ÖNSÖZ Matematik bir yönüyle, karmaşık yapılarda düzen arama bilimidir. Bu tanım, hem matematik bilimine katkı sağlamak hem de matematiğin en verimli bir şekilde nasıl öğretileceği yollarını aramak için gayret gösterenlere yönelik bir mesaj içermektedir. Karmaşık bir yapıdaki düzeni çözümleyebilmek, ona farklı açılardan bakabilme ile kolaylaşır. Bu bağlamda, verimli bir matematik öğretiminin önemli bileşenlerinden birisinin de kavramlara ve olaylara farklı özellikleri açısından bakabilmeyi öğretmek olduğu rahatlıkla söylenebilir. Daha formal bir söylem ile bu yaklaşımı çoklu temsil olarak anabiliriz. GeoGebra, matematiksel kavramların üç önemli temsili olan geometri, cebir ve tablo temsillerini birbirleri ile etkileşimli olarak bünyesinde barındıran bir yazılımdır. Matematik eğitimcileri Dr. Markus Hohenwarter ve Dr. Zsolt Lavicza nın önderliğini yaptığı bir ekip tarafından geliştirilen GeoGebra, dinamik yapısı sayesinde, ilköğretimden yükseköğretime kadar her düzeyde matematiksel deneyler ve keşfetme etkinlikleri tasarlayabilmek için mükemmel bir platform sunmaktadır. Ücretsiz açık kaynak kodlu bir felsefe ile geliştirilmekte olan GeoGebra, 40 ın üzerinde yerel dile çevrilmiştir. Türkçemize de Dr. Mustafa Doğan ın önderliğindeki bir ekip tarafından kazandırılmıştır. Dr. Hohenwarter, uluslararası bir çalışma grubunun (Uluslararası GeoGebra Enstitüsühttp://www.geogebra.org/igi) çatısı altında GeoGebra nın matematik eğitiminde kullanımı üzerine çalışan matematik eğitimcilerini bir araya getirmekte ve GeoGebra nın hem teknik özellikleri hem de matematik eğitiminde kullanımı açısından geliştirilmesini sağlamaktadır. GeoGebra, ülkemizde de iki çalışma grubu tarafından temsil edilmektedir (Ankara GeoGebra Enstitüsü - http://www.ankarageogebra.org ve İstanbul GeoGebra Enstitüsü http://geocebir.org). Birinci Avrasya GeoGebra Toplantısı (First Eurasia Meeting of GeoGebra), 3. Uluslararası Gelecek için Öğrenme Alanında Yenilikler Konferansı (Futurelearning-2010) kapsamında bir ilk olma özelliğini taşımakta olup, Ankara GeoGebra Enstitüsü ve FutureLearning Konferansı düzenleme kurulu ile ortak bir çalışmanın ürünüdür. Bu toplantı kapsamında, GeoGebra nın matematik ve matematik eğitiminde kullanımı konulu yerli ve yabancı bildiriler sunulmasının yanında, ulusal çapta geniş katılımlı olarak düzenlenen bir çalıştay ile Türkiye nin dört bir tarafından gelen 100 civarında matematik öğretmenine GeoGebra kullanımı ile ilgili bir eğitim verilecektir. Bu eğitimde, GeoGebra yı hiç bilmeyen kişilerin GeoGebra nın genel özelliklerini bilen, GeoGebra etkinliklerini sınıflarında öğrencileri ile birlikte kullanabilen, basit düzeyde de olsa özgün etkinlikler tasarlayabilen ve en önemlisi de ulusal veya uluslararası kaynaklardan (forumlar veya uzman kişiler) yararlanarak mevcut bilgi ve yeteneklerini artırabilen bir seviyeye getirilmesi hedeflenmektedir. Bu bildiri kitabı, toplantı süresince sunulmuş olan bildirilerin tam metinleri ile birlikte yukarıda tanımlanan çalıştay için rehber niteliği taşıyan bir bölüm içermektedir. Birinci Avrasya GeoGebra Toplantısı nın bütün katılımcılara yararlı olmasını diler saygılarımızı sunarız. Doç. Dr. Sevinç GÜLSEÇEN FutureLearning-2010 Eş Başkanı İstanbul Üniversitesi Fen Fakültesi Öğretim Üyesi Yrd. Doç. Dr. Tolga KABACA Ankara GeoGebra Enstitüsü Başkanı Pamukkale Üniversitesi Eğitim Fakültesi Öğretim Üyesi

