Scale and Maps: – Interpreting and using scale drawings or maps.

Scale & Maps. Grade 8 Mathematics: Scale and Maps - Interpreting and Using Scale Drawings Subtopics Navigator Introduction to Scale Types of Scale Scale Calculations Reading Maps Scale Drawings Real Applications Cumulative Exercises Conclusion Lesson Objectives Understand the concept of scale and its importance in maps and drawings Identify and use different types of scale representations Perform calculations to convert between actual distances and scaled distances Interpret and use scale drawings and maps accurately Apply scale concepts to solve real-world problems Create simple scale drawings from actual measurements Introduction to Scale Scale is the relationship between the measurements on a map, drawing, or model and the actual measurements they represent. It allows us to represent large objects or areas in a smaller, more manageable size while maintaining accurate proportions. Scale Definition: Scale = Drawing Distance : Actual Distance OR Scale = Drawing Distance ÷ Actual Distance Example: If a map has a scale of 1:50,000, this means: 1 cm on the map represents 50,000 cm in real life 1 cm on the map represents 500 m in real life 1 cm on the map represents 0.5 km in real life Types of Scale Scales can be represented in different ways, each with its own advantages: Type of Scale Description Example Use Case Representative Fraction (RF) Ratio without units (e.g., 1:25,000) 1:50,000 Most maps, architectural drawings Linear Scale Graphical bar showing distances [====|====|====] 0 km 5 km 10 km Road maps, atlases Statement Scale Written description of the scale "1 cm represents 1 km" Simple maps, educational materials Comparative Scale Shows different unit conversions 1 cm = 100 m = 0.1 km Technical drawings, engineering Example 1: Converting Between Scale Types Convert the statement scale "2 cm represents 1 km" to: (a) Representative Fraction (b) Comparative scale showing meters Solution: (a) 2 cm : 1 km = 2 cm : 100,000 cm = 1:50,000 (b) 2 cm = 1 km = 1000 m, so 1 cm = 500 m Exercises (Types of Scale) Convert "1 cm represents 250 m" to representative fraction Convert scale 1:10,000 to a statement scale in meters A map scale is 1:25,000. What actual distance does 4 cm on the map represent? Create a linear scale for 1:50,000 showing 0 km, 2 km, and 4 km Which type of scale would be most useful for a hiking map and why? Scale Calculations To work with scales, we need to be able to convert between actual distances and scaled distances: Key Formulas: Actual Distance = Map Distance × Scale Denominator Map Distance = Actual Distance ÷ Scale Denominator Scale = Map Distance : Actual Distance Example 2: Finding Actual Distance On a map with scale 1:100,000, two towns are 7.5 cm apart. What is the actual distance between them? Solution: Actual Distance = Map Distance × Scale Denominator = 7.5 cm × 100,000 = 750,000 cm = 7,500 m = 7.5 km Answer: The towns are 7.5 km apart. Example 3: Finding Map Distance The actual distance between two cities is 120 km. What would this distance be on a map with scale 1:1,500,000? Solution: First convert 120 km to cm: 120 km = 12,000,000 cm Map Distance = Actual Distance ÷ Scale Denominator = 12,000,000 cm ÷ 1,500,000 = 8 cm Answer: The distance on the map would be 8 cm. Exercises (Scale Calculations) On a 1:50,000 map, a road measures 12 cm. What is its actual length? The actual length of a bridge is 850 m. What length would it be on a 1:25,000 map? Two villages are 15 km apart. How far apart would they be on a 1:100,000 map? A map distance of 6.5 cm represents 32.5 km. What is the scale? If 4 cm on a map represents 2 km, what is the representative fraction? Reading and Using Maps SAMPLE MAP [School]----2.5 cm----[Library]----3 cm----[Park] Scale: 1:20,000 North ↑ Example 4: Using a Map with Multiple Locations Using the sample map above: (a) Find the actual distance from School to Library (b) Find the actual distance from Library to Park (c) Find the total distance from School to Park via Library Solution: (a) School to Library: 2.5 cm × 20,000 = 50,000 cm = 500 m (b) Library to Park: 3 cm × 20,000 = 60,000 cm = 600 m (c) Total distance: 500 m + 600 m = 1,100 m = 1.1 km Answer: (a) 500 m, (b) 600 m, (c) 1.1 km Example 5: Finding Scale from Map Measurements On a map, the distance between two landmarks is 8 cm. The actual distance is 4 km. What is the scale? Solution: Convert both to same units: 4 km = 400,000 cm Scale = Map Distance : Actual Distance = 8 cm : 400,000 cm = 1:50,000 Answer: The scale is 1:50,000 Exercises (Reading Maps) On a 1:25,000 map, a walking trail measures 15 cm. How long is the actual trail? A river is 18 km long. What length would it be on a 1:100,000 map? Two cities are 8.5 cm apart on a map with scale 1:500,000. What is the actual distance? If 3 cm on a map represents 7.5 km, what is the scale? On a map, Town A to Town B is 6 cm and Town B to Town C is 9 cm. The scale is 1:200,000. What is the total distance from A to C? Scale Drawings Scale drawings are used in architecture, engineering, and design to represent objects at a reduced size while maintaining accurate proportions. Example 6: Creating a Scale Drawing A rectangular room measures 6 m by 4 m. Create a scale drawing using scale 1:100. Solution: Scale 1:100 means 1 cm represents 100 cm (1 m) Length on drawing: 6 m = 6 cm Width on drawing: 4 m = 4 cm Draw a rectangle 6 cm by 4 cm to represent the room. Example 