ncetm multiplication problem solving

• Progression Maps

Reasoning Skills

Developing opportunities and ensuring progression in the development of reasoning skills

Achieving the aims of the new National Curriculum:

Developing opportunities and ensuring progression in the development of reasoning skills.

The aims of the National Curriculum are to develop fluency and the ability to reason mathematically and solve problems. Reasoning is not only important in its own right but impacts on the other two aims. Reasoning about what is already known in order to work out what is unknown will improve fluency; for example if I know what 12 × 12 is, I can apply reasoning to work out 12 × 13. The ability to reason also supports the application of mathematics and an ability to solve problems set in unfamiliar contexts.

Research by Nunes (2009) identified the ability to reason mathematically as the most important factor in a pupil’s success in mathematics. It is therefore crucial that opportunities to develop mathematical reasoning skills are integrated fully into the curriculum. Such skills support deep and sustainable learning and enable pupils to make connections in mathematics.

This resource is designed to highlight opportunities and strategies that develop aspects of reasoning throughout the National Curriculum programmes of study. The intention is to offer suggestions of how to enable pupils to become more proficient at reasoning throughout all of their mathematics learning rather than just at the end of a particular unit or topic.

We take the Progression Map for each of the National Curriculum topics, and augment it with a variety of reasoning activities (shaded sections) underneath the relevant programme of study statements for each year group. The overall aim is to support progression in reasoning skills. The activities also offer the opportunity for children to demonstrate depth of understanding, and you might choose to use them for assessment purposes as well as regular classroom activities.

Place Value Reasoning

Addition and subtraction reasoning, multiplication and division reasoning, fractions reasoning, ratio and proportion reasoning, measurement reasoning, geometry - properties of shapes reasoning, geometry - position direction and movement reasoning, statistics reasoning, algebra reasoning.

The strategies embedded in the activities are easily adaptable and can be integrated into your classroom routines. They have been gathered from a range of sources including real lessons, past questions, children’s work and other classroom practice.

Strategies include:

• Spot the mistake / Which is correct?
• True or false?
• What comes next?
• Do, then explain
• Make up an example / Write more statements / Create a question / Another and another
• Possible answers / Other possibilities
• What do you notice?
• Continue the pattern
• Missing numbers / Missing symbols / Missing information/Connected calculations
• Working backwards / Use the inverse / Undoing / Unpicking
• Hard and easy questions
• What else do you know? / Use a fact
• Fact families
• Convince me / Prove it / Generalising / Explain thinking
• Make an estimate / Size of an answer
• Always, sometimes, never
• Making links / Application
• Can you find?
• What’s the same, what’s different?
• Odd one out
• Complete the pattern / Continue the pattern
• Another and another
• Testing conditions
• The answer is…
• Visualising

These strategies are a very powerful way of developing pupils’ reasoning skills and can be used flexibly. Many are transferable to different areas of mathematics and can be differentiated through the choice of different numbers and examples.

Nunes, T. (2009) Development of maths capabilities and confidence in primary school, Research Report DCSF-RR118 (PDF)

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In the Classroom

Materials, guidance and National Curriculum resources

Find what you’re after, and more!

This is where the resources are to help you plan and teach great maths lessons, and to assess your pupils’ knowledge and understanding. Simply use the filter options on this page, if necessary right down to the year group you teach or specific type of resource you’re looking for. Many of the resources here are also suitable for collaborative professional development discussions among small groups of teachers.

If you don’t find what you’re looking for in this section, you can use the main ‘search’ function via the magnifying glass at the top of the page to look across the entire website, which includes the National Curriculum Resource Tool and our Early Years section.

Educational Phase

Resource type, exemplification of ready-to-progress criteria.

79 PowerPoints, each one focusing on one of the ready-to-progress criteria in the new DfE maths guidance for KS1 and KS2

DfE guidance on teaching maths at KS3

What the guidance consists of and how it aligns with other NCETM resources

Numberblocks Support Materials

Materials to support Early Years and Year 1 teachers

• Early Years
• Early Years Materials

Curriculum prioritisation in primary maths

A term-by-term framework to support planning and teaching

• Mastery PD Materials

Checkpoints

Diagnostic maths activities to help teachers develop their assessment of students' prior learning for KS3

KS3 key ideas exemplified

A series of 52 PowerPoints containing activities and professional development stimuli based on key ideas from the core concepts explored in the KS3 PD materials

Using mathematical representations at KS3

Guidance for some of the most useful representations for Key Stage 3 mathematics.

Insights from experienced teachers

Pairs of teachers explore small sections of the mastery materials in depth

Primary ITE professional development materials

Six professional development units designed for ITE tutors to use with primary trainee teachers exploring maths

Planning to teach secondary maths

Videos and resources for teaching Key Stage 3 maths topics

Primary Assessment Materials

Questions, tasks and activities supporting teaching for mastery.

