## Mathematische Software – Geogebra

This is the Android Version of GeoGebra. It is a lot more cumbersome to use than the desktop version, but you can get used to it. Although I will be criticizing it a bit below it is a very nice software. It can do far more than an average student would ever need. If you do not know the power of GeoGebra, have a look at its home page. My own programs Euler Math Toolbox or C.a.R. and others in the same category may be more powerful on many areas and more versatile. But GeoGebra offers an all-in-one package with teacher support and a world community. Over the years, they completed the program and added missing features. They even included 3D constructions and augmented reality. For this, we had to use specialized programs like Archimedes 3D or Cabri 3D, both payware. Moreover, they added JavaScript support on webpages so that constructions can still be embedded after Java has been killed in the browser. This free package is to be recommended.

Instead of writing a review, let me point out the shortcomings of CAS with regard to the learning process. No, I won’t be arguing that software hinders the process of acquiring mathematical skills and math has to be done with pencil and paper. Rather the contrary is true. I will be arguing for more software usage. But it needs to be used in an intelligent way. For that, we need intelligent teaching.

Have a look at the example in the image above. It was the first example I tried on my Android device. We are discussing the function

\(f(x) = e^{5x} \, \sin(x)\)

We want to learn its behavior. It is very difficult to get a really good impression on the Android device. You can try to zoom in and out. But without further information, you will not be able to grasp its structure. The software uses the nicest feature of touch screens, the pinch zoom. So you can zoom right into the interesting region as in the image above. Even then, it looks as if the function was zero left of -1.5.

If you zoom out further you see the following.

One can only understand this image if one knows how the two factors look like and have ever studied a dampened oscillation before. So the plot does not really help without the mathematical background. But, on the other hand, if you have the background the plot can be a huge help in asserting the knowledge and confirming it.

Next, I tried to find the first local minimum on the negative axis. You can solve that numerically in the program by touching the graph in the minimum. The software will then display one of these black dots showing all special points of the plot, and you can read the coordinates below the plot. I think this is a very nice way of grasping math and something EMT cannot do that easily. Of course, you can do it on the command line.

>function f(x) &= exp(5*x)*sin(x) 5 x E sin(x) >plot2d(f,-1,0.2); >xm=solve(&diff(f(x),x),0.2) -0.19739555985 >plot2d(xm,f(xm),>points,>add);

But let us talk about the CAS aspect of this solution. GeoGebra produces a very interesting solution to this problem.

\(\{ x = 2 \tan^{-1}(\sqrt{26}+5), \, x=2 \tan^{-1}(-\sqrt{26}+5) \}\)

There is a switch to evaluate this numerically. If it is pressed four surprising values appear: -191.31°, -11.31°, 168.69°, 348.69° (rounded). The degrees can most likely be avoided by setting the program to radian mode. The values are correct.

>fzeros(&diff(f(x),x),-200°,360°); %->° [-191.31, -11.3099, 168.69, 348.69]

What do we make of all this?

First of all, the computations, algebraic or numerical, do not make much sense without proper explanations and without the mathematical background. The zeros of the sine function and consequently of the function f simply are the multiples of 180°. Between each zero, the function has at least one extremum, alternatingly a minimal and maximal value. So much is easy to see with the naked eye. In fact, there is exactly one extremal point in each interval. This is harder to see, however. By computing the derivative, we get the extremal points as the solutions of

\(\tan(x) = – \frac{1}{5}\)

Every book about trigonometry contains a plot of the tangent function like the following.

I added the line y=-0.2. So you can easily see that the extremal points repeat in distances of 180°. Problem solved.

Why does GeoGebra produce such a complicated answer involving the square root of 26? That is a mathematical problem in itself and well above the capabilities of high school students. The reason may be connected to replacing sine with 1 minus cosine squared and solving a quadratic equation.

And why does it only show four of the infinitely many solutions? I do not know.

We learn from all this that numerical or algebraic software or plots can be useful. But without mathematical background they are useless.