Lesson Note on Intersection of Two Pyramids for SS 2

Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: 1st Term
Week: 10
Age: 16 years
Duration: 45 minutes
Subject: Technical Drawing
Curriculum Theme: Geometrical Drawing
Previous Lesson: Intersection of Two Prisms.
Topic: INTERSECTION OF SOLID
Subject Matter: Two intersecting pyramids

Specific Objectives

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

Cognitive Domain:

• Define the term “intersection of solids”.
• State the principles used in determining the intersection of two solids.

Affective Domain:

• Appreciate the importance of accuracy and neatness in technical drawing.
• Show interest in solving problems involving the intersection of solids.

Psychomotor Domain:

• Draw the orthographic views of two intersecting pyramids.
• Accurately determine and draw the line of intersection between two pyramids.

Social Domain:

• Collaborate effectively with peers during practical drawing exercises.

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. Olawore

Instructional Materials

The teacher will teach this lesson with the aid of:

• Models of intersecting pyramids
• Posters and charts showing examples of intersecting solids
• Drawing board
• T-square
• Set squares (45° and 30°/60°)
• Pencils (HB, H, 2H)
• Eraser
• Dividers/Compasses

Rationale for the Lesson

Understanding the intersection of solids is important for pupils as it helps them visualize and represent complex engineering and architectural designs. This skill is fundamental for careers in design, manufacturing, and construction, enabling accurate communication of spatial relationships.

Prerequisite/Previous Knowledge

Pupils should have basic knowledge of orthographic projection, drawing of solids, and the use of drawing instruments.

Lesson Content/Board Summary

INTERSECTION OF SOLID

Introduction to Intersection of Solids

Intersection of solids refers to the process of determining the common line or curve formed when two or more solid objects penetrate or meet each other. This line or curve is also known as the line of interpenetration.

It is important in technical drawing for accurately representing complex shapes and assemblies in various fields like mechanical engineering, architecture, and product design.

Methods for Determining Intersection Lines

The common methods for finding lines of intersection include:

• Line Method: Involves passing lines through one solid and finding where they intersect the surface of the other solid.
• Cutting Plane Method: Involves passing a series of imaginary cutting planes through both solids to generate points of intersection.

Intersection of Two Pyramids

To determine the line of intersection between two pyramids, we typically use orthographic projection combined with either the line method or the cutting plane method. The process involves identifying points where the edges or surfaces of one pyramid meet the other.

Worked Example: Intersection of Two Square Pyramids

Consider two square pyramids: a vertical square pyramid (Pyramid A) and a horizontal square pyramid (Pyramid B) penetrating Pyramid A.

Procedure for Drawing the Intersection:

1. Step 1: Draw Orthographic Views

   Draw the given views (Front Elevation and Plan) of both pyramids in their specified positions. Ensure the centre lines are clearly marked.

2. Step 2: Identify Key Points and Edges

   Focus on the edges of the penetrating pyramid (Pyramid B) and where they cross or meet the faces of the main pyramid (Pyramid A). Mark these points on the plan view.

3. Step 3: Project Points to Elevation

   From the points identified in the plan view, project vertical lines up to the corresponding faces of Pyramid A in the front elevation. Similarly, project horizontal lines from the edges of Pyramid B to locate intersection points.

4. Step 4: Determine Intersection Points on Faces

   Where the projected lines from Pyramid B’s edges intersect the faces of Pyramid A, mark these points. These are points on the line of intersection. You might need to use auxiliary views or a specific cutting plane if the intersection is complex or to determine true lengths of edges if needed.

5. Step 5: Connect the Points

   Connect the marked intersection points in the correct sequence to form the line of intersection. This line will be visible or hidden depending on its position relative to the observer.
6. Step 6: Show Visible and Hidden Lines

   Distinguish between visible lines (continuous thick lines) and hidden lines (dashed lines) for both the pyramids and the line of intersection to clearly represent the final solid form.

Note: Accuracy in projecting points and drawing lines is essential for a correct representation of the intersection.

