Short division is also sometimes instead referred to as the bus stop method.

The short division method is the most common written method introduced in primary school for dividing larger numbers.

To do the short division method, use the following steps:

1. Write the number being divided and consider each of its digit from left to right.
2. Divide each digit in the number separately by the number being divided by.
3. Write the answer to each of these divisions above each digit.
4. If the division is not exact, write the greatest number of times that the dividing number divides into this digit.
5. Write the remainder as a ten for the next digit along.
6. Continue to divide each digit until you reach the final digit.

We will start by looking at a simple short division example.

Here we have 482 ÷ 2.

We start the method by writing 482 and 2, separated by a line.

The next step in the short division method is to divide each digit by 2.

4 ÷ 2 = 2 and so, we write this ‘2’ above.

The next digit is an 8.

8 ÷ 2 = 4 and so, we write ‘4’ above.

Our final digit is 2.

2 ÷ 2 = 1 and so, we write a ‘1’ above.

482 ÷ 2 = 241.

Here is the full short division process.

Since the answer is a whole number and the division was exact, we say this short division has no remainders.

Here is another example of short division.

We have 585 divided by 5.

The first step is to divide the first digit in 585 by 5.

5 ÷ 5 = 1.

We next divide the second digit in 585 by 5.

This time 5 does not divide exactly into 8. We write down the greatest number of times that 5 goes into 8.

5 goes into 8 once.

8 ÷ 5 = 1 remainder 3.

We have a remainder of 3 because we need 3 more to get from 5 to 8.

We write the ‘1’ above and write the remainder of ‘3’ next to the following digit.

We now treat the ‘3’ and the ‘5’ as a ’35’.

We now have 35 ÷ 5.

35 ÷ 5 = 7.

Here is the full short division method.

Since 5 divides exactly into 585, we say that this is an example of short division without remainders.

In this next example we have 148 ÷ 2.

Our first step is to divide ‘1’ by 2.

2 is larger than 1 and so it does not divide into ‘1’ even once.

We write a ‘0’ to show this and carry the ‘1’ over as a remainder.

The ‘1’ and the ‘4’ can be read as ’14’.

The final digit is ‘8’.

148 ÷ 2 = 74.

Since 2 goes exactly into 148 seventy-four times, we say that this short division has no remainders.

In this next example we have 432 ÷ 4.

4 ÷ 4 = 1 and so we write a ‘1’ above.

4 does not divide into 3 because 4 is larger than 3.

We write a ‘0’ to show that 4 divides into 3 zero times.

We carry the ‘3’ over to meet the ‘2’ in the next column. This is read as ’32’.

32 ÷ 4 = 8 and so, we write an ‘8’ above.

432 ÷ 4 = 108.

How to Tell if a Number is Divisible by 9

To decide if a number is divisible by 9, follow these steps:

1. Add the digits of the number.
2. If the sum of these digits is divisible by 9, the original number is divisible by 9.
3. If the sum of these digits is not divisible by 9, the original number is not divisible by 9.
4. Use steps 1 to 3 again to decide if the sum of the digits of the number are divisible by 9.

If a number is divisible by 9 it means that the number can be divided by 9 exactly leaving no remainder.

The rule for divisibility by 9 is that the number is divisible by 9 only if the sum of its digits is also divisible by 9.

In this example, we will use the rule for divisibility by 9 to test if 8595 is divisible by 9.

The first step is to add the digits of the number.

8 + 5 + 9 + 5 = 27.

The next step is to decide if the sum of the digits of the number are divisible by 9.

27 is divisible by 9 because it is 9 × 3.

If the sum of the digits is divisible by 9, the number itself is divisble by 9.

27 is divisible by 9 and so, 8595 is also divisible by 9.

If we are unsure whether the sum of the digits is divisible by 9, add the digits of this number and decide if the new total is also divisible by 9.

For example, here 27 is divisible by 9 because 2 + 7 = 9.

Here is an example of using the rule for divisibility by 9. We will check if 771 is divisible by 9.

The rule to check divsibility by 9 is to add the digits and decide if this total is divisible by 9.

7 + 7 + 1 = 15

15 is not a multiple of 9. The first two multiples of 9 are 9 and 18.

