nrich maths problem solving ks2

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About NRICH

NRICH is an innovative collaboration between the Faculties of Mathematics and Education at the University of Cambridge which focuses on problem solving and on creating opportunities for students to learn mathematics through exploration and discussion. NRICH provides thousands of free online mathematics resources for ages 3 to 18, covering all stages of early years, primary and secondary school education - completely free and available to all. We aim to:

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Problem Solving

 A selection of resources containing a wide range of open-ended tasks, practical tasks, investigations and real life problems, to support investigative work and problem solving in primary mathematics.

Problem Solving in Primary Maths - the Session

Quality Assured Category: Mathematics Publisher: Teachers TV

In this programme shows a group of four upper Key Stage Two children working on a challenging problem; looking at the interior and exterior angles of polygons and how they relate to the number of sides. The problem requires the children to listen to each other and to work together co-operatively. The two boys and two girls are closely observed as they consider how to tackle the problem, make mistakes, get stuck and arrive at the "eureka" moment. They organise the data they collect and are then able to spot patterns and relate them to the original problem to find a formula to work out the exterior angle of any polygon. At the end of the session the children report back to Mark, explaining how they arrived at the solution, an important part of the problem solving process.

In a  second video  two maths experts discuss some of the challenges of teaching problem solving. This includes how and at what stage to introduce problem solving strategies and the appropriate moment to intervene when children find tasks difficult. They also discuss how problem solving in the curriculum also helps to develop life skills.

Cards for Cubes: Problem Solving Activities for Young Children

Quality Assured Category: Mathematics Publisher: Claire Publications

This book provides a series of problem solving activities involving cubes. The tasks start simply and progress to more complicated activities so could be used for different ages within Key Stages One and Two depending on ability. The first task is a challenge to create a camel with 50 cubes that doesn't fall over. Different characters are introduced throughout the book and challenges set to create various animals, monsters and structures using different numbers of cubes. Problems are set to incorporate different areas of mathematical problem solving they are: using maths, number, algebra and measure.

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

Quality Assured Category: Computing Publisher: Department for Education

These three resources, from the National Strategies, focus on solving problems.

  Logic problems and puzzles  identifies the strategies children may use and the learning approaches teachers can plan to teach problem solving. There are two lessons for each age group.

Finding all possibilities focuses on one particular strategy, finding all possibilities. Other resources that would enhance the problem solving process are listed, these include practical apparatus, the use of ICT and in particular Interactive Teaching Programs .

Finding rules and describing patterns focuses on problems that fall into the category 'patterns and relationships'. There are seven activities across the year groups. Each activity includes objectives, learning outcomes, resources, vocabulary and prior knowledge required. Each lesson is structured with a main teaching activity, drawing together and a plenary, including probing questions.

Primary mathematics classroom resources

Quality Assured Collection Category: Mathematics Publisher: Association of Teachers of Mathematics

This selection of 5 resources is a mixture of problem-solving tasks, open-ended tasks, games and puzzles designed to develop students' understanding and application of mathematics.

Thinking for Ourselves: These activities, from the Association of Teachers of Mathematics (ATM) publication 'Thinking for Ourselves’, provide a variety of contexts in which students are encouraged to think for themselves. Activity 1: In the bag – More or less requires students to record how many more or less cubes in total...

8 Days a Week: The resource consists of eight questions, one for each day of the week and one extra. The questions explore odd numbers, sequences, prime numbers, fractions, multiplication and division.

Number Picnic: The problems make ideal starter activities

Matchstick Problems: Contains two activities concentrating upon the process of counting and spotting patterns. Uses id eas about the properties of number and the use of knowledge and reasoning to work out the rules.

Colours: Use logic, thinking skills and organisational skills to decide which information is useful and which is irrelevant in order to find the solution.

GAIM Activities: Practical Problems

Quality Assured Category: Mathematics Publisher: Nelson Thornes

Designed for secondary learners, but could also be used to enrich the learning of upper primary children, looking for a challenge. These are open-ended tasks encourage children to apply and develop mathematical knowledge, skills and understanding and to integrate these in order to make decisions and draw conclusions.

Examples include:

*Every Second Counts - Using transport timetables, maps and knowledge of speeds to plan a route leading as far away from school as possible in one hour.

*Beach Guest House - Booking guests into appropriate rooms in a hotel.

*Cemetery Maths - Collecting relevant data from a visit to a local graveyard or a cemetery for testing a hypothesis.

