# Multiplicative properties

The multiplicative properties of commutativity, distributivity, and associativity form a powerful mathematical idea in multiplication and division. The properties equip students with a conceptual framework for selecting and using appropriate, flexible, efficient solution strategies.

## Multiplicative properties

The multiplicative properties of commutativity, associativity, and distributivity guide the manipulation of multiplicative expressions and selection of strategies used to make calculations and solve problems for multiplication and division. The idea grows in sophistication from modelling the commutative property of multiplication in the early years to generalising the commutative, distributive, associative properties as algebraic laws and applying them to manipulate and solve algebraic equations in secondary. A deep understanding of the multiplicative properties equip students with a conceptual framework for selecting and using appropriate, flexible, efficient solution strategies.

Recognising that multiplication and division are related, yet the same properties and principles do not apply to both is vital. Students need to learn that commutative and associative properties apply to multiplication but not division and be able to reason as to why.

#### Multiplication & Division

The multiplicative properties of commutativity, distributivity and associativity are powerful mathematical ideas in multiplication and division.
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## Commutative property of multiplication

The order in which numbers are multiplied does not affect the product.

Encountering the Commutative property of multiplication when making and describing arrays and rotating them a quarter turn empowers students with the realisation that they can use one multiplication fact to reveal other facts. Students recognise that they can turn multiplication facts around without changing the product. When they know that $4 \times 6 = 24$, they know that $6 \times 4 = 24$ and therefore $4 \times 6 = 6 \times 4$.

## Associative property of multiplication

The multiplication of a set of numbers gives the same product no matter how the numbers are grouped.

Building different rectangular arrays for the same number provides experiences through which students can identify factors and encounter the Associative property. Representing 12 as 1 row of 12, transforming it into 2 rows of 6 and then into 4 rows of 3 reveals the link between rearranging numbers and factors. Manipulating arrays to model related multiples reveals the power of the Associative property. When investigating relationships between multiplication facts for related multiples students use strategies like repeated doubling and halving. They learn that they can multiply by a number by multiplying by each of its factors, e.g. $7 \times 6 = 7 \times 3 \times 2$, or double 21.

By creating, confirming or completing equivalent multiplication number sentences students generalise the levelling principle of multiplication: to maintain equivalence we apply the inverse operation using the same factor to each number. For division, they generalise the constant quotient principle: to maintain the same quotient we apply the same operation using the same factor.

The associative property can be applied to multiplying larger numbers, by breaking numbers into multiplicative parts. Students learn to apply the associative property to multiply multiples of 10, 100, 1000 by using place value to factorise, and to express composite numbers as products of their prime factors and use the associative property to rearrange and simplify calculations.

## Distributive property of multiplication and division

Factors can be partitioned and distributed to make partial products. The partial products are then added together to find the original total of the multiplication.

Partitioning arrays when building multiples and establishing multiplication facts signposts the Distributive property. Students learn that the next multiple is revealed by adding one more row or column to the array. They use this idea as a strategy with the facts they know to find facts they don’t know. For example, if $2 \times 6 = 12$, then $3 \times 6 = 12$ plus one more group of 6. Therefore $3 \times 6 = 2 \times 6 + 1 \times 6$. Students learn that they can multiply each part by the same number and then combine the results. As new multiplication facts are introduced, students recognise that they can subtract one result from the other to reveal facts like the multiples for 9.

The distributive property can be applied to multiply and divide multi-digit numbers by partitioning in standard form, multiplying or dividing each part, and then combining the results. This develops the foundation for introducing and understanding multiplication and division algorithms. Students apply the distributive property as a strategy and record their thinking using diagrams and models such as the Area Model to record partial products. These models are then transformed into algorithms with understanding.