Sequences Foundations

Overview

In this first lesson of the “Exploring Sequences and Series” unit, students learn what a sequence is, how to describe it, and how to classify sequences into arithmetic or geometric types. They connect definitions to real-life contexts and begin using clear notation.

Learning intentions

WALT: Define sequences and identify their types.

• WALT: Recognise arithmetic sequences and interpret meaning of terms and differences.
• WALT: Recognise geometric sequences and interpret meaning of ratios.
• WALT: Classify given sequences and justify decisions using notation or patterns.
• WALT: Use appropriate representations (term lists and simple rule/description).

Success criteria

• I can define a sequence as an ordered list of terms.
• I can classify a sequence as arithmetic or geometric and explain why.
• I can identify the common difference (arithmetic) or common ratio (geometric).
• I can write a correct notation-style description for the pattern (e.g., “add a constant” or “multiply by a constant”).

Curriculum links

• Mathematics (Algebra): Apply sequences and series in solving problems by selecting methods such as using the general term/structure to classify patterns.
• Mathematics (Algebra): Apply sequences and series using relational thinking by connecting the context to the pattern (difference vs ratio).
• Mathematics (Algebra): Apply sequences and series using appropriate mathematical statements and representations to communicate classification clearly.
• Mathematics curriculum direction: developing mathematical reasoning through pattern recognition and explanation in real-life or mathematical contexts.

Lesson structure (45 minutes)

1. 0–5 min · Hook (pattern spotting). Teacher displays four short number lists and asks “What do you notice that could tell us how to find the next term?” Students record one observation per list in books.

2. 5–10 min · Whole-class introduction to “sequence”. Teacher defines sequence as an ordered list of terms and contrasts it with random sets, then models reading “term 1, term 2, term 3”. Students label terms in one example and write a one-sentence definition of a sequence.

3. 10–18 min · Arithmetic sequences (difference). Teacher presents an arithmetic example from a real context (e.g., weekly pay increases by a fixed amount) and shows how the constant difference appears between consecutive terms. Students compute successive differences for a second list and decide if it fits.

4. 18–26 min · Geometric sequences (ratio). Teacher presents a geometric example from a real context (e.g., population growth modelled as multiplying by a constant factor) and shows how the constant ratio appears between consecutive terms. Students compute successive ratios for a second list (checking for multiplication) and decide if it fits.

5. 26–34 min · Guided classification task (communication focus). Teacher gives 6 “mystery sequences” (some arithmetic, some geometric, some that are neither) and a classification table: Arithmetic / Geometric / Not sure / Neither. Students work in pairs to classify each, and for every “Arithmetic” or “Geometric” they add “common difference” or “common ratio” in words.

6. 34–40 min · Relational thinking check (justify quickly). Teacher selects 2 sequences and asks: “What exact relationship supports your classification?” Students provide short justifications to a partner, then one student per sequence shares a clear explanation using the terms “difference” or “ratio”.

7. 40–45 min · Exit ticket (individual). Students answer: (a) define a sequence in one sentence, and (b) classify one new sequence as arithmetic/geometric and state the common difference/ratio.

Resources

• Slide/board display with 4–8 example term lists and 6 mystery sequences
• Student printed classification worksheet (term lists + answer table)
• Graph paper or lined books for working
• Sticky notes for pair feedback
• Calculator for ratio checks (optional, teacher-controlled)
• Timer for transitions between activities

Assessment

• Formative: teacher circulates during pair classification, checking for correct use of “difference” vs “ratio” and clear reasoning.
• Formative: brief whole-class questioning at the end of arithmetic and geometric sections to confirm understanding.
• Summative-in-mini: exit ticket for individual classification and definition (checked against success criteria).

Differentiation

• Support for diverse learners: provide a sentence starter on the worksheet such as “This is arithmetic because the difference between consecutive terms is constant: …” and “This is geometric because the ratio is constant: …”. Include an example completed with each strategy.
• Support for students needing structure: give a “working template” row for differences and ratios (students fill in the table values, not just the final label).
• Extension for advanced learners: include two “trick” sequences (e.g., alternating signs, or constant differences but not from one step to the next, or zero terms) and ask students to state whether they are arithmetic/geometric and why, using precise language.
• EAL considerations: allow justifications using either words or simple notation; encourage students to underline the “constant difference/ratio” they found and to use the same vocabulary across peers.

Extension

• Challenge 1 (advanced): “Write an arithmetic sequence with common difference 5 that starts at 3. Then write a geometric sequence with common ratio 2 that starts at 3. Compare how you would check each one.”
• Challenge 2 (advanced): For one “neither” sequence, propose what transformation (add a constant, multiply by a constant, or neither) would be needed to turn it into an arithmetic or geometric sequence, and justify.