5 ways to add and subtract in your head

5 ways to add and subtract in your head - "SHARED"

Mental methods are about trying to get to the correct answer in the quickest and easiest way!

There are lots of ways to add and subtract numbers in your head.By the time you have looked through the factsheets in this module you will have used 5 different methods for mental addition and subtraction! Try to remember them by using the word "Shared".

When you've read about them, practise the methods you like and can remember most easily.

Ask other people about the methods they use - and share your methods too!

Keep practising, and HAVE FUN!

Reminders to help you add and subtract

Here are three reminders to help you with mental addition and subtraction.A reminder about place value Have a look at the number 623.

6 is the hundreds digit. 2 is the tens digit. 3 is the units digit.

A reminder about adding Have a look at these sums.

Here you can see that the same numbers added together in a different order will give the same answer.Addition and subtraction are opposites. You can check the answer to a subtraction sum by turning the numbers around and adding them up. Have a look below.

Here you've done the sum 10 - 25 and got the answer 15.To check the answer, turn the sum around to 15 + 10 and see if you get 25.

Splitting up numbers

Splitting up numbers is a good method to use for both addition and subtraction. It is sometimes called partitioning.Addition

Take a look at this addition sum: 80 + 49

To make it easier, split the 49 into 40 + 9. This makes the sum: 80 + 40 + 9 = 129

First, add the first two numbers: 80 + 40 = 120

Then add the result of that sum to the third number to get the answer: 120 + 9 = 129

Subtraction

Take a look at this subtraction sum: 150 - 34

To make it easier, split the - 34 into 30 - 4. This makes the sum: 150 - 30 - 4

First, subtract the 30 from 150: 150 - 30 = 120

Then, subtract the 4 from the 120 to get the answer: 120 - 4 = 116

Hundreds, tens and ones

To make addition sums easier, you can separate the hundreds, tens and units and add them up separately.Have a look at how separating works for this sum: 31 + 22

Using this method you can work out that 31 + 22 = 53.Now look at this sum with hundreds as well as tens and units: 125 + 100 + 235 + 132

Using this method you can work out that 125 + 100 + 235 + 132 = 592.

Rounding - addition

Rounding is a method for mental addition which is useful in many different situations. Imagine you are in a shop and you have to quickly work out an amount.

£3.70 + £1.00 = £4.70. Taking away 10p gives £4.60 Have a look at how this addition sum can be solved with rounding: 75 + 19

First, round the 19 up to 20 and work out the sum: 75 + 20 = 95

As 20 is 1 more than 19, you then need to subtract 1 from the total. 95 - 1 = 94

Then you can see that: 75 + 19 = 94

Further reading. These tables have some more information to help you with addition by rounding.

The first shows methods and examples for adding a number between 11 and 14 to another number.

This table shows methods and examples for adding a number between 15 and 19 to another number.

Rounding - subtraction

Rounding is a method for mental subtraction which is useful in many different situations.Imagine you are shopping and need to work out an amount quickly.

£3.70 - £1.00 = £2.70. Adding 10p gives £2.80. Have a look at how this subtraction sum can be solved with rounding: 64 - 17

First, round the 17 up to 20 and work out the sum: 64 - 20 = 44

As 20 is 3 more than 17, you have taken 3 too many from the total. So you need to add 3: 44 + 3 = 47

So you can see that: 64 - 17 = 47 Further reading. These tables have some more information to help you with subtraction by rounding.

The first shows you a methods and examples for subtracting a number between 11 and 14 from another number.

This table shows methods and examples for subtracting a number between 15 and 19 to another number.

Empty number line for counting on

Counting on using an empty number line is a good method for subtracting numbers mentally.Use this method to find the difference between 37 and 50. This is the same as the sum 50 - 37.

When you have pictured that line, count on from 37 to 40, which makes 3. Keep that 3 in your head. Then, count from 40 to 50, which is 10.Have a look below to see how this works.

Now all you need to do is add the 3 to the 10. This makes 13. So: The difference between 37 and 50 is 13.Or50 - 37 = 13

Doubling

If you are adding together two numbers that are nearly the same, you can double one of them and then adjust the difference.Imagine you are adding together 38 and 35.

Key words for mental addition

Here are some of the words which will crop up when doing addition sums.

Have a look below to see how they can be used in the simple sum 3 + 4 = 7. Add 3 add 4 is 7 Altogether Altogether, 3 and 4 make 7. Increase If you increase 3 by 4 you get 7. More 7 is 3 more than 4. Plus 3 plus 4 is 7. Sum The sum of 3 and 4 is 7. Total The total of 3 and 4 is 7.

Key words for mental subtraction

Here are some of the words which will crop up when doing subtraction sums.

Have a look below to see how they can be used in the simple sum 8 - 5 = 3. Decrease If you decrease 8 by 5 you get 3.

Difference The difference between 8 and 5 is 3. Fewer than 3 is 5 fewer than 8. Less than 3 is 5 less than 8. Minus 8 minus 5 is 3. Reduce If you reduce 8 by 5 you get 3. Subtract 8 subtract 5 is 3. Take away 8 take away 5 is 3.

