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A 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. |
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| This circle has been divided into 2 equal
parts. |
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| This circle has been divided into 3 equal
parts. |
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| This circle has been divided into 4 equal
parts. |
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We can shade a portion of a circle to name a specific part of the whole as
shown below.
| 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:
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| There are two equal parts, giving a denominator of 2. One
of the parts is shaded, giving a numerator of 1. |
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| There are three equal parts, giving a denominator of 3. Two
of the parts are shaded, giving a numerator of 2. |
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| There are four equal parts, giving a denominator of 4. One
of the parts is shaded, giving a numerator of 1. |
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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.
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Why is the number  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
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. |
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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.
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two equal parts |
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three equal parts |
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four equal parts |
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five equal parts |
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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.
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| Example 5 |
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one-half |
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one-third |
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one-fourth |
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one-fifth |
The fractions above all have the same numerator. Each of these fractions is called a unit fraction.
| Definition: |
A 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.
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Example 6
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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. |
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Example 7
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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. |
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Example 8
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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. |
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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.
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Let's look at some more examples.
| Example 9 |
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Example 10 |
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Example 12 |
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Example 11 |
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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.
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.
Remember that a fraction is the number of shaded parts divided by the number of equal parts.
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In example 13, we will write each fraction using words. Place your mouse over the red text
to see if you got it right.
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Example 13 |
| Number |
Words |
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answer 1 |
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answer 2 |
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answer 3 |
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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.
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Exercises
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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.
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| 1. |
What fraction is represented by the shaded rectangle below?
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| 2. |
What fraction is represented by the shaded circle below?
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| 3. |
Write one-sixth as a fraction.
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| 4. |
Write three-sevenths as a fraction.
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| 5. |
Write seven-eighths as a fraction.
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