Maximising and Minimising Quantity

Overview

This 60-minute lesson is designed for Year 11 students in Aotearoa New Zealand to explore optimisation problems involving maximising and minimising quantities. It aligns with the New Zealand Curriculum (NZC) and uses real-world contexts linked to students’ experiences to develop algebraic and graphical reasoning skills.

---

Achievement Objectives (NZC)

• Number and Algebra:

  • Find solutions that maximise or minimise a quantity while meeting constraints, by making lists, tables, graphs and comparing values
  • Form and solve linear and quadratic equations from real-life contexts

• Mathematical Processes:

  • Investigate, represent, and communicate mathematical findings clearly and confidently
  • Reason mathematically and justify solutions

---

Key Competencies Addressed

• Thinking: Developing problem-solving and reasoning skills through optimisation problems.
• Using language, symbols, and texts: Interpreting and representing relationships using graphs and algebraic expressions.
• Relating to others: Encouraging collaboration and respect for diverse approaches.
• Managing self: Building resilience by trial-and-error and perseverance in problem-solving.
• Participating and contributing: Sharing findings and considering multiple perspectives.

---

Cultural Responsiveness

This lesson embraces New Zealand’s bicultural foundation by:

• Using contextual examples relevant to students’ lives in Aotearoa, including situations from marae planning or local environmental design, honouring mātauranga Māori perspectives on space and resource use.
• Encouraging reflection on different worldviews in problem-solving approaches, integrating culturally relevant cooperative group work.
• Recognising the role of language through vocabulary explanations that affirm students’ diverse linguistic backgrounds and the use of clear NZ English spelling.

---

Learning Intentions

Students will be able to:

• Model optimisation problems with algebraic expressions and graphs.
• Use tables and graphs to find maximum or minimum values.
• Solve related linear and quadratic equations.
• Interpret their solutions in context, checking reasonableness.

---

Resources

• Graphing technology or graph paper
• Whiteboard and markers
• Exercise book or worksheets with optimisation problems
• Digital tools (optional, e.g., GeoGebra)

---

Lesson Breakdown

1. Getting Started (10 mins)

• Activate prior knowledge: Brief class discussion — “What does maximising or minimising a quantity mean in real life?” (e.g. maximise area of garden with fixed fencing)
• Introduce everyday examples (e.g. packaging design to hold maximum volume but reduce material).
• Learning intention and success criteria shared clearly. Use simple language and connect to students’ experiences.

2. Exploring and Modelling (20 mins)

• Problem introduction: For example, "Given 40m of fencing, what is the maximum area a rectangular paddock can have?"
• Students work in pairs to:
  • Make tables of length vs area
  • Sketch graphs showing relationship
  • Identify where the area is maximised
• Teacher circulates to prompt reasoning: “What happens if you increase one side? Why does the area change that way?”

3. Applying Algebra and Graphs (15 mins)

• Develop the algebraic formula: Area = length × width and relate to fencing constraint (perimeter formula).
• Solve for one variable and express area as quadratic function.
• Explore how to use completing the square or graphing technology to find maximum area.
• Highlight significance of vertex in parabola as max point.
• Support students to interpret solutions contextually (e.g., dimensions must be positive).

4. Connecting and Reflecting (10 mins)

• Groups share their findings and methods; discuss different approaches (tables, graphs, formula manipulation).
• Reflect on challenges or surprises in the problem-solving process.
• Reinforce vocabulary and concepts such as 'constraints', 'maximum', 'vertex'.
• Summarise how algebra and graphical methods complement each other in optimisation tasks.

5. Wrap-up and Next Steps (5 mins)

• Students self-assess their confidence with optimisation problems.
• Set a simple homework task: find a real-life example of optimisation at home or school and describe how they might model it mathematically.
• Preview next lesson focus (e.g., solving systems of equations in context).

---

Assessment Opportunities

• Observation of group discussions demonstrating reasoning and collaboration.
• Work samples from tables, graphs, and algebraic solutions showing understanding of maximising/minimising under constraints.
• Reflection responses linking maths to real-life application.

---

Differentiation and Extensions

• Provide scaffolded worksheets with guiding questions for students needing additional support.
• Challenge advanced learners with more complex constraints (e.g., fencing multiple paddocks) or solve using technology.
• Use real data or cultural contexts relevant to the students’ communities for richer engagement.

---

References

• New Zealand Curriculum, Mathematics and Statistics, Number and Algebra Strand – Year 11 Achievement Objectives
• NCEA Level 1 Mathematics Achievement Standards guidance – Optimisation and related topics
• Te Mātaiaho Mathematics and Statistics document (draft) January 2025

---

This carefully structured lesson plan ensures learners engage deeply with mathematics through meaningful, authentic contexts while meeting NZ curriculum expectations and embodying culturally responsive teaching. The integration of discussion, practical investigation, and algebraic formalism provides multiple entry points and engages a diverse Year 11 cohort effectively.