# Mental Maths Policy

Last updated on March 5th, 2019 at 11:31 am

LORD BLYTON PRIMARY SCHOOL

MENTAL MATHS POLICY

Revised Oct 2017

Mrs. J. Wales (Mathematics Co-ordinator)

Introduction

This mental maths policy is based on the revised National Curriculum for Mathematics 2014 and should be used alongside our most recent (2014) calculation policy.

It is based on 5 key principles:

• For children to be able to select an efficient method of their choice (whether this be mental or written) they will always do this by asking themselves;
• Can I do this in my head?
• Can I do this in my head and using jottings?
• Do I need to use a standard written method?
• Do I need a calculator? (Level 6 only from 2015)
• Calculating mentally should always be the first choice when presented with any question. For example, if a year 4 pupil had to add 32 and 29 they should not automatically do a column method as they should have appropriate strategies to answer this quicker using mental methods, whether it be partitioning and recombining, using knowledge of near doubles or through rounding and adjusting.
• Mental calculation is much more than mental arithmetic and should be used and applied regularly in a range of contexts for children to develop competence.
• Calculation questions should be presented horizontally or orally so they make their choices based on efficiency and accuracy. If a child sees a column addition set out, they will automatically carry this through.
• Jottings should be encouraged and modelled for pupils so as to support them in the required steps and to provide visual representations of steps.

Typical example of when mental strategies are best:

2009-1997

Attempting the column method is likely to result in children making errors exchanging.

e.g. However, if tackled mentally (with or without jottings) there is less room for error.

e.g. Counting back Counting on

To be secure in mental calculations, pupils need to be taught;

• Key facts they can rapidly recall
• How to use (apply those facts) to solve other questions, e.g;
• ‘3 for free’ –if I know 3+4=7, I also know 4+3=7, 7-4=3 and 7-3=4 and if I know 3×4=12, I also know 4×3=12, 12÷3=4 and 12÷4=3.
• Place value rules, e.g. If I know 4+3=7, then I also know 40+30=70, 400+300=700, 0.4+0.3=0.7, etc.

• The 7 key addition and subtraction mental strategies:
• Counting forwards and backwards
• Re-ordering
• Partitioning-using multiples of 10 and 100
• Partitioning-bridging through multiples of 10
• Partitioning-compensating
• Partitioning-using near doubles
• Partitioning-bridging through numbers other than 10

• The 5 key multiplication and division strategies:
• Knowing multiplication and division facts to 12
• Multiplying and dividing by multiples of 10
• Multiplying and dividing by single digit numbers and multiplying by 2 digit numbers
• Doubling and halving
• Working between fractions, decimals and percentages

How do we help children develop a range of mental strategies?

Individual children will be at varying stages in terms of the number facts they have committed to memory and the strategies they are able to use to work out other related facts. This policy enables teachers to have more clarity and teach the main calculation strategies that all children need to learn. It is essential that a variety of such strategies are modelled by the teacher regularly and that children are made to see the importance of having instant recall and known number facts to enable them to use these with efficiency, e.g. “I knew the answer to 21+22 really quickly because it’s the same as double 21+1” is a much better response than, “I knew the answer to 21+22 because I put 21 in my head and counted on in ones 22 places”.

It is important to explore and trial and test these methods so children do see how inefficient some are for themselves. They cannot just be told it is inappropriate; they must experience this independently if they are to develop in to confident and competent mathematicians.

What might mental calculation sessions look like?

As mental calculation sessions need to involve discussion in order for effective learning to take place they need to be managed appropriately. Asking a question and inviting children to put hands up has several drawbacks:

• It emphasises the rapid, the known, over the derived – children who ‘know’ the answer beat those figuring it out;
• It is unhelpful to those requiring more time as they are distracted and put under pressure by those who already ‘know’ and are ready to answer.

Therefore lessons must be organised to provide thinking time which allow for rapid rather than instant responses and support those children needing longer to figure things out. Successful strategies could be;

• Insisting nobody puts a hand up until a signal has been given
• Children raising a thumb to show they are ready to answer
• Having a sign for children to use to show they aren’t ready/aren’t sure of what to do to get the answer.

