To divide large numbers using the long division method, we set out the calculation as shown below.

We will divide the digits of 8190 by 15.

15 doesn’t divide into 8, so we will divide 81 by 15.

To find out how many times 15 divides into 81, we can list some of the numbers in the 15 times table.

We can stop when we have gone past 81.

75 is the closest to 81 without gong past it.

15 x 5 = 75

So, 81 ÷ 15 = 5

We write the 5 above the 1.

To find the remainder of our division, we subtract 75 from 81.

81 – 75 = 6

The remainder is 6.

We bring down the 9 and continue with the long division method.

Next, we find out how many times 15 divides into 69.

We can see that 4 x 15 = 60. So, 15 divides into 69 four times.

To find the remainder, we subtract 60 from 69.

69 – 60 = 9

The remainder is 9.

We bring down the 0 and continue with the long division method.

From the 15 times table that we have written, we can see that 90 ÷ 15 = 6.

Therefore,

8190 ÷ 15 = 546

How to Find One Half of a Number

To find one half of a number, divide it by 2. For larger numbers, split them into their tens and ones, halve these separately and then add them together.

For example, to find one half of 8, we divide it by 2.

Dividing by 2 means to split the total amount into two equal groups. There will be an equal number in each of the two groups.

We started with 8 counters.

We can see that there are 4 counters in each of the two groups and so, 4 is half of 8.

We say that half of 8 is 4.

We can also say that 8 divided by 2 is 4.

We can write this as 8 ÷ 2 = 4.

How to Teach the Concept of One Half

To teach the concept of one half, use physical items that can be divided easily into two, such as food items or piles of counters. Separate the amount into two equal groups by moving one item at a time. You can also use the phrase, “One for me, one for you” to share an amount in half.

To have one half of an amount, the amount must be split into two groups and there must be the same amount in each of the two groups.

When teaching halving, it is useful to have physical objects that can be used to compare the two groups.

When first identifying one half, objects such as fruit and vegetables can be easily cut into two equal parts. It helps to identify halves when they arise in daily life and show that the two halves are identical and that there are always two parts.

The key point to make when teaching the concept of one half, is that there are always two parts that must be exactly the same size.

Teaching Halving Numbers

To teach the halving of smaller numbers, count out the required amount of counters and move them one by one into two equal piles. For larger numbers, break them down into tens and ones, halve each separately and then add up the result. It may help to write this down.

Start with introducing the idea of halving small numbers. To avoid confusion, only halve even numbers when first introducing the concept of halving. This is because halving odd numbers will result in fractions and decimals which are a more advanced concept that your child may not have understood just yet.

Here are the even numbers. Even numbers are numbers that can be halved exactly without remainder.

Even numbers only end in 0, 2, 4, 6 or 8. When first teaching halving, it is best to pick a number ending in 0, 2, 4, 6 or 8 as it can be halved exactly.

Start with introducing halving with just 2 counters.

Ask your child to move 1 counter to each side to introduce the idea of forming two equally sized groups.

Then move on to 4 counters.

To help your child share an amount in half, use the phrase, “One for me, one for you”.

They should say, “One for me” as they place a counter in the first pile and then say, “One for you” as they place the counter in the other pile.

We can see that half of 4 is 2. Show them that there is exactly 2 counters in each pile. It is important to show that each pile must be the same size if an amount has been halved.

Here is one half of 6.

One half of 6 is 3. Again, count the counters one at a time into two equal groups.

One half of 8 is 4.

One half of 10 is 5.

To halve larger numbers, separate the number into its tens and ones, halve these separately and then add the result together.

Use the following results to halve any larger number:

• One half of 2 is 1.
• One half of 4 is 2.
• One half of 6 is 3.
• One half of 8 is 4.
• One half of 10 is 5.

When teaching halving larger numbers, it is worth learning how to halve each of the even numbers to 10 off by heart. Use these to half the tens and ones parts of larger numbers.

For example, we will find one half of 16.

16 can be partitioned into 10 + 6.

From the list above, we know that half of 10 is 5 and half of 6 is 3.

We simply add 5 and 3 to make 8.

One half of 16 is 8.

In this example, we will find one half of 28.

We can write 28 as 10 + 10 + 8. We then halve each of these values and add up the result at the end.

Half of 10 is 5, half of 10 is 5 and half of 8 is 4.

5 + 5 + 4 = 14 and so, half of 28 is 14.

How to Compare Decimals

To compare decimals, use the following steps:

1. Line up the digits in each place value column directly above another.
2. Compare the digits in each place value column individually from left to right.
3. If the digits are equal, compare the digits in the next column to the right.
4. If a digit is larger than the others, this digit belongs to the largest decimal.

