My 3rd and 4th Graders worked on this visual pattern today, trying to find the 10th, 34th and nth steps. Some stray observations, none of them Earth-shattering:

- With time, most of my students were able to understand this pattern recursively, i.e. they understood and were able to articulate how it grows from the 9th to the 10th step, the (n-1)th to the nth step.
- Many of them articulated a description of the nth step in terms of the bottom row (n+1 long), the height of the leftmost column (n high) and the change from row-to-row (2 less than the bottom row, then 1 less each successive row.)
- This description — the rows and columns in terms of the step — is precisely the sort of description that would have helped for a linear growth pattern. Since the growth is constant, this sort of description turns the laborious addition problem into a multiplication problem.
- When my 4th graders were discussing this pattern in whole-group at the beginning of class, one student noticed that the pattern looks like a pyramid. This is precisely the way of seeing this pattern that is helpful for this sort of visual pattern.
- It seems to me that the strategy that students could eventually develop would be to see a rectangle in this pattern, and patterns like it. This would be a new way of seeing, and could help with lots of non-linear patterns. For example…

…and so on.

In short, the problem space of visual pattern puzzles has a structure, and we can probably specify it.

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