Hw mathematics and calculations policy by Hill West Primary School

MATHS WRITTEN CALCULATIONS POLICY

HILL WEST PRIMARY SCHOOL MATHS WRITTEN CALCULATIONS POLICY PROGRESSION OF WRITTEN CALCULATIONS FROM MENTAL CALCULATIONS TO INFORMAL AND FORMAL WRITTEN CALCULATIONS This policy contains the key pencil and paper procedures that will be taught within our school, based on the Maths Curriculum to be implemented 2014. It has been written to ensure consistency and progression throughout the school and reflects a whole school agreement. It is a working document and supplement to changes due to professional integrity of all staff. Although the focus of the policy is on pencil and paper procedures it is important to recognise that the ability to calculate mentally lies at the heart of the Maths Curriculum. The mental and written methods in the curriculum will be taught systematically from Reception onwards and pupils will be given regular opportunities to develop the necessary skills. However mental calculation is not at the exclusion of written recording and should be seen as complementary to and not as separate from it. In every written method there is an element of mental processing. Sharing written methods with the teacher encourages children to think about the mental strategies that underpin them and to develop new ideas. Therefore written recording both helps children to clarify their thinking and supports and extends the development of more fluent and sophisticated mental strategies. Similarly, emphasis is given on supplementing written and mental methods with visual and kinaesthetic learning. This reflects the pedagogy of the 2014 Maths Curriculum and helps children to visualise and secure a good understanding of the methods taught. During their time at this school children will be encouraged to see mathematics as both a written and spoken language. Teachers will support and guide children through the following important stages:  developing the use of pictures and a mixture of words and symbols to represent numerical activities;  using standard symbols and conventions;  use of jottings to aid a mental strategy;  use of pencil and paper procedures;  explaining mental and written methods. It is important that children do not abandon jottings and mental methods once pencil and paper procedures are introduced. Therefore children will always be encouraged to look at a calculation/problem and then decide which is the best method to choose. Our long-term aim is for children to be able to use an efficient written method, with understanding, that is appropriate for a given task. They will do this by always asking themselves: ‘Can I do this in my head?’ ‘Can I do this in my head using drawings or jottings?’ ‘Do I need to use a pencil and paper procedure?’ Policies/Maths Written Calculations January 2015

Place Value Counters: Ones = Red Tens = Green Hundreds = Blue Thousands = Yellow

Reception Understanding Addition Written: Make own marks or begin to record number sentences. Visual: Looking at pictures alongside the number calculation. Kinaesthetic: Solve practical problems using objects in a real or role play context.

Policies/Maths Written Calculations January 2015

Year 1

Year 2

Year 3

Recording Addition Written: Record simple mental addition in a number sentence using the + and = signs. Visual: Looking at pictures alongside the number calculation. Kinaesthetic: Moving number cards (or number magnets), including operation symbols into the correct place.

Missing Number Problems Written: Be able to complete number sentences where a missing number is shown by a symbol eg. 5+2= ∆ ∆ =5+2 5+∆=7 7=∆+2 ∆ + 2 =7 7=2 +∆ etc. Visual: Looking at pictures alongside the number calculation. Kinaesthetic: Using cubes, place value counters or small objects to represent what they know. Use these to work out missing number.

Begin to record number sentences. Written:

Find own way of recording for addition. (number sentences modelled by adults)

Adding one digit and two digit numbers to 20, including 0. Written: - showing jumps on prepared number lines. - drawing own number line.

eg 6 + 5 = 11

Visual: Showing pictures of objects and counting Policies/Maths Written Calculations January 2015

Missing Number Problems Written: Recognise the use of symbols such as   to stand for unknown numbers and complete number sentences. 9 + ∆ = 13 ∆+ 4 = 13 ∆+ ◊= 13 40 + = 100 +200 = 400 etc Extend to 3 numbers eg 5 + ∆ + 4 = 13 13 = ∆+ ◊+ 3 etc and 13 + 5 = ∆ + 10 12 + ∆ = 14 + 4 etc Visual: Looking at pictures alongside the number calculation. Kinaesthetic: Using cubes, place value counters or small objects to represent what they know. Use these to work out missing number. Adding two digit numbers and ones Written: Use prepared number lines then progress on to drawing own empty number lines to eg - count in ones eg 23 + 3 = 26

+1 25

Writing number sentences i.e. 33 + 5 = 38 Visual:

Missing Number Problems Written: Recognise the use of symbols such as  to stand for unknown numbers and complete number sentences. Eg 19 + ∆ = 33 ∆+ 14 = 33 ∆+ ◊= 33 etc



