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.
\begin{align*}\frac{6}{8}\end{align*} chose vanilla ice cream
\begin{align*}\frac{1}{4}\end{align*} chose chocolate ice cream
\begin{align*}\frac{2}{8}\end{align*} chose sprinkles
\begin{align*}\frac{5}{8}\end{align*} chose hot fudge
\begin{align*}\frac{3}{8}\end{align*} chose caramel
\begin{align*}\frac{2}{4}\end{align*} 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
I. 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
\begin{align*}\frac{1}{4}\end{align*} and \begin{align*}\frac{2}{3}\end{align*}
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
\begin{align*}\frac{1}{5} {\underline{\;\;\;\;\;\;\;\;\;}} \frac{3}{5}\end{align*}
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
\begin{align*}\frac{1}{5} < \frac{3}{5}\end{align*} 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
\begin{align*}\frac{1}{4} \underline{\;\;\;\;\;\;\;\;} \frac{2}{3}\end{align*}
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 4 and 3.
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.
\begin{align*}\frac{1}{4} = \frac{}{12}\end{align*}
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 3 too.
\begin{align*}\frac{1}{4} = \frac{3}{12}\end{align*}
Now we can work on rewriting two-thirds in terms of twelfths.
\begin{align*}\frac{2}{3} = \frac{8}{12}\end{align*}
Now that both fractions have been written in terms of twelfths, we can compare them.
\begin{align*}\frac{3}{12} < \frac{8}{12}\end{align*}
so
\begin{align*}\frac{1}{4} < \frac{2}{3}\end{align*}
Now it is time for you to practice. Rewrite each with a lowest common denominator and compare using <, >, or =.
- \begin{align*}\frac{2}{5} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{6}{10}\end{align*}
- \begin{align*}\frac{2}{3} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{1}{9}\end{align*}
- \begin{align*}\frac{3}{4} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{6}{8}\end{align*}
Take a minute to check your work. Did you choose the correct lowest common denominator? Did you rewrite each fraction accurately?
Use this time to take a few notes about lowest common denominators and comparing fractions.
II. 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
Write in order from least to greatest. \begin{align*}\frac{4}{9},\frac{2}{9},\frac{8}{9},\frac{3}{9},\frac{6}{9}\end{align*}
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 \begin{align*}\frac{2}{9},\frac{3}{9},\frac{4}{9},\frac{6}{9},\frac{8}{9}\end{align*}.
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 \begin{align*}\frac{2}{3},\frac{1}{4},\frac{1}{2}, \frac{5}{6}\end{align*}
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.
\begin{align*}\frac{2}{3}=\frac{8}{12}\\ \frac{1}{4}=\frac{3}{12}\\ \frac{1}{2}=\frac{6}{12}\\ \frac{5}{6}=\frac{10}{12}\end{align*}
Our answer is \begin{align*}\frac{3}{12},\frac{6}{12},\frac{8}{12},\frac{10}{12}=\frac{1}{4},\frac{1}{2},\frac{2}{3},\frac{5}{6}\end{align*}.
Try a few of these on your own.
- Rename \begin{align*}\frac{4}{5},\frac{1}{5},\frac{2}{3}\end{align*}.
- Next write them in order from greatest to least.
Check your answers with your neighbor.
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.
\begin{align*}\frac{6}{8}\end{align*} chose vanilla ice cream
\begin{align*}\frac{1}{4}\end{align*} chose chocolate ice cream
\begin{align*}\frac{2}{8}\end{align*} chose sprinkles
\begin{align*}\frac{5}{8}\end{align*} chose hot fudge
\begin{align*}\frac{3}{8}\end{align*} chose caramel
\begin{align*}\frac{2}{4}\end{align*} 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 4 and 8 is 8.
We only need to change \begin{align*}\frac{2}{4} = \frac{4}{8} \end{align*}.
Now we can write them in order.
\begin{align*}\frac{2}{8},\frac{3}{8},\frac{4}{8},\frac{5}{8}\end{align*}
Now we can write the toppings in order from the most popular to the least popular.
- Hot fudge
- Nuts
- Caramel
- Sprinkles
Terrence is surprised by his findings. He didn’t think that caramel would be more popular than sprinkles!
Technology Integration
Khan Academy Ordering Numeric Expressions
James Sousa Ordering Fractions with Different Denominators
Other Videos:
- http://www.mathplayground.com/howto_comparefractions.html – This is a great video on comparing and ordering fractions. The information is presented very clearly.
Practice Set
Directions: Rename each in terms of tenths.
1. \begin{align*}\frac{1}{5}\end{align*}
2. \begin{align*}\frac{3}{5}\end{align*}
3. \begin{align*}\frac{1}{2}\end{align*}
4. \begin{align*}\frac{4}{5}\end{align*}
Directions: Complete each equal fraction.
5. \begin{align*}\frac{1}{3} = \frac{}{\;9\;}\end{align*}
6. \begin{align*}\frac{2}{3} = \frac{}{18}\end{align*}
7. \begin{align*}\frac{5}{6} = \frac{}{18}\end{align*}
8. \begin{align*}\frac{2}{7} = \frac{}{14}\end{align*}
9. \begin{align*}\frac{4}{9} = \frac{}{36}\end{align*}
10. \begin{align*}\frac{3}{4} = \frac{}{48}\end{align*}
Directions: Identify the lowest common denominator for each pair of numbers.
11. 3 and 6
12. 4 and 10
13. 5 and 3
14. 7 and 2
15. 8 and 4
16. 6 and 4
17. 8 and 5
18. 12 and 5
19. 9 and 2
20. 6 and 7
Directions: Compare the following fractions using <, >, or =
21. \begin{align*}\frac{1}{2} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{1}{3}\end{align*}
22. \begin{align*}\frac{2}{3} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{3}{9}\end{align*}
23. \begin{align*}\frac{4}{6} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{2}{3}\end{align*}
24. \begin{align*}\frac{6}{10} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{4}{5}\end{align*}
25. \begin{align*}\frac{9}{18} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{3}{6}\end{align*}
Directions: Write each series in order from least to greatest.
26. \begin{align*}\frac{5}{6},\frac{1}{3},\frac{4}{9}\end{align*}
27. \begin{align*}\frac{6}{7},\frac{1}{4},\frac{2}{3}\end{align*}
28. \begin{align*}\frac{6}{6},\frac{4}{5},\frac{2}{3}\end{align*}
29. \begin{align*}\frac{1}{6},\frac{4}{5},\frac{2}{4}\end{align*}
30. \begin{align*}\frac{1}{9},\frac{4}{7},\frac{2}{9},\frac{7}{8}\end{align*}
Review
Find equivalent fractions with common denominators in order to compare fractions.