Lesson Note on Construction of Involutes and Cycloids for SS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: First Term
Week: 3
Age: 16 years
Duration: 45 minutes
Subject: Technical Drawing
Curriculum Theme: Technical Drawing Fundamentals
Previous Lesson: Construction of Parabola and Hyperbola.
Topic: SPECIAL CURVES
Subject Matter: Construction of special curves (involutes from circles, cycloids from rolling circles)

Specific Objectives

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

Cognitive Domain:

• Define an involute and a cycloid.
• State the steps involved in constructing an involute from a circle.
• State the steps involved in constructing a cycloid from a rolling circle.

Affective Domain:

• Appreciate the importance of special curves in engineering and design.
• Follow instructions carefully during construction.

Psychomotor Domain:

• Construct an involute from a given circle accurately.
• Construct a cycloid from a given rolling circle accurately.

Social Domain:

• Collaborate with peers to clarify construction steps.

Reference Materials

The following resources were used in planning this lesson:

• Senior Secondary Schools Education Curriculum
• State Unified Scheme of Work
• Technical Drawing for Senior Secondary Schools by A.B. Oluyemi

Instructional Materials

The teacher will teach this lesson with the aid of:

• Drawing instruments (T-square, set squares, compass, protractor, pencils, eraser)
• Drawing board
• Whiteboard or chalkboard
• Computers (for demonstration or visual aids)
• Chart showing examples of involutes and cycloids

Rationale for the Lesson

This lesson helps pupils understand and construct special curves, which are important in various engineering applications such as gear profiles, cam designs, and architectural structures. Understanding these constructions enables pupils to design and analyze mechanical components more effectively.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of basic geometric constructions such as drawing lines, circles, tangents, and dividing lines and circles into equal parts.

Lesson Content/Board Summary

SPECIAL CURVES

Introduction to Special Curves

Special curves are geometric curves that have unique properties and are frequently used in engineering and architectural design. They are often generated by a point on a line or circle rolling on another curve or line.

Construction of an Involute from a Circle

An involute is a curve traced by a point on a taut string as the string is unwound from a circle (or any other curve), or by a point on a straight line rolling without slipping on a circle.

Steps for constructing an involute from a circle:

• Step 1: Draw the base circle and divide it into 12 equal parts. Label the points 0, 1, 2, …, 12.
• Step 2: Draw tangents at each division point on the circle.
• Step 3: Calculate the circumference of the circle (C = πD or 2πR).
• Step 4: Along the tangent at point 0 (or 12), mark off the circumference of the circle. Divide this line into 12 equal parts and label them 1′, 2′, …, 12′.
• Step 5: From point 1 on the circle, mark off a distance equal to 0-1′ on its tangent.
• Step 6: From point 2 on the circle, mark off a distance equal to 0-2′ on its tangent.
• Step 7: Continue this process for all points: from point ‘n’ on the circle, mark off a distance equal to 0-n’ on its tangent.
• Step 8: Draw a smooth curve through all the marked points to obtain the involute.

Construction of a Cycloid from a Rolling Circle

A cycloid is a curve traced by a point on the circumference of a circle as the circle rolls without slipping along a straight line.

Steps for constructing a cycloid from a rolling circle:

• Step 1: Draw a horizontal base line (director line).
• Step 2: Draw the rolling circle (generating circle) touching the base line. Mark a point P on its circumference (usually at the bottom, touching the base line).
• Step 3: Draw a horizontal line through the center of the rolling circle.
• Step 4: Divide the rolling circle into 12 equal parts. From each division point, draw horizontal lines parallel to the base line.
• Step 5: Calculate the circumference of the rolling circle (C = πD). Mark off this distance along the base line from the starting point of the circle.
• Step 6: Divide the circumference distance on the base line into 12 equal parts. Label these points.
• Step 7: From each division point on the base line, draw vertical lines upwards to intersect the center line of the rolling circle. These intersections represent the successive positions of the center of the rolling circle as it rolls.
• Step 8: With each new center position, draw an arc with the radius of the rolling circle, intersecting the corresponding horizontal line from the divided circle. (e.g., from center position 1, draw an arc intersecting horizontal line 1; from center position 2, draw an arc intersecting horizontal line 2, and so on).
• Step 9: Draw a smooth curve through all the intersection points to obtain the cycloid.