For example, we will compare the two decimals 0.1 and 0.3.

The first step is to line up the digits in each place value column. When teaching this, it is useful to start by lining up the decimal points and then writing each digit directly above the others. Square grid paper can be useful for ensuring that the digits are written above another.

The second step is to look at the digits in the place value columns, starting on the left and moving right.

The left-most digits are in the ones column and they are both 0. They are equal.

If the digits are equal, we compare the digits in the next place value column to the right.

3 is greater than 1 and the 3 belongs to the decimal 0.3. Therefore 0.3 is a larger decimal than 0.1.

The two numbers do not have a whole number component as there are 0 ones in both numbers. 0.3 has 3 tenths, whereas 0.1 only has 1 tenth.

We can compare two decimals using a decimal model.

In this model, the whole square represents one whole unit and each column is one tenth.

We can see that 0.1, shown in red, is one tenth and 0.3, shown in yellow, has three tenths. 0.3 is clearly larger than 0.1.

Here is an example of comparing the decimals 0.2 and 0.14.

We first line the digits up above one another.

Again, the ones digit is 0 in both numbers. We move on to compare the digits in the tenths column.

2 is larger than 1 and so stop. 0.2 is larger than 0.14.

As soon as one digit is larger than another digit in the same place value column, this belongs to the larger decimal. There is no need to compare any other digits in the columns to the right.

A common misconception is that the more digits in the decimal, the bigger the number. This is not a rule and it will often result in the wrong answer.

In this example, some people might think that 0.14 is larger than 0.2 because 14 is bigger than 2. We cannot compare decimal numbers by simply reading the decimal numbers as though they were whole numbers.

0.2 is larger than 0.14 because 0.2 contains 2 tenths, whereas 0.14 contains only 1 tenth.

Here is a decimal model to show the comparison in size of these decimals.

We can clearly see that 0.2 is a larger decimal than 0.14.

We say that 0.2 is greater than 0.14 and this can be written mathematically as 0.2 > 0. 14.

Here is another example of comparing two decimal numbers. We have 0.3215 and 0.3211.

We write the digits directly above each other and compare them from left to right.

The digits are the same in the ones column, they are both 0.

The digits are the same in the tenths column, they are both 3.

The digits are the same in the hundredths column, they are both 2.

The digits are the same in the thousandths column, they are both 1.

Finally, in the ten-thousandths column, 5 is larger than 1.

Therefore 0.3215 is larger than 0.3211. It is larger by only 4 ten-thousandths, which is by a very small amount.

The two decimals are shown in a model below.

Comparing Decimals with Models

To compare decimals with models, use base ten blocks to represent each place value. Collect the number of tenths, hundredths and thousandths together and see which amount is the largest.

Here is a visual image for modelling tenths, hundredths and thousandths.

The entire square shown is worth one whole unit. You can use a base ten block flat to represent one whole if teaching this to a child.

Each column rod is worth one tenth because ten of these rods make the whole square.

Each individual unit cube can be worth one hundredth.

For smaller quantities, you may wish to draw the diagrams as they are shown on this page.

Here the two rods in 0.23 and 0.28 are both shown by the two complete vertical columns.

0.23 has 3 more hundredths, which are shown by the three squares.

0.28 has 8 more hundredths, shown by the 8 squares.

When teaching comparing decimals, is useful to introduce decimals using decimal models before then proceeding onto comparing decimals digit by digit. Eventually, once you have grasped the relative size of decimals, it is not necessary to keep drawing out models each time.

How to Subtract Decimals

To subtract decimals, follow these steps:

1. Line up the digits of the number being subtracted below the other number.
2. Put in zero digits to make the numbers the same length.
3. Subtract the digits in each place value column individually, working from right to left.
4. If the upper digit is smaller than the digit below it, regroup by subtracting 10 from the digit to its left and adding 10 to this digit.
5. Write the answer to each digit subtraction below to form the final answer.

Follow these steps if you need to subtract decimal numbers without a calculator.

Here is an example of subtracting decimals without regrouping.

We have 8.5 – 2.1.

The two decimal numbers have the same length, with one digit after the decimal point.

Write the decimal number being subtracted below the other number. We are subtracting 2.1 from 8.5 and so, we write 2.1 below 8.5.

Start by writing the numbers in line with each other by lining up the decimal places. The ones column digits are written in line with each other and the tenths column digits are written in line with each other.

Before doing any subtraction calculations, write the decimal point of the answer in directly below the other decimal points in the question.

