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Problem Solving

Connect 4 Factors

Playing solo: The player must fill the grid with the counters from both boxes. They must avoid lining up four numbers with a common factor (other than one).

Playing duo: Each player has a box of counters to choose from. Take it in turns to drop a counter into the game board. The winner is the first to line up four numbers with a common factor (other than one).

There are nine squares shown. The squares have side lengths of   1 ,  4 ,  7 ,  8 ,  9 ,  10 ,  14 ,  15  and  18  cm. They can be fitted together with no gaps and no overlaps to form a rectangle. How?

Factors and Multiples

The aim of the game is to try and make the longest chain possible by connecting factors and multiples.

Thinking Skills

In this activity your children will be working on capacity. They must create a given amount in one of the containers in the least amount of moves. Sounds simple eh?

The game has different levels to complete and will really get your children thinking.

Super for outside the box thinking.

Puppies Puzzle

The puppies need to go to sleep. There needs to be an even amount of puppies in each basket to make this happen. How many ways can they be sorted?

Broken Calculator

This is a great activity that encourage the children to use limited numbers and functions to make totals.

The children can progress through the levels to challenge themselves even further.

Pets Puzzle

Can you solve the Pet shop Puzzle? Mr Jones sells two kinds of animals and there are 44 animal legs in the shop. Over half of the pets are puppies. How many puppies and budgies are there in the pet shop?

River Puzzle

How can you safely bring the goat, wolf and cabbage  across the river?

The sum of each side of the triangle should equal the number in the centre of the triangular shape.

The first player to make a square on the grid wins. Encourage your children to think about developing winning strategies and movement patterns.

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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

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Play of the Wild

Education is an admirable thing, but it is well to remember from time to time that nothing that is worth knowing can be taught. -Oscar Wilde

Outdoor Problem Solving Activities KS2- Learning Maths

Outdoor Problem Solving Activities KS2 – Learning Maths Outdoors

These are some ideas for outdoor problem solving activities for KS2 to help children with learning maths outdoors. One of the most critical aspects of teaching and learning maths is to be able to solve problems. While teaching maths in school, I found that it can become easy to get overly focused on teaching the rules and procedures for doing maths. These are essential tools that everyone needs, but the actual point is to be able to apply those skills in real life. Therefore, children must be given opportunities to practice problem-solving and make it purposeful.

I hope you find these outdoor problem solving activities for KS2 useful, and perhaps they will also inspire you to try other ideas. If you would like some other ways to take learning maths outdoors, you may also want to see my post, Outdoor Maths Activities KS2 .

Nim is a mathematical strategy game where two players take turns removing objects from a pile. Each player must take at least one object per turn. The goal is to either take or avoid taking the last item from the stack. Children can play nim with a pile of sticks or rocks.

Ordering natural objects by different features

Children can place in order objects such as rocks or pinecones based on the characteristic(s) they decide upon. It might be longest to shortest, least to the greatest circumference or smallest to greatest volume.

Observing the sun & moon

Children can explore and investigate the sun and moon, including changes that take place over time. They can try to figure out some of the following questions, and ask some of their own questions as well.

• What time does the sun set and rise? Does this ever change? How do you know?
• Where on the horizon do you first/last see the sun or the moon? Does this change? How do you know?
• Can you observe the phases of the moon- how does it change? Are there any patterns that you notice?

Measuring circles

Children can measure circles (such as flower pots, tree stumps or other circular objects found outside). They can measure the circumference, radius and diameter and then investigate the relationship between radius/diameter and circumference. What do children notice? Is there a pattern? They may even be able to ‘discover’ pi.

How tall is a tree?

Measure / calculate the height of a tree with the shadow & calculation method, triangle method and/or clinometer method.  If children try more than one method, do they get the same results?  Which method might be more accurate?

(ex. Estimating by height, clinometer method, looking through legs method, pencil method, meter stick method)

Different ways children can help with planning in a garden

Children can figure out how much space is available in the garden and how many different types of plants can be planted. Can they figure out how many of one kind of plant will fit into one planter box? Can they figure out how many different sized plants fit into the same planter box? Children would need to use their measuring skills to calculate the surface area of the garden. Then they would need to find out how much space each plant requires (e.g. from seed packets) and then determine how many and which of the plants can be used.

If a new planter box is purchased, children can help work out how many bags of compost would be needed to fill it. (They would need to measure, calculate the size, etc.) Will there be any leftover soil from one of the bags? How do you know?

