Lesson 2: Analyzing Linear Graphs; Linear vs. Nonlinear; Increasing, Decreasing, Constant

How can you tell if a function is linear just by looking at its graph or its equation?

What does it mean for a function to be "constant" in a real-world context?

How does the rate of change for a nonlinear function differ from that of a linear function?

Linear Function: A function whose graph is a straight line and has a constant rate of change.
Nonlinear Function: A function whose rate of change is not constant; the graph is not a straight line (e.g., curves).
Increasing: As the x-values increase, the y-values also increase (the graph goes "up").
Decreasing: As the x-values increase, the y-values decrease (the graph goes "down").
Constant: A section of a graph where the y-value does not change (m = 0).

8.F.A.3: Identifying linear vs. nonlinear equations (like y = x^2 vs. y = 3x + 4).
Standard 8.F.B.5: Interpreting "distance-time" graphs, which are common on standardized tests to measure a student's ability to translate visual slopes into narrative descriptions.

Description: Students will analyze various graphs to classify them as linear or nonlinear. They will practice "reading" a graph from left to right to describe its behavior (increasing/decreasing) and will sketch graphs based on verbal stories.

Purpose: To move students toward functional modeling. Understanding qualitative features allows students to interpret complex data (like stock market trends or speed) without needing a specific formula first.

DOK Level: Level 2 (Skill/Concept) – Classifying and describing; Level 3 (Strategic Thinking) – Sketching a graph based on a verbal description.

"Left to Right" Rule: Students often forget to read graphs from left to right, misidentifying a decreasing line as "increasing" if they look at it from the bottom up.

Curvature: Students may think any line that isn't horizontal or vertical is "nonlinear."

Constant vs. Zero: Students may struggle to explain that a horizontal line represents a "constant" value (like a car stopped at a light) rather than "no data."

Support: Provide a "Graph Dictionary" with visual examples of increasing, decreasing, and constant slopes.

Scaffolding: Use highlighters to mark different sections of a single graph (e.g., green for increasing, red for decreasing).

Extension: Have students write their own "Graph Story" (e.g., a trip to the park) and exchange it with a partner to see if the partner can sketch the corresponding graph accurately.

Observations

Exit tickets