School Course In Geometry Magdalena Metodieva Petkova

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02 Nov 2017

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University of Ruse Angel Kanchev

Abstract: This article provides an incremental approach to integration of mathematical software in teaching future math teachers. For this purpose is selected the subject School course in Geometry. Emphasis is on developing and presenting some specific tasks in GeoGebra environment through static and dynamic text, animation, LaTeX formulas, the tools - sliders and checkbox.

Keywords: GeoGebra, school course in geometry, approach and tasks

INTRODUCTION

The School Course in Geometry (SCG) is a discipline that is studied by students majoring in mathematics and Informatics in the fifth semester of their preparation for teachers of mathematics.

The goal of the course is deepening the knowledge of students on a wide range of methods for solving tasks in geometry. The SCG is the basis for teaching disciplines such as Methods of Instruction in Mathematics (MIM), Extracurricular Work in Mathematics (EWM), etc.

The curriculum of the course covers the following topics: Logical construction course in geometry; Movements and equality; Figures in the plane; Vectors in the plane and in the space; Similarity and homothety; Trigonometric functions of angles; Metrical dependencies in a triangle, polygon, and circumference; Area of figures; Geometrical constructions; Geometrical transformations; Geometry of complex numbers; Parallelism and perpendicularity in the space area; Coordinate method; Bodies in the space area; Volumes; Surface area; Sections [5].

The development includes exemplary elaborated tasks in GeoGebra environment, -related to the educational content and in accordance with the curriculum in secondary school.

GeoGebra is a software tool that is used all over the world for creating mathematical constructions and dynamic drawings, which are suitable for application in teaching mathematics to students of different age groups. The positive aspects of the use of GeoGebra, which can be listed, are:

Expanding and enriching learning environment through the use of interactive methods and strategies of teaching and learning;

Improvement of the existing static forms of teaching materials and implementation of dynamic (multimedia) developments by integrating image, audio, animation, and text;

Electronic and multimedia teaching materials in the educational process [1, p. 146].

The proposed four basic stages of mastering the knowledge and skills to implement the GeoGebra in teaching mathematics are:

First /Preparatory stage. Formation of interest to software by presenting its characteristics (basic menus, submenus, and toolbars) and tasks that require simplified geometric shapes and structures (in the Mathematical software (MS) that is studied by students in third semester).

Second stage. Study on application of basic commands by developing a system of examples linked to educational content. Development of simple sample assignments and/or structures with the participation of learners. Presentation of tools for advanced learners.

Third stage. Self-study process and preparation on learning resources by using GeoGebra. Presentation of results to appropriate audiences with an Interactive Whiteboard (IWB).

Fourth stage. Training on the methods of mathematics teaching and learning, based on GeoGebra software. Application of the knowledge and the skills in real learning process and evaluating the results of specialist [1, p. 147].

The second and third stages shall be presented to the students during their training at SCG, and the fourth stage – during the Current Pedagogical Practice in Mathematics (CPPM) or MIM.

Importent Terms

Figure 1: Interface

Figure 2: Toolbars

Slider

Parameter that is changing in a predefined interval

Checkbox

Checkbox for hiding and displaying objects

Figure 3: The tools Slider and Checkbox

theoretical knowledge. Developedments

All selected thematic developments are represented through Internet Browser. They have been elaborated in such a way as to provide multi-functional application - in a computer lab, during a presentation, IWB, in an individual work process.

The introduction of mathematical formulas in GeoGebra is performed through LaTEX.

What is LaTEX?

LaTEX is a word processing system, widespread in the scientific and academic communities. It is designed for making scientific documentation such as books, articles, dissertations, electronic textbooks in the areas of mathematics, physics, chemistry, engineering, and others. LaTEX is using the program TEX (established in 1977 by the mathematician Donald Knuth for writing texts containing mathematical formulas) as a mechanism for word processing [2, p. iii], [4, p. 1 - 5].

How mathematical formulas in LaTЕX appear without compiling text visualization?

This command

\frac{а}{b}=\frac{6}{7}

in LaTЕX is equivalent to the command

,

or this command

\int{ \frac{4 (cosx)^{2}+3(sinx)^{2}}{b}dx

have the equivalent of

.

The possibility of using LaTEX in the development of mathematical constructions in GeoGebra by students studying mathematics and informatics could contribute to the preparation of high-quality footage for secondary education.

The specifications of each structure are provided in advance.

Linear function of an independent variable

The following three examples involve two variables a and b which change at predefined interval [-50, 50]. This interval can be changed depending on the condition of the task.

The structure includes a color drawing, title, and two slider, and two input fields value, dynamic and static text.

Figure 4: Problem 1

Figure 5: Problem 2

Figure 6: Problem 3

Excircles of a triangle

The structure includes a basic drawing, dynamic, static text, and four sliders.

Construction:

Construct a random triangle.

Prolong of all the sides.

Construct the three internal bisectors.

Construct the three external bisectors.

The excenters of the excircles are the points , obtained by the intersection of the respective internal and external angle bisectors:

.

Draw the three circumferences, making perpendicular lines from the points to the sides of the triangle. These would be the three radii of the three excircles [3, c.94].

Figure 7: Excircles of a triangle with internal and external angle bisectors

Euler line.

The orthocenter, the centroid and the circumcenter of any triangle are aligned. They belong to the same straight line, called line of Euler [6].

The construction includes drawing, static and dynamic text, four checkbox.

Figure 8: Line of Euler

Figure 9: Problem 1 (acute-angled triangle)

Figure 10: Problem 2 (obtuse-angled triangle)

Figure 11: Problem 3 (right triangle)

Regular hexagon.

The construction contains a drawing, a slider (parameter), settings to start and stop the animation, static text, tracking.

Construction:

A parameter is developed, where is the length of the radii of all seven circumferences in the figure.

The first circumference is the center is drawn, which appears central for the construction with a center and a point on it (point B).

The point B will be the center of the second circumference with radius.

The intersections of the first to the second circumference are exactly those points, which are the centers of the next two circumference with radii equal to.

Continue until all new sections are used as centers of circumferences, i.e. there must be six circumference without central circumference.

The following settings are additional: add color, insert animation, place trail static text, export to html format (suitable for training and implementation in presentations).

Figure 12: Regular hexagon (slider increment 2.00)

Figure 13: Regular hexagon (slider increment 0.02)

CONCLUTION

Working with software GeoGebra helps to raise the level of teaching mathematics. This software can be modified to the needs of the users (students, teachers, researchers). It is an easy tool for organizing examples of current teaching/ presentation and for complex structures, requiring extensive research in a given area.

The study of GeoGebra in SCG contributes to the creative development and improvement of future math teachers during their training.



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