Calculation Policy 2014 The following calculation policy has been devised to meet requirements of the National Curriculum 2014 for the teaching and learning of mathematics, and is also designed to give pupils a consistent and smooth progression of learning in calculations across the school. Please note that early learning in number and calculation in FS1 and FS2 follows the “Development Matters” EYFS document, and this calculation policy is designed to build on progressively from the content and methods established in the Early Years Foundation Stage. Age stage expectations The calculation policy is organised according to age stage expectations as set out in the National Curriculum 2014, however it is vital that pupils are taught according to the stage that they are currently working at, being moved onto the next level as soon as they are ready, or working at a lower stage until they are secure enough to move on. Providing a context for calculation: It is important that any type of calculation is given a real life context or problem solving approach to help build children’ s understanding of the purpose of calculation, and to help them recognise when to use certain operations and methods when faced with problems. This must be a priority within calculation lessons. Choosing a calculation method: Children need to be taught and encouraged to use the following processes in deciding what approach they will take to a calculation, to ensure they select the most appropriate method for the numbers involved.
Transcript
Calculation Policy 2014
The following calculation policy has been devised to meet requirements of the National Curriculum 2014 for the teaching and learning of mathematics, and is also designed to give pupils a consistent and smooth progression of learning in calculations across the school. Please note that early learning in number and calculation in FS1 and FS2 follows the “Development Matters” EYFS document, and this calculation policy is designed to build on progressively from the content and methods established in the Early Years Foundation Stage. Age stage expectations The calculation policy is organised according to age stage expectations as set out in the National Curriculum 2014, however it is vital that pupils are taught according to the stage that they are currently working at, being moved onto the next level as soon as they are ready, or working at a lower stage until they are secure enough to move on. Providing a context for calculation: It is important that any type of calculation is given a real life context or problem solving approach to help build children’s understanding of the purpose of calculation, and to help them recognise when to use certain operations and methods when faced with problems. This must be a priority within calculation lessons. Choosing a calculation method:
Children need to be taught and encouraged to use the following processes in deciding what approach they will take to a calculation, to ensure they select the most appropriate method for the numbers involved.
Reception
Calculation
1. Responds to the vocabulary involved in addition and subtraction in rhymes and games. e.g. 1,2,3 little acorns, Squirrel can you see them? 1,2,3 little acorns, take them to your store… 5 Little leaves so bright and gay were dancing about on the tree one day. The wind came blowing through the town and one little leaf came tumbling down…
2. Recognises differences in quantity when comparing sets of objects. Who has the most lego? Do we have more cups or more plates at snack time?
3. Finds one more or one less from a group of up to ten objects. Role play, pay shop etc…
4. Relates addition by combining two groups. Combine the two pots of bears to see how many there are altogether
5. Relates subtraction to taking away. Taking away fruit, playing Kim’s Game etc…
6. In practical activities and discussion, begins to use the vocabulary involved in adding and subtracting. More, less, take away, added, altogehter…
7. Finds one more or one less than a number from 1 to 20.
There were 10 in the bed and the little one said …
8. Uses developing mathematical ideas and methods to solve practical problems
Uses a range of strategies for addition and subtraction, including some mental recall of number bonds.
9. Count reliably from 1 – 20, places them in order.
10. Using quantities and objects, add and subtract two single digit numbers, counting on and back to find the answer.
11. Solve problems involving doubling, halving and sharing
Year 1
Addition Subtraction Multiplication Division
Relate addition to counting on, recognise that addition can be done in any order, use practical and informal written methods to support the addition of a one-digit number, or a multiple of 10, to a one-digit or two-digit number. Understand subtraction as a “take away” and find a “difference” by counting up. Use practical, informal written methods to support the subtraction of a one-digit number from a one-digit or two-digit number, and a multiple from a two-digit number. Add and subtract one and two digit numbers to 20, including zero. Use the vocabulary of addition and subtraction and symbols to describe and record addition and subtraction number sentences.
Solve practical problems that involve combining groups of 2, 5 or 10, or sharing into equal groups. Count on from and back in zeros in ones, twos, fives and tens. Recognise odd and even numbers to 20. Recall the doubles of numbers to 10. Find simple fractions of objects, numbers and quantities (1/2 and ¼).