• Assessment Materials

Mathematical Prompts for Deeper Thinking videos

Videos showing a teacher working with small groups of students from her Year 8 class

• Classroom Videos

Progression Maps for Key Stages 1 and 2

The progression maps are structured using the topic headings as they appear in the National Curriculum

• Progression Maps

Departmental Workshops

Collaborative CPD sessions for maths departments or other groups of secondary teachers

Primary Calculation Guidance

Recommendations and effective practice teaching ideas

Primary Subject Knowledge Audit

Assess your confidence in teaching the content of the KS1 and KS2 maths curriculum

Secondary Subject Knowledge Audit

Assess your confidence in teaching the content of the KS3 maths curriculum

Secondary ITE professional development materials

Six professional development units designed for ITE tutors to use with secondary trainee maths teachers

Secondary Assessment Materials

Materials to support you and your colleagues in assessing students at KS3

Marking Guidance

Guidance documents to enable effective marking and feedback

Research study modules

Ten study modules enabling you to access, understand and engage with academic research articles

Shaping the Year 7 curriculum: building on Year 6

Ready-to-use training materials, ideal for use in planning effective curriculum transition from primary

Resources for teachers using the mastery materials

Supplementary content to support use of the Secondary Mastery PD Materials

Adapting maths teaching for the Covid-19 period

Guidance for secondary schools – updated November 2020

Training materials for DfE Mathematics guidance

A ready-to-use PowerPoint designed for whole school development, and videos to introduce the guidance

The structure of the number system

Theme 1 comprises four core concepts: place value, estimation and rounding; properties of number; ordering and comparing; simplifying and manipulating expressions, equations and formulae.

Operating on number

Theme 2 comprises two core concepts: arithmetic procedures; solving linear equations.

Multiplicative reasoning

Theme 3 comprises two core concepts: understanding multiplicative relationships; trigonometry.

Sequences and graphs

Theme 4 comprises two core concepts: sequences; graphical representations.

Statistics and probability

Theme 5 comprises three core concepts: statistical representations and measure; statistical analysis; probability.

Theme 6 comprises four core concepts: geometrical properties; perimeter, area and volume; transforming shapes; constructions.

1.1 Place value, estimation and rounding

Core concept 1.1 covers the structure of our place-value system (particularly as it relates to decimals) and rounding numbers to a required number of decimal places or significant figures.

1.2 Properties of number

Core concept 1.2 focuses on factors, multiples and primes, exploring various representations to support understanding of highest common factors and lowest common multiples.

1.3 Ordering and comparing

Core concept 1.3 covers the conversion of decimals to fractions (and vice versa), ordering positive and negative integers, fractions and decimals, and the expression of numbers in standard form.

1.4 Simplifying and manipulating expressions, equations and formulae

Core concept 1.4 concerns the generalisation of number structures, the use of algebraic symbols, and techniques for their manipulation. The rearranging of formulae is also explored.

2.1 Arithmetic procedures

Core concept 2.1 offers guidance on developing understanding of the mathematical structures that underpin standard procedures for calculation with decimals, fractions and directed numbers.

2.2 Solving linear equations

Core concept 2.2 explores how linear equations are effectively the formulation of a series of operations on unknown numbers and how solving them involves undoing these operations.

3.1 Understanding multiplicative relationships

Core concept 3.1 explores fractions, percentages, ratio and proportion (direct and inverse) as contexts in which multiplicative relationships are used.

3.2 Trigonometry

Core concept 3.2 introduces trigonometric functions through the unit circle. Ideas of similarity, scale factor and multiplicative relationships are integrated to explore problem-solving.

4.1 Sequences

Core concept 4.1 covers sequences through exploration of the mathematical structure. It also introduces the idea of the nth term of a linear sequence.

4.2 Graphical representations

Core concept 4.2 introduces x- and y-coordinates as input and output of a function. It explores linear and quadratic functions, and how linear graphs can solve simultaneous equations.

5.1 Statistical representations and measures

Core concept 5.1 covers measures of central tendency, work with grouped data and measures of spread, as well as the construction of scatter graphs and pie charts.

5.2 Statistical analysis

Core concept 5.2 explores making informed choices about which statistical analysis and representation to use for different types of data, as well as the effect on interpretation.

5.3 Probability

Core concept 5.3 introduces probability as a way to quantify, explore and explain likelihood and coincidence, and to reason about uncertainty.

6.1 Geometrical properties

Core concept 6.1 covers angle facts and the geometry of intersecting lines, similarity and congruence, and Pythagoras’ theorem.

6.2 Perimeter, area and volume

Core concept 6.2 covers how the formulae for perimeter, area and volume are derived and connected, and the importance of reasoning mathematically to solve a range of problems.

6.3 Transforming shapes

Core concept 6.3 introduces translation, rotation, reflection and enlargement. It explores the degrees of freedom available, in terms of what does and doesn’t vary for each one.

6.4 Constructions

Core concept 6.4 explores: triangles of given lengths; a perpendicular bisector of a line segment; a perpendicular to a given line through a given point; an angle bisector.

The Bar Model

A representation used to expose mathematical structure

Multiplicative Reasoning – all downloads

Links to all the downloadable folders and files in the Multiplicative Reasoning section

Year 1 curriculum map

The whole of Year 1, split into units

Year 2 curriculum map

The whole of Year 2, split into units

Year 3 curriculum map

The whole of Year 3, split into units

Year 4 curriculum map

The whole of Year 4, split into units

Year 5 curriculum map

The whole of Year 5, split into units

Year 6 curriculum map

The whole of Year 6, split into units

Previous Reception experiences and counting within 100

Unit 1 – 7 weeks

Comparison of quantities and part–whole relationships

Unit 2 – 3 weeks

Numbers 0 to 5

Unit 3 – 2 weeks

Recognise, compose, decompose and manipulate 2D and 3D shapes

Unit 4 – 3 weeks

Numbers 0 to 10

Unit 5 – 3 weeks

Additive structures

Unit 6 – 4 weeks

Addition and subtraction facts within 10

Unit 7 – 3 weeks

Numbers 10 to 100

Unit 1 – 4 weeks

Calculations within 20

Fluently add and subtract within 10.