The next step is to subtract the digits in each place value column, starting with the digits on the right. Subtract the digit below from the digit above.

5 – 1 = 4 and so, we write the digit 4 in the answer space below.

We now subtract the digits in the ones column.

8 – 2 = 6 and so, we write 6 in the space below.

The answer is simply read as 8.5 – 2.1 = 6.4.

When teaching subtracting without regrouping, it is important to ensure that the digits are lined up carefully. Use square grid paper and write each digit in each grid. Use a ruler to draw two lines below the question to write the answer in. To line up decimals correctly, write in the decimal point first, directly below the other decimal points. Then write each place value column in line one by one.

This method is called column subtraction or vertical subtraction and it is the best written method to use to subtract numbers without a calculator.

How to Subtract Decimals with Regrouping

Regrouping is used when the digit being subtracted is larger than the digit above. To regroup, add ten to the digit on top and then subtract one from the digit that is in the place value column immediately to the left of this digit.

With subtraction, the process of regrouping is also sometimes called exchanging or borrowing. Exchanging, borrowing and regrouping are just different names for the same process.

Regrouping will be needed whenever any of the digits of a particular number are larger than the corresponding digits in the same place value columns of the number it is being subtracted from.

For example, here is 0.72 – 0.46.

In the hundredths column, the 6 is larger than the 2 it is being subtracted from.

We cannot subtract 6 from 2 using this method because 6 is larger than 2. We need to regroup by borrowing ten from the tenths column.

To borrow in subtraction, add 10 to a digit and subtract 10 from the digit immediately to its left.

We add 10 to the 2 to make 12 and subtract 1 from the 7 to make 6.

This borrowing works because 1 tenth is worth the same as 10 hundredths. By taking one tenth from the 7, we can add 10 to the number in the hundredths column.

After borrowing, the subtraction process can be completed as normal. 12 – 6 = 6.

We now subtract the digits in the tenths column. We use the new regrouped value of 6 now.

6 – 4 = 2 and so, we write 2 in the answer space below.

0.72 – 0.46 = 0.26. Here is the full process of regrouping by borrowing a tenth shown below.

How to Subtract Decimals with Different Decimal Places

If the numbers contain a different number of digits after the decimal point, write zero digits at the end of the shorter decimal until there are the same number of decimal places in both numbers. Then subtract the digits in each place value column from right to left.

For example, 7.39 – 6.4 is written as 7.39 – 6.40 so that both decimal numbers have two decimal places.

7.39 has 2 decimal places, whereas 6.4 only has 1 decimal place. If the decimals have varying lengths, then zeros are written at the end of the shorter decimal to make it the same length.

Now we subtract the digits in the hundredths column. 9 – 0 = 9, so we write a 9 in the hundredths column of the answer.

In the tenths column, 4 is larger than 3 so we borrow from the ones column. 7 – 1 = 6 and then 3 + 10 = 13.

We can now subtract 4 from 13. 13 – 4 = 9 and so, we write the answer of 9 in the tenths column.

In the ones column, 6 – 6 = 0.

Our answer is 7.39 – 6.4 = 0.99.

We can see that it is necessary to write zeros in when subtracting decimals of varying lengths in order to ensure that numbers can be regrouped correctly.

Here is an example of 5.4 – 2.58. 5.4 has 1 decimal place, whereas 2.58 has 2 decimal places.

To subtract decimals of different lengths, write zero digits on to the end of the shorter decimal until they are the same length.

We write 5.4 – 2.58 as 5.40 – 2.58.

We subtract 1 tenth from the 4 tenths to make 3 tenths.

We add 10 to the 0 hundredths to make 10 hundredths.

Now we can subtract the digits in the hundredths column. 10 – 8 = 2 and so, we write 2 in below.

In the tenths column, 5 is larger than 3 so we need to borrow again.

We subtract 1 unit from 5 units to make 4 units.

We add 10 to 3 tenths to make 13 tenths.

We can now subtract the tenths column. 13 – 5 = 8, which we write below.

In the units column, 4 – 2 = 2.

5.4 – 2.58 = 2.82.

The method of subtraction with regrouping can be extended to all decimal numbers and is the best method to use if subtracting decimals without a calculator.

What are the Different Types of Angle?

There are 7 main different types of angle:

• Zero Angles
• Acute Angles
• Right Angles
• Obtuse Angles
• Straight Lines
Obtuse Angles
• Full Turns

The different types of angle are classified by their size in the table below.

