Session 8, Part B:
Fractions With Cuisenaire Rods

In This Part: Representing Fractions With Rods | Other Denominators
Modeling Operations | Try It Yourself

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." Note 5

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.
Note 6

 Session 8: Index | Notes | Solutions | Video