# Lesson Plan: Graphing and Interpreting Proportional Relationships

## Lesson Objectives

This lesson can be completed in one 50-minute class period.

- Graph proportional relationships
- Interpret the meaning of points on graphs of proportional relationships
- Understand the concept of slope in the context of proportional relationships
- Compare different representations of proportional relationships (graphs, tables, equations)

## Florida BEST Standards

- MA.8.AR.3.1: Determine if a linear relationship is also a proportional relationship.
- MA.8.AR.3.2: Given a table, graph or written description of a linear relationship, determine the slope.
- MA.8.AR.3.3: Given a real-world situation, determine if the relationship between two quantities is linear. Write an equation in slope-intercept form.
- MA.8.AR.1.3: Rewrite algebraic expressions with rational coefficients in different but equivalent forms.
- MA.8.F.1.1: Given a set of ordered pairs, a table, a graph or mapping diagram, determine whether the relationship is a function. Identify the domain and range of the relation.

## Prerequisite Skills

- Plotting points on a coordinate plane
- Understanding of unit rate

## Key Vocabulary

- Slope
- Origin
- Linear relationship
- Coordinate plane

## Warm-up Activity (10 minutes)

Present students with the following data table showing hours worked and total pay for a part-time job:

Hours Worked | Earnings ($) |
---|---|

0 | 0 |

2 | 30 |

4 | 60 |

6 | 90 |

8 | 120 |

Ask students to:

- Plot these points on a coordinate plane
- Determine if they form a straight line through the origin
- Calculate the hourly wage (slope of the line)

Use this Desmos activity:

__https://www.desmos.com/calculator/utzva5dltc__

## Teach (25 minutes)

### Definitions

**Proportional Relationship**: A relationship between two quantities where one quantity is a constant multiple of the other.**Slope**: The steepness of a line, calculated as the change in y divided by the change in x.**Unit Rate**: The rate per one unit of the independent variable.**Origin**: The point (0,0) on a coordinate plane.**Linear Relationship**: A relationship that forms a straight line when graphed.**Coordinate Plane**: A two-dimensional plane using x and y axes to determine the position of points.

Use this slide show to review these and other definitions:

### Instruction

Demonstrate how to graph proportional relationships, explain how the unit rate relates to the slope of the graph, and show how to compare different proportional relationships using graphs.

### Example 1: Graphing a Cell Phone Plan

A cell phone plan charges $0.10 per minute of talk time.

- Create a table of values for minutes used and cost.
- Graph the relationship.
- Identify the unit rate and explain its meaning in context.

Solution:

Minutes (x) | Cost (y) |
---|---|

0 | $0 |

10 | $1 |

20 | $2 |

30 | $3 |

40 | $4 |

50 | $5 |

Graph: A straight line passing through these points and the origin.

Unit rate: $0.10 per minute, represented by the slope of the line.

Slope calculation: Slope = (y2 - y1) / (x2 - x1) = (5 - 0) / (50 - 0) = 5/50 = 0.10

Use this Desmos activity:

__https://www.desmos.com/calculator/kamtpaelpc__

### Example 2: Comparing Running Speeds

Runner A covers 6 miles in 1 hour.

Runner B covers 10 miles in 2 hours.

- Create a table of values for both runners.
- Graph both relationships on the same coordinate plane.
- Compare the speeds of the two runners.

Solution:

Time (hours) | Runner A Distance (miles) | Runner B Distance (miles) |
---|---|---|

0 | 0 | 0 |

1 | 6 | 5 |

2 | 12 | 10 |

3 | 18 | 15 |

Graph: Plot both sets of points on the same coordinate plane.

Runner A's line is y = 6x, Runner B's line is y = 5x

Runner A is faster as their line has a steeper slope (6 vs 5).

Use this Desmos activity:

__https://www.desmos.com/calculator/owuxtaevo1__

### Example 3: Sales Growth in a Startup Business

A tech startup's monthly revenue growth is shown in the following data:

Months since launch | Monthly Revenue ($1000s) |
---|---|

0 | 0 |

1 | 20 |

2 | 40 |

3 | 60 |

4 | 80 |

5 | 100 |

- Graph this relationship on a coordinate plane.
- What does the point (3, 60) represent?
- What is the meaning of the slope in this context?

Solution:

Graph: A straight line passing through these points and the origin.

- The point (3, 60) represents that after 3 months since launch, the company's monthly revenue was $60,000.
- Slope calculation: Slope = (100 - 0) / (5 - 0) = 20
- The slope of 20 means the company's revenue is growing by $20,000 per month.
- This linear relationship indicates a consistent growth rate, which is a positive sign for the startup.

