Lesson Notes By Weeks and Term v3 - Junior Secondary 2
Construction
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Construction
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Subject: General Mathematics
Class: Junior Secondary 2
Term: 3rd Term
Week: 7
Theme: Mensuration And Geometry
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Construct triangles given:(a) 2 sides and a secluded angle(b) 2 angles and a side between them(c) All the 3 sides Bisect any given angles
draw an arc to cut QP at A and QR at B.
3. With A as the center, draw an arc inside the angle.
4. With B as the center and the same radius, draw another arc to intersect the arc drawn from A. Label the intersection point C.
5. Draw a straight line from Q through C.
6. The line QC bisects ∠PQR into two 40° angles. (Visual representation for teacher: An angle PQR. Arc from Q intersecting QP and QR at A and B. Arcs from A and B intersecting at C. Line QC bisecting the angle.) DE, 8 cm long.
2. At point D, use a protractor to construct an angle of 50° (∠EDF). Draw a ray from D.
3. At point E, use a protractor to construct an angle of 60° (∠DEF) on the same side of DE as ∠EDF. Draw a ray from E.
4. The intersection point of the two rays is point F.
5. Triangle DEF is the required construction. (Visual representation for teacher: A line segment DE, then an angle 50 deg opening from D upwards, and an angle 60 deg opening from E upwards. The two angle arms intersect at F.) c. Constructing a Triangle given All Three Sides (SSS - Side-Side-Side)
Concept: If the lengths of all three sides of a triangle are known, a unique triangle can be constructed (provided the triangle inequality theorem holds: the sum of any two sides must be greater than the third side).
Tools: Ruler, compass, sharp pencil.
Steps:
1. Draw the longest side as base: Use the ruler to draw the longest given side as the base. Label its endpoints (e.g., A and B).
2. Draw the first arc: Open the compass to the length of the second given side. Place the compass needle at one endpoint of the base (e.g., A) and draw an arc above the base.
3. Draw the second arc: Open the compass to the length of the third given side. Place the compass needle at the other endpoint of the base (e.g., B) and draw another arc to intersect the first arc. Label the intersection point (e.g., C).
4. Complete the triangle: Connect the endpoints of the base (A and B) to the intersection point (C) with straight lines.
5. Label: Label the triangle clearly.
Worked Example 3: Construct a triangle XYZ with XY = 9 cm, YZ = 7 cm, and XZ = 6 cm.
1. Draw line segment XY, 9 cm long.
2. With X as the center, and radius 6 cm (length of XZ), draw an arc above XY.
3. With Y as the center, and radius 7 cm (length of YZ), draw another arc to intersect the first arc. Label the intersection point Z.
4. Join X to Z and Y to Z with straight lines.
5. Triangle XYZ is the required construction. (Visual representation for teacher: A line segment XY. Arc from X with radius X
Z. Arc from Y with radius Y
Z. Intersection is
Z. Join XZ and YZ.)
2. Bisecting Any Given Angle Concept: To bisect an angle means to divide it into two equal angles. This can be done accurately using only a compass and a ruler.
Tools: Ruler, compass, sharp pencil.
Steps:
1. Draw the angle: Draw the angle to be bisected (e.g., ∠ABC), with vertex B.
2. Draw an arc: Place the compass needle at the vertex (B) and draw an arc that intersects both arms of the angle (BA and BC). Label the intersection points D and E respectively.
3. Draw intersecting arcs: Place the compass needle at D and draw an arc within the angle. Without changing the compass opening, place the needle at E and draw another arc to intersect the first arc. Label the intersection point F.
4. Draw the bisector: Draw a straight line from the vertex (B) through the intersection point
F. This line (BF) is the angle bisector.
5. Verify: The angle ∠ABF should be equal to ∠CB
F. Worked Example 4: Bisect an angle of 80°.
1. Draw an angle of 80° (e.g., ∠PQR) with vertex Q.
2. With Q as the center, draw an arc to cut QP at A and QR at B.
3. With A as the center, draw an arc inside the angle.
4. With B as the center and the same radius, draw another arc to intersect the arc drawn from A. Label the intersection point C.
5. Draw a straight line from Q through C.
6. The line QC bisects ∠PQR into two 40° angles. **(Visual representation for teacher: An angle PQ
R. Arc from Q intersecting QP and QR at A and
B. Arcs from A and B intersecting Geometric Construction: Geometric construction involves drawing geometric figures accurately using only a compass and a straightedge (ruler without markings for measurement). A sharp pencil is essential for precision. It is crucial to maintain accuracy throughout the process, as small errors can accumulate.
Construction of Triangles: A triangle is a three-sided polygon. For a unique triangle to be constructed, specific information about its sides and angles must be provided. This lesson focuses on three primary conditions: a. Constructing a Triangle given Two Sides and a Secluded Angle (SAS - Side-Angle-Side)
Concept: If the lengths of two sides and the measure of the angle between them are known, a unique triangle can be constructed. The "secluded angle" means the included angle.
Tools: Ruler, compass, sharp pencil.
Steps:
1. Draw the first side: Use the ruler to draw one of the given sides as the base. Label its endpoints (e.g., A and B).
2. Construct the secluded angle: At one endpoint of the base (e.g., A), use a protractor (or compass and ruler for specific angles like 60°, 90°, 45°) to construct the given angle. Draw a long ray from A.
3. Mark the second side: Open the compass to the length of the second given side. Place the compass needle at the vertex of the angle (e.g., A) and draw an arc to intersect the ray drawn in step
2. Label the intersection point (e.g., C).
4. Complete the triangle: Connect the unjoined endpoint of the base (e.g., B) to the point marked in step 3 (C) with a straight line.
