Progression towards a standard written method of calculation · The empty number line will then be introduced to aid calculations. ... tens and units in a 3 digit number. 1. 5 6 3

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Lanchester All Saints’ Catholic Primary School

Calculations Policy

Recruitment and Selection Policy Statement All Saints’ Catholic Primary School is committed to safeguarding and promoting the welfare of children and young people and expects all staff and volunteers to share

this commitment.

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Lanchester All Saints’ Catholic Primary School

Contents

Page number

3 Introduction

4 Whole school approach

6 Stages in Addition

7 Stages in Subtraction

8 Stages In Multiplication

10 Stages in Division

13 Informal to standard written calculations

14 Summary

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Progression towards a standard written method of calculation INTRODUCTION The National Numeracy Strategy provides a structured and systematic approach to teaching number. There is a considerable emphasis on teaching mental calculation strategies. Up to the age of 9 (Year 4) informal written recording should take place regularly and is an important part of learning and understanding. More formal written methods should follow only when the child is able to use a wide range of mental calculation strategies. REASONS FOR USING WRITTEN METHODS

To aid mental calculation by writing down some of the numbers and answers involved

To make clear a mental procedure for the pupil

To help communicate methods and solutions

To provide a record of work to be done

To aid calculation when the problem is too difficult to be done mentally

To develop and refine a set of rules for calculation

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Lanchester All Saints’ Catholic Primary School - Whole School Approach We have developed a consistent approach to the teaching of written calculation methods. This will establish continuity and progression throughout the school. Mental methods will be established. These will be based on a solid understanding of place value in number and will include the following:

i. Remembering number facts and recalling them without hesitation e.g. pairs of numbers which make 10 Doubles & halves to 20

ii. Using known facts to calculate unknown facts e.g. 6 + 6 = 12 therefore 6 + 7 = 13 24 + 10 = 34 therefore 24 + 9 = 33

iii. Understanding and using relationships between addition & subtraction to find answers and check results e.g. 14 + 6 = 20 therefore 20 –6 = 14

iv. Having a repertoire of mental strategies to solve calculations e.g. doubles / near doubles bridging 10 / bridging 20 adding 9 by +10 & -1

v. Making use of informal jottings such as blank number lines to assist in calculations with larger numbers e.g.83 – 18 = 65

vi. Solving one-step word problems (either mentally or with jottings) by identifying which operation to use, drawing upon their knowledge of number bonds and explaining their reasoning

vii. Beginning to present calculations in a horizontal format and explain mental steps using numbers, symbols or words

viii. Learn to estimate/approximate first e.g. 29 + 30 (round up to nearest 10, the answer will be near to 60).

Place value will be taught mentally first from Reception class where number tracks are used, progressing to number lines (to 10 or 20 as appropriate) in Years 1 and 2. The empty number line will then be introduced to aid calculations. Subtraction will be taught by counting on and counting back depending on the numbers. Numbers such as 10, 100, 1000 will be called Landmark Numbers.

+ 2 + 60

+ 3

18 20 80 83

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WHEN ARE CHILDREN READY FOR WRITTEN CALCULATIONS? Addition and subtraction

Do they know addition and subtraction facts to 20?

Do they understand place value and can they partition numbers?

Can they add three single digit numbers mentally?

Can they add and subtract any pair of two digit numbers mentally?

Can they explain their mental strategies orally and record them using informal jottings?

Multiplication and division

Do they know the 2, 3, 4, 5 and 10 time table

Do they know the result of multiplying by 0 and 1?

Do they understand 0 as a placeholder?

Can they multiply two and three digit numbers by 10 and 100?

Can they double and halve two digit numbers mentally?

Can they use multiplication facts they know to derive mentally other multiplication facts that they do not know?

Can they explain their mental strategies orally and record them using informal jottings?

The above lists are not exhaustive but are a guide for the teacher to judge when a child is ready to move from informal to formal methods of calculation.

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Stages in Addition 1. Mental method, using partitioning: 47 + 76 = (40 + 70) + (7 + 6) or 47 + 76 = (47 + 70) + 6 2. Introduction to vertical layout, using partitioning

300 + 70 + 8

400 + 80 + 7

700 + 150 + 15 = 865

3. Vertical layout, expanded working, moving to adding the least significant digit

first:

47 47 368 368

+ 76 + 76 + 493 + 493

110 13 700 11

13 110 150 150

123 123 11 700

861 861

4. Compact method:

47 368

+ 76 +493

123 861

1 1 1 1

5. Bigger numbers and decimals (the decimal number should be written in a

square).

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Stages in Subtraction by Decomposition Before the introduction of the formal written method for subtraction (decomposition), children should be able to:

recall all subtraction facts to 20;

subtract multiples of 10 (such as 160 – 70) using the related subtraction fact,16 – 7, and their knowledge of place value;

understand the place value of hundreds, tens and units in a 3 digit number. 1.

5 6 3

- 2 4 1

3 2 2

2. 563 - 278

4 5 15 6 13

- 2 7 8

2 8 5

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Stages in Multiplication Before the introduction of formal written methods for multiplication, children should be

able to:

recall multiplication facts for the tables used;

partition numbers into multiples of one hundred, ten and one;

work out products such as 70 × 5, 70 × 50, 700 × 5 or 700 × 50 using the related

fact 7 × 5 and their knowledge of place value;

add two or more single-digit numbers mentally;

add multiples of 10 (such as 60 + 70) or of 100 (such as 600 + 700) using the

related addition fact, 6 + 7, and their knowledge of place value;

add combinations of whole numbers using the column method.

