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‘AI means maths problem-solving skills are more important than ever’

Cambridge bolsters classroom learning with new 'Problem-Solving Schools' initiative

By Stephen Bevan Published: 16th November 2023

Credit: Phil Boorman

Mathematicians at the University of Cambridge are supporting UK schools to help prioritise problem solving in maths – a key skill that is likely to become ever more critical with the rise of automation and artificial intelligence.

The new Problem-Solving Schools initiative, developed by the University’s Faculty of Mathematics, aims to create ‘a movement of problem-solving schools’ by providing free learning resources and teacher training to refocus attention on the skill.  Along with fluency and reasoning, problem solving has been central to the National Curriculum for maths since it was introduced in 2014, but often does not receive the same amount of attention in the classroom.

In the summer, Ofsted published new guidance encouraging schools to focus more consistently on teaching problem solving, and emphasised the importance of teaching skills that “equip [pupils] for the next stage of education, work and life”.

Dr Ems Lord, Director of NRICH , which provides thousands of free online mathematics resources for ages three to 18, and is launching Problem-Solving Schools, said: “It's fair to say that many schools feel increasingly confident supporting fluency and reasoning skills, and there’s a lot of support out there. What’s been missing is the problem-solving aspect, and that’s been repeatedly picked up by Ofsted. It’s not being prioritised, often because of a lack of training for teachers and a lack of access to sufficient, high-quality resources to support it.

Dr Ems Lord at the University's Maths Faculty. Credit: Nathan Pitt

“Some schools are not covering it as well as others, so it means we’re in this very patchy landscape and at the same time we have AI coming in, with everyone thinking about how that will impact future roles and careers. And it’s looking increasingly likely that students who are good problem solvers, and have good teamwork skills, are the ones who are going to thrive.”

Although AI is developing rapidly, Dr Lord says at present problem solving isn’t one of its strong points. And business analysts believe that in the future jobs which computers cannot perform ­– that require uniquely human skills such as critical thinking ­– will become more significant and those with these skills will be in even more demand.

“I can put our problems into an AI system, some it can solve, some it gives ridiculous answers to. But how would someone know which is which unless they know how to solve the problem themselves – or even know what question to ask to get the answer they’re after?

“Problem-solving is not about memorising facts, it’s about being confronted with something for the first time and thinking, ‘Right, how do I use my skills to approach this?’ And these are transferrable skills, for all aspects of life, which will help children in the future, not just at work but also socially. We want our young people to have the curiosity and confidence to question things, so if they come across some data or a graph in the media, or wherever, they have the experience and skills to know what a good graph looks like, and they can analyse it for themselves.

“It’s such an important area that we have to get right, and at the moment we’re not doing it. The whole point of learning maths is to be able to solve problems.”

Dr Lord says the Problem-Solving Schools initiative aims to help embed the skill in classrooms by providing themed resources and webinar training on how to best use them – to support teachers who might be lacking in confidence themselves, or are unsure how to refocus how they teach the Curriculum.

The webinar series will also include tips on engaging parents with maths so they can help support their children in the subject. In a recent study , NRICH’s Solving Together project, which offers family-friendly homework activities, was found to significantly increase parental involvement in the subject.

'Problem-solving is not about memorising facts, it’s about being confronted with something for the first time and thinking, ‘Right, how do I use my skills to approach this?'

- Dr Ems Lord, Director of NRICH

Pupils using NRICH maths resources. Credit: University of Cambridge

In addition, a Charter for schools to sign up to is also being introduced. It puts problem solving at the heart of maths learning, from the commitment of the school’s leadership team, to values in the classroom – where good problem-solving behaviour is encouraged, and where it’s ok to make mistakes – to how activities can be widened out to the local community.

The NRICH team has developed the programme in consultation with schools, and has actively sought the views of colleagues in the Department for Education, and the National Centre for Excellence in the Teaching of Mathematics – the Government’s maths body set up to improve mathematics teaching in England.

“Many of the resources given to teachers up to this point have focused on fluency, and if a teacher isn’t mathematically trained they tend to revert to where they feel safe, how they were taught,” says Dr Lord. “We need to break the mould on that, we need to make sure there are good resources available for problem-solving learning, and free training, so it isn’t a case of ‘we should be doing this’, but, ‘why wouldn’t we be doing this?’

“We’ve created a complete, wraparound package. We’re looking for schools across the country to sign up to the Charter, create a movement of problem-solving schools and change the agenda.”

Professor Bhaskar Vira, Pro-Vice-Chancellor for Education at the University of Cambridge, said: “Problem-Solving Schools is an exciting initiative that builds on the University’s work to support schools around the country through outreach and learning. NRICH’s high quality resources will help maths teachers embed problem solving in the classroom, as part of Cambridge’s mission to contribute to society through education, learning and research, and equip pupils with this key skill for the future.”

As part of the Problem-Solving Schools launch, NRICH is developing its resources, which have been supporting learners since the outreach programme’s launch 25 years ago , and recently made a huge contribution to the national effort during the COVID-19 lockdowns. Between March and September 2020, nrich.maths.org registered a 95% increase in UK visits compared to the previous year. In the 2020–21 school year alone, the site attracted just under 33 million page views. In spring 2020, the UK Government highlighted NRICH resources to schools and the team contributed to the BBC’s heavily used Bitesize maths resources.

And as the team launches its newest initiative, it continues to support post-pandemic catch-up work, by helping fill gaps in knowledge and focusing on students’ attitude to maths.

“It’s not just about doing the maths, it’s about enjoying it and finding it worthwhile – understanding the applications,” says Dr Lord. “If our materials are just about covering subject knowledge it’s really hard for student to enjoy what they’re doing.

“It’s a bit like having never seen Messi score a goal. If all you’ve done is go to football practice, where the coach puts down markers and tells you to dribble through them for an hour, and you come back the next week and do exactly the same thing, you kind of wonder why you’re doing it.

“But if you go to football practice and then switch on the TV and see a Messi wonder goal – it’s like ‘Aah – that’s what it’s all about!’ And I sometimes think that’s what’s missing when we talk about maths – the sheer moments of awe and wonder that you can have, and that feeling when you solve a problem which is absolutely fantastic!”

Credit: University of Cambridge

The text in this work is licensed under a Creative Commons Attribution 4.0 International License .

The resources on this page will hopefully help you teach AO2 and AO3 of the new GCSE specification - problem solving and reasoning.

This brief lesson is designed to lead students into thinking about how to solve mathematical problems. It features ideas of strategies to use, clear steps to follow and plenty of opportunities for discussion.

The PixiMaths problem solving booklets are aimed at "crossover" marks (questions that will be on both higher and foundation) so will be accessed by most students. The booklets are collated Edexcel exam questions; you may well recognise them from elsewhere. Each booklet has 70 marks worth of questions and will probably last two lessons, including time to go through answers with your students. There is one for each area of the new GCSE specification and they are designed to complement the PixiMaths year 11 SOL.

These problem solving starter packs are great to support students with problem solving skills. I've used them this year for two out of four lessons each week, then used Numeracy Ninjas as starters for the other two lessons.  When I first introduced the booklets, I encouraged my students to use scaffolds like those mentioned here , then gradually weaned them off the scaffolds. I give students some time to work independently, then time to discuss with their peers, then we go through it as a class. The levels correspond very roughly to the new GCSE grades.

Some of my favourite websites have plenty of other excellent resources to support you and your students in these assessment objectives.

@TessMaths has written some great stuff for BBC Bitesize.

There are some intersting though-provoking problems at Open Middle.

I'm sure you've seen it before, but if not, check it out now! Nrich is where it's at if your want to provide enrichment and problem solving in your lessons.

MathsBot  by @StudyMaths has everything, and if you scroll to the bottom of the homepage you'll find puzzles and problem solving too.

I may be a little biased because I love Edexcel, but these question packs are really useful.

The UKMT has a mentoring scheme that provides fantastic problem solving resources , all complete with answers.

I have only recently been shown Maths Problem Solving and it is awesome - there are links to problem solving resources for all areas of maths, as well as plenty of general problem solving too. Definitely worth exploring!

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Problem solving in the United Kingdom

• Original article
• Published: 14 August 2007
• Volume 39 , pages 395–403, ( 2007 )

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• Hugh Burkhardt 1 &
• Alan Bell 1  

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We trace the development of problem solving in UK school mathematics over the last century or so, illuminating our descriptions with task exemplars. This is an informative and cautionary tale of mutual incomprehension between leaders in mathematical education and the public they seek to serve. Intelligent and energetic pioneers have developed and improved ways to teach problem solving, though often with little attention to the challenges these methods present to typical teachers. Political decision makers, despite rhetorical support for “real-world” problem solving, have failed to understand the need for the changes proposed. Currently, there are some hopeful signs but it is doubtful if they will be realized in classroom practice. The challenge of modifying the system dynamics so as to yield large-scale improvements remains an unsolved problem in the UK, as elsewhere; at least, it is now recognized and being worked on.

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Forging New Opportunities for Problem Solving in Australian Mathematics Classrooms through the First National Mathematics Curriculum

Faire vivre une formation à l’enseignement des mathématiques par résolution de problèmes : le cas du cours « Didactique des mathématiques II et laboratoire » à l’Université du Québec à Montréal

Can Mathematical Problem Solving Be Taught? Preliminary Answers from 30 Years of Research

It is said that the Japanese Government once decided that only 2% of students need to learn problem solving.

Though can one, on cultural grounds alone, justify giving mathematics more curriculum time than, say, music?

Alas for both these perspectives, interview data suggest that the overwhelming majority of students see school mathematics as just a route to a necessary qualification.

As the Association for Teaching Aids in Mathematics .

I am grateful to Susie Groves and Kaye Stacey who first introduced me to Consecutive Sums as an investigation in ‘The Burwood Box’, their pioneering work on problem solving in Australian teacher education. HB.

“Evidence” to British government committees is opinion, with or without evidential support beyond anecdotes.

Informally called “Cockcroft Missionaries”, they formed a cadre of expertise whose influence can still be found.

Research on criterion referencing shows that thousands of much more precise statements are needed to define levels unambiguously. The policy makers saw that this was absurd but, not recognizing that the problem is fundamental, restricted the number to ten or so per level by making the statements broad and imprecise.

In contrast, English was allowed process-based attainment targets: Reading, Writing, Speaking and Listening. This probably reflected the deeper understanding of both politicians and civil servants, for whom English is their working toolkit while, in this land of C.P.Snow’s Two Cultures , their mathematical literacy is limited (see Sect. 7).

The appendix of tasks was removed from subsequent revisions of the report on the spurious grounds that there was a different, cross-subject working group on assessment.

Reasoning length is the time a competent student spends on a prompted piece of a task. It differs from task length for tasks with several parts.

“Numeracy” has become a deeply ambiguous term. Its original definition (Crowther 1959) as “the mathematical equivalent of literacy” has been corrupted until, for most people, it means reliable skills in arithmetic (as though literacy were only spelling and grammar).

Imagine: “To reduces greenhouse gases, we will go over to nuclear fusion for electric power next year.” “We will now provide effective cures for cancer to all who need it.” Why has education no sense of feasibility?

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Burkhardt, H., Bell, A. Problem solving in the United Kingdom. ZDM Mathematics Education 39 , 395–403 (2007). https://doi.org/10.1007/s11858-007-0041-4

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Accepted : 01 June 2007

Published : 14 August 2007

Issue Date : October 2007

DOI : https://doi.org/10.1007/s11858-007-0041-4

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Maths Problem Solving At KS2: Strategies and Resources For Primary School Teachers

John Dabell

Maths problem solving KS2 is crucial to succeeding in national assessments. If your Key Stage 2 pupils are still struggling with reasoning and problem solving in Maths, here are some problem solving strategies to try with your classes; all aligned to Ofsted’s suggested primary school teaching strategies.

