Models and Patterns in Art, Architecture and Nature: Scale and Proportion

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1 Models and Patterns in Art, Architecture and Nature: Scale and Proportion EPISD Math Models Team Say Thanks to the Authors Click (No sign in required)

2 To access a customizable version of this book, as well as other interactive content, visit AUTHOR EPISD Math Models Team CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-source, collaborative, and web-based compilation model, CK-12 pioneers and promotes the creation and distribution of high-quality, adaptive online textbooks that can be mixed, modified and printed (i.e., the FlexBook textbooks). Copyright 2017 CK-12 Foundation, The names CK-12 and CK12 and associated logos and the terms FlexBook and FlexBook Platform (collectively CK-12 Marks ) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License ( licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the CC License ), which is incorporated herein by this reference. Complete terms can be found at terms-of-use. Printed: October 15, 2017

3 Chapter 1. Models and Patterns in Art, Architecture and Nature: Scale and Proportion CHAPTER 1 Models and Patterns in Art, Architecture and Nature: Scale and Proportion TEKS 7, 7A), 7(B), 7(D) (see actual TEKS in resources) Learning Objectives To use scale and proportion in art and architecture. To use scale factor in surface area and volume of similar solids. Introduction In this section, students will learn how scale and proportion relates to the real world in art and architecture. They will learn how to change the scales to better adapt drawings or renderings to their original size. Students will also use scales and proportions as scale factors which affect surface area and volume of similar solids. Vocabulary Proportion - Formed when two ratios are set equal to each other. Ratio - A way to compare two numbers, measurements or quantities by dividing one number by the other and expressing the answer as a fraction. Scale - The ratio of the length in a drawing (or model) to the length of the real thing. Scale factor - The ratio of any two corresponding lengths in two similar geometric figures. Surface area - The total area of the exterior surface of a solid. Volume - The total amount of space enclosed in a solid. 1

4 Content What started out as a status symbol but became a classic toy? Dollhouses! Originally, only wealthy nobles could afford these scale models of homes. Today, every toy aisle has a selection of dollhouses, ranging from tiny to huge. Perfect Scale While some dollhouses are toys for toddlers, others are valuable collectibles made by artists who create perfect scale models of real houses, rooms, and furniture. In a high-quality dollhouse, every room, table, doorway and accessory is similar to the real-life item. Dollhouses are usually constructed with scale ratios of 1 12 or 1 24 to the real thing. This means they cover between or of the floor space of the homes they replicate. 2

5 Chapter 1. Models and Patterns in Art, Architecture and Nature: Scale and Proportion To ensure their copies are perfect, miniature artists use tiny versions of real life tools such as lathes, table saws, and drills to build their models. They carefully study each real-life item and then draw up plans for the smaller versions. The final products are almost identical to the full-sized originals. In a close-up photo, it s tough to tell whether you re looking at the original room or at a perfectly similar miniature version. Some of the world s finest miniatures end up in art museums, where crowds of people spend hours marveling at their details. What if you had a 1:80 scale model of the Eiffel Tower. The model stands 4 meters tall. How could you find the height of the actual Eiffel Tower? After completing this Concept, you ll be able to use indirect measurements to solve scale problems like this one. Watch This MEDIA Click image to the left or use the URL below. URL: 3

6 Guidance One place where ratios are often used is in making maps. The scale of a map describes the relationship between distances on a map and the corresponding distances on the earth s surface. These measurements are expressed as a fraction or a ratio. So far we have only written ratios as fractions, but outside of mathematics books, ratios are often written as two numbers separated by a colon (:). For example, instead of 2 3, we would write 2:3. Ratios written this way are used to express the relationship between a map and the area it represents. For example, a map with a scale of 1:1000 would be a map where one unit of measurement (such as a centimeter) on the map would represent 1000 of the same unit (1000 centimeters, or 10 meters) in real life. Example A Anne is visiting a friend in London, and is using the map below to navigate from Fleet Street to Borough Road. She is using a 1:100,000 scale map, where 1 cm on the map represents 1 km in real life. Using a ruler, she measures the distance on the map as 8.8 cm. How far is the real distance from the start of her journey to the end? Solution The scale is the ratio of distance on the map to the corresponding distance in real life. Written as a fraction, it 1 is We can also write an equivalent ratio for the distance Anne measures on the map and the distance in real life that she is trying to find: 8.8 x. Setting these two ratios equal gives us our proportion: = 8.8 x. Then we can cross multiply to get x = That s how many centimeters it is from Fleet Street to Borough Road; now we need to convert to kilometers. There are cm in a km, so we have to divide our answer by = 8.8.