Unit 3 – 1 week

Addition and subtraction of two-digit numbers (1)

Unit 4 – 2 weeks

Introduction to multiplication

Unit 5 – 7 weeks

Introduction to division structures

Unit 6 – 2 weeks

Unit 7 – 2 weeks

Addition and subtraction of two-digit numbers (2)

Unit 8 – 3 weeks

Unit 9 – 1 week

Unit 10 – 2 weeks

Unit 11 – 1 week

Position and direction

Unit 12 – 1 week

Multiplication and division – doubling, halving, quotitive and partitive division

Unit 13 – 3 weeks

Sense of measure – capacity, volume, mass

Unit 14 – 2 weeks

Adding and subtracting across 10

Unit 1 – 2 weeks

Numbers to 1,000

Unit 2 – 10 weeks

Right angles

Manipulating the additive relationship and securing mental calculation.

Unit 4 – 4 weeks

Column addition

Unit 5 – 2 weeks

2, 4, 8 times tables

Unit 6 – 3 weeks

Column subtraction

Unit 7 – 1 week

Review of column addition and subtraction

Unit 1 – 3 weeks

Numbers to 10,000

Unit 2 – 5 weeks

3, 6, 9 times tables

7 times table and patterns.

Unit 5 - 2 weeks

Understanding and manipulating multiplicative relationships

Unit 6 – 5 weeks

Coordinates

Decimal fractions.

Unit 1 – 5 weeks

Unit 2 – 2 weeks

Negative numbers

Short multiplication and short division.

Unit 4 – 6 weeks

Area and scaling

Unit 5 – 5 weeks

Calculating with decimal fractions

Factors, multiples and primes.

Unit 7 – 4 weeks

Unit 8 – 7 weeks

Converting units

Unit 9 - 2 weeks

Unit 10 - 3 weeks

Calculating using knowledge of structures (1)

Unit 1 – 6 weeks

Multiples of 1,000

Numbers up to 10,000,000.

Unit 3 – 4 weeks

Draw, compose and decompose shapes

Multiplication and division.

Unit 5 – 4 weeks

Area, perimeter, position and direction

Fractions and percentages.

Unit 7 – 6 weeks

Unit 8 – 1 week

Ratio and proportion

Unit 9 – 2 weeks

Calculating using knowledge of structures (2)

Unit 10 – 1 week

Solving problems with two unknowns

Unit 11 – 2 weeks

Order of operations

Mean average.

Unit 13 – 1 week

School sees better results through teaching for mastery

Embracing teaching for mastery in maths has seen pupils’ love for the subject contribute to some exceptional end of year outcomes for all year groups

Why Mastery Readiness isn't just 'mastery-lite'!

The Mastery Readiness Programme enables schools to put in place structures, systems and positive mathematical culture so they can successfully engage in teaching for mastery

Don’t abandon counters just because it’s secondary school

At Trinity Academy in Halifax all teachers are now using place value counters with students across the ability range

Teaching for mastery at secondary: is it compatible with putting students in sets?

A visit to two lessons at Tomlinscote School in Frimley, Surrey

Numberblocks – Series 1

Series 1, all episodes

Numberblocks – Series 2

Series 2, all episodes

Video material to support the implementation of the National Curriculum

Multiplication.

Lesson videos on elements of progression in multiplication across the primary school

Subtraction

Lesson videos on elements of progression in subtraction in Key Stage 2

Number Facts

Lesson videos on number facts for all four operations of addition, subtraction, multiplication and division

Number and Place Value

Lesson videos on children developing understanding and fluency with number

Lesson videos on structures of division in terms of grouping and sharing at Key Stage 1; moving into a written algorithm at Key Stage 2 and demonstration of fluency at Key Stage 3

Lesson videos addressing the ability to develop a secure understanding and calculate with fractions

Lesson videos addressing the demands of the New Curriculum to introduce formal algebra into KS2

Multiplicative Reasoning

Lesson videos on representations in terms of the bar model and double number line are used to support reasoning and problem solving

Reasoning Skills

Developing opportunities and ensuring progression in the development of reasoning skills

Gap tasks: maximising the impact of Work Groups

We visit Gorse Hill Primary School Work Group in Swindon, and find out how gap tasks are driving powerful changes in pedagogy

A Year 1 lesson on difference as a form of subtraction

A lesson from 2015/16 led by Clare Christie, maths subject leader for the Ashley Down Schools Federation in Bristol

A Year 3 lesson on multiplication

A set of video clips from 2014/15 centred on one lesson, focusing on the 6-times-table

A Year 4 lesson on place value with decimal numbers

A year 6 lesson on comparing fractions.