Use 3 numbers eg 10 + ∆ + 50 = 100 ∆ + ◊ + O =100 347 + = 447 Visual: Looking at pictures alongside the number calculation. Kinaesthetic: Using cubes, place value counters or small objects to represent what they know. Use these to work out missing number. Adding three digit numbers and ones Written: Writing number sentences i.e. 133 + 5 = 138 Visual: - Using the number square, reinforce moving to the right when adding one. - Arrow cards; notice how the tens and hundreds stay the same but the ones change. - Using fingers to count on.

and

Visual:

Select two groups of objects to make a given total. Kinaesthetic: Solve practical problems using objects in a real or role play context.

how many are there altogether. Kinaesthetic: Using cubes, place value counters or small objects to count together one-digit numbers.

Using the number square, show chn how to move to the right when adding one. Arrow cards; notice how the tens stay the same but the ones change. Using fingers to count on.

Kinaesthetic: Using dienes cubes or place value counters to add the numbers together.

Kinaesthetic: Using dienes cubes or place value counters to add the numbers together. Adding three digit numbers and tens or hundreds Written: Writing number sentences i.e. 133 + 20 = 153, 146 + 200 = 346 Visual: - Arrow cards; notice how the hundreds and ones stay the same when adding multiples of ten. (tens and ones stay the same when adding multiples of a hundred) - Using fingers to count on in multiples of ten or hundred. Kinaesthetic: Using dienes cubes or place value counters to add the numbers together.

Policies/Maths Written Calculations January 2015

Adding 10 Written: Using an empty number line to add 10 to a single digit number. +10

8 18 8 + 10 = 18 Visual: Images of arrow cards to show how the ones stay the same and the tens change. Looking at a number square and understanding how, when ten is added, the movement on the square is downwards. Kinaesthetic: Using place value counters to realise that the ones don’t change when ten is added.

Policies/Maths Written Calculations January 2015

Adding multiples of 10 Written: 36 + 20 = 56 Visual: Number Square, arrow cards Kinaesthetic: Using dienes cubes or place value counters to add multiples of 10.

Adding two three-digit numbers See Primary Maths Curriculum, Appendix 1

Adding two two-digit numbers See Primary Maths Curriculum, Appendix 1

Without bridging 10 135 + 123 = 258

Written: Recording addition in columns supports place value and prepares for formal written methods with larger numbers. (Maths National Curriculum - P12) Expanded column addition (with no bridging 10) i.e. 36 + 23 = 30 + 6 20 + 3 50 + 9 = 59

Written: Number sentences and column addition i.e.

(Starting from Ones first; “5 + 3 = 8, 30 + 20 = 50 and 100 + 100 = 200”)

+ 135 + 123 + 258 With bridging 10 147 + 138 = 185

(Starting from Ones first; “7 + 8 = 15 so 5 goes in the Ones column and the ten is carried into the tens column. 40 + 30 + 10 = 80 and 100 + 100 = 200”)

+ 147 + 138 + 285 + 21

Bridging the tens Written: It is important to secure number bonds to 10 in order to help with bridging the ten. Use a number line to add a pair of single digit numbers to bridge through 10 eg 8 + 5 =13 2

+3 10

So; 8 + 2 + 3 = 10 + 3 = 13 Represent number line calculations in a number sentence Eg +1 +5

Visual: Number Square Arrow cards

Visual: Looking at a number square and, starting at the biggest number, counting on. Identify the movement onto the next line. Showing a marked number line with emphasis on the ten. Kinaesthetic: Physically moving a counter along a number square, starting at the biggest number and counting on the smallest.

Adding three one-digit numbers (See Y1, adding one digit numbers)

Shows 19 + 6 = 25

Policies/Maths Written Calculations January 2015

Kinaesthetic: - Using dienes cubes or place value counters to add count the numbers together. - Children to collect counters together within each column, changing ones for tens where necessary. Set out in columns or using a table: Hundreds Tens Ones

+ 36 + 27 + 13 + 50 + 63

Visual: - Arrow cards set out in the column addition format. - Expanded column addition 100 + 30 + 6 100 + 20 + 3 200 + 50 + 9 = 259

Expanded column addition (with bridging) i.e. 36 + 27 = 30 + 6 20 + 7 50 + 3 = 63 10

Using a bead bar to count on the smallest number. Identifying how 10 is crossed.

Reception Recording Subtraction

Written: Find own way of recording for subtraction eg crossouts.

7–2=5 Visual: Give answers to adult modelled subtraction. Kinaesthetic: Solve practical problems in a real or role play context.

