Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Calculate the area of composite 2D shapes.
• Determine the surface area of cubes and rectangular prisms.
• Calculate the volume of cubes and rectangular prisms.
• Solve real-world problems involving these measurements.
• Convert between different units of area and volume.

Lesson summary

This week, we delve deeper into the world of measurement, focusing on area, surface area, and volume. These are crucial concepts that help us understand and interact with the space around us. From calculating the amount of paint needed for your bedroom wall to determining the capacity of a water tank for your garden, measurement skills are essential in everyday life, particularly in a practical environment like South Africa. Understanding these concepts not only strengthens your mathematical abilities but also equips you with valuable problem-solving skills applicable in various fields like construction, agriculture, and design.

Lesson notes

2.1 Area: Area is the amount of two-dimensional space a shape occupies. It's measured in square units (e.g., cm², m², km²). Remember, area is only for flat, 2D shapes.

Rectangle: Area = length × width (A = l × w)

Square: Area = side × side (A = s²) (Since a square is a special type of rectangle where length = width = side)

Triangle: Area = ½ × base × height (A = ½ × b × h)

Circle: Area = π × radius² (A = πr²) (π is approximately 3.14 or 22/7)

Example 1: Rectangle and Square Imagine you're tiling a rectangular kitchen floor that is 4 meters long and 3 meters wide. What is the area of the floor that needs to be tiled?

Solution: Area = length × width = 4 m × 3 m = 12 m². You will need enough tiles to cover 12 square meters. Consider a square garden plot with sides of 5 meters. What is the area available for planting?

Solution: Area = side × side = 5 m × 5 m = 25 m².

Example 2: Triangle A triangular piece of land has a base of 10 meters and a height of 8 meters. What is its area?

Solution: Area = ½ × base × height = ½ × 10 m × 8 m = 40 m².

Example 3: Circle A circular swimming pool has a radius of 3 meters. What is the area of the pool's surface? (Use π = 3.14)

Solution: Area = π × radius² = 3.14 × (3 m)² = 3.14 × 9 m² = 28.26 m².

Area of Composite Shapes: Composite shapes are made up of two or more simpler shapes. To find their area, break them down into those simpler shapes, calculate the area of each, and then add them together.

Example 4: Composite Shape A shape is made up of a rectangle (length 6cm, width 4cm) and a triangle (base 4cm, height 3cm) on top of the rectangle. Find the total area.

Solution: Area of rectangle = 6 cm × 4 cm = 24 cm² Area of triangle = ½ × 4 cm × 3 cm = 6 cm² Total area = 24 cm² + 6 cm² = 30 cm². 2.2 Surface Area: Surface area is the total area of all the faces of a 3D object. It’s also measured in square units (e.g., cm², m²).

Cube: A cube has 6 identical square faces. Surface Area = 6 × (side²) = 6s² Rectangular Prism (Cuboid): A rectangular prism has 6 rectangular faces. Surface Area = 2 × (length × width + length × height + width × height) = 2(lw + lh + wh)

Example 5: Cube A cube has sides of 2 meters. What is its surface area?

Solution: Surface Area = 6 × (2 m)² = 6 × 4 m² = 24 m².

Example 6: Rectangular Prism A rectangular prism has a length of 5 cm, a width of 3 cm, and a height of 4 cm. What is its surface area?

Solution: Surface Area = 2 × (5 cm × 3 cm + 5 cm × 4 cm + 3 cm × 4 cm) = 2 × (15 cm² + 20 cm² + 12 cm²) = 2 × (47 cm²) = 94 cm². 2.3 Volume: Volume is the amount of three-dimensional space an object occupies. It's measured in cubic units (e.g., cm³, m³, liters).

Cube: Volume = side × side × side = s³ Rectangular Prism: Volume = length × width × height = l × w × h Example 7: Cube A cube has sides of 3 meters. What is its volume?

Solution: Volume = (3 m)³ = 3 m × 3 m × 3 m = 27 m³.

Example 8: Rectangular Prism A rectangular prism has a length of 6 cm, a width of 2 cm, and a height of 5 cm. What is its volume?

Solution: Volume = 6 cm × 2 cm × 5 cm = 60 cm³. 2.4 Conversions: 1 m = 100 cm 1 m² = (100 cm)² = 10,000 cm² 1 m³ = (100 cm)³ = 1,000,000 cm³ Example 9: Conversion Convert 5 m² to cm².

Solution: 5 m² = 5 × 10,000 cm² = 50,000 cm². Convert 2 m³ to cm³.

Solution: 2 m³ = 2 × 1,000,000 cm³ = 2,000,000 cm³. Guided Practice (With Solutions)

Question 1: A rectangular garden bed is 8 meters long and 5 meters wide. What is the area of the garden bed?

Solution: Area = length × width = 8 m × 5 m = 40 m². The area of the garden bed is 40 square meters. This represents the space available for planting.

Question 2: A cube has sides of 4 cm. Calculate its surface area.

Solution: Surface Area = 6 × side² = 6 × (4 cm)² = 6 × 16 cm² = 96 cm². The surface area represents the total area of all the faces of the cube, as if you were wrapping it in paper.

Question 3: A rectangular container is 10 cm long, 5 cm wide, and 3 cm high. Calculate its volume.

Solution: Volume = length × width × height = 10 cm × 5 cm × 3 cm = 150 cm³. The volume represents the amount of space inside the container. If it was filled with water, the volume would be the amount of water it holds.

Question 4: A composite shape is made up of a square (side 3 cm) and a semicircle (radius 1.5 cm) attached to one side of the square. Find the total area. (Use π = 3.14)

Solution: Area of the square = 3 cm × 3 cm = 9 cm² Area of the semicircle = ½ × π × r² = ½ × 3.14 × (1.5 cm)² = ½ × 3.14 × 2.25 cm² ≈ 3.53 cm² Total Area = 9 cm² + 3.53 cm² = 12.53 cm² Question 5: Convert 3 m³ to cm³.

Solution: 3 m³ = 3 × (100 cm)³ = 3 × 1,000,000 cm³ = 3,000,000 cm³. Remember that volume increases cubically when changing from larger to smaller units, so there are a lot more cubic centimeters in a cubic meter. Independent Practice (Questions Only) Calculate the area of a triangle with a base of 12 cm and a height of 7 cm. A circular rug has a radius of 2 meters. What is the area of the rug? (Use π = 3.14) A cube has sides of 6 meters.