Lesson Notes By Weeks and Term v3 - Senior Secondary 1
Special Curves
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Special Curves
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Subject: Technical Drawings
Class: Senior Secondary 1
Term: 3rd Term
Week: 12
Theme: Geometrical Constructions
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Explain the term loci, ellipse, cycloids, and the ir applications. Construct an ellipse using different methods
This section provides in-depth explanations of the key terms and construction methods for special curves, enabling the teacher to deliver the content comprehensively. This section outlines the flow of the lesson, detailing teacher and student engagements.
Teacher Activities: Introduction (10 minutes): Begin by reviewing basic geometric constructions (lines, circles, tangents, divisions of lines/circles). Introduce the concept of 'locus' with simple examples (e.g., path of a door handle as a door opens, path of a ball thrown in the air). Briefly explain that special curves are specific types of loci important in engineering and design.
Explanation of Key Concepts (25 minutes): Loci: Provide a clear definition and simple examples.
Ellipse: Define, explain major/minor axes, foci, centre. Discuss real-world applications relevant to Nigeria (e.g., bridges, architectural features, gears).
Cycloid: Define as the locus of a point on the circumference of a rolling circle. Explain the rolling motion clearly. Discuss applications (e.g., cam profiles, gear teeth).
Trochoid: Define as the locus of a point inside or outside the circumference of a rolling circle. Distinguish between inferior and superior trochoids with clear descriptions. Discuss applications (e.g., Wankel engine rotors, railway wheels). Demonstration of Ellipse Construction (30 minutes): Using a large drawing board, projector, or interactive whiteboard, meticulously demonstrate the construction of an ellipse using the Concentric Circle Method (Auxiliary Circle Method). Explain each step clearly, emphasizing accuracy, division into equal parts, and correct projection. Label all points and axes correctly. Encourage students to ask questions during the demonstration. Demonstration of Cycloid Construction (30 minutes): On the drawing board/projector, demonstrate the construction of a cycloid, detailing each step from drawing the base line and generating circle to connecting the locus points. Emphasize the concept of the rolling circle and how the centre moves. Demonstration of Trochoid Construction (20 minutes): Briefly demonstrate the construction of both an inferior and superior trochoid, highlighting the difference from the cycloid construction (the point P's distance from the centre).
Guided Practice Setup (10 minutes): Provide students with drawing instruments (T-square, set squares, compass, dividers, pencils, drawing paper). Set a specific problem for ellipse construction.
Circulation and Support: Circulate among students, providing individual guidance, checking progress, correcting errors, and answering questions. Emphasize neatness, precision, and correct use of instruments.
Conclusion and Assignment (5 minutes): Summarize the key concepts and construction methods covered. Assign independent practice questions.
Student Activities: Active Listening and Note-Taking: Students will listen attentively to explanations and take detailed notes on definitions, properties, and construction steps.
Questioning: Students will ask clarifying questions during the teacher's explanations and demonstrations.
Observation: Students will closely observe the teacher's demonstrations of construction methods.
Guided Practice: Students will attempt to construct an ellipse using the concentric circle method under the teacher's guidance.
Participation: Students will respond to questions posed by the teacher, discuss concepts with peers, and contribute to class discussions.
Independent Practice: Students will complete assigned construction problems as homework or in-class exercises, applying the learned methods.
Materials: Drawing board/projector Whiteboard/chalkboard T-square, set squares, compass, dividers, protractor, French curves (for demonstration) Drawing paper (A3 or A2 for practicals) Pencils (HB, H, 2H) Erasers Textbooks (for reference, if available, though lesson note is self-contained) --- The teacher should guide students through these problems, step-by-step, ensuring they understand the process.
Question 1: Explanation of Terms Explain the meaning of 'locus' and state two real-life applications of an ellipse in Nigeria.
Solution 1: Locus: A locus is the path traced by a point, line, or surface moving in a specific manner according to given conditions. It represents all possible positions of a point that satisfies a particular geometric rule. Real-life applications of an ellipse in Nigeria:
1. Architectural Design: Elliptical arches are used in the construction of modern buildings, bridges (e.g., pedestrian bridges, overpasses on expressways), and monument designs across Nigeria for structural integrity and aesthetic appeal.
2. Mechanical Engineering: Elliptical gears are used in specific machinery for non-uniform motion transfer, such as in textile machinery or packaging equipment in Nigerian factories. Also, cam profiles can be designed with elliptical segments for controlled motion.
3. Infrastructure: Some road networks or roundabout designs might incorporate elliptical curves for smoother traffic flow or specific aesthetic requirements in urban planning.
Question 2: Ellipse Construction Construct an ellipse given a major axis of 120 mm and a minor axis of 80 mm using the Concentric Circle Method (Auxiliary Circle Method).
Solution 2: (Teacher to demonstrate and guide students)
1. Draw Axes: Draw a horizontal line AB 120 mm long. Mark its midpoint O. Through O, draw a vertical line CD 80 mm long, ensuring O is its midpoint. So, OA = OB = 60 mm, and OC = OD = 40 mm.
