Creating Rules for Patterns

Students learn best when they physically engage with pattern growth, not just see static examples. Manipulating objects helps them feel the difference between adding a fixed number and multiplying by a factor, making abstract rules concrete and memorable.

• 6th Class
• Mathematical Mastery and Real World Reasoning
• 4 activities
• 25 min–45 min

Learning Objectives

1. Formulate the rule for a given numerical pattern using a variable.
2. Compare and contrast additive and multiplicative patterns, explaining the core difference in their growth.
3. Design a novel pattern and clearly articulate its rule in both words and algebraic notation.
4. Evaluate the clarity and efficiency of different rule descriptions for a given pattern.

Peer Teaching

⏱ 30 min·Pairs

Manipulative Build: Additive vs Multiplicative

Provide linking cubes or tiles. Pairs create an additive pattern (add 3 each time) then extend it to a multiplicative one (multiply by 2). They write rules in words and symbols, swap with another pair to verify. Discuss differences in growth speed.

Prepare & details

Explain the difference between an additive and a multiplicative pattern.

Facilitation Tip: In Variable Hunt, encourage partners to switch roles after each turn so both practice substituting values and articulating predictions.

Setup: Presentation area at front, or multiple teaching stations

Materials: Topic assignment cards, Lesson planning template, Peer feedback form, Visual aid supplies

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

Peer Teaching

⏱ 45 min·Small Groups

Pattern Design Challenge: Real-World Rules

Small groups design a pattern based on scenarios like rabbit population growth (multiplicative) or fence post spacing (additive). Formulate rules with variables, draw visuals, and present. Class votes on clearest rule.

Prepare & details

Design a pattern and formulate its rule using a variable.

Setup: Presentation area at front, or multiple teaching stations

Materials: Topic assignment cards, Lesson planning template, Peer feedback form, Visual aid supplies

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

Peer Teaching

⏱ 35 min·Small Groups

Rule Evaluation Stations: Critique and Refine

Set up stations with sample patterns and flawed rules. Groups rotate, identify errors, rewrite rules, and test with numbers. Whole class shares one strong correction.

Prepare & details

Evaluate the effectiveness of different ways to describe a pattern's rule.

Setup: Presentation area at front, or multiple teaching stations

Materials: Topic assignment cards, Lesson planning template, Peer feedback form, Visual aid supplies

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

Peer Teaching

⏱ 25 min·Pairs

Variable Hunt: Partner Prediction Game

Partners take turns giving a pattern rule like 3n + 1, opponent predicts next terms and justifies. Switch roles, then create original rules to challenge each other.

Prepare & details

Explain the difference between an additive and a multiplicative pattern.

Setup: Presentation area at front, or multiple teaching stations

Materials: Topic assignment cards, Lesson planning template, Peer feedback form, Visual aid supplies

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

Teaching This Topic

Experienced teachers start with concrete examples before moving to symbols, using manipulatives to build trust in the rule. They avoid rushing to algebra, instead scaffolding from word descriptions to variables. Research shows that when students create their own patterns and rules, they develop deeper ownership and understanding of variables as placeholders.

What to Expect

By the end of these activities, students should confidently distinguish additive from multiplicative patterns and express rules clearly using both words and algebraic notation. They will also critique descriptions for precision and efficiency.

Watch Out for These Misconceptions

Common Misconception

During Manipulative Build, watch for students who assume all patterns grow by adding the same number. Redirect them by asking, 'What happens if you double instead? Did the rule change?'

What to Teach Instead

Use the manipulative sets to build both additive and multiplicative sequences side by side. Ask groups to compare the visual growth and adjust their initial assumption based on the comparison.

Common Misconception

During Variable Hunt, watch for students who treat 'n' as a fixed placeholder unrelated to the sequence. Redirect them by asking, 'If n is 1, what is the first term? How about n is 2?'

What to Teach Instead

Have partners substitute small values into their partner's rule, recording each result. Discuss how n acts as a counter for each term's position.

Common Misconception

During Rule Evaluation Stations, watch for students who prefer word descriptions over algebraic rules without justification. Redirect them by asking, 'Which rule is faster to use for the 100th term? Why?'

What to Teach Instead

Provide examples where word descriptions become cumbersome, like 'add 3 to the previous term 99 times.' Ask students to rewrite these as algebraic rules and compare efficiency.

Assessment Ideas

Quick Check

After Manipulative Build, present two sequences: 4, 8, 12, 16... and 2, 4, 8, 16... Ask students to identify each as additive or multiplicative and write the rule for each using words.

Exit Ticket

During Pattern Design Challenge, ask students to write the algebraic rule for a given pattern, then create their own simple additive pattern and write its rule before leaving class.

Discussion Prompt

After Rule Evaluation Stations, display two different rules for the same pattern. Ask students, 'Which rule is clearer and why? How could we improve the other rule?' Have them discuss in pairs before sharing with the class.

Extensions & Scaffolding

• Challenge students to design a pattern with both additive and multiplicative parts, such as starting at 2 and then doubling each step after the second term.
• For students who struggle, provide partially completed sequences with missing terms to fill in before writing the rule.
• Deeper exploration: Ask students to research real-world patterns, like Fibonacci in nature, and write rules explaining why the sequence occurs.

Key Vocabulary

Pattern	A sequence of numbers or objects that follows a specific, predictable rule.
Rule	The instruction or relationship that determines how each term in a pattern is generated from the previous term or its position.
Variable	A symbol, usually a letter like 'n' or 'x', used to represent an unknown or changing number in a pattern's rule.
Additive Pattern	A pattern where a constant amount is added to get from one term to the next.
Multiplicative Pattern	A pattern where each term is multiplied by a constant amount to get the next term.