Mental multiplication methods


Here are some of the mental methods you can use. When you've read about them, practise the methods you like and can remember most easily. Multiplying the tens then the units, then adding them together.

Rounding up one number to the nearest 10 and adjusting the answer.

Doubling one number, then halving the answer.

Tip:

Have you ever worried that if it's called 'mental methods' you have to do it in your head?

Well, you don't have to! It can really help to jot down some figures which make the sum easier for you.

Some more mental multiplication methods


Here are two more mental methods you can use. When you've read about them, practise the methods you like and can remember most easily. Changing the order to make the numbers easier to work with

Numbers can be split into factors to make multiplying simpler.

Tip:

Remember! Using mental methods is about choosing the method that works for you and for the numbers you're working with.

Multiplication Glossary

Here are some of the words which will crop up when doing multiplication sums.

Have a look below to see how they can be used in the simple sums 2 x 2 = 4. Factors 2 is a factor of 4. One number is a factor of another number if it divides, or goes into it exactly. Divisible 6 is exactly divisible by 3. 7 is not exactly divisible by 3. Groups of 2 groups of 2 make 4. Lots of 2 lots of 2 make 4. Multiple 4 is a multiple of 2. Multiply If you multiply 2 by 2 you get 4. Product The product of 2 and 2 is 4. Sets of 2 sets of 2 make 4. Times 2 times 2 is 4.

Mental division tips

A division sum can be shown in several different ways.

Estimating When you divide any numbers, it is a good idea to estimate a rough answer first. Your estimate can then be checked against your actual answer.

92 ÷ 3 is approximately90 ÷ 3 which is 30

143 ÷ 7 is approximately140 ÷ 7 which is 20

994 ÷ 5 is approximately1 000 ÷ 5 which is 200 Check by multiplying Multiplication and division are inverses (opposites). Division sums can be checked by multiplying, like this:

81 ÷ 3 = 27 27 x 3 = 81 Jot it down Have you ever worried that if it's called 'mental methods' you have to do it in your head?

Well, you don't have to! It can really help to jot down some figures which make the sum easier for you.

Some mental division methods

Mental methods are about trying to get to the correct answer in the quickest and easiest way.

Here are two of the mental methods you can use. When you've read about them, practise the methods you like and can remember most easily. Splitting the number you're dividing into, to make it simpler.

Numbers can be split into factors to make dividing simpler.

Some more mental division

Mental methods are about trying to get to the correct answer in the quickest and easiest way.

Here are two of the mental methods you can use. When you've read about them, practise the methods you like and can remember most easily. Spacesaver division This is long division without all the written bits! Let's look at the sum 22 972 ÷ 4.

1. 4 into 2 won't go - so carry 2 2. 4 into 22 (5 x 4 = 20) - so carry 2 3. 4 into 29 (7 x 4 = 28) - so carry 1 4. 4 into 17 (4 x 4 = 16) - so carry 1 5. 4 into 12, that will be 3 exactly With this method you're doing a division sum, but all the thinking is multiplication and subtraction! Dividing with even numbers

120 ÷ 40 is the same as: (keep halving both numbers)60 ÷ 2030 ÷ 1015 ÷ 5which is 3 Tip:

Remember! Using mental methods is about choosing the method that works for you and for the numbers you're working with.

If you're in a group you could vote for the most popular methods, then let us know the result!

Division Glossary

Here are some of the words which will crop up when doing division sums.

Have a look below to see how they can be used in the simple sums 6 ÷ 3 = 2 and 7 ÷ 3. Divide If you divide 6 by 3 you get 2. Divisible 6 is exactly divisible by 3. 7 is not exactly divisible by 3. Groups There are 3 groups of 2 in 6. Left over If you divide 7 by 3 the answer is 2 with 1 left over. Remainder If you divide 7 by 3 the answer is 2 with 1 remainder. Share If you share 6 toffees between 3 people, each person gets 2.

What is ratio?

Ratio is a way of comparing amounts of something. It shows how much bigger one thing is than another. For example:

Use 1 measure screen wash to 10 measures water Use 1 shovel of cement to 3 shovels of sand Use 3 parts blue paint to 1 part white

Ratio is the number of parts to a mix. The paint mix is 4 parts, with 3 parts blue and 1 part white.The order in which a ratio is stated is important. For example, the ratio of screenwash to water is 1:10. This means for every 1 measure of screenwash there are 10 measures of water. Mixing paint in the ratio 3:1 (3 parts blue paint to 1 part white paint) means 3 + 1 = 4 parts in all.

3 parts blue paint to 1 part white paint = is ¾ blue paint to ¼ white paint.If the mix is in the right proportions, we can say that it is in the correct ratio.

Understanding direct proportion

Two quantities are in direct proportion when they increase or decrease in the same ratio. For example you could increase something by doubling it or decrease it by halving.If we look at the example of mixing paint the ratio is 3 pots blue to 1 pot white, or 3:1.

