Lesson Notes By Weeks and Term v3 - Junior Secondary 1
Three dimensional figures
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Three dimensional figures
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Subject: General Mathematics
Class: Junior Secondary 1
Term: 2nd Term
Week: 4
Theme: Mensurtion And Geometry
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Identify the properties of cubes and cuboids Identify the properties of pyramids and cones Identity the properties of cylinders and spheres Find volume of a cube and a cuboid.
smoothly from a flat base (usually circular) to a point called the apex or vertex.
Properties: Number of Faces: 2 (1 circular base, 1 curved surface that acts as a 'face')
Number of Edges: 1 (the circular edge of the base)
Number of Vertices: 1 (the apex) Has a circular base and a curved surface.
Examples: Party hats, traffic cones, ice cream cones.
5. Cylinder Definition: A cylinder is a three-dimensional solid with two parallel circular bases of the same size, connected by a curved surface.
Properties: Number of Faces: 3 (2 circular bases, 1 curved lateral surface)
Number of Edges: 2 (the circular edges of the bases)
Number of Vertices: 0 Has two identical circular bases and a curved surface.
Examples: Milo tin, water tanks, Dala Foods tomato paste can, batteries, drums.
6. Sphere Definition: A sphere is a perfectly round three-dimensional object in which every point on its surface is equidistant from its center.
Properties: Number of Faces: 1 (a single curved surface)
Number of Edges: 0 Number of Vertices: 0 It is perfectly round with no flat surfaces.
Examples: A football, a marble, a globe, an orange. --- Introduction to Three-Dimensional Figures (3D Shapes) Three-dimensional figures, or solid shapes, are objects that possess length, width (or breadth), and height (or depth). Unlike two-dimensional (2D) shapes which are flat, 3D shapes occupy space and have volume.
Common Properties of 3D Figures: Most polyhedra (3D shapes with flat faces) share these characteristics: Faces: These are the flat surfaces of the 3D figure.
Edges: These are the lines where two faces meet.
Vertices (singular: Vertex): These are the points or corners where three or more edges meet. Specific Three-Dimensional Figures and Their Properties:
1. Cube Definition: A cube is a three-dimensional solid object bounded by six square faces, with three meeting at each vertex. All its edges are of equal length.
Properties: Number of Faces: 6 (all are squares)
Number of Edges: 12 Number of Vertices: 8 All faces are congruent squares. All edges are of equal length.
Examples: A dice, a sugar cube, a Rubik's cube, some water tanks.
Volume of a Cube: The volume (V) of a cube is found by multiplying its length, width, and height. Since all sides are equal, if 'L' is the length of one edge: `V = L × L × L = L3` Worked Example 1 (Cube Volume): A JSS1 student has a cubic wooden block with an edge length of 5 cm. Calculate the volume of the block.
Solution: Given: Edge length (L) = 5 cm Volume of a cube, V = L3 V = 5 cm × 5 cm × 5 cm V = 125 cm3 The volume of the wooden block is 125 cubic centimeters.
2. Cuboid (Rectangular Prism)
Definition: A cuboid is a three-dimensional solid object with six rectangular faces. It has three pairs of identical opposite faces.
Properties: Number of Faces: 6 (all are rectangles, or some might be squares if specific dimensions are equal)
Number of Edges: 12 Number of Vertices: 8 Opposite faces are identical (congruent) rectangles.
Examples: A matchbox, a brick, a typical textbook, a carton of milk or Indomie noodles.
Volume of a Cuboid: The volume (V) of a cuboid is found by multiplying its length, width, and height. If 'L' is length, 'W' is width, and 'H' is height: `V = L × W × H` Worked Example 2 (Cuboid Volume): A carton of Bournvita has a length of 30 cm, a width of 20 cm, and a height of 15 cm. Calculate the volume of the carton.
