Lesson Notes By Weeks and Term v3 - Senior Secondary 1
Circles and Triangles
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Circles and Triangles
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Subject: Technical Drawings
Class: Senior Secondary 1
Term: 2nd Term
Week: 3
Theme: Geometrical Constructions
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Identify and describe parts of a circle. Identify and describe types of circles. In scribe a circle to a given triangle. Circumscribe a circle to a given triangle. Describe a circle to a given triangle.
Materials: Drawing boards, T-squares, set squares (30/60 and 45 degree), compasses, dividers, pencils (HB, 2H), erasers, clean sheets of A3 or A4 drawing paper, protractor (optional). Chalkboard/whiteboard, markers/chalk.
A. Introduction (10 minutes)
Teacher Activity: Initiate a brief review of basic geometric shapes, lines, and angles from previous lessons. Ask students to identify circular and triangular objects in their surroundings (e.g., fan blades, plates, roof trusses, traffic signs). Introduce the topic "Circles and Triangles" and state the learning objectives for the lesson. Emphasize the importance of accuracy and neatness in technical drawing.
Student Activity: Participate in the review, identifying shapes and objects. Listen attentively to the introduction and objectives.
B. Development of Concepts & Practical Demonstrations (45 minutes)
Teacher Activity: Parts of a Circle (15 mins): Draw a large circle on the board. Systematically identify and label each part (centre, radius, diameter, circumference, chord, arc, segment, sector, tangent, secant) explaining each definition clearly. Emphasize the relationship between radius and diameter.
Types of Circles (10 mins): Demonstrate the construction of concentric circles on the board using a compass (or string and chalk for larger circles). Use real-world examples like an 'odu' (calabash dish) with concentric carvings. Demonstrate eccentric circles, showing centres at different locations.
Constructions (20 mins): Inscribing a Circle: Draw a suitable triangle on the board. Demonstrate, step-by-step, the accurate construction of angle bisectors using a large compass and ruler. Identify the incenter and draw the perpendicular to find the radius. Draw the inscribed circle. Emphasize the tangency property.
Circumscribing a Circle: Draw another triangle. Demonstrate, step-by-step, the accurate construction of perpendicular bisectors of two sides. Identify the circumcenter and measure the radius to any vertex. Draw the circumscribed circle. Emphasize that the circle passes through all vertices.
Escribing a Circle: Draw a third triangle. Demonstrate extending the sides and bisecting the external angles (and one internal if using that method). Identify the excenter and draw the perpendicular to the tangent side for the radius. Draw the escribed circle. Highlight the tangency to one side and the extensions of the other two.
Student Activity: Copy the diagrams and definitions for parts of a circle into their notebooks. Observe teacher demonstrations closely, asking clarifying questions. Take notes on construction steps. Mentally follow the construction processes.
C. Guided Practice (30 minutes)
Teacher Activity: Distribute drawing paper and ensure students have their instruments. Provide specific dimensions for triangles and circles for students to practice. Circulate around the classroom, providing individual guidance, correcting errors in construction, and checking for accuracy and neatness. Encourage peer-to-peer learning and discussion.
Student Activity: Attempt to construct the different types of circles, inscribed, circumscribed, and escribed circles, following the steps demonstrated by the teacher. Seek assistance from the teacher or peers when facing difficulties. Focus on precision in using instruments and neatness in drawing.
D. Conclusion (5 minutes)
Teacher Activity: Review the key concepts covered: parts of a circle, concentric/eccentric circles, and the three types of triangle-related circle constructions. Address any common misconceptions observed during practice. Assign independent practice questions as homework.
Student Activity: Participate in the review. Ask any remaining questions. Note down homework assignments.
Question 1: Identifying Parts of a Circle Construct a circle of radius 40mm. Label its centre (O), a radius (OA), a diameter (BD), a chord (CE) that is not a diameter, a major arc (CDE), a minor segment (formed by CE), and a tangent (FG) at point
A. Solution:
1. Using a compass, draw a circle with centre O and radius 40mm.
2. Mark a point 'A' on the circumference. Draw a line segment from O to A, and label it 'OA' (Radius).
3. Draw a straight line passing through O and extending to two points on the circumference, 'B' and 'D'. Label this 'BD' (Diameter).
4. Draw a straight line connecting two points 'C' and 'E' on the circumference, ensuring it does not pass through O. Label this 'CE' (Chord).