But this amount of paint will only decorate two walls of a room. What if you wanted to decorate the whole room, four walls? You have to double the amount of paint and increase it in the same ratio.If we double the amount of blue paint we need 6 pots.If we double the amount of white paint we need 2 pots.

The amount of blue and white paint we need increase in direct proportion to each other. Look at the table to see how as you use more blue paint you need more white paint:

Pots of blue paint 3 6 9 12

Pots of white paint 1 2 3 4

Have a look at this graph:

Two quantities which are in direct proportion will always produce a graph where all the points can be joined to form a straight line.

Using direct proportion

Understanding proportion can help in making all kinds of calculations. It helps you work out the value or amount of quantities either bigger or smaller than the one about which you have information. Here are some examples:Example 1: If you know the cost of 3 packets of batteries is £6.00, can you work out the cost of 5 packets?To solve this problem we need to know the cost of 1 packet.If three packets cost £6.00, then you divide £6.00 by 3 to find the price of 1 packet. (6 ÷ 3 = 2)Now you know that they cost £2.00 each, to work out the cost of 5 packets you multiply £2.00 by 5. (2 x 5 = 10)So, 5 packets of batteries cost £10.00 Example 2: You've invited friends round for a pizza supper. You already have the toppings, so just need to make the pizza base. Looking in the recipe book you notice that the quantities given in the recipe are for 2 people and you need to cook for 5!Pizza base - to serve 2 people: 100 g flour 60 ml water 4 g yeast 20 ml milk pinch of saltThe trick here is to divide all the amounts by 2 to give you the quantities for 1 serving. Then multiply the amounts by the number stated in the question, 5.For 1 serving, divide by 2: 100 g ÷ 2 = 50 g 60 ml ÷ 2 = 30 ml 4 g ÷ 2 = 2 g 20 ml ÷ 2 = 10 mlFor 5 servings, multiply by 5: 50 g x 5 = 250 g 30 ml x 5 = 150 ml 2 g x 5 = 10 g 10 ml x 5 = 50 mlThe pinch of salt is up to you!

Simplifying ratios

We can often make the numbers in ratios smaller so that they are easier to compare. You do this by dividing each side of the ratio by the same number, the highest common factor. This is called simplifying. Example: In a club the ratio of female to male members is 12:18Both 12 and 18 can be divided by 2. 12 ÷ 2 = 6 18 ÷ 2 = 9So a simpler way of saying 12:18 is 6:9.To make the ratio simpler again, we can divide both 6 and 9 by 3 6 ÷ 3 = 2 9 ÷ 3 = 3So a simplest way of saying 12:18 is 2:3.These are all equivalent ratios, they are in the same proportion. All these ratios mean that for every 2 female members in the club there are 3 males: 12:18 6:9 2:32:3 is easier to understand than 12:18!

Tips for ratio and proportion sums

Ratio can be used to solve many different problems, for example recipes, scale drawing and map work.Changing a ratio A common test question will ask you to change a ratio - the reverse of cancelling down.Example: A map scale is 1 : 25 000. On the map the distance between two shopping centres is 4 cm. What is the actual distance between the shopping centres? Give your answer in km.A scale of 1 : 25 000 means that everything in real life is 25 000 times bigger than on the map.So 4 cm on the map is the same as 4 x 25 000 = 100 000 cm in real life.(Reminder 1 m = 100 cm and 1 km = 1 000 m)Now change the real life distance of 100 000 cm to metres100 000 ÷ 100 = 1 000 mAnd 1 000 m is the same as 1 km.So the shopping centres are 1 km apart.

Keeping things in order When working with ratios keep both the words and the numbers in the same order as they are given in the question.Example: Share a prize of £20.00 between Dave and Adam in the ratio 3:2. The trick with this type of question is to add together the numbers in the ratio to find how many parts there are, divide by the number of parts to find the value of 1 part, then multiply by the number of parts you want to calculate.

First add together the number of parts in the ratio: 3 + 2 = 5 Divide to find out how much 1 part will be: £20.00 ÷ 5 = £4.00 To find Dave's share multiply £4.00 x 3 = £12.00 Adam's share is £4.00 x 2 = £8.00

Dave's £12.00 is of £20.00 (3 of 5 parts). Adam's £8.00 is of £20.00 (2 of 5 parts).

You can check that you have worked out the ratio correctly by adding the shares together. In this sum Dave's and Adam's shares should equal £20.00Let's check: £12.00 + £8.00= £20.00 Correct!

Use the same units Always check that the things you are comparing are measured in the same units. Example: Jenna has 75 pence. Hayley has £1.50 What is the ratio of Jenna's money to Hayley's?.In this problem one amount is in pence, the other in pounds. Before you calculate the ratio you have to make sure they are the same units. We have to convert Hayley's amount into pence first. There are 100 pence to a pound Hayley's £1.50 = 150 pence So the ratio is 75 : 150You can simplify this ratio as both numbers are divisble by 75. The ratio is 1:2.

