Class: Senior Secondary School 1 (SS1, SS 1, SSS1, SSS 1)
Term: 2nd Term
Week: 8
Age: 15 years
Duration: 45 minutes
Subject: Technical Drawing
Curriculum Theme: Technical Drawing
Previous Lesson: Constructing Plane Figures of Equal Areas.
Topic: TANGENTS AND TANGENCY
Subject Matter: principles of tangency, application of tangency, construction of tangent at a point on a circle, construction of tangents to two equal circles, construction of tangents to two unequal circles, tangents of arcs to straight lines

Specific Objectives

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

Cognitive Domain:

Define tangency and state its principles.
Mention at least three applications of tangency in real life.

Affective Domain:

Appreciate the importance of tangency in various technical fields.
Develop an interest in accurate geometrical constructions.

Psychomotor Domain:

Construct a tangent at a given point on a circle.
Construct common tangents to two equal circles.
Construct common tangents to two unequal circles.
Construct tangents of arcs to straight lines.

Work collaboratively with peers during practical construction exercises.

Reference Materials

The following resources were used in planning this lesson:

9 Years Basic Education Curriculum for Technical Drawing.
State Unified Scheme of Work for Technical Drawing SSS 1.
Engineering Drawing by N.D. Bhatt.
https://www.technicaldrawing.org/tangents
https://www.youtube.com/watch?v=tangency_constructions

Instructional Materials

The teacher will teach this lesson with the aid of:

Drawing instruments (set squares, protractor, compasses, ruler).
Drawing board.
Pencils (HB, 2H).
Drawing papers.
Whiteboard/Blackboard and markers/chalk.

Rationale for the Lesson

This lesson helps pupils understand how lines and curves touch each other at a single point without crossing. This knowledge is important for pupils to accurately draw and design objects in engineering, architecture, and other technical fields, making their designs functional and aesthetically pleasing.

Prerequisite/Previous Knowledge

Pupils have basic knowledge of geometrical constructions involving lines, circles, and angles.

Lesson Content/Board Summary

TANGENTS AND TANGENCY

Principles of Tangency

A tangent is a straight line that touches a curve or a circle at only one point, without crossing it. The point where the tangent touches the curve is called the point of tangency.

The following are key principles of tangency:

The radius drawn to the point of tangency is always perpendicular to the tangent line.
When two circles are tangent to each other, the line joining their centres passes through the point of tangency.

Application of Tangency

Tangency principles are used in various fields:

Engineering Design: For designing gears, cams, and other mechanical parts.
Architecture: In designing curved structures, arches, and road layouts.
Manufacturing: For precision cutting and shaping of materials.
Art and Graphics: In creating smooth curves and transitions in designs.

Construction of Tangent at a Point on a Circle

To construct a tangent to a circle at a given point P on its circumference:

Draw a circle with centre O and mark point P on its circumference.
Draw a radius from O to P.
Extend the radius OP beyond P to a suitable length.
At point P, construct a line perpendicular to the extended radius OP. This perpendicular line is the tangent.

Construction of Tangents to Two Equal Circles

To construct common tangents to two equal circles:

Draw two equal circles with centres O1 and O2.
Draw a line connecting their centres O1O2.
For direct (external) tangents: Draw a line parallel to O1O2 at a distance equal to the radius from each circle’s centre, then connect the points of intersection on the circles. Alternatively, bisect O1O2, draw a semicircle, and use it to find the tangent points.
For transverse (internal) tangents: Bisect O1O2. From the midpoint, draw arcs to intersect the circles. Connect these points to form the tangents.

Construction of Tangents to Two Unequal Circles

To construct common tangents to two unequal circles with centres O1 and O2 and radii R1 and R2:

Draw the two circles and connect their centres O1O2.
For direct (external) tangents: From the centre of the larger circle, draw a concentric circle with radius (R1 – R2). Draw tangents from O2 to this new circle. Extend these tangents to meet the larger circle and draw parallel lines from O2 to get the tangent points on the smaller circle.
For transverse (internal) tangents: From the centre of the larger circle, draw a concentric circle with radius (R1 + R2). Draw tangents from O2 to this new circle. Extend these tangents to meet the larger circle and draw parallel lines from O2 to get the tangent points on the smaller circle.

