Lesson Note on Surface Development of Intersecting Solids for SS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: First Term
Week: 11
Age: 16 years
Duration: 45 minutes
Subject: Technical Drawing
Curriculum Theme: Solid Geometry and Surface Development
Previous Lesson: Intersection of Two Pyramids.
Topic: INTERSECTION OF SOLID
Subject Matter: Surface development of intersecting solids (two intersecting cylinders, two intersecting pyramids, two intersecting prisms)

Specific Objectives

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

Cognitive Domain:

• Define intersection of solids.
• Explain the principles involved in developing intersecting surfaces.
• Identify different types of intersecting solids.

Affective Domain:

• Appreciate the practical application of surface development in various industries.
• Show interest in accurately drawing and developing intersecting solids.

Psychomotor Domain:

• Accurately draw the lines of intersection between two solids.
• Develop the surfaces of intersecting cylinders, pyramids, and prisms.
• Use drawing instruments precisely to achieve neat and accurate drawings.

Social Domain:

• Collaborate with peers to discuss and solve surface development problems.

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. Olajide
• https://www.studywithjethro.com/technical-drawing-for-ss1-ss2-and-ss3/
• https://www.pinterest.com/pin/529729860601552594/

Instructional Materials

The teacher will teach this lesson with the aid of:

• Models of intersecting solids (cylinders, pyramids, prisms)
• Posters and charts showing examples of intersecting solids and their developments
• Cardboard for practical demonstration
• Drawing instruments (drawing board, T-square, set squares, compass, dividers, pencils)

Rationale for the Lesson

Understanding the intersection and surface development of solids is important for pupils to visualize and construct objects found in engineering, architecture, and manufacturing. This skill helps in creating patterns for sheet metal work, ducting, and packaging.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of drawing basic solids, orthographic projections, and the surface development of single solids.

Lesson Content/Board Summary

INTERSECTION OF SOLIDS AND SURFACE DEVELOPMENT

Definition of Intersection of Solids

The intersection of solids refers to the line or curve formed when two or more solid objects pass through each other. This line or curve represents the common boundary between the intersecting surfaces.

Purpose of Surface Development of Intersecting Solids

Surface development of intersecting solids involves unfolding the surfaces of the solids to create a flat pattern. This pattern includes the true shape of the cut-out resulting from the intersection, allowing for accurate fabrication of components. This process is essential for manufacturing items such as pipes, ducts, hoppers, and containers in industries like sheet metal work and fabrication.

Types of Intersecting Solids (Examples)

The following are common types of intersecting solids studied in technical drawing:

• Two intersecting cylinders (e.g., a main pipe and a branch pipe)
• Two intersecting pyramids (e.g., a square pyramid intersecting another square pyramid)
• Two intersecting prisms (e.g., a rectangular prism intersecting a triangular prism)

General Procedure for Developing Intersecting Solids

The general steps involved in developing the surfaces of intersecting solids are:

• Step 1: Draw Orthographic Views. Draw the necessary orthographic views (e.g., front view, plan view, side view) of both solids in their correct intersecting positions.
• Step 2: Establish Cutting Planes/Generators. Divide the surfaces of the solids into a sufficient number of points or lines using cutting planes or generating lines.
• Step 3: Locate Intersection Points. Determine the points where the cutting planes or generating lines intersect the surfaces of both solids in all orthographic views.
• Step 4: Project Curve of Intersection. Project these intersection points to accurately establish the curve of intersection in the orthographic views.
• Step 5: Develop Individual Surfaces. Develop the surface of each solid separately, transferring the intersection points onto the respective developments.
• Step 6: Connect Points. Connect the transferred points on each development to show the true shape of the cut-out and the developed line of intersection.

Worked Example: Development of Two Intersecting Cylinders

Consider a main cylinder intersected by a smaller branch cylinder, both with their axes perpendicular.

• Step 1: Draw Orthographic Views.
  • Draw the front view and plan view of both cylinders in their intersecting positions.
  • Ensure the axes of the cylinders are correctly positioned.
• Step 2: Locate Points of Intersection (Orthographic Views).
  • In the plan view, divide the circumference of the branch cylinder into a suitable number of equal parts (e.g., 12).
  • Draw generators (lines) from these division points on the branch cylinder parallel to its axis until they meet the main cylinder.
  • Project these generator lines from the plan view to the front view.
  • Where these projected lines intersect the outline of the main cylinder in the front view, mark points. Connecting these points forms the curve of intersection.
• Step 3: Develop the Main Cylinder.
  • Draw the stretch-out line for the main cylinder, equal in length to its circumference.
  • Project the intersection points from the front view onto the corresponding positions on the stretch-out line, maintaining their heights.
  • Connect these points on the development to show the true profile of the cut-out on the main cylinder.
• Step 4: Develop the Branch Cylinder.
  • Draw the stretch-out line for the branch cylinder.
  • Measure the true lengths of the generators on the branch cylinder from the intersection points found in the orthographic views.
  • Transfer these true lengths onto the corresponding generator lines on the developed surface of the branch cylinder.
  • Connect these points to form the line of intersection on the developed branch cylinder, showing the shape of the opening.
• Step 5: Complete the Development.
  • Add dimensions, fold lines, and any necessary labels to the developed patterns.

