Snow Day Equations: Modeling Linear Relationships

Algebra 1 Grades High School 4:58 Video
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Lesson Description

Learn to create linear equations and graphs to model real-world scenarios, using the example of melting snow.

Video Resource

Modeling with linear equations example 1 | Linear equations and functions | 8th grade | Khan Academy

Khan Academy

Duration: 4:58
Watch on YouTube

Key Concepts

  • Linear equations
  • Slope-intercept form
  • Graphing linear equations
  • Modeling real-world problems

Learning Objectives

  • Students will be able to create a linear equation from a real-world scenario.
  • Students will be able to graph a linear equation based on a given scenario.
  • Students will be able to interpret the meaning of the variables and constants in the context of the problem.

Educator Instructions

  • Introduction (5 mins)
    Begin by discussing real-world situations where quantities change at a constant rate. Ask students for examples (e.g., filling a pool, burning a candle). Briefly review the concept of linear equations and their general form (y = mx + b).
  • Video Viewing and Explanation (15 mins)
    Play the Khan Academy video 'Modeling with linear equations example 1'. Pause at key points to explain the reasoning behind each step. Emphasize how the problem is translated into mathematical terms. Discuss the meaning of the slope and y-intercept in the context of the snow melting problem.
  • Guided Practice (15 mins)
    Work through a similar example problem with the students. For example: 'A taxi charges an initial fee of $3 and $2 per mile. Write an equation and graph to show the total cost as a function of miles driven.' Guide students in defining variables, writing the equation, and creating the graph. Encourage active participation and questioning.
  • Independent Practice (10 mins)
    Provide students with a new real-world scenario to model on their own. For example: 'A plant grows 0.5 inches per week and was initially 2 inches tall. Write an equation and graph to represent the plant's height over time.' Have students work individually and then share their solutions with a partner.

Interactive Exercises

  • Graphing Activity
    Use an online graphing tool (e.g., Desmos) to allow students to experiment with different linear equations and observe how changing the slope and y-intercept affects the graph.
  • Real-World Scenario Creation
    Have students create their own real-world scenarios that can be modeled using linear equations. They should then share their scenarios with the class and challenge others to write the corresponding equation and graph.

Discussion Questions

  • How does the slope of the line relate to the rate of change in the problem?
  • What does the y-intercept represent in the context of a real-world scenario?
  • Can you think of other real-world situations that can be modeled using linear equations?

Skills Developed

  • Problem-solving
  • Mathematical modeling
  • Analytical thinking
  • Graphical representation

Multiple Choice Questions

Question 1:

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

Correct Answer: slope

Question 2:

What is the y-intercept?

Correct Answer: The point where the line crosses the y-axis

Question 3:

A line has a slope of 2 and passes through the point (0, 5). What is the equation of the line?

Correct Answer: y = 2x + 5

Question 4:

Which of the following equations represents a linear relationship?

Correct Answer: y = 3x + 2

Question 5:

What does it mean to model a real-world problem with a linear equation?

Correct Answer: To represent the problem with an equation that shows a constant rate of change.

Question 6:

If a line has a negative slope, what does this indicate?

Correct Answer: The line is decreasing from left to right.

Question 7:

A car travels at a constant speed of 60 miles per hour. Which equation represents the distance (d) traveled after t hours?

Correct Answer: d = 60t

Question 8:

What is the slope of a horizontal line?

Correct Answer: 0

Question 9:

What is the slope of a vertical line?

Correct Answer: Undefined

Question 10:

Which of the following represents slope-intercept form?

Correct Answer: y = mx + b

Fill in the Blank Questions

Question 1:

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

Correct Answer: mx

Question 2:

The y-intercept is the point where the line crosses the _____ axis.

Correct Answer: y

Question 3:

A line with a slope of 0 is a _____ line.

Correct Answer: horizontal

Question 4:

A line with an undefined slope is a _____ line.

Correct Answer: vertical

Question 5:

The _____ of a line is a measure of its steepness.

Correct Answer: slope

Question 6:

In the snow melting example, the amount of snow decreased by a constant rate of _____ inches per day.

Correct Answer: 2

Question 7:

If two lines have the same slope, they are said to be _____

Correct Answer: parallel

Question 8:

The point-slope form of a linear equation is y - y1 = _____ (x - x1)

Correct Answer: m

Question 9:

The x-intercept is the point where the line crosses the _____ axis.

Correct Answer: x

Question 10:

If the slope is positive, the line rises from ____ to _____.

Correct Answer: left, right

Educational Standards

A1.A.4:Analyze real-world and mathematical problems involving linear equations. 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.

Teaching Materials

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