasta (advisor) ; Škorpilová, Martina (referee)
The topic of this diploma thesis are loci of points and their shape in case of using non-Euclidean metrics. The first chapter contains an overview of various loci of points discussed in school geometry as well as theoretical introduction to metrics and metric spa- ces. The second and third chapter describe the same loci of points, but using manhattan and maximum metrics respectively. All of the loci are accompanied by illustrative images. The last chapter of this thesis is dedicated to the application worksheets, that have been created by the author of this thesis on the topic of loci of points in non-Euclidean metrics. The worksheets can be used by both middle and upper school children. Teachers can find not only the answers but also deepining commentaries and possible additional tasks. 1
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A Brief Introduction to Set Theory for High Schools
Weber, David ; Rmoutil, Martin (advisor) ; Škorpilová, Martina (referee)
This thesis gives an explanation of the basic concepts of set theory, focusing primarily on high school students interested in mathematics. The text of the thesis is di- vided into six chapters. The first chapter provides a historical context, mainly explaining the development of the term "infinity" and the reasons for the establishment of axiomatic set theory. The second chapter reminds the reader of propositional logic and gives a sim- plified explanation of predicate calculus. Main focus of this chapter is on the explanation of working with logical quantifiers. The third chapter deals with the Zermelo-Fraenkel set theory axioms and some basic properties about sets they imply. Chapter Four separately introduces relations and related terminology, especially mappings and their properties. The penultimate chapter shows how to establish natural numbers using sets. In its in- troductory part, it is concisely presented a method of such establishment by means of Peano axioms. Further on, the knowledge concerning relations are extended along with the definition of ordering and ordered sets, and some basic properties of natural numbers in context of the described establishment are proved. The last chapter is devoted to the problem of comparing infinite sets. The idea of Hilbert hotel, comparison of sets...
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Arbelos
Horčičková, Klára ; Škorpilová, Martina (advisor) ; Halas, Zdeněk (referee)
The present bachelor thesis proposes several captivating planar objects and their utilization in high school mathematics education. The introductory part of the text is foremostly dedicated to the origin of the term Arbelos, its definition and general characteristics. Subsequently, the terms Archimedean twins and Ar- chimedean circles are presented, as they are closely connected to the Arbelos. In the following part, the reader is introduced to various constructions of Archime- dean circles ordered from oldest to youngest. The thesis acquaints the readers with the construction of the Pappus chain and its construction using circle inversion. The conclusion constitutes of specific problems that arise from the problematics that was dealt with in the text. 1
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Surfaces of engineering practice
Duspivová, Jiřina ; Surynková, Petra (advisor) ; Škorpilová, Martina (referee)
This diploma thesis deals with geometrical surfaces and their engineering aplications. The teoretic parts are focused on the definitions of basic concepts and mathematical descriptions of curves, surfaces and trans- formations. Further, the classification of geometrical surfaces is described as well as examples of their usage in technical practice with deep focus on screws, tools, gears and other parts of machines. The last part describes cre- ation process of four models for education. We used 3D printing technology and 3D modeling in Rhinoceros 3D software for the production of these mo- dels. Source codes and files with the 3D models can be found in electronic attachment. The thesis also includes photographic attachment with pictures of created models. 1
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Dot product in school mathematics
Krejčí, Veronika ; Halas, Zdeněk (advisor) ; Škorpilová, Martina (referee)
Title: Dot product in school mathematics Author: Veronika Krejčí Department: Department of Mathematics Education Supervisor: Mgr. Halas Zdeněk, DiS., Ph.D., Department of Mathematics Edu- cation Abstract: The theme of the bachelor thesis is the dot product in school mathe- matics. The bachelor thesis is adressed to the teachers, who teach at the secondary school. The first part consist of an analysis of the textbooks, the result of the ana- lysis is the reason to write this bachelor thesis and to build a new introduction to the theme of the dot product. In the second part we build the new introduction. The third and the fourth part consist of application of the dot product. Keywords: dot product, hyperplane, distance, angle 1
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An Unusual Approach to Circular Inversion
Šebek, Jakub ; Škorpilová, Martina (advisor) ; Boček, Leo (referee)
This bachelor thesis aims to present the topic of circular inversion in a closer way to the non-standard knowledge of high school students actively competing in Mathematical Olympiads. The first chapter describes the topic of antiparallel lines, a relatively common knowledge among such students. The second chapter introduces an antiparallel mapping which is actually a circular inversion, but deduced solely from the properties of antiparallel lines. We consider this way of introduction to be original and closer to the principle of solving more complex olympiad problems using circular inversion. In the following two chapters the topics of power of a point and cross ratio are described and their connection to antiparallel map is shown. In those chapters the circular inversion itself is also introduced and many of its properties are proven. In the last chapter, we solve the Problem of Apollonius and prove the Feuerbach's theorem using inversion. 1
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Eigenvalues of Matrices and Their Localization
Borzíková, Žofia ; Škorpilová, Martina (advisor) ; Halas, Zdeněk (referee)
The diploma thesis is concerned with the topic of eigenvalues of matrices and their lo- calization in the complex plane. First introducing general theorems concerning eigenvalues, eigenvalues for special classes of matrices are then discussed. After presenting the theory of Jordan and Weyr canonical forms, the connection and relation of these two forms is also explained. The estimates of the localizations of the eigenvalues follows from Gershgo- rin's theorem. This text might be used as a didactic material for college-level students of mathematics, thanks to its form having theoretical parts accompanied by examples with commented solutions. It may also be used as a source of information for anyone interested in extending their knowledge of linear algebra. 1
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Constructions of central projections
Kukučík, Martin ; Surynková, Petra (advisor) ; Škorpilová, Martina (referee)
Title: Constructions of central projections Author: Martin Kukučík Department: Departement of Mathematics Education Supervisor: RNDr. Petra Surynková, Ph.D., Departement of Mathematics Education Abstract: The subject of the thesis is the central projection and construction tasks solved in this type of projection. The bachelor thesis describes an explanation of the basic principle of the central projection based on 12 solved stepped construction tasks. One chapter deals with the Platonic solids - tetrahedron, cube, octahedron, dodecahedron, icosahedron, and one of the chapters is devoted to a sphere. The sample examples include solutions in the 3D space and also in the projection plane. In addition to traditional tasks, the bachelor thesis contains distorted views too. Keywords: central projection, center of projection, vanish point and vanish line
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The Growing Education at the Selected Elementary Schools
Škorpilová, Martina ; Jančaříková, Kateřina (advisor) ; Skýbová, Jana (referee)
This diploma thesis deals with teaching growing at primary schools. In the teoretical part the subject growing is charakterized and there is a brief history of teaching this practical subject. Furthermore, attention is paid to the current representation of growing in school education programs, there is an overview of expected outcomes and content of the subject matter of this course and possibilities of application of cross-curricular themes and interdisciplinary relations within it. At the end of the theoretical part there are spatial possibilities for teaching this subject, possibilities of innovation of subject and its link with food sovereignty and the use of local food. The practical part firstly answering the basic research question: How is the teaching of the subject of cultivation work in selected primary schools? The characteristics of each school are described, the hourly subsidies and forms of teaching of this subject, the equipment of the school for the teaching of the course, and the current, interesting and unconventional methods that can serve as examples of good practice of teaching Cultivation works. These dates were assessed in the form of qualitative research. In addition, the practical part is answering the secondary research questions and the results of the quantitative survey,...
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