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System for support of conic sections teaching
Hejlová, Eliška ; Surynková, Petra (advisor) ; Karger, Adolf (referee)
The work presents own software for geometric drawing aimed to construction of conic sections. It's designed for high school students and their teachers, to use in lessons of descriptive geometry and mathematics. It contains a number of exercises with solutions which are prepared to solve in the program. Next part of this work is a theory about conic sections. We show various definitions, constructions and some basic properties. We also show a construction and properties of tangent in the point of conic section. Theory is supplemented by animations and pictures made in program GeoGebra. There are also proofs of equivalence of presented definitions. 1
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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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Proofs of selected geometric constructions
Vaňková, Marie ; Zamboj, Michal (advisor) ; Jančařík, Antonín (referee)
This bachelor thesis is the summary of the chosen constructions used in descriptive and kinematic geometry. These constructions are always described in detail and proven. The first chapter is devoted to the very concept of curve and curvature. The second chapter is focused on conic sections, ie ellipses, hyperbolas and parabolas. These curves are defined, their main characteristics are described, and their equation is derived. Further o , the chapter contains of the various kinds of constructions of these curves. It is par- ticularly about point structures and structures using osculating circles. The third chapter deals with the cyclic curves, ie cycloid, epicycloid, hypocycloid, peri- cycloid and involute of a circle. For these curves, the motion by which they arise is defined, and the given curve's parametric expression is presented. The following is a description of the construction of this motion and proof that the points of this construction correspond to the parametric expression of the cyclic curve. Finally, the fourth chapter focuses on conchoids, which together with cyclic curves rank among the kinematic curves. Even here the motion by which conchoids are created is first introduced, the construction of this motion is described, and it is proved that the constructed points correspond to the...
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A Collection of Solved Problems in Analytical Geometry
Kvapilová, Babeta ; Hromadová, Jana (advisor) ; Surynková, Petra (referee)
This thesis is intended for teachers and students of high schools and universities. It consists collection of solved problems from plane analytical geometry including various solutions and their comparison. The thesis aims to increase the student knowledge of the topic and to provide different approaches to problems and working materials for lessons for teachers. Pictures for better understanding are added for more difficult problems. The practical part focusing on common mistakes and their elimination is included.
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Conics around us
ŠAFÁŘOVÁ, Denisa
This Bachelor's thesis is focused on curves around us. Primarily emphasizes conics and their presence in the real world. They can appear for example in architecture, engineering or nature. Individual conics and their properties are defined in the first part. There are also definitions of chosen algebraic curves. The text is interleaved by pictures made in GeoGebra. The second part is based on identification of conics and chosen curves on photographs using GeoGebra. The theoretical knowledge like algebraic proof that it is a given conic is used for some photographs.
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Loci of given properties
DVOŘÁKOVÁ, Andrea
This bachelor thesis focuses on the description of the loci of given properties and its use in solving of the simple tasks in high schools. It is also focused on the analysis of more difficult problems related to the conic sections. These problems are solved in planimetric or analytic form. There is also a chapter dealing with the axial affinity. It extends the curriculum at high schools.
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Paper folding as a tool in teaching mathematics
SCHINKOVÁ, Nikol
In my thesis I am dealing with the usage of origami at teaching mathematics. In the first chapters I mention a brief history and kinds of creases such as Huzita axioms etc. In the second part I introduce three chapters concerning folding of structures supported by work-sheets at different difficulty levels. The content of these chapters comprises of conic sections and diameters of trapezoids. The last chapters of the thesis are focused on three kinds of folding: Yoshimura, Miura and modular, which are also used in architecture, house design and astronautics.
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