FOREWORD A genuine definition of Mathematics may be stated as a science of searching pattern in complicated structures. This definition contains a message for both mathematicians, who try to make contributions to mathematics as a pure science and mathematics educators, who are looking for innovative ways of teaching for mathematics. Analyzing the secret pattern of a complicated structure is utilized. In this sense, it can be said that one of the important components of efficient mathematics education is to teach being able to look at concepts and events in multiple ways. Formally, we can name this approach as multiple representation. GeoGebra is software combining geometry, algebra and table representations, which are three important representations of mathematical concepts, interactively with each other thanks to its dynamic structure. GeoGebra improved under the leadership of Dr. Markus Hohenwarter and Dr. Zsolt Lavicza presents a perfect platform to design experiment and exploring activities in levels from primary to higher education. GeoGebra, which is improved in an idea of free open resource code, is translated into more than 40 languages. It is translated in Turkish by a group with the leadership of Dr. Mustafa Doğan. Dr. Hohenwarter brings mathematics educators, who are interested in using GeoGebra in mathematics education, close together in an international working group (International Institute of GeoGebra http://www.geogebra.org/igi) and keeps its development in the way of technical level and educational usage. GeoGebra is represented by two working groups in our country (GeoGebra Institute of Ankara - http://www.ankarageogebra.org and GeoGebra Institute of Istanbul http://geocebir.org) The First Eurasia Meeting of GeoGebra, which is held with the host of 3. International Future- Learning Conference on Innovations in Learning is a co-organization of GeoGebra Institute of Ankara and organization committee of FutureLearning-2010 Conference. In this meeting, paper presentation sessions, which are about both usage of GeoGebra for pure mathematics research and mathematics education research, and two kinds of workshop sessions, will be held. In the first workshop, epistemology of use of GeoGebra in Math education and some advanced properties of GeoGebra will be discussed. The second workshop will be a local workshop for mathematics teachers, who are mostly new in GeoGebra. In this workshop, participants will be trained about fundamental properties of GeoGebra and use of GeoGebra in classroom. Furthermore, this workshop will be the first in Turkey by its large scale participation. This proceedings book contains full texts of paper presentations and a manual for the local training workshop. We wish a useful meeting for all participants of the First Eurasia Meeting of GeoGebra. Assoc. Prof. Dr. Sevinç GÜLSEÇEN Co-Chair of FutureLearning-2010 Faculty member of Faculty of Science Istanbul University, Istanbul-TURKEY Assist. Prof. Dr. Tolga KABACA Chair of GeoGebra Institute of Ankara Faculty member of Education Faculty Pamukkale University, Denizli-TURKEY

CONTENTS Papers New methods of teaching and learning mathematics involved by GeoGebra...1 Valerian Antohe Effect of Using GeoGebra on Students Success: An Example about Triangles...9 Mustafa Doğan, Rukiye İçel Geometric and Algebraic Proofs with GeoGebra...21 Sema İpek, Oylum Akkuş-İspir GeoGebra as a tool for mathematical education in Slovakia...29 Jan Guncaga Motivating students in learning mathematics with GeoGebra...36 Kyeong Sik-Choi Constructing 3D Graphs of Function with GeoGebra(2D)...46 Jeong-Eun Park, Young-Hyun Son, O-Won Kwon, Hee-Chan Yang, Keyong Sik-Choi Çeşitli Parametrik Eğrilerin İnşasının Görselleştirilmesi...56 Murat Taş, Sevinç Gülseçen, Tolga Kabaca Türev konusunun GeoGebra Yardımıyle Görsel Anlatımı...68 Murat Taş, Sevinç Gülseçen, Tolga Kabaca Local Training Workshop For GeoGebra and Using in Mathematics Teaching GEOGEBRA VE GEOGEBRA İLE MATEMATİK ÖĞRETİMİ...79 Tolga Kabaca, Muharrem Aktümen, Yılmaz Aksoy, Mehmet Bulut 1. Giriş...79 1.1 Dinamik matematik ne demektir?...80 1.2 Neden GeoGebra?...81 1.3 Uluslar arası GeoGebra Topluluğu ve Türkiye de GeoGebra...82 2. GeoGebra yı Kullanmaya Hazırlık...84 2.1 GeoGebra ya Ulaşma...84 2.2 Yazılımı Kurma...84

