Lesson Note on Intersection of Two Cylinders for SS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: 1st Term
Week: 8
Age: 16 years
Duration: 45 minutes
Subject: Technical Drawing
Curriculum Theme: Solid Geometry
Previous Lesson: Development of Truncated Solids.
Topic: INTERSECTION OF SOLID
Subject Matter: Two intersecting cylinders (equal diameters, unequal diameters, inclined cylinders)

Specific Objectives

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

Cognitive Domain:

• Define intersection of solids.
• State the methods for determining curves of intersection.
• Identify different types of intersecting cylinders.

Affective Domain:

• Appreciate the importance of accurately drawing intersecting solids in technical applications.
• Develop an interest in solving complex drawing problems.

Psychomotor Domain:

• Draw the orthographic projections of two intersecting cylinders with equal diameters.
• Determine and draw the curve of intersection for two cylinders with unequal diameters.

Social Domain:

• Collaborate with peers to discuss and solve technical drawing challenges.

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. Salami
• https://nigerianinfopedia.com.ng/technical-drawing-ss2-first-term-scheme-of-work/
• https://waec.neconigeria.com/technical-drawing-syllabus/

Instructional Materials

The teacher will teach this lesson with the aid of:

• Models of intersecting cylinders
• Posters and charts showing examples of intersecting solids
• Drawing instruments (T-square, set squares, compass, pencils)
• Drawing boards and papers

Rationale for the Lesson

This lesson helps pupils understand how different solid objects meet and join, which is important for designing and constructing items like pipes, ducts, and machine parts. It enables pupils to accurately represent these complex junctions in technical drawings.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of orthographic projections, drawing of cylinders, and basic geometric constructions.

Lesson Content/Board Summary

INTERSECTION OF SOLIDS

Definition of Intersection of Solids

Intersection of solids refers to the lines or curves formed when two or more solid objects pass through each other. These lines or curves are common to the surfaces of both solids at their point of contact.

Methods for Determining Curves of Intersection

The following are common methods used to determine the curves of intersection:

• Element Method: This involves drawing a series of elements (lines) on the surface of one solid and finding where these elements intersect the surface of the other solid.
• Cutting Plane Method: This involves passing a series of imaginary cutting planes through both solids to find points of intersection on the surfaces.

Intersection of Two Cylinders

The type of curve formed when two cylinders intersect depends on their diameters and the angle of their axes.

Case 1: Two Cylinders with Equal Diameters and Perpendicular Axes

When two cylinders of equal diameter intersect with their axes at right angles, the curve of intersection appears as two straight lines in the plan view, forming an ‘X’ shape. In the elevation and end views, the curve of intersection will be elliptical or appear as straight lines depending on the viewing angle.

Example Steps for Drawing:

1. Draw the orthographic views (plan, front elevation, end elevation) of the two cylinders.
2. Divide the circular plan of one cylinder into a number of equal parts (e.g., 12).
3. Project these divisions to the elevation and end views, creating elements on the cylinder surfaces.
4. Locate the points where the elements of one cylinder intersect the surface of the other cylinder.
5. Connect these points smoothly to form the curve of intersection.

Case 2: Two Cylinders with Unequal Diameters and Perpendicular Axes

When two cylinders of unequal diameters intersect with their axes at right angles, the curve of intersection will generally be an ellipse or a combination of curves in all views where the intersection is visible.

Example Steps for Drawing:

1. Draw the orthographic views (plan, front elevation, end elevation) of the two cylinders.
2. Divide the circular plan of the larger cylinder into equal parts and project elements.
3. Use the cutting plane method or element method to find points where the elements of one cylinder meet the surface of the other.
4. Transfer these points to the other views using projectors.
5. Join the located points with a smooth curve to form the curve of intersection.

Case 3: Two Cylinders with Inclined Axes

When two cylinders intersect with their axes inclined to each other, the curve of intersection becomes more complex, often forming a non-standard curve. The element method or cutting plane method is essential for accurately determining these points.