*Design a Table - Involving diagrams, measurements, scale.

Go Further with Investigations

Quality Assured Category: Mathematics Publisher: Collins Educational

A collection of 40 investigations designed for use with the whole class or smaller groups. It is aimed at upper KS2 but some activities may be adapted for use with more able children in lower KS2. It covers different curriculum areas of mathematics.

Starting Investigations

The forty student investigations in this book are non-sequential and focus mainly on the mathematical topics of addition, subtraction, number, shape and colour patterns, and money.

The apparatus required for each investigation is given on the student sheets and generally include items such as dice, counters, number cards and rods. The sheets are written using as few words as possible in order to enable students to begin working with the minimum of reading.

NRICH Primary Activities

Explore the NRICH primary tasks which aim to enrich the mathematical experiences of all learners. Lots of whole class open ended investigations and problem solving tasks. These tasks really get children thinking!

Mathematical reasoning: activities for developing thinking skills

Quality Assured Category: Mathematics Publisher: SMILE

Problem Solving 2

Reasoning about numbers, with challenges and simplifications.

Quality Assured Category: Mathematics Publisher: Department for Education

The Joint Mathematical Council of the United Kingdom

Addressing the five ‘big questions’ in problem-solving with NRICH

The importance of ensuring learners acquire the problem-solving skills which will enable them to thrive both socially and economically in their increasingly automated world is widely recognised (Luckin et al., 2017). Nevertheless, government inspectors have reported serious concerns about the quality and quantity of problem-solving in our schools (Ofsted, 2015). This summer schools were challenged to reflect on ‘Five big questions for problem-solving’ (EEF, 2021). In this blog, we will consider each of those five questions and explore the ways that the NRICH team is supporting schools to address them.

Question one: Do teachers in your school select genuine problem-solving tasks for which pupils do not already have a ready-made method available?

Too often, learners are presented with routine word problems which merely require the application of a known algorithm. ‘Genuine’ problems enable them to make their own problem-solving decisions by choosing their own strategies and enabling them to compare their approach with those of other learners, thus developing their problem-solving efficiency and flexibility. At NRICH , our award-winning activities allow learners to develop these key skills alongside the confidence to tackle genuine problems. Moreover, our ‘ low threshold, high ceiling ‘ approach enables everyone to get started on the problem while ensuring a suitable level of challenge too, making them ideal for whole-class teaching.

Question two: Are pupils given the opportunity to see – through multiple worked examples – to use, and to compare different approaches to solving a problem?

Many problems can be explored in more than one way. Working flexibly, making connections between different areas of the curriculum and reflecting on various problem-solving approaches are key steps towards becoming a more fluent mathematician. NRICH encourages learners to develop these skills in these two ways:

Our primary , secondary and post-16 Live Problems invite learners to explore and submit their ideas to the team. We review each submission that we receive and publish a selection on our website showcasing different approaches and the reasoning behind them.

Our NRIC H online activities sometimes feature ‘hide and reveal’ buttons showcasing different starting points towards a solution for learners to explore further for themselves. This approach enables learners to widen their range of strategies for solving unfamiliar problems and develop alternative approaches to explore when they get stuck using their first-choice strategy.

Question three: Are pupils encouraged to use visual representations to support them to solve a problem?

One of the most important approaches towards solving an unfamiliar problem is drawing a good diagram. Learning to draw diagrams is a skill which we encourage learners of all ages to develop alongside their other mathematical skills and knowledge. From sketching graphs to drawing a bar model, good diagrams can help learners clarify their understanding and identify possible ways forward.

Our four steps towards problem-solving feature highlights the importance of drawing a diagram to enable young learners to get started on a problem. We often highlight a useful diagram, table or sketch graph in the solutions chosen for publication. As learners progress through their learning, the team model more specific drawing skills, such as sketching a graph to help solve a STEP problem.

Question four: Are pupils supported to monitor, reflect on, and communicate their reasoning and choice of strategies, possibly through the use of prompt questions?

NRICH  encourages learners to reflect on their learning using this approach inspired by the Strands of Mathematical Proficiency model introduced by Kilpatrick et al. (2001).

Our approach uses child-friendly language that teachers and parents can share with students five key ingredients that characterise successful mathematicians. At NRICH , we believe that learning mathematics is about much more than just learning topics and routines. Successful mathematicians understand the curriculum content and are fluent in mathematical skills and procedures, but they can also solve problems, explain their thinking and have a positive attitude about themselves as learners of mathematics.

