Topic 1: Sequences and Series

Three 60-minute lesson plans on the same underlying idea — arithmetic and geometric sequences — pitched at DP AA (SL/HL), MYP Grade 10, and IGCSE Grade 10, so the progression across programmes is visible.

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A. DP Mathematics: Analysis & Approaches — SL/HL, Year 1

Caveat: content below is drawn from the pre-publication draft syllabus (first assessment 2029, Topic A: Number & Algebra). Verify against the final published guide once released.

Anchor concept: General term and sum of arithmetic and geometric sequences (syllabus ref. A1). Taught in Year 1, per this school’s SOW_DP1_AAHL (Number & Algebra is the opening unit).

Prior knowledge assumed: Basic algebraic manipulation; substitution into formulae; solving linear equations (from MYP5/IGCSE).

Learning objectives

• SL+HL: Define arithmetic and geometric sequences; derive and apply the general term formulae (u_n = u_1 + (n-1)d) and (u_n = u_1 r^{n-1}).
• SL+HL: Derive and apply the sum formulae for the first (n) terms of both sequence types, including the sum to infinity of a convergent geometric series ((|r|<1)).
• HL only: Represent a series using sigma notation and convert between sigma notation and expanded form (syllabus ref. A1 — HL extension).

Command terms used

• Find (obtain an answer, showing relevant working) — general terms, specific terms, sums.
• Derive (obtain by a sequence of logical steps, HL) — deriving the sum formula from first principles for the sigma-notation extension task.
• Hence — using an established result (e.g. the general term) to find a sum without re-deriving.

Assessment focus (2029 draft AOs)

• Abstraction — recognising a real context (e.g. savings, population growth) as arithmetic or geometric and translating it into a sequence model.
• Computation — correct application of the general term/sum formulae, with and without GDC.

ATL / skills focus

Thinking (pattern recognition, moving between recursive and closed-form representations); self-management (systematic checking of formula conditions, e.g. (|r|<1) before using the infinite sum).

Starter (≈10 min)

Show two numeric sequences on the board (e.g. 3, 7, 11, 15, … and 3, 6, 12, 24, …). Students identify the pattern and predict the next two terms, then articulate in words how each is generated. Surfaces the distinction between “add a constant” and “multiply by a constant” without naming terms yet.

Main teaching sequence (≈35 min)

1. Formalise arithmetic sequences: common difference (d), general term (u_n = u_1+(n-1)d). Worked example, then a “find (u_1) and (d) given two terms” problem (simultaneous equations — links back to MYP/IGCSE algebra).
2. Formalise geometric sequences: common ratio (r), general term (u_n = u_1 r^{n-1}). Parallel worked example.
3. Introduce sum formulae for both: (S_n = \frac{n}{2}(2u_1+(n-1)d)) and (S_n = \frac{u_1(r^n-1)}{r-1}). Discuss why the geometric sum formula fails at (r=1).
4. Introduce sum to infinity for (|r|<1): (S_\infty = \frac{u_1}{1-r}), with a quick geometric / area-model justification of convergence.
5. HL-only extension (last ~10 min, SL students work independently on mixed practice instead): introduce sigma notation, (\sum_{k=1}^{n} u_k), and have HL students rewrite two of the sums already computed in sigma form and evaluate one sigma expression by expanding it.

Formative assessment / check for understanding

Mini whiteboard check: give (u_1, d) (or (u_1, r)) and ask for (u_5) and (S_5) — whole class holds up answers simultaneously. Circulate during the HL sigma-notation task to check correct index bounds.

Plenary / exit ticket (≈10 min)

Exit ticket: “A ball dropped from 2m rebounds to 60% of its previous height each bounce. Find the total vertical distance it travels before coming to rest.” (Requires recognising a geometric series and — implicitly — summing a doubled infinite series for down+up bounces; a genuine Computation + Abstraction task.)

Resources

GDC sequence mode for verifying (u_n)/(S_n) numerically (see Use of GDC in 2026.pdf); Haese AA textbook chapter M03 (Sequences and Series) under IB DP Mathematics/PPT of SL Mathematics/ for further worked examples and practice sets.

Vertical link note

Builds directly on the MYP5 skill of finding and justifying a general rule (below); the sum formulae and sigma notation are new DP content not present in either MYP or IGCSE.

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B. MYP Mathematics — Grade 10 (MYP Year 5)

Anchor concept: Finding and justifying the general rule for arithmetic and geometric sequences (Numerical & Abstract Reasoning branch, MYP5 Extended).

Prior knowledge assumed: Recognising and continuing simple numeric patterns (MYP3/4); basic algebraic notation.

Learning objectives

• Identify whether a given sequence is arithmetic, geometric, or neither, and state the common difference/ratio.
• Find a general rule (in (n)) for a given sequence using inductive reasoning from specific cases.
• Justify or prove that a proposed general rule is correct for all (n) (MYP5 Extended requirement — moves beyond simply stating the rule).