By the end of year 1, children should know number pairs to 10 and 20, addition and subtraction facts for totals to 20, addition doubles for all numbers to at least 10.
Mental Strategies
Reordering 5 + 13 = is the same as 13 + 5 = 10 + 2 + 10 = is the same as 10 + 10 + 2 = Near Doubles 6 + 7 is double 6 and add 1 or double 7 and subtract 1
+ = signs and missing numbers
3 + 4 = = 3 + 4
3 + = 7 7 = + 4
+ 4 = 7 7 = 3 +
+ = 7 7 = + Promoting covering up of operations and numbers. Partition small numbers 8 + 3 = 8 + 2 + 1
Informal Written Methods
Number lines (numbered) 7 + 4
Recording by drawing jumps on prepared lines and constructing own lines ( Teacher model number lines with missing numbers)
Mental Strategies
Pictures / marks Sam spent 4p. What was his change from 10p?
Counting forwards and backwards 42 – 39 count on in ones from 39 57 – 3 count back in ones from 57 72 – 50 count back in tens from 72 - = signs and missing numbers
7 - 3 = = 7 - 3
7 - = 4 4 = - 3
- 3 = 4 4 = 7 -
- = 4 4 = -
Informal Written methods
Number lines (numbered) 11 – 7 (Counting back)
The difference between 7 and 11 (Counting up)
Recording by - drawing jumps on prepared lines - constructing own lines
Pictures and symbols
There are 3 sweets in one bag. How many sweets are there in 5 bags?
(Recording on a number line modelled by the teacher when solving problems) Use of bead strings to model groups of. There is no need to use mathematical symbols for multiplication and division at this stage.
Understand multiplication as arrays
4 x 2 or 4 + 4
With the support of the teacher.
Identify doubles as two lots of.
Pictures / marks
12 children get into teams of 4 to play a game. How many teams are there?
With the support of the teacher.
0 1 2 3 4 5 6 7 8 9 10 11 12
10 11 12 0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9 10 11 12
Year 2
Addition Subtraction Multiplication Division
Add and subtract numbers using concrete objects, pictorial representations and mentally including: a two digit number and ones, a two digit number and tens, two two-digit numbers, adding three one digit numbers. Understand that subtraction is the inverse of addition and vice-versa; use this to derive and record related addition and subtraction number sentences, to check calculations and solve missing number problems.
Count in steps of 2, 3 and 5 from 0 and in 10’s from any number, forward and backward. Represent repeated addition and arrays as multiplication, and sharing and repeated subtraction (grouping) as division; use practical and informal written methods and related vocabulary to support multiplication and division including calculations with remainders. They use commutativity and inverse relations to develop multiplicative reasoning (4x5 = 20 and 20 ÷5 = 4). Find 1/3, ¼, 2/4 and ¾ of a length, shape, set of objects or quantity.
Use the symbols, + , -, x, ÷ and = to record and interpret number sentences involving all four operations; calculate the value of an unknown in a number sentence. By the end of Year 2 children should know number bonds to 20, addition and subtraction facts for all numbers fluently to 20, all pairs of multiples of 10 that total 100, addition doubles for all numbers to 20 and multiples of 10 to 50, derive and use related facts up to 100.
Mental Strategies
Counting forwards and backwards 23 + 5 count on in ones from 23 27 + 60 count on in tens from 27 Reordering 5 + 34 is the same as 34 + 5 5 + 7 + 5 is the same as 5 + 5 + 7 Partition into tens and ones and recombine 12 + 23 = 10 + 2 + 20 + 3 = 30 + 5 = 35 refine to partitioning the second number only: 23 + 12 = 23 + 10 + 2 = 33 + 2 = 35 Compensating 34 + 9 is the same as 34 + 10 – 1 34 + 21 is the same as 34 + 20 + 1
Mental Strategies
Counting forwards and backwards 42 – 39 count on in ones from 39 57 – 3 count back in ones from 57 72 – 50 count back in tens from 72 Compensating 35 – 9 is the same as 35 – 10 + 1 35 – 21 is the same as 35 – 20 - 1
Partition into tens and ones and recombine (partition second number only) 37 – 12 = 37 – 10 – 2 = 27 – 2 = 25
Bridging through multiples of 10 (Using known number facts) 12 – 7 is the same as 12 – 2 – 5 43 – 6 is the same as 43 – 3 - 3
By the end of Year 2 children should know the 2, 5 and 10 times table and begin to know and use other times tables and recall facts. Doubles of all numbers to 20, doubles of multiples of 10 to 50 and corresponding halves. Understand multiplication as arrays and repeated addition
4 x 2 or 4 + 4
or repeated addition/counting in groups of 2 + 2 + 2 + 2
Doubling using partitioning 15 x 2 = (10 x 2) + (5 x 2) =
Understand division as sharing and grouping Sharing – 6 sweets are shared between 2 people. How many do they have each?
