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