Lesson Notes By Weeks and Term v3 - Senior Secondary 1
Constructions
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Constructions
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Subject: General Mathematics
Class: Senior Secondary 1
Term: 1st Term
Week: 5
Theme: Geometry
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Construct a triangle with given sides Bisect a given line segment bisect a given angle Bisect special angle (30°, 45°, 60° and 90°) Construct equivalent angles Construct 4 - sides plane figures. Construct locus of moving point equidistant from;2 point,2 lines, a fixed point etc.
Geometric constructions involve creating geometric figures using only two tools: an unmarked ruler (straightedge) and a compass. The ruler is used to draw straight lines, and the compass is used to draw arcs and circles and to transfer lengths. Precision is paramount.
Teacher Activities: Introduction (10 mins): Engage learners by asking about tools used by builders, tailors, or engineers to draw accurate shapes. Introduce the topic "Constructions" and the specific tools (ruler, compass). Emphasize that no measurements (graduations) on the ruler are to be used, only its straight edge. Briefly review basic geometry terms like line segment, angle, vertex. Demonstration & Explanation (40-50 mins, distributed): Demonstrate each construction method step-by-step on the chalkboard/whiteboard using large-scale instruments (large ruler, string compass if available, or carefully drawn arcs). For each construction, state the objective, list steps clearly, and explain the geometric principle behind it (e.g., perpendicular bisector property, SSS congruence for triangle).
Model good practice: emphasize precision, sharp pencil, firm grip on compass, light construction lines. Guide learners through the first few steps of each construction (e.g., "Now, everyone, draw a line segment `AB` of 6cm for our first construction"). Supervised Practice (40-50 mins, distributed): Provide simple practice tasks for each construction after demonstration. Circulate around the classroom, observe learners' work, and offer immediate feedback and corrections. Address common errors (e.g., changing compass radius unintentionally, arcs not intersecting, faint lines). Encourage peer-to-peer assistance in groups.
Guided Inquiry & Discussion (10 mins): Ask probing questions: "Why must the compass radius be greater than half for line bisection?" "What would happen if the arcs don't intersect?" Facilitate a short discussion on real-life applications relevant to the local environment.
Conclusion: Summarize the key constructions learned. Assign independent practice and homework.
Learner Activities: Tool Preparation: Ensure they have a sharp pencil, an unmarked ruler, and a functional compass.
Active Observation: Pay close attention to teacher demonstrations, taking notes on the steps.
Replication: Follow teacher's instructions to perform each construction individually or in pairs.
Practice: Attempt various construction problems provided by the teacher.
Participation: Ask questions for clarification, contribute to discussions, and demonstrate their constructions when called upon.
Peer Learning: Work with classmates, compare constructions, and help each other understand and rectify errors.
Problem Solving: Apply learned construction techniques to solve given problems, including those involving quadrilaterals and loci. --- Constructing and bisecting 90°:
1. Draw a line segment `QR`. Mark a point `Q` on it.
2. With `Q` as the centre, draw arcs of equal radius to cut `QR` on both sides of `Q` at `S` and `T`.
3. With `S` as the centre, and a radius greater than `SQ`, draw an arc above `QR`.
4. With `T` as the centre and the same radius, draw another arc to intersect the previous arc. Label the intersection `P`.
5. Join `Q` to `P`. `∠PQR = 90°`.
6. To bisect `∠PQR`: With `S` as centre (or another point formed by an arc from `Q` cutting `QP` and `QR`), draw an arc inside `∠PQR`. With `P` (or the point on `QP`) as centre and the same radius, draw another arc to intersect the previous one. Label the intersection `V`. Join `Q` to `V`. `∠RQV = 45°`. (
Commentary: This demonstrates constructing a 90° angle at a point on a line and its bisection to produce 45°.)
Question 4: Constructing a Quadrilateral and Locus (a) Construct a rectangle `WXYZ` with `WX = 8 cm` and `XY = 5 cm`. (b) A point `M` moves such that it is equidistant from two points `A` and `B` which are `10 cm` apart. Construct the locus of `M`.
Solution: (a) Constructing Rectangle `WXYZ`:
1. Draw a line segment `WX = 8 cm`.
2. At point `W`, construct a 90° angle. On the perpendicular line from `W`, measure `5 cm` using the compass and mark point `Z`.
3. At point `X`, construct a 90° angle. On the perpendicular line from `X`, measure `5 cm` using the compass and mark point `Y`.
4. Join `Z` to `Y` with a straight line. 5. `WXYZ` is the constructed rectangle. (
Commentary: This uses the 90° angle construction multiple times and accurate length transfer. Learners should ensure all angles are exactly 90°.) (b) Constructing the locus of `M`:**
1. Draw a line segment `AB = 10 cm`.
2. With `A` as the centre, open the compass to a radius greater than `5 cm`. Draw an arc above and an arc below `AB`.
3. With `B` as the centre and the same radius, draw another set of arcs to intersect the first two arcs. Label the intersection points `C` and `D`.
4. Join `C` to `D` with a straight line.
5. The line `CD` is the perpendicular bisector of `AB`, and it represents the locus of point `M`. (
Commentary: This applies the perpendicular bisector concept as a locus. Emphasize the definition of locus as a path satisfying a given condition.)* --- Instructions: Learners should attempt these constructions using only a ruler and a compass.
Question 1: Constructing a Triangle Construct a triangle `ABC` such that `AB = 7 cm`, `BC = 6 cm`, and `AC = 5 cm`.
