Teacher Guide: Ordering Fractions

Learn to put three or more fractions in order from least to greatest or greatest to least, and to pick the fastest method every time.

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Worksheet Answer key

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10 questions on Comparing & Ordering. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Order three to five fractions in ascending and descending order
Choose the fastest method: same denominator, same numerator, LCD, decimals or benchmarks
Rewrite fractions with the least common denominator to order them
Use the benchmarks 0, 1/2 and 1 to sort fractions into groups before ordering
Order fractions in real contexts such as recipes, race distances and survey results

Prerequisites

• Comparing Fractions with the Same Denominator
• Comparing Fractions with the Same Numerator
• Comparing Fractions by Cross Multiplication
• Comparing Fractions Using Benchmarks
• Finding equivalent fractions and the least common multiple

Discussion Starters

1. Why does a common denominator work better than cross multiplication when you have four fractions to order?
2. When would you rather convert to decimals than find a common denominator?
3. Can you order $\frac{1}{9}$ , $\frac{5}{9}$ and $\frac{8}{9}$ without writing anything down? How did you know?
4. Which list is easier to order: $\frac{2}{3}, \frac{2}{5}, \frac{2}{7}$ or $\frac{2}{3}, \frac{3}{5}, \frac{5}{7}$ ? Why?

Common Misconceptions

A bigger denominator always means a bigger fraction

Remediation: Show fraction bars for $\frac{1}{2}$ , $\frac{1}{4}$ and $\frac{1}{8}$ side by side. Ask: if a pizza is shared by 8 people instead of 2, who gets the bigger slice?

After rewriting with a common denominator, the answer is the list of new fractions

Remediation: Insist that the final answer uses the original fractions. The rewritten fractions are a tool, like scratch paper.

There is one correct method for every list

Remediation: Let groups order the same list with different methods and compare the results and the time they needed. All correct methods give the same order.

Differentiation Ideas

For Struggling Students:

• Start with lists of three fractions that share a denominator or a numerator
• Use fraction bars or the number line to check every order
• Give the LCD and ask only for the rewritten numerators

For On-Level Students:

• Order four fractions with different denominators using the LCD
• Name the fastest method before solving each list
• Solve recipe and race problems that ask for the second smallest or second largest fraction

For Advanced Students:

• Order five fractions that are all close to one benchmark
• Find a fraction that fits between two neighbors in an ordered list
• Compare the work for the LCD method and the decimal method on the same list

Standards Alignment

4.NF.A.2 (CCSS.MATH.CONTENT.4.NF.A.2)
Compare two fractions with different numerators and different denominators by creating common denominators or numerators, or by comparing to a benchmark fraction
5.NF.A.1 (CCSS.MATH.CONTENT.5.NF.A.1)
Replace given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators
6.NS.C.7 (CCSS.MATH.CONTENT.6.NS.C.7)
Understand ordering and absolute value of rational numbers

Lesson Resources

visualFraction Number Line
Place all the fractions of a list on one number line from 0 to 1 and read them from left to right
activityHuman Number Line
Each student gets a fraction card and the class lines up from least to greatest, then explains each swap
worksheetPick the Method
For each list, students first name the fastest method, then order the fractions

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

Ordering fractions means putting three or more fractions in a row by size.

Ascending order (least to greatest): $\frac{1}{4} < \frac{1}{2} < \frac{3}{4}$
Descending order (greatest to least): $\frac{3}{4} > \frac{1}{2} > \frac{1}{4}$

You already know how to compare two fractions. To order several, pick the method that fits the whole list:

1.Same denominator: order the numerators. $\frac{2}{9} < \frac{4}{9} < \frac{7}{9}$
2.Same numerator: the larger the denominator, the smaller the fraction. $\frac{3}{10} < \frac{3}{8} < \frac{3}{5}$
3.Common denominator: rewrite every fraction with the least common denominator (LCD), then order the numerators.
4.Decimals: divide numerator by denominator and order the decimals.
5.Benchmarks $0$ , $\frac{1}{2}$ and $1$ : sort the fractions into groups first, then order inside each group.

