Unit 1 Scale Drawings — Unit Plan

Title	Assessment
Lesson 1 What Are Scaled Copies?	Scaling L Are any of the figures B, C, or D scaled copies of figure A? Explain how you know. Show Solution Only figure C is a scaled copy of figure A. Sample reasoning: In figure C, the length of each segment of the letter L is twice the length of the matching segment in A. Figures B and D are not enlarged evenly. In B, some segments increase and some stay the same. In D, some segments are double in length and some are not.
Lesson 2 Corresponding Parts and Scale Factors	Comparing Polygons $ABCD$ and $PQRS$ Polygon $PQRS$ is a scaled copy of polygon $ABCD$ . Segment lengths are given in units. A, B is vertical 1 down. B, C is a diagonal down 1, right 1. C, D is a diagonal right 1, up 2. D, C is horizontal left 2. P, Q is vertical 1.5 down. Q, R is a diagonal down 1.5, right 1.5. R, S is a diagonal right 1.5, up 3. S, P is horizontal left 3. Name the angle in the scaled copy that corresponds to angle $ABC$ . Name the segment in the scaled copy that corresponds to segment $AD$ . What is the scale factor from polygon $ABCD$ to polygon $PQRS$ ? Show Solution Angle $PQR$ corresponds to angle $ABC$ . Segment $PS$ corresponds to segment $AD$ . The scale factor is $\frac{3}{2}$ because $PS = 3$ and $AD = 2$ .
Lesson 3 Making Scaled Copies	More Scaled Copies Create a scaled copy of $ABCD$ using a scale factor of 4. Triangle Z is a scaled copy of Triangle M. M Select all the sets of values that could be the side lengths of Triangle Z. 8, 11, and 14. 10, 17.5, and 25. 6, 9, and 11. 6, 10.5, and 15. 8, 14, and 20. Show Solution B, D, E
Lesson 4 Scaled Relationships	Corresponding Polygons Here are two polygons on a grid. Two five-sided polygons on a grid. The polygon on the left is labeled ABCDE. The vertices from A going counterclockwise are as follows. Vertex B is 1 unit to the left and 3 units down. Vertex C is 1 unit down. Vertex D is 2 units to the right. Vertex E is 1 unit up. The polygon on the right is labeled PQRTS. The vertices from P going counterclockwise are as follows. Vertex Q is 2 units to the left and 5 units down. Vertex R is 1 unit down. Vertex S is 4 units right. Vertex T is 1 unit up. 1 unit=1 square on the grid. Is $PQRST$ a scaled copy of $ABCDE$ ? Explain your reasoning. Show Solution No. Sample reasoning: $PQRST$ is not a scaled copy of $ABCDE$ because we need to use different scale factors when comparing corresponding lengths (1 for corresponding segments $BC$ and $QR$ and 2 for corresponding segments $CD$ and $RS$ ). Also, not all of their corresponding angles are the same size. Angle $A$ and angle $P$ are not the same size.
Lesson 5 The Size of the Scale Factor	Scaling a Rectangle A rectangle that is 2 inches by 3 inches will be scaled by a factor of 7. What will the side lengths of the scaled copy be? Suppose you want to scale the copy back to its original size. What scale factor should you use? Show Solution 14 inches by 21 inches, because $2 \boldcdot 7 = 14$ and $3 \boldcdot 7 = 21$ . $\frac 17$ , because it is the reciprocal of 7.
Lesson 6 Scaling and Area	Enlarged Areas Lin has a drawing with an area of 20 in². If she increases all the sides by a scale factor of 4, what will the new area be? Noah enlarged a photograph by a scale factor of 6. The area of the enlarged photo is how many times as large as the area of the original? Show Solution 320 in², Sample responses: $20 \boldcdot 4^2 = 320$ If the rectangle is 4 inches by 5 inches, the scaled copy will be $4 \boldcdot 4$ inches by $4\boldcdot 5$ inches and $(4 \boldcdot 4) \boldcdot (4\boldcdot 5) = 16 \boldcdot 20 = 320$ . If the rectangle is 2 inches by 10 inches, the scaled copy will be $4 \boldcdot 2$ inches by $4 \boldcdot 10$ inches and $(4\boldcdot 2) \boldcdot (4\boldcdot 10) = 8 \boldcdot 40 = 320$ . 36 times as large, because $6^2 = 36$ .
