Some of the trickiest things to understand about fractions are that they are only comparable if they are defined by the same whole. For example, if I eat a half a pizza and you eat a third of a pizza, we don't know who ate more since I never said anything about the sizes of our pizzas!
To help students understand this concept, I've been using colored rods. The red rod is half of a purple rod, but it isn't "half" by its nature. The red rod is also a third of a brown rod or a fourth of a blue rod.
I can also ask student questions such as, "If the blue rod is 2/3 of the whole, which rod is the whole? Which rod is 1/3 of the whole?"
We can use them to explore and model equivalency and create our own challenges to demonstrate deeper understanding.
Some of the students are working on a great challenge problem by Kathy Fosnot:
A fourth-grade class traveled on a field trip in four separate vehicles. The school provided a lunch of submarine sandwiches for each group. When they stopped for lunch, the subs were cut and shared as follows:
The first group had 3 people and shared 2 subs equally.
x
The second group had 4 people and shared 3 subs equally.
x
The third group had 9 people and shared 6 subs equally.
x
The last group had 6 people and shared 4 subs equally.
When they returned from the field trip, the children began to argue that the portion of the sandwiches they received was not fair, that some children got more to eat than others. Were they right? Or did everyone get the same amount?
