Ordering fractions

Here you will learn about ordering fractions from least to greatest and greatest to least.

Students will first learn about ordering fractions as part of numbers and operations of fractions in 4th grade.

What is ordering fractions?

Ordering fractions is arranging a set of fractions from least to greatest (ascending order) or from greatest to least (descending order).

To do this you rewrite the fractions so that they have common denominators and then compare the numerators.

For example,

If you order these fractions from least to greatest…

The order becomes…

You can prove this order is correct by creating common denominators. All the denominators (2, 4 and 8) have a common multiple of 8, so you can use 8 as the common denominator.

Now the equivalent fractions are: \; \cfrac{2}{8}, \cfrac{3}{8}, \cfrac{4}{8}, \cfrac{5}{8}, \cfrac{6}{8}.

Since the fractions have the same denominators all the parts are the same size. You can use the size of the numerators to order the fractions. The fractions are in order from least to greatest, because the numerators are in order from least to greatest.

What is ordering fractions?

Common Core State Standards

How does this relate to 4th grade math?

• Grade 4: Numbers and Operations – Fractions (4.NF.A.2)
  Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as \cfrac{1}{2}. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

How to order fractions

In order to order fractions:

1. Create equivalent fractions that have a common denominator.
2. Order the fractions by the size of the numerator.
3. Rewrite the fractions in their original form.

In order to order mixed numbers:

1. Convert mixed numbers to improper fractions.
2. Create equivalent fractions that have a common denominator.
3. Order the fractions by the size of the numerator.
4. Rewrite the fractions in their original form.

Ordering fractions examples

Example 1: ordering proper fractions

Order the fractions from least to greatest.

1. Create equivalent fractions that have a common denominator.

All the denominators (2, 4, 6 and 12) have a multiple of 12, so you can use 12 as the common denominator.

2Order the fractions by the size of the numerator.

Here are the equivalent fractions with the common denominator of 12 :

Now that the denominators are like, you can order them from least to greatest by the numerator:

3Rewrite the fractions in their original form.

Example 2: ordering proper fractions

Order the fractions from greatest to least.

Create equivalent fractions that have a common denominator.

All the denominators (3, 4, 6 and 8) have a multiple of 24, so you can use 24 as the common denominator.

Order the fractions by the size of the numerator.

Here are the equivalent fractions with the common denominator of 24 :

Now that the denominators are like, you can order them from greatest to least by the numerator:

Rewrite the fractions in their original form.

Example 3: ordering improper fractions and mixed numbers

Order the numbers from greatest to least.

Convert mixed numbers to improper fractions.

You can write all the numbers as fractions (proper and improper) or as mixed numbers. To use the same steps as above, write them all as fractions.

For 1\cfrac{1}{2}, break the whole into halves, then count the total halves.

There are 3 halves, so 1 \cfrac{1}{2} = \cfrac{3}{2}.

For 1 \cfrac{2}{3}, break the whole into thirds, then count the total thirds.

There are 5 thirds, so 1 \cfrac{2}{3} = \cfrac{5}{3}.

Create equivalent fractions that have a common denominator.

All the denominators (2, 3, 5 and 10) have a multiple of 30, so you can use 30 as the common denominator.

Order the fractions by the size of the numerator.

Here are the equivalent fractions with the common denominator of 30 :

Now that the denominators are like, you can order them from greatest to least by the numerator:

Rewrite the fractions in their original form.

Example 4: ordering improper fractions and mixed numbers

Order the numbers from least to greatest.

Convert mixed numbers to improper fractions.

You can write all the numbers as fractions (proper and improper) or as mixed numbers. To use the same steps as above, write them all as fractions.

For 2 \cfrac{1}{5}, break the wholes into fifths, then count the total fifths.

There are 11 fifths, so 2 \cfrac{1}{5} = \cfrac{11}{5}.

For 2 \cfrac{3}{10}, break the whole into tenths, then count the total tenths.

There are 23 tenths, so 2 \cfrac{3}{10} = \cfrac{23}{10}.

Create equivalent fractions that have a common denominator.

