Opening the book…
So far, we have been thinking about a fraction as the answer to a division problem. For example, \(\frac{2}{3}\) 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.
Discussion:
Developing a conceptual understanding of fractions is fundamental for solving problems at any level. For example, a bad representation can lead to inaccurate answers, such as justifying that \(\frac{2}{3} > \frac{3}{4}\) with the following representations.
Discuss what issues you see these representations have, and how this can help you think about the definition of a fraction.
Similarly, someone can fail to understand that when two rectangles of the same area are split in three equal parts, then each part is a third. This must ensure the pieces have equal area, but not necessarily the same shape. For example, the following rectangles are both representing \(\frac{1}{3}\).
While it is easier to conclude that both figures represent \(\frac{1}{3}\), it could be confusing to conclude that both figures represent the same area. Why do you think someone might struggle with this concept?
The following views of fractions will provide conceptual approaches to reflect on these questions.
The Whole as the Unit
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 \(\dfrac{1}{2}\) in a problem, you may 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, which are known as different parts of different wholes.
"Which number is shown by the shaded region?"
Solution
Because there is no clarity on what the whole is, the answers will vary. Some correct answers are:
- 5, if we consider each circle to be a whole (1).
- \(2 \frac{1}{2}\), if we consider two circles to be a whole (1).
- \(\frac{5}{8}\), if we consider eight circles to be a whole (1).
- \(1 \frac{1}{4}\), if we consider four circles to be a whole (1).
Discussion:
Consider the following figure:
- If the shaded region represents \(3 \frac{2}{3}\), what is the whole?
- Find three other numbers that could be represented by the shaded region, and explain what the whole is for each answer.
Fraction Comparison and Number Line Representation
If we think of fractions as portions of a segment, we can talk about their location on a number line. To locate a fraction on a number line, the unit is the distance between 0 and 1.
This measurement model makes it much easier to tackle questions about the relative size of fractions based on where they are 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 larger.
Notice that \(\dfrac{3}{5}\) and \(\dfrac{5}{8}\) are very close, but \(\dfrac{5}{8}\) is bigger. To compare fractions, we can use intuitive methods such as benchmarks, relying on intuition to determine how a fraction compares to a number you know well.
Benchmark Method:
A fraction is greater than 1 if its numerator is greater than its denominator. This is a consequence of the denominator representing how many pieces are in one whole (one unit), and the numerator representing how many pieces are in your portion. So if the numerator is bigger, that means you have more than the number of pieces needed to make one whole.
Another common benchmark is \(\frac{1}{2}\). A fraction is greater than \(\frac{1}{2}\) if the numerator is more than half the denominator. It is equivalent to check that a fraction is greater than \(\frac{1}{2}\) if twice the numerator is bigger than the denominator.
Discussion:
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. \[\frac{25}{23}, \quad \frac{4}{7}, \quad \frac{17}{35}, \quad \frac{2}{9}, \quad \frac{14}{15} \ldotp \nonumber \]
You probably used your creative thinking to organize these numbers by considering the relative sizes of fractions. The next discussion allows you to expand your problem-solving skills by exploring different strategies before we officially introduce them. Use your creative thinking to solve the problem in the discussion.
Discussion:
Arrange each group of fractions in ascending order. Keep track of your thinking and your methods.
- \[\frac{7}{17}, \quad \frac{4}{17}, \quad \frac{12}{17} \ldotp \nonumber \]
- \[\frac{5}{6}, \quad \frac{10}{11}, \quad \frac{2}{3} \ldotp \nonumber \]
Methods Using Denominators or Numerators Comparison:
If two fractions have the same denominator, look at the numerators—in this case, the larger numerator appears in the larger fraction. 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.
If the fractions have the same numerators, pay attention to the denominator—the larger denominator actually makes the smaller fraction. For example, \(\frac{3}{4} > \frac{3}{5}\). Notice that \(\frac{3}{4}\) and \(\frac{3}{5}\) both have the same number of pieces, but the \(\frac{3}{4}\) are bigger pieces.
If the numerator is the denominator\(-1\), you can easily compare them. 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 \(\frac{10}{11} = 1 - \frac{1}{11}\).
You can extend the test above to fractions whose numerators are the denominator minus a constant. 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:
Using the denominator or numerator comparisons is not always possible, so you may need to find equivalent fractions of the given fractions before you use these comparisons. To find equivalent fractions, you can use the Key Fraction Rule.
\[\frac{xa}{xb} = \frac{a}{b}, \nonumber \]
(at least for positive whole numbers \(x\)).
We say that \(\frac{xa}{xb}\) and \(\frac{a}{b}\) are equivalent fractions.
(at least for positive whole numbers ).
We say that and are equivalent fractions.
Arithmetic Sequences
Consider the following pattern:
\[5, \quad 8, \quad 11, \quad 14, \quad 17, \quad 20, \quad 23, \quad 26, \ldots \nonumber \]
Discussion:
Work with a classmate to answer the following questions:
- Can you predict the next five numbers of the pattern above?
- Can you predict the \(100^{th}\) number?
- Describe the pattern in words.
The pattern above is an arithmetic sequence, because you can get the next number by adding the same amount each time.
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is a constant.
Consider the following sequences:
A. \(1, 2, 3, 4, 5, \dots\)
B. \(2, 4, 6, 8, 10, \dots\)
C. \(\dfrac{1}{3}, 1, \dfrac{5}{3}, \dfrac{7}{3}, 3, \dots \)
Verify that these are arithmetic sequences. How do you know?
Solution
To verify that these are arithmetic sequences, we need to find the difference between the consecutive terms.
Pattern A: \[\begin{split} \underbrace{1, \quad} \underbrace{2, \quad} \underbrace{3, \quad} \underbrace{4, \quad} 5, & \ldots \\ +1\; \; +1\; \; +1\; \; +1 \quad & \end{split} \nonumber \]
Pattern B: \[\begin{split} \underbrace{2, \quad} \underbrace{4, \quad} \underbrace{6, \quad} \underbrace{8, \quad} 10, & \ldots \\ +2\; \; +2\; \; +2\; \; +2 \qquad & \end{split} \nonumber \]
Pattern C: \[\begin{split} \underbrace{\frac{1}{3}, \quad} \underbrace{1, \quad} \underbrace{\frac{5}{3}, \quad} \underbrace{\frac{7}{3}, \quad} 3, & \ldots \\ + \frac{2}{3}\; \; + \frac{2}{3}\; + \frac{2}{3}\; \; + \frac{2}{3}\; \quad & \end{split} \nonumber \]
Discussion:
Make up two fraction sequences of your own, one that is an arithmetic sequence and one that is not an arithmetic sequence.
Exchange your sequences with a partner, but do not tell your partner which is which.
When you get your partner’s sequences, decide which is an arithmetic sequence and which is not. Check if you and your partner agree.
This page titled 3.4: The Whole as a Unit- Fraction Comparison and Arithmetic Sequences was last modified on Tue, 16 Jun 2026 02:38:02 GMT and is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Michelle Manes via source content that was edited to the style and standards of the LibreTexts platform.
The book could not be loaded just now.