Fractions on Number Line

We know that fractions are expressions that represent part of a whole. Just like whole numbers and integers, fractions can also be plotted on number line. A fraction on a number line is a way to visually show fractions, by plotting them as points on a line. In this representation, the denominator tells us how many equal parts the interval should be divided into, and the numerator tells how many of those equal parts to count from the left to locate the fraction. 
 

By plotting fractions on a number line, we can understand where fractions lie in relation to whole numbers, and shows that fractions are real numbers placed at precise distances on the number line. 
 

For example, to plot \(\frac{3}{5}\) on a number line, draw a number line from 0 to 1, divide it into 5 equal parts, then count 3 parts from 0, and that point is \(\frac{3}{5}\). 
 

To plot fractions on a number line, use the steps listed below:

Step 1: Draw the horizontal number line. 
Decide the range you require. Usually you can take from 0 to 1 for proper fractions, or between relevant whole numbers for mixed or improper fractions. 
 

Step 2: Identify the type of fraction. 
Identify whether the fraction is proper, improper or mixed. 

• For a proper fraction, the numerator will be less than the denominator (numerator < denominator). For example, \(\frac{3}{4}, \frac{7}{10}\). It lies between 0 and 1 on a number line. 
   
• For an improper fraction, the numerator will be greater than or equal to the denominator (numerator ≥ denominator). It will be greater than or equal to 1. For example, \(\frac{7}{5}, \frac{11}{6}\), etc. 
   
• A mixed fraction consists of a whole number plus a proper fraction. For example, \(2\frac{2}{5}\). It also lies between two consecutive whole numbers. 
   

Step 3:  Mark the relevant whole numbers on the line. 
For a proper fraction, mark 0 and 1. For a mixed fraction or improper fraction, mark the two whole numbers between which the fraction lies. For example, for \(\frac{7}{5}\), which can be reduced to \(1\frac{2}{5}\), mark 1 and 2. 
 

Step 4: Divide the segment into equal parts according to the denominator. 
Between two integers, divide the interval into as many equal segments as the denominator of the fraction. For example, if the fraction is \(\frac{2}{5}\), divide the interval into 5 equal segments. 

Step 5: From the left, count forward by the numerator. 
Starting from the left endpoint, which is either 0 or the lower whole number in a mixed or improper fraction, move right by as many segments as the numerator. 

Step 6: Mark the point. 
Place a dot or mark at that position and label it with the fraction. That point shows the fraction’s value on the number line. 
 

We know that a number line arranges numbers including fractions in ascending order from left to right. As the number line is a visual representation, it's easier to compare fractions, even with different denominators, by simply seeing their positions. 

How to compare two or more fractions using a number line? 
 

Step 1: Plot each fraction on the same number line. 

• For proper fractions, draw a number line from 0 to 1, divide the segment into equal parts to the denominator, then mark each fraction at the correct point. 
   
• For mixed numbers or improper fractions, extend the number line beyond 1, mark the relevant whole numbers, divide intervals and plot the points. 

Step 2: Compare the positions.
The fraction whose point lies further to the right is the greater fraction. The one to the left is the smaller. 

Step 3: Use visual comparison if needed. 
Rather than always converting to common denominators or decimals, number line placement helps in easy comparison. 
 

There are different ways to represent fractions on a number line based on their type:

• Proper Fractions: These fractions are less than 1. They are placed between 0 and 1 on the number line.
   
• Improper Fractions: These are greater than or equal to 1. They appear to the right of 1 on the number line.
   
• Mixed Fractions: These consist of a whole number and a proper fraction. They are plotted by marking the whole number first, then moving ahead by the fractional part.
   
• Equivalent Fractions: Fractions that represent the same value appear at the same position on the number line. 

To represent equivalent fractions on a number line, follow the steps below:

Step 1: Express all the given fractions in their simplest form.

Step 2: Check if the fractions have the same simplest form to confirm they are equivalent.

Step 3: Now, we plot the fractions on the number line.

