Do You Know The 33 Mental Maths Strategies (KS1 and KS2) You Should Be Teaching In Years 1, 2, 3, 4, 5 and 6?

In this article we explain what mental mental maths is, why it’s important that KS1 and KS2 pupils develop it and which mental maths strategies your pupils should have clearly embedded by the end of Year 6, and why they are needed at KS1 and KS2.

We also provide a clear progression for teaching mental maths strategies with specific tips on how to teach mental maths at every stage of primary school – from Year 1 right the way through to the mental arithmetic skills required in Year 6 for the KS2 SATs to enable pupils to recall and apply number knowledge rapidly and accurately.

To help you find the part of the blog that’s most relevant to you, we’ve included some jump links below which will take you to the right place straight away!

What is mental maths

Mental maths is the ability to work out calculations of all sorts in your head (not just addition sums). It means being able to give an answer to a question without having to write down every step of the calculation, although making the odd note is acceptable.

Good mental maths skills in primary school generally show that a child has grasped what numbers represent, can spot patterns, and has developed excellent fluency and recall.

What are mental strategies in maths?

Mental strategies in maths are accepted ways of working maths out in your head that help us to take shortcuts and get to the correct answer in an efficient way.

Why are mental maths strategies important?

Mental strategies are the foundations for most of the areas of mathematics that use numbers.

Without efficient mental strategies, children can often struggle to quickly and fluently calculate.

It is also worth noting that mental strategies are also the foundation of any written or formal method in mathematics.

Added to all of this is the fact that we can’t ignore that mental maths appears six times in the content domain breakdowns for the government’s KS2 Mathematics Test Framework; it’s one of the factors that the national assessments are trying to assess so let’s make sure we know how to teach it!

Ultimate Mental Maths PowerPoint!

Download this editable mental maths ppt which is packed with everything you need to teach the ultimate mental maths lesson including; procedural variation, conceptual variation and dozens of prepared questions!

Developing true fluency in mathematics

Effective mental strategies are important if children are to develop ‘true’ fluency.

‘True’ fluency can be best defined as children being able to confidently use and apply their knowledge of number relationship, number facts and our number system in order to calculate and solve problems.

It is worth remembering that fluency in maths is not simply restricted to being able to ‘recall’ known facts. More accurately, it is how children can use and apply these facts, including through a range of mental maths strategies, that are important.

“Low achievers are often low achievers not because they know less, but because they don’t use numbers flexibly.” – Jo Boaler

Be careful not to mislabel mental maths skills

One important thing to remember when working to develop ‘true’ fluency is that accuracy is not the same as fluency.

For example, consider the following scenarios, which while accurate, may not necessarily be classed as fluent:

  • A Year 2 child calculating 40 + 8 by counting in ones;
  • A Year 4 child calculating 1003 – 998 using a formal written method;
  • A Year 6 child calculating 41.79 + 25.3 + 25.7 – 41.79 by adding the first three numbers and then subtracting the fourth.

This extract from the research paper, Developing Computational Fluency with Whole Numbers published in 2000 by S J Russell, remains one of the best explanations of fluency:

Fluency rests on a well-built mathematical foundation with three parts

  • an understanding of the meaning of the operations and their relationships to each other — for example, the inverse relationship between multiplication and division;
  • the knowledge of a large repertoire of number relationships, including the addition and multiplication “facts” as well as other relationships, such as how 4 × 5 is related to 4 × 50;
  • and a thorough understanding of the base ten number system, how numbers are structured in this system, and how the place value system of numbers behaves in different operations — for example, that 24 + 10 = 34 or 24 × 10 = 240.

Rapid recall vs mental calculations with jottings

When we discuss mental calculations in maths at Key Stage 2, we need to be clear about the distinction between facts that children should be able to rapidly recall vs the types of calculations that children should be able to calculate mentally, sometimes with the support of jottings.

Rapid recall of number facts is important because if children are able to recall number facts automatically (in other words, they gain automaticity in these facts) it allows them to free up their working memory when faced with more complex questions and word problems across the whole maths curriculum, including 2-step and multi-step word problems as they progress up the school.