The teaching of mental strategies should be taught in whole class lessons and be reinforced during mental starters or other mental calculation slots, e.g. basic skills sessions.

Teaching the 7 Addition and Subtraction Strategies

i) Counting forwards and backwards

Children first encounter counting by beginning at 1 and counting in ones. Their sense of number is extended by beginning at different numbers and counting both forwards and backwards in steps of increasing difficulty. They will also learn that while addition is commutative (can be done in any order) it is more efficient to count on from the larger number. Eventually ‘counting on’ will be replaced by more efficient methods.

Progression:

Step 1

4+8

7-3

13+4

15-3

18-6

Count on in ones from 4 or in ones from 8

Count back in ones from 7

Count on from 13

Count back in ones from 15

Count back in twos

Step 2

14+3

27-4

18-4

30+3

Count on in ones from 14

Count on or back in ones from any 2 digit no.

Count back in twos from 18

Count on in ones from 30

Step 3

40+30

90-40

35-15

Count on in tens from 40

Count back from 90 or count on from 40 (in tens)

Count on in steps of 5

Step 4

73-68

86-30

570+300

960-500

Count on 2 to 70 the 3 to 73

Count back in tens from 86 or on in tens from 30

Count on in hundreds from 300

Count back in hundreds from 960 or count on in hundreds from 500

Step 5

1 ½ = ¾

Count on in quarters

Step 6

1.7+0.5

Count on in tenths

ii)  Reordering

Sometimes a calculation can be more easily worked out by changing the order of the numbers. The way in which children rearrange numbers in a particular calculation will depend on which number facts they have instantly available to them.

It is important for children to know when numbers can be reordered (eg 2 + 5 + 8 = 8 + 2 + 5 or

15 + 8 – 5 = 15 – 5 + 8 or 23 – 9 – 3 = 23 – 3 – 9) and when they can not (eg 8 – 5 ≠ 5 – 8).

The strategy of changing the order of numbers only really applies when the question is written down. It

is difficult to reorder numbers if the question is presented orally.

Progression:

Step 1

5 + 13 = 13 + 5

3 + 4 + 7 = 3 + 7 + 4

Step 2

2 + 36 = 36 + 2

5+7+5 = 5+5+7 (spotting a number bond)

Step 3

23+54 = 54+23

12-7-2 = 12-2-7 (getting to 10)

Step 4

6+13+4+3 = 6+4+13+3

17+9-7 = 17-7+9

28+75 = 75+28 (thinking of 28 as 25+3)

Step 5

3+8+7+6+2 = 3+7+8+2+6

25+36+75 = 75+25+36

200+567 = 567+200

1.7+2.8+0.3 = 1.7+0.3+2.8

Step 6

180+650 = 650+180 (thinking of 180 as 150+30)

4.8+2.5-1.8 = 4.8-1.8+2.5

iii) Partitioning-using multiples of 10 and 100

Children need to know that numbers can be partitioned into, for example, hundreds, tens and ones, so that 475=400+70+5. This enables them to see numbers as wholes rather than just single digits in columns. This type of partitioning is a useful strategy for addition and subtraction. Both numbers involved can be partitioned this way, though it is often better to keep the first one whole and just partition the number being added or subtracted.

Step 1

30+47

78-40

25+14

30+40+7

70-40+8

20+10+5+4

Step 2

23+45

68-32

40+20+5+3

60-30+8-2

Step 3

55+37

55+30+7=85+7

Step 4

43+28+51

5.6+3.7

4.7-3.5

40+20+50+3+8+1

5.6+3+0.7=8.6+0.7

4.7-3-0.5

Step 5

540+280

276-153

540+200+80

276-100-50-3

iv) Partitioning – Bridging through multiples of 10

An important aspect of having an appreciation of number is to know when a number is close to 10 or a multiple of 10: to recognise, for example, that 47 is 3 away from 50. The use of an empty number line where the multiples of ten are seen as landmarks is helpful and enables children to have an image of jumping forwards or backwards to these landmarks.

