There are four basic operations in mathematics, i.e. the ways in which numbers are combined to create more numbers. There’s addition, subtraction, multiplication, and division.
The long division method incorporates the other three operations, making it an excellent way for children to solve problems whilst also becoming comfortable with mathematics.
Introduced by British mathematician Henry Briggs circa 1600 AD, long division is a tried and tested method that has been in use for centuries. It involves breaking down a division problem into smaller steps.
But what is the method, how does it work, and why do we still teach it to this day? These are questions that we’ll answer in this article.
The long division method is a step-by-step method used to divide a large number by another large number. Typically, it’s used to divide three-digit numbers by two-digit numbers but can be used for larger or smaller numbers. The long division process is incredibly similar to short division or the bus stop method, both of which are taught to kids at a very young age in primary school, but builds on them for more advanced division problems.
There are two methods when it comes to using long division. There’s the formal long division method and the chunking long division method. Both will give you the same final answer but will have slightly different ways of getting there. We’ll explore these different division methods in more detail later on in this article.
Definition of long division terminology
Before we get into examples of long division in action, there are some words and terminology that you will want to familiarise yourself with so you don’t get confused.
- Dividend is the number that we are dividing.
- Divisor is the number that we are using to divide the dividend with.
- Quotient is the result of dividing the dividend with the divisor. In other words, it’s the answer as a whole number.
- Remainder is the leftover number that cannot be divided any further. It is often represented as a decimal point when looking at the quotient as it indicates that the divisor does not go into the remaining number anymore.
To show this in action, let’s take a look at the following examples:
Example 1: 4 ÷ 2 = 2
With a simple question of 4 ÷ 2, the number 4 is the dividend since it’s the large number that’s being divided. This means that the number 2 is the divisor since it’s the number that divides the dividend. Lastly, we have the number 2 again, which is the answer, also known as the quotient. In this example, we do not have a remainder. Let’s take a look at another example.
Example 2: 22 ÷ 4 = 5.5
In this example, 22 is the dividend. The number 4 is the divisor and the number 5 is the quotient. This means that 4 goes into 22 a total number of 5 times (equaling 20). This leaves us with a remainder of 2 (22 - 20). This is represented as 0.5 since 0.5 x 4 = 2, which makes up the remainder. Typically, the divisor multiplied by all the digits after decimal points will equal the remainder.
These examples also show the long division equation. The long division equation is as follows, “Dividend = (Divisor x Quotient) + Remainder”. If we plug this equation into the second example, we can see this works:
22 = (4 x 5) + 2
The long division method allows us to divide large numbers by other numbers in easy-to-remember steps. But it's not done using typical division methods where you use the division symbol or division slash, it uses a division tableau.
This is where the divisor is written to the left of the dividend, but it is separated by a vertical bar [|] or right parenthesis [)]. The quotient is written above the dividend, and it is separated by an overbar (straight line above the dividend). The vertical bar or right parenthesis, in combination with the overbar, is known as the division bracket or the long division symbol.
As mentioned earlier, the long division method follows a step-by-step process. It is as follows:
1. Check if the divisor is equal to or lower than the first digit of the dividend. If not, combine the first and second digits of the dividend together.
2. Divide the dividend by the divisor and write the first digit of the quotient at the top
3. After subtracting the result from the digit, write the difference below the original number
4. Bring the next digit of the dividend down
5. Repeat the same process until you have your final answer
It can seem unclear when it’s described like that, but when you see it in action, it will begin to make sense. Below are some practice examples that can help you understand the concept and show you how to use the long division process to solve division problems.
In this example, the first digit of the dividend is either a greater or equal number to the divisor. Let’s consider the practice question 642 ÷ 3. Following the long division method steps outlined above, the process is as follows:
1. The first digit of 6 is greater than the divisor of 3.
2. So 6 ÷ 3 = 2. We can write 2 above the number 6 at the top as it’s the first digit of the quotient.
3. We can put another number 6 (3 x 2) below the 6 in the dividend and subtract them together. 6 - 6 = 0, meaning we add a 0 below both sixes.
4. We can then bring the second digit of the dividend down next to the 0, which creates the number 4. 3 x 1 = 3, which is lower than 4, and 3 x 2 = 6, which is greater than 4. This shows us that the highest number of times 3 goes into 4 is 1. Therefore we can write 1 above the 4 at the top as it’s the second digit of the quotient.
