Unit 1 Lesson 7 Scale Drawings Answer Key

Studnets work with scale drawings in grade 7 draws on earlier work with geometry and geometric measurement. Students began to learn about two- and three-dimensional shapes in kindergarten, and continued this work in grades 1 and 2, composing, decomposing, and identifying shapes. Students' work with geometric measurement began with length and continued with area. Students learned to "structure two-dimensional space," that is, to see a rectangle with whole-number side lengths as an array of unit squares, or rows or columns of unit squares. In grade 3, students distinguished between perimeter and area. They connected rectangle area with multiplication, understanding why (for whole-number side lengths) multiplying the side lengths of a rectangle yields the number of unit squares that tile the rectangle. They used area diagrams to represent instances of the distributive property. In grade 4, students applied area and perimeter formulas for rectangles to solve real-world and mathematical problems, and learned to use protractors. In grade 5, students extended the formula for the area of a rectangle to include rectangles with fractional side lengths. In grade 6, students built on their knowledge of geometry and geometric measurement to produce formulas for the areas of parallelograms and triangles, using these formulas to find surface areas of polyhedra.

In this unit, students study scaled copies of pictures and plane figures, then apply what they have learned to scale drawings, e.g., maps and floor plans. This provides geometric preparation for grade 7 work on proportional relationships as well as grade 8 work on dilations and similarity.

Students begin by looking at copies of a picture, some of which are to scale, and some of which are not. They use their own words to describe what differentiates scaled and non-scaled copies of a picture. As the unit progresses, students learn that all lengths in a scaled copy are multiplied by a scale factor and all angles stay the same. They draw scaled copies of figures. They learn that if the scale factor is greater than 1, the copy will be larger, and if the scale factor is less than 1, the copy will be smaller. They study how area changes in scaled copies of an image.

Next, students study scale drawings. They see that the principles and strategies that they used to reason about scaled copies of figures can be used with scale drawings. They interpret and draw maps and floor plans. They work with scales that involve units (e.g., "1 cm represents 10 km"), and scales that do not include units (e.g., "the scale is 1 to 100"). They learn to express scales with units as scales without units, and vice versa. They understand that actual lengths are products of a scale factor and corresponding lengths in the scale drawing, thus lengths in the drawing are the product of the actual lengths and the reciprocal of that scale factor. They study the relationship between regions and lengths in scale drawings. Throughout the unit, they discuss their mathematical ideas and respond to the ideas of others (MP3, MP6). In the culminating lesson of this unit, students make a floor plan of their classroom or some other room or space at their school. This is an opportunity for them to apply what they have learned in the unit to everyday life (MP4).

In the unit, several lesson plans suggest that each student have access to ageometry toolkit. Each toolkit contains tracing paper, graph paper, colored pencils, scissors, centimeter ruler, protractor (clear protractors with no holes that show radial lines are recommended), and an index card to use as a straightedge or to mark right angles. Providing students with these toolkits gives opportunities for students to develop abilities to select appropriate tools and use them strategically to solve problems (MP5). Note that even students in a digitally enhanced classroom should have access to such tools; apps and simulations should be considered additions to their toolkits, not replacements for physical tools.

Note that the study of scaled copies is limited to pairs of figures that have the same rotation and mirror orientation (i.e. that are not rotations or reflections of each other), because the unit focuses on scaling, scale factors, and scale drawings. In grade 8, students will extend their knowledge of scaled copies when they study translations, rotations, reflections, and dilations.

Lesson 1-6: Scaled Copies

Students will learn about scaling shapes. An image is ascaled copy of the original if the shape is stretched in a way that does not distort it. For example, here is an original picture and five copies. Pictures C and D are scaled copies of the original, but pictures A, B, and E are not.

In each scaled copy, the sides are a certain number of times as long as the corresponding sides in the original. We call this number thescale factor. The size of the scale factor affects the size of the copy. A scale factor greater than 1 makes a copy that is larger than the original. A scale factor less than 1 makes a copy that is smaller.

Here is a task to try with your student:
1. For each copy, tell whether it is a scaled copy of the original triangle. If so, what is the scale factor?
2. Draw another scaled copy of the original triangle using a different scale factor.
Solution:
1. 1. Copy 1 is a scaled copy of the original triangle. The scale factor is 2, because each side in Copy 1 is twice as long as the corresponding side in the original triangle. 5 ⋅ 2 = 10 , 4 ⋅ 2 = 8 , ( 6.4 ) ⋅ 2 = 12.8
  2. Copy 2 is a scaled copy of the original triangle. The scale factor is 1 2 or 0.5, because each side in Copy 2 is half as long as the corresponding side in the original triangle. 5 ⋅ ( 0.5 ) = 2.5 , 4 ⋅ ( 0.5 ) = 2 , ( 6.4 ) ⋅ ( 0.5 ) = 3.2
  3. Copy 3 is not a scaled copy of the original triangle. The shape has been distorted. The angles are different sizes and there is not one number we can multiply by each side length of the original triangle to get the corresponding side length in Copy 3.
2. Answers vary. Sample response: A right triangle with side lengths of 12, 15, and 19.2 units would be a scaled copy of the original triangle using a scale factor of 3.
Lesson 7-12: Scale Drawings

Students will be learning about scale drawings. Ascale drawingis a two-dimensional representation of an actual object or place. Maps and floor plans are some examples of scale drawings.

Thescale tells us what some length on the scale drawing represents in actual length. For example, a scale of "1 inch to 5 miles" means that 1 inch on the drawing represents 5 actual miles. If the drawing shows a road that is 2 inches long, we know the road is actually 2 ⋅ 5 , or 10 miles long.

Scales can be written with units (e.g. 1 inch to 5 miles), or without units (e.g., 1 to 50, or 1 to 400). When a scale does not have units, the same unit is used for distances on the scale drawing and actual distances. For example, a scale of "1 to 50" means 1 centimeter on the drawing represents 50 actual centimeters,1 inch represents 50 inches, etc.

Here is a task to try with your student:

Kiran drew a floor plan of his classroom using the scale 1 inch to 6 feet.
1. Kiran's drawing is 4 inches wide and 5 1 2 inches long. What are the dimensions of the actual classroom?
2. A table in the classroom is 3 feet wide and 6 feet long. What size should it be on the scale drawing?
3. Kiran wants to make a larger scale drawing of the same classroom. Which of these scales could he use?
  1. 1 to 50
  2. 1 to 72
  3. 1 to 100
Solution:
1. 24 feet wide and 33 feet long. Since each inch on the drawing represents 6 feet, we can multiply by 6 to find the actual measurements. The actual classroom is 24 feet wide because 4 ⋅ 6 = 24 . The classroom is 33 feet long because 5 1 2 ⋅ 6 = 5 ⋅ 6 + 1 2 ⋅ 6 = 30 + 3 = 33 .
2. 1 2 inch wide and 1 inch long. We can divide by 6 to find the measurements on the drawing. 6 ÷ 6 = 1 and 3 ÷ 6 = 1 2 .
3. A, 1 to 50. The scale "1 inch to 6 feet" is equivalent to the scale "1 to 72," because there are 72 inches in 6 feet. The scale "1 to 100" would make a scale drawing that is smaller than the scale "1 to 72," because each inch on the new drawing would represent more actual length. The scale "1 to 50" would make a scale drawing that is larger than the scale "1 to 72," because Kiran would need more inches on the drawing to represent the same actual length.