Application of GeoGebra for Teaching Mathematics (2014) - PDF

Advances in Science and Technology Research Journal Volume 8, No. 24, Dec. 2014, pages 51–54 Original Article DOI: 10.12913/22998624/567 APPLICATION OF GEOGEBRA FOR TEACHING MATHEMATICS Dariusz Majerek1 1 Department of Applied Mathematic...

Advances in Science and Technology Research Journal Volume 8, No. 24, Dec. 2014, pages 51–54 Original Article DOI: 10.12913/22998624/567 APPLICATION OF GEOGEBRA FOR TEACHING MATHEMATICS Dariusz Majerek1 1 Department of Applied Mathematics, Lublin University of Technology, Nadbystrzycka 38, 20-618 Lublin, Poland, e-mail: [email protected] Received: 2014.09.25 ABSTRACT Accepted: 2014.10.25 This paper shows how GeoGebra can be helpful in teaching mathematics. GeoGe- Published: 2014.12.01 bra is an interactive geometry, algebra, statistics and calculus application, intended for learning and teaching mathematics and science from primary school to university level. It can be used for active and problem oriented teaching and fosters mathematical experiments and discoveries both in classroom and at home. In this work we show the sketch of using the above-mentioned software to build, solve and illustrate mathemati- cal problems. Keywords: dynamic geometry, algebra, calculus. INTRODUCTION GeoGebra depends on software licensed under the GNU General Public License (GPL), Today’s guidelines for teaching mathematics the LGPL, the Apache license and others. The indicate the important role of visualization tech- software is licensed under the «GeoGebra Non- niques. As a response for these needs many of Commercial License Agreement», which asserts software application were created to build geo- that while the «source code is licensed […] under metric constructions and solving analytical and the terms of the GNU General Public License», algebraic problems. One of the best application the translation files, installers, and web services designed to construct and illustrate some math- are licensed under non-GPL-compatible terms. ematical issues is GeoGebra. Commercial use is prohibited without the pur- It was created by Markus Hohenwarter chase of a separate license which prevents the in 2001/2002 as part of his master’s thesis in resulting combined work from being considered mathematics education and computer science free software. at the University of Salzburg in Austria. Sup- GeoGebra is available on multiple platforms ported by the Austrian Academy of Science he with its desktop applications for Windows, Mac was able to develop the software as a part of his OS and Linux, with its tablet apps for Android, PhD project in mathematics education. Mean- iPad and Windows, and with its web application while GeoGebra received many international based on HTML5 technology. awards, and was translated by mathematics instructors and teachers all over the world to more than 25 languages. Since 2006 GeoGebra THE MAIN IMPEDIMENTS IN TEACHING is supported by the Austrian Ministry of Edu- MATHEMATICS cation to maintain the free availability of the software for mathematics education at schools Today young people are accustomed to pic- and universities. In July 2006, GeoGebra found torial culture through a wide access to social me- its way to the US, where its development con- dia such as facebook, twitter or youtube. A mul- tinues at Florida Atlantic University in the NSF titude of different forms of media are involved, project Standard Mapped Graduate Education including, for example text, audio, graphics, ani- and Mentoring. mation, video, and virtual reality. Additionally, 51 Advances in Science and Technology Research Journal Vol. 8 (24) 2014 the development and rapid growth of the Inter- All this makes the GeoGebra a great tool for net in combination with its increasing accessi- teaching and learning mathematics. Since all ob- bility for the public has opened up a whole new jects in GeoGebra are dynamic, students can see digital world. This makes that in learning pro- how it’s changing when he change the parameters cess students are increasingly inclined to accept of the problem. In geometrical constructions all the content given to them in this way. Especially the objects such as points, sections, circles and in mathematics where a number of issues require lines can be moved in any way. This makes the a lot of imagination, students are discouraged constructions more clearer. In addition all the con- to learn the subject when it is not provided in a structions can be made by point and click tech- modern and accessible way. nique or introducing them through command line. The main obstacles in teaching mathematic- sematics are: Concepts without an adequate illustration, INTERFACE Mathematical graphs are static in classical way of teaching mathematics i.e. drawing it GeoGebra has a very clear and intuitive in- on a piece of paper, terface divided into parts corresponding to the Static objects do not allow for generalization algebra and geometry (Figure 1). Depending on of the concept. your needs it can be freely modified to suit the considered issue. The National Council of Teachers of Math- We have several views: ematics (NCTM), which is the world’s largest as- algebraic view, sociation of mathematics teachers, declared tech- geometric view, nology as one of their six principles for school spreadsheet view, mathematics. CAS (Computer Algebra System) view, Technology is essential in teaching and protocol design view, learning mathematics; it influences the math- command line. ematics that is taught and enhances students’ learning. All these views are linked with each other, that is, if we introduce an object in one of the views, it will appear in others in an appropriate form. So MAIN FEATURES OF GEOGEBRA for example, if we put a function in