Type of Angle	Size in Degrees
Zero Angle	Exactly 0°
Acute Angle	Less than 90°
Right Angle	Exactly 90°
Obtuse Angle	Larger than 90° and less than 180°
Straight Angle	Exactly 180°
Reflex Angle	Larger than 180° and less than 360°
Full Turn	Exactly 360°

How to Classify an Angle

To classify an angle, first measure its size in degrees. Then compare this angle to the following values:

• If the angle is less than 90°, it is an acute angle.
• If the angle is exactly 90°, it is a right angle.
• If the angle is between 90° and 180°, it is an obtuse angle.
• If the angle is exactly 180°, it is a straight line.
• If the angle is exactly 360°, then it is a full turn.

Although you can turn through an angle that is larger than 360°, there is no new name for an angle that is larger than 360°.

For example, turning 360° completes ones full turn and completing 720° is two full turns because 2 lots of 360° is 720°.

Here is a summary table of the different angle types.

Here are some examples of identifying angle types.

The angle in the top left is 315°. This angle is larger than 180° but it is less than 360°. This means that it opens wider than a straight line but it is not quite a full turn. The angle is a reflex angle.

The angle in the top right shows a turn that has gone all the way around back to where it started. It shows a full turn. A full turn is 360°.

The angle in the middle is a right angle. This is because it is exactly 90°. It is also marked with a square in the corner of the angle which tells us that it is a right angle.

The angle at the bottom left is a straight line. Straight lines are exactly 180°.

The angle in the bottom middle is 150°. 150° is larger than 90° but it is less than 180°. It is larger than a right angle but it is not as large as a straight line angle. This is an obtuse angle.

The angle in the bottom right is 40°. 40° is less than 90° and so, this angle is an acute angle.

Here is another example of classifying an angle.

This angle is 60°.

This angle is less than 90° and so, this angle is classified as an acute angle.

Apart from a zero angle, there is no smaller angle than an acute angle. Since zero angles look like a straight line, they are difficult to work with and are not commonly found in typical angle classification questions.

Here is another example of identifying an angle type.

Here we have a 280° angle.

280° is larger than a straight line angle of 180° and it is less than a full turn of 360°. It is a reflex angle.

It is important to make sure that you measure the correct angle when measuring reflex angles. We are measuring the marked arrow shown in red on the diagram. We have rotated all the way around past a straight line to get to this point.

Acute Angles

Acute angles refer to any angle that is less than 90°. This means that an acute angle is smaller than a right angle.

Here are some examples of acute angles.

When teaching acute angles, we can remember that these are the smallest type of angle classification and so, they are ‘a cute’ angle. Linking the idea of being small and being cute can make this name easy to remember.

Acute angles along with right angles are probably the easiest type of angles to learn first because they are most commonly seen in day-to-day life.

Right Angles

Right angles measure exactly 90°. They are a quarter of a full turn and are shown with a small square in the corner of the angle.

Right angles are commonly found around the home such as on the corners of tables, shelves, books and boxes. For this reason, they can be one of the easiest types of angle to teach first.

The name of right angle orignally comes from the latin words for it,’angulus rectus’. Rectus means upright. When measured from horizontal position, a right angle is upright and hence the name right angle.

Obtuse Angles

An obtuse angle is larger than 90° but less than 180°. Therefore an obtuse angle is larger than a right angle but is not as large as a straight line.

Here are some examples of obtuse angles.

A common real life example of an obtuse angle is the tip of a roof of a house.

Straight Angles

A straight angle is exactly 180° and is called this because it looks like a straight line. When an object rotates through a straight angle, it has turned through 180° and therefore it has reversed its original direction.

Here is a picture of a straight angle.

A straight angle, or straight line must be exactly 180. If it is slightly more or less than this, then there will be a deviation at the corner of the angle and it will not be a straight line.

Straight lines are found everywhere and children may find it difficult to think of straight line angles as angles rather than lines. To teach straight line angles, it is helpful to mark the centre of the straight line as the corner of the angle. It is also useful to show an angle rotating around between 179° and 181°. In between must be 180°.

Reflex Angles

Reflex angles are larger than 180° and less than 360°. This means that reflex angles open wider than a straight line but are not large enough to complete a full turn.

Here are some examples of reflex angles.

Reflex angles are sometimes confused in name with obtuse angles. These are the two most commonly mixed up names when teaching angle classification. It is useful to show examples of both reflex and obtuse angles when teaching them. It can be helpful to compare both angles to a straight line, perhaps using a ruler, which can help us decide if the angle is less than or larger than 180°.

Full Turns

A full turn is exactly 360°. After completing a full turn, an object is facing the same direction that it was originally.

Here is a diagram showing a full rotation.

Full rotations can sometimes hard to draw or explain since the two arms of the angle are directly on top of each other.