Lesson Note on Construction of Parabola and Hyperbola for SS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: First Term
Week: 2
Age: 16 years
Duration: 45 minutes
Subject: Technical Drawing
Curriculum Theme: Geometric Constructions
Previous Lesson: Meaning and Applications of Special Curves.
Topic: SPECIAL CURVES
Subject Matter: Construction of special curves (parabola by rectangle method, hyperbola by asymptote method)

Specific Objectives

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

Cognitive Domain:

• Define parabola and hyperbola.
• State the methods for constructing a parabola and a hyperbola.

Affective Domain:

• Appreciate the importance of special curves in various engineering designs.
• Show accuracy and neatness in their technical drawings.

Psychomotor Domain:

• Construct a parabola using the rectangle method.
• Construct a hyperbola using the asymptote method.

Social Domain:

• Collaborate with peers when discussing construction techniques.

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. A. Opoku

Instructional Materials

The teacher will teach this lesson with the aid of:

• Drawing board
• T-square
• Set squares (45 degrees and 30/60 degrees)
• Pencils (HB, H, 2H)
• Eraser
• Drawing compass
• Drawing instruments
• Computers (for digital demonstration, if available)
• Prepared charts showing examples of constructed special curves

Rationale for the Lesson

This lesson enables pupils to understand and apply the principles of constructing special curves like parabolas and hyperbolas. These curves are important in various fields such as architecture, engineering, and optics for designing structures and components.

Prerequisite/Previous Knowledge

Pupils should have basic knowledge of geometric constructions, including dividing lines into equal parts, drawing parallel lines, and using drawing instruments accurately.

Lesson Content/Board Summary

SPECIAL CURVES

Parabola

A parabola is a plane curve which is symmetrical and approximately U-shaped. It is the locus of a point that moves such that its distance from a fixed point (focus) is equal to its distance from a fixed straight line (directrix).

The following are steps for constructing a parabola using the rectangle method:

• Step 1: Draw the axis and directrix. Mark the vertex (V) and focus (F).
• Step 2: Construct a rectangle of the desired width and height for the parabola. The vertex V should be the midpoint of the base of the rectangle.
• Step 3: Divide the vertical side of the rectangle (from V upwards) into a number of equal parts (e.g., 4 or 5). Label these divisions 1, 2, 3, etc.
• Step 4: Divide the horizontal half of the top side of the rectangle (from V outwards) into the same number of equal parts. Label these 1′, 2′, 3′, etc.
• Step 5: Draw lines from the vertex V to the points 1′, 2′, 3′ on the top side.
• Step 6: From points 1, 2, 3 on the vertical side, draw lines parallel to the axis of the parabola.
• Step 7: The intersection of the line from V to 1′ and the horizontal line from 1 gives a point on the parabola. Repeat for 2′ and 2, 3′ and 3, etc.
• Step 8: Draw a smooth curve through the vertex V and all the intersection points to complete one half of the parabola. Repeat for the other half.

Hyperbola

A hyperbola is a type of smooth curve lying in a plane, consisting of two pieces, called connected components or branches, that are mirror images of each other and resembling two infinite bows. It is the locus of a point that moves such that the difference of its distances from two fixed points (foci) is constant. Asymptotes are lines that the curve approaches but never touches.

The following are steps for constructing a hyperbola using the asymptote method:

• Step 1: Draw the two asymptotes (lines that intersect at a point). Let the angle between them be suitable (e.g., 60 to 90 degrees).
• Step 2: Mark a point P on the hyperbola. This point usually defines the position of one branch.
• Step 3: From point P, draw lines parallel to each asymptote to meet the other asymptote. This forms a small parallelogram.
• Step 4: Draw a series of lines from the origin (intersection of asymptotes) passing through the region where the hyperbola will lie.
• Step 5: For each line, draw lines parallel to the asymptotes from the points where it intersects the asymptotes. The intersection of these parallel lines will give points on the hyperbola.
• Step 6: Alternatively, draw a line from P to one asymptote, and from the point of intersection on that asymptote, draw a line parallel to the other asymptote. This generates a series of points.
• Step 7: Draw a smooth curve through point P and the generated points to form one branch of the hyperbola. The other branch can be drawn by symmetry.

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 if they know what special curves are or where they might have seen them in real life (e.g., parabolic antennas, cooling towers).
Pupils’ Activity: Pupils respond by sharing their ideas and observations.
Learning Point: Pupils are introduced to the concept of special curves and their relevance.

Step 2: Explanation of Parabola

Time: 7 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines a parabola and explains the rectangle method for its construction, referring to the board summary.
Pupils’ Activity: Pupils listen attentively, take notes, and ask questions for clarification.
Learning Point: Pupils understand the definition and basic principle of constructing a parabola by the rectangle method.

Step 3: Demonstration of Parabola Construction

Time: 10 minutes
Teaching Skill: Demonstration/Modelling
Teacher’s Activity: The teacher uses a drawing board and instruments to demonstrate the step-by-step construction of a parabola using the rectangle method.
Pupils’ Activity: Pupils observe the demonstration carefully, noting the techniques and precision used by the teacher.
Learning Point: Pupils visualize the practical steps involved in constructing a parabola.

Step 4: Explanation of Hyperbola

Time: 7 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines a hyperbola, explains the concept of asymptotes, and describes the asymptote method for its construction, referring to the board summary.
Pupils’ Activity: Pupils listen, take notes, and ask questions about hyperbolas and asymptotes.
Learning Point: Pupils understand the definition and basic principle of constructing a hyperbola by the asymptote method.

Step 5: Demonstration of Hyperbola Construction

Time: 7 minutes
Teaching Skill: Demonstration/Modelling
Teacher’s Activity: The teacher uses a drawing board and instruments to demonstrate the step-by-step construction of a hyperbola using the asymptote method.
Pupils’ Activity: Pupils observe the demonstration, paying attention to how the asymptotes and points are used to form the curve.
Learning Point: Pupils visualize the practical steps involved in constructing a hyperbola.

Step 6: Evaluation/Review

Time: 5 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Define a parabola.
2. List three steps involved in constructing a parabola using the rectangle method.
3. What are asymptotes in the context of a hyperbola?
4. State two steps in constructing a hyperbola by the asymptote method.

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

Step 7: Conclusion

Time: 4 minutes
Teaching Skill: Summarization/Consolidation
Teacher’s Activity: The teacher summarizes the key learning points, emphasizing the importance of accuracy in constructing special curves. The teacher assigns homework for pupils to practice constructing both curves.
Pupils’ Activity: Pupils listen to the summary and copy down the homework assignment.
Learning Point: Pupils consolidate their understanding and are encouraged to practice.

Lesson Keywords

• Parabola – A symmetrical U-shaped curve where every point is equidistant from a fixed point (focus) and a fixed straight line (directrix).
• Hyperbola – A curve with two separate branches, where the difference of the distances from two fixed points (foci) is constant.
• Asymptote – A line that a curve approaches as it heads towards infinity.
• Vertex – The turning point of a parabola.
• Focus – A fixed point used in defining a conic section.
• Directrix – A fixed straight line used in defining a conic section.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex design problems requiring the application of these curves. For pupils needing more support, the teacher can provide individual guidance during construction practice and simplified diagrams to follow.

Note for teachers using this lesson plan

Ensure all drawing instruments are available and in good working condition. Emphasize neatness and accuracy in all constructions. Provide ample practice time for pupils to master the construction methods. Encourage pupils to identify real-world applications of these special curves to enhance their understanding and interest.