Thread: 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?

Originally Posted by Veronica1999

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

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

Originally Posted by Veronica1999

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

This should come out the same as your answer:

$\displaystyle mn - (m-2)(n-2)$

$\displaystyle = mn - (mn - 2m - 2n +4)$

$\displaystyle = mn - mn + 2m + 2n - 4$

$\displaystyle =2m+2n-4 = 2(m+n)-4$

Originally Posted by Veronica1999

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

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

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?

???

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