What is a Fraction?
A fraction is used to represent a whole number that is divided into equal parts. Numbers like 0, 1, 2, 3, 4, and so on are known as whole numbers.
So, fractions are the numbers that exist between each of these whole numbers. There are an infinite number of fractions between any two whole numbers.
Fractions are used mostly in the context of division. In general, a fraction \(\frac{a}{b}\) can be interpreted as \(a\div~b\). But division is not restricted to numbers. We can perform division on things, shapes, and a lot more!
For example, a pizza can be divided into 2 parts. Similarly, shapes like circles, squares, and triangles can also be divided into equal parts. In all these cases, fractions can be used to represent a part of the whole.
How do we express Fractions?
Fractions are expressed in the \(\frac{a}{b}\) form, where \(b\neq 0\).
It is pretty easy to predict where a fraction lies on the number line. If \(b> a\) in \(\frac{a}{b}\) (when the denominator is greater than the numerator), then the fraction lies between 0 and 1. These fractions are called proper fractions. For example, \(\frac{1}{2}\) lies between 0 and 1. The denominator 2 divides the space between 0 and 1 on the number line into two equal parts.
The numerator 1 denotes that the fraction lies exactly in the middle of 0 and 1.
Note that if the numerator is 0 \(\left (\frac{0}{2} \right )\), the fraction is equivalent to 0, and if the numerator is 2 \(\left (\frac{2}{2} \right )\), the fraction is equivalent to 1.
If a > b in \(\left (\frac{a}{b} \right )\) ( when the numerator is greater than the denominator), the fraction lies outside the space between 0 and 1. These fractions are called improper fractions. Improper fractions can be converted into a mixed number having a whole number part, and a fractional part.
In this way, we can easily identify where the number is located on the number line.
For example,\(\frac{3}{2} = \frac{2 + 1}{2} \) Conversion to a mixed number
\( ~~~~~~~~~~~~~~~~~~~~~~~= \frac{2}{2} + \frac{1}{2} \) Simplification
\( ~~~~~~~~~~~~~~~~~~~~~~~~= 1+\frac{1}{2} = 1\frac{1}{2}\)
So, \( \frac{3}{2}\) lies between 1 and 2.
Representing Fractions using models
Fractions can be visually represented in multiple ways. Two of the most effective methods of representing fractions are tape diagrams and area models.
Consider a fraction\( \frac{5}{6}\). We know that this number lies between 0 and 1 as the numerator is smaller than the denominator. We can represent these fractions using a tape diagram as follows.
First, we divide “5 wholes” and divide each of these wholes into 6 equal parts. Each of these small divisions now represents \( \frac{1}{6}\). The fraction \( \frac{5}{6}\) can be obtained by selecting 5 such divisions at a time.
Let’s consider \( 3\div~6\) and express the fraction using the area model. In this case, “3 wholes” are divided into 6 parts. Let’s represent this using the area model.
There are 3 wholes, and each whole is divided into 6 divisions. Each part gets 16 of the whole.
So, \( 3\div~6 = 3\times~\frac{1}{6} = \frac{3}{6}\)
Solved Examples
Example 1: Use a model to represent \( 1\div~4\).
Solution:
\( 1\div~4\) is the same as \( \frac{1}{4}\). To represent this using a tape diagram, we need to divide 1 whole into 4 equal parts.
Each division of the whole represents the fraction \( \frac{1}{4}\). So, \( 1\div~4 = \frac{1}{4}\)
Example 2: Divide \( 4\div~5\) and use a model to represent the result.
Solution:
As we have learned already, \( 4\div~5\) is the same as \( \frac{4}{5}\). We can represent \( \frac{4}{5}\) on a tape diagram by taking “4 wholes” and dividing each of those wholes into 5 divisions.
When we divide 4 wholes into 5 parts, each part is \( \frac{1}{5}\) of a whole. And supposed we take 4 parts out of 5, we obtain, \(4\div~5 = \frac{4}{5}\).
Example 3: A pumpkin pie and an apple pie is shared among 6 people. If both pies are of the same size, use an area model to find the fraction of pie shared with each person.
Solution:
Since the two pies are of the same size, and are shared equally among six people, each person will get an equal share of both pies.
Here “2 wholes” are divided into 6 divisions. Each person gets \(\frac{1}{6}\) of each whole. That means each person gets \(\frac{1}{6}\) of the share of each pie.
Example 4: Suppose you are sharing 3 different chocolate bars with 2 of your friends. Use an area model to find the fraction of chocolate bars that the three of you receive.
Solution:
You are sharing 3 chocolate bars with 2 of your friends. So, “3 wholes” are divided into 3 divisions. The three divisions stand for the two friends and yourself. The tape diagram looks like this:
Here, 3 wholes are divided into 3 divisions. Each part gets \(\frac{1}{3}\) of the whole. So, each person gets \(\frac{1}{3}\) of each chocolate bar.
A fraction \(\frac{a}{b}\) is the same as \(a\div~b\), as a fraction is essentially the same as dividing two numbers.
Fractions and decimals are the same. Fractions can be expressed as decimals, and decimals can be expressed as fractions. They are just two different ways of expressing the numbers between two whole numbers.
A tape diagram helps us in representing a single fraction. On the other hand, the area model helps us visualize real-life applications of fractions. concepts. By looking at what your child is doing correctly and which concepts they understand, you can determine ways to practice areas that they are still developing.