What do surveyors, mapmakers, architects, engineers, and builders all have
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1 Exploring Scale Drawings Learning Goals In this lesson, you will: Work with applications of similarity and scale factor. Use scale drawings and maps. Key Term scale drawings What do surveyors, mapmakers, architects, engineers, and builders all have in common? All of these people use scale drawings. Scale drawings are representations of real objects or places that are in proportion to the real objects or places they represent. The scale in a scale drawing is given as a ratio. Maps and blueprints are examples of scale drawings. Why do you think scale drawings are important? 11.3 Exploring Scale Drawings 563
2 Problem 1 The term scale drawing is described as the representation of real objects or places that are in proportion to the real objects or places they represent. Students will write a sentence to describe the meaning of various drawings that have different scales. A scale is written as two numbers separated by a colon, where the first number represents the length of the drawing and the second number represents the actual length of the object. Grouping Ask a student to read the introduction to Problem 1 aloud. Discuss the worked example as a class. Problem 1 Scale Drawings The purpose of a scale drawing is to represent either a very large or very small object. The scale of a drawing might be written as: 1 cm : 4 ft Drawing Actual Length Length This scale means that every 1 centimeter of length in the drawing represents 4 feet of the length of the actual object. The scale of a map might look like this: 1 in. : 200 mi Map Actual Distance Distance This scale means that every 1 inch of distance on the map represents 200 miles of actual distance. 564 Chapter 11 Scale Drawings and Scale Factor
3 Grouping Have students complete Question 1 with a partner. Then share the responses as a class. Share Phase, Question 1 When describing the scale on a drawing, what does the first number represent? When describing the scale on a drawing, what does the second number represent? If the first number is larger than the second number, what does this mean in terms of the size of the drawing and the actual size of the object? If the second number is larger than the first number, what does this mean in terms of the size of the drawing and the actual size of the object? 1. Write a sentence to describe the meaning of each. a. A scale on a map is 1 in. : 2 ft This scale means that for every 1 inch on the map there are 2 feet of actual distance. b. A scale on a drawing is 1 cm : 4 cm This scale means that for every 1 centimeter of length on the drawing there are 4 centimeters of length on the actual object. c. A scale on a drawing is 2 in. : 1 in. This scale means that for every 2 inches on the drawing there is 1 inch of the actual object. d. A scale on a drawing is 1 cm : 1 cm. This scale means that for every 1 cm of the drawing, there is 1 centimeter of the actual object. The drawing and object are the same size Exploring Scale Drawings 565
4 Problem 2 A partial street map of Washington D.C. and a map key is provided. Students use the key to estimate distances between locations. Miles are used in the key as the unit of measurement and all answers are approximations. Materials Ruler Problem 2 A Map of Washington, D.C. A partial map of Washington, D.C., is provided. A scale is included on the map. Arlington National Cemetery Visitors Center LINCOLN MEMORIAL 1 in mi POTOMAC RIVER THOMAS JEFFERSON MEMORIAL THE WHITE HOUSE WASHINGTON MONUMENT NATIONAL MALL UNION STATION U.S. CAPITOL Grouping Have students complete Questions 1 through 5 with a partner. Then share the responses as a class. 1. Complete the table to help tourist groups plan their visits to our nation s capital. Sights Approximate Distance Using Roads and Paths White House to Lincoln Memorial miles Lincoln Memorial to Arlington Cemetery (Visitor Center) miles Arlington Cemetery (Visitor Center) to Jefferson Memorial miles Jefferson Memorial to Washington Monument 1 mile Washington Monument to U.S. Capitol 2 miles U.S. Capitol to Union Station 1 2 mile 566 Chapter 11 Scale Drawings and Scale Factor