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 reviews their previous knowledge on orthographic projections and drawing of basic solids like cylinders.
Pupils’ Activity: Pupils respond to the greetings and answer questions related to previous topics.
Learning Point: Pupils recall prior knowledge relevant to the lesson.

Step 2: Introduction to Intersection of Solids

Time: 7 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher introduces the topic “Intersection of Solids” and uses models of intersecting cylinders to illustrate what the curve of intersection represents.
Pupils’ Activity: Pupils observe the models and listen attentively to the teacher’s explanation.
Learning Point: Pupils understand the concept of intersection of solids.

Step 3: Explanation of Methods

Time: 7 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher explains the element method and the cutting plane method for determining the curve of intersection, highlighting their applications.
Pupils’ Activity: Pupils listen and ask questions for clarification on the methods.
Learning Point: Pupils identify the methods used to determine curves of intersection.

Step 4: Intersection of Equal Diameter Cylinders

Time: 8 minutes
Teaching Skill: Demonstration
Teacher’s Activity: The teacher demonstrates on the board or chart how to draw the intersection of two cylinders with equal diameters and perpendicular axes, showing step-by-step construction.
Pupils’ Activity: Pupils observe the demonstration, taking notes on the drawing steps.
Learning Point: Pupils learn how to draw the intersection of equal diameter cylinders.

Step 5: Intersection of Unequal Diameter Cylinders

Time: 8 minutes
Teaching Skill: Demonstration/Guidance
Teacher’s Activity: The teacher demonstrates the procedure for drawing the intersection of two cylinders with unequal diameters and perpendicular axes, emphasizing the use of elements or cutting planes.
Pupils’ Activity: Pupils follow the demonstration and attempt to sketch the process on their drawing pads.
Learning Point: Pupils understand the process for drawing the intersection of unequal diameter cylinders.

Step 6: Discussion on Inclined Cylinders

Time: 3 minutes
Teaching Skill: Discussion
Teacher’s Activity: The teacher briefly discusses the complexity of intersecting cylinders with inclined axes and explains that the same principles of element or cutting plane method apply, but require more careful construction.
Pupils’ Activity: Pupils participate in the discussion and ask questions about inclined intersections.
Learning Point: Pupils grasp the concept of inclined cylinder intersection.

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 intersection of solids.
2. State two methods for determining the curve of intersection.
3. Describe how the curve of intersection appears for two equal diameter cylinders with perpendicular axes in plan view.
4. What is the general shape of the curve of intersection for two unequal diameter cylinders with perpendicular axes?

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 points of the lesson and assigns homework, which includes practicing drawing various intersecting cylinder scenarios.
Pupils’ Activity: Pupils listen to the summary and copy down the homework assignment.
Learning Point: Pupils consolidate their learning and prepare for further practice.

Lesson Keywords

• Intersection – The point or line where two or more things cross or meet.
• Solids – Three-dimensional objects with length, width, and height.
• Cylinders – Solids with two parallel circular bases and a curved surface.
• Orthographic Projection – A method of representing three-dimensional objects in two dimensions.
• Curve of Intersection – The line or curve formed on the surface of solids where they meet.
• Elements – Imaginary lines drawn on the surface of a solid to aid in projection.
• Axis – An imaginary line passing through the center of a solid.

Differentiation

For pupils who grasp the concepts quickly, the teacher can provide more complex intersection problems, such as cylinders intersecting at oblique angles. For pupils who require more support, the teacher will provide simplified examples and one-on-one guidance during practical drawing sessions, focusing on basic cases first.

Note for teachers using this lesson plan

Emphasize practical drawing and accuracy in constructions. Encourage pupils to use their drawing instruments correctly. Provide ample time for hands-on practice, as technical drawing is a skill-based subject. Visual aids like physical models and step-by-step charts are highly beneficial.