15 is not divisible by 9 and so, 771 is not divisible by 9. A number is only divisible by 9 if its digits add to a number that is divisible by 9.

It does not matter how large the number is. The rule for divisibility by 9 always works without exception.

For example, here is 7,719,984.

7 + 7 + 1 + 9 + 9 + 8 + 4 = 45

45 is divisible by 9 and so, 7,719,984 is also divisible by 9.

Here is a large example of a number not divisible by 9. Here is 529,943.

5 + 2 + 9 + 9 + 4 + 3 = 32

32 is not divisible by 9 and so, 529,943 is not divisible by 9

Why Does the Rule for Divisibility by 9 Work?

The rule for divisibility by 9 works because numbers are written in base 10. Each digit of a number represents a multiple of 9 plus the value of that digit. The multiple of 9 is divisible by 9 so only the value of the digits needs to be checked. A number is divisible by 9 if the sum of its digits is divisible by 9.

For example, 7236 can be written as 7 × 1000 plus 2 × 100 plus 3 × 10 plus 6.

7 × 1000 is the same as 7 × 999 + 7, 2 × 100 is the same as 2 × 99 + 2 and 3 × 10 is the same as 3 × 9 + 3.

The multiples of 999, 99 and 9 are divisible by 9 and so, we just need to check the remaining digits.

If 7 + 2 + 3 + 6 is divisible by 9, then the whole number is.

7 + 2 + 3 + 6 = 18, which is in the 9 times table. Therefore 7236 is divisible by 9.

List of Numbers Divisible by 9

There are 11 numbers less than 100 that are divisible by 9:

9, 18, 27, 36, 45, 63, 72, 81, 90 and 99.

The numbers divisible by 9 between 1 and 100 are easy to remember because their digits add up to 9.

In this lesson we use the column addition method to add 3-digit numbers.

To add numbers using the column addition method, line up the digits of each number above each other and add them separately. The sum of each digit is written below, with one digit per box.

We use the method of regrouping when more than one digit is produced after adding the digits in a place value column.

In maths, regrouping is the process of moving values between place value columns. When carrying out an addition, regrouping is often used to move a group of ten to the next place value column up. Some examples of regrouping are replacing 10 ones with 1 group of ten, or 10 tens with 1 group of one hundred.

When teaching addition, some teachers use the word 'carrying' as well as the word 'regrouping'.

Carrying and regrouping mean the same in addition. Carrying is the movement of a ten over to the next place value column. Regrouping is also used in subtraction, where it is sometimes known as borrowing.

We will learn about regrouping in this lesson by considering some examples of addition involving regrouping.

Here is an example of 246 + 173:

The number 246 has 6 units (or ones). These are shown by the six purple counters in the units column.

It has 4 tens. These are shown by the four groups of ten purple counters in the tens column.

It also has >2 hundreds, which are shown by the two groups of one hundred purple counters in the hundreds column.

We want to add 173.

The number 173 has 3 units, which are shown by the three green counters in the units column.

It has 7 tens. These are shown by the seven groups of ten green counters in the tens column.

It also has 1 hundred. This is shown by the one group of one hundred green counters in the hundreds column.

Now that we have lined up the digits of each number in their place value columns, we can add the digits in each column.

We have a total of 9 counters in the units column.

Next, we look at the tens column. We have a total of 11 tens.

The number 11 has two digits.

We can’t have more than one digit in each place value column.

We therefore need to regroup 100 counters to carry over to the hundreds column.

In the hundreds column, we can have groups of one hundred. Therefore, we can regroup one hundred counters from the tens column and carry them over to the hundreds column.

Now that we have carried 1 hundred over to the hundreds column, we are left with 1 ten in the tens column.

Finally, we look at the hundreds column. We have 4 groups of one hundred counters.

Therefore,

246 + 173 = 419

When teaching regrouping, it is useful to introduce the concept visually as above. You can use counters or Dienes blocks to help teach this. Once the idea of regrouping is understood, column addition is best taught methodically with written digits.

We use regrouping whenever the sum of the digits in each column of the column addition is larger than 9. We regroup ten ones to make 1 group of ten or we can regroup ten groups of ten to make one group of one hundred.

When using column addition, the regrouping process is often called carrying as the extra digit is carried across to be added with the next column of digits.

