Rounding off a number to the nearest ten means to write down the multiple of ten that the given number is nearest to.

To teach rounding off to the nearest ten, it is helpful to introduce rounding using a number line. This helps to build an understanding of the relative sizes of each number and then to fully understand the process of rounding.

To round off to the nearest ten using a number line place the number on the number line and count how far away the closest multiple of ten is above or below this number. The closest number in the ten times table is the answer.

For each number there is a choice of whether to round down or to round up.

Rounding down to the nearest ten means to write down the number in the ten times table that is below the given number. This will be the number of tens that the given number contains.

Rounding up to the nearest ten means to write down the number in the ten times table that is above the given number. This will be the ten that is one more than the ten that the given number contains.

Below is the number 23. 23 contains ‘2’ tens, which is 20.

The choice is whether to round down to 20 or to round up to the next ten along, which is 30.

On the number line, label the 23, 20 and 30.

When teaching rounding off using a number line, you will most likely be using a number line that is already completed with all whole numbers written on it. In this case we can simply find the number 23 and then look to the left and right to find the next numbers that have a ‘0’ digit in the units column.

Counting how far away 23 is from 20, we count: 22, 21 and 20.

23 is 3 whole numbers away from 20.

Counting how far away 23 is from 30, we count: 24, 25, 26 ,27, 28, 29 and 300.

23 is 7 whole numbers away from 30.

23 is nearer to 20 than it is to 30 and so, 23 rounds off downwards to 20.

We can see that 25 is the midway point between 20 and 30 and 23 is on the left of this point. We can see that because it is less than 25, it will round down.

We will now look at the example of 26. 26 contains a ‘2’ in the tens column. 2 tens are 20.

The choice is whether to round down to 20 or whether to round up to 30.

Counting down to 20 we have: 25, 24, 23, 22, 21 and 20.

26 is 6 numbers larger than 20.

Counting up to 30 we have: 27, 28, 29 and 30.

26 is 4 numbers below 30.

26 is nearer to 30 than it is to 20 and so, 26 rounds up to 30.

Remember that 25 is the midway point between 20 and 30. We can see that 26 is nearer to 30 than 20 because it is on the right hand side of 25.

So we can see that on the number line, we can look at the midpoint between two tens and look at the position of the given number in comparison to this midpoint.

The midpoint between any two tens will always by a number that ends in a 5 in the units digit.

When teaching rounding off using a number line, you can look at different tens and then mark the number that is directly in the middle to show this.

We need to decide how to round off 25, which is directly in between 20 and 30.

Do we decide to round down to 20 or round up to 30? We have to choose one of these multiples of ten.

We decide to round 25 up.

The rule is that if a number ends in 5 or more then round it up and if a number ends in 4 or less, round it down.

The reason we round down numbers which end in 4 or less, is that they are on the left of the midpoint between two tens.

Any number ending in 6 or more is on the right of the midpoint and we round it up.

The reason we also round up numbers with 5 in the units digit is because as soon as we have any extra decimal digits attached, the number will move to the right of the midway point.

For example, 25.1 is now nearer to 30 than it is to 20.

By including 25 in the rule to round up means that we also correctly round up all numbers such as 25.1, 25.9 and 25.001.

Once we have learnt how to round off numbers using a number line, it is helpful to teach the link between the digits of a number, whilst still referring to the number line.

We will look at using the rounding off rule for the example of 57.

The rounding off rule states that if the whole number ends in a 5 or more, then round up, otherwise round down.

57 ends in a ‘7’ digit.

We have the choice of whether to round down to 50 or up to 60.

Since 7 is 5 or more it rounds up.

57 rounds off to 60 when written to the nearest ten.

57 is nearer to 60 than it is to 50.

When first teaching this rounding off rule, remember to refer to the number line to build the understanding of why the rule works.

We will look at the example of rounding off 83 below.

We look at the digit in the units column, which is a ‘3’.

The rounding off rule says that if this digit is 4 or less, then round down.

Since the number ends with a 3 in the units column, it rounds down.

83 rounds off to 80.

83 is nearer to 80 than it is to 90 because it is left of the midway point of 85.

Here is an example of rounding 95 off to the nearest ten.

On a number line, 95 is directly in between 90 and 100.

90 and 100 are the first two numbers either side of 95 that are in the ten times table. 100 is 10 lots of 10.

Remember that the rounding off rule says that any number with a 5 or more in the units column will round up.

Even though 95 is directly in between 90 and 100, we round it up.

