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Learning Math: Number and Operations

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.

In This Session:
Part A: Interpreting Fractions, Units, and Unitizing
Part B: Fractions With Cuisenaire Rods
Part C: Absolute and Relative Reasoning

In this session, we will look at ways to interpret, model and work with rational numbers. We will examine various ways to determine the “unit” of the ratio we’re expressing with a rational number and to explore the basics of proportional reasoning.

For information on required and/or optional materials, see See Note 1 below.

In this session, you will do the following:
Understand fractions in both part-part and part-whole interpretations
Understand units and unitizing
Learn how to use Cuisenaire Rods to represent and model computation with fractions
Understand the differences between absolute and relative thinking and their relationships to mathematical operations
• Learn that proportional reasoning is an example of relative or multiplicative thinking

Key Terms

Previously Introduced:

rational number

Rational numbers are numbers that can be expressed as a quotient of two integers; when expressed in a decimal form they will either terminate (1/2 = 0.5) or repeat (1/3 = 0.333…)


New in This Session:

absolute comparison

An absolute comparison is an additive comparison between quantities. In an absolute comparison, 7 out of 10 is considered to be larger than 4 out of 5, since 7 is larger than 4.

part-part interpretation

Part-part interpretation of a fraction is the notion of comparing one quantity within a whole to another quantity within that whole. For example, if, on a field trip, there are 3 adults for every 10 students, the part-part interpretation of this relationship would be 3/10 (which could also be written as 3:10).

part-whole interpretation

Part-whole interpretation of a fraction represents one or more parts of a single unit. For example, the fraction 4/5 represents the part-whole relationship in the following phrase: “Four out of five dentists prefer Blasto toothpaste.”

relative comparison

A relative comparison is a multiplicative, or proportional, comparison between quantities. In a relative comparison, 4 out of 5 is considered to be larger than 7 out of 10, since 4/5 is larger than 7/10.


Materials Needed:

  • Cuisenaire Rods (optional)

You can use the cutouts of the rods from the Rod Template.

Series Directory

Learning Math: Number and Operations


Produced by WGBH Educational Foundation. 2003.
  • Closed Captioning
  • ISBN: 1-57680-678-2

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