Lesson Notes By Weeks and Term v5 - Grade 11
Measurement: scale, maps and plans – Week 6 focus
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Measurement: scale, maps and plans – Week 6 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Subject: Mathematical Literacy
Class: Grade 11
Term: 3rd Term
Week: 6
Theme: General lesson support
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This week, we delve into the practical application of measurement through scales, maps, and plans. Understanding these concepts is crucial for navigating our world, whether planning a trip across South Africa, interpreting building plans for a house, or understanding distances and areas represented on a map. Being able to interpret and use scale allows us to translate measurements from a representation (like a map) to real-world dimensions and vice versa. This skill is essential not only in academics but also in many professions and everyday life situations.
2.1 Understanding Scale: A scale is the ratio that compares a measurement on a map, plan, or model to the corresponding measurement on the actual object or in the real world. It's a way to represent large distances or areas in a smaller, more manageable format.
There are three main types of scales: Ratio Scale: Expressed as a ratio, such as 1:50,
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0. This means that 1 unit on the map represents 50,000 of the same units in reality. For instance, 1 cm on the map represents 50,000 cm (or 500 meters or 0.5 km) in reality.
Word Scale: States the relationship between map distance and ground distance in words, such as "1 cm represents 1 kilometer." Bar Scale (Graphical Scale): A visual representation of the scale using a line divided into segments representing specific distances on the ground. This type of scale remains accurate even if the map is enlarged or reduced. 2.2 Converting Between Scales: It’s crucial to be able to convert between these different types of scales.
Ratio to Word Scale: If the ratio scale is 1:100,000, it means 1 cm on the map represents 100,000 cm in reality. Convert 100,000 cm to kilometers: 100,000 cm = 1000 m = 1 km. So, the word scale is "1 cm represents 1 km." Word to Ratio Scale: If the word scale is "1 cm represents 5 km," convert 5 km to centimeters: 5 km = 5000 m = 500,000 cm. So, the ratio scale is 1:500,000. 2.3 Calculating Actual Distances: To calculate the actual distance between two points, you need to measure the distance on the map and then use the scale to convert it to real-world distance.
Formula: Actual Distance = Map Distance x Scale Factor The scale factor is the number that map distance is multiplied by to get the real-world distance.
For a scale of 1:50,000, the scale factor is 50,
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0. Example 1: Using a Ratio Scale A map has a scale of 1:250,
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0. The distance between Johannesburg and Pretoria on the map is 2 cm. What is the actual distance between the two cities?
Map Distance: 2 cm Scale: 1:250,000 Scale Factor: 250,000 Actual Distance = 2 cm x 250,000 = 500,000 cm Convert cm to km: 500,000 cm = 5000 m = 5 km.
Therefore, the actual distance between Johannesburg and Pretoria is 5 km (based on this simplified example, actual distance is closer to 56km in reality).
Example 2: Using a Word Scale A floor plan has a scale of "1 cm represents 0.5 meters." A room on the plan is 6 cm long. What is the actual length of the room?
Map Distance: 6 cm Word Scale: 1 cm represents 0.5 meters Scale Factor: 0.5 meters per cm Actual Length = 6 cm x 0.5 meters/cm = 3 meters 2.4 Calculating Areas: When dealing with areas, you need to square the scale factor.
If the scale is 1:X for lengths, the area scale is 1:X².
Formula: Actual Area = Map Area x (Scale Factor)² Example 3: Calculating Area A farmer has a field that measures 4 cm x 5 cm on a map with a scale of 1:10,
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0. What is the actual area of the field in square meters?
Map Area: 4 cm x 5 cm = 20 cm² Scale: 1:10,000 Scale Factor: 10,000 Actual Area = 20 cm² x (10,000)² = 20 x 100,000,000 cm² = 2,000,000,000 cm² Convert cm² to m²: 2,000,000,000 cm² = 2,000,000,000 / (100 x 100) m² = 200,000 m² Therefore, the actual area of the field is 200,000 m². 2.5 Choosing an Appropriate Scale: The choice of scale depends on the size of the area you want to represent and the level of detail you need to show. Large Scale Maps (e.g., 1:10,000): Show small areas with a lot of detail. Suitable for town plans or site plans. Small Scale Maps (e.g., 1:1,000,000): Show large areas with less detail. Suitable for maps of countries or continents. When creating a map or plan, you need to consider the size of the paper/display area and the real-world dimensions. The scale should allow you to fit the entire area onto the paper/display while still providing sufficient detail.
Example 4: Determining a Suitable Scale You want to create a map of your school grounds. The school grounds are 200 meters long and 150 meters wide. You want to fit the map onto an A4 page (21 cm x 29.7 cm). What would be a suitable scale? First, convert meters to centimeters: 200 m = 20,000 cm and 150 m = 15,000 cm. You need to fit 20,000 cm onto 29.7 cm (length of A4) and 15,000 cm onto 21 cm (width of A4).
For the length: 20,000 cm / 29.7 cm ≈ 673.
4. Round up to
7
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0. For the width: 15,000 cm / 21 cm ≈ 714.
3. Round up to
8
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0. To ensure the entire school grounds fit on the A4 page, choose the larger number, which is
8
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0. A suitable scale would be approximately 1:
8
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0. You might need to adjust this slightly to simplify the scale to something like 1:1000 or 1:500, and then decide how much of the paper it fills. Guided Practice (With Solutions)
Question 1: A map of the Kruger National Park has a scale of 1:500,
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0. The distance between Skukuza Rest Camp and Satara Rest Camp is 8 cm on the map. Calculate the actual distance between these two camps in kilometers.