What is Regrouping in Maths?

Regrouping is the process of exchanging values between the place value columns of a number. Explained simply, regrouping is moving digits from one place value column to another.

We regroup in groups of ten when moving between the ones, tens, hundreds and thousands place value columns.

For example:

• 10 individual ones (units) can be regrouped as 1 ten.





• 10 individual tens can be regrouped as 1 hundred.





• 10 individual hundreds can be regrouped as 1 thousand.



Teaching Regrouping

When teaching regrouping numbers it is useful to first show this concept using physical manipulatives, such as place value blocks (sometimes called Base 10 blocks, Dienes blocks or Multibase Arithmetic Blocks).

This lesson uses base ten blocks to teach regrouping and it might be useful to work along with us using your own base ten blocks. If you do not have base ten blocks, you can download and print some along with a base ten place value chart above.

When starting to teach regrouping it is easiest to begin by regrouping units to tens.

Start with a collection of units (ones) base ten blocks and some tens rods. We can count the number of blocks in each ten rod to see that there are 10.

We will take some units blocks in the example below and see if we can regroup some as a ten.



We can line the blocks up and count until we make ten.

Remember that 10 individual one cubes can be regrouped as a single ten rod.

Regrouping the ten individual ones as a 10 rod, we have three left over.

10 + 3 = 13

Regrouping physical objects can make it easier to count them.

It is easier to see that we have 3 more than 10 when we have 1 tens rod and the 3 one cubes left over.

We will look at a regrouping example to practise regrouping 10 tens as a hundred. Again it is useful to try this yourself, using a collection of tens rods and hundreds flats (if you do not own Dienes blocks, remember that you can print some above).

Here are a group of tens rods. We will regroup some of the tens as hundreds.



We can count the tens rods until we get to 10.

10 tens rods can be exchanged for a one hundred flat base ten block.

Once the 10 tens have been regrouped as a one hundred, there are 5 tens remaining.

We have 100 (from the hundreds flat piece) plus another 5 tens.

100 + 50 = 150.

Again, it is easier to count these tens now that they have been regrouped. We can see that there are only 5 tens more than one hundred. Regrouping makes it easier to count.

Now we will look at how regrouping is used to move between place value columns.

When teaching regrouping numbers it is useful to represent the numbers on a place value chart with base ten blocks.

Here is an example of regrouping on a place value chart.

Remember that regrouping is used whenever there is more than 9 in a place value column.

Looking at the chart below, there are more than 9 ones (units).



We can circle 10 of these units. When teaching regrouping with base ten block images, a strategy is to use a pen or pencil to circle groups of ten.

We take these ten unit blocks away and replace them with a single ten rod, which is worth the same. However since it is a ten rod, it belongs in the tens column.



Here is the complete regrouping process.



To see the value of the number, we write down the number of blocks in each place value column.



We have 133 made up of ‘1’ hundred, ‘3’ tens and ‘3’ ones.

Here is another example of regrouping numbers on a place value chart.

Looking at the units column, there are 6 ones.

We need ten or more to regroup so we do not regroup these.



In the tens column we have circled a group of ten rods.

We will regroup 10 tens as 1 hundred.

We remove the circled tens from the chart and replace them with a hundred flat, which is placed into the hundreds column.



We now read the number of blocks in each column of the place value chart.



We have 346 represented by these base ten blocks.



Regrouping in Addition (Carrying) and Subtraction (Borrowing)

Regrouping is needed in both addition and subtraction. It is important to teach regrouping because it is necessary for understanding written methods of addition and subtraction. It is particularly the case for adding or subtracting numbers that contain more than 1 digit.

When performing addition, regrouping is often called carrying. Carrying is regrouping up through the place value columns, moving from units to tens to hundreds.

When performing addition, regrouping is used when the addition produces a digit larger than 9 in a place value column.

Here is an example of carrying / regrouping in addition.



When adding 38 + 16, we start by adding the ones (units) column.

8 + 6 = 14, which has two digits.

We write down the ‘4’ in the units but carry the ‘1’ over to the tens column using regrouping.



We add this carried ten as part of the tens column addition.

3 + 1 + 1 = 5



And so, 38 + 16 = 54.

In subtraction, regrouping is called borrowing. Borrowing is regrouping down through the place value columns, moving from hundreds to tens to units.