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, reviews the previous lesson on basic geometric constructions, and introduces the topic of special curves by asking pupils if they know any curves used in engineering designs.
Pupils’ Activity: Pupils respond to the teacher’s questions and listen attentively.
Learning Point: Pupils are prepared for the new topic and recall prior knowledge.

Step 2: Explanation of Involute

Time: 5 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines an involute and explains its principle using a simple analogy of unwinding a string from a circular object. The teacher also states the steps for its construction.
Pupils’ Activity: Pupils listen, take notes, and ask questions for clarification.
Learning Point: Pupils understand the definition and concept of an involute.

Step 3: Demonstration of Involute Construction

Time: 8 minutes
Teaching Skill: Demonstration/Modeling
Teacher’s Activity: The teacher demonstrates the step-by-step construction of an involute from a given circle on the board, ensuring all lines and points are clearly visible and labeled.
Pupils’ Activity: Pupils observe the demonstration carefully, noting the procedure and techniques.
Learning Point: Pupils visualize the construction process and understand the practical steps.

Step 4: Pupils Practice Involute Construction

Time: 5 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher guides pupils as they attempt to construct an involute from a circle in their drawing books, providing assistance where needed.
Pupils’ Activity: Pupils practice constructing the involute, following the demonstrated steps.
Learning Point: Pupils gain practical experience in constructing an involute.

Step 5: Explanation of Cycloid

Time: 5 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines a cycloid and explains its principle using an analogy of a point on a rolling wheel. The teacher also states the steps for its construction.
Pupils’ Activity: Pupils listen, take notes, and ask questions for clarification.
Learning Point: Pupils understand the definition and concept of a cycloid.

Step 6: Demonstration of Cycloid Construction

Time: 8 minutes
Teaching Skill: Demonstration/Modeling
Teacher’s Activity: The teacher demonstrates the step-by-step construction of a cycloid from a rolling circle on the board, emphasizing the division of the circle and the base line.
Pupils’ Activity: Pupils observe the demonstration carefully, paying attention to the details of the rolling motion.
Learning Point: Pupils visualize the construction process and understand the practical steps for a cycloid.

Step 7: Evaluation/Review

Time: 7 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Define an involute.
2. List three steps involved in constructing an involute from a circle.
3. Define a cycloid.
4. Outline the main steps to construct a cycloid from a rolling circle.

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
Teacher’s Activity: The teacher summarizes the key learning points about involutes and cycloids and their construction. The teacher then assigns homework for pupils to practice constructing both curves with different given dimensions.
Pupils’ Activity: Pupils listen to the summary and copy down the homework.
Learning Point: Pupils reinforce their understanding and are prepared for further practice.

Lesson Keywords

• Involute – A curve traced by a point on a taut string as it is unwound from a circle.
• Cycloid – A curve traced by a point on the circumference of a circle as the circle rolls along a straight line.
• Tangent – A straight line that touches a curve or curved surface at a single point.
• Rolling circle – The circle that generates a cycloid by rolling along a line.
• Locus – The set of all points satisfying a given condition.

Differentiation

For pupils who grasp the concepts quickly, the teacher can challenge them to construct an involute from a square or a spiral. For those needing more support, the teacher can provide additional one-on-one guidance during practice sessions and simplify the initial construction steps.

Note for teachers using this lesson plan

Ensure that all drawing instruments are available and in good condition. Emphasize accuracy in measurements and drawing lines. Allow sufficient time for practical demonstration and pupil practice, as these constructions require precision. Encourage pupils to ask questions and provide constructive feedback on their work.