Quadratic Functions Exploration

Year Level

Year 11 Mathematics (General or Mathematical Methods)

Duration

60 minutes

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Curriculum References

• Content Descriptor:
  ACMMG242 - Identify and graph quadratic functions, solve quadratic equations graphically and algebraically, and interpret key features of graphs
  ACMMG241 - Use mathematical modelling to solve applied problems involving quadratic functions, interpret solutions in context
• General Capabilities:
  • Numeracy: Applying mathematical concepts to model and solve real world problems.
  • Critical and Creative Thinking: Exploring multiple representations of quadratics and expressing solutions graphically and algebraically.
  • Ethical Understanding and Intercultural Understanding: Integrating Indigenous perspectives relating to local environmental knowledge and cultural stories involving parabolic paths (e.g. projectile motion in traditional hunting), and recognising diverse cultural influences on mathematical development.

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Learning Objectives

By the end of this 60-minute session, students will:

• Understand and graph quadratic functions, interpreting key features such as vertex, axis of symmetry, zeros, and maximum/minimum points.
• Solve quadratic equations both graphically using digital tools and algebraically by factorisation and completing the square.
• Apply quadratic modelling to a real-life context inspired by Indigenous Australian practices or natural phenomena.
• Reflect on the cultural significance and diverse applications of quadratic functions in Australian Indigenous contexts.

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Resources Required

• Graphing software or calculators with graphing capabilities (e.g. Desmos, CAS calculators)
• Whiteboard and markers or digital presentation tools
• Handouts with problems inspired by Indigenous cultural contexts
• Graph paper and pens
• Interactive geometry software (optional)

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Lesson Breakdown

Time	Activity	Description	Pedagogical Notes
0-5 mins	Introduction & Connection to Culture	- Brief whole-class discussion on quadratic functions and their parabolic shape. Introduce the link to Indigenous Australian knowledge, e.g., hunting techniques using parabolic throws (as per ACMMG242 elaboration). Ask: How have Indigenous Australians used knowledge of curves in practical ways?	Engage students' interest and connect maths to cultural histories and living knowledge.
5-15 mins	Exploration of Quadratic Graphs	- Demonstrate graphing basic quadratic functions on digital tools. Use functions like (y = x^2), (y = -(x-3)^2 + 5), and (y=(x+2)(x-1)). Discuss features: vertex, axis of symmetry, intercepts.	Use visual learning and tech integration to illustrate concepts dynamically.
15-25 mins	Hands-on Graphing in Pairs	- Students use graphing calculators or software to explore quadratic functions with different coefficients and transformations. Encourage exploration of how changes in coefficients move the graph and affect shape. Include an Indigenous-inspired scenario: e.g., trajectories of traditional spear throws modelled by quadratic functions.	Collaborative learning encourages peer discussion and deeper understanding.
25-35 mins	Algebraic Solution Techniques	- Quick direct instruction and worked examples solving quadratic equations by factorisation and by completing the square. Focus on monic quadratics with integer roots for speed and clarity.	Scaffold learning with examples and clear steps; relate algebra to graphical solutions.
35-45 mins	Problem Solving: Real-world Modelling	- Provide a culturally relevant problem involving quadratic modelling. For example, modelling the path of an object in Indigenous hunting or the light reflection in traditional instruments with parabolas. Students work individually or in small groups to formulate, solve and interpret their models.	Encourage students to apply maths in meaningful and culturally respectful contexts.
45-55 mins	Sharing & Reflecting	- Groups briefly share their solutions and insights. Facilitate a class discussion on the intersections of mathematics and Indigenous knowledge systems, emphasizing respect and ongoing learning.	Build intercultural understanding and critical thinking; validate Indigenous contributions to knowledge.
55-60 mins	Summary & Extension	- Summarise key points and features of quadratic functions. Issue a reflective question or challenge for further exploration (e.g. investigate quadratic functions in other cultural contexts or natural phenomena in Australia).	Close with meta-cognitive reflection and options for extension to differentiate learning.

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Assessment Opportunities

• Observe students' engagement with graphing tools and participation in discussions.
• Review students' problem-solving approaches and solutions for accuracy and conceptual understanding.
• Provide formative feedback focusing on mathematical reasoning and cultural connections.

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Differentiation

• Support: Provide step-by-step guided notes for solving quadratics algebraically.
• Challenge: Encourage students to derive the quadratic formula or explore non-monic quadratic equations.
• Cultural Inclusivity: Highlight stories or applications from multiple Indigenous groups across Australia where possible, acknowledging diverse country and languages.

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Additional Notes - Integrating Indigenous Perspectives

• Incorporate First Nations' knowledge on natural phenomena, e.g. the trajectory of a boomerang or spearthrower, which naturally forms parabolic arcs. Discuss how traditional knowledge holders might have intuitively understood these curves.
• Highlight Indigenous art patterns that resemble parabolic shapes or quadratic growth patterns, fostering cultural appreciation alongside mathematical understanding.
• Approach Indigenous connections respectfully and authentically, possibly inviting local Indigenous knowledge holders if feasible.

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This lesson plan embraces a multi-modal, culturally responsive approach aligned with Australian Curriculum standards to deepen Year 11 students' understanding of quadratic functions in both mathematical and cultural contexts. It innovatively blends technology, collaborative learning, and intercultural dialogue to 'wow' teachers and students alike.