Mathematics · Grade 8

Active learning ideas

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.

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

• 25–45 min
• Pairs → Whole Class
• 3 activities

Activity 01

• 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.

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.

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

Activity 02

• 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.

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.

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

Activity 03

• 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.

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.

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

A few notes on teaching this unit

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.

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.

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Watch Out for These Misconceptions

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

  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.

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

  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.

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Methods used in this brief

• Concept Mapping