Reasoning and problem solving are widely understood to be one of the most important activities in school mathematics. As far back as 1982,  The Cockcroft Report , stated:

‘The ability to solve problems is at the heart of mathematics. Mathematics is only “useful” to the extent to which it can be applied to a particular situation and it is the ability to apply mathematics to a variety of situations to which we give the name “problem solving”. […] At each stage […] the teacher needs to help pupils to understand how to apply the concepts and skills which are being learned and how to make use of them to solve problems. These problems should relate both to the application of mathematics to everyday situations within the pupils’ experience, and also to situations which are unfamiliar.’

Thirty plus years later and problem solving is still the beating heart of the Maths curriculum and – along with fluency and reasoning – completes the triad of aims in the 2014 New National Curriculum.

Ofsted’s view on problem solving in the Maths curriculum

Despite its centrality, Ofsted report that ‘ problem solving is not emphasised enough in the Maths curriculum ’. Not surprisingly, problem solving isn’t taught that well either because teachers can lack confidence, or they tend to rely on a smaller range of tried and tested strategies they feel comfortable with but which may not always ‘hit home’. If you’re looking to provide further support to those learners who haven’t yet mastered problem solving, you probably need a range of different strategies, depending on both the problem being attempted and the aptitude of the pupil.

We’ve therefore created a free KS2 resource aimed at Maths Coordinators and KS2 teachers that teaches you when and how to use 9 key problem solving techniques:  The Ultimate Guide to Problem Solving Techniques

The context around KS2 problem solving

According to Jane Jones, former HMI and National Lead for Mathematics, in her presentation at the Jurassic Maths Hub:

• Problems do not have to be set in real-life contexts, beware pseudo contexts.
• Providing a range of puzzles and other problems helps pupils to reason strategically to approach problems, sequence unfolding solutions, and use recording to help their mathematical thinking for next steps.
• It is particularly important that teachers and TAs stress reasoning, rather than just checking whether the final answer is correct.
• Pupils of all ability need to learn how to solve problems – not just the high attainers or fastest workers.

The Ultimate Guide to Problem Solving Techniques

9 ready-to-go problem solving techniques with accompanying tasks to get KS2 reasoning independently

How to approach KS2 maths problems

So what do we do? Well Ofsted advice is pretty clear on what to do when teaching problem solving. Jane Jones says we should:

• Set problems as part of learning in all topics for all pupils.
• Vary the ways in which you pose problems.
• Try to resist prompting pupils too soon and focusing on getting ‘the answer’ – pupils need to build their confidence, skills and resilience in solving problems, so that they can apply them naturally in other situations.
• Make sure you discuss alternative approaches with pupils to help develop their reasoning.
• Ensure that problems for high attainers involve demanding reasoning and problem-solving skills, not just harder numbers.

Perhaps more than most topics in Maths, teaching pupils how to approach problem solving questions effectively requires a systematic approach. Pupils can face any number of multi-step word problems throughout their SATs and they will face them without our help. To truly give pupils the tools they need to approach problem solving in Maths we must ingrain techniques for  approaching  problems.

With this in mind, below are some methods and techniques for you to consider when teaching problem solving in your KS2 Maths lessons. For greater detail and details on how to teach this methods, download the  Ultimate Guide to Problem Solving Techniques

Models for approaching KS2 problem solving

Becoming self-assured and capable as a problem solver is an intricate business that requires a range of skills and experience. Children need something to follow. They can’t just pluck a plan of attack out of thin air which is why models of problem solving are important especially when made memorable. They help establish a pattern within pupils so that, when they see a problem, they feel confident in taking the steps towards solving it.

Find out how we encourage children to approach problem solving independently in our blog: 20 Maths Strategies KS2 That Guarantee Progress for All Pupils.

The most commonly used model is that of George Polya (1973), who proposed 4 stages in problem solving, namely:

• Understand the problem
• Devise a strategy for solving it
• Carry out the strategy
• Check the result

Many models have followed the Polya model and use acronyms to make the stages stick. Which model you use can depend on the age of the children you are teaching and sometimes the types of word problems they are trying to solve. Below are several examples of Polya model acronyms:

C – Circle the question words U – Underline key words B – Box any key numbers E – Evaluate (what steps do I take?) S – Solve and check (does my answer make sense and how can I double check?)

R – Read the problem correctly. I – Identify the relevant information. D – Determine the operation and unit for expressing the answer. E – Enter the correct numbers and calculate

I – Identify the problem D – Define the problem E – Examine the options A – Act on a plan L – Look at the consequences

R – Read and record the problem I – Illustrate your thinking with pictures, models, number lines etc C – Compute, calculate and check E – Explain your thinking

R – Read the question and underline the important bits U – Understand: think about what to do and write the number sentences you will need C –  Choose how you will work it out S – Solve the problem A – Answer C – Check

Q – Question – read it carefully U – Understand – underline or circle key elements A – Approximate – think about the size of your answer C – Calculate K – Know if the answer is sensible or not

T – Think about the problem and ponder E – Explore and get to the root of the problem A – Act by selecting a strategy R – Reassess and scrutinise and evaluate the efficiency of the method

The idea behind these problem solving models is the same: to give children a structure and to build an internal monitor so they have a business-like way of working through a problem. You can choose which is most appropriate for the age group and ability of the children you are teaching.

The model you choose is less important than knowing that pupils can draw upon a model to follow, ensuring they approach problems in a systematic and meaningful way. A far simpler model – that we use in the   Ultimate Guide to KS2 Problem Solving Techniques  – is UCR: Understand the problem, Communicate and Reflect.

You then need to give pupils lots of opportunities to practice this! You can find lots of FREE White Rose Maths aligned maths resources, problem solving activities and printable worksheets for KS1 and KS2 pupils in the Third Space Learning Maths Hub .

You might also be interested in:

• 25 Fun Maths Problems For KS2 And KS3 (From Easy To Very Hard!)
• 30 Problem Solving Maths Questions And Answers For GCSE
• Why SSDD Problems Are Such An Effective Tool To Teach Problem Solving At KS3 & KS4

What’s included in the guide?

After reading the  Ultimate Guide to KS2 Problem Solving Techniques , we guarantee you will have a new problem solving technique to test out in class tomorrow. It provides question prompts and activities to try out, and shows you step by step how to teach these 9 techniques

• Open ended problem solving
• Using logical reasoning

Working backwards

Drawing a diagram

Drawing a table

Creating an organised list

Looking for a pattern

Acting it out

Guessing and checking

Cognitive Activation: getting KS2 pupils in the lightbulb zone

If you need more persuasion, pupils who use strategies that inspire them to think more deeply about maths problems are linked with higher Maths achievement. In 2015 The  National Education Research Foundation  (NFER) published ‘ PISA in Practice: Cognitive Activation in Maths ’. This shrewd report has largely slipped under the Maths radar but it offers considerable food for thought regarding what we can do as teachers to help mathematical literacy and boost higher mathematical achievement.

Cognitive Activation isn’t anything mysterious; just teaching problem solving strategies that pupils can think about and call upon when confronted by a Maths problem they are trying to solve. Cognitive It encourages us as teachers to develop problems that can be solved in more than one way and ‘may require different solutions in different contexts’. For this to work, exposing children to challenging content and encouraging a culture of exploratory talk is key. As is:

• Giving pupils maths problem solving questions that require them to think for an extended time.
• Asking pupils to use their own procedures for solving complex problems.
• Creating a learning community where pupils are able to make mistakes.
• Asking pupils to explain how they solved a problem and why they choose that method.
• Presenting pupils with problems in different contexts and ask them to apply what they have learned to new contexts.
• Giving pupils problems with no immediately obvious method of solution or multiple solutions.
• Encouraging pupils to reflect on problems.

Sparking cognitive activation is the same as sparking a fire – once it is lit it can burn on its own. It does, however, require time, structure, and the use of several techniques for approaching problem solving. Techniques, such as open-ended problem solving, are usually learned by example so we advise you create several models to go through with pupils, as well as challenge questions for independent work. Many examples exist and we encourage you to explore more (e.g. analysing and investigating, creating a tree diagram, and using simpler numbers).

Read these:

• How to develop maths reasoning skills in KS2 pupils
• FREE CPD PowerPoint: Reasoning Problem Solving & Planning for Depth
• KS3 Maths Problem Solving

That time, effort, and planning will – however – be well spent. Equipping pupils with the tools to solve problems they have never seen before is more akin to teaching for life than teaching for Maths. The skills they gain from being taught problem solving successfully will be skills they use and hone for the rest of their life – not just for their SATs.

For a range of problem solving techniques, complete with explanations, contextual uses, example problems and challenge questions – don’t forget to download our free  Ultimate Guide to KS2 problem solving and reasoning techniques  resource here.

KS2 problem Solving FAQs

Here are some techniques to teach problem solving to primary school pupils: Open ended problem solving Using logical reasoning Working backwards Drawing a diagram Drawing a table Creating an organised list Looking for a pattern Acting it out Guessing and checking

Ofsted say that teachers can encourage problem-solving by: Setting problems as part of learning in all topics for all pupils. Varying the ways in which you pose problems. Trying to resist prompting pupils too soon and focusing on getting ‘the answer’ – pupils need to build their confidence, skills and resilience in solving problems, so that they can apply them naturally in other situations. Making sure you discuss alternative approaches with pupils to help develop their reasoning. Ensuring that problems for high attainers involve demanding reasoning and problem-solving skills, not just harder numbers.

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Mathematics: Problem Solving and Investigations

A list of resources containing activities which could be integrated into your scheme of work with the purpose of integrating problem solving and investigational maths work into the natural process of learning mathematics and not just as a bolt-on added extra.

The first few resources are aimed at primary mathematics with the remainder aimed at Key Stages 3 & 4.

Making Molecules

Quality Assured Category: Mathematics Publisher: cre8ate maths

This activity asks students to investigate the structure of hydrocarbons.  A printed sheet representing carbon and hydrogen atoms is cut up and used by students to build as many isomers as they can for two chemical formulae.

Students are then challenged to discover a 3-D shape with 60 vertices, made from regular pentagons and hexagons. Teachers may choose to give 12 pentagons and 20 hexagons to groups of students for them to build the model or make up the shape from its net.

Problem Cards

Quality Assured Collection Category: Mathematics Publisher: Nuffield Foundation

This collection contains two packs of problem cards that have been designed for use by students in conjunction with the main work described in the Teachers' Guides. It is intended that the majority of students should at least be able to 'have a go' at most of the questions, but should also be encouraged, to the full extent of their individual abilities, to think around a problem and to devise alternatives and generalisations.

There are two packs of problem cards, purple and red, each accompanied by a teachers' guide.

Starting Investigations

Quality Assured Category: Mathematics Publisher: Collins Educational

A book of mathematical investigations aimed at students working at National Curriculum levels 1 to 3.

Contains forty simple investigations covering topics such as Odds and Evens, Place Value, Number patterns and many more.

Badger Maths problem solving

Quality Assured Category: Mathematics Publisher: Badger

A series of resources aimed at years one through to six aimed at developing problem solving skills. Each resource gives an example of a four step problem solving approach and task cards split into levels.

Problem solving with EYFS, Key Stage One and Key Stage Two children

Quality Assured Category: Computing Publisher: Department for Education

A set of resources from the National Strategies aimed at years one to six designed to help students become proficient problem solvers in mathematics.

