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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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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

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 19\frac{1}{9}, 59\frac{5}{9} and 89\frac{8}{9} without writing anything down? How did you know?
  • 4. Which list is easier to order: 23,25,27\frac{2}{3}, \frac{2}{5}, \frac{2}{7} or 23,35,57\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 12\frac{1}{2}, 14\frac{1}{4} and 18\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): 14<12<34\frac{1}{4} < \frac{1}{2} < \frac{3}{4}
  • Descending order (greatest to least): 34>12>14\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. 1.Same denominator: order the numerators. 29<49<79\frac{2}{9} < \frac{4}{9} < \frac{7}{9}
  2. 2.Same numerator: the larger the denominator, the smaller the fraction. 310<38<35\frac{3}{10} < \frac{3}{8} < \frac{3}{5}
  3. 3.Common denominator: rewrite every fraction with the least common denominator (LCD), then order the numerators.
  4. 4.Decimals: divide numerator by denominator and order the decimals.
  5. 5.Benchmarks 00, 12\frac{1}{2} and 11: 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: 59\frac{5}{9}, 29\frac{2}{9}, 79\frac{7}{9}, 49\frac{4}{9}
1

Check the denominators

All four fractions are ninths → Same denominator: compare the numerators
2

Order the numerators

2<4<5<72 < 4 < 5 < 7 → 2,4,5,72, 4, 5, 7
3

Write the fractions in that order

Keep the denominator 99 → 29<49<59<79\frac{2}{9} < \frac{4}{9} < \frac{5}{9} < \frac{7}{9}

Common Mistakes

Ordering unit fractions by their denominators: writing 13,14,16\frac{1}{3}, \frac{1}{4}, \frac{1}{6} as least to greatest because 3<4<63 < 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 16<14<13\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 512\frac{5}{12} after 23\frac{2}{3} because 5>25 > 2

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

Correct: Rewrite first: 23=812\frac{2}{3} = \frac{8}{12}, and 512<812\frac{5}{12} < \frac{8}{12}, so 512<23\frac{5}{12} < \frac{2}{3}.

Changing only the denominator when rewriting: 34=312\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: 3×34×3=912\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 34\frac{3}{4} cup of flour, 23\frac{2}{3} cup of sugar and 12\frac{1}{2} cup of milk. Which ingredient do you need most?
  • Sports: runners cover 710\frac{7}{10} km, 34\frac{3}{4} km and 45\frac{4}{5} km in training. Who ran farthest?
  • Surveys: 35\frac{3}{5} of one class and 58\frac{5}{8} of another walk to school. Which class walks more?
  • Measuring: drill bits and wrenches are labeled 316\frac{3}{16}, 14\frac{1}{4} and 532\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 34\frac{3}{4} cup of flour, 13\frac{1}{3} cup of sugar and 12\frac{1}{2} cup of milk. With benchmarks: 13<12<34\frac{1}{3} < \frac{1}{2} < \frac{3}{4}, so flour is the largest amount.

1Try It Yourself

A granola recipe uses 34\frac{3}{4} cup of flour, 23\frac{2}{3} cup of sugar, 12\frac{1}{2} cup of milk and 56\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 35\frac{3}{5} km, 34\frac{3}{4} km and 310\frac{3}{10} km. The numerators are the same, so 310<35<34\frac{3}{10} < \frac{3}{5} < \frac{3}{4}.

2Try It Yourself

In training, runner A ran 710\frac{7}{10} km, runner B ran 34\frac{3}{4} km, runner C ran 45\frac{4}{5} km and runner D ran 1320\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, 14\frac{1}{4} of the students chose music, 12\frac{1}{2} chose sports and 18\frac{1}{8} chose art. Same numerators: 18<14<12\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 35\frac{3}{5}, class B 58\frac{5}{8}, class C 1120\frac{11}{20}, class D 1625\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 00, 12\frac{1}{2} and 11 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 00, 12\frac{1}{2} and 11 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 24\frac{2}{4} and 36\frac{3}{6} take the same place in the order. Write them next to each other with an equals sign: 14<24=36<56\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 00, 12\frac{1}{2} or 11 used to sort fractions quickly
Equivalent fractions
Fractions that name the same amount, like 34\frac{3}{4} and 912\frac{9}{12}

Formula Card

Same denominator

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

Order the numerators

Same numerator

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

The larger denominator gives the smaller fraction

Rewriting with a common denominator

ab=a×kb×k\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

ab=a÷b\frac{a}{b} = a \div b

Divide the numerator by the denominator, then order the decimals

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