Lesson 7.5 Scale Drawings 5-7 Scale Drawings and Scale Models Course 2 5-7 Scale Drawings and Scale Models Lesson 7.5 Scale Drawings Students will be able to understand ratios and proportions in scale drawings. Students will be able to use ratios and proportions with scale.

Homework (4/17/17)  Worksheet: Angle Sum of Triangles and Quadrilateral (front and back) (4/18/17)  Complementary and Supplementary Worksheet 2 (4/19/17)  Chapter 7 Practice Test (4/20/17)  Study Guide, Ch 7 hw packet

Scale Drawings and Scale Models Course 2 5-7 Scale Drawings and Scale Models Write each fraction in the simplest form. 4 48 1 12 9 135 1 15 2. 1. Convert the following measurements. 3. 192 inches = feet 16 4. 18.5 feet = inches 222 5. 324 inches = feet 27

Ratio A ratio is a comparison of two quantities

Scale A scale is a ratio between two sets of measurements. Examples: Drawings: ¼ inch = 1 foot Maps: 1 inch = 250 miles

Scale Drawing (Model) A scale drawing (model) is a drawing that uses a scale to make an object smaller than (reduction) or larger than (enlargement) the real object.

Scale Factor A scale factor is a ratio used to enlarge or reduce similar figures. Examples: enlarging a piece of candy for a drawing…or the Willy Wonka Factory…

The Scale Factor The scale factor is the amount that you enlarge or reduce an object by. Enlarge: A scale factor that is larger than 1 will make the shape get bigger. Reduce: A scale factor that is smaller than 1 but larger than 0 will make the shape get smaller. Explain that in scale drawings objects are usually scaled down so that they fit on the page. Occasionally a small object can be scaled up as shown in this example. To find the actual size of the coin we have to divide by 12.2 cm by 0.5 to give 24.4 mm. Remember, the shape of the object does not change, only its size!

The lengths and widths of objects of a scale drawing or model are proportional to the lengths and widths of the actual object.

Example 1: Finding a Scale Factor Course 2 5-7 Scale Drawings and Scale Models Example 1: Finding a Scale Factor Identify the scale factor. Room Blueprint Length (in.) 144 18 Width (in.) 108 13.5 blueprint length room length 18 144 Write a ratio using one of the dimensions. = 1 8 = Simplify. 1 8 The scale factor is .

Insert Lesson Title Here Course 2 5-7 Scale Drawings and Scale Models Insert Lesson Title Here Try This: Example 1 Identify the scale factor. Model Aircraft Blueprint Length (in.) 12 2 Wing span (in.) 18 3 blueprint length aircraft length 2 12 = Write a ratio using one of the dimensions. 1 6 = Simplify. The scale factor is . 1 6

Additional Example 2: Using Scale Factors to Find Unknown Lengths Course 2 5-7 Scale Drawings and Scale Models Additional Example 2: Using Scale Factors to Find Unknown Lengths A photograph was enlarged and made into a poster. The poster is 20.5 inches by 36 inches. The scale factor is . Find the size of the photograph. 5 1 poster photo 5 1 Think: = 36 L 5 1 = Write a proportion to find the length L. 5L = 36 Find the cross products. L = 7.2 Divide.

Additional Example 2 Continued Course 2 5-7 Scale Drawings and Scale Models Additional Example 2 Continued A photograph was enlarged and made into a poster. The poster is 20.5 inches by 36 inches. The scale factor is . Find the size of the photograph. 5 1 poster photo 5 1 Think: = 20.5 w 5 1 = Write a proportion to find the width w. 5w = 20.5 Find the cross products. w = 4.1 Divide. The photo is 7.2 in. long and 4.1 in. wide.

Scale Drawings and Scale Models Course 2 5-7 Scale Drawings and Scale Models Try This: Example 2 Mary’s father made her a dollhouse which was modeled after the blueprint of their home. The blueprint is 24 inches by 45 inches. The scale factor is . Find the size of the dollhouse. 1.5 1 dollhouse blueprint 1.5 1 Think: = L 45 = 1.5 1 Write a proportion to find the length l. Find the cross products. L = 45 · 1.5 L = 67.5 Multiply.

