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 with 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 the other rods in terms of these two rods:

The next-longest rod with a three-car one-color train is the dark-green rod, which also contains a red train, a light-green train, and a white train. Note that the thirds in this case are the 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 also contains a light-green train and a white train:

In fact, we can show that every rod with a three-car one-color train also contains a light-green train. In order to represent thirds using rods, the rod length must be divisible by 3, which in our original Cuisenaire configuration is represented by the light-green rod.

Consequently, if we want to deal with halves and thirds at the same time, we need a rod that has both a red train and a 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, light green is 1/2, and all the other rods can be named in terms of these rods.

  Using the rods to model operations