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    Adding and Subtracting Fractions: Made Easy

    The Fear of Fractions!

    Fraction is a new concept with an entirely different notation which makes it difficult for students to understand. A part of the reason why students find it strenuous is also that they have only ever worked with whole numbers. So naturally, fractions seem unfamiliar to them.

    The struggles and challenges with fractions are real.

    The National Assessment of Educational Progress, 2005 states that “The mathematics education literature is resounding in its findings that understanding fractions is a challenging area of mathematics for North American students to grasp”. It’s this initial fear of fractions that propels further and makes comprehending the addition and subtraction of fractions burdensome. 

    This article will help students understand how to add and subtract fractions with easy steps and visual models. We’ll also look at some tricks to simplify adding and subtracting fractions.

    Read on to make sure that fractions are your kids’ friends and not a foe! 

    Table of Contents

    What are Fractions?

    Fractions - dividing a whole into equals
    Fractions - a part of whole
    Fractions - shaded parts of wholes

    Most curriculums focus on fractions as shaded regions or areas, but they are beyond that. Students often see fractions as an act of dividing and shading parts of shapes, so they miss out on the fact that fractions are numbers between the whole numbers. 

    Take a close look at the ruler image below. The marking between 0 and 1 represent numbers that are different from whole numbers. These are called fractions. 

    Fractions are numbers between whole numbers
    Fractions – numbers between whole numbers

    It’s important to emphasize that fractions are numbers that help us be precise and accurate with amounts. We don’t always have one full kilogram or one full inch while measuring. Therefore, it becomes critical to find a way to represent these “parts” and that is where fractions take charge.

    Fractions are written as a/b where “a” is the number on the top and is called the numerator. While “b” is the number at the bottom and is called the denominator.

    Numerator and Denominator of a fraction

    For example: For the fraction 1/2; 1 is the numerator and 2 is the denominator.

    Students often confuse the numerator with the denominator and vice versa. An easy way for these confusing terms is to remember them like –  D for down and D for the denominator. So the Denominator always goes down!

    TYPES OF FRACTIONS:

    The most common types of fractions are:

    Types of Fractions
    Types of fractions

    The most commonly used fractions while adding and subtracting fractions are :

    Like and Unlike Fractions.

    Fractions with the same denominators are called Like fractions.

    Example: 

    Like fractions have same denominators
    Like fractions

    Fractions with different denominators are called unlike fractions.

    Example: 

    Unlike fractions have different denominators
    Unlike Fractions

    Another type of fraction that students must know before learning to add and subtract fractions is: Equivalent Fraction

    Fractions that have different numerators and denominators but are equal to the same value are called Equivalent Fractions.

    Example:

    All fractions 1/2, 2/4, 3/6, and 4/8 represent the same value which is ‘HALF’

    As fractions are like any other numbers, they can be represented through different models. 

    Models for Representing Fractions:

    The most common models for representing fractions are:

    a) Area Model:

    In the area model, fractions are represented as parts of an area or a region.

    Circular and rectangular fraction sets are used to develop an understanding that fractions are parts of a whole.

    Examples of Area Model:

    Area Model of Representing Fractions
    Area Model

    b) Linear Model:

    In the linear model, fractions are considered as lengths instead of areas. The number line is an important linear model for students to grasp fractions as numbers.

    The linear model for the fraction ¾ will be:

    Linear Model of Representing Fractions
    Linear Model

    c) Discrete Model:

    In the discrete model, the whole is understood to be a set of discrete objects. Subsets of this whole make up the fractional parts.

    For example half of the class, 1/3 of a tray of eggs. Counters, marbles, cubes or any other set of objects that can be counted can be used as a manipulative to model fractions.

    Discrete Model of Representing Fractions

    How to Make Adding and Subtracting Fractions Easy?

    Doing fractions > Learning fractions

    The first and foremost thing that students need to do whilst learning to add and subtract fractions is understanding fractions better. They need to practice fractions to make connections to the real world, i.e, they need to fold papers, cut parts, color shapes, etc.

    One major challenge with fractions is that it’s not always concrete. We start teaching fractions through visuals but when it comes to operations with fractions we switch to rules and procedures for the same. Rote memorization of the steps leads to more confusion.

    In fraction addition, we add the numerators but not the denominators. But in fraction multiplication, we multiply the denominators as well as the numerators. This puzzles the students even more.

    It’s best advised to use visual models to show the addition of fractions which will help students understand the steps rather than mugging the steps without conceptual clarity.

    Doing fractions > Learning fractions

    Activity Suggestion:  

    To add 1/8 and 3/8, take a pizza carved in 8 slices

    Pizza activity
    Activity to understand fractions

    ⅛ is one slice carved out of the total 8 slices,

    And 2/8 are 2 slices out carved of the total 8 slices,

    Now if we add or put together both the fractions/pizza slices, we get 3 slices of the 8 pizza slices which imply 3/8.

