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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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Mascheroni constructions
Hatschbachová, Jana ; Hromadová, Jana (advisor) ; Škorpilová, Martina (referee)
This bachelor's thesis concerns with the topic of Mascheroni constructions. These are constructions done with compass alone. The thesis contains three chapters, the first is devoted to a brief summary of the history of Mascheroni constructions. The second chapter presents Mohr-Mascheroni theorem and its proof. The third chapter contains a description of several basic Mascheroni constructions including proofs. Part of this work is also a GeoGebra book - Mascheroni constructions, which is divided into two parts. In the first part, there are stepped solutions for all constructions mentioned in the third chapter. The second part contains interactive exercises for each construction.
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Advanced Topics in Plane Geometry
Hajmová, Kateřina ; Štěpánová, Martina (advisor) ; Moravcová, Vlasta (referee)
The aim of this thesis is to introduce a series of knowledge from advanced planimetry, which can be proved using the knowledge of high school geometry. Selected theorems deal with characteristic of squares with a common vertex (Finsler-Hadwiger theorem, Theorem of Four Squares, Bottema's theorem), significant points of plane entities (Gergonne point theorem, Švrček point theorem, Simson's Theorem, Miquel's theorem), Feuerbach's circle and its relation to the Euler's line. In this thesis, there is also mentioned Reim's theorem, Napoleon's theorem, and Thebault's theorem. This thesis contains a lot of illustrations created in the Geogebra Mathematical Software, which are available online in interactive form.
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Using mathematical calculations to solve problems in sciences
Gáborová, Andrea ; Novotná, Jarmila (advisor) ; Jančařík, Antonín (referee)
This work deals with word problems from natural sciences whose main mathematical apparatus are circle, sphere and their parts. The work is divided into three main parts - part about word problems, part about used mathematical apparatus in word problems and part containing verbal tasks from sciences. The first part defines the word problems and describes how to solve the problems. The second part contains the mathematical apparatus used in the word problems in part three. The third part is a collection of verbal tasks from the sciences, divided into the taskes solved at primary and secondary schools. The individual vocabulary is presented as a necessary theoretical basis of sciences. In the third part, I created most of the vocabulary myself.
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Length measurement
Pecinová, Iva ; Šarounová, Alena (advisor) ; Surynková, Petra (referee)
My bachelor thesis Length measurement deals with the euclidean length measurement. It deals with the history of the length measurement in the Czech country, the emergence of the basic unit of length - meter and especially the circumference. The work is intended primarily for high school teachers of mathematics and lovers of measurements which are expected to at least high school knowledge of mathematics. This text is good for primary schools teachers of mathematics which is intended Handbook for small meter. The component part of my bachelor thesis is an enclosed CD, where is found the bachelor thesis in an electronic form and the already mentioned Handbook for small meter - a version for printing.
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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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Algebraic curves in history and school
Fabián, Tomáš ; Kvasz, Ladislav (advisor) ; Vondrová, Naďa (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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Taxicab Metric in Teaching-Learning Process at Basic School
Bruna, Jiří ; Zhouf, Jaroslav (advisor) ; Vondrová, Naďa (referee)
This master's thesis explores the possibility of including Taxicab metric as a subject matter into instruction at lower secondary level of education and it does so in several ways. Firstly, it looks into a curricular document of state level (Framework Educational Programme) and discusses instances at which the subject matter and the concept of lower secondary education are in agreement. Secondly, this thesis analyses a selected series of textbooks with respect to exercises that can be seen as linked to non-Euclidean metrics. Furthermore an experiment is described and evaluated, whose purpose, as a part of this thesis, was to find out if selected pupils can successfully solve problems in the context of the Taxicab metric and if related instruction influenced pupils' understanding of the concept of line segment and circle in a desired way. The teaching material which constituted an integral part of the experiment is presented as well.
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Plane geometry teaching at secondary schools
Machovcová, Lucie ; Zhouf, Jaroslav (advisor) ; Dvořák, Petr (referee)
This work compares and evaluates several math's textbooks for secondary school where we can find schoolwork from plane geometry. The aim of this work is drawing the main advantages and disadvantages of those textbooks, evaluating whether all textbooks contain themes which are required by School Curriculum and interpretation of a questionnaire survey among teachers of mathematics. In the last chapter, it is described how a new ideal textbook of plane geometry wouldlook like. Those were taken on the grounds of gained information from my questionnaire survey. Themes which cannot be found in compared textbooks for secondary schools are a part of recently made textbook since it is not necessary to know any new terms for their understanding. Key words: geometry, teaching geometry, textbooks, polygon, circle
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