36 What is a Fraction? Revisited
So far, we have been thinking about a fraction as the answer to a division problem. For example, is the result of sharing two pies among three children.
Of course, pies do not have to be round. We can have square pies, or triangular pies or squiggly pies or any shape you please.
This “Pies Per Child Model” has served us perfectly well in thinking about the meaning of fractions, equivalent fractions, and even adding and subtracting fractions.
However, there is no way to use this model to make sense of multiplying fractions! What would this mean?
So what are fractions, if we are asked to multiply them? We are forced to switch models and think about fractions in a new way.
This switch is fundamentally perturbing. Think about students learning this for the first time. We keep switching concepts and models, and speak of fractions in each case as though all is naturally linked and obvious. None of this is obvious, it is all absolutely confusing. This is just one of the reasons that fractions can be such a difficult concept to teach and to learn in elementary school!
Think / Pair / Share (What’s wrong here?)
For each of the following visual representations of fractions, there is a corresponding incorrect symbolic expression.
- Why is the symbolic representation incorrect?
- What might elementary students find confusing in these visual representations?
Units and unitizing
In thinking about fractions, it is important to remember that there are always units attached to a fraction, even if the units are hidden. If you see the number in a problem, you should ask yourself “half of what?” The answer to that question is your unit, the amount that equals 1.
So far, our units have been consistent: the “whole” (or unit) was a whole pie, and fractions were represented by pies cut into equal-sized pieces. But this is just a model, and we can take anything, cut it into equal-sized pieces, and talk about fractions of that whole.
One thing that can make fraction problems so difficult is that the fractions in the problem may be given in different units (they may be “parts” of different “wholes”).
Example (Everyone is right!)
Mr. Li shows this picture to his class and asks what number is shown by the shaded region.
- Kendra says the shaded region represents the number 5.
- Dylan says it represents .
- Kiana says it represents .
- Nate says it is .
Mr. Li exclaims, “Everyone is right!”
Think / Pair / Share
- How can it be that everyone is right? Justify each answer by explaining what each student thought was the unit in Mr. Li’s picture.
- Now look at this picture:
- If the shaded region represents , what is the unit?
- Find three other numbers that could be represented by the shaded region, and explain what the unit is for each answer.
Example (Segments)
This picture
represents . The whole segment (the unit) is split into three equal pieces by the tick marks, and two of those three equal pieces are shaded.
Think / Pair / Share
For each pi