Comparing the value of fractions with the same denominators

A Year 6 lesson on line graphs

A lesson from 2015/16 led by James Berry, subject leader and deputy head at Chesterton Primary School in Wandsworth

Year 2 Shanghai showcase lesson

Ms Lynn, part of the the England–China exchange, leads a Year 2 lesson at Tubbenden Primary School

Year 5 Shanghai showcase lesson

Ms Dai, part of the the England–China exchange, leads a Year 5 lesson at Tubbenden Primary School

Comparison of quantities and measures

Spine 1: Number, Addition and Subtraction – Topic 1.1

Introducing 'whole' and 'parts': part–part–whole

Spine 1: Number, Addition and Subtraction – Topic 1.2

Composition of numbers: 0–5

Spine 1: Number, Addition and Subtraction – Topic 1.3

Composition of numbers: 6–10

Spine 1: Number, Addition and Subtraction – Topic 1.4

Additive structures: introduction to aggregation and partitioning

Spine 1: Number, Addition and Subtraction – Topic 1.5

Additive structures: introduction to augmentation and reduction

Spine 1: Number, Addition and Subtraction – Topic 1.6

Addition and subtraction: strategies within 10

Spine 1: Number, Addition and Subtraction – Topic 1.7

Composition of numbers: multiples of 10 up to 100

Spine 1: Number, Addition and Subtraction – Topic 1.8

Composition of numbers: 20–100

Spine 1: Number, Addition and Subtraction – Topic 1.9

Composition of numbers: 11–19

Spine 1: Number, Addition and Subtraction – Topic 1.10

Addition and subtraction: bridging 10

Spine 1: Number, Addition and Subtraction – Topic 1.11

Subtraction as difference

Spine 1: Number, Addition and Subtraction – Topic 1.12

Addition and subtraction: two-digit and single-digit numbers

Spine 1: Number, Addition and Subtraction – Topic 1.13

Addition and subtraction: two-digit numbers and multiples of ten

Spine 1: Number, Addition and Subtraction – Topic 1.14

Addition: two-digit and two-digit numbers

Spine 1: Number, Addition and Subtraction – Topic 1.15

Subtraction: two-digit and two-digit numbers

Spine 1: Number, Addition and Subtraction – Topic 1.16

Composition and calculation: 100 and bridging 100

Spine 1: Number, Addition and Subtraction – Topic 1.17

Composition and calculation: three-digit numbers

Spine 1: Number, Addition and Subtraction – Topic 1.18

Securing mental strategies: calculation up to 999

Spine 1: Number, Addition and Subtraction – Topic 1.19

Algorithms: column addition

Spine 1: Number, Addition and Subtraction – Topic 1.20

Algorithms: column subtraction

Spine 1: Number, Addition and Subtraction – Topic 1.21

Composition and calculation: 1,000 and four-digit numbers

Spine 1: Number, Addition and Subtraction – Topic 1.22

Composition and calculation: tenths

Spine 1: Number, Addition and Subtraction – Topic 1.23

Composition and calculation: hundredths and thousandths

Spine 1: Number, Addition and Subtraction – Topic 1.24

Addition and subtraction: money

Spine 1: Number, Addition and Subtraction – Topic 1.25

Composition and calculation: multiples of 1,000 up to 1,000,000

Spine 1: Number, Addition and Subtraction – Topic 1.26

Negative numbers: counting, comparing and calculating

Spine 1: Number, Addition and Subtraction – Topic 1.27

Common structures and the part–part–whole relationship

Spine 1: Number, Addition and Subtraction – Topic 1.28

Using equivalence and the compensation property to calculate

Spine 1: Number, Addition and Subtraction – Topic 1.29

Composition and calculation: numbers up to 10,000,000

Spine 1: Number, Addition and Subtraction – Topic 1.30

Numberblocks – Series 3

Series 3, all episodes

Problems with two unknowns

Spine 1: Number, Addition and Subtraction – Topic 1.31

Counting, unitising and coins

Spine 2: Multiplication and Division – Topic 2.1

Structures: multiplication representing equal groups

Spine 2: Multiplication and Division – Topic 2.2

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

Spine 2: Multiplication and Division – Topic 2.3

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

Spine 2: Multiplication and Division – Topic 2.4

Commutativity (part 2), doubling and halving

Spine 2: Multiplication and Division – Topic 2.5

Structures: quotitive and partitive division

Spine 2: Multiplication and Division – Topic 2.6

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

Spine 2: Multiplication and Division – Topic 2.7

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

Spine 2: Multiplication and Division – Topic 2.8

Guidance on the teaching of fractions in Key Stage 1

Spine 3: Fractions – Topic 3.0

Preparing for fractions: the part–whole relationship

Spine 3: Fractions – Topic 3.1

Unit fractions: identifying, representing and comparing

Spine 3: Fractions – Topic 3.2

Non-unit fractions: identifying, representing and comparing

Spine 3: Fractions – Topic 3.3

Adding and subtracting within one whole

Spine 3: Fractions – Topic 3.4

Working across one whole: improper fractions and mixed numbers

Spine 3: Fractions – Topic 3.5

Multiplying whole numbers and fractions

Spine 3: Fractions – Topic 3.6

Finding equivalent fractions and simplifying fractions

Spine 3: Fractions – Topic 3.7

Common denomination: more adding and subtracting

Spine 3: Fractions – Topic 3.8

Multiplying fractions and dividing fractions by a whole number

Spine 3: Fractions – Topic 3.9

Linking fractions, decimals and percentages

Spine 3: Fractions – Topic 3.10

Times tables: 7 and patterns within/across times tables

Spine 2: Multiplication and Division – Topic 2.9

