Integer Bars: Learning Fractions

The word fraction means a portion of a whole. For example, if you have a whole pizza cut into 8 equal slices, then each slice is a fraction of the whole pizza. A group of slices (for example, 2 or 3 slices) is also a fraction of the whole pizza. Here is a picture of a whole pizza cut into 8 equal slices:

Unit - In previous pages we have always used the white integer bar to represent 1 unit. In this case the whole pizza is 1 unit. The 1 unit is divided into 8 equal slices. Each slice is less than 1 unit. In this case one slice is 1/8 (one eighth) of the whole pizza. Here are some examples to show this. In this first example we have 4 slices each of which is 1/8 of the whole pizza.

You will also notice that this is exactly one half of the whole pizza. This can be written as:

If you divide the 4 by 4 you get a 1 and if you divide the 8 by 4 you get a 2. That is how we convert the fraction 4/8 to the fraction 1/2. Both fractions are equal. Converting a fraction this way is called simplification. The fractions 4/8 and 1/2 are called equivalent fractions because they both represent the same number of slices of the pizza. In the next example we have 6 slices or 6/8 of the whole pizza:

Following the same method used to simplify 4/8 into 1/2, we can also simplify 6/8 by dividing each number by 2 and get the answer 3/4. The above picture shows 3/4 of the whole pizza. One last example where we have 2 slices of the whole pizza:

The 2/8 can be simplified to 1/4 by dividing each number by 2. In the last example we have 1/4 (one fourth) of the whole pizza.

Some fractions can not be simplified because both numbers can not be evenly divided by the same number. For example, in the fraction 5/8 the 8 can be evenly divided by either 2, 4, or 8 but the 5 can not be evenly divided by any of these numbers. Therefore, 5/8 can not be simplified. The fraction 10/16 can be simplified by dividing both numbers by 2, which equals 5/8 which can not be simplified any further.

Proportional Fractions - At the beginning of this page we defined the word fraction as a portion of the whole. We have shown that there is more than one way to write the portion that is half of the pizza. One half can be written as 4/8, 2/4 or 1/2. We say that these equivalent fractions are proportional fractions because they are the same portion of the pizza. Here are some different examples of proportional fractions using the integer bars. All of them are equivalent.

Names used in fractions - Let's introduce how the numbers in a fraction are called.

Using integer bars to work with fractions - Now we will show some more examples of fractions using the integer bars. Let's start with one whole orange bar. The orange bar is 1 unit. We will show some examples of different fractions of the orange bar.

Since we are able to fit two yellow bars to match the orange bar, that means that each yellow bar is one half or 1/2 unit. If we add the two yellow bars together we have:

Whenever the numerator and the denominator are the same then the fraction value is 1 unit.

In this example five red bars equal one orange bar, therefore each red bar is one fifth or 1/5 unit. Once again, if we add the five red bars together we have:

Any bar can be defined as one unit. In this next example we will define the dark green bar to be 1 unit.

As you can see, three red bars equal one dark green bar, so each red bar is one third or 1/3 unit. We can add all three red bars together as follows:

In the previous two examples the same red bar is a different fraction because we are using a different size bar to represent one unit. The same red bar is 1/5 (of the orange bar) and 1/3 (of the dark green bar). It depends on what is chosen to be 1 unit.

Four white bars equal one purple bar so each white bar is one fourth or 1/4 unit. Adding the four white bars together gives us the following equation:

The following table shows you how to pronounce the different fractions from 1/2 to 1/10.

In most of our past examples we have used a numerator of 1. Now we will show some examples where the numerator is other than one. The black bar which is size 7 represents 1 unit therefore each white bar of size 1 is 1/7 unit. As you can see in the following two pictures what we have is 3/7 which can be represented with three white bars or one light green bar.

The following example uses the blue bar of size 9 to represent 1 unit. Each white bar represents 1/9 so the five white bars represent 5/9. The yellow bar, which is equivalent to the five white bars, also represents 5/9.

Proper Fractions - when the numerator is smaller than the denominator, as in the previous two examples, it is called a proper fraction.

Improper Fractions - when the numerator is larger than the denominator, as in the following two examples, it is called an improper fraction.

Mixed Fractions - The value of an improper fraction can also be written as a combination of an integer and a fraction. This is called a mixed fraction.

On this next example the dark green bar is defined as 1 unit, so one red bar is 1/3 unit. The fact that the following picture has four red bars means that the fraction is 4/3. This is a case where the numerator is greater than the denominator so it is called an improper fraction. We know that 3/3 equals 1 unit, therefore an improper fraction has a fraction value greater than 1, in this case 4/3. The numerator can also be represented by a single brown bar which is the same size as four red bars.

The three red bars equal 1 unit and the fourth red bar is 1/3, therefore the value of 4/3 can be written as the following mixed fraction,

Using the yellow bar as 1 unit, the following example shows another improper fraction using both 9 white bars as well as a single blue bar. Both of these have a value of 9/5.

In this last example of a mixed fraction the light green bar is defined as 1 unit. Each white bar is 1/3 unit. This example show 13 white bars which is 13/3.

Each group of three white bars is 1 unit so we have a total of four units and an extra white bar. This can be written as the mixed fraction 4 1/3. Here is the equation:

Exercises - Use the integer bars to show each of the following fraction values and tell if it is a proper or improper fraction then write any improper fractions as mixed fractions.

Fraction	Integer Bars Picture	Proper or Improper	Mixed Fraction
1/4
3/9
8/3
7/5
2/7
12/3
3/10
15/4
23/7

After you have completely filled in the table then you can check your answers.

1/2	one half
1/3	one third
1/4	one fourth
1/5	one fifth
1/6	one sixth
1/7	one seventh
1/8	one eighth
1/9	one ninth
1/10	one tenth