Numbers are composed of smaller numbers. For example, 5 can be thought of as ‘2 and 3’ or ‘4 and 1’. Part-part-whole understanding—that is, recognising that a whole is made of smaller parts—is foundational to the manipulation of numbers.
Students need to appreciate that place value is based on a part-part-whole relationship. Any number can be represented as the sum of the place values of its digits.
The powerful mathematical idea that a number is the sum of its parts is applied when we use expanded notation to record a number. For example, the value of the number $48096$ is the sum of the place values of its digits: $48 096 = 4 \times 10000 + 8 \times 1000 + 9 \times 10 + 6 \times 1$.
Recognising the place value parts of a number and decomposing a number into these parts is an essential skill for mental and formal calculation. Consider $24 + 15$ and $18 \times 5$ as two examples. Decomposing each number into their place value parts facilitates easy computation.
$$24 + 15 = (20 + 4) + (10 + 5) = (20 + 10) + (4 + 5) = 30 + 9 = 39$$
$$18 \times 5 = (10 + 8) \times 5 = (10 \times 5) + (8 \times 5) = 50 + 40 = 90$$
Foundation
In Foundation students link numeral names to symbols and quantities and recognise the cardinality principle: the last number we say when counting is the number for the collection. The ability to conserve number (to know that a number remains the same even when its arrangement or appearance changes) is an essential foundation for working with place value and understanding that a number is the sum of its parts.
Year 1
In Year 1 students encounter the powerful idea that a number is the sum of its parts as they describe collections grouped as ‘tens and ones’.
The word ‘and’, when talking about a number like 31 represented as 3 tens and 1 one, makes the structure of 2-digit numbers salient. Students learn that the total represented by a 2-digit number is the sum of the place values of its digits. 31 is made of 3 tens and 1 one. It is $30 + 1$.
Students start to deepen their understanding of the idea that a number is the sum of its parts as they identify and record the equivalence between different representations of the same number when representing numbers flexibly. For example, $45 = 40 + 5$ and when we exchange tens for ones we can see that $40 + 5 = 30 + 15$.
Year 2
In Year 2 students extend the idea that a number is the sum of its parts as they work with numbers to 1 000. Students use expanded notation to rename and regroup numbers, and learn that these different representations are equivalent in value. For example, students recognise that 245 can be represented as 2 hundreds + 4 tens + 5 ones, and that it can also be represented as 24 tens and 5 ones or 23 tens and 15 ones.
Year 3
In Year 3 students continue to apply the powerful idea that a number is the sum of its parts, as they work with numbers with numbers with up to 6-digits. When viewing a number like 475 512, students recognise that the total value of the number is the sum of the place values of its digits, and can express 475 512 as $(4 \times 100 000) + (7 \times 10 000) + (5 \times 1000) + (5 \times 100) + (1 \times 10) + (2 \times 1)$.
Year 4
In Year 4 students apply the powerful idea that a number is the sum of its parts to understand the structure of decimals and partition them in standard and non-standard forms. Using the idea that the value represented by any numeral is the sum of the place values of its digits, students record decimals using expanded notation. For example, students recognise that $2.54 = 2 + 5/10 + 4/100$.
Year 5
In Year 5 students partition, represent, order and compare decimals with up to 3 decimal places. They record numbers with up to 3 decimal places using expanded notation. The powerful idea that a number is the sum of its parts provides the foundation for recognising equivalence between decimals and fractions. For example, students learn to recognise representations such as $5.048$ and $5 48/1000$ as synonymous.
Year 6
In Year 6 students apply the idea that a number is the sum of its parts to partition, interpret and position negative integers.
Year 7 and 8
In Year 7 students apply their understanding that a number is the sum of its parts to represent natural numbers in expanded notation, using place value and powers of 10. They explain connections between place value and expanded notation, for example $3750 = 3 \times 10^3 + 7 \times 10^2 + 5 \times 10^1$.
Year 9 and 10
In Year 9 students apply the properties of place value as they represent decimals in exponential form. For example, they represent $0.475 = 4/10 + 7/100 + 5/1000$ as $0.475 = 4 \times 10^{-1} + 7 \times 10^{-2} + 5 \times 10^{-3}$, or represent decimals such as $0.00023$ as $23 \times 10^{-5}$.