Problem Solving Tasks

Quality Assured Collection Category: Mathematics Publisher:

The Spode Group have produced a number of resources to support problem solving in mathematics through real life problems.

The resources contain a wide range of open-ended tasks, practical tasks, investigations and real life problems still useful today to place the mathematics learnt in the classroom into a real world context.

Bowland Maths: Assessment Tasks

Quality Assured Collection Category: Mathematics Publisher: Bowland Charitable Trust

Bowland Maths includes over thirty tasks designed to help assess students’ achievements and progression against key processes. To help with this assessment, each task contains sample work, and a 'progression table' showing how students’ work on the task can provide evidence of their progress with the four key processes: representing, analysing, communicating and reflecting.

These materials are also ideal for formative assessment that concentrates on providing the types of rich feedback that have been proven to help students improve their reasoning.

Graded Assessment in Mathematics (GAIM)

Quality Assured Collection Category: Mathematics Publisher: Nelson Thornes

GAIM is a teacher assessment scheme for Key Stages Three and Four. The scheme is designed to encourage teaching and learning through practical problem solving and investigations,  involve students in all assessment and record keeping and introduce continuous assessment into normal classroom practice.

Investigative and Problem-Solving Approaches to Mathematics and Their Assessment

Quality Assured Category: Mathematics Publisher: Institute of Physics

This resource was written in response to the Cockroft Report to address the recommendation that all mathematics teaching should include opportunities for exposition by the teacher, discussion between teachers and students and between students themselves, practical work, consolidation and practice, problem solving and investigational work. The purpose of the pack was to address these issues and support their introduction and assessment.

Design a Board Game

Quality Assured Category: Mathematics Publisher: Shell Centre for Mathematical Education

In this Shell Centre module, groups carefully design and produce their own board games. These games are then played and evaluated by other class members.

Plan a Trip

In this Shell Centre module students plan and undertake a class trip using costings, scheduling, surveys and everyday arithmetic.

Produce a Quiz Show

In this Shell Centre modlue, Produce a Quiz Show, students devise, schedule, run and evaluate their own classroom quizzes.

Pure Investigations

These two books from the Shell centre focus on the pure investigations.

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• Department for Education

National curriculum in England: mathematics programmes of study

Updated 28 September 2021

Applies to England

© Crown copyright 2021

This publication is licensed under the terms of the Open Government Licence v3.0 except where otherwise stated. To view this licence, visit nationalarchives.gov.uk/doc/open-government-licence/version/3 or write to the Information Policy Team, The National Archives, Kew, London TW9 4DU, or email: [email protected] .

Where we have identified any third party copyright information you will need to obtain permission from the copyright holders concerned.

This publication is available at https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study

Purpose of study

Mathematics is a creative and highly interconnected discipline that has been developed over centuries, providing the solution to some of history’s most intriguing problems. It is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment. A high-quality mathematics education therefore provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject.

The national curriculum for mathematics aims to ensure that all pupils:

• become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately
• reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
• can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions

Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. The programmes of study are, by necessity, organised into apparently distinct domains, but pupils should make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They should also apply their mathematical knowledge to science and other subjects.

The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.

Information and communication technology (ICT)

Calculators should not be used as a substitute for good written and mental arithmetic. They should therefore only be introduced near the end of key stage 2 to support pupils’ conceptual understanding and exploration of more complex number problems, if written and mental arithmetic are secure. In both primary and secondary schools, teachers should use their judgement about when ICT tools should be used.

Spoken language

The national curriculum for mathematics reflects the importance of spoken language in pupils’ development across the whole curriculum – cognitively, socially and linguistically. The quality and variety of language that pupils hear and speak are key factors in developing their mathematical vocabulary and presenting a mathematical justification, argument or proof. They must be assisted in making their thinking clear to themselves as well as others, and teachers should ensure that pupils build secure foundations by using discussion to probe and remedy their misconceptions.

School curriculum

The programmes of study for mathematics are set out year-by-year for key stages 1 and 2. Schools are, however, only required to teach the relevant programme of study by the end of the key stage. Within each key stage, schools therefore have the flexibility to introduce content earlier or later than set out in the programme of study. In addition, schools can introduce key stage content during an earlier key stage, if appropriate. All schools are also required to set out their school curriculum for mathematics on a year-by-year basis and make this information available online.

Attainment targets

By the end of each key stage, pupils are expected to know, apply and understand the matters, skills and processes specified in the relevant programme of study.

Schools are not required by law to teach the example content in [square brackets] or the content indicated as being ‘non-statutory’.

Key stage 1 - years 1 and 2

The principal focus of mathematics teaching in key stage 1 is to ensure that pupils develop confidence and mental fluency with whole numbers, counting and place value. This should involve working with numerals, words and the 4 operations, including with practical resources [for example, concrete objects and measuring tools].

At this stage, pupils should develop their ability to recognise, describe, draw, compare and sort different shapes and use the related vocabulary. Teaching should also involve using a range of measures to describe and compare different quantities such as length, mass, capacity/volume, time and money.

By the end of year 2, pupils should know the number bonds to 20 and be precise in using and understanding place value. An emphasis on practice at this early stage will aid fluency.

Pupils should read and spell mathematical vocabulary, at a level consistent with their increasing word reading and spelling knowledge at key stage 1.

Year 1 programme of study

Number - number and place value.

Pupils should be taught to:

• count to and across 100, forwards and backwards, beginning with 0 or 1, or from any given number
• count, read and write numbers to 100 in numerals; count in multiples of 2s, 5s and 10s
• given a number, identify 1 more and 1 less
• identify and represent numbers using objects and pictorial representations including the number line, and use the language of: equal to, more than, less than (fewer), most, least
• read and write numbers from 1 to 20 in numerals and words

Notes and guidance (non-statutory)

Pupils practise counting (1, 2, 3…), ordering (for example, first, second, third…), and to indicate a quantity (for example, 3 apples, 2 centimetres), including solving simple concrete problems, until they are fluent.

Pupils begin to recognise place value in numbers beyond 20 by reading, writing, counting and comparing numbers up to 100, supported by objects and pictorial representations.

They practise counting as reciting numbers and counting as enumerating objects, and counting in 2s, 5s and 10s from different multiples to develop their recognition of patterns in the number system (for example, odd and even numbers), including varied and frequent practice through increasingly complex questions.

They recognise and create repeating patterns with objects and with shapes.

Number - addition and subtraction

• read, write and interpret mathematical statements involving addition (+), subtraction (−) and equals (=) signs
• represent and use number bonds and related subtraction facts within 20
• add and subtract one-digit and two-digit numbers to 20, including 0
• solve one-step problems that involve addition and subtraction, using concrete objects and pictorial representations, and missing number problems such as 7 = ? − 9

Pupils memorise and reason with number bonds to 10 and 20 in several forms (for example, 9 + 7 = 16; 16 − 7 = 9; 7 = 16 − 9). They should realise the effect of adding or subtracting 0. This establishes addition and subtraction as related operations.

Pupils combine and increase numbers, counting forwards and backwards.

They discuss and solve problems in familiar practical contexts, including using quantities. Problems should include the terms: put together, add, altogether, total, take away, distance between, difference between, more than and less than, so that pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly.

Number - multiplication and division

• solve one-step problems involving multiplication and division, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher

Through grouping and sharing small quantities, pupils begin to understand:

• multiplication and division
• doubling numbers and quantities
• finding simple fractions of objects, numbers and quantities

They make connections between arrays, number patterns, and counting in 2s, 5s and 10s.

Number - fractions

• recognise, find and name a half as 1 of 2 equal parts of an object, shape or quantity
• recognise, find and name a quarter as 1 of 4 equal parts of an object, shape or quantity

Pupils are taught half and quarter as ‘fractions of’ discrete and continuous quantities by solving problems using shapes, objects and quantities. For example, they could recognise and find half a length, quantity, set of objects or shape. Pupils connect halves and quarters to the equal sharing and grouping of sets of objects and to measures, as well as recognising and combining halves and quarters as parts of a whole.

Measurement

• lengths and heights [for example, long/short, longer/shorter, tall/short, double/half]
• mass/weight [for example, heavy/light, heavier than, lighter than]
• capacity and volume [for example, full/empty, more than, less than, half, half full, quarter]
• time [for example, quicker, slower, earlier, later]
• lengths and heights
• mass/weight
• capacity and volume
• time (hours, minutes, seconds)
• recognise and know the value of different denominations of coins and notes
• sequence events in chronological order using language [for example, before and after, next, first, today, yesterday, tomorrow, morning, afternoon and evening]
• recognise and use language relating to dates, including days of the week, weeks, months and years
• tell the time to the hour and half past the hour and draw the hands on a clock face to show these times

The pairs of terms: mass and weight, volume and capacity, are used interchangeably at this stage.

Pupils move from using and comparing different types of quantities and measures using non-standard units, including discrete (for example, counting) and continuous (for example, liquid) measurement, to using manageable common standard units.

In order to become familiar with standard measures, pupils begin to use measuring tools such as a ruler, weighing scales and containers.

Pupils use the language of time, including telling the time throughout the day, first using o’clock and then half past.

Geometry - properties of shapes

• 2-D shapes [for example, rectangles (including squares), circles and triangles]
• 3-D shapes [for example, cuboids (including cubes), pyramids and spheres]

Pupils handle common 2-D and 3-D shapes, naming these and related everyday objects fluently. They recognise these shapes in different orientations and sizes, and know that rectangles, triangles, cuboids and pyramids are not always similar to each other.

Geometry - position and direction

• describe position, direction and movement, including whole, half, quarter and three-quarter turns

Pupils use the language of position, direction and motion, including: left and right, top, middle and bottom, on top of, in front of, above, between, around, near, close and far, up and down, forwards and backwards, inside and outside.

Pupils make whole, half, quarter and three-quarter turns in both directions and connect turning clockwise with movement on a clock face.

Year 2 programme of study

• count in steps of 2, 3, and 5 from 0, and in 10s from any number, forward and backward
• recognise the place value of each digit in a two-digit number (10s, 1s)
• identify, represent and estimate numbers using different representations, including the number line
• compare and order numbers from 0 up to 100; use <, > and = signs
• read and write numbers to at least 100 in numerals and in words
• use place value and number facts to solve problems

Using materials and a range of representations, pupils practise counting, reading, writing and comparing numbers to at least 100 and solving a variety of related problems to develop fluency. They count in multiples of 3 to support their later understanding of a third.

As they become more confident with numbers up to 100, pupils are introduced to larger numbers to develop further their recognition of patterns within the number system and represent them in different ways, including spatial representations.

Pupils should partition numbers in different ways (for example, 23 = 20 + 3 and 23 = 10 + 13) to support subtraction. They become fluent and apply their knowledge of numbers to reason with, discuss and solve problems that emphasise the value of each digit in two-digit numbers. They begin to understand 0 as a place holder.

• using concrete objects and pictorial representations, including those involving numbers, quantities and measures
• applying their increasing knowledge of mental and written methods
• recall and use addition and subtraction facts to 20 fluently, and derive and use related facts up to 100
• a two-digit number and 1s
• a two-digit number and 10s
• 2 two-digit numbers
• adding 3 one-digit numbers
• show that addition of 2 numbers can be done in any order (commutative) and subtraction of 1 number from another cannot
• recognise and use the inverse relationship between addition and subtraction and use this to check calculations and solve missing number problems

Pupils extend their understanding of the language of addition and subtraction to include sum and difference.

