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 $\frac{\Delta x}{\Delta y}$
 A: 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  $x$. 
A: 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.
A: Graph of a linear function
I think it has something to do with how we write from left to right and therefore interpret that graph and the definition of slope "from left to right". Look at graph 1 normally and you will observe a slope of 3, RISE over RUN. Our reference axis is x.
Same graph but from a rotated view
Look at graph 2 now and if you read the axis from right to left instead, with the positive and negative inverted, you can instead observe a slope of 1/3, RUN over RISE. Our reference axis here is y.
Hope that helped.
A: Just like a linear function $y=ax+b$ shows the relation between the two variables (which is dependent is determined by the problem), the slope of the line (linear function) shows the relation between changes of the two variables: $\frac{\Delta y}{\Delta x}$ or $\frac{\Delta x}{\Delta y}$.
The key is which variable is dependent. 
For $y=ax+b$, if $y$ is dependent on $x$, then the slope is:
$$\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}=\frac{(ax_2+b)-(ax_1+b)}{x_2-x_1}=a;$$
if $x$ is dependent on $y$, then the slope is:
$$\frac{\Delta x}{\Delta y}=\cdots=\frac{1}{a}.$$