6 2 can be modelled as: Grouping – There are 6 sweets. How many people can have 2 each? (How many 2’s make 6?)
-10
+1
25 35 26
37 25 27
-2 -10
0 1 2 3 4 5 6 7 8
0 2 4 6
Bridging through multiples of 10 (Using known number facts) 5 + 8 is the same as 5 + 5 + 3 65 + 7 is the same as 65 + 5 + 2 Near Doubles 13 + 14 is the same as double 13 and + 1 or double 14 and – 1 39 + 40 is the same as double 40 and – 1
Written Methods
Empty Number Lines Children need to be able to partition numbers in ways other than into tens and ones to help them make multiples of ten by adding in steps. 8 + 7
Written Methods
Empty Number Lines Counting Back 15 – 7 =
Counting up 74 – 27 =
35 45 44
+10
-1
Year 3
Addition
Subtraction Multiplication Division
Add or subtract mentally including: a three digit number and ones, a three digit number and tens, a three digit number and hundreds. Develop and use formal written methods of columnar addition and subtraction to record, support or explain + and – of numbers up to 3-digit numbers.
Count from 0 in multiples f 4, 8, 50 and 100. Find 10 or 100 more or less than a given number. Multiply one-digit and two digit numbers by 10 or 100 and describe the effect. Use practical and informal written methods to x and ÷ 2-digit numbers (e.g. 13x3, 50÷4). Understand that division is the inverse of multiplication, and vice versa; use this to derive and record related multiplication and division number sentences. Recognise, find and write fractions of a discrete set of objects: unit fractions and non-unit fractions with small denominators.
By the end of Year 3 children should know number bonds to 100, sums and differences of multiples of 10, addition doubles of multiples of 10 to 100.
Mental Strategies
Counting forward and backwards 50 + 38 count on in tens and then ones from 50 35 + 15 count on in steps of 5 from 35 Reordering 23 + 54 is the same as 54 + 23 13 + 21 + 13 is the same as 13 + 13 + 21 Partition into tens and ones and recombine Partition both numbers and recombine. Refine to partitioning the second number only e.g. 36 + 53 = 53 + 30 + 6 = 83 + 6 = 89
Compensating 53 + 18 is the same as 53 + 20 – 2 Bridging through Multiples of Ten 49 + 32 is the same as 49 + 1 + 31
Mental Strategies
Counting forward and backwards
90 – 27 count back in tens and then ones from 90
Counting on to find the difference. 90 – 27 is the same as 27 + 3 + 60
Compensating
84 – 18 is the same as 84 – 20 + 2
Partition into tens and ones and recombine (second number only)
Continue as in Year 2 but with appropriate numbers e.g. 97 – 15 = 97 -10 – 5 = 87 – 5 = 82
By the end of Year 3 children should know the 3, 4 and 8 times table, and the corresponding division facts. Doubles of multiples of 10 to 100 and corresponding halves.
Mental Strategies
Understand multiplication as arrays and repeated addition Continue to understand multiplication as repeated addition and continue to use arrays (as in Year 2). Doubling multiples of 10 up to 100 by using known facts Double 4 is 8 so double 40 is 80. Use Near Doubles Double 19 is the same as double 20 -2 Double by partitioning Double 54 is the same as double 50 and then double 4 and recombine. Multiply one and two digit numbers by 10 and 100 Explain how the digits move when multiplying. Commutativity 4 x 12 x 5 is the same as 4 x 5 x 12 So is the same as 20 x 12 = 240
Mental Strategies
Use the relationship between multiplication and division. 32 ÷ 4 I know 4 x 8 is 32 so 32 ÷ 4 = 8 Halving using partitioning Half of 102 is the same as half of 100 and then half of 2 and recombine. Use partitioning to divide TU by U 68 ÷ 4 is the same as (40 ÷ 4) + (28 ÷ 4) = 10 + 7 = 17
Informal Written Method
Understand division as sharing and grouping – repeated subtraction 15 ÷ 3 can be modelled as: Sharing – 15 shared between 3 OR 0 3 6 9 12 15
Or 18 ÷ 3 can be modelled as: Sharing – 18 shared between 3 (see Year 2 diagram) Grouping - How many 3’s make 18?
