Teaching Math with the KDE Interactive Geometry Program
I've written quite a bit about using Linux to help educate people. In the past, I've discussed using Linux to teach astronomy, programming and computer logic design. So today, I'm writing about using the KDE Interactive Geometry (Kig) program to teach mathematics. Kig allows you to use various tools to diagram and demonstrate different mathematical concepts. With Kig, you can draw points, lines, line segments, half lines, vectors, circles and various other conic sections. When Kig refers to a “half line”, it means what I was taught was a ray—essentially a line with one endpoint. Drawing hyperbolic curves on the computer sure beats getting dry-erase marker all over yourself or sneezing because of chalk dust. Even more important though, Kig diagrams are interactive, which means that once you create a diagram, you can move various elements around and see how they behave (more on that later).
Kig's user interface can be a bit deceptive at times. When you first start the program, you are presented with a grid and a group of tools used to create various diagram elements. At this point, you begin to think that the interface is intuitive and that you already “know” how to use it. Then, for a brief moment, you run into trouble. For example, if you try to use a tool to construct a circle given a center point and a line segment as a radius, the program is expecting that the line segment already has been created; you can't create the line segment as part of constructing the circle. After you've used the program for a few minutes, you begin to understand how it “thinks” and things go pretty smoothly.
Thankfully I read the documentation that came with Kig, otherwise, I would have missed out on some of its more powerful features. For example, if you select a curve, say a parabola, from your diagram, you can use the point tool to create a point on that curve. Later, you can drag that point around with the mouse, and it won't leave the curve to which it was constrained. Then, you can use that point to construct other curves, such as a tangent to the curve.
Without reading the documentation, I would have completely overlooked the Add Text Label function that is available by right-clicking on a curve. This function doesn't merely add text to your diagram; there's a text tool for that. The Add Text Label function lets you display information about a curve, such as slope, equation, focus and so on. Once the label has been added to the diagram, you can change various parts of the curve, and the label will reflect those changes.
For example, if you created a parabola through three points, you can add a label that displays the equation of that curve. You also can create a label that displays the coordinates of the points. Then, you can move the points around with your mouse, and see the labels change.
So, what can you do with Kig? Is it just a geometry teaching tool? Kig would be interesting if it were only for teaching geometry, but it can be used for much more. I can easily see how to use Kig to teach algebra, geometry, trigonometry, physics, analysis and calculus.
Let's start with algebra. Figure 1 shows two lines on the Cartesian (X,Y) coordinate plane. Each line is defined by two points, and the coordinates of those points are displayed nearby. The red point in the center is the intersection of the two lines. The equations of the lines also are displayed. By dragging the points around, you can change the lines and explore concepts such as slope, Y-intercept, the solution of an equation and the solution of a system of two equations.
I vaguely remember something from high-school geometry that said “the opposite interior angles formed by two parallel lines crossed by a third, are equivalent.” Figure 2 demonstrates this statement for the case where the angles happen to be 90 degrees. Kig makes it easy to construct two parallel lines. Then, you create a third line that crosses the other two. Finally, you tell Kig to label the angle formed by the various lines. Once this is done, you or your student can change the angles, the distance between the parallel lines and so on—the theorem still holds.
I also remember from when I was in school how obtuse mathematical statements tend to be (particularly those statements related to triangles and the size of their sides and angles), but quite often a picture easily much explains them. Kig would allow you to create an interactive demonstration of each the Euclidean geometry theorems.
Measurements of 30-60-90 do not make a very attractive super model, but they do make a great triangle (at least in Euclidean space, but let's not warp things too much here). Figure 3 demonstrates that the sum of all angles in a triangle is 180 degrees. Once this diagram is constructed, students can drag the angles around and explore any triangle they please.
This also would be a great way to demonstrate the various trigonometric ratios, such as sin, cos and tan. In this case, you simply would construct a right angle, and let the student manipulate the lengths of the sides. Kig could be asked to display the angles and lengths of the sides, and the student then could calculate and verify the various ratios.
But, this is where I found what I think to be one of the weaknesses of Kig. Perhaps I just don't know how, but I was unable to create a triangle and explicitly configure two of the angles. I could drag points to the approximate position I wanted, but I was unable to construct an exact 30-60-90 triangle. It seemed like Kig was keeping the lengths of the sides constant, and thus, when I tried to change the angles, they just didn't “fit”. I think this is important, so if it can be done, please let me know how.
Many concepts in physics can be expressed with what's known as a “vector”. Such things as position, velocity and acceleration can all be expressed as a vector. Kig has rudimentary support for vectors, including vector addition. Figure 4 shows a hypothetical physics problem expressed as the sum of two vectors. Here we have an object moving along a given vector. This object also is being affected by a force, expressed as another vector—gravity in this case. The resulting motion is found by adding the two vectors, as shown.
In analysis, the student begins to learn the features of various curves. Figure 5 gives an example of a hyperbolic curve. I've also included the calculated asymptotes as well as the equation of the curve in both the Cartesian and Polar coordinate systems. Of course, this diagram is completely interactive. Finally, Figure 6 shows a parabola and a tangent line. You also can see the equation of the tangent and watch the equation change as you move the tangent point along the parabola. This could be a nice way to introduce the concept of the differential and, eventually, the derivative in calculus.
Despite a few weaknesses, Kig is a very powerful tool for teaching upper-level mathematics. After climbing a little bit of a learning curve (yes, it's all about curves), both students and teachers can use Kig to have fun learning and teaching mathematics.