Year 1

Year 2

Year 3

Recording Subtraction

Visual: Looking at pictures alongside the number calculation. There were 8 cakes on a plate. Mary ate 3 of them. How many were left?

Visual: Looking at pictures alongside the number calculation.

Written: Record simple subtraction in a number sentence using the – and = signs.

Kinaesthetic: Moving number cards (or number magnets), including operation symbols into the correct place. Missing Number Problems Written:

Be able to complete number sentences where a missing number is shown by a symbol eg. 6-2=∆ 6-∆=4 ∆ -2=4 etc. Policies/Maths Written Calculations January 2015

∆ =6-2 4=∆-2 4=О-∆

Written: Record mental subtraction in a number sentence using the – and = signs.

Kinaesthetic: Moving number cards (or number magnets), including operation symbols into the correct place.

Missing Number Problems Written:

Written: Record mental subtraction in a number sentence using the – and = signs, using appropriate numbers. Visual: Looking at pictures alongside the number calculation. Kinaesthetic: Moving number cards (or number magnets), including operation symbols into the correct place. Missing Number Problems

Recognise the use of symbols such as or to stand for unknown numbers and complete number sentences. 13 - ∆ = 9 ∆ - 4 = 9 ∆ - ◊= 9 etc

Written: Recognise the use of symbols such as or to stand for unknown numbers and complete number sentences 36 – 17 =∆ ∆ - 15 = 19 ∆ - ◊= 19 20 - ∆ - ◊= 5 etc

Extend to:

13 + 5 = ∆ - 10 etc Visual: Looking at pictures alongside the number calculation.

Visual: Looking at pictures alongside the number calculation.

Visual: Looking at pictures alongside the number calculation. Kinaesthetic: Using cubes, place value counters or small objects to represent what they know. Use these to work out missing number. Subtracting one-digit numbers and two digit numbers to 20, including 0. Written: Use a marked, partly marked or empty number line to count back in jumps going under the line. Eg. 18 – 5 (counting back) - marked line 9



10 11 12

13 14 15

What is the difference between 13 and 18? (counting on) – empty line

Kinaesthetic: Using cubes, place value counters or small objects to represent what they know. Use these to work out missing number.

Subtracting ones from a two-digit number.

Subtracting ones from a three-digit number.

Written: Use marked, partly marked or empty number lines to count back (take away)

Written: Write number sentences e.g. 156 – 5 = 151

Children should be secure in the knowledge that the smallest number must be taken away from the biggest; it can’t be done in any order. (Primary Maths Curriculum p12)

56 – 5 = 51

Children need to begin to understand when it is sensible to count back eg 18 – 5 Visual: Pictures e.g. 26 flowers (10 in two flower beds and 6 in one flowerbed) with 5 crossed out. Looking at a number square and Policies/Maths Written Calculations January 2015

Visual: Arrow cards. Pictures e.g. 26 flowers (10 in two flower beds and 6 in one flowerbed) with 5 crossed out. Looking at a number square and understanding how, when one is subtracted, the movement on the square is to the left. Kinaesthetic: Using place value counters

Children should be secure in the knowledge that the smallest number must be taken away from the biggest; it can’t be done in any order. (Primary Maths Curriculum p12)

Visual: Arrow cards. Kinaesthetic: Using place value counters to subtract the ones. Putting largest number in their head and counting back on fingers.

understanding how, when one is subtracted, the movement on the square is to the left.

to subtract the ones. Putting largest number in their head and counting back on fingers.

Kinaesthetic: Using place value counters to subtract the ones. Putting largest number in their head and counting back on fingers. Moving a counter along a numbered or blank numberline.

Policies/Maths Written Calculations January 2015

Subtracting 10 Written: Using an empty number line to subtract 10 from a two digit number. -10

18 -10

18 + 10 = 18 Visual: Images of arrow cards to show how the ones stay the same and the tens change. Looking at a number square and understanding how, when ten is subtracted, the movement on the square is upwards. Kinaesthetic: Using place value counters to realise that the ones don’t change when ten is subtracted

Policies/Maths Written Calculations January 2015

Subtracting multiples of ten from a twodigit number. Children should recognise that the ones don’t change when a multiple of 10 is subtracted.

Subtracting multiples of ten from a three-digit number. Children should recognise that the ones don’t change when a multiple of 10 is subtracted.

Written: Record as number sentences i.e. 56 – 20 = 36

Written: Record as number sentences i.e. 156 – 20 = 136

Visual:

Arrow cards. Pictures e.g. 26 flowers (10 in two flower beds and 6 in one flowerbed) with a bed of 10 crossed out. Looking at a number square and understanding how, when ten is subtracted, the movement on the square is upwards.