When performing subtraction, borrowing is used when the digit being subtracted is larger than the digit being subtracted from.

Here is an example of borrowing / regrouping in subtraction.

We have 36 – 18. The 8 is larger than the 6 and so we need to use regrouping.



1 of the tens is worth 10 ones. We can borrow a ten from the ‘3’ and it becomes a ‘2’. We can now write this ten in the ones column. Adding 10 to 6, we get 16.



16 – 8 = 8

And we can subtract the tens column now.

2 – 1 = 1.



36 – 18 = 18

What are Base 10 Blocks?

Base ten blocks are also known as Dienes blocks, Multibase Arithmetic Blocks (MAB Blocks for short) and Place Value blocks.

Base 10 blocks are objects that are used to represent Units (Ones), Tens, Hundreds and Thousands. They are made up of small cubes.

There are four types of base ten blocks, which are called the unit block, the rod, the flat and the cube block.

The smallest value is one cube unit block, which is worth 1.

A row of ten unit cubes are joined together to make a rod.

The rod (or long) is worth 10.

Ten rows of these rods can be joined together to make a flat sheet of 100 unit cubes.

The flat is worth 100.

Ten flats can be joined together to make a cube block.

The cube block is worth 1000. This is the MAB base ten block with the largest value.

Below is a printable teaching poster that may be useful for teaching the Dienes block values.

Dienes blocks (or MAB blocks) are also called base 10 blocks because they follow our number and place value system. This system is called the base ten system because the numbers are built from 10 different digits: 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0. This means that each place value column works in groups of ten. The value of each place value column is ten times greater than the one to its right. We could also say that the value of each place value column is ten times smaller than the one to its left.

Why are Base 10 Blocks Important?

Base 10 blocks are a useful way to enable children to visualise numbers and to see how they relate to each other in our number system. Base ten Dienes blocks are a physical manipulative that children can used to build and represent numbers. The advantage of Dienes blocks is that they can easily represent larger numbers when compared to other manipulatives such as counters.

MAB base ten blocks allow children to see how ten unit blocks in the units column can be replaced with one ten rod, which we can then move into the tens column. This helps children to build a stronger understanding of the regrouping process which is used in addition and subtraction.

How to use Base 10 Blocks to Represent, Model and Build Numbers

Base ten MAB blocks can be used to represent numbers by placing them in groups onto a blank place value chart. The number of each types of base ten block is the same as the digits of the number being represented.

When teaching base ten place value, it is best to start with the units column and work through the columns looking at larger and larger numbers.

Here are some unit cube MAB blocks used to represent the first few counting numbers.

Here are 3 unit blocks.

Units are the smallest value of the Dienes blocks and we put these three blocks in the units column of the place value chart.

The 3 units are simply worth 3.

Here is an example of drawing the number 6 with base ten blocks.

Which number is represented by the base ten block image example below?

We put the Dienes blocks onto a blank place value chart. The rods are worth ten and so they belong in the tens column.

The number drawn in the image above contains 4 rods in the tens column. We write a ‘4’ in the tens column.

There are no blocks in the units column and so we put a ‘0’.

We have 40 represented with the base ten blocks.

Here are some examples of numbers drawn in base ten blocks for you to work out the value of.

We have 3 ten rods drawn as base ten blocks.

3 tens are worth 30.

These MAB base ten blocks are worth 30.

Here are some Dienes rod blocks drawn on a place value chart. What is the value of these blocks?

We have 5 ten rods. 5 tens are 50.

We will now try drawing numbers with base ten blocks.

Try drawing 500 using base ten blocks.

Looking at the digits from right to left, we have ‘0’, ‘0’ and ‘5’.

We need ‘0’ units, ‘0’ tens but ‘5’ hundreds. We pick up 5 of the hundred flats.

5 hundred flat Dienes blocks are worth 500.

What is the value of the MAB base ten blocks shown in the following place value chart?

We have 7 hundred flat MAB blocks.

7 hundreds are worth 700.

We can write this number with a 7 in the hundreds column because there are 7 separate Dienes blocks in the hundreds column.

We write a zero in the other columns because there is nothing in them.

We will now look at some examples of drawing and modelling numbers that are built from more than one type of base ten block.

What is the 3-digit number represented by this set of base ten blocks?

