Classifying Number Sequences

Overview

Students investigate how first differences, second differences and common ratios reveal whether a numerical sequence is linear, quadratic or exponential. The lesson builds on prior work with patterns, substitution and simple algebraic rules, while emphasising that a short finite sequence may fit more than one possible rule.

Learning intentions

• WALT use first differences, second differences and common ratios to classify sequences.
• WALT justify a classification using accurate mathematical language and working.
• WALT write or verify a suitable nth-term rule for straightforward examples.
• WALT recognise that a short sequence does not always uniquely determine its rule.

Success criteria

• I can calculate and interpret first and second differences.
• I can test for a constant ratio.
• I can explain why a sequence is linear, quadratic or exponential.
• I can write or check an nth-term rule for a familiar sequence.

Curriculum links

• Algebra and patterns: recognise sequences governed by rules, identify patterns, and represent or generalise them algebraically.
• Mathematical reasoning: use appropriate methods, concepts, terms and representations; communicate a logical chain of reasoning.
• Mathematical methods in context: apply number and algebraic processes accurately and explain assumptions or limitations.
• NZ Curriculum capabilities: thinking; using language, symbols and texts; managing self; participating and contributing.

Prior knowledge: Students should be able to continue simple sequences, subtract consecutive terms, multiply or divide by a constant, substitute values into an algebraic rule, and use index notation.

Key vocabulary: sequence, term, position, common difference, first difference, second difference, common ratio, linear, quadratic, exponential, nth term, rule, classify, justify.

Lesson structure (60 minutes)

1. 0–5 min · Hook and diagnostic. Display the sequences 4, 7, 10, 13 and 3, 6, 12, 24 using the hook and diagnostic slides. Ask, “What is changing, and how could you prove it?” Students individually write the next two terms and one observation, then compare with a partner. Quickly identify subtraction and multiplication strategies.

2. 5–15 min · Explicit teaching and modelling. Use the classification flowchart and worked examples to model the following.

• Linear: 5, 9, 13, 17 has constant first difference +4, so a linear rule is possible. Since the first term is 5, the rule is (T_n=4n+1).
• Quadratic: 2, 6, 12, 20 has first differences 4, 6, 8 and constant second difference +2, so it is quadratic. Students verify (T_n=n^2+n).
• Exponential: 3, 6, 12, 24 has common ratio 2, so it is exponential: (T_n=3(2^{n-1})).

Stress the classification test: constant first differences indicate linear behaviour; constant second differences indicate quadratic behaviour; a constant ratio indicates exponential behaviour. Students annotate the examples and explain the evidence to a partner.

3. 15–22 min · Guided comparison. Model a checking routine: write the terms in a row, calculate first differences, calculate second differences if needed, then test ratios where appropriate. Use 7, 11, 15, 19 and 2, 5, 10, 17. Students complete the difference tables on the guided examples and classification table and hold up L, Q or “not enough information”. Discuss that “not constant first differences” does not automatically mean exponential.

4. 22–42 min · Pair investigation. Place students in pairs, with one pair of students working as an observer/checker if needed. Distribute the sequence investigation and classification table. Pairs classify eight straightforward sequences, recording first differences, second differences or ratios, a justified conclusion, and an nth-term rule where requested. Include:

• 8, 12, 16, 20
• 1, 4, 9, 16
• 5, 10, 20, 40
• 20, 17, 14, 11
• 2, 5, 10, 17
• 81, 27, 9, 3
• 6, 10, 14, 18
• 3, 7, 13, 21

Students must use a sentence such as, “The sequence is quadratic because its second differences are constant at ___.” Circulate, question reasoning and check that students distinguish a difference from a ratio.

5. 42–50 min · Discuss and challenge assumptions. Review selected answers using the investigation discussion and misconception slides. Present the short sequence 2, 4, 6, 8 and ask whether it must continue with 10. Explain that many rules can produce the same first few terms; classification is a useful model supported by the available evidence, not proof of a unique rule from a short list. Students suggest another possible continuation and explain what extra information would help.

6. 50–56 min · Independent reasoning check. Students complete the final two worksheet questions without partner support: classify 4, 9, 16, 25 and write a rule; then verify whether (T_n=2n+3) generates 5, 7, 9, 11. Teacher checks responses and conferences with students who have incomplete difference tables.

7. 56–60 min · Exit ticket and recap. Students answer on the final section of the independent check and exit ticket:

8. Classify 6, 10, 14, 18 and state the evidence.

9. Classify 2, 6, 18, 54 and write an nth-term rule.

10. Why might four terms not uniquely determine a sequence rule?

Answers: 1. Linear; first differences are +4; (T_n=4n+2). 2. Exponential; ratio is 3; (T_n=2(3^{n-1})). 3. Different rules can agree for the first few terms, so more information or a stated model is needed.

Resources

• the complete sequence classification slide deck
• the sequence investigation worksheet
• Whiteboard and markers
• Calculators, if routinely used by the class
• Exercise books and pens
• Mini-whiteboards or paper for diagnostic responses
• Visible timer

Assessment

• Listen for precise explanations during the hook, guided comparison and pair investigation; ask, “What stayed constant?” and “How does that prove your classification?”
• Check worksheet difference tables, ratio calculations, rule verification and use of units-free mathematical notation.
• Use the exit ticket to identify students needing further practice with second differences, ratios, or indexing from (n=1).

Differentiation

• Support: provide a partially completed table with columns for terms, first differences, second differences and ratios; keep the classification flowchart visible; allow students to use sentence starters such as “I classified this as ___ because…”.
• Support for EAL learners and students needing literacy assistance: pair visual notation with the key vocabulary, read instructions aloud, and accept an oral justification before students record it.
• Targeted support: begin with constant-difference and constant-ratio examples, then add second differences; conference with students who subtract terms in the wrong direction or compare non-consecutive terms.
• Extension: ask students to find two different rules that fit 2, 4, 6, 8, explain why the sequence is not uniquely determined, and identify what additional term would distinguish their rules.

Common misconceptions to address: A linear sequence need not start at zero; quadratic sequences do not have constant first differences; a constant increase is not the same as a constant ratio; negative or fractional ratios are still possible; and the position rule must be checked from (n=1), not just from the visible terms.