Lesson Note on True Shapes of Truncated Solids for SS 2

Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: First Term
Week: 6
Age: 16 years
Duration: 45 minutes
Subject: Technical Drawing
Curriculum Theme: Solid Geometry and Projections
Previous Lesson: Plotting Loci in Link Mechanisms.
Topic: TRUE SHAPES
Subject Matter: Meaning of true shapes (actual shape of cut surfaces), true shapes of truncated solids (cones, prisms, pyramids, cylinders)

Specific Objectives

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

Cognitive Domain:

• Define true shape of a surface.
• Explain the method for determining the true shape of a truncated solid.

Affective Domain:

• Appreciate the importance of true shapes in engineering and construction.
• Develop patience and precision when constructing true shapes.

Psychomotor Domain:

• Accurately construct the true shape of a truncated prism.
• Construct the true shape of a truncated cylinder or cone using projection methods.

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
• N.D. Bhatt, Engineering Drawing. Charotar Publishing House.
• A. W. Boundy, Engineering Drawing. McGraw-Hill.

Instructional Materials

The teacher will teach this lesson with the aid of:

• Models of truncated solids (prisms, cylinders, cones)
• Posters illustrating true shape constructions
• Drawing instruments (set squares, compass, ruler, protractor)
• Cardboard or drawing paper
• Whiteboard/chalkboard and markers/chalk

Rationale for the Lesson

Understanding true shapes is important for accurately representing and manufacturing components with cut surfaces. This lesson helps pupils to visualize and determine the actual dimensions of inclined surfaces, which is essential in various technical fields.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of orthographic projections, types of geometrical solids (prisms, pyramids, cylinders, cones), and how to draw sections of solids.

Lesson Content/Board Summary

TRUE SHAPES

Meaning of True Shapes

The true shape of a surface is its actual size and form as seen when looking perpendicularly at that surface. When a solid is cut by an inclined plane, the resulting section is not usually seen in its true shape in the standard orthographic views (plan, elevation). To obtain the true shape, the section must be projected onto a plane parallel to the cutting plane.

Methods of Determining True Shapes

The true shape of a truncated solid is determined by projecting the cut surface onto an auxiliary plane that is parallel to the cutting plane. This method involves:

• Drawing the orthographic views (plan and elevation) of the solid with the cutting plane.
• Drawing a reference line (X1Y1) parallel to the cutting plane.
• Projecting lines from the points on the cut surface perpendicular to the X1Y1 line.
• Transferring distances from the plan view (for elevation-based cutting planes) or elevation view (for plan-based cutting planes) to the auxiliary view.
• Connecting the transferred points to obtain the true shape.

Steps to Determine True Shape of a Truncated Prism (Worked Example)

To find the true shape of a truncated square prism cut by an inclined plane:

1. Step 1: Draw Orthographic Views

   Draw the plan and elevation of the square prism, showing the cutting plane intersecting the prism. Label the points of intersection on the edges in both views (e.g., 1, 2, 3, 4 in elevation and 1′, 2′, 3′, 4′ in plan).

2. Step 2: Draw Auxiliary Reference Line

   Draw a new reference line (X1Y1) parallel to the cutting plane in the elevation view. This line represents the auxiliary plane.

3. Step 3: Project Perpendicular Lines

   From each point on the cutting plane in the elevation view (1, 2, 3, 4), draw projector lines perpendicular to the X1Y1 line into the auxiliary view area.

4. Step 4: Transfer Distances

   Measure the perpendicular distances of the corresponding points from the X-Y reference line in the plan view (e.g., distance of 1′ from X-Y, 2′ from X-Y, etc.). Transfer these distances along the projector lines from the X1Y1 line in the auxiliary view. For example, mark point 1” on the projector from 1, such that its distance from X1Y1 is equal to the distance of 1′ from X-Y in the plan.

5. Step 5: Connect Points

   Connect the marked points (1”, 2”, 3”, 4”) in the auxiliary view in the correct sequence. The resulting polygon is the true shape of the truncated surface.

