Unlocking Proportional Relationships: Writing Equations from Rates

Algebra 1 Grades High School 2:10 Video
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Lesson Description

Learn how to identify the constant of proportionality and write proportional equations from real-world scenarios involving rates.

Video Resource

Writing proportional equations | Rates & proportional relationships | 7th grade | Khan Academy

Khan Academy

Duration: 2:10
Watch on YouTube

Key Concepts

  • Proportional Relationship
  • Constant of Proportionality
  • Writing Equations

Learning Objectives

  • Identify proportional relationships from word problems.
  • Determine the constant of proportionality (k) from given information.
  • Write a proportional equation in the form y = kx.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a proportional relationship and its key characteristics (constant ratio between two quantities, graph is a straight line through the origin). Briefly discuss real-world examples of proportional relationships (e.g., distance and time at a constant speed).
  • Video Viewing and Guided Practice (15 mins)
    Play the Khan Academy video "Writing proportional equations | Rates & proportional relationships | 7th grade | Khan Academy." Pause at key points to ask students clarifying questions and ensure understanding. Encourage students to take notes on the example problems presented in the video.
  • Worked Example and Application (15 mins)
    Present a worked example problem similar to the one in the video. Guide students through the process of identifying the proportional relationship, determining the constant of proportionality, and writing the equation. Then, provide students with a similar problem to solve independently or in pairs. Emphasize the importance of defining variables clearly.
  • Class Discussion and Wrap-up (10 mins)
    Lead a class discussion on the strategies used to solve the problems. Address any remaining questions or misconceptions. Summarize the key concepts of proportional relationships, constant of proportionality, and writing equations. Preview the next lesson, which will build upon this foundation.

Interactive Exercises

  • Real-World Scenarios
    Present students with several real-world scenarios and ask them to identify whether or not they represent proportional relationships. For those that are proportional, have them determine the constant of proportionality and write the corresponding equation.
  • Graphing Proportional Relationships
    Provide students with proportional equations and have them graph the relationships on a coordinate plane. Discuss the connection between the constant of proportionality and the slope of the line.

Discussion Questions

  • What are the characteristics of a proportional relationship?
  • How do you identify the constant of proportionality in a word problem?
  • Why is it important to define your variables when writing an equation?

Skills Developed

  • Problem-solving
  • Analytical Thinking
  • Equation Writing

Multiple Choice Questions

Question 1:

Which of the following equations represents a proportional relationship?

Correct Answer: y = 5x

Question 2:

In the equation y = kx, what does 'k' represent?

Correct Answer: The slope

Question 3:

If y is proportional to x and y = 12 when x = 3, what is the constant of proportionality?

Correct Answer: 4

Question 4:

A car travels 150 miles in 3 hours. Assuming a constant speed, what is the constant of proportionality (miles per hour)?

Correct Answer: 50

Question 5:

Which statement best describes a proportional relationship?

Correct Answer: The ratio between two variables is constant.

Question 6:

If y = 7x, and x = 4, what is the value of y?

Correct Answer: 28

Question 7:

A recipe calls for 2 cups of flour for every 1 cup of sugar. If 'f' represents cups of flour and 's' represents cups of sugar, which equation represents this relationship?

Correct Answer: f = 2s

Question 8:

Which graph represents a proportional relationship?

Correct Answer: A line passing through the origin

Question 9:

What is the y-intercept of a proportional relationship?

Correct Answer: 0

Question 10:

In a proportional relationship, if one quantity doubles, what happens to the other quantity?

Correct Answer: It doubles

Fill in the Blank Questions

Question 1:

In a proportional relationship, the ratio between y and x is always a ________.

Correct Answer: constant

Question 2:

The equation y = kx represents a ________ relationship.

Correct Answer: proportional

Question 3:

The constant of proportionality is also known as the ________ of the line.

Correct Answer: slope

Question 4:

If y = 8x, then the constant of proportionality is ________.

Correct Answer: 8

Question 5:

A graph of a proportional relationship always passes through the ________.

Correct Answer: origin

Question 6:

In the equation y = kx, 'y' is the ________ variable.

Correct Answer: dependent

Question 7:

If 5 apples cost $2.50, the cost per apple represents the ________ of proportionality.

Correct Answer: constant

Question 8:

The equation d = 65t represents the distance (d) traveled in terms of time (t). The constant 65 represents the ________.

Correct Answer: speed

Question 9:

A table of values represents a proportional relationship if the quotient of y/x is ________ for all pairs of x and y.

Correct Answer: equal/same

Question 10:

If y is directly proportional to x and x increases, then y will also ________.

Correct Answer: increase

Educational Standards

A1.A.4:Analyze real-world and mathematical problems involving linear equations. A1.A.4.3:Write the equation of the line given its slope and y-intercept, slope and one point, two points, x- and y-intercepts, or a set of data points. A1.F.1:Understand functions as descriptions of covariation (how related quantities vary together) in real-world and mathematical problems. A1.F.1.3:Write linear functions, using function notation, to represent mathematical models.

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