    Concrete -> Contextual -> Computational 

    Concrete learning to contextual learning to computational learning

    To introduce a new and tricky concept like fractions, it is critical to provide students with ample opportunities to concretely absorb it. This means immersing students in experiences like paper cutting, paper folding, drawing, paper pizzas, apples, chocolate bars, etc. Only by using such experiences will they be able to see, touch, and feel the concept of fractions and make their own discoveries.

    Using concrete method to understand fractions


    Once students get enough of this concrete exposure, they would start making connections between these objects and the real world. 

    A lot of time should be spent creating, exploring, folding, and visualizing fractions before moving them to fractional problems involving bare numbers.

    For example: Before asking ¼ of 20 =? , we can present a real-world scenario like this:

    “Ron had 20 dollars. He spent ¼ of it. How much money is he left with?”

    This will help even a student with little understanding of fractions to start making connections.

    Show them “WHY” the rules work

    If the notion of fractions as numbers is developed well then it is easier to understand operations on fractions too. Just like we add and subtract numbers, fractions too can be added and subtracted. It is important to emphasize conceptual understanding along with the procedural knowledge of the steps.

    For example:

    We know, 1 apple + 1 apple = 2 apples

    1 + 1 =2

    Similarly, we can add fractions too.

    1 half + 1 half = 2 halves

    We also know 2 apples + 3 apples = 5 apples

    2+3 = 5

    Similarly, we can add fractions too:

    2/6 + 3/6 = 5/6

    To know the sum of 2/6 and 3/6; consider it as two one-sixths and three one-sixths which will add up to five one-sixths just like we add numbers. 

    Now let’s take a subtraction example:

    We know 5 apples – 2 apples = 3 apples

    5 - 2 = 3

    On similar lines, we can subtract fractions too:

    5/6 - 2/6 = 3/6

    In the case of “like fractions,” we can just add/subtract the numerators and keep the denominator the same. But we can’t add unlike fractions as we add like fractions. 

    The reason behind this is very simple.


    Just like we can’t add 2 apples and 3 oranges and say the sum is 5 applonges, we can’t add fractions with different denominators. It is, thus, important to first define addition as the combination of two or more like-unit quantities. Similarly, subtraction is taking away like-unit quantities.


    So, how do we add/subtract unlike fractions? Let’s look into that next!

    STEPS TO ADD AND SUBTRACT FRACTIONS


    Now, let’s look at the steps we can follow to add or subtract fractions:


    Step 1: Make denominators the same


    Step 2: Add or Subtract the numerators (keeping the denominator the same)


    Step 3: Simplify the fraction

    To add or subtract unlike fractions, the first step is to make denominators the same so that numerators can be added just like we do for like fractions.

    STEP 1: SAME DENOMINATORS


    How do we make denominators the same?


    In the case of like fractions, the denominators will already be the same, so you can skip Step 1 and move to Step 2.

    For Unlike fractions,  there are 2 possibilities:

    i) If one denominator is a multiple of the other denominator

    Example: ½ + ¾

    Making denominators same - like fractions

    In this case – 4 is a multiple of 2. We can multiply 2 by 2 to make it 4 and as a result, the denominators will become the same. So, the larger number becomes the common denominator.

    Making the denominator same - like fractions

    Applying : 

    1/2 = 2/4

     

    So the problem now becomes:

    Now the problem becomes 2/4 + 3/4

    If one denominator is a multiple of the other, we can multiply the smaller denominator by a number (say k) that creates the larger denominator. Then the larger denominator becomes the common denominator.

    Denominator with common multiple

    ii) If both the denominators have no common factor

    Example: ¼ + ⅗ 

    Making the same denominators

    In this case, 4 and 5 have no common factors. We can simply multiply the denominators to get a common denominator.

    4 X 5 = 20 so 20 is the common denominator for both the fractions.


    If there is no common factor to both the denominators then you multiply both the denominators to get the common denominator.


    Let’s see how we will get 20 as a common denominator for both the fractions:

    If there is no common factor in the denominator

    If we multiply the denominator, we have to multiply the numerator as well to get an equivalent fraction.

    So the problem now becomes:

    Result with no common factor in denominator

    STEP 2: ADD/SUBTRACT NUMERATORS

    This step is pretty simple and very straightforward. We have to add/subtract the numerators and the result of their sum/difference is the new numerator. The common denominator (as discussed in Step 1) remains the same.


