Fractions
Objectives
This lesson presents the background material which
will allow you to do the following:
- Given a diagram showing part of a whole, find the associated fraction.
- Given a fraction, find the associated part of a whole in a diagram.
- Recognize equivalent fractions.
- Express a fraction in simplest form.
- Convert mixed numbers to improprer fractions and vice versa.
Introduction
If we wish to measure the number of marbles or pennies that we have in a jar, then natural numbers (1,2,3,...) are all that we need. However, if we wish to measure how much cake we were given after dinner, we probably weren't given 1,2 or 3 cakes but rather a slice that represents a part of 1 whole cake. In this case we need to consider a new set of numbers that can represent parts of a whole. Fractions are such numbers.
Definition
A fraction consists of two numbers
organized vertically with a line between them. The upper number is
called the numerator and the lower number the denominator.
Meaning of the Denominator
The denominator of a fraction
refers to the number of equal parts that a whole has been divided
into.
For example, if we have a pie and
we divide it into two pieces, our associated fraction would
contain 2 in the denominator.
If we divide our pie into three pieces,
our associated fraction would contain 3 in the denominator.
If we divide our pie into four
pieces, our associated fraction would contain 4 in the
denominator.
Meaning of the Numerator
We have learned that the
denominator of a fraction refers to the number of pieces that a
whole quantity has been divided into. The numerator of a fraction
refers to the number of these pieces that we have.
For example, if we divide a pie
into four pieces, our associated fraction would contain 4 in the
denominator to indicate that we have divided the pie into four
pieces. If we have three of these pieces, our numerator would
contain 3.
If we divide a pie into three
pieces, our associated fraction would contain 3 in the denominator
to indicate that we have divided the pie into three pieces. If we
have two of these pieces, our numerator would contain 2.
Click below to practice
visualizing the relationship between numerators, denominators and
the associated fraction.
Click below to practice visually
obtaining the correct fraction of a whole.
Equivalent Fractions
Given a pie, we can use the
meanings of the numerator and the denominator to obtain the
quantity associated with the following fractions.
We can see that these fractions
refer to the same quantity.
Given a pie, we can use the
meanings of the numerator and the denominator to obtain the
quantity associated with the following fractions.
Once again, we can see that these
fractions refer to the same quantity.
As refers to two
wholes or 2, it is worth noting that whole numbers have fractional
forms.
Fractions in Simplest Form
We have seen that there are
numerous ways of expressing the same quantity using fractions.
Given all the possible representations for a quantity, if we
select the fraction where the the numerator and denominator have
the smallest possible values, the fraction is said to be in
simplest form.
For example: Given
We can express this quantity with
the following fractions.
Thus, in
simplest form would be .
Practice:
Mixed Numbers and Improper Fractions
Improper fractions and mixed numbers are used in cases when we have one or more wholes and a fraction. Consider the case where we have 3 and cakes. There are two ways to perceive this. First we can perceive this as three whole cakes and two extra slices where each slice is one third of a cake.
The expression consistent with this perspective is called a mixed number which consists of a whole number followed by a fraction. In this case, a mixed number to represent three and two third cakes is .
The second way we can perceive this is as 11 slices of cake, where each slice is of the whole cake.
The expression consistent with this perspective is called an improprer fraction. Using the defintion of the numerator and the denominator above, we can quickly find that is the appropriate improper fraction to represent three and two thirds cakes.
The following table presents other examples of mixed numbers with their associated improper fractions.
Mixed number |
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Associated Improper Fraction |
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Click below to practice converting between mixed numbers and improper fractions:
Summary
Now that you have finished this lesson, you should be able to do the following:
- Given a diagram showing part of a whole, find the associated fraction.
- Given a fraction, find the associated part of a whole in a diagram.
- Recognize equivalent fractions.
- Express a fraction in simplest form.
- Convert mixed numbers to improprer fractions and vice versa.
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