Unit Plan 14 (Grade 8 Math): Linear vs. Nonlinear
8th graders learn to distinguish linear from nonlinear functions by analyzing rules, graphs, and tables. They identify constant rates of change, recognize curved patterns, and justify conclusions using slope evidence.
Focus: Distinguish linear patterns from curves; recognize constant rate of change vs. varying rates.
Grade Level: 8
Subject Area: Mathematics (Functions)
Total Unit Duration: 5 sessions (one week), 45–60 minutes per session
I. Introduction
This week builds a strong conceptual line between linear and nonlinear relationships. Students decide whether a relation shows a constant rate of change (linear) or a varying rate (nonlinear) across rules, graphs, and tables. They connect “straight line” to the form y = mx + b and practice spotting nonlinear forms such as y = x^2, y = 3^x, y = k/x, y = |x|, and area = pi r^2. The emphasis is on evidence: equal differences in tables, straight vs. curved graphs, and function rules that imply constant or changing rates.
Essential Questions
- How do I know if a function is linear just by looking at its rule, graph, or table?
- What does constant rate of change look like in each representation?
- Why does a curved graph or changing table increments signal a nonlinear function?
II. Objectives and Standards
Learning Objectives — Students will be able to…
- Decide if a function is linear (y = mx + b) or nonlinear by examining rules, graphs, and tables.
- Explain constant rate of change using slope on graphs and equal differences over equal input intervals in tables.
- Provide examples of nonlinear functions (quadratic, exponential, inverse variation, absolute value, area formulas) and describe why their rates vary.
- Communicate conclusions clearly with units, slope/rate language, and a brief justification tied to the representation.
Standards Alignment — CCSS Grade 8
- 8.F.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, for example, the area of a circle with radius r is not linear in r because the area is pi r^2.
Success Criteria (student-friendly)
- I can point to equal differences in a table (for equal x-steps) and say it’s linear.
- I can look at a graph and state whether it’s a straight line (linear) or curved/stepped (nonlinear).
- I can read a rule and tell if it fits y = mx + b or shows varying rate (for example, x^2, 3^x, k/x, |x|).
- I can explain my decision with evidence and in context.