Number
Multiplication and
division
Teaching for mastery in
primary maths
Contents
Introduction3
01. Introduction to multiplication and division
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02. Multiplication and division4
03. Exploring multiplication and division
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04. Short multiplication5
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05. Multiplication by two-digit numbers and short division
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06. Long multiplication and long division
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Number: Multiplication and division
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Number: Multiplication and division
Introduction
The operations of multiplication and division have many different underlying
structures and can be applied in a wealth of contexts.
Fluency with multiplication facts (times tables) is vital for success in mathematics,
particularly in algebra. As such, these facts must be understood, memorised and
represented in a variety of contexts.
Throughout this learning journey, pupils will build their understanding of multiplication
and division using varied concrete and pictorial representations. They should use these
representations to reason and explain answers to problems.
Mental methods should be encouraged at all opportunities so that pupils do not become
overly reliant on formal methods, which are not always the most efficient way to solve a
problem.
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Number: Multiplication and division
Chapter 1:
Introduction to multiplication and division
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.
Chapter 2:
Multiplication and division
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.
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Chapter 3:
Exploring multiplication and division
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.
Chapter 4:
Short multiplication
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.
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Chapter 5:
Multiplication by two-digit numbers and short division
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.
Chapter 6:
Long multiplication and long division
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.
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