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Ratio Rally: Graphing Proportions

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David Kindt

Tier 1
For Schools

Lesson Plan

Ratio Rally: Graphing Proportions

Students will be able to represent proportional relationships in tables and equations, analyze their characteristics, graph them on a coordinate plane, and solve real-world problems involving ratios and proportions.

Understanding ratios and proportional relationships is crucial for making sense of many real-world situations, from calculating recipes to understanding scale models and financial decisions. This lesson equips students with practical problem-solving skills.

Audience

7th Grade

Time

30 minutes

Approach

Interactive discussion, guided practice, and independent application.

Materials

  • Ratio Rally Slide Deck, - Warm-Up: Ratio Brainstorm, - Proportional Problem-Solving Worksheet, - Proportional Problem-Solving Answer Key, - Whiteboard or Projector, and - Markers or Pens

Prep

Prepare Materials

10 minutes

  • Review the Ratio Rally Slide Deck and familiarize yourself with the content.
    - Print copies of the Warm-Up: Ratio Brainstorm (one per student).
    - Print copies of the Proportional Problem-Solving Worksheet (one per student).
    - Review the Proportional Problem-Solving Answer Key.
    - Ensure whiteboard/projector and markers are ready.

Step 1

Warm-Up: Ratio Brainstorm

5 minutes

  • Distribute the Warm-Up: Ratio Brainstorm.
    - Ask students to individually brainstorm examples of ratios they encounter in daily life.
    - Briefly discuss student responses as a class to activate prior knowledge.

Step 2

Introduction to Proportional Relationships

8 minutes

  • Present the Ratio Rally Slide Deck (Slides 1-4).
    - Introduce the concept of proportional relationships, defining key terms like ratio, rate, and constant of proportionality.
    - Explain how to identify proportional relationships from tables and equations.
    - Guide students through an example of creating a table and equation from a real-world scenario.

Step 3

Graphing Proportional Relationships

7 minutes

  • Continue with the Ratio Rally Slide Deck (Slides 5-7).
    - Demonstrate how to graph proportional relationships, emphasizing that they form a straight line through the origin.
    - Work through an example together, plotting points and drawing the line.
    - Discuss the significance of the origin (0,0) in proportional relationships.

Step 4

Real-World Problem Solving & Practice

7 minutes

  • Distribute the Proportional Problem-Solving Worksheet.
    - Present the Ratio Rally Slide Deck (Slide 8) to introduce the practice problems.
    - Have students work independently or in pairs on the worksheet.
    - Circulate to provide support and answer questions.

Step 5

Wrap-Up and Review

3 minutes

  • Briefly review the key concepts covered: identifying, representing, graphing, and solving proportional relationships.
    - Quickly go over one or two answers from the Proportional Problem-Solving Worksheet using the Proportional Problem-Solving Answer Key as time permits or collect for grading.
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Slide Deck

Welcome to Ratio Rally!

Today, we're exploring the exciting world of Ratios and Proportional Relationships!

Get ready to:

  • Understand what makes relationships proportional
  • Learn how to represent them
  • Graph them like a pro
  • Solve real-world problems!

Welcome students and introduce the exciting topic of ratios and proportional relationships! Ask students if they can think of where they've seen ratios before (e.g., ingredients in a recipe, car speed, map scales).

What's a Proportional Relationship?

A Ratio compares two quantities. (e.g., 3 apples for every 2 bananas)

A Proportional Relationship exists when two quantities have a constant ratio between them.

This constant ratio is called the Constant of Proportionality (often represented by 'k').

If y is proportional to x, then y = kx

Define what a ratio is and provide a few simple examples. Then, introduce the idea of a proportional relationship as two ratios that are equivalent. Emphasize that in a proportional relationship, the ratio between two quantities is always constant.

Finding Proportions in Tables

Look at the ratio of y to x (y/x) for each pair of values.
If the ratio is constant, it's proportional!

Example: Lemonade Mix

Cups of Sugar (x)Liters of Water (y)y/x
266/2 = 3
399/3 = 3
41212/4 = 3

Here, k = 3. So, y = 3x.

Explain how to identify proportional relationships in tables. Key point: the ratio y/x must be the same for all pairs. Go through the example, showing the calculation of k for each pair.

Proportional Equations

A proportional relationship can always be written in the form:

y = kx

Where 'k' is the constant of proportionality.

Examples:

  • y = 5x (k=5)
  • d = 60t (distance = 60 mph * time; k=60)

Not Proportional Examples:

  • y = 2x + 1 (Why?)
  • y = x² (Why?)

Show how equations like y = kx directly represent proportional relationships. Give a non-example (y = 2x + 1) and explain why it's not proportional (doesn't pass through the origin when x=0, and the y/x ratio isn't constant).

Graphing Proportional Relationships

When you graph a proportional relationship, it always has two special features:

  1. It is a straight line.
  2. It passes through the origin (0,0).

Why (0,0)? If you have '0' of something, you also have '0' of the related quantity!

Transition to graphing. Emphasize the two key characteristics: straight line and passing through the origin. Briefly explain what the origin represents in this context (e.g., 0 items cost $0, 0 hours driven is 0 miles).

Let's Graph an Example!

Scenario: A baker uses 2 cups of flour for every 1 batch of cookies.

Table:

Batches (x)Cups of Flour (y)(x, y)
00(0,0)
12(1,2)
24(2,4)
36(3,6)

Graphing Steps:

  1. Plot the points from the table.
  2. Draw a straight line connecting the points, extending it through the origin.

Go through a step-by-step example of plotting points from a table and drawing the line. Highlight how the graph confirms the relationship is proportional.

Analyzing the Graph

From a graph, you can:

  • Verify Proportionality: Is it a straight line? Does it go through (0,0)?

