Polygons, Interior and Exterior Angles and 3D Figure Properties for Primary 6

Class: Primary 6
Term: 3rd Term
Week: 7
Age: 11 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Geometry and Measurement
Previous Lesson: Scale Drawing, Ratios, Conversions, Worked Examples and Uses.
Topic: POLYGONS
Subject Matter: Names and features of 2-dimensional polygons, sum of interior angles of a regular polygon, exterior angles of regular polygons, describing 3-dimensional figures and examples, properties of 3-dimensional figures such as prism pyramid cube cone, quantitative aptitude on 2D and 3D figures.

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

• Identify and name different 2-dimensional polygons and their features.
• Calculate the sum of interior angles of a regular polygon.
• Calculate the measure of an exterior angle of a regular polygon.
• Describe 3-dimensional figures and provide examples.
• State the properties (faces, edges, vertices) of common 3-dimensional figures.
• Solve quantitative aptitude problems involving 2D and 3D figures.

Affective Domain:

• Appreciate the importance of polygons in real-life objects and structures.
• Develop an interest in exploring the properties of geometric shapes.

Psychomotor Domain:

• Draw simple 2D polygons.
• Construct simple 3D figures using available materials (if applicable).

Social Domain:

• Collaborate with peers to solve problems related to polygons.

Reference Materials

The following resources were used in planning this lesson:

• 9 Years Basic Education Curriculum (Primary Mathematics)
• State Unified Scheme of Work (Primary 6 Mathematics)
• New General Mathematics for Primary Schools, Book 6
• Textbook
• Workbook
• Formula book
• Charts

Instructional Materials

The teacher will teach this lesson with the aid of:

• Textbook
• Workbook
• Formula book
• Charts showing various 2D and 3D shapes
• Models of cubes, pyramids, prisms, and cones

Rationale for the Lesson

This lesson helps pupils understand the basic shapes that make up objects around them, both flat and solid. Learning about polygons enables pupils to identify, describe, and work with geometric figures, which is important for problem-solving in mathematics and understanding the physical world.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of basic plane shapes like squares, rectangles, triangles, and circles, and understand the concept of angles.

Lesson Content/Board Summary

POLYGONS

Names and Features of 2-Dimensional Polygons

A polygon is a closed plane figure made up of three or more straight line segments. These segments are called sides, and they meet at points called vertices.

The following are common 2-dimensional polygons and their number of sides:

• Triangle – 3 sides
• Quadrilateral – 4 sides
• Pentagon – 5 sides
• Hexagon – 6 sides
• Heptagon – 7 sides
• Octagon – 8 sides
• Nonagon – 9 sides
• Decagon – 10 sides

Sum of Interior Angles of a Regular Polygon

An interior angle is an angle inside a polygon, formed by two adjacent sides.

The sum of the interior angles of a polygon with ‘n’ sides is given by the formula:

Sum of interior angles = (n – 2) × 180°

Example 1: Find the sum of the interior angles of a pentagon.

Solution:

Step 1: Identify the number of sides (n). For a pentagon, n = 5.

Step 2: Apply the formula: Sum = (n – 2) × 180°

Sum = (5 – 2) × 180°

Sum = 3 × 180°

Sum = 540°

Example 2: A polygon has a sum of interior angles of 720°. How many sides does it have?

Solution:

Step 1: Write down the formula: (n – 2) × 180° = Sum

Step 2: Substitute the given sum: (n – 2) × 180° = 720°

Step 3: Divide both sides by 180°: n – 2 = 720° / 180°

n – 2 = 4

Step 4: Add 2 to both sides: n = 4 + 2

n = 6

The polygon is a hexagon.

Exterior Angles of Regular Polygons

An exterior angle is an angle formed by one side of a polygon and the extension of an adjacent side.

The sum of the exterior angles of any convex polygon is always 360°.

For a regular polygon with ‘n’ sides, each exterior angle is given by the formula:

Each exterior angle = 360° / n

Example 1: Find the measure of one exterior angle of a regular octagon.

