Lesson Note on Auxiliary View of Geometrical Solids: Truncated Pyramids and Prisms for SS 2

Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: Second Term
Week: 5
Age: 16 years
Duration: 45 minutes
Subject: Technical Drawing
Curriculum Theme: Technical Drawing
Previous Lesson: Auxiliary View of Geometrical Solids: Full Cones, Pyramids and Cylinders.
Topic: AUXILIARY VIEW OF GEOMETRICAL SOLIDS
Subject Matter: Auxiliary views of truncated geometrical solids (truncated pyramids, truncated prisms)

Specific Objectives

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

Cognitive Domain:

• Define auxiliary view and explain its purpose.
• Explain the concept of truncation in geometrical solids.
• Identify types of truncated geometrical solids.

Affective Domain:

• Appreciate the importance of auxiliary views in representing true shapes.
• Develop patience and precision when drawing auxiliary views.

Psychomotor Domain:

• Construct the auxiliary view of a truncated pyramid.
• Construct the auxiliary view of a truncated prism.

Social Domain:

• Work cooperatively with peers to solve drawing problems.

Reference Materials

The following resources were used in planning this lesson:

• Senior Secondary Schools Education Curriculum
• State Unified Scheme of Work
• Engineering Drawing by N.D. Bhatt
• https://www.technicaldrawing.org/auxiliary-views
• https://www.drawingskills.com/truncated-solids

Instructional Materials

The teacher will teach this lesson with the aid of:

• Drawing board
• T-square and set squares
• Pencils (HB, 2H) and eraser
• Drawing paper
• Models of truncated pyramids and prisms
• Whiteboard/chalkboard and markers/chalk

Rationale for the Lesson

This lesson helps pupils understand how to obtain the true shape and size of inclined surfaces on geometrical solids. This skill is important for accurate representation and fabrication in various engineering and technical fields.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of orthographic projection, true lengths, and the basic construction of geometrical solids.

Lesson Content/Board Summary

AUXILIARY VIEWS OF TRUNCATED GEOMETRICAL SOLIDS

Introduction to Auxiliary Views

An auxiliary view is an orthographic projection of an object onto any plane other than the standard horizontal, frontal, or profile planes. Its main purpose is to show the true shape and size of an inclined or oblique surface that is not parallel to any of the principal planes of projection.

Truncated Geometrical Solids

A truncated geometrical solid is a solid whose apex or top portion has been cut off by a plane. The cutting plane can be inclined or parallel to the base.

The following are types of truncated geometrical solids:

• Truncated Pyramid: A pyramid cut by a plane, resulting in a new top surface.
• Truncated Prism: A prism cut by a plane, resulting in an inclined top surface.

General Steps for Drawing Auxiliary Views of Truncated Solids

To draw an auxiliary view of a truncated solid, follow these steps:

• Step 1: Draw the given orthographic views (plan and elevation) of the solid and the cutting plane.
• Step 2: Project lines from the points of intersection on the cutting plane in the elevation to the corresponding edges in the plan view.
• Step 3: Draw a reference line (X1-Y1) parallel to the cutting plane in the elevation. This line represents the auxiliary plane.
• Step 4: Project lines perpendicular to the X1-Y1 reference line from all relevant points on the cutting plane and the base of the solid in the elevation.
• Step 5: Transfer distances from the plan view (or elevation, depending on the auxiliary view) to the new projection lines. These distances are measured from the X-Y reference line in the plan view to the points on the solid.
• Step 6: Connect the transferred points to form the true shape of the truncated surface or the auxiliary view of the solid.

Worked Example 1: Auxiliary View of a Truncated Square Pyramid

Problem: Draw the auxiliary view showing the true shape of the cut surface of a square pyramid, base 40mm side, height 60mm, cut by a plane inclined at 45 degrees to the horizontal, bisecting the axis.

• Step 1: Draw the plan (square base) and elevation (triangle) of the square pyramid. Locate the cutting plane on the elevation, bisecting the axis and inclined at 45 degrees.
• Step 2: Mark the points where the cutting plane intersects the edges of the pyramid in the elevation (e.g., 1, 2, 3, 4). Project these points horizontally to the corresponding edges in the plan view.
• Step 3: Draw a reference line X1-Y1 parallel to the 45-degree cutting plane in the elevation.
• Step 4: From points 1, 2, 3, 4 on the cutting plane in the elevation, draw projectors perpendicular to the X1-Y1 line.
• Step 5: Measure the perpendicular distances of the corresponding points on the plan view from the original X-Y reference line. Transfer these distances onto the projectors in the auxiliary view, measuring from the X1-Y1 line. For example, if point ‘A’ in the plan is ‘x’ distance from X-Y, then point ‘A” in the auxiliary view will be ‘x’ distance from X1-Y1 along its projector.
• Step 6: Connect the transferred points (1′, 2′, 3′, 4′) in the correct sequence to obtain the true shape of the cut surface.

Worked Example 2: Auxiliary View of a Truncated Triangular Prism

Problem: A triangular prism, base 50mm side, height 70mm, is cut by a plane inclined at 30 degrees to the horizontal, passing through a point 20mm from the top of the prism along one vertical edge. Draw the auxiliary view of the truncated portion.