53 83 89
+30 +6
97 82
87
- 5 - 10
Written Methods
Empty Number lines 48 + 36 =
Or
Expanded Column Method
47 40 7
76 70 6
110 13 123
Progressing to columnar addition only when the children are secure with the expanded method.
358 + 73 431
11
Written Methods
Only move onto formal written methods once the children are secure with a mental strategy. Empty Number Lines Counting up 326 – 178 =
Expanded Column Method Progressing to columnar subtraction only when the children are secure with the expanded method. 74 – 27 =
70 4 7060 14
4 76
414
20 7 20 7 2 7
40 7 4 7
Informal Written Method
Partitioning to multiply TU x U 32 x 3 = (30 x 3) + (2 x 3) 90 + 6 = 96
Written Methods
Partitioning using the Grid Method
38 × 7 = (30 × 7) + (8 × 7) =
210 + 56 = 266
Remainders 16 ÷ 3 = 5 r1 Sharing - 16 shared between 3, how many left over? Grouping – How many 3’s make 16, how many left over?
0 3 6 9 12 15 18
Year 4
Addition
Subtraction Multiplication Division
Count in multiples of 6, 7, 9, 25 and 100. Count backwards through zero to include negative numbers. Find 1000 more or less than a given number. Add or subtract mentally including: a three digit number and ones, a three digit number and tens, a three digit number and hundreds. Add or subtract mentally pairs of two-digit whole numbers (e.g. 47 + 58, 91 – 35) Refine and use efficient written methods to + or – of up to 4-digit whole numbers and £.p
Multiply and divide numbers to 1000 by 10 and then 100 (whole number answers), understanding the effect; relate to scaling up and scaling. Develop and use written methods to record, support and explain x and ÷ of 2-digit and 3 digit numbers by a 1-digit number, including ÷ with remainders (e.g. 15x9, 98 ÷6). Solve problems involving increasingly harder fractions to calculate quantities and fractions to divide quantities, including non-unit fractions when the answer is a whole number.
By the end of year 4 children should know sums and differences of multiples of 10, 100 and 1000. Addition doubles of numbers 1 to 100 and the corresponding halves, pairs of fractions that total 1.
Mental Strategies
Counting forward and backwards 47 + 58 count on 50 from 47, then 3 to 100 then 5 to 105 570 + 300 cont on in hundreds from 570 Reordering 6 + 13 + 4 + 3 is the same as 6 + 4 + 13 + 3 28 + 75 is the same as 75 + 28 Partition into hundreds, tens and ones and recombine Either partition both numbers and recombine or partition the second number only e.g. 55 + 37 = 55 + 30 + 7 = 85 + 7 = 92
Bridging through multiples of ten 57 + 34 is the same as 57 + 3 + 31 Compensating
Mental Strategies
Counting forwards and backwards 73 – 68 count up from 68, counting 2 to 70 then 3 to 73 960 – 500 count back in hundreds from 960 Reordering 17 + 9 – 7 is the same as 17 – 7 + 9 Partition into hundreds, tens and ones and recombine (second number only) 365 – 42 = 365 – 40 = 325 – 2 = 323 Bridging through multiples of ten 92 – 25 = 92 – 2 – 20 -3 Compensating 64 – 28 is the same as 64 – 30 + 2
By the end of Year 4 children should know times tables up to 12x12, and the corresponding division facts. Use known and derived facts to x and ÷ mentally, including x 3 numbers together. Double 2 digit numbers and find corresponding halves.