Pupils practise addition and subtraction to 20 to become increasingly fluent in deriving facts such as using 3 + 7 = 10; 10 − 7 = 3 and 7 = 10 − 3 to calculate 30 + 70 = 100; 100 − 70 = 30 and 70 = 100 − 30. They check their calculations, including by adding to check subtraction and adding numbers in a different order to check addition (for example, 5 + 2 + 1 = 1 + 5 + 2 = 1 + 2 + 5). This establishes commutativity and associativity of addition.

Recording addition and subtraction in columns supports place value and prepares for formal written methods with larger numbers.

• recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers
• calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication (×), division (÷) and equals (=) signs
• show that multiplication of 2 numbers can be done in any order (commutative) and division of 1 number by another cannot
• solve problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and division facts, including problems in contexts

Pupils use a variety of language to describe multiplication and division.

Pupils are introduced to the multiplication tables. They practise to become fluent in the 2, 5 and 10 multiplication tables and connect them to each other. They connect the 10 multiplication table to place value, and the 5 multiplication table to the divisions on the clock face. They begin to use other multiplication tables and recall multiplication facts, including using related division facts to perform written and mental calculations.

Pupils work with a range of materials and contexts in which multiplication and division relate to grouping and sharing discrete and continuous quantities, to arrays and to repeated addition. They begin to relate these to fractions and measures (for example, 40 ÷ 2 = 20, 20 is a half of 40). They use commutativity and inverse relations to develop multiplicative reasoning (for example, 4 × 5 = 20 and 20 ÷ 5 = 4).

• choose and use appropriate standard units to estimate and measure length/height in any direction (m/cm); mass (kg/g); temperature (°C); capacity (litres/ml) to the nearest appropriate unit, using rulers, scales, thermometers and measuring vessels
• compare and order lengths, mass, volume/capacity and record the results using >, < and =
• recognise and use symbols for pounds (£) and pence (p); combine amounts to make a particular value
• find different combinations of coins that equal the same amounts of money
• solve simple problems in a practical context involving addition and subtraction of money of the same unit, including giving change
• compare and sequence intervals of time
• tell and write the time to five minutes, including quarter past/to the hour and draw the hands on a clock face to show these times
• know the number of minutes in an hour and the number of hours in a day

Pupils use standard units of measurement with increasing accuracy, using their knowledge of the number system. They use the appropriate language and record using standard abbreviations.

Comparing measures includes simple multiples such as ‘half as high’; ‘twice as wide’.

Pupils become fluent in telling the time on analogue clocks and recording it. They become fluent in counting and recognising coins. They read and say amounts of money confidently and use the symbols £ and p accurately, recording pounds and pence separately.

• identify and describe the properties of 2-D shapes, including the number of sides, and line symmetry in a vertical line
• identify and describe the properties of 3-D shapes, including the number of edges, vertices and faces
• identify 2-D shapes on the surface of 3-D shapes, [for example, a circle on a cylinder and a triangle on a pyramid]
• compare and sort common 2-D and 3-D shapes and everyday objects

Pupils handle and name a wide variety of common 2-D and 3-D shapes including: quadrilaterals and polygons and cuboids, prisms and cones, and identify the properties of each shape (for example, number of sides, number of faces). Pupils identify, compare and sort shapes on the basis of their properties and use vocabulary precisely, such as sides, edges, vertices and faces.

Pupils read and write names for shapes that are appropriate for their word reading and spelling. Pupils draw lines and shapes using a straight edge.

• order and arrange combinations of mathematical objects in patterns and sequences
• use mathematical vocabulary to describe position, direction and movement, including movement in a straight line and distinguishing between rotation as a turn and in terms of right angles for quarter, half and three-quarter turns (clockwise and anti-clockwise)

Pupils should work with patterns of shapes, including those in different orientations.

Pupils use the concept and language of angles to describe ‘turn’ by applying rotations, including in practical contexts (for example, pupils themselves moving in turns, giving instructions to other pupils to do so, and programming robots using instructions given in right angles).

• interpret and construct simple pictograms, tally charts, block diagrams and tables
• ask and answer simple questions by counting the number of objects in each category and sorting the categories by quantity
• ask-and-answer questions about totalling and comparing categorical data

Pupils record, interpret, collate, organise and compare information (for example, using many-to-one correspondence in pictograms with simple ratios 2, 5,10).

Lower key stage 2 - years 3 and 4

The principal focus of mathematics teaching in lower key stage 2 is to ensure that pupils become increasingly fluent with whole numbers and the 4 operations, including number facts and the concept of place value. This should ensure that pupils develop efficient written and mental methods and perform calculations accurately with increasingly large whole numbers.

At this stage, pupils should develop their ability to solve a range of problems, including with simple fractions and decimal place value. Teaching should also ensure that pupils draw with increasing accuracy and develop mathematical reasoning so they can analyse shapes and their properties, and confidently describe the relationships between them. It should ensure that they can use measuring instruments with accuracy and make connections between measure and number.

By the end of year 4, pupils should have memorised their multiplication tables up to and including the 12 multiplication table and show precision and fluency in their work.

Pupils should read and spell mathematical vocabulary correctly and confidently, using their growing word-reading knowledge and their knowledge of spelling.

Year 3 programme of study

• count from 0 in multiples of 4, 8, 50 and 100; find 10 or 100 more or less than a given number
• recognise the place value of each digit in a 3-digit number (100s, 10s, 1s)
• compare and order numbers up to 1,000
• identify, represent and estimate numbers using different representations
• read and write numbers up to 1,000 in numerals and in words
• solve number problems and practical problems involving these ideas

Pupils now use multiples of 2, 3, 4, 5, 8, 10, 50 and 100.

They use larger numbers to at least 1,000, applying partitioning related to place value using varied and increasingly complex problems, building on work in year 2 (for example, 146 = 100 + 40 + 6, 146 = 130 +16).

Using a variety of representations, including those related to measure, pupils continue to count in 1s, 10s and 100s, so that they become fluent in the order and place value of numbers to 1,000.

• a three-digit number and 1s
• a three-digit number and 10s
• a three-digit number and 100s
• add and subtract numbers with up to 3 digits, using formal written methods of columnar addition and subtraction
• estimate the answer to a calculation and use inverse operations to check answers
• solve problems, including missing number problems, using number facts, place value, and more complex addition and subtraction

Pupils practise solving varied addition and subtraction questions. For mental calculations with two-digit numbers, the answers could exceed 100.

Pupils use their understanding of place value and partitioning, and practise using columnar addition and subtraction with increasingly large numbers up to 3 digits to become fluent (see mathematics appendix 1 (PDF, 248KB) ).

• recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables
• write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental and progressing to formal written methods
• solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems and correspondence problems in which n objects are connected to m objects

Pupils continue to practise their mental recall of multiplication tables when they are calculating mathematical statements in order to improve fluency. Through doubling, they connect the 2, 4 and 8 multiplication tables.

Pupils develop efficient mental methods, for example, using commutativity and associativity (for example, 4 × 12 × 5 = 4 × 5 × 12 = 20 × 12 = 240) and multiplication and division facts (for example, using 3 × 2 = 6, 6 ÷ 3 = 2 and 2 = 6 ÷ 3) to derive related facts (30 × 2 = 60, 60 ÷ 3 = 20 and 20 = 60 ÷ 3).

Pupils develop reliable written methods for multiplication and division, starting with calculations of two-digit numbers by one-digit numbers and progressing to the formal written methods of short multiplication and division.

Pupils solve simple problems in contexts, deciding which of the 4 operations to use and why. These include measuring and scaling contexts, (for example 4 times as high, 8 times as long etc) and correspondence problems in which m objects are connected to n objects (for example, 3 hats and 4 coats, how many different outfits?, 12 sweets shared equally between 4 children, 4 cakes shared equally between 8 children).

• count up and down in tenths; recognise that tenths arise from dividing an object into 10 equal parts and in dividing one-digit numbers or quantities by 10
• recognise, find and write fractions of a discrete set of objects: unit fractions and non-unit fractions with small denominators
• recognise and use fractions as numbers: unit fractions and non-unit fractions with small denominators
• recognise and show, using diagrams, equivalent fractions with small denominators

• compare and order unit fractions, and fractions with the same denominators
• solve problems that involve all of the above

Pupils connect tenths to place value, decimal measures and to division by 10.

They begin to understand unit and non-unit fractions as numbers on the number line, and deduce relations between them, such as size and equivalence. They should go beyond the [0, 1] interval, including relating this to measure.

Pupils understand the relation between unit fractions as operators (fractions of), and division by integers.

They continue to recognise fractions in the context of parts of a whole, numbers, measurements, a shape, and unit fractions as a division of a quantity.

Pupils practise adding and subtracting fractions with the same denominator through a variety of increasingly complex problems to improve fluency.

• measure, compare, add and subtract: lengths (m/cm/mm); mass (kg/g); volume/capacity (l/ml)
• measure the perimeter of simple 2-D shapes
• add and subtract amounts of money to give change, using both £ and p in practical contexts
• tell and write the time from an analogue clock, including using Roman numerals from I to XII, and 12-hour and 24-hour clocks
• estimate and read time with increasing accuracy to the nearest minute; record and compare time in terms of seconds, minutes and hours; use vocabulary such as o’clock, am/pm, morning, afternoon, noon and midnight
• know the number of seconds in a minute and the number of days in each month, year and leap year
• compare durations of events [for example, to calculate the time taken by particular events or tasks]

Pupils continue to measure using the appropriate tools and units, progressing to using a wider range of measures, including comparing and using mixed units (for example, 1 kg and 200g) and simple equivalents of mixed units (for example, 5m = 500cm).

The comparison of measures includes simple scaling by integers (for example, a given quantity or measure is twice as long or 5 times as high) and this connects to multiplication.

Pupils continue to become fluent in recognising the value of coins, by adding and subtracting amounts, including mixed units, and giving change using manageable amounts. They record £ and p separately. The decimal recording of money is introduced formally in year 4.

Pupils use both analogue and digital 12-hour clocks and record their times. In this way they become fluent in and prepared for using digital 24-hour clocks in year 4.

• draw 2-D shapes and make 3-D shapes using modelling materials; recognise 3-D shapes in different orientations and describe them
• recognise angles as a property of shape or a description of a turn
• identify right angles, recognise that 2 right angles make a half-turn, 3 make three-quarters of a turn and 4 a complete turn; identify whether angles are greater than or less than a right angle
• identify horizontal and vertical lines and pairs of perpendicular and parallel lines

Pupils’ knowledge of the properties of shapes is extended at this stage to symmetrical and non-symmetrical polygons and polyhedra. Pupils extend their use of the properties of shapes. They should be able to describe the properties of 2-D and 3-D shapes using accurate language, including lengths of lines and acute and obtuse for angles greater or lesser than a right angle.

Pupils connect decimals and rounding to drawing and measuring straight lines in centimetres, in a variety of contexts.

• interpret and present data using bar charts, pictograms and tables
• solve one-step and two-step questions [for example ‘How many more?’ and ‘How many fewer?’] using information presented in scaled bar charts and pictograms and tables

Pupils understand and use simple scales (for example, 2, 5, 10 units per cm) in pictograms and bar charts with increasing accuracy.

They continue to interpret data presented in many contexts.

Year 4 programme of study

• count in multiples of 6, 7, 9, 25 and 1,000
• find 1,000 more or less than a given number
• count backwards through 0 to include negative numbers
• recognise the place value of each digit in a four-digit number (1,000s, 100s, 10s, and 1s)
• order and compare numbers beyond 1,000
• round any number to the nearest 10, 100 or 1,000
• solve number and practical problems that involve all of the above and with increasingly large positive numbers
• read Roman numerals to 100 (I to C) and know that over time, the numeral system changed to include the concept of 0 and place value

Using a variety of representations, including measures, pupils become fluent in the order and place value of numbers beyond 1,000, including counting in 10s and 100s, and maintaining fluency in other multiples through varied and frequent practice. They begin to extend their knowledge of the number system to include the decimal numbers and fractions that they have met so far.

