Problem: Shade 3/8 of the area of the rectangle.

As he constructed the diagram (see Devon's First Step), Devon explains: "First I broke the rectangle into four parts. There would be six columns in each half and then three columns in each quarter."

"Once I had four parts," he continued, "I had to split the fourths in half to get eighths (see Devon's Second Step). So 1½ columns would make an eighth." Devon concluded, "Then I shaded in three of the eighths" (see Devon's Third Step).

I then asked Devon what percent of the grid was shaded. He said that he wasn't sure. I asked him to think about what he already knew. I asked him what percent one half of the rectangle would be. He responded, "50 percent." I then asked what percent one-fourth of the rectangle would be. He responded, "25 percent." I then asked how this information could help solve the problem. Devon appeared to think about the question for a minute and responded, "Each of the shaded sections was 1/8 which was half of one fourth. So the percent must be half of 25 percent. So it is 12½ percent." I asked if that was the answer. Devon said that he had three pieces that were each 1/8 or 12½% which would be 12½ + 12½ + 12½ or 37½%. I then asked the class how we could write this as a decimal. A small chorus of voices responded with, "Point three seven five." (p. 21-22)

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This case is taken from Smith, M. S., Stein, M.K., & Silver,
E. A. (2005). *
Improving
instruction in rational numbers and proportionality: Using cases to
transform mathematics teaching and learning (Volume 1).* New York:
Teachers College Press.