Lesson 1: Introduction to Proportional and Non-Proportional Relationships
Duration of Days: 3
Lesson Objective
Students will be able to distinguish between proportional and non-proportional linear relationships across multiple representations (graphs, tables, equations, and verbal contexts) and construct equations in the form y=mx+b to model them.
What structural features of an equation, a coordinate graph, or an input-output table tell us immediately whether a relationship is proportional or non-proportional?
How does changing the initial value (y-intercept) of a linear function alter its graph and real-world meaning without changing its constant rate of change?
In a real-world comparison scenario, what does the graphical intersection point of two linear models represent?
Proportional Relationship,
Non-Proportional Relationship,
Linear Equation,
Constant Rate of Change,
Slope (m),
Initial Value,
y-intercept (b),
Origin (0,0),
Intersection Point,
System of Equations
8.F.A.3: Interpret the equation y=mx+b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
8.F.B.4: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph.
8.EE.B.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
Description: This 3-day introductory block establishes the core algebraic mechanics for building and graphing single-variable linear models.
Purpose: To explicitly ground the concept of linear structures before students advance to formal systems of equations. It connects abstract function patterns directly to coordinate lines.
DOK Level: Level 2 (Skill / Concept Application)
The Origin Confusion: Assuming any graph displaying a straight line must represent a proportional relationship, overlooking whether or not the y-intercept passes through the origin (0,0).
Variable Attachment Error: Reversing the placement of values when assembling the equation y=mx+b (e.g., writing the flat initial fee next to the independent variable x instead of isolating it as the constant b).
Sign Inversion on Decay: Dropping the negative indicator when modeling a decreasing rate of change, such as writing +m for an asset that is steadily losing value over time.
Color code the equations
Exit Tickets