Maximising Practical Problems

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Level

Year 11 (ages approximately 15–16)
Class size: 30 students
Duration: 60 minutes

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Learning Area

Mathematics and Statistics (Te Mātaiaho)
Focus: Number and Algebra — Applied optimisation and problem solving

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Achievement Objectives (New Zealand Curriculum)

• Algebra (Year 11)
  • Form and solve linear and quadratic equations and inequalities, including contextual problems.
  • Make and test conjectures about relationships and patterns in linear, quadratic, and exponential functions.
  • Find optimal solutions that maximise or minimise quantities within constraints, using tables, graphs, and algebraic methods.
• Number (Year 11)
  • Perform operations involving rates, ratios, percentages, and compound interest, and apply these to real-life contexts.
• Key Competencies
  • Thinking: Apply critical and creative thinking to solve problems and make decisions in unfamiliar contexts.
  • Using language, symbols, and texts: Use mathematical language and representations accurately to communicate and reason.
  • Participating and contributing: Collaborate on problem-solving tasks respecting different perspectives and knowledges.

(Referenced and adapted from Te Mātaiaho Mathematics and Statistics Draft 2025 and NZ Curriculum senior secondary maths documents) .

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Learning Intentions

Students will:

• Understand how to model real-world optimisation problems mathematically.
• Use tables, graphs, and algebraic expressions to explore and find maximum or minimum values under given constraints (e.g., maximise area given a fixed perimeter).
• Develop reasoning skills by linking algebraic methods with contextual problem-solving.
• Collaborate respectfully, valuing diverse approaches and drawing from New Zealand’s bicultural context.

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Cultural Responsiveness

• Bicultural perspective:
  Incorporate contexts relevant to Aotearoa New Zealand, such as land use (māra kai/garden design), or traditional navigational or construction problems related to Māori knowledge systems (e.g., maximising resource efficiency on a waka).
• Multicultural inclusion:
  Encourage students to share and compare problem-solving strategies from their cultural backgrounds, fostering a learning community where multiple perspectives are valued.
• Use examples and language that respect Māori and Pasifika worldviews and encourage students to work collaboratively, building on whanaungatanga (relationships).
• Acknowledge mathematical knowledge as a taonga (treasure) passed through generations in various cultures, including indigenous traditions.

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Equipment and Resources

• Graph paper or digital graphing tools (e.g., GeoGebra).
• Calculators or computers/tablets with spreadsheet or graphing capability.
• Whiteboard and markers.
• Problem task sheets with real-life contexts (see attached).
• Measuring tools or materials (optional physical models such as cardboard to represent shapes).

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Lesson Outline

Time	Activity	Description
0-10	Engage & Activate Prior Knowledge	Anchor discussion with real-life contexts (e.g., fencing a rectangular paddock for maximum area with fixed fence length). Ask questions to elicit students’ prior experience with perimeter and area. Use short “think-pair-share” to brainstorm strategies. Introduce key vocabulary.
10-25	Explore — Modelling the Problem	Introduce a practical optimisation problem: Maximise the area of a paddock with a fixed perimeter of 100m. Guide students to: • Define variables (e.g., length = x, width = y). • Write an equation for the perimeter constraint. • Express area as a function of one variable. Use tables or spreadsheets to calculate and compare areas for different x values.
25-35	Explain — Algebraic Approach and Graphing	Demonstrate forming the quadratic equation for area. Sketch or use technology to graph area vs length. Discuss the parabola shape and vertex representing maximum area. Connect graph features to algebraic solutions.
35-45	Elaborate — Collaborative Problem Solving	In small groups, students solve a related problem (e.g., designing rectangular gardens or containers with different constraints). They create tables, equations, and graphs to find optimal dimensions. Emphasise use of mathematical language and reasoning. Teacher circulates, prompting reflection and explanation.
45-55	Reflect and Connect	Whole-class discussion on strategies, findings, and how they relate to real-world contexts like design and budgeting. Encourage students to explain their reasoning and listen to peers’ approaches. Highlight connections to wider contexts and other curriculum strands (like measurement and geometry).
55-60	Summary and Formative Assessment	Recap key ideas: modelling constraints, using tables and graphs for optimisation. Quick individual quiz or exit question: “Explain why finding the vertex of a parabola is important in these problems.” Preview next steps in algebra and real-world applications.

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Assessment Opportunities

Formative:

• Observation of group collaboration and reasoning.
• Students create and explain tables and graphs showing optimisation.
• Exit quiz questions assessing understanding of maximum/minimum in quadratic contexts.

Summative:

• Can be linked with NCEA Level 1 assessments on quadratic equations, functions, and optimisation problems involving context .

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Teacher Tips and Extensions

• Use digital graphing tools to allow dynamic manipulation of variables and visual connection to algebraic forms.
• Link with financial maths, e.g., maximise profit given costs and income constraints.
• Use kinaesthetic models (string, cardboard) to make the shapes tangible for learners needing concrete learning experiences.
• Encourage use of Te Reo Māori terms when naming shapes or measurement units where appropriate.

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Specialist Vocabulary

• Variable, function, constraint, quadratic equation, parabola, vertex, maximum, optimise, coefficient
• Perimeter, area, dimensions, table of values, graph, axis

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Mātauranga Māori Connection Examples

• Explain geometry and measurement concepts through kōwhaiwhai patterns and weaving, where patterns and optimisation play a role in design.
• Explore historical and contemporary Māori applications of measurement and geometry in waka construction, pā building, and land division.
• Encourage students to discuss or research how traditional Māori navigators calculated optimal routes and distances—applying similar maths concepts in practical ways.

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This lesson plan draws directly on the New Zealand Curriculum’s 2025 draft for Mathematics and Statistics at Year 11, emphasising authentic contexts, making links across maths strands, and embedding cultural responsiveness consistent with the bicultural foundations and multicultural environment of New Zealand classrooms .

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If you would like, I can prepare detailed student worksheets and exemplar solutions to accompany this plan.