Class: Senior Secondary School 1 (SS1, SS 1, SSS1, SSS 1)
Term: Second Term
Week: 7
Age: 15 years
Duration: 45 minutes
Subject: Technical Drawing
Curriculum Theme: Technical Drawing
Previous Lesson: Equal Areas: Theorems of Similar Figures.
Topic: EQUAL AREAS OF SIMILAR FIGURES
Subject Matter: construction of triangles of equal area, construction of quadrilaterals of equal area, construction of polygons of equal area, application of equal area theorems in construction tasks

Specific Objectives

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

Cognitive Domain:

Define the principle of equal areas in geometric constructions.
State the theorems related to constructing figures of equal areas.

Affective Domain:

Appreciate the importance of accuracy in technical drawing constructions.
Show interest in solving practical geometric problems using equal area principles.

Psychomotor Domain:

Construct triangles of equal area to given figures.
Construct quadrilaterals of equal area to given figures.
Construct polygons of equal area to given figures.

Collaborate effectively with peers during practical construction exercises.

Reference Materials

The following resources were used in planning this lesson:

9 Years Basic Education Curriculum for Senior Secondary Schools.
State Unified Scheme of Work for Technical Drawing (SSS 1).
Oluyemi, A.B. (2010). Technical Drawing for Senior Secondary Schools. University Press PLC.
Geometric drawing charts and posters.

Instructional Materials

The teacher will teach this lesson with the aid of:

Drawing board or T-square
Set squares (45 and 60 degrees)
Compass
Ruler
Pencils (HB, 2H)
Eraser
Drawing paper
Charts showing examples of constructions of equal areas
Posters illustrating equal area theorems

Rationale for the Lesson

This lesson helps pupils understand how to convert complex shapes into simpler ones while maintaining the same area, which is important for various design and engineering applications. It enables pupils to develop precision and problem-solving skills in geometric constructions, skills that are useful in many technical fields.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of basic geometric constructions, properties of triangles, quadrilaterals, and polygons, and the use of drawing instruments.

Lesson Content/Board Summary

EQUAL AREAS OF SIMILAR FIGURES

Principle of Equal Areas

The principle of equal areas states that certain geometric figures can have the same area even if their shapes are different. A common application involves transforming a polygon into a triangle of equal area.

Key principles include:

Triangles on the same base and between the same parallel lines have equal areas.
Triangles with equal bases and equal heights have equal areas.

Construction of Triangles of Equal Area

To construct a triangle of equal area to another, one can use the principle that triangles on the same base and between the same parallel lines have equal areas.

Steps to construct a triangle equal in area to a given triangle (e.g., ABC) with a new base (e.g., AD):

Draw a line parallel to the new base AD passing through the vertex C.
Extend side AB to meet the parallel line at a point (say, E).
Triangle ADE will have an equal area to triangle ABC.

Construction of Quadrilaterals of Equal Area

A quadrilateral can be converted into a triangle of equal area. This is useful for finding the area of the quadrilateral by calculating the area of the equivalent triangle.

Steps to convert a quadrilateral (e.g., ABCD) into a triangle of equal area:

Draw a diagonal (e.g., AC).
From the opposite vertex (D), draw a line parallel to the diagonal AC to intersect the extended base AB at a point (e.g., E).
Connect C to E.
Triangle BCE will have an equal area to the quadrilateral ABCD.

Construction of Polygons of Equal Area

Any irregular polygon can be converted into a triangle of equal area by systematically reducing the number of sides.

General steps to convert a polygon into a triangle of equal area:

Select a vertex and draw a diagonal to a non-adjacent vertex.
From the intermediate vertex, draw a line parallel to the diagonal to intersect the extended side.
This process removes one vertex, reducing the number of sides by one, while maintaining the area.
Repeat the process until a triangle is formed.

Application of Equal Area Theorems in Construction Tasks

The principles of equal areas are applied in various technical drawing tasks, such as:

Calculating the area of irregular plots of land.
Redrawing complex shapes into simpler forms for analysis.
Designing components where area equivalence is important.
Dividing land or surfaces into equal parts.

