Progression in Mental and �Written Strategies
PRIMARY
SECONDARY
POST 16
ALTERNATIVE
Mental Calculations
Please note that they highlight progression but are not necessarily matched with the national curriculum.
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Written strategies
Can I do it in my head using a mental strategy?
Could I use some jottings to help me?
Should I use a formal written method to work it out?
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Mental Calculations
23 + 5
3 + 5 = 8
so 23 + 5 = 28
Using number bonds
Adding a single-digit number to a 2-digit number
Using partitioning
28 + 5
28 + 5 = 28 + 2 + 3
28 + 5 = 33
Using compensation
28 + 9
28 + 9 = 28 + 10 – 1
28 + 9 = 37
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28 – 5
8 – 5 = 3
So 28 – 5 = 23
Using partitioning
Subtracting a single-digit number from a 2-digit number
Using number bonds
Using compensation
23 – 8
23 – 8 = 23 – 3 – 5
23 – 8 = 15
37 – 9
37 – 9 = 37 – 10 + 1
37 – 9 = 28
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37 + 10
37 + 10 = 47
Adding 10
Adding two 2-digit numbers
Using partitioning with no bridging through a multiple of 10
Using partitioning, bridging through a multiple of ten
23 + 14
23 = 2 tens and 3 ones
14 = 1 ten and 4 ones
23 + 14 = 3 tens and 7 ones
23 + 14 = 37
35 + 17
35 + 17 = 35 + 10 + 7
Adding multiples of 10
23 + 30
23 + 30 = 53
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37 + 19
37 + 19 = 37 + 20 – 1
Using compensation
Adding two 2-digit numbers
Using near doubles (consecutive)
45 + 46
45 + 46 = 45 + 45 + 1
45 + 46 = double 45 + 1
45 + 46 = 90 + 1 45 + 46 = 91
Using near doubles (non consecutive)
25 + 27
25 + 27 = 25 + 25 + 2
25 + 27 = double 25 + 2
25 + 27 = 50 + 2 25 + 27 = 52
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52 – 10
52 – 10 = 42
Subtracting 10
Subtracting two 2-digit numbers
Using partitioning with no bridging through a multiple of 10
38 – 23
38 = 3 tens and 8 ones
23 = 2 tens and 3 ones
38 – 23 = 1 ten and 5 ones
38 – 23 = 15
Subtracting multiples of 10
43 – 20
43 – 20 = 23
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43 – 25
In total 10 + 5 + 3 = 18 has been added to 25 to equal 43.
43 – 25 = 18
Using counting on to find the difference
Subtracting any two 2-digit numbers
Using counting backwards, bridging through a multiple of ten
43 – 25
25 = 20 and 5
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35 – 19
35 – 19 = 35 – 20 + 1
Using compensation
Subtracting two 2-digit numbers
Using adjustment or constant difference
34 – 16
34 – 16 could be rewritten as 38 – 20. The numbers have been adjusted to make the calculation simpler.
The difference between 34 and 16 is the same as the difference between 38 and 20 because 4 has been added to each number.
38 – 20 = 18
34 – 16 = 18
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4.2 + 3.4
3.4 = 3 + 0.4
4.2 + 3.4 = 4.2 + 3 + 0.4
4.2 + 3.4 = 7.6
Adding numbers with decimals
Using compensation
Using partitioning with no bridging through whole numbers
Using partitioning, bridging
through whole numbers
4.7 + 3.5
3.5 = 3 + 0.5
3.4 + 0.9
3.4 + 0.9 = 3.4 + 1 – 0.1
3.4 + 0.9 = 4.3
2.5 + 1.9
2.5 + 1.9 = 2.5 + 2 – 0.1
2.5 + 1.9 = 4.4
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8 + 6 + 2
Using number bonds
Adding three single-digit numbers
8 + 6 + 2 = 16
3 + 3 + 4
Using consecutive numbers
3 + 4 + 5
Using doubles
3 + 3 + 4
3 + 3 + 4 = double 3 + 4
3 + 3 + 4 = 6 + 4
3 + 3 + 4 = 10
3 + 4 + 5 = 4 + 4 + 4
3 + 4 + 5 = 12
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40 x 3
Multiplying a single-digit number by multiples of 10
Multiplying by 10, 100 and 1,000 and multiples of 10
40 x 3 is 10 times bigger than 4 x 3 so the answer will be 10 times bigger
40 x 3 = 120
46 x 10 = 460
Multiplying a number by 10, 100
and 1,000
Multiplying a number with decimals by 10, 100 and 1,000
0.34 x 1,000 = 340
0.34 x 100 = 34
0.34 x 10 = 3.4
46 x 1,000 = 46,000
46 x 100 = 4,600
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Multiplying by multiples of 10 and 100
Multiplying multiples of 10 by multiples of 10
30 x 40
Each step is 10 times bigger.
