Graphing Ratios: Visualizing Proportional Relationships

Mathematics Grades 7th Grade 5:12 Video
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Lesson Description

Learn how to represent and interpret ratios on a coordinate plane, connecting tables, graphs, and real-world scenarios to understand proportional relationships.

Video Resource

Ratios on coordinate plane

Khan Academy

Duration: 5:12
Watch on YouTube

Key Concepts

  • Ratio
  • Proportional Relationship
  • Coordinate Plane
  • Unit Rate
  • Constant of Proportionality

Learning Objectives

  • Students will be able to represent a ratio as an ordered pair on a coordinate plane.
  • Students will be able to identify proportional relationships from graphs and tables.
  • Students will be able to interpret the meaning of points on a graph in the context of a real-world scenario.
  • Students will be able to determine the unit rate or constant of proportionality from graphs and tables.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a ratio and proportional relationship. Ask students for real-world examples of ratios they encounter in their daily lives (e.g., ingredients in a recipe, speed of a car). Briefly discuss the coordinate plane and how it is used to represent data.
  • Video Viewing (10 mins)
    Play the Khan Academy video "Ratios on coordinate plane." Instruct students to take notes on the examples presented in the video, paying attention to how ratios are transformed into ordered pairs and plotted on the coordinate plane.
  • Guided Practice (15 mins)
    Work through examples similar to those in the video as a class. Start with a simple ratio (e.g., 2 apples for every 1 orange). Create a table of values, plot the points on a coordinate plane, and discuss the relationship between the ratio and the graph. Emphasize how the straight line represents a proportional relationship.
  • Independent Practice (15 mins)
    Provide students with a worksheet containing ratio problems. Students will need to create tables, plot points on a coordinate plane, and answer questions about the relationships shown in the graphs. Include problems with varying levels of difficulty.
  • Wrap-up & Assessment (5 mins)
    Review the key concepts of the lesson and address any remaining questions. Administer a short quiz to assess student understanding of graphing ratios and identifying proportional relationships.

Interactive Exercises

  • Graphing Challenge
    Present students with a real-world scenario involving a ratio (e.g., the cost of pizza slices). Have them work in pairs to create a table of values, plot the points on a coordinate plane using online graphing tools or graph paper, and analyze the proportional relationship.
  • Ratio Detective
    Provide graphs of different scenarios (some proportional, some not). Students analyze each graph and identify which ones represent proportional relationships and explain why or why not.

Discussion Questions

  • How can you tell if a graph represents a proportional relationship?
  • What does the slope of a line representing a proportional relationship tell you?
  • How can you use a graph to find the value of a ratio for a value not explicitly listed in a table?

Skills Developed

  • Graphing
  • Data Analysis
  • Proportional Reasoning
  • Problem-Solving
  • Critical Thinking

Multiple Choice Questions

Question 1:

Which of the following points would be on the graph of a proportional relationship where y is twice the value of x?

Correct Answer: (2, 4)

Question 2:

A baker uses 3 cups of sugar for every 5 cups of flour. Which ordered pair represents this ratio on a coordinate plane?

Correct Answer: (3, 5)

Question 3:

If a graph shows a proportional relationship, what must be true about the line?

Correct Answer: It must pass through the origin.

Question 4:

What does the constant of proportionality represent on a graph of a proportional relationship?

Correct Answer: The slope of the line

Question 5:

A car travels 60 miles in 1 hour. If 'x' represents hours and 'y' represents miles, which equation represents this proportional relationship?

Correct Answer: y = 60x

Question 6:

Which table represents a proportional relationship?

Correct Answer: x: 1, y: 5; x: 2, y: 10; x: 3, y: 15

Question 7:

What is the unit rate in the proportional relationship represented by the point (2, 10) on a graph?

Correct Answer: 5

Question 8:

If McKenna earns $6 for every time she shovels snow, and 'x' is the number of times she shovels, and 'y' is her total earnings, what is the ordered pair if she shovels 5 times?

Correct Answer: (5, 30)

Question 9:

In a graph of a proportional relationship, if one point is (3, 9), what is the value of y when x is 1?

Correct Answer: 3

Question 10:

If a graph does NOT pass through the origin, can it represent a proportional relationship?

Correct Answer: No, never

Fill in the Blank Questions

Question 1:

A relationship where two quantities have a constant ratio is called a ________ relationship.

Correct Answer: proportional

Question 2:

The point (0, 0) on a coordinate plane is called the ________.

Correct Answer: origin

Question 3:

The constant ratio in a proportional relationship is also known as the ________ ________.

Correct Answer: unit rate

Question 4:

The equation y = kx represents a proportional relationship, where k is the ________ of ________.

Correct Answer: constant, proportionality

Question 5:

On a graph, a proportional relationship is represented by a ________ line.

Correct Answer: straight

Question 6:

If 4 apples cost $2, then 8 apples will cost $________, assuming a proportional relationship.

Correct Answer: 4

Question 7:

The ________ is the ratio of the vertical change to the horizontal change between two points on a line.

Correct Answer: slope

Question 8:

In the ordered pair (x, y), 'x' represents the ________ axis and 'y' represents the ________ axis.

Correct Answer: horizontal, vertical

Question 9:

If a graph goes through the point (1,5) and represents a proportional relationship, the constant of proportionality is ________.

Correct Answer: 5

Question 10:

A ________ shows the relationship between two sets of numbers.

Correct Answer: ratio

Educational Standards

7.A.1:Explain the concept of proportionality in mathematical models and situations and distinguish between proportional and non-proportional relationships. 7.A.1.1:Identify a relationship between two varying quantities, x and y, as proportional if it can be expressed in the form y = kx; distinguish proportional relationships from non-proportional relationships. 7.A.1.2:Recognize that the graph of a proportional relationship is a line through the origin and the coordinate (1, r), where r is the slope and the unit rate (constant of proportionality, k). 7.A.2.1:Represent proportional relationships with tables, verbal descriptions, symbols, and graphs; translate from one representation to another. Determine and compare the unit rate (constant of proportionality, slope, or rate of change) given any of these representations.

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