2.2.1 Kurulum İçin Gerekli Dosyalar.84 2.2.2 GeoGebra yı Çalıştırma.84 3. GeoGebra ya Giriş...84 3.1 Geometrik İnşalar...85 3.2 Cebirsel Giriş ve Komutlar...86 3.3 Hesap Çizelgesi Girişleri...86 3.4 Kullanıcı Arayüzünü ve Araç Çubuğunu Özelleştirme...87 3.5 GeoGebra nın Araç Çubuklarını Tanıyalım...87 3.5.1 Taşı 87 3.5.2 Yeni Nokta.88 3.5.3 İki Noktadan Geçen Doğru 88 3.5.4 Dik doğru...88 3.5.5 Çokgen...89 3.5.6 Merkez ve bir noktadan geçen çember..89 3.5.7 Elips...90 3.5.8 Açı.90 3.5.9 Nesneyi doğruda yansıt.90 3.5.10 Sürgü 90 3.5.11 Çizim tahtasını taşı..91 4. Etkinlik Örnekleri...91 4.1 Çemberde Açı ve Uzunluklar...91 4.2 Parabolün denklemi ile eğrisi arasındaki ilişki...96 4.3 İletki ile açı ölçmenin görselleştirilmesi...99 4.4 Etkinliğin Konusu: Tavşanın zıplamasını modelleyelim...100 4.5 Simetri Kavramını Keşfedelim...104 4.6 Dizi komutundan ve tablodan yararlanma (Çemberin çevresi ve π sayısı)...106 4.7 Fraktalar ve yeni bir araç oluşturma...109 4.8 Pergel ve Çizgilik ile Geometrik çizim uygulamaları...113 4.9 Geometrik İspatları Görselleştirme...114

New methods of teaching and learning mathematics involved by GeoGebra Valerian N. Antohe GeoGebra Institute of Timisoara Pedagogical High School D.P.Perpessicius, Braila, Romania valerian_antohe@yahoo.com Abstract: Archimedes drew his figures on beach sand, mud or ash on a floor or put on his body, previously anointed with oil; on his body the figures were drawn with nails. When the Roman general Marcellus conquered in 212 BC Syracuse in Sicily, the city of Archimedes, a Roman soldier came across this genius contemplating his drew circles on the sand. "Nolite turbare circulos meos!" (Do not break my circles) Archimedes told to the soldier, but this, irritated, stabbed him with the sword, killing him. Today the place of the geometric constructions is on dynamic platforms supported by specialized software. One of these platforms is GeoGebra software. As the inventor stated, GeoGebra is dynamic mathematics software for all levels of education that joins arithmetic, geometry, algebra and calculus. It offers multiple representations of objects in its graphics, algebra, and spreadsheet views that are all dynamically linked. While other interactive software (e.g. Cabri Geometry, Geometer's Sketchpad) focus on dynamic manipulations of geometrical objects, the idea behind GeoGebra is to connect geometric, algebraic, and numeric representations in an interactive way. You can do constructions with points, vectors, lines, conic sections as well as functions and change them dynamically afterwards. Furthermore, GeoGebra allows you to directly enter and manipulate equations and coordinates. Thus you can easily plot functions, work with sliders to investigate parameters, find symbolic derivatives, and use powerful commands like Root or Sequence, (www.geogebra.org). Because some of us (teachers of different subjects), see the computers in terms of "IT specialist", producers of software and not as a user, we propose this ongoing project in order to bring the computer in the context of approaching successful teaching. We believe that the teacher, regardless of the specialty they teach, shouldn t know what's in the magic box (called generic computer), but he ll have to know that the magic box will help him to assist the student to learn, the route to knowledge becoming pleasant! This paper is an invitation to solve math problems in a natural didactic way using the GeoGebra platform. Keywords: Innovation, Geometric Locus, New Way of Learning. Introduction There is a struggle to integrate the computer in school. This approach must be read: "the integration of educational software in education". This is a related desire to implement new teaching methods in mathematics. An "educational software application is something that everyone could use on a computer, without having advanced knowledge about computers and programming. Draw, build, unite, and investigate properties, change shape and size. Properties remain the same? Why? Can you formulate the theorem from this investigation? Prove it rigorously! Experience should not only be lived, but shared. When the action will become more global, this software that I called 1

