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1 SCALE CITY The Road to Proportional Reasoning: Miniature Land Handouts TABLE OF CONTENTS Click on a title to go directly to the handout. Handout 1: Are Toys Proportional? Exploring how toys and their accessories follow rules of proportionality Handout 2: Scales from Greetings from Miniature Land A viewing guide to the video Handout 3: Practice Scaling Using architectural scale to determine the dimensions of the Empire State Building Handout 4: Scaling Miniatures Additional practice scaling during class time or for homework Handout 5: Scale Challenge Additional practice scaling during class time or for homework Handout 6: Surface Area and Volume Practice Problems Using information from the online interactive to determine surface area Handout 7: Surface Area and Volume Homework Problems Additional practice determining surface and volume Handout 8: Scale of Planets Comparing the sizes of two objects, the sun and moon, to calculate the dimensions of a planetary model Handout 9: Game: Race to the Top of the Washington Monument Game where students answer three sets of problems (multiple choice, short answer, and true/false) to scale a model drawing of the Washington Monument

2 MINIATURE LAND: HANDOUT 1 Are Toys Proportional? Name: Date: 1. Select a toy and some accessories that go with the toy. Choose a toy that directly represents a person or a real object. 2. Measure several dimensions of the toy and accessories. Record the measurements. 3. Compare the items to the objects actual sizes. You may need to research to find the standard sizes for items like refrigerators, cars, etc. 4. Present your findings to the class. Use the following presentation template: The toy examined is a. The appearance of the object does or does not look proportional in these details. (Describe at least two observations.) A. B. In comparing the object s measurements with the actual object, I (we) discovered that We think that proportional reasoning was used or not used because

3 Are Toys Proportional? Self-Assessment Checklist: I completed all four steps of the project. I carefully measured and recorded several dimensions of the object. I researched to find measurements of an actual object. I used mathematics to compare the size of the object to an actual object. I presented my findings using the presentation template. My overall work shows good effort and attention to detail. Group Assessment Checklist: We quietly and efficiently gathered materials for the project. We worked well as a team. We focused on the task and used time efficiently. We completed all four steps in the process. We used math to compare this object to an actual object. We managed materials well as a group and cleaned up quickly. Our overall work shows good effort and attention to detail.

4 MINIATURE LAND: HANDOUT 2 Scales from Greetings from Miniature Land Handout 2: Scales from Greetings from Miniature Land Name: Date: There are three scales mentioned in the video Greetings from Miniature Land. Please record those scales. 1. Kentucky Gateway Museum in Maysville, Kentucky 2. Behringer Crawford Museum in Covington, Kentucky 3. Huntertown Elementary School architectural model Handout 2: Scales from Greetings from Miniature Land Name: Date: There are three scales mentioned in the video Greetings from Miniature Land. Please record those scales. 1. Kentucky Gateway Museum in Maysville, Kentucky 2. Behringer Crawford Museum in Covington, Kentucky 3. Huntertown Elementary School architectural model Handout 2: Scales from Greetings from Miniature Land Name: Date: There are three scales mentioned in the video Greetings from Miniature Land. Please record those scales. 1. Kentucky Gateway Museum in Maysville, Kentucky 2. Behringer Crawford Museum in Covington, Kentucky 3. Huntertown Elementary School architectural model

5 MINIATURE LAND: HANDOUT 3 Practice Scaling Name: Date: 1. The Empire State Building continues to be an icon of New York City. Built in 1931, it was the tallest building in the world for many years. Even today, it is still among the tallest buildings in the world. Using an architectural scale of 1:192 and the information in the table below, determine what the measurements of a model of the Empire State Building would be. Calculate in feet and inches (correct to the nearest 1/16 of an inch). Show your work below and then record your answer on the table.. Dimension Metric English 1:192 scale Height of roof 381 meters 1250 feet Height to top of tower 449 meters 1472 feet Maximum length 129 meters 424 feet Maximum width 57 meters 187 feet A. Determine the model s height from ground level to the roof. B. Determine the model s height from ground level to the top of the tower. C. Determine the maximum length of the model. D. Determine the maximum width of the model. 2. Another way to write the architectural scale is 1/16 of an inch for every foot. Why does it seem that the architectural scale is suited for the English system instead of the metric system?

6 KEY: MINIATURE LAND: HANDOUT 3 Practice Scaling Note to Teacher: There are a variety of ways in which students might make these calculations. For example, they might convert the actual feet to inches and then divide the product by 192. Then they could divide their quotient by 12 to get an answer. Or they might set up equivalent fractions or proportions as outlined below. Regardless of the approach, to obtain measurements that are exact as possible for the model, students must convert decimal values to inches and fractions of inches (sixteenths). At this point, you may want to encourage them to use a calculator to facilitate converting decimals with several places to inches and fractions of inches. In the first calculation, you might ask students to estimate what feet would be in inches. They will probably say roughly half a foot or six inches. Ask them if they think this approximate value is accurate enough to create an architectural model or if the extra 1/8 inch makes a significant difference. How much does 1/8 inch represent in the 1/192 scale? (Two feet) 1. The Empire State Building continues to be an icon of New York City. Built in 1931, it was the tallest building in the world for many years. Even today, it is still among the tallest buildings in the world. Using an architectural scale of 1:192 and the information in the table below, determine what the measurements of a model of the Empire State Building would be. Calculate in feet and inches (correct to the nearest 1/16 of an inch). Show your work below and then record your answer on the table.. Dimension Metric English 1:192 scale Height of roof 381 meters 1250 feet 6 feet 6 1/8 inches Height to top of tower 449 meters 1472 feet 7 feet 8 inches Maximum length 129 meters 424 feet 2 feet 2 1/2 inches Maximum width 57 meters 187 feet 11 and 11/16 inches A. Determine the model s height from ground level to the roof. 1/192 = x feet/ 1250 feet 1/192 = x/1250 1/ = x 1250/192 = x x = feet Convert to inches (the most accurate result will be obtained by leaving the entire number in the calculator) = inches 125/1000 = simplifies to 1/8 Height of model at roof in feet and inches = 6 feet 6 1/8 inches