Retrieval practice is one way of helping pupils gain this automaticity.

They are also able to more efficiently and accurately solve problems, reason and make connections if they are not having to repeatedly calculate the same ‘basic’ facts.

“In teaching procedural and factual knowledge, ensure the students get to automaticity. Explain to students that automaticity [with key number facts] is important because it frees their minds to think about concepts.” –

Daniel Willingham- cognitive scientist, in Is it true that some people just can’t do maths?

However there are only a limited number of number facts that should be expected to be rapidly recalled.

Teaching the maths facts conceptually first at KS2

The first thing to do is to teach the underlying maths for these facts conceptually (i.e. number bonds to 10 are conceptually taught). Then, only when children have a secure conceptual understanding, should rapid recall/automaticity be attempted.

There are many other calculations that, depending on their year group, children should be able to calculate mentally, sometimes with the support of jottings, and these calculations are underpinned by the number facts which they can rapidly recall.

For example, by Year 6 most children should be able to calculate 34 x 5 mentally (30 x 5 + 4 x 5) by partitioning and using their knowledge of the distributive law supported by basic jottings.

Using mental strategies in the SATs arithmetic test

Efficient mental calculation strategies are key to success in the KS2 SATs Paper 1- Arithmetic.

In the Arithmetic paper, over 80% of questions are designed to be able to be solved mentally, or through jottings.

However, many children attempt the majority of the paper using formal written methods, which leads to them running out of time and not completing the paper.

The answer to how to improve mental maths skills is quite simple: start by learning the number facts needed for rapid recall in each primary school year.

Mental maths strategies for Year 1 to Year 6

Here is your comprehensive list of mental maths facts and strategies you should teach each year group with the aim of rapid recall.

Mental Maths Year 1

There is such a wide range of maths ability in Year 1 that it’s generally not practical to expect a whole class to have achieved rapid recall of all the Year 1 maths facts so we’re starting with Year 1.

Mental Maths Year 2

Here are the mental maths facts and strategies to teach in Year 2.

By the end of Year 2, and by the time children reach KS2, they should have developed rapid recall in mental maths for their basic addition and subtraction facts.

This includes all number bonds to 20, and doubles to 20.

The table below shows the basic addition facts that children should know as part of their Year 2 mental maths practice by the time they reach Year 3.

It is also worth noting that pupils should also be able to rapidly recall the related subtraction facts.

Year 1 & 2 mental maths rapid recall grid
The basic addition facts children should know by Year 3.

These facts must be kept current and practised and used regularly.

Without this regular practice, it often becomes the case that you can ask a Year 2 child 4 + 7= ? and they will give you the answer within a few seconds, whereas  if you pose the same question to a Year 6 child they will take much longer as the practice hasn’t been there!


Mental Maths Year 3

Here are the mental maths facts and strategies to teach in Year 3.

This table shows the times table facts that children should be able to rapidly recall by the end of Year 3. You will see that in Year 3 mental maths knowledge continues to play an important role.

It is also important to note that, as well as the multiplication facts, children should be able to rapidly recall the related division facts as well. For example, for 4 x 6 = 24, children should know 24 ÷ 6 = 4 and 24 ÷ 4 = 6.

In addition to the facts above, and their basic addition facts, Year 3 children should be able to answer the following mental maths questions:

  • Addition and subtraction of multiples of 10 where the answer is between 0 and 100 (e.g. 70 + 30 = 100, 20 + 40 = 60)
  • Double and halves of multiples of 10 to 100 (e.g. double 60 = 120)
  • Multiplying two-digit number by 10. (e.g. 24 x 10 = 240)

Mental Maths Year 4

Here are the mental maths facts and strategies to teach in Year 4.

By the end of Year 4, children should be able to rapidly recall the multiplication and division facts for the 1-12 x tables.

The government is introducing the multiplication check for Year 4 pupils from 2020, with a ‘taster’ year in 2019 to underline how important rapid recall of multiplication facts truly is.