Solution:

Step 1: Identify the number of sides (n). For an octagon, n = 8.

Step 2: Apply the formula: Each exterior angle = 360° / n

Each exterior angle = 360° / 8

Each exterior angle = 45°

Example 2: A regular polygon has an exterior angle of 60°. How many sides does it have?

Solution:

Step 1: Write down the formula: 360° / n = Each exterior angle

Step 2: Substitute the given exterior angle: 360° / n = 60°

Step 3: Multiply both sides by n: 360° = 60° × n

Step 4: Divide both sides by 60°: n = 360° / 60°

n = 6

The polygon is a hexagon.

Describing 3-Dimensional Figures and Examples

3-dimensional (3D) figures are solid shapes that have length, width, and height. They occupy space.

Examples of 3D figures include:

• Cube: A 3D shape with 6 identical square faces.
• Cuboid: A 3D shape with 6 rectangular faces.
• Prism: A 3D shape with two identical and parallel bases (polygons) and rectangular side faces.
• Pyramid: A 3D shape with a polygon base and triangular faces that meet at a single point (apex).
• Cylinder: A 3D shape with two identical circular bases and a curved side.
• Cone: A 3D shape with a circular base and a curved surface that tapers to a single point (apex).
• Sphere: A perfectly round 3D object where every point on its surface is equidistant from its center.

Properties of 3-Dimensional Figures

3D figures have faces, edges, and vertices.

• Face: A flat surface of a 3D shape.
• Edge: A line segment where two faces meet.
• Vertex (plural: vertices): A point where three or more edges meet.

The following are properties of some 3-dimensional figures:

• Cube:
  • 6 faces (all squares)
  • 12 edges
  • 8 vertices
• Cuboid:
  • 6 faces (all rectangles or some squares and some rectangles)
  • 12 edges
  • 8 vertices
• Triangular Prism:
  • 5 faces (2 triangles, 3 rectangles)
  • 9 edges
  • 6 vertices
• Square Pyramid:
  • 5 faces (1 square, 4 triangles)
  • 8 edges
  • 5 vertices
• Cylinder:
  • 3 faces (2 circular, 1 curved surface)
  • 2 edges (circular edges)
  • 0 vertices
• Cone:
  • 2 faces (1 circular, 1 curved surface)
  • 1 edge (circular base edge)
  • 1 vertex (apex)

Quantitative Aptitude on 2D and 3D Figures

This involves solving problems that require understanding the relationships between different shapes or applying formulas in practical scenarios.

Example: A square has a side length of 5 cm. If it is used as the base of a cube, what is the total surface area of the cube?

Solution:

Step 1: Identify the shape and given information. The base is a square with side 5 cm, meaning the cube has side length (l) = 5 cm.

Step 2: Recall the formula for the area of one face of a cube. Area of one face = l × l = 5 cm × 5 cm = 25 cm².

Step 3: Recall the number of faces in a cube. A cube has 6 identical faces.

Step 4: Calculate the total surface area. Total surface area = 6 × Area of one face = 6 × 25 cm² = 150 cm².

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Set Induction

Teacher’s Activity: The teacher greets the pupils and asks them to name some shapes they see around the classroom (e.g., blackboard, desk, window). The teacher then asks what makes these shapes different from each other.

Pupils’ Activity: Pupils respond by naming shapes and discussing their observations.

Learning Point: Pupils recall prior knowledge of basic shapes and are introduced to the concept of polygons.

Step 2: Names and Features of 2-Dimensional Polygons

Time: 8 minutes

Teaching Skill: Explanation/Illustration

Teacher’s Activity: The teacher defines what a polygon is and uses charts to display various 2D polygons (triangle, quadrilateral, pentagon, hexagon, etc.). The teacher guides pupils to identify the number of sides and vertices for each polygon.

Pupils’ Activity: Pupils observe the charts, listen to the definitions, and identify the features of different polygons.

Learning Point: Pupils learn to identify and name 2D polygons based on their number of sides.

