Lesson Note on True Length and Angles of a Line in Space for SS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: Second Term
Week: 9
Age: 16 years
Duration: 45 minutes
Subject: Technical Drawing
Curriculum Theme: Orthographic Projections
Previous Lesson: Traces of a Point and Line in Space: Projection and Tracing.
Topic: TRUE LENGTH AND ANGLES OF A LINE IN SPACE
Subject Matter: Meaning of true length, meaning of true angle, determination of true length and angles of a line in space

Specific Objectives

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

Cognitive Domain:

• Define true length of a line.
• Explain the concept of true angle of a line to a plane.
• State the methods used to determine true length and angles.

Affective Domain:

• Appreciate the importance of determining true lengths and angles in engineering drawings.
• Develop patience and precision in drawing techniques.

Psychomotor Domain:

• Accurately determine the true length of a given line in space using the auxiliary view method.
• Accurately determine the true angle of a given line to a principal plane using the auxiliary view method.
• Draw auxiliary views to find true lengths and angles of lines.

Social Domain:

• Collaborate with peers to solve technical drawing problems.
• Share ideas and techniques for effective drawing.

Reference Materials

The following resources were used in planning this lesson:

• Senior Secondary Schools Education Curriculum
• State Unified Scheme of Work
• Olagbemi, S. L. (2018). Technical Drawing for Senior Secondary Schools. Ibadan: University Press PLC.

Instructional Materials

The teacher will teach this lesson with the aid of:

• Video clips showing true length and angle determination
• Projector
• Models of lines in space
• Protractor
• Drawing instruments (T-square, set squares, pencils, drawing paper)

Rationale for the Lesson

Understanding true length and angles of lines is important for accurately representing three-dimensional objects in two-dimensional drawings. This knowledge helps pupils correctly interpret and create technical drawings, which is essential in various engineering and design fields.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of orthographic projections, auxiliary views, and basic geometric constructions.

Lesson Content/Board Summary

TRUE LENGTH AND ANGLES OF A LINE IN SPACE

Meaning of True Length

The true length of a line is its actual length in three-dimensional space. A line shows its true length only when it is parallel to the plane of projection. If a line is inclined to all principal planes (HP and VP), its projections on these planes will appear shorter than its actual length.

Meaning of True Angle

The true angle of a line to a plane is the actual angle that the line makes with that plane in three-dimensional space. This angle is seen in its true size when the plane is viewed as an edge and the line is projected onto a plane parallel to its true length projection.

Methods for Determining True Length and Angles

The most common methods for determining true length and true angles of a line in space are:

• Auxiliary View Method
• Rotation Method

This lesson will focus on the Auxiliary View Method.

Determination of True Length (Auxiliary View Method)

To determine the true length of a line AB, an auxiliary view is taken such that the auxiliary plane is parallel to the line AB.

Procedure:

• Step 1: Draw the given front view (FV) and top view (TV) of the line AB.
• Step 2: Draw a reference line (X1Y1) parallel to one of the projected views of the line (e.g., parallel to the front view a’b’ or top view ab).
• Step 3: Project perpendicular lines from the ends of the projected view (e.g., from a’ and b’ if X1Y1 is parallel to a’b’) across the X1Y1 line.
• Step 4: Transfer the distances from the other view (e.g., from the top view ab, measure the perpendicular distances of a and b from the XY line) onto the new projection lines, measuring from the X1Y1 line.
• Step 5: Connect the transferred points to obtain the true length of the line.

Example 1: Determine the true length of line AB given its front view (a’b’) and top view (ab).

Given:
Front View: a'(20, 50), b'(60, 20)
Top View: a(20, 70), b(60, 40)
(Assume coordinates are in mm relative to an origin).

Solution Steps:

1. Step 1: Draw the XY reference line. Plot points a’, b’, a, b to form the front and top views.
2. Step 2: Draw an auxiliary reference line X1Y1 parallel to the front view a’b’.
3. Step 3: Project lines perpendicular from a’ and b’ to intersect X1Y1.
4. Step 4: From the top view, measure the distance from point ‘a’ to the XY line and transfer it from X1Y1 along the projector from a’ to get point A1. Similarly, measure the distance from ‘b’ to the XY line and transfer it from X1Y1 along the projector from b’ to get point B1.
5. Step 5: Connect A1 and B1. The length of A1B1 is the true length of the line AB.

Determination of True Angle (Auxiliary View Method)

To determine the true angle of a line with a principal plane (e.g., Horizontal Plane – HP), the line must first be shown in its true length, and the principal plane must be viewed as an edge.