7 KEY: Practice Scaling B. Determine the model s height from ground level to the top of the tower. 1/192 = x feet/ 1250 feet 1/192 = x feet/1472 feet 1/ = x 1472/192 = x x = feet Convert feet to inches. 2/3 12 = 8 inches Height of model at top of tower in feet and inches = 7 feet 8 inches C. Determine the maximum length of the model. 1/192 = x feet/424 feet 1/192 = x/424 1/ = x 424/192 = x x = feet Convert to inches = 2.5 or 2 1/2 inches Maximum length of model = 2 feet 2 1/2 inches D. Determine the maximum width of the model. 1/192 = x feet/424 feet 1/192 = x/187 1/ = x 187/192 = x x = Convert to inches = inches 6875/10000 = 11/16 inches Maximum width of model = 11 and 11/16 inches 2. Another way to write the architectural scale is 1/16 of an inch for every foot. Why does it seem that the architectural scale is suited for the English system instead of the metric system? Ask your students to discuss different theories. Students may realize that 192 is the product of 12 times 16. There are 12 inches in a foot, and inches are measured in sixteenths. With metric measurements it would be easier to use a scale divisible by 10 or 100 so that conversion to different metric units is easily accomplished.

8 MINIATURE LAND: HANDOUT 4 Scaling Miniatures Name: Date: 1. In her job for the Toyland Wonder Company, Sela is proposing a line of action figures modeled after law enforcement officers and their equipment. She chooses to use a scale of 1:12. Estimate to determine if the numbers are reasonable. Determine how large the clipboard and flashlight should be. Answer with fractions in simplest terms. Correct any proposed measurements that are not reasonable. Actual Measurements Proposed Toy Measurements Height of Police Officer (Male) 6 feet or 72 inches 6 inches Height of Police Officer (Female) 5 feet 6 inches or 66 inches 6 1/2 inches Length of Squad Car 15 feet long or 180 inches 15 inches Clipboard 9 inches by 12 inches Length of Cell Phone 3 1/5 inches 1/4 inch Length of Flashlight 20 inches 2. The Builder s Club is creating a display of favorite structures. Using the architectural scale of 1:192 or 1/16 of an inch for every foot, how tall would the models of the following structures be? Answer in feet and inches, and round to the nearest 1/16 of an inch. Do the calculations on the bottom and back of the paper. Show your work. Structure Height in Feet Model Height 1:192 Big Ben 316 feet Eiffel Tower 1052 feet Statue of Liberty 305 feet 1 inch Gateway Arch Seattle Space Needle Washington Monument 630 feet 555 feet 5 1/8 inches 605 feet

9 Scaling Miniatures 3. How can you show the following scales are equal? Explain and show all your work. A. 1 inch for every 16 feet is equal to 1/16 of an inch for every foot B. 1/16 of an inch for every foot is equal to 1:192

10 KEY: MINIATURE LAND: HANDOUT 4 Scaling Miniatures 1. In her job for the Toyland Wonder Company, Sela is proposing a line of action figures modeled after law enforcement officers and their equipment. She chooses to use a scale of 1:12. Estimate to determine if the numbers are reasonable. Determine how large the clipboard and flashlight should be. Answer with fractions in simplest terms. Correct any proposed measurements that are not reasonable. Actual Measurements Proposed Toy Measurements Height of Police Officer (Male) 6 feet or 72 inches 6 inches Height of Police Officer (Female) 5 feet 6 inches or 66 inches 5 1/2 inches Length of Squad Car 15 feet long or 180 inches 15 inches Clipboard 9 inches by 12 inches 3/4 inch x 1 inch Length of Cell Phone 3 1/5 inches 1/4 inch Length of Flashlight 20 inches 1 2/3 inches 2. The Builder s Club is creating a display of favorite structures. Using the architectural scale of 1:192 or 1/16 of an inch for every foot, how tall would the models of the following structures be? Answer in feet and inches, and round to the nearest 1/16 of an inch. Do the calculations on the bottom and back. Show your work. Structure Height in Feet Model Height 1:192 Big Ben 316 feet 1 foot 7 3/4 inches Eiffel Tower 1052 feet 5 feet 5 3/4 inches Statue of Liberty 305 feet 1 inch 1 foot 7 1/16 inches Gateway Arch 630 feet 3 feet 3 3/8 inches Seattle Space Needle 555 feet 5 1/8 inches 2 feet 10 11/16 inches Washington Monument 605 feet 3 feet 1 13/16 inches Note to Teacher: Here are two possible approaches that students might use to complete the table. 1) Use the 1:192 scale to set up equivalent ratios and solve for the unknown value (1/192 = model height/actual height). 2) Use the 1/16 inch = 1 foot scale. Divide the height in feet by 16 to get the total number of inches in the model and then convert to feet and inches. (For example, 316 divided by 16 equals 19 3/4 or 1 foot 7 3/4 inches). Encourage students to share their reasoning and methods for solving the problems. Students will still need to convert decimal remainders from feet to inches and inch remainders from decimals to 1/16 inches.

