I have a class of grade nine students this semester that are part of a stretch program. This essentially means that they get 160 hours to complete a 120 hour course. The class is designed to accommodate the transition from elementary school (Grades 1-8) into high school (Grades 9-12) for those students who feel uncomfortable with their math ability.
It also affords me a few extra days here and there to stress certain topics. One of my foci this semester has been pattern modelling. Essentially, we work with various patterns and develop generic rules to describe their behaviour. Linear relations will be our finish line, but I am making sure to provide ample concreteness before abstracting into notations.
This week we began the topic of coordinate graphs; we did this without formal variables. We didn’t graph functions, relations, or even tables of values–we graphed life.
The first time students saw a grid was on Monday morning. As they filtered into the room, I had a blank GeoGebra file open. Unannounced to them, I plotted each student as they came into the room. If a student left, the graph decreased. If a pack entered, I plotted them quickly. It didn’t take long for some to take notice and begin to make sense of the situation.
Soon questions began to fly:
“Which dot am I?”
“What are you drawing?”
“What is that clump of dots?”
Very preliminary. I kept my mouth shut and continued. If the students were going to figure this out, they were going to do it as a collective. Some began hypothesizing that the graph must somehow be a graph of grades. A piece of me died inside when I realized that some students saw this as the only valid function of a graph, but I remained silent.
The bell rang, and the national anthem began. At our school, every student stops and stands until the anthem is complete. This means that students who are late for period one must remain in the hallway until the anthem is completed. This provided a great talking point because during the anthem, the graph stopped.
Once it had finished, more dots appeared. One student then shouted out:
“You’re graphing us!”
I finally answered:
“What do you mean, graphing you?”
“You are using that grid to keep track of who is in the room.”
This was a pretty good explanation, but I countered:
“How am I supposed to know which dot is who?”
“That’s not important. The graph just tracks how many people are in the room now.”
Two things impressed me about this comment. First, he used the word graph. I liked that after they kept calling “points”, “dots”. Some part of my OCD math brain found that incredibly offensive. Second, it opened up the class for some interpretation questions. By now, the graph looked like this:
|Please excuse the awful quality
I addressed the class and began to identify certain points as people. I said that point A (the first one) was a certain student. From that, the class deduced that the graph could not represent grades because this student would not be at the bottom. One girl pointed out that he always came early. From there, the class was reasonably sure that the graph represented time.
I asked what two points almost directly on top of one another meant. They correctly deduced that it represented two people entering together.
At that moment, another student walked in. I plotted them. The class took notice.
We talked about what happened when the points lowered. Someone must have left. Why was there a pause? Students knew that it was the time every student was idle during the anthem.
Things were going very well, and I was about to move on, when a student bolted out of his desk and out into the hall. I lowered the graph. He then poked his head in, and jumped through the door. I plotted him again. He left; I plotted. He entered; I plotted.
He was proving to himself, quite ingeniously, that their interpretation was correct. As the class went on, I made sure to plot every late student. If a student had to go to the bathroom, they plotted themselves, and explained how.
The next day, students entered with a fresh file on the board. They were much quicker to get to deducing. When I spoke to them, the graph looked like this:
It was a graph that compared the number of boys to the number of girls. A girl counted as a positive on the y-axis, while a boy counted as a negative. It took them a while to decipher.
During our process, the same student tried his door trick. He discovered that the graph was effected. Soon after, a group of girls came late. This messed things up because the graph was supposed to fall when someone entered the room.
A girl then got up and tried the door trick. The class soon deduced the graph. I then could ask questions:
“What do these three points in a straight line mean?”
“How many people are in the class?”
“Are there more girls or boys in the room?”
“Am I included on the graph?”
The real-time strategy helped students gain an orientation of the coordinate plane. We have since graphed real-world relations, used Dan Meyer’s Graphing Stories, created our own graphs, and began a study on the shapes of linear relations.
I have also graphed “Number of Students in Desks” vs. “Time” and “Number of Students at Desks” vs. “Number of Students at Tables” in real time. Their power is not in their complexity. In fact, they present quite simple graphs. It comes from their relevance; they place the students as active pieces in the lesson. It gives them the unique role of being both problem creators and solvers.