Solution:
1. Draw a line segment `AB` of length `7 cm`.
2. With `A` as the centre, and radius `5 cm` (length `AC`), draw an arc above `AB`.
3. With `B` as the centre, and radius `6 cm` (length `BC`), draw another arc to intersect the first arc. Label the intersection point `C`.
4. Join `A` to `C` and `B` to `C` with straight lines.
5. Triangle `ABC` is constructed. (
Commentary: This reinforces the SSS construction method, ensuring learners can transfer lengths accurately using a compass.)
Question 2: Bisecting a Line Segment and an Angle (a) Draw a line segment `PQ` of length `8.4 cm` and construct its perpendicular bisector. (b) Draw an angle `DEF = 110°` and bisect it.
Solution: (a) Bisecting line segment `PQ`:
1. Draw a line segment `PQ = 8.4 cm`.
2. With `P` as the centre, open the compass to a radius greater than half of `8.4 cm` (i.e., > `4.2 cm`). Draw an arc above and an arc below `PQ`.
3. With `Q` as the centre and the same radius, draw another set of arcs to intersect the first two arcs. Label the intersection points `X` and `Y`.
4. Join `X` to `Y` with a straight line. Line `XY` is the perpendicular bisector of `PQ`. (
Commentary: This combines drawing a specific length with the bisection technique. Emphasize that the bisector should be perpendicular and pass through the midpoint.) (b) Bisecting angle `DEF = 110°`:
1. Draw a line `EF`. At point `E` on `EF`, use a protractor to draw `∠DEF = 110°` (
Note: While we can construct many angles, `110°` is not one of the "special angles" usually constructed with just ruler and compass, so for the initial angle, a protractor is implied here for practical classroom purposes, but the bisection must be done with compass and ruler).
2. With `E` as the centre and any convenient radius, draw an arc to cut `ED` at `G` and `EF` at `H`.
3. With `G` as the centre, draw an arc inside `∠DEF`.
4. With `H` as the centre and the same radius, draw another arc to intersect the previous arc. Label the intersection `K`.
5. Draw a straight line from `E` through `K`. Line `EK` bisects `∠DEF`, so `∠DEK = ∠KEF = 55°`. (
Commentary: This demonstrates the general angle bisection method. The teacher should note the use of a protractor to draw the initial 110° angle, as the focus here is on bisecting it with compass and ruler.)
Question 3: Constructing and Bisecting Special Angles (a) Construct `∠ABC = 60°`. Then, bisect it to obtain a `30°` angle. (b) Construct `∠PQR = 90°`. Then, bisect it to obtain a `45°` angle.
Solution: (a) Constructing and bisecting 60°:
1. Draw a line segment `BC`.
2. With `B` as the centre and any convenient radius, draw an arc to cut `BC` at `X`.
3. With `X` as the centre and the same radius, draw another arc to intersect the first arc. Label the intersection `A`.
4. Join `B` to `A`. `∠ABC = 60°`.
5. To bisect `∠ABC`: With `X` as centre, draw an arc inside `∠ABC`. With `A` as centre and the same radius, draw another arc to intersect the previous one. Label the intersection `Y`. Join `B` to `Y`. `∠CBY = 30°`. (
Commentary: This reinforces the fundamental 60° construction and its bisection to produce 30°.) (b) Constructing and bisecting 90°:
1. Draw a line segment `QR`. Mark a point `Q` on it.
2. With `Q` as the centre, draw arcs of equal radius to cut `QR` on both sides of `Q` at `S` and `T`.
3. With `S` as the centre, and a radius greater than `SQ`, draw an arc above `QR`.
4. With `T` as the centre and the same radius, draw another arc to intersect the previous arc. Label the intersection `P`.
5. Join `Q` to `P`. `∠PQR = 90°`.
6. To bisect `∠PQR`: With
Community Planning and Land Demarcation (Surveying): Application: When dividing a piece of communal land for farming or housing, surveyors use construction principles to create accurate boundaries. For example, to ensure fair distribution, a boundary might be the perpendicular bisector of a line connecting two specific landmarks or a locus equidistant from two converging streams. Constructing regular quadrilaterals (squares, rectangles) for building plots ensures optimal use of space and ease of construction.
Nigerian Context: In rural and urban areas, accurate land demarcation prevents disputes. Geometric constructions are fundamental in creating surveys and cadastral maps used by land registries and town planners.
Architecture and Building Construction: Application: Architects use constructions to design floor plans, ensuring walls are perpendicular (90° angles) or parallel, and that rooms are of specific dimensions. Builders use these concepts on-site for setting out foundations, ensuring corners are square, and establishing true vertical and horizontal lines. Constructing a perfect 90-degree corner for a house foundation is a direct application.
Nigerian Context: From modern skyscrapers in Lagos to traditional housing designs, geometric accuracy is vital for structural integrity and aesthetics. Carpenters rely on accurate constructions for making roof trusses and furniture joints. Crafts and Design (Tailoring, Art, Engineering Fabrication): Application: Tailors and fashion designers use construction techniques to draft patterns for clothing, ensuring symmetry, correct angles for collars, sleeves, and fitting. Artists employ geometric principles for perspective drawing and creating intricate patterns (e.g., in Adire fabric designs). Engineers use these for drawing technical blueprints of mechanical parts, ensuring precise angles and dimensions.
Nigerian Context: The vibrant textile industry (e.g., ankara, tie-dye) and various local craft industries (e.g., woodwork, metalwork) rely on precision derived from geometric principles for quality and appeal. ---