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Working Systematically is part of our Thinking Mathematically collection.
Mathematicians often talk about the importance of working systematically. This means that rather than working in a haphazard and random way, there is a methodical, organised and logical approach. The problems below will challenge you to work systematically, and will help you appreciate the benefits of working in this way.
Scroll down to see our complete collection of problems, or explore the two sub-collections.
The key to solving these problems is to notice patterns or properties. Organising your work systematically allows you to notice what might not otherwise be obvious.
Finding All Solutions
These problems challenge you to find all possible solutions. One of the best answers to "How do you know you have found them all" is to be able to say "I worked systematically!"
Two and Two
How many solutions can you find to this sum? Each of the different letters stands for a different number.
Summing Consecutive Numbers
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
1 Step 2 Step
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
Can They Be Equal?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
How many different symmetrical shapes can you make by shading triangles or squares?
Shifting Times Tables
Can you find a way to identify times tables after they have been shifted up or down?
Peaches Today, Peaches Tomorrow...
A monkey with peaches, keeps a fraction of them each day, gives the rest away, and then eats one. How long can his peaches last?
Charlie's Delightful Machine
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Triangles to Tetrahedra
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Making a Difference
How many different differences can you make?
Where Can We Visit?
Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.
By selecting digits for an addition grid, what targets can you make?
Squares in Rectangles
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Each clue in this Sudoku is the product of the two numbers in adjacent cells.
Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
Consecutive Negative Numbers
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?
The clues for this Sudoku are the product of the numbers in adjacent squares.
Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?
The items in the shopping basket add and multiply to give the same amount. What could their prices be?
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
Warmsnug Double Glazing
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Use the differences to find the solution to this Sudoku.
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Exploring problem solving with 5th and 6th class learning path.
This learning path gives a brief overview of how to develop a classroom culture of sharing knowledge, valuing mistakes and providing cognitively challenging tasks. Problem solving strategies can be explicitly taught using this learning path with an introduction to the concept of Low Threshold High Ceiling Tasks also. 11 resources in this Learning Path
Developing Maths problem solving
a pre assessment to see how children are able to respond and complete problem
How it maps to the curriculum
Strand unit: Operation: Addition
Suggestions for use: Assessment of student learning
Same But Different
The images on this website can be used for developing classroom discussion, critical thinking and collaboration. Key components for problem solving.
Strand: Useful Websites
Suggestions for use: This can website can be used by teachers as a mini problem-solving lessons (10 mins approx.) to develop a classroom climate conducive to problem-solving. Teachers should encourage all pupils to ‘have a go’ and ‘value all contributions’. Promoting higher order skills of reasoning and discussion. Success is based on effort and skills rather than answers.
Low Threshold High Ceiling
A low threshold high ceiling task is one which is designed to be mathematically accessible, and to have built-in extension opportunities. In other words, everyone can get started and everyone can get stuck. In this updated feature, NRich brings together their favourite low threshold high ceiling tasks, as well as two articles which will support you in creating a low threshold high ceiling classroom.
Suggestions for use: Developing a positive classroom climate in maths through mathematical discussion and collaboration
Problem Solving with 5th and 6th Class - Sandwiches (Trial and Improvement)
The problem is particularly valuable as it gives students an opportunity to work on a proof to explain why something is impossible.
Suggestions for use: Using number to predict, generalise and verify
Problem Solving with 5th and 6th Class - Tea Cups (Working Systematically)
The problem lends itself for small group work, so that the learners have an opportunity to decide on approaches. Learners can collaborate, share and discuss the different solutions, and each method's strengths and weaknesses.
Suggestions for use: Allowing learners to read the accompanying story with the problem can support student’s comprehension and decision making about which pieces of information are relevant. Development of higher order maths skills: Applying and problem-solving, Communicating and expressing, Integrating and connecting, Reasoning
Problem Solving with 5th and 6th Class - Tables without tens (Pattern spotting)
This problem provides an interesting way of revising multiplication tables. It is also very useful for getting learners to predict what they think they will find out and spot pattern between times tables.
Strand unit: Number Theory
Content objective: This resource should enable a child to:
- identify common factors and multiples
- identify factors and multiples
- identify simple prime and composite numbers
Suggestions for use: times tables
Problem Solving with 5th and 6th Class - Counting Cards (Working Backwards)
While this problem provides the “trick” element to it, it is firmly rooted in mathematical concepts and problem solving strategies.
Problem Solving with 5th and 6th Class - Baravelle (Visualising)
The aim of this problem is to encourage discussion about the different ways of seeing, and to pose questions that can form the focus of further investigation.
