B. Number · Number Sense: Numbers to 100 000, Fractions, Decimals and Percents

Comparing and Ordering Mixed Numbers and Improper Fractions

B1.4About 35 minutes

Learning goals

I can compare mixed numbers by looking at the whole numbers first.
I can change improper fractions to mixed numbers to compare them.
I can order a list of fractions, mixed numbers and whole numbers.
I can place mixed numbers on a number line.

Words to know

mixed number
A whole number and a fraction together, like 2 1/4.

improper fraction
A fraction equal to or greater than 1, like 9/4.

between
In the middle of two numbers. 2 1/2 is between 2 and 3.

Let's start on the trail

A provincial park has three hiking trails:

Maple Loop: 2 1/2 km
Beaver Pond: 9/4 km
Lookout: 2 2/3 km

Which trail is the shortest? Which is the longest? The numbers are written in different ways, so we need a plan.

A wooden boardwalk crosses a beaver pond with a beaver lodge, surrounded by red and orange autumn maples. — A boardwalk trail beside a beaver pond in a park. Trail lengths are often given in kilometres, sometimes as fractions or mixed numbers. AI-generated image

The plan

Change any improper fractions into mixed numbers.
Compare the whole numbers first. A bigger whole number means a bigger amount.
If the whole numbers are the same, compare the fraction parts using the strategies from the last lesson.

For the trails: 9/4 = 2 1/4. All three trails are 2-and-a-bit km. Now compare 1/4, 1/2 and 2/3. In twelfths: 3/12, 6/12, 8/12.

A number line from 2 to 3, marked in twelfths. Beaver Pond (2 1/4), Maple Loop (2 1/2), Lookout (2 2/3).

So the trails from shortest to longest are Beaver Pond (9/4 km), Maple Loop (2 1/2 km), Lookout (2 2/3 km).

Worked example

Example 1: Different whole numbers

Compare 3 1/5 and 2 4/5.

Look at the whole numbers first: 3 > 2.
The fraction parts don't matter: 2 4/5 is still less than 3.

3 1/5 > 2 4/5

Worked example

Example 2: Improper fraction vs mixed number

Compare 7/3 and 2 1/2.

Change 7/3: 6 thirds make 2 wholes, with 1 left. 7/3 = 2 1/3.
Same whole number (2). Compare 1/3 and 1/2: thirds are smaller than halves.
So 2 1/3 < 2 1/2.

7/3 < 2 1/2

Worked example

Example 3: Order a mixed list

Order from least to greatest: 13/4, 3, 2 7/8, 3 1/2.

Change 13/4: 12 fourths make 3, with 1 left. 13/4 = 3 1/4.
2 7/8 is the only number less than 3. It goes first.
Then 3, then 3 1/4, then 3 1/2 (1/4 < 1/2).

2 7/8, 3, 13/4, 3 1/2

The list from Example 3 on a number line marked in eighths. Only 2 7/8 is less than 3; the others are 3 or between 3 and 4, in the order 3, 13/4, 3 1/2.

Worked example

Example 4: Both improper

Compare 7/4 and 5/3.

7/4 = 1 3/4 and 5/3 = 1 2/3.
Same whole number. Compare 3/4 and 2/3: in twelfths, 9/12 and 8/12.
9/12 > 8/12.

7/4 > 5/3

Whole numbers first

Before comparing, ask: "Which whole numbers is each amount between?" If two amounts are between different whole numbers, you're already done!

Just for fun

Before you lace up your hiking boots, change every improper fraction to a mixed number. Otherwise you might pick the "long" 9/4 km trail and be back before your snack is even unwrapped!

Hands-on activity

Which glass has the most?

You need: three clear glasses or jars that are all the same size, measuring cups (1 cup, 1/2 cup, 1/3 cup and 1/4 cup), and water. Do this at the sink.

Fill glass A with 5/4 cups (five 1/4 cup scoops).
Fill glass B with 1 1/2 cups (one 1 cup scoop and one 1/2 cup scoop).
Fill glass C with 1 1/3 cups (one 1 cup scoop and one 1/3 cup scoop).
Before you look, predict the order from least to greatest. Write your prediction.
Line up the glasses and check the water heights.

Check: 5/4 = 1 1/4, so the order is A (1 1/4), C (1 1/3), B (1 1/2). The heights of A and C are close, so look carefully!

Your turn

Use >, < or =: 4 1/3 ___ 3 2/3

Use >, < or =: 1 5/8 ___ 1 3/8

Use >, < or =: 5/2 ___ 2 1/4

Use >, < or =: 9/3 ___ 3

Use >, < or =: 11/6 ___ 2

Order from least to greatest: 2 1/2, 1 3/4, 2 1/4

Which is greater: 7/4 or 5/3?

More practice: warm-up

Use >, < or =: 2 1/2 ___ 3 1/2

Use >, < or =: 5 1/4 ___ 4 3/4

Use >, < or =: 8/8 ___ 1

Which is greater: 6/5 or 4/5?

Between which two whole numbers is 7/2?

More practice: on your own

Compare 10/4 and 2 3/4.

Order from greatest to least: 3 1/6, 17/6, 3

Jo skied 2 3/4 km. Pat skied 2 5/8 km. Who skied farther?

Compare 1 2/3 and 1 5/6.

Write a mixed number between 4 1/2 and 5.

Challenge

Order from least to greatest: 7/4, 1 5/8, 1 1/2, 2

In spring, a family collects maple sap. Bucket A has 11/3 L, bucket B has 3 3/4 L and bucket C has 3 1/2 L. Order the buckets from most to least sap.

Find the improper fraction with denominator 4 that is between 2 1/2 and 3.

Check your understanding

Quiz

Which is greater?

Compare 9/4 and 2 1/2.

Which is equal to 3?

Which list is in order from least to greatest?

What is the best first step to compare 3 1/8 and 2 7/8?

Which amount is between 4 and 5?

For parents and teachers

Big idea: This lesson finishes B1.4: comparing and ordering fractions including improper fractions and mixed numbers. The main habit is "whole numbers first, then the fraction parts".

Common difficulties:

Being fooled by big fraction parts (thinking 2 7/8 > 3 1/8 because 7/8 > 1/8).
Comparing an improper fraction by its numerator (thinking 9/4 > 2 1/2 because 9 is big). Change it to a mixed number first.
Comparing fraction parts with different denominators. Use equivalent fractions (twelfths are useful for halves, thirds, fourths and sixths).

How to help: Use measuring cups when cooking and ask, "Which is more, 5/4 cups or 1 1/3 cups?" Then pour to check. Maple syrup season (sap buckets, litres of syrup) is a fun Canadian context for comparing amounts.