Understanding Functions

Functions are abstract, so active learning helps students build concrete understanding. Engaging with functions through hands-on simulations and visual representations like graphs makes the one-to-one input-output relationship tangible and easier to grasp.

• Grade 8
• Mathematics
• 3 activities
• 25 min–45 min

Concept Mapping

⏱ 30 min·Pairs

Function Machine Simulation

Students work in pairs to create 'function machines' using envelopes and index cards. One student writes a rule (e.g., 'add 3') on the inside of the envelope and gives inputs on cards. The other student calculates the output and verifies it. They then switch roles.

Prepare & details

Explain how a pattern in a table of values can be described using an algebraic rule.

Facilitation Tip: During the Function Machine Simulation, encourage pairs to explicitly state the rule they are using for their machine and test each other's inputs and outputs.

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

⏱ 45 min·Small Groups

Graphing Real-World Scenarios

Provide students with scenarios like 'cost of buying apples at $0.50 each' or 'distance traveled at 60 km/h'. Students create tables of values, determine the algebraic rule, and then graph the relationship, identifying the slope and y-intercept.

Prepare & details

Identify whether a relationship between two variables is linear based on first differences in a table.

Facilitation Tip: For Graphing Real-World Scenarios, circulate to ensure students are correctly labeling axes and interpreting the slope in the context of the scenario, not just as a number.

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·Small Groups

Is it a Function? Sort

Prepare cards with various relationships represented as tables, graphs, and equations. Students sort these cards into 'function' and 'not a function' categories, justifying their decisions based on the definition of a function.

Prepare & details

Analyze how the rate of change in a pattern connects to the slope of its graph.

Facilitation Tip: During the Is it a Function? Sort, prompt students to explain their reasoning for placing each card, especially focusing on the vertical line test for graphs and unique outputs for tables and equations.

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

Experienced teachers approach functions by emphasizing the core definition: for every input, there is exactly one output. They use multiple representations—tables, graphs, and equations—to show the interconnectedness of these ideas, moving from concrete examples to abstract rules. Avoid simply presenting the vertical line test; ensure students understand *why* it works by connecting it back to the definition.

What to Expect

Students will be able to confidently identify and represent functions using tables, graphs, and simple rules. They will articulate why a given relationship is or isn't a function, and interpret the meaning of rate of change in real-world contexts.

Watch Out for These Misconceptions

Common Misconception

During the Is it a Function? Sort, students may confuse any pattern with the specific rule of a function (one output per input).

What to Teach Instead

Redirect students to look at the input column for tables or the x-values for graphs and ensure no input is repeated with a different output; for equations, check if substituting a single input value yields only one output value.

Common Misconception

During the Graphing Real-World Scenarios activity, students may assume that the graph of a function must always go up.

What to Teach Instead

Ask students to consider scenarios with a negative rate of change, such as a car losing fuel or a price decreasing over time, and to graph these relationships, observing how the line slopes downwards.

Assessment Ideas

Quick Check

During the Is it a Function? Sort, observe student groupings and listen to their justifications as they sort cards to quickly identify misunderstandings about the definition of a function.

Exit Ticket

After Graphing Real-World Scenarios, ask students to write a sentence explaining what the slope of their graph represents in the context of the scenario.

Peer Assessment

During the Function Machine Simulation, have students swap roles and have the 'tester' provide feedback to the 'creator' on the clarity of the function rule and the accuracy of the outputs.

Extensions & Scaffolding

• Challenge: Ask students to create their own real-world scenario that represents a non-linear function and graph it.
• Scaffolding: Provide partially completed tables or graphs for students who are struggling to start.
• Deeper Exploration: Have students research real-world phenomena that are modeled by linear functions and present their findings.