5 Share Phase, Questions 1 through 5 What is 1 of 1 of a mile? 2 2 What is 1 of of 1 2 of a mile? What fractional part of a mile is used to approximate the distance between the White House and the Lincoln Memorial? What fractional part of a mile is used to approximate the distance between the Lincoln Memorial and the Arlington Cemetery? What fractional part of a mile is used to approximate the distance between the Arlington Cemetery and the Jefferson Memorial? What fractional part of a mile is used to approximate the distance between the Jefferson Memorial and the Washington Monument? What fractional part of a mile is used to approximate the distance between the Washington Monument and the U.S. Capitol? What fractional part of a mile is used to approximate the distance between the U.S. Capitol and the Union Station? 2. Why does it make sense to use roads and paths instead of measuring directly from one sight to the next sight? To actually travel from one sight to the next, roads and paths must be followed. The direct path may involve walking through buildings or areas that are not accessible. 3. Explain how you estimated the distances between sights. I used the edge of a sheet of paper and marked the distances as I followed a path from one sight to the next. Then, I placed the edge of the paper showing the total distance against the scale. If the distance was more than one mile, I marked one mile on the edge of the paper and went back to the beginning of the scale to estimate the additional distance. 4. Why are your answers approximate distances? My answers are approximate values because it is difficult to get an accurate distance from one sight to the next. Also, the scale is in quarters of a mile, so I could estimate only distances that fall between the quarter marks. 5. What is the total miles traveled between sights? The total distance traveled between the sights is approximately miles Exploring Scale Drawings 567
6 Problem 3 A map of the United States and a map key is provided. Students use the key to estimate distances between different cities. Kilometers and miles are used in the key as the units of measurement and all answers are approximations. Students will express each answer in both kilometers and miles. Problem 3 A Map of the United States A map of the United States is shown. A scale is included on the map. Seattle Augusta Materials Ruler San Franciscoco Los Angeles Chicago Washington, D.C. Grouping Have students complete Questions 1 through 9 with a partner. Then share the responses as a class. Share Phase, Questions 1 and 2 How many kilometers are on the scale? How many miles are on the scale? How do kilometers compare to miles? Which is larger? How much larger? Will the scale on the map be helpful in determining a reasonable estimation of the distance between locations? Explain. Austin 600 km 600 mi Determine the approximate distances between the locations. State the distances in miles and kilometers. 1. Washington, D.C., to San Francisco, California The distance from Washington, D.C., to San Francisco is approximately 2700 miles or 4500 kilometers. 2. Washington, D.C., to Seattle, Washington The distance from Washington, D.C., to Seattle is approximately 2800 miles or 4500 kilometers. 568 Chapter 11 Scale Drawings and Scale Factor
7 Share Phase, Questions 3 through 7 Will the distance between locations be helpful in determining how long it will take you to travel? Is it easier to use the kilometer scale or the miles scale? Why? Is the number of kilometers between locations given on any road signs in the United States? Why or why not? 3. Washington, D.C., to your state capital Answers will vary. 4. Chicago, Illinois, to Los Angeles, California The distance from Chicago to Los Angeles is approximately 2000 miles or 3200 kilometers. 5. Augusta, Maine, to Austin, Texas The distance from Augusta to Austin is approximately 2100 miles or 3300 kilometers. 6. Which is longer, a mile or a kilometer? How can you tell? A mile is longer than a kilometer. According to the scale, 600 miles is a longer distance than 600 kilometers. 7. How many kilometers make one mile? Explain how you determined your answer. Dividing the number of kilometers by the number of miles, I always got a calculation of approximately 1.6. Therefore, 1.6 kilometers is approximately 1 mile. For example, 2220 km km 1380 mi. 1 mi 11.3 Exploring Scale Drawings 569
