Relations vs. Functions: Core Concepts

Active learning helps students grasp the distinctions between relations and functions by engaging multiple representations and collaborative reasoning. When students manipulate physical materials or discuss analogies, they move beyond abstract definitions to concrete understanding.

• Grade 11
• Mathematics
• 3 activities
• 20 min–40 min

Learning Objectives

1. Classify a given set of ordered pairs, mapping diagram, or graph as either a relation or a function.
2. Explain, using the definition of a function, why a specific relation fails to meet the criteria for a function.
3. Analyze how restricting or expanding the domain of a relation can change its status as a function.
4. Compare and contrast the graphical representations of relations that are functions with those that are not, using the vertical line test.
5. Demonstrate the relationship between independent and dependent variables in a given real-world scenario, identifying if it represents a function.

Stations Rotation

⏱ 40 min·Small Groups

Stations Rotation: Function or Relation?

Set up four stations with different representations: a set of ordered pairs, a mapping diagram, a table of values, and a graph. Small groups rotate through stations, identifying if each is a function and justifying their choice using specific vocabulary like domain and range.

Prepare & details

How does the vertical line test communicate the fundamental definition of a function?

Facilitation Tip: During Station Rotation, place the ordered pairs station first, as it builds foundational skills before moving to mapping diagrams or graphs.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: The Vending Machine Analogy

Students use the analogy of a vending machine to explain functions (one button leads to one specific snack). They work in pairs to create their own real world analogies, such as a person's height over time or a social insurance number, and present them to the class.

Prepare & details

Why is it useful to differentiate between a relation and a function in mathematical modeling?

Facilitation Tip: For The Vending Machine Analogy, ask students to sketch their own vending machine examples before discussing as a class.

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

Inquiry Circle

⏱ 30 min·Small Groups

Inquiry Circle: Domain Constraints

Groups are given a set of physical constraints, such as the height of a ball over time, and must determine the appropriate domain and range. They then swap their scenarios with another group to see if the mathematical model holds up under peer review.

Prepare & details

Analyze how changing the domain of a relation affects its status as a function.

Facilitation Tip: In Collaborative Investigation, assign each group a different constraint to explore, then have them present their findings to compare domain effects.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Teaching This Topic

Start with clear definitions but immediately move to examples where students must apply them. Avoid overemphasizing rules like the vertical line test without connecting it to the definition of a function. Research shows that students benefit from classifying examples before creating their own, so prioritize analysis over production in early stages.

What to Expect

By the end of these activities, students will confidently classify examples as relations or functions and justify their reasoning using precise mathematical language. They will also recognize when a relation fails to meet function criteria and propose corrections.

Watch Out for These Misconceptions

Common Misconception

During Station Rotation, watch for students who classify all graphs as functions without testing the vertical line test.

What to Teach Instead

Direct students to the graph station first and require them to physically place a ruler vertically on the graph to check for multiple outputs before making their decision.

Common Misconception

During Think-Pair-Share: The Vending Machine Analogy, watch for students who confuse the vending machine's output with its operation.

What to Teach Instead

Ask students to describe what happens when they press the same button twice; emphasize that the same input must always produce the same output for it to be a function.

Assessment Ideas

Quick Check

After Station Rotation, provide students with three different representations: a set of ordered pairs, a mapping diagram, and a graph. Ask them to label each as 'Relation only' or 'Function' and write one sentence justifying their choice for at least two of them.

Discussion Prompt

During Think-Pair-Share: The Vending Machine Analogy, present the scenario: 'The number of hours you study and your test score.' Ask students in pairs to discuss whether this is always a function, then share with the class. Follow up by asking how restricting the domain (e.g., only students who studied between 1 and 3 hours) might change the classification.

Exit Ticket

After Collaborative Investigation, give students a graph that fails the vertical line test. Ask them to: 1. Write down two ordered pairs from the graph that demonstrate why it is not a function. 2. Sketch a slight modification to the graph that would make it a function.

Extensions & Scaffolding

• Challenge students to create a graph that passes the vertical line test but has a domain restriction that makes it not a function.
• For struggling students, provide a partially completed mapping diagram with blanks for inputs and outputs to scaffold their understanding.
• Deeper exploration: Have students research and present on real-world phenomena that are relations but not functions, such as temperature over time or population growth models.

Key Vocabulary

Relation	A set of ordered pairs, where each pair represents a relationship between an input and an output value.
Function	A special type of relation where each input value is associated with exactly one output value.
Domain	The set of all possible input values (x-values) for a relation or function.
Range	The set of all possible output values (y-values) for a relation or function.
Vertical Line Test	A graphical method to determine if a relation is a function; if any vertical line intersects the graph at more than one point, it is not a function.