1 Visualization of the Concept of Polar Defined Function by Using Maple Tolga KABACA *, Muharrem AKTÜMEN ** * Uşak University, Faculty of Arts and Science, Department of Mathematics, Uşak, Turkey ** Kastamonu University, Faculty of Education, Department of Primary Mathematics Education, Kastamonu, Turkey

2 Computer algebra systems have great potential for professional engineers, scientists and technologists. CAS can reduce the tedium of routine manipulation required in some mathematical models and procedures, give a greater assurance of accuracy, and even enable users to apply techniques which they might find difficult without such assistance [1]. Also (CAS) potentially offer students a powerful tool which is used in mathematics education. COMPUTER ALGEBRA SYSTEM

3 According to Majewski (1999) CAS should help us to: 1. Expand our computational power 2. Explore mathematical concepts 3. Experiment with mathematical concepts 4. Visualize mathematical objects 5. Prepare and publish teaching materials 6. Improve communication between teacher and student 7. Present concepts online and support distance education 8. Test and train students COMPUTER ALGEBRA SYSTEM

4 In our study; We have observed that one of the challenging issues, in calculus courses, is polar coordinates and graphing polar functions’ curves. We have developed an illustration to make this concept clearer. In general, students believe that polar coordinates and Cartesian coordinates are completely different concepts. Especially, traditional method of drawing a polar defined function does not make any sense in students’ mind. COMPUTER ALGEBRA SYSTEM

5 I mean, by traditional method, finding some polar points and connecting them continuously. Since we don’t use same method while drawing a Cartesian defined function, the relationship, between Cartesian and polar defined function, can not be realized clearly. Actually, both of them are very similar. Only difference is identifying style of a point in IR plane. Computers’ method of drawing method of a curve can help us. Because Computers use same method while drawing all kind of functions. POLAR COORDINATES 2

6 Let’s try to make our students meet a contradiction POLAR COORDINATES

7 COORDINATES

8 WHAT IS THE DIFFERENCE BETWEEN TWO GRAPHS? y x0 (x, y) LET’S OBSERVE HOW THE GRAPHS ARE BEING FORMED…..

9 CARTESIAN COORDINATES

10 POLAR COORDINATES

11 (x, f(x)) y x0 (x, y)  r = (rcos , rsin  ) since the mission of variables is not same let’s call them differently… COORDINATES P P

12  Let’s define Cartesian function y = f(x) = 1 + sin(x) in polar coordinates and observe the following graphs. Let’s define Cartesian function y = f(x) = 1 + sin(x) in polar coordinates and observe the following graphs.  Let’s also define the polar function r = f(  ) = 1 + sin(  ) in Cartesian coordinates and observe the following graphs. Let’s also define the polar function r = f(  ) = 1 + sin(  ) in Cartesian coordinates and observe the following graphs. Now, we can provide an opportunity to observe how a polar or a cartesian function is being formed for our students.COORDINATES

13 Polar Function in Cartesian Coordinates

14 Cartesian Function in Polar Coordinates

15 Student Worksheet

16 Conclusion Even if students know the difference of the role of the variables (x,y) and ( ,r) theoritically, the application was confusing for them. At the end of this visualisation, we observed a deeper insight. We gave some complicated polar curves (e.g. rose curves) to students to draw them by using paper and pencil techique. By the help of last worksheet, our students find a lot of opportunity of cheking their graphs and observing the relationship between the independent variable  and dependent variable r interactively. Especially engineering students had showed great attraction for this interactive application.

17 References 1.Clements, R. (1999). Essential Mathematical Concepts Needed by User of Computer Algebra. Teaching Mathematics and its Applications, Volume 18. No 4. 2.Majewski. M., (1999). Pitfalls and Benefits of the use of Technology in Teaching Mathematics, Proceedings of the Asian Technology Conference in Mathematics,

18 Thank you for your Attention! For further information please contact Tolga KABACA;