# Illustrative Mathematics Grade 8, Unit 2, Lesson 1: Projecting and Scaling

Learning Targets:

• I can decide if one rectangle is a dilation of another rectangle.
• I know how to use a center and a scale factor to describe a dilation.

Related Pages
Illustrative Math

#### Lesson 1: Projecting and Scaling

Let’s explore scaling.

Illustrative Math Unit 8.2, Lesson 1 (printable worksheets)

#### Lesson 1 Summary

The following diagram shows how to decide if one rectangle is a dilation of another rectangle. #### Lesson 1.1 Number Talk: Remembering Fraction Division

Find each quotient. Write your answer as a fraction or a mixed number.

#### Lesson 1.2 Sorting Rectangles

Rectangles were made by cutting an 8 1/2-inch by 11-inch piece of paper in half, in half again, and so on, as illustrated in the diagram. Find the lengths of each rectangle and enter them in the appropriate table.

1. Some of the rectangles are scaled copies of the full sheet of paper (Rectangle A). Enter the measurements of those rectangles in the table.
2. Some of the rectangles are not scaled copies of the full sheet of paper. Enter the measurements of those rectangles in the table.
3. Look at the measurements for the rectangles that are scaled copies of the full sheet of paper. What do you notice about the measurements of these rectangles? Look at the measurements for the rectangles that are not scaled copies of the full sheet. What do you notice about these measurements?
4. Stack the rectangles that are scaled copies of the full sheet so that they all line up at a corner, as shown in the diagram. Do the same with the other set of rectangles. On each stack, draw a line from the bottom left corner to the top right corner of the biggest rectangle. What do you notice?
5. Stack all of the rectangles from largest to smallest so that they all line up at a corner. Compare the lines that you drew. Can you tell, from the drawn lines, which set each rectangle came from?

#### Are you ready for more?

In many countries, the standard paper size is not 8.5 inches by 11 inches (called “letter” size), but instead 210 millimeters by 297 millimeters (called “A4” size). Are these two rectangle sizes scaled copies of one another?

8.5 inches = 215.9 mm
11 inches = 279.4 mm
We can divide to find whether the scale factors for the sides are the same.
215.9 ÷ 210 = 1.03
279.4 ÷ 297 = 0.94
Since the scale factors are not the same, the two rectangle sizes are not scaled copies of one another.

#### Lesson 1.3 Scaled Rectangles

Here is a picture of Rectangle R, which has been evenly divided into smaller rectangles. Two of the smaller rectangles are labeled B and C.

1. Is B a scaled copy of R? If so, what is the scale factor?
2. Is C a scaled copy of B? If so, what is the scale factor?
3. Is C a scaled copy of R? If so, what is the scale factor?

#### Lesson 1 Practice Problems

1. Rectangle A measures 12 cm by 3 cm. Rectangle B is a scaled copy of Rectangle A. Select all of the measurement pairs that could be the dimensions of Rectangle B.
A. 6 cm by 1.5 cm
B. 10 cm by 2 cm
C. 13 cm by 4 cm
D. 18 cm by 4.5 cm
E. 80 cm by 20 cm
2. Rectangle A has length 12 and width 8. Rectangle B has length 15 and width 10. Rectangle C has length 30 and width 15.
a. Is Rectangle A a scaled copy of Rectangle B? If so, what is the scale factor?
b. Is Rectangle B a scaled copy of Rectangle A? If so, what is the scale factor?
c. Explain how you know that Rectangle C is not a scaled copy of Rectangle B.
d. Is Rectangle A a scaled copy of Rectangle C? If so, what is the scale factor?
3. Here are three polygons.
a. Draw a scaled copy of Polygon A with scale factor 1/2.
b. Draw a scaled copy of Polygon B with scale factor 2.
c. Draw a scaled copy of Polygon C with scale factor 1/4.
4. Which of these sets of angle measures could be the three angles in a triangle?
5. In the picture lines AB and CD are parallel. Find the measures of the following angles. Explain your reasoning.

The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. 