Melting Ice: Constructing Linear Functions from Real-World Data

Algebra 1 Grades High School 6:34 Video
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Lesson Description

Learn how to create linear functions from real-world scenarios, like the melting of an ice sheet, using slope-intercept form. This lesson reinforces understanding of rate of change and initial values.

Video Resource

Constructing linear functions example 1 | Algebra I | Khan Academy

Khan Academy

Duration: 6:34
Watch on YouTube

Key Concepts

  • Linear Functions
  • Slope-Intercept Form (y = mx + b)
  • Rate of Change (Slope)
  • Initial Value (y-intercept)
  • Function Notation (S(t))

Learning Objectives

  • Students will be able to identify the slope and y-intercept from a word problem.
  • Students will be able to write a linear function in slope-intercept form given two points or a rate of change and initial value.
  • Students will be able to interpret the meaning of the slope and y-intercept in the context of a real-world scenario.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of linear functions and their representation in slope-intercept form (y = mx + b). Briefly discuss the meaning of slope (m) and y-intercept (b) in a general context. Introduce function notation, emphasizing that S(t) represents the thickness of the ice as a function of time.
  • Video Viewing (10 mins)
    Play the Khan Academy video "Constructing linear functions example 1 | Algebra I | Khan Academy." Instruct students to take notes on the problem setup, the identification of the slope and y-intercept, and the final equation derived.
  • Guided Practice (15 mins)
    Work through the ice sheet problem step-by-step, mirroring the video's explanation. Emphasize the following points: * Identifying the initial value (S(0) = 2) as the y-intercept. * Calculating the slope (rate of change) using the two given data points (0, 2) and (3, 1.25). * Writing the equation in slope-intercept form (S(t) = -0.25t + 2). * Discussing the meaning of the slope (-0.25 meters per week) in the context of the problem.
  • Independent Practice (15 mins)
    Present students with similar word problems that require them to construct linear functions. Examples: * A candle burns at a rate of 0.5 inches per hour. It starts at 10 inches tall. Write a function to represent the candle's height after t hours. * A taxi charges an initial fee of $3 plus $2 per mile. Write a function to represent the cost of a taxi ride for m miles. * A phone loses 10% of its battery every hour. Write a function representing the remaining battery life as a percentage.
  • Wrap-up and Assessment (5 mins)
    Review the key concepts and learning objectives. Administer a short quiz (multiple choice or fill-in-the-blank) to assess student understanding.

Interactive Exercises

  • Graphing the Function
    Have students graph the ice sheet function (S(t) = -0.25t + 2) using graphing calculators or online graphing tools. Discuss how the graph visually represents the melting of the ice sheet and how the slope and y-intercept are reflected in the graph.
  • Partner Problem Solving
    Divide students into pairs and provide each pair with a different word problem. Have them work together to construct the linear function and explain their reasoning to each other. Each student has to write the equation in their own words, explaining the rate of change, slope, y-intercept, and the meaning of the function.

Discussion Questions

  • How does the slope represent the rate of change in a real-world scenario?
  • What does the y-intercept represent in a linear function?
  • How can you identify the slope and y-intercept from a word problem?
  • Why is it important to define your variables when writing a function?
  • Can all real world scenarios be modeled using a linear function? Why or why not?

Skills Developed

  • Problem-solving
  • Mathematical modeling
  • Critical thinking
  • Interpreting data
  • Algebraic reasoning

Multiple Choice Questions

Question 1:

What does 'm' represent in the slope-intercept form of a linear equation (y = mx + b)?

Correct Answer: The slope

Question 2:

In the ice sheet problem, what does S(t) represent?

Correct Answer: The ice sheet's thickness in meters

Question 3:

What is the y-intercept in the equation y = 3x - 5?

Correct Answer: -5

Question 4:

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

Correct Answer: 2

Question 5:

If a function is decreasing, what type of slope does it have?

Correct Answer: Negative

Question 6:

A phone loses 5% of battery per hour, and starts at 100%. Which is the slope?

Correct Answer: -5

Question 7:

A function is y=3x+5. What is the y value when x=0?

Correct Answer: 5

Question 8:

If the point (0, b) on a graph, what does b represent?

Correct Answer: The y-intercept

Question 9:

A line is horizontal. What is the slope?

Correct Answer: Zero

Question 10:

If a line is moving up, what type of slope does it have?

Correct Answer: Positive

Fill in the Blank Questions

Question 1:

The slope-intercept form of a linear equation is y = ___ + b.

Correct Answer: mx

Question 2:

The rate of change of a linear function is represented by the ____.

Correct Answer: slope

Question 3:

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

Correct Answer: y-intercept

Question 4:

In the ice sheet problem, the initial thickness of the ice is the ____.

Correct Answer: y-intercept

Question 5:

A function written as f(x) is in ____ notation.

Correct Answer: function

Question 6:

A line has a slope of 2, and a y-intercept of 4. In slope intercept form, the line is y=2x ____ 4.

Correct Answer: +

Question 7:

A line is moving down. The slope is therefore ____.

Correct Answer: negative

Question 8:

In the equation y = mx +b, b means ____.

Correct Answer: y-intercept

Question 9:

The slope is also called ____.

Correct Answer: rate of change

Question 10:

If a line is vertical, the slope is ____.

Correct Answer: undefined

Educational Standards

A1.F.1.3:Write linear functions, using function notation, to represent mathematical models. 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.3.1:Identify and generate equivalent representations of linear functions, graphs, tables, and real-world situations.

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