squareCircleZ

GeoGeobra is an intelligent graphing software that allows the user to interactively explore Cartesian & Euclidean geometry – as well as calculus. Best of all – it’s a free offering!

You can download it from http://www.geogebra.org/.

From their blurb:

GeoGebra is a free and multi-platform dynamic mathematics software for schools that joins geometry, algebra and calculus.

Building an Interactive Document in GeoGebra

Let’s go through the process of creating a document in GeoGebra. Our document will allow the end-user to explore the changing slope of a polynomial as x changes value.

First, we enter the function at the bottom of the screen, using the carat “^” for powers of x:

The program converts the function display (see under “Free objects”) so that it is more easily read by a human. I have scaled the y-axis by clicking on the “Move” tool (the one on the far top right) and simply dragging the axis to the desired scale.

Next, we are going to add a tangent line to our curve. We add a new point on the function using the “New point” tool:

Note the other tools that are available on this tool item:

• Intersection of 2 two curves
• Midpoint between 2 points

Now we choose the “Tangent” tool as follows:

Note the other tools that are available on this drop-down. You can construct:

• Perpendicular line through a point
• Parallel line through a point
• Perpendicular bisector
• Bisector of an angle
• Diameter line of a conic section

We now have a tangent line. The exploratory activity we can do now is to drag the point “A” to any position on the curve and the tangent line follows along. Even better, we can get a readout of the actual slope as we move around the curve, by typing in:

s = Slope[t]

Rather than just giving a numerical value for the slope, it actually gives a triangle with base length 1 unit, indicating more clearly what a slope at a point really means. The result for one part of the curve is as follows:

Visual Calculus

Let’s now add the curve of the first derivative to our existing plot. (Of course, we expect the first derivative curve to be a parabola, since it will be a polynomial of degree 2).

We achieve this by entering:

Derivative[f]

The green curve is the first derivative curve (a parabola, as expected):

We can trace the locus of a point (B) moving on the first derivative curve, as follows:

You can use GeoGebra to examine critical points like local maximums and minimums on the curve and the point of inflection (point A) illustrated above.

Other Tools

GeoGebra is a feature-rich offering. The other tools available in GeoGebra that I have not already mentioned include:

• Rotate an object around a point
• Draw line segments
• Draw vectors
• Draw polygons (including regular polygons)
• Construct various circles, arcs and sectors
• Angles, distances and areas
• You can add text and images
• You can zoom in and out on objects

Split Functions

GeoGebra will draw piece-wise functions (with a little coaxing). You can achieve the following (with a vector thrown in):

Output

I like the variety of output options. You can either:

• Save your file for later use (it will have a .GGB extension)
• Save the graph as an image in PNG, EPS, SVG or EMF format
• Save the graph to the clipboard (for manipulation in an image editing program or for pasting into a document)
• (The best one) Save as an interactive Web page

I will be using this tool to create interactive explorations in the Interactive Mathematics site. But for now, here is the above example in a Web page that you can play with. (You don’t see the toolbars, of course, just the Drawing Pad:

Cubic polynomial with first derivative exploration
[Caution - needs Java which you may need to download]

Euclidean Geometry

GeoGebra allows you to easily create angles, polygons and conics.

As you can see in the regular dodecagon above, GeoGebra allows you to measure angles, including internal angles.

Gripes

You need Java 1.4.2 or later to run GeoGebra.

I’ve never been a big fan of Java-based software, including those based on Web plugins, because the Java run-time environment is a huge memory hog and it takes a very long time to load. Compared to Adobe’s Flash, Java is very bloated.

The size of the associated files in Geogebra is quite crazy:

• GeoGebra-parabola.ggb (2 kB) This is the file that I created above (with the parabola and first derivative. Nice and small)
• geogebra_gui.jar (313 kB) – the GeoGebra interface
• geogebra_cas.jar (506 kB) – the GeoGebra computer algebra system – or inbuilt ‘calculator’
• geogebra.jar (649 kB) – ??
• geogebra_export.jar (670 kB) – also ??
• geogebra_properties.jar (340 kB) – why so big? A properties file should consist of a small amount of text, surely?

For comparison, a typical Flash file on the Interactive Mathematics site is around 30 kB and the required plugin is around 2 MB (but this does not get downloaded each time you access a Flash file, just once).

However, being Java-based, GeoGebra will run on any operating system and that is a big plus.

Conclusion

GeoGebra is an impressive geometry and calculus exploratory tool. I will use it over the next few months to create more interactives for the Interactive Mathematics site.

GeoGebra is more intelligent than MS Math 3.0, which I reviewed recently (see Microsoft Math 3.0 Review), but the audience for each product is not exactly the same. Having both products (for under US$20), you will have some excellent tools for exploring mathematics.

Do yourself a favour – download GeoGebra now!

Footnote: There are some good resources written using GeoGebra at Index of Interactive Materials For Classroom Demonstrations.

Related posts:

1. How to import Geogebra files into JSXGraph GeoGebra is a good tool for creating interactive java applets....
2. Graphs using free math software Do we still need to graph on paper, or should...
3. Autograph 2-D and 3-D graph plotter: a review Autograph is a well-designed 2-D and 3-D plotting application...
4. Free math software downloads Wanting to use some math software but find it’s too...
5. Explore the slope of the cos curve What is the value of the slope of the cosine...