We will look at this example again, but this time we will use numbers to represent each digit and set out the calculation using the column addition method.

We line up the digits of each number according to their place values.

We begin by adding the digits in the units column.

6 + 3 = 9

We write the 9 in the units column, underneath the 6 and the 3.

Next, we add the digits in the tens column.

4 + 7 = 11

The number 11 has 1 unit, which we keep in the tens column, and 1 ten, which we carry over to the hundreds column and write below the line.

Finally, we add the digits in the hundreds column. We must also remember to add the 1 that we carried over.

2 + 1 + 1 = 4

Therefore,

246 + 173 = 419

We will look at some further examples of column addition involving regrouping.

Here we have the 3-digit numbers 417 + 235.

We set the numbers out with the column addition method, with one digit per box and one number above the other.

We will add the units first.

7 + 5 = 12

We cannot have two digits written in one box or one place value column and so, we know we need to regroup.

We write down the units digit of '12', which is '2'.

The '1' ten is carried over to the tens column. We do not write it between the lines. Instead we can write it below the line.

We need to add this '1' that has been carried in the tens column.

1 + 3 + the '1' which was carried = 5

Adding the digits in the hundreds column we have 4 + 2 = 6.

In this example of adding three-digit numbers we have 863 + 524.

We set the numbers out in the column addition method with the first number above the second.

We can add the units column digits easily since 3 + 4 = 7.

In the tens column we have 6 + 2 = 8.

Adding the hundreds digits we have 8 + 5 = 13.

13 contains two digits and so, we must carry the '1' into the next column.

Ten hundreds make 1 thousand.

We write the '3' in the hundreds column and the '1' is carried into the thousands column on the left.

3-Digit Column Addition with Regrouping Twice

Once the regrouping process has been taught, it can be applied multiple times within the same question. Some 3-digit addition problems will involve regrouping twice within one sum.

Here is the example of 365 + 187.

Starting with 5 + 7 in the units column, we get 12.

12 contains two digits and so we carry the '1' over to the tens column.

We then add the tens column digits.

Adding the digits in the tens column, we have 6 + 8 from the two numbers given, plus the '1' that we carried during the regrouping process.

6 + 8 + 1 = 15.

Again 15 contains two digits, so we need to regroup.

We carry the '1' over from the tens to the hundreds column, leaving the '5' in the tens column.

Finally, we add the hundreds column digits. We have 3 + 1 in the two numbers given and we need to add the '1' that was carried.

3 + 1 + 1 = 5.

We have 365 + 187 = 552.

We carried twice in this sum, regrouping from the ones column to the tens column and also from the tens column to the hundreds column.

Here is another example with regrouping twice.

We are adding the 3-digit numbers 728 + 644.

We set the digits out above each other as per the column addition method and add the digits in the units (ones) column.

8 + 4 = 12, which is a two-digit number. We need to carry the '1' across into the tens column, leaving the '2' in the units column.

Now we add the tens column digits, plus the '1' that was carried.

In the tens column we have, 2 + 4 plus the one that we carried.

2 + 4 + 1 = 7, which only contains one digit. We can simply write '7' in the tens column.

Now we add the hundreds digits.

We have 7 + 6, which equals 13.

We cannot write '13' in the hundreds column alone because it contains 2-digits.

We carry the '1' into the thousands column.

We can regroup ten hundreds to make one thousand.

Since there is no other digit in the thousands column, we simply write the '1' here.

We have added two 3-digit numbers to make a 4-digit number because we regrouped from the hundreds column into the thousands column.

Here is another example of adding two 3-digit numbers to make a 4-digit number in the thousands.

We have 876 + 552.

Adding the units column, we have 6 + 2, which equals 8.

There is no need to regroup.

We move onto the tens column. 7 + 5 = 12 and so, we regroup the 10 tens to make '1' hundred.

We write the '1' below the hundreds column.

Now adding the hundreds column we have 8 + 5 plus the '1' that we carried from the tens column.

8 + 5 + 1 = 14.

We need to regroup these hundreds into a thousand.

We carry the '1' over into the thousands column and leave the '4' hundreds behind in the hundreds column.

We have no other digits in the thousands column and so, we write the '1' in the thousands column.

