Rational numbers are a "ratio" of one value to another. It's common to think of a fraction as a statement of some number of parts of a particular whole. When working with fractions, it's helpful to think about how to define that "whole" so that various fractional parts can be seen on a common scale.

To help you visualize this, in this section you will learn to represent fractions with Cuisenaire Rods and then see how to use these rods to perform operations with fractions. As you do the problems, think about how you are approaching each problem, what difficulties you are encountering, and how you are communicating your thinking.

Here is a set of Cuisenaire Rods:

In order to represent fractions with these rods, you need to choose a rod to serve as a unit (in other words, to represent the whole, or the value "1"). The rule is that you must be able to represent the rod you choose with at least one single-color two-car "train" of the same length, built out of shorter rods (with no pieces left over). This allows you to use the rods to do computations with fractions.

For example, if you want to do computations with halves, the shortest rod you can use to represent "1" is red -- you can make a two-car one-color train out of white rods that is the same length as a red. In this case, each white rod represents a half:

The next-longest rod with a two-car one-color train is the purple rod; that rod also contains a red train and a white train:

The next longest rod to satisfy the requirement is the dark-green rod; it also contains a light-green train, a red train, and a white train. Note that the halves in this case are the light-green rods. If we name the dark-green rod 1, then the light-green rod is 1/2, the red rod is 1/3, and the white rod is 1/6.

In fact, we could show that every rod with a two-car one-color train also contains a red train. In order to represent halves using rods, the rod length must be divisible by two, which in our original Cuisenaire configuration is represented by the red rod.

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