Solution: Given: Length (L) = 30 cm, Width (W) = 20 cm, Height (H) = 15 cm Volume of a cuboid, V = L × W × H V = 30 cm × 20 cm × 15 cm V = 600 cm2 × 15 cm V = 9000 cm3 The volume of the Bournvita carton is 9000 cubic centimeters.
3. Pyramid Definition: A pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and the apex form a triangular face (lateral face). Pyramids are named by the shape of their base (e.g., square pyramid, triangular pyramid). Properties (for a square-based pyramid, common example): Number of Faces: 5 (1 square base, 4 triangular lateral faces)
Number of Edges: 8 (4 base edges, 4 lateral edges)
Number of Vertices: 5 (4 base vertices, 1 apex vertex)
Examples: Ancient Egyptian pyramids, some types of roofing structures in Nigeria.
4. Cone Definition: A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.
Properties: Number of Faces: 2 (1 circular base, 1 curved surface that acts as a 'face')
Number of Edges: 1 (the circular edge of the base)
Number of Vertices: 1 (the apex) Has a circular base and a curved surface.
Examples: Party hats, traffic cones, ice cream cones.
5. Cylinder Definition: A cylinder is a three-dimensional solid with two parallel circular bases of the same size, connected by a curved Teacher Activities: Introduction (10 minutes): Display a collection of real-life 3D objects or clear pictures of them (e.g., a matchbox, a Milo tin, a football, a pyramid model, a party hat). Ask students to identify the objects and discuss what makes them different from flat drawings (2D shapes). Guide them to understand they occupy space and have depth. Introduce the term "three-dimensional figures" or "solid shapes." Exploring Properties (20 minutes): Cube and Cuboid: Present physical models of a cube and a cuboid (e.g., a dice, a textbook, a brick). Guide students to count and identify the faces, edges, and vertices for each shape using the models. Facilitate a discussion on the differences and similarities between cubes and cuboids (e.g., all faces of a cube are squares, while a cuboid has rectangular faces).
Pyramid and Cone: Display a pyramid model (preferably square-based) and a cone (e.g., a party hat or traffic cone). Guide students to identify the base, apex, and the nature of their faces (flat triangles for pyramid, curved for cone). Count faces, edges, and vertices for the pyramid. Explain why a cone has only one circular edge and one vertex.
Cylinder and Sphere: Present a cylinder (e.g., Milo tin) and a sphere (e.g., football). Guide students to identify the circular bases and curved surface of the cylinder. Count its faces, edges, and vertices. Discuss the unique nature of a sphere – a single curved surface with no distinct faces, edges, or vertices.
Introducing Volume (15 minutes): Explain the concept of volume as the amount of space occupied by a 3D object. Introduce the standard units of volume (cubic centimeters, cubic meters). Present the formula for the volume of a cube (L3) and a cuboid (L × W × H). Work through Worked Example 1 and 2 on the board, explaining each step clearly and ensuring units are correctly applied. Guided Practice and Classwork (15 minutes): Provide students with practice questions targeting identification of properties and calculation of volume for cubes and cuboids. Walk around the classroom, observe students' work, and provide immediate feedback and support.
Student Activities: Observation and Identification: Students will observe the displayed 3D objects and identify them by their common names.
Hands-on Exploration: Students will handle physical models (if available) of cubes, cuboids, pyramids, cones, cylinders, and spheres, and physically count their faces, edges, and vertices under teacher guidance.
Drawing: Students will attempt to sketch simple representations of the 3D shapes.
Note-taking: Students will copy key definitions, properties, and volume formulas into their notebooks.
Problem Solving: Students will solve guided practice questions in their exercise books, applying the learned formulas and concepts.
Discussion: Students will actively participate in class discussions, asking and answering questions about the properties of the shapes. --- The teacher should present these questions on the board and guide students through the solutions, emphasizing the steps and reasoning.
Question 1: A typical JSS1 Mathematics textbook is shaped like a cuboid. a) How many faces does it have? b) How many edges does it have? c) How many vertices does it have?