Key words for ratio and proportion

Ratio is a way in which quantities can be divided or shared.Example: Share £20 between two people in a ratio of 3:1.A ratio of 3 + 1 = 4 parts, the money needs to be divided into 4 parts. 20 ÷ 4 = £5.If one person is getting three parts they will have 3 x 5 = £15 The other person will have one part, £5.

Simplest form. Ratios can be simplfied by finding common factors.Direct proportion. Ratios are in direct proportion when they increase or decrease in the same ratio.Equivalent ratios. When both sides of a ratio can be multiplied or divided by the same number to give an equivalent ratio.Example: In a group there are 15 males and 12 females. What is the ratio of males to females? Give your example in its simplest form.So the ratio of males to females is 15:12. However, both sides of the ratio can be divided by 3. Dividing 15 and 12 by 3 gives 5:4. 5:4 is the ratio in its simplest form. 5:4 and 15:12 are equivalent ratios.

Factor The factors of a number are those numbers which divide into it exactly.Example: 1 x 12 = 12 2 x 6 = 12 3 x 4 = 12So the factors of 12 are 1, 2, 3, 4, 6 and 12.

What is rounding?

Rounding is a way of simplifying numbers. If the driveway of a house is 5 metres and 7 cm long we would usually just say it is 5 m long. Saying it's 5 m long will be close enough most of the time. Here is another example. The picture shows a stick of rock next to a ruler. The ruler has only got the 10 cm points marked on it.

We can't see exactly how long the rock is. But we can see to the nearest 10 cm. The end of the rock is close to the 20 cm mark. So we say that the rock is 20 cm long to the nearest 10 cm. What about this longer stick. How long is it to the nearest 10 cm?

It is closer to 30 cm than 20 cm. So we say it is 30 cm long to the nearest 10 cm. Rounding numbers to the nearest 10 means finding which 10 they are nearest to.

Rounding a number to the nearest hundred or to the nearest thousand can be done in the same kind of way. There is more about this in the other factsheets.

Rules for rounding

Example A stick of rock is 27 cm long. How long is it to the nearest 10 cm? Answer 27 cm is between 20 cm and 30 cm. So 27 cm will get rounded to either 20 cm or 30 cm. To get the right answer we need to decide whether 27 is nearer to 20 or 30.

You can see from the picture that it is closer to 30.So 27 cm is rounded up to 30 cm.So the stick of rock is 30 cm long when we measure to the nearest 10 cm. For the same reasons 26, 27, 28 and 29 all get rounded up to 30.And 21, 22, 23 and 24 all get rounded down to 20

What about 25? It's exactly half way between 20 and 30. It has to be rounded one way or the other. The rule that everyone usually follows is that 25 gets rounded up to 30. The Rules In this way we get the rules about rounding up and down.

1, 2, 3 and 4 get rounded down 5, 6, 7, 8 and 9 get rounded up These rules work for all numbers, whether you are using tens, hundreds or thousands (or anything else). There is more about these rules in the other factsheets.

Rounding tens, hundreds and thousands

Rounding a number is another way of writing a number approximately. We often don't need to write all the figures in a number, as an approximate one will do.For a population of 27 653 the number is large and will change daily. It is better to round up and say 28 000.

Rounding to the nearest ten

To round a number to the nearest 10, you have to decide if the number is nearest to 10, 20 30 etc. To do this you follow a rule.

Is 37 nearer to 30 or to 40?

As the unit figure is 7, you round up to 40.

Rounding to the nearest 10 can help you estimate the cost of your shopping.

Rounding to the nearest hundred

To round a number to the nearest 100, you have to decide if the number is nearest to 100, 200, 300 etc. The rule is the same as for rounding to the nearest 10, but this time look at the tens figure.

Is 236 nearer to 200 or to 300?

As the tens figure is 3, you round down to 200.Rounding to the nearest 100 can help you estimate your yearly spending on rent or mortgage.

Rounding to the nearest thousand

To round a number to the nearest 1 000, you have to decide if the number is nearest to 1 000, 2 000, 3 000 etc. Follow the rules as above now looking at the hundreds figure.

Is 8 572 nearer to 8 000 or to 9 000?

As the hundreds figure is 5, follow the rule and round up to 9 000.

When a figure is halfway between two hundreds, the rule is to round up.Rounding to the nearest 1 000 can help you estimate the number of people who attended a pop concert or football match.

Example

If 43 715 tickets were sold for a football match, that number could be rounded to the nearest ten, hundred or thousand:

rounding 43 715 to the nearest 10 would give 43 720 rounding 43 715 to the nearest 100 would give 43 700 rounding 43 715 to the nearest 1,000 would give 44 000.

