Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Calculate perimeter and area of 2D shapes.
• Calculate surface area and volume of 3D objects.
• Convert between metric units.
• Solve problems with compound shapes.
• Interpret scale drawings and maps.

Lesson summary

This week, we delve deeper into the practical application of measurement, focusing on perimeter, area, and volume. This is a crucial skill in Mathematical Literacy because it empowers you to make informed decisions in various real-world scenarios, from managing household expenses to understanding property values and planning construction projects. In South Africa, where resource management and cost-effectiveness are vital, understanding measurement principles is essential for both personal and community development.

Lesson notes

2.1 Perimeter: Perimeter is the total distance around the outside of a two-dimensional shape. To find the perimeter, you simply add up the lengths of all the sides. The units of perimeter are units of length (e.g., mm, cm, m, km).

Example: A rectangular vegetable garden is 5 meters long and 3 meters wide. To find the perimeter, you add up the lengths of all four sides: 5m + 3m + 5m + 3m = 16m.

Therefore, the perimeter of the garden is 16 meters. This means you need 16 meters of fencing to enclose the garden. 2.2 Area: Area is the amount of surface a two-dimensional shape covers. Area is measured in square units (e.g., mm², cm², m², km²).

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

Square: Area = side × side (A = s²)

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

Circle: Area = π × radius² (A = πr²) (where π ≈ 3.142)

Example 1: Calculate the area of a rectangular plot of land that is 20 meters long and 15 meters wide. Area = length × width = 20m × 15m = 300 m². This means 300 square meters of lawn can be planted on this plot.

Example 2: Calculate the area of a circular swimming pool with a radius of 3 meters. Area = π × radius² = 3.142 × (3m)² = 3.142 × 9m² ≈ 28.278 m². This means approximately 28.278 square meters of pool liner would be needed. 2.3 Volume: Volume is the amount of space a three-dimensional object occupies. Volume is measured in cubic units (e.g., mm³, cm³, m³).

Cube: Volume = side × side × side (V = s³)

Rectangular Prism (Cuboid): Volume = length × width × height (V = l × w × h)

Cylinder: Volume = π × radius² × height (V = πr²h)

Sphere: Volume = (4/3) × π × radius³ (V = (4/3)πr³)

Example 1: A rectangular water tank is 2 meters long, 1.5 meters wide, and 1 meter high. Calculate the volume of the tank. Volume = length × width × height = 2m × 1.5m × 1m = 3 m³. This means the tank can hold 3 cubic meters of water, which is equivalent to 3000 liters (since 1 m³ = 1000 liters).

Example 2: A cylindrical silo has a radius of 2.5 meters and a height of 10 meters. Calculate the volume of the silo. Volume = π × radius² × height = 3.142 × (2.5m)² × 10m = 3.142 × 6.25m² × 10m ≈ 196.375 m³. 2.4 Surface Area: Surface area is the total area of all the surfaces of a three-dimensional object. We need to calculate the area of each surface and add them together.

Rectangular Prism (Cuboid): Surface Area = 2(lw + lh + wh)

Cube: Surface Area = 6s² Cylinder: Surface Area = 2πr² + 2πrh Sphere: Surface Area = 4πr²

Example: Find the surface area of a rectangular box that measures 5cm long, 3cm wide and 2cm high. Surface Area = 2(lw + lh + wh) = 2((5cm x 3cm) + (5cm x 2cm) + (3cm x 2cm)) = 2(15cm² + 10cm² + 6cm²) = 2(31cm²) = 62cm². 2.5 Compound Shapes and Volumes: Many real-world objects are made up of combinations of basic shapes. To find the area or volume of a compound shape, divide the shape into simpler shapes, calculate the area or volume of each, and then add or subtract the results as needed.

Example: A building is shaped like a rectangle with a triangle on top. To find the total area of the wall, you would find the area of the rectangle and the area of the triangle separately, then add them together. 2.6 Scale Drawings and Maps: Scale drawings and maps use a scale to represent real-world objects or areas in a smaller format. The scale indicates the relationship between the distance on the drawing/map and the corresponding distance in reality. Understanding scales allows you to calculate actual dimensions and areas from the drawing/map.

Example: A map has a scale of 1:50,

0

0

0. This means that 1 cm on the map represents 50,000 cm (or 500 meters or 0.5 km) in reality. If two towns are 4 cm apart on the map, the actual distance between them is 4 cm × 50,000 = 200,000 cm = 2000 meters = 2 km. Guided Practice (With Solutions)

Question 1: A farmer wants to fence a rectangular field that is 80 meters long and 60 meters wide. Fencing costs R45 per meter. How much will it cost the farmer to fence the entire field?

Solution: Calculate the perimeter: Perimeter = 2 × (length + width) = 2 × (80m + 60m) = 2 × 140m = 280m Calculate the total cost: Cost = Perimeter × Cost per meter = 280m × R45/m = R12,600 Answer: It will cost the farmer R12,600 to fence the field.

Question 2: A circular tablecloth has a diameter of 1.8 meters. Calculate the area of the tablecloth.

Solution: Calculate the radius: Radius = Diameter / 2 = 1.8m / 2 = 0.9m Calculate the area: Area = π × radius² = 3.142 × (0.9m)² = 3.142 × 0.81m² ≈ 2.545 m² Answer: The area of the tablecloth is approximately 2.545 m².

Question 3: A cylindrical water tank has a radius of 1 meter and a height of 2 meters. Calculate the volume of the tank in liters.

Solution: Calculate the volume in cubic meters: Volume = π × radius² × height = 3.142 × (1m)² × 2m = 3.142 × 1m² × 2m ≈ 6.284 m³ Convert cubic meters to liters: 1 m³ = 1000 liters, so Volume in liters = 6.284 m³ × 1000 liters/m³ = 6284 liters Answer: The volume of the tank is approximately 6284 liters.