Pupils will begin year 4 with an understanding of some of the individual concepts covered in this criterion, but they need to leave year 4 with a coherent understanding of multiplicative relationships, and how multiplication and division equations relate to the various multiplicative structures. Pupils need to be able to apply the commutativite property of multiplication in 2 different ways. The first can be summarised as ‘1 interpretation, 2 equations’. Here pupils must understand that 2 different equations can correspond to one context, for example, 2 groups of 3 is equal to 6 can be represented by 2 × 3 = 6 and by 3 × 2 = 6. Spoken language can support this understanding.
“2 groups of 3 is equal to 6.” “3, two times is equal to 6.” “2 groups of 3 is equal to 3, two times.”
“2 groups of 7 is equal to 14.” “7 groups of 2 is equal to 14.” “2 groups of 7 is equal to 7 groups of 2.”
“factor times factor is equal to product.” “The order of the factors does not alter the product.”
“If we swap the values of the divisor and quotient, the dividend remains the same.”
The second way that pupils must understand commutativity, can be summarised as ‘one equation, two interpretations’. Here, pupils must understand that a single equation, such as 2 × 7 = 14, can be interpreted in two ways. Pupils should understand that both interpretations correspond to the same total quantity (product). An array is an effective way to illustrate this.
Pupils should be able to bring together these ideas to understand that either of a pair of multiplication equations can have two different interpretations. Pupils must understand that, because division is the inverse of multiplication, any multiplication equation can be rearranged to give division equations. The value of the product in the multiplication equation becomes the value of the dividend in the corresponding division equations. 2 × 7 = 14 7 × 2 = 14 14 ÷ 2 = 7 14 ÷ 7 = 2 This means that the commutative property of multiplication has a related property for division.
As with multiplication, any division equation can be interpreted in two different ways, and these correspond to quotitive and partitive division. 14 ÷ 7 = 2. Pupils need to be able to fluently move between the different equations in a set, and understand how one known fact (such as 7 × 2 = 14) allows them to solve 4 different calculations each with two possible interpretations. You can find out more about fluency in manipulating multiplication and division equations here in the calculation and fluency section: 4MD–2.