Lesson Note on Auxiliary View of Geometrical Solids: Full Cones, Pyramids and Cylinders for SS 2

Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: 2nd Term
Week: 4
Age: 16 years
Duration: 45 minutes
Subject: Technical Drawing
Curriculum Theme: Orthographic and Auxiliary Projections
Previous Lesson: Auxiliary View of Geometrical Solids: Drawing Auxiliary Plans and Elevations.
Topic: AUXILIARY VIEW OF GEOMETRICAL SOLIDS
Subject Matter: Auxiliary views of full geometrical solids (cones, pyramids, cylinders)

Specific Objectives

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

Cognitive Domain:

• Define an auxiliary view.
• State the principles of auxiliary projection.

Affective Domain:

• Appreciate the importance of auxiliary views in technical design.
• Show interest in drawing complex geometrical shapes.

Psychomotor Domain:

• Draw the auxiliary view of a given cone.
• Draw the auxiliary view of a given pyramid.

Social Domain:

• Collaborate with peers when sharing drawing instruments and ideas.

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, Book 2

Instructional Materials

The teacher will teach this lesson with the aid of:

• Video clips showing auxiliary projections
• Posters and charts illustrating auxiliary views of solids
• Drawing instruments (drawing board, T-square, set squares, compass, protractor, pencils)

Rationale for the Lesson

This lesson helps pupils understand how to represent the true shape and size of inclined surfaces on geometrical solids. This skill is important for accurately communicating design intentions in engineering and manufacturing fields.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of orthographic projections, basic geometrical solids, and the use of drawing instruments.

Lesson Content/Board Summary

AUXILIARY VIEW OF GEOMETRICAL SOLIDS

Definition of Auxiliary View

An auxiliary view is an orthographic projection of an object onto a plane that is not parallel to any of the principal projection planes (horizontal, frontal, or profile planes). It is used to show the true size and shape of inclined or oblique surfaces that would appear foreshortened in the principal views.

Principles of Auxiliary Projection

The main principle of auxiliary projection is that the projection lines are drawn perpendicular to the inclined surface. The auxiliary plane is positioned parallel to the inclined surface, allowing the true shape and size of that surface to be seen.

Steps for Drawing Auxiliary Views of Geometrical Solids

• Draw the given orthographic views (e.g., plan and front elevation) of the solid.
• Identify the inclined surface whose true shape is required.
• Draw a reference line (X1-Y1) parallel to the inclined surface in the front view. This line represents the auxiliary plane.
• Project lines perpendicular from all critical points of the inclined surface in the front view to the auxiliary plane (X1-Y1).
• Transfer distances from a previous view (usually the plan view) to the auxiliary view. These distances are measured perpendicular from the main reference line (X-Y) to the points in the plan view, and then transferred perpendicular from the auxiliary reference line (X1-Y1) to the corresponding projected lines.
• Connect the transferred points to form the true shape of the inclined surface.
• Add hidden lines and centre lines where necessary.

Worked Example 1: Auxiliary View of a Cone

Given the orthographic views of a cone with an inclined cutting plane, draw the auxiliary view showing the true shape of the cut surface.

Procedure:

1. Step 1: Draw the front elevation and plan of the cone, including the inclined cutting plane intersecting the cone.
2. Step 2: Divide the base circle in the plan into an equal number of parts (e.g., 12) and project these points to the base of the cone in the front elevation. Connect these points to the apex.
3. Step 3: Mark the points where the inclined cutting plane intersects the generator lines of the cone in the front elevation.
4. Step 4: Draw a new reference line (X1-Y1) parallel to the inclined cutting plane in the front elevation.
5. Step 5: Project lines perpendicular from each intersection point on the cutting plane in the front elevation to cross the X1-Y1 line.
6. Step 6: From the plan view, measure the perpendicular distances of each corresponding point from the main X-Y reference line.
7. Step 7: Transfer these measured distances to the auxiliary view, marking them perpendicular from the X1-Y1 line along the projected lines.
8. Step 8: Connect the marked points with a smooth curve to obtain the true shape of the cut surface (an ellipse).

Worked Example 2: Auxiliary View of a Pyramid

Given the orthographic views of a square-based pyramid with an inclined cutting plane, draw the auxiliary view showing the true shape of the cut surface.

