Table of Contents
- What are Fractions?
- Models to understand and represent fractions
- Using models to add Fractions
- Using models to subtract Fractions
- Using models to add Mixed numbers
- Using models to subtract Mixed numbers
- A number line to represent Fractions
- How do we add Fractions on a number line?
- How do we subtract Fractions on a number line?
- Frequently Asked Questions
What are Fractions?
Fractions are represented in mathematics as a numerical value that defines a part of a whole.
A fraction is a part or section of any quantity taken from the whole, and it can be any number, a specific value, or a thing.
For example:
There is a pizza with eight slices and you have one slice from it. Write the fraction to represent the pizza slice you had.
We can write the given conditions as:
- one-eighth, or,
- \( \frac{1}{8}\)
\( \frac{\square }{\square}\)- In this form of representing a fraction, we write the total number of parts below the horizontal line, also known as the denominator. The number of equal parts written above the horizontal line, is known as the numerator.
If we talk about one pizza slice out of eight, total pieces, then the fraction representing this value will have a denominator that represents the total number of slices in the pizza and the numerator represents the slice you had. So the fraction will be \( \frac{1}{8}\) .
Models to Understand and Represent Fractions
- Fraction strips
Fraction strips are rectangular pieces either shown on a digital device or printed on paper and represent different parts of a larger whole.
They can be cut apart and manipulated to see how different parts can be combined to form the whole or to compare fractional amounts for equality.
Using Models to Add Fractions
Modeling fractions using a strip
When the denominator of a fraction is the same, we simply add the numerator and represent it by using a fraction strip.
As shown in the above image, we have to simply add the numerator.
So, \( \frac{3}{5} ~+~ \frac{1}{5}~ =~ \frac{4}{5} \)
Using Models to Subtract Fractions
When the denominator of the fraction is the same, we simply subtract the numerator, and represent it by using a fraction strip.
As shown in the above image, we simply have to subtract one numerator from the other.
So, \( \frac{5}{6}~ -~ \frac{1}{6}~ =~ \frac{4}{6} \)
Using Models to Add Mixed Numbers
There are two parts to mixed fractions: a whole number and a proper fraction. On a fraction strip, we must first mark two strips: the whole number part on the top and its successor on the bottom, in order to represent mixed fractions.
Example: Add \( 1~\frac{2}{5} ~+~ \frac{1}{5} \)
As shown in the above image, we have to simply add the numerator after converting the mixed fraction into a proper fraction.
So, \( \frac{7}{5}~ +~ \frac{1}{5}~=~ \frac{8}{5} \)
Using Models to Subtract Mixed Numbers
As we have already seen earlier, there are two parts to mixed fractions: a whole number and a proper fraction. On a fraction strip, we must first mark two parts, the whole number part on the top and its successor on the bottom, to represent mixed fractions.
When the denominator is not the same, we have to break it into smaller fractions until the denominator becomes the same. After that, we simply subtract the terms and represent the fraction.
Example: Subtract \( 1~\frac{1}{4}~ -~ \frac{1}{2} \)
As shown in the above image, we have to write the mixed number as a fraction and subtract the numerators.
We have: \( 1~\frac{1}{4}~ -~ \frac{1}{2} \)
So, \( \frac{5}{4} ~-~ \frac{1}{2} ~=~\frac{3}{4} \)
There are multiple ways of modeling fractions. We saw how to model fractions using a strip. Now, let us explore another way of modeling fractions using a number line.
A Number Line to Represent Fractions
We can model or represent fractions and mixed numbers on a number line. A number line is the best pictorial representation of numbers as it helps us explore more and learn new aspects related to numbers.
Here is an example of how we can represent numbers, fractions, and mixed numbers on a number line.
How Do We Add Fractions on a Number Line?
Create the number line and mention the addend with a dot. Then we add the part of the other addend from the number line using a curve.
Example: Add \( \frac{4}{8} ~+~ \frac{1}{8} \) and write the answer.
Draw the number line to represent the addend.
Now the addend is \( \frac{1}{8} \). It means that we have to add one part out of 8. Because we are adding one part, we will have to “jump” onwards one step from the maximum.
So, \( \frac{4}{8} ~+~ \frac{1}{8}~ =~ \frac{5}{8} \)
How Do We Subtract Fractions on a Number Line?
Draw the number line and mark the minuend with a dot. We then subtract the part of the subtrahend by making jumps to the left. Let us understand this by looking at an example:
Example: Subtract \( \frac{4}{8} ~- ~\frac{1}{8} \) and write the answer on a number line.
Draw the number line to represent the minuend.
Now the subtrahend is \( \frac{1}{8} \). This means we have to subtract one part out of \( \frac{4}{8} \). Since we are removing one part, we will have to “jump” one step to the left from the minuend.
So, \(\frac{4}{8}~ -~ \frac{1}{8}~ = ~\frac{3}{8} \)
Fraction strips (also known as fraction bars or fraction tiles) allow students to see how the same “whole” can be divided into multiple equal-sized parts. Students can imagine fractional amounts by moving the strips and placing them side by side.
To add and subtract fractions with like denominators, follow these steps:
- If needed, we convert mixed numbers to improper fractions.
- Subtract or add the numerators.
- Write the new numerator above the denominator.
Students must be able to demonstrate that a fraction can be written as a sum of unit fractions if the denominator is kept constant (and must be greater than 0). The denominator can be greater than the numerator.