a command line, its graph appears in geometric view. All the he main idea of using GeoGebra into every- changes of the parameters of the function are im- day teaching and learning is to provide opportuni- mediately shown on graph. ties for students of different mathematical skills Beside the basic capabilities of GeoGebra and levels for better understanding concepts and like drawing figures, lines and function graphs we fostering them to doing mathematics in new at- can also calculate or measure angles, points of in- tractive way. terceptions, lengths, fields, circumferences, maxi- Here are the main features of Geogebra: mum and minimum of a function, derivatives and free for noncommercial use, integrals. Obviously, GeoGebra can be used as an multiplatform, advanced calculator but not only. It can operate clear and easy understanding graphical user on vectors, matrices and even solve a system of interface, linear equations. rich database of ready-made examples, technical documentation in many languages, marking objects follow the mathematical syntax, EXAMPLES ability to save a project in multiple formats, We show the capabilities of GeoGebra in works with LaTeX, some examples. all objects in GeoGebra are dynamic, possibility to publish the work on the website Example 1 through javascript, program is translated into many foreignlan- Suppose we want to construct a circle de- guages.​ scribed on the triangle (Figure 2). We know that 52  Advances in Science and Technology Research Journal Vol. 8 (24) 2014 Fig. 1. GeoGebra interface Fig. 2. Circle described on a triangle Fig. 3. Tangent line and derivative of the function 53  Advances in Science and Technology Research Journal Vol. 8 (24) 2014 the center of the circle must be an interception of This example helps students understand the bisectors. Radius of the circle is the sector from meaning of the derivative of a function. Obvi- intersection of two bisectors to one of the vertex. ously we can change the formula of f and then Construction of a circle is performed in the fol- all points and lines changes automatically, so lowing steps: student can see this phenomenon for different draw any triangle ABC, functions. construct two bisectors of any two sides, find the interception of bisectors and mark it by D, CONCLUSION draw a circle with center in D and radius DA. We show the benefits of introduction dynamic mathematics software into teaching and learning ample 2 Example 2 process. We conclude that all students, from any ample 2 example we want to show the relationship between the slope of a In this In this example we want to show the relation- level of mathematical knowledge can be encour- In this example ngent line of a ship we function want to show the andthederivative relationship of this line between function. the firstslope In the aged to step ofwe study amathematics by using this applica- between slope of a tangent of a func- ngent line of ve to draw a tion some function and forderivative and derivative function, of this of example this function. function. In the tion. In the first firstCurrent step we trends in the teaching of science call ve to draw some step we have for function, to draw some function, for example: example for the use of visualization techniques, and Geo- 2 𝑓𝑓(𝑥𝑥) = (𝑥𝑥 − 1) − 1. Gebra fits perfectly this trend. 𝑓𝑓(𝑥𝑥) = (𝑥𝑥 − 1)2 − 1. e next step is to put Thea point next stepon is thetograph put a point on the𝐴𝐴. of 𝑓𝑓 called graph Then draw a tangent e next step is of to f put called a A. point Then on draw the a graph tangent of 𝑓𝑓 line in called point 𝐴𝐴. A draw Then REFERENCES a tangent e in point 𝐴𝐴. This line3).has (Figure Thisslope line has andslope weandcallweitcall 𝑚𝑚. itIf m. weIf put a point 𝑃𝑃 = e in 𝑚𝑚) (𝐴𝐴), point on 𝐴𝐴. This line a graph has slope and of we callpoint it 𝑚𝑚. If do we put1.a Hohenwarter point 𝑃𝑃 = M., Preiner J.: Dynamic mathematics we putand set Pthe a point option = (x(A), m) on this a graph andto set the trace ifwith it moves, GeoGebra. The Journal of Online Mathemat- (𝐴𝐴), 𝑚𝑚) on a graph option and set of𝐴𝐴this the𝑃𝑃option point to doalso ofbe tracethis itpoint ifmoved moves, to then do trace if itwhich moves, en together with point point will and trace a line is ics and Its Applications, No. 7, 2007. en together with point together 𝐴𝐴 withpoint point 𝑃𝑃 entical with derivative of 𝑓𝑓 given by formulawill A also point P be willmoved also beand moved trace a line which is 2. International GeoGebra Institute, „GeoGebra” [Avail- and trace entical with derivative ofa𝑓𝑓line which given by is identical with derivative formula able: https://www.geogebra.org/license, 15.10.2014]. of f given by formula: 𝑓𝑓 ′ (𝑥𝑥) = 2𝑥𝑥 − 2. 3. NCTM, „Principles and Standards for School 𝑓𝑓 ′ (𝑥𝑥) = 2𝑥𝑥 − 2. Mathematics”, NCTM, 2000. Rysunek 3. Tangent line and derivative of the function Rysunek 3. Tangent line and derivative of the function This example helps students to understand the meaning of the derivative This example a function. helpswe Obviously students to understand can change theofmeaning the formula of the 𝑓𝑓 and then derivative all points and a function. Obviously we can change the formula of 𝑓𝑓 and then all points es changes automatically, so student can see this phenomenon for different and es changes automatically, so student can see this phenomenon for different nctions. nctions. mmary mmary In this short article we show the benefits of introduction dynamic In this software athematics short article into we the show theand teaching benefits of process. learning introduction dynamic We conclude athematics software at all students, 54 any from intolevel theofteaching and learning mathematical process. knowledge can beWe conclude encouraged at all students, from any level of mathematical knowledge can be encouraged