Procedure:

1. Step 1: Draw the front elevation and plan of the square-based pyramid, including the inclined cutting plane intersecting the pyramid.
2. Step 2: Mark the points where the inclined cutting plane intersects the edges of the pyramid in the front elevation.
3. Step 3: Draw a new reference line (X1-Y1) parallel to the inclined cutting plane in the front elevation.
4. Step 4: Project lines perpendicular from each intersection point on the cutting plane in the front elevation to cross the X1-Y1 line.
5. Step 5: From the plan view, measure the perpendicular distances of each corresponding point from the main X-Y reference line.
6. Step 6: Transfer these measured distances to the auxiliary view, marking them perpendicular from the X1-Y1 line along the projected lines.
7. Step 7: Connect the marked points with straight lines in sequence to obtain the true shape of the cut surface (a polygon).

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, checks attendance, and asks pupils to recall what orthographic projection is and its importance.
Pupils’ Activity: Pupils respond to the greetings and answer questions about orthographic projection.
Learning Point: Pupils recall previous knowledge on orthographic projection.

Step 2: Explanation of Auxiliary Views

Time: 7 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines auxiliary views, explaining their purpose, especially for showing true shapes of inclined surfaces, using charts and examples.
Pupils’ Activity: Pupils listen attentively, ask questions for clarification, and take notes.
Learning Point: Pupils understand the definition and purpose of an auxiliary view.

Step 3: Principles of Auxiliary Projection

Time: 7 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains the principles of auxiliary projection, emphasizing that projection lines are perpendicular to the inclined surface and that distances are transferred from a previous view.
Pupils’ Activity: Pupils observe the explanations and diagrams, relating them to the definition.
Learning Point: Pupils grasp the fundamental principles of drawing auxiliary views.

Step 4: Demonstration – Auxiliary View of a Cone

Time: 8 minutes
Teaching Skill: Demonstration/Modelling
Teacher’s Activity: The teacher demonstrates step-by-step how to draw the auxiliary view of a cone with an inclined cut on the board or using a projector, guiding pupils through each stage.
Pupils’ Activity: Pupils observe the demonstration carefully, asking questions about the drawing process.
Learning Point: Pupils learn the practical steps for drawing an auxiliary view of a cone.

Step 5: Practice – Auxiliary View of a Cone

Time: 8 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher provides a simple problem for pupils to attempt drawing the auxiliary view of a cone, offering individual assistance where needed.
Pupils’ Activity: Pupils practice drawing the auxiliary view of a cone using their drawing instruments.
Learning Point: Pupils apply the learned steps to draw the auxiliary view of a cone.

Step 6: Demonstration – Auxiliary View of a Pyramid

Time: 5 minutes
Teaching Skill: Demonstration/Modelling
Teacher’s Activity: The teacher briefly demonstrates how to draw the auxiliary view of a pyramid with an inclined cut, highlighting similarities and differences from the cone example.
Pupils’ Activity: Pupils observe and note the steps for drawing the auxiliary view of a pyramid.
Learning Point: Pupils understand the procedure for drawing an auxiliary view of a pyramid.

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 an auxiliary view?
2. State two reasons why auxiliary views are drawn.
3. Briefly explain the principle of transferring distances when drawing an auxiliary view.
4. List the major steps involved in drawing the auxiliary view of a geometrical solid.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: Remaining time (approx. 0 minutes)
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key learning points of the lesson and assigns homework: “Draw the auxiliary view of a cylinder cut by an inclined plane at 45 degrees to its axis.”
Pupils’ Activity: Pupils listen to the summary and copy the homework assignment.
Learning Point: Pupils consolidate their understanding and receive further practice.

Lesson Keywords

• Auxiliary View – A view projected onto a plane not parallel to the principal planes.
• True Shape – The actual size and form of a surface as seen perpendicular to it.
• Inclined Surface – A surface that is not parallel to any of the principal projection planes.
• Projection – The process of representing a 3D object on a 2D plane.
• Geometrical Solids – Three-dimensional shapes like cones, pyramids, and cylinders.

Differentiation

For pupils who grasp the concept quickly, the teacher can provide more complex solids or multiple inclined surfaces to draw. For pupils needing more support, the teacher can offer simplified examples or provide pre-drawn orthographic views to start from, focusing only on the auxiliary projection steps.

Note for teachers using this lesson plan

Ensure that pupils have access to proper drawing instruments. Emphasize neatness and accuracy in drawing, as these are critical in technical drawing. Use visual aids extensively to help pupils visualize the 3D objects and their projections. Encourage peer learning and correction during practical sessions.