True Shape of a Truncated Cylinder

For a truncated cylinder, the process is similar but requires dividing the circular base into a number of equal parts (e.g., 8 or 12 divisions). Project these points to the elevation and follow the same steps of drawing an auxiliary reference line, projecting perpendiculars, and transferring distances to obtain the elliptical true shape.

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 sections of solids in the previous lesson. The teacher also introduces the topic by showing a model of a truncated solid and asking if the cut surface is seen in its actual shape in the regular views.
Pupils’ Activity: Pupils respond to the questions and observe the model.
Learning Point: Pupils recall prior knowledge and are introduced to the concept of true shapes.

Step 2: Meaning of True Shapes

Time: 7 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines what a true shape is, emphasizing that it is the actual size and form of a surface seen perpendicular to it. The teacher explains why true shapes are necessary in technical drawing.
Pupils’ Activity: Pupils listen attentively and take notes on the definition of true shapes.
Learning Point: Pupils understand the meaning and importance of true shapes.

Step 3: Explanation of the Concept and Method

Time: 10 minutes
Teaching Skill: Explanation/Illustration
Teacher’s Activity: The teacher explains the general method for determining true shapes using an auxiliary view. The teacher uses posters and diagrams to illustrate the concept of projecting onto a plane parallel to the cutting plane.
Pupils’ Activity: Pupils observe the illustrations and ask questions for clarification.
Learning Point: Pupils grasp the theoretical basis and general steps for finding true shapes.

Step 4: Demonstration (Truncated Prism)

Time: 10 minutes
Teaching Skill: Demonstration/Modelling
Teacher’s Activity: The teacher demonstrates on the whiteboard or using a projector how to construct the true shape of a truncated square prism, following the step-by-step procedure outlined in the board summary.
Pupils’ Activity: Pupils carefully watch the demonstration, asking questions about each step.
Learning Point: Pupils see a practical application of the true shape construction method.

Step 5: Practice (Truncated Cylinder/Cone)

Time: 5 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher briefly explains how the method applies to truncated cylinders or cones, mentioning the need for dividing the circular base. The teacher encourages pupils to think about how they would approach such a problem.
Pupils’ Activity: Pupils discuss and suggest approaches for constructing true shapes of curved solids.
Learning Point: Pupils extend their understanding to different types of solids.

Step 6: Class Activity

Time: 3 minutes
Teaching Skill: Application
Teacher’s Activity: The teacher assigns a simple problem for pupils to attempt individually or in pairs, such as sketching the initial steps for a truncated triangular prism.
Pupils’ Activity: Pupils attempt the assigned problem using their drawing instruments.
Learning Point: Pupils apply the learned concepts in a practical exercise.

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 a true shape in technical drawing?
2. State two reasons why it is important to determine true shapes.
3. Briefly explain the main principle used to determine the true shape of a section.
4. List the key steps involved in constructing the true shape of a truncated prism.

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/Assignment
Teacher’s Activity: The teacher summarizes the key points of the lesson, reiterating the importance of true shapes. The teacher then gives homework, instructing pupils to draw the true shape of a given truncated solid.
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

• True shape – The actual size and form of a surface.
• Truncated – Cut short or cut off, referring to a solid whose apex or end has been removed by a cutting plane.
• Auxiliary view – A view projected onto a plane that is not parallel to any of the principal planes of projection.
• Cutting plane – An imaginary plane used to cut through an object to reveal its internal features or a section.
• Orthographic projection – A method of representing three-dimensional objects in two dimensions, typically using multiple views (plan, elevation).

Differentiation

For pupils who grasp the concept quickly, the teacher can provide more complex truncated solids (e.g., hexagonal prism, inclined cutting plane at an unusual angle). For pupils who need more support, the teacher can provide pre-drawn orthographic views and guide them through the projection and transfer of distances step-by-step.

Note for teachers using this lesson plan

Ensure pupils have a strong foundation in orthographic projections before introducing true shapes. Emphasize precision in drawing and measurement. Use physical models to help pupils visualize the cut surfaces and the concept of true shape. Encourage step-by-step working and labeling of points for clarity.