Section A Check Section A Checkpoint	Problem 1 Select all the pairs of figures that are scaled copies of each other. (Note: All measurements are rounded to the nearest whole number.) A. B. C. D. E. Show Solution D, E Problem 2 Create a scaled copy of the triangle using a scale factor of 2. Show Solution

Lesson 7 Scale Drawings	Length of a Bus and Width of a Lake A scale drawing of a school bus has a scale of $\frac12$ inch to 5 feet. If the length of the school bus is $4\frac12$ inches on the scale drawing, what is the actual length of the bus? Explain or show your reasoning. A scale drawing of a lake has a scale of 1 cm to 80 m. If the actual width of the lake is 1,000 m, what is the width of the lake on the scale drawing? Show Solution 45 ft. Sample reasoning: There are 9 groups of $\frac12$ in $4\frac12$ . If $\frac12$ inch represents 5 feet, then $4\frac12$ inches represents $9 \boldcdot 5$ or 45 feet. 12.5 cm. Sample reasoning: Since every 80 m is represented by 1 cm, 1,000 m is represented by $1, 000 \div 80$ or 12.5 cm.
Lesson 9 Creating Scale Drawings	Drawing a Pool A rectangular swimming pool measures 50 meters in length and 25 meters in width. Make a scale drawing of the swimming pool where 1 centimeter represents 5 meters. What are the length and width of your scale drawing? Show Solution
Lesson 10 Changing Scales in Scale Drawings	Window Frame Here is a scale drawing of a window frame that uses a scale of 1 cm to 6 inches. Create another scale drawing of the window frame that uses a scale of 1 cm to 12 inches. Show Solution Scaled copy of the drawing where each length is half as long as in the original.
Lesson 11 Scales without Units	Scaled Courtyard Drawings Andre drew a plan of a courtyard at a scale of 1 to 60. On his drawing, one side of the courtyard is 2.75 inches. What is the actual measurement of that side of the courtyard? Express your answer in inches and then in feet. If Andre made another courtyard scale drawing at a scale of 1 to 12, would this drawing be smaller or larger than the first drawing? Explain your reasoning. Show Solution 165 in, which is 13.75 ft. Sample reasoning: $2.75 \boldcdot 60=165$ . $165 \div 12 = 13.75$ . It would be larger. Sample reasoning: A scale of 1 to 12 means the length on paper is $\frac{1}{12}$ of the original length, so the drawing would be larger than one drawn at $\frac{1}{60}$ the original length.
Lesson 12 Units in Scale Drawings	Drawing the Backyard Lin and her brother each created a scale drawing of their backyard, but at different scales. Lin used a scale of 1 inch to 1 foot. Her brother used a scale of 1 inch to 1 yard. Express the scales for the drawings without units. Whose drawing is larger? How many times as large is it? Explain or show your reasoning. Show Solution Lin’s scale of 1 inch to 1 foot can be written as 1 to 12. Her brother’s scale of 1 inch to 1 yard can be written as 1 to 36. Lin’s drawing is larger. Sample reasonings: The lengths on Lin's plan are 3 times the corresponding lengths on her brother's drawing. The area of Lin's drawing is 9 times the area of her brother's drawing. Since 1 yard equals 3 feet, the scale of Lin’s brother’s drawing is equivalent to 1 inch to 3 feet. Each inch on his drawing represents a longer distance than on Lin’s drawing, so his drawing will require less space on paper. At 1 inch to 1 foot, Lin’s drawing will show $\frac{1}{12}$ of the actual the distances. At 1 inch to 1 yard, or 1 inch to 3 feet, her brother’s drawing will show $\frac{1}{36}$ of the actual distances. Since $\frac{1}{12}$ is larger than $\frac{1}{36}$ , Lin's drawing will be larger.
Section B Check Section B Checkpoint	Problem 1 Here is a map of Yellowstone National Park. Use the map to answer the questions. About how long is the south border of the actual park? Estimate the actual area of the park. Explain your reasoning. A different map of Yellowstone National Park uses the scale 1 inch to 4 miles. How long is the south border of the park on that map? Show Solution about 50 mi about 3,000 sq mi. Sample reasoning: The park is roughly a rectangle. The height of the park is about 60 mi and the width is about 50 mi. $50 \boldcdot 60 = 3,000$ Sample response: 12.5 inches (if 50 mi is used as the actual length) Note: The answer here should be $\frac14$ of whatever value students have for the actual length of the south border of the park.