95 rounds off to 100 when written to the nearest 10.

How to Tell if a Number is Divisible by 4

To decide if a number is divisible by 4, follow these steps:

1. Look at the last two digits in the tens and ones columns of the number.
2. If this 2-digit number is divisible by 4, then the original number is divisible by 4.
3. All 2-digit numbers are divisible by 4 if they can be halved and halved again to give a whole number.

If a number is divisible by 4, this means that it is a multiple of 4. A number that is divisible by 4 is in the 4 times table and can be divided exactly by 4 leaving no remainder.

A number is divisible by 4 if its last 2 digits are divisible by 4. There is no need to look at the preceding digits. All numbers are divisible by 4 if they can be halved and halved again to give a whole number.

For example, we will test if 732 is divisible by 4.

The first step is to look at the last 2 digits of the number.

The last two digits of 732 are 32.

The next step if to decide if the last 2 digits are divisible by 4.

32 can be halved and then halved again to give a whole number. 32 ÷ 2 = 16 and then 16 ÷ 2 = 8. 32 is a multiple of 4 because it is 4 × 8.

32 is divisible by 4 and so, 732 is divisible by 4.

This means that 732 can be divided exactly by 4.

732 divided by 4 is 183.

The divisibility by 4 rule only tells us if a number is divisible by 4 but it does not tell us the answer to the division.

Here is an example of using the divisibility by 4 rule to prove that a number is not divisible by 4.

We have 44,422.

Looking at the last two digits of 44,422, we have 22.

22 is not a multiple of 4 and so, the number 44,422 is not a multiple of 4 either.

We know that 22 is not a multiple of 4 because it cannot be halved and halved again to leave a whole number.

Half of 22 is 11 and half of 11 is 5.5.

Here is another example of using the rule to test for divisibility by 4.

Here is 3740.

The last two digits are 40.

40 is a multiple of 4 and so, 3740 is too.

We can see that 40 can be halved and halved again to make 10. 40 is 10 lots of 4.

Why Does the Divisibility by 4 Rule Work?

The rule for divisibility by 4 works because 4 divides exactly into all multiples of 100. The hundreds part of the number is always divisible by 4 so only the digits in the tens and ones columns need checking. If the last 2 digits are divisible by 4, the number is divisible by 4.

All multiples of 100 are divisible by 4. This is because 4 × 25 = 100. If 4 divides exactly into 100, it divides exactly into all multiples of 100 such as 200 and 300.

All numbers can be partitioned into their hundreds, tens and units. For example 116 can be written as 100 + 16.

100 is divisible by 4 and so we just need to check if 16 is divisible by 4 too. If both 100 and 16 are divisible by 4, then 116 is also divisible by 4.

We just check the last two digits of 16.

16 is divisible by 4 because we can halve it and halve it again to leave a whole number.

16 is 4 lots of 4.

16 is a multiple of 4 and so, 116 is also a multiple of 4.

All three digit numbers and greater can be written as a multiple of 100 plus a 2-digit number. The multiple of 100 is divisible by 4 so if the last 2 digits are divisible by 4, the number itself is divisible by 4.

For example, 52,164 can be written as 52,100 + 64. 52,100 is a multiple of 100 and so, it is divisible by 4. We just need to decide if 64 is a multiple of 4.

64 can be halved to get 32 and halved again to get 16. 64 is 16 × 4.

64 is divisible by 4 and so, 52,164 is also divisible by 4.

Halve and Halve Again Strategy

The halve then halve again strategy is a method used to divide larger numbers by 4. Dividing by 4 is the same as dividing by 2 and then dividing by 2 again. This method is used to break down more complicated divisions into more manageable steps.

For example to divide 20 by 5, we halve 20 and then halve it again.

Half of 20 is 10. 10 is an even number and we can halve it again.

Half of 10 is 5. Therefore 20 ÷ 4 = 5.

Although the halve and halve again strategy has 2 steps, they are generally easier to manage than simply dividing a number by 4. This means that this strategy is useful for dividing by 4 mentally.

A simple way to test if a number is divisible by 4, we simply need to be able to halve it and halve it again. If after halving once, the number is even, then it can be halved again. If a number gives an even number when halved then it is divisible by 4.

Here is 60 ÷ 4.

Half of 60 is 30. We can see that 30 ends in a 0 and so, it is even.

60 must be divisible by 4 because it gives an even number when halved.

Half of 30 is 15 and so, 4 × 15 = 60. 4 divides into 60 fifteen times.

