Quantitative Reasoning A 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 MTE 494 Arizona State University 1 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 2 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 1 Understanding the quantities embedded in the situation and 2 Understanding how these quantities are related to each MTE 494 3 other Arizona State University 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 4 Arizona State University Temporal dimensions of these relationships MTE 494 Arizona State University 5 This situation 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 MTE 494 Arizona State University 7 Reasoning about quantities and solving by reasoning We also know about DB s before and after weights 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 8 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 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 lbs MTE 494 Arizona State University 9
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