7 Chapter 1. Models and Patterns in Art, Architecture and Nature: Scale and Proportion The distance from Fleet Street to Borough Road is 8.8 km. In this problem, we could have just used our intuition: the 1 cm = 1 km scale tells us that any number of cm on the map is equal to the same number of km in real life. But not all maps have a scale this simple. You ll usually need to refer to the map scale to convert between measurements on the map and distances in real life! Example B Antonio is drawing a map of his school for a project in math. He has drawn out the following map of the school buildings and the surrounding area He is trying to determine the scale of his figure. He knows that the distance from the point marked A on the baseball diamond to the point marked B on the athletics track is 183 meters. Use the dimensions marked on the drawing to determine the scale of his map. Solution We know that the real-life distance is 183 m, and the scale is the ratio distance on map distance in real life. To find the distance on the map, we use Pythagoras Theorem: a 2 + b 2 = c 2, where a and b are the horizontal and vertical lengths and c is the diagonal between points A and B = c = c = c = c c So the distance on the map is about cm. The distance in real life is 183 m, which is cm. Now we can divide: The scale of Antonio s map is approximately 1:1100. Scale = Another visual use of ratio and proportion is in scale drawings. Scale drawings (often called plans) are used extensively by architects. The equations governing scale are the same as for maps; the scale of a drawing is the distance on diagram ratio distance in real life. 5

8 Example C Oscar is trying to make a scale drawing of the Titanic, which he knows was 883 ft long. He would like his drawing to be at a 1:500 scale. How many inches long does his sheet of paper need to be? Solution We can reason intuitively that since the scale is 1:500, the paper must be = f eet long. Converting to inches means the length is 12(1.766) = inches. Oscar s paper should be at least 22 inches long. Guided Practice The Rose Bowl stadium in Pasadena, California measures 880 feet from north to south and 695 feet from east to west. A scale diagram of the stadium is to be made. If 1 inch represents 100 feet, what would be the dimensions of the stadium drawn on a sheet of paper? Will it fit on a standard inch sheet of paper? Solution Instead of using a proportion, we can simply use the following equation: (distance on diagram) = (distance in real distance on diagram life) (scale). (We can derive this from the fact that scale = distance in real life.) Plugging in, we get height on paper = 880 f eet 1 inch 100 f eet = 8.8 inches width on paper = 695 f eet 1 inch 100 f eet = 6.95 inches The scale diagram will be 8.8 in 6.95 in. It will fit on a standard sheet of paper. Explore More 1. A restaurant serves 100 people per day and takes in $908. If the restaurant were to serve 250 people per day, how much money would it take in? 2. The highest mountain in Canada is Mount Yukon. It is the size of Ben Nevis, the highest peak in Scotland. Mount Elbert in Colorado is the highest peak in the Rocky Mountains. Mount Elbert is the height of Ben Nevis and the size of Mont Blanc in France. Mont Blanc is 4800 meters high. How high is Mount Yukon? 3. At a large high school it is estimated that two out of every three students have a cell phone, and one in five of all students have a cell phone that is one year old or less. Out of the students who own a cell phone, what proportion owns a phone that is more than one year old? For 4-6, suppose a map of Ratio City has a scale of 1:1,000,000, where 1 centimeter on the map represents 10 kilometers in real life. Use that scale to determine the real-life distances in kilometers. 4. The distance on the map between city hall and high school is 1.2cm. 5. The distance on the map between city hall and the main library is 0.6cm. 6. The distance on the map between the main library and the high school is 0.4cm. For 7-10, use the map in Example A. Using the scale printed on the map, determine the distances (rounded to the nearest half km) between: 6 7. Points 1 and 4 8. Points 22 and Points 18 and Tower Bridge and London Bridge

9 Chapter 1. Models and Patterns in Art, Architecture and Nature: Scale and Proportion Vocabulary TEKS 6(A), 7(B) (see actual TEKS in resources) Learning Objectives Understand the difference between a relation and a function. Introduction Do you know how to identify a function? Do you know what a relation is? How about graphing - do you know how to graph functions? Well, inputs, outputs, functions and relations are all words that are building blocks of Algebra. Once you understand them, you can analyze, calculate, graph and work with functions. Vocabulary Relation - A relation is a set of values, usu Content 7

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