They connect estimation and rounding numbers to the use of measuring instruments.

Roman numerals should be put in their historical context so pupils understand that there have been different ways to write whole numbers and that the important concepts of 0 and place value were introduced over a period of time.

• add and subtract numbers with up to 4 digits using the formal written methods of columnar addition and subtraction where appropriate
• estimate and use inverse operations to check answers to a calculation
• solve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and why

Pupils continue to practise both mental methods and columnar addition and subtraction with increasingly large numbers to aid fluency (see mathematics appendix 1 (PDF, 248KB) ).

• recall multiplication and division facts for multiplication tables up to 12 × 12
• use place value, known and derived facts to multiply and divide mentally, including: multiplying by 0 and 1; dividing by 1; multiplying together 3 numbers
• recognise and use factor pairs and commutativity in mental calculations
• multiply two-digit and three-digit numbers by a one-digit number using formal written layout
• solve problems involving multiplying and adding, including using the distributive law to multiply two-digit numbers by 1 digit, integer scaling problems and harder correspondence problems such as n objects are connected to m objects

Pupils continue to practise recalling and using multiplication tables and related division facts to aid fluency.

Pupils practise mental methods and extend this to 3-digit numbers to derive facts, (for example 600 ÷ 3 = 200 can be derived from 2 x 3 = 6).

Pupils practise to become fluent in the formal written method of short multiplication and short division with exact answers (see mathematics appendix 1 (PDF, 248KB) ).

Pupils write statements about the equality of expressions (for example, use the distributive law 39 × 7 = 30 × 7 + 9 × 7 and associative law (2 × 3) × 4 = 2 × (3 × 4)). They combine their knowledge of number facts and rules of arithmetic to solve mental and written calculations for example, 2 x 6 x 5 = 10 x 6 = 60.

Pupils solve two-step problems in contexts, choosing the appropriate operation, working with increasingly harder numbers. This should include correspondence questions such as the numbers of choices of a meal on a menu, or 3 cakes shared equally between 10 children.

Number - fractions (including decimals)

• recognise and show, using diagrams, families of common equivalent fractions
• count up and down in hundredths; recognise that hundredths arise when dividing an object by 100 and dividing tenths by 10
• solve problems involving increasingly harder fractions to calculate quantities, and fractions to divide quantities, including non-unit fractions where the answer is a whole number
• add and subtract fractions with the same denominator
• recognise and write decimal equivalents of any number of tenths or hundreds
• find the effect of dividing a one- or two-digit number by 10 and 100, identifying the value of the digits in the answer as ones, tenths and hundredths
• round decimals with 1 decimal place to the nearest whole number
• compare numbers with the same number of decimal places up to 2 decimal places
• solve simple measure and money problems involving fractions and decimals to 2 decimal places

Pupils should connect hundredths to tenths and place value and decimal measure.

They extend the use of the number line to connect fractions, numbers and measures.

Pupils continue to practise adding and subtracting fractions with the same denominator, to become fluent through a variety of increasingly complex problems beyond one whole.

Pupils are taught throughout that decimals and fractions are different ways of expressing numbers and proportions.

Pupils’ understanding of the number system and decimal place value is extended at this stage to tenths and then hundredths. This includes relating the decimal notation to division of whole number by 10 and later 100.

They practise counting using simple fractions and decimals, both forwards and backwards.

Pupils learn decimal notation and the language associated with it, including in the context of measurements. They make comparisons and order decimal amounts and quantities that are expressed to the same number of decimal places. They should be able to represent numbers with 1 or 2 decimal places in several ways, such as on number lines.

• convert between different units of measure [for example, kilometre to metre; hour to minute]
• measure and calculate the perimeter of a rectilinear figure (including squares) in centimetres and metres
• find the area of rectilinear shapes by counting squares
• estimate, compare and calculate different measures, including money in pounds and pence
• read, write and convert time between analogue and digital 12- and 24-hour clocks
• solve problems involving converting from hours to minutes, minutes to seconds, years to months, weeks to days

Pupils build on their understanding of place value and decimal notation to record metric measures, including money.

They use multiplication to convert from larger to smaller units.

Perimeter can be expressed algebraically as 2(a + b) where a and b are the dimensions in the same unit.

They relate area to arrays and multiplication.

• compare and classify geometric shapes, including quadrilaterals and triangles, based on their properties and sizes
• identify acute and obtuse angles and compare and order angles up to 2 right angles by size
• identify lines of symmetry in 2-D shapes presented in different orientations
• complete a simple symmetric figure with respect to a specific line of symmetry

Pupils continue to classify shapes using geometrical properties, extending to classifying different triangles (for example, isosceles, equilateral, scalene) and quadrilaterals (for example, parallelogram, rhombus, trapezium).

Pupils compare and order angles in preparation for using a protractor and compare lengths and angles to decide if a polygon is regular or irregular.

Pupils draw symmetric patterns using a variety of media to become familiar with different orientations of lines of symmetry; and recognise line symmetry in a variety of diagrams, including where the line of symmetry does not dissect the original shape.

• describe positions on a 2-D grid as coordinates in the first quadrant
• describe movements between positions as translations of a given unit to the left/right and up/down
• plot specified points and draw sides to complete a given polygon

Pupils draw a pair of axes in one quadrant, with equal scales and integer labels. They read, write and use pairs of co-ordinates, for example (2, 5), including using co-ordinate-plotting ICT tools.

• interpret and present discrete and continuous data using appropriate graphical methods, including bar charts and time graphs
• solve comparison, sum and difference problems using information presented in bar charts, pictograms, tables and other graphs

Pupils understand and use a greater range of scales in their representations.

Pupils begin to relate the graphical representation of data to recording change over time.

Upper key stage 2 - years 5 and 6

The principal focus of mathematics teaching in upper key stage 2 is to ensure that pupils extend their understanding of the number system and place value to include larger integers. This should develop the connections that pupils make between multiplication and division with fractions, decimals, percentages and ratio.

At this stage, pupils should develop their ability to solve a wider range of problems, including increasingly complex properties of numbers and arithmetic, and problems demanding efficient written and mental methods of calculation. With this foundation in arithmetic, pupils are introduced to the language of algebra as a means for solving a variety of problems. Teaching in geometry and measures should consolidate and extend knowledge developed in number. Teaching should also ensure that pupils classify shapes with increasingly complex geometric properties and that they learn the vocabulary they need to describe them.

By the end of year 6, pupils should be fluent in written methods for all 4 operations, including long multiplication and division, and in working with fractions, decimals and percentages.

Pupils should read, spell and pronounce mathematical vocabulary correctly.

Year 5 programme of study

• read, write, order and compare numbers to at least 1,000,000 and determine the value of each digit
• count forwards or backwards in steps of powers of 10 for any given number up to 1,000,000
• interpret negative numbers in context, count forwards and backwards with positive and negative whole numbers, including through 0
• round any number up to 1,000,000 to the nearest 10, 100, 1,000, 10,000 and 100,000
• solve number problems and practical problems that involve all of the above
• read Roman numerals to 1,000 (M) and recognise years written in Roman numerals

Pupils identify the place value in large whole numbers.

They continue to use number in context, including measurement. Pupils extend and apply their understanding of the number system to the decimal numbers and fractions that they have met so far.

• add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and subtraction)
• add and subtract numbers mentally with increasingly large numbers
• use rounding to check answers to calculations and determine, in the context of a problem, levels of accuracy
• solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why

Pupils practise using the formal written methods of columnar addition and subtraction with increasingly large numbers to aid fluency (see mathematics appendix 1 (PDF, 248KB) ).

They practise mental calculations with increasingly large numbers to aid fluency (for example, 12,462 – 2,300 = 10,162).

• identify multiples and factors, including finding all factor pairs of a number, and common factors of 2 numbers
• know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers
• establish whether a number up to 100 is prime and recall prime numbers up to 19
• multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers
• multiply and divide numbers mentally, drawing upon known facts
• divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context
• multiply and divide whole numbers and those involving decimals by 10, 100 and 1,000
• recognise and use square numbers and cube numbers, and the notation for squared (²) and cubed (³)
• solve problems involving multiplication and division, including using their knowledge of factors and multiples, squares and cubes
• solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign
• solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates

Pupils practise and extend their use of the formal written methods of short multiplication and short division (see mathematics appendix 1 (PDF, 248KB) ). They apply all the multiplication tables and related division facts frequently, commit them to memory and use them confidently to make larger calculations.

They use and understand the terms factor, multiple and prime, square and cube numbers.

Pupils use multiplication and division as inverses to support the introduction of ratio in year 6, for example, by multiplying and dividing by powers of 10 in scale drawings or by multiplying and dividing by powers of a 1,000 in converting between units such as kilometres and metres.

Distributivity can be expressed as a(b + c) = ab + ac.

They understand the terms factor, multiple and prime, square and cube numbers and use them to construct equivalence statements (for example, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 9² x 10).

Pupils use and explain the equals sign to indicate equivalence, including in missing number problems (for example 13 + 24 = 12 + 25; 33 = 5 x ?).

Number - fractions (including decimals and percentages)

• compare and order fractions whose denominators are all multiples of the same number
• identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths

• add and subtract fractions with the same denominator, and denominators that are multiples of the same number
• multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams

• recognise and use thousandths and relate them to tenths, hundredths and decimal equivalents
• round decimals with 2 decimal places to the nearest whole number and to 1 decimal place
• read, write, order and compare numbers with up to 3 decimal places
• solve problems involving number up to 3 decimal places
• recognise the per cent symbol (%) and understand that per cent relates to ‘number of parts per 100’, and write percentages as a fraction with denominator 100, and as a decimal fraction

Pupils should be taught throughout that percentages, decimals and fractions are different ways of expressing proportions.

They extend their knowledge of fractions to thousandths and connect to decimals and measures.

Pupils connect equivalent fractions > 1 that simplify to integers with division and other fractions > 1 to division with remainders, using the number line and other models, and hence move from these to improper and mixed fractions.

Pupils connect multiplication by a fraction to using fractions as operators (fractions of), and to division, building on work from previous years. This relates to scaling by simple fractions, including fractions > 1.

Pupils practise adding and subtracting fractions to become fluent through a variety of increasingly complex problems. They extend their understanding of adding and subtracting fractions to calculations that exceed 1 as a mixed number.

Pupils continue to practise counting forwards and backwards in simple fractions.

Pupils continue to develop their understanding of fractions as numbers, measures and operators by finding fractions of numbers and quantities.

Pupils extend counting from year 4, using decimals and fractions including bridging 0, for example on a number line.

Pupils say, read and write decimal fractions and related tenths, hundredths and thousandths accurately and are confident in checking the reasonableness of their answers to problems.

They mentally add and subtract tenths, and one-digit whole numbers and tenths.

They practise adding and subtracting decimals, including a mix of whole numbers and decimals, decimals with different numbers of decimal places, and complements of 1 (for example, 0.83 + 0.17 = 1).

Pupils should go beyond the measurement and money models of decimals, for example, by solving puzzles involving decimals.

