How many squares does the diagonal of this rectangle go through? 
I have a rectangle made of tiles measuring $9$ by $12$ tiles to get $108$ tiles. A diagonal line is cut through the top left corner down to the bottom right corner. How many tiles does the diagonal go through?

I had this question on a test today and I want to know the answer. I did this by literally drawing it up, and shading the squares the line went through. I got 14 squares. Now I searched this up on MSE and found this answer which asks the same question. However, I don't quite understand the answer, and I don't get what the $(N,M)$ part of the last line meant either. (I'm only a Year 7). So, can someone give me a way how to find out the answer using techniques a Year 7/8 would know. Also, please keep formulas to a minimum. (We are not allowed to use a formula in our working out, in which we have to show). Thank you

The diagonal $d$ will go through two grid points which divide it into three equal parts. Each part $d'$ is a diagonal of a $4\times3$ grid rectangle $R$. It intersects three horizontal and two vertical interior grid lines of $R$. These $5$ intersection points partition $d'$ into $6$ parts. It follows that $d'$ traverses $6$ tiles, hence $d$ traverses $3\cdot 6=18$ tiles.