We count the number of MAB blocks in each place value column.

In the hundreds column we have 2 blocks.

In the tens column we have 3 blocks.

In the units / ones column we have 5 blocks.

The number represented by these base ten blocks is 235.

Here is another example of modelling a number using Dienes bas ten blocks.

There are 4 hundred flats, 2 ten rods and 8 unit cubes.

The number that has been built is 428.

We will now try drawing the number 890 using place value blocks.

Looking at the digits from right to left, 890 contains ‘0’ unit blocks, ‘9’ tens rods and ‘8’ hundred flats.

What number is drawn using the Dienes base ten blocks shown?

There are ‘7’ unit blocks, ‘0’ tens rods and ‘6’ hundred flats.

We have 607.

We have a printable place value chart for teaching place value using base ten blocks to work along with all of these examples.

What are some other uses for base 10 blocks?

Base 10 blocks are useful when introducing addition and subtraction using written methods such as column addition. They can help to visualise the size of the numbers and are a useful teaching tool for introducing these topics.

Our lessons using place value blocks for teaching addition and subtraction can be found at:

• 2-Digit Column Addition
• Column Subtraction without Borrowing / Regrouping

Place value blocks are also useful for understanding the regrouping process used in addition and subtraction. Regrouping with base 10 blocks has been seen already where we exchange 10 units for a tens rod or 10 tens rods for a one hundred flat piece.

Base 10 blocks can also be used to introduce volume. The unit cubes can be used to show the unit value (simplest value), which would be one centimetre cubed (cm³). These can then be used to show how they can be combined to make different sizes of cubes, cuboids or other abstract shapes. The volume of the new shape can be found by counting up the total number of cubes.

How to Write a Ratio as Fractions

To write a ratio as fractions, add the total parts in the ratio to find the denominators and write each part of the ratio as the individual numerators.

When writing ratios as fractions, the number of fractions that are formed is the same as the number of parts in the ratio.

Here is a step-by-step guide to writing a ratio as a fraction:

1. Count the number of different terms in the ratio to find the number of fractions that will be made.
2. Add the different terms in the ratio to find the denominator that will be written on the bottom of all of the fractions.
3. Write each individual term of the ratio as the numerator for each of the separate fractions.

Here is an example of converting a ratio to two fractions.

In this example, there are 4 blue counters to 2 purple counters. The counters are in a ratio of 4:2.

There are two terms in the ratio, 4 and 2. Therefore, we can write this ratio as two separate fractions: the fraction of counters that are blue and the fraction that are purple.

The first step in writing this ratio as a fraction is to add the two parts in the ratio to find the denominator, which is the number on the bottom of the fraction.

We add together the number of blue counters and the number of purple counters.

4 + 2 = 6

So, the denominator of each fraction is 6.

The denominator tells us the total number of items in the ratio. In this example it tells us the total number of counters.

Next, we will find the fraction of the total counters that are blue.

According to the ratio 4:2, four of the counters are blue. This means that four out of six counters are blue. We take the 4 from the ratio and write it as the numerator on top of the fraction.

Therefore, the fraction of counters that are blue is   ⁴ / ₆.

Now, we’ll find the fraction of counters that are purple.

If we look at the ratio, we can see that two of the counters are purple. This means that two out of the six counters are purple. We take the 2 from the ratio and write it as the numerator of the fraction.

Therefore, the fraction of counters that are purple is   ² / ₆.

The sum of the parts in the ratio tells us the denominator of the fractions.

Each part in the ratio tells us the numerators of the fractions.

To write a ratio as a fraction in its simplest form, add the parts of the ratio to form the denominator, write the individual parts of the ratio as the numerators and then divide the numerators and denominators of the fraction by the same amount if possible. Only divide the numerators and denominators if they can be divided exactly to produce a whole number.

The fractions that are formed from this ratio can be simplified by dividing both the numerators and denominators by 2.

The fraction   ⁴ / ₆   can be simplified to   ² / ₃  and the fraction   ² / ₆  can be simplified to   ¹ / ₃ .

Here is another example of converting a ratio to a fraction.

We have boys and girls in a ratio of 5:8. We want to write the fraction of the total amount that are boys and the fraction of the total amount that are girls.

We follow the steps as before.

5 + 8 = 13, so the denominators on the bottoms of the fractions will be 13.

