Writing equations this early in the year can cause problems in the classroom, especially when you can have any mix of 4th – 12th grade students in class. This year I have a large group of older students who have experience with equations (yet don’t quite remember how to do it- I wonder how this always happens?). So after a quick failure in writing an equation (that’s easy Mr. A! Wait, what? I’m lost!), I turned to Fawn.
What do you notice? That’s EZ Mr. A! You are adding 2 trees each time. I wait, I don’t respond- this scares them. I don’t want to justify their quick response, I want them to really look and think. I let the silence linger, they start looking at the problem, I see the gears spinning in their heads- they are starting to do what I need them to do and examine things a little more critically. One student says, There is always one black tree Mr. A! I stay silent. Another says, You are stacking trees. Another tells me we have two rows of trees and you are adding one tree to each row. I just casually lean on the board and wait. Things are really heating up without me saying a word, I love it.
Sometimes the best thing that can happen in class is when a student points out something that may or may not be related to the problem or what you are doing, but it shifts student thinking. By staying silent for so long I got them to shift out of cruise mode and analyzing, but one of my younger students said, You are making a stairs out of the green trees. Now I have used this pattern quite a few times before now but I have never heard that one. I had to break the silence. I asked him what he meant. You are stacking the trees on one another and if you look at only the green trees, they kinda look like stairs. He came forward to the whiteboard and drew this:
Great observation! How does this help us? Silence. I wait. Another student said, Well Mr. A, you are having us do a graph for our pumpkins, what if I drew a line over the top of our trees? So I did, and waited. (This year I have not had my students graph VPs, we have just looked at patterns and practiced writing equations for them. I had intended to take this next step with them on this problem, but was interested to see what direction this was taking.) I had some of my older students look at it and tell me that was an easy line, it went up one and over one. It’s slope was one. But my young student came through again, But our pattern was 2 trees, not 1! So they modified the slope to 2/1 or 2. They quickly wrote down the equation of y=2x + 1, identifying the black tree as a constant in the picture. How does that relate to our pumpkin graphs? Students identified 1 as the y-intercept, and groaned that they had to find out where all their lines would cross the y-axis. They had the idea so I set them off to work on a sample pumpkin so they wouldn’t fear they were messing their own work up.
Here are a couple of examples of what students got done today.
Not the road I was intending, but it did lead to where I wanted to go. Once again math proves to me that there are many different ways to think and process about a problem. Some are fast, some are slow but they all get us to the end.