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 recall what they learned about developing surfaces of single solids. The teacher then introduces the topic of intersecting solids.
Pupils’ Activity: Pupils respond to the questions and listen attentively to the introduction.
Learning Point: Pupils connect previous knowledge to the new topic and are prepared for the lesson.

Step 2: Definition and Purpose of Intersection of Solids

Time: 7 minutes
Teaching Skill: Explanation/Discussion
Teacher’s Activity: The teacher defines intersection of solids and explains its purpose, using real-life examples like pipe joints. The teacher uses charts and models to illustrate.
Pupils’ Activity: Pupils listen, observe the models, and ask questions for clarification.
Learning Point: Pupils understand what intersection of solids means and why it is important.

Step 3: Types of Intersecting Solids

Time: 7 minutes
Teaching Skill: Illustration/Categorization
Teacher’s Activity: The teacher displays models and charts showing different types of intersecting solids, such as two cylinders, two pyramids, and two prisms, highlighting their unique features.
Pupils’ Activity: Pupils identify and differentiate between the various types of intersecting solids shown.
Learning Point: Pupils can recognize and name common types of intersecting solids.

Step 4: General Procedure for Developing Intersecting Solids

Time: 8 minutes
Teaching Skill: Explanation/Instruction
Teacher’s Activity: The teacher explains the general step-by-step procedure for developing the surfaces of intersecting solids, emphasizing the importance of orthographic views and locating intersection points.
Pupils’ Activity: Pupils listen carefully and take notes on the procedural steps.
Learning Point: Pupils grasp the systematic approach required for surface development of intersecting solids.

Step 5: Demonstration of Intersecting Cylinders

Time: 10 minutes
Teaching Skill: Demonstration/Practical
Teacher’s Activity: The teacher demonstrates the development of two intersecting cylinders on the whiteboard or drawing board, following the steps outlined in the board summary. The teacher explains each projection and transfer of points clearly.
Pupils’ Activity: Pupils observe the demonstration, paying close attention to the drawing techniques and projections.
Learning Point: Pupils observe the practical application of the development procedure for a specific type of intersecting solid.

Step 6: Guided Practice

Time: 5 minutes
Teaching Skill: Coaching/Supervision
Teacher’s Activity: The teacher provides a simple exercise, asking pupils to sketch the orthographic views and locate initial intersection points for a given intersecting solid on their drawing sheets.
Pupils’ Activity: Pupils attempt the exercise, applying the steps demonstrated by the teacher.
Learning Point: Pupils begin to apply the learned procedures and identify areas needing further clarification.

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 the intersection of solids.
2. State two purposes of developing the surfaces of intersecting solids.
3. Mention three types of intersecting solids.
4. Briefly explain the first two steps in developing the surfaces of two intersecting cylinders.

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
Teacher’s Activity: The teacher summarizes the key learning points of the lesson, reiterating the importance of accuracy in drawing and development. The teacher assigns homework which includes drawing and developing a given intersecting solid.
Pupils’ Activity: Pupils listen to the summary and copy the homework assignment.
Learning Point: Pupils consolidate their understanding and prepare for independent practice.

Lesson Keywords

• Intersection – The line or curve formed when two surfaces or solids meet or pass through each other.
• Solids – Three-dimensional objects such as cylinders, pyramids, and prisms.
• Development – The process of unfolding the surfaces of a 3D object into a 2D plane.
• Cylinder – A solid or hollow body with a circular base and straight parallel sides.
• Pyramid – A solid figure with a polygonal base and triangular faces that meet at a common point (apex).
• Prism – A solid geometric figure whose two ends are similar, equal, and parallel rectilinear figures, and whose sides are parallelograms.
• Orthographic Projection – A method of representing three-dimensional objects in two dimensions, typically through front, top, and side views.

Differentiation

For pupils who are struggling, the teacher will provide simplified examples and offer more one-on-one guidance during the practice session. Advanced pupils will be challenged with more complex intersection problems or asked to explore intersections of solids with different axis orientations.

Note for teachers using this lesson plan

Ensure that pupils have a strong foundation in orthographic projection and development of single solids before introducing intersecting solids. Ample visual aids and practical demonstrations are essential for this topic. Encourage pupils to use their drawing instruments precisely and neatly. Provide opportunities for hands-on practice, perhaps using cardboard models to help visualize the developments.