The GeoGebra Language will not only be a working method but also a step in opening a viable way to exchange ideas, and the investigations will become constructions of new methods of investigation of math phenomena. Different methods of math teaching have been proposed and knowledge of these methods may help in working out a better teaching strategy. It is not appropriate for a teacher to commit to one particular method. A teacher should adopt a teaching approach after considering the nature of the students, their interests and maturity and the resource available. Every method has certain benefits and few flaws and it is the teacher s work to decide which method is the best. The investigation above will present an implementation of GeoGebra in order to solve some math problems. Bringing the computer in the context of approaching successful teaching It is often said more about using interactive methods in teaching mathematics and about their implementation in the curriculum. However we appreciate that the first step must be done when the blackboard and chalk are replaced with dynamic image of mathematical phenomena, integrated in dynamic software like GeoGebra. There are no barriers to this and only the wish to use the system can produce the desired success. If the implementation conformity with the curriculum seems to be difficult, we accept that GeoGebra platform will be a challenge for beginning and math teachers, if they will accept an innovative method to transmit information, they will encourage students, to spark interest to investigate, to discover the phenomenon mathematically and to justify the results found in rigorous mathematical sense in the end. These were the desires of the new Institute GeoGebra from Romania, to propose to our fellow colleagues a teaching approach, making the invitation to use the GeoGebra platform for dynamic representation of mathematical results. Ultimately, the whole curricula within mathematics could be structured in terms of sequences of GeoGebra using topics with associated learning resources. Students could form teams to explore these sequences, just in the same way they now explore levels within video-game environments, [Gerry Stahl and All, 2010]. We can do constructions with points, vectors, segments, lines, conic sections as well as functions and change them dynamically afterwards. On the other hand, equations and coordinates can be entered directly. Thus, GeoGebra has the ability to deal with variables for numbers, vectors and points, finds derivatives and integrals of functions, [Valerian Antohe, 2009]. Math problems solved with the GeoGebra platform A step by step construction, which represents the visual interpretation of the mathematical context, a problem of a geometric locus will follows the next steps: constructing geometric figures based on hypotheses, applying geometric transformation, (move the point, move the point along the line, move the line preserving the direction or modify the figure preserving the measure of some angles, etc.). Understanding the relationship between Euclidian construction and proof, we can create demonstration that involves animation and action button, find out geometrically and algebraically connections in a rigorous proof, [Gabriela-Simona Antohe, 2009]. The statement of the first problem: On the fixed line d, are considered the fixed points A and B in the plane and the mobile point M. In the plane are built regular polygons of [AM] and [AN] sides, with m, respectively n 2

number of sides, where m, n N, m, n 3. The circumscribed circles of those polygons are intersected in M and P, (Fig.1). Is required the geometric locus of P, [A. Dafina, 2003]. Figure 1: Geometric locus of P for m=5 and n=4 After the investigation with GeoGebra software, some different results could be raised. The GeoGebra application shows in a real time all the changes and will allow the changing of m and n for different values with the sliders, (Fig.2). Figure 2: Geometric locus of P for m=5 and n=4 The statement of the second problem: Miquel's Five-Circle Theorem is among a sequence of wonderful theorems in plane geometry bearing his name. Let P1, P2, P3, P4 and P5 be five points. Let Q1=P2P5 P1P3, Q2=P1P3 P2P4, Q3=P2P4 P3P5, Q4=P3P5 P1P4, and Q5=P1P4 P2P5. Let the other intersections of the consecutive circumscribed circles of triangles Q5Q1P1, Q1Q2P2, Q2Q3P3, Q3Q4P4, and Q4Q5P5 be M1, M2, M3, M4, and M5 respectively. Prove that M1, M2, M3, M4 and M5 are cyclic, (Fig.3). There are a lot of interesting proofs of this theorem. Miquel's Five-Circle Theorem is difficult to prove algebrically, [Hongbo Li, 2004]. 3