876 + 552 = 1428.

What is Addition Without Regrouping

Addition with regrouping is when one of the place value columns add up to a number that is 10 or greater. Only the ones digit is written down and the tens digit of the answer is regrouped to be added to the next column.

Addition without regrouping is when the digits add up to a number that is 9 or less. The answer can simply be written below each place value column. There is no carrying of tens or hundreds.

When talking about adding numbers, regrouping means the same as carrying. Carrying was originally a more commonly used word that applies only to adding numbers, however, the phrase regrouping is becoming more popular. Both words mean the same thing when performing addition.

In subtraction, regrouping means the same as borrowing.

How to Add without Regrouping

To add numbers without regrouping, follow these steps:

1. Write the numbers directly above each other, lining up the place value columns.
2. Add the digits in each column, starting on the right and moving left.
3. Write the answer to each addition in the answer space directly below each column of digits.

For example, here is 352 + 215.

When teaching written addition methods, square grid paper is useful to ensure that the digits are lined up above each other.

The rule for addition without regrouping is to only write one digit in each position on the grid. Never write two digits in one place.

Step 1 is to write the numbers directly above each other.

Step 2 is to add the digits in each column.

We start on the right in the units column.

2 + 5 = 7

We now add the digits in the tens column.

5 + 1 = 6

We now move left to add the digits in the hundreds column.

3 + 2 = 5

We have now added the total number of ones, tens and hundreds by adding each digit separately.

The answer to the addition is found by combining the digits in the answer space.

352 + 215 = 567.

We were able to add all digits without arriving at an answer larger than 9 for any digit. We did not have to use the regrouping process.

Regrouping is not always used. Regrouping is only used when the sum of digits in a place value column is ten or more. This is because we cannot write two digits in one column and we need groups of ten for the regrouping process.

When first teaching written addition, it is recommended to begin by looking at problems without regrouping, such as in this lesson.

Here is another example of adding two three-digit numbers using the column addition method.

Again, this example will not involve regrouping because when we add each digit we will not get an answer that is bigger than 9.

We have the three-digit addition of 531 + 248.

We line up the digits that are in the same place value columns above each other and then start adding the digits starting from the right.

Adding the digits in the ones column we have:

1 + 8 = 9

Now we add the digits in the tens column.

3 + 4 = 7

Finally we move left onto adding the digits in the hundreds column.

5 + 2 = 7

We finally read our answer from left to right from the answer space.

531 + 248 = 779

Here is the full written addition method, adding the two three digit numbers.

Here is an example of addition without regrouping. 514 + 372 = 886. We can simply add the digits in each place value column.

We have 514 + 372.

We set out the column addition written method with the units, tens and hundreds digits written above each other.

We will add the digits separately from right to left, starting in the ones column.

4 + 2 = 6

We will now add the tens column digits.

1 + 7 = 8

Now we will add the hundreds digits.

We can see the full written addition method below.

514 + 372 = 886

Don’t forget to download our PDF worksheets above for more practice of adding three-digit numbers without regrouping.

How to Teach Addition Without Regrouping

Addition without regrouping can be taught using base ten blocks and arranging them into place value columns. The total number of each type of block in each column will represent the digits of the answer.

A digit is one of the numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9 that we can combine together to write numbers.

To add numbers, line up each digit in each number above each other in place value columns. Then add each digit separately from right to left.

In this lesson, we are adding numbers without regrouping. This means that when we add the digits in each place value column together, we will not get an answer in each place value column that is larger than 9.

Here is our first example of adding two three-digit numbers.

We have 352 + 215.

We can partition the numbers into their hundreds, tens and units columns.

352 is made up of 3 groups of one hundred, 5 groups of ten and 2 ones.

215 is made up of 2 groups of one hundred, 1 group of ten and 5 ones.

When teaching column addition it can be helpful to first show the size of the numbers with counters or dienes blocks as in this example. It can help to remind us of the size of the number that each digit represents.

We now add the ones, tens and hundreds separately.

We start with the rightmost column, which is the units (or ones) column.

2 + 5 = 7

We now add the groups of counters in the tens column.

5 + 1 = 6

We now add the groups of hundreds in the hundreds column.

