Checkerboard Borders

Learning Focus

Use variables to describe ways of seeing a pattern in a diagram.

Big idea

When a border surrounds an $N \times N$ square, different groupings (corners, sides, layers) lead to equivalent expressions for the same count.

Worked example

A one-tile-wide border around an $N \times N$ center can be counted as:

• $4N$ for the four sides, then adjust if corners were double-counted, or
• $4(N+1)$ depending on how the border is drawn

The key is: write an expression in $N$, then check it on a concrete size (like $N=5$ or $N=6$).

Takeaways

• Diagrams can be counted in more than one correct way.
• Variables let a contractor (or you) generalize beyond one diagram.
• Equivalent expressions name the same quantity differently.