This test covers just multiplication facts, and you can find out all you need to know about the multiplication check in this blog post.

However, as the table below shows, by the time children get to their Year 4 mental maths, they only have 38 new multiplication (and related division) facts left that they need to develop the ability to rapidly recall. 

Year 4 Mental Maths Rapid Recall Chart
The 38 new multiplication (and division) facts that children need to know by the end of Year 4.

In addition to the facts above, and their basic addition facts, Year 4 children should be able to answer the following mental maths questions:

  • Addition and subtraction of multiples of 10 (e.g. 70 + 30 = 100, 50 + 60 = 110, 20 + 40 = 60);
  • Addition and subtraction of multiples of 100 where the answer is 1,000 or less (e.g. 300 + 400 = 700, 400 + 600 = 1,000);
  • Double and halves of multiples of 10 to 100 (e.g. double 60 = 120, half 50 = 25);
  • Multiplying two-digit numbers by 10 (e.g. 24 x 10 = 240);
  • Halves of any even number to 100 (e.g. half of 22 = 11);
  • And multiplying any two and three-digit number by 10 and 100 (e.g. 24 x 100 = 2,400)

Mental Maths Year 5

Here are the mental maths facts and strategies to teach in Year 5

Together with the 1-12 x multiplication and division facts, and their basic addition facts, children should be able to answer the following Year 5 mental maths questions:

  • Addition and subtraction of multiples of 10 (e.g. 70 + 30 = 100, 50 + 60 = 110, 20 + 40 = 60);
  • Addition and subtraction of multiples of 100 (e.g. 300 + 400 = 700, 400 + 600 = 1,000, 800 + 500 = 1,300);
  • Addition and subtraction of multiples of 1000 (e.g. 3000 + 4000 = 7000);
  • Double and halves of multiples of 10 to 100 (e.g. double 60 = 120, half 50 = 25);
  • Quadruples (x4) of all numbers to 10 (e.g. 6 x 4 = 24);
  • Multiplying two-digit number by 10. (e.g. 24 x 10 = 240);
  • Halves of any number to 100 (e.g. half of 22 = 11, half of 51 = 25.5);
  • Multiplying and dividing any number by 10 and 100 (e.g. 24 x 100 = 2,400, 45 ÷ 100 = 0.45, 3.4 x 10 = 34);
  • Squares of all number up to 12;
  • And cubes of 2,3,4 and 5.

Mental Maths Year 6

Here are the mental maths facts and strategies to teach in Year 6.

Together with the 1-12 x multiplication and division facts, and their basic addition facts, children should be able to answer the following Year 6 mental maths questions:

  • Addition and subtraction of multiples of 10 (e.g. 70 + 30= 100, 50 + 60 = 110, 20 + 40 = 60);
  • Addition and subtraction of multiples of 100 (e.g. 300 + 400 = 700, 400 + 600 = 1,000, 800 + 500 = 1,300);
  • Addition and subtraction of multiples of 1000 (e.g. 3000 + 4000 = 7000);
  • Double and halves of multiples of 10 to 100 (e.g. double 60 = 120, half 50 = 25);
  • Quadruples (x4) of all numbers to 10 (e.g. 6 x 4 = 24);
  • Multiplying two-digit number by 10 (e.g. 24 x 10 = 240);
  • Halves of any number up to 100 (e.g. half of 22 = 11, half of 51 = 25.5);
  • Multiplying and dividing any number by 10 and 100 (e.g. 24 x 100 = 2,400, 45 ÷ 100 = 0.45, 3.4 x 10 = 34);
  • Multiplication of multiples of 10 and 100 based on known facts (e.g. 40 x 40 = 1,600);
  • Squares of all number up to 12;
  • And cubes of 2,3,4 and 5.

KS2 Mental Maths Calculations Years 3 To Year 6

The rapid recall number facts above go hand in hand with a wider set of mental maths calculations that are needed from Year 3 to Year 6.