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 what they understand by “solids” in technical drawing and if they can imagine what happens when two solids meet or pass through each other.
Pupils’ Activity: Pupils respond by mentioning examples of solids like cubes, pyramids, cones, and discuss what might happen when they intersect.
Learning Point: Pupils recall prior knowledge of solids and are introduced to the concept of intersection.

Step 2: Explanation of Intersection of Solids

Time: 5 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines “intersection of solids” and explains its importance using real-world examples from engineering and architecture, referring to models and charts.
Pupils’ Activity: Pupils listen attentively, observe the models, and take notes on the definition and importance.
Learning Point: Pupils understand the definition and practical relevance of intersecting solids.

Step 3: Principles and Methods of Intersection

Time: 8 minutes
Teaching Skill: Explanation/Illustration
Teacher’s Activity: The teacher explains the general principles and methods (line method, cutting plane method) used to determine lines of intersection between solids, focusing on how these apply to pyramids.
Pupils’ Activity: Pupils observe the illustrations on charts and listen to the explanations, asking questions for clarification.
Learning Point: Pupils grasp the theoretical basis for finding intersection lines.

Step 4: Demonstration – Drawing Intersecting Pyramids (Part 1)

Time: 10 minutes
Teaching Skill: Demonstration/Modelling
Teacher’s Activity: The teacher demonstrates on the board how to draw the orthographic views of two intersecting square pyramids, step-by-step, as outlined in the board summary.
Pupils’ Activity: Pupils carefully observe the teacher’s demonstration, paying attention to the initial setup and projection of views.
Learning Point: Pupils see the practical application of drawing intersecting pyramids.

Step 5: Demonstration – Drawing Intersecting Pyramids (Part 2)

Time: 7 minutes
Teaching Skill: Demonstration/Guided Practice
Teacher’s Activity: The teacher continues the demonstration, showing how to identify and connect the points of intersection to form the final line of interpenetration, distinguishing between visible and hidden lines.
Pupils’ Activity: Pupils continue to observe the completion of the drawing, noting how the intersection line is formed and represented.
Learning Point: Pupils understand the full process of determining and drawing the line of intersection.

Step 6: Guided Practice

Time: 5 minutes
Teaching Skill: Coaching/Supervision
Teacher’s Activity: The teacher provides a simple intersection problem for pupils to attempt on their drawing sheets, offering guidance and correction as needed.
Pupils’ Activity: Pupils begin drawing the orthographic views and attempt to locate some initial intersection points under the teacher’s supervision.
Learning Point: Pupils apply the learned skills in a practical setting.

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. What is meant by the “intersection of solids”?
2. Mention two methods used to determine the line of intersection between solids.
3. Describe the first three steps you would take to draw the intersection of two square pyramids.
4. Why is accuracy important when drawing the intersection of solids?

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key learning points about the intersection of pyramids and assigns homework, which involves drawing another example of intersecting pyramids.
Pupils’ Activity: Pupils listen to the summary and note down the homework assignment.
Learning Point: Pupils reinforce their understanding and are prepared for further practice.

Lesson Keywords

• Intersection – The common line or curve formed when two solids meet or penetrate each other.
• Solids – Three-dimensional geometric shapes like pyramids, prisms, cones, and cylinders.
• Pyramids – Solids with a polygonal base and triangular faces that meet at an apex.
• Orthographic Projection – A method of representing three-dimensional objects in two dimensions, typically using plan, front, and side views.
• Line of Interpenetration – Another term for the line of intersection.
• Cutting Plane Method – A technique involving imaginary planes to find intersection points.

Differentiation

For pupils who are struggling, the teacher will provide pre-drawn orthographic views to help them focus only on finding the intersection points. Advanced learners will be challenged with more complex intersection problems, such as pyramids with inclined axes or different base shapes, and encouraged to explore more advanced methods.

Note for teachers using this lesson plan

Ensure that all pupils have their complete drawing instruments. Emphasize the importance of neatness and precision throughout the demonstration and practical exercises. Encourage pupils to ask questions and provide individual guidance during the guided practice session. Using physical models will greatly aid visualization.