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

Schools regularly state that mental maths skills are an area that they need to develop in their pupils, especially at KS2. But on occasion this is met with confusion, ambiguity and lots of questions such as:

What are mental maths strategies?

How do we teach them?

What should children be able to do mentally?

How can I improve mental arithmetic speed in my classroom?

In this blog post, we explain what mental mental maths strategies are, 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 You’ll Find Inside This Mental Maths Strategies Blog

So, without further ado, let’s take a more in depth look into the world of mental maths, and find out how you can improve mental maths in your school!.

#### 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 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 KS2, 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 questions and problems across the whole maths curriculum. 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.

For more detail on the KS2 SATS questions in the 2018 Arithmetic Paper, see our Maths Sats 2018 Breakdown

#### So How Can You Improve Pupils’ Mental Maths Skills?

The answer is quite simple, and it is to start with the number facts needed for rapid recall.

So, what number facts should children be able to rapidly recall or gain automatically with?

And and what mental maths questions should we be asking them?

##### Mental Maths Year 1 and Mental Maths Year 2: What To Teach For Rapid Recall

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.

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

It is important however that these facts are 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!

#### What Is The Best Resource For Improving Mental Maths Strategies In KS1 And KS2

This isn’t just our opinion as it has been downloaded and used by thousands of teachers!

Our Fluent in Five resource has been designed specifically to help develop the skills needed for success in the 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.

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 3, 4, 5, 6

#### Mental Maths Year 3: What To Teach For Rapid Recall

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: What To Teach For Rapid Recall

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.

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: What To Teach For Rapid Recall

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 In Year 6: What To Teach For Rapid Recall

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.
##### Specific KS2 Mental Maths Calculations Needed Throughout Years 3 To Year 6

The rapid recall number facts above go hand in hand with a wider set of mental maths calculations.

What follows is a list of those mental maths calculations.

#### Mental Addition Strategies and Mental Subtraction Strategies

The tables 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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.

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.

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

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

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

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

##### Using Multiplication And Division Facts

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

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.

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

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

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.

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

So, 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:

• 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.
• 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.
• 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.
• 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.
• 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.
• 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.

We are also always on the lookout for other tricks for mental maths that you may use in your school, so if you have any that you would like to share then please let us know via our Facebook or Twitter pages.

If you are looking for further help with teaching mental maths, or any other element of KS2 maths in your school, take a look at our how our 1-to-1 interventions can make a difference for your target pupils.

References

Russell, Susan Jo (2007). Developing  Computational Fluency with

Whole Numbers in the Elementary Grades

#### How Third Space Learning Helps Prepare Pupils For KS2 SATs

Over three quarters of the 8,000 pupils we provide our weekly 1 to 1 maths support to every year are in Year 6. Most schools choose to put these pupils on our KS2 SATS Booster Programme which has been carefully refined from the feedback of thousands of pupils and teachers to provide the most efficient and effective way to revise for SATs.

We follow a structured sequence of lessons working through both arithmetic and reasoning questions and when additional gaps are identified we plug them then and there for that individual. The beauty of the one-to-one tuition is that each child receives the lesson personalised to their own needs. If this sounds like something that would work well in your school, find out more about our SATs and Catch up programmes here.

##### 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.

#### 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!