2. Draw Concentric Circles: With O as the centre, draw a larger circle (major auxiliary circle) with radius 60 mm (half of the major axis) and a smaller circle (minor auxiliary circle) with radius 40 mm (half of the minor axis).
3. Divide Circles: Divide both circles into 12 equal parts using a protractor (360/12 = 30 degrees per division) or a pair of dividers. Draw radial lines from O through these division points. Label points on the major circle 1, 2, ..., 12 and corresponding points on the minor circle 1', 2', ..., 12'.
4. Project from Major Circle: From each point on the major circle (e.g., point 1), draw a vertical line (parallel to CD) towards the horizontal axis.
5. Project from Minor Circle: From the corresponding point on the minor circle (e.g., point 1' on the same radial line as point 1), draw a horizontal line (parallel to AB) towards the vertical axis.
6. Locate Ellipse Points: The intersection of the vertical line from point 1 and the horizontal line from point 1' gives a point on the ellipse (P1). Repeat this process for all 12 division points (P1, P2, ..., P12).
7. Draw the Ellipse: Carefully connect all the plotted points (P1 to P12, including A, B, C, D which are also points on the ellipse) with a smooth, continuous curve using a French curve or freehand.
Commentary: Accuracy in dividing the circles and drawing parallel projection lines is crucial for a smooth and accurate ellipse.
Question 3: Cycloid Construction Construct one arch of a cycloid, given that the diameter of the generating circle is 50 mm.
Solution 3: (Teacher to demonstrate and guide students)
1. Radius and Base Line: The radius R = 50/2 = 25 mm. Draw a horizontal base line, say AB. The length of one complete roll is the circumference = πD = 3.142 50 mm = 157.1 mm. Mark this length on the base line.
2. Generating Circle: Draw a circle with radius 25 mm tangent to the base line at point
A. Mark its centre C.
3. Divide Circle: Divide the generating circle into 12 equal parts. Label them 0, 1, 2, ..., 11,
1
2. Point 0 and 12 are at the bottom (tangent to the base line).
4. Divide Base Line: Divide the 157.1 mm length on the base line into 12 equal parts. Label these points 0', 1', 2', ..., 12'.
5. Locus of Centre: Draw a line parallel to the base line through the centre C of the initial circle. This is the locus of the centre of the rolling circle. 6. *Mark Centre C.
3. Divide Circle: Divide the generating circle into 12 equal parts. Label them 0, 1, 2, ..., 11,
1
2. Point 0 and 12 are at the bottom (tangent to the base line).
4. Divide Base Line: Divide the 157.1 mm length on the base line into 12 equal parts. Label these points 0', 1', 2', ..., 12'.
5. Locus of Centre: Draw a line parallel to the base line through the centre C of the initial circle. This is the locus of the centre of the rolling circle.
6. Mark Centre Positions: From each division point on the base line (1', 2', ...), draw vertical lines to intersect the locus of the centre line. These are the positions of the centre (C1, C2, ... C12) at different stages of rolling.
7. Draw Horizontal Lines from Circle: From each division point on the generating circle (1, 2, ...), draw horizontal lines parallel to the base line.
8. Locate Cycloid Points: From centre C1, with radius R (25mm), draw an arc to intersect the horizontal line from point
1. This is P
1. From C2, with radius R, draw an arc to intersect the horizontal line from point
2. This is P
2. Continue this for all points (P0, P1, ..., P12). Note that P0 is at A, and P12 is at B.
9. Draw the Cycloid: Connect the points P0 to P12 with a smooth, continuous curve.
Commentary: Ensure the horizontal lines from the circle divisions and the vertical lines from the base line divisions are accurately drawn. The arcs from the centre positions must be drawn with the exact radius of the generating circle. ---
This section highlights how the concepts of special curves are relevant in various aspects of Nigerian life and development. Architecture and Construction (Bridges and Buildings): Ellipses are extensively used in the design of aesthetically pleasing and structurally sound arches for bridges, overhead passes, and modern building entrances across Nigerian cities like Lagos, Abuja, and Port Harcourt. For instance, some pedestrian bridges or decorative elements in public squares may feature elliptical curves. Understanding their construction allows future architects and civil engineers to design robust and visually appealing infrastructure. Mechanical Engineering and Manufacturing (Machinery and Vehicles): Cycloids and trochoids are crucial in designing specific types of gear teeth profiles for smooth and efficient power transmission in machinery used in Nigerian industries (e.g., agricultural processing machines, cement factories, textile mills). They are also applied in cam mechanisms for converting rotary motion into specific reciprocating or oscillating motions, vital in various Nigerian manufacturing and automation processes.
Furthermore, the understanding of trochoids can be seen in the motion of components within internal combustion engines, including those assembled or manufactured locally.
Industrial Design and Product Development: The elegant forms of ellipses, and sometimes cycloids or trochoids, are incorporated into the design of various products, furniture, and consumer goods in Nigeria. From elliptical tabletops and mirrors to the curved aesthetics of modern vehicles or electronic gadgets, these curves contribute to both functionality and visual appeal. This knowledge empowers aspiring Nigerian product designers to create innovative and ergonomic designs that meet local market demands. ---