SCALE 2013 Judo Math Inc.
7 th grade Geometry Discipline: Yellow Belt Training Order of Mastery: Scale 1. What is scale (tie to ratio) (7G1) 2. Art with scale and skewed sale (7G1) 3. Scaling down (7G1) 4. Scaling up (7G1) 5.Reproduce a scale drawing with a different scale (7G1) Welcome to the Yellow Belt Scale See the picture of the skyscraper below? How tall is it in real life? Having trouble answering that question? That s because you are missing some key information a scale! Depending on the scale I give you, this could be a picture of a toy skyscraper or a building downtown! It all hinges on that scale. Without a scale, we wouldn t be able to represent really big or really small things on a regular sized paper. Small things you ask? Yes, small things can be represented with scale too! Things like bacteria and cells that you have to look at through a microscope can be seen with the naked eye and investigated because of scale! And that map of the United States? It wouldn t be worth very much without a scale because we would have no idea how far apart anything is! So embrace the scale, my friend, because it s going to be helping you forever no matter what you decide to do in the future! Good luck grasshopper. 2013 Judo Math Inc.
1. What is scale (tie to ratio) Way back in the first discipline black belt, you became a master of ratios and rates. You may have forgotten, but at the tail end of that packet, you dabbled a little bit in geometry as you used ratios to determine the missing side of similar figures. You may recall these shapes We were able to determine the missing side of the big rectangle by writing two equivalent ratios: 12 4 =? 8 and we can see that the? is simply 24! How could these two rectangles connect to the idea of scale that we talked about in the intro? Discuss your answer above with someone around you some key words you could include in your explanation would be: ratio, similar, enlarge, reduce, etc. 1
In order to solve problems with scale, it is very important to identify the scale as a ratio. You will then do some work with equivalent ratios which can also make use of tools such as ratio tables, double number lines or ribbon diagrams. 1. A map has a scale of 1 in: 15 mi. If Mount Pleasant and Santa Cruz are 7 in apart on the map, then how far apart are the real cities? 2. A model motorcycle has a scale of 1 in:3ft. If the model motorcycle is 3.5 in long, then how long is the real motorcycle? 3. An elephant that is 8.2 feet tall casts a shadow that is 6 ft long. Find the length of the shadow that a 6 foot feeding station casts at the same time of day. 4. A map has a scale of 1 cm:11km. If sun Valley and Midway are 3.7 cm apart on the map then how far apart are the real cities? 2
5. A particular giraffe is 16 feet tall. A model was built of it with a scale of 1 in: 4ft. How tall is the model? 6. An igloo is 12 ft wide. A model of it was built with a scale of 1 in: 3 ft. How wide is the model? Explain to your brother: What method do you use to solve straightforward scale prolems like the ones above? Explain in writing how you would teach your 5 th grade brother how to visualize and then solve these types of problems You may want to write and solvea sample problem to show him in the space below. 3
A clever costume 1. For your Halloween costume, you want to dress up like your iphone so you are trying to create a scaled version of your phone that covers most of your body except for your head. How wide and tall must you make your cutout for your costume? The iphone 5 s dimensions are: Height 4.87 inches (123.8 mm) Width: 2.31 inches (58.6 mm) 2. Each of the aps on an iphone are 3/8 of an inch across. How big will they be on your costume? 3. Go online (or ask your teacher to give you) the dimensions of the newest ipad and the ipad mini. Are the screens of each of these products similar rectangles? Would your costume have to change if you decided to dress up as one of these instead? 4
2. Art with scale and skewed scale (7G1) The idea of things being drawn to scale beings up an interesting question what do things look like that are not drawn to scale? What types of problems could arise if a drawing was not drawn to scale? In a practical sense: If plans that architects draw up are not to scale, an entire building could be constructed incorrectly which could have dangerous implications. If your map didn t have a scale, you might incorrectly estimate how long it would take you to get somewhere and you could be late!!! In an artistic sense, you could make some pretty funny looking drawings or pictures! Check out the four images below and write about what is going on with the SCALE in that picture (example) In this picture, the soda bottle and the man are very_ out of scale. I would_ estimate that either the soda bottle needs to be decreased by a scale of about 15:1 or the man needs to be scaled up by a scale of 1:15. 5
In the space below, draw a picture of your face but try to change the scale of some of your features in the following way: Scale up your nose by a ratio of 1:2 (for every 1 inch you nose is, make it two inches in the drawing) Scale down your mouth by a ratio of 3:1 (for every 3 inches that your mouth is, make it 1 inch in the drawing) Scale up your ears by a ratio of 1:4 My Skewed Scale Self-Portrait 6
Art connection (mini-project): Now using some of the pictures as examples from the two pages ago, set up a situation with a friend where you create a skewed scale image something that would really make people go what in the world?! when they look at it! This could either be done using pictures you find online and then using a program like photoshop to edit, or by setting up a situation using distance to make something just look unreasonable for example in this picture, the man is standing far away from the Eiffel tower and the picture is taken at just the right angle to make it look as though he is taller than it! Use the space below to brainstorm some ideas for your project! 7
