Lesson Notes By Weeks and Term v3 - Junior Secondary 3
Construction
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Construction
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Subject: General Mathematics
Class: Junior Secondary 3
Term: 3rd Term
Week: 8
Theme: Mensuration And Geometry
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construct angle 45o Construct angles 300 Use a pair of compasses to copy a given angle Construct simple shapes.
say R.
5. Measure the distance between points A and B on the given angle using the compass (place the compass point at A and adjust the pencil point to B).
6. With R as the centre and the same radius (AB distance), draw an arc to intersect the long arc drawn in step 4 at a point, say S.
7. Draw a straight line from P through S.
8. The angle ∠SPQ is a copy of ∠XYZ. 2.6 Construction of Simple Shapes This involves applying angle and line segment constructions to create polygons.
Equilateral Triangle: A triangle with all three sides equal and all three angles equal to 60°. Steps for constructing an equilateral triangle with side length 's':
1. Draw a line segment AB of length 's'.
2. With A as the centre and radius 's', draw an arc above AB.
3. With B as the centre and the same radius 's', draw another arc to intersect the first arc at point C.
4. Join A to C and B to
C. Triangle ABC is an equilateral triangle.
Square: A quadrilateral with four equal sides and four right (90°) angles. Steps for constructing a square with side length 's':
1. Draw a line segment AB of length 's'.
2. Construct a 90° angle at point A (using steps from 2.1), extending upwards. Mark a point D on this vertical line such that AD = 's'.
3. Construct a 90° angle at point B (using steps from 2.1), extending upwards.
4. With D as the centre and radius 's', draw an arc.
5. With B as the centre and radius 's', draw an arc to intersect the line extending from B upwards (from step 3) at point C. (Alternatively, from D, draw an arc of radius 's', and from C, draw an arc of radius 's' to intersect at a fourth point, say C, but this is less direct).
6. More practically for a square: From point D, with radius 's', draw an arc. From point B, with radius 's', draw an arc. The intersection of these arcs will be point
C. Join B to C and C to
D. ABCD is a square. Worked Example (Constructing 45°): Problem: Construct an angle of 45° at point P on a line X
Y. Solution:
1. Draw a line segment X
Y. Mark a point P on it. ``` X-----------P-----------Y ```
2. Construct 90° at P: With P as centre, draw arcs intersecting XY at A and B. With A and B as centres (radius > AP), draw arcs intersecting at C above
P. Draw P
C. Angle CPX = 90°. ``` C | X---A-P-B---Y ```
3. Bisect 90°: With P as centre, draw an arc intersecting PA and PC at D and E respectively. With D and E as centres (radius > DE/2), draw arcs intersecting at F within angle CPX. Draw PF. ``` C /| / | E F | / | D----P---Y X-----A----- ``` Angle FPX = 45°. This section provides detailed steps for the constructions covered in this lesson. Teachers should demonstrate each step clearly on the board or with an overhead projector. 2.1 Construction of Angle 90° (Right Angle) The construction of 45° relies on the ability to construct 90°.
Steps:
1. Draw a straight line segment, say AB.
2. Choose a point P on the line AB.
3. With P as the centre and any convenient radius, draw arcs to intersect the line AB at two points, say X and Y.
4. With X as the centre and a radius greater than PX, draw an arc above P.
5. With Y as the centre and the same radius (as in step 4), draw another arc to intersect the previous arc at a point, say C.
6. Draw a straight line from C to P. The angle ∠CPB (or ∠CPA) is 90°. 2.2 Construction of Angle 45° Angle 45° is half of 90°.
Therefore, it is constructed by bisecting a 90° angle.
Steps:
1. First, construct an angle of 90° (e.g., ∠CPB) as described above.
2. With P as the centre and any convenient radius, draw an arc to intersect the arms of the 90° angle (line PB and line PC) at two points, say D and E, respectively.
3. With D as the centre and a radius greater than half the distance DE, draw an arc in the interior of ∠CPB.
4. With E as the centre and the same radius, draw another arc to intersect the previous arc at a point, say F.
5. Draw a straight line from P through F.
6. The angle ∠FPB is 45°. 2.3 Construction of Angle 60° The construction of 30° relies on the ability to construct 60°.
Steps:
1. Draw a straight line segment, say AB.
2. Choose a point O on the line AB.
3. With O as the centre and any convenient radius, draw an arc to intersect the line AB at a point, say C.
4. With C as the centre and the same radius (as in step 3), draw another arc to intersect the first arc at a point, say D.