List of Numbers Divisible by 4

There are 25 numbers between 0 and 100 that are divisible by 4:

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100.

Because 100 ends in a 0 the pattern repeats again for the next 100 numbers.

We have:

104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200.

There are 25 multiples of 4 for each hundred numbers. If you know the first 25 two-digit multiples of 4, then these same two digits are in every multiple of 4.

How to Convert a Decimal to a Fraction

To convert a decimal to a fraction follow these steps:

1. Write the decimal digits as the numerator of the fraction.
3. Write the place value column of the last decimal digit as the denominator of the fraction.
4. Simplify the fraction if necessary.

The decimal place value columns from left to right are the tenths, hundredths and thousandths.

The last digit of the decimal number determines what the denominator on the bottom of the fraction is.

For example, the last digit of 0.9 is in the tenths column and so the fraction will be out of 10.

The only digit of the decimal number is 9 and so, 9 is the numerator of the fraction.

The decimal number 0.9 is written as a fraction as   ⁹ / ₁₀  .

If there is only one decimal place in the decimal number, the fraction is written out of 10.

Here is an example of writing a decimal number in the hundredths as a fraction.

The last digit of 0.93 is in the hundredths column and so, the fraction is out of 100.

The decimal digits are 93, so the fraction is   ⁹³ / ₁₀₀  .

If there are two decimal places in the decimal number, the fraction is written out of 100.

Here is an example of writing a decimal number in the thousandths as a fraction.

The last digit of 0.239 is in the thousandths column, so the fraction is out of 1000.

We write the decimal digits 239 as the numerator.

The decimal number 0.239 as a fraction is   ²³⁹ / ₁₀₀₀  .

If there are three decimal places in the decimal number, the fraction is written out of 1000.

Here is another example of writing the decimal 0.043 as a fraction.

The last digit of 0.043 is in the thousandths column, so the fraction is out of 1000.

We only look at the digits after the first zero and so, the decimal digits of 43 go as the numerator.

0.043 is written as a fraction as   ⁴³ / ₁₀₀₀  .

Here is the decimal number 0.101.

The last digit of 0.101 is in the thousandths column so the fraction is written out of 1000.

The decimal digits are 101. We write 0.101 as a fraction as   ¹⁰¹ / ₁₀₀₀  .

Converting Decimals to Mixed Numbers

To convert a decimal to a mixed number follow these steps:

1. Write the digits before the decimal point down as the whole number part of the answer.
2. Write the digits after the decimal point as the numerator in the fraction of the answer.
3. Write the place value column of the last decimal digit as the denominator.

A mixed number is a combination of a whole number and a fraction written alongside another.

When the decimal number is larger than 1, it will convert into a mixed number.

For example, here is the decimal number of 5.33.

The first step is to write the digit before the decimal point as the whole number part of the answer. We write 5 down as the whole number part of the answer.

Next we form the fraction by looking at the decimal digits.

The last digit is in the hundredths column and so, the fraction is out of 100.

The decimal digits after the decimal point are 33, so we write 33 out of 100.

The decimal 5.33 written as a mixed number is 5   ³³ / ₁₀₀  .

Here is another example of writing a decimal as a mixed fraction. We have 7.03.

We first write the 7 as the whole number part of the mixed number.

The last decimal digit is in the hundredths column and so, the fraction is out of 100.

The first non-zero digit after the decimal point is 3 and so, the numerator is 3.

7.03 is written as a fraction as 7   ³ / ₁₀₀  .

Here is an example of writing the decimal 2.607 as a mixed number.

We first write the whole number 2.

The last digit is in the thousandths column and so the fraction is out of 1000.

The numerator is 607.

2.607 written as a mixed number is 2   ⁶⁰⁷ / ₁₀₀₀  .

How to Write a Decimal as a Fraction in Simplest Form

To write a decimal as a fraction in simplest form, write the decimal digits above the place value column of the last decimal digit. To simplify, divide the numerator and denominator by a number that divides exactly into both of them.

For example, here is the decimal number 3.2. We will write it as a fraction in simplest form.

We first write the whole number part of 3 separately.

The last digit of 3.2 is in the tenths column and so, the fraction is out of 10.

We put the decimal digit of 2 as the numerator.

3.2 is written as a fraction as 3   ² / ₁₀  .

However, both 2 and 10 are even and can be divided by 2.

Dividing 2 by 2 gives us a numerator of 1 and then dividing 10 by 2 gives us a denominator of 5.