3 + 2 = 5

When teaching addition, it can be useful to start with physical objects and simply count them. Once the idea is undertood, it is helpful to move on to a written method and regularly practise following the procedure through and setting it out with one digit per box.

Particular focus should be given to lining up the digits in the same place value columns.

We simply read our answer from the digits in the answer space between the lines.

It tells us how many hundreds, tens and units we have in total.

352 + 215 = 567

To multiply a number by 1000, move each digit in that number three place value columns to the left.

All digits in the original number remain in the same order.

To multiply a whole number by one thousand, a simple trick is to add three zero digits to the end of that whole number.

Here is an example of multiplying a whole number by 1000.

In the example above, the 2 has been moved three places to the left. It has moved from the units column to the thousands column.

As there are no longer any digits in either the hundreds column, the tens column or the units column, we write a zero in each space to show that they are worth zero.

You may have already found that 2 x 1000 = 2000 by “adding three zeros”. This only works when multiplying whole numbers.

When teaching multiplying by 1000, it is best to show the digits moving three place value columns first. This is what is actually happening because 1000 is made of 10 x 10 x 10, which is the same as multiplying by 10 three times.

Each time we multiply by ten we move our digit one place value column to the left because our place value columns are in base ten and represent a multiplication by ten.

Most people will still ‘add three zeros’ when multiplying a whole number by 1000 and this is the easiest and best way to multiply a whole number by 1000 as it can be done quickly and mentally.

This trick should be shown after showing the digits moving in their place value columns and it should be emphasised that this only works with whole numbers.

It is important to understand how multiplying by 1000 works because not all numbers that we will be multiplying will be whole numbers.

Here is an example of multiplying a decimal number by 1000:

To multiply the decimal number 0.05 by 1000, we move the 5 three places to the left. We move it from the hundredths column to the tens column. We must make sure that we show that the units column is worth zero by writing a zero in this space.

If there are no digits to the left of the decimal point, we always write a single zero digit in front of the decimal point to show this.

When multiplying a number with more than one digit by 1000, we need to make sure that we move every digit three places to the left.

For example:

We begin by moving the 3 three places to the left. We move it from the tenths column to the hundreds column. The 9 can then follow. We move the 9 from the hundredths column to the tens column.

We must make sure that we show that the units column is worth zero by writing a zero in this space.

And so, 0.39 x 1000 = 390.

We can see the the digits in the original number, which are ‘3’ and ‘9’, remain in the same order in the answer and that they are still next to each other.

We need to be more careful when dealing with numbers where there is a zero between two other digits.

For example:

We must make sure that we move every digit in 7.04 three places to the left, including the zero. We move the 7 from the units column to the thousands column, we move the 0 from the tenths column to the hundreds column and we move the 4 from the hundredths column to the tens column.

Because there are no longer any digits in the units column, we must write in a zero to show that it is worth zero.

The digits in the answer are in the same order as the digits in the original question. We have a ‘7’, followed by a ‘0’, followed by a ‘4’.

We can us this to help us multiply numbers by 1000 quickly and mentally.

We know that when we are multiplying by 1000 any digits in the units column will move to the thousands column.

We have a ‘7’ in the units column, so we know that when we multiply by 1000, the answer will be 7 thousand and something.

We simply move the 7 to the thousands column and repeat the following digits in that same order. So we put the ‘0’ and the ‘4’ immediately afterwards.

When teaching multiplication by 1000, we can start by showing the place value columns and with practice, knowing that the digits will remain in the same order, we can speed this up to doing it mentally.

When we multiply a number by 100 we move each digit in that number two place value columns to the left.

Multiplying a whole number by 100 is the same as adding two zeros to the end of the number.

In the example below we will multiply the whole number 8 by one hundred.

In the example above, the 8 has been moved two places to the left. It has moved from the units column to the hundreds column. As there are no longer any digits in either the tens column or the units column, we write a zero in both spaces to show that they are worth zero.

8 is a whole number with no digits after the decimal point. Therefore we can use the trick of “adding two zeros” to the end of the number. This trick only works when multiplying whole numbers by one hundred.

It is important to understand how multiplying by 100 works because not all numbers that we will be multiplying will be whole numbers.

