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Intro to slope

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:
A coordinate plane. The x-axis runs from 0 to 10 and is scaled by 1. The y-axis runs from 0 to 10 and is scaled by 1. A graph of a line intersects the points (3, 2) and (5, 8). Closed points are plotted at (3, 2) and (5, 8).
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):
A coordinate plane. The x-axis runs from 0 to 10 and is scaled by 1. The y-axis runs from 0 to 10 and is scaled by 1. A graph of a line intersects the points (3, 2) and (5, 8). Closed points are plotted at (3, 2) and (5, 8). There is a horizontal segment from (3, 2) to (5, 2) that is labeled 2. There is a vertical segment from (5, 2) to (5, 8) that is labeled 6.
Slope=Change in yChange in x=62=3
Use the graph below to find the slope of the line that goes through the points (1,2) and (6,6).
A coordinate plane. The x-axis runs from 0 to 10 and is scaled by 1. The y-axis runs from 0 to 10 and is scaled by 1. A graph of a line intersects the points (1, 2) and (6, 6). Closed points are plotted at (1, 2) and (6, 6).
Slope=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

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).
A coordinate plane. The x-axis runs from 0 to 10 and is scaled by 1. The y-axis runs from 0 to 10 and is scaled by 1. A graph of a line intersects the points (2, 7) and (5, 1). Closed points are plotted at (2, 7) and (5, 1). There is a horizontal segment from (2, 7) to (5, 7) that is labeled 3. There is a vertical segment from (5, 7) to (5, 1) that is labeled negative 6.
Slope=Change in yChange in x=63=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).
A coordinate plane. The x-axis runs from 0 to 10 and is scaled by 1. The y-axis runs from 0 to 10 and is scaled by 1. A graph of a line intersects the points (1, 9) and (4, 0). Closed points are plotted at (1, 9) and (4, 0).
Slope=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

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).
A coordinate plane. The x-axis runs horizontally and has no tick marks. The y-axis runs vertically and has no tick marks. A graph of a line increases at a constant rate as it moves left to right and intersects 2 closed points. There is a horizontal segment from from the first closed point that is labeled Run. There is a vertical segment from the second closed point that is labeled negative Rise.
Slope=Change in yChange in x=RiseRun

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).
A coordinate plane. The x-axis runs from 0 to 10 and is scaled by 1. The y-axis runs from 0 to 10 and is scaled by 1. A graph of a line intersects the points (3, 2) and (7, 4). Closed points are plotted at (3, 2) and (7, 4).
Slope=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

2) Use the graph below to find the slope of the line that goes through the points (6,9) and (2,1).
A coordinate plane. The x-axis runs from negative 8 to 10 and is scaled by 1. The y-axis runs from 0 to 10 and is scaled by 1. A graph of a line intersects the points (negative 6, 9) and (2, 1). Closed points are plotted at (negative 6, 9) and (2, 1).
Slope=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

3) Use the graph below to find the slope of the line that goes through the points (8,3) and (4,6).
A coordinate plane. The x-axis runs from negative 10 to 10 and is scaled by 1. The y-axis runs from negative 10 to 2 and is scaled by 1. A graph of a line intersects the points (negative 8, negative 3) and (4, negative 6). Closed points are plotted at (negative 8, negative 3) and (4, negative 6).
Slope=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

4) Use the graph below to find the slope of the line that goes through the points (4,5) and (9,5).
A coordinate plane. The x-axis runs from 0 to 10 and is scaled by 1. The y-axis runs from 0 to 10 and is scaled by 1. A graph of a line intersects the points (4, 5) and (9, 5). Closed points are plotted at (4, 5) and (9, 5).
Slope=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

5) Use the graph below to choose the slope of the line that goes through the points (3,2) and (3,8).
A coordinate plane. The x-axis runs from 0 to 10 and is scaled by 1. The y-axis runs from 0 to 10 and is scaled by 1. A graph of a line intersects the points (3, 2) and (3, 8). Closed points are plotted at (3, 2) and (3, 8).
Slope=
Choose 1 answer:

Challenge problems

See how well you understand slope by trying a couple of true/false problems.
6) A line with a slope of 5 is steeper than a line with a slope of 12
Choose 1 answer:

7) A line with a slope of 5 is steeper than a line with a slope of 12
Choose 1 answer:

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