Connecting multiplication and division, and the distributive law

Spine 2: Multiplication and Division – Topic 2.10

Times tables: 11 and 12

Spine 2: Multiplication and Division – Topic 2.11

Division with remainders

Spine 2: Multiplication and Division – Topic 2.12

Calculation: multiplying and dividing by 10 or 100

Spine 2: Multiplication and Division – Topic 2.13

Multiplication: partitioning leading to short multiplication

Spine 2: Multiplication and Division – Topic 2.14

Division: partitioning leading to short division

Spine 2: Multiplication and Division – Topic 2.15

Multiplicative contexts: area and perimeter 1

Spine 2: Multiplication and Division – Topic 2.16

Structures: using measures and comparison to understand scaling

Spine 2: Multiplication and Division – Topic 2.17

Using equivalence to calculate

Spine 2: Multiplication and Division – Topic 2.18

Calculation: ×/÷ decimal fractions by whole numbers

Spine 2: Multiplication and Division – Topic 2.19

Multiplication with three factors and volume

Spine 2: Multiplication and Division – Topic 2.20

Factors, multiples, prime numbers and composite numbers

Spine 2: Multiplication and Division – Topic 2.21

Combining multiplication with addition and subtraction

Spine 2: Multiplication and Division – Topic 2.22

Multiplication strategies for larger numbers and long multiplication

Spine 2: Multiplication and Division – Topic 2.23

Division: dividing by two-digit divisors

Spine 2: Multiplication and Division – Topic 2.24

Using compensation to calculate

Spine 2: Multiplication and Division – Topic 2.25

Mean average and equal shares

Spine 2: Multiplication and Division – Topic 2.26

Scale factors, ratio and proportional reasoning

Spine 2: Multiplication and Division – Topic 2.27

Combining division with addition and subtraction

Spine 2: Multiplication and Division – Topic 2.28

Decimal place-value knowledge, multiplication and division

Spine 2: Multiplication and Division – Topic 2.29

Multiplicative contexts: area and perimeter 2

Spine 2: Multiplication and Division – Topic 2.30

Mindset: why is it so important in teaching for mastery?

Liam Colclough, a headteacher in Sheffield, is convinced that to introduce teaching for mastery effectively, a shift in mindset is required

Teaching for mastery: isn’t it just 'good teaching'?

Emma Patman, a Year 4 teacher, explains how her teaching has been transformed by her exposure to teaching for mastery pedagogy

Using a high quality textbook to support teaching for mastery

We visit the classroom of Hannah Gray, a primary teacher involved in trialling new Singapore-style textbooks, designed to support a mastery approach

Meeting the needs of all without ability setting

Removing setting from maths teaching in a three-form entry primary school

TRGs: Collaborative professional development in action

Green shoots of Teacher Research Groups (TRGs) are slowly spreading their fronds of mastery exploration across the country

How can teaching for mastery work in a mixed age class?

A group of schools in Northamptonshire have been addressing this issue, with encouraging results

Cardinality and Counting

Understanding that the cardinal value of a number refers to the quantity, or ‘howmanyness’ of things it represents

Understanding that comparing numbers involves knowing which numbers are worth more or less than each other

Composition

Understanding that one number can be made up from (composed from) two or more smaller numbers

Looking for and finding patterns helps children notice and understand mathematical relationships

Shape and Space

Understanding what happens when shapes move, or combine with other shapes, helps develop wider mathematical thinking

Comparing different aspects such as length, weight and volume, as a preliminary to using units to compare later

Teacher collaboration supports mixed-attainment classes

A London school models one possible way to make the transition

Key Stage 1 - Multiplication 1 video lessons

Video lessons on multiplication for children in Years 1 and 2

• Video Lessons

Key Stage 1 - Multiplication 2 video lessons

Key stage 1 - multiplication 3 video lessons, key stage 1 number, addition and subtraction video lessons.

Video lessons on number, addition and subtraction for children in Years 1 and 2

Lower Key Stage 2 - Fractions 1 video lessons

Video lessons on fractions for children in Years 3 and 4

Lower Key Stage 2 - Fractions 2 video lessons

Lower key stage 2 - fractions 3 video lessons, upper key stage 2 - fractions video lessons.

Video lessons on fractions for children in Years 5 and 6

Upper Key Stage 2 - Number, Addition and Subtraction video lessons

Video lessons on number, addition and subtraction for children in Years 5 and 6

Upper Key Stage 2 - Linking fractions, decimals and percentages video lessons

Video lessons on linking fractions, decimals and percentages for children in Years 5 and 6

Numberblocks at home

Resources to accompany the CBeebies Numberblocks series, designed for parents to use at home with children

What maths to teach for the rest of 2020/21

Post-16 Professional Development and Resources

Opportunities for developing teachers and maths departments and resources to support all post-16 maths teaching

Review of fractions

Fractions greater than 1.

Unit 9 – 5 weeks

Symmetry in 2D shapes

Unit 12 – 2 weeks

Unit fractions

Unit 8 – 5 weeks

Non-unit fractions

Unit 9 – 4 weeks

Parallel and perpendicular sides in polygons

Numbers 0 to 20.