1. Mental method using partitioning multiplying tens first: 38 x 7 38 x 7 = (30 x 7) + (8 x 7) = 210 + 56 = 266 2. Grid layout 38 x 7

x 30 8

7 210 56 266

3. Grid layout - extend to bigger numbers i.e. 238 x 7

x 200 30 8

7 1400 210 56 1666

Extend to ThHTU 4. Extend to bigger numbers: 56 x 27 56 x 27 = (50 + 6) x (20 + 7)

x 50 6

20 1000 120 = 1120

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7 350 42 = 392 +

1512

5. Vertical format, expanded working

Extend to HTU x U Long multiplication

6. Vertical format, compact working

7. Children should progress to using the formal written layout for multiplication

problems.

38 x 7

210 (30x7) 56 (8x7)

266

56 x 27

1000 (50 x 20) 120 (6 x 20) 350 (50 x 7)

42 (6 x 7)

1512

56 x 27

1120 (56 x 20) 392 (56 x 7)

1512

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Stages in Division To progress towards short division, children need to be able to:

understand and use the vocabulary of division;

partition two-digit and three-digit numbers into multiples of 100, 10 and 1 in different ways;

recall multiplication and division facts for the tables used;

recognise multiples of one-digit numbers and divide multiples of 10 or 100 by a single-digit number using their knowledge of division facts and place value;

know how to find a remainder working mentally, e.g. find the remainder when 48 is divided by 5;

understand and use multiplication and division as inverse operations. 1. Number lines & grouping

2.

TU ÷ U, no remainder and no carrying, e.g. 69 3

3. TU ÷ U, remainder but no carrying, e.g. 68 ÷ 3

T U

2 2 R 2

3 6 8

4. TU ÷ U, carrying from T to U but no remainder, e.g. 76 ÷ 4. When dealing with

carrying figures, relate to knowledge of place value.

T U

1 8

4 7 36

23 3 69

0 8 4 2 6

2 2 2 2

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5. TU ÷ U with carrying and remainder, e.g. 96 ÷ 7

T U

1 3 R 5

7 9 26

Continue to develop the short division method as follows:-

1. No remainder, no carrying, e.g. 844 ÷ 4 2. Remainder, no carrying, e.g. 486 ÷ 4 3. No remainder, carrying from T to U, e.g. 860 ÷ 4 4. No remainder, carrying from H to T, e.g. 928 ÷ 4 5. No remainder, carrying from H to T and T to U, e.g. 984 ÷ 4 6. Remainder and carrying, e.g. 743 ÷ 4 7. Examples where consideration needs to be given to the placing of the

quotient,

e.g. 387 ÷ 4

0 9 6 r 3 4 ) 3 38 27

Initially pupils should cross out the hundreds digit and carry it over to the tens as well as placing zero in the quotient. 8. Examples where there are zeros in the quotient, e.g. 818 ÷ 4, 5609 ÷ 8

2 0 4 r 2 0 7 0 1 r 1 4 ) 8 1 18 8 ) 5 56 0 9

Emphasise zero as place holder. 9. Express remainders as fractions, e.g. 387 ÷ 4

0 9 6 ¾ 4 ) 3 38 27

10. Decimals 11. Measures

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12. Divide amounts of money,

e.g. Grandma emptied her money box. There was £12.46. She shared it

equally between her five grandchildren. How much did they each get and how

much was left over?

0 2 ∙ 4 9 r 1 5 ) 1 12 ∙24 46 Each child got £2.49 and there was 1p left over.

13. Varied contexts where the remainder is expressed as a whole number, or a

fraction, or a decimal.

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Informal to Standard Written Calculations As a general rule the majority of the class should cover these stages, in this order, during Key Stage 2. If a child cannot grasp a method, go back & consolidate the previous method before trying again. Statutory requirements for KS2

Addition Subtraction Multiplication Division

Y 3

Add numbers with up to three digits, using formal written methods of columnar addition

Subtract numbers with up to three digits, using formal written methods of columnar subtraction

Recall and use multiplication facts for the 3, 4 and 8 multiplication tables

Write and calculate mathematical statements for multiplication using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental and progressing to formal written methods

Recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables

Write and calculate mathematical statements for division using mental and progressing to formal written methods

Y 4

Add numbers with up to 4 digits using the formal written methods of columnar addition where appropriate

Subtract numbers with up to 4 digits using the formal written methods of columnar subtraction where appropriate

Multiply two-digit and three-digit numbers by a one-digit number using formal written layout

No statutory requirements. They are expected to carry on with methods from Y3 and, if appropriate, move onto methods from Y5,

Y 5

Add whole numbers with more than 4 digits, including using formal written methods (columnar addition)

Add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and subtraction)

Multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers

Multiply whole numbers and those involving decimals by 10, 100 and 1000

Divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context.

Y 6

There are no formal statutory requirements for addition in Y6. They are expected to continue with the formal written methods and apply this to problem solving.

There are no formal statutory requirements for subtraction in Y6. They are expected to continue with the formal written methods and apply this to problem solving.

Multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication

Divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context Divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context

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Summary

children should always estimate first

always check the answer, preferably using a different method eg. the inverse operation

always decide first whether a mental method is appropriate

pay attention to language - refer to the actual value of digits

children who make persistent mistakes should return to the method that they can use accurately until ready to move on

children need to know number and multiplication facts by heart

discuss errors and diagnose problem and then work through problem - do not simply re-teach the method

when revising or extending to harder numbers, refer back to expanded methods. This helps reinforce understanding and reminds children that they have an alternative to fall back on if they are having difficulties.

This policy was agreed October 2015 and will be reviewed annually.