Estimating using rounding

We can use rounding numbers to get a rough idea or an estimate. An estimate might be a little more or a little less than the actual amount.By carrying out an estimate we can check that the answers to problems are sensible.If you were buying 9 identical shirts for the school's sports team that cost £7.80 each, to get a rough idea of the total cost you could round up £7.80 to £8.00. You could also round up 9 shirts to 10 shirts. Your calculation would then be10 x £8.00 = £80.00

The actual cost would be9 x £7.80 = £70.20 Notice that the actual cost of £70.20 is a little less than our £80.00 estimate. This is because we rounded up. When using a calculator it is a good idea to estimate the answer first in case you make keying errors.

Example

To estimate the cost of 11 pens at 95p each, you could round down 11 to 10 pens and round up 95p to £1.00 The estimated cost would then be10 x £1.00 = £10.00

Key words for rounding and estimating

Here are some words that you'll come across when rounding and estimating.For example with the sum:197 – 50 = 147Rounding To write a number to a given amount of accuracy. Rounding 197 to the nearest 100 would be 200.Estimate To give a rough answer that may be a little less or a little more than the actual result. To estimate 197 – 50 you may instead work out 200 – 50 to give an estimate of 150.Approximate An answer that is not exactly correct but is close enough to be useful in working out a sum. To approximate 197 – 50 you may instead work out 200 – 50 to give an estimate of 150.To the nearest.. rounding off.. A guide to how accurate your rounding needs to be. Rounding 197 to the nearest 100 would be 200. Rounding off 147 to the nearest 10 would be 150.Actual The correct answer to a sum. The estimate of 197 – 50 is 150. The actual answer is 147.

Writing big numbers

We come across large numbers in our everyday life so it is important to be able to read them. To help with numbers that have more than five figures, which might be difficult to read, we use place value.

Place value is the idea that a figure has a different value when used in different places.

Below is a place value table with the numbers 7 853 and 5 387.Note: Each column can only contain one figure from 0 to 9.

In the number 7 853 (seven thousand eight hundred and fifty three) the 7 has the value 7 thousand. This number is 7 000 + 800 + 50 + 3.In the number 5 387 (five thousand three hundred and eighty seven) the 7 has the value 7 units. This number is 5 000 + 300 + 80 + 7. In these two numbers the 7 stands for different values when it is in different places.

Writing numbers up to a million

Using the place value table can help you to write large numbers.Look at the following numbers:

Numbers in figures Numbers in words

10 Ten

100 Hundred

1 000 Thousand

10 000 Ten thousand

100 000 Hundred thousand

1 000 000 Million

You will notice that the numbers are grouped in three figures. There is a space between each group of three figures (counting from right to left). You will sometimes see a comma used to separate the three figures. (If there is no comma in a large number and you have problem saying it, try putting in the comma.)This grouping can help you to say the number 405 000.The first group of three figures is four hundred and five and the last three figures

show thousands (since there are three zeros in a thousand).The number is four hundred and five thousand.

This example shows how important zero is. In 405 000 the zero between the 4 and 5 keeps the place for the missing tens of thousands. Without the zero, the number is forty five thousand (45 000).

Writing numbers in words in figures

There are times when you may need to write down a large number in figures that someone has told you in words. Newspaper stories often have large numbers written in figures that may be difficult to make sense of unless you can say them in words.Examples 1. Write five thousand, three hundred and six in figures.Put the 5 in the thousands column and the 3 in the hundreds column. The 6 should go in the units column so make sure you fill the tens column with a 0 to show no tens.

Th H T U

5 3 0 6

The number is 5 306. 2. Write twenty six thousand, seven hundred and fifty in figures.For this number we would start with the tens of thousands column. Any number larger than 9 999 would have 5 figures. Start with the 2 in the tens of thousands column and continue by putting the 6 in the thousands column, the 7 in the hundreds column and the 5 in the tens column. The units column must have a 0 to show no units.

TTh Th H T U

2 6 7 5 0

The number is 26 750. 3. Write 58 432 in words.

TTh Th H T U

5 8 4 3 2

Again by writing in the letters TTh, Th, H, T and U above the number can help with writing it in words. Grouping the numbers in threes from right to left will let you know that the number must be fifty eight thousand and something. The last three figures can be read on their own as four hundred and thirty two. The number is fifty eight thousand, four hundred and thirty two.4. Write 1 200 in words.

Th H T U

1 2 0 0

Start by writing the letters Th, H, T and U above the number you have been given. You can then see that the 1 is in the thousands column and the 2 is in the hundreds column. The number is one thousand, two hundred.Note: Another way of writing this number would be twelve hundred. Although this is not incorrect if you think writing numbers such as these as twelve hundred might confuse you, stick to writing them in terms of thousands.

Ordering large numbers

When you have a series of large numbers, which are not in number order, it is sometimes difficult to make sense of them.Here is a table showing the daily profits of a supermarket written in order of days.

If these numbers were put into a place value table, it would be easier to arrange them in order.

Look at each column in turn. The figures for Friday and Saturday will be the largest as these have figures in the tens of thousands column. Looking at the thousands column shows that since there is a 4 in the thousand column for Saturday and a 0 in the thousands column for Friday that Saturday has the largest number. Carry on for each of the other numbers.

