Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Calculate perimeter and area of composite 2D shapes.
• Calculate volume of 3D objects like prisms and cylinders.
• Solve practical problems involving measurement.
• Convert between different units of measurement.
• Estimate and justify the reasonableness of results.

Lesson summary

Measurement is a fundamental skill in Mathematical Literacy, vital for navigating everyday life and participating effectively in various aspects of South African society. This week, we delve deeper into applying perimeter, area, and volume calculations in practical contexts. From calculating the amount of fencing needed for a garden to determining the capacity of a water tank, these skills empower you to make informed decisions and solve real-world problems. We build upon the foundational understanding established in Week 1, focusing on more complex shapes and scenarios.

Lesson notes

2.1 Perimeter: The perimeter is the total distance around the outside of a two-dimensional shape. It is a linear measurement, expressed in units like meters (m), centimeters (cm), millimeters (mm), kilometers (km). For regular shapes, perimeter can be calculated by adding the lengths of all sides.

Rectangle: P = 2(length + width) = 2(l + w)

Square: P = 4 side = 4s Circle: P = Circumference = 2πr (where r is the radius and π ≈ 3.14159) 2.2 Area: Area is the amount of surface a two-dimensional shape covers. It is a two-dimensional measurement, expressed in square units like square meters (m²), square centimeters (cm²), square millimeters (mm²), hectares (ha), or square kilometers (km²).

Rectangle: A = length width = l * w Square: A = side side = s² Triangle: A = 1/2 base height = 1/2 b * h Circle: A = πr² (where r is the radius and π ≈ 3.14159) 2.3 Volume: Volume is the amount of space a three-dimensional object occupies. It is a three-dimensional measurement, expressed in cubic units like cubic meters (m³), cubic centimeters (cm³), or liters (L). Note that 1 cm³ = 1 ml and 1 m³ = 1000

L. Rectangular Prism (Cuboid): V = length width height = l w * h Cube: V = side side * side = s³ Cylinder: V = πr²h (where r is the radius of the base and h is the height)

Triangular Prism: V = (1/2 base height_triangle) height_prism. This is the area of the triangular base multiplied by the height of the prism. 2.4 Composite Shapes: Composite shapes are made up of two or more simpler shapes. To find the perimeter or area of a composite shape, you need to break it down into its individual component shapes, calculate the perimeter or area of each, and then add or subtract as needed. For perimeter, be careful not to include interior lengths that aren't part of the outer boundary of the composite shape. 2.5 Unit Conversions: Understanding unit conversions is crucial for accurate calculations.

Here are some common conversions: 1 m = 100 cm 1 km = 1000 m 1 cm = 10 mm 1 m² = (100 cm)² = 10000 cm² 1 m³ = (100 cm)³ = 1000000 cm³ 1 liter (L) = 1000 milliliters (ml) = 1000 cm³ 1 m³ = 1000 L 2.6 Worked

Examples: Example 1 (Perimeter & Area - Composite Shape): A homeowner in Soweto wants to fence a garden that is a rectangle with a semi-circular extension. The rectangle is 8m long and 5m wide. The semi-circle is attached to one of the 5m sides. Calculate the length of fencing required.

Solution: Perimeter of rectangle (excluding the side used for the semi-circle): 8 + 5 + 8 = 21m Diameter of semi-circle = 5m, so radius = 5/2 = 2.5m Circumference of full circle = 2πr = 2 π 2.5 = 5π Circumference of semi-circle = (5π) / 2 ≈ 7.85m Total fencing needed: 21m + 7.85m = 28.85m Example 2 (Volume & Capacity - Rectangular Prism): A farmer in KwaZulu-Natal needs a water tank to store rainwater. He wants a tank that is 2m long, 1.5m wide, and 1.2m high. How many liters of water can the tank hold?

Solution: Volume of the tank: V = l w h = 2m 1.5m 1.2m = 3.6 m³ Since 1 m³ = 1000 L, the tank can hold 3.6 * 1000 = 3600 liters of water.

Example 3 (Unit Conversion & Area): A contractor is quoting for tiling a kitchen floor. The floor measures 3.5 meters by 2.8 meters. The tiles are square and measure 20cm by 20cm. How many tiles are needed to cover the floor?

Solution: Convert floor dimensions to cm: 3.5m = 350cm, 2.8m = 280cm Area of the floor: 350cm * 280cm = 98000 cm² Area of one tile: 20cm * 20cm = 400 cm² Number of tiles needed: 98000 cm² / 400 cm² = 245 tiles Guided Practice (With Solutions)

Question 1: A rectangular garden is 12 meters long and 8 meters wide. A path of width 1.5 meters is built around the garden. Calculate the area of the path.

Solution: Area of the garden: 12m * 8m = 96 m² Dimensions of the garden including the path: Length = 12m + 2(1.5m) = 15m; Width = 8m + 2(1.5m) = 11m Area of the garden including the path: 15m * 11m = 165 m² Area of the path: Area (garden + path) – Area (garden) = 165 m² - 96 m² = 69 m²

Commentary: We first calculated the area of the inner rectangle (garden). Then we found the dimensions of the larger rectangle formed by the garden and the path. The area of the path is the difference between these two areas.

Question 2: A cylindrical water tank has a radius of 1.4 meters and a height of 3 meters. Calculate the volume of water the tank can hold, in liters.

Solution: Volume of the cylinder: V = πr²h = π (1.4m)² 3m ≈ 18.47 m³ Conversion to liters: 18.47 m³ * 1000 L/m³ = 18470 L

Commentary: We used the formula for the volume of a cylinder. Then, we converted the volume from cubic meters to liters using the conversion factor 1 m³ = 1000

L. Question 3: A farmer wants to build a kraal in the shape of a square with sides of 15 meters. He needs to buy fencing. Fencing costs R45 per meter. How much will the fencing cost?

Solution: Perimeter of the square kraal: P = 4 s = 4 15m = 60m Total cost of fencing: 60m * R45/m = R2700

Commentary: The perimeter of the square represents the total length of fencing required.