Sequence Foundations

Overview

In this first lesson of “Mastering Series and Sequences”, students build the core idea of a sequence as an ordered pattern with a rule. They begin with real contexts and then classify sequences as arithmetic or geometric using structure and reasoning.

Learning intentions

WALT: Understand what sequences are, and identify key features that distinguish arithmetic and geometric sequences.

Success criteria

• I can define a sequence as an ordered list of terms generated by a rule.
• I can give an example of a sequence and describe how each term is produced.
• I can classify a sequence as arithmetic, geometric, or neither, with a justification.
• I can represent a sequence using words (and, when appropriate, algebraic notation).

Curriculum links

• Algebra — AS91258: Apply sequences and series in solving problems (beginning with selecting and using methods to model patterns).
• NZ Curriculum Mathematics (Year 12-aligned algebraic thinking): pattern recognition, generalisation, and using mathematical representations to communicate reasoning.
• Mathematical competencies: using mathematical ideas to explain patterns, and communicating thinking clearly using appropriate representations.

Lesson structure (45 minutes)

1. 0–5 min · Hook (pattern or sequence?). Teacher displays three short term sets and asks “Is this a sequence? Why/why not?” Students quickly vote and justify to a partner using “ordered terms” and “rule/pattern”.

2. 5–15 min · Direct teaching (what sequences are). Teacher models the idea of terms, positions (term numbers), and a generating rule; gives one arithmetic and one geometric example and highlights the constant difference/ratio. Students complete a quick guided worksheet: label terms, state the rule in words, and identify the type.

3. 15–25 min · Interactive classification stations (relational thinking). Teacher sets up 3 mini-stations (or one whole-class set with groups of 5) with 6–9 examples on cards: arithmetic, geometric, and “neither”. Students work in groups to classify each card and write a one-sentence justification using “difference is constant” or “ratio is constant”.

4. 25–35 min · Whole-class discussion (justify & correct). Teacher selects two “tricky” items where students often misclassify (e.g., ratio changes sign, difference changes, or alternating patterns). Students contribute reasoning steps; teacher prompts “What stays the same? What changes? What would the next term be and why?”

5. 35–43 min · Guided practice (from rule to structure). Teacher gives a short practice set of 3 prompts: (a) identify the type, (b) predict the next term, (c) write the rule in words. Students work independently, then check with a partner; teacher circulates for misconceptions (constant difference vs constant ratio).

6. 43–45 min · Exit ticket (assessment for learning). Teacher collects exit tickets with 2 questions: define “sequence” in one sentence and classify one new example with justification. Students submit, showing their reasoning in writing.

Resources

• Printed card set of example sequences (arithmetic, geometric, neither)
• Guided notes sheet (definition, terms, rule, classification checklist)
• Practice worksheet (3 short questions)
• Exit ticket slips
• Timer and whiteboard/markers
• Student notebooks or one-page sequence template

Assessment

• Formative checks during hook voting and station justifications (teacher listens for correct language: ordered, rule, term pattern).
• Teacher observation during guided practice for misconceptions (e.g., confusing addition with multiplication, or assuming constant ratio when it is not).
• Exit ticket: definition of sequence and classification justification.

Differentiation

• Support (guided notes): Provide a “Classification checklist” with sentence starters:
• “It is arithmetic because the difference between consecutive terms is…”
• “It is geometric because the ratio between consecutive terms is…”
• “It is neither because…” Also include a worked example on the notes sheet.
• Support (structured practice): For students needing extra scaffolding, allow them to fill a table of terms and compute differences/ratios using calculators if needed, focusing on conceptual correctness.
• Extension (advanced learners): Give a challenge sequence that is not purely arithmetic/geometric but has a detectable rule (e.g., squares, or an alternating arithmetic rule). Students must (1) define the rule, (2) generate the next two terms, and (3) explain why it is not arithmetic/geometric using a counter-example to “constant difference/ratio”.
• EAL/SEN considerations: Encourage use of visual supports (difference/ratio tables) and provide word banks for describing patterns (constant, consecutive, ratio, difference, ordered). Pair learners strategically so they can rehearse oral reasoning before writing.

Extension (optional)

• Create and present an original sequence: using a clear rule, students write the first 5 terms and a one-sentence explanation of how to generate them; they can share in a short “Gallery Walk” after the exit ticket if time allows.