Mathematics · Grade 7

Active learning ideas

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.

• Ontario Curriculum Expectations
• 8.F.A.1

• 20–35 min
• Pairs → Whole Class
• 4 activities

Activity 01

• 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.

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.

What to look for

Provide students with a set of ordered pairs (e.g., {(1, 5), (2, 7), (3, 9), (1, 6)}). Ask them to circle the pair that shows this relation is NOT a function and explain why in one sentence.

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

Activity 02

• 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.

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.

What to look for

On an index card, ask students to write one real-world example of a function and identify the input, output, and the rule connecting them. For example: Input = number of hours worked, Output = total pay, Rule = hourly wage times hours worked.

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

Activity 03

• 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.

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.

What to look for

Present two mapping diagrams: Diagram A shows each student assigned to one homeroom. Diagram B shows each student assigned to multiple clubs. Ask students: 'Which diagram represents a function? Explain your reasoning using the terms input and output.'

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

Activity 04

• 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.

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.

What to look for

Provide students with a set of ordered pairs (e.g., {(1, 5), (2, 7), (3, 9), (1, 6)}). Ask them to circle the pair that shows this relation is NOT a function and explain why in one sentence.

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

A few notes on teaching this unit

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.

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

• 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.'

  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.

• 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.

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

• 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.

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

---

Methods used in this brief

• Concept Mapping