Use this Desmos activity:

__https://www.desmos.com/calculator/pmft2bgx4q__

### Example 4: Education and Income Relationship

Let's examine the relationship between years of education and median weekly earnings using real data from the U.S. Bureau of Labor Statistics (2020):

Years of Education | Median Weekly Earnings ($) |
---|---|

8 | 626 |

12 | 789 |

14 | 938 |

16 | 1305 |

- Graph this relationship on a coordinate plane.
- What does the point (16, 1305) represent?
- Is this a proportional relationship? Why or why not?

Solution:

Graph: Plot the points on a coordinate plane. The resulting line will not pass through the origin. (Note: The graph is a linear regression line.)

- The point (16, 1305) represents that individuals with 16 years of education (typically a bachelor's degree) had median weekly earnings of $1,305.
- This is not a proportional relationship because:
- The graph does not pass through the origin (0, 0).
- The rate of change (slope) is not constant between all points.
- The range of this graph would be for x ≥ 0, since it makes no sense that 0 years of education resulting in any earnings.

- However, there is a clear positive correlation between years of education and median weekly earnings.
- We can calculate the average rate of change: Slope = (1305 - 626) / (16 - 8) = 679 / 8 ≈ 84.88 This means that, on average, each additional year of education is associated with an increase of about $84.88 in median weekly earnings.

Use this Desmos activity, which includes a regression line:

__https://www.desmos.com/calculator/q2gs51jf5f__

## Review (10 minutes)

Group activity: Students analyze and graph real-world data from arts, technology, and sports. Use this slide show to support the following three examples, which include data sets and lines of best fit. The slope as a unit rate can potentially be used to make predictions:

__https://www.media4math.com/library/slideshow/applications-proportional-reasoning-linear-graphs__

### Example 1. Arts: Broadway Show Ticket Prices (2010-2019)

This data set shows the changing price of Broadway theater ticket prices.

Data from The Broadway League:

Year | Average Ticket Price ($) |
---|---|

2010 | 86.21 |

2013 | 98.42 |

2016 | 109.21 |

2019 | 123.87 |

Students graph this data and interpret the slope as the average yearly increase in ticket prices. Here is a Desmos activity you can use as a companion to this review activity. Note that it includes a line of best fit, which you can briefly explain to students:

__https://www.desmos.com/calculator/mc1y6t4jwe__

### Example 2. Technology: Global Smartphone Sales (2016-2019)

This data set on Smartphone sales can be used to generate a line of best fit.

Data from Statista:

Year | Smartphones Sold (millions) |
---|---|

2016 | 1470 |

2017 | 1506 |

2018 | 1556 |

2019 | 1486 |

Students graph this data and discuss why this relationship might not be proportional. Use this Desmos activity, which includes a regression line. Discuss the advantages and disadvantages of this linear model:

__https://www.desmos.com/calculator/t6nctgqhcp__

### Example 3. Sports: NBA Player Height vs. Points per Game (2020-2021 season)

This data set looks at the relationship between height and average points per game for NBA players.

Data from NBA statistics:

Height (inches) | Avg Points per Game |
---|---|

72 | 14.2 |

75 | 15.8 |

78 | 17.3 |

81 | 18.9 |

84 | 20.1 |

Students graph this data and interpret the slope as the average increase in points scored per inch of height. Here is a Desmos activity to use:

__https://www.desmos.com/calculator/1hsko7f67q__

## Assess (5 minutes)

Administer this 10-question quiz.

## Quiz

- Graph the proportional relationship y = 3x. What is the slope of this line?

- If a line passes through the points (0,0) and (4,12), what is the unit rate?

- Compare the steepness of y = 2x and y = 0.5x. Which has a greater rate of change?

- What does the point (1, 5) represent on a graph of a proportional relationship?

- Graph y = 4x and y = 2x on the same coordinate plane. How do they compare?

- If a car travels 240 miles in 4 hours at a constant speed, what is the slope?

- Does the graph of y = 3x + 2 represent a proportional relationship? Why or why not?

- On a graph of distance vs. time for a constant speed, what does the slope represent?

- If two people are running at different constant speeds, how would their distance-time graphs compare?

- What is the y-intercept of all proportional relationships when graphed?

## Answer Key

- 3
- 3
- y = 2x has a greater rate of change
- For every 1 unit of x, y increases by 5 units
- y = 4x has a steeper slope
- Slope is 60 (miles per hour)
- No, because it doesn't pass through (0,0)
- Speed or velocity
- Different slopes, steeper slope indicates faster speed
- (0,0)

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