Step 4: Equivalent fractions occupy the same position on the number line, even though they may look different.
 

Represent proper fractions on a number line using the following steps:

Step 1: Since the numerator is less than the denominator, we represent it on a number line between 0 and 1.

Step 2: Divide the section between 0 and 1 into equal parts based on the denominator.

Step 3: We now count to the numerator from the left to represent the proper fraction on the number line.
 

Represent mixed fractions on a number line using the following steps:

Step 1: Identify the fractional part and whole part of the mixed fraction.

Step 2: Start plotting from the whole number part of the mixed fraction.

Step 3: Based on the denominator of the fractional part, divide the number line into equal parts.

Step 4: Now, mark the given mixed fraction on the number line.
 

Represent improper fractions on a number line using the following steps:

Step 1: If the given fraction is an improper fraction, convert it into a mixed fraction.

Step 2: Next, plot the mixed fraction on the number line.

Step 3: Improper fractions have a numerator greater than or equal to the denominator.

Step 4: If the numerator and denominator are equal, the fraction equals 1 and should be represented on the number line.

Step 5: When the numerator is greater than the denominator, the fraction needs to be converted into its equivalent mixed fraction before plotting.
 

Fractions on a number line are a complex topic in mathematics. Therefore, some tips and tricks are mentioned below to help us master the topic.
 

• Understand equal intervals: Always divide the space between two whole numbers into equal parts based on the denominator. For example, to plot \(\frac{3}{4} \) divide the section between 0 and 1 into four equal parts.
   
• Use the numerator as a counter: The numerator tells you how many parts to move from 0. For \(\frac{3}{4} \)​, move 3 parts to the right after dividing the segment into 4 equal intervals.
   
• Convert improper fractions to mixed numbers: Before plotting an improper fraction like \(\frac{3}{4} \)​, convert it into a mixed number 1\(\frac{3}{4} \) to easily locate it between 1 and 2 on the number line.
   
• Remember negative fractions move left: For negative fractions, move the same number of equal parts to the left of 0. For instance, \(\frac{-3}{4} \) lies three parts to the left of 0.
   
• Practice with benchmarks: Use common fractions like \(\frac{1}{2} \), \(\frac{1}{3} \)and \(\frac{3}{4} \) as reference points. These help you quickly estimate where other fractions fall on the number line.
   
• Use visual aids and everyday objects: Parents and teachers can demonstrate fraction on number line to students by folding paper, cutting fruit or using measuring cups. Link these visuals to positions on a number line to strengthen conceptual understanding. 
   
• Emphasize equal spacing: Remind learners that the spaces between marks represent equal parts. Make use of graph papers or rulers to help children make accurately spaced intervals. 
   
• Start with benchmarks: Teach students to compare fractions using familiar anchor points like \(\frac{1}{2}\) or 1. This helps them in understanding easily before moving to complex examples. 
   
• Teach the idea of counting parts instead of jumps: Remind students that if a fraction is \(\frac{3}{4}\), they count three intervals, not points. Parents and teachers can teach them slowly and clearly. 
   
• Connect improper fractions to mixed numbers: Show both improper and mixed fractions on number line for students. So they realize that both \(\frac{7}{4}\) and \(1\frac{3}{4}\) fall at the same position. This helps in understanding equivalence of fractions. 
   

Fractions play an important role in our everyday life. Visualizing fractions on a number line helps us measure the accurate portion of a whole amount. Here, we will look at a few real-life examples of fractions:

• While cooking, we use fractions to measure the quantity of ingredients, such as\(\frac{1}{2} \)cup of sugar.

• We can use fractions to manage time by dividing it into sections. For example: 30 minutes = \(\frac{1}{2} \) of an hour.

• With a fraction number line, students can determine the distance traveled and the distance remaining during a travel journey.

• When shopping, we may come across discounts that involve fractions. For example: \(\frac{1}{2} \) off an item.

• Fractions on a number line help us measure the remaining time or distance in a fitness routine. 
  For example: \(\frac{1}{4} \) of the workout routine.