As individuals, the team explored the question:
For any n x m rectangle such that GCD(n,m) = 1, find a rule for determining the number of unit squares (1 x 1) that a diagonal passes through. The rectangles are a) 2 by 7 and b) 3 by 4. In order to come up with an appropriate solution, one must consider the greatest common divider and its possibilities. The numbers need to be considered, are they prime or composite numbers? Is the slope a factor in finding the answer? The first answer provided was to find out the number of squares the diagonal line goes through, first divide the L (length) of the rectangle by the W (width). This will not be a whole number so rounding up necessary, then multiply that number by the W, which provides an answer. For example if the first rectangle W=2 and L=7 when seven is divided by two the answer is 3.5. Round that number up to 4 and then multiply by (2 width) to get 8. If the W=3 and L=4 then divide four by three and get 1.33 and round up to two. Multiply two by three and get six.
This theory could work, however, one must ask how would you write an equation to tell someone to round up. If the answer yielded two even numbers then rounding would not be necessary. On the other hand, rounding may be needed for odd integers. Another theory to solve this question could be to multiply m by n and divide by two. This doesn’t work because in the 2×7 rectangle the diagonal passes through 8 squares. There are instances when this theory might work, but it is not a valid theory because it is not sure to work every time.