Session 8, Part C:
Absolute and Relative Reasoning (30 minutes)

Rational numbers or fractions can be used in many different ways. One source of confusion, especially with fractions, is the difference between absolute and relative reasoning. In Part A, we used a rational number to compare a part to a whole. It's important to understand, however, that there is more than one way to make a comparison.

Here is a situation that you can think about numerically in at least two different ways: A baby and an adult both gain two pounds in one month.

 • You could think about the fact that each of them gained an equal amount of weight -- two pounds. • You could think about the fact that the baby's gain was greater, because the gain was a greater percentage of the baby's original weight than of the adult's original weight.

These are examples of two types of reasoning. The first uses absolute reasoning, which refers to a quantity by itself, without respect to its relation to other quantities (each gains two pounds, period). In contrast, the second uses relative reasoning, which compares that quantity to the originals to see how they relate to one another (the baby's gain is greater with respect to its original weight).

We can relate these two types of reasoning to operations. Absolute thinking is additive: Two boys each grew two inches last year. (Add two inches to their original heights.) In contrast, relative thinking is multiplicative -- the two inches might be 1/10 of the infant boy's prior height but only 1/24 of the first grader's prior height. Note 7

Problem C1

 a. Think about the meaning of the term "more." Make a list of several situations using the term "more." Which of these situations use absolute reasoning, and which use relative reasoning? b. Use the term "more" in four different problems, one for each of the four basic operations.

 Problems C2-C5 discuss ratio as a comparative index, requiring relative and multiplicative thinking. As you do these problems, think about the ways in which they use both relative and absolute reasoning. Note 8

 Problem C2 Which of these rectangles is most square: 75' by 114', 455' by 494', or 284' by 245'?

 Video Segment In this segment, Vicki and Nancy explore several different methods for solving the problem of which rectangle is more square. They settle for relative reasoning but then go on to explore yet another, more visual method. Watch this segment after you've completed Problem C2. Which method did you come up with to solve this problem? If you are using a VCR, you can find this segment on the session video approximately 17 minutes and 58 seconds after the Annenberg Media logo.

 Problem C3 Each carton below contains some white eggs and some brown eggs. Which has more brown eggs?

 Problem C4 Describe how you would decide which ski ramp is steeper, Ramp A or Ramp B:

 Problem C5 What kind of information is necessary to describe the "crowdedness" of an elevator?

 Problems C2-C5 adapted from Lamon, Susan J. Teaching Fractions and Ratios for Understanding: Essential Content Knowledge and Instructional Strategies for Teachers (pp. 17-19). © 1999 by Lawrence Erlbaum Associates. Used with permission. All rights reserved.

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 Session 8: Index | Notes | Solutions | Video