Unit 8 – 4 weeks

Unitising and coin recognition

Self-audit questions for a teacher to assess confidence in the teaching of number in KS1 and KS2

Additive reasoning

Self-audit questions for a teacher to assess confidence in the teaching of additive reasoning in KS1 and KS2

Self-audit questions for a teacher to assess confidence in the teaching of multiplicative reasoning in KS1 and KS2

Self-audit questions for a teacher to assess confidence in the teaching of fractions in KS1 and KS2

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A collaborative national network developing and spreading excellent practice, for the benefit of all pupils and students.

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• Year 3 curriculum map

2, 4, 8 times tables

• Covid Recovery

Unit 6 – 3 weeks

The PowerPoint file contains slides you can use in the classroom to support each of the learning outcomes for this unit, listed below. The slides are comprehensively linked to associated pedagogical guidance in the  NCETM Primary Mastery Professional Development materials . There are also links to the ready-to-progress criteria detailed in the  DfE Primary Mathematics Guidance 2020 .

Classroom slides for this unit

All spring term units, learning outcomes, related pages.

• Curriculum prioritisation in primary maths

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A collaborative national network developing and spreading excellent practice, for the benefit of all pupils and students.

About this website

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Or search by topic

Number and algebra

• The Number System and Place Value
• Calculations and Numerical Methods
• Fractions, Decimals, Percentages, Ratio and Proportion
• Properties of Numbers
• Patterns, Sequences and Structure
• Algebraic expressions, equations and formulae
• Coordinates, Functions and Graphs

Geometry and measure

• Angles, Polygons, and Geometrical Proof
• 3D Geometry, Shape and Space
• Measuring and calculating with units
• Transformations and constructions
• Pythagoras and Trigonometry
• Vectors and Matrices

Probability and statistics

• Handling, Processing and Representing Data
• Probability

Working mathematically

• Thinking mathematically
• Mathematical mindsets
• Cross-curricular contexts
• Physical and digital manipulatives

For younger learners

• Early Years Foundation Stage

Advanced mathematics

• Decision Mathematics and Combinatorics
• Advanced Probability and Statistics

Published 2011 Revised 2020

Rich Mathematical Tasks

What is a rich mathematical task?

Why would I want to use rich tasks in my maths lessons?

Where can I find rich mathematical tasks for primary children?

I wonder whether you have ever asked yourself any of the above questions.  I am hearing from more and more primary teachers who would like to inject something 'extra' into their maths lessons.  They each have an underlying reason or reasons to get in touch with NRICH:

• Some feel that they need a change of approach to reinvigorate their mathematics teaching generally;
• Some report that the children in their school do well, but find it difficult to apply their mathematical knowledge and skills to new situations;
• Some would like their pupils to enjoy mathematics more;
• Some are worried that they are not stretching the higher-attaining children;
• Some are concerned that the lower-attaining children are 'turned-off' maths, lack confidence and have almost given up. 

Of course this is not an exhaustive list.  What you might find surprising is that the professional development we offer at NRICH for all of the above scenarios has a common focus: rich mathematical tasks. 

In this article, I will describe the start of a project, which began in the spring term of 2010.  Pete Hall, the NCETM East of England Regional Coordinator, contacted me to tell me about a number of £1000 grants on offer to schools who wanted to develop their understanding and use of rich mathematical tasks. The application form was relatively straightforward to complete, requiring some detail about the theme (what it was and why it had been chosen); who would be involved and a commitment to contribute to the NCETM rich tasks community.  Schools were required to give a breakdown of how the money would be spent and they promised to submit a short written report to NCETM on completion of the project.

Four schools in the east of England were successfully awarded a grant:  Clover Hill Infants' School in Norwich, Harrold Lower School near Bedford, Lakenham Primary School also in Norwich and St Philip's Primary School in Cambridge.  Each school decided to spend at least some of their money on professional development run by NRICH and I hope by describing what we have achieved so far, you may feel able to lead one or more staff meetings in your own school without necessarily paying for NRICH support!

At all four schools, I have led an initial workshop, varying in length from a staff meeting to a half day.  In all cases, we have begun with having a go at an activity altogether.  I feel it is important for everyone to engage in some mathematics - it reminds us what it is like to be a learner and it gives us a common experience (to some extent), which aids subsequent discussion.  The problem that I have used in all four schools is Magic Vs .  (Do have a go at it if you do not know it.  The approach I took with the teachers is exactly the same as that suggested in the teachers' notes on the website.)  We spent anything from about thirty to forty-five minutes actually working on the problem itself, with me taking the role of 'teacher', just as I would if I was with a group of children.

Having reached a suitable pausing point, we reflected on what we had done.  What mathematical 'content' knowledge did we use as we tackled this problem? By this I mean the aspects of number, calculation, shape and space, data handling and/or measures I needed to know, or I came to know.  In terms of the Magic Vs problem, the following list reflects the range of suggestions:

• Odd/even numbers
• Addition/subtraction
• Number bonds
• Consecutive numbers
• Multiplication/division (perhaps)
• Factors/multiples (perhaps)

Next, we asked ourselves what problem-solving strategies we found useful.  Here are those that came up frequently:

• Using trial and improvement
• Noticing and explaining patterns
• Working systematically
• Making conjectures
• Tweaking/altering/varying
• Testing ideas
• Generalising
• Talking to each other

I often find it helpful to reflect on mathematical activities in this way, that is considering the 'content' and processes separately.  In terms of the Magic Vs problem, it is interesting to note that the 'content' we used was fairly basic, possibly not going beyond that usually met in Key Stage 1.  However, we used a vast range of strategies to solve the problem, some of which are rather sophisticated. 