Kinaesthetic:

Using place value counters or dienes cubes to subtract the tens.

Arrow cards.

Kinaesthetic:

Using place value counters or dienes cubes to subtract the tens.

Bridging the tens Written: It is important to secure number bonds to 10 in order to help with bridging the ten. Use a number line to subtract a pair of single digit numbers to bridge through 10 eg 13 - 5 =8 3

Subtracting a two-digit number from a two-digit number. See Primary Maths Curriculum, Appendix 1

Subtracting a three-digit number from a three-digit number. See Primary Maths Curriculum, Appendix 1

Children should develop their understanding of when it is sensible to count back and when to count on eg

 

93 – 12 (count back) 93 - 88 (count on)

10 -2

13 -3

So; 13 - 3 - 2 = 10 - 2 = 8 Visual: Looking at a number square and, starting at the biggest number, counting back. Identify the movement onto the line above. Showing a marked number line with emphasis on the ten. Kinaesthetic: Physically moving a counter along a number square, starting at the biggest number and counting back.

Written: Recording addition in columns supports place value and prepares for formal written methods with larger numbers. (Maths National Curriculum - P12)

Written: Pupils practise using columnar subtraction with increasingly large numbers up to three digits to become fluent (Maths National Curriculum - P19)

Column Subtraction (Not bridging 10) 77 – 34 =

- 77 - 34 - 43

- 177 - 134 - 143

(Starting from Ones first; “7 – 4 = 3 and 70 – 30 = 40”)

Column Subtraction (Bridging 10) 54 – 28 = 41 - 54 - 28 - 26

(Starting from Ones first; “4 – 8 can’t be done so I need to exchange a ‘ten’ for ten ‘ones’. So then 14 – 8 = 6 and 40 – 20 = 20.”)

Policies/Maths Written Calculations January 2015

193 – 12 (count back) 193 - 188 (count on)

(Starting from Ones first; “7 – 4 = 3 and 70 – 30 = 40. 100 – 100 = zero.”)

Column Subtraction (Bridging 10) 254 – 128 = 241 - 254 - 128 - 126

(Starting from Ones first; “4 – 8 can’t be done so I need to exchange a ‘ten’ for ten ‘ones’. So then 14 – 8 = 6, 40 – 20 = 20 and 200 – 100 = 100.”)

Policies/Maths Written Calculations January 2015

Visual: Pictures of diene cubes, place value counters or arrow cards set up in column subtraction format.

Kinaesthetic: - Masking tape columns on the floor. Coloured pieces of card, representing ones/tens for children to move. - Using place value counters or diene cubes set up in column subtraction format. Children to move counters away that are being subtracted. Set out in columns or using a table: Tens Ones

Kinaesthetic: - Masking tape columns on the floor. Coloured pieces of card, representing ones/tens/hundreds for children to move. - Using place value counters or diene cubes set up in column subtraction format. Children to move counters away that are being subtracted. Set out in columns or using a table: Hundreds Tens Ones

ADDITION All children should be taught, and learn to, apply these methods when solving a variety of number-based problems. They should use written addition methods confidently and with deep understanding. They should also develop their language of addition so that they are able to recognise when to use these methods. Alongside the written methods, mental methods should continue to be taught so that children know when it is appropriate to use mental methods and when formal written methods are required.

Year 4

Year 5

Bridging hundreds and thousands Written: Number sentences and column addition 2147 + 3168 = 5315

+ 2147 + 3168 + 5315 + 11 Visual: - Pictures of place value counters. - Arrow cards set out in the column addition format. - Broken down Column addition: 2147 + 3168 15 110 200 5000 5325 Kinaesthetic: Using dienes cubes or place value counters set out in the column addition format. Set out in columns or using a table: Thousands Hundreds Tens Ones

(Starting from Ones first; “7 + 8 = 15 so 5 goes in the Ones column and the ten is carried into the tens column. 40 + 60 + 10 = 110 so 10 goes in the tens column and the hundred is carried over to the hundreds column, 100 + 100 + 100 = 300 and 2,000 + 3,000 = 5,000”)

Policies/Maths Written Calculations January 2015

Year 6

Ensure proficiency of formal written methods as shown in Appendix 1 of the Primary National Curriculum.

Adding numbers with up to four digits See Primary Maths Curriculum, Appendix 1

Adding whole numbers with 4 or more digits See Primary Maths Curriculum, Appendix 1

Written: Number sentences and column addition i.e.