7: Interpreting a Scale Drawing A scale drawing of a car uses scale 1:50. If the drawing shows the car as 8 cm long, how long is the actual car? Solution: Actual length = Drawing length × Scale denominator = 8 cm × 50 = 400 cm = 4 m Answer: The actual car is 4 meters long. Exercises (Scale Drawings) A house plan uses scale 1:100. If a room measures 3.5 cm by 2.8 cm on the plan, what are its actual dimensions? A model car is built at scale 1:24. If the actual car is 4.8 m long, how long is the model? A garden measures 12 m by 8 m. What would its dimensions be on a scale drawing at 1:200? On a scale drawing at 1:50, a table is drawn as 3.6 cm long. How long is the actual table? A building is 45 m tall. What height would it be on a scale drawing at 1:500? Real-World Applications Example 8: Planning a Journey Sarah is planning a bike ride using a map with scale 1:75,000. On the map, her route measures 28 cm. If she cycles at an average speed of 15 km/h, how long will her journey take? Solution: Actual distance = 28 cm × 75,000 = 2,100,000 cm = 21 km Time = Distance ÷ Speed = 21 km ÷ 15 km/h = 1.4 hours = 1 hour 24 minutes Answer: The journey will take 1 hour and 24 minutes. Example 9: Architecture and Construction An architect creates a floor plan at scale 1:100. On the plan, a wall is drawn as 8.5 cm long. If building materials cost $15 per meter, how much will it cost to build this wall? Solution: Actual wall length = 8.5 cm × 100 = 850 cm = 8.5 m Cost = 8.5 m × $15/m = $127.50 Answer: It will cost $127.50 to build the wall. Cumulative Exercises A map has scale 1:50,000. Two towns are 12.5 cm apart on the map. What is the actual distance between them in kilometers? The actual distance between two cities is 84 km. On a map, they are 14 cm apart. What is the scale of the map? A scale model of a building is made at 1:200. If the actual building is 48 m tall, how tall is the model in centimeters? On a map with scale 1:25,000, a road measures 18 cm. What is its actual length in meters? A rectangular field measures 450 m by 300 m. What would its dimensions be on a scale drawing at 1:2,500? Two villages are 6.5 km apart. How far apart would they be on a map with scale 1:65,000? A map distance of 8 cm represents an actual distance of 4.8 km. What is the scale as a representative fraction? A model airplane is built at scale 1:72. If the model is 25 cm long, how long is the actual airplane in meters? On a hiking map with scale 1:40,000, a trail measures 22.5 cm. How long is the actual trail in kilometers? A house plan uses scale 1:150. If a room measures 4.2 cm by 3.6 cm on the plan, what is the actual area of the room in square meters? Show/Hide Solutions Problem 1: A map has scale 1:50,000. Two towns are 12.5 cm apart on the map. What is the actual distance between them in kilometers? Solution: Actual Distance = Map Distance × Scale Denominator = 12.5 cm × 50,000 = 625,000 cm = 6,250 m = 6.25 km Answer: 6.25 km Problem 2: The actual distance between two cities is 84 km. On a map, they are 14 cm apart. What is the scale of the map? Solution: 84 km = 8,400,000 cm Scale = Map Distance : Actual Distance = 14 cm : 8,400,000 cm = 1:600,000 Answer: 1:600,000 Problem 3: A scale model of a building is made at 1:200. If the actual building is 48 m tall, how tall is the model in centimeters? Solution: 48 m = 4,800 cm Model height = Actual height ÷ Scale denominator = 4,800 cm ÷ 200 = 24 cm Answer: 24 cm Problem 4: On a map with scale 1:25,000, a road measures 18 cm. What is its actual length in meters? Solution: Actual length = 18 cm × 25,000 = 450,000 cm = 4,500 m Answer: 4,500 m Problem 5: A rectangular field measures 450 m by 300 m. What would its dimensions be on a scale drawing at 1:2,500? Solution: Scale 1:2,500 means 1 cm represents 2,500 cm (25 m) Length: 450 m ÷ 25 m/cm = 18 cm Width: 300 m ÷ 25 m/cm = 12 cm Answer: 18 cm by 12 cm Problem 6: Two villages are 6.5 km apart. How far apart would they be on a map with scale 1:65,000? Solution: 6.5 km = 650,000 cm Map distance = 650,000 cm ÷ 65,000 = 10 cm Answer: 10 cm Problem 7: A map distance of 8 cm represents an actual distance of 4.8 km. What is the scale as a representative fraction? Solution: 4.8 km = 480,000 cm Scale = 8 cm : 480,000 cm = 1:60,000 Answer: 1:60,000 Problem 8: A model airplane is built at scale 1:72. If the model is 25 cm long, how long is the actual airplane in meters? Solution: Actual length = 25 cm × 72 = 1,800 cm = 18 m Answer: 18 m Problem 9: On a hiking map with scale 1:40,000, a trail measures 22.5 cm. How long is the actual trail in kilometers? Solution: Actual length = 22.5 cm × 40,000 = 900,000 cm = 9,000 m = 9 km Answer: 9 km Problem 10: A house plan uses scale 1:150. If a room measures 4.2 cm by 3.6 cm on the plan, what is the actual area of the room in square meters? Solution: Actual length = 4.2 cm × 150 = 630 cm = 6.3 m Actual width = 3.6 cm × 150 = 540 cm = 5.4 m Area = 6.3 m × 5.4 m = 34.02 m² Answer: 34.02 m² Conclusion/Recap In this lesson, we've explored the fundamental concepts of scale and their applications in maps and drawings. We've learned how to interpret different types of scales, perform calculations to convert between actual and scaled distances, and apply these skills to real-world situations. Understanding scale is essential for reading maps, creating scale drawings, and solving practical problems in geography, architecture, and everyday life. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c