nrich problem solving tasks

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• Authentic tasks
• F - 10 Resources

Authentic tasks are designed to help students see mathematics as worthwhile and important. When students understand the purpose of a given problem in mathematics, they are more likely to persist when challenged. Authentic tasks generally have an ‘open middle’ which means that students can use different representations and solutions to communicate their knowledge and reasoning.

These curated links provide MAV members with access to nine authentic tasks from some of our primary consultants’ favourite resources. The 11 criteria provide MAV members with a research-informed context to consider each task’s potential impact on student thinking, ways of working, attitudes towards mathematics, their knowledge and understanding.

The following criteria was used to select the tasks based on their potential:

Used with permission © Martin Holt Educational Consultant 2017

If you would like to learn more about this approach to assessing or using tasks contact [email protected]

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Measurement and geometry, number and algebra.

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Number and algebra

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

Geometry and measure

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

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• Handling, Processing and Representing Data
• Probability

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

For younger learners

• Early Years Foundation Stage

Advanced mathematics

• Decision Mathematics and Combinatorics
• Advanced Probability and Statistics

Interactive Tasks and Games

Interactive tasks and games.

Mr Barton's Rich Tasks

On this page I have collected together my favourite rich tasks that I have used over the last 11 years, in an accessible, easy to use format. I also invite teachers to share their ideas for interesting probing questions and lines of inquiry for students to investigate.

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For me, a rich task is one that both engages and challenges students with a wide level of mathematical ability. They need to be “low barrier, high ceiling”, by which I mean students need to have found success/made progress with the task within the first 30 seconds, but there is still enough meat left to keep them thinking 30 minutes (or even 3 lessons) later.

I feel activities like these are crucial for students’ mathematical development. They allow them to be creative, and work together in meaningful and positive ways. When developing our Scheme of Work (you can read my series of blog posts about it), we decided to include a compulsory rich task for all students each topic unit, and many of those can be found below.

The key to a good rich task are the questions that accompany it. This is where effective differentiation happens. All students begin the task in exactly the same way, but once an initial stage has been reached, students (individual or in groups) are free to pursue different investigations, probing questions and lines of inquiry. These can be provided by the teacher, or even by the students themselves.

The strength of the rich task lies in these questions. So here is my plan: I am going to share as many of my favourite rich tasks as possible, and hopefully teachers from around the world are going to provide the questions. These can be lines of inquiry, investigations, prompts, hypotheses, extensions, simplifications, modifications, whatever you like. Crucially, you do not need to know the answer yourself. Just throw it out there! There will be space for these in the Comments section at the bottom of each TES Resource page, and I will always get the ball rolling with a few questions of my own.

Please join in. Please spread the word. Please just share even one question. And then the tasks will keep getting better, and better, and better. 

The Rich Tasks keyboard_arrow_up Back to Top

Task 1 - Positive Differences Brief Description: Students build simple number pyramids by taking the positive difference of pairs of numbers Potential Skills Involved: Arithmetic, Fractions, Decimals, Writing Expressions, Proof

Task 2 - The Factors and Multiples Game Brief Description: Students play a strategic game on a 1-100 number grid, crossing off factors and multiples Potential Skills Involved: Arithmetic, Factors, Multiples, Primes, Proof

Task 3 - Choose 3 Numbers Brief Description: Students try to guess each other's starting numbers by working backwards from the sums of pairs of numbers Potential Skills Involved: Arithmetic, Writing expressions, Solving Equations

Task 4 - Will they meet? Brief Description: Can you help Romeo and Juliet get back together in my first ever romantic maths activity? Potential Skills Involved: Enlargement, Vectors, Similar Shapes, Rotation

Task 5 - Number Shacks Brief Description: Can you figure out how the numbers of these shacks are formed and use this to predict answers and spot patterns? Potential Skills Involved: Arithmetic, Writing Expressions

Task 6 - Averaging it out Brief Description: What happens when we continually take the mean of sets of numbers? Potential Skills Involved: Averages, ICT

