Most showed me the pattern growing along the diagonal but were able to see it growing in other ways. They figured out that you needed to add 3 blocks, then 4 blocks, then 5 blocks... I asked if this was a linear pattern and some said yes while others said no. I asked what a graph of the number of the blocks vs. step number would look like.
After they thought about that for a while I pulled up Desmos and plotted the points.
They could see that the points did not form a line and someone said that this was quadratic (yay!). I asked how we have found the equation to represent a quadratic pattern so far, and they said that they have used graphing calculators. I then told them that we could figure out the rule without using graphing calculators. We talked a little about how rearranging blocks to form a square plus some blocks can be helpful with quadratic patterns, but that it didn't work here. Admittedly, I was stuck on this yesterday, but thanks to Dave Lanovaz, I was able to share a fantastic strategy with my class today. He suggested doubling the pattern like this:
We now have rectangles whose side lengths we can relate to the step number:
Now that we have a rule for this pattern, we can divide by 2 to get the rule for the original pattern:
I love this! I love that I learned a new way of finding a rule for a quadratic pattern and that it is accessible to all of my students.
We spent the rest of today's class working on the trig matching activity. There was an error in my original fine - the corrected version is here. They are very slowly working through these problems and will continue tomorrow.