Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Calculate real-life distances and areas from maps using scale.
• Interpret and convert between different types of scales.
• Use maps and plans to determine directions and positions.
• Apply scale to create simple scaled drawings.
• Solve problems using site plans, floor plans, and maps.

Lesson summary

This week, we delve into the essential skill of interpreting and using scale, maps, and plans. This is a critical skill in Mathematical Literacy because it allows us to understand and interact with the world around us, from navigating our neighbourhoods to planning home improvements. Being able to read and interpret maps and plans empowers us to make informed decisions related to travel, construction, design, and resource allocation – skills highly valuable in the South African context where spatial awareness is essential for development and economic participation.

Lesson notes

What is Scale? Scale is the ratio between the distance on a map or plan and the corresponding distance on the ground (real life). It is how we represent a large area or object in a manageable size. Understanding scale is crucial because it allows us to translate measurements from a map or plan to real-world measurements, and vice versa. There are three main ways to express scale: Ratio Scale: Expressed as a ratio, e.g., 1:

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0. This means that 1 unit of measurement on the map represents 100 units of the same measurement in real life. For example, 1 cm on the map represents 100 cm (or 1 meter) in reality.

Word Scale: Expressed in words, e.g., "1 cm represents 5 km". This is straightforward and easy to understand.

Bar Scale (Graphical Scale): A line divided into segments, representing specific distances. This is useful because it remains accurate even if the map or plan is enlarged or reduced in size. Converting Between Scale Types Being able to convert between scale types is a vital skill.

Example 1: Converting a Ratio Scale to a Word Scale Suppose we have a ratio scale of 1:50,

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0. This means 1 cm on the map represents 50,000 cm in real life.

Let's convert this to kilometers: 50,000 cm = 500 meters (divide by 100 to convert cm to meters) 500 meters = 0.5 km (divide by 1000 to convert meters to kilometers) Therefore, the word scale is: "1 cm represents 0.5 km".

Example 2: Converting a Word Scale to a Ratio Scale Suppose we have a word scale of "1 inch represents 2 miles". Let's convert this to a ratio scale. The goal is to have both sides of the ratio in the same units, preferably centimeters. 1 mile = 1.60934 kilometers (approximately) 2 miles = 2 * 1.60934 km = 3.21868 km 3.21868 km = 3218.68 meters (multiply by 1000) 3218.68 meters = 321868 cm (multiply by 100) Therefore, the ratio scale is approximately 1:321,

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8. We often round this for simplicity to 1:320,000 or 1:300,

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0. Using Scale to Calculate Real-World Distances The formula to remember is: Real-World Distance = Map/Plan Distance × Scale Factor Where the "scale factor" is the number on the right side of the ratio. For example, if the scale is 1:1000, the scale factor is

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0. Example 3: Calculating Distance with a Map of Durban On a map of Durban with a scale of 1:25,000, the distance between the Moses Mabhida Stadium and uShaka Marine World measures 8 cm. What is the actual distance in kilometers? Real-World Distance = 8 cm × 25,000 = 200,000 cm Convert cm to meters: 200,000 cm / 100 = 2,000 meters Convert meters to kilometers: 2,000 meters / 1000 = 2 km Therefore, the actual distance between the Moses Mabhida Stadium and uShaka Marine World is 2 km. Using Scale to Calculate Areas When dealing with areas, remember that the scale factor needs to be squared. Real-World Area = Map/Plan Area × (Scale Factor)^2 Example 4: Calculating Area with a Site Plan A rectangular plot of land is represented on a site plan with a scale of 1:

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0. On the plan, the plot measures 10 cm by 6 cm. What is the actual area of the plot in square meters? Area on the plan = 10 cm × 6 cm = 60 cm² Real-World Area = 60 cm² × (500)² = 60 cm² × 250,000 = 15,000,000 cm² Convert cm² to m²: 15,000,000 cm² / (100 cm/m)² = 15,000,000 cm² / 10,000 cm²/m² = 1,500 m² Therefore, the actual area of the plot is 1,500 m². Interpreting Maps and Plans Understanding the symbols and conventions used on maps and plans is critical.

These conventions can include: Contour lines: On topographical maps, these lines show elevation. Closer lines indicate steeper slopes.

Symbols: Maps use various symbols to represent features like schools, hospitals, rivers, and roads. A legend (or key) will explain these symbols.

North Arrow: Indicates the direction of North, which is vital for orientation.

Grid References: Help to locate specific points on a map. Scale and Plans Plans, such as building plans or floor plans, also use scales. These scales are crucial for construction workers, architects, and homeowners to accurately build structures or make renovations.

Example 5: Using a Floor Plan for Room Dimensions A floor plan of a house has a scale of 1:

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0. On the plan, the living room measures 8 cm by 6 cm. What are the actual dimensions of the living room in meters?

Length of living room: 8 cm × 50 = 400 cm = 4 meters Width of living room: 6 cm × 50 = 300 cm = 3 meters Therefore, the living room is 4 meters long and 3 meters wide. Guided Practice (With Solutions)

Question 1: A map has a scale of 1:100,

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0. Two towns are 4.5 cm apart on the map. What is the actual distance between the towns in kilometers?

Solution: Real-World Distance = 4.5 cm × 100,000 = 450,000 cm Convert cm to meters: 450,000 cm / 100 = 4,500 meters Convert meters to kilometers: 4,500 meters / 1000 = 4.5 km

Commentary: This question directly applies the formula for calculating real-world distance. The key is remembering to convert the units appropriately.

Question 2: A rectangular garden is represented on a plan with a scale of 1 cm = 2 meters.