Mathematics · Year 6

Active learning ideas

Identifying Linear Patterns and Rules

Active learning helps students see how input-output rules connect to real patterns. When students move, talk, and test ideas in small groups, they move beyond guessing to proving relationships between numbers. This builds the foundation for algebraic reasoning they will use in later years.

• ACARA Content Descriptions
• AC9M6A01

• 30–45 min
• Pairs → Whole Class
• 3 activities

Activity 01

• Inquiry Circle
• 45 min · Small Groups

Inquiry Circle: Pattern Detectives

Groups are given a series of 'growing patterns' made of matchsticks. They must create a table of values, identify the rule (e.g., multiply by 2, add 1), and predict the 10th and 100th term.

How can we predict the hundredth term in a sequence without calculating every step?

Facilitation Tip

During Pattern Detectives, circulate and ask each group, 'How did you find the jump between numbers?' to push thinking toward constant differences.

What to look for

Provide students with the sequence: 3, 7, 11, 15. Ask them to: 1. Identify the type of pattern (additive or multiplicative). 2. State the rule for the sequence. 3. Calculate the 10th term.

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Activity 02

• Stations Rotation
• 40 min · Small Groups

Stations Rotation: Function Machines

Students rotate through stations where one student acts as the 'machine' with a secret rule. Others provide an 'input' number, and the machine provides the 'output' until the rule is guessed.

What is the difference between an additive pattern and a multiplicative pattern?

Facilitation Tip

Set up Function Machines with input-output pairs that require both addition and multiplication so students practice both types of rules.

What to look for

Display a table with two columns, 'Input' and 'Output', showing values like (1, 5), (2, 10), (3, 15). Ask students to write down the rule connecting Input to Output and predict the Output for an Input of 7.

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Activity 03

• Gallery Walk
• 30 min · Whole Class

Gallery Walk: Visualizing Rules

Students create posters showing a pattern, its table of values, and its algebraic rule. They walk around the room and try to match rules to patterns created by their peers.

How can a table of values help us identify a hidden rule?

Facilitation Tip

For the Gallery Walk, provide sticky notes and ask students to write 'What rule could fit all the points?' on each poster they visit.

What to look for

Present two sequences: Sequence A (2, 4, 6, 8) and Sequence B (2, 4, 8, 16). Ask students: 'Which sequence has a linear rule? How do you know? What is the rule for Sequence A? Can you find a rule for Sequence B?'

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

A few notes on teaching this unit

Teachers should model how to organize data in a two-column table before students attempt to find rules. Avoid rushing to the answer; instead, ask students to test their own rules by predicting the 20th term. Research shows that students who construct their own tables and test predictions develop stronger algebraic thinking than those who follow a pre-made formula.

Students will explain their rules clearly, use tables to organize data, and recognize linear patterns by identifying constant differences. They will also distinguish between patterns that grow by addition and those that require multiplication or other operations.

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Watch Out for These Misconceptions

• During Pattern Detectives, watch for students who only describe the recursive rule (add 3 each time) but cannot find the functional rule (multiply by 3). Redirect by asking, 'If the first term is 3, what would the 0th term be?' to show the need for a rule that connects term number to value.

  During Function Machines, display a table with 'Term Number' and 'Value' columns. Ask students to fill in the values for terms 1 through 5, then challenge them to find a rule that connects 'Term Number' to 'Value' for any term.

• During Gallery Walk, watch for students who assume all patterns are linear and try to force a constant difference where none exists.

  During Gallery Walk, place at least one non-linear poster (like square numbers) among linear ones. Ask students to explain why some posters have constant differences and others do not.

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Methods used in this brief

• Inquiry Circle
• Stations Rotation
• Gallery Walk