8 8. How many days would it take to travel from Washington, D.C., to San Francisco, California, traveling at 60 miles per hour for 8 hours per day? Show your work. A trip from Washington, D.C., to San Francisco, California, would take about 5.6 days < 45 hours < 5.6 days. 9. Does your response to Question 8 seem realistic? Explain your reasoning. Student explanations may vary. I think the number of days is too low. The number of miles is an underestimate because there is no direct route from Washington, D.C., to San Francisco, California. Also, if I was traveling with my family that far, we would probably want to make stops and enjoy the trip. Problem 4 Students compare scales that have different units of measure to determine which scale would produce the largest and smallest drawing of an object. They will answer questions to determine whether a scale drawing is larger or smaller than the actual figure. The movie Honey, I Shrunk The Kids is used as the context where students will calculate the sizes of the models built by the special effects teams, given the actual sizes in real life. Microscopes, architectural drawings, billboards, and statues are used as the context for students to calculate either the size of the actual object or the size of the scale drawing of the object. Problem 4 Interpreting Scales Grouping 1. Which scale would produce the largest scale drawing of an object when compared to the actual object? Explain your reasoning. 1 in. : 25 in. 1 cm : 1 m 1 in. : 1 ft If each scale is written with the same units for both terms, the scales would be 1 in. : 25 in., 1 cm : 100 cm, and 1 in. : 12 in. The largest scale would be 1 : 12 or 1 in. : 1 ft. In this case, every 1 inch represents 12 inches. This would make a larger diagram than the other ratios. Have students complete Questions 1 through 4 with a partner. Then share the responses as a class. 570 Chapter 11 Scale Drawings and Scale Factor
9 Share Phase, Questions 1 through 4 How can you compare centimeters to meters? How can you convert both to the same unit of measure? How can you compare inches to feet? How can you convert both to the same unit of measure? How can you compare millimeters to meters? How can you convert both to the same unit of measure? How can you compare centimeters to millimeters? How can you convert both to the same unit of measure? What does the numerator of a scale factor represent? What does the denominator of a scale factor represent? 2. Which scale would produce the smallest scale drawing of an object when compared to the actual object? Explain your reasoning. 1 in. : 10 in. 1 cm : 10 cm 1 mm : 1 m If each scale is written with the same units for both terms, the scales would be 1 in. : 10 in., 1 cm : 10 cm, and 1 mm : 1000 mm. The smallest scale would be 1 : 1000 or 1 mm : 1 m. In this case, every 1 millimeter represents 1000 millimeters. This would make a smaller diagram than if every unit represented 10 units. 3. The scale of a drawing is 6 cm : 1 mm. Is the scale drawing larger or smaller than the actual object or place? Explain your reasoning. The scale drawing is larger than the actual object or place. Scales are written as drawing length : actual length. In this case, the first value, the drawing length, is larger than the second value, the actual length. 4. Given a scale of 5, explain how you can tell whether the drawing is bigger or smaller 4 than the actual object. The drawing is bigger than the actual object. The top value represents a length in the drawing, and the bottom value represents a length of the actual object Exploring Scale Drawings 571
10 Grouping Have students complete Question 5 with a partner. Then share the responses as a class. Have students complete Questions 6 through 13 with a partner. Then share the responses as a class. 5. In the 1989 movie Honey I Shrunk the Kids, a professor accidentally shrinks his kids to 1 of an inch with a 4 shrink ray. The kids then get accidentally sent out to the backyard. To the tiny kids, the backyard seems to have giant ants, giant bees, and grass as tall as trees! Each ant and bee were actually these sizes in real life: Length Height Width Share Phase, Question 5 Can you compare millimeters to inches? How? How does the length of a bee compare to the length of an ant? Which is longer? How does the height of a bee compare to the height of an ant? Which is taller? How does the width of a bee compare to the width of an ant? Which is wider? What does the scale 1:40 mean? length of the ant model? length of the bee model? height of the ant model? height of the bee model? width of the ant model? width of the bee model? Ant 12 mm 3 mm 1 mm Bee 0.5 in in in. The special effects team used a scale of 1 : 40 to create models of giant ants and bees. One unit of actual length corresponded to 40 units of length on each model. Complete the table to show the sizes of the models built by the team. Length Height Width Ant 480 mm or 48 cm 120 mm or 12 cm 40 mm or 4 cm Bee 20 in. 10 in. 10 in. 6. A microscope has a scale of 100 : 1. A microorganism appears to be 0.75 inch in length under the microscope. a. How long is the microorganism? Show your work The microorganism is actually inch long. b. A microorganism is millimeter long. How long will it appear under the microscope? Show your work The microorganism will appear to be 8.5 millimeters long under the microscope. 572 Chapter 11 Scale Drawings and Scale Factor
