2.5 Ordering Fractions

Learning Objectives

Introduction

The Ice Cream Sundae Fundraiser

The students in the sixth grade have decided to add a “Make Your Own Ice Cream Sundae” to the sixth grade social. They figure if they charge $1.50 per sundae, then they can make some money for the next sixth grade social.

On Friday night, the first eight students came in and made their sundaes. They looked delicious!

Here is what the students chose for their ice cream and toppings.

 chose vanilla ice cream

 chose chocolate ice cream

 chose sprinkles

 chose hot fudge

 chose caramel

 chose nuts

Terrence wants to figure out which toppings were the most popular and which toppings were the least popular. You are going to help him do this.

In this lesson, you will learn all about ordering fractions. When you see this problem again at the end of the lesson, you will know how to help Terrence write the toppings in order from the most popular or greatest to the least popular.

Guided Learning

Compare Fractions Using Lowest Common Denominator

If you think back to our last lesson on equivalent fractions, you may have noticed that some fractions have different denominators. Remember that when we are talking about the denominator we are talking about the bottom number of the fraction. The numerator refers to the top number of the fraction.

Example A

 and 

Here we have two different fractions with two different denominators. Remember that the denominator lets us know how many parts one whole has been divided into. Here the first fraction, one fourth, has been divided into four parts. The second fraction, two thirds, has been divided into three parts. In this example, we have two different fractions to compare.

How do we compare fractions?

When we compare two fractions, we want to figure out which fraction is larger and which one is smaller. If we have two fractions with the same denominator, it becomes easier to determine which fraction is greater and which one is less.

Example B

We want to use greater than >, less than < or equal to = to compare these two fractions. This one is easy because our denominators are the same. They have common or like denominators. Think about this in terms of pizza.

If both pizzas were divided into five pieces and one person has one-fifth of the pizza and the other person has three-fifths of the pizza, who has more pizza?

The person with three-fifths of the pizza has more pizza. Therefore, we can compare those fractions like this.

Example C

How do we compare fractions that do not have common or like denominators?

When we are trying to compare two fractions that do not have like denominators, it helps to rewrite them so that they have a common denominator.

Let’s look at the two fractions we had earlier.

Example D

We want to compare these fractions, but that is difficult because we have two different denominators. We can rewrite the denominators by finding the least common multiple of each denominator. This least common multiple becomes the lowest common denominator.

First, write out the multiples of four and three.

4, 8, 12

3, 6, 9, 12

I can stop there because twelve is the lowest common denominator of both 4 and 3.

Next, we rewrite the each fraction in terms of twelfths. This means we make a fraction equivalent to one fourth in terms of twelfths, and we make a fraction equivalent to two thirds in terms of twelfths.

Remember back to creating equal fractions? We multiplied the numerator and the denominator by the same number to create the equal fraction. Well, half of our work is done for us here. Four times three is twelve. We need to complete our equal fraction by multiplying the numerator by three too.

Now we can work on rewriting two thirds in terms of twelfths.

Now that both fractions have been written in terms of twelfths, we can compare them.

so

Guided Practice

Now it is time for you to practice. Rewrite each with a lowest common denominator and compare using <, >, or =.

Take a minute to check your work. Did you choose the correct lowest common denominator? Did you rewrite each fraction accurately?

Solution:

  1. <
  2. >
  3. =

Use this time to take a few notes about lowest common denominators and comparing fractions.

Order Fractions Using Lowest Common Denominator

We just learned how to compare fractions with different denominators. Sometimes, we need to write fractions in order from least to greatest or from greatest to least. If we have fractions with common denominators, this becomes very simple.

Let’s look at an example.

Example E

Write in order from least to greatest.

Since all of these fractions are written in ninths, the common denominator, we can use the numerators and arrange them in order from the smallest numerator to the largest numerator.

Our answer is .

How do we order fractions that do not have a common denominator?

To do this, we will need to rewrite the fractions using a common denominator. This is the best way to know how to order the fractions.

Example F

If we wanted to write these fractions in order from least to greatest, we would need to rewrite them so that they have a common denominator.

We can use the lowest common denominator (LCD) for 3, 4, 2, and 6. That number would be 12.

Next, we rewrite each fraction in terms of twelfths.

Our answer is .

Guided Learning

Try a few of these on your own.

  1. Rename  (using the lowest common denominator).
  2. Next write them in order from least to greatest.

Check your answers with your neighbor.

Solution:

  1. 12/15, 3/15, 10/15
  2. ⅕, ⅘, ⅔

Real Life Example Completed

The Ice Cream Sundae Fundraiser

Now that you have learned all about comparing and ordering fractions, it is time to help Terrence. Here is the problem once again.

The students in the sixth grade have decided to add a “Make Your Own Ice Cream Sundae” to the sixth grade social. They figure if they charge $1.50 per sundae, then they can make some money for the next sixth grade social.

On Friday night, the first eight students came in and made their sundaes. They looked delicious!

Here is what the students chose for their ice cream and toppings.

 chose vanilla ice cream

 chose chocolate ice cream

 chose sprinkles

 chose hot fudge

 chose caramel

 chose nuts

Terrence wants to figure out which toppings were the most popular and which toppings were the least popular. You are going to help him do this.

First, let’s underline the important information.

Since we are only interested in ordering the toppings, we don’t need to underline the ice cream flavors. The topping that is the most popular is the greatest fraction and the topping that is the least popular is the smallest fraction.

To order these fractions, we will need to rewrite them all with the same lowest common denominator. The lowest common denominator for four and eight is eight.

We only need to change .

Now we can write them in order.

Now we can write the toppings in order from the most popular to the least popular.

  1. Hot fudge
  2. Nuts
  3. Caramel
  4. Sprinkles

Terrence is surprised by his findings. He didn’t think that caramel would be more popular than sprinkles!

Review

Video Resources

Ordering Fractions Video