Figure 3: Miquel's Five-Circle Theorem When n = 3, the three vertices of a triangle are on a unique circle, which can be taken as the unique circle determined by the three edges of the triangle, called the Miquel 3-circle, (Fig.4). When n = 4, the 4 edges of a quadrilateral form 4 distinct 3-tuples of edges, each determining a Miquel 3-circle, and Miquel's 4-Circle Theorem says that the 4 Miquel 3-circles pass through a common point (i.e., are concurrent), called the Miquel 4-point, (Fig. 5). This combination of perspectives allows the teacher to demonstrate, in front of students and together with them, strategies revealing the "behavior" of figures. Connections between different representations of math concepts will accomplish here the necessary background for better understanding, steady knowledge of mathematical literature. One appreciates the pedagogical implications of exploring geometry in a dynamic environment, both as an investigation tool and as a demonstration one, the connection between math educators and specialists in informatics being one of the best and a challenge at the same time. The term of Dynamic-Info-Geometry could be a method of math teaching and the start of future investigations in applied mathematics, [Gabriela Antohe, 2009]. Figure 4: Miquel's 3-Circle Theorem 4

Figure 5: Miquel's 4-Circle Theorem. The position of A, B, F, could be modified but the four circles will have a common point P, the Wallace point The third problem statement: One of the current problems of education is multidisciplinary treatment of some problems of mathematical modeling. In a study of mathematical modeling of surface water quality I investigated the problem of water quality characteristics of the Danube, such as the evolution of dissolved oxygen concentration. For this we considered the mathematical modeling of these with Spline functions. The data referring to the dissolved oxygen have been processed in Matlab taking into consideration the determination of a function with the help of spline functions. The same data were processed in GeoGebra in order to obtain a polynomial function which could describe the data evolution. This example could be a result to the question of the Weierstras problem in order to find the appropriate function which could describe the reality better, find the function and identify methods of interpolation. Weierstrass Law, which shows that any continuous function f can be approached with quite a good precision on a close given interval by a polynomial forms. Unfortunately this theorem does not offer any practical criterion of finding the right polynomial form [Cline K.S., 2007]. With GeoGebra, the polynomial form will be: P(x)=-0.00001 x 11 + 0.0007 x 10-0.02265 x⁹ + 0.425 x⁸ - 5.10931 x⁷ + 41.1428 x⁶ - 225.28496 x⁵ + +833.75065 x⁴ - 2028.84107 x³ + 3063.25584 x² - 2551.097 x + 880.81. This form it is obtained by GeoGebra using the command Polynomial [A,B,C,,K, L], (Fig. 6). 5

Figure 6: GeoGebra Polynomial form demonstrate that the DO concentration is too small in January 2007 Figure 7: Matlab interpolation with Cubic Spline Function more appropriated to reality. The program analyzes the Dissolved Oxygen concentration at the Danube River, SGA Braila, Km 219 Analyzing the polynomial function of 11-th degree graph, the great anomaly could be seen during January- February 2007. GeoGebra shows that a polynomial form could not have modelled that reality. This is a convergence to the idea that other function like harmonic function must be analyzed. The interpolation with Spline function, (Fig. 7), was more relevant and the graph represented the hypothesis that the evolution of the quality parameter, [V.Antohe, C.Stanciu, 2009]. The fourth problem statement: Let ABC be a triangle and M be an inner point of the triangle so that AM=BC. Shoe that max{(bm/ac);(cm/ab)} 2 1/2-1 and this is the best possible constant. 6

