Sequences, Series, and Sigma

Overview

An 85-minute IB Mathematics Analysis and Approaches SL lesson introducing number patterns and sigma notation. Students move from recognizing sequence types to fluently interpreting and evaluating sums written in sigma notation.

Learning Intentions

• Students will be able to distinguish between a sequence and a series, and between finite and infinite examples.
• Students will be able to classify number patterns as arithmetic, geometric, recursive, or neither.
• Students will be able to interpret sigma notation, identifying the index variable, lower and upper limits, and the general term.
• Students will be able to expand and evaluate sums expressed in sigma notation, including linear and geometric general terms.

Success Criteria

• I can explain the difference between a sequence and a series using a specific example.
• I can identify the type of pattern in a given sequence and justify my classification.
• I can correctly read a sigma expression and write out its expanded form.
• I can calculate the value of a finite sum written in sigma notation.

Curriculum Links

• IB Mathematics AA SL — sequences and series, including arithmetic and geometric sequences
• IB Mathematics AA SL — use of sigma notation to represent series
• IB Mathematical Practices — generalization, representation, and communication of mathematical structure
• IB Approaches to Learning — thinking skills: pattern recognition, classification, and logical reasoning

Lesson Structure (85 minutes)

[0–8 min] Prior Knowledge Review Display five number sequences on the board (e.g., 2, 5, 8, 11, … and 3, 6, 12, 24, …). Ask students to write down the next two terms and describe the rule in one sentence on mini whiteboards or scratch paper. Cold-call three to four students to share, using responses to surface existing understanding of common differences and common ratios before introducing formal vocabulary.

[8–20 min] Inquiry Opening Present the question: "A theater has 20 seats in the first row, 23 in the second, 26 in the third — how many seats are in the entire theater if there are 30 rows?" Allow pairs two minutes to estimate, then three minutes to attempt a method. Take two or three approaches from the class, deliberately leaving the question of efficiency open. Use this to motivate the need for concise notation and systematic summation tools.

[20–38 min] Direct Instruction Formally define sequence vs. series with notation (u₁, u₂, uₙ) and the partial sum Sₙ. Classify the four pattern types — arithmetic (constant difference d), geometric (constant ratio r), recursive (each term depends on previous terms, e.g., Fibonacci), and neither. Introduce sigma notation: Σ symbol, index variable (typically i or k), lower limit, upper limit, and general term. Work through three carefully sequenced examples on the board:

1. Expand Σ (i = 1 to 5) of (2i + 1), then evaluate.
2. Expand Σ (i = 0 to 4) of 3 · 2^i and connect to geometric series.
3. Write the series 4 + 7 + 10 + 13 + 16 in sigma notation.

Pause after each example for a 30-second turn-and-talk: "What does each part of the notation tell us?" Keep pace brisk but allow two or three student questions per example.

[38–55 min] Guided Practice Distribute the structured practice worksheet (six problems, graduated difficulty). Problems 1–2: identify sequence type and state d or r. Problems 3–4: expand and evaluate given sigma expressions. Problems 5–6: write a described or listed series in sigma notation. Students work individually; circulate and use targeted questioning rather than correcting directly — ask "What does the index represent here?" or "What value does i take first?" After 12 minutes, review problems 3 and 5 whole-class, annotating a projected version of the worksheet.

[55–78 min] Collaborative Activity Groups of four or five receive a card-sort set of 12 cards containing: series written out term by term, equivalent sigma expressions, computed values, and verbal descriptions. Students must match each series to its sigma notation, its expanded form, and its sum. One card in each set is deliberately miswritten with an error in the limits; groups must identify and correct it. Each group records their reasoning on a shared sheet. With five minutes remaining, two groups share one match they found challenging and explain their reasoning to the class.

[78–85 min] Exit Ticket Students independently complete three questions on a half-sheet:

1. Write the series 5 + 8 + 11 + 14 + 17 in sigma notation.
2. Evaluate Σ (k = 1 to 4) of k².
3. State whether Σ (i = 1 to ∞) of (1/2)^i is finite or infinite and explain in one sentence.

Collect before dismissal. Use responses to identify misconceptions around index starting values and infinite convergence, informing the next lesson on infinite geometric series.

Resources

• Mini whiteboards or scratch paper for the opening review
• Projected slideshow with theater-row context and worked examples
• Structured practice worksheet (printed, one per student)
• Card-sort sets (one per group of four or five, pre-cut and bagged)
• Exit ticket half-sheets (printed, one per student)
• Board or document camera for annotating guided practice solutions

Assessment

• Formative observation during guided practice — circulate and note common errors with limit notation or index substitution.
• Card-sort group recording sheets — reviewed after class to assess accuracy and reasoning quality.
• Exit ticket — provides individual written evidence of each learning intention; used to plan the opening of the next lesson.

Differentiation

• Support: provide a partially completed reference card defining each component of sigma notation with a labeled diagram; reduce card-sort to eight cards.
• Extension: challenge early finishers to express the Fibonacci sequence recursively in formal notation and research whether an infinite geometric series with |r| < 1 always converges.
• EAL/SEN: pre-teach key vocabulary (index, limit, term, expand) with visual annotations before the lesson; allow use of a bilingual mathematics glossary during independent work.