If children grow crops such as pumpkins, they can do things like ordering them from heaviest to lightest. This way they might see which is the ‘prize-winning pumpkin’ in a harvest festival.

They might also consider how much each pumpkin could be sold for.  This would involve calculating each pumpkin’s cost, based on a price of £1.00 per kilo, £2.00 per kilo (or whatever reasonable price is determined). They might also try to figure out if larger pumpkins (or other crops like corn) always weigh more than smaller ones? *The children will have to define what is the larger, longer, or wider circumference.

Children could get involved with selling the crops they grow.  This will give them plenty of opportunities to use mathematical skills and handle money in real-life contexts. They will also need to plan the pricing of different vegetables based on weight, the number of vegetables, or selling them in combinations. Children could even try to figure out the appropriate amount to charge for each crop based on the cost of the seeds, the growing time, grocery store costs or any other factors.

If they do sell some of what they grow after school or possibly at a market, then the children can apply their maths skills to figure out how much each customer will need to pay when buying any particular fruit or vegetable.  

Monitoring Plant Growth

See if children can figure out how quickly different plants grow. Can they determine the rate of growth (e.g. mm per week)? Which seedlings / plants grow fastest? Is it a steady rate of growth, or does it change?

*To take this further, children might also conduct an experiment to compare different groups. They might compare the growth or the growth rate of plants whose seeds have been frozen vs those that have not, or something else.  

Making Shapes

How many shapes can you make (including shapes within shapes) using a set number of sticks (ex. 6 large sticks and 6 small sticks)? Is there a way to make more or fewer shapes using the same number of sticks?

Planning and holding a bake sale

(Some parts of this activity take place inside and some outside. This activity can be linked further to learning maths outdoors if children use some ingredients grown from a school garden).

Baking for a bake sale is a great way to give children hands-on practice solving problems in real contexts. For a baking project they will need to follow recipes and accurately use measuring cups and weighing scales. They might also want to make larger quantities and scale up recipes by doubling, tripling, or quadrupling them. The children will need to plan ahead and calculate how much of each ingredient they need to buy to make a set number of cookies, cupcakes, etc. They must also determine how many batches would be needed to make 200 cupcakes if the recipe makes 2 dozen.

Then when they hold an actual sale they will be using their maths skills to calculate how much to charge people.  They must also learn how to give people the proper change (just like selling vegetables above). They can setup a stall outside of school and sell them after school one day. 

Organizing and running a track & field event

Children can get involved in organizing a sports day or track and field event. They can first decide which events to include and then figure out how much space is needed for each activity (e.g. measuring the appropriate length and width required). The children can then help measure and set up the activities.

Once races are held, that data can be used to make calculations. For example – for a jumping event, children can measure how far people jump, and after several tries, figure out if there is improvement and how much (finding the difference). Children can time how far it takes them to run certain distances, e.g. 500 meters versus ½ a mile. They can figure out how fast they are running. They might then calculate how fast they ran different races (e.g. miles per hour) and then figure out when they ran faster or slower.  

Finding ways to approximate measurements

See if children can find different ways to measure the approximate distance between two far points with a meter stick and string. This might be the length of the playground or the distance between two trees, etc. Children might compare different ways of measuring the approximate distance such as measuring the length it takes them to take one step and then counting the number of steps between two points. They might also use the string to go between the two points and measure the string’s length. See if they can find any other ways to find the approximate distances.

I hope that you find these outdoor problem solving activities for KS2 helpful and it helps you take teaching and learning maths outdoors!

Arithmetic , Gardening , Geometry , Maths , Measurement , Number & Place Value , School Age , Sticks

hands-on learning , Learning Outdoors , learning outside , outdoor learning , Sticks

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Developing Critical Thinking Skills At KS2 Using Same Surface Different Depth Problems: How I Wish I’d Taught Maths (5)

Clare Sealy

Clare Sealy looks at the struggles primary school pupils can have when implementing critical thinking skills when subject knowledge is lacking, and the effect this can have on their attempts at problem solving activities in KS2.

This article is part of a series published to help primary school teachers and leaders implement some of the insights and teaching techniques derived from Craig Barton’s bestselling book How I Wish I’d Taught Maths . Links to the other 5 articles appear at the end.

In the introduction to this series, I outlined how Craig Barton, in his book How I wish I’d taught maths,  described how he had changed his teaching as the result of reading research around learning and memory , in particular cognitive load theory in the classroom . In the latter part of his book, the focus turns to helping pupils use what they know.