    Let’s take our previous examples and continue from there:

    1. ½ + ¾

    After making the denominators the same, this problem looks like this:

     Step 1 - Add/Subtract Numerators

    Now, we need to add the two numerators together (2+3 =5) to get the new numerator while the denominator(4) remains the same. 

    Step 1 - add/subtract numerators

    So the answer will be 5/4.

    1.   ¼ + ⅗.

    After making the denominators the same, this problem looks like this:

    Example 2- add/subtract numerators

    Now, we need to add the numerators together (5 + 12 = 17) to get the new numerator while the denominator(20) remains the same. 

    So the answer will be 17/20.

    STEP 3: SIMPLIFY THE FRACTION

    The answers we have derived above are correct, but we may simplify the fraction further till there are no common factors in the numerator and the denominator other than 1.

    The fraction can be reduced to its simplified form by removing the common factors.


    Let’s continue with our previous examples:

    Simplifying the fraction - example 1

    And,

    Simplifying the fractions - example 2

    In the above cases –  5/4 and 17/20 are already simplified fractions as the numerator and the denominator have no common factors.

    Let’s take some examples of fractions that can be simplified:

    4/12 is not simplified. 

    4 is a common factor in both the numerator and the denominator so it can be reduced to its simplified form as follows:

    Please note: We will not be taking up mixed numbers as a separate case because these are also fractions written in the different (mixed) form.

    Let’s summarize these steps by taking one example of addition and subtraction each. 

    Time to practice all three steps together! 

    Addition Problem: 3/4 + 1/12

    Step 1: Make the denominators the same


    Denominators are 4 and 12. 12 is a multiple of 4 so the common denominator will be 12.

    Making denominators the same

    Step 2: Add/Subtract the numerators (keeping the denominator as same)

    adding the numerators

    Step 3: Simplify the fraction


    To simplify 10/12, the common factor is 2

    simplifying the fraction

    So 10/12= 5/6 (in simplified form)

    Hence, 3/4 + 1/12= 5/6

    Now let’s look at a subtraction problem:


    Example: 2/5 1/3 


    Step 1: Make the denominators the same


    Common denominator will be 3 x 5 = 15

    Common denominator for subtraction

    Step 2: Add/Subtract the numerators (keeping the denominator the same)

    Subtraction in fraction

    Step 3: Simplify the fraction

    1/15 is in its simplified form already as 1 and 15 have no common factors.

    The Butterfly Method 

    A very interesting and useful method to quickly add/subtract fractions is the butterfly method.

    Butterfly method to add/subtract fractions
    Source 

    In this method, we draw the butterfly wings to imply which two numbers are to be multiplied together. We then go on to write the result in the respective antenna. The denominators are multiplied and the result of that is written below in the abdomen.

    In the end, we simply add/subtract the antenna and write it over the abdomen to get the result.

    Adding/Subtracting Fractions – Common Mistakes

    Teachers need to ensure that students are taken gradually from the concrete to the contextual to computational level.

    Students need to be given a lot of examples to help them get over some of the common mistakes and misconceptions:  

    1.  Adding/subtracting numerators and denominators


    One important thing to pay close attention to while adding/subtracting fractions is the way students represent them:

    The most common mistake in adding fractions is to add both the numerators and the denominators individually just as we add whole numbers.

    For example: When adding 2/3 and 1/4, a common mistake is to represent each fraction as shown above and then put them together to form 3/7 as the answer.

    When students add, combine or find the sum by putting together the wholes and both the fractional parts, it only seems reasonable that they look at each fraction independently.

    So, it is important to emphasize that fractional parts cannot be manipulated independently from their whole.

    That is why it is essential to have a common denominator. In the case of a common denominator, the fractions can be interpreted on the same diagram and combined. Let’s look at this same example when done on the common whole.

    In this case, the common denominator for 2/3 and 1/4 will be 12.

    Now consider the whole made of 12 parts.

    a whole made of 12 parts

    Now let’s find ⅔ and ¼.


    To find 2/3 first divide the whole into 3 equal parts:

    Dividing a whole of 12 parts into 3 equal parts

    2/3 will be 2 of these 3 equal parts:

    Finding 2/3

                                                                     2/3 = 8/12 

    Similarly, to find 1/4 we need to divide the whole into 4 equal parts:

    Dividing a whole into 4 equal parts

    1/4 will be:

    1/4 = 3/12

                                                             1/4 = 3/12

    Now to add 2/3 and 1/4 we can combine 8 parts and 3 parts whose sum equals 11 parts.

    2/3 + 1/4 = 8/12+ 3/12 = 11/12

    Similarly, this same mistake is observed in subtracting fractions as well. Both the numerators and the denominators are subtracted individually just as we subtract whole numbers.

    For example: 5/6 – 1/3 =  (5-1)(6-3) = 4/3

    Let’s find the correct way of solving  5/6 – 1/3 =?