• convert between different units of metric measure [for example, kilometre and metre; centimetre and metre; centimetre and millimetre; gram and kilogram; litre and millilitre]
• understand and use approximate equivalences between metric units and common imperial units such as inches, pounds and pints
• measure and calculate the perimeter of composite rectilinear shapes in centimetres and metres
• calculate and compare the area of rectangles (including squares), including using standard units, square centimetres (cm²) and square metres (m²), and estimate the area of irregular shapes
• estimate volume [for example, using 1 cm³ blocks to build cuboids (including cubes)] and capacity [for example, using water]
• solve problems involving converting between units of time
• use all four operations to solve problems involving measure [for example, length, mass, volume, money] using decimal notation, including scaling

Pupils use their knowledge of place value and multiplication and division to convert between standard units.

Pupils calculate the perimeter of rectangles and related composite shapes, including using the relations of perimeter or area to find unknown lengths. Missing measures questions such as these can be expressed algebraically, for example 4 + 2b = 20 for a rectangle of sides 2 cm and b cm and perimeter of 20cm.

Pupils calculate the area from scale drawings using given measurements.

Pupils use all 4 operations in problems involving time and money, including conversions (for example, days to weeks, expressing the answer as weeks and days).

• identify 3-D shapes, including cubes and other cuboids, from 2-D representations
• know angles are measured in degrees: estimate and compare acute, obtuse and reflex angles
• draw given angles, and measure them in degrees (°)
• angles at a point and 1 whole turn (total 360°)
• angles at a point on a straight line and half a turn (total 180°)
• other multiples of 90°
• use the properties of rectangles to deduce related facts and find missing lengths and angles
• distinguish between regular and irregular polygons based on reasoning about equal sides and angles

Pupils become accurate in drawing lines with a ruler to the nearest millimetre, and measuring with a protractor. They use conventional markings for parallel lines and right angles.

Pupils use the term diagonal and make conjectures about the angles formed between sides, and between diagonals and parallel sides, and other properties of quadrilaterals, for example using dynamic geometry ICT tools.

Pupils use angle sum facts and other properties to make deductions about missing angles and relate these to missing number problems.

• identify, describe and represent the position of a shape following a reflection or translation, using the appropriate language, and know that the shape has not changed

Pupils recognise and use reflection and translation in a variety of diagrams, including continuing to use a 2-D grid and coordinates in the first quadrant. Reflection should be in lines that are parallel to the axes.

• solve comparison, sum and difference problems using information presented in a line graph
• complete, read and interpret information in tables, including timetables

Pupils connect their work on coordinates and scales to their interpretation of time graphs.

They begin to decide which representations of data are most appropriate and why.

Year 6 programme of study

• read, write, order and compare numbers up to 10,000,000 and determine the value of each digit
• round any whole number to a required degree of accuracy
• use negative numbers in context, and calculate intervals across 0
• solve number and practical problems that involve all of the above

Pupils use the whole number system, including saying, reading and writing numbers accurately.

Number - addition, subtraction, multiplication and division

• multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication
• divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context
• divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context
• perform mental calculations, including with mixed operations and large numbers
• identify common factors, common multiples and prime numbers
• use their knowledge of the order of operations to carry out calculations involving the 4 operations
• solve problems involving addition, subtraction, multiplication and division
• use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy

Pupils practise addition, subtraction, multiplication and division for larger numbers, using the formal written methods of columnar addition and subtraction, short and long multiplication, and short and long division (see mathematics appendix 1 (PDF, 248KB) ).

They undertake mental calculations with increasingly large numbers and more complex calculations.

Pupils continue to use all the multiplication tables to calculate mathematical statements in order to maintain their fluency.

Pupils round answers to a specified degree of accuracy, for example, to the nearest 10, 20, 50, etc, but not to a specified number of significant figures.

Pupils explore the order of operations using brackets; for example, 2 + 1 x 3 = 5 and (2 + 1) x 3 = 9.

Common factors can be related to finding equivalent fractions.

Number - Fractions (including decimals and percentages)

• use common factors to simplify fractions; use common multiples to express fractions in the same denomination
• compare and order fractions, including fractions >1
• add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions

• identify the value of each digit in numbers given to 3 decimal places and multiply and divide numbers by 10, 100 and 1,000 giving answers up to 3 decimal places
• multiply one-digit numbers with up to 2 decimal places by whole numbers
• use written division methods in cases where the answer has up to 2 decimal places
• solve problems which require answers to be rounded to specified degrees of accuracy
• recall and use equivalences between simple fractions, decimals and percentages, including in different contexts

Pupils should use a variety of images to support their understanding of multiplication with fractions. This follows earlier work about fractions as operators (fractions of), as numbers, and as equal parts of objects, for example as parts of a rectangle. Pupils use their understanding of the relationship between unit fractions and division to work backwards by multiplying a quantity that represents a unit fraction to find the whole quantity (for example, if quarter of a length is 36cm, then the whole length is 36 × 4 = 144cm).

They practise calculations with simple fractions and decimal fraction equivalents to aid fluency, including listing equivalent fractions to identify fractions with common denominators.

Pupils can explore and make conjectures about converting a simple fraction to a decimal fraction (for example, 3 ÷ 8 = 0.375). For simple fractions with recurring decimal equivalents, pupils learn about rounding the decimal to three decimal places, or other appropriate approximations depending on the context. Pupils multiply and divide numbers with up to 2 decimal places by one-digit and two-digit whole numbers. Pupils multiply decimals by whole numbers, starting with the simplest cases, such as 0.4 × 2 = 0.8, and in practical contexts, such as measures and money.

Pupils are introduced to the division of decimal numbers by one-digit whole numbers, initially, in practical contexts involving measures and money. They recognise division calculations as the inverse of multiplication.

Pupils also develop their skills of rounding and estimating as a means of predicting and checking the order of magnitude of their answers to decimal calculations. This includes rounding answers to a specified degree of accuracy and checking the reasonableness of their answers.

Ratio and proportion

• solve problems involving the relative sizes of 2 quantities where missing values can be found by using integer multiplication and division facts
• solve problems involving the calculation of percentages [for example, of measures and such as 15% of 360] and the use of percentages for comparison
• solve problems involving similar shapes where the scale factor is known or can be found
• solve problems involving unequal sharing and grouping using knowledge of fractions and multiples

Pupils recognise proportionality in contexts when the relations between quantities are in the same ratio (for example, similar shapes and recipes).

Pupils link percentages or 360° to calculating angles of pie charts.

Pupils should consolidate their understanding of ratio when comparing quantities, sizes and scale drawings by solving a variety of problems. They might use the notation a:b to record their work.

• use simple formulae
• generate and describe linear number sequences
• express missing number problems algebraically
• find pairs of numbers that satisfy an equation with 2 unknowns
• enumerate possibilities of combinations of 2 variables

Pupils should be introduced to the use of symbols and letters to represent variables and unknowns in mathematical situations that they already understand, such as:

• missing numbers, lengths, coordinates and angles
• formulae in mathematics and science
• equivalent expressions (for example, a + b = b + a)
• generalisations of number patterns
• number puzzles (for example, what 2 numbers can add up to)
• solve problems involving the calculation and conversion of units of measure, using decimal notation up to 3 decimal places where appropriate
• use, read, write and convert between standard units, converting measurements of length, mass, volume and time from a smaller unit of measure to a larger unit, and vice versa, using decimal notation to up to 3 decimal places
• convert between miles and kilometres
• recognise that shapes with the same areas can have different perimeters and vice versa
• recognise when it is possible to use formulae for area and volume of shapes
• calculate the area of parallelograms and triangles
• calculate, estimate and compare volume of cubes and cuboids using standard units, including cubic centimetres (cm³) and cubic metres (m³), and extending to other units [for example, mm³ and km³]

Pupils connect conversion (for example, from kilometres to miles) to a graphical representation as preparation for understanding linear/proportional graphs.

They know approximate conversions and are able to tell if an answer is sensible.

Using the number line, pupils use, add and subtract positive and negative integers for measures such as temperature.

They relate the area of rectangles to parallelograms and triangles, for example, by dissection, and calculate their areas, understanding and using the formulae (in words or symbols) to do this.

Pupils could be introduced to compound units for speed, such as miles per hour, and apply their knowledge in science or other subjects as appropriate.

• draw 2-D shapes using given dimensions and angles
• recognise, describe and build simple 3-D shapes, including making nets
• compare and classify geometric shapes based on their properties and sizes and find unknown angles in any triangles, quadrilaterals, and regular polygons
• illustrate and name parts of circles, including radius, diameter and circumference and know that the diameter is twice the radius
• recognise angles where they meet at a point, are on a straight line, or are vertically opposite, and find missing angles

Pupils draw shapes and nets accurately, using measuring tools and conventional markings and labels for lines and angles.

Pupils describe the properties of shapes and explain how unknown angles and lengths can be derived from known measurements.

These relationships might be expressed algebraically for example, d = 2 × r; a = 180 − (b + c).

• describe positions on the full coordinate grid (all 4 quadrants)
• draw and translate simple shapes on the coordinate plane, and reflect them in the axes

Pupils draw and label a pair of axes in all 4 quadrants with equal scaling. This extends their knowledge of one quadrant to all 4 quadrants, including the use of negative numbers.

Pupils draw and label rectangles (including squares), parallelograms and rhombuses, specified by coordinates in the four quadrants, predicting missing coordinates using the properties of shapes. These might be expressed algebraically for example, translating vertex (a, b) to (a − 2, b + 3); (a, b) and (a + d, b + d) being opposite vertices of a square of side d.

• interpret and construct pie charts and line graphs and use these to solve problems
• calculate and interpret the mean as an average

Pupils connect their work on angles, fractions and percentages to the interpretation of pie charts.

Pupils both encounter and draw graphs relating 2 variables, arising from their own enquiry and in other subjects.

They should connect conversion from kilometres to miles in measurement to its graphical representation.

Pupils know when it is appropriate to find the mean of a data set.

Key stage 3

Introduction.

Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. The programme of study for key stage 3 is organised into apparently distinct domains, but pupils should build on key stage 2 and connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They should also apply their mathematical knowledge in science, geography, computing and other subjects.

Decisions about progression should be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content in preparation for key stage 4. Those who are not sufficiently fluent should consolidate their understanding, including through additional practice, before moving on.

