The following prompt presented at Twitter Math Camp by the Mighty  Max Math Forum (aka Max Ray) has been rattling around in my brain for the last few weeks.  Here a grid representing streets in Ursala’s town:

The problem-solving session, masterfully orchestrated by Max, allowed each group of teachers to develop their own representation of the situation and think about what questions could be asked. For example, if Ursala is at point 1 and needs to get to point 19 along the line segments, without backtracking, how many ways are there for her to travel?  Lots of discussion ensued at our table including the definition of backtracking.

I’ve been at school the last few days and anyone who has sat near me at a meeting in the last few weeks has seen me doodle this scenario, I’m sure wondering what my nerdy math-brain was concocting:

Simplifying the grid and turning it into a pattern expanded the questions that I wanted to ask.  For instance, how many line segments (or streets) in Ursala’s case) are used in step n?

What I’m still grappling with is how to expand my wonder about this scenario past the algebraic representations.  In talking with other teachers recently, it seems as though many of us have been programmed to solve these, and many other problems algebraically.  I recognize that many students won’t reach for the algebraic aid.  So my next step is to try to see this situation in other ways, sans algebra to better understand how my students are likely to see it.