## Why is this skill important and what is it?

This skill is part of what is probably the single most important thing that we need children to understand in the whole of their mathematics education. It is essential that they understand how numbers work!
What do we mean by this?

Well, the beauty of our Hindu-Arabic number system is that, with just ten basic symbols (0-9), we can write, and use, any number, no matter how large or small it is! The important points to know about this system are what we call PLACE VALUE and the use of ZERO.

• ‘Place value’ means that a number’s value (how much it is worth) depends upon its place. So in 383, the first ‘3’ has a different value from the second ‘3’ because of its place in the number.
• ‘Zero’ is very important because it can hold the place for a value that has nothing in it. So in 303, the zero tells us there are no tens; 343 would have told us there were 4 tens, i.e. 40. We say that zero acts as a ‘place holder’ in that it enables us to distinguish between 305 and 35. Three hundred and five needs ‘0’ to tell us there are no tens in this number, although there are ones and hundreds.

Children need to understand exactly how this system of numbers works. If they are not secure in their understanding here, they are simply unable to progress in terms of calculation skills. Many of the difficulties that children have later in learning mathematics are found, upon investigation, to come back to a misunderstanding of the number system. It is for this reason that teachers are really concerned that children develop a robust understanding of place value between 7 and 9 years of age, and that they are not rushed on to do more complex calculations without ensuring that this basic foundation is in place.

## So how is this skill taught?

Children need both what they see (visual images) and what they hear (aural patterns) to help them with understanding the way numbers work. ‘Place value cards’, such as those below, help:

When each of these three cards is overlaid on another, the three-digit number appears:

If one digit (or ‘place value card’) is removed, it is clear what is being taken away, e.g. if we remove the ‘3’ from 536, we are taking away ‘30’ as we can see from the card. And if we now add a thousands card, such as the 7000 card below, we can add a digit before the ‘5’ in 500.

If we now place the four cards on top of each other, we then get the whole number, 7536.  But each part can clearly be identified as its correct value: 7000  as 7 thousands, 500 as 5 hundreds, 30 as 3 tens and 6 as 6 ones.

As well as using cards like this to lay out numbers, we can use this same idea to split numbers into their constituent parts. Thus, 7536 can be split into 7000 + 500 + 30 + 6.  This process is known as partitioning.

It is very important that children can do ‘place value additions’ like these: 20 + 6 = 26, 300 + 70 = 370, 200 + 4 = 204, 100 + 50 + 9 = 159, and so on. These do not require that children do any adding or counting on. They should simply ‘know’ the answer from understanding how the numbers work.  For this reason, we call 20 + 6 a ‘no-work-sum’ as we should not have to do any work to do it!

Summarising then, we teach children to partition numbers into thousands, hundreds, tens and ones. 23,631 can be partitioned into 23,000 + 600 + 30 + 1.  This is especially important when there is a zero acting as a place holder, e.g. in 15,089 where there are no hundreds.

• 23,631 = 23,000 + 600 + 30 + 1
• 15,089 = 15,000 + 80 + 9 (there are no hundreds)

Similarly, we can generate four-digit and five-digit numbers using place value additions.

• 4000 + 300 + 20 + 7 = 4,327
• 18,000 + 200 + 9 = 18,209

A very useful activity which helps reinforce these skills involves asking children to ‘zap’ a particular digit using a calculator.

• Enter a four-digit number into the calculator, e.g. 6284
• Decide which digit to ‘zap’ (this means we change that digit to zero!) e.g. we decide to zap the ‘2’ in 6284.
• Enter the appropriate subtraction to zap that digit, e.g.
6284 – 200
• Press = If you are correct, that digit should now be zero, e.g. 6084.
• Repeat this.

A combination of all these activities should help to reinforce children’s understanding of this important skill.

Practise Together: These activities are intended to be shared. Read the Explanation of the skill being practised and then play the game or share the task. Watch out for the points highlighted in the Explanation and if necessary, help your child, following the advice in ‘How this skill is taught’ section. Shared activities are not only more fun – they enable you to actively support your child’s learning.

Explanation & Worksheets: Having practised a skill together using the shared activities, children can then rehearse the skill using the ‘Child alone’ sheets. These are presented in order of difficulty 1-5 and should only be given to the child AFTER the Practise Together activities. In this way you can be sure that the child has acquired this skill first. We cannot rehearse what something have not yet learned!

Test: Take a test, questions from this area

Counting in sequence: Count any sequence of numbers from 1 to 10,000 forward or back with confidence

Place value: Understand that 4392 is made up of 4000 + 300 + 90 + 2 and that 4092 has no hundreds

Money: Begin to understand that £6.54 is six pounds and 54 pence and that £6.04 is six pounds and 4p while £6.40 is six pounds and 40p

Counting in tens & hundreds: Count in tens or hundreds forward and back from any number, e.g. 284, 294, 304, 314, etc. understanding how to cross a multiple of 10, 100 or 1000

Count multiples: Count in (add or subtract) multiples of 10, 100 or 1000 (800+300)

Writing fractions: Understand how fractions are written, e.g. ½ and ¾ and begin to realise that ½ is the same as 2/4 or 3/6 etc.

Number Concepts: Count in different ways, understand how numbers work, become fluent in the ways of numbers

Adding and Subtracting: Mentally add or subtract numbers with confidence and develop written ways of adding and subtracting larger numbers or more of them!

Multiplying and Dividing: Know the times tables and use these to perform mental multiplication and divisions; develop written methods for multiplication and division.

7-9: Lower Juniors

9-11: Upper Juniors