What follows is a list of those mental maths calculations.

Mental Addition Strategies and Mental Subtraction Strategies

The entries below help show the progression in the types of addition and subtraction calculations that children should be able to calculate mentally, often supported by jottings.

These are organised by key mental calculation strategies.

These progressions start at the start of Year 3, so it is important to note that there are progression steps that should have occurred in KS1 before children are able to carry out the calculations below mentally.

A child’s ability to fluently carry out these types of calculations mentally depends on their confidence in the rapid recall of the number facts set out above.

This means that these skills are therefore best looked at as a progression, rather than a set of year group expectations.

Counting Forwards And Backwards

Counting forwards and backwards is first encountered in KS1, beginning at one and counting on in ones.

Pupils’ sense of number is extended by beginning at different numbers and counting forwards and backwards in steps, not only of ones, but also of twos, fives, tens, hundreds, tenths and so on.

Mental Maths Progression: Counting Forwards And Backwards

These are the ways you can help your class to progress with counting forwards and backwards:

Counting on or back in tens from any number– e.g. working out 27 + 60= ? by counting on in tens from 27

Progression in mental maths at KS2

Counting on or back in fives from any multiple of 5– e.g. 35+15=? by counting on in steps of 5 from 35.

Progression in mental maths at KS2

Counting on or back in hundreds from any number e.g. 570 + 300= ? by counting on in hundreds from 570.

Progression in mental maths at KS2

Counting on or back in tenths and/or hundredths- e.g. 3.2 + 0.6 = ? by counting on in tenths. 1.7 + 0.55=? by counting on in tenths and hundredths.

Partitioning

Partitioning strategies teach children how to break up large numbers into smaller ones.

It is important that children are aware that numbers can be partitioned- both along the place value boundaries (canonically) and in other ways (non-canonically).

They can then use their partitioning to help them calculate addition and subtraction calculations. This can be extended as children progress through KS2.

Mental Maths Progression: Partitioning

These are the ways you can help your class to progress with partitioning:

Calculations with whole numbers which do not involve crossing place value boundaries- e.g. 23 + 45= ? by 40 + 5 +20 + 3 or 40 + 23 + 5

Progression in mental maths at KS2

Calculations with whole numbers which involves crossing place value boundaries e.g. 49 – 32= ? by 49 – 9 – 23 or 57 + 34=? by 57 + 3 + 31

Progression in mental maths at KS2

Calculations with decimal numbers which do not involve crossing place value boundaries 5.6 + 3.7= ? by 5.6 + 3 +0.7  or 540 + 380= ? by 540 + 300 + 80 or 540 + 360 + 20

Progression in mental maths at KS2

Calculations with decimal numbers which involve crossing place value boundaries e.g. 1.4 + 1.7= ? by 1.4 + 0.6 + 1.1 and 0.8 + 0.35= ? by 0.8 + 0.2 + 0.15

Compensating And Adjusting

Compensation involves adding more than you need and then subtracting the extra off that you have added.

This strategy is useful for adding numbers that are close to a multiple of 10, such as numbers that end in 1 or 2, or 8 or 9.

The number to be added is rounded to a multiple of 10 plus or minus a small number.

For example, adding 9 is carried out by adding 10, then subtracting 1. A similar strategy works for adding decimals that are close to whole numbers.

Mental Maths Progression: Compensating And Adjusting

These are the ways you can help your class to progress with compensating and adjusting:

Compensating and adjusting to 10– e.g. 34 + 9=? by 34 + 10 – 1 or 34 – 11= ? by 34 – 100 – 1 = ?

Progression in mental maths at KS2

Compensating and adjusting multiples of 10 e.g. 38 + 68= ? by 38 + 70 – 2 or 45 – 29 = 45 – 30 + 1

Progression in mental maths at KS2

Compensating and adjusting multiples of 10 or 100 e.g. 138 + 69= ? by 138 + 70 – 1 or 299 – 48 = 300 – 48 – 1

Progression in mental maths at KS2

Compensating and adjusting multiples with decimals e.g

2 ½ + 1 ¾ by 2½ + 2 – ¼  or 5.7 + 3.9 by 5.7 + 4.0 – 0.1

Calculating Using Near Doubles

When children have an automatic recall of basic double facts, they can use this information when adding two numbers that are very close to each other.