Figure 8: The problem of a geometric inequality relation After the geometrical context was done, (Fig.8), the values of the left and on the right hand were analyzed. The value of the difference between the left member and the right member of the inequality appear in the horizontal line, the graph of p(x). Even if this modelling does not give a demonstration on GeoGebra rigorous analysis of the successive positions of the peaks of the triangle, keeping the requirements of the problem will emphasize such inequality in borderline cases, cases that can get out of context. Of course for educated this exercise of geometric and algebra realization of context is a good exercise, a first step to understanding, analyzing and demonstrating the problem rigorously. Other two similar problems are presented in these imagines, (Fig. 8). Conclusion Figure 8: Other problems of a geometric inequality relation GeoGebra provides good opportunity for students to work in pairs and talk through the project together. Attractive presentations prepared in advance, not only capture students attention but also may lessen the immediate cognitive load for educated and educators. In addition to what is traditionally recognized as benefits, a lot of teachers often use real world models. In order to enhance the image mathematics by creating a halo effect, the proposed efficient space for this will be the GeoGebra platform. The teachers who use GeoGebra must be more specific, more "open minded", willing to allow for experimentation, and give more guidance at the start of any GeoGebra experiment. Dynamic geometry offers opportunities to bring the real world into the classroom, adding visualization, color and animation. This would not be possible in a traditional classroom. This GeoGebra thinking is expected in 7

various topics of the curriculum but, if they are not found there, we shall connect the GeoGebra thinking with topics and other different experiences, in a model of more efficient curricula. References Gerry Stahl, Murat Perit Çakir, Stephen Weimar, Baba Kofi Weusijana and Jimmy Xiantong Ou, (2010), Enhaging Mathematical Communication for Virtual Math Teams, Acta Didactica Napocensia, Vol5, Nr.2, 2010 Valerian Antohe, (2009), Limits of Educational Soft GeoGebra in a Critical Constructive Review, Annals. Computer Science Series, 7-th Tome, 1-st Fascicle, Tibiscus University of Timisoara, Romania Gabriela-Simona Antohe, (2009), Modeling a geometric locus problem with GeoGebra, Annals. Computer Science Series, 7-th Tome, 2-nd Fascicle, Tibiscus University of Timisoara, Romania Andrei Dafina, (2003), Consideration about a geometric locus problem, Axioma Journal No.22, SSM Prahova, Romania Cline K.S., (2007), Secrets of the Mathematical Contest in Modeling, Journal of Multivariate Analysis, 1.6. Valerian Antohe, Constantin Stanciu, (2009), Modeling and Simulation of Quality Indicators of Surface Water, Environmental Engineering and Management Jurnal, Vol. 8, No.6, George Asachi Tehnical University of Iasi, Romania Hongbo Li, (2004), On Miquel s Five-Circles Theorem, MM Research Preprints, 166-177, MMRC, AMSS, Academia Sinica, No. 23, December 2004 8

Effect of Using GeoGebra on Students Success: An Example about Triangles Mustafa DOĞAN Department of Primary Mathematics Education University of Selcuk, Konya,TURKEY mudogan@selcuk.edu.tr Rukiye İÇEL Department of Primary Mathematics Education University of Selcuk, Konya,TURKEY Abstract :This study aimed to observe effects of using GeoGebra based activities about triangles on 8 th grade students achievement. Two different classrooms from a primary school in Konya have been selected for the study. A pre-test has been applied to the both classes. The pre-test results show that there was not any statistically significant difference between the groups. One of the groups selected as experiment and the other as control group. Teaching and learning activities for the experiment group were mainly prepared with GeoGebra which were based on the Ministry of National Education Teacher Guide book. Simultaneously, the control group continued their usual teaching and learning process as guided by the Ministry. A post-test has been applied to the both groups after two weeks of teaching. Furthermore, one month after the application a recall test was applied as well. Possible comparisons between the tests and the groups have been performed. The results show that computer based applications (GeoGebra) have positive effects on students' learning and achievement. It has also been observed that it improves students motivation with positive impact. This paper presents the GeoGebra prepared activities including Pythagoras relation. Keywords: Dynamic Geometry, GeoGebra, Students Success, Triangles Introduction Nowadays, transferring information to learning environments has mainly depends on technological developments in many ways. The use of technology in learning environments provides both execution of education in accordance with the requirements of the era, and opportunities to train more qualified individuals. Most commonly used technologies in the learning environments are computers and their software. Training activities become increasingly more complex day by day parallel with the developments in science and the change in the nature of the knowledge. Thus, such many factors require and force use of computers in contemporary education. It is a general agreement that the traditional methods force students to learn mathematics by memorization ending up with a falling success and imposing a feeling of being unsuccessful in mathematics. However, the nature of mathematics requires 9