11 Share Phase, Questions 6 through 11 What does the microscope scale 100:1 mean? What does the microscope scale 1000:1 mean? Which microscope is more powerful, one that has a scale of 100:1 or one that has a scale of 1000:1? Explain. power of the microscope? How do you compare centimeters to millimeters? How do you compare inches to feet? size of the original poster? 7. A different microscope has a scale of 1000 : 1. An amoeba has a length of 25 millimeters under the microscope. What is the actual length of the amoeba? Show your work mm The actual length of the amoeba is millimeter. 8. A centimeter-long paramecium appears to be 17.5 millimeters long under a microscope. What is the power of the microscope? Show your work mm cm With like units, this can be rewritten as 17.5 mm mm The microscope magnifies all images to 50 times their actual size. 9. The height of a building in an architectural drawing is 12 inches. The actual height of the building is 360 feet. What is the scale of the drawing? Show your work. The scale of the drawing is 12 inches : 360 feet. With like units, the scale of the drawing would be 1 foot : 360 feet. The scale would be 1 : A poster was enlarged and made into a billboard. The billboard was 20.5 feet by 36 feet. The scale used was 5 : 1. What was the size of the original poster? Explain your reasoning. The original poster was 4.1 feet by 7.2 feet. The way I determined the original poster size was that I took the dimensions of the poster and divided by 5. So, feet, and feet. 11. How do you determine the scale if a statue is 60 feet high and its scale drawing shows the height as 1 foot high? The scale is written as a ratio of the size of the scale model to the size of the original object. In this case, the scale would be 1 : 60 or Exploring Scale Drawings 573
12 Share Phase, Questions 12 and 13 If the scale on a map was expressed in kilometers, how would you determine the number of miles between two locations? If the scale on a map was expressed in miles, how would you determine the number of kilometers between two locations? How did you decide the scale for the drawing of your math classroom? Did everyone use the same scale? How were the scales different? Can different scales be used to draw your math classroom? 12. Explain how to calculate the actual distance between two cities if you know the distance between them on a map and the scale of the map. First I would take the scale and set up a proportion comparing the scale to the distance measured. Then, I would solve the proportion to come up with the actual distance. For example if the scale was 1 cm : 4 mi and the distance measured was 4 cm my work would look like this: 1 cm 4 cm : 4 mi x mi x = 16 mi 13. Draw a scale drawing of your math classroom. Give the dimensions of the room and the scale. Answers will vary. 574 Chapter 11 Scale Drawings and Scale Factor
13 Problem 5 An example of a blueprint is provided. Students are given certain specifications and they will create a design for a school courtyard using the blueprint and a scale of 1 inch 5 1 foot. 8 Problem 5 Blueprints A blueprint is a technical drawing, usually of an architectural or engineering design. An example of a blueprint is shown. Materials Ruler Grouping Have students complete the blueprint with a partner. Then share the responses as a class. 40 SCALE 1/8 = 1 Share Phase, Problem 5 What is a key? What is usually included in a key? What are some of the features you used? Did you include your features in a key? Did your classmates use different features? How can you tell if your blueprint is accurate? Would it have been easier to draw the blueprint if the scale was written in a metric unit? Why or why not? 1. Design a courtyard for your school using this blueprint and the scale 1 inch 5 1 foot. 8 Include: features appropriate for a courtyard that would enhance the environment features that would be popular for students, teachers, and parents at least 10 features in the space provided (multiples of the same feature are acceptable) All features should be: drawn to scale positioned on the blueprint keeping scale in mind drawn directly on the blueprint or cut out of paper and taped to the blueprint labeled, either directly on the item or by using a key Be prepared to share your solutions and methods Exploring Scale Drawings 575
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