Teaching for mastery in primary maths

In this chapter, pupils focus on equal grouping and sharing, and represent situations involving multiplication and division.

At this early stage, pupils do not write formal statements using multiplication and division signs. Instead, they develop familiarity with different contexts for each concept. They use a variety of concrete objects and pictorial representations to do this, with a particular focus on arrays.

In this chapter, pupils demonstrate that multiplication of two numbers can be done in any order (commutative), but that division of one number by another cannot.

Pupils develop fluency with multiplication and division facts for the 2, 5 and 10 times tables. They represent and make connections between these tables using concrete objects and pictures. They connect the 10 times table to their understanding of place value (eg, that 3 tens are equal to 30).

Pupils read, write and interpret mathematical statements involving multiplication (×), division (÷) and the equals sign (=). They secure their understanding of the concepts of multiplication and division using manipulatives and pictorial representations, and further develop this through problem solving.

Problems either relate to the grouping and sharing of discrete and continuous quantities, or show that multiplication is the same as repeated addition or that the multiplication of two numbers can be done in any order (commutative) but that division of one number by another cannot. They begin to relate these problems to fractions and measures (eg, 40 ÷ 2 = 20, 20 is a half of 40). They use commutativity and inverse calculations to develop multiplicative reasoning (eg, 4 × 5 = 20 and 20 ÷ 5 = 4).

It is important that pupils use objects and representations as opportunities to deepen understanding of multiplication and division, rather than as a crutch to solve problems. They should be encouraged to use mathematical vocabulary to describe their actions.

In this chapter, pupils solve multiplication and division problems in different contexts, including missing number problems.

Problems include measuring and scaling contexts, (eg, 4 times as high, 8 times as long etc.) and correspondence problems, in which m objects are connected to n objects (eg, if there are 3 hats and 4 coats, how many different outfits are possible?).

Through problem solving, pupils develop efficient mental methods and further understand the concepts of commutativity and associativity (eg, 4 × 12 × 5 = 4 × 5 × 12 = 20 × 12 = 240), as well as multiplication and division facts (eg, using 3 × 2 = 6, 6 ÷ 3 = 2 and 2 = 6 ÷ 3) to derive related facts (eg, 30 × 2 = 60, 60 ÷ 3 = 20 and 20 = 60 ÷ 3).

Pupils continue to read, write and interpret mathematical statements involving multiplication (×), division (÷) and the equals sign (=), starting with calculations involving one and two-digit numbers for the multiplication tables they know, and progressing to the formal written methods of short multiplication and division. The development of mental methods should be encouraged at all opportunities.

In this chapter, pupils solve two-step problems in different contexts. They choose the appropriate operation and work with numbers of increasing size.

When investigating and solving problems, including correspondence questions, pupils become fluent in the formal written methods of short multiplication. They recall multiplication and division facts up to the 12 times table, which complements their conceptual understanding of multiplication and division more generally.  

Building on their understanding of place value, pupils use known and derived facts to multiply and divide mentally, including multiplying by 0 and 1 and dividing by 1. They extend this to three-digit numbers (eg, 600 ÷ 3 = 200 can be derived from 2 x 3 = 6). Later, pupils multiply two and three-digit numbers by a one-digit number using the formal written layout.

In this chapter, pupils multiply numbers with up to four digits by a one or two-digit number using formal written methods, including long multiplication for two-digit numbers. They are also introduced to short division.

Pupils use multiplication facts (times tables) to multiply whole numbers by 10, 100 and 1000. While this is primarily useful in long multiplication, these mental skills must also be practised and applied beyond this. Pupils should have numerous opportunities draw on known facts to multiply and divide numbers mentally, as well as to multiply and divide whole numbers and those involving decimals by 10, 100 and 1,000.

They begin to use formal written methods for short division and use representations, such as place value counters, to divide numbers with up to four digits by a one-digit number, and interpret remainders appropriately for the context.

Pupils solve multiplication and division problems, including those involving scaling by simple fractions and rates. For a given multiplication or division calculation, they decide on the most appropriate method, be it mentally or through the use of informal jottings or formal written methods.

In this chapter, pupils use formal written methods to multiply and divide numbers with up to four digits by a two-digit whole number.

Pupils use the formal written method of long division and interpret remainders as appropriate to the context. They may be whole numbers, fractions or rounded.

Long multiplication and long division should be used only when mental methods or informal jottings are the least efficient options.