All the denominators (2, 4, 5 and 10) have a common multiple of 20, so you can use 20 as the common denominator.

Order the fractions by the size of the numerator.

Here are the equivalent fractions with the common denominator of 20 :

Now that the denominators are like, you can order them from least to greatest by the numerator:

Rewrite the fractions in their original form.

Example 5: ordering improper fractions and mixed numbers

Order the numbers from greatest to least.

Convert mixed numbers to improper fractions.

You can write all the numbers as fractions (proper and improper) or as mixed numbers. To use the same steps as above, write them all as fractions.

For 1\cfrac{1}{3}, break the whole into thirds, then count the total thirds.

There are 4 thirds, so 1\cfrac{1}{3} = \cfrac{4}{3}.

For 1\cfrac{1}{2}, break the whole into halves, then count the total halves.

There are 3 halves, so 1\cfrac{1}{2} = \cfrac{3}{2}.

Create equivalent fractions that have a common denominator.

All the denominators (2, 3, 6 and 9) have a common multiple of 18, so you can use 18 as the common denominator.

Order the fractions by the size of the numerator.

Here are the equivalent fractions with the common denominator of 18 :

Now that the denominators are like, you can order them from greatest to least by the numerator:

Rewrite the fractions in their original form.

Teaching tips for ordering fractions

• Before ordering multiple fractions, students should be comfortable comparing two fractions. Comparing fractions can be done through common numerators, common denominators or with benchmark fractions. Although not modeled here, these strategies can also be used for ordering fractions.

• Fraction work in 3rd grade centers around understanding through models; particularly area models and number lines. To build on this in 4th grade, always have physical or digital models available for students to use when necessary.

• While ordering fractions worksheets have their place, look for ways to incorporate real world activities as practice as well. For example, students can order the ingredients of a recipe or they can measure multiple items in inches and then order the measurements.

Our favorite mistakes

• Assuming that an improper fraction is smaller than a mixed number
  Both improper fractions and mixed numbers have wholes, they are just represented in different ways. Improper fractions include wholes as part of the numerator, so it is not always as obvious how many wholes there are. Because of this it can be easy to think that a mixed number is greater, but that is not always true.
  
  For example,
  \cfrac{8}{3} is larger than 2 \cfrac{1}{3}, because \cfrac{8}{3} = 2 \cfrac{2}{3}.

• Assuming the fraction with the smallest numbers is the smallest fraction
  It is the relationship between the numerator and denominator that matters, not just thinking about the numbers individually.
  
  For example,
  \cfrac{3}{4} may seem smaller than \cfrac{68}{100}, since 3 and 4 are smaller numbers, but if you use a common denominator to compare them, you see that \cfrac{3}{4} is the larger fraction:
  
  \cfrac{3 \; \times \; 25}{4 \; \times \; 25}=\cfrac{75}{100} \; and \; \cfrac{75}{100}>\cfrac{68}{100}.

• Mixing up the order
  Pay attention to whether the order should be least to greatest or greatest to least. It is easy to confuse the two and often multiple choice tests have both orders listed as answers, but the correct answer is the order that was asked in the question.

• Not recording the answer as the original fractions
  It is easy to do the work and then forget to write the original numbers as your final answer.
  
  For example,
  Write these numbers in order of size:
  \cfrac{5}{6} \quad \cfrac{7}{12} \quad \cfrac{3}{4}
  
  With the same denominator:
  \cfrac{10}{12} \quad \cfrac{7}{12} \quad \cfrac{9}{12}
  
  In order of size:
  \cfrac{7}{12} \quad \cfrac{9}{12} \quad \cfrac{10}{12}
  
  The final answer is:
  \cfrac{7}{12} \quad \cfrac{3}{4} \quad \cfrac{5}{6}

Related lessons on fractions

• Fractions
• Comparing fractions
• Simplifying fractions
• Improper fraction to mixed number
• Equivalent fractions
• Fractions of numbers
• Mixed number to improper fraction
• Numerator and denominator
• Types of fractions

The next lessons are

• Fractions operations
• Decimals
• Exponents
• Adding fractions
• Subtracting fractions

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