11 KEY: Scaling Miniatures Big Ben 1/192 = x feet/316 feet x = 316/192 x = Convert the decimal remainder to inches = 7 3/4 inches Eiffel Tower 1/192 = x feet/1052 feet 192x = 1052 x = 1052/192 x = inches Convert decimal remainder to inches x 12 = 5 3/4 inches Statue of Liberty 305 feet 1 inch Set up equivalent fractions and convert feet to inches. 1/192 = x feet/(305 feet x 12)+1 192x = 3661 inches x = 3661/192 x = or 1 foot 7 1/16 inches Gateway Arch 1/192 = x/630 feet 192x = 630 x = 630/192 x = inches Convert decimal remainder to inches x 12 = 3 3/8 inches Seattle Space Needle 555 feet 5 1/8 inches Set up equivalent fractions and convert feet to inches. 1/192 = x feet/[(555 feet x 12) +5 inches +(1/8) inch] 1/192 = x/6665 1/8 inches /192 = x x = 34 11/16 inches Washington Monument 605 feet 1/192 = x feet/605 feet 192x = 605 x = 605/192 x = 3 13/16 inches 3. How can you show the following scales are equal? Explain and show all your work. A. 1 inch for every 16 feet is equal to 1/16 of an inch for every foot Here are a few possible ways students might demonstrate that the scales are equal. Convert both ratios to inches 1/(16 12) = (1/16)/12 inches 1/192 = 1/ /192 = 1/16 1/12 1/192 = 1/192

12 KEY: Scaling Miniatures Narrow the measurement by counting backwards and dividing by 2 each time. 1 inch for every 16 feet 1/2 inch for every 8 feet 1/4 inch for every 4 feet 1/8 inch for every 2 feet 1/16 inch for every foot B. 1/16 of an inch for every foot is equal to 1:192 Here are a few ways to examine these values. You or your students may have other good ways of checking these values by using models or symbols. Follow the definition for a proportion and test it. If a/b = c/d then ad = b., 1: 192 is the same as 1/192, a = 1/16, b = 12(12 inches in a foot), c = 1, d = 192 (1/16)/12 = 1/192 1/ = = 12 These are equivalent ratios, so the scales are the same. Simplify the most complicated fraction. Write the problem out in symbols. 1/16 of an inch for every foot (1/16)/12 = (1/16) 12 = 1/16 12/1 = 1/16 1/12 = 1/192 1/192 is another method of writing 1:192 Set up the two values as equivalent ratios leaving one unknown value to determine if they are equal. 1:192 can be written as 1/192 1/192 = (1/16)/x solve for x x = 192 1/16 x = 12 so the two ratios are equal or proportional.

13 MINIATURE LAND: HANDOUT 5 Scale Challenge Name: Date: 1. For a family reunion, one of the activities is a board game for a homemade Family Heritage Trivia Game. To construct a board game three times the dimensions of most board games, compute the following dimensions and enter them in the chart. A. Determine the larger sizes. Height of Player Piece Dimensions of Board Dice Smaller Size 1.5 inches 19 1/2 inches by 19 1/2 inches 1 inch x 1 inch x 1 inch Larger Size B. What would the dimensions of the large board be in feet and inches? C. If the paint used to cover the board requires four ounces per square foot, how much paint is needed to cover one side of the larger board? Round your answer to the nearest tenth place. (Hint: First determine the area.) D. Use the computation from c to determine how many bottles of paint you ll need to buy to paint the large board if the paint comes in 16-ounce bottles. 2. Randy thinks the wheelchair ramp at the old community center is too steep. He and his parents looked into the guidelines of the Americans with Disabilities Act. The chart below indicates that for existing structures a slope steeper than 1:8 is not acceptable. This slope ratio is calculated as the rise (height) of the ramp to the run (length) of the ramp. For every three inches of rise at least a two-foot run is required. Since two feet equals 24 inches, the ratio is 3:24 or 1:8. Allowable Ramp Dimensions for Construction in Existing Sites, Buildings, and Facilities Slope* Maximum Rise Maximum Run Steeper than 1:10 but no steeper than 1:8 3 in or 75 mm 2 ft or 0.6 m Steeper than 1:12 but no steeper than 1:10 6 in or 150 mm 5 ft or 1.5 m * A slope steeper than 1:8 not allowed. The rise of the wheelchair ramp at the community center is seven inches. The run of the ramp is 58 inches. Is this slope steeper than 1:8?

14 Scale Challenge 3. The Americans with Disabilities Act requires ramps in new buildings to have a slope of 1:12 or less. This means that for every inch of height (rise) of the ramp, there needs to be 12 inches of length (run). The recommended ramp slope is 1:16. A. For a rise of five inches, how long should the ramp be if it meets the 1:12 requirements? B. For a rise of five inches, how long should the ramp be if it meets the 1:16 recommendations? 4. Here are the heights of some of the world s tallest buildings in meters. Building Country Top of Spire Top of Building Taipei 101 Taiwan 508 meters 448 meters Shanghair World Financial Center China 492 meters Petronas Towers Malaysia 452 meters Sears Tower U.S.A meters Jin Mao Tower China meters A. To create a scale drawing comparing the heights of these buildings, the class chose to use a scale of 1 cm for every 100 meters. Fill in the chart with the values they would use for their drawing. Building Country Top of Spire Top of Building Taipei 101 Taiwan Shanghai World Financial Center China Petronas Towers Malaysia Sears Tower U.S.A. Jin Mao Tower China B. How would you express a scale or ratio of 1 cm for every one hundred meters as a ratio in which you only use centimeters? (Hint: You need to convert meters to centimeters.) 5. Using the scale of 1:192, determine what the dimensions would be for a model of your school or a model of your classroom.

15 KEY: MINIATURE LAND: HANDOUT 5 Scale Challenge 1. For a family reunion, one of the activities is a board game for a homemade Family Heritage Trivia Game. To construct a board game three times the dimensions of most board games, compute the following dimensions and enter them in the chart. A. Determine the larger sizes. Smaller Size Larger Size Height of Player Piece 1.5 inches 4.5 inches Dimensions of Board 19 1/2 inches by 19 1/2 inches 58.5 inches by 58.5 inches Dice 1 inch x 1 inch x 1 inch 3 inches x 3 inches x 3 inches B. What would the dimensions of the large board be in feet and inches? 58.5/12 = feet or 4 feet 10 1/2 inches C. If the paint used to cover the board requires four ounces per square foot, how much paint is needed to cover one side of the larger board? Round your answer to the nearest tenth place. (Hint: First determine the area.) = square feet square feet 4 oz/square foot= ounces or 95.1 ounces D. Use the computation from c to determine how many bottles of paint you ll need to buy to paint the large board if the paint comes in 16-ounce bottles. 95.1/16 = , so you will need six bottles of paint. Teaching Tip: You might ask students to estimate how much paint it would take to paint the smaller board, which is on a scale of 1:3 with the larger board. If they have done the Mural Math lesson and interactive, they ll know that it s much less than onethird the amount (actually one-ninth). 2. Randy thinks the wheelchair ramp at the old community center is too steep. He and his parents looked into the guidelines of the Americans with Disabilities Act. The chart below indicates that for existing structures a slope steeper than 1:8 is not acceptable. This slope ratio is calculated as the rise (height) of the ramp to the run (length) of the ramp. For every three inches of rise at least a two-foot run is required. Since two feet equals 24 inches, the ratio is 3:24 or 1:8. Allowable Ramp Dimensions for Construction in Existing Sites, Buildings, and Facilities Slope* Maximum Rise Maximum Run Steeper than 1:10 but no steeper than 1:8 3 in or 75 mm 2 ft or 0.6 m Steeper than 1:12 but no steeper than 1:10 6 in or 150 mm 5 ft or 1.5 m * A slope steeper than 1:8 not allowed.