• Step 1: Draw the plan (equilateral triangle base) and elevation (rectangle) of the triangular prism. Indicate the cutting plane in the elevation, inclined at 30 degrees, passing through the specified point.
• Step 2: Identify the points where the cutting plane intersects the vertical edges of the prism in the elevation (e.g., A, B, C). Project these points horizontally to the corresponding edges in the plan view.
• Step 3: Draw a reference line X1-Y1 parallel to the 30-degree cutting plane in the elevation.
• Step 4: From points A, B, C on the cutting plane in the elevation, draw projectors perpendicular to the X1-Y1 line.
• Step 5: Measure the perpendicular distances of the corresponding vertical edges from the original X-Y reference line in the plan view. Transfer these distances onto the projectors in the auxiliary view, measuring from the X1-Y1 line.
• Step 6: Connect the transferred points (A’, B’, C’) to obtain the true shape of the truncated surface.

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 orthographic projections and true lengths in previous lessons. The teacher then introduces the topic of auxiliary views by showing models of truncated solids and asking how one might get the actual shape of the cut surface.
Pupils’ Activity: Pupils respond to the questions and observe the models, linking them to prior knowledge.
Learning Point: Pupils recall prior knowledge and are introduced to the concept of auxiliary views and truncated solids.

Step 2: Explanation of Auxiliary Views

Time: 7 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines auxiliary views, explaining their purpose in showing the true shape and size of inclined surfaces. The teacher emphasizes that standard orthographic views do not always show true shapes.
Pupils’ Activity: Pupils listen attentively and take notes on the definition and purpose of auxiliary views.
Learning Point: Pupils understand what an auxiliary view is and why it is used.

Step 3: Explanation of Truncated Solids

Time: 7 minutes
Teaching Skill: Explanation/Classification
Teacher’s Activity: The teacher explains what truncated solids are, using the models to illustrate how a cutting plane modifies a solid. The teacher specifically mentions truncated pyramids and prisms.
Pupils’ Activity: Pupils observe the models and listen to the explanation, asking questions for clarification.
Learning Point: Pupils understand the concept of truncated solids and identify common types.

Step 4: Demonstration of Drawing Steps

Time: 10 minutes
Teaching Skill: Demonstration/Instruction
Teacher’s Activity: The teacher demonstrates the general steps for constructing an auxiliary view of a truncated solid on the board, emphasizing the projection lines and transfer of distances. The teacher draws a simple example to illustrate the process.
Pupils’ Activity: Pupils observe the demonstration carefully, noting the sequence of steps and asking relevant questions about the procedure.
Learning Point: Pupils learn the systematic procedure for drawing auxiliary views.

Step 5: Worked Example 1: Truncated Square Pyramid

Time: 8 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher guides pupils through the step-by-step construction of the auxiliary view for a truncated square pyramid, as outlined in the board summary. The teacher encourages pupils to sketch along or follow the steps mentally.
Pupils’ Activity: Pupils actively participate by following the teacher’s instructions and asking questions about specific drawing techniques.
Learning Point: Pupils practice drawing the auxiliary view of a truncated square pyramid.

Step 6: Worked Example 2: Truncated Triangular Prism

Time: 7 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher guides pupils through the step-by-step construction of the auxiliary view for a truncated triangular prism, reinforcing the concepts taught in the previous example.
Pupils’ Activity: Pupils continue to practice by following the teacher’s guidance, applying the principles learned to a different solid.
Learning Point: Pupils reinforce their understanding by drawing the auxiliary view of a truncated triangular prism.

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, and what is its primary purpose?
2. Explain what is meant by a truncated geometrical solid.
3. Mention two types of truncated geometrical solids.
4. List the general steps involved in drawing an auxiliary view of a truncated solid.

Pupils’ Activity: Pupils answer orally and in writing.
Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 1 minute
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key points of the lesson, reiterating the importance of auxiliary views in technical drawing. The teacher assigns homework, which is to draw the auxiliary view of a truncated hexagonal prism.
Pupils’ Activity: Pupils listen to the summary and copy down the homework.
Learning Point: Pupils review the main concepts and receive homework.

Lesson Keywords

• Auxiliary view – A projection of an object onto a plane other than the standard principal planes, used to show true shape.
• Truncation – The act of cutting off the top or apex of a geometrical solid by a plane.
• Geometrical solids – Three-dimensional shapes such as pyramids, prisms, cones, and cylinders.
• Pyramid – A solid with a polygonal base and triangular faces that meet at an apex.
• Prism – A solid with two identical and parallel polygonal bases and rectangular faces.
• True shape – The actual shape and size of a surface, as seen when looking perpendicular to that surface.
• Orthographic projection – A method of representing three-dimensional objects in two dimensions, typically using multiple views (e.g., plan, elevation).

Differentiation

For pupils who grasp the concepts quickly, the teacher can provide more complex truncated solids (e.g., truncated cones or cylinders) or ask them to determine the development of the truncated surfaces. For pupils who need more support, the teacher can provide pre-drawn orthographic views to start their auxiliary view constructions and offer one-on-one guidance during the drawing exercises.

Note for teachers using this lesson plan

Teachers should ensure they have adequate drawing materials and models for demonstration. Practical demonstration on the board or using a projector is important for this topic. Encourage pupils to use proper drawing instruments and maintain accuracy. Emphasize the systematic approach to drawing auxiliary views rather than memorization.