Describing Linear Sequences

Active learning helps Year 6 students grasp linear sequences because movement and discussion make abstract patterns concrete. Handling cards or racing to find sequences keeps pupils engaged while they internalise the idea of a constant difference between terms.

• Year 6
• Mathematics
• 4 activities
• 20 min–35 min

Learning Objectives

1. Calculate the nth term for a given linear sequence using algebraic notation.
2. Compare and contrast arithmetic and geometric sequences by identifying their defining rules.
3. Explain the algebraic rule for a linear sequence in terms of its starting term and common difference.
4. Create a linear sequence given an algebraic rule, demonstrating understanding of the relationship between notation and number patterns.

Inquiry Circle

⏱ 25 min·Pairs

Pair Sort: Arithmetic vs Geometric Cards

Prepare cards with sequence starts like 2,4,6 or 2,4,8. Pairs sort them into arithmetic or geometric piles, justify with differences or ratios, then write nth terms for arithmetic ones. Share one example with the class.

Prepare & details

Differentiate between an arithmetic sequence and a geometric sequence.

Facilitation Tip: During Pair Sort, remind pairs to verbalise why each card belongs in the arithmetic or geometric pile using the visible differences and ratios.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Inquiry Circle

⏱ 35 min·Small Groups

Small Groups: Sequence Hunt Relay

Groups visit school areas to find linear patterns, like tiles in corridors or steps on stairs. They photograph, note common differences, and derive nth terms. Regroup to present and critique findings.

Prepare & details

Explain how sequences help us model real-life patterns like growth or savings.

Facilitation Tip: For Sequence Hunt Relay, circulate with a timer visible so teams feel the urgency but know the focus is accuracy, not speed.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Inquiry Circle

⏱ 20 min·Whole Class

Whole Class: nth Term Challenge Board

Project sequences on the board. Class calls out differences; teacher reveals nth term. Pupils vote on corrections for flawed rules, discussing as a group before confirming.

Prepare & details

Critique a given rule for a sequence and suggest improvements if necessary.

Facilitation Tip: On the nth Term Challenge Board, ask students to explain their formulas aloud before writing them down to catch early overcounting errors.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Inquiry Circle

⏱ 30 min·Individual

Individual: Design Your Sequence

Each pupil creates a linear sequence from a real-life scenario, like plant pots in rows. They write the nth term and three peer-critique questions. Swap and respond.

Prepare & details

Differentiate between an arithmetic sequence and a geometric sequence.

Facilitation Tip: In Design Your Sequence, remind students to include the nth term rule on their posters before swapping for peer review.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Teaching This Topic

Teach linear sequences by pairing concrete actions with symbolic rules. Use number lines and counters to model why the nth term formula adjusts by one position. Avoid rushing to the formula; let students discover the pattern first through guided exploration and error analysis. Research shows that self-correction through peer feedback deepens understanding more than teacher correction alone.

What to Expect

Pupils will confidently identify common differences, extend sequences correctly, and write nth term rules without overcounting. They will also distinguish arithmetic from geometric sequences through clear reasoning.

Watch Out for These Misconceptions

Common Misconception

During Pair Sort, watch for pupils assuming all increasing sequences are arithmetic.

What to Teach Instead

Have pairs physically group cards and justify each placement using written differences or ratios on the back of the cards. Ask them to create one geometric sequence deliberately to highlight the difference in growth.

Common Misconception

During nth Term Challenge Board, watch for pupils writing the nth term as first term plus difference times n without adjustment.

What to Teach Instead

Provide number lines and counters for students to model the sequence step-by-step, marking each jump. Ask them to compare their starting position to the formula, prompting them to adjust by subtracting or adding the extra step.

Common Misconception

During Sequence Hunt Relay, watch for pupils ignoring geometric sequences as irrelevant to linear work.

What to Teach Instead

Require teams to classify every card they collect as arithmetic or geometric, using ratio checks for the latter. After the relay, facilitate a quick discussion on why recognising ratios matters even when focusing on linear sequences.

Assessment Ideas

Quick Check

After Pair Sort, give students three sequences to classify as arithmetic or geometric and state the common difference or ratio for each. Collect responses to check their ability to differentiate between patterns.

Exit Ticket

After nth Term Challenge Board, hand out exit tickets with the rule 3n + 4. Ask students to write the first four terms and explain what the '3' and the '+ 4' represent in the sequence’s pattern.

Discussion Prompt

During Sequence Hunt Relay, pause the game to ask: 'If a sequence starts with 12 and has a common difference of -2, is it linear or geometric? How do you know? What is the nth term rule?' Listen for students to justify their answers using the sequence values and differences.

Extensions & Scaffolding

• Challenge: Ask early finishers to create a sequence where the nth term formula includes a negative common difference and explain how the pattern changes.
• Scaffolding: Provide partially completed sequence tables with gaps for students to fill, alongside a number line to mark jumps.
• Deeper exploration: Introduce a sequence with a fractional common difference and ask students to predict non-integer terms, linking to prior work on fractions.

Key Vocabulary

Sequence	A set of numbers or objects in a specific order or pattern.
Term	Each individual number or element within a sequence.
Linear Sequence	A sequence where the difference between consecutive terms is constant. Also known as an arithmetic sequence.
Common Difference	The constant amount added or subtracted to get from one term to the next in a linear sequence.
nth term	An algebraic expression that describes any term in a sequence based on its position (n).