Teacher Guide: Ordering Mixed Numbers

Learn how to put three to five mixed numbers and improper fractions in order, from least to greatest or from greatest to least.

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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 mixed numbers and improper fractions from least to greatest and from greatest to least
Change improper fractions into mixed numbers to compare whole parts first
Order fraction parts with a common denominator, benchmarks or decimals
Solve real problems that ask for the largest, the smallest or a position in the order

Prerequisites

• Introduction to Mixed Numbers
• Converting Improper Fractions to Mixed Numbers
• Comparing Mixed Numbers
• Comparing Fractions Using Benchmarks

Discussion Starters

1. Why is $1\frac{9}{10}$ smaller than $2\frac{1}{10}$ , even though $\frac{9}{10}$ is much bigger than $\frac{1}{10}$ ?
2. How can you tell quickly that $\frac{13}{6}$ is greater than 2?
3. When would you rather use decimals, and when a common denominator?
4. In a race, is the winner the greatest or the smallest number? Why?

Common Misconceptions

Improper fractions are always smaller than mixed numbers because they have no whole part

Remediation: Have students change $\frac{13}{6}$ into $2\frac{1}{6}$ and place it on the number line next to $1\frac{3}{4}$ . The whole part was there all along, only hidden.

The number with the biggest fraction part is the biggest number

Remediation: Ask whether $1\frac{9}{10}$ m of rope is longer than $2\frac{1}{10}$ m. Mark both on the number line: whole parts decide first.

A bigger denominator means a bigger fraction

Remediation: Cut two equal strips into 5 and 8 pieces and compare one piece of each. Then compare $2\frac{1}{5}$ and $2\frac{1}{8}$ .

Differentiation Ideas

For Struggling Students:

• Start with lists where all whole parts are different
• Use the number line visual for every list before writing symbols
• Give only fraction parts with the same denominator at first

For On-Level Students:

• Order four numbers mixing mixed numbers and improper fractions
• Choose between common denominators, benchmarks and decimals
• Solve word problems that ask for the second largest or second smallest value

For Advanced Students:

• Order five or more numbers with denominators like 7, 9 and 12
• Find a mixed number that fits between two given numbers in the order
• Write their own ordering puzzles where two numbers are equal

Standards Alignment

4.NF.A.2 (CCSS.MATH.CONTENT.4.NF.A.2)
Compare two fractions with different numerators and different 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 mixed numbers and improper fractions between 0 and 5 and read the order from left to right
activityHuman Number Line
Each student holds a card with a mixed number or improper fraction, and the class lines up in order
worksheetOrder the Measurements
Lists of plank lengths, race times and recipe amounts to put in order

Lesson Content

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

Definition

To order a list of numbers means to write them from least to greatest (ascending order, with

<

) or from greatest to least (descending order, with

>

). When the list mixes mixed numbers like

2\frac{1}{3}

and improper fractions like

\frac{11}{4}

, use this plan:

1.Write every number in the same form. Change each improper fraction into a mixed number: $\frac{11}{4} = 2\frac{3}{4}$ , because $11 = 2 \times 4 + 3$ .
2.Sort by the whole parts. A number with a bigger whole part is always bigger: every $3\frac{\square}{\square}$ is greater than every $2\frac{\square}{\square}$ .
3.Break ties with the fraction parts. Numbers with the same whole part are ordered by their fraction parts. Use a common denominator or a benchmark like $\frac{1}{2}$ .
4.Write the order with $<$ or $>$ , using the numbers exactly as they were given.

Another way: change every number into a decimal (

\frac{11}{4} = 2.75

2\frac{1}{3} \approx 2.33

) and order the decimals.

