I-MATH
Graphing on the Coordinate Plane
I-MATH
Graphing on the Coordinate Plane
Mathematics has a long and interesting history. Every culture, and every decade has contributed new ideas to mathematics. It is important that students know this, and that they realize that mathematics is a living and growing body of knowledge. Hundreds of men and women have contributed, over the centuries, to our mathematical knowledge. The Math Forum has a wonderful page on the internet that will give you an alphabetical list of all of these mathematicians, with links to information about them and their contributions:
From this list, René Descartes is described as "a philosopher whose work, "La Géométrie" includes his application of algebra to geometry from which we now have Cartesian geometry". Descartes created the idea of what we now call the Cartesian coordinate plane, which is included in many geometry courses, and is essential in much of algebra 1 and algebra 2 and the mathematics that follows these courses. A simple idea: every position on a plane can be described by 2 numbers, representing the distance from the distance of a point from a vertical line, and the distance of the point from a horizontal line!
More about the history of mathematics can be found at the following web page:
and a bit more about the coordinate system on the next page:
The National Council of Teachers of Mathematics Standards for School Mathematics in geometry says:
"Instructional programs from pre-kindergarten through grade 12 should enable all students to specify locations and describe spatial relationships using coordinate geometry and other representational systems."
The Geometer's Sketchpad has some interesting features that can be used in teaching coordinate systems for algebra and geometry. If you have access to Sketchpad, open a new GSP file, and choose Show Axes. You will now see a coordinate grid, with tick marks on the x- and y-axes. You can also choose to "Show Grid" if you like.
Use the point tool to construct a point and then show the label. Click on the point and choose Coordinates in the Measure menu. Now drag the point around and observe what happens!
Next, use the line tool (not the segment tool) to construct a line, through your previous point or not through it, as you please. Now select the line (not the points) and choose Equation in the Measure menu. Drag the line in two ways: first by clicking on either one of the two points and then dragging - what does this do to the equation? Now click on the line itself and drag it - what effect does this have on the equation?
Explore the Graph and Measure menus, and if you teach algebra or coordinate geometry, you might want to consider creating some activities using the coordinate features of GSP, for your classes, and/or as your I-MATH project. Perhaps the following diagram might give you some ideas to explore. In this diagram, n is the altitude to line FH. What conclusions can you make about the equations of n and line FH? If you could drag point G, what might happen to the line, the triangle, and the equations? How could you find the coordinates of point J?