Task 7 - Fraction Arrangement Brief Description: Can you order different digits to produce the biggest and smallest possible answers for these fraction problems? Potential Skills Involved: Operations with fractions

Task 8 - Diffy Brief Description: The first lesson our new bunch of Year 7s experience, and one of my all time favourites Potential Skills Involved: Arithmetic, Writing Expressions, Proof

Task 9 - Simultaneous Equations Staircase Brief Description: Why does everyone get the same answer to these simultaneous equation problems? Potential Skills Involved: Simultaneous Equations, Proof

Task 10 - How many angles? Brief Description: Using a geoboard, how many angles between 10 and 180 can you make? Potential Skills Involved: Angle Facts, Circle Theorems

Task 11 - Number Reverse Brief Description: What happens when we reverse the digits of numbers and perform operations on them? Potential Skills Involved: Arithmetic, Writing expressions, Proof

Task 12 - Multiplication Reduction Brief Description: Follow the rule to reduce a number in size using multiplication. Does anything interesting happen? Potential Skills Involved: Arithmetic, Writing Expressions

Task 13 - How many quadrilaterals? Brief Description: Using a geoboard, how many different quadrilaterals can you make? Potential Skills Involved: Properties of shapes, Angle facts

Task 14 - 1089 Brief Description: Why is the number 1089 so special? Potential Skills Involved: Arithmetic, Writing expressions, Proof

Task 15 - Square Co-ordinates Brief Description : What do the co-ordinates of the corners of squares have in common? Potential Skills Involved: Co-ordinates, Properties of shapes, Vectors, Proof

Task 16 - Polar Bears Brief Description : Can you figure out how to get the totals in this dice game? Potential Skills Involved: Arithmetic

Task 17 - Pascal's Triangle Brief Description : What maths can you discover hiding in Pascal's triangle? Potential Skills Involved: Sequences

Task 18 - Entrapment Brief Description : A fun strategy game using all of the transformations Potential Skills Involved: Reflection, Rotation, Translation, Enlargement

Task 19 - Fire Hydrants Brief Description : Where is the optimum position to place these fire hydrants to maximise their coverage? Potential Skills Involved: Geometrical Reasoning

Task 20 - Diagonals of Rectangles Brief Description : How many squares does the diagonal of a rectangle pass through? Potential Skills Involved: Arithmetic, Sequences, Factors, Multiples, Primes

Task 21 - T-totals Brief Description : How can you work out the T-number in this classic piece of maths coursework? Potential Skills Involved: Arithmetic, Writing Expressions, Proof

Task 22 - Number Snakes Brief Description : What is the longest number snake you can make using these simple rules? Potential Skills Involved: Arithmetic, Properties of Numbers, Writing Expressions

Task 23 - Summing Consecutive Numbers Brief Description : Which numbers can be made using the sums of consecutive numbers? Potential Skills Involved: Arithmetic, Writing Expressions

Task 24 - NIM Brief Description : The wonderful strategy game using piles of counters Potential Skills Involved: Strategy, Factors, Multiples, Primes

Task 25 - Function Machines Brief Description : Why do these function machines seem to give the same difference? Potential Skills Involved: Arithmetic, Order of Operations, Writing Expressions, Expanding Brackets

Task 26 - Leap Frog Brief Description : If you leap over this set of 3 points enough times, what do you notice? Potential Skills Involved: Co-ordinates, Construction, Vectors

Task 27 - Solving Linear Equations Brief Description : By arranging sets of digits, what types of solutions can you generate to these simple linear equation problems? Potential Skills Involved: Solving linear equations

Task 28 - Decimal Arithmetic Brief Description : By arranging sets of digits, can you make the biggest and smallest decimal totals possible? Potential Skills Involved: Arithmetic, Decimals, Place Value

Task 29 - 24 Cubes Brief Description : What different 3D objects can you make with 24 cubes and what do you notice about their properties? Potential Skills Involved: Surface Area, Volume, Similarity

