Here is today's visual pattern:
The version the students were working with was in grayscale so the patterns formed by the different coloured circles were not evident. They recognized that this was a quadratic pattern, but when I asked how they knew many said that it was because the number of circles was not increasing by the same amount. So I threw some numbers (something like (1,5), (2,15), (3,23), (4,63)) on the board and asked if they represented a quadratic pattern and again, got lots of "Yes, they don't go up by the same amount.". Yikes! I pushed further and asked them to tell me what they knew about this "pattern". They found the first differences and were even more convinced. Someone said something about second differences so we found those. Someone said that the second differences needed to not be constant and I decided to just correct that misconception. We could have looked at other known quadratic patterns and verified one way or the other, but I wanted to keep moving forward. We looked at the number of circles in each step, adding the 4th step to get more data, and showed that the 2nd differences were, in fact, constant.
Then we looked at different ways of seeing the pattern growing and found a rule.
This all took quite a bit of time which is why I decided to keep the ropes for tomorrow. I explained the goals for the ropes. Firstly, to find a relationship between thickness and diameter so that we could extrapolate to find the length of a thicker rope needed for a given number of knots. Secondly, given two different ropes, determine how to make them the same length with the same number of knots. The latter will require students to solve a system of linear equations which we have not done for a little while. So the rest of today's class was spent practicing solving systems with this handout that my colleague put together (thanks, Michelle!). Many students could not remember how to solve these, but with a few questions from me (What's the same in the two orders? What's different? What does that mean?) they did quite well.