Scale Drawings and Scale Models Course 2 5-7 Scale Drawings and Scale Models Try This 2 Continued Mary’s father made her a dollhouse which was modeled after the blueprint of their home. The blueprint is 24 inches by 45 inches. The scale factor is . Find the size of the dollhouse. 1.5 1 dollhouse blueprint 1.5 1 Think: = w 24 1.5 1 = Write a proportion to find the width w. w = 24 · 1.5 Find the cross products. w = 36 Multiply. The dollhouse is 67.5 inches long and 36 inches wide.

Additional Example 3: Measurement Application Course 2 5-7 Scale Drawings and Scale Models Additional Example 3: Measurement Application On a road map, the distance between Pittsburgh and Philadelphia is 7.5 inches. What is the actual distance between the cities if the map scale is 1.5 inches = 60 miles? Let d be the actual distance between the cities. 1.5 60 7.5 d = Write a proportion. 1.5 · d = 60 · 7.5 Find the cross products. 1.5d = 450 Multiply. 1.5d 1.5 = 450 1.5 Divide. d = 300 The distance between the cities is 300 miles.

Insert Lesson Title Here Course 2 5-7 Scale Drawings and Scale Models Insert Lesson Title Here Try This: Example 3 On a road map, the distance between Dallas and Houston is 7 inches. What is the actual distance between the cities if the map scale is 1 inch = 50 kilometers? Let d be the actual distance between the cities. 1 50 7 d = Write a proportion. 1 · d = 50 · 7 Find the cross products. 1d = 350 Multiply. d = 350 The distance between the cities is 350 kilometers.

Scale Drawings and Scale Models Insert Lesson Title Here Course 2 5-7 Scale Drawings and Scale Models Insert Lesson Title Here Identify the scale factor. 1. Statue of Liberty Model Height (in.) 1,824 8 1 228 2. On a scale drawing, a kitchen wall is 6 inches long. The scale factor is . What is the length of the actual wall? 1 24 144 inches, or 12 feet

Scale: 3𝑐𝑚 90𝑘𝑚 3𝑐𝑚 90𝑘𝑚 = 11𝑐𝑚 𝑥 330 km 5-7 Course 2 5-7 Scale Drawings and Scale Models Insert Lesson Title Here 3. On a road map, the distance from Green Bay to Chicago is 11 cm. What is the actual distance between the cities if the map scale is 3 cm = 90 km? Scale: 3𝑐𝑚 90𝑘𝑚 3𝑐𝑚 90𝑘𝑚 = 11𝑐𝑚 𝑥 330 km

A set of landscape plans shows a flower bed that is 6. 5 inches wide 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? What is the scale factor? 1𝑖𝑛 4𝑓𝑡 = 6.5 𝑖𝑛 𝑥 26 feet 1in/4feet

What is the scale factor? The central chamber of the Lincoln memorial, which features a marble statue of Abraham Lincoln, has a height of 60 feet. Suppose a scale model of the chamber has a height of 4 inches. What is the scale of the model? Write a ratio of the height of the model to the actual height of the statue? What is the scale factor? 60ft 4in 15ft 1in 1 inch = 15 feet 1in/15feet

Antonio is designing a room that is 20 feet long and 12 feet wide Antonio is designing a room that is 20 feet long and 12 feet wide. Make a scale drawing of the room. Use the scale 0.25 inches = 4 feet. STEP 1: Find the room’s length on the drawing (let x = length) STEP 2: Find the room’s width on the drawing (let w = width) 20 x 4 .25 = Length = 1.25 inches 12 x 4 .25 Width = .75 inches =

Find the missing lengths The second picture is an enlargement of the first picture. What are the missing lengths? 11.2 cm 5.6 cm Pupils should notice that the second picture is twice the size of the first picture (because 11.2 cm is double 5.6 cm) and use this to find the missing lengths. Link: N8 Ratio and proportion – scale factors 6.7 cm ? 6.7 cm 13.4 cm 2.9 cm 5.8 cm 5.8 cm ?

