Slope.

Step 1 of 4

Rise and run

In the last Unit each side of an equation was a line, such as $y = 2x + 1$⁠. Now look at the line itself. A line is written in the form

$y = mx + n$⁠

The number $m$⁠ is the slope: how much $y$⁠ rises for every single step to the right in $x$⁠. The board shows $y = 2x + 1$⁠. Start at the point where $x = 1$⁠ and move $2$⁠ steps to the right. How much has the line risen on the way?

Step 2 of 4

Anywhere on the line

A line rises at the same rate all along its length. So you can measure the slope anywhere, with a step of any length: the slope is the rise divided by the run. Writing the rise as $\Delta y$⁠ (how much $y$⁠ changed) and the run as $\Delta x$⁠ (how much $x$⁠ changed):

$m = \frac{\Delta y}{\Delta x}$$m =$$\frac{\Delta y}{\Delta x}$⁠

The green triangle on the board shows the run across and the rise up. Run is now $1$⁠.

Step 3 of 4

Negative slope

A line that goes down as you move right “rises” by a negative amount, so its slope is negative. The board shows a line that falls $3$⁠ for every $2$⁠ steps to the right. Once you have answered, choose Rise over run under Look inside.

Step 4 of 4

Which is steeper?

The sign of the slope says which way the line tilts: positive goes up, negative goes down. Its size, without the sign, says how steep it is. A slope of $0$⁠ is a horizontal line. Move Slope m from $-4$⁠ to $4$⁠ and see.