Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Solve multi-step measurement problems.
• Calculate optimal packing and material usage.
• Interpret scale drawings and maps.
• Analyse measurement systems and choose appropriate methods.
• Calculate costs for measurement-based projects.

Lesson summary

This week, we delve deeper into measurement, focusing on complex applications in real-life contexts. Measurement is not just about knowing the length of a table; it's a fundamental skill that impacts daily decisions, from budgeting for groceries to planning a community event. In South Africa, where resources are often constrained, accurate measurement and informed estimation become crucial for efficient planning and responsible resource allocation, benefiting families, communities, and businesses. This week builds on your prior knowledge of measurement units and conversions, exploring how to apply these skills in more intricate and practical scenarios.

Lesson notes

2. 1.

Optimal Packing and Material Usage: Many real-world problems involve finding the most efficient way to pack items into a container or to use materials to minimize waste. These problems often require calculating volumes, areas, and perimeters.

Volume: The amount of space a 3D object occupies. Units include cm³, m³, litres (L), and millilitres (mL). Recall that 1 L = 1000 cm³ and 1 m³ = 1000

L. Area: The amount of surface a 2D shape covers. Units include cm², m², and hectares (ha). Remember that 1 ha = 10 000 m².

Perimeter: The total distance around the outside of a 2D shape. Units include cm, m, and km.

Formulas for Common Shapes: It is crucial to remember formulae for volume, area, and perimeter for common shapes, such as: Cube: Volume = side³, Area = 6 side² Rectangular Prism: Volume = length width height, Area = 2(length width + length height + width height)

Cylinder: Volume = π radius² height, Area = 2π radius height + 2π radius² Sphere: Volume = (4/3) π radius³, Area = 4π radius² Rectangle: Area = length width, Perimeter = 2(length + width)

Circle: Area = π radius², Circumference (Perimeter) = 2π * radius 2.

2. Scale Drawings and Maps: Scale drawings and maps represent real-world objects and locations in a smaller, proportional size. The scale indicates the ratio between the drawing/map distance and the actual distance. For instance, a scale of 1:100 means that 1 cm on the drawing represents 100 cm (or 1 meter) in reality.

Calculating Actual Distances: To find the actual distance, multiply the distance on the drawing/map by the scale factor.

Calculating Map Distances: To find the distance on the drawing/map, divide the actual distance by the scale factor. 2.

3. Cost Analysis and Budgeting: Many real-world measurement problems involve calculating costs associated with materials, labor, and other expenses. It is essential to understand unit costs, quantity discounts, and how to calculate total costs.

Unit Cost: The cost per single unit of a product or service.

Quantity Discount: A reduction in the unit cost when purchasing a large quantity of a product.

Total Cost: The sum of all expenses associated with a project or task. 2.4 Optimization Problems: Optimization problems deal with finding the "best" solution to a problem, often involving maximizing or minimizing a certain quantity (e.g., maximizing volume while minimizing surface area, or minimizing material waste while meeting specific structural requirements).

Worked example

Example 1: Optimal Packing

A small business in Durban produces boxes of rusks. Each box is 20 cm long, 15 cm wide, and 10 cm high. The business wants to ship these boxes in larger containers that are 1.2 m long, 0.9 m wide, and 0.6 m high.

a) How many rusk boxes can fit into one container?

b) If each rusk box costs R35 to produce, what is the total value of the rusk boxes in one container?