# More: In-Depth Discussion of the Reasoning Activities in "Teaching Fractions and Ratios for Understanding"

By Susan J. Lamon | Go to book overview

Chapter 13
Rates
Do this investigation.
 a. The following rectangle is divided into 1/4 -inch squares. Shade 2/5 of the small squares in the following figure. What is the ratio of shaded to unshaded squares?Click for a larger version b. Reunitize the same area using a chunk consisting of two 1/4 -inch squares; that is, in terms of 1/2 -inch by 1/4 -inch rectangles. (Hint: You can reunitize by tracing the rectangles with a red pen in the figure above. You should trace 10 small rectangles on the figure.) What is the ratio of shaded to unshaded rectangles? c. Reunitize the same area using a chunk consisting of 4 of the 1/4 -inch squares, that is, in 1/2 -inch squares. What is the ratio of shaded to unshaded squares? d. Based on your drawings in a, b, and c, identify three equivalent ratios. How do you know they are equivalent? e. Graph the ratios by putting number of shaded squares or rectangles on the vertical axis and unshaded on the horizontal axis. f. Determine the coordinates of other points on the line. g. Determine if these points lie on the graph: (3, 7), (13, 8.67), (3/4, 1 1/2), (6, 4), (96, 64). h. What is the constant associated with this equivalence class? i. What is the slope of the line? j. Using the same grid, make a graph showing comparisons that may be described by a 2:1 relationship. k. What is the constant associated with {2: 1} l. Looking at the graphs you have drawn, can you predict where the line for {2:4} would lie? m. How did you decide it location?

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