Each calculation is multiplied by 10, so the answer is becoming 10 times bigger each time.
30 x 40 is 100 times bigger
than 3 x 4
Multiplying multiples of 100 by multiples of 10
700 x 50
30 x 40 = 1,200
700 x 50 is 1,000 times
bigger than 7 x 5
700 x 50 = 35,000
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0.3 x 2 is 10 times smaller than 3 x 2 so the answer will be 10 times smaller
0.3 x 2 = 0.6
Multiplying by tenths
Multiplying a single-digit number by tenths
0.4 x 6 is 10 times smaller than 4 x 6
the answer will be 10 times smaller
Multiplying tenths by tenths
0.3 x 2
0.4 x 0.6
0.4 x 6 is 10 times smaller than 4 x 6
so the answer will be 10 times smaller
0.4 x 0.6 is 10 times smaller than 0.4 x 6
so the answer will be 10 times smaller
0.4 x 0.6 is 100 times smaller than 4 x 6
0.4 x 0.6 = 0.24
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180 ÷ 3
Identifying division facts
Dividing by 10 and 100 and multiples of 10 by a single-digit number
180 ÷ 3 is 10 times bigger than 18 ÷ 3 so the answer is 10 times bigger
180 ÷ 3 = 60
600 ÷ 10 = 60
Dividing a number by 10, 100 and 1,000
Dividing a number with decimals by 10, 100 and 1,000
624 ÷ 10 = 62.4
624 ÷ 100 = 6.24
600 ÷ 100 = 6
6,240 ÷ 1,000 = 6.24
6,000 ÷ 1,000 = 6
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17 x 3
Using partitioning
Multiplying 2-digits by a single-digit
Multiplying by 5
Using doubling
Multiplying by 9 using compensation
24 x 5
24 x 10 = 240
Half of 240 = 120
24 x 5 = 120
17 x 3 = (10 x 3) + (7 x 3)
17 x 3 = 30 + 21 = 51
17 x 3 = 51
To multiply a number by 5, multiply it by 10 and then halve it.
Multiplying a number by 4 is the same as doubling it, then doubling it again.
14 x 4
14 x 4 = 14 x 2 x 2
14 x 4 = 56
14 x 9
14 x 9 = ( 14 x 10 ) – 14
14 x 9 = 126
Instead of multiplying a number by 9,
multiply it by 10 then subtract one group
to be left with 9 groups.
