Learn on PengiReveal Math, AcceleratedUnit 9: Linear Relationships

Lesson 9-1: Describe the Slope of a Line

In this Grade 7 lesson from Reveal Math, Accelerated (Unit 9), students learn to describe and calculate the slope of a line as the ratio of rise to run using the formula slope = (y₂ − y₁) / (x₂ − x₁). Through real-world contexts like weekly savings and cycling speed, students connect slope to unit rate and the constant of proportionality in proportional relationships. The lesson builds students' ability to interpret the steepness of a line on a graph and apply slope to solve problems involving constant rates of change.

Section 1

Constant of Proportionality

Property

The constant of proportionality, often called the unit rate ( $r$ ), is the constant ratio in a proportional relationship.
If quantities $x$ and $y$ are proportional, their relationship can be described by the equation $y = rx$ .
The constant $r$ can be found by calculating the ratio $r = \frac{y}{x}$ for any corresponding pair $(x, y)$ where $x \neq 0$ . On a graph, the unit rate is represented by the point $(1, r)$ .

Examples

A car travels 180 miles in 3 hours. The constant of proportionality is $\frac{180}{3} = 60$ miles per hour. The equation is $d = 60h$ .
A graph of a proportional relationship between cost and pounds of bananas passes through the point $(1, 0.55)$ . The constant of proportionality is 0.55 dollars per pound.
A table shows that 4 movie tickets cost 52 dollars. The unit rate (constant of proportionality) is $\frac{52}{4} = 13$ dollars per ticket.

Explanation

The constant of proportionality is the 'secret multiplier' that connects the two quantities. It tells you how much of the second quantity you get for exactly one unit of the first quantity, like price per item or miles per hour.

Section 2

Slope of a Line

Property

The slope of a line is a rate of change that measures the steepness of the line.

It is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate. In symbols:

m = \frac{\Delta y}{\Delta x} = \frac{\text{change in } y\text{-coordinate}}{\text{change in } x\text{-coordinate}}

where

\Delta x

is positive if we move right and negative if we move left, and

\Delta y

is positive if we move up and negative if we move down.

Examples

A line passes through the points $(3, 5)$ and $(7, 13)$ . Its slope is $m = \frac{13-5}{7-3} = \frac{8}{4} = 2$ .
For a line containing points $(-2, 9)$ and $(4, 6)$ , the slope is $m = \frac{6-9}{4-(-2)} = \frac{-3}{6} = -\frac{1}{2}$ .
The slope of a line passing through $(100, 50)$ and $(120, 40)$ is $m = \frac{40-50}{120-100} = \frac{-10}{20} = -0.5$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Constant of Proportionality

Property

Examples

A car travels 180 miles in 3 hours. The constant of proportionality is $\frac{180}{3} = 60$ miles per hour. The equation is $d = 60h$ .
A graph of a proportional relationship between cost and pounds of bananas passes through the point $(1, 0.55)$ . The constant of proportionality is 0.55 dollars per pound.
A table shows that 4 movie tickets cost 52 dollars. The unit rate (constant of proportionality) is $\frac{52}{4} = 13$ dollars per ticket.

Explanation

Section 2

Slope of a Line

Property

The slope of a line is a rate of change that measures the steepness of the line.

It is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate. In symbols:

m = \frac{\Delta y}{\Delta x} = \frac{\text{change in } y\text{-coordinate}}{\text{change in } x\text{-coordinate}}

where

\Delta x

is positive if we move right and negative if we move left, and

\Delta y

is positive if we move up and negative if we move down.

Examples

A line passes through the points $(3, 5)$ and $(7, 13)$ . Its slope is $m = \frac{13-5}{7-3} = \frac{8}{4} = 2$ .
For a line containing points $(-2, 9)$ and $(4, 6)$ , the slope is $m = \frac{6-9}{4-(-2)} = \frac{-3}{6} = -\frac{1}{2}$ .
The slope of a line passing through $(100, 50)$ and $(120, 40)$ is $m = \frac{40-50}{120-100} = \frac{-10}{20} = -0.5$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Constant of Proportionality

Property

Examples

A car travels 180 miles in 3 hours. The constant of proportionality is $\frac{180}{3} = 60$ miles per hour. The equation is $d = 60h$ .
A graph of a proportional relationship between cost and pounds of bananas passes through the point $(1, 0.55)$ . The constant of proportionality is 0.55 dollars per pound.
A table shows that 4 movie tickets cost 52 dollars. The unit rate (constant of proportionality) is $\frac{52}{4} = 13$ dollars per ticket.

Explanation

Section 2

Slope of a Line

Property

The slope of a line is a rate of change that measures the steepness of the line.

It is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate. In symbols:

m = \frac{\Delta y}{\Delta x} = \frac{\text{change in } y\text{-coordinate}}{\text{change in } x\text{-coordinate}}

where

\Delta x

is positive if we move right and negative if we move left, and

\Delta y

is positive if we move up and negative if we move down.

Examples

A line passes through the points $(3, 5)$ and $(7, 13)$ . Its slope is $m = \frac{13-5}{7-3} = \frac{8}{4} = 2$ .
For a line containing points $(-2, 9)$ and $(4, 6)$ , the slope is $m = \frac{6-9}{4-(-2)} = \frac{-3}{6} = -\frac{1}{2}$ .
The slope of a line passing through $(100, 50)$ and $(120, 40)$ is $m = \frac{40-50}{120-100} = \frac{-10}{20} = -0.5$ .