So, this led on to further discussion:  what makes this Magic Vs problem so 'rich'?  Suggestions included:

• It is easy to get started but also has the potential to be taken to high levels of mathematics (what NRICH terms ' low threshold high ceiling ')
• It has more than one answer
• It is 'open-ended', in the sense that although there are some answers, you can go on asking, and pursuing, your own questions
• The way to go about solving the problem is not immediately obvious
• It can be approached in many different ways
• It requires you to use a range of knowledge and skills
• It leads to generalisations
• It might deepen our understanding of odd/even numbers
• It is non-threatening (perhaps linked to the fact everyone can begin to have a go)

By specifically talking about these characteristics, the idea is not to suggest that every problem we use in the classroom should tick all these boxes.  Instead, by raising awareness of a set of characteristics, we can understand how resources we already use might be tweaked to make them 'richer'. 

This in turn leads to another important point.  Although in each session, the participating teachers came up with reasons for Magic Vs being a rich mathematical task, these are not necessarily inherent in the problem itself.  Would the teachers have thought it was rich if I had simply handed each one of them a piece of paper with the problem written on it and demanded they work in silence?  Some may have reached similar conclusions, but I suspect some would not.  So, the potential of a task to be rich is not enough in my opinion.  There are two other elements (at least!).

If we want children to get better at solving mathematical problems, then we need to encourage them to think in a mathematical way and to have a range of strategies at their fingertips, which they can draw upon.  Therefore, the questions and prompts we use, in conjunction with the tasks we provide, are crucial.  In this first session with the teachers, I showed them the ' Primary Questions and Prompts for Mathematical Thinking ' book, published by the ATM and give them a taster of its contents.  I am a huge fan of this book.  The authors define certain activities, which 'typify mathematical thinking', and suggest questions and prompts to encourage these.  These suggestions are entirely context-free, in other words they could be used when children are working on any topic, from number to calculating to shape to measuring to data handling.  So, the first element to consider in conjunction with using rich tasks is the way we question learners in the classroom.

The second element is what I term the classroom 'culture'.  Rich tasks and good questioning will thrive in a classroom where children are encouraged to talk to each other, where they are happy to offer ideas without the fear of being wrong, where their opinions are welcomed.  The culture of your classroom reflects your values so in all four schools, we discussed what we value in mathematics and how this affects the way we work in class. 

Reflecting on all three of these inter-related aspects of mathematics teaching (rich tasks, questioning and classroom culture) is a lot to cram into a staff meeting, let alone half a day.  And I threw in a quick tour of the NRICH website too!  Along the way, we talked about the benefits of such an approach. Many children who are currently 'high-attaining' may feel uncomfortable when presented with such tasks.  They may not be used to being challenged in mathematics.  They may be used to knowing immediately what to do when faced with a problem.  However, surely as teachers we have an obligation to equip our children with skills that will carry them in good stead as they get older?  Encouraging an ethic of perseverance and the idea of relishing a challenge is part and parcel of mathematics teaching, although it is something that perhaps feels rather daunting as a teacher. 

We arranged a date for me to return to each school so there was time for everyone to mull over the first session.  All the participating teachers agreed to try out at least one rich task with their children in the intervening weeks.  They will come to the second session prepared to talk about their experience:  the things that went well and those that didn't go well; the surprises and the lessons learned.  We hope then to find some ways forward for each school so that they can build on their achievements and continue to go from strength to strength.

Building relational understanding with the Core Competencies and NCETM’s Big Ideas

What do the Maths — No Problem! Core Competencies and NCETM’s Big Ideas have in common? They’re both important maths mastery principles that work together to build relational understanding.

The NCETM’s Five Big Ideas are lesson design principles for teaching maths for mastery and the Five Core Competencies are attributes that help learners develop deeper thinking.

Seems fairly straightforward. But how do the Big Ideas and the Core Competencies align? Do they serve different purposes? What are the implications for teaching?

To save yourself hours of digging through the internet, keep reading.

What are the NCETM’s Five Big Ideas?

Let’s start by looking at the Five Big Ideas in a bit more detail.

The NCETM Five Big Ideas were created to enhance teaching for mastery. These research-based principles frame the lesson studies, professional dialogue and lesson design process within Teacher Research Groups (TRGs).

Mastery specialists from regional maths hubs have helped spread the Five Big Ideas through TRGs.

So, what are the Five Big Ideas?

• Representation and structure: representations used in lessons expose the mathematical structure so that students can do the maths without needing the representation.
• Mathematical thinking: if taught ideas are to be understood deeply, they musn’t be passively received but must be worked on by the student: thought about, reasoned with and discussed with others.
• Fluency: quick and efficient recall of facts and procedures and the flexibility to move between different contexts and representations of mathematics.
• Variation: Variation is twofold. It is firstly about how the teacher represents the concept, often in more than one way, to draw attention to critical aspects, and to develop deep and holistic understanding. It is also about the sequencing of the episodes, activities and exercises used within a lesson and follow-up practice, paying attention to what stays the same and what changes, to connect the mathematics and draw attention to mathematical relationships and structure.
• Coherence: breaking lessons down into small connected steps that gradually unfold the concept, providing access for all children and leading to a generalisation of the concept and the ability to apply the concept to a range of contexts.

