A Suggestion of Using Geogebra at College Level: the Case of Finding Arc-Length

A Suggestion of Using GeoGebra at College Level: The Case of Finding Arc-Length of a Curve

Tolga KABACA*, Mehmet BULUT**

*Pamukkale University, Education Faculty, Department of Teaching Mathematics, DENIZLI-TURKEY

**Gazi University, Education Faculty, Department of Teaching Mathematics, ANKARA-TURKEY

Members of GeoGebra Institute of Turkey

It is certain that, GeoGebra is a powerful tool to teach mathematics. Its dynamic feature, which allows constituting a bridge between algebra and geometry, may be used to design a discovery learning environment. In this kind of environment, we can see the student as a GeoGebra user, while we are a GeoGebra programmer. I want to call your attention to another possible use of computers in teaching math. Students should be also seen as a programmer.

Computer programming makes the computers can think and act about something perfect and fluent. Desired mathematical thinking can be thought as a computer program. If the algorithm of the program is perfect and the syntax is completely correct, the program works fluently and operates the correct answer for us, completely like our mathematical thinking structure. If our thinking structure on a certain concept is a well-constructed one, we can make any generalization by the help of that concept.

At this point, we should ask ourselves the question “what do we need to understand by computer programming in the sense of teaching math?” we need to focus on only making a perfect algorithm of programming issue. After the making algorithm, any programming language may be used. But our software is not need to be even a programming language. It is enough that using a software that allows to run our algorithm. Moreover, the software, which we used, should be simple to learn and to use. By running the algorithm, we can provide an opportunity to check thinking structure for the math learners.

Let’s consider on the concept “finding arc-length of a curve” the make more clear what I mean. We can assign two tasks for the students. First, they should construct an algorithm that can evaluate the arc-length of every kind of curve, which is represented by a function. Then, they need to run this algorithm by using GeoGebra.

The algorithm may be something like follow, after an enough discussion environment; 1- plot the graph of the function f(x), 2- determine some points on the curve of f(x) for x = a from x = b if you want to evaluate the arc-length in the interval x  [a, b] 3- Constitute the line segments, attaching every successive point on the curve. 4- Find the sum of the length of all line segments, which obtained at the previous step. 5- When we increase the number of points on the curve, we will approach to the real value of arc-length of the curve.

Now, we should help our students while running this algorithm on GeoGebra. This kind of approach may be considered as an alternative way instead of presenting ready GeoGebra applications. By this way, students may realize their own thinking structure by simulating GeoGebra’s way of running algorithm. Let’s see how GeoGebra can execute the algorithm above;  First, we need to assign necessary free objects (inputs); - Function f(x). I defined as “f(x)” here. - Initial point and end point. I defined as “a” and “b” here. - Difference of the every successive point on the x-axis. I defined as “delta” here. . Actually, “delta” object is not free. It is dependent to the variable “n” which represents the number of difference between “a” and “b”. Therefore; delta=(b-a)/n

 Then, it is needed to obtain a sequence of points on the curve by following GeoGebra command;

pointlist=Sequence[(a + i delta, f(a + i delta)), i, 0, n]

 Let’s construct the line segments, attaching every successive points in the sequence above;

segmentlist=Sequence[Segment[Element[pointlist, i], Element[pointlist, i + 1]], i, 1, n]

 Last, we should to evaluate the sum of the every line segment in the list above;

arclength=Sum[segmentlist,n]

When a student constructs the GeoGebra application above, he or she can change the function, initial and end points and number of difference. So, he or she can either make some experiments or check his or her thinking structure.

Let’s make a trial;

Choose the function f( x )= 1 - x2 and initial and end points as -1 and 1. Three different screen views are displayed as follows;

Table.1: Finding the Sum of the seven segments on the curve Table.2: Finding the Sum of the 68 segments on the curve

Since the function represents a half circle with the radius 1, we found the arc-length as 3.14, which is close to the real number .

We can also construct the algebraic result by using the command lines as an extension of our GeoGebra application. GeoGebra helps us by its dynamic properties. So, we found our result easily. That is, GeoGebra automatically evaluates the length of every line segments. If we encourage our students to study on finding the length of every line segment algebraically, they may recognize a famous integral formula that evaluates the arc length of a curve.

If we rewrite the GeoGebra command “arclength=Sum[segmentlist,n]” in terms of mathematics; n

arclength= segmentlistk k =1 Since GeoGebra evaluates every segment’s length dynamically, we reached the result easily. We should ask our students to construct every segment’s length algebraically. This will direct our students to the integral formula of arc-length. So, they will have constructed a bridge between visualization and abstract concept. Let’s see the suggestion of constructing the integral formula;

The GeoGebra command “Segment[Element[pointlist, i], Element[pointlist, i + 1]] ” represents the length of line segment attaching two successive points on the curve and means mathematically;

Length of the line segment between the points

(a + i*delta, f(a + i*delta)) and (a + (i+1)*delta, f(a + (i+1)*delta)) that means

length=( a +( i + 1)* delta - a - i * delta)2 +( f ( a + ( i + 1)* delta ) - f ( a + i * delta )) 2 =delta2 +( f( a + ( i + 1)* delta ) - f ( a + i * delta ))2

To find the total length we need to evaluate following sum instead of the GeoGebra command Sum[segmentlist,n];

n delta2 +( f( a + ( i + 1)* delta ) - f ( a + i * delta ))2 i=0 Finally, as we understand from the visualization, the actual arc-length is

n limdelta2 +( f ( a + ( i + 1)* delta ) - f ( a + i * delta ))2 n i=0

Since as n goes to infinity delta goes to 0 and delta is the difference in independent variable x, the term ( f( a+ ( i + 1)* delta ) - f ( a + i * delta )) will be the difference in dependent variable f(x). Let’s represent it as deltaf(x). Besides, do not forget x  [a, b], therefore;

n b Arclength=lim deltax2 +( deltaf ( x ))2 = dx 2 + df ( x ) 2 n i=0 a

b 2 骣df( x ) =1 + 琪 dx a 桫 dx

And, this is the final result. Let’s check the arc length of the semicircle by this formula;

2 1 骣 2 Arclength = 1+琪( 1 -x) dx = p (Evaluated by Maple) -1 桫

As a result, we discovered the famous integral formula with our students. With the help of a well designed GeoGebra assisted course, all calculus concepts may be programmed and made more concrete.