Whatever the age of the children we teach, many find it hard to transfer what they know how to do in one context to another. This is most evident when it comes to maths problem solving , or in the SATs reasoning papers. They know the maths, they just can’t work out which bit of maths they need in this specific circumstance.

What are critical thinking skills?

At the basic level, critical thinking is the ability to analyse facts presented to us to form a judgement about a topic. It is an incredibly important skill to have at higher education and beyond, and is one of the key factors in astute decision making. 

Most of us explicitly encounter critical thinking and other higher order thinking skills such as metacognition in the classroom at either GCSE or A-Level, but laying their foundations at primary school is becoming more and more common and valued. 

However, “critical thinking and the ability to solve problems is not a generic skill that can be taught and that children can transfer from one problem to another. While there are some metacognitive strategies that can help a bit, what is really crucial is having a very secure understanding of the actual maths – the domain specific knowledge – that lies at the heart of the problem.” Daniel Willingham (2006)

Critical thinking (as well as scientific thinking and other domain-based types of thinking) is not a skill. There is not a set of critical thinking skills that can be acquired and deployed regardless of context. There are, however, metacognitive strategies that (once learned) make the critical thinking process more likely, and make up a key part of many quality first teaching methods .

The ability to think critically (to actually do what the metacognitive strategies call for) depends on domain knowledge and practice. For teachers, the situation is not hopeless, but no one should underestimate the difficulty of teaching pupils to think critically.

This said, metacognition is an incredibly valuable skill for pupils to have for any number of reasons, from helping low-ability students catch up to their peers to helping the whole class minimise the impact of the summer slide .

The metacognitive strategies mentioned involve reflecting on what you are doing during problem solving activities in KS2, asking yourself questions such as:

‘What am I doing?’

‘Why am I doing this?’

‘How does it help me?’

This is all very well if you have secure domain knowledge and can answer these questions. However, if you lack this knowledge, the questions are just frustrating.

Crib Sheet for How I Wish I'd Taught Primary Maths

Download the key findings from research; share with your staff, your SLT, and at your next job interview!

How to help your pupils develop critical thinking skills for problem solving activities throughout KS2

There are of course  problem solving strategies  we can give pupils to help them become better critical thinkers. For example, underlining the important words. However, this relies on pupils understanding what the important word are in the first place.

Often, irrelevant surface features seem important to pupils whereas we experts can see they are completely irrelevant, because our domain knowledge and experience of answering many, many questions means we can spot the deep underlying structure a mile off.

It’s the same with other strategies such as setting work out systematically (you have to know what system is likely to be helpful), working backwards (you have to know whether this is likely to be useful in this situation) or even using a bar model. Bar models can be so helpful, but you have to know whether or not this kind of question is suitable for the bar model treatment.

‘Same surface, different deep’ or SSDD Problems

Problem solving maths questions usually have an arbitrary surface structure and a deep structure. The surface structure involves the context in which the problem is set and has nothing to do with the actual maths; for example, in a question about buying tickets to a funfair, the funfair and ticket are part of the surface structure.

They are but the wrapper in which the real maths is wrapped. Pupils can get fixated on this ‘wrapper’, rather than the underlying deep mathematical structure held within it.

I recall a SATs question about paving inside a greenhouse. The child thought that they couldn’t do it because they didn’t know what a greenhouse was! Whereas I immediately knew that this was going to be an area question. The surface structure was transparent to me whereas, it was thoroughly opaque to the pupil.

All the underlining, systematic working or bar modelling in the world wouldn’t get past this erroneous latching onto surface features.

How to get past the surface features 

To overcome this hurdle, Craig recommends teaching children to recognise the deep structure of maths problems and how to identify and then disregard surface features.

It should go without saying that children need to be thoroughly secure in the underlying maths before attempting problem solving.

It is a mistake to think that maths problem solving is a good way of consolidating learning, let alone using it in the initial knowledge acquisition phase. Problem solving is about transferring learning from one context to another.

Problem solving at KS2 is about using your critical thinking skills to generalise

It therefore comes at the end of learning to do something, not mid- way and definitely not at the beginning.

But what is more, if at the end of a unit on, say division, we give children a load of division problems, this will not help them work out what the deep structure is. They already know; it’s division! This is fine, but it won’t help children learn to decide whether or not a particular problem requires division or not.

As well as problem solving activities at the end of units, teachers also need to allocate separate times where children have to work out what the deep structure of a problem actually is, regardless of surface features.  