Mental Maths Progression: Near Doubles

These are the ways you can help your class to progress with near doubles:

Near doubles to numbers under 20 e.g. 18 + 16 is double 18 and subtract 2 or double 16 and add 2.

Progression in mental maths at KS2

Near doubles to multiples of 10 e.g. 60 + 70 is double 60 and add 10 or double 70 and subtract 10 or 75 + 76 is double 76 and subtract 1 or double 75 and add 1.

Progression in mental maths at KS2

Decimal near doubles to whole numbers e.g. 2.5 + 2.6 is double 2.5 add 0.1 or double 2.6 subtract 0.1.

Mental Multiplication Strategies And Mental Division Strategies

Place Value Multiplication Strategies

Children should be able to build upon their rapid recall of 1-12 x multiplication and division facts, and multiplication and division facts for multiples of 10 and 100 to calculate an increasing range of multiplication questions mentally.

Mental Maths Progression: Place Value  

These are the ways you can help your class to progress with place value:

Multiply a 2-digit number by a single digit by partitioning– e.g. 26 x 3 = 20 x 3 + 6 x 3

Progression in mental maths at KS2

Multiply a decimal number with up to 2 decimal places by a single digit by partitioning– e.g. 3.42 x 4 = 3 x 4 + 0.4 x 4 + 0.02 x 4

Doubling And Halving Strategies

Children should be able to recognise halving as the inverse of doubling and be able to rapidly calculate doubles and halves of numbers.

Some double and half facts are rapid recall rather than ones that children should need to calculate each time, and these are covered in the lists above.

Mental Maths Progression: Doubling And Halving

These are the ways you can help your class to progress with doubling and halving:

Find the doubles and halves of any two-digit number and any multiple of 10 or 100– e.g. half 680 or double 73

Progression in mental maths at KS2

Multiply and divide by 4 by doubling/halving twice and 8 by doubling/halving again. – e.g. 34 x 4 = 34 x 2 x 2.

Progression in mental maths at KS2

Find the doubles and halves of any number up to 10,000 by partitioning – e.g. half of 32,202 by halving 3,000, 2000, 200 and 2.

Progression in mental maths at KS2

Multiply by 50 by multiplying by 100 and halving e.g. 8 x 50= 8 x 100 divided by 2

Progression in mental maths at KS2

Divide a multiple of 25 by 25 dividing by 100 then multiplying by 4 (by doubling and doubling again) e.g. 350 ÷ 25 = 350 ÷ 100 x 2 x2

Progression in mental maths at KS2

Divide a multiple of 50 by 50 by dividing by 100 then doubling- e.g. 450 ÷ 50= 450 ÷ 10 x 2

Progression in mental maths at KS2

Double and half decimal number with up to one decimal place by portioning – e.g. half of 8.4 by halving 8 and halving 0.4

Mental Calculation Strategies For Fractions, Decimals and Percentages

Children should be able to develop their understanding of fractions, decimals and percentages and how they are related to division.

They should therefore be able to use their rapid recall multiplication and division facts to calculate some questions involving fractions, decimals and percentages mentally.

Mental Maths Progression: Fractions, Decimals And Percentages

These are the ways you can help your class to progress with fractions, decimals and percentages:

Mentally find fractions of numbers in the 2,3,4,5 and 10 times table using known multiplication and division facts– e.g. 3/5 of 45 by 45 ÷ 5 x 3.

Progression in mental maths at KS2

Recall percentage equivalents to ½, 1/3, ⅕, ⅙, 1/10 and 1/100 – e.g. ¼ = 25%

Progression in mental maths at KS2

Find 10% or multiples of 10% of whole numbers and quantities- e.g. 30% of 50 by 50 ÷ 10 x 3

Progression in mental maths at KS2

Mentally find 50% by halving and 25% by dividing by 4 or 2 of numbers and quantities- e.g. 25% of 150 by 150 ÷ 4

The tweet below is something that you may have seen going around twitter in early 2019, but it represents a useful strategy to help work out tricky percentages.