With this in mind, we have created  this self assessment tool  to help learners recognise where their mathematical strengths and weaknesses lie. We hope learners will explore NRICH activities and then take time to reflect on their own mathematical capabilities using our model.

Question 5: Is professional development time allocated to develop teachers’ pedagogical understanding of problem-solving, with particular support for early career teachers?

NRICH supports teachers to maximise the potential of our activities by offering free, regular professional development for teachers .  Each session is delivered online, enabling teachers to access the support wherever they are based, reducing teacher travel and cover costs for schools. We also record the sessions and upload them to our website so that schools can access them for future professional development days or staff/department meetings in their settings.

The live sessions are led by NRICH team members and they link directly to our latest primary , secondary and post-16 Live Problems. This approach enables teachers to consider the possibilities of the activities with the NRICH team before exploring them the next day with their own classes. Later, they are invited to share their classwork with our team for possible publication on the NRICH website.

The five ‘big questions’ provide excellent starting points for evaluating the teaching and learning of problem-solving in different settings. I hope that this blog shares an insight into the different ways that NRICH can support schools to address the five questions for themselves by engaging with our activities, Live Problems and teacher webinars.

Dr Ems Lord FCCT

Director of NRICH

Centre for Mathematical Sciences

University of Cambridge

Selected references

EEF. (2021). EEF Blog: Integrating evidence into maths teaching – guiding problem-solving. Accessed from https://educationendowmentfoundation.org.uk/news/eef-blog-integrating-evidence-into-mathematics-guiding-problem-solving /

Kilpatrick, J. Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics (Vol. 2101). J. Kilpatrick, & National research council (Eds.). Washington, DC: National Academy Press.

Luckin, R., Baines, E., Cukurova, M., Holmes, W., & Mann, M. (2017). Solved! Making the case for collaborative problem-solving. Accessed from http://oro.open.ac.uk/50105/1/solved-making-case-collaborative-problem-solving.pdf

Ofsted. (2015). Better Maths Conference Spring Keynote 2015. Accessed here https://www.slideshare.net/Ofstednews/better-mathematics-keynote-spring-2015

The Nature of Problem-solving in Primary Maths

The Educational Recording Agency, in collaboration with NRICH, has published the first set of investigative Maths lessons inspired by nature, on the Natural Curriculum website. The aim is to utilise the wealth of outstanding BBC Natural History footage in a creative, exploratory approach to teaching Mathematics that feeds off children’s wonder of the natural world.

Credit: BBC/John Downer Productions Ltd/Stephen Downer

Mathematics is a fundamental tool for solving problems and understanding the world around us. However, for many students, mathematics can be an intimidating subject, and they struggle to understand its abstract concepts.

With a quarter of a million children entering secondary school without basic Maths and English , and roughly 175,000 children failing their Maths GCSE every year , addressing the prevailing negative attitudes towards the subject has become the key issue facing Maths educators, according to Dr Ems Lord, Director of NRICH .

One way in which this can be achieved is by making Maths more relevant and relatable to children’s experiences and interests from a young age.

NRICH is a collaboration between the Faculties of Mathematics and Education at the University of Cambridge, which focuses on problem solving and on creating opportunities for students to learn Mathematics through exploration and discussion. Its aim is to enrich and enhance the experience of the Mathematics curriculum for all learners.  Dr Lord explains :

“At NRICH, we actively seek engaging contexts so that learners want to explore the Maths, derive enjoyment from the tasks and are encouraged to persevere with solving problems and finding solutions,”

It seems that we only need to look outside the classroom window for an answer.

The natural world is an excellent source of inspiration for learning Mathematics. It is something that children can see, touch, and experience in their everyday lives. When students can see how mathematical concepts apply to the world around them, they are more likely to be interested and engaged in the learning process.

One teacher who preaches this sentiment is David Millington, the man behind the Natural Curriculum, which originally launched in 2020 with an online collection of KS2 grammar lessons inspired by clips from the BBC’s ground-breaking natural history programmes. When David approached the Educational Recording Agency about using clips under the terms of the ERA Licence, their value as a means of inspiring and engaging pupils was immediately evident to the CEO, Helena Djurkovic, and the ERA Board agreed to fund the development of the website. Its launch in 2020 was just in time to make an invaluable contribution to the remote education landscape during lockdown.

