Exploring Quadratic Patterns

Overview

In this 60-minute lesson designed for Year 11 students in New Zealand, learners will explore quadratic patterns, form and solve quadratic equations, and interpret their graphs. This lesson integrates key elements from the New Zealand Curriculum (NZC), specifically achievement objectives from the Mathematics and Statistics learning area relevant to Year 11. The activities encourage conceptual understanding, procedural fluency, and critical thinking while embedding culturally responsive practices reflective of Aotearoa New Zealand’s bicultural and multicultural context.

Learning Objectives

Achievement Objectives (NZ Curriculum - Mathematics and Statistics - Level 6)

Recognise the relationships between elements of a quadratic pattern; write an equation to represent the rule for the pattern, and use the equation to make conjectures.
Form and solve quadratic equations and explain why there are different numbers of solutions.
Interpret key features of the graph of a quadratic function, linking solutions of quadratic equations to the graph of a parabola.
Use tables and graphs to represent quadratic functions and connect these with linear relationships.
Develop procedural fluency and conceptual understanding through problem-solving and reasoning.

Key Competencies Addressed

Thinking: Developing critical and creative thinking by investigating patterns and forming conjectures.
Using language, symbols, and texts: Interpreting and producing mathematical symbols, graphs, and equations.
Relating to others: Collaborative problem solving and sharing reasoning.
Managing self: Persisting through challenges and reflecting on learning strategies.
Participating and contributing: Discussing ideas including different cultural perspectives to develop collective understanding.

Cultural Responsiveness

In recognition of Aotearoa New Zealand’s bicultural foundations and rich multicultural society:

Incorporate Māori contexts where possible, such as patterns found in traditional weaving (raranga) or architecture (wharenui designs), highlighting the mathematical relationships within these cultural artefacts.
Encourage students to share and explore how different cultures represented patterns and solved mathematical problems historically.
Promote a whakatauākī (proverb) to frame the lesson, such as “Nāu te rourou, nāku te rourou, ka ora ai te iwi” (“With your basket and my basket, the people will thrive”) to emphasize the collaborative nature of learning.
Facilitate group work that respects diverse perspectives and communication styles, ensuring all voices are heard and valued.

Resources Required

Whiteboard and markers
Graphing calculators or graphing software/applications (e.g., GeoGebra)
Printed handouts with quadratic table values and pattern sequences
Paper for working calculations and sketching graphs
Visual examples of quadratic patterns, including Māori weaving patterns or natural parabola shapes

Lesson Breakdown (60 minutes)

1. Getting Started (10 minutes)

Activate prior knowledge: Begin with a brief discussion recalling linear patterns and their equations. Pose an engaging question: “What patterns do you notice here?” showing a simple quadratic pattern such as 1, 4, 9, 16, 25... (squares of natural numbers).
Hook activity: Present a culturally relevant visual pattern (e.g., a section of tukutuku panel pattern) and ask students to observe and describe patterns they see.
Set the purpose: Explain that today’s focus is on understanding quadratic patterns and how we can represent these using equations and graphs, connecting mathematics to real-world and cultural experiences.

2. Working Session (40 minutes)

Part A: Exploring Quadratic Patterns (15 minutes)

Provide students with sequences representing quadratic patterns.
Invite them to:
- Create tables of corresponding terms.
- Identify differences between terms to distinguish quadratic from linear patterns.
- Derive the quadratic equation representing the pattern, e.g., (y = x^2) or (y = ax^2 + bx + c).
Facilitate paired discussion to share their findings and conjectures.

Part B: Solving Quadratic Equations (15 minutes)

Model how to form and solve quadratic equations derived from patterns, including explanation of why equations can have different numbers of solutions (e.g., (x^2 = 9) has two solutions).
Introduce the zero product property and solving by factorisation.
Provide scaffolded practice problems where students solve quadratic equations in pairs or small groups.

Part C: Graphing and Interpretation (10 minutes)

Demonstrate graphing quadratic functions using technology or by plotting points from a table.
Discuss key features such as vertex, axis of symmetry, roots (x-intercepts), and y-intercept.
Invite students to interpret graphs in context, e.g., discussing what the roots mean in a real-life problem.

3. Connecting and Reflecting (10 minutes)

Organise a class discussion reflecting on:
- How quadratic patterns differ from linear.
- The usefulness of graphs, tables, and equations in representing patterns.
- Different cultural perspectives on patterns and problem solving.
Summarise key learning points and articulate learning goals for the next lesson.
Highlight examples of perseverance and critical thinking observed.

Assessment & Feedback

Use observational records during activities to note student engagement and understanding.
Provide formative feedback, emphasising both conceptual understanding and procedural skills.
Encourage self and peer-assessment, where students check and explain answers to each other, reinforcing key vocabulary and concepts.

Extension and Differentiation

For advanced learners: Challenge them to explore more complex quadratic equations or contextualise in optimisation problems (maximising/minimising areas).
For learners needing support: Provide step-by-step scaffolds, manipulatives to represent patterns physically, and opportunities for one-on-one support.

References

New Zealand Curriculum: Mathematics and Statistics Learning Area, Level 6 (Years 11-12) Achievement Objectives.
Te Mātaiaho – Mathematics and Statistics in the New Zealand Curriculum (Years 9–13), Draft January 2025.
NCEA Level 1 Mathematics achievement standards and learning outcomes.
Culturally Responsive Practices in Mathematics, NZ Ministry of Education guidance.

This lesson plan aims to go beyond traditional lecture methods by fostering active inquiry, cultural connections, and collaborative learning, adhering to the principles and aspirations embedded in the New Zealand Curriculum. It supports students to see themselves as capable mathematicians contributing to the learning community.

If you would like, I can also provide detailed student handouts or suggested assessment rubrics for this lesson!

Teaching Instructions