5. Much like we did in step 3, we can add 3 (3 x 1) below the 4, and then subtract these numbers together. 4 - 3 = 1, which goes below these numbers. Unfortunately, 3 does not go into 1. Therefore we can bring the 2 from 642 down. The number left is 12.
6. By working up the 3 times table, we can see that 3 goes into 12 exactly 4 times. We can write 4 next to the 1 at the top, creating the number 214. This is our final answer: 642 ÷ 3 = 214.
In this example, the first digit of the dividend is less than the divisor. Let’s consider the practice question of 768 ÷ 32. Following the long division method steps outlined above, the process is as follows:
1. The first digit of 7 cannot be divided by 32 as a whole number. Therefore, we combine it with the second digit to make the number 76.
2. 32 x 2 = 64 which is less than 76, and 32 x 3 = 96 which is greater than 76. This shows us that the highest number of times 32 goes into 76 is 2. Therefore, we can write 2 above the 6 at the top as it’s the first digit of the quotient.
3. Below the 76, we can put 64 (32 x 2) and subtract these numbers together. This gives us a difference of 12, which we can put below the 64. Unfortunately, 32 does not go into 12, which means that we can proceed to bring down the 8, creating the number 128.
4. Here, we then repeat the steps until we reach our answer. By working up the 32 times table, we know that 32 goes into 128 precisely 4 times. We simply put the 4 next to the 2 at the top, creating the number 24. This is our final answer: 768 ÷ 32 = 24.
These examples showcase the formal written method of long division, which is what we come to think of when we say ‘long division’. However, there’s also a chunking method of long division. It’s a relatively new method and has only begun being taught in schools over the last decade or so. With that being said, let’s explore the chunking method in more detail.
Long division chunking method
The chunking division method is very similar to the formal long division method, except it takes a bit of a shortcut. Instead of going through each dividend digit one by one, the chunking method encourages kids to think about how multiplication can make the process easier. It involves trying to estimate how many times the divisor is likely to go into the dividend, thereby looking at how big of a chunk of the divisor we can subtract in one go. Let’s look at this in practice.
Suppose we are trying to calculate 396 ÷ 18. We know that 18 x 10 = 180 (it’s often easy to start with the five or 10 times tables when trying to chunk). This is considered one chunk. 180 is less than 396, so we can minus 396 - 180 = 216, noting that we used a chunk of 10 (18 x 10 = 180).
Then, we can follow the previous step and see if we can chunk the 18 times tables into our new number, 216. Fortunately, we can take off another chunk of 10 (=180) and subtract that number from 216, leaving us with 216 - 180 = 36. We know that we cannot use another chunk of 10 since 180 is greater than 36. But, if we use the 18 times table, we can see that 18 goes into 36 exactly 2 times, which can be considered our final chunk.
Adding the three chunks together, we are left with 22 (10 + 10 + 2) which is our final answer: 396 ÷ 18 = 22
As you can see, by chunking together the subtractions, we are able to reach the final answer quicker than if we had used the formal written method.
Some parents wonder why long division is still taught in schools, especially when we have calculators on mobile phones. It is thought that this negates the need to teach such a method since all answers can be found using a calculator in a fraction of the time. But there are a few reasons why teaching long division at schools is still an essential part of Key Stage 2 curriculums.
Firstly, children learn the ability to break down big problems into smaller problems. Large numbers can be pretty intimidating even for adults, let alone kids, so practising dividing numbers using small steps can help overcome this. This skill can also be translated to other parts of life, such as setting goals or planning, which is why it’s essential for kids to learn this sooner rather than later.
Secondly, long division incorporates three of the four basic mathematical operations – addition, subtraction, and multiplication. This gives children the ability to practice the other operations, all within a division problem. It can also help children in practising to see the relationship between the different mathematical operations, even when looking at other problems that aren’t division ones.
Thirdly, this method can help save students a lot of time in their exams. Division problems are prevalent in Key Stage 2 SATs maths exams, and using the long division method to break down a complex problem into smaller bitesize chunks, will not only make it easier for students but will also save them a great deal of time.
Lastly, combining all of these together, is that children develop their problem-solving skills and ability to think in a structured manner. Thinking in a logical and structured way to solve a problem is a tremendous skill that is transferable to all kinds of situations, both in school and in their personal life. Regardless of whether we all have a calculator at our fingertips or not, there are fundamental skills and benefits to learning these methods which shouldn’t be overlooked, particularly for developing children.