Writing figures in words

0 zero 10 ten 20 twenty

1 one 11 eleven 30 thirty

2 two 12 twelve 40 forty

3 three 13 thirteen 50 fifty

4 four 14 fourteen 60 sixty

5 five 15 fifteen 70 seventy

6 six 16 sixteen 80 eighty

7 seven 17 seventeen 90 ninety

8 eight 18 eighteen 100 hundred

9 nine 19 nineteen 1 000 thousand

1 000 000 million

Big numbers glossary

Here are some of the words you may come across to do with big numbers.Place value A figure has a different value when used in different places. For example, in these three numbers, the 4 stands for a different value:45 The number 4 has a value of 40 (4 tens)405 The number 4 has a value of 400 (4 hundreds)54 The number 4 has a value of 4 (4 units)Digit A figure or a number. 45 is a two-digit number whereas 405 is a three-digit number.Billion When we talk about a billion we mean a thousand million or 1 000 000 000. If you see a billion in a news story it is referring to a thousand million.Such big numbers can be difficult to imagine.

Numerical order The order that you would write numbers if you were counting from the lowest up. 405, 406 and 407 are in numerical order.Unit The word unit means one. It is the smallest number and is always on the right-hand side of a whole number:5 This number has 5 units72 This number has 2 units591 This number has 1 unit

Multiples

Multiples of a number can be made by multiplying the number by any whole number. The first four multiples of 2 are 2, 4, 6 and 8. You get them by doing 2 x 1, 2 x 2, 2 x 3 and 2 x 4 The numbers you find in the 2-times table are all multiples of 2.Reminder: when you do multiplication you can write the numbers in any order and get the same answer. 6 x 2 is the same as 2 x 6.Here is how to make multiples of 10. Just multiply 10 by a whole number each time. 1 x 10 = 10, 2 x 10 = 20, 3 x 10 = 30, 4 x 10 = 40, 5 x 10 = 50, 6 x 10 = 60, and so on ...The first six multiples of 10 are 10, 20, 30, 40, 50 and 60.Example 1 Is 12 a multiple of 3?If you multiply 3 by 4 you get 12, so 12 is a multiple of 3.Example 2 20 is a multiple of 5 because 4 x 5 = 20.20 is a multiple of 4 too, because 5 x 4 = 20.Example 3 Is 15 a multiple of 3?3 x 5 = 15. So 15 is a multiple of 3, (and also of 5).Example 4 Is 21 a multiple of 6?21 is not a multiple of 6 because you can't make 21 by multiplying 6 by any whole number.6 x 3 = 18 and 6 x 4 = 24 but there is no whole number between 3 and 4 that could give us an answer of 21.Example 5 Is 30 a multiple of 15?30 = 2 x 15, so 30 is a multiple of 15.You can also see that 2 x 3 x 5 = 30 so 30 is a multiple of 2, 3 and 5.And 30 = 3 x 10 so 30 is a multiple of 10.Also 30 = 5 x 6 so 30 is a multiple of 6 too.

Factors

In arithmetic, a factor is a whole number that divides exactly into another whole number.For example, what are the factors of 12? Try making 12 in different ways.Your answer should look like this: 6 x 2 = 12 12 x 1 = 12 4 x 3 = 12

Remember that you can write your numbers in any order you like for a

multiplication so: 2 x 6 is the same as 6 x 2 1 x 12 is the same as 12 x 1 3 x 4 is the same as 4 x 3.

The full list of factors of 12 is 1, 2, 3, 4, 6, and 12.

Some numbers have many factors, so it is a good idea to work in an organised way or you may miss some. Don't forget to include 1 and the number itself in your list. Here is one way to find the factors of 48. Start with 1 and pair off your numbers.1 x 48, 2 x 24, 3 x 16, 4 x 12 and 6 x 8 all make 48.Write the list in order: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Here is another way: Write your first pair of factors with a reasonable space between them, then move on to the next pair until you have them all. (You don't need to put in the lines.) This way, when you get to the 6,8 pair, you can stop because 7 is not a factor and you already have 8 in your list.

Sequences

A sequence is a set of numbers arranged in order according to a rule. Each number in a sequence is called a term. Multiplication tables give good examples of sequences. For example the 2-times table gives you the sequence 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ... and so onEach term comes from the 2-times table. The rule for this sequence is 'add 2' each time.The first three terms of the four times table are 4, 8, 12. You can see that each term in the sequence increases by four. If you carried on with this sequence you would eventually reach 92 (try it!).

What is the next term after 92? Using the rule of adding four each time gives you the next term, 96 (because 92 + 4 = 96).Example What is the next term in the sequence 35, 32, 29, 26, ...?This time each term is three less than the one before it. Using this rule (take three away each time) gives the fifth term as 23, because 26 - 3 = 23. Example A sequence begins 64, 32, 16. What are the next two terms?The numbers are decreasing, but not by equal amounts. The rule for this sequence is 'Divide by two'. The next term will be 8, because 16 ÷ 2 = 8. The term after that will be 4, because 8 ÷ 2 = 4.