Strand: Shape & Space
Strand unit: 2-D Shapes
- make informal deductions about 2-D shapes and their properties
- plot simple co-ordinates and apply where appropriate
- tessellate combinations of 2-D shapes
- use 2-D shapes and properties to solve problems
Suggestions for use: Allows for creative pattern design. A good follow up problem can be found here https://nrich.maths.org/2132/index (Inside 7 squares)
Problem Solving with 5th and 6th Class - Planning a school trip (Reasoning)
This problem will encourage learners to organise information, identify redundant information and to check their work.
Suggestions for use: The activity lends itself to collaborative working, both for children who are inexperienced at working in a group and children who are used to working in this way. By working together on this problem, the task is shared and therefore becomes more manageable than if working alone. A number of cross curricular links can be used to extend this lesson.
Problem Solving with 5th and 6th Class - Magic V's (Conjecture)
Task card involving placing numbers 1 to 5 in the V shape so that the two arms of the V have the same total. Short task.
Strand unit: Operations: Addition & Subtraction
Suggestions for use: Print and display on maths station or use separately in a teacher led problem solving activity. Magic V gives opportunities for children to make conjectures, prove these conjectures and make generalisations. They will be practising addition and subtraction, and applying their knowledge of odd/even numbers. Supporting video clip on the problem can be found at this link https://www.youtube.com/watch?v=-JrZcMbsNdA
Strand unit: Operations
- add and subtract whole numbers and decimals (to three decimal places) without and with a calculator
Problem Solving with 5th and 6th Class - Money Bags (Conjecture)
This problem is a good example of a challenge which does not require high-level mathematics, but does need a systematic approach. It also lends itself to a focus on different ways of recording and learners can discuss on the merits of the different ways of recording findings.
Strand unit: Money - Euro
- compare 'value for money' using unitary method
- explore value for money
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Roughwood Primary School
Where a love of learning grows.
Working systematically with NRICH maths challenges in Class 6
As part of our reasoning and applying our knowledge within mastery maths sessions, the children in Year 5 have completed another investigative NRICH reasoning investigation using their systematic thinking and recording.
The investigation generated lots of discussion and at the end, children wanted to investigate further links between divisors, like 3, 6 and 9, to see if they could see a similar pattern there too.
Using our school learning characters, we’ve needed to be like professor thought shower and captain team work to apply our prior knowledge and understanding.
Look at how the children recorded their answers systematically.
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- Problem Solving and Mastery Ideas
- Teachers' Area
- Resources to Support Reasoning Mathematically
- MASTERY An excellent resource with ideas to support teaching 'mastery' you will need to set up Oxford Owls, which is free to do.
- NRICH A useful article with links to appropriate tasks
- Topmarks: KS2 Lots of Links to good interactive lesson support materials
- Topmarks KS1 Links to lesson support materials and activities
- Investigating number bonds
- Trial and Improvement Task NRICH KS1 Useful article with links to appropriate tasks
- Developing tasks to promote problem solving skills with your class NRICH With good links to tasks to encourage working systematically and other skills
- Conjecturing and Generalising in KS1
Useful Problem Solving Tasks
- Finding all possibilities.pdf
- Finding rules and describing patterns.pdf
- Four Goodness Sake.docx
- Ladybirds in the Garden.docx
- Logic problems.pdf
- Noah 136.pdf
- Pairs of Numbers.docx
- Squares Key Stage 1.zip
- Zios and Zepts.docx
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Year Four's Blog
Beech, cedar and hawthorne show off, problem-solving, a puzzling problem.
Cedar solve the dreaded Coded Hundred Square!
Today we have been solving the riddle of the Coded Hundred Square. Well done to all the children who persevered and tried to find a solution. Here are two examples from William and Alex and Hala and Nabil.
Live Problem: Which Scripts?
If you feel like a challenge, why not try solving this problem? Remember, if you’re finding it hard, keep searching for a solution; draw the problem; discuss it with someone, but don’t give up straight away! On the left-hand side of the page there are some tips for getting started on the problem.
When you have found a solution, you can also submit it to the Nrich website. You never know; the people at Nrich may publish your solution on the website where it will remain forevermore. Good luck!
Tea Cup Challenge!
‘Mathematics is not about numbers, but about life. It is about the world in which we live. It is about ideas. And far from being dull and sterile, as it is so often portrayed, it is full of creativity.’
Some light reading for any interested parents out there: The Elephant in the Classroom: Helping Children Learn & Love Maths by Jo Boaler
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Problem solving - Working Systematically Bundle - Reasoning and fluency (Yr 3 &4)
Age range: 7-11
Resource type: Worksheet/Activity
28 January 2018
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