Find the value of each expression mentally. 

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.

1. 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

2. 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)

3. Look at the measurements for the rectangles that are scaled copies of the full sheet of paper. What do you notice?
4. Look at the measurements for the rectangles that are not scaled copies of the full sheet. What do you notice?

5. 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?

   

6. 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?

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?