16 KEY: Scale Challenge The rise of the wheelchair ramp at the community center is seven inches. The run of the ramp is 58 inches. Is this slope steeper than 1:8? 7/58=1/x 7x=58 x = 58/7 x= No, the slope is not steeper than 1:8. (This problem is tricky since a higher number in the ratio means a less steep slope. It s a great opportunity to review what a ratio means and how larger and smaller numbers would change the proportion.) For more information about these requirements, visit 3. The Americans with Disabilities Act requires ramps in new building to have a slope of 1:12 or less. This means that for every inch of height (rise) of the ramp, there needs to be 12 inches of length (run). The recommended ramp slope is 1:16. A. For a rise of five inches, how long should the ramp be if it meets the 1:12 requirements? 5/x= 1/12 5/60 = 1/12 The answer is 60 inches or 5 feet. B. For a rise of five inches, how long should the ramp be if it meets the 1:16 recommendations? 5/x = 1/16 5/80 = 1/16 The answer is 80 inches or 6 feet 8 inches. 4. Here is a chart of the heights of some of the world s tallest structures in meters: Building Country Top of Spire Top of Building Taipei 101 Taiwan 508 meters 448 meters Shanghair World Financial Center China 492 meters Petronas Towers Malaysia 452 meters Sears Tower U.S.A meters Jin Mao Tower China meters A. To create a scale drawing comparing the heights of these buildings, the class chose to use a scale of 1 cm for every 100 meters. Fill in the chart with the values they would use for their drawing. Building Country Top of Spire Top of Building Taipei 101 Taiwan 5.08 cm 4.48 cm Shanghai World Financial Center China 4.92 cm Petronas Towers Malaysia 4.52 cm Sears Tower U.S.A cm Jin Mao Tower China cm B. How would you express a scale or ratio of 1 cm for every one hundred meters as a ratio in which you only use centimeters? (Hint: You need to convert meters to centimeters.) 1 cm for every 100 meters 100 cm in a meter 100 x 100 = 10,000 cm in 100 meters 1 cm for every 10,000 cm 1:10, Using the scale of 1:192, determine what the dimensions would be for a model of your school or a model of your classroom. (various answers)

17 MINIATURE LAND: HANDOUT 6 Surface Area and Volume Practice Problems Name: Date: Formulas: Surface Area (SA) of a Cube s = side SA = 6s 2 Volume (V) of a Cube V = s 3 Surface Area (SA) of a Rectangular Prism l = length, w = width, h = height SA = 2lw + 2lh + 2wh Volume (V) of a Rectangular Prism V = lwh 1. You sometimes need to know the area of the interior walls of a house or a room, for example, when you re trying to determine the amount of paint or wallpaper you need to cover the walls. To calculate this area you need to first determine the area of the four walls, the area of the doors, and area of the windows. Next, you subtract the areas of the doors and windows from the walls. Using the chart from the interactive, determine the areas of the interior walls of the little house (LH) and big house (BH). (Remember, both houses have five windows.) House of Scales Measurement LH BH Length (in) Width (in) Height (in) Floor Area (in 2 ) Door Width (in) Door Height (in) Door Area (in 2 ) Window Width (in) Window Height (in) Window Area (in 2 ) House Volume (in 3 ) Increase the dimensions of a 2-inch cube by four times. Step 1: Find the length of the new sides. Step 2: Find the surface area of the new cube. Step 3: Find the volume of the new cube.

18 Surface Area and Volume Practice Problems 3. Decrease by 1/2 the dimensions of a rectangular box that has a length of 30 cm, a width of 12 cm, and a height of 8 cm. Step 1: Find the new length, width, and height of the box. Step 2: Find the surface area of the new box. Step 3: Find the volume of the new box.

19 KEY: MINIATURE LAND: HANDOUT 6 Surface Area and Volume Practice Problems Formulas: Surface Area (SA) of a Cube s = side SA = 6s 2 Volume (V) of a Cube V = s 3 Surface Area (SA) of a Rectangular Prism l = length, w = width, h = height SA = 2lw + 2lh + 2wh Volume (V) of a Rectangular Prism V = lwh 1. You sometimes need to know the area of the interior walls of a house or a room, for example, when you re trying to determine the amount of paint or wallpaper you need to cover the walls. To calculate this area you need to first determine the area of the four walls, the area of the doors, and area of the windows. Next, you subtract the areas of the doors and windows from the walls. Using the chart from the interactive, determine the areas of the interior walls of the little house (LH) and big house (BH). (Remember, both houses have five windows.) House of Scales Measurement LH BH Length (in) Width (in) Height (in) Floor Area (in 2 ) Door Width (in) Door Height (in) Door Area (in 2 ) Window Width (in) Window Height (in) Window Area (in 2 ) House Volume (in 3 ) Little House Area of 4 walls Area of Door Area of Window = Area of interior walls for painting Area of 4 walls = 2(Height Width) +2(Height Length) = 2 (3 3) + 2 (3 3) = = 36 in 2 Area of 4 walls Area of Door (Area of Window 5) = Area of interior walls for painting 36 in 2-1 in 2 - ( in 2 ) = in 2

