1 visualization of the concept of polar defined function by using maple tolga kabaca *, muharrem...

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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

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• 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 SYSTEMCOMPUTER ALGEBRA SYSTEM

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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 SYSTEMCOMPUTER ALGEBRA SYSTEM

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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 SYSTEMCOMPUTER ALGEBRA SYSTEM

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• 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 COORDINATESPOLAR COORDINATES

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Let’s try to make our students meet a contradiction

POLAR COORDINATESPOLAR COORDINATES

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COORDINATESCOORDINATES

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WHAT IS THE DIFFERENCE BETWEEN WHAT IS THE DIFFERENCE BETWEEN TWO GRAPHS?TWO GRAPHS?

y

x0

(x, y)

LET’S OBSERVE HOW THE GRAPHS ARE BEING FORMED…..

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CARTESIAN COORDINATESCARTESIAN COORDINATES

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POLAR COORDINATESPOLAR COORDINATES

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(x, f(x))

y

x0

(x, y)

r

= (rcos, rsin)

since the mission of variables is not same let’s call them differently…

COORDINATESCOORDINATES

P

P

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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.

Now, we can provide an opportunity to observe how a polar or a cartesian function is being formed for our students.

COORDINATESCOORDINATES

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Polar Function in Cartesian CoordinatesPolar Function in Cartesian Coordinates

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Cartesian Function in Polar CoordinatesCartesian Function in Polar Coordinates

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Student WorksheetStudent Worksheet

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ConclusionConclusion

• 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.

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References1. 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, 52-59.

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Thank you for your Attention!

For further information please contact Tolga KABACA;

tolgakabaca@yahoo.com