Grade 8

Intermediate

8 min

Lesson 3 of 12

Calculating Slope

Not Started

Master the concept of slope as the 'steepness' of a line and learn how to calculate it.

Why is a black diamond ski run more terrifying than a green circle? It all comes down to one mathematical secret: the slope.

Learning Objectives

1.Calculate the slope of a line passing through two given points using the slope formula
2.Interpret slope as the ratio of vertical change (rise) to horizontal change (run)
3.Distinguish between positive, negative, zero, and undefined slopes

The Concept of Rise over Run

In mathematics, slope is a number that describes both the direction and the steepness of a line. Think of it like walking up a staircase. To get to the next step, you must move up (vertical change) and move forward (horizontal change). We call the vertical change the rise and the horizontal change the run. The slope is simply the ratio of these two movements:

\text{slope} = \frac{\text{rise}}{\text{run}}

. If a line is very steep, it has a large slope. If it is nearly flat, it has a small slope. By looking at a graph, you can pick any two points and count the squares to find these values.

Imagine a line on a graph that passes through $(0,0)$ and $(2,6)$ . 1. Start at $(0,0)$ . 2. To get to the next point, move up 6 units. This is your rise ( $6$ ). 3. Move right 2 units. This is your run ( $2$ ). 4. Divide the rise by the run: $\frac{6}{2} = 3$ . The slope of this line is $3$ .

Quick Check

If a line rises 10 units for every 2 units it runs, what is its slope?

Divide the rise (10) by the run (2).

Answer

The Slope Formula

What if you don't have a graph to count squares? We use the slope formula. If you have two points,

(x_1, y_1)

and

(x_2, y_2)

, the slope (represented by the letter m) is calculated by finding the difference in the

y

-coordinates and dividing it by the difference in the

x

-coordinates:

m = \frac{y_2 - y_1}{x_2 - x_1}

. This formula is just a formal way of saying 'change in

y

divided by change in

x

'. It is vital to keep your points in the same order; if you start with

y_2

on top, you must start with

x_2

on the bottom!

Find the slope of the line passing through $A(1, 2)$ and $B(4, 8)$ . 1. Identify your coordinates: $x_1=1, y_1=2$ and $x_2=4, y_2=8$ . 2. Plug them into the formula: $m = \frac{8 - 2}{4 - 1}$ . 3. Simplify the top and bottom: $m = \frac{6}{3}$ . 4. Final answer: $m = 2$ .

Quick Check

Using the formula, what is the slope of a line passing through (2, 5) and (4, 5)?

Subtract the y-values first: 5 - 5 = ?

Answer

The Four Faces of Slope

Slopes tell a visual story. A positive slope ( $m > 0$ ) goes up from left to right, like a plane taking off. A negative slope ( $m < 0$ ) goes down from left to right, like a plane landing. But there are two special cases: 1. Zero Slope: A horizontal line. There is no 'rise,' so the top of our fraction is $0$ . 2. Undefined Slope: A vertical line. There is no 'run,' so the bottom of our fraction is $0$ . Since we cannot divide by zero in math, we say the slope is undefined.

Calculate the slope for points $(3, 10)$ and $(3, 5)$ . 1. Apply the formula: $m = \frac{5 - 10}{3 - 3}$ . 2. Simplify: $m = \frac{-5}{0}$ . 3. Because the denominator is zero, the slope is undefined. This represents a perfectly vertical line.

Key Takeaways

1Slope measures the steepness of a line using the ratio $\frac{\text{rise}}{\text{run}}$ .
2The standard formula for slope between two points is $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
3Positive slopes go up, negative slopes go down, horizontal lines have a slope of 0, and vertical lines have an undefined slope.

Quick Check

Multiple Choice

What is the slope of a line passing through $(1, 3)$ and $(2, 7)$ ?

Multiple Choice

Which type of line has a slope of 0?

True/False

A line that moves down from left to right has a positive slope.

What's Next?

Review Tomorrow

In 24 hours, try to write down the slope formula from memory and sketch the four types of slopes (positive, negative, zero, undefined).

Practice Activity

Find a set of stairs in your house or school. Measure the height of one step (rise) and the depth of one step (run). Calculate the slope of the staircase!

Browse

Browse

Calculating Slope

Learning Objectives

The Concept of Rise over Run

The Slope Formula

The Four Faces of Slope

Key Takeaways

Quick Check

What's Next?

Calculating Slope

Learning Objectives

The Concept of Rise over Run

The Slope Formula

The Four Faces of Slope

Key Takeaways

Quick Check

What's Next?