# Intro to slope

CCSS Math: 8.F.B.4
Walk through a graphical explanation of how to find the slope from two points and what it means.
We can draw a line through any two points on the coordinate plane.
Let's take the points $(3,2)$ and $(5, 8)$ as an example:
The slope of a line describes how steep a line is. Slope is the change in $y$ values divided by the change in $x$ values.
Let's find the slope of the line that goes through the points $(3,2)$ and $(5, 8)$:
$\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD6}{\greenD 2} = 3$
Use the graph below to find the slope of the line that goes through the points $(1,2)$ and $(6,6)$.
$\text{Slope} =$

$\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD4}{\greenD 5}$
Notice that both of the lines we've looked at so far have been increasing and have had positive slopes as a result. Now let's find the slope of a decreasing line.

## Negative slope

Let's find the slope of the line that goes through the points $(2,7)$ and $(5, 1)$.
$\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD{-6}}{\greenD3} = -2$
Wait a minute! Did you catch that? The change in $y$ values is negative because we went from $7$ down to $1$. This led to a negative slope, which makes sense because the line is decreasing.
Use the graph below to find the slope of the line that goes through the points $(1,9)$ and $(4,0)$.
$\text{Slope} =$

$\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD{-9}}{\greenD 3} = -3$

## Slope as "rise over run"

A lot of people remember slope as "rise over run" because slope is the "rise" (change in $y$) divided by the "run" (change in $x$).
$\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}=\dfrac{\goldD{\text{Rise}}}{{\greenD{\text{Run}}}}$

## Let's practice!

Heads up! All of the examples we've seen so far have been points in the first quadrant, but that won't always be the case in the practice problems.
1) Use the graph below to find the slope of the line that goes through the points $(7,4)$ and $(3,2)$.
$\text{Slope} =$

$\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD2}{\greenD 4} = \dfrac12$
2) Use the graph below to find the slope of the line that goes through the points $(-6,9)$ and $(2,1)$.
$\text{Slope} =$

$\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD{-8}}{\greenD 8} = -1$
3) Use the graph below to find the slope of the line that goes through the points $(-8,-3)$ and $(4,-6)$.
$\text{Slope} =$

$\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD{-3}}{\greenD {12}} = \dfrac{-1}4$
4) Use the graph below to find the slope of the line that goes through the points $(4,5)$ and $(9,5)$.
$\text{Slope} =$

This is a tricky one because there is $\goldD 0$ change in $y$.
$\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD0}{\greenD 5} = 0$
In fact, the slope of a horizontal line is always $0$.
5) Use the graph below to choose the slope of the line that goes through the points $(3,2)$ and $(3,8)$.
$\text{Slope} =$

This is a tricky one because there is $\greenD 0$ change in $x$.
$\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD6}{\greenD 0} =$ undefined
Remember that we can't divide by $0$, which is why the slope is undefined.
In fact, the slope of a vertical line is always undefined. Interestingly, sometimes the slope of a vertical line is also said to be infinity.

## Challenge problems

See how well you understand slope by trying a couple of true/false problems.
6) A line with slope of $5$ is steeper than a line with a slope of $\dfrac12$
True! For increasing lines, lines with greater slopes have greater changes in $y$ relative to the same change in $x$.
7) A line with slope of $-5$ is steeper than a line with a slope of $-\dfrac12$
A line with a slope of $\blueD{-5}$ is steeper than a line with a slope of $\maroonD{-\dfrac12}$ because $|\blueD{-5}|$ is greater than $|\maroonD{-\dfrac12}|$.