Step 3: Sum of Interior Angles of a Regular Polygon

Time: 8 minutes

Teaching Skill: Calculation/Demonstration

Teacher’s Activity: The teacher introduces the formula for the sum of interior angles of a polygon (n-2) × 180°. The teacher works through examples on the board, showing step-by-step calculations for finding the sum and finding the number of sides when the sum is given.

Pupils’ Activity: Pupils copy the formula and examples, actively participate in solving problems, and ask questions for clarification.

Learning Point: Pupils understand how to calculate the sum of interior angles and determine the number of sides of a polygon.

Step 4: Exterior Angles of Regular Polygons

Time: 7 minutes

Teaching Skill: Explanation/Problem-solving

Teacher’s Activity: The teacher explains what an exterior angle is and states that the sum of exterior angles is 360°. The teacher then presents the formula for one exterior angle (360°/n) and solves examples on the board for both finding the angle and finding the number of sides.

Pupils’ Activity: Pupils listen, copy the formulas and examples, and practice solving similar problems.

Learning Point: Pupils learn to calculate the measure of exterior angles and determine the number of sides of a regular polygon.

Step 5: Describing 3-Dimensional Figures and Examples

Time: 6 minutes

Teaching Skill: Visualisation/Categorisation

Teacher’s Activity: The teacher introduces 3D figures, explaining they have length, width, and height. Using models and charts, the teacher shows examples like cubes, pyramids, prisms, and cones, asking pupils to describe them based on their appearance.

Pupils’ Activity: Pupils observe the models and charts, describe the 3D figures, and give real-life examples.

Learning Point: Pupils are able to describe and give examples of various 3D figures.

Step 6: Properties of 3-Dimensional Figures

Time: 5 minutes

Teaching Skill: Identification/Listing

Teacher’s Activity: The teacher defines faces, edges, and vertices. Using the 3D models, the teacher guides pupils to count and state the number of faces, edges, and vertices for figures like a cube, cuboid, and pyramid.

Pupils’ Activity: Pupils actively count the faces, edges, and vertices on the models and state their properties.

Learning Point: Pupils identify and state the properties of 3D figures.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Define a polygon.
2. State the formula for the sum of interior angles of a polygon with ‘n’ sides.
3. Calculate the sum of the interior angles of a hexagon.
4. What is the sum of the exterior angles of any convex polygon?
5. If a regular polygon has 9 sides, what is the measure of one of its exterior angles?
6. Mention two examples of 3-dimensional figures.
7. How many faces, edges, and vertices does a cube have?

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 1 minute

Teacher’s Activity: The teacher summarizes the key points of the lesson, emphasizing the importance of understanding 2D and 3D shapes in everyday life. The teacher assigns homework: “Find the sum of interior angles of a nonagon. A regular polygon has an exterior angle of 40°. How many sides does it have?”

Pupils’ Activity: Pupils listen to the summary and copy the homework assignment.

Learning Point: Pupils consolidate their learning and apply it through homework.

Home Task: Solve three new exercises on POLYGONS. Show each step and check each answer.

Lesson Keywords

• Polygon – A closed plane figure made of straight line segments.
• Interior Angle – An angle inside a polygon formed by two adjacent sides.
• Exterior Angle – An angle formed by one side of a polygon and the extension of an adjacent side.
• 3-Dimensional Figure – A solid shape that has length, width, and height.
• Face – A flat surface of a 3D shape.
• Edge – A line segment where two faces meet.
• Vertex – A point where three or more edges meet.

Differentiation

The teacher will provide additional support to struggling pupils by offering simplified examples and one-on-one guidance. More advanced pupils will be challenged with complex quantitative aptitude problems or asked to explore properties of less common polygons/polyhedra.

Note for teachers using this lesson plan

Ensure that pupils have a clear understanding of the difference between 2D and 3D shapes. Use physical models and visual aids as much as possible to help pupils grasp the concepts, especially for 3D figures. Encourage active participation and peer learning. Review multiplication and division skills as they are essential for angle calculations.