It’s been a super hectic few months, so Explore MTBoS is exactly what I need to get the blog back in gear. Here we go!
One of my absolutely favorite open-ended problems is The Border Problem, snagged directly from the Boaler & Humphreys’ Connecting Mathematical Ideas. I use the problem as a way to talk with students (and parents) about how we can represent our thinking using numerical expressions, and how being attentive to how we are showing our thinking numerically can help someone else to understand how we are “seeing” or thinking about a particular problem. It’s also super useful as a way to introduce algebraic expressions.
This is a 10×10 grid. Without counting, talking or writing anything down, determine how many squares are in the border.
Humphreys is super deliberate in how she poses the problem, and it was great to read about the amount of thought that had been put in to thinking through every aspect its framing. I follow her advice and deliberately do NOT print out a copy for each student (since this increases the chance that kids will count squares). Once I’ve given a good 20-30 seconds or so for kids to come up with their idea I ask them to share only their answer with the people at their table (there will always be some kids who will get 40). I then have students write down a numerical expression to show how they thought about it and I pass out smaller 10×10 grids so that they can also show their strategy geometrically. Once students have had a while to familiarize themselves with their strategy and those of their group members I show them a 6×6 grid and ask them to “see” it both using their own strategy and one of two others. I do the same for a 15×15, then ask them what it would be for a much bigger grid (24×24, 231×231, etc.), eventually building to an NxN.
Getting all the class’ strategies up on the board and discussing them together makes for an awesome conversation. Students want to name the different strategies (2Ls, Cut Corners, etc.), are super willing to give credit to one another for cool ways of looking at the problem that they hadn’t seen, and can’t help but making connections between the different ways of thinking. All good stuff!