Mental Strategies
Doubling 2 digit numbers by partitioning Double 52 is double 50 add double 2 Use Near Doubles Double 29 is the same as double 30 -2 Multiply one and two digit numbers by 10 and 100 Explain how the digits move when multiplying. Partitioning to multiply TU x U 32 x 3 = (30 x 3) + (2 x 3) 90 + 6 = 96 Partitioning to multiply U x U 6 x 7 is the same as (6 x 2) + (6 x 5) Using known facts to multiply a multiple of 10 by a single digit 40 x 6
Mental Strategies
Use the relationship between multiplication and division. 32 ÷ 4 I know 4 x 8 is 32 so 32 ÷ 4 = 8 Using known facts If 2 x 3 = 6 then 600 ÷ 3 = 200. Halve by partitioning Half of 246 is the same as half of 200, then half of 46 and recombine. Find one quarter by halving and halving again One quarter of 48 is the same as half of 48 = 24 and then half of 24 = 12 Divide numbers to 1000 by 10 and 100 (whole number answers) Explain how the digits move when dividing. Divide by 4 and 8 by halving 104 ÷ 4 = half of 104 and then half of 52 104 ÷ 8 = half of 104, half of 52 and then half of 26
55 85 92
+30 +7
38 + 68 is the same as 38 + 70 - 2
Near Doubles
76 + 75 is double 76 and – 1 or double 75 and + 1
Written Methods
Expanded Column Method
358 + 73 = either 300 + 50 + 8 + 70 + 3 300 +120 + 11 = 431 Or 358 + 73 300 120 11 451 Progressing to columnar addition only when the children are secure with the expanded method.
358 + 73 431
11
Written Methods
Expanded Column Method Progressing to columnar subtraction only when the children are secure with the expanded method.
74 – 27 =
70 4 7060 14
4 76
414
20 7 20 7 2 7
40 7 4 7
I know that 4 x 6 = 24 so 40 x 6 is 240
Multiplying by 4, 8, 5 and 20 using doubling and halving X 4 by doubling and doubling again X 8 by doubling three times X by 5 by multiplying by 10 then halving X by 20 by x by 10 and then doubling
Written Methods
Partitioning using the Grid Method
38 × 7 = (30 × 7) + (8 × 7) =
210 + 56 = 266
Expanded Short Multiplication
30 8
7
210 30 7 210
56 8 7 56
266
Progressing to short multiplication only when the children are secure with the expanded method. Short Multiplication
5
38
7
266
The step here involves adding 210 and 50 mentally with only the 5 in the 50 recorded. This highlights the need for children to be able to add a multiple of 10 to a two-digit or three-digit number
Written Methods
Repeated subtraction on a number line
30 ÷ 6 can be modelled as: grouping – groups of 6 taken away and the number of groups counted. Including questions with remainders 41 ÷ 4 = 10 r 1
Chunking Taking groups of the divisor away
6 196
60 6 10
136
60 6 10
76
60 6 10
16
12 6 2
4 32
Answer: 32R 4
or
-1
-40
10 x 4
mentally before they reach this stage.
6 196
180 6 30
16
12 6 2
4 32
Answer: 32R 4
Progressing to short division only when the children are secure with the chunking method. Short division for HTU ÷ U
2
9 7
3 2 9 1
Year 5
Addition
Subtraction Multiplication Division
Count forwards and backwards in steps of powers of 10 for any given number up to 1,000,000. Extend mental methods for whole number calculations with increasingly large numbers (12,462 – 2300 = 10,162). Use efficient written methods to + and – numbers with more than 4 digits and decimals with up to 2 places.
Extend mental methods for whole number calculations drawing upon known facts; to multiply a two-digit number by a one-digit number (e.g. 12 x 9), to multiply by 25 (e.g. 16 x 25) Use understanding of place value to multiply and divide whole numbers and decimals by 10, 100 or 1000. Refine and use efficient written methods to multiply numbers up to 4 digits by a one or two digit number. Refine and use efficient written methods to divide numbers up to 4 digits by a one digit number (short division).
By the end of year 5 children should know sums and differences of decimals, doubles and halves of decimals, what must be added to any four digit number to make the next multiple of 1000.
