1. Quantitative Reasoning A quantity is anythingan object, event, or quality thereofthan 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. 1 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. 2 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 MTE 494 Arizona State University 3 Understanding a problem situation quantitatively means: 1.Understanding the quantities embedded in the situation, and 2.Understanding how these quantities are related to each other

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? MTE 494 Arizona State University 4 Some relevant quantities embedded within this scenario: Dieter As weight before the diet; Dieter As weight after the diet Dieter Bs weight before the diet; Dieter Bs 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

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

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

7. MTE 494 Arizona State University 7 Reasoning about quantities and solving-by-reasoning So DBs after weight is [(7 x 19) + 2] lbs and that is 5/6 as much as his before weight. This means that DBs 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.