Keen to expand on the resource’s initial impact, David, together with the educational resource team at ERA, sought to maximise the potential that the natural world (supported by compelling footage from the BBC), could have in core-subject learning, with Mathematics being the natural (pun intended) next step. The Bristol-based teacher recounts:

“The first ever natural world film clip that I used in a Maths lesson was from the BBC Blue Planet series. In the clip, Sir David Attenborough reels off some impressive number facts about the size of the mighty blue whale. With my then Year 5 class captivated, it was the perfect way to kick start a unit on measurement and length.” 

David Millington, teacher and content creator for Natural Curriculum

In the academic year 2021/22 the Natural Curriculum joined forces with NRICH to explore the best way of using natural history clips to teach primary Maths.

Four online problem-solving Mathematics lessons were developed and piloted with primary-aged learners from 9 schools out of 70 that applied to take part. The findings of the study revealed that bringing the natural world into the classroom was a popular teaching and learning approach among many Year 3 and Year 4 learners, with more than three-quarters of the participants (75.6%) reporting that they had learnt something new about nature, and a significant number mentioning that they had learnt something new about Maths.

Building on the success of the pilot project, the team has designed a detailed range of new resources for practising Maths problem-solving in areas such as measurement, geometry and statistics, with inspiring BBC natural history clips featuring fascinating animals such as narwhals, armadillos and bears.

“NRICH is delighted to be collaborating with the Natural Curriculum team. Our team has drawn inspiration from the BBC's world-class library of natural history footage. We have worked closely with the Natural Curriculum to develop exciting and engaging problem-solving challenges for young learners which link to their school curriculum.”

Dr Lord continues , “Our initial pilot project reinforced the value of bringing together Mathematics and the natural world in the classroom, and we were thrilled to expand this collaboration further and release this new suite of resources.”

Click above to visit the Natural Curriculum & NRICH project

The Natural Curriculum and NRICH partnership leverages world-class content from David Attenborough and the BBC to capture that fascination with nature and promotes an enjoyment of Mathematics that is so vital to instil from an early age. The first lessons for Year 3 are now available for free on the Natural Curriculum website with more in development.

These resources support the wider movement now evident in schools to include the natural world in learning. Speaking at the Association for Science Education Conference back in 2011, David Attenborough told teachers:

“Bringing nature into the classroom can kindle a fascination and passion for the diversity of life on earth and can motivate a sense of responsibility to safeguard it.”

Wise words. A new Natural History GCSE will finally launch in 2025 – just in time for the many thousands of young pupils who have been inspired to respect and care for their planet by the Natural Curriculum and are keen to continue on their educational journey.

Bringing nature into the classroom can kindle a fascination and passion for the diversity of life on earth and can motivate a sense of responsibility to safeguard it.

Bringing the Outside In: The Natural Curriculum

Bristol-based primary school teacher David Millington talks about his inspiring website, the Natural Curriculum, made in partnership with ERA, which provides free grammar lessons for Years 4-6.

Natural Curriculum

Grammar and maths lessons for KS2 centred on captivating clips from the BBC’s natural history archives.

Tackling Climate Change with Young Climate Warriors

Young Climate Warriors have curated a series of video playlists to teach primary aged pupils about the causes, impacts and solutions of climate change.

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• 3D Geometry, Shape and Space
• Measuring and calculating with units
• Transformations and constructions
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• Handling, Processing and Representing Data
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Working mathematically

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• Mathematical mindsets
• Cross-curricular contexts
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For younger learners

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Problem-solving Schools

Problem-solving Schools supports learners embed key problem-solving strategies as they progress through their schooling, backed by our Problem-solving Schools' Charter .

Want to raise the profile of mathematical problem-solving in your school?

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What is the Problem-solving Schools initiative?

We aim to help you raise the profile of mathematical problem-solving in your school. Our Charter offers a framework intended to inform policy and practice.

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Use our Charter to help you reflect on your current practice and to agree on areas for development.

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Our support material and webinars aim to help your school move forward on its problem-solving journey.

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14 November 2023

NRICH are launching an exciting new FREE initiative for schools this autumn.

Part of the University of Cambridge, the NRICH project is launching a new programme to support the teaching of problem-solving in schools. NRICH is inviting schools to sign up to the new Problem-solving Schools charter and become a recognised NRICH Problem-solving School!

High-quality problem solving resources and dedicated teacher CPD will also be shared with schools in the new year.

Schools can  find out more and register for free:  nrich.maths.org   

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