Teaching Methods/Instructional Techniques

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

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction/Review
Teacher’s Activity: The teacher reviews previous knowledge on basic geometric constructions and asks pupils to recall how to find the area of simple shapes like triangles and rectangles. The teacher then introduces the idea of transforming shapes without changing their area.
Pupils’ Activity: Pupils respond to questions and actively participate in the review.
Learning Point: Pupils recall prior knowledge and are prepared for the new topic.

Step 2: Presentation of Topic

Time: 5 minutes
Teaching Skill: Information Giving
Teacher’s Activity: The teacher clearly states the topic for the day: “EQUAL AREAS OF SIMILAR FIGURES” and writes it on the board. The teacher briefly explains what will be covered in the lesson.
Pupils’ Activity: Pupils listen attentively and copy the topic into their notebooks.
Learning Point: Pupils know the lesson’s focus.

Step 3: Explanation of Equal Area Principle

Time: 7 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains the fundamental principle of equal areas using simple diagrams on the board, emphasizing that triangles on the same base and between parallel lines have equal areas. The teacher also explains how this principle is used to convert shapes.
Pupils’ Activity: Pupils observe the diagrams, listen to the explanation, and ask clarifying questions.
Learning Point: Pupils understand the theoretical basis for equal area constructions.

Step 4: Construction of Triangles of Equal Area

Time: 8 minutes
Teaching Skill: Demonstration/Guided Practice
Teacher’s Activity: The teacher demonstrates step-by-step on the board or a large drawing sheet how to construct a triangle of equal area to a given triangle, using the principle of parallel lines. The teacher ensures pupils follow along with their instruments.
Pupils’ Activity: Pupils carefully follow the teacher’s demonstration and attempt the construction on their drawing sheets.
Learning Point: Pupils acquire the skill to construct triangles of equal area.

Step 5: Construction of Quadrilaterals of Equal Area

Time: 8 minutes
Teaching Skill: Demonstration/Guided Practice
Teacher’s Activity: The teacher demonstrates the process of converting a given quadrilateral into a triangle of equal area. The teacher emphasizes drawing the diagonal and the parallel line from the opposite vertex.
Pupils’ Activity: Pupils observe the demonstration and practice the construction of converting a quadrilateral to a triangle of equal area.
Learning Point: Pupils learn to convert quadrilaterals into equivalent triangles.

Step 6: Construction of Polygons of Equal Area

Time: 7 minutes
Teaching Skill: Demonstration/Guided Practice
Teacher’s Activity: The teacher explains and demonstrates how to convert a more complex polygon (e.g., a pentagon) into an equivalent triangle by progressively reducing its sides using the equal area principle.
Pupils’ Activity: Pupils pay close attention to the step-by-step reduction process and begin practicing the construction.
Learning Point: Pupils develop the ability to convert polygons into equivalent triangles.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

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

Define the principle of equal areas in geometric constructions.
List the steps to convert a quadrilateral into a triangle of equal area.
State one practical application of equal area theorems in technical drawing.
Demonstrate the construction of a triangle of equal area to a given irregular polygon (on paper).

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 3 minutes
Teaching Skill: Summarization/Assignment
Teacher’s Activity: The teacher summarizes the key learning points of the lesson, reiterating the importance of equal area constructions. The teacher then gives homework, asking pupils to practice converting various polygons into triangles of equal area.
Pupils’ Activity: Pupils listen to the summary and copy the homework assignment.
Learning Point: Pupils consolidate their learning and are given tasks for further practice.

Lesson Keywords

Equal Areas – The concept of two or more geometric figures having the same surface measure, regardless of their shape.
Transformation – The process of changing a geometric figure’s shape or position without altering its fundamental properties, such as area.
Parallel Lines – Lines that are equidistant from each other and never meet, important for constructing equal area triangles.
Equilateral Triangle – A triangle with all three sides of equal length. (Not directly covered but a basic shape)
Quadrilateral – A polygon with four sides.
Polygon – A closed plane figure bounded by straight lines.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex polygons for conversion or challenge them to find alternative construction methods. For pupils who struggle, the teacher can provide one-on-one assistance, simplify the shapes, or use pre-drawn diagrams as templates to guide their constructions.

Note for teachers using this lesson plan

Ensure that all pupils have access to complete drawing instruments. Emphasize neatness and accuracy in all constructions. Encourage pupils to explain their steps as they work, fostering deeper understanding. Practical demonstrations are key for this topic; use a large board or projector if available.