Mental Strategies
Counting forwards and backwards 3.2 + 0. 6 count on in tenths
Using known number facts 4.5 + 0.7 I know that 5 + 7 = 12 so 0.5 + 0.7 = 1.2 4 + 1.2 = 5.2
Reordering
25 + 36 + 75 is the same as 25 + 75 + 36 1.7 + 2.8 + 0.3 is the same as 1.7 + 0.3 + 2.8 Partition into thousands, hundreds, tens and ones and recombine
Either partition both numbers and recombine or partition the second number only e.g. 358 + 73 = 358 + 70 + 3 = 428 + 3 = 431
Mental Strategies
Reordering 58 + 47 – 38 is the same as 58 – 38 + 47 Partition into thousands, hundreds, tens and ones and recombine (second number only) 4.7 – 3.5 = 4.7 – 3 – 0.5 Finding the difference 3027 – 2996 = Start with the smallest number and + 4, then + 27 = 31 Bridging through multiples of 10 6070 – 4987 = 4987 + 13+ 1000 + 70 Compensating 405 – 399 is the same as 405 – 400 + 1
By the end of year 5 children should be able to apply all the multiplication tables and related facts frequently, commit them to memory and use them confidently to make larger calculations. Recognise and use square numbers and cube numbers.
Mental Strategies
Doubling numbers by partitioning Double 752 is double 700, add double 50, add double 2 Use Near Doubles Double 629 is the same as double 630 minus 2 Multiply whole numbers and decimals numbers by 10, 100 and 1000 Explain how the digits move when multiplying. Partitioning to multiply TU x U 32 x 3 = (30 x 3) + (2 x 3) 90 + 6 = 96 Using known facts to multiply a multiple of 10 or 100 by a single digit
Mental Strategies
Use the relationship between multiplication and division. 32 ÷ 4 I know 4 x 8 is 32 so 32 ÷ 4 = 8 Use known facts to divide multiples of 10 by single digits 270 ÷ 3 I know 27 ÷ 3 = 9 so the answer must be 90 Divide whole numbers and decimals to 1000 by 10, 100 and 1000 Explain how the digits move when dividing. Use partitioning to divide TU by U 68 ÷ 4 is the same as (40 ÷ 4) + (28 ÷ 4) = 10 + 7 = 17
358 428 431
+70 +3
Bridging through multiples of 10 675 + 235 = 675 + 25 + 210 Compensating 2458 + 79 is the same as 2458 + 80 - 1 Near Doubles 1250 + 1260 is double 1250 then add 10
Written Methods
Column Addition (Only once the children are secure with the expanded method). 358 + 73 431
11
Extend to numbers with more than four digits 3587 + 675 = 4262 3587 + 675 4262 111
. Extend to decimals (same number of decimals places) and adding several numbers (with different numbers of digits).
Written Methods
Expanded Column Method 741 – 367 =
700 40 1 700 600 130
4011
1 76
413
111
300 60 7 300 60 7 3 6 7
300 70 4 3 7 4
Progressing to columnar subtraction only when the children are secure with the expanded method. Extend to numbers with more than four digits
and to decimals (same number of decimals places).
400 x 6 I know that 4 x 6 = 24 so 400 x 6 is 2400 Multiply by 25 and 50 48 x25 = 48 x 100 ÷ 4 62 x 50 = 62 x 100 ÷ 2
Written Methods
Partitioning using the Grid Method
56 × 27 is approximately
60 × 30 = 1800.
Expanded Short Multiplication
30 8
7
210 30 7 210
56 8 7 56
266
Short Multiplication
5
38
7
266
The step here involves adding 210 and 50 mentally with only the 5 in the 50 recorded. This highlights the need for children to be able to add a multiple of 10 to a two-digit or three-digit number mentally before they reach this stage. Long Multiplication
Written Methods
Chunking or long division Taking groups of the divisor away
6 196
180 6 30
16
12 6 2
4 32
Answer: 32R 4
Short division for numbers up to ThHTU ÷ U
2
9 7
3 2 9 1
24 x 16 2
2 4 x 1 6 2 4 0 1 4 4 3 8 4 124 x 26 1 2
1 2 4 X 2 6 2 4 8 0 7 4 4 3 2 2 4 1 1
Year 6
Addition
Subtraction Multiplication Division
Calculate mentally with integers and decimals: U.t + U.t, U.t – U.t Use efficient written methods to + and – integers and decimals. Extend mental methods for whole number calculations with increasingly large numbers (12,462 – 2300 = 10,162).
Calculate mentally with integers and decimals: TU x U, TU ÷ U, U.t x U, U.t ÷ U. Multiply 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. Divide numbers up to 4 digits by a two digit number using the formal written method of short division. Use written division methods in cases where the answer has up to 2 decimal places.