This means setting a range of SSDD problems sharing the same surface features – for example a shopping problem involving apples and pears – but which each have a different deep structure .

Read more: KS2 Problem Solving and KS3 Maths Problem Solving

Translating this to a primary school context.

Let’s return to the question about stickers from the 2017 KS2 SATs paper we considered when considering goal free problems:

The surface feature here is stickers.

As experts, we know straightaway that we could substitute packs of stickers with boxes of apples or packets of balloons or even a family ticket to the cinema.

In fact, in a variation of Craig’s SSDD technique for a primary context, I’d suggest also doing DSSD problems (different surface, same deep) problems too, asking children to cross out the words ‘pack of stickers’ and replace with suitable alternative, and then repeat the problem to understand that the surface features do not change the underlying maths at all.

Then I’d suggest moving on to SSDD problems, with appropriate differentiation in the classroom . Let’s stick with stickers as our ‘same surface’.  

The deep structure of our original question involved knowing that you had to multiply to find the price of 12 separate stickers and then subtracting to find the difference. But we could ask mathematically different questions while keeping the context and visual look of the problem the same.

For example:

• How much does one sticker cost? (though I’d adapt the price so the division came out as a whole number of pence)
• Stickers are 8cm wide and 6cm high. Ally sticks 3 stickers in a row, without any gaps. What is the perimeter of the shape she has now made?
• Ally buys 7 packs of stickers a month, Jack buys 3 packs of stickers a month and Chen buys 5 packets a month. What is the average number of packets bought by the 3 children in one month?
• Ally buys a pack of 12 stickers. She has spent 15% of her birthday money. How much birthday money has Ally got left? (again, I would adjust the price into something more workable)

Another great way to translate problem solving into a primary context is through topical maths investigations.

Extension ideas for problem solving activities in KS2

Extending both ideas, we could make a grid where the rows contained questions with a different surface structure and the columns contained questions with the same deep structure. This grid could be cut into individual boxes with pupils having to sort each box accordingly, to reconstruct the grid.

Tigers, Cake or Money? A unique approach to critical thinking

One questioning in the classroom strategy for helping children understand the deep structure of division problems, is to ask children if this is a tiger, cake or money sort of division question.

What this means is, could we swap the surface features of the problem we are given to one involving tigers, or cake or money?

Why these three I hear you ask?

This is because, where division problems do not divide exactly, it is really useful to:

• Be able to decide if you need to round up or down (These are the tiger questions. If you haven’t got enough cages for your tigers you might get eaten)
• Have a remainder that’s a fraction (These are the cake questions as we can each have 1 and a half cakes)
• Or have a remainder expressed as a decimal (These are the money questions as we can have £2.47 each)

An example of a Tiger question

This is a great example of a tiger question. With 4 spare tigers, you need to have an extra box! Having 2/3 of a box wouldn’t work, neither would having 0.666 of a box. Rewriting this as a tiger question helps understand the deep structure.

A cage holds 6 tigers

How many cages are needed to hold 52 tigers?

How to make a trickier Tiger question

Here is a slightly harder ‘tiger’ problem:

Let’s rewrite this:

A zookeeper has 7,600 tigers (!)

Cages can contain 500 tigers.

How many cages does the zookeeper need?

The answer:

15.2 cages is obviously not enough to stop the keeper from being eaten.

Answers requiring a decimal answer are usually money questions already, or calculations rather than word problems. Hence they are easy to categorise.

Some children find ‘tiger’ type questions particularly hard, and give answers that don’t make sense because they haven’t rounded up or down. So in the brick example above, they give the answer as 15.2 because they haven’t recognised that doesn’t make sense.

By naming certain deep structures, children are more able to identify them when they arise, and this is a fantastic way to help children with problem solving activities throughout KS2.

Sources of Inspiration

• Willingham, D.T. (2006) ‘How knowledge helps: it speeds and strengthens reading comprehension, learning and thinking’. American Educator 30 (1) p.30

This is the fifth blog in a series of 6 adapting the book How I Wish I’d Taught Maths for a primary audience. Some have already been mentioned in this post, but if you wish to read the remaining blogs in the series, check them out below:

• Direct Instruction: How I Wish I’d Taught Maths (2)
• Deliberate Practice In Education: How I Wish I’d Taught Maths (4)
• How Retrieval Practice Helps Long-Term Maths Skills: How I Wish I’d Taught Maths (6)

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