20 KEY: Surface Area and Volume Practice Problems Big House Area of 4 walls Area of Door Area of Window = Area of interior walls for painting Area of 4 walls = 2(Height Width) +2(Height Length) = 2 (30 30) + 2 (30 30) = = = 3600 in 2 Area of 4 walls Area of Door (Area of Window 5) = Area of interior walls for painting 3600 in in in 2 = Increase the dimensions of a 2-inch cube by four times. Step 1: Find the length of the new sides. Sides = 8 inches Step 2: Find the surface area of the new cube. SA = 384 in 2 Step 3: Find the volume of the new cube. V = 512 in 3 3. Decrease by 1/2 the dimensions of a rectangular box that has a length of 30 cm, a width of 12 cm, and a height of 8 cm. Step 1: Find the new length, width, and height of the box. 1 = 15 cm, w = 6 cm, and h = 4 cm Step 2: Find the surface area of the new box. SA = 348 cm 2 Step 3: Find the volume of the new box. V = 360 cm 3

21 MINIATURE LAND: HANDOUT 7 Surface Area and Volume Homework Problems Name: Date: Formulas: Surface Area (SA) of a Cube s = side SA = 6s 2 Volume (V) of a Cube V = s 3 Surface Area (SA) of a Rectangular Prism l = length, w = width, h = height SA = 2lw + 2lh + 2wh Volume (V) of a Rectangular Prism V = lwh 1. A cube has sides that measure 4 cm. Complete the chart. Please show work on a separate sheet of paper. Side length Surface area Volume Original cube 4 cm 1/2 sides of original 2 cm 1/4 sides of original 1 cm Increase sides 2 times 8 cm Increase sides 3 times 12 cm Increase sides 4 times 16 mm 2. A rectangular prism has the following measurements: l = 15 inches, w = 9 inches, h = 6 inches. Complete the chart. Please show work on a separate sheet of paper. Original figure 15 in 9 in 6 in 1/3 each side 5 in 3 in 2 in 2/3 each side 10 in 6 in 4 in 2 times each side 30 in 18 in 12 in 4 times each side 60 in 36 in 24 in 6 times each side 90 in 54 in 36 in Length Width Height Surface Area Volume

22 Surface Area and Volume Homework Problems 3. A miniature house has the following dimensions: l = 18 inches, w = 12 inches, and h = 14 inches. The windows in the house have a length of three inches and a width of two inches (the house has five windows). The doors in the house have a height of 10 inches and a width of 3 inches (the house has two doors). Using the dimensions, calculate the area of the interior walls of the house. A. Determine the area of the walls. Area of walls = 2 (length height) + 2 (width height) 2 ( ) + 2 ( ) = B. Determine the area of the windows. Length of window width of window = area of window = Number of windows is Number of windows x area of window = total area of windows = C. Determine the area of the doors. Height of door width of door = area of door = Number of doors is Number of doors area of door = total area of doors = D. Determine the area of the interior walls. Area of walls Area of windows Area of doors = Area of interior walls - - =

23 KEY: MINIATURE LAND: HANDOUT 7 Surface Area and Volume Homework Problems 1. A cube has sides that measure 4 cm. Complete the chart. Please show work on a separate sheet of paper. Side length Surface area Volume Original cube 4 cm 96 cm 2 64 cm 3 1/2 sides of original 2 cm 24 cm 2 8 cm 3 1/4 sides of original 1 cm 6 cm 2 1 cm 3 Increase sides 2 times 8 cm 384 cm cm 3 Increase sides 3 times 12 cm 864 cm 2 1,728 cm 3 Increase sides 4 times 16 mm 1,536 cm 2 4,096 cm 3 2. A rectangular prism has the following measurements: l = 15 inches, w = 9 inches, h = 6 inches. Complete the chart. Please show work on a separate sheet of paper. Length Width Height Surface Area Volume Original figure 15 in 9 in 6 in 558 in in 3 1/3 each side 5 in 3 in 2 in 62 in 2 30 in 3 2/3 each side 10 in 6 in 4 in 248 in in 3 2 times each side 30 in 18 in 12 in 2,232 in 2 6,480 in 3 4 times each side 60 in 36 in 24 in 8,928 in 2 51,840 in 3 6 times each side 90 in 54 in 36 in 20,088 in 2 174,960 in 3 3. A miniature house has the following dimensions: l = 18 inches, w = 12 inches and h = 14 inches. The windows in the house have a length of 3 inches and a width of 2 inches (the house has 5 windows). The doors in the house have length of 10 inches and a width of 3 inches (the house has 2 doors). Using the dimensions, calculate the area of the interior walls of the house. A. Determine the area of the walls. 2 (18 14) + 2 (12 14) = = 840 in 2 B. Determine the area of the windows. 3 2 = 6 Number of windows is = in 2 is total area of windows C. Determine the area of the doors = 30 in 2 Number of doors is = in 2 is total area of doors D. Determine the area of the interior walls. 840 in 2 30 in 2 60 in 2 = 750 in 2

24 MINIATURE LAND: HANDOUT 8 Scale of Planets Name: Date: In this worksheet, we ll be using data from our solar system to compare scale. The sun s mean equatorial radius (the distance from its geometric center to the surface) is 695,500 km. The earth s mean equatorial radius is 6, kilometers. 1. How many times bigger is the sun than the earth? (Answer to four decimal places.) 2. The moon s equatorial radius is 1,737.4 kilometers. What is the decimal value of the ratio of the earth s radius to the moon s radius? 3. Determine the scale comparison of the planets given the following information and fill in the chart below. (Answer to four decimal places.) Planet Radius (km) Scale Comparison to Earth Ratio of Planet to Earth Mercury 2, Earth to 1 Venus 6,051.8 Earth 6, Mars 3,397 Jupiter 71,492 Saturn 60,268 Uranus 25,559 Neptune 24, How could you use the scale comparison to determine the size for models of planets in the solar system? Provide an example of how you could do the calculations.