Cross multiplication works for two fractions at a time, but four fractions already need six pairs. A common denominator puts the whole list on one scale in one go.

Worked Examples

Order from least to greatest: $\frac{5}{9}$ , $\frac{2}{9}$ , $\frac{7}{9}$ , $\frac{4}{9}$

Check the denominators

All four fractions are ninths → Same denominator: compare the numerators

Order the numerators

$2 < 4 < 5 < 7$ → $2, 4, 5, 7$

Write the fractions in that order

Keep the denominator $9$ → $\frac{2}{9} < \frac{4}{9} < \frac{5}{9} < \frac{7}{9}$

Answer: $\frac{2}{9}, \frac{4}{9}, \frac{5}{9}, \frac{7}{9}$

Common Mistakes

Ordering unit fractions by their denominators: writing $\frac{1}{3}, \frac{1}{4}, \frac{1}{6}$ as least to greatest because $3 < 4 < 6$

Why it's wrong: With the same numerator, a larger denominator means the whole is cut into more, smaller pieces.

Correct: Least to greatest is $\frac{1}{6} < \frac{1}{4} < \frac{1}{3}$ : the largest denominator gives the smallest fraction.

Comparing only the numerators when the denominators differ, e.g. putting $\frac{5}{12}$ after $\frac{2}{3}$ because $5 > 2$

Why it's wrong: Numerators can only be compared directly when the pieces have the same size.

Correct: Rewrite first: $\frac{2}{3} = \frac{8}{12}$ , and $\frac{5}{12} < \frac{8}{12}$ , so $\frac{5}{12} < \frac{2}{3}$ .

Changing only the denominator when rewriting: $\frac{3}{4} = \frac{3}{12}$

Why it's wrong: Multiplying only the denominator makes the pieces smaller without adding more of them, so the value changes.

Correct: Multiply the numerator and the denominator by the same number: $\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$ .

Mixing up ascending and descending order

Why it's wrong: Students order correctly from least to greatest, but the question asked for greatest to least (or the other way round).

Correct: Underline the words in the question first. Ascending starts with the smallest fraction ( $<$ ), descending starts with the greatest ( $>$ ).

Why It Matters

Ordering fractions is something you do whenever several amounts have to be ranked:

Cooking: a recipe lists $\frac{3}{4}$ cup of flour, $\frac{2}{3}$ cup of sugar and $\frac{1}{2}$ cup of milk. Which ingredient do you need most?
Sports: runners cover $\frac{7}{10}$ km, $\frac{3}{4}$ km and $\frac{4}{5}$ km in training. Who ran farthest?
Surveys: $\frac{3}{5}$ of one class and $\frac{5}{8}$ of another walk to school. Which class walks more?
Measuring: drill bits and wrenches are labeled $\frac{3}{16}$ , $\frac{1}{4}$ and $\frac{5}{32}$ inch, and you must find them in size order.

It is also the basis for ordering decimals, percentages and negative numbers later on.

Real World Applications

Recipes

Recipes often give ingredients as fractions of a cup, and you may want to know which ingredient you need the most or the least of.

Example:

A cake uses $\frac{3}{4}$ cup of flour, $\frac{1}{3}$ cup of sugar and $\frac{1}{2}$ cup of milk. With benchmarks: $\frac{1}{3} < \frac{1}{2} < \frac{3}{4}$ , so flour is the largest amount.

1Try It Yourself

A granola recipe uses $\frac{3}{4}$ cup of flour, $\frac{2}{3}$ cup of sugar, $\frac{1}{2}$ cup of milk and $\frac{5}{6}$ cup of oats.

Put the amounts in order from least to greatest.

Step 1: Choose the right answer

Rewrite all four amounts in twelfths and pick the right order:

Race Distances

Coaches compare training distances that are often given as fractions of a kilometer or a mile.

Example:

Three runners covered $\frac{3}{5}$ km, $\frac{3}{4}$ km and $\frac{3}{10}$ km. The numerators are the same, so $\frac{3}{10} < \frac{3}{5} < \frac{3}{4}$ .

