Modeling Spending with Linear Functions

Algebra 1 Grades High School 5:22 Video
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Lesson Description

Learn how to model real-world scenarios, like spending money, using linear functions and graphs. This lesson covers creating tables of values, graphing linear equations, and interpreting the graph to solve application problems.

Video Resource

Linear function example: spending money | Linear equations and functions | 8th grade | Khan Academy

Khan Academy

Duration: 5:22
Watch on YouTube

Key Concepts

  • Linear functions
  • Slope and y-intercept
  • Modeling real-world problems with equations and graphs

Learning Objectives

  • Students will be able to create a table of values for a linear equation.
  • Students will be able to graph a linear equation based on a real-world scenario.
  • Students will be able to interpret a graph of a linear equation to solve application problems.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of linear functions and their general form (y = mx + b). Discuss how linear functions can be used to model real-world situations where there is a constant rate of change. Introduce the problem context: Jill receiving $40 and spending a fixed amount each day.
  • Video Viewing and Explanation (10 mins)
    Play the Khan Academy video "Linear function example: spending money." Pause at key points to explain the steps the presenter is taking. Emphasize how the equation y = 40 - 2.5x represents the situation, explaining that 40 is the initial amount (y-intercept) and -2.5 is the rate of spending per day (slope).
  • Creating a Table of Values (10 mins)
    Guide students through the process of creating a table of values. Ask them to calculate 'y' (amount of money left) for different values of 'x' (number of days). Focus on choosing 'x' values that result in integer or easily manageable 'y' values. Discuss why choosing multiples of 2 simplifies the calculation in this specific problem.
  • Graphing the Equation (10 mins)
    Demonstrate how to plot the points from the table of values on a coordinate plane. Explain that the x-axis represents the number of days and the y-axis represents the amount of money. Connect the points to create the line representing the linear function. Discuss the importance of labeling the axes and choosing appropriate scales.
  • Interpreting the Graph (5 mins)
    Show how to use the graph to estimate the amount of money Jill has left after a certain number of days. Reinforce the connection between the graph, the equation, and the real-world situation. For example, ask: "Where on the graph do we find the answer to how much money she has after 8 days?"
  • Practice Problems (10 mins)
    Provide students with similar word problems involving linear functions. Have them work in pairs or small groups to create tables of values, graph the equations, and answer questions based on the graphs.

Interactive Exercises

  • Spending Scenario Worksheet
    Provide a worksheet with several spending scenarios. Students will need to write a linear equation to model each scenario, create a table of values, and graph the equation. They will then answer questions about the scenario based on their graph.
  • Graphing Applet
    Use an online graphing applet (e.g., Desmos, GeoGebra) to allow students to graph the linear equations and visualize the relationship between the equation, the table, and the graph. Have them manipulate the equation and observe how the graph changes.

Discussion Questions

  • How does the slope of the line relate to Jill's spending habits?
  • What does the y-intercept represent in this context?
  • How can we use the equation to find the amount of money Jill has left after any given number of days?
  • What are some other real-world scenarios that can be modeled using linear functions?

Skills Developed

  • Algebraic reasoning
  • Graphing linear equations
  • Problem-solving
  • Interpreting graphs

Multiple Choice Questions

Question 1:

A linear function is represented by the equation y = 5x + 3. What does the '5' represent?

Correct Answer: The slope

Question 2:

In the equation y = mx + b, what does 'b' represent?

Correct Answer: The y-intercept

Question 3:

A line passes through the points (0, 4) and (1, 6). What is the slope of the line?

Correct Answer: 2

Question 4:

Which of the following equations represents a linear function?

Correct Answer: y = 2x - 3

Question 5:

If Jill starts with $50 and spends $5 per day, which equation represents her spending?

Correct Answer: y = -5x + 50

Question 6:

The y-intercept of a graph represents?

Correct Answer: Where the line crosses the y-axis.

Question 7:

A line has a slope of 3 and passes through the point (0, -2). What is the equation of the line in slope-intercept form?

Correct Answer: y = 3x - 2

Question 8:

Which of the following describes a negative slope?

Correct Answer: The line falls from left to right.

Question 9:

If you graph the equation y = x, what will the slope be?

Correct Answer: 1

Question 10:

What is the value of y when x= 4 in the equation y = -2x + 10?

Correct Answer: 2

Fill in the Blank Questions

Question 1:

The general form of a linear equation is y = ____ + b.

Correct Answer: mx

Question 2:

The slope of a line is defined as rise over ____.

Correct Answer: run

Question 3:

The point where a line intersects the y-axis is called the ____.

Correct Answer: y-intercept

Question 4:

A line with a slope of zero is a ____ line.

Correct Answer: horizontal

Question 5:

If a line has a positive slope, it slopes ____ from left to right.

Correct Answer: upward

Question 6:

Two lines that never intersect are called _____ lines.

Correct Answer: parallel

Question 7:

The variable that we can freely change in an equation is called the ______ variable.

Correct Answer: independent

Question 8:

Using the equation y = 3x + 2, when x is zero, y will be ____.

Correct Answer: 2

Question 9:

The change in y divided by the change in x is the _____ of a line.

Correct Answer: slope

Question 10:

The y-intercept is the value of y when x equals ______.

Correct Answer: 0

Educational Standards

A1.A.4:Analyze real-world and mathematical problems involving linear equations. A1.F.1:Understand functions as descriptions of covariation (how related quantities vary together) in real-world and mathematical problems. A1.F.3:Represent functions in multiple ways and use the representation to interpret real-world and mathematical problems. A1.A.4.5:Analyze and interpret associations between graphical representations and written scenarios.

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