5. Highlight the arc from C through D to E as a thicker line or with an arrow. Label it 'Major Arc CDE'.
6. Shade the region bounded by chord CE and the minor arc CE. Label this 'Minor Segment'.
7. At point A, draw a line perpendicular to OA (radius). Label this line 'FG' (Tangent). (Teacher to draw this on the board step-by-step for students to copy and label)
Question 2: Constructing Concentric and Eccentric Circles a) Construct two concentric circles with radii 30mm and 50mm respectively. b) Construct two eccentric circles. The first circle has a radius of 40mm with centre O
1. The second circle has a radius of 25mm with centre O2, such that O2 is 60mm to the right of O
1. Solution: a)
Concentric Circles:
1. Mark a point 'O' as the common centre.
2. Set the compass to 30mm radius, place the needle at O, and draw the first circle.
3. Without moving the compass needle from O, set the compass to 50mm radius and draw the second circle. (Teacher to demonstrate) b)
Eccentric Circles:
1. Mark a point 'O1' on your paper.
2. Set the compass to 40mm radius, place the needle at O1, and draw the first circle.
3. From O1, measure 60mm horizontally to the right and mark the point 'O2'.
4. Set the compass to 25mm radius, place the needle at O2, and draw the second circle. (Teacher to demonstrate)
Question 3: Inscribing a Circle in an Equilateral Triangle Inscribe a circle in an equilateral triangle of side length 70mm.
Solution:
1. Draw the triangle: Construct an equilateral triangle ABC with each side measuring 70mm. (Draw base AB=70mm, then with A and B as centers and 70mm radius, draw arcs to intersect at C).
2. Bisect angles: Using a compass, bisect angle A and angle B. From A, draw an arc to cut AB and AC. From these points, draw two arcs to intersect. Draw the bisector. Repeat for angle B.
3. Locate Incenter: The point where the two angle bisectors intersect is the incenter, I.
4. Find Radius: From I, drop a perpendicular to one side, say AB. (Place compass at I, draw an arc cutting AB at two points. Bisect the segment between these two points. The distance from I to the midpoint of this segment is the radius). Let the point of perpendicularity be
D. ID is the radius.
5. Draw Inscribed Circle: With I as the centre and ID as the radius, draw the circle. It should touch all three sides of the triangle. (Teacher to perform this construction on the board, explaining each step clearly)
Question 4: Circumscribing a Circle around a Scalene Triangle Circumscribe a circle around a triangle with sides measuring 60mm, 80mm, and 100mm.
Solution:
1. Draw the triangle: Construct the triangle ABC with sides 60mm, 80mm, and 100mm. (Draw base of 100mm, then use compass to mark 60mm and 80mm arcs from the ends to find the third vertex).
2. Bisect sides perpendicularly: Using a compass, construct the perpendicular bisectors of any two sides, for example, AB and B
C. For AB: Open compass more than half of AB. From A, draw arcs above and below AB. From B, with the same radius, draw arcs intersecting the first ones. Draw the line connecting the intersections. Repeat for BC. 3. *Locate the triangle ABC with sides 60mm, 80mm, and 100mm. (Draw base of 100mm, then use compass to mark 60mm and 80mm arcs from the ends to find the third vertex).
2. Bisect sides perpendicularly: Using a compass, construct the perpendicular bisectors of any two sides, for example, AB and B
C. For AB: Open compass more than half of AB. From A, draw arcs above and below AB. From B, with the same radius, draw arcs intersecting the first ones. Draw the line connecting the intersections. Repeat for BC.
3. Locate Circumcenter: The point where the two perpendicular bisectors intersect is the circumcenter, C.
4. Find Radius: The distance from the circumcenter C to any vertex (e.g., CA, CB, or CC) is the radius. Measure this distance.
5. Draw Circumscribed Circle: With C as the centre and the measured distance (CA) as the radius, draw the circle. It should pass through all three vertices (A, B, and C). (Teacher to perform this construction on the board, emphasizing accuracy for the circumcenter and radius)
Question 5: Escribing a Circle to a Right-Angled Triangle Construct a right-angled triangle with sides 50mm, 120mm, and 130mm. Describe a circle tangent to the 50mm side and the extensions of the other two sides.
Solution:
1. Draw the triangle: Construct a right-angled triangle ABC, with the right angle at
B. Let AB = 120mm, BC = 50mm, and AC = 130mm (hypotenuse).
2. Extend sides: Extend side AB beyond B and side CB beyond B. (This is to get the external angles).
Let's be explicit: Extend side AB past B and side AC past C. (We want the excircle opposite side AC, so we extend AB and BC). More accurately, if the 50mm side is BC, we want the excircle tangent to B
C. So, extend AB beyond A, and CB beyond
C. Correction for Escribing Logic: To describe a circle tangent to side BC (the 50mm side), we need to extend sides AB and A
C. Extend side AB past B to a point
D. Extend side AC past C to a point E.
3. Bisect External Angles: Bisect the external angle at B (angle formed by line DB and BC). Bisect the external angle at C (angle formed by line EC and BC).
