Linear vs. Non-Linear Functions

Active learning works well for linear vs. non-linear functions because students need to physically examine equations, graphs, and tables to spot patterns in rates of change. Abstract ideas like ‘straight line’ and ‘constant rate’ become concrete when learners sort, compare, and justify their choices with real data.

• 8th Grade
• Mathematics
• 3 activities
• 20 min–30 min

Learning Objectives

1. Classify given functions as linear or non-linear based on their graphical representation.
2. Calculate the rate of change from a table of values to determine if a function is linear.
3. Compare the algebraic forms of linear and non-linear functions to identify key differences.
4. Explain how a constant rate of change defines a linear function.
5. Predict whether a real-world scenario, such as distance traveled at a constant speed versus the area of a growing square, will yield a linear or non-linear function.

Carousel Brainstorm

⏱ 30 min·Small Groups

Sorting Activity: Linear or Not?

Provide groups with a set of cards showing functions as equations, graphs, and tables. Students sort them into linear and non-linear categories, then for each card explain which feature of that representation told them the answer. Sorting by representation type (all graphs together, all tables together) as a second pass deepens analysis.

Prepare & details

Differentiate between linear and non-linear functions based on their graphs and equations.

Facilitation Tip: During the Sorting Activity, provide rulers so students can check straightness on printed graphs before deciding on their classification.

Setup: Charts posted on walls with space for groups to stand

Materials: Large chart paper (one per prompt), Markers (different color per group), Timer

• Remember
• Understand
• Analyze
• Relationship Skills
• Social Awareness

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: Constant Rate Check

Give each pair a table of values with equal x-increments. Students calculate the differences in y-values, determine if the rate is constant, and classify the function. Pairs then predict what the graph of their function would look like before checking on graph paper or desmos.

Prepare & details

Explain how the rate of change determines the linearity of a function.

Facilitation Tip: In the Think-Pair-Share, require students to calculate at least two consecutive differences before sharing, so the constant-rate test becomes procedural habit.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Carousel Brainstorm

⏱ 25 min·Small Groups

Real-World Prediction: Linear or Curve?

Present five real-world scenarios: simple interest earned over time, compound interest, constant speed, accelerating car, population growth at a fixed percentage. Students predict linear or non-linear for each, sketch a rough graph shape, and compare predictions in small groups before researching or computing to verify.

Prepare & details

Predict whether a real-world scenario will produce a linear or non-linear function.

Facilitation Tip: For Real-World Prediction, ask students to sketch both a line and a curve on the same axes to visualize how the curve’s steepness changes while the line’s does not.

Setup: Charts posted on walls with space for groups to stand

Materials: Large chart paper (one per prompt), Markers (different color per group), Timer

• Remember
• Understand
• Analyze
• Relationship Skills
• Social Awareness

Teaching This Topic

Teach this topic by building the concept of rate of change through multiple representations first. Avoid rushing to definitions; instead, let students discover the constant-rate rule themselves through guided tables and graphs. Research shows that students who generate their own examples grasp the distinction faster than those who only receive teacher explanations.

What to Expect

By the end of these activities, students will confidently label functions as linear or non-linear and support their decisions with evidence from equations, graphs, or tables. You’ll see clear justifications and correct use of terms like ‘constant rate’ and ‘curve’.

Watch Out for These Misconceptions

Common Misconception

During the Sorting Activity, watch for students who judge a graph as linear merely because it looks ‘almost straight’ at the scale shown.

What to Teach Instead

Have students zoom in on the graph using graphing software or trace a small segment with a ruler to reveal any curvature. Require them to adjust their classification if the zoom shows a curve.

Common Misconception

During the Think-Pair-Share, watch for students who believe a steeper line is automatically non-linear because the rate seems larger.

What to Teach Instead

Ask students to compare a very steep linear graph (e.g., y = 100x) to a shallow curve (e.g., y = x^2/10) side by side. Have them calculate the differences in the table for both and note that the line’s differences stay constant while the curve’s do not.

Assessment Ideas

Quick Check

After the Sorting Activity, provide each pair with five new mixed cards. Ask them to sort two cards correctly and justify their placement to you before moving on.

Exit Ticket

After the Think-Pair-Share, ask students to answer the exit-ticket question comparing a steady-speed scenario to a growing-area scenario, using the constant-rate test from the activity.

Discussion Prompt

During the Real-World Prediction activity, pose the discussion question about table values and constant differences. Listen for students to explain that the y-values must change by the same amount each time for the function to be linear.

Extensions & Scaffolding

• Challenge early finishers to create an equation, table, and graph for a non-linear function that starts with a steep slope but flattens quickly.
• Scaffolding: For struggling students, provide pre-labeled cards with color-coded edges (blue for linear, green for non-linear) to reduce cognitive load during sorting.
• Deeper exploration: Invite students to research quadratic and exponential functions, then prepare a short presentation comparing their rate-of-change patterns to linear ones.

Key Vocabulary

Linear Function	A function whose graph is a straight line. It has a constant rate of change.
Non-Linear Function	A function whose graph is not a straight line. Its rate of change varies.
Rate of Change	The measure of how much one quantity changes with respect to another quantity. For linear functions, this is constant.
Constant Rate of Change	The rate of change that remains the same between any two points of a function, indicating a linear relationship.