The slope of a line

Warm-up

Three quick questions from earlier lessons.

Discover

Two slope triangles sit on the same line: one is 2 across and 3 up, the other 4 across. How far up is the second one?

6
5

Change the run. Divide the rise by the run each time.

It is always 1.5
It grows with the triangle

Key idea

Every slope triangle on a line gives the same ratio.

The slope of a line is

\dfrac{\text{vertical change}}{\text{horizontal change}} = \dfrac{y_2 - y_1}{x_2 - x_1}

. All slope triangles on a line are similar, so the slope does not depend on which two points you use. A line that goes down from left to right has a negative slope.

Worked example

Through $(1, 2)$ and $(5, 8)$ .

8 - 2 = 6

Vertical change.

5 - 1 = 4

Horizontal change.

Slope

\tfrac{3}{2}

6 ÷ 4.

Your turn, guided

Your turn: the slope through $(0, 7)$ and $(2, 1)$ .

-3

(1 − 7) ÷ (2 − 0).

Practice ★

Find the slope of the line through $(6, 2)$ and $(9, 8)$ .
Find the slope of the line through $(4, 5)$ and $(9, 7)$ .

A slope triangle on a line has a horizontal side of 6 and a vertical side of 2. Another slope triangle on the same line has a horizontal side of 24. How long is its vertical side?

Watch out!

Practice ★★

Find the slope of the line through $(6, 5)$ and $(8, 0)$ .
Find the slope of the line through $(0, 11)$ and $(7, 6)$ .

Boss problem. At the same moment, a 6 ft post casts a 4 ft shadow and a tree casts a 8 ft shadow. The post with its shadow and the tree with its shadow form similar right triangles.

Exit ticket

Three questions to check you have got it.

Find the slope of the line through $(6, 2)$ and $(9, 8)$ .
Find the slope of the line through $(4, 5)$ and $(9, 7)$ .
Find the slope of the line through $(1, 3)$ and $(6, 5)$ .

Summary card

Slope

Rise over run, the same for every slope triangle on the line.

Watch out: down from left to right means negative.

About this lesson