1. The Border Problem
Without counting, use the information given in the figure above (exterior is 10 x 10 square; interior is an 8 x 8 square; the border is made up of 1x1 squares) to determine the number of squares needed for the border. If possible, find more than one way to describe the number of border squares

2. Numerical Expressions Geometric Explanation 10^2-8^2
Area of large (exterior) square minus area of the small (interior) square.
4(10)-4
There are four sides of length 10 minus the four overlapping squares.
2(10)+2(8)
The are two sides of length 10 and two sides of length 8. Notice that for the two sides of length 10 that all corners of the border are counted.
4(8)+4
There are four sides of length 8. We must add 4 for the four corners.
4(9)
There are four sides of length 9. Each side includes a different corner. Each of the four corners is counted exactly once.
There is one side of length 10 (which includes two corners). There are two sides of length 9 (which includes a total of 2 corners). There is a side of length 8 (which includes no corners)

3. Video Discussion Why without talking? Why without writing?
Why without counting one by one?
Why not give them each a grid to facilitate their thinking?
Why did the teacher act as the recorder for the arithmetic expressions?
Boaler, J. & Humphreys, C. (2005). Building on student ideas: The border problem, part I. Connecting mathematical ideas: Middle school video cases to support teaching and learning (pp.13-39). New Hampshire: Heineman Publications.

4. What about a 6 in by 6 in grid?
What about an n inch by n inch grid?
Create a verbal representation
Use the verbal representation to introduce the notion of variable
If n represents the number of unit squares on one side, give an algebraic expression for the number of unit squares in the border.
Develop understanding of function, variables (independent and dependent) and graphing.