Multiplicative properties and principles

The commutative, associative, and distributive properties describe fundamental relationships within multiplication and division. These properties, together with the constant quotient principle and the inverse relationship between multiplication and division, make clear the multiplicative structure and support efficient calculation.

The commutative property of multiplication tells us that the order in which two numbers are multiplied does not change the product. For example, $5 \times 4$ gives the same result as $4 \times 5$. The commutative property aids calculation and the recall of multiplication facts. Division is not commutative.

The distributive property of multiplication tells us that when multiplying a number, we can partition it into smaller parts, multiply each part separately by the same multiplier, and then recombine the parts. For example, $17 \times 24 = (10 \times 20) + (10 \times 4) + (7 \times 20) + (7 \times 4)$. The distributive property is the basis of the standard multiplication algorithm. It aids mental and written calculation strategies by breaking numbers into smaller parts.

The distributive property can be applied to division. We can break the dividend into smaller parts, divide each part by the divisor and then combine the results. For example, $72 ÷ 6 = (60 ÷ 6) + (12 ÷ 6)$.

The associative property of multiplication tells us that we can rearrange and regroup numbers and their factors when multiplying without changing the product. For example, $5 \times (2 \times 6) = (5 \times 2) \times 6$, which means that $5 \times 12 = 10 \times 6$. Rearranging and regrouping factors aids mental calculation by forming numbers that are easier to multiply. Division is not associative.

The associative property helps us to make sense of the levelling principle of multiplication. The levelling principle tells us that we can generate equivalent multiplication expressions by multiplying one number by a given factor and dividing the other number by the same factor. For example, $5 \times 12$ gives the same result as $10 \times 6$ because one number was doubled and the other number was halved, that is:

$$12 \times 5 = (6 \times 2) \times 5 = 6 \times (2 \times 5) = 6 \times 10$$

Understanding this principle aids calculation, by enabling us to rearrange the numbers in unknown facts into known facts.

The constant quotient principle tells us that we can generate equivalent division expressions by multiplying or dividing both numbers by the same factor. For example, $24 ÷ 6$ gives the same result as $48 ÷ 12$ (both numbers were multiplied by a factor of 2) and $24 ÷ 6$ gives the same result as $8 ÷ 2$ (both numbers were divided by a factor of 3). Understanding this principle aids calculation by simplifying the numbers we are working with.

Multiplication and division are inverse operations. This means that each multiplication fact can be undone by working in reverse using division. For example, $3 \times 8 = 24$ so $24 ÷ 8 = 3$. The array offers a powerful model for recognising the inverse relationship between multiplication and division and multiplication and division fact families.

Year 2

In Year 2, students recognise the inverse relationship between doubling and halving and how multiplication facts for 2 can be used to determine division facts for 2. They encounter the commutative and distributive properties of multiplication as they construct and deconstruct arrays. For example, when representing an array for 15, students can describe the same array as 5 rows of 3 and as 3 columns of 5: 5 threes is equivalent to 3 fives. This is an example of the commutative property.

Students encounter the distributive property as they develop strategies for determining the total represented by an array. For example, when determining the total for 3 rows of 5, students recognise that they can partition the array to show 3 fives as “2 fives + 1 more five”. Rearranging arrays introduces the levelling principle and how it can be used to reveal equivalent multiplication facts. For example, by starting with an array for 12 represented as 4 rows of 3 and rearranging it to form 2 rows of 6, students can see that when they halve the number of rows the number in each row doubles.

Year 3

In Year 3, students continue to develop awareness of the inverse relationship between multiplication and division and how multiplication facts for 3, 4, 5 and 10 can be used to determine division facts for dividing by 3, 4, 5 and 10. Students encounter the commutative, distributive and associative properties of multiplication as they develop and record calculation strategies. For example, to represent 6 x 4 students might use informal recording and drawings to show how:

• they could use the commutative property to turn the fact around and represent $6 \times 4$ as $4 \times 6$.
• they could use the associative property to represent $6 \times 4$ as $(6 \times 2) \times 2$.
• they could use the distributive property to represent $6 \times 4$ as $(5 \times 4) + 4$.

Year 4

In Year 4, students develop and record strategies for using known facts to calculate unknown facts. They use doubling and halving, commutativity, or adding or subtracting one more group to/from a known fact. Multiplication facts for 2, 3, 5, and 10 provide a foundation for establishing facts for 4, 6, 7, 8, and 9. For example, to calculate $9 \times 4$, the expression can be transformed using the distributive property:

$$9 \times 4 = (10 \times 4) − 4$$

The commutative, associative, and distributive properties of multiplication are applied to support flexible thinking. The levelling principle is also used to simplify calculations. For instance, the cost of five scooters at $96 each can be calculated in several ways:

• halving 96 and multiplying the result by 10 (levelling principle & associative property):

$$ 5 \times 96 = 5 \times (2 \times 48) = (5 \times 2) \times 48 = 10 \times 48$$

• rounding 96 to 100, calculating $5 \times 100$ and subtracting $5 \times 4$ (distributive property):

$$ 5 \times 96 = (5 \times 100) − (5 \times 4)$$

• partitioning 96 as $90 + 6$, multiplying the 90 and the 6 by 5, and combining the results (distributive property):

$$ 5 \times 96 = (5 \times 90) + (5 \times 6)$$

Year 5

In Year 5, students recognise and explain the connection between multiplication and division as inverse operations and use this to develop families of number facts. They apply the distributive property to reason that all multiples can be formed by partitioning, regrouping and combining. For example, all multiples of 7 can be formed by combining a multiple of 2 with the corresponding multiple of 5: $3 \times 7 = 3 \times 2 + 3 \times 5$; $4 \times 7 = 4 \times 2 + 4 \times 5$; $5 \times 7 = 5 \times 2 + 5 \times 5$…

Students apply the associative and distributive properties to develop, select and use efficient, effective calculation strategies involving larger numbers. For example, students solve $17 \times 24$ by partitioning each number into place value parts and adding all partial products formed.

Partitioning using place value lays the foundation for using formal algorithms.

Students apply the distributive property to solve division problems. They learn that the dividend can be partitioned into smaller parts and then divided by the divisor. For example, $72 ÷ 6 = (60 ÷ 6) + (12 ÷ 6)$.

Students write and recognise equivalent division expressions using the fact that equivalent division expressions result if both numbers in a division expression are divided or multiplied by the same factor (constant quotient principle).

Year 6

In Year 6, students select and apply the commutative, associative and distributive properties of multiplication flexibly to solve multiplication problems efficiently. They use, compare and evaluate strategies for multiplying larger numbers, and apply properties and principles to multiply decimals. Students learn to use brackets and the order of operations to write and solve number sentences, recognising that there is no hierarchy between multiplication and division as they move from left to right. They apply knowledge of inverse operations and properties and principles of multiplication and division to find unknown numbers in equivalent number sentences.

Year 7 to Year 8

In Years 7 and 8, students use the commutative and associative properties appropriately to develop efficient strategies for solving multiplication and division problems involving fractions and decimals. They generalise the commutative, associative and distributive properties of multiplication by expressing the properties algebraically, and apply them to manipulate and solve algebraic equations.