How do you find the slope between two points?

 
Relevant MM Lessons:

Overview: Slope

This question has its own dedicated lesson ». Make sure to check it out if you need a more thorough read, or want to become an expert on how slope shows up on tests.In Algebra, when we discuss the concept of slope in the Coordinate Plane, we are always talking about linear slope - and linear slope can always be described distinctly by using two points that a slope line could go through, since it takes two points to determine a line.In English terms, we say the slope between two points in the Coordinate Plane is the ratio of its $y$ direction change to its $x$ direction change. For this reason, we end up with several common phrases and fractions that students and teachers use to describe slope, such as
  • "rise over run"
  • "change in $y$ over change in $x$
  • "Delta" $y$ over "delta" $x$
Many students are programmed to go directly to the slope formula, which not only reminds us what slope is, but also calculates the answer for us:$$m = \frac{y_2 - y_1}{x_2 - x_1}$$Note that we often use the letter $m$ to represent slope (I have no idea why, tbh).To avoid the common mistakes I see many students make, be careful of two things:1. Line up your points correctly!If you are measuring the slope between the points $(4,3)$ and $(6, -5)$, you must call one point "Point 1" and the other "Point 2", and then proceed to put Point 1's $x$ component in the formula for $x_1$, not for $x_2$. Same with the $y$ components. Mixing one of the points up and reversing the order of subtraction will get you the wrong answer!2. Use extra parenthesis to avoid easy-to-make negative mistakes!When using the formula with negative coordinates, as we often have to do, students tend to make negative sign errors with the actual calculation. When I see my students do this, it's almost always because they aren't using enough parenthesis to understand what exactly is going on. My number one recommendation on slope calcualtions is to set up the formula with blank parenthesis first, and then plug in the values after.Example:Find the slope between $(-1,-6)$ and $(-6, -11)$.First, set it up with blanks:$$m = \frac{(\,\,) - (\,\,)}{(\,\,) - (\,\,)}$$Now, put things where they belong. You'll be much less likely to forget that you're working with subtracting negative numbers now!$$\begin{align} m & = \frac{(-11)-(-6)}{(-6)-(-1)} \\ & = \frac{-5}{-5} = 1 \end{align}$$