  • Find the Constant of Proportionality (k): This is the slope of the line!
    k = y/x for any point (x,y) on the line (except the origin).

    Think: For every 1 unit increase in x, how many units does y increase?

Summarize how to analyze a graph to determine proportionality and to find the constant of proportionality (slope).

Your Turn: Problem Solving!

Now it's time to put your proportional skills to the test!

Work through the problems on your worksheet:

  1. Identify proportional relationships.
  2. Represent them with tables and equations.
  3. Graph them.
  4. Solve real-world scenarios.

Remember to show your work and use what you've learned!

Explain that students will now apply what they've learned to solve problems on a worksheet. Encourage them to use tables, equations, and graphs as tools.

lenny

Warm Up

Warm-Up: Ratio Brainstorm

Name: ____________________________

Date: ____________________________

Think about your everyday life. Where do you see ratios? A ratio is a comparison of two quantities.

Brainstorm at least three (3) different examples of ratios you encounter or can think of. For each example, briefly describe the quantities being compared.

Example 1:







Example 2:







Example 3:







Bonus Question: Why do you think understanding ratios might be important in these situations?






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lenny

Worksheet

Proportional Problem-Solving Worksheet

Name: ____________________________

Date: ____________________________

Part 1: Identifying Proportional Relationships

Determine if each relationship is proportional. Explain your reasoning.

  1. Scenario: A cyclist rides at a constant speed of 15 miles per hour.

    Time (hours)Distance (miles)
    115
    230
    345

    Is it proportional?





    Explain:




  2. Scenario: A membership to a gym costs $20 plus $5 per visit.

    VisitsTotal Cost
    1$25
    2$30
    3$35

    Is it proportional?





    Explain:




Part 2: Representing and Graphing Proportional Relationships

For the following scenario, complete the table, write an equation, and graph the relationship.

  1. Scenario: A recipe calls for 3 cups of water for every 2 cups of rice.

    Table:

    Cups of Rice (x)Cups of Water (y)
    0
    2
    4
    6

    Equation (y = kx):




    Graph: (Sketch a graph on the coordinate plane below. Label your axes!)

























Part 3: Solving Real-World Problems

Solve the following problems using what you know about proportional relationships.

  1. A car travels 210 miles on 7 gallons of gas. At this rate, how many miles can the car travel on 10 gallons of gas?











  2. Sarah earns $12 per hour babysitting. If she babysits for 4.5 hours, how much money will she earn?











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Answer Key

Proportional Problem-Solving Answer Key

Part 1: Identifying Proportional Relationships

  1. Scenario: A cyclist rides at a constant speed of 15 miles per hour.

    Time (hours)Distance (miles)y/x (Distance/Time)
    11515/1 = 15
    23030/2 = 15
    34545/3 = 15

    Is it proportional? Yes
    Explain: The ratio of distance to time (y/x) is constant (15) for all pairs of values. This means the constant of proportionality (k) is 15, and the relationship can be written as y = 15x.

  2. Scenario: A membership to a gym costs $20 plus $5 per visit.

    Visits (x)Total Cost (y)y/x (Cost/Visits)
    1$2525/1 = 25
    2$3030/2 = 15
    3$3535/3 ≈ 11.67

    Is it proportional? No
    Explain: The ratio of total cost to visits (y/x) is not constant. Also, if there are 0 visits, the cost is $20 (y-intercept), which means the relationship does not pass through the origin (0,0), a key characteristic of proportional relationships. The equation would be y = 5x + 20.

Part 2: Representing and Graphing Proportional Relationships

  1. Scenario: A recipe calls for 3 cups of water for every 2 cups of rice.

    Table:

    Cups of Rice (x)Cups of Water (y)
    00
    23
    46
    69
    Explanation: For every 2 cups of rice, there are 3 cups of water. The constant of proportionality (k) is 3/2 or 1.5.

    Equation (y = kx): y = (3/2)x or y = 1.5x

    Graph: (See example graph below)
    The graph should show a straight line passing through the origin (0,0), (2,3), (4,6), and (6,9). The x-axis should be labeled "Cups of Rice" and the y-axis "Cups of Water".

    graph TD
    A[Origin (0,0)] --> B(Point 1: (2,3))
    B --> C(Point 2: (4,6))
    C --> D(Point 3: (6,9))

    (Self-correction: Cannot use mermaid. Will describe the graph clearly in text format)

    Graph Description:

    • A coordinate plane with the x-axis labeled "Cups of Rice" and the y-axis labeled "Cups of Water."
    • Points plotted at (0,0), (2,3), (4,6), and (6,9).
    • A straight line drawn connecting these points, extending from the origin.

Part 3: Solving Real-World Problems

  1. A car travels 210 miles on 7 gallons of gas. At this rate, how many miles can the car travel on 10 gallons of gas?
    Thought Process: First, find the constant of proportionality (miles per gallon). Then use that constant to find the distance for 10 gallons.

    • Step 1: Find the rate (k). k = miles/gallons = 210 miles / 7 gallons = 30 miles per gallon.
    • Step 2: Set up the equation. Let y = miles and x = gallons. So, y = 30x.
    • Step 3: Solve for 10 gallons. y = 30 * 10 = 300 miles.
      Answer: The car can travel 300 miles on 10 gallons of gas.

  2. Sarah earns $12 per hour babysitting. If she babysits for 4.5 hours, how much money will she earn?
    Thought Process: Identify the constant rate (earnings per hour) and multiply by the number of hours.

    • Step 1: Identify the constant of proportionality (k). k = $12 per hour.
    • Step 2: Set up the equation. Let y = money earned and x = hours worked. So, y = 12x.
    • Step 3: Solve for 4.5 hours. y = 12 * 4.5 = $54.
      Answer: Sarah will earn $54 for babysitting 4.5 hours.
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