Procedure for True Angle with HP:

• Step 1: Obtain the true length projection of the line (e.g., A1B1) as described above.
• Step 2: Draw a new auxiliary reference line (X2Y2) perpendicular to the true length projection (A1B1).
• Step 3: Project perpendicular lines from the ends of the true length projection (A1, B1) across the X2Y2 line.
• Step 4: Transfer the distances from the view perpendicular to the true length view (e.g., from the top view, measure the perpendicular distances of a and b from the XY line) onto the new projection lines, measuring from the X2Y2 line.
• Step 5: Connect the transferred points. The angle this projection makes with the X2Y2 line is the true angle of the line with the Horizontal Plane (HP).

Example 2: Determine the true angle of line AB (from Example 1) with the Horizontal Plane (HP).

Solution Steps:

1. Step 1: Follow steps 1-5 from Example 1 to find the true length A1B1.
2. Step 2: Draw a reference line X2Y2 perpendicular to the true length projection A1B1.
3. Step 3: Project lines perpendicular from A1 and B1 across X2Y2.
4. Step 4: From the front view (a’b’), measure the distance from point a’ to the XY line and transfer it from X2Y2 along the projector from A1 to get point A2. Similarly, measure the distance from b’ to the XY line and transfer it from X2Y2 along the projector from B1 to get point B2.
5. Step 5: Connect A2 and B2. The angle that A2B2 makes with the X2Y2 line is the true angle of the line AB with the Horizontal Plane (HP).

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 asks pupils if they remember how to draw orthographic projections and what an auxiliary view is. The teacher then introduces the topic by explaining that sometimes we need to find the actual size and angle of lines in space.
Pupils’ Activity: Pupils respond to questions about orthographic projections and auxiliary views.
Learning Point: Pupils recall prior knowledge related to the topic.

Step 2: Meaning of True Length

Time: 5 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines “true length” of a line and explains why projected lengths are often shorter than the true length, using a simple model of a line inclined to a plane.
Pupils’ Activity: Pupils listen attentively and ask questions for clarification.
Learning Point: Pupils understand the meaning of true length.

Step 3: Determination of True Length (Auxiliary View Method)

Time: 10 minutes
Teaching Skill: Demonstration/Explanation
Teacher’s Activity: The teacher demonstrates the step-by-step procedure for determining the true length of a line using the auxiliary view method on the board or with a projector, explaining each step clearly.
Pupils’ Activity: Pupils observe the demonstration, take notes, and follow along with their drawing instruments if possible.
Learning Point: Pupils learn the procedure for finding true length.

Step 4: Worked Examples (True Length)

Time: 8 minutes
Teaching Skill: Problem Solving/Application
Teacher’s Activity: The teacher guides pupils through one or two worked examples for determining true length, ensuring they understand how to apply the steps.
Pupils’ Activity: Pupils attempt to solve the examples in their notebooks, seeking assistance when needed.
Learning Point: Pupils apply the true length determination procedure.

Step 5: Meaning of True Angle

Time: 5 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines “true angle” of a line to a plane and explains when it appears in its true size, again using visual aids.
Pupils’ Activity: Pupils listen and understand the concept of true angle.
Learning Point: Pupils understand the meaning of true angle.

Step 6: Determination of True Angle (Auxiliary View Method)

Time: 7 minutes
Teaching Skill: Demonstration/Explanation
Teacher’s Activity: The teacher demonstrates the step-by-step procedure for determining the true angle of a line to a principal plane using the auxiliary view method, building upon the true length determination.
Pupils’ Activity: Pupils observe the demonstration, take notes, and try to replicate the steps.
Learning Point: Pupils learn the procedure for finding true angle.

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 the true length of a line?
2. When does a line show its true length on a projection plane?
3. State two methods used to determine the true length of a line.
4. Explain how to determine the true angle of a line to the Horizontal Plane using the auxiliary view method.

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 true length and angles and assigns homework, which involves drawing problems to find true lengths and angles.
Pupils’ Activity: Pupils listen to the summary and copy down the homework.
Learning Point: Pupils consolidate their understanding and prepare for practice.

Lesson Keywords

• True Length – The actual length of a line in space.
• True Angle – The actual angle a line makes with a plane.
• Auxiliary View – A projection onto a plane that is not parallel to any of the principal planes.
• Projection – The representation of an object on a plane.

Differentiation

For pupils who grasp the concepts quickly, the teacher can provide more complex problems involving lines in different orientations or angles to multiple planes. For pupils needing more support, the teacher can provide additional guided practice, simpler examples, and one-on-one assistance, focusing on the basic steps of the auxiliary view method.

Note for teachers using this lesson plan

Ensure pupils have a solid understanding of orthographic projections before introducing true length and angles. Emphasize precision in drawing and measurement. Use physical models to help visualize the concepts of lines in space and their projections. Encourage pupils to practice regularly to develop their drawing skills.