Step 1

6+7 = 6+4+3, 23-9 = 23-3-6, 15+7 = 15+5+2

Step 2

49+32 = 49+1+31

Step 3

57+14 = 57+3+11 or 57+13+1

Step 4

3.8+2.6 = 3.8+0.2+2.4

Step 5

584-176 = 584-184+8

v) Partitioning-Compensating (also known as rounding & adjusting)

This strategy is useful for adding numbers close to a multiple of 10, those ending in 1,2,8 or 9 in particular. Instead of adding 9 for example, add ten then subtract 1. A similar strategy works for decimals, where numbers are close to whole numbers or a whole number of tenths.

Progression:

Step 1

5+9 = 5+10-1

Step 2

34+9 = 34+10-1, 70-9 = 70-10+1

Step 3

53+11 = 53+10+1, 84-19 = 84-20+1

Step 4

38+69 = 38+70-1, 64-19 = 64-20+1

Step 5

138+69 = 138+70-1, 405-399 = 405-400+1, 2.5+1.75 = 2.5+2-0.25

Step 6

5.7+3.9 = 5.7+4-0.1

vi) Partitioning – Using near doubles

If children have instant recall of doubles they can use this information when adding numbers very close to each other, e.g. their knowledge of double 6 enables them to quickly calculate 6+7 rather than using a counting on strategy.

They must be secure in doubling and halving to do this effectively.

Progression:

Step 1

5 + 6

Double 5 add 1 or double 6 subtract 1

Step 2

40 + 39

Double 40 subtract 1

Step 3

18 + 16

Double 18 subtract 2 or double 16 add 2

Step 4

380 + 380

Double 400 subtract 20 twice

Step 5

1.5 + 1.6

Step 6

421 + 387

Double 400 add 21 then subtract 13

vii) Partitioning-Bridging through numbers other than 10

Time is a universal measure that is non-metric so children need to learnt hat bridging through 10 or 100 is not always appropriate. A digital clock displaying 9:59 will, in two minutes time, read 10:01 (not 9:61 which is a common misconception). It is therefore sometimes necessary to bridge through 60, and with hours and days, through 24.

Progression:

Step 1

How long is it from 2pm to 6pm?

It is half past 7. What time was it three hours ago?

It is 7am. How many hours to midday?

Step 2

How many minutes from 10:30 to 10:45?

From 9:50 to 10:15?

Step 3

40 minutes after 3:30?

50 minutes before 1pm?

It is 10:40. How long until 12:00?

Step 4

It is 8:35. How long until 9:17?

Step 5

It is 11:30. How long until 15:29?

Teaching the 5 Multiplication & Division Strategies

i) Knowing multiplication & division facts to 12

Instant recall of multiplication and division facts is a key objective in developing children’s numeracy skills. Learning these facts and being fluent at recalling them quickly is a gradual process which takes place over time and which relies on regular opportunities for practice in a variety of situations; not just rote learning.

Progression:

Step 1

Count in twos to 20

Count in tens to 50

Count in fives to 20 or more

Step 2

Count in fives to at least 30

Recall the two times table up to 2×10

Recall the ten times table up to 10×10

Recall division facts for the two and ten times tables

Step 3

Count in threes to 30

Count in fours to 40

Recall 5 times table up to 5×10

Recall the corresponding division facts

Step 4

Count in sixes, sevens, eights and nines

Recall the 3 times table up to 3X10

Recall the 4 times table up to 4×10

Recall the corresponding division facts

Step 5

Know square numbers up to 10×10

Recall 6,8, 9 and 7* times tables up to x10

Recall the corresponding division facts

*This order recognises that children can use their knowledge of 3 and 4 times tables to help them with their 6 and 8 times tables before moving on to 9 and 7.

Step 6

Recall 11 and 12 times tables

Recall the corresponding division facts

Know the squares of 11 and 12 (i.e. 11×11 and 12×12)

ii) Multiplying and dividing by multiples of 10

Being able to multiply by 10 and multiples of 10 depends on a secure understanding of place value and is fundamental to being able to multiply and divide larger numbers.

