The Cartesian coordinate system is a fundamental necessity to mathematics. Anyone beyond elementary school knows this. However, when a student first encounters the system, it can be overwhelming. Two number lines- one horizontal and one vertical- at the same time? Negative numbers? This can be a difficult system to master for many students.

Helpings students recognize the usefulness of a mathematical concept or system, before they are introduced to the technical terms and procedures, can be very helpful. Surely students could learn about the Cartesian coordinate system by having the teacher stand in the front of class and lecture on about the origin, x-axis, y-axis, procedure for graphing, etc. However, in the case, students do not fully recognize the utility of the system. Additionally, all the terms, definitions, and procedures can scare students away. Admittedly, this is how I taught it last year, resulting in low engagement and effectiveness. Instead, having students discover the Cartesian system layer by layer would allow them to realize its beauty, before learning the ins-and-outs.

To do this, I had two students volunteer come to my interactive white board (IWB), which was simply a white screen. I had Student A face the board, while Student B turned the other direction. With student A watching, I marked the board in the upper-left hand corner.

I then created the same exact page, before I had marked the board, which in this case was simply another blank white page. Student A would now give directions to Student B on where to mark the board, trying to get as close as possible to where I had marked it before. However, Student A now had to turn around and face away from the board. This forced Student A to only give verbal clues. Student B would use these clues to make their best guess, resulting something similar to this:

Most cases resulted in the the students getting vaguely close, but never spot on. As a class, we reflected and discussed how difficult this was.

I then had the students repeat the process, though this time I added an x-axis and y-axis (explained to them as a horizontal and vertical line). Again, I marked the board with Student A watching, and Student B facing away:

Now a new clean x-axis and y-axis page, it was time for Student A to give the verbal directions to Student B. This example illustrates one of the resulting marks from Student B:

Typically, Student B’s mark was closer to my mark. Furthermore, our ensuing discussion focused on how much easier it was for Student A to give Student B the clues. Students used words like “lower left-hand corner” and “bottom half.” Little did they know they were discussing the different quadrants.

Still digging deeper, I gave Student A and Student B another chance. This time, I added the horizontal and vertical lines:

Again, a turned around Student A gave Student B verbal clues. This time they achieved the following:

Success! Student B had finally hit my mark. Their sense of self-satisfaction was unmatched. Also, our discussion keyed in on how extremely easy this time was. This was especially highlighted on how Student A told Student B to start in the middle (the origin) and to move right 3 spaces and then up 3 spaces.

I had the students perform this procedure one last time, in this case with the positive and negative numbers on the graph:

Student B was able to use Student A’s verbal directions to get the answer:

This time, the students moved even quicker. Student A told Student B, “Start where the 0 is [the origin] and move right to the 2, then down to the negative 7.” The class discussion was very positive, with students reflecting on how much easier this example was, compared the previous three.

Through this lesson, students were introduced to the important pieces of the Cartesian coordinate system. *However*, they were not simply told the different characteristics and procedures. This is something I told them after that fact; the origin is the center of the graph, that we have ordered pairs, the first number is the x coordinate and gives horizontal direction and distance, the second number is the y coordinate and gives vertical direction and distance, etc. Through this method, students were not overwhelmed or scared away by the coordinate grid. Instead, they comfortably introduced to the different layers of the system. In the end, they put it all together to fully grasp how the system worked. They even viewed this as a game, asking me several times since “Can we play that coordinate grid game?” The effectiveness of having students discover and identify the mathematics themselves can not be underestimated.

What a brilliant idea! I’m always looking for and thinking about how to help students to develop conceptual, deep understanding of math concepts – this lesson does it in spades!

I love the way you scaffolded the task so that students easily became more successful with each attempt, while showing graphically *why* their results were improving, leading to a nice, easily understood, mathematical conclusion. Great stuff!