Unit 2, Lesson 8
Projecting and Scaling
Accelerated 7
8.2
Activity
Rectangles are made by cutting an -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.
An image of an 8 point 5 by 11 inch rectangle. A dashed line divides the entire rectangle in half and one rectangle is labeled B. A dashed line divides the other half of B and one rectangle is labeled C. A dashed line divides the other half of C and one rectangle is labeled D. A dashed line divides the other half of D and both rectangles are labeled E.
- Some of the rectangles are scaled copies of Rectangle A (the full sheet of paper). Record the measurements of those rectangles in this table.
rectangle length of short side (inches) length of long side (inches) A 11 - Some of the rectangles are not scaled copies of Rectangle A (the full sheet of paper). Record the measurements of those rectangles in this table.
rectangle length of short side (inches) length of long side (inches) - Look at the measurements for the rectangles that are scaled copies of the full sheet of paper. What do you notice?
- Look at the measurements for the rectangles that are not scaled copies of the full sheet. What do you notice?
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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?
- 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? What do you notice?
8.3
Activity
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.
- Is B a scaled copy of R? If so, what is the scale factor?
- Is C a scaled copy of B? If so, what is the scale factor?
- Is C a scaled copy of R? If so, what is the scale factor?
Student Lesson Summary
In this diagram, the larger rectangle is a scaled copy of the smaller one, and the scale factor is because and . Scaled copies of rectangles have another interesting property: the diagonal of the large rectangle contains the diagonal of the smaller rectangle. This is the case for any two scaled copies of a rectangle if we line them up as shown. If two rectangles are not scaled copies of one another, then their diagonals would not match up.
A small rectangle inside a larger rectangle with same lower left point and a dashed diagonal line from that lower left point through both upper points. The small rectangle has length 4 and height 2. The large rectangle has length 6 and height 3.
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