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Mapping Your Way - Scaling and Distance Estimation

Grade 5CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Maps and Perspective: A map is a bird’s-eye view of a place, showing it as if you are looking straight down from the sky. Visually, maps appear flat and use simplified symbols rather than realistic pictures to represent objects like buildings, roads, and parks.

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The Concept of Scale: Since large areas like cities cannot be drawn in their actual size on paper, they are reduced using a 'Scale'. A scale tells us how much 1 cm1\text{ cm} on the map represents in real life. Visually, this is often shown as a small horizontal bar at the bottom of the map labeled 1 cm=10 km1\text{ cm} = 10\text{ km}.

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Directions on a Map: To navigate a map, we use the four cardinal directions. North is always at the top, South at the bottom, East to the right, and West to the left. Visually, maps often include a 'Compass Rose' which is a cross-shaped icon showing these directions.

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Grid Lines and Location: Maps are often divided into a grid of squares, similar to graph paper. Each square can be identified by its position (like a coordinate). These grids help in measuring the area occupied by a place and in finding exact locations by looking at where horizontal and vertical lines intersect.

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Scaling Up and Scaling Down: This is the process of making a drawing larger or smaller using grids. If you move a drawing from a 1 cm1\text{ cm} grid to a 2 cm2\text{ cm} grid, you are 'Scaling Up' or enlarging it. The shape stays the same, but it looks bigger because each part of the drawing now occupies larger squares.

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Map Symbols and Legend: Maps use symbols to save space and keep the drawing clean. For example, a blue wavy line represents a river, and a small black circle represents a city. The 'Legend' or 'Key' is a small box on the side of the map that explains what each of these visual symbols means.

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Distance Estimation: We can estimate the real-world distance between two cities by measuring the distance between them on the map with a ruler in centimeters and then multiplying that number by the map's scale factor.

📐Formulae

Actual Distance=Map Distance×Scale Value\text{Actual Distance} = \text{Map Distance} \times \text{Scale Value}

Map Distance=Actual DistanceScale Value\text{Map Distance} = \frac{\text{Actual Distance}}{\text{Scale Value}}

Scale Factor=New Grid Side LengthOriginal Grid Side Length\text{Scale Factor} = \frac{\text{New Grid Side Length}}{\text{Original Grid Side Length}}

New Drawing Side=Original Side×Scale Factor\text{New Drawing Side} = \text{Original Side} \times \text{Scale Factor}

💡Examples

Problem 1:

On a map of a city, the scale is given as 1 cm=5 km1\text{ cm} = 5\text{ km}. If the distance between a school and a library on the map is 3.5 cm3.5\text{ cm}, find the actual distance between them.

Solution:

  1. Identify the scale: 1 cm=5 km1\text{ cm} = 5\text{ km}.
  2. Identify the map distance: 3.5 cm3.5\text{ cm}.
  3. Use the formula: Actual Distance=Map Distance×Scale Value\text{Actual Distance} = \text{Map Distance} \times \text{Scale Value}.
  4. Calculate: 3.5×5=17.5 km3.5 \times 5 = 17.5\text{ km}.
  5. The actual distance is 17.5 km17.5\text{ km}.

Explanation:

To find the real-world distance, we multiply the number of centimeters measured on the map by the kilometers represented by each single centimeter.

Problem 2:

A picture of a leaf is drawn on a 1 cm1\text{ cm} square grid. If the leaf is 3 cm3\text{ cm} long in this drawing, how long will it be if it is redrawn on a 2 cm2\text{ cm} square grid?

Solution:

  1. Original length on 1 cm1\text{ cm} grid = 3 cm3\text{ cm} (which is 33 grid squares).
  2. The new grid size is 2 cm2\text{ cm}.
  3. Calculate the Scale Factor: Scale Factor=2 cm1 cm=2\text{Scale Factor} = \frac{2\text{ cm}}{1\text{ cm}} = 2.
  4. New length = Original length×Scale Factor=3×2=6 cm\text{Original length} \times \text{Scale Factor} = 3 \times 2 = 6\text{ cm}.
  5. The new length of the leaf will be 6 cm6\text{ cm}.

Explanation:

When the grid square side is doubled from 1 cm1\text{ cm} to 2 cm2\text{ cm}, every dimension of the drawing also doubles. The shape remains proportional but becomes larger.