Today in my 6^{th} grade general classes, we explored coordinates in detail. Instead of using a definition based approach or piecing together horizontal and vertical number lines, I borrowed an idea from Dan Meyer and added to it to create an intellectual need in my students.

Earlier this week, my students plotted points on number lines as part of our integer unit. Today, I began the lesson with the following image projected on the board:

I asked students to pick any point on the board and write a detailed description of the location of the point. After a couple minutes, I chose one student to read her description and another student to use the description to find the point. Since the first student picked the top-left point, it was pretty easy to find. I switched the rolls for these students and things got interesting. Since the new reader chose a point near the center-left, it took about 2 minutes for the new searcher to find the point.

I asked for another pair of students and repeated the process. Then, I asked for another pair. Finally, a fourth pair of students went through the activity. Not surprisingly, similar struggles developed for each new point that was being described.

I asked, “What would help us find points easily?”

Some students pondered for a few moments, then they offered answers. One student suggested color-coding the points. Another student suggested breaking up the board into different sections. A third student suggested using letters. Finally, a student suggested using numbers.

I asked students why they offered these suggestions. The consistent answer is labels will help us know which point is which.

Fantastic! Conveniently, I just happened to have this next slide:

Now, I changed the game a bit. I asked students to choose two points, then write a description for moving from the first point they chose to the other point. In order to make the activity have more value, I required that students not include the name of the second point.

After a couple minutes, I repeated a process like I did earlier and asked one student to follow another student’s directions. After the directions were followed and the second point was found, the students swapped rolls. By this point, students were loving the activity. They were realizing how hard it is to write a good set of directions and everyone wanted to try giving and following directions. During my second period, I even had a student say, “This is fun! Can we do it all day?”

After about 5 pairs of students completed the process in my first period, I sensed a growing awareness in the room that our descriptions would never be completely accurate.

I prompted students to think through the question, “What could we do to write easy and accurate directions to move from one point to another?”

One student offered up drawing lines to connect all of the points and labeling them (an interesting thought I had not anticipated popping up in the conversation). Another student suggested making some sort of chart. Finally, a student offered up the thought, “Couldn’t we just use a grid?”

YES! In fact, I just happened to have the ability to throw a grid on the picture. See below:

From this point, I asked students to revise their descriptions from the last activity by using directions (left/right, up/down) and distance (the number of spaces they need to move in a certain direction). The best part was students were eager to tell their new directions for another student to follow, so we ran through that process a handful of times. They wanted to prove they had good directions. They wanted to use the mathematical tool. They had an intellectual need that was now being met with the coordinate plane!

I proceeded to use the rest class to introduce the formal definitions (axis names, ordered pairs, quadrant names) and set students loose to plot some points (they did well with it!). As I ended the day, I asked students if they saw why coordinates are helpful and to share their thoughts. I got answers that varied in detail, but one thought really stuck with me as I went through the rest of the day. A student in my first period said, “The coordinates are better than words because you don’t have to guess about what they mean.”

What are your thoughts? Are there any holes in this method? Is it helpful in developing the utility value of the plane? Couldn’t I just as easily made the entire slide the first quadrant? Yes, but where would the fun be in that?