Without bridging 10 3135 + 2123 = 5258

+ 3135 + 2123 + 5258

With bridging 10 2147 + 3138 = 5185

With bridging 10 2147 + 3138 = 5158

+ 2147 + 3138 + 5285 + 21 Visual: - Pictures of place value counters. - Arrow cards set out in the column addition format. - Expanded column addition 3000 +100 + 30 + 6 3000 + 100 + 20 + 3 6000 + 200 + 50 + 9 = 6259 Kinaesthetic: Using place value counters set out in the column addition format. Set out in columns or using a table. Support carrying by exchanging 10 ones for a ten. Thousands Hundreds Tens Ones

(Starting from Ones first; “5 + 3 = 8, 30 + 20 = 50, 100 + 100 = 200 and 2,000 + 3,000 = 5,000”)

(Starting from Ones first; “7 + 8 = 15 so 5 goes in the Ones column and the ten is carried into the tens column. 40 + 30 + 10 = 80, 100 + 100 = 200 and 2,000 + 3,000 = 5,000”)

Policies/Maths Written Calculations January 2015

(Starting from Ones first; “5 + 3 = 8, 30 + 20 = 50, 100 + 100 = 200 and 2,000 + 3,000 = 5,000”)

(Starting from Ones first; “7 + 8 = 15 so 5 goes in the Ones column and the ten is carried into the tens column. 40 + 30 + 10 = 80, 100 + 100 = 200 and 2,000 + 3,000 = 5,000”)

Ensure proficiency of formal written methods as shown in Appendix 1 of the Primary National Curriculum.

SUBTRACTION All children should be taught, and learn to, apply these methods when solving a variety of number-based problems. They should use written subtraction methods confidently and with deep understanding. They should also develop their language of subtraction so that they are able to recognise when to use these methods. Alongside the written methods, mental methods should continue to be taught so that children know when it is appropriate to use mental methods and when formal written methods are required.

Year 4

Year 5

Year 6

Bridging hundreds and thousands See Primary Maths Curriculum, Appendix 1

Written: 6254 – 3628 = 5141 - 6254 - 3628 - 2626

Visual: Pictures of place value counters or arrow cards set up in column subtraction format.

Kinaesthetic: Masking tape columns on the floor. Coloured pieces of card, representing ones/tens/hundreds/thousands for children to move. Using place value counters set up in column subtraction format. Children to take the relevant amounts away from the columns. Subtracting a whole number up to four digits. See Primary Maths Curriculum, Appendix 1

Written:

(Starting from Ones first; “4 – 8 can’t be done so I need to exchange a ‘ten’ for ten ‘ones’. So then 14 – 8 = 6, 40 – 20 = 20, 200 – 600 can’t be done so I need to exchange again. This time, I’m going to exchange a thousand for ten hundreds. So 1,200 – 600 = 600 and 5,000 – 3,000 = 2,000.”)

Policies/Maths Written Calculations January 2015

Ensure proficiency of formal written methods as shown in Appendix 1 of the Primary National Curriculum.

Pupils practise using columnar subtraction with increasingly large numbers up to aid fluency (Maths National Curriculum – P25)

Column Subtraction (Not bridging 10) 77 – 34 =

- 6177 - 3134 - 3043

Column Subtraction (Bridging 10) 6254 – 3128 = 241 - 6254 - 3128 - 3126

Visual: Pictures of place value counters or arrow cards set up in column subtraction format.

Set out in columns or using a table:

(Starting from Ones first; “7 – 4 = 3 and 70 – 30 = 40. 100 – 100 = zero and 6,000 – 3,000 = 3,000.”)

(Starting from Ones first; “4 – 8 can’t be done so I need to exchange a ‘ten’ for ten ‘ones’. So then 14 – 8 = 6, 40 – 20 = 20, 200 – 100 = 100 and 6,000 – 3,000 = 3,000.”)

Policies/Maths Written Calculations January 2015

(Starting from Ones first; “7 – 4 = 3 and 70 – 30 = 40. 100 – 100 = zero and 6,000 – 3,000 = 3,000.”)

(Starting from Ones first; “4 – 8 can’t be done so I need to exchange a ‘ten’ for ten ‘ones’. So then 14 – 8 = 6, 40 – 20 = 20, 200 – 100 = 100 and 6,000 – 3,000 = 3,000.”)

