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Quantitative Reasoning

Quantitative Reasoning. A quantity is anything—an object, event, or quality thereof—than can be measured or counted. . A value of a quantity is its measure or the number of items that are counted . A value of a quantity involves a number and a unit of measure. .

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Quantitative Reasoning

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  1. Quantitative Reasoning Aquantity is anything—an object, event, or quality thereof—than can be measured or counted. • Avalue of a quantity is its measure or the number of items that are counted. A value of a quantity involves a number and a unit of measure. MTE 494 Arizona State University

  2. An important distinction: A quantity is not the same thing as a number or a value of the quantity One can think of a quantity without knowing its value. For example: the amount of snowfall on a given day is a quantity, regardless of whether someone actually measured this amount. One can think/speak about the amount of snowfall without knowing a value of this amount. MTE 494 Arizona State University

  3. Quantitative Analysis • Analyzing problem situations is key to be a skilled problem solver • Quantitative analyses of problem situations should be a first step toward helping students develop a deep understanding of such situations Understanding a problem situation quantitatively means: • Understanding the quantities embedded in the situation, and • Understandinghow these quantities are related to each other MTE 494 Arizona State University

  4. Example: Two dieters were overheard having the following conversation at a Weight Watchers meeting: • Dieter A: “I lost 1/8 of my weight. I lost 19 lbs.” • Dieter B: “I lost 1/6 of my weight, and now you weigh 2 pounds less than I do.” • How much weight did Dieter B lose? Some relevant quantities embedded within this scenario: • Dieter A’s weight before the diet; Dieter A’s weight after the diet • Dieter B’s weight before the diet; Dieter B’s weight after the diet • The amount of weight lost by Dieter A; The amount of weight lost by Dieter B • The difference in their weights before the diets; The difference in their weights after the diets MTE 494 Arizona State University

  5. This scenario can be seen as having a quantitative structure depicted below: MTE 494 Arizona State University

  6. Reasoning about quantities and solving-by-reasoning • We want to know how much weight Dieter B (DB) lost—it is the difference between his before-and-after diet weights. • We know about DA’s before and after weights: • DA losing 1/8 of his weight means that his after weight must be 7/8 as much as his before weight. • We also know that DA lost 19 lbs, which is the amount equal to 1/8 of his before weight. Since 7/8 of his weight is 7 times as much as 1/8 of it, DA’s after weight must equal (7 x 19) lbs • We also know about Dieter B’s (DB) weight loss: • DB losing 1/6 of his weight means that his after weight is 5/6 as much as his before weight • DA’s after weight being 2 lbs less than DB’s after weight means that DB’s after weight must be 2 lbs more than DA’s after weight, or [(7 x 19) + 2] lbs MTE 494 Arizona State University

  7. Reasoning about quantities and solving-by-reasoning • So DB’s after weight is [(7 x 19) + 2] lbs and that is 5/6 as much as his before weight. • This means that DB’s after weight is 5 times as much as 1/6 of his before weight. • So it must be that 1/6 of his before weight is 1/5 as much as his after weight (using our meaning of fractions). • That is, DB lost (1/5) x [(7 x 19) + 2] lbs = 27 lbs. MTE 494 Arizona State University

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