Solution 1: a) A cuboid has 6 faces. (There's a front, back, top, bottom, left side, right side) b) A cuboid has 12 edges. (4 on the top face, 4 on the bottom face, and 4 connecting the top and bottom faces) c) A cuboid has 8 vertices. (4 on the top face, 4 on the bottom face)
Question 2: Consider a model of a square-based pyramid. a) How many faces does it have? b) How many edges does it have? c) How many vertices does it have?
Solution 2: a) A square-based pyramid has 5 faces. (1 square base and 4 triangular lateral faces) b) A square-based pyramid has 8 edges. (4 edges for the square base and 4 lateral edges connecting the base vertices to the apex) c) A square-based pyramid has 5 vertices. (4 vertices on the square base and 1 vertex at the apex)
Question 3: Describe the properties of the following common objects found in Nigeria: a) A Milo tin (consider it a perfect geometric shape). b) A football.
Solution 3: a) A Milo tin is a cylinder. It has 3 faces (2 flat circular bases and 1 curved surface). It has 2 edges (the circular boundaries of the top and bottom bases). It has 0 vertices. It has two identical circular bases and a smooth, curved lateral surface. b) A football is a sphere. It has 1 face (a single continuous curved surface). It has 0 edges. It has 0 vertices. It is perfectly round, with all points on its surface equidistant from its center.
Question 4: A cubic sugar cube has an edge length of 2.5 cm. Calculate its volume.
Solution 4: Given: Edge length (L) = 2.5 cm Formula for volume of a cube: V = L3 V = (2.5 cm)3 V = 2.5 cm × 2.5 cm × 2.5 cm V = 6.25 cm2 × 2.5 cm V = 15.625 cm3 The volume of the sugar cube is 15.625 cubic centimeters.
Question 5: A standard Nigerian brick measures 22 cm in length, 10 cm in width, and 7 cm in height. Find the volume of one such brick.
Solution 5: Given: Length (L) = 22 cm, Width (W) = 10 cm, Height (H) = 7 cm Formula for volume of a cuboid: V = L × W × H V = 22 cm × 10 cm × 7 cm V = 220 cm2 × 7 cm V = 1540 cm3 The volume of one brick is 1540 cubic centimeters. ---
Construction and Building Design: Application: Architects and builders constantly work with 3D shapes. Houses are typically cuboid structures, pillars are cylindrical, and some roofs are pyramidal. Understanding the properties of these shapes is crucial for structural stability, aesthetic design, and efficient use of space.
Nigerian Context: In Nigeria, knowledge of 3D figures helps masons and engineers calculate the volume of materials like sand, gravel, and cement needed to mix concrete for foundations, walls, and beams. For example, knowing the volume of a cuboid-shaped water tank helps a family determine its water storage capacity, which is vital in areas with inconsistent water supply.
Packaging and Logistics: Application: Manufacturing companies design packaging for their products using 3D shapes. Cuboids are common for cartons (e.g., Indomie noodles, beverages), while cylinders are used for tins (e.g., Milo, tomato paste). The ability to calculate volume is critical for optimizing packaging size, fitting maximum products into shipping containers, and efficient stacking in warehouses or market stalls.
Nigerian Context: Traders in markets like Onitsha Main Market or Balogun Market need to understand how cuboid-shaped goods (cartons of provisions) can be arranged efficiently within their stalls or during transportation in lorries to maximize profit and minimize transport costs. Similarly, companies like Dangote Cement package their products in cuboid bags, and the volume determines the quantity.
Agriculture and Storage: Application: Farmers and agricultural businesses use 3D shapes for storage facilities. Grain silos are often cylindrical, while barns or granaries are typically cuboid. Calculating the volume of these structures helps farmers determine their storage capacity for harvests like maize, rice, or groundnuts, planning for surpluses or shortages.
Nigerian Context: A farmer in Kano State, harvesting a large quantity of groundnuts, would need to know the volume of their cylindrical or cuboid storage facility to estimate how much of their produce can be stored effectively before selling, preventing spoilage and ensuring food security. ---