Lesson 3: Linear Functions in Context
Duration of Days: 2
Lesson Objective
Students will be able to interpret and model real-world situations using linear functions, specifically identifying the meaning of the slope as a constant rate of change and the y-intercept as an initial value within a given context.
In a real-world scenario, what physical quantity does the "slope" represent (e.g., speed, cost per item, unit rate)?
How can you tell from a word problem if the y-intercept is zero (proportional) or a non-zero constant (non-proportional)?
If a linear graph represents a journey, what does a negative slope tell you about the direction or speed?
Initial Value: The starting point of a scenario (the y-intercept).
Rate of Change: The speed or cost at which something increases or decreases (the slope).
Independent Variable: The "input" (typically time, quantity, or distance).
Dependent Variable: The "output" (typically total cost, total distance, or remaining balance).
Contextual Modeling: The process of translating a narrative into a mathematical equation.
8.EE.B.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph.
Target C.2: The student interprets the unit rate as the slope of the graph of a proportional relationship.
Target C.3: The student compares two different proportional relationships represented in different formats.
Description: Students will practice extacting data from stories.
Purpose: To move students from "solving for x" to using math as a tool for decision-making and analysis.
DOK Level: Level 3 (Strategic Thinking) – Students must interpret, analyze, and justify their mathematical models based on situational context.
Subscription Services: Comparing flat-rate vs. pay-per-use models (e.g., gym memberships or streaming services).
Fundraising: Modeling "Pizza Problems" or "Teddy Bear Sales" scenarios.
Gig Economy: Analyzing earnings for ride-share drivers (base fare + price per mile).
Variable Misidentification: Students may confuse the "start-up fee" (y-intercept) with the "recurring cost" (slope).
Misinterpreting Zero: Thinking that if a scenario starts at zero, it cannot be linear.
Unit Confusion: Struggling to identify the correct units for slope (e.g., saying "5" instead of "5 per hour").
Support: Provide structured graphic organizers to separate the "Starting Amount" from the "Moving Amount."
Scaffolding: Provide sentence stems: "The initial value is ____ because ____. The rate of change is ____ per ____ because ____."
Extension: Use complex multi-step problems that involve comparing two different linear models.
Exit Tickets