# Fractions

## Visualize Fractions


### Understand the Meaning of Fractions

Andy and Bobby love pizza. On Monday night, they share a pizza equally. How much of the pizza does each one get? Are you thinking that each boy gets half of the pizza? That’s right. There is one whole pizza, evenly divided into two parts, so each boy gets one of the two equal parts.
In math, we write  to mean one out of two parts.
On Tuesday, Andy and Bobby share a pizza with their parents, Fred and Christy, with each person getting an equal amount of the whole pizza. How much of the pizza does each person get? There is one whole pizza, divided evenly into four equal parts. Each person has one of the four equal parts, so each has  of the pizza.
On Wednesday, the family invites some friends over for a pizza dinner. There are a total of  people. If they share the pizza equally, each person would get  of the pizza.
A fraction is a way to represent parts of a whole. The denominator  represents the number of equal parts the whole has been divided into, and the numerator  represents how many parts are included. The denominator,  cannot equal zero because division by zero is undefined.
In , the circle has been divided into three parts of equal size. Each part represents  of the circle. This type of model is called a fraction circle. Other shapes, such as rectangles, can also be used to model fractions.
What does the fraction  represent? The fraction  means two of three equal parts.
In  and , we used circles and rectangles to model fractions. Fractions can also be modeled as manipulatives called fraction tiles, as shown in . Here, the whole is modeled as one long, undivided rectangular tile. Beneath it are tiles of equal length divided into different numbers of equally sized parts.
We’ll be using fraction tiles to discover some basic facts about fractions. Refer to  to answer the following questions:
It takes  twenty-fourths, so 
This leads us to the Property of One.
What if we have more fraction pieces than we need for  whole? We’ll look at this in the next example.

### Model Improper Fractions and Mixed Numbers

In  (b), you had eight equal fifth pieces. You used five of them to make one whole, and you had three fifths left over. Let us use fraction notation to show what happened. You had eight pieces, each of them one fifth,  so altogether you had eight fifths, which we can write as  The fraction  is one whole,  plus three fifths,  or  which is read as one and three-fifths.
The number  is called a mixed number. A mixed number consists of a whole number and a fraction.
Fractions such as  and  are called improper fractions. In an improper fraction, the numerator is greater than or equal to the denominator, so its value is greater than or equal to one. When a fraction has a numerator that is smaller than the denominator, it is called a proper fraction, and its value is less than one. Fractions such as  and  are proper fractions.

### Convert between Improper Fractions and Mixed Numbers

In , we converted the improper fraction  to the mixed number  using fraction circles. We did this by grouping six sixths together to make a whole; then we looked to see how many of the  pieces were left. We saw that  made one whole group of six sixths plus five more sixths, showing that 
The division expression  (which can also be written as ) tells us to find how many groups of  are in  To convert an improper fraction to a mixed number without fraction circles, we divide.
In , we changed  to an improper fraction by first seeing that the whole is a set of five fifths. So we had five fifths and four more fifths.
Where did the nine come from? There are nine fifths—one whole (five fifths) plus four fifths. Let us use this idea to see how to convert a mixed number to an improper fraction.

### Model Equivalent Fractions

Let’s think about Andy and Bobby and their favorite food again. If Andy eats  of a pizza and Bobby eats  of the pizza, have they eaten the same amount of pizza? In other words, does  We can use fraction tiles to find out whether Andy and Bobby have eaten equivalent parts of the pizza.
Fraction tiles serve as a useful model of equivalent fractions. You may want to use fraction tiles to do the following activity. Or you might make a copy of  and extend it to include eighths, tenths, and twelfths.
Start with a  tile. How many fourths equal one-half? How many of the  tiles exactly cover the  tile?
Since two  tiles cover the  tile, we see that  is the same as  or 
How many of the  tiles cover the  tile?
Since three  tiles cover the  tile, we see that  is the same as 
So,  The fractions are equivalent fractions.

### Find Equivalent Fractions

We used fraction tiles to show that there are many fractions equivalent to  For example,  and  are all equivalent to  When we lined up the fraction tiles, it took four of the  tiles to make the same length as a  tile. This showed that  See .
We can show this with pizzas, too. (a) shows a single pizza, cut into two equal pieces with  shaded. (b) shows a second pizza of the same size, cut into eight pieces with  shaded.
This is another way to show that  is equivalent to 
How can we use mathematics to change  into  How could you take a pizza that is cut into two pieces and cut it into eight pieces? You could cut each of the two larger pieces into four smaller pieces! The whole pizza would then be cut into eight pieces instead of just two. Mathematically, what we’ve described could be written as:
These models lead to the Equivalent Fractions Property, which states that if we multiply the numerator and denominator of a fraction by the same number, the value of the fraction does not change.
When working with fractions, it is often necessary to express the same fraction in different forms. To find equivalent forms of a fraction, we can use the Equivalent Fractions Property. For example, consider the fraction one-half.
So, we say that  and  are equivalent fractions.

### Locate Fractions and Mixed Numbers on the Number Line

Now we are ready to plot fractions on a number line. This will help us visualize fractions and understand their values.
Let us locate  and  on the number line.
We will start with the whole numbers  and  because they are the easiest to plot.
The proper fractions listed are  and  We know proper fractions have values less than one, so  and  are located between the whole numbers  and  The denominators are both  so we need to divide the segment of the number line between  and  into five equal parts. We can do this by drawing four equally spaced marks on the number line, which we can then label as  and 
Now plot points at  and 
The only mixed number to plot is  Between what two whole numbers is  Remember that a mixed number is a whole number plus a proper fraction, so  Since it is greater than  but not a whole unit greater,  is between  and  We need to divide the portion of the number line between  and  into three equal pieces (thirds) and plot  at the first mark.
Finally, look at the improper fractions  and  Locating these points will be easier if you change each of them to a mixed number.
Here is the number line with all the points plotted.
In Introduction to  Integers, we defined the opposite of a number. It is the number that is the same distance from zero on the number line but on the opposite side of zero. We saw, for example, that the opposite of  is  and the opposite of  is 
Fractions have opposites, too. The opposite of  is  It is the same distance from  on the number line, but on the opposite side of 
Thinking of negative fractions as the opposite of positive fractions will help us locate them on the number line. To locate  on the number line, first think of where  is located. It is an improper fraction, so we first convert it to the mixed number  and see that it will be between  and  on the number line. So its opposite,  will be between  and  on the number line.

### Order Fractions and Mixed Numbers

We can use the inequality symbols to order fractions. Remember that  means that  is to the right of  on the number line. As we move from left to right on a number line, the values increase.

### Key Concepts

1. Property of One
2. Mixed Numbers
3. Proper and Improper Fractions
4. Convert an improper fraction to a mixed number.
5. Convert a mixed number to an improper fraction.
6. Equivalent Fractions Property

### Practice Makes Perfect

In the following exercises, name the fraction of each figure that is shaded.
In the following exercises, shade parts of circles or squares to model the following fractions.
In the following exercises, use fraction circles to make wholes using the following pieces.
In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.
In the following exercises, draw fraction circles to model the given fraction.
In the following exercises, rewrite the improper fraction as a mixed number.
In the following exercises, rewrite the mixed number as an improper fraction.
In the following exercises, use fraction tiles or draw a figure to find equivalent fractions.
In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.
In the following exercises, plot the numbers on a number line.
In the following exercises, order each of the following pairs of numbers, using  or 

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.



ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.