Assessment criteria addressed

• Criterion B (Investigating patterns) — primary focus: students are given an open, under-scaffolded task and must choose their own technique to find and justify a rule.
• Criterion C (Communicating) — using correct notation ((u_n), (d), (r)) and producing a clear, logically ordered justification.

Command terms used

Verify (provide evidence to validate a result), Justify (give valid reasons/evidence for a conclusion — distinct from simply explaining), Describe (state the characteristics of a pattern in words).

ATL / skills focus

Thinking skills (generating a general rule from specific instances; testing and refining a conjecture); communication skills (structuring a written justification).

Starter (≈10 min)

Matchstick-pattern task: show a growing sequence of shapes built from matchsticks (e.g. triangles sharing a side). Students count matchsticks for the first 3–4 shapes and predict the 10th without drawing it.

Main teaching sequence (≈35 min)

1. Formalise the matchstick pattern as an arithmetic sequence; extract (u_1) and (d) from the context and write the general term.
2. Contrast with a geometric context (e.g. paper-folding: 1, 2, 4, 8 layers) — extract (u_1) and (r) and write the general term.
3. Present a “broken” investigation: a sequence with a less obvious rule (e.g. differences that are themselves not constant, or a sequence requiring two steps to see the pattern) for Standard-level students to describe in words and Extended-level students to express algebraically and justify.
4. Explicit instruction on what “justify” requires here: showing the rule holds by substitution for at least the first few terms and explaining why the reasoning generalises — not just restating the pattern.
5. Independent/paired investigation task: students generate their own pattern (e.g. from a real-life context: tiling a growing patio border) and produce a general rule with justification.

Formative assessment / check for understanding

Peer-check in pairs: swap justifications and identify whether the partner’s argument would convince a stranger the rule holds for all (n), or only for the terms tested.

Plenary / exit ticket (≈10 min)

One sequence given with no context; students state whether it is arithmetic, geometric, or neither, and justify their classification in two sentences.

Resources

Physical/digital matchstick or paper-folding manipulatives; GDC or spreadsheet for generating and checking terms of self-created sequences.

Vertical link note

This is the direct precursor to the DP A1 general-term/sum work above — MYP5 stops at finding and justifying the rule; DP formalises the sum and (HL) sigma notation. It builds on IGCSE’s nth-term work (below) by adding the justification/proof expectation.

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C. IGCSE Mathematics — Grade 10 (Extended tier)

Caveat: IGCSE content below follows standard Cambridge (0580) / Edexcel International GCSE (4MA1) Extended-tier syllabus content. No official board specification file was found on this machine — cross-check against the current specification before formal use.

Anchor concept: Finding the nth term of linear, quadratic, and geometric sequences.

Prior knowledge assumed: Substitution into algebraic expressions; solving linear and simple quadratic equations; expanding brackets.

Learning objectives

• Find the nth term of a linear sequence from given terms.
• Find the nth term of a simple quadratic sequence using the second-difference method.
• Find the nth term (general term) of a geometric sequence and use it to find a specified term.

Assessment objectives addressed

AO1 (recall and apply standard techniques — applying the nth-term formulae); AO3 (solve problems in unfamiliar contexts — e.g. working backwards from a term value to find (n)).

Command terms used

Calculate, Find, Show that, Write down — standard Cambridge/Edexcel command-term conventions.

Starter (≈10 min)

Quick-fire: given 4–5 terms of a sequence, students state whether the sequence is going up by a constant amount, a constant multiple, or neither, and predict the next term.

Main teaching sequence (≈35 min)

1. Linear sequences: derive (nth\ term = dn + (a-d)) from the common difference (d) and first term (a); practice with several sequences including negative common difference.
2. Quadratic sequences: introduce the second-difference method (halve the constant second difference to get the coefficient of (n^2)), then find the remaining linear part by subtraction. Work through one full example step by step.
3. Geometric sequences: derive the general term (u_n = ar^{n-1}); practice finding a specified term and, as an extension, finding (r) or (a) given two terms (requires solving an exponential/algebraic equation — stretch task for stronger students).
4. Mixed practice set requiring students to first classify the sequence type before applying the correct method — deliberately unscaffolded to build the same “identify before solve” instinct MYP and DP both require.

Formative assessment / check for understanding

Whole-class mini-whiteboards: give a sequence, ask for its nth-term formula; check simultaneously.

Plenary / exit ticket (≈10 min)

“Show that the nth term of the sequence 2, 7, 14, 23, 34, … is (n^2+2n-1).” (Requires the second-difference method plus verification by substitution — a genuine “show that” task.)

Resources

Past-paper style nth-term questions (Extended tier); GDC or table/spreadsheet to generate sequences and check derived formulae numerically.

Vertical link note

Sets up the general-term concept that MYP5 then requires students to justify/prove, and that DP AA extends with sum formulae, infinite sums, and (HL) sigma notation.