common factors problem solving ks2

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Number Year 6 Block 2 Week 5 - factors and common factors

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

4 April 2022

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Factors, common factors and the highest common factor for Year 6 from Block 2 - Week 5

Differentiated worksheets with reasoning and problem solving for the rest of this week can be found here: https://www.tes.com/teaching-resource/number-year-6-block-2-week-5-factors-multiples-primes-prime-factors-square-numbers-and-cubes-12181791

In this lesson, pupils will gain fluency practice when finding factors, common factors and the highest common factor.

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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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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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KS2 factors: guide for parents

By Atom | Nov 13, 2023, 3:56 PM

As your child moves up through Key Stage 2, they’ll start to learn about factors and prime numbers in maths lessons. If you’re looking to help your child learn at home but not sure where to start, we’re here to help!

Keep reading to:

Learn about factors and prime numbers

Understand what your child will be taught at school

Plus, download free factors worksheets for Years 5 and 6!

What is a factor?

A factor of a number defines the integers (whole numbers) which divide exactly into that number with no remainder. 5 is a factor of 35 because 35 ÷ 5 = 7 .

Every number can be represented as a product of 2 of its factors. For example, we can represent the number 12 in 6 different ways:

1 group of 12

2 groups of 6

3 groups of 4

4 groups of 3

6 groups of 2

12 groups of 1

The reason for this is that the number 12 has 6 different factors: 1, 2, 3, 4, 6 and 12.

What are common factors?

Common factors describe the factors which two different numbers can both be divided by. For example, take the factors of 16 and 18:

Factors of 16: 1, 2 , 4, 8, 16

Factors of 18: 1, 2 , 3, 6, 9, 18

The only common factor (other than 1) between these two numbers is the number 2. Since 16 isn't a factor of 18, both numbers also have factors which are unique to them. 12 and 16 do not share all their factors, but still have 3 common factors: 4, 2 and 1.

In Key Stage 2 maths tests, your child might see factor questions involving a Venn diagram . Here’s an example of one of these types of questions on Atom .

In the right (red) circle, we see all the factors of 12. These are 1, 2, 3, 4, 6 and 12.

All the numbers in the left (blue) circle are in the intersection, meaning there are no factors of the left number that are not factors of 12. This means that all the factors of the left number are also factors of 12 – so the number itself must be a factor of 12.

The factors in the intersection are all the factors of 6, so the left number is 6. The correct answer is 6 & 12 .

Your child might also come across longer worded questions. Take a look at this example from Atom :

To solve this problem, we need to find a common factor of 3 , 6 and 15 . This will give us the number of boxes.

The factors of 3 are 1 and 3

The factors of 6 are 1, 2, 3 and 6

The factors of 15 are 1, 3, 5 and 15

The only common factor of 3 , 6 and 15 is 3 . Ron can pack 3 identical boxes. To find out how many pastries are in each identical box, we divide their total by 3. Each box will have 1 chocolate muffin (3÷3), 2 red velvet cakes (6÷3), and 5 swiss rolls (15÷3).

So the answer is C – 5 swiss rolls.

Free factors worksheets for Years 5–6

Support your child’s understanding of factors with these free Key Stage 2 worksheets! Includes factors worksheets and spelling lists for Years 5 and 6.

What is a prime number?

A prime number is a whole number that has exactly two factors: 1 and itself. When a prime number is divided by 1 or itself, the result is a whole number with no remainders, decimals or fractions.

1 is not a prime number because it can only be divided by one number: itself.

The first prime number is 2, because it can be divided by two numbers: 1 and itself. 2 is also the only even prime number, as all even numbers can be divided by 2.

The opposite to a prime number is a composite number . A composite number has more factors than 1 and itself. For example, 6 is a composite number because it can also be divided by 2 and 3.

Prime factor trees

All integers (whole numbers) can be expressed as a product of their prime factors . A prime factor is a prime number that is a factor of a number.

All integers can be broken down into prime factors. We do this by repeatedly dividing the number by integers until it becomes a series of prime numbers. A prime factor tree can help us find them.

Below are two prime factor trees for 30. The prime factors are shown in light blue, at the end of the branches.

It doesn't matter which way you split up the number – the prime factors will always be the same.

What will my child learn about factors in Key Stage 2?

Factors and prime numbers are generally not taught until Year 5 – the start of Upper Key Stage 2. Your child will learn:

How to find all the factor pairs of a number, and common factors of two numbers

The vocabulary of prime numbers, prime factors and composite numbers

How to work out whether a number up to 100 is prime

To recall prime numbers up to 19

How to use factors to solve problems involving multiplication and division

In Year 6, your child will continue to build upon their knowledge of factors and prime numbers from Year 5. They will be expected to identify common factors, common multiples and prime numbers, and use common factors to simplify fractions.

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Factors and Multiples Worksheets

Welcome to our Factors and Multiples Worksheets. Here you will find a wide range of free Math Worksheets which will help your child to learn to use multiples and factors at a 4th Grade/ 5th Grade level.

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Factors and Multiples Worksheet

These sheets have been designed to support your child with their learning of multiples and factors.

The sheets are graded in order of difficulty with the easiest sheet coming first in each section.

Using these sheets will help your child to:

• Know and understand what multiples and factors are;
• apply knowledge of multiples and factors to solve problems;
• Develop and practice their mental calculation skills.

Want to test yourself to see how well you have understood this skill?.