    In this case, the common denominator for 5/6 and 1/3 will be 6 (common multiple).

    Now consider the whole made of 6 parts:

    Whole made of 6 parts

    5/6 will be:

    For 1/3 we divide the whole into 3 equal parts:

    dividing a whole of 6 into 3 equal parts

    1/3 will be one of these three equal parts:

                                     1/3 = 2/6

    Now to subtract 1/3 from 5/6, we will take away 2 parts from the 5 parts of the same whole.

    5/6 – 1/3 =   5/6 – 2/6  =    3/6

    2.  Adding/subtracting numerators while ignoring the denominators

    As students struggle to see fractions as different operations, they often treat them as whole numbers.

    12/13 + 7/8 = 19, because 12 + 7 = 19

    12/13 – 7/8 = 5, because 12 – 7 = 5

    The key here is to make their understanding of fractions concrete, right from the very beginning. 

    3.  Adding/subtracting denominators while ignoring the numerators

    Some students add only the denominators and ignore the numerators. They look at the denominators as two whole numbers and add/subtract them.

    12/13 + 7/8 = 21, because 13 + 8 = 21

    12/13 – 7/8 = 5, because 13 – 8 = 5

    Interesting Activities for Addition and Subtraction of fractions


    Incorporate simple activities like cutting real-world objects to develop interest and a better grasp of fractions. Activities like Tic tac toe, BINGO, or matching activities can be done to make fractions problems more fun.

    Fraction activity with apples

    Fraction in Everyday Life


    Parents should encourage talking about fractions with kids in everyday life. Encourage kids to apply fractions in day-to-day tasks like equal division of things, in measurement, or while cooking their favorite new recipes. We can also include fractions while talking about time or the marks kids scored in school!

    Talking about fractions evades the fear of it and kids are more likely to enjoy practicing them.

    Games on Fractions

    Games encourage students to practice a lot of questions which they generally don’t like doing otherwise. Check out these fun fraction games at SplashLearn

    Fractions Models/Manipulatives


    Using manipulatives like fraction strips, area models, lego blocks and number lines make fractions interesting and engaging for the kids. These manipulatives help kids visualize fractions and hence understand them better.

    Fraction strip

    Fraction Word problems


    Solving contextual problems helps students relate fraction learning to real-life situations. They understand the significance, the need, and importance of fractions and how to apply them to solve problems. You can check out these fraction word problems games on SplashLearn and make fractions a lot simpler!
     

    To Summarize:

    • It’s important for students to recognize that fractions are beyond shading and coloring. 
    • Fractions are numbers that are used to represent the numbers between any two consecutive whole numbers.
    • Help students practice fractions by cutting, pasting, coloring to help them understand fractions better and have fun with them. 
    • Use fraction models to help them visualize and grasp addition and subtraction of fractions.
    • Make sure they know why the steps of adding/subtracting fractions work rather than just blindly applying the steps.
    • As a parent,use fractions in everyday conversations, relate fractions to real-life contexts and provide them instances from daily life activities like cooking, baking, time and measurement.

    Teach Fractions with SplashLearn

    Loved by 40+million parents, SplashLearn is a one-stop solution for your kids’ math and reading journey. You can make your little learners play games on adding and subtracting fractions or learn through worksheets. With curriculum-aligned teaching, SplashLearn provides a personalized experience that adapts to every child’s needs.

    Frequently Asked Questions (FAQs)

    1. What are fractions?

    Fractions are numbers between the whole numbers (0,1,2,3,4). They represent a portion/part of an entire thing. A fraction has two parts – numerator and denominator.

    2. What are the steps for adding and subtracting fractions?

    Step 1: Make denominators the same
    Step 2: Add/Subtract the numerators (keeping the denominator as same)
    Step 3: Simplify the fraction

    3. How do you add and subtract fractions with different denominators?

    Fractions with different denominators can be added by converting them to fractions with the same denominators with the help of equivalent fractions.

    4. How do you add and subtract mixed numbers?

    First, write the mixed numbers as fractions. Now add/subtract just like you do with fractions. In the end, don’t forget to convert the answer (fraction) to a mixed number!

    5. How do you add and subtract negative fractions?

    Negative fractions can be considered as fractions with numerators as negative. The steps to add and subtract the negative fractions remain the same as fractions except now kids would need to add negative or positive numerators.

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    AUTHOR
    Era Kaila
    Era has been in the math education domain for 7 years and currently works as curriculum designer with SplashLearn. Her interest lies in designing games and activities that make math interesting and engaging for kids. Era is determined in making the subject fun for anyone who perceives it to be scary or boring! She loves travelling and exploring new places along with trying different cuisines.

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