Quadratic Functions and Graphs

Overview

This 60-minute lesson is designed for Year 11 students in New Zealand and focuses on understanding quadratic functions through their patterns, equations, and graphing. It aligns with the New Zealand Curriculum (NZC) and incorporates cultural responsiveness, especially reflecting New Zealand’s bicultural and multicultural context. The lesson addresses key achievement objectives in algebra and graphical representation and develops critical key competencies including thinking, relating to others, and using language, symbols, and texts.

Curriculum Links

Achievement Objectives (NZC: Mathematics and Statistics, Years 9-11)

• Algebra: Recognise the relationships between elements of linear or quadratic patterns, write equations to represent pattern rules, and use equations to make conjectures.
• Algebra: Form and solve quadratic equations; explain the solution properties and relate them to parabolas.
• Graphing: Make tables and graphs of quadratic functions, interpret key information and features of their graphs.
• Key Competencies: Thinking; Using language, symbols, and texts; Relating to others.

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Learning Outcomes

By the end of this lesson, students will be able to:

1. Identify and describe the elements of quadratic patterns.
2. Formulate quadratic equations from given patterns or tables.
3. Sketch and interpret graphs of quadratic functions, including identifying key features such as vertex and intercepts.
4. Discuss real-world or culturally relevant contexts where quadratic functions apply.
5. Collaborate effectively to support learning and share mathematical reasoning.

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Resources

• Whiteboard and markers
• Graph paper
• Student workbooks or notebooks
• Calculators (graphing if available)
• Digital graphing tools (optional)
• Pattern cards with numeric sequences representing quadratic relations
• Culturally relevant data sets or examples (e.g., Māori navigation patterns, environmental growth examples)

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Lesson Structure

1. Getting Started (10 minutes)

• Warm-up Activity (5 min): Quick mental arithmetic and pattern recognition challenge with number sequences (linear and quadratic).
• Introduction to Quadratic Patterns (5 min): Present examples of quadratic sequences and their differences — encourage students to notice the constant second difference.
• Use clear, concise language and mathematical vocabulary, modelling correct usage.

2. Working Session (35 minutes)

Activity 1: Identify and Formulate (15 min)

• Students work in pairs to examine pattern cards showing quadratic sequences.
• Task: Write an equation for the pattern given the sequence differences.
• Teacher circulates, scaffolding with probing questions to link the pattern, formula, and graph shape.
• Emphasise use of mathematical language and reasoning.

Activity 2: Graph and Interpret (20 min)

• In groups of 3-4, students create tables of values for their quadratic equations.
• Sketch their graphs on graph paper.
• Use digital tools if available to graph the functions.
• Groups analyse key features: vertex, intercepts, symmetry.
• Encourage discussion on how these features relate to the equation components.

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3. Connecting and Reflecting (10 minutes)

• Groups present one key finding or challenge to the class.
• Teacher facilitates whole-class discussion on similarities and differences between quadratic and linear functions.
• Reflect on cultural relevance — e.g., how quadratic patterns might relate to natural phenomena, architecture, or other contexts meaningful to students including Māori and Pasifika perspectives.
• Reinforce key mathematical vocabulary and concepts discussed.

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4. Plenary and Next Steps (5 minutes)

• Summarise the lesson’s key points and stress the connection between pattern recognition, algebraic representation, and graphical interpretation.
• Encourage students to reflect on how mathematical patterns appear in their community or cultural narratives.
• Set a short homework task: find and describe a quadratic pattern or situation at home or in the media.

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Cultural Responsiveness

• Bicultural Foundations: Integrate concepts of whakapapa (genealogy) and natural growth cycles from Māori perspectives to contextualise quadratic growth or patterns (e.g., growth of plants or tidal patterns).
• Multicultural Inclusion: Encourage students to share examples from their own cultures that connect with patterns and functions, creating an inclusive environment recognising diverse knowledge systems.
• Use a "mathematics as a language" approach valuing the different ways of expressing mathematical ideas.
• Ensure groupings encourage collaboration across different backgrounds and abilities, fostering mutual respect and shared learning.

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Assessment for Learning

• Observe students during pair and group activities for evidence of forming and solving quadratic equations and graph interpretation.
• Formative questions during activities to gauge understanding.
• Collect and review student graphs and equations for accuracy and use of mathematical language.
• Encourage self-assessment reflection on the cultural contexts connection.

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Teacher Notes

• Scaffold carefully, particularly with students new to quadratic concepts.
• Use multiple representations: numeric patterns, algebraic expressions, and graphs.
• Promote peer teaching in groups to build confidence.
• Model persistence and valuing mistakes as learning opportunities.
• Connect mathematics to students’ real life and cultural contexts to increase engagement and relevance.

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This lesson plan aligns with New Zealand Curriculum standards, NCEA Level 1 expectations, and incorporates culturally responsive pedagogical strategies to engage Year 11 students in mathematics learning with depth and relevance .