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 $\frac{\Delta x}{\Delta y}$

If you write the equation of a line as $$y = mx+b$$ then it comes from this naturally. That's because if you move along the line a little bit from the point $(x,y)$ to the point $(x+\Delta x, y+\Delta y)$, then the new point satisfies the equation too. So you have that $$y+\Delta y = m(x+\Delta x) + b = mx + m\,\Delta x + b$$ Subtracting these gives $$\Delta y = m\,\Delta x$$ which means that $$\frac{\textrm{rise}}{\textrm{run}} =\dfrac{\Delta y}{\Delta x} = m$$ You can call it what you want, but the coefficient of $x$ in the original equation is the rise over the run. If you want to call run/rise the slope, then its reciprocal would be the coefficient of  $x$. 
