Learning Math: Number and Operations
Rational Numbers and Proportional Reasoning 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. See Note 7 below.
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. See Note 8 below.
Which of these rectangles is most square: 75′ by 114′, 455′ by 494′, or 284′ by 245′?
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?
You can find this segment on the session video approximately 17 minutes and 58 seconds after the Annenberg Media logo.
Each carton below contains some white eggs and some brown eggs. Which has more brown eggs?
Describe how you would decide which ski ramp is steeper, Ramp A or Ramp B:
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.
To learn more about absolute and relative reasoning, go to Learning Math: Patterns, Functions, and Algebra, Session 4.
The difference between absolute and relative reasoning is critical to the study of proportions. It is important to understand that any additive situation is absolute and cannot be a proportion. All proportions are relative and relate the change to the original; thus, they are multiplicative. For example, “Jill has three more brothers than Kim” is an absolute and additive relationship and is not a proportion. “Kari has twice as many brothers as Leanne” is a relative and multiplicative relationship and is a proportion.
Answers will vary. Here are some possibilities:
a. Joan has $100. Maria has $25 more than Joan does. This situation uses the word “more” in an absolute sense.
Kids who eat sugar have 25% more cavities than kids who don’t. This situation uses the word “more” in a relative sense.
b. Addition: Nina had six plants in her garden before she planted three more. How many does she have now?
Subtraction: Jen has 100 stickers, 20 more than Mike. How many does Mike have?
Multiplication: Millie has done three times more homework problems than Suzy, who has done five. How many problems did Millie finish?
Division: Millie has done 15 homework problems, three times more than Suzy. How many has Suzy done?
Each of the rectangles has one dimension that is 39 feet longer than the other. However, 39 feet is relatively large for the 75′ by 114′ rectangle and relatively small for the 455′ by 494′ rectangle. This means that in the 455′ by 494′ rectangle, the difference in dimension will be least important; therefore, this rectangle is the most square in a relative sense.
Using absolute reasoning, there are more brown eggs in the first carton (seven) than in the second (five). However, using relative reasoning, there are more brown eggs in the second carton (5/12) than in the first (7/18). So, depending on which way we look at it, either carton could have “more” brown eggs.
Though the second ski ramp is taller (an absolute comparison), it may or may not be steeper (a relative comparison). You would need to know the lengths of A and B and then look at the fractions 7/A and 10/B. The greater fraction corresponds to the steeper ski ramp.
Probably the two most important things you need to know are the capacity (or size) of the elevator and the number of people in it. An elevator with four passengers may be very crowded or relatively empty, depending on how many people it is intended to carry comfortably. A relative comparison is required.
Session 1 What Is a Number System?
Understand the nature of the real number system, the elements and operations that make up the system, and some of the rules that govern the operations. Examine a finite number system that follows some (but not all) of the same rules, and then compare this system to the real number system. Use a number line to classify the numbers we use, and examine how the numbers and operations relate to one another.
Session 2 Number Sets, Infinity, and Zero
Continue examining the number line and the relationships among sets of numbers that make up the real number system. Explore which operations and properties hold true for each of the sets. Consider the magnitude of these infinite sets and discover that infinity comes in more than one size. Examine place value and the significance of zero in a place value system.
Session 3 Place Value
Look at place value systems based on numbers other than 10. Examine the base two numbers and learn uses for base two numbers in computers. Explore exponents and relate them to logarithms. Examine the use of scientific notation to represent numbers with very large or very small magnitude. Interpret whole numbers, common fractions, and decimals in base four.
Session 4 Meanings and Models for Operations
Examine the operations of addition, subtraction, multiplication, and division and their relationships to whole numbers. Work with area models for multiplication and division. Explore the use of two-color chips to model operations with positive and negative numbers.
Session 5 Divisibility Tests and Factors
Explore number theory topics. Analyze Alpha math problems and discuss how they help with the conceptual understanding of operations. Examine various divisibility tests to see how and why they work. Begin examining factors and multiples.
Session 6 Number Theory
Examine visual methods for finding least common multiples and greatest common factors, including Venn diagram models and area models. Explore prime numbers. Learn to locate prime numbers on a number grid and to determine whether very large numbers are prime.
Session 7 Fractions and Decimals
Extend your understanding of fractions and decimals. Examine terminating and non-terminating decimals. Explore ways to predict the number of decimal places in a terminating decimal and the period of a non-terminating decimal. Examine which fractions terminate and which repeat as decimals, and why all rational numbers must fall into one of these categories. Explore methods to convert decimals to fractions and vice versa. Use benchmarks and intuitive methods to order fractions.
Session 8 Rational Numbers and Proportional Reasoning
Begin examining rational numbers. Explore a model for computations with fractions. Analyze proportional reasoning and the difference between absolute and relative thinking. Explore ways to represent proportional relationships and the resulting operations with ratios. Examine how ratios can represent either part-part or part-whole comparisons, depending on how you define the unit, and explore how this affects their behavior in computations.
Session 9 Fractions, Percents, and Ratios
Continue exploring rational numbers, working with an area model for multiplication and division with fractions, and examining operations with decimals. Explore percents and the relationships among representations using fractions, decimals, and percents. Examine benchmarks for understanding percents, especially percents less than 10 and greater than 100. Consider ways to use an elastic model, an area model, and other models to discuss percents. Explore some ratios that occur in nature.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants) who have adapted their new knowledge to their classrooms.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants) who have adapted their new knowledge to their classrooms.