Basic algebra problem

1. A flat, rectangular board is built by gluing together a number of
square pieces of the same size. The board is m squares wide and n
squares long. Using the letters m and n, write expressions for

(b) the total number of 1 × 1 squares with free edges (the number of
1 × 1 squares that are not completely surrounded by other squares);

I just doubled the perimeter 2(m + n) and then subtracted 4 to correct the overcounting.
Teacher says I am missing the point of the problem and should have done

mn - (m - 2)(n - 2) = mn - mn - 2m - 2n + 4

Whats wrong with getting it simpler?
What is the point of the redundant way?

Quote:

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Originally Posted by Veronica1999

I just doubled the perimeter 2(m + n) and then subtracted 4 to correct the overcounting.

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You didn't double the perimeter, you found the perimeter. The perimeter is the length of the entire path around the rectangle.

Quote:

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Originally Posted by Veronica1999

mn - (m - 2)(n - 2) = mn - mn - 2m - 2n + 4

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This should come out the same as your answer:

Quote:

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Originally Posted by Veronica1999

Whats wrong with getting it simpler?
What is the point of the redundant way?

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This I cannot answer. Your solution seems fine to me.

Quote:

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Originally Posted by Veronica1999

1. A flat, rectangular board is built by gluing together a number of
square pieces of the same size. The board is m squares wide and n
squares long. Using the letters m and n, write expressions for

(b) the total number of 1 × 1 squares with free edges (the number of
1 × 1 squares that are not completely surrounded by other squares);

I just doubled the perimeter 2(m + n) and then subtracted 4 to correct the overcounting.
Teacher says I am missing the point of the problem and should have done

mn - (m - 2)(n - 2) = mn - mn - 2m - 2n + 4

Whats wrong with getting it simpler?

Nothing wrong, in my opinion ... some teachers prefer you do it their way.

What is the point of the redundant way?

???

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one final observation, note that mn - (m - 2)(n - 2) = mn - mn + 2m + 2n - 4 , not mn - mn - 2m - 2n + 4