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Pythagorean triangles
Sláma, Michal ; Jančařík, Antonín (advisor) ; Kvasz, Ladislav (referee)
Title: Pythagorean triangles Author: Michal Sláma Department: Department of mathematics and mathematical education Supervisor: RNDr. Antonín Jančařík, Ph.D. Abstract: The thesis considers usable problems concerning pythagorean triangles. In the first part, there is described an expression of a parametrization of pythagorean triangles. Next parts are dedicated to an expression of properties of edges and radii of the incircle and excircles. Next chapters describe Heronian triangles and their decomposition to pythagorean triangles, as a way to solution of the problem of a Heronian triangle where all heights are integer numbers. In addition are given some examples of triangles for school practice. Keywords: Pythagorean triangles, Heronian triangles, incircle and excircles, triangle heights as integer numbers
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Elementary functions and domain
Vitásek, Tomáš ; Pilous, Derek (advisor) ; Jančařík, Antonín (referee)
The aim of my bachelor thesis is to show the teachers and the students of educational mathematics the relation between the elementary functions and their domains and the methods of creating problems focused on finding the domain. In the chapter called "Zobrazení", I try to define and explain the terminology closely connected to elementary functions and their doma- ins which I use in the following parts. Crucial chapters are "Definiční obory elementárních funkcí" and "Návrhy úloh". In the first of those two chapters I thoroughly explain the algorithm of solving problems focused on finding the domain of elementary functions and I discuss what the domains of various elementary functions are like. In the second of those two chapters, I suggest functions fitting a concrete domain. The aim is to suggest functions in a way that a student taking their maturita exam in mathematics at high school would be able to solve the problem of finding the domain. Keywords: elementary function, domain, function composition.
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Models of Lobachevskij's geometry and possibilities of their use at secondary school
Kosina, Jan ; Kvasz, Ladislav (advisor) ; Jančařík, Antonín (referee)
This thesis Models of Lobachevskij's geometry and the possibilities of their use at secondary school focuses on one kind of non-Euclidean geometries, the Bolyai - Lobachevskij's geometry. The first chapter describes the history of non-Euclidean geometry, shows difficulties of understanding of one publication dedicated to these problems by current students of secondary schools and shows some chosen methods in the didactics of mathematics, especialy the constructivist method. The second chapter is dedicated to elemental concepts of projective geometry, Bolyai - Lobachevskij's geometry and it shows its basic models. It further analyses the specific features of this kind of geometry in Beltrami - Klein's model, especially mutual positions of straight lines. This theses further contains a set of gradual tasks. The third chapter is dedicated to the description of a didactical experiment. In this experiment were students of secondary school acquainted with this theory and tasks, which they solved. Student's solution were writen down and than analysed in the constructivist methodology term in the didactics of mathematics.
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Tandemat - didactic game for teaching of mathematics at the secondary school
Šilhánová, Lucie ; Vondrová, Naďa (advisor) ; Jančařík, Antonín (referee)
Tandemat - didactic game for the teaching of mathematics at the secondary school The thesis concerns a didactic game called Tandemat which I have created as a complement for the teaching of mathematics at secondary schools. The goal of the thesis is to find out and describe the potential of this game for the teaching of mathematics at this school level. First, the importance of games in our lives and also in teaching is mentioned. Next, some resources concerning games in the teaching of mathematics are described. The core of the thesis consists of the game Tandemat which I have elaborated and which is inspired by the popular game Activity. The results of the pilot studies of the game with the preliminary name Aktivity which were realized at elementary school and university helped to improve the game into its present version Tandemat. This has been tested by five groups of pupils in four classes of a secondary grammar school. The experiments were recorded and the acquired data were analyzed using the method based on the grounded theory approach. Moreover, a questionnaire was administered to pupils, concerning their grades in mathematics and the Czech language and their remarks on the game itself. The results of the analyses, observations and questionnaires show the potential and the limits of...