Tangents of Arcs to Straight Lines

This involves constructing an arc that is tangent to a given straight line at a specific point, or tangent to two lines, or tangent to a line and a circle. The general principle involves locating the centre of the tangent arc by using perpendiculars and offset lines.

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 if they know what happens when a moving car wheel just touches the ground at one point. The teacher then introduces the topic “Tangents and Tangency” and states the lesson objectives.
Pupils’ Activity: Pupils respond to the questions and listen attentively to the teacher.
Learning Point: Pupils are introduced to the concept of tangency and the lesson’s goals.

Step 2: Presentation of Principles of Tangency

Time: 7 minutes
Teaching Skill: Explanation/Illustration
Teacher’s Activity: The teacher explains the definition of a tangent and the point of tangency. The teacher illustrates the two main principles of tangency on the board, emphasizing that the radius at the point of tangency is perpendicular to the tangent.
Pupils’ Activity: Pupils listen, take notes, and ask questions for clarification.
Learning Point: Pupils understand the fundamental definition and principles of tangency.

Step 3: Presentation of Application of Tangency

Time: 5 minutes
Teaching Skill: Explaining/Relating
Teacher’s Activity: The teacher discusses various real-world applications of tangency, such as in designing gears, railway curves, and road layouts. The teacher encourages pupils to suggest other examples.
Pupils’ Activity: Pupils contribute ideas and learn about the practical relevance of tangency.
Learning Point: Pupils recognize the practical importance of tangency in everyday life and various industries.

Step 4: Demonstration of Tangent at a Point on a Circle

Time: 8 minutes
Teaching Skill: Demonstration/Modelling
Teacher’s Activity: The teacher demonstrates on the board how to construct a tangent at a given point on a circle using appropriate drawing instruments. The teacher explains each step clearly.
Pupils’ Activity: Pupils observe the demonstration carefully and attempt to follow the steps in their drawing books.
Learning Point: Pupils learn the step-by-step procedure for constructing a tangent at a point on a circle.

Step 5: Demonstration of Tangents to Equal and Unequal Circles

Time: 8 minutes
Teaching Skill: Demonstration/Guided Practice
Teacher’s Activity: The teacher demonstrates the construction of common direct and transverse tangents to two equal circles, and then briefly explains the principles for unequal circles. The teacher guides pupils as they attempt these constructions.
Pupils’ Activity: Pupils pay close attention and practice the constructions with their drawing instruments.
Learning Point: Pupils gain practical skills in constructing tangents between two circles of equal and unequal radii.

Step 6: Demonstration of Tangents of Arcs to Straight Lines

Time: 7 minutes
Teaching Skill: Demonstration/Problem-solving
Teacher’s Activity: The teacher demonstrates how to construct an arc tangent to a straight line or two lines, explaining the logic behind locating the centre of the tangent arc. The teacher provides a simple example.
Pupils’ Activity: Pupils observe and try to understand the construction logic, attempting basic sketches.
Learning Point: Pupils understand the method for constructing tangent arcs to straight lines.

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 tangency and state one of its principles.
Mention two applications of tangency in engineering.
Describe the first two steps in constructing a tangent at a point on a circle.
Explain the difference between direct and transverse tangents to two circles.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes
Teaching Skill: Summarizing/Assignment
Teacher’s Activity: The teacher summarizes the key learning points about tangents and tangency. The teacher gives pupils homework to practice all the tangent constructions taught in class.
Pupils’ Activity: Pupils listen to the summary and copy down the homework.
Learning Point: The lesson is concluded, and pupils are given practice tasks.

Lesson Keywords

Tangent – A straight line that touches a curve or circle at exactly one point.
Tangency – The state of being tangent; the property of touching at a single point.
Point of Tangency – The single point where a tangent line touches a curve or circle.
Radius – A line segment from the centre of a circle to any point on its circumference.
Perpendicular – At an angle of 90 degrees to a given line or surface.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex tangent construction challenges, such as constructing an arc tangent to two other arcs. For pupils who require more support, the teacher will provide one-on-one guidance during practical sessions and simplify the steps for basic constructions.

Note for teachers using this lesson plan

Ensure that all pupils have complete and functional drawing instruments. Emphasize accuracy during demonstrations and practical exercises. Encourage pupils to practice repeatedly to master the construction techniques. Provide individual feedback on their drawings.