By the end of Year 6 children should know addition and subtraction facts for multiples of 10 to 1000, what must be added to a decimal with units, tenths and hundredths to make the next whole number.
Mental Strategies
Counting forwards and backwards 1.7 + 0.55 count on in tenths and hundredths Reordering 1004 + 607 + 706 is the same as 1004 + 706 + 607 Partition into thousands, hundreds, tens, ones and decimal fractions and recombine
Either partition both numbers and recombine or partition the second number only e.g. 35.8 + 7.3 = 35.8 + 7 + 0.3 = 42.8 + 0.3 = 43.1
Bridging through multiples of 10 0.8 + 0.35 is the same as 0.8 + 0.2 + 0.15 Compensating 5.7 + 3.9 is the same as 5.7 + 4 – 0.1
Mental Strategies
Counting forwards and backwards
0.5 – 0.31 count on in hundredths to 0.4 and then tenths to 0.5 Reordering 4.7 + 5.6 – 0.7 is the same as 4.7 – 0.7 + 5.6 Partition into thousands, hundreds, tens, ones and decimal fractions and recombine (second number only )
276 – 153 = 276 – 100 – 50 – 3 Bridging through multiples of 10 8.34 – 2.8 is the same as 0.2 + 5 + 0.34 Compensating 6.8 – 4.9 is the same as 6.8 – 5.0 + 0.1
By the end of Year 6 children should continue to use all the multiplication tables to calculate and maintain their fluency. Use place value and x facts to derive related facts involving decimals.
Mental Strategies
Doubling 2 digit numbers by partitioning Double 52 is double 50 add double 2 Use Near Doubles Double 29 is the same as double 30 -2 Double decimals to 1dp by partitioning Double 7.2 is double 7 and double 0.2 Multiply whole numbers and decimals numbers by 10, 100 and 1000 Explain how the digits move when multiplying. Partitioning to multiply TU x U 32 x 3 = (30 x 3) + (2 x 3) 90 + 6 = 96
Mental Strategies
Use the relationship between multiplication and division. 32 ÷ 4 I know 4 x 8 is 32 so 32 ÷ 4 = 8 Use known facts to divide multiples of 10 by single digits 270 ÷ 3 I know 27 ÷ 3 = 9 so the answer must be 90 Divide by 25 and 50 by dividing by 100 and multiplying 460 ÷ 25 is the same as 460 ÷ 100 = 4.6 x 4 = 18.4 270 ÷ 50 is the same as 270 ÷ 100 = 2.7 x 2 = 5.4 Divide whole numbers and decimals to 3 decimal places by 10, 100 and 1000 Explain how the digits move when dividing. Use partitioning to divide TU by U 68 ÷ 4 is the same as (40 ÷ 4) + (28 ÷ 4) = 10 + 7 = 17
35.8 42.8 43.1
+7 +0.3
Near Doubles 2.5 + 2.6 is double 2.5 add 0.1 or double 2.6 and subtract 0.1
Written Methods
Column Method Extend to numbers with any number of digits and decimals with 1 and 2 decimal places. 124.9 + 117.25 = 242.15 124.9 + 117.25 242.15
11
Written Methods
Expanded Column Method
400
500 0 3 400 90 13 500
400100
0
903
3
134
5 09 13
3
200 70 8 200 70 8 200 70 8 2 7 8
200 20 5 200 20 5 2 2 5
Progressing to columnar subtraction only when the children are secure with the expanded method. Extend to numbers with any number of digits and decimals with 1 and 2 decimal places.
Using known facts to multiply a multiple of 10 or 100 by a single digit 400 x 6 I know that 4 x 6 = 24 so 400 x 6 is 2400 because my answer needs to be 100 times bigger. Multiply by 25 and 50 48 x25 = 48 x 100 ÷ 4 62 x 50 = 62 x 100 ÷ 2
Written Methods
Partitioning using the Grid Method
286 × 29 is approximately
300 × 30 = 9000.
Short Multiplication
5
38
7
266
The step here involves adding 210 and 50 mentally with only the 5 in the 50 recorded. This highlights the need for children to be able to add a multiple of 10 to a two-digit or three-digit number mentally before they reach this stage. Long Multiplication 24 x 16 2
