Exploring Quadratics

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Overview

Unit: Functions and Relations Unveiled
Lesson: 8 of 12
Topic: Introduction to Quadratic Functions
Key Stage: KS4 – Year 10
Curriculum Reference: GCSE Mathematics (UK), covering topics from the Number, Algebra, and Graphs strands; specifically, this lesson aligns with:

• A12: Sketch graphs of quadratic functions, identifying roots, intercepts and turning points.
• A9: Generate and interpret graphs of functions.
• A4: Substitute numerical values into formulae and expressions.

Duration: 50 minutes
Class Size: 25 students
Teacher Notes: This lesson assumes learners have a solid grasp of linear functions, graphs, and basic algebraic manipulation. Designed to be visually interactive and conceptually rich whilst remaining accessible for learners exploring non-linear functions for the first time.

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Learning Objectives

By the end of the lesson, students will:

1. Understand the standard form of a quadratic function:
   f(x) = ax² + bx + c

2. Describe key features of a quadratic graph:

   • U-shaped structure (parabola)
   • Axis of symmetry
   • Vertex (turning point)
   • Direction of opening (upward/downward)
   • y-intercept

3. Compare linear and quadratic graphs visually and algebraically.

4. Use technology and graphing reasoning to explore patterns and make conjectures.

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Success Criteria

✔ I can recognise the standard form of a quadratic function.
✔ I can identify and describe the shape and key features of a quadratic graph.
✔ I can explain how quadratic graphs differ from linear functions.
✔ I can predict the impact of changing coefficients a, b, and c.

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Resources

• Whiteboard and markers
• Graphing calculators or access to Desmos or GeoGebra (prepared with templates)
• Printed card sets (Linear vs Quadratic Matching Cards)
• Mini whiteboards for students
• Visualiser
• Pre-drawn blank axes on A3 sheets for group activity
• Exit tickets

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Lesson Outline

⏱️ Starter (0–8 minutes) – “Linear or Not?”

Goal: Activate prior knowledge and transition students from linear to quadratic thinking.

• Display 4 graphs: two linear and two quadratic (without equations).
• Students use mini whiteboards to categorise them – "Linear" or "Not Linear".
• Discuss as a class:
  • "What makes a function linear? What’s different about the others?"
  • Highlight the shape of parabolas and how they visually distinguish quadratic functions.

Teacher Tip: Use probing questions to encourage detailed justifications:
“Why do you think this curve is not linear?” or “What happens as x gets larger or smaller?”

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⏱️ Main Teaching Input (8–20 minutes) – Introducing the Parabola

Goal: Define and explore key components of a quadratic function.

1. Present the Standard Form

• Write on board:
  f(x) = ax² + bx + c
  Emphasise:
  • a controls the "steepness" and direction
  • b shifts the vertex sideways
  • c determines the y-intercept

2. Show Visual Variations

Use the visualiser or projector to show plotted graphs:

• f(x) = x²
• f(x) = -x²
• f(x) = x² + 3
• f(x) = x² + 3x - 4

Student Thinking Prompt: “What’s changing? What stays the same?”
Encourage use of mathematical vocabulary: vertex, axis of symmetry, intercept.

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⏱️ Guided Practice (20–32 minutes) – Graph Discovery Activity

Goal: Students explore how values of a, b, and c affect the parabola.

Activity: Quadratic Coefficient Playground
In pairs, students use graphing software (or graphing calculators) with prepared sliders for a, b, and c.

Tasks:

1. Set b = 0, c = 0; change a:
   • Record observations when a > 0, a < 0, and a = 1, 2, 0.5.
2. Set a = 1, c = 0; change b:
   • Observe horizontal shifting and axis of symmetry.
3. Set a = 1, b = 0; change c:
   • Record the position of the y-intercept.

Scaffold Worksheet Includes:

• Fill-in-the-blank descriptions
• A sketch section
• Prediction before testing

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⏱️ Independent Activity (32–40 minutes) – Match, Sort, Justify

Goal: Concretise understanding of quadratic vs linear functions.

Task: Matching Card Game
Provide envelopes of 12 sets of mixed equations & graphs (6 linear, 6 quadratic functions – unlabeled).

Instructions:

• As a group of 3–4, match each function to its correct graph.
• Sort functions into two piles: linear and quadratic.
• Justify your matches verbally: “We think this is quadratic because the graph curves, and a squared term is present...”

Challenge Extension:
Can students order the quadratics by steepness or identify the ones with a negative leading coefficient?

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⏱️ Whole-Class Review (40–46 minutes) – Graph Gallery Walk

Goal: Peer insight, clarification, pattern spotting.

Stick A3 graph sheets (completed during Guided Practice) around the room. Students circulate in pairs with a “Wonder Card”:

“We wonder why graph X has a vertical stretch compared to Y”
“We noticed this graph has the same y-intercept as another – what does that tell us about the c value?”

Teacher facilitates discussion, confirms key terminology. Address misconceptions subtly noted during activities.

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⏱️ Consolidation & Exit Ticket (46–50 minutes) – 3-2-1 Reflection

Distribute Exit Tickets at the door:

• 3 things I learned about quadratic graphs
• 2 differences between linear and quadratic functions
• 1 question I still have

Collect for planning next lesson.

Optional Plenary Challenge (for fast-finishers):

If f(x) = x², sketch the effect of f(x) + 5 and f(x - 3). No calculator. Justify algebraically.

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Differentiation

• Support: Provide partially completed graphs or visual equation hints; peer pairing will be intentional.
• Stretch: Students to explore vertex form: f(x) = a(x - h)² + k and relate to transformations.
• ELL: Use visual cues extensively; template vocabulary sheets provided.

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Assessment Opportunities

• Targeted questioning during starter and gallery walk
• Observation of group activity discussions
• Exit tickets for formative assessment
• Matching card task to assess misconceptions

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Teacher Reflection Prompts (Post-lesson)

• Were students able to explain the role of each coefficient?
• Did the guided exploration produce enthusiasm and genuine curiosity?
• How effective was the card-sort in reinforcing graphical intuition?
• What responses on the exit ticket surprised you?

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Looking Ahead

Next Lesson (Lesson 9): Plotting Quadratics from Scratch
Students will begin to construct quadratic graphs by completing tables of values and will explore symmetry and vertex using real-world contexts (e.g., projectiles). This builds fluency and sets foundation for solving quadratic equations graphically.

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This lesson plan brings quadratic functions alive through visual play, interactivity, and discovery — honouring both UK maths standards and the teenage imagination.