Thousands

Hundreds

Tens

Ones

Policies/Maths Written Calculations January 2015

Thousands

Hundreds

Tens

Ones

MULTIPLICATION All children should be taught, and learn to, apply these methods when solving a variety of number-based problems. They should use written addition methods confidently and with deep understanding. They should also develop their language of multiplication so that they are able to recognise when to use these methods as well as relating the multiplication to addition. Children should develop fluency of their multiplication tables so that they can use them when solving problems.

Reception

Year 1 Oral counting in small steps Eg. 2’s, 5’s and 10’s Written: Creating a link to repeated addition i.e. 2+2+2+2 = 8 Shown on repeated jumps on a number line. +2 0

+2 2

+2 4

+2 6

Visual: - Using a number square, show the pattern of counting on in equal steps. - Arrays - Pictures.

Year 2

Year 3

Recording Multiplication Written: Record multiplication calculations for the 2, 5 and 10’s in a number sentence using the x and = signs.

Visual: Looking at pictures alongside the number calculation. Looking at a number square or number line.

Kinaesthetic: Moving number cards (or number magnets), including operation symbols into the correct place.

Kinaesthetic: Moving number cards (or number magnets), including operation symbols into the correct place. Understanding Commutativity Knowing that multiplication can be done in any order. Written: 5 x 4 = 20 and 4 x 5 = 20 4 x 12 x 5 = 4 x 5 x 12 = 20 x 12 = 240

Kinaesthetic: Using ones counters to create arrays. Using cubes to make columns, groups or arrays. Moving a counter across a number square.

Policies/Maths Written Calculations January 2015

Visual: Arrays turned in different directions. Jumping on a number line in different equal groups e.g. 4’s then 5’s Kinaesthetic: Moving a counter on a number line. Creating arrays with cubes or counters.

Multiplying numbers equal to or less than 10. They connect the 10 multiplication tables to place value, 5 multiplication tables to divisions on a clock face and 2 multiplication tables to even numbers. (Primary Maths Curriculum P13) Written: 5+5+5+5 = 20 or 4 x 5 = 20

Number line +5 +5 +5 5

-Array ●●●●● ●●●●● ●●●●● ●●●●● 4 x 5 = 20

30 + 12 = 42 (Applying column addition where appropriate)

+5 15

Visual: - Array using place value counters

5 x 4 = 20

Show that multiplication can be done in any order (commutative) (Primary Maths Curriculum P13) Demonstrate the array turned 90o; The children should notice that the answer is the same. -

Policies/Maths Written Calculations January 2015

Number Square. Using coloured squares to show the patterns.

Kinaesthetic: Place value counters. Moving a counter on a number square. Dienes cubes. Cubes -

Written: As an interim step to understanding the short method of multiplication, children will become familiar with ‘grid multiplication’. Grid Multiplication i.e. 3 x 14 = x 10 4 3 30 12

Visual: - Marks or tallies llll llll llll llll = 20 -

Multiplying two-digit numbers. Through doubling, they connect the 2, 4 and 8 multiplication tables.

Counting in equal steps on a number square.

Kinaesthetic: Place value counters. Moving a counter on a number square. Dienes cubes. - Cubes -

Policies/Maths Written Calculations January 2015

DIVISION Linking division facts to multiplication facts will provide the foundations for developing written methods of division. They should begin to relate division to fractions and measures.

Reception

Year One

Year Two

Oral counting back in small steps Eg. 2’s, 5’s and 10’s Written: Creating a link to repeated subtraction i.e. 8 – 2 - 2 - 2 - 2 Shown on repeated jumps on a number line. 0

2 -2

4 -2

6 -2

8 -2

Visual: - Using a number square, show the pattern of counting back in equal steps. - Arrays - Pictures.

Year Three

Recording Division Written: Record division calculations using the ÷ and = signs.

Visual: Looking at pictures alongside the number calculation. Looking at a number square or number line.

Kinaesthetic: Moving number cards (or number magnets), including operation symbols into the correct place.

Kinaesthetic: Using ones counters to create arrays. Using cubes to make columns, groups or arrays. Moving a counter backwards across a number square.

Policies/Maths Written Calculations January 2015

Dividing a 2-digit number

Children understand that, unlike multiplication, division can only be done in a certain order. (Primary Maths Curriculum p13)

Written: 20 ÷ 4 = 5

Dividing a 2-digit number by a 1- digit number using short division Written: 23 3/69 (3 into 6 goes 2 times so I’ll put ‘2’ in the ‘tens column’. Then 3 into 9 goes 3 times so 3 goes in the ones column. The answer is 23.)

