Sequence Foundations

Overview

In this first lesson of Sequences and Series Mastery, students develop the language and notation needed to describe sequences as ordered lists. They connect familiar patterns to explicit and recursive rules, preparing for later work with arithmetic and geometric sequences and series.

Learning intentions

Students will describe, represent and generate sequences using explicit and recursive rules, and explain the relationship between term position and term value.

Success criteria

• I can explain what a sequence and a term are.
• I can calculate a specified term using (u_n).
• I can write an explicit rule for a simple pattern.
• I can generate terms recursively and explain the difference between the two rule types.

Curriculum links

• Sequences and series — modelling and solving problems using arithmetic and geometric sequences and series.
• Algebraic relationships — representing relationships between quantities algebraically and graphically to solve problems and make predictions.
• Number and patterns — recognising, describing and extending numerical patterns.
• Mathematical reasoning — explaining why a rule works and whether a rule is uniquely determined.

Lesson structure (60 minutes)

1. 0–7 min · Diagnostic warm-up. Open with the opening pattern challenge and display the sequences (4, 7, 10, 13,\ldots), (2, 4, 8, 16,\ldots), and (1, 4, 9, 16,\ldots). Students use mini-whiteboards to write the next two terms, describe the pattern and identify what information is still needed to define the sequence completely. Quickly scan responses to identify misconceptions about term number and common difference.

2. 7–18 min · Direct teaching and modelling. Use the notation and modelling slides to define a sequence as an ordered list, with the first term written (u_1), the second (u_2), and the (n)th term (u_n). Model (u_n=3n+1), finding (u_1), (u_4) and (u_{10}), then model the recursive definition (u_1=4,\ u_{n+1}=u_n+3). Students annotate the sequence notation worksheet and answer brief check questions on their whiteboards.

3. 18–28 min · Guided comparison. Display the same sequence in both forms: (5, 8, 11, 14,\ldots), (u_n=3n+2), and (u_1=5,\ u_{n+1}=u_n+3). Think aloud while comparing what each rule tells us and how the fourth term is obtained. Students complete a two-column comparison on the worksheet, then pair-check: “An explicit rule tells me…” and “A recursive rule requires…”. Address the common error of writing a recursive rule without a starting value.

4. 28–42 min · Pattern investigation. In pairs, students investigate four sequences on the pattern investigation questions: a constant-increase sequence, a constant-multiplier sequence, a square-number sequence and a less familiar sequence with missing terms. They record the first six terms, term positions and an explicit rule where possible, then write a recursive definition. Students plot term number against term value using the digital plotting instructions in Desmos or GeoGebra, or use the spreadsheet template. Circulate and question: “How do you know your rule works for every displayed term?”

5. 42–53 min · Think-pair-share and reasoning. Present the sequence (2, 5, 8, 11,\ldots) and ask whether the first four terms determine one unique rule. Pairs propose and test two possible rules, using the initial terms to justify their reasoning. Share selected responses, reinforcing that a finite list of terms can fit more than one rule unless a pattern family or context is specified. Extension students may compare (u_n=3n-1) with an alternative rule that agrees initially but diverges later.

6. 53–60 min · Exit ticket and review. Use the plenary and exit-ticket prompt. Students independently complete: (a) define a sequence; (b) for (u_n=4n-1), find (u_6); (c) write a recursive rule for (7, 10, 13, 16,\ldots); and (d) state one difference between explicit and recursive notation. Collect responses to group students for the next lesson.

Resources

• the complete sequence foundations slide deck
• the sequence notation and pattern investigation worksheet
• Mini-whiteboards and pens
• Desmos or GeoGebra
• Spreadsheet template for term plotting
• Calculators
• NSW syllabus and HSC-style sequence questions for teacher reference

Assessment

• Check warm-up whiteboards for correct term ordering, pattern descriptions and misconceptions.
• Question pairs during modelling and investigation, listening for accurate use of “term”, “position”, “explicit” and “recursive”.
• Use the exit ticket to assess notation, substitution, recursive generation and conceptual distinction; reteach missing prerequisite skills before Lesson 2.

Differentiation

• Support students with a term table labelled (n) and (u_n), pattern prompts such as “add ___ each time”, and partially completed rules on the worksheet.
• Provide worked examples, colour-coded term positions and sentence starters: “The first term is…”, “To find the next term, I…”.
• Allow students to work with a partner and use a calculator or spreadsheet to reduce arithmetic load while maintaining the algebraic reasoning.
• Challenge confident students to compare multiple rules for the same initial terms and justify whether a rule is unique, including testing their rules beyond the displayed terms.
• For EAL/D and students requiring additional support, pre-teach the words “ordered”, “term”, “position”, “explicit” and “recursive” with symbols and repeated oral rehearsal.