Introduction to Functions and Mappings

Functions and mappings can feel abstract to students, so active learning helps them see inputs and outputs in concrete ways. Working with visuals and hands-on tasks makes the one-output-per-input rule clearer than abstract definitions alone.

• Year 10
• Mathematics
• 4 activities
• 20 min–35 min

Learning Objectives

1. Compare a given set of ordered pairs to determine if it represents a function or a general relation.
2. Identify the domain and range of a function from its mapping diagram or rule.
3. Construct a mapping diagram for a given function rule and a specified domain.
4. Explain, using examples, why a function requires each input to have only one output.

Stations Rotation

⏱ 25 min·Pairs

Card Sort: Functions vs Relations

Prepare cards with input-output pairs: some functions, some not. In pairs, students sort into two piles and justify choices using mapping rules. Follow with class share-out to discuss edge cases like empty sets.

Prepare & details

Differentiate between a function and a general relation using examples.

Facilitation Tip: During the Card Sort, circulate and listen for students to justify their groupings with clear language about inputs and outputs.

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

Stations Rotation

⏱ 35 min·Small Groups

Mapping Diagram Construction

Provide sets of domain and range values on cards. Small groups draw arrows to create valid functions, then swap to critique others' mappings for one-to-one properties. Extend by inventing their own sets.

Prepare & details

Explain the significance of domain and range in defining a function.

Facilitation Tip: When students construct mapping diagrams, ask them to pause after each arrow and state, 'This input maps to exactly one output.'

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

Stations Rotation

⏱ 30 min·Whole Class

Function Machine Relay

Set up a 'machine' with hidden operations. Whole class relays inputs through, predicting outputs aloud. Reveal the rule at end and graph results to identify domain restrictions.

Prepare & details

Construct a mapping diagram for a given function and identify its properties.

Facilitation Tip: In the Function Machine Relay, time each student’s turn to keep energy high and ensure all students contribute at least once.

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

Stations Rotation

⏱ 20 min·Individual

Domain-Range Hunt

List everyday scenarios like temperatures to Celsius. Individuals identify domains and ranges, then pairs verify with mapping sketches. Class votes on trickiest examples.

Prepare & details

Differentiate between a function and a general relation using examples.

Facilitation Tip: For the Domain-Range Hunt, provide calculators and colored pencils to help students visualize restrictions and avoid rushing through the scenarios.

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

Teaching This Topic

Experienced teachers start with the visual and tactile—mapping diagrams and card sorts—before moving to abstract notation. They avoid defining functions too early and instead let students discover the uniqueness rule through examples. Research shows that letting students debate borderline cases (like piecewise or step functions) deepens understanding of the domain-range relationship.

What to Expect

Students will confidently identify functions and relations, explain why mapping diagrams show valid functions, and distinguish domain and range using both visual and numerical examples. They will use precise function notation in their reasoning and discussion.

Watch Out for These Misconceptions

Common Misconception

During Card Sort: Functions vs Relations, watch for students who group all relations with multiple outputs into the function pile without checking the one-output rule.

What to Teach Instead

During the Card Sort, give each pair red pens to mark any mappings that fail the one-output-per-input test. Require them to revisit those pairs and re-sort them into the relation pile, explaining their reasoning aloud to a partner.

Common Misconception

During Domain-Range Hunt, watch for students who assume all real numbers are allowed in the domain without checking for restrictions like division by zero or square roots of negatives.

What to Teach Instead

During the Domain-Range Hunt, have students highlight any numbers or operations in the scenario that might limit inputs. Ask them to write a sentence explaining why each restriction matters before finalizing their domain lists.

Common Misconception

During Mapping Diagram Construction, watch for students who list every possible output in the range even when some values aren’t actually achieved by the function.

What to Teach Instead

During Mapping Diagram Construction, ask students to trace each arrow and only list the outputs that appear in their diagram. Encourage them to cross out any predicted outputs that don’t actually occur in their mappings.

Assessment Ideas

Quick Check

After Card Sort: Functions vs Relations, provide students with three sets of ordered pairs. Ask them to circle the sets that represent functions and underline the sets that represent relations. Then, ask them to identify the domain and range for one of the function sets.

Exit Ticket

After Mapping Diagram Construction, give students the function rule f(x) = 3x - 2 and the domain {1, 2, 3}. Ask them to: 1. Construct a mapping diagram for this function. 2. List the range of the function.

Discussion Prompt

During Domain-Range Hunt, present a scenario: 'A student council election where each student (input) can vote for only one candidate (output).' Ask students: 'Is this a function? Why or why not? What would make it NOT a function?' Listen for references to the one-output-per-input rule in their reasoning.

Extensions & Scaffolding

• Challenge: Give students a set of ordered pairs with one ambiguous output. Ask them to adjust either the output or the domain so the set becomes a function, then justify their choice.
• Scaffolding: Provide partially completed mapping diagrams with missing arrows or outputs. Ask students to fill in only the missing pieces to complete the function.
• Deeper exploration: Introduce piecewise functions using real-world examples (e.g., postage costs). Have students create mapping diagrams and list domain restrictions for each piece.

Key Vocabulary

Function	A rule that assigns to each input exactly one output. It is a special type of relation.
Relation	A set of ordered pairs, where each input can be associated with one or more outputs.
Domain	The set of all possible input values (often represented by x) for a function.
Range	The set of all possible output values (often represented by y or f(x)) that result from the domain of a function.
Mapping Diagram	A visual representation of a function or relation using two columns of ovals, one for inputs and one for outputs, with arrows showing the connections.