Progression:

Step 1

7X10, 60 ÷10

Step 2

6×100, 26×10, 700÷100

Step 3

4×60, 3×80, 351×10, 79×100, 580÷10

Step 4

9357×100, 9900÷10, 737÷10, 2060÷100

Step 5

23×50, 637.6×10, 135.4÷100

iii) Multiplying & dividing by single digit numbers and multiplying by 2 digit numbers

Progression:

Step 1

9×2, 5×4, 18÷2, 16÷4

Step 2

7×3, 4×8, 35÷5, 23×2, 46÷2

Step 3

13×9, 32×3, 36÷4, 93÷3

Step 4

428×2, 154÷2, 47×5, 3.1×7

Step 5

13×50, 14×15, 153÷51, 8.6×6, 2.9×9, 45.9÷9

iv) Doubling and halving

The ability to double numbers is a fundamental tool for multiplication. Historically, all multiplication was calculated by a process of doubling and adding. Most people find doubles the easiest multiplication facts to remember so it is important that children are taught to use them to simplify other calculations. Sometimes it can be helpful to halve one of the numbers in a product and double the other.

Progression:

Step 1

7+7 is double 7

Step 2

7+7=7×2

Half of 14 is 7

Half of 30 is 15

Step 3

18 add 18 is double 18

Half of 18 is 9

60×2 is double 60

Half of 120 is 60

Half of 900 is 450

Step 4

14×5=14×10÷2

12×20=12x10x2

60×4=60x2x2 (double then double again)

Step 5

36×50=36×100÷2

Quarter of 64=64÷2÷2 (halve it then halve it again)

15×6=30×3 (double one and halve the other to keep it balanced)

Step 6

26×8=26x2x2x2

20% of £15=10% of £15×2

36×25=36×100÷4

1.6÷2=0.8

v) Fractions, decimals and percentages

Children need an understanding of how fractions, decimals and percentages relate to each other. For example, if they know that ½ , 0.5 and 50% are all ways of representing the same part of a whole then calculations such as these can be seen as different versions of the same calculation:

½ x 40

40 x 0.5

50% of £40.00

Sometimes it might be easier to work with fractions, sometimes with decimals and sometimes with percentages. There are strong links between this section and the earlier section, ‘Multiplying & Dividing by Multiples of 10’.

Progression:

Step 1

Find half of 8, find half of 30

Step 2

Find one third of 18, one tenth of 20, one fifth of 15

Step 3

Find half of 9, expressing as 4 ½ , know 0.7 is 7 tenths, know that 0.5 is ½ , find half of 150, find half of £21.60

Step 4

Know that 27/100 = 0.27, know that 75/100 is ¾ or 0.75, know 3 hundredths is 3/100 or 0.03, know that 10% is 0.1 or 1/10, find 25% of £100, find 70% of 100cm

Step 5

Know that 0.007 is 7/1000, 0.1×26, 0.01×17, 7×8.6, know 43% is 0.43 or 43/100, find 17.5% of 5250

Rapid Recall Facts

Children should be able to rapidly recall the following:

Step 1

All pairs that total 10 e.g. 6+4

Addition & subtraction facts for all numbers to at least 5

Addition doubles of all numbers to at least 5

Step 2

Addition & subtraction facts for all numbers to at least 10

All pairs that total 20 e.g. 13+7

All pairs of multiples of 10 with a total of 100 e.g. 90+10

Multiplication facts for 2 and 10 times tables and corresponding division facts

Doubles of all numbers to 10 and the corresponding halves

Multiplication facts up to 5×5 e.g. 4×3

Step 3

Addition & subtraction facts for all numbers to 20

All pairs of multiples of 100 with a total of 1000

All pairs of multiples of 5 with a total of 100

Multiplication facts and corresponding division facts for 2,5 and 10 times tables

Step 4

Multiplication facts for 2,3,4,5 and 10 times tables and the corresponding division facts

Step 5

Multiplication facts to 12×12 and corresponding division facts

Step 6

Squares of all integers from 1-12

*Please remember that mental maths skills don’t just develop naturally; they have to be taught, revised and practised regularly and children must be given plenty of opportunity to apply them in real life, everyday situations. The steps are just a guide and it is quite common for children to find the methods classed as ‘more complex’ easier than those they are expected to master first. Remember that what we are aiming for is to teach our children to be competent and efficient with their mental methods so it is not a necessity that they ‘tick all of the boxes’, but rather that they have very good recall of number facts and put this to good use when selecting their strategies for mental calculation.