Defining Functions

Active learning transforms abstract function definitions into concrete experiences. Students need to see, touch, and test the one-output-per-input rule to move beyond memorizing words to owning the concept. These activities turn tables, graphs, and verbal rules into hands-on evidence.

• 8th Grade
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

1. Classify relations as functions or non-functions given a set of ordered pairs, a table, a graph, or a mapping diagram.
2. Explain the 'vertical line test' and demonstrate its application to identify functions from graphs.
3. Compare and contrast the characteristics of a function versus a general relation, citing specific examples.
4. Justify why a rule assigning exactly one output to each input is essential for predictable mathematical models.

Stations Rotation

⏱ 30 min·Small Groups

Card Sort: Relations vs. Functions

Prepare cards showing tables, graphs, and mappings: half functions, half not. In small groups, students sort into two piles and justify each choice with evidence from the representation. Groups share one example with the class.

Prepare & details

Differentiate between a relation and a function using various representations.

Facilitation Tip: For Card Sort, model the first two examples aloud so students notice the pattern before working in pairs.

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

⏱ 25 min·Whole Class

Human Graph: Vertical Line Test

Assign students coordinates to stand as graph points. Use a long rope as a 'vertical line' slid across; if it hits multiple points, discuss why it is not a function. Switch roles and graph new relations.

Prepare & details

Explain how to identify a function from a table, graph, or mapping diagram.

Facilitation Tip: During Human Graph, position students physically so they can step aside when a vertical line would hit multiple points, making the test visual and memorable.

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·Pairs

Function Machine

Pairs designate one student as the 'machine' with a secret rule. The other inputs numbers verbally; the machine outputs one value per input. Switch and guess rules, then graph results to verify functions.

Prepare & details

Justify why the concept of a function is fundamental to mathematical modeling.

Facilitation Tip: In Function Machine, vary inputs with decimals and negatives to push students beyond whole numbers and simple patterns.

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

⏱ 40 min·Small Groups

Mapping Diagram Build

Provide input-output cards. Small groups connect inputs to outputs with string on a board, ensuring one per input. Test by tugging strings; multiples break the function rule. Record valid mappings.

Prepare & details

Differentiate between a relation and a function using various representations.

Facilitation Tip: When building Mapping Diagrams, ask students to justify why two arrows from one input means it’s not a function, not just to label it.

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

Teach functions by starting with clear contrasts: show a relation that is a function and one that isn’t, then ask students to articulate the difference in their own words before formalizing the definition. Avoid rushing to the textbook rule. Use real-world examples like taxi fares or amusement park ticket prices to anchor the idea that outputs must be predictable. Research shows students grasp functions better when they experience the breakdown of unpredictability firsthand, so design activities where incorrect outputs feel visibly wrong.

What to Expect

Successful learning looks like students confidently distinguishing functions from non-functions across all representations. You will hear them using precise language like 'each input has exactly one output' and see them applying the vertical line test or checking mapping arrows without prompts.

Watch Out for These Misconceptions

Common Misconception

During Card Sort, watch for students grouping all relations together without checking for multiple outputs per input.

What to Teach Instead

Have students physically separate the sets and then verbally explain why a relation with two outputs for one input cannot be a function, using their sorted cards as evidence.

Common Misconception

During Function Machine, watch for students assuming that repeating outputs disqualify a function.

What to Teach Instead

Ask them to input 3 and 5 into the machine and observe the same output. Then ask them to explain whether the machine is still a function, focusing on the input-output relationship rather than output repetition.

Common Misconception

During Human Graph, watch for students deciding a graph is a function based on smoothness or familiarity.

What to Teach Instead

Ask them to walk through the vertical line test step by step, using their bodies to block lines and confirm whether any vertical line touches the graph more than once.

Assessment Ideas

Quick Check

After Card Sort, provide 5-7 ordered pairs on the board. Ask students to write 'Function' or 'Not a Function' next to each and justify their choice in one sentence, using language from their sorting work.

Exit Ticket

After Mapping Diagram Build, give students a table, a mapping diagram, and a simple graph. Ask them to identify which representation is NOT a function and explain why, referencing the definition of a function they practiced during the activity.

Discussion Prompt

During Function Machine, pose this question: 'Imagine you are designing a system for ordering pizza online. Why is it crucial that the price of a pizza (output) is a function of its size and toppings (inputs)?' Listen for discussions of predictability and customer satisfaction tied to the one-output-per-input rule.

Extensions & Scaffolding

• Challenge students to create a function that produces the same output for every input and defend why it is still a function.
• For struggling students, provide partially filled tables or mapping diagrams to reduce cognitive load while they practice identifying functions.
• Deeper exploration: Have students design a function machine that uses two operations (e.g., multiply by 2 then add 3) and challenge peers to find the rule.

Key Vocabulary

Relation	A set of ordered pairs, where each pair consists of an input and an output value. A relation shows how inputs and outputs are connected.
Function	A special type of relation where each input value is paired with exactly one output value. It follows a specific rule.
Input	The value that is put into a function or relation, often represented by 'x'. Each input should have only one corresponding output in a function.
Output	The value that results from applying a function or relation to an input, often represented by 'y'. In a function, an output can be associated with multiple inputs, but an input cannot be associated with multiple outputs.
Mapping Diagram	A visual representation showing the relationship between sets of inputs and outputs using arrows. It clearly illustrates if each input maps to only one output.