We then put 5 and 8 as the numerators on the tops of the fractions.

⁵ / ₁₃  of the people are boys and   ⁸ / ₁₃  of the people are girls.

We cannot simplify these fractions any further because 13 is a prime number and cannot be divided exactly by any smaller number.

Here is another example of writing a ratio as a fraction.

We have cat owners to dog owners in the ratio 12:7.

The first step is to add the parts in the ratio together to find the denominators of the fractions.

12 + 7 = 19 and so, 19 is the denominator on the bottom of the fractions.

The next step is to write each separate part of the ratio as numerators on the tops of the fractions.

We have   ¹² / ₁₉  and   ⁷ / ₁₉  .

This means that   ¹² / ₁₉  are cat owners, while   ⁷ / ₁₉  are dog owners.

19 is a prime number and so it cannot be divided exactly by any number apart from 1 and itself. We cannot simplify these fractions any further.

How to Write Fractions as a Ratio

To write fractions as a ratio, first write the fractions as equivalent fractions that have the same denominator. Then write the numerators of the fractions, each separated by a colon.

Here is an example of converting fractions into a ratio.

Of the following 5 counters,  ³ / ₅   are blue and   ² / ₅   are purple.

The

of 5 mean that there are 5 counters in total.

The

of 3 in   ³ / ₅   means that three of the counters are blue.

The numerator of 2 in   ² / ₅   means that two of the counters are purple.

To write this as a ratio, we write the number of blue counters and number of purple counters.

There are three blue counters and two purple counters.

We separate the numbers in a ratio with a colon.

So, the ratio is written as 3:2.

We can see that there is a simple way to write the two fractions as a ratio, by comparing the numerators.

Since the fractions are both out of 5, we can simply look at the numerators.

The numerator of 3 tells us that there are three blue counters, so we can write 3 in the ratio.

The numerator of 2 tells us that there are two purple counters, so we can write 2 in the ratio.

Here is another example of writing two fractions as a ratio.

We have the fractions of   ² / ₇   and   ⁵ / ₇  .

Since the denominators are both 7, this tells us that there are 7 parts in total.

We can simply write the numerators as a ratio.

We have 2 : 5.

There is a simple check to see if the fractions have been converted to a ratio correctly. Add the numbers in the ratio to see if they add up to the denominator of the fraction.

For example, 2 + 5 = 7, which is the denominator of both fractions in the question.

Here is another example of writing fractions in a ratio.

We have   ⁷ / ₁₁   who are boys and   ⁴ / ₁₁   who are girls.

The fractions have the same denominator, so it is easy to write them as a ratio.

We write the 7 and 4 as a ratio, separated by a colon.

We have the ratio 7 : 4. In this example, this means that there are 7 boys to every 4 girls.

Here is an example of comparing the fractions of the amounts of cats and dogs owned by a group of people.

We have   ⁹ / ₂₃   cats and   ¹⁴ / ₂₃  dogs.

Since the denominators of the fraction are the same, we can simply write the numerators in a ratio.

We have 9 : 14, which means that for every 9 cats there are 14 dogs.

How to Convert Fractions to Ratios when the Denominators are Different

If the denominators of the fractions are not the same, then we need to use equivalent fractions to make them have a common denominator before we can write them as a ratio.

Here is an example with more than one fraction with different denominators.

We have   ² / ₃   ,   ¹ / ₄   and   ¹ / ₁₂   .

The lowest common multiple of 3, 4 and 12 is 12. This means that 12 is the first number that can be divided exactly by 3, 4 and 12.

We will use 12 as our denominator.

We multiply both the numerators and denominators by the same value to write the fractions as equivalent fractions.

We write   ² / ₃   as   ⁸ / ₁₂   by multiplying the numerator and denominator by 4.

We write   ¹ / ₄   as   ³ / ₁₂   by multiplying the numerator and denominator by 3.

¹ / ₁₂   is already out of 12, so we do not need to change it.

Now that the fractions have the same denominator, we can write them as a ratio using their numerators.

We can write   ⁸ / ₁₂   ,   ³ / ₁₂   and   ¹ / ₁₂   as 8 : 3 : 1.

We can check our answer by adding the values in the ratio.

8 + 3 + 1 = 12, which is our denominator on the bottom of the fractions.