Graphing Proportional Relationships: Is That Line Really Telling the Truth?

Algebra 1 Grades High School 3:05 Video
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Lesson Description

Learn how to identify proportional relationships from graphs, focusing on linearity and the importance of the origin. This lesson reinforces the concept of proportionality and strengthens your graph analysis skills.

Video Resource

Identifying proportional relationships from graphs | 7th grade | Khan Academy

Khan Academy

Duration: 3:05
Watch on YouTube

Key Concepts

  • Proportional Relationships
  • Linearity
  • Origin (0,0)

Learning Objectives

  • Identify whether a graph represents a proportional relationship.
  • Explain why a line must pass through the origin to represent a proportional relationship.
  • Determine the constant of proportionality from a graph or table.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a proportional relationship (y = kx, where k is the constant of proportionality). Briefly discuss the characteristics of linear equations and their graphical representation.
  • Video Viewing (10 mins)
    Play the Khan Academy video 'Identifying proportional relationships from graphs | 7th grade | Khan Academy'. Instruct students to take notes on the key points discussed, especially the two conditions for a graph to represent a proportional relationship: it must be a line, and it must pass through the origin.
  • Guided Practice (15 mins)
    Work through examples similar to those in the video. Present graphs and tables, and guide students through determining whether they represent proportional relationships. Emphasize the importance of verifying that the relationship is linear and passes through the origin. Work through finding the constant of proportionality using various points on the graph. Discuss the meaning of the constant of proportionality in context (e.g., 'For every target hit, Natalie gets 3 points').
  • Independent Practice (15 mins)
    Provide students with worksheets or online exercises containing various graphs and tables. Have them independently identify proportional relationships and justify their answers. Encourage them to work in pairs to discuss their reasoning.
  • Wrap-up and Assessment (5 mins)
    Administer the multiple-choice and fill-in-the-blank quizzes to assess understanding of the concepts covered.

Interactive Exercises

  • Graphing Tool Activity
    Use an online graphing tool (like Desmos) to allow students to manipulate lines and observe how changing the equation affects whether it passes through the origin. Have them create lines that represent proportional relationships and lines that do not.

Discussion Questions

  • Why is it important for a graph to be a straight line to represent a proportional relationship?
  • What does it mean if a line on a graph representing a relationship does NOT pass through the origin?
  • How can you determine the constant of proportionality from a graph?

Skills Developed

  • Graph Analysis
  • Proportional Reasoning
  • Mathematical Justification

Multiple Choice Questions

Question 1:

Which of the following is a characteristic of a graph that represents a proportional relationship?

Correct Answer: It is a straight line that passes through the origin.

Question 2:

If a graph is a straight line but does not pass through the origin, what does this tell you about the relationship it represents?

Correct Answer: It is not a proportional relationship.

Question 3:

The constant of proportionality in a graph can be determined by:

Correct Answer: Calculating the slope (rise over run).

Question 4:

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

Correct Answer: The constant of proportionality

Question 5:

Which graph represents a proportional relationship?

Correct Answer: A line that goes through (0,0)

Question 6:

In a proportional relationship, if x doubles, what happens to y?

Correct Answer: y doubles

Question 7:

Which of the following equations does NOT represent a proportional relationship?

Correct Answer: y = x + 1

Question 8:

What is the y-intercept of a line that represents a proportional relationship?

Correct Answer: 0

Question 9:

If a graph shows the relationship between hours worked (x) and money earned (y), and it is a proportional relationship, what does the slope represent?

Correct Answer: The hourly wage

Question 10:

Given a table of values for x and y, how can you determine if they represent a proportional relationship?

Correct Answer: Check if the ratio y/x is constant for all pairs of values.

Fill in the Blank Questions

Question 1:

A proportional relationship must be represented by a _______ line.

Correct Answer: straight

Question 2:

For a graph to represent a proportional relationship, it must pass through the ______.

Correct Answer: origin

Question 3:

The equation for a proportional relationship is y = ____, where k is the constant of proportionality.

Correct Answer: kx

Question 4:

The constant of proportionality can be found by dividing y by ____.

Correct Answer: x

Question 5:

If the ratio between two variables is constant, they have a _______ relationship.

Correct Answer: proportional

Question 6:

In a proportional relationship, the y-intercept is always _______.

Correct Answer: zero

Question 7:

If two quantities are proportional, their graph will be a straight line with a slope of ____ k.

Correct Answer: constant

Question 8:

If y = 5x, then the constant of proportionality is ____.

Correct Answer: 5

Question 9:

A graph that curves is _______ a proportional relationship.

Correct Answer: not

Question 10:

In the example with Natalie hitting targets, the number of points she gets is _______ proportional to the targets hit.

Correct Answer: directly

Educational Standards

A1.A.4.1:Analyze, use and apply mathematical models and other data sets (e.g., graphs, equations, two points, a set of data points) to calculate and interpret slope and the x- and y-intercepts of a line. A1.F.2.1:Distinguish between linear and nonlinear (including exponential) functions. Understand that linear functions grow by equal intervals (arithmetic) and that exponential functions grow by equal factors over equal intervals (geometric). A1.F.3.1:Identify and generate equivalent representations of linear functions, graphs, tables, and real-world situations.

Teaching Materials

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