Students will be able to make sense of the Border Problem and appreciate various methods of figuring out the total number of border squares. Students will look for structure and use this structure to figure out the total number of border squares in other figures.

## Number of Players

Individual or Group

Without counting one by one, how many squares are in the colored border of the 10 x 10 grid?

- Turn to your partner and talk to each other about what you got and how you figured it out.
- How many squares are there?
- Did any of you get a different answer?
- How many of you got 40 the first time? Can you explain how you got 40?
- Did any of you get 38 the first time? Can you explain how you got 38?
- Let’s see some different methods for getting 36.
- How many of you understand __________’s method? Can you paraphrase it?
- How about a different method?
- How are these methods alike and different? Various Methods:
- (4 x 10) – 4
- 10 + 9 + 8 + 8
- 10+ 10 +8 +8
- (10 x 10)-(8 x 8)
- 4 x 8+4

- How many colored squares are in Figure 1?Explain how you know.
- How many colored squares are in figure 1,006? Explain how you know.
- How many colored squares are in figure 1 x 10 to the power of 6? Explain how you know.
- How can you determine how many colored squares will be in any figure? Justify your ideas.

- If you have 64 squares can you use all of them to make a square border? If you can, what is the side length of the square? Explain your thinking with mathematical justification.
- If you have any number of squares, how can you determine if you can make the square border using all of the squares?
- If you have 256 squares can you use all of them to make a square border? If you can, what is the side length of the square? Explain you thinking with mathematical justification.

## Reference

Licensed under Creative Commons Attribution 3.0 by Yekaterina Milvidskaia and Tiana Tebelman