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.

The following PDF offers an overview of the complete Multiplication and Division spine, including a synopsis of each segment.

**Easy-download:** Spine 2 overview

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 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.

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## 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 the three, six and nine 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.

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**Easy-download**: Year 3 (All of the teacher guides and representations for segments

2.7–2.9 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: using measures and comparison to understand 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 in one easy-to-download zip file.)

## 2.18 Using equivalence to calculate

Develop efficiency in calculation by using equivalence, through adjusting the factors (in multiplication) and the dividend and divisor (in division).

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## 2.19 Calculation: ×/÷ decimal fractions by whole numbers

Develop strategies for multiplying and dividing decimal fractions by whole numbers, including combining known facts with unitising, multiplying and dividing by 10 and 100, and using adjusting strategies.

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## 2.20 Multiplication with three factors and volume

Use multiplication to calculate the volume of cuboids and shapes comprised of several cuboids; use division to solve associated inverse problems. Use associativity and commutativity to solve abstract multiplication problems with three factors.

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

Identify properties of factors and multiples including square and prime numbers, composite numbers, common and prime factors, and common multiples. Use factor pairs to solve problems efficiently.

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## 2.22 Combining multiplication with addition and subtraction

Learn to combine multiplication with addition or subtraction. Learn to use brackets to change the order of operations. Build on knowledge of the distributive law.

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**Easy-download:** Year 5 (All of the teacher guides and representations for segments

2.18–2.22 in one easy-to-download zip file.)

## 2.23 Multiplication strategies for larger numbers and long multiplication

Develop strategies for multiplying two numbers with two or more digits, including adjusting strategies when multiplying by a power of ten, partitioning followed by multiplication and addition of partial products, and long multiplication.

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## 2.24 Division: dividing by two-digit divisors

Learn to divide by two-digit divisors, recording calculations using either the short or long division algorithm. Represent remainders in an appropriate way, according to the context, including using the short or long division algorithm to express remainders as decimal fractions.

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## 2.25 Using compensation to calculate

Learn how multiplication and division calculations are affected when one element of the calculation is multiplied or divided by a scale factor.

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## 2.26 Mean average and equal shares

Understand the concept of mean average and learn how to find the mean of a set of data. Use the mean to compare sets of data and learn when it is appropriate to use the mean.

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## 2.27 Scale factors, ratio and proportional reasoning

Use bar modelling and ratio grids to reason about multiplicative relationships between two or more cardinal quantities, and explore correspondence problems. Extend understanding of scaling measures to make and interpret maps and scale/compare the dimensions of similar shapes.

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## 2.28 Combining division with addition and subtraction

Learn to combine division with addition or subtraction. Revisit the use of brackets to change the order of operations. Build on knowledge of the distributive law.

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## 2.29 Decimal place-value knowledge, multiplication and division

Develop efficient calculation strategies, and connect knowledge of multiplying and dividing by 10/100/1,000 to understanding of place value, including application to conversion between metric units of measure.

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## 2.30 Multiplicative contexts: area and perimeter 2

Build on earlier knowledge of area and perimeter. Learn to find the area of parallelograms and triangles by identifying the perpendicular height. Compare areas and perimeters and apply scale factors to side-length, perimeter and area.

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**Easy-download:** Year 6 (All of the teacher guides and representations for segments

2.23–2.30 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**: Joanna Caisová, Adrian Cannell, Clare Christie, Katie Crozier, Jonathan East, Claire Gerrard, Rebecca Holland, Alison Hopper, Elizabeth Lambert, Conor Loughney, Suzanne Mathews, Debbie Morgan, Sally O'Brien, Alexandra Parry, Georgia Ryan, Cat Stone, Debbie Weible, Andrew Whitehead

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

**Editorial and production**: Liam Benson, Design and Define; Nicholas Bromley; Charlotte Christensen; Tracey Cowell, Passion for Publishing; Rachel Houghton; Jalita Jacobsen; Bill Mantovani; David Mantovani; Sam Radford; Rosie Stewart, Swales & Willis; Cheryl Stirr; Yvette Sturdy, DiGiV8 Ltd; Rachel Trolove, Trolove Engineering Solutions Ltd; Andrew Young

**Illustrations**: Steve Evans; Alphablocks Ltd

JDRicketts28 - there aren't any plans for further Spines to be added.