Visual: Understand the operation of division as Sharing or grouping equally Sharing: 6÷2 = 3

Visual: Using place value counters displayed as: Tens Ones OOOOOO OOOOOOOOO Kinaesthetic: Using place value counters or dienes cubes in a grid similar to above. Dividing a 2-digit number by a 1- digit number using short division, with remainders Written: 1 4r1 7/929

Grouping: 15÷5 = 3

(7 goes into 9 once, so I’ll put 1 in the ‘tens column’ but there is 2 left over so I’ll carry that 2 over to the ones column. 7 goes into 29 4 times with one left over so I’ll put 4 in the ‘ones column’. That leaves me with an answer of 14 remainder 1.)

●●●●● ●●●●● ●●●●● ●●●●●

Policies/Maths Written Calculations January 2015

Visual: Using place value counters displayed as: Tens Ones OOOOOOOOO OOOOOOOOO

20 ÷ 5 = 4

Pictures to represent division calculations. Kinaesthetic: Moving place value counters, cubes or diene cubes into groups. Using objects/resources to complement division calculations.

The two left over are then carried over into the ones column and exchanged for the appropriate number of ones. (20) Kinaesthetic: Using place value counters or dienes cubes in a grid similar to above.

Year 4 Multiplying three numbers. Knowing that multiplication can be done in any order. Written: 4 x 12 x 5 = 4 x 5 x 12 = 20 x 12 = 240 Visual: Arrays turned in different directions. Jumping on a number line in different equal groups e.g. 4’s then 5’s Kinaesthetic: Moving a counter on a number line. Creating arrays with cubes or counters.

Year 5

Year 6

Multiplying two 2-digit or 3-digit numbers Written: Long Multiplication (See Appendix 1 of the Primary National Curriculum)

124 x 26 2480 744 3224

1 1

(20 x 4 = 80, 20 x 20 = 400 and 20 x 100 = 2000 which makes 2480 so far. Then, 6 x 4 = 24 so I need to put the 20 in the tens column at the top. 6 x 20 = 120 = the 20 at the top makes 140. The 100 needs to go in the hundreds column and 6 x 100 = 600, add the 100 from the top makes 700. Altogether I have got 744. Finally, I’m going to add these two ‘sub totals’ together using column addition methods. So 0 + 4 = 4, 80 + 40 = 120 so I’ll put 100 in the hundreds column at the bottom. Then 400 + 700 + 100 = 1200. Again, 1000 will need to carry over to the thousands column. Lastly, 2000 + 1000 = 3000 and the final answer is 3224.)

Visual: Grid multiplication so that children can see where the answer has come from. x 100 20 4 20 2000 400 80 = 2480 (See example above) 6 600 120 24 = 744 (See example above)

Kinaesthetic Place value counters arranged as arrays. Piecing together a long multiplication template using number cards. Policies/Maths Written Calculations January 2015

Kinaesthetic Place value counters arranged as arrays. Piecing together a long multiplication template using number cards.

Multiplying by numbers with 2 or 3 digits. By the end of year 4, all children should know their multiplication facts to 12x12. (Primary Maths Curriculum P26) Written: As an interim step to understanding the short method of multiplication, children will become familiar with ‘grid multiplication’. Grid Method: i.e. 3 x 24 = x 20 4 3 60 12 60 + 12 = 72 (Applying column addition where appropriate) Pupils practise to become fluent in the formal written method of short multiplication with exact answers – see Mathematics appendix 1 (Primary Maths Curriculum p26) Short Multiplication: 24 x 3 72 1

Visual: - Array using place value counters

Policies/Maths Written Calculations January 2015

Multiplying by numbers with up to 4 digits.

Written: Short Multiplication: 2324 x 3 6972

47 x 5 = (40 x 5) + (7 x 5) = 200 + 35 = 235

Visual: Grid multiplication so that children can see where the answer has come from. x 2000 300 20 4 3 6000 900 60 12

Kinaesthetic: Place value counters arranged as arrays. Piecing together a short multiplication template using number cards.

Multiplying by decimals

Extend to decimals, with up to 2-decimal places, multiplied by a single digit. Eg

Written:

4.92 x 3 (ans approx: 15) 4.92 x 3 = (4.0 x 3) + (0.9 x 3) + (0.02 x 3) = 12 + 2.7 + 0.06 = 14.76

2.4 x 3 7.2 1

Visual: Grid multiplication so that children can see where the answer has come from. x 2 0.4 3 6 1.2 6 + 1.2 = 7.2

Ensure proficiency of formal written methods as shown in Appendix 1 of the Primary National Curriculum.

Counting in equal steps on a number square.

