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.

• Year 6
• Mathematics
• 3 activities
• 30 min–45 min

Learning Objectives

1. Identify the constant difference in a linear number sequence.
2. Formulate a rule for a linear pattern using algebraic notation.
3. Calculate the value of any term in a linear sequence given its rule.
4. Compare additive and multiplicative patterns to determine linearity.
5. Analyze tables of values to predict future terms in a sequence.

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.

Prepare & details

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.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

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

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.

Prepare & details

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.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

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

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.

Prepare & details

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.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

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

Teaching This Topic

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.

What to Expect

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.

Watch Out for These Misconceptions

Common Misconception

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.

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Assessment Ideas

Exit Ticket

After Pattern Detectives, give students 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.

Quick Check

During Function Machines, display a table with 'Input' and 'Output' columns 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.

Discussion Prompt

After Gallery Walk, 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?'

Extensions & Scaffolding

• Challenge: Ask students to create their own linear pattern and trade with a partner, who must find the rule and the 15th term.
• Scaffolding: Provide partially filled tables with the first two inputs and outputs completed for students who need a starting point.
• Deeper: Introduce two-step rules like 'multiply by 2 then add 1' and ask students to find the rule and the 100th term.

Key Vocabulary

Sequence	A set of numbers that follow a specific order or pattern.
Term	Each individual number within a sequence.
Linear Pattern	A pattern where the difference between consecutive terms is constant, indicating a steady rate of change.
Rule	A mathematical expression or statement that describes the relationship between the position of a term and its value in a sequence.
Position	The place of a term in a sequence, often represented by 'n' or 'x'.