Introduction to Functions

Functions can feel abstract to middle schoolers, so active tasks ground the concept in concrete actions like sorting, building, and graphing. Moving, discussing, and testing ideas helps students internalize that a function’s rule must give exactly one output per input, not sometimes two.

• Grade 7
• Mathematics
• 4 activities
• 20 min–35 min

Learning Objectives

1. Classify relations as functions or non-functions based on the one-to-one or many-to-one input-output rule.
2. Identify the independent and dependent variables in given real-world scenarios and explain their relationship.
3. Analyze mapping diagrams, tables of values, and coordinate graphs to determine if they represent a function.
4. Create a real-world scenario that can be modeled by a function, defining the input, output, and rule.

Concept Mapping

⏱ 30 min·Small Groups

Card Sort: Relations vs Functions

Prepare cards with mappings, tables, and graphs: some functions, some not. In small groups, students sort into categories, justify choices with vertical line test sketches, then share one non-function example with the class.

Prepare & details

Explain the difference between a relation and a function.

Facilitation Tip: During Card Sort: Relations vs Functions, circulate and listen for students’ reasoning about why a pairing with two outputs fails the function test, then pause the class to share key arguments.

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

⏱ 25 min·Pairs

Function Machine Build

Pairs create a 'function machine' from a box with input slot, rule inside (multiply by 2, add 3), and output chute. They input numbers, predict outputs, then swap machines to test and graph results.

Prepare & details

Analyze real-world examples that can be modeled as functions.

Facilitation Tip: While students build the Function Machine, ask them to test their machine with three inputs and record outputs in a table before claiming it is a function.

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

Real-World Input-Output Hunt

Individually, students list 5 daily scenarios (e.g., pizzas ordered to total cost), create tables labeling variables, then pairs verify if functions and graph one on grid paper.

Prepare & details

Differentiate between independent and dependent variables in a functional relationship.

Facilitation Tip: For Real-World Input-Output Hunt, pair students to debate labels for inputs and outputs before allowing them to move to the next scenario.

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

⏱ 20 min·Whole Class

Graphing Relay: Vertical Line Test

Whole class lines up by teams. Teacher projects graphs; first student draws vertical line to test, tags next if function. Rotate until all tested, discuss patterns.

Prepare & details

Explain the difference between a relation and a function.

Facilitation Tip: In Graphing Relay: Vertical Line Test, assign roles so every student sketches one line and explains whether it passes or fails, then rotate papers for peer verification.

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 hands-on tasks before formal definitions. A function’s rule is only reliable if it consistently maps each input to one output, so sorting physical cards and building machines make that reliability visible. Avoid rushing to graphing; let students discover the vertical line test’s purpose through trial, error, and peer critique.

What to Expect

Students will confidently distinguish functions from general relations, identify inputs and outputs in real contexts, and use the vertical line test with clear reasoning. They will support claims with evidence from tables, diagrams, and graphs and explain their thinking to peers.

Watch Out for These Misconceptions

Common Misconception

During Card Sort: Relations vs Functions, watch for students who accept every relation as a function. Redirect by asking them to test each card’s mapping with the phrase 'Does this input have only one output? If not, it’s not a function.'

What to Teach Instead

Prompt pairs to debate each card that shows multiple outputs for one input, using the vertical line sketches on their desks to visualize the failure.

Common Misconception

During Graphing Relay: Vertical Line Test, watch for students who assume all functions must increase. Redirect by handing them a step-function graph and asking them to explain why outputs can stay the same or drop without breaking the function rule.

What to Teach Instead

Have students test diverse graphs in small groups, including constant and decreasing segments, and share counterexamples to the 'must increase' claim.

Common Misconception

During Real-World Input-Output Hunt, watch for students who always label the horizontal axis as the input. Redirect by asking them to read the scenario aloud and decide which quantity depends on the other before labeling.

What to Teach Instead

Require pairs to justify their input-output labels in writing before moving to the next scenario, using the scenario’s wording as evidence.

Assessment Ideas

Quick Check

After Card Sort: Relations vs Functions, show a set of ordered pairs on the board and ask students to circle the pair that shows this relation is NOT a function and explain why in one sentence.

Exit Ticket

During Function Machine Build, have students write one real-world function example on their machine’s output table, labeling the input, output, and the rule connecting them before leaving class.

Discussion Prompt

After Graphing Relay: Vertical Line Test, present two mapping diagrams and ask students: 'Which diagram represents a function? Explain your reasoning using the terms input and output.' Circulate to listen for precise use of these terms.

Extensions & Scaffolding

• Challenge students to create a step-function machine that models a real pricing scenario, like a parking garage, and write its rule algebraically.
• For students who struggle, provide partially completed input-output tables or mapping diagrams with one blank per table to focus their attention on the single-output rule.
• Deeper exploration: Ask students to design a non-function relation that mimics a function on most inputs but breaks at one critical point, then graph it and explain why it fails the vertical line test.

Key Vocabulary

Function	A rule that assigns each input value to exactly one output value. It's a special type of relation.
Relation	A set of ordered pairs, which can show any connection between inputs and outputs, not necessarily a unique one.
Input	The value that is put into a function or relation, often represented by 'x'. Each input must have only one output.
Output	The value that results from applying the function's rule to an input, often represented by 'y'. Each output corresponds to a specific input.
Independent Variable	The variable that can be changed or controlled; its value does not depend on another variable. It is typically the input.
Dependent Variable	The variable that is measured or observed; its value depends on the independent variable. It is typically the output.