Patterns, Tables and Rules

Overview

In this 60-minute lesson, students generalise growing patterns by connecting visual models, tables, verbal descriptions and algebraic rules. They build on prior work with sequences and substitution, comparing recursive rules (how a pattern grows) with explicit rules (how to find any term).

Learning intentions

• WALT identify and describe patterns using tables.
• WALT distinguish the position number from the term value.
• WALT formulate, test and explain an nth-term rule.
• WALT connect visual, verbal, tabular and algebraic representations.

Success criteria

• I can identify how a pattern changes and generate further terms.
• I can label position number and term value correctly in a table.
• I can write a recursive rule and an explicit algebraic rule.
• I can test my rule and justify why it works for every term.

Curriculum links

• Mathematics and Statistics — Mathsteasers: higher-order thinking questions that challenge advanced learners and deepen understanding.
• Mathematics and Statistics — Mathsteasers / Alignment: connecting challenge tasks with relevant mathematical content.
• Mathematics and Statistics — Mathsteasers / Additional resources for advanced learners: using rich pattern investigations to extend reasoning.
• Key competencies: thinking; using language, symbols and texts; managing self; relating to others.

Lesson structure (60 minutes)

1. 0–7 min · Hook and notice. Display a visual pattern that grows by adding three tiles each time using the visual pattern hook. Ask, “How could we predict the 20th figure without drawing all 20?” Students think independently, share observations with a partner, and describe what changes and what stays constant.

2. 7–17 min · Connect representations. Model a four-column table with position number, term value, change and rule using the representation slides. Explicitly distinguish “position 3” from “term 3”, then demonstrate a recursive description such as “start at 4 and add 3” and an explicit rule such as (T(n)=3n+1). Students complete a partially filled example and explain which number represents position and which represents the term.

3. 17–30 min · Match and sort. In pairs, students match visual patterns, tables, verbal rules and algebraic rules from the pattern representation matching task. They must test at least two matches by calculating terms, then use think-pair-share to defend one match. Circulate and ask, “What evidence proves these descriptions represent the same pattern?” Address common errors, including using the term value as (n) or matching only the first term.

4. 30–45 min · Investigate a growing pattern. Distribute matchsticks or tiles and direct pairs to investigate one pattern from the growing pattern investigation instructions. Students build or sketch the first four figures, complete a table, describe the recursive rule, and find an explicit nth-term rule. They test their rule for a later term using a calculator or spreadsheet, recording how they know it works. Provide graph paper or a table template where needed.

5. 45–54 min · Generality and comparison. Invite selected pairs to present different rules for the same pattern. Students compare methods, including finding the constant difference, and explain why a rule works for every position rather than only the first few. Introduce the possibility that some patterns are non-linear: students who finish early examine a pattern whose differences are not constant and describe what further evidence would be needed.

6. 54–60 min · Independent assessment and plenary. Students complete the “find and justify the nth-term rule” task on the independent nth-term task. They state the rule, test it with two terms, and write one sentence explaining its generality. Finish by asking students to share one difference between a recursive rule and an explicit rule.

Resources

• the Patterns, Tables and Rules slide deck
• the pattern practice and assessment worksheet
• Matchsticks or linking tiles
• Graph paper
• Calculators or spreadsheet tool
• Pattern cards for teacher-created matching sets
• Whiteboards and pens
• Table templates and partially completed examples
• Coloured pencils

Assessment

• Listen during the hook and matching sort for accurate use of “position”, “term”, “difference” and “rule”.
• Check tables and questioning during the investigation, particularly whether students test rules beyond the examples used to create them.
• Collect the independent task and assess whether students identify the pattern, generate terms, formulate an nth-term rule, substitute correctly and justify why the rule is general.

Differentiation

• Support learners with physical tiles, a completed first row, highlighted column headings, a table template and sentence starters: “The pattern increases by…”, “The term at position (n) is…”, and “I know this works because…”.
• Offer dyslexia-friendly reading options: uncluttered worksheets, clear sans-serif font, generous spacing, symbols alongside key words, oral instructions, and the option to explain reasoning verbally before writing.
• Pair students strategically and allow calculator or spreadsheet checking. Model one example slowly, then remove scaffolds as confidence increases.
• Extend advanced learners with non-linear patterns, such as patterns with increasing differences, or ask them to prove why two different-looking rules are equivalent for all positive integer positions. They may also design a pattern for another pair to solve.