1\frac{7}{8} < 2\frac{1}{4} < \frac{7}{3} < \frac{5}{2}

Worked Examples

Order from least to greatest: $3\frac{1}{2}$ , $1\frac{4}{5}$ , $2\frac{1}{3}$

Look at the whole parts

$3\frac{1}{2}$ has 3, $1\frac{4}{5}$ has 1, $2\frac{1}{3}$ has 2 → Whole parts: 3, 1, 2

Check for equal whole parts

All three whole parts are different, so the fraction parts do not matter → Sort by whole parts only

Sort the whole parts from least to greatest

$1 < 2 < 3$ → $1\frac{4}{5} < 2\frac{1}{3} < 3\frac{1}{2}$

Answer: $1\frac{4}{5} < 2\frac{1}{3} < 3\frac{1}{2}$ . Even though $\frac{4}{5}$ is the biggest fraction part, $1\frac{4}{5}$ is the smallest number.

Common Mistakes

Thinking an improper fraction like $\frac{13}{6}$ is small because it looks like an ordinary fraction

Why it's wrong: A fraction whose numerator is bigger than its denominator is more than 1. $\frac{13}{6}$ is more than 2.

Correct: Change it first: $\frac{13}{6} = 2\frac{1}{6}$ , which is greater than $1\frac{3}{4}$ .

Ordering by the fraction parts and ignoring the whole parts, for example $2\frac{1}{4} < 1\frac{7}{8}$

Why it's wrong: The fraction part is always less than 1, so it can never make up for a smaller whole part.

Correct: Sort by whole parts first: $1\frac{7}{8} < 2\frac{1}{4}$ because $1 < 2$ .

Thinking a bigger denominator means a bigger number, for example $2\frac{1}{8} > 2\frac{1}{5}$

Why it's wrong: With the same numerator, a bigger denominator means smaller pieces.

Correct: $\frac{1}{8} < \frac{1}{5}$ , so $2\frac{1}{8} < 2\frac{1}{5}$ .

Writing the order in the wrong direction

Why it's wrong: The question asked for greatest to least (or for the fastest time first), but the list was written from least to greatest out of habit.

Correct: Read the question again before writing the answer: greatest to least uses $>$ , least to greatest uses $<$ . The fastest time is the smallest number.

Why It Matters

Putting several measurements in order is something you do all the time:

Woodwork: Planks of $2\frac{5}{8}$ m, $\frac{17}{6}$ m and $3\frac{1}{10}$ m must be sorted from longest to shortest before cutting.
Sports: Race times like $\frac{11}{4}$ hours and $2\frac{2}{3}$ hours are ordered to find the winner and the podium.
Cooking: Recipes list amounts like $1\frac{3}{4}$ cups and $\frac{5}{3}$ cups, and you may want to know which ingredient you need most of.
Science: Results of an experiment are ranked to see a pattern.

In the lesson Comparing Mixed Numbers you compared two numbers. Here you will order a whole list at once, even when some numbers are written as improper fractions.

Real World Applications

Woodwork and Building

Carpenters sort boards by length before cutting, so the longest boards are used for the longest parts.

Example:

Boards of $2\frac{1}{2}$ m, $\frac{9}{4}$ m and $2\frac{3}{5}$ m: since $\frac{9}{4} = 2\frac{1}{4}$ , the order from shortest to longest is $\frac{9}{4} < 2\frac{1}{2} < 2\frac{3}{5}$ .

1Try It Yourself

Three planks are $3\frac{3}{8}$ m, $\frac{10}{3}$ m and $3\frac{2}{5}$ m long.

Which plank is the longest?

Step 1: Write the mathematical expression

Write $3\frac{3}{8}$ as the whole part plus the fraction part, so a calculator can turn it into a decimal:

Race Results

Race organizers rank finishing times from smallest to largest to find the winner and the podium.

Example:

Times of $1\frac{1}{2}$ h, $\frac{7}{5}$ h and $1\frac{3}{4}$ h: since $\frac{7}{5} = 1.4$ , $1\frac{1}{2} = 1.5$ and $1\frac{3}{4} = 1.75$ , the winner is the runner with $\frac{7}{5}$ h.

