Quantitative ReasoningAn important distinction:Quantitative AnalysisExample: Two dieters were overheard having the following conversation at a Weight Watchers meeting:Slide 5This situation can be seen as having a quantitative structure depicted below:Slide 7Slide 8Reasoning about quantities and solving-by-reasoningQuantitative ReasoningA quantity is an attribute of something (an object, event, etc) 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. 1MTE 494 Arizona State UniversityAn important distinction:A quantity is not the same thing as a number or a value of the quantityOne 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.2MTE 494 Arizona State UniversityQuantitative 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 situationsMTE 494 Arizona State University3Understanding a problem situation quantitatively means:1.Understanding the quantities embedded in the situation, and 2.Understanding how these quantities are related to each otherExample: 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? MTE 494 Arizona State University4Some 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 dietsMTE 494 Arizona State University5Temporal dimensions of these relationshipsThis situation can be seen as having a quantitative structure depicted below:MTE 494 Arizona State University6MTE 494 Arizona State University7Reasoning about quantities and solving-by-reasoningWe 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.MTE 494 Arizona State University8Reasoning about quantities and solving-by-reasoningDB losing 1/6 of his weight means that his after weight is 5/6 as much as his before weightDA’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 We also know about DB’s before and after weights:MTE 494 Arizona State University9Reasoning about quantities and solving-by-reasoningSo 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. Therefore it must be that 1/6 of DB’s 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
View Full Document