Example What are the next two terms in the sequence 1, 2, 4, 8, ?The rule is 'multiply by 2 each time'. The next two terms are 8 x 2 = 16 and then 16 x 2 = 32.Example Look at this sequence: 3, 5, 8, 12, ...It doesn't follow any of the rules above. But if you look at the differences between each pair of terms, you can see that they are 2, 3 and 4. The next difference will be 5 and so the fifth term is 17, because 12 + 5 = 17.

Number patterns

Some sequences can be shown as number patterns, for example:

The difference between one term and the next is 2. So the next term in the sequence will be 10. (We don't need the pattern to work it out.)Example Here the patterns are made from circles.

The differences between the terms are 2, 3 and 4. The next difference will be 5, so the fifth term is 15. Square Numbers You square a number by multiplying it by itself. For example 5 squared is 5 x 5 = 25 and not 5 x 2 = 10. It is an easy mistake to make!The first four terms of the sequence of square numbers are 1, 4, 9 and 16. They are worked out by squaring the numbers 1, 2, 3 and 4 like this:

These can be written 1², 2², 3² and 4². We say this as 'one squared', 'two squared' and so on. The tenth term will be 'ten squared' which is written 10². That is 10 x 10 = 100

What are digits?

Numbers are made from combinations of the digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9For example 816 is a three digit number. It has 3 digits. You might also call it a three figure number.

Some examples The number 3 538 has four digits 3 5 3 and 8 The number 276 has three digits 2 7 and 6The number 41 has two digits 4 and 1The number 5 has only one digit 5

Multiplying by 10, 100 and 1 000

Multiplying by 10 When you multiply a decimal number by 10 you move all the digits one place to the left. The number becomes 10 times bigger.Example 2.63 x 10 = 26.3You can see that the digits move along to the left. Units move to Tens, and the others follow like this:

Multiplying by 100 When you multiply a decimal number by 100 you move all the digits two places to the left. The number becomes 100 times bigger.Example 2.63 x 100 = 263

Multiplying by 1 000 When you multiply a number by 1 000 you move all the digits three places to the left. The number becomes 1 000 times bigger.Example 2.63 x 1 000 = 2 630

Dividing by 10, 100 and 1 000

Dividing by 10 When you divide a decimal number by 10 you move all the digits one place to the right. The number becomes 10 times smaller.Example 3 502 ÷ 10 = 350.2You can see that the digits move along to the right. Thousands move to Hundreds, and the others follow like this:

Dividing by 100 When you divide a decimal number by 100 you move all the digits two places to the right. The number becomes 100 times smaller.Example 3 502 ÷ 100 = 35.02

Dividing by 1 000 When you divide a decimal number by 1 000 you move all the digits three places to the right. The number becomes 1 000 times smaller.Example 3 502 ÷ 1 000 = 3.502

Shortcuts

When multiplying by 10, 100 and 1 000 there's a pattern that can help you get the right answer very quickly. This method moves the decimal point rather than the digits.

Multiplying number Number of places to move the decimal point

10 1

100 2

1 000 3

10 000 4

The zeros in the multiplying number tell you how many places to move the decimal point.Example Multiply 2.341 by 100100 has two zeros. Make 2.34 bigger by moving the decimal point two places.

It's the same when you divide except you have to remember to move the decimal point the other way. Remember when you divide you are making the number smaller.Example Divide 761.2 by 1010 has one zero. Make 761.2 smaller by moving the decimal point 1 place.

When multiplying (or dividing) by 10, 100, 1 000, etc count the zeros to find out how much bigger (or smaller) your number must be.

Make sure you move the digits (or the decimal point) in the correct direction!

Working with metric and decimal units

Being able to multiply or divide by 10, 100 and 1 000 is useful when you want to convert between units. Here are some rules and examples, starting with pounds and pence .

£ 1 = 100 p

Multiply by 100 to change pounds into pence.Divide by 100 to change pence into pounds.Examples £2 = 2 x 100 = 200 p £15.38 = 15.38 x 100 = 1 538 p 139 p = 139 ÷ 100 = £1.39 225 p = 225 ÷ 100 = £2.25

1 kg = 1 000 g

Multiply by 1 000 to change kilograms into grams.Divide by 1 000 to change grams into kilograms.Examples 3 kg = 3 x 1 000 = 3 000 g 2.5 kg = 2.5 x 1 000 = 2 500 g 4 000 g = 4 000 ÷ 1 000 = 4 kg 1 500 g = 1 500 ÷ 1 000 = 1.500 kg = 1.5 kg

1 litre = 1 000 millilitres

Multiply by 1 000 to change litres into millilitres.Divide by 1 000 to change millilitres into litres.Examples 6 l = 6 x 1 000 = 6 000 ml 3.25 l = 3.25 x 1 000 = 3 250 ml 10 000 ml = 10 000 ÷ 1 000 = 10 l 750 ml = 750 ÷ 1 000 = 0.750 l = 0.75 l