2Try It Yourself

In training, runner A ran $\frac{7}{10}$ km, runner B ran $\frac{3}{4}$ km, runner C ran $\frac{4}{5}$ km and runner D ran $\frac{13}{20}$ km.

Which distance is the second shortest?

Step 1: Write the mathematical expression

Rewrite every distance in twentieths and give the second shortest one as a fraction:

Survey Results

Survey results from groups of different sizes are often reported as fractions, and ranking them shows which group answered yes most often.

Example:

In a survey, $\frac{1}{4}$ of the students chose music, $\frac{1}{2}$ chose sports and $\frac{1}{8}$ chose art. Same numerators: $\frac{1}{8} < \frac{1}{4} < \frac{1}{2}$ .

3Try It Yourself

Four classes were asked whether they walk to school. Yes answers: class A $\frac{3}{5}$ , class B $\frac{5}{8}$ , class C $\frac{11}{20}$ , class D $\frac{16}{25}$ .

Which class has the largest share of students who walk?

Step 1: Write the mathematical expression

Write each fraction as a decimal and give the largest one as a fraction:

Key Takeaways

1Ascending order goes from least to greatest ( $<$ ), descending order from greatest to least ( $>$ )
2Same denominator: order the numerators
3Same numerator: the larger the denominator, the smaller the fraction
4Different denominators: rewrite every fraction with the least common denominator, order the numerators, then write the original fractions
5Decimals are fastest when the LCD is large and the denominators divide easily (2, 4, 5, 8, 10, 20, 25)
6Benchmarks $0$ , $\frac{1}{2}$ and $1$ split the list into groups, so you only compare fractions inside each group

Frequently Asked Questions

How do I choose the fastest method?

Look at the whole list. Same denominators or same numerators? Use that rule. Do the benchmarks

0

\frac{1}{2}

and

1

split the fractions into clear groups? Sort first. Otherwise use the LCD when it is small (like 12 or 24), or decimals when the LCD is large or you have a calculator.

What if two fractions in the list are equal?

Equivalent fractions such as

\frac{2}{4}

and

\frac{3}{6}

take the same place in the order. Write them next to each other with an equals sign:

\frac{1}{4} < \frac{2}{4} = \frac{3}{6} < \frac{5}{6}

Do I have to use the least common denominator?

No, any common denominator works. Multiplying all the denominators together always gives one. The least common denominator just keeps the numbers small, which means fewer mistakes.

How can I check my order?

Turn every fraction into a decimal with a calculator. In ascending order each decimal must be larger than the one before it, in descending order smaller.

Glossary

Ascending order: Arranged from least to greatest, written with $<$
Descending order: Arranged from greatest to least, written with $>$
Common denominator: A denominator shared by several fractions after they are rewritten as equivalent fractions
Least common denominator (LCD): The smallest common denominator, equal to the least common multiple of the denominators
Benchmark fraction: A familiar reference value such as $0$ , $\frac{1}{2}$ or $1$ used to sort fractions quickly
Equivalent fractions: Fractions that name the same amount, like $\frac{3}{4}$ and $\frac{9}{12}$

Formula Card

Same denominator

\frac{a}{d} < \frac{b}{d} \text{ when } a < b

Order the numerators

Same numerator

\frac{n}{a} < \frac{n}{b} \text{ when } a > b

The larger denominator gives the smaller fraction

Rewriting with a common denominator

\frac{a}{b} = \frac{a \times k}{b \times k}

Multiply numerator and denominator by the same number k until every fraction has the LCD

Fraction to decimal

\frac{a}{b} = a \div b

Divide the numerator by the denominator, then order the decimals

Teacher Guide: Ordering Fractions

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Recipes

Race Distances

Survey Results

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Same denominator

Same numerator

Rewriting with a common denominator

Fraction to decimal

More in This Topic

Comparing Fractions with the Same Denominator

Comparing Fractions with the Same Numerator

Fractions on a Number Line

Comparing Mixed Numbers