Diagonals - Square Madness

I ended up doing the Diagonals work sheet today just because I wanted to see if I could figure out the answer, and I thought I did for a second until I tried a larger number of rows and columns

In class I thought I had  figured it out but that was because of the way that I had gone about trying to solve it. I was doing random sample squares with varying rows and columns which doesn't help to see any patterns. I still haven't figured it out since my answer seems to only work up to a certain size of square. This is just kind of a way to see the process I took for the next time I look at the problem.

The Steps:

Understand the Problem:
Find a mathematical expression that can determine the number of shaded squares

Devise a Plan:
When nothing works, looking for a pattern always gives some inkling of information. So here I just did a systematic approach of drawing squares keeping the rows constant and changing the columns. During class the issue I had in class was that we didn't keep a constant row or column so we didn't see what was staying constant throughout.

Carry out the Plan:

Look Back:
Well I thought I had figured it out for a second time but then tried it out with a random larger m and n ( 5 x 7) and it failed to give back the correct number. I think I'm a bit stuck for now and decided to clear my head after writing this. Another day to attempt it tomorrow.