Introduction to Functions

Active learning helps students grasp the abstract concept of functions by making relationships between quantities concrete. Working with visuals like mapping diagrams and real-world examples lets students see how inputs connect to outputs, which builds intuition before formal definitions. This hands-on approach reduces confusion about domain, range, and the vertical line test.

• 9th Grade
• Mathematics
• 3 activities
• 20 min–30 min

Learning Objectives

1. Classify a given relation as a function or not a function, justifying the classification using the definition of a function.
2. Determine the domain and range of a function represented by a graph, equation, or table of values.
3. Compare and contrast relations and functions, identifying key characteristics of each.
4. Create a mapping diagram that represents a function and another that represents a relation, explaining the difference.
5. Evaluate the validity of a relation as a function using the vertical line test on its graph.

Concept Mapping

⏱ 30 min·Individual

Mapping Diagram Gallery: Function or Not?

Post 10 mapping diagrams around the room, some representing functions and some representing relations that are not functions. Students circulate with sticky notes, marking each as 'function' or 'not a function' and writing one justification sentence. The class then reviews each station's results and resolves disagreements.

Prepare & details

Differentiate between a relation and a function using various representations.

Facilitation Tip: For Mapping Diagram Gallery, provide a mix of function and non-function examples and have students physically sort them into labeled bins.

Setup: Tables with large paper, or wall space

Materials: Concept cards or sticky notes, Large paper, Markers, Example concept map

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: Design a Function

Ask students individually to draw a mapping diagram with exactly 4 inputs that is a function, then revise it to make it not a function by changing only one arrow. Partners compare their diagrams and explain what change violated the function definition. Pairs share the most interesting cases with the class.

Prepare & details

Explain how to determine the domain and range of a function from a graph or equation.

Facilitation Tip: During Think-Pair-Share, ask students to sketch their function examples first before explaining to partners to reinforce the input-output relationship.

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

Concept Mapping

⏱ 30 min·Small Groups

Vertical Line Test Station Rotation

Set up four stations with different graphs (linear, parabola, circle, scattered points). Students apply the vertical line test at each station using a piece of string or a ruler as a physical vertical line, record the domain and range visually, and determine whether the graph represents a function. Groups discuss any cases they disagree on.

Prepare & details

Justify the importance of the vertical line test in identifying functions.

Facilitation Tip: For Vertical Line Test Station Rotation, set up four stations with different graph types and require students to rotate with a recording sheet to document their findings.

Setup: Tables with large paper, or wall space

Materials: Concept cards or sticky notes, Large paper, Markers, Example concept map

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

Teaching This Topic

Teach functions by starting with concrete, relatable examples like temperature or movie recommendations so students see why functions matter. Use multiple representations—ordered pairs, mapping diagrams, equations, and graphs—so students connect all forms. Avoid rushing to the formal definition; let students discover the rules through guided exploration and discussion.

What to Expect

Students will confidently identify functions from ordered pairs, mapping diagrams, and graphs. They will explain why a relation is or is not a function using domain, range, and the vertical line test. Discussions and designs will show they understand how one quantity depends on another.

Watch Out for These Misconceptions

Common Misconception

During Mapping Diagram Gallery, watch for students who incorrectly label a relation as not a function because two inputs share the same output.

What to Teach Instead

Direct students back to the definition using the gallery’s examples: ask them to point to the inputs and outputs and explain why outputs can repeat but inputs cannot.

Common Misconception

During Think-Pair-Share: Design a Function, watch for students who assume equations like y = x squared are automatically functions without checking.

What to Teach Instead

Have students graph their designed function and apply the vertical line test immediately, using the graph to confirm or correct their initial assumption.

Assessment Ideas

Quick Check

After Mapping Diagram Gallery, provide 3-4 sets of ordered pairs on a handout. Ask students to write 'Function' or 'Not a Function' next to each set and explain why.

Exit Ticket

After Vertical Line Test Station Rotation, give students a graph of a relation. Ask them to: 1. State whether it is a function and explain using the vertical line test. 2. List the domain and range using inequality notation.

Discussion Prompt

During Think-Pair-Share: Design a Function, pose the following to small groups: 'Can your movie recommendation system be a function? Why or why not? What would the domain and range be?' Listen for explanations that connect inputs (viewing history) to outputs (recommendations) and address whether an input can have multiple outputs.

Extensions & Scaffolding

• Challenge: Ask students to create a function where the output is always greater than the input and graph it to verify.
• Scaffolding: Provide partially completed mapping diagrams with some inputs or outputs missing for students to fill in.
• Deeper exploration: Introduce piecewise functions and have students design one that models a real-world scenario, such as a cell phone data plan.

Key Vocabulary

Relation	A set of ordered pairs, where each first element (input) is related to each second element (output) in some way.
Function	A special type of relation where each input has exactly one output.
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.
Ordered Pair	A pair of numbers written in a specific order, typically (x, y), representing a point on a graph or a specific input-output relationship.