## Reflection: Connection to Prior Knowledge Seeing Structure and Defining Problems - Section 3: Another Problem Without Words

This problem is foundational to the idea of systematic lists.  One of my favorite math textbooks, Problem Solving Strategies: Crossing the River with Dogs, dedicates an early chapter to helping students explore how creating systematic lists is itself a strategy.

The dot problem we're investigating today provides the first four steps in a sequence.  As I help students make sense of what they see, I ask them to partition each solution into subsets.  Look at "Step 3".  There are five solutions here: one with two black dots, three with one black dot, and one with none.  If we only consider the three with one dot filled in, can we arrange them in a sequence?  See how we can think about that one black dot "moving," step-by-step from its position at the front to the back.

When students are stuck, I show them how to think like that, and then I ask them to practice by partitioning the solution set for Step 4 in a similar way.  Right away, we can see that there is one more way to have one black dot.  We see that there's still just one way to have no black dots, and then we count a few ways to fill in two dots.  We also notice that we can further sub-divide the solutions with two black dots into those with one white dot in between or two.

When it comes time to figure out what the 13 solutions to Step 5 will look like, I tell students to think about the subgroups they've already discovered on the previous problems, and by then, everyone is able to access the problem.

Systematic Lists
Connection to Prior Knowledge: Systematic Lists

# Seeing Structure and Defining Problems

Unit 6: Mini Unit: Patterns, Programs, and Math Without Words
Lesson 2 of 10

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