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Double 30
30 = 3 tens
Double 3 = 6
Double 3 tens = 6 tens
Double 30 = 60
Doubling 2-digit numbers
Doubling multiplies of 10
Doubling
Doubling multiplies of 5
Double 35
35 = 30 + 5
Double 30 = 60
Double 5 = 10
Double 35 = 60 + 10
Double 35 = 70
Double 37
37 = 30 + 7
Double 30 = 60
Double 7 = 14
Double 37 = 60 + 14
Double 37 = 74
Double 4.7
4.7 = 4 + 0.7
Double 4 = 8
Double 0.7 = 1.4
Double 4.7 = 8 + 1.4
Double 4.7 = 9.4
Doubling numbers with decimals
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Half of 60
60 = 6 tens
6 ÷ 2 = 3
6 tens ÷ 2 = 3 tens
Half of 60 = 30
Halving 2-digit numbers
Halving multiplies of 10 with an even number of tens
Halving
Half of 72
72 = 70 + 2
Half of 70 = 35
Half of 2 = 1
Half of 72 = 35 + 1
Half of 72 = 36
Half of 158
158 = 100 + 50 + 8
Half of 100 = 50
Half of 50 = 25
Half of 8 = 4
Half of 158 = 50 + 25 + 4
Half of 158 = 79
Halving 3-digit numbers
Halving multiplies of 10 with an odd number of tens
Half of 50
50 = 40 + 10
Half of 40 = 20
Half of 10 = 5
Half of 50 = 20 + 5
Half of 50 = 25
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Written calculations: Addition and Subtraction
whole | |
part | part |
Tens Frame
Number line
thousands | hundreds | tens | ones |
| | | |
100
1
10
1000
Place Value grid
Base ten
part
part
whole
Part-whole models
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Additive Structure
addend
addend
sum
sum | |
addend | addend |
addend + addend = sum
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Additive Structure
subtrahend
difference
minuend
minuend | |
subtrahend | difference |
minuend - subtrahend = difference
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Reception ELGs | |
Addition and Subtraction | |
Number | Numerical Patterns |
|
|
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Composition of numbers
5 + 0
4 + 1
3 + 2
Explore and represent patterns
within numbers up to 10
Doubling facts
Subitise up to 5
7
Automatically recall number bonds up to 5
Explore and represent patterns with odds
and evens
Reception ELGs
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Year 1 |
Addition and Subtraction |
|
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Understanding teen numbers as a 10 and ones
Can add a one-digit number to a two-digit number within 20
12 + 7 = 19
10 + 1 = 11
10 + 2 = 12
10 + 3 = 13
10 + 4 = 14
0
5
10
15
20
Year 1 addition
? | |
12 | 7 |
12
7
?
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17 – 8
Taking away and counting back (used only for subtracting small numbers)
Year 1 subtraction
0
5
10
15
20
8
?
Can subtract a one-digit number from a two-digit number
20 – 8
20 | |
8 | ? |
20
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Year 2 |
Addition and Subtraction |
|
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tens | ones |
| |
| |
tens | ones |
| |
| |
44
54
64
74
75
76
77
+10
+10
+10
+1
+1
+1
44 + 33
30 + 3
16
+4
+3
20
23
16 + 7
4 + 3
27 + 30
+10
+10
+10
27
37
47
57
Can add a two-digit number and 1s using concrete objects or pictorial representations
Can add a two-digit number and 10s using concrete objects or pictorial representations
Can add 2 two-digit numbers using concrete objects or pictorial representations
Year 2 addition
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tens | ones |
| |
37 – 4
Can subtract a two-digit number and 1s using concrete objects or pictorial representations
Year 2 subtraction
0
5
10
15
20
25
30
35
40
32 – 5
– 2 – 3
Can subtract a two-digit number and 1s using concrete objects or pictorial representations
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Year 2 subtraction
Can subtract a two-digit number and 10s using concrete objects or pictorial representations
tens | ones |
| |
53 – 40
13
53
Can subtract a two-digit number using concrete objects or pictorial representations
54 – 23
– 20 – 3
31
54
– 3
– 10
– 10
– 10
– 10
– 10
– 10
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Year 3 |
Addition and Subtraction |
|
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hundreds | tens | ones |
| | |
| | |
245
250
255
260
+4
+4
246 + 8
+ 4 + 4
Can add a three-digit number and 10s
Can add a three-digit number and 1s mentally using partitioning
Can add a three-digit number and
100s
Year 3 addition
235 + 40
hundreds | tens | ones |
| | |
| | |
+ 10
+ 10
+ 10
+ 10
hundreds | tens | ones |
| | |
| | |
243 + 300
243
343
443
543
+ 100
+ 100
+ 100
275
265
255
245
235
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tens | ones |
| |
| |
hundreds | tens | ones |
| | |
| | |
42 + 31
367 + 185
47 + 36
tens | ones |
| |
| |
Can add numbers with up to three digits using the expanded column method