Find the missing lengths The second shape is an enlargement of the first shape. What are the missing lengths? 6 cm 4 cm 4 cm ? 4.5 cm ? 4.5 cm 3 cm 9 cm 6 cm The second shape is 11/2 times bigger than the first shape (because 9 cm is 11/2 × 6 cm). Multiplying the lengths in the first shape by 11/2 will give the lengths in the second shape, whereas dividing the lengths in the second shape by 11/2 (or multiplying by 2/3) will give the lengths in the first shape. Link: N8 Ratio and proportion – scale factors 5 cm ? 5 cm 7.5 cm

Find the missing lengths The second cuboid is an enlargement of the first. What are the missing lengths? 10.5 cm 3.5 cm 3.5cm ? Pupils should notice that the lengths in the second cuboid are 3 × the lengths in the first cuboid (because 5.4 cm is 3 × 1.8 cm) and use this to find the missing lengths. Ask pupils if the volume of the second cuboid is 3 × the volume of the first cuboid. The volume of the first cuboid is 7.56 cm3 and the volume of the second cuboid is 204.12 cm3. Allow pupils to use calculators to verify that the volume is, in fact, 27 × more (in other words 33 × more). If there is time, pupils could investigate the relationship between the enlargement of the lengths and the enlargement of the volume for other cuboids or, in two dimensions, the relationship between the enlargement of the lengths and the enlargement of the area for rectangles. Link: N8 Ratio and proportion – scale factors S8 Perimeter, area and volume – area and volume 1.2 cm 1.8 cm 3.6 cm ? 3.6cm 5.4 cm

Enlargement A’ A Shape A’ is an enlargement of shape A. The length of each side in shape A’ is 2 × the length of each side in shape A. Stress the every side in the shape has to be 2 × bigger to enlarge the shape by a scale factor of 2. Ask pupils to tell you the difference between the angles in the first shape (the object) and the angles in the second (the image). Tell pupils that when we enlarge a shape the lengths change but the angles do not. The original shape and its image are not congruent but they are similar. We say that shape A has been enlarged by scale factor 2.

A’B’ AB B’C’ BC A’C’ AC = = = the scale factor 6 4 12 8 9 6 = = = 1.5 6 cm 4 cm 6 cm 9 cm B B’ C 8 cm 12 cm C’ Remind pupils that we can write ratios as fractions as well as using the ratio notation. The notation that we use depends on the context of the problem. AC : A’C’ is the same ratio as AC/A’C’, written in a different way. The result means that the ratio of any two corresponding lengths in the object and the image can be used to find the scale factor. Click to reveal actual lengths on the diagram and reveal how these can be used to find the scale factor. A’B’ AB B’C’ BC A’C’ AC = = = the scale factor 6 4 12 8 9 6 = = = 1.5

Find the scale factor What is the scale factor for the following enlargements? B’ B Deduce that the scale factor for the enlargement is 3 by counting squares. Show that that the ratios of any of the corresponding lengths on the image and in the object are equal to the scale factor. Scale factor 3

Find the scale factor What is the scale factor for the following enlargements? C’ C Deduce that the scale factor for the enlargement is 2 by counting squares. Scale factor 2

Find the scale factor What is the scale factor for the following enlargements? D’ Deduce that the scale factor for the enlargement is 3.5 by counting squares. Point out that it doesn’t matter which lengths we compare. They are all enlarged by the same scale factor. For example, in the second shape it is difficult to determine the length of the sides. Instead we can compare the widths of the shapes. The width of the first shape is 2 units and the width of the second shape is 7 units. 7 ÷ 2 gives us the scale factor 3.5. D Scale factor 3.5

Find the scale factor What is the scale factor for the following enlargements? E E’ The lengths in the second shape are ½ the size of the lengths in the first shape and so the scale factor is 0.5 or ½. Point out that this is still called an enlargement even though the shape has been made smaller. Scale factor 0.5