Working mathematically

Through the mathematics content, pupils should be taught to:

Develop fluency

• consolidate their numerical and mathematical capability from key stage 2 and extend their understanding of the number system and place value to include decimals, fractions, powers and roots
• select and use appropriate calculation strategies to solve increasingly complex problems
• use algebra to generalise the structure of arithmetic, including to formulate mathematical relationships
• substitute values in expressions, rearrange and simplify expressions, and solve equations
• move freely between different numerical, algebraic, graphical and diagrammatic representations [for example, equivalent fractions, fractions and decimals, and equations and graphs]
• develop algebraic and graphical fluency, including understanding linear and simple quadratic functions
• use language and properties precisely to analyse numbers, algebraic expressions, 2-D and 3-D shapes, probability and statistics

Reason mathematically

• extend their understanding of the number system; make connections between number relationships, and their algebraic and graphical representations
• extend and formalise their knowledge of ratio and proportion in working with measures and geometry, and in formulating proportional relations algebraically
• identify variables and express relations between variables algebraically and graphically
• make and test conjectures about patterns and relationships; look for proofs or counter-examples
• begin to reason deductively in geometry, number and algebra, including using geometrical constructions
• interpret when the structure of a numerical problem requires additive, multiplicative or proportional reasoning
• explore what can and cannot be inferred in statistical and probabilistic settings, and begin to express their arguments formally

Solve problems

• develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems
• develop their use of formal mathematical knowledge to interpret and solve problems, including in financial mathematics
• begin to model situations mathematically and express the results using a range of formal mathematical representations
• select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems

Subject content

• understand and use place value for decimals, measures and integers of any size
• order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥
• use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property
• use the 4 operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative
• use conventional notation for the priority of operations, including brackets, powers, roots and reciprocals
• recognise and use relationships between operations including inverse operations
• use integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 and distinguish between exact representations of roots and their decimal approximations
• interpret and compare numbers in standard form A x 10 n 1≤A<10, where n is a positive or negative integer or 0

• define percentage as ‘number of parts per hundred’, interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express 1 quantity as a percentage of another, compare 2 quantities using percentages, and work with percentages greater than 100%
• interpret fractions and percentages as operators
• use standard units of mass, length, time, money and other measures, including with decimal quantities
• round numbers and measures to an appropriate degree of accuracy [for example, to a number of decimal places or significant figures]
• use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a<x≤b
• use a calculator and other technologies to calculate results accurately and then interpret them appropriately
• appreciate the infinite nature of the sets of integers, real and rational numbers
• ab in place of a × b
• 3y in place of y + y + y and 3 × y
• a² in place of a × a, a³ in place of a × a × a; a²b in place of a × a × b

• coefficients written as fractions rather than as decimals
• substitute numerical values into formulae and expressions, including scientific formulae
• understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors
• collecting like terms
• multiplying a single term over a bracket
• taking out common factors
• expanding products of 2 or more binomials
• understand and use standard mathematical formulae; rearrange formulae to change the subject
• model situations or procedures by translating them into algebraic expressions or formulae and by using graphs
• use algebraic methods to solve linear equations in 1 variable (including all forms that require rearrangement)
• work with coordinates in all 4 quadrants
• recognise, sketch and produce graphs of linear and quadratic functions of 1 variable with appropriate scaling, using equations in x and y and the Cartesian plane
• interpret mathematical relationships both algebraically and graphically
• reduce a given linear equation in 2 variables to the standard form y = mx + c; calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically and algebraically
• use linear and quadratic graphs to estimate values of y for given values of x and vice versa and to find approximate solutions of simultaneous linear equations
• find approximate solutions to contextual problems from given graphs of a variety of functions, including piece-wise linear, exponential and reciprocal graphs
• generate terms of a sequence from either a term-to-term or a position-to-term rule
• recognise arithmetic sequences and find the nth term
• recognise geometric sequences and appreciate other sequences that arise

Ratio, proportion and rates of change

• change freely between related standard units [for example time, length, area, volume/capacity, mass]
• use scale factors, scale diagrams and maps
• express 1 quantity as a fraction of another, where the fraction is less than 1 and greater than 1
• use ratio notation, including reduction to simplest form
• divide a given quantity into 2 parts in a given part:part or part:whole ratio; express the division of a quantity into 2 parts as a ratio
• understand that a multiplicative relationship between 2 quantities can be expressed as a ratio or a fraction
• relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions
• solve problems involving percentage change, including: percentage increase, decrease and original value problems and simple interest in financial mathematics
• solve problems involving direct and inverse proportion, including graphical and algebraic representations
• use compound units such as speed, unit pricing and density to solve problems

Geometry and measures

• derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms (including cylinders)
• calculate and solve problems involving: perimeters of 2-D shapes (including circles), areas of circles and composite shapes
• draw and measure line segments and angles in geometric figures, including interpreting scale drawings
• derive and use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle); recognise and use the perpendicular distance from a point to a line as the shortest distance to the line
• describe, sketch and draw using conventional terms and notations: points, lines, parallel lines, perpendicular lines, right angles, regular polygons, and other polygons that are reflectively and rotationally symmetric
• use the standard conventions for labelling the sides and angles of triangle ABC, and know and use the criteria for congruence of triangles
• derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures [for example, equal lengths and angles] using appropriate language and technologies
• identify properties of, and describe the results of, translations, rotations and reflections applied to given figures
• identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids
• apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles
• understand and use the relationship between parallel lines and alternate and corresponding angles
• derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons
• apply angle facts, triangle congruence, similarity and properties of quadrilaterals to derive results about angles and sides, including Pythagoras’ Theorem, and use known results to obtain simple proofs
• use Pythagoras’ Theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles
• use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3-D
• interpret mathematical relationships both algebraically and geometrically

Probability

• record, describe and analyse the frequency of outcomes of simple probability experiments involving randomness, fairness, equally and unequally likely outcomes, using appropriate language and the 0-1 probability scale
• understand that the probabilities of all possible outcomes sum to 1
• enumerate sets and unions/intersections of sets systematically, using tables, grids and Venn diagrams
• generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities
• describe, interpret and compare observed distributions of a single variable through: appropriate graphical representation involving discrete, continuous and grouped data; and appropriate measures of central tendency (mean, mode, median) and spread (range, consideration of outliers)
• construct and interpret appropriate tables, charts, and diagrams, including frequency tables, bar charts, pie charts, and pictograms for categorical data, and vertical line (or bar) charts for ungrouped and grouped numerical data
• describe simple mathematical relationships between 2 variables (bivariate data) in observational and experimental contexts and illustrate using scatter graphs

Key stage 4

This programme of study specifies:

• the mathematical content that should be taught to all pupils, in standard type
• additional mathematical content to be taught to more highly attaining pupils, in braces { }

Together, the mathematical content set out in the key stage 3 and key stage 4 programmes of study covers the full range of material contained in the GCSE Mathematics qualification. Wherever it is appropriate, given pupils’ security of understanding and readiness to progress, pupils should be taught the full content set out in this programme of study.

Through the mathematics content pupils should be taught to:

• consolidate their numerical and mathematical capability from key stage 3 and extend their understanding of the number system to include powers, roots {and fractional indices}
• select and use appropriate calculation strategies to solve increasingly complex problems, including exact calculations involving multiples of π {and surds}, use of standard form and application and interpretation of limits of accuracy
• consolidate their algebraic capability from key stage 3 and extend their understanding of algebraic simplification and manipulation to include quadratic expressions, {and expressions involving surds and algebraic fractions}
• extend fluency with expressions and equations from key stage 3, to include quadratic equations, simultaneous equations and inequalities
• move freely between different numerical, algebraic, graphical and diagrammatic representations, including of linear, quadratic, reciprocal, {exponential and trigonometric} functions
• use mathematical language and properties precisely
• extend and formalise their knowledge of ratio and proportion, including trigonometric ratios, in working with measures and geometry, and in working with proportional relations algebraically and graphically
• extend their ability to identify variables and express relations between variables algebraically and graphically
• make and test conjectures about the generalisations that underlie patterns and relationships; look for proofs or counter-examples; begin to use algebra to support and construct arguments {and proofs}
• reason deductively in geometry, number and algebra, including using geometrical constructions
• explore what can and cannot be inferred in statistical and probabilistic settings, and express their arguments formally
• assess the validity of an argument and the accuracy of a given way of presenting information
• develop their use of formal mathematical knowledge to interpret and solve problems, including in financial contexts
• make and use connections between different parts of mathematics to solve problems
• model situations mathematically and express the results using a range of formal mathematical representations, reflecting on how their solutions may have been affected by any modelling assumptions
• select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems; interpret their solution in the context of the given problem

In addition to consolidating subject content from key stage 3, pupils should be taught to:

• apply systematic listing strategies, {including use of the product rule for counting}
• {estimate powers and roots of any given positive number}
• calculate with roots, and with integer {and fractional} indices
• calculate exactly with fractions, {surds} and multiples of π {simplify surd expressions involving squares [for example √12 = √(4 × 3) = √4 × √3 = 2√3] and rationalise denominators}
• calculate with numbers in standard form A × 10n, where 1 ≤ A < 10 and n is an integer
• {change recurring decimals into their corresponding fractions and vice versa}
• identify and work with fractions in ratio problems
• apply and interpret limits of accuracy when rounding or truncating, {including upper and lower bounds}
• factorising quadratic expressions of the form x 2 + bx + c, including the difference of 2 squares; {factorising quadratic expressions of the form ax 2 + bx + c}
• simplifying expressions involving sums, products and powers, including the laws of indices
• know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments {and proofs}
• where appropriate, interpret simple expressions as functions with inputs and outputs; {interpret the reverse process as the ‘inverse function’; interpret the succession of 2 functions as a ‘composite function’}
• use the form y = mx + c to identify parallel {and perpendicular} lines; find the equation of the line through 2 given points, or through 1 point with a given gradient
• identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically {and turning points by completing the square}

• {sketch translations and reflections of the graph of a given function}
• plot and interpret graphs (including reciprocal graphs {and exponential graphs}) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration
• {calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts}
• {recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point}
• solve quadratic equations {including those that require rearrangement} algebraically by factorising, {by completing the square and by using the quadratic formula}; find approximate solutions using a graph
• solve 2 simultaneous equations in 2 variables (linear/linear {or linear/quadratic}) algebraically; find approximate solutions using a graph
• {find approximate solutions to equations numerically using iteration}
• translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or 2 simultaneous equations), solve the equation(s) and interpret the solution
• solve linear inequalities in 1 {or 2} variable {s}, {and quadratic inequalities in 1 variable}; represent the solution set on a number line, {using set notation and on a graph}
• recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions (r n where n is an integer, and r is a positive rational number {or a surd}) {and other sequences}
• deduce expressions to calculate the nth term of linear {and quadratic} sequences.
• compare lengths, areas and volumes using ratio notation and/or scale factors; make links to similarity (including trigonometric ratios)
• convert between related compound units (speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts

• interpret the gradient of a straight line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion
• {interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of instantaneous and average rate of change (gradients of tangents and chords) in numerical, algebraic and graphical contexts}
• set up, solve and interpret the answers in growth and decay problems, including compound interest {and work with general iterative processes}
• interpret and use fractional {and negative} scale factors for enlargements
• {describe the changes and invariance achieved by combinations of rotations, reflections and translations}
• identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment
• {apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results}
• construct and interpret plans and elevations of 3D shapes
• interpret and use bearings
• calculate arc lengths, angles and areas of sectors of circles
• calculate surface areas and volumes of spheres, pyramids, cones and composite solids
• apply the concepts of congruence and similarity, including the relationships between lengths, {areas and volumes} in similar figures
• apply Pythagoras’ Theorem and trigonometric ratios to find angles and lengths in right-angled triangles {and, where possible, general triangles} in 2 {and 3} dimensional figures
• know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°; know the exact value of tan θ for θ = 0°, 30°, 45°, 60°

• describe translations as 2D vectors
• apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors; {use vectors to construct geometric arguments and proofs}
• apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to 1
• use a probability model to predict the outcomes of future experiments; understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size
• calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions
• {calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams}
• infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling
• interpret and construct tables and line graphs for time series data
• {construct and interpret diagrams for grouped discrete data and continuous data, ie, histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use}
• appropriate graphical representation involving discrete, continuous and grouped data, {including box plots}
• appropriate measures of central tendency (including modal class) and spread {including quartiles and inter-quartile range}  
• apply statistics to describe a population
• use and interpret scatter graphs of bivariate data; recognise correlation and know that it does not indicate causation; draw estimated lines of best fit; make predictions; interpolate and extrapolate apparent trends whilst knowing the dangers of so doing.

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The UK still has a problem with maths but it’s not attitude

• Ravi Gurumurthy

About Nesta

Nesta is an innovation foundation. For us, innovation means turning bold ideas into reality and changing lives for the better. We use our expertise, skills and funding in areas where there are big challenges facing society.

