Why is slope rise/run? What makes slope rise over run?  What makes the standard equation for a line use a slope of rise over run as opposed to run over rise? 
What would the standard equation of a line look like if m was equal to run/rise $\frac{\Delta x}{\Delta y}$

We conventionally draw graphs like this:
$$
\begin{array}{c|ccccccc} 
10 & & & & & & & \bullet \\
9 \\
8 & & & & & & \bullet \\
7 \\
6 \\
5 & & & & \bullet & \bullet \\
4 \\
3 & & & \bullet \\
2 \\
1 & \bullet \\
0 & & \bullet \\  \hline
& \text{sunday} & \text{monday} & \text{tuesday} & \text{wednesday} & \text{thursday} & \text{friday} & \text{saturday}
\end{array}
$$
The slope should measue how steep the graph is, which in this example is how fast this changing quantity is changing as time passes.  As you see, this quantity went from $1$ up to $10$ in $6$ days.  That means its average rate of change is $9/6=1.5\text{ per day}$.  That is the slope of the line through the first and the last of the plotted points.  That is how steep it is, and thus it is how fast the quantity changes.