25 Scale of Planets Scale of Planets: Ball Models Using your scale comparisons and rounding them to the nearest tenth place, choose one ball to represent Earth and select at least three other balls to represent three other planets on the same scale. You will not be able to provide exact models, but you should choose balls that represent the dimensional differences as well as possible. Object Gumball Baseball Softball Tennis Ball Racquet Ball Ping Pong Ball Soccer Ball Basketball Beach Ball Exercise Ball Volleyball Marble Beads Playground Ball Giant Balls Diameter 1 inch 3 inches 4 inches 2.5 inches 2 1/4 inches 1.5 inches 8 inches 9 inches 12 inches, 16 inches 18 inches, 21 inches, 25 inches, 29 inches 8 inches 5/8 inch 3/16 inch, 1/2 inch 6 inches, 10 inches, 8.5 inches 36 inches, 40 inches 1. Object Planet Scale to Earth Diameter

26 Scale of Planets Scale of Planets: One to One Hundred Million To create a two-dimensional scale model of the solar system in 1/100,000,000 scale: First, determine how big the sun would be. The sun s mean equatorial radius is 695,500 km. 695,500/100,000,000 = km 1 km = 1000 meters = meters 1 meter = 100 centimeters = centimeters Diameter is twice the radius = 1391 centimeters Size of Model Planets on a 1:100,000,000 Scale (Fill in the chart below.) Planet Radius (km) Radius (m) Radius (cm) Diameter (cm) Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Based on the scale of 1/100,000,000, which model planets have a diameter that is less than the length of a pencil? Which two planets are the closest in size?

27 Scale of Planets Planets Distances from the Sun on a 1:100,000,000 Scale (Fill in the chart below.) Planet Distance from Sun (km) 1 to 100 million scale (km) Miles km (rounded to 4 decimal places) Mercury 57,909,175 Venus 108,208,930 Earth 149,597,890 Mars 227,936,640 Jupiter 778,412,020 Saturn 1,426,725,400 Uranus 2,870,972,200 Neptune 4,498,252,900 What is the difficulty in setting up a scale model of the solar system that includes both distance from the sun and the diameters of planets on a 1 to 100 million scale? Source: solarsystem.nasa.gov/planets/charchart.cfm

28 KEY: MINIATURE LAND: HANDOUT 8 In this worksheet, we ll be using data from our solar system to compare scale. Scale of Planets The sun s mean equatorial radius (the distance from its geometric center to the surface) is 695,500 km. The earth s mean equatorial radius is 6, kilometers. 1. How many times bigger is the sun s equatorial radius than the earth s? (Answer to four decimal places) The moon s equatorial radius is 1,737.4 kilometers. What is the decimal value of the ratio of the earth s radius to the moon s radius? Determine the scale comparison of the planets given the following information and fill in the chart below. (Answer to four decimal places.) Size of Model Planets on a 1:100,000,000 Scale Planet Radius (km) Scale Comparison to Earth Ratio of Planet to Earth Mercury 2, Earth to 1 Venus 6, Earth to 1 Earth 6, Earth 1 to 1 Mars 3, Earth to 1 Jupiter 71, Earth to 1 Saturn 60, Earth to 1 Uranus 25, Earth to 1 Neptune 24, Earth to 1 4. How could you use the scale comparison to determine the size for models in the solar system? Provide an example of how you could do the calculations. (various answers)

29 KEY: Scale of Planets Scale of Planets: Ball Models Using your scale comparisons and rounding them to the nearest tenth place, choose one ball to represent Earth and select at least three other balls to represent three other planets on the same scale. You will not be able to provide exact models, but you should choose balls that represent the dimensional differences as well as possible. Object Diameter Object Diameter Gumball 1 inch Baseball 3 inches Softball 4 inches Tennis Ball 2.5 inches Racquetball 2 1/4 inches Ping Pong Ball 1.5 inches Soccer Ball 8 inches Basketball 9 inches Beach Ball 12 inches, 16 inches Exercise Ball 18 inches, 21 inches, 25 inches, 29 inches Volleyball 8 inches Marble 5/8 inch Beads 3 1/6 inch, 1/2 inch Playground Ball 6 inches, 10 inches, 8.5 inches Giant Balls 36 inches, 40 inches Sample Answers: Here are just a few possibilities. Many different answers are possible. If you would like to simplify this activity, you might tell students which ball to use for Earth, and then ask them to decide which balls might represent the other planets and why. Object Planet Scale to Earth Diameter Gumball Earth inch Bead Mars 1 1/2 1/2 inch Basketball Saturn inches Softball Uranus inches Object Planet Scale to Earth Diameter Baseball Earth inches Ping Pong Ball Mars /2 inches Exercise Ball Saturn inches Beach Ball Neptune inches Object Planet Scale to Earth Diameter Playground Ball Earth inches Softball Mercury inches Basketball Venus inches Giant Ball Neptune inches

30 KEY: Scale of Planets Object Planet Scale to Earth Diameter Marble Earth 1 5/8 5/8 inches Bead Venus 0.9 5/8 1/2 inch Tennis Ball Uranus 4 5/8 2 1/2 inches Bead Mars 0.5 5/8 1/4 inch Object Planet Scale to Earth Diameter Softball Earth inches Giant Ball Saturn inches Rackquetball Mars 1/ /4 inches Beach Ball Uranus inches Scale of Planets: One to One Hundred Million To create a two-dimensional scale model of the solar system in 1/100,000,000 scale: First, determine how big the sun would be. The sun s mean equatorial radius is 695,500 km. 695,500/100,000,000 = km 1 km = 1000 meters = meters 1 meter = 100 centimeters = centimeters Diameter is twice the radius = 1391 centimeters Size of Model Planets on a 1:100,000,000 Scale Planet Radius (km) Radius (m) Radius (cm) Diameter (cm) Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Based on the scale of 1/100,000,000, which model planets are smaller than the length of a pencil? Mercury, Mars, Venus, and Earth Which two planets are the closest in size? Venus and Earth

