This year I have had the privilege of being part of a support group for math teachers. It is actually called a Professional Development Collaborative, but it feels more like a support group. It includes teachers from our district that teach math at some level from elementary school to high school. We meet once a month to do math, plan for instruction, debrief on how lessons went after our last meeting, and to peruse each other’s blog posts from the last month. It has been a great opportunity to learn a ton of math, and to collaboratively plan lessons that are being taught k-12 in our district.
At our last meeting, we worked together to predict future steps in this visual pattern from Visual Patterns:
We spent about an hour as a group trying to find an expression that could be used to predict any step in this pattern, and I will be honest, I was in a bit over my head. High school algebra is still a bit of a bad dream for me. The beauty of this group of teachers is, however, that it is a safe place to try this problem out. With a bit of a headache (and a firm belief that this would be beyond my 4th graders students), the group took a look at the problem that we would do with our students. We chose this problem:
After spending so much time on the triangle problem, we didn’t have a ton of time to plan the lesson at that meeting. Thankfully I co-teach this class with another member of this support group, and we spent about 40 minutes one afternoon doing the math, writing expressions, and anticipating what students would say. We asked each other questions like, “do we expect them to write an expression with a variable? What questions will we need to ask to get them to see the pattern? What kinds of patterns do we think they will see? What tools should we give them? Graph paper? Color tiles? Do we need to show them a ratio table so that they can see the pattern numerically?”. We decided that the big idea we wanted them to explore was equivalence. That the different ways they may see and record the pattern were all equivalent.
We began the lesson with a Google Slides presentation with the images and questions we wanted them to consider, and a few thoughts about what they might do and how we would get them to the big idea of equivalence.
I was surprised to see many students noticed, not only that the tower was increasing by 1, but that there were other ways they could describe the pattern they were seeing. We had decided to name the steps for them (step 1, step 2, step 3). Students were able to build step 4 without much trouble. When we asked them to build step 6 without building step 5, there were a few groans, but for the most part, they were able to try it. Many were quickly able to move on to finding a rule that would help them to predict any step. We had two ways students recorded the pattern they were seeing, mainly step + 3, and step + 2 + 1. Though they didn’t all count this increase in the same way. Some saw it this way, and considered the “tower” on the back to be growing by 3 with each step.
Others saw it this way, considering the red circled blocks as the constant 3, and the tower growth to be a number equal to the step.
Still others saw it as only growing 2 with each step, and then they had to count that one little block on the bottom.
The things that surprised us during this lesson:
- Their ability to access this problem without much struggle;
- The variety of interpretations of the pattern
- The students that struggled. There was at least one group that never developed a rule, and it was a group that does not typically struggle in math. They were trying to apply a multiplicative rule to this pattern, and couldn’t see why it wasn’t working until the end of our time together.
Next up? Try a multiplicative pattern! I was so encouraged at how they attacked this problem that I plan to try a pattern with a multiplicative relationship so that they can compare the two patterns and their growth. Stay tuned…