Lesson Overview

Duration: 60 minutes
Class Size: 25 students
Unit: Polygon Angles Uncovered
Lesson Title: Exploring Interior Angles of Polygons
National Curriculum Links:

• Mathematics Program of Study: Geometry – Properties of Shapes (Year 11)
• Calculate and interpret the sums of interior angles of polygons (NC Objectives: GCSE Geometry - Properties of shapes, A1.1)
• Apply angle facts, properties of polygons, and geometric reasoning to solve problems (NC Objectives: GCSE Geometry - Reasoning and Problem Solving)

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Learning Objectives

By the end of the lesson, students will be able to:

1. Explain the concept of interior angles in polygons.
2. Derive and apply the formula for the sum of interior angles of an n-sided polygon: ((n-2) \times 180^\circ).
3. Calculate the sum of interior angles for triangles, quadrilaterals, pentagons, and other polygons.
4. Use reasoning to solve problems involving missing interior angles in polygons.
5. Communicate mathematically, creating a clear visual presentation of their findings using PowerPoint.

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Resources

• Whiteboard and markers
• Geometry sets (protractors, rulers)
• Worksheets with polygons for angle calculation practice
• Laptops/tablets with PowerPoint or similar presentation software for each group
• Pre-prepared polygon cut-outs (triangles, quadrilaterals, pentagons, hexagons)
• Interactive powerpoints slides to support explanation
• Mini-whiteboards for formative assessment

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Lesson Structure

1. Starter Activity (10 minutes)

Objective: Activate prior knowledge of basic polygon properties and angles.

• Begin by showing images of different polygons (triangle, quadrilateral, pentagon, hexagon).
• Ask students to shout out (or jot down) how many sides and angles each polygon has.
• Brief discussion: What do they know about the sum of interior angles of these shapes?
• Quick formative quiz: Using mini-whiteboards, students estimate sum of interior angles for a triangle and quadrilateral.
• Highlight that today’s focus will be on understanding and applying a formula for any polygon.

National Curriculum Alignment: Recall and use facts about angles, including those within polygons.

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2. Main Activity Part 1 – Guided Explanation & Investigation (15 minutes)

Objective: Understand and derive the formula ((n-2) \times 180^\circ) for interior angles sum.

• Using a large equilateral triangle cut-out, demonstrate how the triangle’s interior angles sum to 180°.
• Progress by joining triangles to form quadrilaterals and pentagons, showing the sum grows by 180° with each additional side.
• Guide students to observe the pattern: the polygon can be split into triangles, each with 180°, so the number of triangles = (n-2).
• Write and explain the formula step-by-step on the board.
• Check understanding: each student writes down the sum of interior angles of a polygon with 7 sides.

Differentiation:

• Support by scaffolding with polygon-to-triangle decomposition diagrams.
• Challenge: extend to exterior angle sum understanding briefly if students grasp quickly.

National Curriculum Alignment: Identify properties of polygons and use geometric facts to derive angle formulas.

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3. Main Activity Part 2 – Group Problem Solving (20 minutes)

Objective: Apply the formula to calculate missing interior angles and sum of interior angles.

• Students split into 5 groups of 5.
• Each group receives polygons (triangles, quadrilaterals, pentagons) with some interior angles marked and others missing.
• Task: calculate missing angles using the sum formula and reasoning (angle sum property of polygons).
• Encourage students to discuss reasoning, annotate polygons, and write explanations.
• Each group records their solutions and reasoning in a shared PowerPoint slide they create.

Creative Element: Groups include diagrams, colour-coding, and brief written explanations in their slides.

Teacher Role: Circulate to support, challenge misconceptions, and encourage precise mathematical language.

National Curriculum Alignment: Apply angle facts and problem-solving skills to unfamiliar polygonal shapes.

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4. Class Discussion and Presentation (10 minutes)

Objective: Consolidate understanding through peer teaching and visual communication.

• Each group presents their PowerPoint slide(s) to the class showcasing their problem, solution, and reasoning.
• Class provides constructive feedback emphasising clarity, correctness, and mathematical vocabulary.
• Teacher highlights strong reasoning and presentation elements, reinforcing key concepts.

National Curriculum Alignment: Present mathematical arguments and proof confidently.

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5. Plenary and Assessment (5 minutes)

Objective: Formative assessment of the learning objectives and reflection.

• Quick quiz on mini-whiteboards:
  • Calculate interior angles sum of polygons with 10 and 12 sides.
  • Find missing angle in an octagon given the other interior angles.
• Exit ticket: One sentence describing what they learned about polygon interior angles, one question they still have.

Teacher Notes: Review exit tickets post-lesson to tailor next steps or revision.

National Curriculum Alignment: Demonstrate knowledge of geometric properties and problem solving.

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Differentiation and Challenge

• Support: Additional scaffolding with visual aids and guided examples; one-to-one support if needed.
• Mainstream: Access to all activities with peer collaboration to promote deeper reasoning.
• Challenge: Investigate the angle sum of regular polygons and link to exterior angles/formulas for future lessons.

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Assessment

• Formative: Mini-whiteboard responses during starter and plenary, group problem solving explanations in slides.
• Summative Preparation: PowerPoint presentations document evidence of learning to be used for informal assessment and revision.

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Cross-Curricular Links

• ICT: Developing presentation and digital communication skills.
• English: Use of precise mathematical language and articulation during presentations.

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Reflection and Next Steps

This lesson is designed to build solid foundational understanding and encourage mathematical communication through technology. Next lessons can build on this by exploring exterior angles and applying polygon angle knowledge to real-world contexts and more complex polygons.