Lesson Notes By Weeks and Term v5 - Grade 12
Measurement: complex applications in real-life contexts – Week 1 focus
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Measurement: complex applications in real-life contexts – Week 1 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Subject: Mathematical Literacy
Class: Grade 12
Term: 2nd Term
Week: 1
Theme: General lesson support
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This week, we delve into the practical and complex applications of measurement in real-life contexts. Measurement is not just about knowing formulas; it's about applying that knowledge to solve problems encountered daily, from managing household finances and planning home improvements to interpreting data related to public health and economic trends. Understanding measurement allows us to make informed decisions, analyze situations critically, and participate meaningfully in our communities. In South Africa, with diverse socio-economic realities, the ability to measure accurately and apply measurement concepts is crucial for financial literacy, resource management, and community development.
2.1 Area, Perimeter, and Volume of Composite Shapes A composite shape is a shape made up of two or more simpler shapes. To find the area, perimeter, or volume of a composite shape, we need to break it down into its simpler components, calculate the area, perimeter, or volume of each component, and then add (or subtract, if a component is a hole) the results.
Area: The amount of surface a 2D shape covers. Units are typically square meters (m²), square centimeters (cm²), etc.
Perimeter: The total distance around the outside of a 2D shape. Units are typically meters (m), centimeters (cm), etc.
Volume: The amount of space a 3D object occupies. Units are typically cubic meters (m³), cubic centimeters (cm³), litres (L), etc.
Formulas to Remember: Rectangle: Area = length x width; Perimeter = 2(length + width)
Square: Area = side x side; Perimeter = 4 x side Triangle: Area = 1/2 x base x height Circle: Area = πr²; Circumference (Perimeter) = 2πr (where r is the radius)
Cube: Volume = side³ Rectangular Prism: Volume = length x width x height Cylinder: Volume = πr²h (where h is the height)
Example 1: A farmer in KwaZulu-Natal wants to build a chicken coop. The base will be a rectangle with a semicircle on one end. The rectangle is 5m long and 3m wide. The semicircle is attached to the 3m side. Calculate the total area of the base of the chicken coop.
Step 1: Identify the shapes. We have a rectangle and a semicircle.
Step 2: Calculate the area of the rectangle. Area = length x width = 5m x 3m = 15 m² Step 3: Calculate the area of the semicircle. The diameter of the semicircle is 3m, so the radius is 1.5m. Area of a circle = πr² = π(1.5m)² ≈ 7.07 m². Area of the semicircle = 7.07 m² / 2 ≈ 3.54 m² Step 4: Add the areas together. Total area = 15 m² + 3.54 m² = 18.54 m² Example 2: A homeowner in Gauteng wants to install a small swimming pool in their backyard. The pool is shaped like a rectangular prism with dimensions 4m long, 3m wide, and 1.5m deep. Calculate the volume of water needed to fill the pool completely.
Step 1: Identify the shape. The pool is a rectangular prism.
Step 2: Apply the formula for the volume of a rectangular prism. Volume = length x width x height = 4m x 3m x 1.5m = 18 m³ Step 3: Convert cubic meters to litres. 1 m³ = 1000 L.
Therefore, the volume of water needed is 18 m³ x 1000 L/m³ = 18000 L 2.2 Unit Conversions Being able to convert between different units of measurement is crucial.
Here are some common conversions: Length: 1 m = 100 cm; 1 km = 1000 m; 1 inch = 2.54 cm; 1 foot = 30.48 cm; 1 mile = 1.609 km Area: 1 m² = 10000 cm²; 1 hectare = 10000 m² Volume: 1 L = 1000 mL; 1 m³ = 1000 L Mass: 1 kg = 1000 g; 1 ton = 1000 kg Currency: Depends on current exchange rates (e.g., 1 USD = ZAR). Always check the latest exchange rate.
Example 3: A South African company imports tiles from Italy. The price of the tiles is €25 per square meter. If the current exchange rate is €1 = ZAR 20, calculate the price of the tiles in ZAR per square meter.
Step 1: Identify the given information. Price = €25/m²; Exchange rate = €1 = ZAR 20 Step 2: Multiply the price in euros by the exchange rate. Price in ZAR = €25/m² x ZAR 20/€ = ZAR 500/m² 2.3 Scale Drawings and Maps Scale drawings and maps represent real-world objects or areas proportionally. The scale indicates the ratio between the distance on the drawing/map and the corresponding distance in reality.
Example 4: 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 m or 0.5 km) in reality. If the distance between two towns on the map is 8 cm, calculate the actual distance between the towns.
Step 1: Understand the scale. 1 cm on the map = 0.5 km in reality Step 2: Multiply the distance on the map by the scale factor. Actual distance = 8 cm x 0.5 km/cm = 4 km 2.4 Cost Calculations Many real-life measurement problems involve calculating costs related to materials, labor, and waste.
Example 5: A builder needs to tile a rectangular floor that is 6m long and 4m wide. The tiles cost ZAR 120 per square meter. He estimates that 10% of the tiles will be wasted due to cutting and breakage. The labor cost is ZAR 80 per square meter. Calculate the total cost of tiling the floor.
Step 1: Calculate the area of the floor. Area = length x width = 6m x 4m = 24 m² Step 2: Calculate the amount of tiles needed, including wastage. Wastage = 10% of 24 m² = 0.10 x 24 m² = 2.4 m². Total tiles needed = 24 m² + 2.4 m² = 26.4 m² Step 3: Calculate the cost of the tiles. Tile cost = 26.4 m² x ZAR 120/m² = ZAR 3168 Step 4: Calculate the labor cost. Labor cost = 24 m² x ZAR 80/m² = ZAR 1920 Step 5: Calculate the total cost. Total cost = Tile cost + Labor cost = ZAR 3168 + ZAR 1920 = ZAR 5088 2.5 Data Interpretation Analyzing measurement data presented in tables, graphs, and charts is an essential skill.
Example 6: A municipality monitors water consumption in a suburb.