Lesson 13 Draw It to Scale	No cool-down
Unit 1 Assessment End-of-Unit Assessment	Problem 1 Quadrilateral $EFGH$ is a scaled copy of quadrilateral $ABCD$ . Select all of the true statements. Quadrilateral EFGH is a scaled copy of quadrilateral ABCD. AB = 3, BC = 4, CD = 6, and DA = 8. In EFGH, EF corresponds to AB, FG corresponds to BC, GH corresponds to CD, and HE corresponds to DA. GH = 12. A. Segment $EF$ is twice as long as segment $AB$ . B. Segment $CD$ is twice as long as segment $FG$ . C. The measure of angle $HEF$ is twice the measure of angle $DAB$ . D. The length of segment $EH$ is 16 units. E. The area of $EFGH$ is twice the area of $ABCD$ . Show Solution A, D Problem 2 Rectangle A measures 9 inches by 3 inches. Rectangle B is a scaled copy of Rectangle A. Select all of the measurement pairs that could be the dimensions of Rectangle B. A. 4.5 inches by 1.5 inches B. 8 inches by 2 inches C. 10 inches by 4 inches D. 13.5 inches by 4.5 inches E. 90 inches by 30 inches Show Solution A, D, E Problem 3 A scale drawing of a rectangular park is 5 inches wide and 7 inches long. The actual park is 280 yards long. What is the area of the actual park, in square yards? A. 35 B. 200 C. 1,400 D. 56,000 Show Solution 56,000 Problem 4 Here is a polygon. Draw a scaled copy of the polygon using a scale factor of $\frac 1 2$ . A polygon drawn on a grid. Horizontal base 1, the top of the polygon, is 8 units. Vertical side 1, the left side of the polygon, is 6 units. Vertical side 2, the right side of the polygon, is 4 units. Horizontal base 2, the bottom of the polygon, from left to right is over 4 horizontally, up 2 vertically, and over 4 horizontally. Show Solution Minimal Tier 1 response: Work is complete and correct. Sample: See above. Acceptable errors: Figure is somehow in a different orientation; figure overlaps original. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Sample errors: Correct drawing with scale factor 2 instead of scale factor $\frac 1 2$ ; minor error in determining dimensions of figure, such as a pair of segments 1 unit longer than they should be. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: Drawing shows lack of use of any scale factor; multiple errors in determining dimensions; incorrect attempt at drawing with scale factor 2. Problem 5 The scale of a map says that 4 cm represents 5 km. What distance on the map represents an actual distance of 10 km? What actual distance is represented by 10 cm on the map? Show Solution 8 cm (or equivalent) 12.5 km (or equivalent) Problem 6 Tyler has two different maps of Ohio. On the first map, the scale is 1 cm to 10 km. The distance from Cleveland to Cincinnati is 40 cm. On the second map, the scale is 1 cm to 50 km. What is the distance from Cleveland to Cincinnati on the second map? Explain your reasoning. Show Solution The distance is 8 cm. Sample explanations: On the 1 cm : 50 km scale map, each centimeter represents 5 times as much actual distance as on the 1 cm: 10 km map. That means that on the 1 cm : 50 km map the distance from Cleveland to Cincinnati will be one fifth as much, 8 cm. The actual distance from Cleveland to Cincinnati is 400 km, because $40 \boldcdot 10 = 400$ . The distance on the second map that represents 400 km is 8 cm, because $400 \div 50 = 8$ . Minimal Tier 1 response: Work is complete and correct. Sample: Lengths on the second map are five times smaller because 1 cm represents 50 km instead of 10 km. Divide 40 cm by 5 to get 8 cm. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Sample errors: Multiplication or division errors in otherwise correct work; work involves a correct substantive intermediate step (such as the actual distance from Cleveland to Cincinnati) but goes wrong after that; one mistake involving an “upside down” scale factor (or multiplying when division is called for); a correct answer without explanation. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: Work does not involve proportional reasoning; an incorrect answer without explanation, even if close; multiple mistakes that involve inversion of scale factors. Problem 7 Elena's bedroom is a rectangle with length 5 meters and width 3 meters. Make a scale drawing of Elena’s bedroom, using a scale of 1 to 50. Be sure to label the dimensions of your drawing. Elena’s bedroom door is 0.8 meters wide. How wide should the door be on the scale drawing? Explain how you know. Elena’s bed measures 4 centimeters by 3 centimeters on the scale drawing. What are the actual measurements of her bed? Show Solution A rectangle labeled with length 10 cm and width 6 cm (or equivalent units, such as 0.1 m and 0.06 m) 1.6 cm (or equivalent units). Sample reasoning: 0.8 m is 80 cm. At 1 to 50, the width of the door is 1.6 cm, because $80 \div 50 = 1.6$ . 200 cm by 150 cm (or equivalent units). Actual measurements are 50 times as long as the corresponding measurements on the drawing. Minimal Tier 1 response: Work is complete and correct, with complete explanation or justification. Sample: A rectangle, labeled 10 cm and 6 cm, with the larger side labeled 10 cm. 1.6 cm. Because the scale is 1 to 50, the door’s 80 cm width becomes $\frac{80}{50}$ cm in the scale drawing. 2 m by 1.5 m Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. Sample errors: One calculation or conversion error; clear error in relative shape of rectangle; incomplete explanation of 1.6 cm calculation; describing instead of building and labeling the scale drawing. Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors. Sample errors: Two or more error types from Tier 2 response; multiple calculation and conversion errors; scaling in wrong direction (multiplying or dividing when inappropriate); using incorrect scale factor. Tier 4 response: Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery. Sample errors: Two or more error types from Tier 3 response; adding or subtracting when working with scale factor; misunderstanding of the meaning and use of “1 to 50.”