# Cuisenaire Rods (Whole Numbers)

## What are Cuisenaire Rods?

Cuisenaire Rods, sometimes called Number Rods or Relational Rods, are a set of 10 rectangular rods in 10 distinguishing colors. Each rod and its color correspond to a specific length. White rods are the shortest and denote one unit. The orange rod is the longest. Each rod in the sequence is one unit (white rod) more than the previous rod. This pattern is made visible by what is often called the “staircase.” You can reinforce your students’ understanding of this pattern by topping each step in the staircase with a white rod. Other staircases can be formed using different rods for the steps.

The Brainingcamp virtual version further extends the collection of rods by adding an 11th rod. This supplementary rod is adjustable and can be sized for further numeric explorations. However, for the purpose of this blog post, I will stick to the original set of rods.

## Rod Relationships

So, are these rods intended only for Elementary students or early math concepts? Is the use of these rods too “babyish” for topics beyond counting or cardinality? Well, no, they are not. The relationships between the rods make them valuable sense-making devices for many math concepts.

As earlier defined, the length of the rods differs by one unit or white rod, and we can define the white rod as the unit or 1. However, what if for today, let’s refer to Figure 1, you’ll see the brown rod is now 1. What does the purple rod represent? The red rod?

It’s a general convention that the rods are labeled by the first letter of the names of their colors (except for blue, brown, and black, which are named by their final letters).

Using letters as demonstrated in Figure 2, means we can start thinking symbolically and begin to provide visual models for students for abstract representations such as n = 2p. How does the red rod compare to the other rods in Figure 3?

When rods are placed end to end horizontally, it is referred to as a “train.” Trains are used to show equivalencies between representations. Let’s take a look at Figure 4, if I build a train of 3 purple rods, what other one-color trains can I make that match the purple train? Which rods work? Which rods don’t? How are they related to the original purple train?

In Figure 11, n ÷ r asks what we have to multiply by r to get n, or ? × r = n. How many 2s make up 8? Or, in terms of repeated subtraction, how many times can we subtract r from n.

In this simple case, the division can be modeled using 2 individual rods, brown and red. However, in the case when the dividend is a larger number, it would need to be represented with multiple rods.

Division with remainders is done by matching a train built with as many of the divisor rods that will fit. The “remainder” is filled in with a single rod that fills in the train. The remainder, of course, can be written as a whole number or a fraction by comparing it with the divisor, as seen in Figure 12.

## By Mark Schmit

Mark is a dedicated and passionate educator, content, and product developer whose had his finger on the pulse of Math Education since 1987. Mark strives to make student learning deeper and teacher instruction better.