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Algebraic Curves in History and School
Fabián, Tomáš ; Kvasz, Ladislav (advisor) ; Jančařík, Antonín (referee)
TITLE: Agebraic Curves in History and School AUTHOR: Bc. Tomáš Fabián DEPARTMENT: The Department of mathematics and teaching of mathematics SUPERVISOR: prof. RNDr. Ladislav Kvasz, Dr. ABSTRACT: The thesis includes a series of exercises for senior high school students and the first year of university students. In these exercises, students will increase their knowledge about conics, especially how to draw them. Furthermore, students can learn about two unfamiliar curves: Conchoid and Quadratrix. All these curves are afterwards used for solving other problems - some Apollonius's problems, Three impossible constructions etc. Most of the construction is done in GeoGebra software. All the tasks are designed for students to learn how to work with this software. The subject discussed is put into historical context, and therefore the exercises are provided with historical commentary. The thesis also includes didactic notes, important or interesting solutions of exercises, possible issues, mistakes and another relevant notes. KEYWORDS: conic, circle, ellipse, parabola, hyperbole, conchoid, quadratrix, trisecting an angle, squaring the circle, rectification of the circle, doubling a cube, Apollonius's problem, GeoGebra
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Beginnings of Greek mathematics - Thales, Pythagoras and Euclid
Ebelová, Gabriela ; Kvasz, Ladislav (advisor) ; Jančařík, Antonín (referee)
Title: The beginnings of Greek mathematics - Thales, Pythagoras and Euclid Author: Gabriela Ebelová Department: Department of Matematics and Mathematical Education Supervisor: Prof. RNDr. Ladislav Kvasz, Ph.D. The work deals with a new look at the history of mathematics in Greece and its use in teaching mathematics today. The first part presents the three mathematicians Thales, Pythagoras and Euclid. It describes their lives and important works. The second part of the book is devoted to the analysis Arpad Szabo The Begginings of Greek Mathematics. There is presented a different view of the history of Greek mathematics. Content of the third and last part is the use of history in the teaching of mathematics. Also, comparing teaching methods in historical Greece and currently. Klíčová slova: history of mathematics, Arpád Szabó, teaching of mathematics
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Beneficial utilisation of GeoGebra software in geometric constructions at the lower secondary level of basic school
Krejčíčková, Klára ; Holíková, Marie (advisor) ; Jančařík, Antonín (referee)
Diplomová práce se zabývá využitím programu GeoGebra ve výuce ma- tematiky, konkrétně v tématu konstrukce trojúhelník·. Cílem práce bylo na- vrhnout přípravu na hodiny výuky konstrukce trojúhelník· s využitím pro- gramu GeoGebra a následně ověřit, zda využití softwaru dynamické geometrie mělo vliv na zvládnutí nového učiva. Teoretická část obsahuje použité pojmy z planimetrie. V dalších dvou kapitolách se čtenář seznámí s programem GeoGebra a GeoTest a některými výzkumy, které jsou zaměřené na použití program· dynamické geometrie ve výuce matematiky. Následuje kapitola se- znamující s úlohami, které jsou v experimentu využity, a popisuje přípravu jednotlivých hodin. Poslední část je věnovaná pr·běhu experimentu, srovnání výsledk· žák· pomocí test· s kontrolní třídou. Získaná data jsou zpracována na základě pozorování a test·. Výsledky pozorování a analýzy potvrzují, že GeoGebra je vhodným doplňkem výuky konstrukce trojúhelník·. 1
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Gauss and the constructability of regular polygons with ruler and compass
Sedláčková, Veronika ; Kvasz, Ladislav (advisor) ; Jančařík, Antonín (referee)
1 ABSTRACT The bachelor thesis deals with chosen Euclidean constructions of regular polygons and summarizes their historical development. It focuses on the mathematician who is essentially adherent to this theme, his name is Carl Friedrich Gauss. In the first part of the thesis the important statements from Gauss's life and particularly from his scientific publications are given. Then the idea of algebraic formulation of the constructions with ruler and compass is characterized and the main theorems about these constructions are proved here. Further Gauss's theorem about constructability of regular polygons is given and proved while using Galois Theory. The next part is focused on Gauss's construction of the regular 17-gon, which is described in details. At the same time the thesis explains other interesting constructions from various authors created during the 19th century and in the beginning of the 20th century.
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Mathematics on the chess board
Šperl, Jiří ; Jančařík, Antonín (advisor) ; Pilous, Derek (referee)
TITTLE: Mathematics on the chessboard AUTHOR: Jiří Šperl DEPARTMENT: The Department of mathematics and the teaching of mathematics SUPERVISOR: RNDr. Antonín Jančařík, Ph.D. ABSTRACT: The main subject of my thesis is mathematical problems on the chessboard using chess pieces. The work aims to demonstrate how a secondary school student would approach and solve several typical mathematical tasks of this nature. Consequently, it outlines ways to incorporate chessboard mathematical problems and exercises in mathematical classes. Moreover, the thesis includes a compact collection of solved problems on the chessboard that can serve as an inspiring source of unconventional mathematical tasks in conventional mathematical education. My own mathematical research forms a major part of the thesis. The research was conducted as a series of tests in three school classes. In order to achieve a high de- gree of objectivity classes of students with different specializations were selected to take part in the tests. The participating classes were also of different age groups. The theoretical part of the thesis takes a look at the past of the subject and presents several interesting historical problems concerning the mathematics on the chess- board. Last but not least, the thesis contains a discussion of solutions of the...
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