Cuisenaire Rods are a versatile collection of rectangular rods of 10 colors with each color corresponding to a different length. The shortest rod, the white, is 1 centimeter long; the longest, the orange, is 10 centimeters long. One set of rods contains 74 rods: 4 each of the orange (σ), blue (e), brown (n), black (k), dark green (d), and yellow (y); 6 purple (p); 10 light green (g); 12 red (r); and 22 white (w). One special aspect of the rods is that, when they are arranged in order of length in a pattern commonly called a “staircase,” each rod differs from the next by 1 centimeter, which is the length of the shortest white rod.
Unlike Color Tiles, which provide a discrete model of numbers, Cuisenaire Rods, because of their different yet related lengths, provide a continuous model. This means they allow you to assign a value to 1 rod and then assign values to the other rods by using the relationships among the rods.
Cuisenaire Rods can be used to develop a wide variety of mathematical ideas at many different levels of complexity. Initially, however, students use the rods to explore spatial relationships by making flat designs that lie on a table or by making three-dimensional designs by stacking the rods. The intent of students’ designs, whether to cover a certain amount of a tabletop or to fill a box, will lead students to discover how some combinations of rods are equal in length to other, single rods. Students’ designs can also provide a context for investigating symmetry. Older students who have no previous experience with Cuisenaire Rods may explore by comparing and ordering the lengths of the rods and then recording the results on grid paper to visualize the inherent “structure” of the design. In all their early work with the rods, students have a context in which to develop their communication skills through the use of grade-appropriate arithmetic and geometric vocabulary.
Though students need to explore freely, some may appreciate specific challenges, such as being asked to make designs with certain types of symmetry or certain characteristics, such as different colors representing different fractional parts.
One of the basic uses of Cuisenaire Rods is to provide a model for the numbers 1 to 10. If the white rod is assigned the value of 1, the red rod is assigned the value of 2 because the red rod has the same length as a “train” of two white rods. Similarly, the rods from light green through orange are assigned values from 3 through 10, respectively. The orange and white rods can provide a model for place value. To find the length of a certain train, students can cover the train with as many orange rods as they can and then fill in the remaining distance with white rods; so a train covered with 3 orange rods and 4 white rods is 34 white rods long. The rods can be placed end to end to model addition. For example 2 plus 3 can be found by first making a train with a red rod (2) and a light green rod (3) and then finding the single rod (yellow) whose length (5) is equal in length to the 2-rod train. This model corresponds to addition on a number line.
The rods can also be used for acting out subtraction as the search for a missing addend. For example, 5 minus 2 can be found by placing a red rod (2) on top of a yellow (5), then looking for the rod which, when placed next to the red, makes a train equal in length to the yellow.
Multiplication, such as 5 times 2, is interpreted as repeated addition by making a train of 5 red rods or of 2 yellow rods.
Division, such as 10 divided by 2, may be interpreted as repeated subtraction (“How many red rods make a train as long as an orange rod?”) or as sharing (“Two of what color rod make a train as long as an orange rod?”).
Cuisenaire Rods also make effective models for decimals and fractions. If the orange rod is designated as the unit rod, then the white, red, and light green rods represent 0.1, 0.2, and 0.3, respectively. If the dark green rod is chosen as the unit, then the white, red, and light green rods represent 1⁄6, 2⁄6 (1⁄3), and 3⁄6 (1⁄2), respectively. Once the unit rod has been established, addition, subtraction, multiplication, and division of decimals and fractions can be modeled in the same way as the operations with whole numbers.
Cuisenaire Rods are suitable for a variety of geometric and measurement investigations. Once students develop a sense that the white rod is 1 centimeter long, they have little difficulty in accepting and using centimeters as units of length. Since the face of the white rod has an area of 1 square centimeter, the rods are ideal for finding area in square centimeters. Since the volume of the white rod is 1 cubic centimeter, the rods can exemplify the meaning of volume as students use rods to fill up boxes. Students may even develop a sense of a milliliter as the capacity of a container that holds exactly one white rod. Cuisenaire Rods offer many possibilities for forming and discovering number patterns both through creating designs that are growing according to some pattern and through finding the number of ways in which a rod can be made as the sum of other rods. This second scenario can lead to the concept of factors of a number and prime numbers.
The rods also provide a context for building logical reasoning skills. For example, students can use 2 loops of string to create a Venn Diagram showing the multiples of both red and light green rods (which represent 2 and 3, respectively) by placing the multiples of red (red, purple, dark green, brown, and orange) in 1 loop, the multiples of light green (light green, dark green, and blue) in the other loop, and then creating an overlap of the 2 loops and placing the rod representing the common multiple (dark green which represents 6) in the overlap.
Cuisenaire Rods are wonderful tools for assessing students’ mathematical thinking. Watching students work with Cuisenaire Rods gives you a sense of how they approach a mathematical problem. Their thinking can be “seen”, in that thinking is expressed through the way they construct, recognize, and continue spatial patterns. When a class breaks up into small working groups, you are able to circulate, listen, and raise questions, all the while focusing on how individuals are thinking. Here is a perfect opportunity for authentic assessment. Having students describe their designs and share their strategies and thinking with the whole class gives you another opportunity for observational assessment. Furthermore, you may want to gather students’ recorded work or invite them to choose pieces to add to their math portfolios.