Comparing Functions

Active learning works for comparing functions because students must engage with multiple representations side-by-side. This mirrors real-world problem solving, where data rarely fits neatly into one format. By moving, matching, and discussing, students confront the need to extract and compare key properties like rate of change and initial value directly.

• 8th Grade
• Mathematics
• 3 activities
• 20 min–35 min

Learning Objectives

1. Compare the rates of change for two functions presented in different formats (graphical, tabular, algebraic, verbal).
2. Analyze which of two functions, represented differently, is more suitable for a given real-world scenario.
3. Justify conclusions about the relative properties (e.g., rate of change, initial value) of two functions using evidence from their distinct representations.
4. Calculate the rate of change and initial value for functions presented in tabular or graphical formats.
5. Translate between different representations of a function (e.g., from a table to an equation, from a graph to a verbal description).

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: Which Is Faster?

Give each pair two functions in different formats (one as a graph, one as a table). Each student independently determines the rate of change for their assigned function, then the pair compares and identifies which grows faster. Pairs share their reasoning process with the class.

Prepare & details

Compare the rates of change of two functions presented in different formats.

Facilitation Tip: During Think-Pair-Share, assign roles: one student explains the graph, one explains the table or equation, then they compare findings.

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

Decision Matrix

⏱ 30 min·Small Groups

Matching Activity: Same Function, Different Look

Provide a set of cards showing the same linear function in four formats (equation, graph, table, verbal description). Students match complete sets and then compare two different functions to identify which has the steeper rate of change and the higher initial value. Groups must cite specific evidence from each format.

Prepare & details

Analyze which function would be more suitable for a particular real-world application.

Facilitation Tip: For the Matching Activity, require students to annotate each pair with slope and y-intercept before deciding if they represent the same function.

Setup: Groups at tables with matrix worksheets

Materials: Decision matrix template, Option description cards, Criteria weighting guide, Presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Gallery Walk

⏱ 35 min·Small Groups

Gallery Walk: Real-World Function Comparison

Post six stations, each showing two functions representing real scenarios (e.g., two phone data plans, two savings accounts). Students determine which function represents the better deal under given conditions, writing their reasoning on a response sheet. Debrief focuses on how representation format affected difficulty.

Prepare & details

Justify conclusions about two functions based on their various representations.

Facilitation Tip: During the Gallery Walk, have students leave written feedback on sticky notes for each pair’s comparison, focusing on clarity and evidence.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Teaching This Topic

Teach students to convert all representations to the same form before comparing. Use color-coding to highlight corresponding parts of different formats. Avoid rushing to conclusions about which function is 'better'—focus first on equivalence. Research shows that students benefit from repeated practice comparing the same pair of functions in different orders, which builds flexibility.

What to Expect

Successful learning looks like students accurately comparing functions across formats using precise language. They identify equivalent properties in different representations and justify their comparisons with evidence. Partners and small groups should reach agreement on which function grows faster or has a higher starting value.

Watch Out for These Misconceptions

Common Misconception

During Think-Pair-Share, watch for students who assume the visually steeper graph always represents the faster growth rate without checking axis scales.

What to Teach Instead

Have students measure the rise and run on both graphs and calculate the slope before sharing their initial impressions, then revisit their conclusion.

Common Misconception

During Matching Activity, watch for students who compute unit rate from the table but interpret it as a different quantity than the visual slope of the graph.

What to Teach Instead

Require students to write the slope as a fraction (rise/run) for both representations and annotate the numerator and denominator on each before matching.

Assessment Ideas

Exit Ticket

After Think-Pair-Share, collect students’ written comparisons of the two functions, noting whether they correctly identified rate of change and initial value in both representations.

Quick Check

During Matching Activity, circulate and listen for students explaining why matched pairs represent the same function, using evidence from both representations.

Discussion Prompt

After Gallery Walk, ask students to present one comparison from the walk and justify their conclusion, holding peers accountable for evidence.

Extensions & Scaffolding

• Challenge: Ask students to create a fourth representation (verbal description or equation) for an unmatched function from the Matching Activity, then find its match.
• Scaffolding: Provide a partially completed table or graph with labeled axes to help students fill in missing values before comparing.
• Deeper: Have students research and bring in two real-world functions (e.g., population growth, savings plans) in different formats, then compare them in a written analysis.

Key Vocabulary

Rate of Change	The constant rate at which the dependent variable changes with respect to the independent variable. It is often represented as 'rise over run' or 'change in y over change in x'.
Initial Value	The value of the dependent variable when the independent variable is zero. For linear functions, this is the y-intercept.
Linear Function	A function whose graph is a straight line. Its rate of change is constant.
Representation	A way of showing a mathematical relationship, such as an equation, a graph, a table of values, or a verbal description.