Unit 5: Linear Equation Introduction

Proportional vs. Non-Proportional Relationships: Proportional relationships can be modeled by the equation y=mx (starting at the origin, (0,0)), while non-proportional relationships include a non-zero initial starting value and are modeled by y=mx+b.

Structural Elements of Linear Models: In the equation y=mx+b, the coefficient m represents the constant rate of change (slope) and the constant b represents the starting value or initial amount (y-intercept).

Interpreting Representations Contextually: How to read, isolate, and locate the initial value and constant rate of change across varying representations, including graphs, input-output data tables, and real-world verbal scenarios (such as tracking bank accounts, draining water tanks, or paying gym entry fees).

Systems of Equations Concepts: The point where two linear graphs intersect represents a shared coordinate pair where both independent and dependent quantities are exactly equal for both scenarios.

Identify Initial Values and Rates: Find the initial amount and the rate of change directly from graphs, coordinate tables, and written problem scenarios.

Construct Linear Equations: Write explicit linear equations in slope-intercept form (y=mx+b) to represent proportional and non-proportional growth or decay problems.

Graph Linear Models: Plot coordinate sets from a data table or trace lines from real-world contexts onto a coordinate plane using starting points and rates.

Analyze Graphical Intersections: Pinpoint the precise intersection coordinates of two lines on a graph and explain the real-world contextual meaning of that intersection.

Predict and Calculate Values: Use linear equations to solve for unknown dependent outputs given a specific independent input (e.g., calculating total savings after a specific number of weeks or months).

• Linear Analysis and Comparison Task: Students are given a written prompt comparing two real-world scenarios. They must write equations for both individuals, sketch their corresponding lines on a coordinate plane, locate their point of intersection, and write a paragraph explaining what that intersection point means in the context of the problem.

• Multi-Representation Matching Assessment: Students interpret data tables and graph segments representing real-world processes), accurately calculating the y-intercept and slope, describing their real-world meaning, and converting them into a verified functional equation.