2Try It Yourself

In a trail race, Lea finishes in $\frac{9}{4}$ h, Tom in $2\frac{1}{3}$ h and Sam in $2\frac{1}{6}$ h.

What is the winning (smallest) time?

Step 1: Write the mathematical expression

Write Lea's time $\frac{9}{4}$ as a division you can type into a calculator:

Cooking and Baking

Cooks compare the amounts in a recipe to see which ingredients they need most of and what to buy first.

Example:

A bread recipe uses $\frac{7}{2}$ cups of flour, $1\frac{1}{4}$ cups of water and $\frac{3}{4}$ cup of seeds: $\frac{3}{4} < 1\frac{1}{4} < \frac{7}{2}$ , so flour is the biggest amount.

3Try It Yourself

A granola recipe uses $\frac{11}{8}$ cups of oats, $1\frac{1}{3}$ cups of milk, $1\frac{1}{2}$ cups of flour and $\frac{5}{4}$ cups of nuts.

What is the second smallest amount?

Step 1: Write the mathematical expression

Write the oats amount $\frac{11}{8}$ as a division you can type into a calculator:

Key Takeaways

1Ascending order goes from least to greatest ( $<$ ), descending order goes from greatest to least ( $>$ )
2Change every improper fraction into a mixed number first, for example $\frac{11}{4} = 2\frac{3}{4}$
3Sort by whole parts: a bigger whole part always means a bigger number
4When whole parts are equal, compare the fraction parts with a common denominator or with the benchmark $\frac{1}{2}$
5Changing every number into a decimal is another reliable way to order them
6Write the final order with the numbers exactly as they were given, and check the direction the question asks for

Frequently Asked Questions

Do I have to change improper fractions into mixed numbers?

It is the quickest way to see the whole part. You can also change every mixed number into an improper fraction (for example

2\frac{1}{3} = \frac{7}{3}

) and compare with a common denominator, but the numbers get bigger.

What if two numbers in the list are equal?

Then they share the same place in the order. For example

\frac{9}{4} = 2\frac{1}{4}

, so you write

2\frac{1}{4} = \frac{9}{4}

next to each other.

When are decimals better than a common denominator?

Decimals are handy when the denominators share no small common multiple, like 7, 8 and 10, or when you have a calculator. A common denominator is better for halves, thirds, quarters and sixths, because it gives exact answers.

Where does a whole number like 3 go?

A whole number is a mixed number with fraction part 0. So

2\frac{9}{10} < 3 < 3\frac{1}{9}

Glossary

Mixed number: A whole number and a proper fraction together, like $2\frac{3}{4}$
Improper fraction: A fraction whose numerator is greater than or equal to its denominator, like $\frac{11}{4}$
Ascending order: From least to greatest, written with $<$
Descending order: From greatest to least, written with $>$
Common denominator: A denominator shared by several fractions, so their numerators can be compared directly
Benchmark fraction: A well known fraction such as $\frac{1}{2}$ used as a reference point to compare other fractions quickly

Formula Card

Improper fraction to mixed number

\frac{a}{b} = q\frac{r}{b}

where

a = q \times b + r

Divide the numerator by the denominator: the quotient $q$ is the whole part, the remainder $r$ is the new numerator.

Whole parts first

q_1 < q_2 \Rightarrow q_1\frac{r_1}{b_1} < q_2\frac{r_2}{b_2}

A smaller whole part always gives a smaller number, whatever the fraction parts are.

Equal whole parts

q\frac{r_1}{b} < q\frac{r_2}{b} \iff r_1 < r_2

With the same whole part and a common denominator, compare the numerators.

Teacher Guide: Ordering Mixed Numbers

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

Woodwork and Building

Race Results

Cooking and Baking

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Improper fraction to mixed number

Whole parts first

Equal whole parts

More in This Topic

Comparing Fractions with the Same Denominator

Comparing Fractions with the Same Numerator

Fractions on a Number Line

Comparing Mixed Numbers