• Try our NEW quick quiz at the bottom of this page.

Quicklinks to ...

Multiples and Factors Help

Multiples worksheets, factors worksheets.

• Factors and Multiples Riddles
• Easier/Harder Worksheets
• More related resources

Factors and Multiples Online Quiz

If a number is a multiple of another number, it means that it can be made out of adding groups of the other number together.

12 is a multiple of 4 because 4 + 4 + 4 = 12 (or 4 x 3 = 12)

27 is a multiple of 9 because 9 + 9 + 9 = 27 (or 9 x 3 = 27)

17 is not a multiple of 4 because it cannot be made by adding groups of 4 together.

A factor is a number that divides into another number with no remainder.

In other words every number is divisible by each of its factors.

1 is a factor of every whole number, because every integer is divisible by one.

3 and 7 are both factors of 21 because 3 x 7 = 21

10 and 6 are both factors of 60 because 10 x 6 = 60

7 is not a factor of 24 because 24 is not divisible by 7 (24 ÷ 7 = 3 remainder 3).

Multiples and Factors are connected with each other:

• if we know that 3 is a factor of 12, then 12 is a multiple of 3
• if we know that 33 is a multiple of 11, then 11 is a factor of 33.
• also, if we know that 24 is not a multiple of 7, then 7 is not factor of 24.

The example below shows the relationship visually.

If we know that 3 is a factor of 24, then 24 must be a multiple of 3.

If we know that 24 is a multiple of 3, then 3 must be a factor of 24.

Multiples and Factors Worksheets

We have split our worksheets into 3 different sections:

• the first section contains only worksheets about Multiples
• the second section contains only worksheets about Factors
• the third section contains worksheets with both Factors and Multiples
• Multiples Sheet 4:1
• PDF version
• Multiples Sheet 4:2
• Multiples Sheet 4:3
• Multiples Sheet 4:4

We have two worksheets on finding Factor Pairs up to 100.

We have two worksheets which involve finding all the factors of different numbers.

• Factor Pairs Worksheet 1
• Factor Pairs Worksheet 2
• Factors Worksheet 4:1
• Factors Worksheet 4:2
• Factors and Multiples Worksheet 4:1
• Factors and Multiples Worksheet 4:2
• Factors and Multiples Worksheet 4:3

Factor and Multiples Riddles

Using riddles is a great way to get children to apply their knowledge of factors and multiples to solve problems.

It is also a good way to get children working collaboratively and talking about the language together.

Each riddle consists of some clues and a selection of possible answers.

Solving the clues gradually eliminates all the incorrect answers leaving just one solution.

• Factors and Multiples Riddles 1
• Factors and Multiples Riddles 2

Looking for some easier Multiples Sheets

The sheets in this section cover similar areas to the worksheets on this page but are at an easier level.

• round a number to the nearest 10, 100 or 1000;
• use the > and < symbols correctly for inequalities;
• use multiples and apply them to solve problems.
• Rounding Inequalities Multiples Worksheets

Looking for some harder Factors and Multiples Worksheets

We also have some more advanced worksheets about multiples and factors.

The worksheets below are more suitable for 6th graders and above.

• Greatest Common Factor Worksheets
• Least Common Multiple Worksheets
• Factor Tree Worksheets (easier)
• Prime Factorization Worksheets (harder)

More Recommended Math Resources

Take a look at some more of our worksheets similar to these.

Divisibility Rules 1-10 Chart

We have a range of charts which can help you determine whether a number between 1 and 10 is a factor of a number.

• Divisibility Rules 1-10 Charts

Balancing Math Equations Worksheets

The sheets in this area will help your child understand the use and purpose of the equals sign (=) in an equation.

It will also help children learn to start manipulating and calculating numerical expressions so that they are equivalent.

This will stand them in good stead for when they start to learn algebra, and manipulate algebraic equations.

• Balancing Math Equations

Sieve of Erastosthenes

The Sieve of Erastosthenes is a method for finding what is a prime numbers between 2 and any given number.

Eratosthenes was a Greek mathematician (as well as being a poet, an astronomer and musician) who lived from about 276BC to 194BC.

If you want to find out more about his sieve for finding primes, and print out some Sieve of Eratosthenes worksheets, use the link below.

• Sieve of Eratosthenes page

Want to find out more about primes?

Take a look at our Prime Number page which clearly describes what a prime numbers is and what they are not.

There are also many different questions about prime numbers answered, as well as information about the density of primes.

• What is a Prime Number

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This quick quiz tests your knowledge and skill with factors and multiples here!

Fun Quiz Facts

• This quiz was attempted 1,569 times last academic year. The average (mean) score was 14.8 out of 20 marks.
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Common Factors — Party Bag Problem Solving: Foundation (Year 6)

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Enhance children's knowledge and understanding of common factors with this party-bag themed problem solving worksheet. Children must look at the list of factors and work out the common factors for the pairs. Can they work out how many of each item needs to go in each party bag?

This worksheet is available in different levels. Please select the drop down menu to select another level.

• Key Stage: Key Stage 2
• Subject: Maths
• Topic: Multiplication & Division
• Topic Group: Calculations
• Year(s): Year 6
• Media Type: PDF
• Resource Type: Worksheet
• Last Updated: 23/11/2023
• Resource Code: M2WFT14810
• Curriculum Point(s): Identify common factors, common multiples and prime numbers.

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