8.5: What is a Fraction? Revisited
Models
We have been thinking about a fraction as the answer to a division problem. For example, \(\frac{2}{3}\) is the result of dividing 2 into 3 portions. Like 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. An issue arises, however, when we recall that the division should be fair. For example, in the image below, are we sure that the two shaded pieces shown for that child really are 2/3 of a square?
So, we have stuck with circular pies to make this a little easier. That said, depending on the numbers of children and pies, even a circular pie is not easy to divide. See if you can draw the amount of pie each child gets if 4 pies are shared among 11 children.
Despite this, when the numbers are friendly, the “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?
This is a good time to remind ourselves that there is no reason to force ourselves to stick with one model. We can switch models and think about fractions in a new way. Of course, there is actually a reason we tend to try sticking with one model. The more familiar a model is, the more comfortable we are using it! 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! Let's get in the mindset of someone just learning about fractions for the first time. Try to answer the two questions below and explain your ansers to another person. Does that person "buy" your answers?
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?
\[\frac{1}{3} \qquad \qquad \frac{2}{3} > \frac{3}{4} \nonumber \]
\[\frac{1}{3} \neq 13 \nonumber \]
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. In other words, you can think of "\(\frac{1}{2}\)" as actually shorthand for "half of one ".
For our pie model, our units have been consistent: the “whole” (or unit) was one whole pie (which we treated as single circle), 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”).
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 \(2 \frac{1}{2}\).
- Kiana says it represents \(\frac{5}{8}\).
- Nate says it is \(1 \frac{1}{4}\).
Mr. Li exclaims, “Everyone is right!”
Do you believe Mr. Li in the example above? His statement is accurate. See if you can justify each answer by explaining what each student thought was the unit in Mr. Li’s picture. Try the following and share your answers with another person. Did you and the other person get the same answers?
Now look at this picture:
- If the shaded region represents \(3 \frac{2}{3}\), 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.
What other models could we use? How about line segments? Suppose the picture below represents \(\frac{2}{3}\). Let's make sense of that claim. There are three important characterisitcs of the picture. There is the black horizontal line segment, there are two vertical line segments, and there is a thick grey horizontal line segment. The black horizontal line segment can be used as the whole (the unit ) because it is split into three equal-length pieces by the vertical line segments and two of those three equal pieces correspond to the length of the thick grey horizontal line segment.
For each picture below, say what fraction it represents and how you know you are right. Share with a friend and see if they accept your answers.
Ordering Fractions
If we think about fractions as “portions of a line segment,” then we can talk about their locations on a number line. We can start to treat fractions like numbers on their own. In the back of our minds, we should remember that fractions are always relative to some unit. But on a number line, the unit is clear: it is the distance between 0 and 1.
This number line model makes it much easier to tackle questions about the relative size of fractions: the sizes are based on where they appear on the number line. We can mark off different fractions as parts of the unit segment. Just as with whole numbers, fractions that appear farther to the right are greater.
3/5 and 5/8 are very close, but 5/8 is just a bit bigger. Try to do the following on your own and then share with another person. Did your answers make sense to them?
- What quick method can you use to determine if a fraction is greater than 1?
- What quick method can you use to determine if a fraction is greater than \(\frac{1}{2}\)?
- Organize these fractions from smallest to largest using benchmarks : 0 to \(\frac{1}{2}\), \(\frac{1}{2}\) to 1, and greater than 1. Justify your choices. \[\frac{25}{23}, \quad \frac{4}{7}, \quad \frac{17}{35}, \quad \frac{2}{9}, \quad \frac{14}{15} \ldotp \nonumber \]
- Arrange each group of fractions in ascending order . Keep track of your thinking and your methods.
- \[\frac{7}{17}, \quad \frac{4}{17}, \quad \frac{12}{17} \ldotp \nonumber \]
- \[\frac{3}{7}, \quad \frac{3}{4}, \quad \frac{3}{8} \ldotp \nonumber \]
- \[\frac{5}{6}, \quad \frac{7}{8}, \quad \frac{3}{4} \ldotp \nonumber \]
- \[\frac{8}{13}, \quad \frac{12}{17}, \quad \frac{1}{6} \ldotp \nonumber \]
- \[\frac{5}{6}, \quad \frac{10}{11}, \quad \frac{2}{3} \ldotp \nonumber \]
Fraction Intuition
Below there are a lot of different "rules" you may have discovered over the years by just noticing patterns. But it is also likely that you never really wrote them down. For each one, think about if you ever discovered this on your own and, if so, would you have described it the same way.
Greater than 1: A fraction is greater than 1 if its numerator is greater than its denominator. How can we see this? Well, the denominator represents how many pieces in one whole (one unit). The numerator represents how many pieces in your portion. So if the numerator is bigger, that means you have more than the number of pieces needed to make one whole.
Greater than \(\frac{1}{2}\): A fraction is greater than \(\frac{1}{2}\) if the numerator is more than half the denominator. Another way to check (which might be an easier calculation): a fraction is greater than \(\frac{1}{2}\) if twice the numerator is greater than the denominator.
Why? See if you can justify this to yourself.
Same denominators: If two fractions have the same denominator, just compare the numerators. The fractions will be in the same order as the numerators . For example, \(\frac{5}{7} < \frac{6}{7}\). Why? Well, the pieces are the same size since the denominators are the same. If you have more pieces of the same size, you have a bigger number.
Same numerators: If the numerators of two fractions are the same, just compare the denominators. The fractions should be in the reverse order of the denominators . For example, \(\frac{3}{4} > \frac{3}{5}\). The justification for this one is a little trickier: The denominator tells you how many pieces make up one whole. If there are more pieces in a whole (if the denominator is bigger), then the individual pieces must be smaller. And if you take the same number of pieces (same numerator), then the collection with the bigger pieces is greater.
Numerator = denominator\(-1\): You can easily compare two fractions whose numerators are both one less than their denominators. The fractions will be in the same order as the denominators. Think of each fraction as a pie with one piece missing. The greater the denominator, the smaller the missing piece, so the greater the amount remaining. For example, \(\frac{6}{7} < \frac{10}{11}\), since \(\frac{6}{7} = 1 - \frac{1}{7}\) and \(\frac{10}{11} = 1 - \frac{1}{11}\).
Numerator = denominator − constant: You can extend the test above to fractions whose numerators are both the same amount less than their denominators. The fractions will again be in the same order as the denominators, for exactly the same reason. For example, \(\frac{3}{7} < \frac{7}{11}\), because both are four “pieces” less than one whole, and the \(\frac{1}{11}\) pieces are smaller than the \(\frac{1}{7}\) pieces.
Equivalent fractions: Find equivalent fractions that lets you compare numerators or denominators, and then use one of the above "rules".