## Learning Math: Number and Operations

# Rational Numbers and Proportional Reasoning Part B: Fractions With Cuisenaire Rods (45 minutes)

**In This Part: Representing Fractions With Rods**

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. See Note 4 below.

To help you visualize this, in this section you will learn how to represent fractions with Cuisenaire Rods and then see how to use the rods to perform operations with fractions.

Here is a set of Cuisenaire Rods:

In order to represent fractions with these rods, you need to choose one rod to serve as a unit (in other words, to represent the whole, or value “1”). The rule to follow is that you must also be able to represent the rod you choose with at least one single-color “train” of the same length, built out of shorter rods. This way you will be able 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. That’s because 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 represents a half:

The next-longest rod that has a two-car, one-color train is the purple rod, and that rod has an all-red train, as well as an all-white train:

The next-longest rod to satisfy the requirement is the dark-green rod, and it also has an all-red train, as well as a light-green and a white train. Notice that the halves in this case are 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 that has a two-car, one-color train also has an all-red train. This means that in order to represent halves using rods, the rod length must be divisible by 2, which in our original Cuisenaire configuration is the red rod.

**In This Part: Other Denominators
**Similarly, if you want to represent thirds, you should choose the shortest rod that has a three-car, one-color train and name that rod “1.”

The shortest rod that has a three-car, one-color train is the light green rod. If light green is “1,” then white is 1/3, and we could name all other rods in terms of these two rods:

The next-longest rod that has a three-car, one-color train is the dark-green rod. It has a three-car all-red train, as well as a light green, and a white train. Notice that the thirds in this case are red 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.

The next-longest rod to satisfy the requirement is the blue rod, which has a three-car, light-green train, and also a white train:

In fact, we can show that every rod that has a three-car, one-color train also has a light-green train, so any time we want to deal with thirds, we must choose a rod with an all-light-green train to represent “1.”

Consequently, if we want to deal with halves and thirds at the same time, we need a rod that has both an all-red train and an all-light-green train. As we’ve seen above, this is the dark-green rod. If we call the dark-green rod “1,” then white is 1/6, red is 1/3, and light green is 1/2, and all other rods can be named in terms of these rods.

**In This Part: Modeling Operations
**Here is the model for adding halves and thirds, using dark green to represent “1.” Note how we can assign fractional values to all rods shorter than dark green. In this case, each rod represents a fraction of the dark green rod. (The fractional values of rods would change if we were to change the rod that represents the unit.)

You can now model addition and subtraction, as well as multiplication and division, by “making trains.” See Note 5 below.

Think of addition as a merging of different “cars” of the trains. For example, since red = 1/3 and light green = 1/2, you can model 1/3 + 1/2 with a red-and-light-green train:

**This length is equal to a yellow, whose value is 5/6.**

Similarly, you can think of subtraction as a missing addend. For example, you can model 1/2 – 1/3 by finding the rod you would need to add to the red rod to make a train the length of the light green.

**This is the white rod, whose value is 1/6. **

If you think of multiplication by a fraction as evaluating a part of a group, you can model 1/2 • 1/3 by “counting” 1/2 of the rod that represents 1/3. Red represents 1/3, and 1/2 of a red is a white, whose value is 1/6:

**One-half of a red is a white.**

Similarly, to model 1/3 • 1/2, “count” 1/3 of the rod that represents 1/2. Light green represents 1/2, and 1/3 of a light green is a white, whose value is 1/6:

**One-third of a light green is a white.**

Division by a fraction must be thought of as a quotative situation. You can model it by asking, “How many of this rod are there in that rod?” For example, 1/2 1/3 asks, “How many reds (1/3) are there in a light green (1/2)?”

**There are 1 1/2 reds in a light green.**

Similarly, 1/3 1/2 asks, “How many light greens (1/2) are there in a red (1/3)?”

**There are 2/3 of a light green in a red.**

Here’s another example: To model 1/21/6, we need to ask, “How many whites (1/6)are there in a light green (1/2)?”

**There are three whites in a light green.**

Similarly, to divide 1/6 by 1/2, we need to ask, “How many light greens (1/2) are there in a white (1/6)?”

**There is 1/3 of a light green in a white.**

See Note 6 below.

**In This Part: Try It Yourself
**Use the following Interactive Activity to work on the problems below, which let you use the rods to try out representations and operations with fractions.

You can model a fraction by stacking two or more Cuisenaire® Rods.

Drag the Cuisenaire Rods onto the grid

###### Cuisenaire® is a trademark of ETA hand2mind. All rights reserved.

Video Segment

In this video segment, Rhonda and Andrea use rods to model multiplication and division with thirds and fourths. First they must figure out what their model is going to be in order to do their computations. Watch this segment after you’ve completed Problems B2 and B3.

You can find this segment on the session video approximately 11 minutes and 42 seconds after the Annenberg Media logo.

### Notes

**Note 4
**Many people who have trouble with fractions and computations with fractions do not have a mental image of what a fraction represents, which makes it very difficult to do computations. This section gives a concrete representation of the fractions and helps you understand why the shortcuts for computations work.

**Note 5
**To learn more about different meaning of operations, go to Session 4, Part A.

**Note 6**

The Cuisenaire Rods model illustrates why the algorithms for adding and subtracting fractions work — namely, that you cannot add the fractions until they are expressed in the same units. It also shows why the alternative algorithm for dividing fractions (finding a common denominator and then dividing the numerators) works. It does not, however, illustrate why the multiplication algorithm (multiplying the numerators and multiplying the denominators) works.