When teaching about the Cartesian Co-ordinate system I fear that most teachers focus on the how and not enough on the why. The how is taught, traditionally, by the teacher showing the students two number lines, perpendicular to each other, labeling the intersection of the lines (0,0) and then labeling the positive and negative x and y axes. After, the students are then given many different points to label, of varying difficulty, and then given a word problem. The lesson ends with the students working on page XX and completing either the odds or the evens.
This method, I believe, does not allow the students to understand why the Cartesian Co-ordinate system is extremely valuable. Here is how I introduce this concept:
Breaking the class up into groups of 4, each group then picks a wall in my classroom, or in the hallway. Each group of 4 is then divided up into two pairs of 2, and one pair remains at the wall, while the other pair leaves the wall for 5 minutes. I then walk to each wall and point to a particular section on the wall and leave an erasable mark on it. The following task is then given to the pair of students who are waiting at the wall:
You must create a set of instructions, which you leave at your wall, for the other 2 students to read. These instructions can include whatever you would like but the goal of the instructions is to assist your other group members in finding the point. Once you complete your instructions you are to take a picture of the wall, with the mark showing, and then erase the mark.
I supply my students with flipcams, ipods, and iPads to take the picture.
Once this part is complete, the other 2 students are to return, read the instructions and make a mark of their own. Their mark and the picture is then compared. Afterward we do some debriefing as a class. I ask for groups read their instructions, and share on why their instructions worked or didn't work. Every single time I have done this with my class I always get the same remark:
We need some sort of similar explanation which we all understand before hand
It is only then that I introduce the Cartesian co-oridinate system and tell this story:
Some mathematics historians claim it may be that Descartes's inspiration for the coordinate system was due to his lifelong habit of staying late in bed. According to some accounts, one morning Descartes noticed a fly walking across the ceiling of his bedroom. As he watched the fly, Descartes began to think of how the fly's path could be described without actually tracing its path. His further reflections about describing a path by means of mathematics led to La Géometrie and Descartes's invention of coordinate geometry.