What is a scale drawing? http://www.basic-mathematics.com/scale-drawings.html Scale Drawing • A drawing that shows a real object with accurate sizes except they have all been reduced or enlarged by a certain amount (called the scale). The scale is shown as the length in the drawing, then a colon (":"), then the matching length on the real thing. Example: this drawing has a scale of "1:10", so anything drawn with the size of "1" would have a size of "10" in the real world, so a measurement of 150mm on the drawing would be 1500mm on the real horse. Investigating Scale Click on the Butterfly Below… • The scale indicates how many units of length of the actual object are represented by each unit of length in the drawing. Scale • A scale of 1:1 implies that the drawing of the grasshopper is the same as the actual object. • The scale 1:2 implies that the drawing is smaller (half the size) than the actual object (in other words, the dimensions are multiplied by a scale factor of 0.5). • The scale 2:1 suggests that the drawing is larger than the actual grasshopper -- twice as long and twice as high (we say the dimensions are multiplied by a scale factor of 2). • If no units are listed in the scale, then you can assume that the drawing and the object are measured using the same units. For example, the scale 1:2 might represent 1 cm:2 cm or 1 in.:2 in. Teacher’s Domain videos Island of the Little (48 s) • http://www.teachersdomain.org/resource/vtl07.math.number.rat.islandlitt/ Island of the Giants (3min 33s) • http://www.teachersdomain.org/resource/vtl07.math.number.rat.landgiants/ Vocabulary • Scale drawing/scale model: is used to represent an object that is too large or too small to be drawn or built at actual sizes • Scale: gives the relationship between the measurements on the drawing or model and the measurements of the real object • Scale factor: the ratio of a length on a scale drawing or model to the corresponding length on the real object Ratios • Rates are often written with a slash rather than the word per: – such as mi/h for miles per hour – $2/dozen for $2 per dozen – a car traveling 30 miles per hour – making a long-distance telephone call that costs 20¢ per minute – skating at an ice rink that costs $10 for 2 hours Proportions A statement that shows two ratios are equivalent. • A proportion is often used when one ratio is known and only part of a second ratio is known, such as: Example “The ratio of girls to boys in a class is 6:8 and there are 12 boys in the class.” A proportion can be set up and solved to find how many girls there are in the class. • Set up the proportion. Solving Proportions _6_ = _X_ 8 12 cross multiply: 6 x 12 = 8 x X 72 = 8x isolate the variable: 72 = 8x 8 8 X = 9 girls By using proportions, you can find lengths needed to make a scale drawing or can find the actual lengths of an object based on a given scale drawing. map scales • If 1 cm on a map represents a distance of 250 km, what is the approximate distance of a length represented by 2.7 cm? We can set up a proportion to show: 1 cm = 2.7cm 250 km X • Solving the equation for x, we get x = 250 • 2.7 = 675 km. 3 ways scale can be expressed 1. 1 cm = 1 km 2. ______ = 1 km 3. 1: 50,000 Practice Problem • A student has a map on which the scale is 2 cm = 5 km. Having measured the distance between two points on the map to be 7.5 cm how do you calculate the real world distance from this measurement? Scale Factor Example: Suppose a scale model has a scale of 2 inches = 16 inches. The scale factor is 2 or 1 16 8 The lengths and widths of objects of a scale drawing or model are proportional to the lengths and widths of the actual object. Your Turn, Again! In an illustration of a honey bee, the length of the bee is 4.8 cm. The actual size of the honeybee is 1.2 cm. What is the scale of the drawing? 4.8 cm = 1cm 1.2 cm x cm 4.8x = 1.2 x = .25 The scale of the drawing is 1 cm = .25cm Example 1: Find Actual Measurements A set of landscape plans shows a flower bed that is 6.5 inches wide. The scale on the plans is 1 inch = 4 feet. What is the width of the actual flower bed? Let x represent the actual width of the flower bed. Write and solve a proportion. Plan width----> 1 inch = 6.5 inches<---plan width Actual width--> 4 feet x feet <-----actual width 1x = 46.5 cross products x= 26 The actual flower bed width is 26 feet. From the last example, what is the scale factor? To find the scale factor, write the ratio of 1 inch to 4 feet in simplest form. 1inch = 1 inch Convert 4 feet 4 feet 48 inches to inches The scale factor is 1 . That is , each 48 measurement on the plan is 1 the actual measurement. 48 Example 2: Determine the Scale In a scale model of a roller coaster, the highest hill has a height of 6 inches. If the actual height of the hill is 210 feet, what is the scale of the model? Model height---> 6 inches = 1 inch <--model height Actual height--->210 feet x feet <--actual height 6x = 210 6x = 210 x= 35 6 6 So, the scale is 1” = 35 feet Your Turn! On a set of architectural drawings for an office building, the scale is 1/2” = 3 feet. Find the actual length of each room. .5” = 2” Lobby: 2 inches 3ft x ft .5x = 6 x = 12 The actual length of the lobby is 12 ft Cafeteria: 8.25 inches .5” = 8,25” 3ft x ft .5x = 24.75 x = 49.5 The actual length of the cafeteria is 49.5 feet Practice Problem • A mural of a dog was painted on a wall. The enlarged dog was 45 ft. tall. If the average height for this breed of dog is 3 ft., what is the scale factor of this enlargement? Can you express this scale in more than one way? • The scale factor is 45:3. This can be simplified to 15:1 or expressed in other ways, such as 7.5:0.5 Make a scale map of your desk Place three or four objects on their desk. Orient the objects parallel to the edges of the desk. Use a 1:10 scale, with 1 cm on the map representing 10 cm of the desk top. To help students appreciate what the scale is doing and how the numbers are used in calculating, the teacher may give students a 10 cm × 25 cm rectangle of paper to be one of the objects on the desk. This gives students one object for which it is easy to work out what the scaled version is; they may be able to generalize this to their other objects with more awkward dimensions. A second map using a different scale could then be produced, perhaps 2 cm = 5 cm (which is 1:2.5). Graph paper may help students in drawing their maps.