Slope
(KS3, Year 8)

The slope of a line is its steepness. The larger the slope, the steeper the line.

Understanding the Slope of a Line

The slope of a line is how far the line goes up (or down) divided by how far a line goes across (left to right). Look at the line below:

If we draw a triangle under the line and measure how far it goes up and across...

• The line goes up by 3...
• ...and across by 3.

The slope of the line is 1.

Real Examples of the Slope of a Line

• Lines that slope from the bottom-left up to the top-right have a positive slope.
  
  The line above has a slope of 2 because it goes up by 2 squares and across by 1.
• Lines that slope from the top-left up to the bottom-right have a negative slope.
  
  The line above has a slope of −2 because it goes down by 2 squares and across by 1.
• Lines that do not slope up nor down have a slope of 0.
  
  
• Lines that do not slope across have an undefined slope.
  
  

Formulas to Find the Slope of a Line

We can find the slope of a line if we know how far it goes up and how far it goes across. In Cartesian coordinates, the y-axis measures how far up a line is and the x-axis measures how far across a line is. This lets us define a formula to find the slope of a line:


finding the slope of a line We can find the slope if we know two points (in Cartesian coordinates) on the line.


finding the slope between two points

Rise Over the Run

An easy way to remember the meaning of slope is "rise over run". Rise is how far up (or down) a line goes. Run is how far across a line goes.

A Slope of 1

A slope of 1 gives a 45° line. It goes up and across by the same amount.

Fractional Slopes

A slope can be a fraction such as ½ or ¾.

A fractional slope is still positive (slopes from bottom-left to top-right) but it is less steep than a slope of 1.