We simplify the fraction   ² / ₁₀   to   ¹ / ₅  .

Therefore the decimal number of 3.2 can be written as a simplified fraction as 3   ¹ / ₅  .

How to find the Numerator and Denominator of a Fraction

The numerator is the number on top of the fraction, above the dividing line. The denominator is the number on the bottom of the fraction, below the dividing line.

To find a numerator, read the number on the top of the fraction.

To find a denominator, read the number on the bottom of the fraction.

For example in the fraction   ¹ / ₂   , the numerator is 1 and the denominator is 2.

We can find a missing numerator or denominator in an equivalent fraction.

An equivalent fraction is made by multiplying or dividing the numerator and denominator of a fraction by the same amount.

For example,   ¹ / ₃   is equivalent to   ² / ₆  .

To create an equivalent fraction multiply the numerator and denominator by the same amount.

In this example, we have multiplied the numerator by 2 and the denominator by 2.

When first teaching finding missing numbers in fractions, it is often helpful to include the arrows from each numerator and denominator as shown in the image above.

This method helps children to stick carefully to the idea that we are multiplying or dividing by the same values on the top and on the bottom of our fraction.

Here is the same process for filling in a blank of a fraction shown in reverse. Dividing the numerator and denominator by the same amount will make an equivalent fraction.

We have divided 6 and 2 in half.

How to Find the Missing Numerator of a Fraction

To find a missing numerator, follow these steps:

1. Find the amount that the first denominator is multiplied or divided by to get to the denominator that is missing its numerator.
2. Multiply or divide the known numerator by this same amount to find the unknown numerator.

Here is an example of finding a missing numerator. We have   ³ / ₄   =   / ₃₂  .

We need to fill in the blank for this fraction.

In simple terms, use the following steps to find an unknown numerator:

• Step 1: What do we multiply the denominators by?
• Step 2: Multiply the known numerator by this same value to find the missing numerator.

We have multiplied 4 by 8 to get 32.

To find the amount that one denominator has been multiplied by in an equivalent fraction, divide the largest denominator by the smallest denominator.

32 ÷ 4 = 8 and so, the numerator and denominator in this equivalent fraction have been multiplied by 8.

We multiply 3 by 8 to get 24.

The missing numerator in this question is 24.

To find a missing numerator, look at the denominators of the fractions.

One fraction has both a numerator and denominator. Find the number that this denominator is multiplied by to get to the denominator that is missing its numerator.

Multiply the known numerator by this number to find the unknown numerator.

How to Find the Missing Denominator of a Fraction

To find a missing denominator, follow these steps:

1. Find the amount that the first numerator is multiplied or divided by to get to the numerator that is missing its denominator.
2. Multiply or divide the known denominator by this same amount to find the unknown denominator.

For example, here we have a question with a fraction missing its denominator.   ¹ / ₃   =   ³ /  .

We need to fill in the blank for this fraction.

In simple terms, use the following steps to find an unknown denominator:

• Step 1: What do we multiply the numerators by?
• Step 2: Multiply the known denominator by this same value to find the missing denominator.

We have multipled the first numerator by 3.

To find the value that a numerator has been multiplied by in an equivalent fraction, divide the largest numerator by the smallest numerator.

9 ÷ 3 = 3 and so, the first fraction has been multiplied by 3.

The next step is to multiply the known denominator by 3 to find the unknown denominator.

3 multiplied by 3 is 9 and so, the missing denominator is 9.

To find the missing denominator, first look at the numerators of the fractions.

One fraction has both a numerator and denominator. Find the number that this numerator is multiplied by to get to the numerator that is missing its denominator.

Multiply the known denominator by this number to find the unknown denominator.

Here is another example of finding a missing denominator. In this question, we are dividing to find the unknown denominator.

We have   ⁴ / ₂₀   =   ¹ /  .

We can see that the 4 has been divided by 4 to get a numerator of 1 in the fraction on the right.

Since the numerator of 4 has been divided by 4, the denominator of 20 is also divided by 4.

20 ÷ 4 = 5 and so, the unknown denominator is 5.

The missing denominator is 5.

When teaching filling in the blanks of equivalent fractions to your child, the arrows can be useful for introducing the procedure and idea of equivalency, however it is normal that with practice that they will be able to do these questions without using the arrows.

The understanding of this procedure can be impeded by slow times table recall and I would recommend a times table warm up first using our Multiplication Grid. Then follow up this topic by practising with the 'Equivalent Fractions: Finding a Missing Numerator or Denominator' worksheets above.