Our number system uses a Base 10 place value system to represent quantities. Using a place value system based on 10 came about because we have 10 fingers; we record how many we have in a collection using the 10 digits on our hands to help us to quantify.

Every number, from the extremely large to the infinitesimally small, can be represented using 10 unique symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The value of a digit is dependent on where it is situated in a number, and the value of that place.

The content on the following tabs unpacks three powerful mathematical ideas associated with place value:

- 10 of these equals 1 of those
- Place gives value
- Renaming

#### In Conversation with Dr Ange Rogers

*This transcript is not currently available.*

#### 10 of these are one of those

The power of our number system lies in its place value structure. Two key mathematical ideas organise the structure of our base 10 place value system.

The first is that “ten of these is equal to 1 of those”. In this instance 10 ones are equal to a unit of 1 ten. Children build a sense of 10 as a unit through bundling, stacking, and grouping, and through counting large collections using groups of 10 which visually models the total in the collection. Through repeated experiences children build the idea that 10 ones are equal to 1 ten and that 10 tens are equal to 1 hundred.

The inverse is also true, that 1 of these is equal to 10 of those. One is the same as 10 tenths, one tenth is the same as 10 hundredths, one hundredth is the same as 10 thousandths.

The second key mathematical idea in place value is that 1000 of these is equal to 1 of those: 1000 ones are equal to 1 thousand, 1000 hundreds are equal to 1 million, 1000 millions are equal to 1 billion, and 1000 billions is equal to a trillion.

This pattern helps us to name really large numbers. Let’s take a closer look – 152 million, 673 thousand, 428.

The power of our number system lies in its place value structure. The mathematical ideas of 10 of these is equal to 1 of those and 100 of these is equal to 1 of those organise the structure of our base 10 number system.

### Base 10

Our number system uses a Base 10 place value system. We organise collections into groups of ten, which we call a ‘unit of ten’. On the place value chart, we record the number of units of ten in the ‘tens column’. For example, if we are counting a collection of 56, we group this into units of ten and record the number of tens in the ‘tens’ column, which here would be 5. The remaining ones are recorded in the ‘ones’ column as 6. The number that represents the whole amount is written as 56 which, when using Base 10, translates as 5 tens and 6 ones, or 50 and 6.

To represent a collection of 123 items using the Base 10 system, we once again group into units of ten, which would be 12 tens and 3 ones. However, the conventions of the Base 10 system do not allow us to record more than 9 units in any one column. Therefore, we must group the ‘groups of ten’ to make a unit which is ten times bigger. In this example, we group the 10 tens to form a unit of 1 hundred and record this quantity in the ‘hundreds’ column. We record the remaining 2 tens in the ‘tens’ column and the 3 ones in the ‘ones’ column. The number that represents the whole amount is written as the number 123, which when using Base 10, translates as 1 hundred, 2 tens and 3 ones, or 100, 20 and 3.

### Power of ten

Our number system is based on a pattern which is known as the Base 10 property of place value. We can see this property in a number, because as we look at the whole number from right to left, the value of each column increases by a power of ten. Therefore, the value of a digit in the ‘hundreds’ column is 10 times larger than the same digit in the ‘tens’ column. This is because a digit in the ‘hundreds’ column is actually representing a count of ‘10 groups of ten’.

Understanding the Base 10 property provides us with a simple way to multiply and divide by powers of ten. For example, if we multiply 45 by 10, the 4 tens become 10 times bigger and become 4 hundreds, and the 5 ones become 10 times bigger and become 5 tens. Placing the digit 0 in the ‘ones’ column shows that there are no extra ones. It looks like we simply move the digit 4 to the ‘hundreds’ column (where it automatically becomes ten times larger) and the digit 5 to the ‘tens’ column (where it automatically becomes ten times larger). Indeed, this simple act is more efficient than completing an algorithm and is an application of the Base 10 property, when students understand that each column to the left is bigger to the power of 10.

Our Base 10 place value system is simple but elegant. Beneath this simplicity lies a complex multiplicative structure that is difficult, but critical, for students to understand.

## Place gives value

A powerful idea in place value is that *place gives value. *This means that the value of a digit is determined by its place in a number. This is an abstract idea for students to comprehend, and highlights the complexities of our number system for novice learners.

### Meaning of ‘place’

‘Place’ refers to the* location of an object.*

Using the context of our number system, the object we are referring to is a *digit*. We may say, ‘the digit 3 is in the ‘tens' column’, which means the digit 3 is sitting, or located, in the ‘tens’ column. Therefore, the *place* of a digit in a number determines its value. This demonstrates the *positional property* of our number system.

### Meaning of ‘value’

‘Value’ refers to the *worth of an object.*

In our number system, the same digit can have a different ‘value’ according to its position in a number. It is the *multiplicative property *that allows us to calculate the value of each digit.

For example, in the number 333, the value of the digit 3 in the ‘hundreds’ column is ‘3 hundreds’, while the value of the digit 3 in the ‘tens’ column is ‘3 tens’, and the value of the digit 3 in the ‘ones’ column is ‘3 ones’. To calculate the value of a digit in our number system, we multiply the face value of the digit by its place value. The sum of each digit's value is the magnitude of the whole number.

## The importance of renaming

When a number is rearranged into a different form without changing its value, this is called ‘renaming’. Numbers can be renamed in standard and non-standard ways. For example, 385 can be renamed in a standard way as 38 tens and 5 ones. It can be renamed in a non-standard way as 37 tens and 15 ones. The value of the number remains the same but its form changes. Renaming can be particularly difficult to master as students need to be able to use multiplicative thinking and to have a deep understanding of our place value system, where “10 of these is one of those”.

### Smaller units are nested within larger units

Renaming numbers requires students to understand that there are smaller units nested within larger units. For example, to rename 320 in terms of tens, a student needs to not only understand that there are 2 tens in 20, but that there are also 30 tens ‘inside’ the 3 hundreds. When asked how many tens are in 320 *altogether*, students may focus their attention on the ‘tens’ column, as they believe this is the only place the tens ‘live’. Students with such a narrow focus struggle to notice the multiplicative relationship that exists *between* columns, which makes it impossible to rename numbers.

It is very important for teachers to support the teaching of renaming with visual models, so that students can develop their understanding of what nested numbers look like. Using Russian Babushka dolls is one model which shows students how smaller objects can be hidden inside larger objects. This can help students to make similar connections within place value, where smaller units, like tens, ‘hide inside’ larger units of hundreds or thousands.

We want students to understand that when we rename a number its overall value does not change. It has only been written in a different form. Whether a number is written as *32 tens* or *29 tens and 30 ones*, its value of 320 remains the same.

**Contexts for renaming**

It is important that students understand that there are contexts where renaming is useful. Using a formal algorithm to calculate is a genuine context which uses renaming purposefully. For example, rather than talking about ‘borrowing’ or ‘carrying’, referring to ‘renaming’ a unit, is a more accurate description of what is happening when modelling how to calculate using the algorithm. Consistently referring to ‘renaming’ in this way connects the algorithm with the renaming work where students have used concrete materials to understand place value ideas. It is important students appreciate that when we rename in an algorithm, we are* not* changing a number’s value; we are writing it in a form that makes our computation less cumbersome.

Renaming has important links with converting units of measurement. For example, if we are going to convert 230 centimetres into metres, we are considering a unit which is 10 times bigger. Our focus is not on the unit of a centimetre, but on the unit of a metre. 1 metre has the equivalent value of 100cm. 230cm has the same value as 2m and 30 cm, or 2.3 m. This is renaming.

The more we can help students to recognise these applications of renaming, the deeper their understandings will become.