Introduction to Functions

Active learning helps students grasp the abstract concept of functions by allowing them to manipulate and compare multiple representations at once. When students physically sort, draw, or simulate functions, they build mental models that connect the formal definition to concrete examples.

• Grade 10
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

1. Classify given sets of ordered pairs, graphs, and mapping diagrams as either relations or functions.
2. Analyze graphical representations to determine if they represent a function using the vertical line test.
3. Calculate and state the domain and range for linear functions, including those with contextual restrictions.
4. Compare and contrast the characteristics of relations and functions across different representations.

Ready-to-Use Activities

Concept Mapping

⏱ 25 min·Pairs

Card Sort: Relations vs Functions

Prepare cards showing tables, graphs, arrow diagrams, and equations. In pairs, students sort cards into 'function' or 'relation' piles and justify choices with evidence from each representation. Follow with a class share-out to resolve disagreements.

Prepare & details

Differentiate between a relation and a function using various representations.

Facilitation Tip: During Card Sort: Relations vs Functions, ensure each group has at least one example where an input maps to multiple outputs so students can directly observe the distinction.

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

Concept Mapping

⏱ 30 min·Small Groups

String Vertical Line Test

Print graphs on large paper and provide yarn or string. Small groups lay string vertically across graphs to check for multiple intersections, classifying each as a function or not. Record findings and discuss edge cases like vertical lines.

Prepare & details

Explain how the vertical line test helps identify a function from its graph.

Facilitation Tip: For the String Vertical Line Test, have students work in pairs to test their own graphs, fostering discussion about why the test works.

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

Concept Mapping

⏱ 35 min·Individual

Domain-Range Detective

Give scenarios like height vs age or cost vs items bought. Individuals list possible domain and range values, then pairs create mapping diagrams. Share and refine as a class to emphasize real-world restrictions.

Prepare & details

Analyze the importance of domain and range in defining a function's behavior.

Facilitation Tip: In Domain-Range Detective, require students to justify restrictions in domain or range using the context of each example, not just the graph or table.

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

Concept Mapping

⏱ 40 min·Pairs

Function Machine Simulation

One student acts as the 'machine' for a secret function rule. Pairs input x-values verbally and receive y-outputs, then graph points to guess the rule. Switch roles and verify with domain-range analysis.

Prepare & details

Differentiate between a relation and a function using various representations.

Facilitation Tip: During Function Machine Simulation, have students rotate roles to ensure everyone participates in input-output testing and rule discovery.

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

Start with concrete examples before formal definitions, using real-world scenarios like ticket pricing or temperature conversion. Avoid rushing to the vertical line test until students understand why repeated outputs per input break the function rule. Research shows that students grasp functions better when they first experience the concept through multiple representations before abstracting the definition.

What to Expect

Students will confidently distinguish functions from relations, apply the vertical line test correctly, and identify domain and range with accuracy. They will explain their reasoning using multiple representations, showing deep understanding beyond memorization.

Watch Out for These Misconceptions

Common Misconception

During Card Sort: Relations vs Functions, watch for students who classify a relation with repeated x-values as a function simply because it 'looks like a graph'.

What to Teach Instead

Prompt students to check the table or mapping for that relation, asking them to count how many outputs correspond to the repeated input. Have them physically separate the cards and re-sort based on the one-output rule.

Common Misconception

During Domain-Range Detective, watch for students who assume domain and range always include all real numbers unless told otherwise.

What to Teach Instead

Ask students to revisit their scenario cards and explain why restrictions exist, such as square roots or real-world constraints like age or money. Have them adjust their domain and range lists based on these justifications.

Common Misconception

During String Vertical Line Test, watch for students who apply the test mechanically without connecting it to the function definition.

What to Teach Instead

After the string test, have students trace their finger along the graph to show the input-output relationship for each point the string touches. Ask them to explain why the test catches broken rules using their own words.

Assessment Ideas

Quick Check

After Card Sort: Relations vs Functions, provide a mixed set of five graphs, tables, and mappings. Ask students to label each as a 'function' or 'relation' and write one sentence explaining their choice for each, using specific features from the representation.

Exit Ticket

After String Vertical Line Test, give students a blank index card to sketch one graph that passes the test and one that fails. Below each graph, they must write the domain and range for the function graph and explain why the non-function graph does not meet the definition.

Discussion Prompt

During Function Machine Simulation, pose the question: 'What would happen if your vending machine gave two different snacks for the same coin?' Guide students to articulate why a function requires one output per input in their own machine designs.

Extensions & Scaffolding

• Challenge students to create their own function machine with at least three different operations and trade with peers to solve.
• For students who struggle, provide pre-sorted pairs of examples and ask them to explain why each belongs to its category.
• Deeper exploration: Have students research and present on how functions appear in technology, such as algorithms or coding, connecting classroom math to real-world tools.

Key Vocabulary

Relation	A set of ordered pairs that describes a connection between two sets of values. It does not require each input to have only one output.
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 (often represented by 'x') for a relation or function.
Range	The set of all possible output values (often represented by 'y') for a relation or function.
Vertical Line Test	A graphical method used 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.