What are the Maths — No Problem! Five Core Competencies?

Now, let’s take a look at what MNP brings to the table. The Maths — No Problem! Five Core Competencies are the attributes you want to see in your learners as you teach for mastery.

Learners who show these core competencies are set up for maths success.

The Five Core Competencies are:

• Visualisation: ask learners to show ‘how they know’ at every stage of solving the problem.
• Generalisation: challenge learners to dig deeper by finding proof.
• Communication: encourage learners to answer in full sentences. Try asking learners to talk about the work they’re doing or use structured tasks centred around a class discussion.
• Number sense: a learner’s ability to work fluidly and flexibly with numbers .
• Metacognition: teach learners to think about how they are thinking. This helps learners solve multi-step tasks and promotes the ability to keep complex information in mind.

How the Big Ideas and Core Competencies work together to build relational understanding

The last couple of Big Ideas don’t match up quite as neatly with the Core Competencies. But the ones that do align like this:

When you apply the lesson design principles from Big Ideas to develop the Core Competencies, you help learners build relational understanding.

Relational understanding focuses on not just knowing a rule, but understanding why it works and establishing connections.

So, how do the Ideas and Competencies work together to develop relational understanding?

‘Visualisation’ and ‘Representation and structure’

Relational understanding is all about visualising and understanding the underlying structure behind problems. To build relational understanding, try using Ban Har-style questioning like:

“Can you see?” “Can you imagine?”

It’s essential to allow learners space for visualisation before offering explanations. Also, try to avoid too much pencil on paper.

Using manipulatives helps learners to visualise and allows teachers to expose the structure of the mathematics at hand. But it’s vital to use manipulatives as tools — not toys.

How should you get started? I often build in time to allow learners to just play at first, especially if the resource is completely new to them. Manipulatives are a good way of promoting flexible thinking by asking those learners quick to arrive at an abstract solution to prove their thinking in a different way.

It’s worth noting that the Education Endowment Fund recommends removing manipulatives once understanding is secure to avoid over-reliance or procedural use of one particular model.

A critical component of scaffolding is making sure you carefully consider which representation to use. This helps provide access for all learners. When designing lessons, consider what to record on the board — even down to how colour-coding may aid understanding.

‘Generalisation’, and ‘Communication’

All three principles are about making connections, spotting links, noticing patterns and reasoning — which all help to build a connected body of knowledge.

Supporting learners’ generalisation skills can include getting them to explore whether statements are always, sometimes, or never true (or false — using the idea of negative variation).

Another good strategy is using peer discussion. Here, learners establish consensus around rules, examples and counterexamples (or non-examples). Encourage them to explain, describe and justify their methods and results, and reflect on their conclusions.

‘Fluency’ and ‘Number sense’

Fluency and number sense are closely related: partitioning facts, times tables facts and using connected facts like equivalent fractions. When learners are fluent, they can use the known to work out the unknown — an important component of relational understanding.

For me, number sense and fluency are all about noticing patterns, checking to see whether an answer is reasonable, and selecting efficient and appropriate methods of calculation. Sound number sense avoids emphasising procedural recall and rehearsal.

Developing relational understanding relies on lessons that encourage learners to make connections and delve deeper. Teaching relational understanding is demanding, but it’s worth it! By building a connected body of knowledge and skills, your learners can become true mathematical thinkers.

Joe Jackson-Taylor

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Year 4 - NCETM Unit 5 - 7 Times Table and Patterns

Subject: Mathematics

Age range: 7-11

Resource type: Unit of work

Last updated

12 January 2024

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This is a unit of work that is inspired by the planning from NCETM Year 4 Unit 5. The learning outcomes that the NCETM state are covered in this unit if planning are as follows:

1 Pupils represent counting in sevens as the 7 times table 2 Pupils explain the relationship between adjacent multiples of seven 3 Pupils use their knowledge of the 7 times table to solve problems 4 Pupils identify patterns of odd and even numbers in the times tables 5 Pupils represent a square number 6 Pupils use knowledge of divisibility rules to solve problems

In this unit of work the learning objectives are the following:

• Revise column method for + and -
• Count in and multiply by 7
• Solve 7 times table problems
• Identify patterns of odd and even numbers in times tables
• Identify square numbers
• Find divisibility rules

Each lesson follows the Powerpoint slide structure of: Revisit Address Misonceptions Vocabulary Focus Guided Practice TalkTask Deepen Independent Practice

These lessons follow a mastery approach and have been commented on by OFSTED as being a great tool for great maths teaching. Print-outs are provided. The following elements are mastery teaching are used throughout: stem sentences, generalisations, high-level vocabulary, the slides allow for a ping-pong approach, clear and varied visual representations, opportunities for using resources etc.

Worksheets provided follow the structure of: Fluency, Problem-solving and Reasoning to ensure progression.

3 of these lessons are not intended to be in books and do not come with ‘next steps’ but the rest do. This is to allow next steps and progress to be een within books for that lesson.

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