Note: This is part 2 in a multi-part series on fractions. In part 1, I discussed two different meanings for fractions, I recommend you start there.
The Whole Story
Discussing fractions without discussing their associated whole (also called referent unit) can be problematic. The quantity is assumed to refer to some referent unit. However, I can show students a picture such as this
Each response is possibly correct. If the first student was thinking of one of the bars as the whole, than she is correct that the shaded region is of that whole. If instead, a student perceived both bars together as a whole, then 2 shaded boxes would represent of that whole. Thus, it is incredibly important to clearly define the whole when discussing fractions in school mathematics.
Typically, teachers are not very clear on what the referent unit is when discussing fractions. This may lead to some student confusion. For example, if I ask a student which is bigger, or , it is typically assumed that they are referring to the same whole and so is indeed less than . However, if I add in a context then this may not actually be the case. For example, I might compare of a mouse’s weight and of an elephant’s weight. This might cause some confusion because students are typically given blanket statements that is always less than without stating that this is assuming that they are referring to the same whole.
Another reason why understanding the whole is essential is that some operations with fractions involve consistent referent units (addition and subtraction) while others do not (multiplication and division). For example, in the equation x + y = z (where x, y and z are fractions) x, y, and z each refer to the same referent unit. However, in the equations xy = z, the referent unit for x differs from y and z. I will explore this further in part 3.