31 KEY: Scale of Planets Planets Distances from the Sun on a 1:100,000,000 Scale Planet Distance from Sun (km) 1 to 100 million scale (km) miles (km x.6214) Mercury 57,909, Venus 108,208, Earth 149,597, Mars 227,936, Jupiter 778,412, Saturn 1,426,725, Uranus 2,870,972, Neptune 4,498,252, What is the difficulty in setting up scale models of the solar system using distance from the sun and the diameters of planets on a 1 to 100 million scale? While the planet models are easy to create and view, the distance between the models makes it impossible to look at them all at once. Source: solarsystem.nasa.gov/planets/charchart.cfm

33 Race to the Top of the Washington Monument Multiple Choice Questions 1. The giant oak in town was 80 feet tall. The giant oak in the model railroad scene was about 4 5/8 inches tall. The scale used in the railroad scene was the A. O Scale, 1:48 B. N Scale, 1:160 C. HO Scale, 1:187 D. Z Scale, 1: A mansion measures 80 by 200 feet. On a scale of 1:24, a model of the mansion would be A. 4 by 10 feet B. 3 2/3 by 9 2/3 feet C. 3 1/3 by 8 1/3 feet D. 2 1/2 by 6 1/2 feet 3. Lego builders commonly use a minifig scale of 1:48. On this scale, a rural post office building measuring 12 by 12 feet would be A. 2 by 2 inches B. 3 by 3 inches C. 4 by 4 inches D. 6 by 6 inches 4. One model building was four inches taller than all the other buildings. If the scale used was 1:192, the model building was actually A. 48 feet taller than the rest B. 64 feet taller than the rest C. 192 feet taller than the rest D. 768 feet taller than the rest 5. When comparing the volume of a 3-cm cube and a 6-cm cube, the volume of the 3-cm cube is A. half the volume of the 6-cm cube B. 1/3 the volume of the 6-cm cube C. 1/4 the volume of the 6-cm cube D. 1/8 the volume of the 6 cm cube 6. When comparing the surface area of a 3-cm cube and a 6-cm cube, the surface area of the 3-cm cube is A. 54 cm 2 (the larger cube s surface area is 216 cm 2 ) B. 36 cm 2 (the larger cube s surface area is 216 cm 2 ) C. 27 cm 2 (the larger cube s surface area is 216 cm 2 ) D. 9 cm 2 (the larger cube s surface area is 216 cm 2 ) 7. The most efficient formula for surface area of a cube is A. 6s 2 B. 2lw + 2lh + 2wh C. s 2 D. lwh

34 8. The most efficient formula to find the volume of a cube is A. 6s 2 B. 2lw + 2lh + 2wh C. s 3 D. lwh 9. The formula to find the surface area of a rectangular prism is A. 6s 2 B. 2lw + 2lh + 2wh C. s 3 D. lwh 10. The formula to find the volume of a rectangular prism is A. 6s 2 B. 2lw + 2lh + 2wh C. s 3 D. lwh Race to the Top of the Washington Monument Multiple Choice Questions

35 Race to the Top of the Washington Monument Short Answer Questions 1. A school bus is 39.5 feet long. To order one for a model railroad scene on the HO Scale (1:187), how long should the school bus be? Answer to four decimal places. 2. 1/16 of an inch is used to represent each foot. How tall would the model representation of a 6-foot tall person be? Answer in simplest form. 3. A small collector car scale is 1:64. Using this scale, how long would a toy of a 9-foot electric car be? Answer to the nearest 1/16 of an inch. 4. If the ramp has a 14-inch rise and a run of 126 inches, what is the slope ratio, also called the rise over the run, in simplest terms? 5. A ramp with a rise of 15 inches and a run of 210 inches is planned for the new school. What is the ratio of the rise over the run for this ramp? 6. Under the building guidelines, the steepest slope allowed for an access ramp for a new building is 1:12. Would a slope of 1:15 be allowed for a new building? 7. Using the architectural scale of 1:192, how much taller would a model of Toronto s CN Tower be than a model of Shanghai s Oriental Pearl Tower? Toronto CN Tower Shanghai Oriental Pearl Tower meters 468 meters to the spire

36 Race to the Top of the Washington Monument Short Answer Questions 8. Chelsea is donating her dollhouse for a fundraising auction. She wants to add wallpaper to the living room. The living room is 12 inches deep, 10 inches wide, and 14 inches high. There is one door that is 9 inches by 3 inches. Two windows each measure 3 by 5 inches. The dollhouse living room has three walls and is open at the side of the width so you can see inside. What is the total surface area of the walls for wallpaper? Area of Windows: 3 in 5 in = 2 windows = Area of Door: 9 in 3 in = Total square area of door and windows = Area of wall one = 12 in 14 in = Area of wall two = 10 in 14 in = Area of wall three = 12 in 14 in = Total possible area of walls = Total wall area total square area of door and windows = 9. Sunny and Lena are mailing boxes full of T-shirts and school supplies to children in another country. Sunny s box is 5 inches by 3 inches by 3 inches. Lena s box is 10 inches by 6 inches by 6 inches. How many times greater is the volume of Lena s box than the volume of Sunny s box? 10. Sunny s box is 3 inches by 3 inches by 5 inches. Lena s box is 6 inches by 6 inches by 10 inches. How many times greater is the surface area of Lena s box than the surface area of Sunny s box?

