Which is a better treatment of that problem with the rectangle's diagonal? How are you defining better? Better for what purpose? Help me out here.
From Alan Schoenfeld's 1994 Math 67 midterm:
The diagonal of the 3 x 5 rectangle below passes through the interiors of 7 of the 15 squares that comprise it. In general, consider an N x M rectangle. Through how many of the NM squares that comprise the N x M rectangle does the diagonal pass?
From Kate Nowak's blog:
Draw a 9 by 3 rectangle on a square grid. Draw one diagonal. How many squares does the diagonal pass through? Draw some non-similar rectangles with one diagonal. How many squares does the diagonal pass through? Develop a rule to determine the number of squares a diagonal passes through for any rectangle of any size.
My own treatment, submitted for review, correction, and debate:
How many squares will the diagonal of the large rectangle cut through? [This question added because it wasn't clear I'd ask it - dm]
I'll follow up in the comments at some point on the decisions that went into my redesign.