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Spatial generalizations of the properties of the triangle
Šrubař, Jiří ; Karger, Adolf (advisor) ; Boček, Leo (referee) ; Lávička, Miroslav (referee)
TITLE Spatial generalizations of the properties of the triangle AUTHOR Jirˇı' Sřubarˇ SUPERVISOR Prof. RNDr. Adolf Karger, DrSc. DEPARTMENT Department of mathematics education ABSTRACT The present thesis describes various interesting properties of a triangle. The aim is to find and prove similar properties of its spatial generalization - a tetrahedron. Even though both synthetic and computational methods are used for proving spatial relations, synthetic approach is preferred whenever possible. The thesis is divided into two parts. In the first part, the properties of the tetrahedron analogous to the centroid and the orthocenter of the triangle are described. Also, conditions on the existence of the orthocenter of the tetrahedron are derived. Moreover, for tetrahedrons without an orthocenter, the so-called Monge point is introduced as its generalization. In the second part of the thesis, some further properties of the triangle are studied - - the Simson line, the de Longchamps point, the nine-point circle, the Euler line, the Lemoine point, the isodynamic points, the Lemoine axis and the Brocard axis. As the main contribution of the present thesis we define and prove the existence of spatial analogues of the above mentioned properties for the tetrahedron - the de Longchamps point, the twelve-point and...
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Jan Sobotka - the Inspiration After Hundred Years
Zuščák, Tomáš ; Kuřina, František (advisor) ; Boček, Leo (referee) ; Nádeník, Zbyněk (referee)
Inspirativnost díla Jana Sobotky pro učitele matematiky. Život a dílo Jana Sobotky, charakteristika jeho působení. Stav geometrického bádání v českých zemích a ve světě v 19. a na počátku 20. století. Rozbor vybraných prací Jana Sobotky. Apolloniova úloha a příbuzné úlohy - Apolloniova úloha v euklidovské rovině, na kulové ploše a v trojrozměrném euklidovském prostoru, izogonální sféra. Podnětnost Sobotkova díla ilustrovaná na několika úlohách. Význam kalkulů v geometrii.
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Graphs and graphical communication
Klemová, Hana ; Šarounová, Alena (advisor) ; Boček, Leo (referee)
This text is dedicated to the teachers of not only math and natural sciences. I wanted to show the readers a general view on the problems of graphical imagery and thematic cartography. In the beginning of the text I mentioned a brief historical prologue. In the next part I desribed reasons of using graphs and their significancy for today's society. To know the ropes in classification of graphs by their types is a headstone of understanding the graphical imagery. Thematic cartography is one of the parts of graphical communication, which is often forgotten, even though we come across it daily. In the end of my work I presented samples of graph utilisation in a non-mathematical subjects of high school tuition.
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Web application for teaching basic knowledge of mathematics at secondary school
Pavlicová, Vladimíra ; Boček, Leo (referee) ; Robová, Jarmila (advisor)
Title: Web application for teaching basic knowledge of mathematics at secondary school Author: Vladimíra Pavlicová Department: Department of Mathematics Education Supervisor: RNDr. Jarmila Robová, CSc. Supervisor's e-mail address: robova@karlin.mff.cuni.cz Abstract: The presented work is intended to serve as a teaching material for pupils of the first year of secondary school, focusing on the basic knowledge of mathematics. Specifically, it deals with the subject matter of powers and polynomials. It is composed of two sub-sections - an educational text, in which the subject matter is clearly explained and illustrated through solved examples, and exercises to practice, which allow pupils to check, eventually improve, their knowledge. Owing to the fact that the work has a form of a web application, the emphasis is placed on interactive elements (for example, step-by-step solutions, multiple choice tests), which increase the attractiveness of the presented curriculum for pupils. Pupils are therefore able to learn and practice the curriculum by enjoyable way. The fact that the application is freely available on the Internet can also increase pupils' interest in it. Keywords: basic knowledge of mathematics, powers, polynomials
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Moebius geometry
Kovaříková, Dana ; Karger, Adolf (referee) ; Boček, Leo (advisor)
The main task of this diploma work was to write a teaching text for a seminar of grammar school's students, which would explain principals of circular geometry in a simplified manner. The text contains an interpretation of the concept of a directed circular curve. It examines the curves corresponding touch point and also the inference of transformations in circular inversion, similarity and dilatation. There are some problems of Apollonios with pre-assigned directed elements that are analytically solved. The text and problems are completed with pictures.
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Mappings in geometry
Trkovská, Dana ; Boček, Leo (referee) ; Kubát, Václav (advisor)
This diploma dissertation is dedicated to applications of geometrical mappings. It is intended as a tuitional material specially for students of the third year of the mathematics teachers programm at Mathematical and Physical faculty of Charles University in Prague. The text can be used as a supplementary material for a seminar at secondary school as well. It is based on lectures of the course Geometry II. Students are familiar with the term mapping already during the lessons at elementary and secondary schools. Therefore in the diploma dissertation we at first give only a summary of basic knowledge about mappings in geometry, in the language of mathematics textbooks. Next part of this thesis includes theoretical knowledge about mappings in geometry in the form of definitions and propositions together with their proofs. A great part is dedicated to characterization of affine mappings, specially isometries and similarities. At the end circular inversion is explained as an example of a mapping that is not affine. For better imagination the whole text is complemented with a number of figures. Theoretical part is followed by a collection of exercises. Of course, solutions of all exercises are given.
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