63 Using GeoGebra as an Expressive Modeling Tool: Discovering the Anatomy of the Cycloid’s Parametric Equation Tolga KABACA 1 , Muharrem AKTUMEN 2 1 PhD. Pamukkale University, Faculty of Education, Denizli, TURKEY 2 PhD. Ahi Evran University, Faculty of Education, Kirsehir, TURKEY tkabaca@pau.edu.tr aktumen@gmail.com ABSTRACT. In Greek geometry, curves were defined as objects, which are geometric and static. For example, a parabola is defined as the intersection of a cone and plane like other conics, which are first introduced by Apollonius of Perga (262 BC – 190 BC). Alternatively, 17 th century European mathematicians have preferred to define the curves as the trajectory of a moving point. In his Dialogue Concerning Two New Science of 1638, Galileo found the trajectory of a canon ball. Assuming a vacuum, the trajectory is a parabola (Barbin, 1996). We can understand that some of the scientists, who studied on curves, actually were interested in the problems of applied science, like Galileo as an astronomer and a physician, Nicholas of Cusa as an astronomer etc. Some of the scientist, who lived approximately in the same century, took further the research on the curves as a mathematician (e.g. Roberval, Mersenne, Descartes and Wren). 1. Introduction In our time, dynamically capable software GeoGebra provides us an innovative opportunity to investigate and understand the curves described as dynamically. GeoGebra can be thought as an innovative mathematical modeling tool. After a comprehensive literature synthesis about modeling by using technology, Doer and Pratt propose two kinds of modeling according to the learners’ activity; “exploratory modeling” and “expressive modeling” (Doer and Pratt, 2008).