Exploring Quadratic Functions

Overview

This 60-minute lesson is designed for Year 11 Mathematics students in New Zealand. It focuses on deepening students’ understanding of quadratic functions and their real-world applications, in alignment with the New Zealand Curriculum (NZC) and NCEA Level 1 expectations. The lesson blends algebraic, graphical, and contextual approaches to quadratic functions, encouraging critical thinking and engagement through varied activities.

Curriculum Connections

Learning Area: Mathematics and Statistics
Level: Year 11 / NCEA Level 1 Mathematics
Strands: Algebra, Measurement and Geometry
Key Competencies:

Thinking: Develop understanding of quadratic relationships and their properties.
Using language, symbols, and texts: Interpret and construct mathematical representations such as equations, tables, and graphs.
Relating to others: Collaborate to solve problems and discuss reasoning.
Managing self: Take ownership of learning through problem-solving and reflection.
Participating and contributing: Share and evaluate multiple problem-solving strategies.

Achievement Objectives:

Recognise relationships in quadratic patterns and represent them with algebraic expressions and graphs.
Form and solve quadratic equations, explaining connections between solutions and the graph of a parabola.
Interpret and describe key features of linear and quadratic graphs, including vertex, intercepts, and symmetry.
Apply these understandings to optimise contextual problems (e.g., maximising area or profit).

Reference:

Te Mātaiaho Mathematics and Statistics Draft Curriculum, Years 9–13 (2025)
NCEA Level 1 Mathematics Learning Outcomes

Learning Intentions

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

Identify and describe the characteristics of quadratic functions from tables, graphs, and equations.
Formulate quadratic equations from patterns and solve them using factorisation and graph interpretation.
Apply quadratic functions to solve real-world problems, including optimisation tasks.

Resources

Whiteboard and markers
Graphing calculators or digital graphing tools (e.g., GeoGebra)
Student handouts with patterns, tables, and contextual problems
Rulers, squared paper
Projector for modelling examples

Lesson Breakdown

1. Introduction and Motivation (10 minutes)

Contextual Starter: Present a real-world scenario where quadratic relationships appear, e.g., the trajectory of a ball or maximum area fencing problem.
Ask students to discuss with a partner what they notice about the shape or pattern involved.
Review prior knowledge: briefly recap linear vs quadratic sequences and graphs.
State the learning intentions clearly.

2. Explore Patterns and Form Equations (15 minutes)

Present a number pattern or table illustrating quadratic growth (e.g., 1, 4, 9, 16, ...).
In pairs, students identify the relationship and attempt to write an equation for the pattern.
Teacher models connecting a quadratic pattern to its standard form equation ( y = ax^2 + bx + c ), focusing on identifying (a), (b), and (c) using differences or substitution.
Highlight how this links to graph features such as vertex and intercepts.

3. Graphing Quadratic Functions (15 minutes)

Using graphing calculators or digital tools, students create tables of values and plot quadratic functions (e.g., ( y = x^2 - 4x + 3 )).
Identify key features: vertex, axis of symmetry, x- and y-intercepts.
Discuss transformations: How changes in (a), (b), and (c) affect the graph’s shape and position.
Teacher questions to reinforce understanding: What do the roots represent in context? Why might the parabola open upwards or downwards?

4. Solving Quadratic Equations and Optimisation (15 minutes)

Introduce solving quadratic equations by factorisation where possible, connecting zeros of the graph to equation solutions.
Pose an optimisation problem, e.g., “Find the dimensions of a rectangle with maximum area given a fixed perimeter.”
Students work in small groups to model the problem with a quadratic function, solve it, and interpret the solution in context.
Teacher circulates to support reasoning and identify misconceptions.

5. Reflect and Connect (5 minutes)

Whole-class discussion on how quadratic functions relate to real-world problems and patterns in other subjects (e.g., physics trajectories, economics for profit maximising).
Encourage students to articulate any new understandings or strategies.
Recap achievement of learning intentions.

Cultural Responsiveness

In line with New Zealand’s bicultural foundations and multicultural society, this lesson:

Integrates contexts familiar and relevant to students’ diverse backgrounds, such as local environmental or sports examples, to make maths meaningful.
Encourages students to share their interpretations and methods, respecting multiple ways of understanding and solving problems.
Uses collaborative group work promoting whakawhanaungatanga (building relationships) and manaakitanga (care for others) in mathematical discussions.
Recognises and values students’ cultural narratives and perspectives when exploring patterns, promoting mathematical thinking as a collective, inclusive activity that acknowledges Māori pedagogical principles such as ako (reciprocal learning) and pūmanawa (capability).

Assessment Opportunities

Formative observation during paired and group tasks to assess students’ understanding of quadratic relationships, ability to write equations, and interpret graphs.
Targeted questioning to delve into students’ reasoning and identify misconceptions (e.g., mixing up linear and quadratic patterns).
Review of students’ written solutions for the optimisation problem, checking context-appropriate interpretation and correct use of mathematical notation.

Extensions and Differentiation

For advanced learners: Introduce the discriminant concept to predict solutions’ number and nature, and challenge students to explore non-integer roots graphically.
For students needing support: Provide scaffolded handouts with partially completed tables and guided steps for writing equations and graphing. Use physical manipulatives for visualising parabolas.

Teacher Reflection Guide

Were the students able to connect patterns to algebraic expressions and graphs effectively?
Did the contextual problems engage different learning styles and cultural backgrounds?
How did peer discussions support conceptual understanding?
What misconceptions arose, and how were they addressed?
How can technology integration be improved next time to enhance students’ agency and understanding?

This lesson plan embraces the holistic approach promoted by the NZ curriculum, building deep mathematical understanding through situated, connected learning experiences, while fostering cultural respect and collaborative engagement.

If you would like, I can also provide a complementary worksheet or sample assessment task aligned to this lesson.

Teaching Instructions