Exploring Representation
 Try It Yourself: Typical Week | Equations Related to Data | Problem Reflection | Your Journal

 Question: How does this task encourage the use of representations? Show Answer
 Our Answer: The amount of time spent on various activities was first represented by numbers of hours. These numbers could easily be compared by subtraction to make observations, such as, "2.5 more hours were spent on entertainment than on chores." Fractions and percents could also be used to represent relationships. The percents, however, always indicated the amount of hours relative to the whole (week), rather than an absolute number of hours.
 Question: How are part-whole graphs helpful as representations of percents (showing percents as parts of a region)? What are some advantages of using this kind of graph? Show Answer
 Our Answer: The fractions and percents can be shown as parts of a region, such as a long rectangular strip. This type of graph is a visual representation where the area for a category is a proportional part of the whole; in this case, it is based on the fraction of the week devoted to that category. These labeled part-whole graphs help show that a fraction (such as 20/120) and a percent (17%) represent the same amount, to the nearest whole percent. One advantage of the part-whole graph is that the colored areas can be compared even if you don't know any of the numbers. The numeric ratio E:C, for example, would equal the ratio of the area of the E part to the C part.
 Question: How might this problem help students develop their understanding of circle graphs? How do they compare to the rectangles or strip graphs? Show Answer
 Our Answer: The hourly strip graph helps record the data but does not produce a representation that is easily analyzed or compared to other such data sets. The percentage strip graph is more comparable to circle graphs. Both are useful when all the data come from separate categories that represent all of the possibilities for one whole survey, population, or data set. In both percentage strip graphs and circle graphs, attention is drawn to the relative sizes of the different parts. The amounts, compared to the whole, may be represented by percents, fractions, or decimals.
 Question: How does the use of verbal statements and equations help students analyze the data? Show Answer
 Our Answer: Verbal statements and equations prompt the use of percents in meaningful statements. In the previous examples, once two parts of the data are compared as a ratio, the related percent can be found by dividing. These percents are then given clear meaning when we make statements about the data, such as, "I spend about 71.4% as much time on work as on sleep," based on the ratio W:S = 40:56.

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