Linear Versus Non-Linear

Overview

This 60-minute lesson is designed for Year 7 students, following the National Curriculum for England in mathematics, focusing on Number – sequences. It forms part 5 of 7 in the unit "Mastering Sequences in Maths," based on White Rose Maths Year 7 v3 guidance and aimed at developing strong conceptual understanding of linear and non-linear sequences.

---

Curriculum Links and Learning Objectives

National Curriculum references:

• Pupils should be taught to:
• Generate and describe linear number sequences (including those involving fractions and decimals).
• Explore and continue non-linear sequences such as quadratic sequences.
• Recognise the connection between the position-to-term rule and the sequences produced.
• Key stage 3 programmes of study (Mathematics, Number – Sequences and Relationships).

Learning Objectives (WALT):

• WALT: Distinguish between linear and non-linear sequences.

Success Criteria:

• I can identify and describe the rule of a linear sequence.
• I can spot the differences in how non-linear sequences change compared to linear sequences.
• I can classify at least five sequences correctly as linear or non-linear.

---

Resources

• Whiteboards and pens for quick formative assessments.
• Graphic organisers showing characteristics of linear vs non-linear sequences (e.g. tables, position-to-term rule diagrams).
• Printed sets of sequence cards (linear and non-linear examples).
• Projector/slideshow to show real-life data sequences.
• Worksheets for classification and reasoning tasks.
• Calculators (optional for extension).

---

Lesson Breakdown

Introduction (10 minutes)

1. Starter – Quick Recap (5 min):

• Begin with two simple sequences on the board: 2, 4, 6, 8... and 1, 4, 9, 16...
• Ask students to spot any patterns and share what they notice about the differences in how the numbers change.
• Briefly remind/reinforce the definition of linear sequences (constant difference) vs non-linear sequences (difference varies).

2. Set WALT and success criteria (2 min): Write these clearly on the board for reference during the lesson.

3. Introduce key vocabulary and concepts (3 min):

• Terms: linear, non-linear, term-to-term rule, position-to-term rule, nth term.
• Use a graphic organiser projected on the board contrasting key features.

---

Main Activities (35 minutes)

Activity 1: Classify Sequence Cards in Mixed-Ability Groups (15 min)

• Organise the class into mixed-ability groups of 4-5.
• Hand out sets of sequence cards—some linear (e.g. 3, 7, 11, 15...), some non-linear (e.g. 2, 4, 8, 16...) and others trickier like sequences involving decimals or squares.
• Provide graphic organisers for each group to record each sequence, identify the pattern, and classify it linear or non-linear.
• Encourage students to explain their reasoning using the vocabulary introduced.
• Teacher circulates, scaffolds, and challenges thinking with probing questions such as “How do you know this is non-linear?” or “What is the rule for this sequence?”

Activity 2: Position-to-Term Rule Exploration (10 min)

• Using sequences from Activity 1, model finding nth terms for linear sequences, reinforcing constant difference concept.
• Contrast with attempts to find nth terms in non-linear sequences and explain why it differs.
• Students attempt a similar task on worksheet segments: write the position-to-term rule or explain why it can’t be linear.

Activity 3: Real-Life Data Sequence Analysis - Extension (10 min)

• Project real-world examples (e.g., plant growth measurements, ticket sales over days, or population data).
• Challenge students to work in pairs to analyse data to decide if the sequence appears linear or non-linear.
• Encourage them to justify their classification by discussing patterns in differences or ratios.
• Advanced learners may use calculators to experiment with quadratic differences or graph plotting on grids for visualisation.

---

Plenary (10 minutes)

Class Discussion and Self-Assessment:

• Invite groups to share one example of a linear and a non-linear sequence they classified, explaining their reasoning.
• Refer back to success criteria; students self-assess their understanding using thumbs up/down or a quick exit ticket:
• Write one sentence explaining how to tell a linear from a non-linear sequence.
• Address any lingering misconceptions as needed.

---

Differentiation

• Support: Use graphic organisers to visualise sequence patterns; offer sentence starters like “This sequence is linear because…” for discussion; group lower-attaining students with confident peers for scaffolding.
• Challenge: Extension tasks involve analysing real datasets and exploring quadratic sequences; encourage formulation of algebraic expressions for complex nth terms.
• SEN: Visual aids and step-by-step guidance; simplified sequences; allow verbal responses instead of written where appropriate.

---

Assessment Opportunities

• Formative checks during group discussions and card classification.
• Written work on position-to-term rules and sequence classification.
• Plenary exit ticket provides individual evidence of learning.
• Observation of reasoning and vocabulary use to ensure concept understanding.

---

Reflection Points for Teacher

• Were all groups able to confidently distinguish linear from non-linear sequences?
• Did the extension challenge high achievers appropriately?
• Was the mix of visual, verbal, and hands-on tasks effective for differing learning styles?
• Note any students needing extra support for follow-up lessons.

---

By engaging students with hands-on classification, algebraic exploration, and real-life data analysis, this lesson meets the expectations of the National Curriculum for Year 7, ensuring a secure understanding of types of sequences and their properties while encouraging mathematical communication and reasoning.