Exploring Quadratics Creatively

Overview

Duration: 60 minutes
Class Size: 30 students
Year Level: Year 11 (around 16–17 years old)
Subject: Mathematics
Curriculum: Australian Curriculum – Mathematics Standard (Version 9.0)
Strand & Sub-Strand:

• Strand: Algebra
• Sub-Strand: Functions
• Content Description Code: ACMSM084 – Explore the features of quadratic functions, including symmetry, intercepts, maxima, minima, and the effect of dilations, reflections and translations using digital tools and symbolic manipulation.

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Learning Intentions

By the end of this lesson, students will:

1. Understand the key features of quadratic functions from both algebraic and graphical perspectives.
2. Explore transformations that affect the shape and position of parabolas.
3. Use digital technology creatively to investigate quadratic behaviour.
4. Interpret and construct mathematical representations with increasing sophistication.

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Success Criteria

Students will be able to:

• Identify and describe key features of quadratic graphs (vertex, axis of symmetry, intercepts).
• Apply transformations to sketch and interpret variations of quadratic functions.
• Collaboratively use digital tools to model and analyse quadratic behaviour.
• Communicate mathematical ideas using appropriate terminology and notation.

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Resources Required

• Mini whiteboards and markers (one per student)
• Graphing calculators or devices with Desmos Graphing Calculator or GeoGebra pre-loaded
• Grid paper handouts
• Pre-cut coloured transparent overlays (red, blue, green)
• Virtual whiteboard or projector
• Printed 'Quadratic Transformations Code Challenge' pack

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Lesson Structure

⏰ 0–10 min: Tuning In – Mini Whiteboard Challenge

Activity: Students enter to see a large graph of a basic quadratic function on the board:
y = x²

Teacher quickly calls out equations and students sketch graph shapes on mini whiteboards:

1. y = (x - 3)²
2. y = x² - 4
3. y = -2x²

Goal: Activate prior knowledge of transformations.

Engagement Strategy: Students show boards at the same time for rapid feedback. Encourage noticing patterns:

• Horizontal shifts
• Vertical shifts
• Reflections
• Stretching/Shrinking

Teacher Tip: Use this as informal diagnostic assessment.

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⏰ 10–25 min: Explicit Instruction – Quadratic Transformers

Method: Using teacher laptop and projector, guide students through exploring how each part of the function y = a(x - h)² + k transforms the graph.

• a = opens up/down, stretches/shrinks
• h = moves left/right
• k = moves up/down

Concrete Demonstration: Overlay printed parabolas on coloured transparency sheets. Stack them to compare shapes visually before showing digitally.

Student Note-Taking Prompts:

• What role does each constant play in affecting the graph?
• Use terminology: vertex, dilate, reflect, translate.

Visual Anchor: Display Desmos/GeoGebra simulations. Let students predict before clicking to reveal.

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⏰ 25–40 min: Creative Exploration – Transform It Tournament

Activity: Students work in pairs on devices to complete an investigation where they must:

1. Start with y = x²
2. Match given graphs using transformations
3. Use sliders on technology to test predictions and refine conjectures
4. Record successful equations in their workbook

Pairs will attempt to complete the “Quadratic Transformations Code Challenge”. Each task unlocks part of a final "code".

Differentiation: Advanced students are given a more complex graph and asked to derive possible equations.

Teacher Role: Rotate around, facilitate questioning:

• "What happens if you change 'a' to a decimal?"
• "How would you undo this transformation?"

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⏰ 40–50 min: Synthesis – Redesign the Ride (Real World Task)

Scenario: "You’re engineering a new rollercoaster drop. Parabolas are everywhere in rollercoaster design!"

Task: Students are asked to:

• Sketch a rollercoaster drop that follows a parabola using grid paper
• Label key features (vertex, axis of symmetry, maximum/minimum point)
• Write a possible equation for the trajectory

Use coloured overlays to simulate different designs quickly and overlay corrected versions for class discussion.

Link to ACMSM084: Promotes understanding of contextual application of quadratic features and transformations.

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⏰ 50–60 min: Reflection and Mini Assessment

Plenary:

• Students complete a short quiz:
  • 3 multiple choice transformation questions
  • 1 short answer: “Describe how the graph of y = (x – 2)² – 3 was transformed from y = x².”

Reflection Exit Slip (on sticky notes):

• “One thing I learned about quadratics today was…”
• “A question I still have is…”

Stick on the classroom door as they leave.

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Assessment Opportunities

• Visual checks and feedback during mini whiteboard activity
• Observations during partner digital work
• Exit slip responses
• Formal mini quiz at the conclusion

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Modifications and Differentiation

For EAL/D Learners:

• Provide a visual glossaries page of key terms with icons
• Use sentence starters for graph descriptions

For High Achievers:

• Investigate piecewise functions
• Design a multi-segment rollercoaster combining linear, quadratic, and exponential pieces

For Students Requiring Support:

• Work with concrete transformations using graph overlays first
• Pair with a peer tutor in Desmos exploration

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Extension Activity (Homework or Optional Task)

Create a 30-second animation or visual slide set modelling a real-world quadratic application (fountain trajectory, sports shot, satellite dish). Include the function and an explanation of the meaning of each parameter.

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Cross-Curricular Links

• STEM/Physics – Modelling projectile motion
• Visual Arts – Symmetry and parabolic designs
• Digital Technologies – Coding sliders in Desmos or Graphing tools

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Teacher Reflection Prompts

• Did students demonstrate stronger understanding using digital tools?
• Were students able to describe transformations with confident language?
• Could this approach be extended to other types of functions (exponential, trigonometric)?

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Final Thought

This lesson blends concrete visual tools, abstract algebra manipulation, creative design, and collaborative reasoning. It fuses mathematical rigour with accessible innovation—all directly tied to the Australian Mathematics Curriculum and designed to foster deep conceptual understanding.