Here is an example of multiplying a decimal number by 100:

To multiply the decimal 0.8 by 100, we move the 8 two places to the left. We move it from the tenths column to the tens column. We must also make sure that we show that the units column is worth zero by writing a zero in that space.

And so, 0.8 x 100 = 80.

In the following example, we are asked to multiply the decimal 0.08 by 100.

To do this, we move the 8 two places to the left, from the hundredths column to the units column. And so, 0.08 x 100 = 8

When multiplying a number with more than one digit by 100, we need to make sure that we move every digit two places to the left.

For example:

To multiply 5.2 by 100, we begin by moving the 5 two places to the left. It moves from the units column to the hundreds column. The 2 can then follow. We move the 2 from the tenths column to the tens column.

Because there are no longer any digits in the units column, we must write in a zero to show that the column is worth zero. Therefore, 5.2 x 100 = 520.

We need to be more careful when dealing with numbers where there is a zero between two other digits.

For example:

We must make sure that we move every digit in 3.901 two places to the left, including the zero.

We move the 3 from the units column to the hundreds column, we move the 9 from the tenths column to the tens column, we move the 0 from the hundredths column to the units column and we move the 1 from the thousandths column to the tenths column.

Therefore, 3.901 x 100 = 390.1.

To multiply a number by 10 move each digit in that number one place to the left.

If there is no longer a digit in any of the columns to the left of the decimal point, put a zero in these columns. These are the units columns, tens columns, hundreds columns and so on.

To multiply a whole number by ten, add a zero to the end of the number.

This is a commonly taught technique for multiplying by ten, however this will only work for whole numbers and will not work for decimals.

For example, we will multiply the whole number 9 by ten.

In the example above, the 9 has been moved one place value column to the left. The digit of ‘9 is moved from the units column into the tens column.

Because there is now a space in the units column, a zero is written in to show that the units column contains nothing.

Since 9 is a whole number, it is much easier to put a zero on the end of the digit ‘9’ to make ’90’.

We might also know from our times tables that 9 x 10 = 90.

However, it is important to understand how multiplying by 10 works because not all numbers that we will be multiplying will be whole numbers and we will use this same method for decimal numbers.

Here is an example of multiplying a decimal by 10:

To multiply the decimal 0.9 by 10, we move the digit of 9 one place to the left. So we move the nine from the tenths column to the units column. Therefore, 0.9 x 10 = 9.

We do not write any zeros after a decimal point unless there are other non-zero digits that come after the decimal point.

In the following example, we are asked to multiply the decimal number 0.009 by 10.

We begin by moving the digit of 9 one place to the left, from the thousandths column to the hundredths column. We then make sure that we show that the units column and the tenths column are both worth zero by writing a zero in each space.

We always put zeros in between the decimal point and our first digit that is not zero.

We also always write the zero before the decimal point if there are no digits to the left of the decimal point at all.

For instance we wrote 0.09 rather than .09.

When multiplying a number with more than one digit by 10, we need to make sure that we move every digit one place value column to the left and that the digits remain in the exact same order.

For example:

To multiply the decimal number of 4.1 by 10, we begin by moving the 4 one place to the left. So we move it from the units column to the tens column.

The digit of 1 can then follow. We move the 1 from the tenths column to the units column. And so, 4.1 x 10 = 41.

Notice how the two digits in 4.1 were ‘4’ and ‘1’ in that order and they remain next to each other in the answer of 41.

Since the answer of 41 is a whole number, we do not need to write a zero after the decimal point.

We need to be more careful when multiplying by 10 where there is a zero between two other digits.

For example:

We must make sure that we move every digitin the decimal 3.08 one place to the left, including the zero. We move the 3 from the units column to the tens column, we move the 0 from the tenths column to the units column and we move the 8 from the hundredths column to the tenths column.

The digits remain in the same order. Therefore, 3.08 x 10 = 30.8.

A trick for teaching multiplying by ten is that instead of moving the digits one place to the left, we move the decimal point one place to the right.

Technically, this is not what is happening, however this can be used as a trick to help students who find the original process more challenging. This is because it can be easier to move one decimal point right rather than moving every digit to the left in a longer number.

When multiplying a decimal number by ten mentally, moving the decimal point may be a suitable process to use, however it is worth remembering that it is actually the digits that are moving.