Task 30 - Tilted Squares Brief Description : How many squares with area 1-20 can you create? Potential Skills Involved: Area, Pythagoras

Frequently Asked Questions keyboard_arrow_up Back to Top

Are you saying we should be doing this every lesson? No. Definitely not. I am acutely aware of the need for students to gain practise in key mathematical skills. But I do strongly believe that regular lessons like this are just as important for a student's mathematical development and to increase their enjoyment in the subject. They should not be seen as one-offs. Both students and teachers should value them as highly as any type of lesson. What do I do if a child doesn't engage with a particular question or line of inquiry? The simple answer is that I give them another one! I have to make a judgment call as to whether the student has genuinely tried and not just given up too easily. But if, for whatever reason, a probing question or line of inquiry hasn't resonated with a student, then I will set them off on something else. Indeed, the beauty of having lots of questions up your sleeve is that you are far more likely to find ones that engage your students than if you just have one line of investigation that the whole class is following. Would you do this type of lesson for an observation? Yes, I definitely would. 100%. Sure, such lessons are a little bit on the risky side as you don't know what is going to happen. But they are also incredibly flexible. Imagine you had meticulously planned a lesson with a 40 slide PowerPoint and 5 beautifully prepared, differentiated worksheets. And then you find that the students don't understand even the basics. Or, they understand far more than you anticipated. You might be in a bit of trouble. But with a lesson rammed full of probing questions, you can just try them out on another line of inquiry. Or, better still, get them to come up with their own.

Or search by topic

Number and algebra

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

Geometry and measure

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

Probability and statistics

• Handling, Processing and Representing Data
• Probability

Working mathematically

• Thinking mathematically
• Developing positive attitudes
• Cross-curricular contexts
• Physical and digital manipulatives

Advanced mathematics

• Decision Mathematics and Combinatorics
• Advanced Probability and Statistics

For younger learners

• Early Years Foundation Stage

The Problem-Solving Schools' Charter

The NRICH team has developed this Charter to help you reflect on how you currently promote mathematical problem-solving in your school

Values and ethos

We have a shared belief that:

• Mathematical ability is not fixed: everyone can learn and make progress
• Problem-solving often involves taking wrong turns and making mistakes: every learner has the right to struggle and the right to enjoy success
• Everyone should have the opportunity to develop the skills and attitudes necessary to become confident problem-solvers
• Problem-solving can motivate learners to learn new mathematics, apply previous learning and make mathematical connections

Leadership and professional development

In our setting:

• Our staff promote positive attitudes towards problem-solving
• Time is set aside to discuss problem-solving in our meetings
• Our displays, newsletters, website, and social media content celebrate problem-solving for all
• Our monitoring system ensures that priority is given to problem-solving and mathematical thinking
• We engage with printed, online and face-to-face professional development opportunities offered by subject organisations

Curriculum, pedagogy and assessment

We are committed to:

• Regularly embedding non-standard problem-solving opportunities in our maths curriculum for all
• Ensuring that problems, and classroom support, offer opportunities for all to experience both struggle and success
• Allocating time to developing key problem-solving skills and positive attitudes
• Including non-standard problems in our internal/formative assessments
• Liaising with other subjects so that meaningful cross-curricular links can be made

Classroom culture

• Create a safe environment in which learners explore, take risks, and appreciate the value of learning from their mistakes
• Celebrate multiple approaches to solving problems and discuss the merits of the different strategies offered
• Provide frequent opportunities for individual and collaborative problem-solving, where learners are given both thinking time, and opportunities to share ideas and insights
• Celebrate the mathematical thinking of every learner

Problem-solving beyond the classroom/school

We encourage:

• Learners to engage with school Maths Club(s) and high quality maths books, ideally stocked by the school library
• Learners to take advantage of printed, online and off-site mathematical enrichment opportunities
• Parents and carers to engage with problem-solving through family homeworks and in-school events, while recognising that not every adult has had a positive experience of maths
• Our learners to appreciate, and learn more about, the achievements of a diverse range of mathematicians

Become a Problem-Solving School