37 Race to the Top of the Washington Monument True or False Questions 1. True or False: Calculating volume will tell you how much wrapping paper you need for a box. 2. True or False: The following scales are equal: 1/2 inch for every foot 1:24 3. True or False: Two ratios are proportional if the decimal values of the ratios are equal. 4. True or False: The following are equal: 1/64 1/4 inch for every foot 5. True or False: If you are creating a model of the Empire State Building in limited space, 1:24 is a better scale than 1: True or False: On a scale of 1:12, three inches of a model represent two feet of the actual object. 7. True or False: The Great Pyramid stands feet high. Using a 1:24 scale, a model of the Great Pyramid would be 18 feet 8 3/4 inches tall. 8. True or False: You can use the formulas for surface area and volume of a rectangular prism for a cube. 9. True or False: If the length, width, and height are three times greater for a larger box, then the volume will be three times the volume of the smaller box. 10. True or False: There are two cubes. One cube s edges are four times as long as the edges of the other cube. The surface area of the larger cube will be four times the surface area of the smaller cube.

38 KEY: MINIATURE LAND: HANDOUT 9 Race to the Top of the Washington Monument KEY Multiple Choice Questions Point Value for Game Questions 1-3: 2 points Questions 4-5: 3 points Question 6: 2 points Questions 7-10: 1 point Answer Key 1. D, 2.C, 3. B, 4. B, 5. D, 6. A, 7. A, 8. C, 9. B, 10. D KEY Short Answer Questions 2 points 1. A school bus is 39.5 feet long. To order one for a model railroad scene on the HO Scale (1:187), how long should the school bus be? Answer to four decimal places inches 2 points 2. 1/16 of an inch is used to represent each foot. How tall would the model representation of a 6-foot tall person be? Answer in simplest form. 3/8 inch or inches 2 points 3. A small collector car scale is 1:64. Using this scale, how long would a toy of a 9-foot electric car be? Answer to the nearest 1/16 of an inch. 1 and 11/16 inches 2 points 4. If a ramp has a 14-inch rise and a run of 126 inches, what is the slope ratio, also called the rise over the run, in simplest terms? 1:9 2 points 5. A ramp with a rise of 15 inches and a run of 210 inches is planned for the new school. What is the ratio of the rise over the run for this ramp? 1/14 or 1:14

39 KEY Short Answer Questions (continued) KEY: Race to the Top of the Washington Monument 1 point 6. Under the building guidelines, the steepest slope allowed for an access ramp for a new building is 1:12. Would a slope of 1:15 be allowed for a new building? Yes. 1:12 is the steepest allowable slope in a new building. A less steep slope is acceptable. 1:15 is less steep than 1:12. 3 points 7. Using the architectural scale of 1:192, how much taller would a model of Toronto s CN Tower be than a model of Shanghai s Oriental Pearl Tower? Toronto CN Tower meters meters Shanghai Oriental Pearl Tower, 468 meters to the spire meters meters 4 points 8. Chelsea is donating her dollhouse for a fundraising auction. She wants to add wallpaper to the living room. The living room is 12 inches deep, 10 inches wide, and 14 inches high. There is one door that is 9 inches by 3 inches. Two windows each measure 3 by 5 inches. The dollhouse living room has three walls open at the side of the width. What is the total surface area of the walls for wallpaper? Area of Windows: 3 in 5 in = 15 in 2 2 windows = 30 in 2 Area of Door: 9 in 3 in = 27 in 2 Total square area of door and windows = 57 in 2 Area of wall one = 12 in 14 in = 168 in 2 Area of wall two = 10 in 14 in = 140 in 2 Area of wall three = 12 in 14 in = 168 in 2 Total possible area of walls = 476 in 2 Total wall area total square area of door and windows = 476 in 2 57 in 2 = 419 in 2 3 points 9. Sunny and Lena are mailing boxes full of T-shirts and school supplies to children in another country. Sunny s box is 5 inches by 3 inches by 3 inches. Lena s box is 10 inches by 6 inches by 6 inches. How many times greater is the volume of Lena s box than the volume of Sunny s box? Volume of Sunny s box = 45 in 3 Volume of Lena s box = 360 in = 8 The volume of Lena s box is eight times greater than the volume of Sunny s. 3 points 10. Sunny s box is 3 inches by 3 inches by 5 inches. Lena s box is 6 inches by 6 inches by 10 inches. How much greater is the surface area for Lena s box than the surface area of Sunny s box? Surface area of Sunny s box = 78 in 2 Surface area of Lena s box = 312 in = 4 The surface area of Lena s box is four times greater than the surface area of Sunny s box.

40 KEY: Race to the Top of the Washington Monument KEY True or False Questions 1 point 1. True or False: Calculating volume will tell you how much wrapping paper you need for a box. False. Calculating surface area will tell you how much wrapping paper you need for a box. Calculating volume will tell you how much the box will hold. 2 points 2. True or False: The following scales are equal; 1/2 inch for every foot 1:24 True. Write in all inches (1/2)/[(1 foot) 12 inches/foot] = 0.5/12 Is 0.5/12 equal to 1:24 or 1/24? a = 0.5, b = 12, c = 1, d = /12 = 1/24 Test by the definition for a proportion a/b = c/d if ad = bc (0.5)24 = 12 (1) 12 = 12 Since ad is equal to bc, these scales are equal. 1 point 3. True or False: Two ratios are proportional if the decimal values of the ratios are equal. True. The decimal value of the ratios is determined by dividing the numerator by the denominator when the ratio is written as a fraction. 2 points 4. True or False: The following are equal: 1/64 1/4 inch for every foot False. 1/64 is not equal to (1/4 inch or 0.25 inches)/[(1 foot) (12 inches/foot)] a = 1, b = 64, c = 0.25, d = 12 inches 1/64 = 0.25/12 inches if ad = bc 1 12 = is not equal to 16 so these two ratios are not equal. 1/64 is equal to 1/4 inch for every 16 inches. 2 points 5. True or False: If you are creating a model of the Empire State Building in limited space, 1:24 is a better scale than 1:192. False. 1:24 of the Empire State Building would be very large (1472 feet 24 or feet tall). The 1:192 model would be 7 feet 8 inches tall. 2 points 6. True or False: On a scale of 1:12, three inches of a model represent two feet of the actual object. False. 24 inches = 2 feet 2 inches/24 inches simplifies to 1:12 3 inches/24 inches simplifies to 1:8

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