Fractions

Fractions

whole circleA circle is a geometric shape that we have seen in other lessons. The circle to the left can be used to represent one whole. We can divide this circle into equal parts as shown below.

 

 

 


circle halves

This circle has been divided into 2 equal parts.


circle thirds

This circle has been divided into 3 equal parts.


circle fourths

This circle has been divided into 4 equal parts.


We can shade a portion of a circle to name a specific part of the whole as shown below.

circle one half red

shaded_1over2.gif


circle two thirds pink

shaded_2over3.gif


circle one fourth blue

shaded_1over4.gif


examples of fractions

Definition: A fraction names part of a region or part of a group. The top number of a fraction is called its numerator and the bottom part is its denominator.

So a fraction is the number of shaded parts divided by the number of equal parts as shown below:

number of shaded parts   numerator

number of equal parts      denominator

Looking at the numbers above, we have:

circle one half red

There are two equal parts, giving a denominator of 2. One of the parts is shaded, giving a numerator of 1.

example1_1.gif

1over2.gif


circle two thirds pink

There are three equal parts, giving a denominator of 3. Two of the parts are shaded, giving a numerator of 2.

example2_part2.gif

2over3.gif


circle one fourth blue

There are four equal parts, giving a denominator of 4. One of the parts is shaded, giving a numerator of 1.

example3_part1.gif

1over4.gif


Note that the fraction bar means to divide the numerator by the denominator. Let's look at some more examples of fractions. In examples 1 through 4 below, we have identified the numerator and the denominator for each shaded circle. We have also written each fraction as a number and using words.

Example 1:

circle one half red

example1_1.gif

             1over2.gif

         One-half


Example 2:

circle one third pink          circle two thirds pink

example2_part1.gif            example2_part2.gif

              1over3.gif                                      2over3.gif

        One-third                             two-thirds


Example 3:

circle one fourth blue     circle two fourths blue     circle three fourths blue

example3_part1.gif        example3_part2.gif        example3_part3.gif

              1over4.gif                                 2over4.gif                                 3over4.gif

        One-fourth                    Two-fourths                   Three-fourths


Example 4:

circle one fifth green  circle two fifths green  circle three fifths green  circle four fifths green

example4_part1.gif    example4_part2.gif     example4_part3.gif     example4_part4.gif

              1over5.gif                             2over5_1.gif                              3over5.gif                               4over5.gif

         One-fifth                   Two-fifths                   Three-fifths                  Four-fifths


Why is the number 3/4ths written as three-fourths? We use a hyphen to distinguish a fraction from a ratio. For example, "The ratio of girls to boys in a class is 3 to 4." This ratio is written a 3 to 4, or 3:4. We do not know how many students are in the whole class. However, the fraction 3/4 is written as three-fourths (with a hyphen) because 3 is 3/4 of one whole. Thus a ratio names a relationship, whereas, a fraction names a number that represents the part of a whole. When writing a fraction, a hyphen is always used.

It is important to note that other shapes besides a circle can be divided in equal parts. For example, we can let a rectangle represent one whole, and then divide it into equal parts as shown below.

rectangle halves white two equal parts

rectangle thirds white three equal parts

rectangle fourths white four equal parts

rectangle fifths white five equal parts

Remember that a fraction is the number of shaded parts divided by the number of equal parts. In the example below, rectangles have been shaded to represent different fractions.

Example 5:

rectangle one half red 1over2.gifOne-half

rectangle one third pink 1over3.gifOne-third

rectangle one fourth blue 1over4.gifOne-fourth

rectangle one fifth green 1over5.gifOne-fifth

The fractions above all have the same numerator. Each of these fractions is called a unit fraction.

Definition: unit fraction is a fraction whose numerator is one. Each unit fraction is part of one whole (the number 1). The denominator names that part. Every fraction is a multiple of a unit fraction.

In examples 6 through 8, we will identify the fraction represented by the shaded portion of each shape.

Example 6:

In example 6, there are four equal parts in each rectangle. Three sections have been shaded in each rectangle, but not the same three. This was done intentionally to demonstrate that any 3 of the 4 equal parts can be shaded to represent the fraction three-fourths.

A. rectangle three fourths blue 3over4.gif

B. rectangle three fourths blue nonroutine 3over4.gif


Example 7:

In example 7, each circle is shaded in different sections. However, both circles represent the fraction two-thirds. The value of a fraction is not changed by which sections are shaded.

circle two thirds pink  circle two thirds pink rotated

              2over3.gif                              2over3.gif


Example 8:

In example 8, each rectangle is shaded in different sections. However, both rectangles represent the fraction two-fifths. Once again, the value of a fraction is not changed by which sections are shaded.

A. rectangle two fifths green 2over5_2.gif

B. rectangle two fifths green 2over5_2.gif

In the examples above, we demonstrated that the value of a fraction is not changed by which sections are shaded. This is because a fraction is the number of shaded parts divided by the number of equal parts.

Let's look at some more examples.

Example 9:

In example 9, the circle has been shaded horizontally; whereas, in example 10, the circle was shaded vertically. The circles in both examples represent the same fraction, one-half. The positioning of the shaded region does not change the value of a fraction.

circle one half red1over2.gif

Example 10:

circle one half red rotated1over2.gif

Example 11:

In example 11, the rectangle is positioned horizontally; whereas in example 12, the rectangle is positioned vertically. Both rectangles represent the fraction four-fifths. The positioning of a shape does not change the value of the fraction it represents.

rectangle four fifths green4over5.gif

Example 12:

rectangle four fifths green vertical

   4over5.gif

Remember that a fraction is the number of shaded parts divided by the number of equal parts.

In example 13, we will write each fraction using words. Place your mouse over the red text to see if you got it right.

Example 13
Number Words
3over5_0.gif answer 1
2over7.gif answer 2
5over6.gif answer 3
3over8.gif answer 4

Summary: A fraction names part of a region or part of a group. A fraction is the number of shaded parts divided by the number of equal parts. The numerator is the number above the fraction bar, and the denominator is the number below the fraction bar.


Exercises

In Exercises 1 through 5, click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. Note: To write the fraction two-thirds, enter 2/3 into the form.

1. What fraction is represented by the shaded rectangle below?
 
rectangle three fifths small
ANSWER BOX:   

RESULTS BOX: 

2. What fraction is represented by the shaded circle below?
circle five eighths orange small
ANSWER BOX:   

RESULTS BOX: 

3. Write one-sixth as a fraction.
ANSWER BOX:   

RESULTS BOX: 

4. Write three-sevenths as a fraction.
ANSWER BOX:   

RESULTS BOX: 

5. Write seven-eighths as a fraction.
ANSWER BOX:   

RESULTS BOX: