Introduction to Functions and Their Representations

Active learning helps students grasp abstract ideas like limits by moving beyond symbols into visual, hands-on experiences. When students manipulate graphs, tables, and equations together, they build mental models that persist beyond symbolic manipulation. This topic demands spatial reasoning and pattern recognition, which are strengthened through collaborative problem-solving and movement-based activities.

• 12th Grade
• Mathematics
• 3 activities
• 20 min–30 min

Learning Objectives

1. Compare and contrast relations and functions, identifying the defining characteristic of a function across graphical, algebraic, and tabular representations.
2. Analyze how restrictions on the domain and range of a function affect its graph and potential real-world applications.
3. Construct a function from a given real-world scenario, representing it accurately using algebraic notation and a graph.
4. Identify the domain and range of a function given its graphical or algebraic representation.

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: The Mystery of 0/0

Students receive three different rational functions that all result in 0/0 at x=2. In pairs, they use tables and graphs to determine if the limit exists and why the outcomes differ. They then share their findings with the class to categorize types of removable discontinuities.

Prepare & details

Differentiate between a relation and a function using various representations.

Facilitation Tip: During The Mystery of 0/0, pause after students share their initial thoughts and ask them to sketch what they think is happening at the hole on the graph before revealing the correct behavior.

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

Gallery Walk

⏱ 30 min·Small Groups

Gallery Walk: Limit Notation and Graphs

Stations around the room display complex graphs with various jumps, holes, and asymptotes. Small groups move between stations to write the formal limit notation for specific x-values and infinity. They leave sticky notes with justifications for their answers for the next group to review.

Prepare & details

Analyze how domain and range restrictions impact the behavior of a function.

Facilitation Tip: Set a timer during the Gallery Walk so students have time to analyze each station fully, and remind them to jot down questions on sticky notes to post near challenging graphs.

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

Inquiry Circle

⏱ 25 min·Small Groups

Inquiry Circle: End Behavior Race

Groups compete to match polynomial and rational functions with their corresponding horizontal asymptotes. They must use algebraic manipulation to prove their matches. The first group to correctly justify all pairings wins.

Prepare & details

Construct a function from a real-world scenario and represent it graphically.

Facilitation Tip: In the End Behavior Race, assign roles (recorder, grapher, timer) so every student contributes visibly and the group stays coordinated during the quick-paced activity.

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

Teachers should emphasize that limits are about proximity, not equality, and use analogies like approaching a doorway without entering it. Avoid rushing to formal epsilon-delta definitions too early—build intuition first. Research shows that students benefit from repeated exposure to limits in different contexts (tables, graphs, equations) over several lessons, not in a single block.

What to Expect

Students will confidently explain what a limit represents and justify their conclusions using multiple representations. They will identify when limits exist or do not exist, describe end behavior, and connect graphical, numerical, and algebraic evidence. Discussions will focus on precision of language and clarity of reasoning.

Watch Out for These Misconceptions

Common Misconception

During The Mystery of 0/0, watch for students who assume that because the function is undefined at a point, the limit cannot exist.

What to Teach Instead

Use the 'zoom-in' feature on the graphing calculator to show that as x approaches the point, the y-values get closer and closer to a specific number, even if the function skips that number at the point itself.

Common Misconception

During End Behavior Race, listen for students who say a function 'reaches' infinity or that infinity is a big number.

What to Teach Instead

Have peers compare growth rates by examining tables and graphs, pointing out that infinity describes unending growth, not a destination, and that different functions grow at different speeds.

Assessment Ideas

Quick Check

After Gallery Walk, display three new representations (table, graph, equation) and ask students to determine if each represents a function and justify their answer using the vertical line test or definition of a function.

Discussion Prompt

After The Mystery of 0/0, present the scenario 'the relationship between a student's height and their age' and facilitate a class discussion on whether it is a function, its domain and range, and how real-world restrictions might apply.

Exit Ticket

After End Behavior Race, give each student a graph of a relation and ask them to write the domain and range, state whether it is a function, and provide one sentence of justification.

Extensions & Scaffolding

• Challenge early finishers to create their own function with a removable discontinuity and explain its limit behavior using a graph, table, and equation.
• For students who struggle, provide a partially completed table or graph where they fill in missing values to determine the limit.
• Give extra time groups a limit scenario involving a piecewise function and ask them to graph it and justify the limit at the boundary point.

Key Vocabulary

Function	A relation where each input value (from the domain) corresponds to exactly one output value (in the range).
Domain	The set of all possible input values for which a function is defined.
Range	The set of all possible output values that a function can produce.
Relation	A set of ordered pairs, where each input may correspond to one or more output values.
Vertical Line Test	A graphical test used to determine if a curve represents a function; if any vertical line intersects the graph more than once, it is not a function.