Kinaesthetic: Place value counters. Moving a counter on a number square. Dienes cubes. - Cubes

Policies/Maths Written Calculations January 2015

Kinaesthetic: Piecing together a short multiplication template using number cards.

DIVISION Linking division facts to multiplication facts will provide the foundations for developing written methods of division. They should begin to relate division to fractions and measures.

Year 4

Year 5

Year 6

Dividing a 3-digit number by a 1- digit number using short division

Dividing a 4-digit number by a 1- digit number using short division

Continue to embed understanding of short division with and without remainders (See Y5)

Written: 144 3/41312

Written: 21 4 4 3/ 641312

Dividing a whole number by a 2-digit number using long division.

(3 into 4 goes once so I’ll put ‘1’ in the ‘hundreds column’ and carry the remaining ‘1’ into the tens column . Then 3 into 13 goes 4 times with 1 left over so 4 goes in the ‘tens column’ and the 1 carrys over to the ones column. Finally, 3 into 12 goes 4 times so I can put 4 in the ones column. The answer is 144.) (3 is the divisor, 432 is the dividend and the number on top is the quotient.)

(3 into 6 goes twice so I’ll put 2 in the ‘thousands column’. 3 then goes into 4 once so I’ll put ‘1’ in the ‘hundreds column’ and carry the remaining ‘1’ into the tens column . Then 3 into 13 goes 4 times with 1 left over so 4 goes in the ‘tens column’ and the 1 carrys over to the ones column. Finally, 3 into 12 goes 4 times so I can put 4 in the ones column. The answer is 144.) (3 is the divisor, 432 is the dividend and the number on top is the quotient.)

Written:

Visual: Using place value counters or diene cubes displayed as: Hundreds Tens Ones OOOO OOO OO Carry over where necessary.

Visual: Using place value counters or diene cubes displayed in a Th, H, T and O table.

Kinaesthetic: Using place value counters or dienes cubes in a grid similar to above. Moving number cards on a short division template. Dividing a 3-digit number by a 1- digit number using short division, with remainders

Kinaesthetic: Using place value counters or diene cubes displayed in a Th, H, T and O table. Moving number cards on a short division template.

28 15/ 432 300 (15 x20) 132 120 (8 x20) 12 (15 goes into 4 times with 4 remaining, so that gets carried over to the tens column. 15 into 43 goes twice with 13 remaining so I’ll write 2 above in the tens column. 15 multiplied by 20 is 300 so I’ll write that under the dividend to help me keep track. 432 – 300 = 132. 15 into 132 goes 8 times with 12 left over.) (15 is the divisor, 432 is the dividend and the number on top is the quotient.)

The remainder could be extended i.e. 12/15 = 4/5 = 0.8 Dividing a 4-digit number by a 1- digit number using short division, with remainders

Answer = 28r12 or 28 4/5 or 28.8

Written: 8 6r2 5/4332

Written: 1 2 8 6 r2 5/6144332

Visual: Using place value counters or diene cubes displayed in a Th, H, T and O table.

(5 doesn’t go into 4, so I’m going to carry that 4 over to the tens column. 5 goes into 43 8 times with 3 remaining. Carrying that 3 over will make 32 and 5 goes into 32 6 times with 2 remaining.)

(5 goes into 6 once, with 1 left over so that will get carried over into the ‘hundreds column’. 5 then into 14 goes twice with 4 left over so that 4 gets carried over into the ‘tens column’. 5 into 43 goes 8 times with 3 carried over into the ones column. Finally, 5 into 32 goes 6 times with 2 remaining.)

Policies/Maths Written Calculations January 2015

Visual: Using place value counters or diene cubes displayed as:

Visual: Using place value counters or diene cubes displayed in a Th, H, T and O table.

Kinaesthetic: Using place value counters or diene cubes displayed in a Th, H, T and O table. Moving number cards on a short division template.

Kinaesthetic: Using place value counters or diene cubes displayed in a Th, H, T and O table, carrying over where necessary. Moving number cards on a short division template.

(Adding in the hundreds column as necessary.) Kinaesthetic: Using place value counters or dienes cubes in a grid similar to above. Recording Division Written: Record division calculations using the รท and = signs.

Recording Division Written: Record division calculations using the รท and = signs.

Visual: Looking at pictures alongside the number calculation. Looking at a number square or number line.

Kinaesthetic: Moving number cards (or number magnets), including operation symbols into the correct place.

Policies/Maths Written Calculations January 2015

Signed

A. Staples

Date

28th January 2015

Signed

B.M. Clarke

Date

28th January 2015

Policies/Maths Written Calculations January 2015

(For the Local Governing Body)

(Head Teacher)