11-16 year olds show passion for maths in a poll, raising new questions about the UK’s underwhelming performance in international league tables for numeracy

As the first GCSE students sit down to take their maths exam today, a survey commissioned by founding partners of the Maths Mission programme, Nesta , the innovation foundation, and Tata, a global enterprise, shows:

• 30 per cent say they love maths and work hard at it, 16% say they love maths and find it easy and 15 per cent say they’re a natural
• Encouragingly 92 per cent believe a person can improve at maths
• 67 per cent say their maths ability is good or very good - and just 7 per cent say their maths ability is poor or very poor

Notably, boys have more confidence in their maths ability:

• One in five boys say they love maths and find it easy, compared to just over one in 10 girls
• Nearly twice as many boys as girls claim they are ‘a natural’ at maths (19 per cent vs. 10 per cent)
• Girls were much more likely to admit to trying and failing at maths, with 22 per cent of girls saying they ‘try at maths but just can’t do it’, compared with just 14 per cent of boys

Overall this confidence is reassuring but contrasts sharply with some indicators of maths attainment: UK teenagers are ranked just 27th in the OECD's most recent global tests when it comes to their maths skills, a fall from 26th place in 2012.

The jobs market and demand for skills is changing in line with shifting economies and new technologies. 54 per cent of young people say that maths is used in most jobs - from business to building and 40 per cent recognise that the understanding of maths is just as important in the creative industries as it is in maths, science and technology. 39 per cent say an understanding of maths is becoming more and more important as digital technology develops.

Jed Cinnamon, from Nesta’s education team who leads the Maths Mission programme, said, “Today’s secondary school students say they enjoy and even love maths - in stark comparison to the fear/ dread 18 per cent of adults said they had for the subject at school. It’s great to see an indication of positive attitudes to the subject, but we need to explore how this links to attainment and choices to study maths beyond the age of 16.

“The poll indicates a mismatch between passion, performance and the pipeline of maths graduates - and it’s high time to explore new ways to tackle this critical issue. Nesta and Tata’s Maths Mission is testing if the answer could lie outside the formal curriculum, by linking maths to practical problems, parental conversations and even peer support.”

Marcus du Sautoy OBE FRS, Simonyi Professor for the Public Understanding of Science and Professor of Mathematics at the University of Oxford , says, "It is really encouraging to see that so many students recognise the importance of mathematics to modern life, that mathematics is not only important in technology and science but also to the creative arts. One of the tragedies of our education system is its propensity to teach subjects in isolation.

“Finding ways to bring mathematics out of its silo and celebrate its connections to subjects like history, art, music as well as its importance to the sciences will allow more students the chance to see how relevant this fundamental language is to our modern lives. Mathematics has some great stories to tell but students often get bogged down in the technical grammar of the language. I am very supportive of any initiatives that will help more students to hear the big ideas. Maths is useful but it is also beautiful and full of fascinating intellectual journeys that humans have travelled on throughout history. Let’s not cheat our students out of those exciting ideas. The results of the survey reveal that many students are up for the challenge."

David Landsman, Executive Director, Tata Limited said, “We all know that the UK underperforms in maths compared with some of our most important competitors. But the interesting findings of this survey show that the causes are complex. Young people are enthusiastic and confident about maths, and there are many excellent teachers. Business can play an important role in linking what young people learn at school with what they’ll need in the world of work. We at Tata know that maths is essential to all the things we do, so we’re proud to be working with NESTA on imaginative ways to help improve the UK’s maths performance.”

Nesta and Tata will be hosting an event at Hay Festival on Saturday 2 June 2018 to discuss some of these issues and to showcase new Maths teaching concepts.

For more information contact:

Juliet Grant in the Nesta media team: [email protected] / +44 (0) 20 7438 2668

NOTES TO EDITOR

The research was conducted by Censuswide, with 2,000 respondents aged 16+ in GB between 11.05.18-14.05.18 and 1,002 11-16 year olds between 10.05.18-14.05.18. Censuswide abide by and employ members of the Market Research Society which is based on the ESOMAR principles.

About the Maths Mission pilots

Tata and Nesta are working together to run three pilots focused on nurturing an interest in - and love of - maths and problem-solving outside the formal educational curriculum. Each of the three pilots looks to tackle the issue working with a key group - whether it be parents, students or teachers, and has a strong focus on developing collaborative skills through peer-to-peer activities. It aims to boost attitudes and increase attainment in maths, promoting the subject as a practical, engaging and problem-solving tool. The pilots are:

• Cracking the Code : an open youth challenge centred around ‘escape-rooms’ aimed at changing student’s perception of maths, through classroom experiences, live events and collaborative group-work;
• Solving together: using parental SMS messaging to improve maths problem solving, working with the Behavioural Insights Team ;
• Young maths mentors: developing pupil maths and peer mentoring skills in schools, working with both Funkey Maths and Franklin Scholars .

About Nesta:

Nesta is a global innovation foundation. We back new ideas to tackle the big challenges of our time, making use of our knowledge, networks, funding and skills. We work in partnership with others, including governments, businesses and charities. We are a UK charity that works all over the world, supported by a financial endowment. Find out more about Nesta .

Nesta is a registered charity in England and Wales 1144091 and Scotland SC042833.

Tata is one of the world’s most dynamic business groups. In Europe our operations span a diverse portfolio of 19 companies with over 65,000 employees. These companies include iconic brands such as Jaguar Land Rover and Tetley Tea as well as leading Tata businesses such as Tata Steel, TCS, Tata Communications, Tata Interactive Systems and Indian Hotels Company. Tata businesses in the UK are fully committed to our groups’ global commitment to nurturing learning and skills and inspiring academic learning and technical excellence. That’s why Tata is proud to be global partner of what is one of the UK’s best established and most respected educational events.

About Hay Festival

hayfestival.org

Hay Festival brings readers and writers together to share stories and ideas in sustainable events around the world. The festivals inspire, examine and entertain, inviting participants to imagine the world as it is and as it might be.

Featuring over 600 of the world’s greatest writers, global policy makers, pioneers and innovators in 800 events across 11 days, Hay Festival Wales (24 May-3 June) showcases the latest ideas in the arts, sciences and current affairs, alongside a rich schedule of music, comedy and entertainment. A galaxy of literary stars gathers to launch new work, while the biggest ever HAYDAYS and #HAYYA programmes give young readers the opportunity to meet their heroes and get creative.

Explore the 2018 programme and book tickets at hayfestival.org or call 01497 822 629.

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This page contains question sheets which are sent out to new students by many colleges before they arrive to start their undergraduate degree. These questions make suitable bridging material for students with single A-level Mathematics as they begin university - the material is partly revision, partly new material. All 11 sheets cover material relevant to the Mathematics, Mathematics & Statistics and Maths and Philosophy courses; sheets 8, 9 and 10 are not relevant to the Mathematics and Computer Science degree.

For each sheet the subject matter is briefly described, and there is some recommended reading material; the chapter numbers refer to the fourth edition of D.W.Jordan and P.Smith's book Mathematical Techniques, published by Oxford University Press in 2008.

• Sheet 1: Standard Functions and Techniques,  PDF Reading: §§ 1.3, 1.6-1.8, 1.10-1.16
• Sheet 2: Differentiation, PDF Reading: Chapter 2
• Sheet 3: Further Differentiation,  PDF Reading: §§ 3.1-3.5, 3.9-3.10
• Sheet 4: Applications of Differentiation, PDF Reading: §§ 4.1-4.4
• Sheet 5: Taylor Series, PDF Reading: §§ 5.1-5.4
• Sheet 6: Complex Numbers, PDF Reading: Chapter 6
• Sheet 7: Matrices, PDF Reading: Chapter 7
• Sheet 8: Vectors, PDF Reading: §§ 9.1-9.4, 9.6
• Sheet 9: The Scalar 'Dot' Product, PDF Reading: §§ 10.1-10.3, 10.9
• Sheet 10: The Vector 'Cross' Product, PDF Reading: §§ 11.1-11.2
• Sheet 11: Integration, PDF Reading: §§ 14.1-15.4, 15.8
• All the above 11 sheets as one file: PDF
• All the above 11 sheets as one webpage: Questions
• Induction 1: PDF Reading: R.B.J.T. Allenby Numbers and Proof , Chapter 7
• Induction 2:  PDF Reading: R.B.J.T. Allenby Numbers and Proof , Chapter 7
• Algebra 1: PDF Reading: No pre-requisites
• Algebra 2: PDF Reading: Chapters 7 and 8
• Calculus 1 - Curve Sketching: PDF Reading: §§ 4.1-4.4
• Calculus 2 - Numerical Methods and Estimation: PDF Reading: §4.6, §5.2
• Calculus 3 - Techniques of Integration: PDF Reading: §§17.5-17.7
• Calculus 4 - Differential Equations: PDF Reading: §§ 22.3-22.4, Chapter 18
• Calculus 5 - Further Differential Equations:  PDF Reading: Chapter 19, §22.5
• Complex Numbers: PDF Reading: Chapter 6
• Geometry: PDF Reading: §10.1, §10.9, §11.1, §16.1
• The second 11 sheets as one file:  PDF
• The second 11 sheets as one webpage: More challenging Questions
• Dynamics 1 - Basic Definitions. Newton's Second Law  PDF
• Dynamics 2 - Oscillations and Further Examples.  PDF
• These two sheets as one webpage: Further Sheets

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Only smartest people can solve maths puzzle in seconds – and '99% fail' test

If you want to see how good you are at maths, try this tricky maths puzzle that's been leaving adults on TikTok scratching their heads. Set a timer and get to work solving it

• 14:12, 27 AUG 2024

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Put your maths skills to the test by trying this tricky question that's left adults stumped.

The puzzle was posted onto the IQ Test US TikTok page. While it may seem very easy at first glance, it's been leaving people feeling perplexed.

The brainteaser first tells you that A+A = 2, and that A+B = 3. Using that information, you then have to figure out the formula "A + B x 2 = ?"

Apparently, only 1% of people are able to solve the equation in a matter of seconds. So why not set a timer and see how long it takes you to solve.

Even if you don't get it done instantly, we'll give you a pass for getting it correct! If you need some help, don't fret. First, you'll need to first work out what the value of A and B is.

By doing this, you can fill in the equation with numbers instead of letters. Then, there's actually a special trick to working out the sum – and failing to use it will likely result in you getting an incorrect answer.

The trick is to use the rule of BIDMAS, which we will explain here. Alternatively, scroll down to see the answer in full.

BIDMAS is an acronym used to help remember the order of operations in mathematics. It stands for:

• I ndices (also known as exponents or powers)
• D ivision and M ultiplication (from left to right)
• A ddition and S ubtraction (from left to right)

Here's a brief explanation of each step:

Brackets : Solve anything inside parentheses or other types of brackets first.

• Example: In the expression ( (2 + 3) \times 4 ), solve ( 2 + 3 ) first because it's inside the brackets.

Indices : Then, calculate any exponents or powers.

• Example: In the expression ( 3^2 + 4 ), calculate ( 3^2 ) (which equals 9) before adding 4.

Division and Multiplication : Next, perform all division and multiplication operations from left to right.

• Example: In the expression ( 6 \div 2 \times 3 ), perform the division ( 6 \div 2 ) (which equals 3), then multiply by 3.

Addition and Subtraction : Finally, perform all addition and subtraction from left to right.

• Example: In the expression ( 7 - 2 + 5 ), subtract 2 from 7 (to get 5), and then add 5.

It’s important to follow the BIDMAS rules to ensure that mathematical expressions are solved correctly. Misordering the operations can lead to incorrect results.

The answer is five. If you worked out the easy sums correctly, you'll know that A is 1 and B is 2, making the final sum "1 + 2 x 2 = ?".

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