1 km = 1 000 m

Multiply by 1 000 to change kilometres into metres.Divide by 1 000 to change metres into kilometres.Examples 7 km = 7 x 1 000 = 7 000 m 3.3 km = 3.3 x 1 000 = 3 300 m 1 250 m = 1 250 ÷ 1 000 = 1.250 km = 1.25 km 750 m = 750 ÷ 1 000 = 0.750 km = 0.75 km

1 m = 100 cm

Multiply by 100 to change metres into centimetres.Divide by 100 to change centimetres into metres.Examples 3 m = 3 x 100 = 300 cm 2.51 m = 2.51 x 100 = 251 cm 345 cm = 345 ÷ 100 = 3.45 m 902 cm = 902 ÷ 100 = 9.02 m

1 cm = 10 mm

Multiply by 10 to change centimetres into millimetres.Divide by 10 to change millimetres into centimetres.Examples 4 cm = 4 x 10 = 40 mm 31 cm = 31 x 10 = 310 mm 105 mm = 105 ÷ 10 = 10.5 cm 50 mm = 50 ÷ 10 = 5 cm

Place value

All numbers use one or more of these ten digits 1 2 3 4 5 6 7 8 9 0For example 816 is a three digit number. It has 3 digits. You might also call it a three figure number.

Digits can be used on their own to give us small numbers like 2 and 4. They can be used together to make bigger numbers, like 27, 431 and 2 146 Question Is the digit 4 always worth 4?Answer No. For example 4 is worth a different amount in each of these numbers: 4, 40, 400, 4 000 Because we only have ten digits, the same ten have to be used in such a way that we always know whether a 4 stands for four, forty, four hundred or four thousand. Place value helps with this. Understanding place value tells us whether we have been given a bill for four pounds, forty pounds or four hundred pounds: £4, £40, £400Place value is vital. It means putting digits into columns. These columns are always in the same order.

thousands hundreds tens units (ones)

The value of a digit depends on which column it is in - units, hundreds, tens, thousands, etc.Let's look at the number 4 444. There are four 4s, but each 4 means something different. How much is each 4 worth?


4 4 4 4

4 444 is worth four thousand four hundred and forty four.

Place holders

Look at the number 4 040. How much is it worth?


4 0 4 0

This number is four thousand and forty. We must use zeros to keep the digits in the correct columns. If we missed out the zeros from the number above we would have 44 and that is a very different number from 4 040. 44 = four tens and four units


4 4

4 040 is much bigger than 44. 4 040 = four thousands, no hundreds, four tens and no units


4 0 4 0

Both the fours and the zeros are important in this number. Zero is called a place holder. It is not worth anything on its own, but it changes the value of other digits. In this case zeros change the number 44 to the much larger number 4 040. The digit on the right of any number must always go into the units column. If there are no units there will be a zero. For example in the number 20 there is a zero in the units place.


0 0 2 0

Common mistakes

Whenever we work with numbers we must always remember to use place value. If we don't our answers will be wrong. Have a go

at working out this sum 142 + 56

Did you put the digits into columns?

Did you put the digits on the right into the units column?If you don't put the digits into the

correct columns you will get a wrong answer. Forgetting to use place holders is another common error. How would you write the number six thousand, three hundred and nine? It should look lke this

thousands hundreds tens units

6 3 0 9

That's 6 309. If you missed out the place holder (the zero) you would have written 639. That would be six hundred and thirty-nine, which would be a completely different number.Another common error is mistaking big digits for big numbers. For example 1 111 may look smaller than 999 because it is made up of small digits, but put them into columns to see that 1 111 is bigger.

thousands hundreds tens units

1 1 1 1

9 9 9

Inequalities - more than and less than

You will use a lot of different symbols when you are working with numbers. You probably know some of them already. + - x ÷ = These are all mathematical symbols. There are also symbols to show 'less than' and 'more than'.What symbol would you use to could show 1p is less than £1 ? 1p ? £1

The symbol to show 'less than' is <. It looks like an equals sign squashed to a point at one end.The two ends are not equal to show that the numbers are not equal.

So to say that 1p is less than £1 we write 1p < £1 What symbol could show that £1 is more than 1p ? £1 ? 1p We can use the < sign if we turn it around. This is the greater than sign > Then we can use it like this £1 > 1p The widest part of the arrow is always next to the largest amount. The pointed end is on the small side.

Seven digit numbers

How do you write one million pounds in numbers? Like this £1 000 000 The number 1 000 000 has seven digits. These are the seven columns

How would you write one million five hundred thousand?

That is 1 500 000. How would you write one million, three hundred and twenty thousand and fifty four? Put place holder zeros into the empty columns like this

That number is 1 320 054. It would be very different if it was 1 302 054

That is one million, three hundred and two thousand and fifty four, which is smaller than 1 320 054 1 302 054 < 1 320 054

5 ways to add and subtract in your head

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