Visualizing Expressions with the Border Problem

Here’s a problem that captures a number of nuances in understanding expressions. Take a moment to try it out, and then read on to see how students thought about it.

Without counting each individual square, how many red squares are there in the figure? How did you figure that out?

Gathered around a small easel, students shared how they saw the 36 red squares. First, they went through all the ideas with just numbers, like 10+10+8+8 = 36 or 4*9 = 36. Understanding the different methods was the first cognitive task of the lesson.

After that, students generalized the equations using variables. This was the next cognitive task. We had to define a variable as n = the length of the inside blue square (n=8 in this case):  

10+10+8+8 = 36 translates to 2(n+2) + 2n = 36

4*9 = 36 translates to 2(n+1) = 36

A few days later, we revisited the students’ equations to confirm that they all simplify to the same basic formula, 2n+4. Recognizing that even though these expressions appear different, they all represent the same value. This was the third cognitive task, and it involved the use of the distributive property, which has been another focus so far this year.

The final cognitive challenges sent kids further into the realm of generalization. If these formulas are true, they should tell us the border tiles for any case (n=0, n=1, n=2, n=3, etc …). Furthermore, they should tell us how large the border will be for partial cases (n = 2½)  or even negative cases (n = -1).

The visualizations students included in these forays show the development of a deep understanding of the meaning of expressions.