5. Draw a straight line from O through D.
6. The angle ∠DOB is 60°. 2.4 Construction of Angle 30° Angle 30° is half of 60°.
Therefore, it is constructed by bisecting a 60° angle.
Steps:
1. First, construct an angle of 60° (e.g., ∠DOB) as described above.
2. With O as the centre and any convenient radius, draw an arc to intersect the arms of the 60° angle (line OB and line OD) at two points, say E and F, respectively.
3. With E as the centre and a radius greater than half the distance EF, draw an arc in the interior of ∠DOB.
4. With F as the centre and the same radius, draw another arc to intersect the previous arc at a point, say G.
5. Draw a straight line from O through G.
6. The angle ∠GOB is 30°. 2.5 Copying a Given Angle This technique allows replication of an angle without knowing its measure.
Steps:
1. Given: An angle ∠XYZ.
2. To copy: Draw a new line segment, say PQ, which will be one arm of the copied angle.
3. With Y as the centre (vertex of the given angle) and any convenient radius, draw an arc to intersect arms YX and YZ at points A and B respectively.
4. With P as the centre (vertex of the new angle) and the same radius (as in step 3), draw a long arc intersecting PQ at a point, say R.
5. Measure the distance between points A and B on the given angle using the compass (place the compass point at A and adjust the pencil point to B).
6. With R as the centre and the same radius (AB distance), draw an arc to intersect the long arc drawn in step 4 at a point, say S.
7. Draw a straight line from P through S.
8. The angle ∠SPQ is a copy of ∠XYZ. 2.6 Construction of Simple Shapes This involves applying angle and line segment constructions to Materials: Whiteboard/Blackboard, markers/chalk, ruler, pair of compasses, protractor (for verification only), drawing sheets/exercise books for students.
Introduction (10 minutes): Teacher Activity: Engage students by asking about tools used by carpenters, architects, or tailors to measure and mark precise angles or shapes. Introduce the concept of geometric construction using only a compass and straightedge, explaining its historical significance and modern relevance (e.g., in drafting, design). Remind students of the importance of accuracy and neatness.
Student Activity: Participate in the discussion, sharing observations about measurement tools. Listen attentively and understand the objective of the lesson.
Development (50 minutes): Activity 1: Construction of Angle 45° (15 minutes)
Teacher Activity: Demonstrate step-by-step the construction of a 90° angle on the board, explaining each arc and line. Proceed to demonstrate the bisection of the 90° angle to construct 45°, emphasizing the use of the compass for equal radii. Circulate, observe students' progress, and provide immediate feedback and assistance.
Student Activity: Follow the teacher's demonstration carefully. Construct a 90° angle in their exercise books. Subsequently, bisect the 90° angle to construct 45° precisely. Attempt to measure their constructed angles with a protractor to verify accuracy (emphasize that the construction should be accurate without protractor).
Activity 2: Construction of Angle 30° (15 minutes)
Teacher Activity: Demonstrate the construction of a 60° angle, highlighting how it arises naturally from drawing arcs with the same radius. Proceed to demonstrate the bisection of the 60° angle to construct 30°, reiterating the principles of angle bisection. Encourage students to articulate the steps as the teacher demonstrates.
Student Activity: Construct a 60° angle in their exercise books. Bisect the 60° angle to construct 30°. Compare results with peers and discuss any discrepancies.
Activity 3: Copying a Given Angle (10 minutes)
Teacher Activity: Draw an arbitrary angle (e.g., 70° or 110°) on the board. Demonstrate the steps for copying this angle onto a new baseline, emphasizing the critical steps of measuring arc lengths with the compass. Provide a specific angle for students to copy (e.g., an angle drawn on a worksheet).
Student Activity: Observe the teacher's demonstration keenly. Attempt to copy the given angle onto a new line segment in their books. Use a protractor to verify if their copied angle matches the original.
Activity 4: Construction of Simple Shapes (10 minutes)
Teacher Activity: Demonstrate the construction of an equilateral triangle with a given side length (e.g., 5 cm). Discuss how the 60° angle is implicitly used in an equilateral triangle. Guide students through the steps to construct a square with a given side length (e.g., 4 cm), linking back to the 90° angle construction.
