The *Multiplication and Division* spine is divided into 30 segments. For each of these segments we have produced a detailed teacher guide, including text and images. The images are also presented as animated PowerPoint slides, which further enhance teacher knowledge and can be used in the classroom (for best results, please view these in ‘Slideshow’ view; for some slides, supporting notes are provided in the ‘Notes’ section). These materials are not, in any way, lesson plans, but they can be used in conjunction with a high-quality mastery textbook to support planning. Click on the ‘+’ signs below to view the materials.

Year 1 | Year 2 | Year 3 | Year 4 | Year 5 | Year 6

## 2.1 Counting, unitising and coins

Explore the concept of unitising by counting in units of two, five or ten; investigate how objects can be counted efficiently by counting in units other than one; apply unitising in the context of the low-denomination coins (1 p, 2 p, 5 p and 10 p).

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## 2.2 Structures: multiplication representing equal groups

Explore how objects can be arranged in equal groups, and how the number of groups and the size of the groups can be described; represent equally grouped objects with addition and multiplication expressions, connecting multiplication to repeated addition.

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## 2.3 Times tables: groups of 2 and commutativity (part 1)

Build up the two times table by combining children’s experience of counting in units of two and of representing equal groups; explore how, in a multiplication equation, the factors can appear in either order and the product remains the same.

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## 2.4 Times tables: groups of 10 and of 5, and factors of 0 and 1

Build up the ten and five times tables, combining children’s experience of counting in units of five or ten and of representing equal groups; explore patterns in the ten and five times tables, and generalise about the product when one factor is zero or one.

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## 2.5 Commutativity (part 2), doubling and halving

Explore how one multiplication equation can have two different grouping interpretations (e.g., an equation from the two times table can be interpreted in terms of groups of two, or two equal groups); make connections between the two times table, doubling and halving.

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## 2.6 Structures: quotitive and partitive division

Introduce the quotitive and partitive structures of division; skip count using the divisor, or use known multiplication facts, to find the quotient; generalise about the quotient when dividend = 0, dividend = divisor, or divisor = 1.

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**Easy-download**: Year 2 (All of the teacher guides and representations for segments 2.2-2.6 will be available in one easy-to-download zip file.)

## 2.7 Times tables: 2, 4 and 8, and the relationship between them

Build up the four/eight times table; using different structures/interpretations of multiplication and division, solve problems related to these tables; explore connections between the two, four and eight times tables.

*Segment materials in preparation*

## 2.8 Times tables: 3, 6 and 9, and the relationship between them

Build up the three/six/nine times table; using different structures/interpretations of multiplication and division solve problems related to these tables; explore connections between these times tables.

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## 2.9 Times tables: 7 and patterns within/across times tables

Build up the seven times table and solve associated multiplication and division problems; explore times table patterns including generalising about the product in terms of odd/even factors, reviewing divisibility rules, and exploring square numbers.

*Segment materials in preparation*

**Easy-download**: Year 3 (All of the teacher guides and representations for segments 2.7–2.9 will be available in one easy-to-download zip file.)

## 2.10 Connecting multiplication and division and the distributive law

Explore why multiplication is commutative while division is not. Build on understanding of the difference between adjacent multiples to explore the distributive law, and apply it to derive multiplication facts.

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## 2.11 Times tables: 11 and 12

Build up the eleven and twelve times tables using the distributive law, and solve associated multiplication and division problems. Combine known six times table facts with doubling facts and strategies to multiply by twelve.

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## 2.12 Division with remainders

Explore how some quantities can be split into equal groups with a remainder, and express this using mathematical notation; practise interpreting the meaning of the remainder in different contexts.

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## 2.13 Calculation: multiplying and dividing by 10 or 100

Use place-value knowledge to develop strategies for multiplying/dividing by 10 and 100. Generalise about the product or quotient when a factor or the dividend is made 10 or 100 times bigger/smaller.

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## 2.14 Multiplication: partitioning leading to short multiplication

Introduce the short multiplication algorithm, using it to multiply two-/three-digit numbers by single-digit numbers; explore regrouping where necessary.

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## 2.15 Division: partitioning leading to short division

Introduce the short division algorithm, using it to divide two-/three-digit numbers by single-digit numbers; explore exchange where necessary.

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## 2.16 Multiplicative contexts: area and perimeter 1

Use addition and multiplication to solve problems about the perimeter of irregular and regular 2D shapes, and to find the area of rectilinear and composite rectilinear shapes; use division to solve associated inverse problems.

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## 2.17 Structures: multiplication and division as scaling

Build on segment 2.13 to introduce the scaling structure of multiplication and division; use known multiplication and division strategies to solve problems about scaling/comparison problems.

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**Easy-download:** Year 4 (All of the teacher guides and representations for segments 2.10–2.17 will be available in one easy-to-download zip file.)

## 2.18 Equivalence and multiplication/division structures for calculation

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## 2.19 Calculation: ×/÷ decimal fractions by single-digit numbers

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## 2.20 Multiplication with three factors and volume

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## 2.21 Factors, multiples, prime numbers and composite numbers

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## 2.22 Combining multiplication with addition and subtraction

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**Easy-download:** Year 5 (All of the teacher guides and representations for segments 2.18–2.22 will be available in one easy-to-download zip file.)

## 2.23 Multiplication strategies for larger numbers and long multiplication

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## 2.24 Division: dividing by two-digit divisors

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## 2.25 Calculation: developing efficient division strategies

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## 2.26 Mean average and equal shares

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## 2.27 Scale factors, ratio and proportional reasoning

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## 2.28 Combining division with addition and subtraction

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## 2.29 Decimal place value knowledge, multiplication and division

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## 2.30 Multiplicative contexts: area and perimeter 2

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**Easy-download:** Year 6 (All of the teacher guides and representations for segments 2.23–2.30 will be available in one easy-to-download zip file.)

The NCETM and Maths Hubs would like to thank the following for their contribution to the materials:

**Primary mathematics specialists**: Adrian Cannell, Clare Christie, Katie Crozier, Jonathan East, Claire Gerrard, Joanna Caisová, Rebecca Holland, Conor Loughney, Suzanne Mathews, Sally O'Brien, Alexandra Parry, Georgia Ryan, Cat Stone, Debbie Weible, Andrew Whitehead

**Educational consultant**: Dr Alf Coles, University of Bristol

**Editorial and production**: Nicholas Bromley; Tracey Cowell, Passion for Publishing; Rachel Houghton; Jalita Jacobsen; Yvette Sturdy, DiGiV8 Ltd.; Rachel Trolove, Trolove Engineering Solutions Ltd.

**Illustrations**: Steve Evans; Alphablocks Ltd.