Mental Maths Tips and Tricks
This viral tweet is a very good example of a mental maths trick. It’s also a great way to use mental maths tricks to impress your friends!

Top Mental Maths Tips: How You Can Teach Mental Maths Strategies

We’ve dealt with the ‘what’ in significant detail- but how do we actually go about teaching mental strategies? Here is a summary of our top tips:

  1. Teach mental maths strategies and mental calculation techniques– don’t just rely on children ‘picking them up’- it is important that lesson time is devoted to teaching strategies conceptually and supporting children to make connections between their known facts and mental calculations. This is best achieved through modelling and the use of manipulatives etc.
  2. Engage children in discussion– children should be encouraged to discuss their mental strategies with each other and as a class, and adults in the classroom should join in this discussion. Children will see and approach calculations mentally in different (equally as valid) ways- and through sharing these they expose each other to different ways of thinking about and ‘seeing’ a calculation.
  3. Provide regular mental maths practice– children should have regular mental maths practice that focuses on mental calculation strategies. Alongside teaching the strategies in the main maths lesson, schools where children have a high level of competency and fluency in mental strategies often devote 15-20 minutes a day to the practise and development of mental strategies and rapid recall outside of the main maths lesson.
  4. Don’t think that timed testing is the only way to achieve rapid recall- timed testing has been shown by many research studies to be one of the least effective ways of developing rapid recall. Instead, ensure children have plenty of opportunities to use, apply and recall the facts that you want them to be able to recall rapidly.
  5. Play games and create opportunities for meaningful activities – If the activities are fun and meaningful children will be supported in developing number sense and fluency in an increasing range of calculations.
  6. Ensure ‘basic’ number facts are practised– It’s important that you do not neglect ‘basic’ number facts- for example, number bonds within 10, 20 and 100 and the 1-12x multiplication table. Often facts such as number bonds are only practised at KS1, but it is vital that these are practised and children are encouraged to use these facts in their mental calculations- remember- If you don’t provide the opportunity for them to use it, they will lose it!

To help pupils practise and develop these mental maths skills don’t forget to download our free Ultimate Mental Maths Powerpoint for KS2.

The best resource for improving mental maths strategies in KS1 and KS2

Our Fluent in Five resource has been designed specifically to help develop the skills needed for success in the maths SATs Arithmetic paper, with a key focus of the resource being the identification of questions which would be more efficient to be solve with a mental or written method.

For Years 5 and Year mental maths worksheets are included as part of the Fluent in Five resource. For Years 1 to 4, the mental maths questions are all on slides only.

Fluent in Five is already used in thousands of schools across the UK, so why not try the first 6 weeks for free here: Fluent in Five for Year 1,2,3, 4, 5, 6

Read more

References

Russell, Susan Jo (2007). Developing  Computational Fluency with Whole Numbers in the Elementary Grades

Do you have pupils who need extra support in maths?
Every week Third Space Learning’s maths specialist tutors support thousands of pupils across hundreds of schools with weekly online 1-to-1 lessons designed to plug gaps and boost progress. Since 2013 we’ve helped over 60,000 primary school pupils become more confident, able mathematicians. You can learn more about our interventions or request a personalised quote for your school to speak to us about your school’s needs and how we can help.

Tim Handley
Tim Handley
Third Space Maths Consultant
Author
Tim Handley is a maths consultant and author, working as part of the Third Space Learning team to create resources and blog posts.
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Ultimate Mental Maths PowerPoint!

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Downloadable resource

Ultimate Mental Maths PowerPoint!

Download this editable mental maths ppt which is packed with everything you need to teach the ultimate mental maths lesson including; procedural variation, conceptual variation and dozens of prepared questions!

Download Free Now!