high-level of mental processes such as critical thinking, reasoning, imagination and considering many different features with related facts. To achieve this, it is not enough to use only pencil-drawn shapes on paper or board. In particular, along with the constructivist approach, mathematics courses need to be addressed with different emphases which make them enjoyable, understandable and constructible in terms of students. Developed technology in the last century turned people to computers and their use in many areas. At the primary age, children mainly use computer for entertainment especially spending more time for gaming. It is accepted that computer and software use in Primary education is promising and may improve mathematics education remarkably, if it is directed to teaching and learning process. In this respect, computer based mathematics courses are offered as an alternative. Geometric constructions acquire dynamic properties with the computer (dragging, transforming, rotating, symmetry, opening and closing of a prism, or a pyramid etc.) so that students can make observations as well as the imagination. Using computer in geometry teaching is implemented with the new elementary mathematics curriculum in our country and has become indispensable (MEB, 2005). The most important role of computers in primary mathematics education is stated as making the learning of abstract concepts easier in the curriculum. Some previous researches in the area reported that computer use is more effective than the traditional approach to learning, especially, in transformation geometry, polygons, prisms and pyramids. This research aimed to observe possible effects of GeoGebra based activities on 8 th grade students achievements for triangles and Pythagoras relation. Literature Review It is officially stated that two main purposes of primary education in our country are to prepare individuals to higher education and to life (MEB, 2005). Reasoning, critical thinking and problem solving are considered as necessary mental skills to achieve these purposes effectively. Mathematics plays an important role in developing these skills. This importance installs primary responsibilities for everybody in the field of mathematics education (Baykul, 1999). Parallel with the recent developments in all over the world, primary and secondary schools mathematics curriculum have been reconstructed in Turkey as well. It is especially indicated in the new mathematics curriculum that Computer Based Mathematics Teaching (BDMÖ) provides meaningful learning experiences of mathematics for students. Therefore, it has to be integrated into mathematics courses (Çakıroğlu et al, 2008). Geometry is called as "it examines figures and their movements in the elementary mathematics curriculum. It is stressed in the curriculum that while the geometrical thinking is developing also knowledge acquired in geometry activities have to provide visual and analytical reasoning and inference with a hierarchical order within the required attention respectively. The results of student s reasoning with intuition are called conjecture. Producing information via inference called conclusion, although very few students may produce information via inference. It is also highlighted that while the students achieve targets about related areas of geometry, special attention and importance should be given for processing of specific skills, affective features, psychomotor skills and selfregulation. In this context, they are especially stated in the Ministry's own textbooks that the dynamic geometry software have to be used and experience should be shared with the students. Sulak (2002) studied effects of computer-based instruction on student achievement and attitudes in mathematics courses. In the study, the computer-based teaching was found to be better when compared to the 10