4. Locate Excenter: The intersection point of these two external angle bisectors is the excenter (I_a).
5. Find Radius: From I_a, drop a perpendicular to side BC (the 50mm side). Let the point of perpendicularity be
F. The length I_aF is the radius.
6. Draw Escribed Circle: With I_a as the centre and I_aF as the radius, draw the circle. It should be tangent to BC and the extensions of AB and AC. (Teacher to demonstrate this complex construction carefully, ensuring students understand the extensions and angle bisectors)* Differentiation (for diverse learners): Visual Aids: Provide printed handouts with clear diagrams and construction steps for visual learners or those who struggle with copying from the board.
Group Work: Form mixed-ability groups where stronger students can support struggling peers during practical activities. Varied
Examples: Offer different types of triangles (e.g., acute, obtuse, right-angled, isosceles, equilateral) for the constructions, allowing students to tackle challenges at their own pace.
Remediation (for struggling learners): Simplified Tasks: Start with basic constructions: drawing a circle with given radius/diameter, identifying and labeling parts on a pre-drawn circle. Provide partially completed construction steps as a guide for in/circumscribing circles, requiring students to complete the missing steps. Focus on one type of construction at a time (e.g., only inscribing for one session) before moving to others.
One-on-One Support: Offer individualized coaching during practical sessions, demonstrating steps physically on their drawing paper.
Re-teaching: Conduct a mini-lesson for a small group of struggling learners, focusing on the specific areas they find challenging (e.g., angle bisectors, perpendicular bisectors).
Peer Tutoring: Pair students with strong understanding with those who need help to explain steps and monitor their practice.
Extension (for high-achieving learners): Complex Constructions: Challenge them to construct more complex figures that combine circles and triangles, such as designing a logo that incorporates inscribed and circumscribed elements, or drawing a three-leaf clover using arcs and tangents. Introduce them to constructing common external/internal tangents to two circles.
Design Project: Assign a mini-design project, e.g., "Design a plan for a small recreational park for your community, incorporating circular pathways, triangular flower beds, and possibly a circular water fountain." Students would need to apply the constructions learned creatively.
Research & Presentation: Encourage research into specific Nigerian architectural or craft designs that heavily feature circles and triangles, and have them present their findings, including how these geometric principles might have been applied.
Problem Solving: Provide challenging word problems that require applying circle and triangle properties in an engineering or design context. For instance, determining the optimal placement of a circular stage within a triangular event space.
Architectural Design and Construction (Nigerian Housing & Infrastructure): Circular Huts/Buildings: In many parts of Nigeria, particularly in rural areas, traditional houses or storage facilities are circular (e.g., certain Igbo, Yoruba, or Hausa/Fulani architectural styles). The concepts of centre, radius, and circumference are fundamental in their planning and construction.
Dome Structures and Arches: Modern architecture in Nigeria often incorporates circular elements like dome roofs (e.g., mosques, auditoriums) or arched doorways and windows. Designing these requires precise application of circle properties.
Road Design (Roundabouts): Urban planning in Nigerian cities frequently employs roundabouts to manage traffic flow. The construction of concentric and eccentric circles is essential for designing multi-lane roundabouts, road markings, and landscaping features within these junctions. Mechanical Engineering and Craftsmanship (Nigerian Industries & Artisans): Gears, Wheels, Pulleys: Industries in Nigeria that involve manufacturing and machinery (e.g., automobile repair, agro-processing equipment) heavily rely on circular components. Understanding the geometry of circles is critical for designing and fabricating gears, sprockets, and pulleys used in vehicles, grinding machines, or textile looms.
Calabash Decoration/Woodwork: Local artisans in Nigeria who decorate calabashes (e.g., in Oyo, Kwara states) or work with wood often create intricate patterns using circular and triangular motifs. The principles of inscribing and circumscribing circles help in achieving symmetric and aesthetic designs.
Jewellery and Textile Design: Nigerian jewellery (e.g., beadwork, metal crafts) and traditional textile patterns (e.g., Adire, Aso Oke) frequently incorporate geometric shapes. The ability to construct perfect circles and integrate them within triangular frameworks is a valuable skill for designers in these sectors.
Surveying and Land Use Planning: Borehole Influence Zones: In hydrogeology, the area of influence of a water borehole is often modelled as a circle. When planning the optimal placement of multiple boreholes in a community, understanding concentric (for varying pump capacities) or eccentric circles (for spatially distributed boreholes) is important to ensure efficient water extraction without interference.
Mapping and Layout: Surveyors use geometric principles to map land and define boundaries. The ability to accurately draw and relate circles and triangles is crucial in creating precise site plans for construction projects, farmlands, or community layouts across Nigeria.