3. Scaling down (7G1) A map cannot be of the same size as the area it represents. So, the measurements are scaled down to make the map of a size that can be conveniently used by motorists, cyclists, etc. A scale drawing of a building (or bridge) has the same shape as the real building (or bridge) that it represents but a different size. Builders use scaled drawings to make buildings and bridges. A ratio is used in scale drawings of buildings to show: DRAWING LENGTH: ACTUAL LENGTH A ratio is used in maps to show: MAP SCALE: ACTUAL DISTANCE Sometimes scale is shown by just a picture (like to the left). In this case it is best to use a ruler to translate the scaled distance onto the map to determine the actual distance Example using buildings: Martha made a scale drawing of the auditorium. The scale of the drawing was 1 inch = 2 feet. The stage is 24 inches in the drawing. How long is the actual stage? Stage 24 inches Scale: Actual: Since the scale is being multiplied by 24 to get the length of the side, I 1in 24 in = know that I need to multiply the 2ft x ft numerator by 24 so I end up with the actual stage being 48 feet long. Example in maps: Vivian is taking an SAT prep class at the community center in Lancaster. The community center is 3 centimeters away from Vivian's house on a city map. The map uses a scale of 1 centimeter = 2 kilometers. What is the actual distance between Vivian's house and the community center? Community Center Vivian s house 1cm = 3cm 2km x km To get the 3 in the numerator, I need to multiply by 3. In order to keep the ratios equal I need to multiply the denominators by 3 as well which gives me 6km from Vivian s house to the community center. 8
Above is a map of Costa Rica. Using the scale in the bottom left corner of the map, estimate the distances between the following cities. (note, you may want to use a ruler on this one and pay attention to if your answer is in miles or km!) 1. Liberia to Santa Cruz km 2. Santa Cruz to Tortuguero mi 3. San Jose to the nearest border with Panama mi 4. Puerto Limon to Puerto Jimenez km 5. The approximate width of Costa Rica at its widest spot mi 6. The scale for this map is not written as a ratio, but rather as a drawing. Please write a ratio scale below that is accurate for the map above: 9
Some more practice. Please make sure that you create a drawing and are careful with your units as you set up equivalent ratios: 1. Anne made a scale drawing of a house. The dining room, which is 6 meters long in real life, is 2 centimeters long in the drawing. What is the scale of the drawing? 2. Rodolfo and his friends are visiting chocolate shops in South Plains. They take a cab from one chocolate shop to another one that is 4 miles away. On a map, the two are separated by 2 inches. What scale does the map use? 3. After a long hike in a state park, Pamela decides to go relax at the beach. The parking permit she purchased allows her to park at any state beach without paying again. The nearest state beach is 8 1 kilometers away from the park. 2 How far apart are the state park and the closest state beach on a map with a scale of 1 centimeter = 2 kilometers? 4. Durand, a high school student, is also enrolled in a class at the local junior college. The college is 8 kilometers away from the high school, and these two are 2 1 4 centimeters away on a map of the area. What is the map's scale? 10
5. Al measured the elementary school and made a scale drawing. He used the scale 1 centimeter = 6 meters. The actual width of the school yard is 57 meters. How wide is the school yard in the drawing? 6. 7. Here is a map of san Diego. Determine how many miles it is from Chula Vista to Encinitas and then create a ratio scale for the map using the drawing in the bottom left corner. 11
8. Below is a map of the United States. It has two different types of scales on it. a) Why do you think they put the two different scales on this map? b) Use the scales to determine the distance from Los Angeles to New York. 12
4. Scaling up (7G1) In the last section, we looked at lots of examples where really large things like buildings and maps were scaled down to fit on a small piece of paper. Scaling can also be used in opposite situations where really small things are scaled up to be represented to the naked eye. A lot of things that we would want to Scale up would be looked at on a microscope and will be measured in µm or micrometers. The conversion rate for the micrometer is : 1 µm =.ooooo1 m Brainstorm Box: What are the tiniest things that you can think of that might be able to represented in more detailing by being scaled up Write them around this thought bubble. Try it out! To the left is a picture taken through a microscope of a fossil that was found in the depths of the ocean. 1 µm is shown by a short line. Use that line to estimate how many µm across this image is. Using the conversion rate above (µm =.ooooo1 m) state how many meters across this fossil is. 13
1. Termites: To the left is a termite under a microscope. Using the scale on the drawing, how many mm across is a termite? Use a ruler to draw a line that length here: 2. Hair Follicles: Use the scale at the bottom of the image to estimate the width of one of the hairs in this image. 3. Ant larvae: The line at the bottom of this image represents 1 µm. About how big is this larvae in meters? 14
5. Reproduce a scale drawing with a different scale (7G1) ON MAPS sometimes the scale is written like this: 1 : 100,000 means that the real distance is 100,000 times the length of 1 unit on the map or drawing. Sometimes you will be given a scale with units, and you can re-write it without units by doing some small conversions: For example: Write the scale without units:? 1 in : 9 yd First convert the 9 yards to feet. (9yards*3=27 feet) then to inches: (27 feet * 12 inches = 324in) So the final scale is 1in: 324 in OR 1: 324 Now you try 1in : 14 yd 1cm : 20m 4ft: 3 miles 5 in: 100 yd 15
For each of the scales below, write what the given scale means in words and then write a possible scale with units (there will be multiple answers). Give each scale units and simplify: 16
Reproducing a cartoon with another scale! Scale (or grid) drawing method: 1. Using a ruler, draw a grid (horizontal and vertical lines) covering the entire drawing. Determine an appropriate size (and units) for the side of each square in the grid. Record the size they are using for the length of each square here 2. On another sheet of paper, reproduce the original grid using a scalar factor of 2. With the enlarged grid complete, reproduce the portion of Note, if you have a cartoon cutout that you would rather create a scale drawing go ahead and use it! 17