Exploring Quadratics Creatively

Overview

Duration: 60 minutes
Class size: 30 Year 11 students
Curriculum Area: Mathematics & Statistics
Curriculum Level: Level 6 – New Zealand Curriculum
Achievement Objectives (Algebra strand):

• Form and solve quadratic equations.
• Reason logically and analytically when solving problems and modelling situations using algebraic methods.
• Interpret and communicate mathematical and statistical ideas effectively.

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Learning Intentions

By the end of the lesson, students will:

• Understand how quadratic expressions and equations are formed.
• Identify and apply the key features of quadratic graphs (vertex, axis of symmetry, intercepts).
• Solve and analyse quadratic equations using factoring, completing the square, and the quadratic formula.
• Connect algebraic and graphical representations in meaningful real-world contexts.

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Success Criteria

Students will be able to:

• Solve quadratic equations with at least two methods.
• Match quadratic graphs to their corresponding equations.
• Explain real-world applications of quadratics confidently in small group discussions.
• Collaborate in creative group work to present a mathematical model involving quadratic relationships.

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Materials Required

• Graph paper and pencils
• Laptops/tablets (if available) with graphing software like GeoGebra or Desmos (offline if needed)
• Printed scenario handouts (“Parabolic Pathways”)
• Blu-tack or magnets for whiteboard
• Colour markers / highlighters
• Printed card sets with quadratic equations and graph segments
• 10-metre measuring tape, masking tape, and small beanbags for motion activity

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Lesson Outline

1. Hook Activity – Quadratics in the Air (10 minutes)

Objective: Engage students with a real-world context involving projectile motion.

• Pose the scenario: "What shape does a thrown beanbag create in the air?"
• Take students outside or into a large space. Create a taped starting line.
• Invite 3 students to throw a beanbag gently and observe its motion.
• Ask the class: "What type of graph do you think models this? Why?"
• Back in class, sketch an estimated parabolic curve on the whiteboard as students describe the motion.

🔹 Curriculum Integration: Apply mathematics to model and understand physical phenomena.

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2. Explicit Teaching – Anatomy of a Quadratic (15 minutes)

Objective: Demonstrate how quadratic equations are structured and interpreted.

• Review general form: 𝑦 = ax² + bx + c
• Explore how changes in 'a', 'b', and 'c' affect the graph:
  • Positive vs negative 'a': direction of parabola
  • Value of 'c': y-intercept
  • Effect of 'b' on the axis of symmetry
• Display three different quadratic equations and their corresponding graphs (printed or on projector).
• Engage class with diagnosing what changed and predicting new graph shapes.
• Students use mini whiteboards or exercise books to sketch approximations of example graphs.

🔹 WOW Moment: Reveal a graph where 'a' = 0 and ask, “What shape is this now?” (linear connection)

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3. Collaborative Learning – Quadratic Detective (15 minutes)

Objective: Deepen understanding by analysing graph-equation pairs.

• In groups of 3, students receive a set of 6 cards: 3 with quadratic graphs, 3 with equations.
• Task: Match each equation to a graph with reasoning.
• Extension: Design a new quadratic graph and write a matching equation.
• Circulate to prompt mathematical vocabulary: vertex, axis of symmetry, factor, root.

🔹 Key Competencies: Managing self, participating and contributing, thinking critically.
🔹 Assessment for Learning: Listen to reasoning, check matches, provide feedback.

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4. Applied Modelling Task – Parabolic Pathways (15 minutes)

Objective: Create and interpret a real-world quadratic model.

• Each group receives a fictional scenario (e.g., designing an arch bridge, modelling a skateboard ramp, calculating water trajectory from a fountain).
• Students identify:
  • What variables are involved?
  • What kind of quadratic fits the model?
  • What does the vertex represent?
  • What do the x-intercepts represent in context?

Students sketch a rough sketch and write a sample quadratic equation using estimated values.
Optional: Students use GeoGebra to graph and validate their model.

🔹 Integration: Literacy (interpreting real-world contexts), Science (physics of motion)

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5. Quickfire Challenge – Solve It Your Way (5 minutes)

Objective: Reinforce multiple methods for solving quadratic equations.

• Display: x² – 5x + 6 = 0
• Challenge students to solve it using BOTH:
  • Factoring
  • Quadratic formula
• Reward teams who show both methods and compare for verification.

🔹 Stretch & Challenge: Offer a non-factorable trinomial to those that finish early.

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6. Reflection & Wrap-up (5 minutes)

Objective: Consolidate learning and gather formative feedback.

• Students complete a “Quadratic Snapshot” slip:
  • One key thing I learnt today was...
  • One thing I’m still unsure about is...
• Teacher collects these to inform mini-lessons or groupings next lesson.
• End with a teaser: "Next week, we’ll explore how quadratics underpin the flight of basketballs and the architecture of Auckland’s Sky Tower!"

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Differentiation & Support

• Extension:

  • Students write a real-world quadratic problem and swap with peers to solve.
  • Explore complex roots via discriminants.

• Support:

  • Keyword banks available.
  • Visual aids showing steps for solving quadratics.
  • Peer partnerships for scaffolding.

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Assessment Opportunities

• Formative:

  • Observation during group tasks
  • “Solve It Your Way” responses
  • Exit slips (“Quadratic Snapshot”)

• Summative (future lesson):

  • Draft or model project presentation of a real-world quadratic situation

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Teacher Reflection Suggestions

After the lesson, consider:

• Which students engaged best with the modelling task?
• Did students grasp the connection between algebra and graphs?
• How could the game element be extended in future?
• Could digital modelling tools enhance understanding more effectively next time?

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Final Note

This lesson aligns with the vision of the NZ Curriculum by empowering learners to become confident, connected, actively involved, and lifelong learners. Mathematics is presented as both functional and imaginative—connecting with curious minds through movement, modelling, and meaning.