## Place value

Understanding place value involves building connections between key ideas of place value, such as quantifying sets of objects by grouping by 10 and treating the groups as units… and using the structure of written notation to capture this information about groupings.

Hiebert & Wearne 1992, p. 99

The power of our number system lies in its base-10 (decimal) place value structure. Base-10 is just one base that can be used to structure number; computers use base-2 (binary), while base-16 (hexadecimal) is used in web colour codes. Our base-10 place value system uses just 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) and enables us to write numbers that are infinitely big or infinitesimally small.

All place value systems have four properties in common (Rogers, 2014):

- the Positional property
- the Base property
- the Additive property
- the Multiplicative property

The patterns we use in our place value system are multiplicative and based on powers of ten. There is a multiplicative relationship between adjacent places, where the value of a digit becomes 10 times the value with each place it is moved to the left, and one tenth of the value with each place it is moved to the right.

#### In conversation with Dr Ange Rogers

#### References

Hiebert, J., & Wearne, D. (1992). Links between teaching and learning place value with understanding in first grade. *Journal for Research in Mathematics Education, 23*, 98-122.

Rogers, A. (2014). Investigating whole number place value in Years 3-6: creating an evidence-based developmental progression (Doctoral dissertation, RMIT University).

## 10 of these is 1 of those

#### 10 of these is 1 of those

10 is a very important number in our place value number system. This is why it is called a * dec*imal place value system. Our system’s multiplicative patterns are based on powers of ten. When we group

*10 of these to make 1 of those*, we are transforming 10 individual ones into a larger unit that is ten times the value.

When we have 10 ones, we can group these to become a unit of 1 ten. This powerful idea allows us to treat ‘ten’ as a *composite unit*, or countable unit of ten, which is more efficient than counting in ones.

Initially, students regard ten as a count of 10 ones, or a *numerical composite unit*, which can be seen when they skip count by multiples of 10 e.g. 10, 20, 30 etc.

When students comprehend 10 as a unit of 1 ten, they see it as an *abstract composite unit*, where they count the number of units e.g. 1 ten, 2 tens 3 tens (Cobb, 1995).

When the number of tens reaches 10, this powerful idea continues to grow, as *10 of these is 1 of those* is referring to 10 tens as 1 hundred (10 x 10 = 100).

In the same way that students understand ten as an abstract composite unit, they develop their understanding of a hundred as a composite unit of a composite unit, in that 10 tens is 1 hundred.

This pattern continues, when the number of hundreds reaches 10 hundreds, where ten of ten of ten is 1 thousand (100 x 10 = 1000, or (10 x 10) x 10 = 1000).

#### Reference

Cobb, P. (1995). Cultural tools and mathematical learning. A case study. *Journal for Research in Mathematics Education, 26*(4), 362-385.

## 1000 of these is 1 of those

The way we say large numbers is based on powers of a thousand, with the pattern of the initial ones, tens and hundreds being repeated.

Department of Education, WA, 2013 p. 41

#### Place Value: 1000 of these is 1 of those

The relationship between the hundred, tens and ones places reflects the understanding that* 10 of these is 1 of those*. However, we need a more efficient way of using place value in order to read, write and say larger whole numbers, than referring to units of ten of ten of ten. The powerful idea that *1000 of these is 1 of those* helps us to structure the way we read, write and say very large numbers.

When we use whole numbers that are greater than 3-digits, an important cyclical pattern in our place value system makes it easier to do this. This second place-value system reuses sets of hundreds, tens and ones, so that large numbers can be read in syllables of hundreds, tens and ones. Each time the hundreds, tens and ones pattern repeats, it is 1000 times greater in value.

This means that if a student is able to read a 3-digit number then they can use that pattern to read any number. For example, students may find the number 23 532 604 very hard to say when it is written as numbers. However, by separating the number at each second place-value boundary, it becomes easier to read and say.

### Reference

Department of Education, Western Australia. First Steps in Mathematics: Book 1. https://myresources.education.wa.edu.au/programs/first-steps-mathematics/number

## Position give place value

Place value is the value a digit has because of its position in a numeral.

Siemon et al 2021, p. 278

The position of a digit in a number gives it its unit value, while the digit itself represents the number of units in that position. This means that the digit 5 in any position in any number represents five units, but the value of the unit alters; it may be worth ones, tens, hundreds, thousands, etc., depending on its position in the number.

Consider the two numbers 538 and 657 as examples. The 5 is worth 5 units in both numbers. However, the value of those units differs in each number:

- 538 à the 5 is worth 5 units and each unit is 1 hundred, so its value is 5 hundreds
- 657 à the 5 is worth 5 units and each unit is 1 ten, so its value is 5 tens

The value of place value units increases by a power of 10 with each place moved to the left. It decreases by a power of 10 with each place moved to the right. Therefore, a digit’s value increases or decreases by a power of 10 when it moves to a place which is either side of its current position.

This power of 10 pattern continues beyond whole numbers, when a digit is moved one place to the right of the decimal point. In a number, each digit becomes one tenth of its value with each place move it makes to the right.

- When ten is divided by 10, its value becomes one because it is one-tenth of its original value of ten. The position to the right of the tens place is a tenth of the value of ten; this is the ones place.
- When one is divided by ten, its value becomes a tenth of one. The position which is to the right of the ones place is a tenth of the value of one; this is the tenths place.
- When a tenth is divided by ten, its value becomes a tenth of a tenth. The position to the right of the tenths place is a tenth of the value of a tenth; this is the hundredths place.

Each time the digit moves one place to its right, it continues to become a tenth of the value of its previous position.

### Reference

Siemon, D., Warren, E., Beswick, K., Faragher, R., Miller, J., Horne, M., Jazby, D., Breed, M., Clark, J. & Braddy, K. (2021). *Teaching mathematics: Foundations to middle years*. Oxford University Press.