Student Activity: Construct an equilateral triangle with a specified side length (e.g., 6 cm). Construct a square with a specified side length (e.g., 5 cm). Measure the sides and angles of their constructed shapes to check for accuracy.
Conclusion (5 minutes): Teacher Activity: Review the key constructions covered: 45°, 30°, copying angles, and simple shapes. Summarize the importance of accurate use of construction tools. Assign homework.
Student Activity: Participate in the review. Ask clarifying questions. Note down homework assignment.
Question 1: Construct an angle of 45°.
Solution: Draw a line segment A
B. Mark a point P on A
B. Construct 90° at P: With P as center, draw an arc intersecting AB at X and Y. With X and Y as centers (radius > PX), draw arcs intersecting at
C. Draw line P
C. Angle CPB = 90°.
Bisect angle CPB: With P as center, draw an arc intersecting PB at D and PC at E. With D and E as centers (radius > DE/2), draw arcs intersecting at
F. Draw line P
F. Angle FPB = 45°.
Commentary: This construction combines the ability to create a right angle and then bisect it, demonstrating understanding of angle bisection.
Question 2: Construct an angle of 30°.
Solution: Draw a line segment O
P. Construct 60° at O: With O as center, draw an arc intersecting OP at Q. With Q as center and the same radius, draw an arc intersecting the first arc at
R. Draw line O
R. Angle ROP = 60°.
Bisect angle ROP: With O as center, draw an arc intersecting OP at S and OR at T. With S and T as centers (radius > ST/2), draw arcs intersecting at
U. Draw line O
U. Angle UOP = 30°.
Commentary: This involves the foundational construction of 60° (an equilateral triangle angle) and then bisecting it precisely.
Question 3: Copy a given angle ∠ABC, where ∠ABC is an acute angle. (Teacher draws an acute angle ∠ABC on the board for students to copy)
Solution: Given: Angle ∠ABC. Draw a new line segment PQ. This will be one arm of the copied angle. With B (vertex of ∠ABC) as center, draw an arc intersecting BA at X and BC at Y. With P (vertex of the new angle) as center and the same radius used in step 3, draw a long arc intersecting PQ at R. Measure the distance XY using the compass. With R as center and the same radius (XY), draw an arc intersecting the long arc from step 4 at
S. Draw a line from P through
S. The angle ∠SPQ is a copy of ∠AB
C. Commentary: This exercise tests the student's ability to transfer angular measure accurately using only a compass.
Question 4: Construct an equilateral triangle with sides 7 cm.
Solution: Draw a line segment AB of length 7 cm. With A as center and radius 7 cm, draw an arc above AB. With B as center and the same radius 7 cm, draw another arc to intersect the first arc at point
C. Join A to C and B to
C. Triangle ABC is an equilateral triangle with sides 7 cm.
Commentary: This demonstrates the practical application of basic arc construction to form a polygon with specific properties. Students should verify that all angles are 60°.
Architectural Design and Building Construction: In Nigeria, architects and masons often use the principles of geometric construction to lay out foundations, mark perpendicular walls, or determine angles for roof structures. For example, constructing a 90° angle is essential for ensuring that the corners of a room or building are perfectly square, preventing structural instability and ensuring aesthetic appeal. The construction of 45° angles is useful for roof pitches or decorative elements. Tailoring and Fashion Design (e.g., Ankara and Aso-Oke): Nigerian fashion designers and tailors frequently use geometric principles to cut fabrics and create patterns. Constructing precise angles allows them to design intricate patterns for garments, cut collars, sleeves, or produce culturally significant designs on Ankara or Aso-Oke fabrics, ensuring symmetry and proper fit. Copying angles is essential when replicating design elements or adapting patterns.
Carpentry and Furniture Making: Carpenters in Nigeria apply construction techniques to measure and cut wood accurately for furniture, door frames, or roof trusses. For example, when building a table, a carpenter must ensure all legs are perpendicular to the tabletop (90° angles) and that any angled supports are cut precisely (e.g., 30° or 45° angles for bracing). Constructing simple shapes ensures that components fit together perfectly.
Surveying and Land Use Planning: Surveyors in Nigeria utilize geometric construction principles to measure and map land, especially in rural areas or for urban development. The ability to construct accurate angles is fundamental for delineating property boundaries, subdividing plots for housing projects, or planning road networks within communities.