5. Label: Label the triangle clearly.
Worked Example 1: Construct a triangle PQR such that PQ = 7 cm, QR = 6 cm, and ∠PQR = 70°.
1. Draw line segment PQ, 7 cm long.
2. At point Q, use a protractor to construct an angle of 70° (∠PQR). Draw a ray from Q.
3. With Q as the center, and radius 6 cm (length of QR), draw an arc to cut the ray constructed in step
2. Label the intersection point R.
4. Join point P to point R with a straight line.
5. Triangle PQR is the required construction. (Visual representation for teacher: A line segment PQ, then an angle 70 deg opening from Q, with R marked on the angle arm 6cm away. P and R are joined.) b. Constructing a Triangle given Two Angles and a Side Between Them (ASA - Angle-Side-Angle)
Concept: If the measures of two angles and the length of the side between them are known, a unique triangle can be constructed.
Tools: Ruler, compass, sharp pencil, protractor (for general angles).
Steps:
1. Draw the included side: Use the ruler to draw the given side as the base. Label its endpoints (e.g., X and Y).
2. Construct the first angle: At one endpoint of the base (e.g., X), use a protractor (or compass for specific angles) to construct the first given angle. Draw a long ray from X.
3. Construct the second angle: At the other endpoint of the base (e.g., Y), use a protractor (or compass) to construct the second given angle on the same side of the base as the first angle. Draw a long ray from Y.
4. Identify the third vertex: The point where the two rays (from step 2 and 3) intersect is the third vertex of the triangle (e.g., Z).
5. Label: Label the triangle clearly.
Worked Example 2: Construct a triangle DEF such that DE = 8 cm, ∠EDF = 50°, and ∠DEF = 60°.
1. Draw line segment DE, 8 cm long.
2. At point D, use a protractor to construct an angle of 50° (∠EDF). Draw a ray from D.
3. At point E, use a protractor to construct an angle of 60° (∠DEF) on the same side of DE as ∠EDF. Draw a ray from E.
4. The intersection point of the two rays is point F.
5. Triangle DEF is the required construction. **(Visual representation for teacher: A line segment DE, then an angle 50 deg opening from D upwards, and an angle 60 deg opening from Teacher Activities: Introduction (5 minutes): Briefly review basic geometric terms (line segment, ray, angle, vertex). Introduce the concept of geometric construction and the tools required (ruler, compass, sharp pencil). Emphasize precision. State the performance objectives for the lesson, connecting them to real-world scenarios. Demonstration of SSS Triangle Construction (10 minutes): Using a large drawing board, projector, or clear visual aids, demonstrate step-by-step the construction of a triangle given its three sides. Clearly verbalize each step, explaining why it is done. Emphasize accurate measurement with the compass. Demonstration of SAS Triangle Construction (10 minutes): Similarly, demonstrate the construction of a triangle given two sides and the secluded angle. Show how to draw an angle using a protractor (for arbitrary angles) or how to construct basic angles (60°, 90°, 45°) with a compass, if relevant to students' prior knowledge. Demonstration of ASA Triangle Construction (10 minutes): Demonstrate the construction of a triangle given two angles and the included side. Stress the importance of drawing both angles on the same side of the base. Demonstration of Angle Bisection (5 minutes): Demonstrate the steps for bisecting a given angle, clearly showing the arcs from the vertex and intersection points. Guided Practice and Supervision (15 minutes): Distribute worksheets with guided practice questions (similar to those in Section 4). Move around the classroom, providing individual assistance, correcting techniques, and reinforcing proper use of tools. Encourage students to articulate their steps.
Review and Summary (5 minutes): Recap the different methods of constructing triangles and the process of angle bisection. Address any common misconceptions or difficulties observed during practice.
Student Activities: Preparation: Ensure they have their mathematical sets (ruler, compass, sharp pencil, protractor) ready and paper.
Observation: Closely observe the teacher's demonstrations of each construction method.
Active Participation: Ask clarifying questions during demonstrations.
Practice: Attempt the guided practice questions immediately after each demonstration or during the guided practice session.
Collaboration: Discuss steps and compare results with peers (under supervision).
Accuracy Check: Learn to measure their constructed sides and angles to check for accuracy using their ruler and protractor.
Town Planning and Surveying (Land Demarcation in Rural/Urban Areas): Nigerian surveyors frequently use principles of triangle construction to divide land into specific plots for residential, commercial, or agricultural use. For instance, when demarcating a new housing estate in Abuja or a farm plot in Kano, surveyors create triangular segments to measure and map irregular boundaries, ensuring fair and accurate allocation of land based on measured sides and angles. The construction of accurate scale drawings from field measurements is critical for official land titles and property development. Local Artisanship (Carpentry and Tailoring): Carpentry: A Nigerian carpenter building a roof truss for a house or constructing a triangular shelf unit for a market stall in Lagos applies SAS, SSS, and ASA principles. They need to cut wood to precise lengths and angles to ensure stability and fit. For example, constructing a strong A-frame roof often requires constructing congruent triangles, for which knowledge of these construction methods is essential.
Tailoring: Local tailors and fashion designers in Aba or Onitsha use geometric construction to cut fabric for complex garment patterns, such as sleeves, collars, or decorative inserts. They often need to bisect angles for perfectly symmetrical designs or construct triangular patterns for specific garment aesthetics, ensuring that the final product is balanced and visually appealing. Infrastructure Development (Bridge Building and Road Design): Engineers constructing bridges or complex road interchanges across Nigeria, for example, over the Niger Delta, rely heavily on accurate geometric constructions. Triangular frameworks are inherently stable and are fundamental to bridge design. Constructing these triangles with precise angles and side lengths ensures the structural integrity and safety of the infrastructure. Similarly, designing curves and junctions in roads involves constructing and bisecting angles to ensure smooth traffic flow.