traditional methods in terms of both achievement and attitudes. Similarly, Aktümen and Kaçar (2008) have investigated possible effects of computer algebra system (Mapple) on students attitudes toward mathematics. They reported that the students who use Mapple in learning environments have more positive attitudes towards mathematics. Güven and Karatas (2003) aimed to determine students views about computer-based learning environment created by dynamic geometry software Cabri. At the end of the study, the students views have changed positively for mathematics in general and geometry in particular. The students also find dynamic geometry environment very useful. Furthermore, it is reported that the students gain more confidence by exploratory mathematical activities. Memişoğlu (2005) has investigated the effects of network research on 6 th grade students achievement in mathematics teaching. It has been reported that the network research method is more effective than the traditional method. Moreover, it was detected that almost all of the students have positive thoughts about the network research. Karakus (2008) intended to determine possible effects of computer-based teaching on student achievement for transformation geometry subjects. In the experimental study, there was significant difference in favor of experiment group. All students of the experiment group have achieved high attainment level with computer-based instruction in teaching of transformation geometry. Moreover, this difference becomes more significant and gets higher for successful students in the subjects of reflection and rotation. However, there is not any significant difference between experiment and control groups for low successful students; it has been observed that computer-based instruction increased the experimental group success. Similarly, Faydacı (2008) investigated how the new subject of transformation geometry in elementary mathematics curriculum effects students conceptions and how the students construct the knowledge about it. A specially designed teaching program was developed (with the help of Wingeomtr software) for technology-supported teaching of the subject. Students handling of transformation geometry subjects, their conceptualization of the concepts and ways of making knowledge meaningful for themselves have been analyzed through the study. During these analyses, they have focused on the source of students perceptions. Main focus was whether the perceptions based on seeing the drawings from computer screen, or with the underlying mathematics of the movements. Results of the study showed that the prepared program taking into account of the principles of constructivist approach (for example, assimilation etc.) contributed to the students learning by doing thought-provoked mathematical abstraction. In addition, it has been identified that the use of technology in learning process has an active role in transition from drawing to the figure of an object. Üstün and Ubuz (2005) performed an experimental study to compare traditional educational environments with the dynamic learning environments (Geometer's Sketchpad used). According to the results of the study, there was a significant difference in favor of the experiment group on the recall (permanence) test. The most important reason for this significant difference is identified as students explorations of geometrical shapes to see possible connections by manipulating the computer based environment. Bedir and colleagues (2005) approved that using Geometer's Sketcpad software on teaching of "Angles and Triangles" topic is more effective than the traditional education in the students achievements. As a dynamic mathematics software, use of is getting increasingly more common all over the world. In addition to construct geometry dynamically, it also provides, as a key element of learning geometry, visualization, 11

estimation, conjecture, construction, discovery, proof and etc. This study is about using GeoGebra for the subjects of triangles in eighth grade. Methodology This experimental study is conducted in the fall semester of 2009-2010 academic year. Two eighth grade classes from a primary school have been selected as experiment (n=20) and control (n=20) groups. Before the classroom activities, a pre-test was applied to the both groups to determine the students attainment level. The questions covered seventh grade objectives for the subject. The pre-test results show that there was not any statistically significant difference between the groups. Therefore, one of the groups selected as experiment and the other as control group. It aimed to observe effects of computer-based learning environment (GeoGebra software) on students success. A two weeks course (total of 12 hours) which contained twelve main GeoGebra activities and many other practices about the stated achievements has been planned in accordance with the official mathematics curriculum. Then the activities were constructed with GeoGebra for the experiment group. The GeoGebra prepared activities aimed to make the subject more dynamic, concrete and visual. GeoGebra software was introduced in introductory hour of the course. In all of the other sessions, the GeoGebra prepared activities were shared with the students both with visual and dynamic features. Furthermore, examples and drawings on the textbooks were constructed with the GeoGebra during the sessions. In the official curriculum (MEB, 2005) teaching of triangle for eighth grade takes total of fifteen hours with eight different objectives. These objectives are stated as follows in the mathematics curriculum. 1. Determines the relationship between the sum or difference of two sides lengths of a triangle and the length of the third side (1 activity). 2. Determines the relationship between the sides lengths of a triangle and corresponding angles degrees between the sides (3 activities). 3. Draws a triangle with given measures of the sufficient elements (3 activities) 4. Able to construct mediator (1 activity), perpendicular bisector (1 activity), angle bisector (2 activities) and altitude of a triangle. 5. Able to construct Pythagoras relation (1 activity). 6. Explains the equality terms associated with triangles. 7. Explains the similarity terms associated with triangles. 8. Determines trigonometric ratios of acute angles for right triangles. Thus, a total of twelve main activities have been prepared with GeoGebra and then used in the classroom for this study. Simultaneously, the control group continued their formal teaching and learning procedure as guided by the Ministry. A post-test was applied simultaneously to the both groups after two weeks of teaching. The post-test contained questions about all the stated objectives for the eighth grade. The post-test has been used to see possible effects of GeoGebra on students success. Furthermore, one month after the application a recall test was applied as well. 12