Can add numbers with two digits using the expanded column method
(no regrouping)
Can add numbers with two digits using the expanded column method (regrouping)
Year 3 addition
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235
240
246
250
– 2
– 6
246 – 8
– 6 – 2
Can subtract a three-digit number
and 10s
Can subtract a three-digit number and 1s mentally using decomposition (bridging ten)
Can subtract a three-digit number and
100s
Year 3 subtraction
625 – 300
273 – 40
325
425
525
625
– 10
– 10
– 10
– 10
– 100
– 100
– 100
273
263
253
243
233
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53 – 35
543 – 324
Can subtract numbers with up to three digits using the expanded column method (with exchange)
Can subtract numbers with two digits using the expanded column method (with exchange)
Year 3 subtraction
tens | ones |
| |
75 – 31
Can subtract numbers with two digits using the expanded column method (no exchange)
tens | ones |
| |
hundreds | tens | ones |
| | |
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Year 4 |
Addition and Subtraction |
|
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Thousands | hundreds | tens | ones |
| | | |
| | | |
3,475 + 1,244
152 + 163
Can add three-digit numbers using the formal column addition method
Can add four-digit numbers using the formal column addition method
Year 4 addition
hundreds | tens | ones |
| | |
| | |
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Year 4 subtraction
Can subtract three-digit numbers using the formal column subtraction method
Can subtract four-digit numbers using the formal column subtraction method
thousands | hundreds | tens | ones |
| | | |
362 – 124
hundreds | tens | ones |
| | |
3,317 – 1,254
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Year 5 |
Addition and Subtraction |
|
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Year 5 addition
Ten thousands | thousands | hundreds | tens | ones |
| | | | |
| | | | |
Can add numbers with more than four-digits
37,056 + 4,863
Can add numbers with wholes and decimals up to three decimal places
13.78 + 8.16
tens | ones | . | tenths | hundredths |
| | . | | |
| | . | | |
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thousands
Can subtract numbers with more than four-digits
Year 5 subtraction
| thousands | hundreds | tens | ones |
| | | | |
48,294 – 13,478
ten
thousands
Can subtract numbers with wholes and decimals up to three decimal places
tens | ones | . | tenths | hundredths |
| | . | | |
17.65 – 6.82
−
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Year 6 |
Addition and Subtraction |
|
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Year 6 addition and subtraction
Can subtract numbers with wholes and decimals up to three decimal places with different numbers of digits
Can add numbers with wholes and decimals up to three decimal places with different numbers of digits
57.3 + 32.908
67.26 – 23.917
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Written calculations: Multiplication and Division
Multiplicative Structure
product | ||||
multiplicand | multiplicand | multiplicand | multiplicand | multiplicand |
Multiplier (number of equal parts)
multiplicand x multiplier = product
The multiplicand and the multiplier are both factors of the product.
factor x factor = product
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Multiplicative Structure
dividend | ||||
quotient (value of part) | quotient (value of part) | quotient (value of part) | quotient (value of part) | quotient (value of part) |
divisor (number of equal parts)
dividend ÷ divisor = quotient
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Multiplicative Structure
dividend | |||||
quotient (value of part) | quotient (value of part) | quotient (value of part) | quotient (value of part) | quotient (value of part) | remainder |
divisor (number of equal parts)
dividend ÷ divisor = quotient plus a remainder
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Quotative division
Quotative and partitive division
There are 32 pupils in a class.
They are put into groups of 4.
There are 32 pupils in a class.
They are put into 4 groups.
There are 4 pupils in each group.
There are 8 groups.
There are 4 groups.
There are 8 pupils in each group.
Partitive division
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EYFS |
Multiplication and Division |
|
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Can double numbers and quantities up to 10 using practical objects
Half of 6
Half of 6 = 3
Half of 6 = 3
Half of 8 = 4
Half of 8
Can halve numbers and quantities up to 10 using practical objects with equal sharing
16 = 8 + 8
Reception ELGs
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Year 1 |
Multiplication and Division |
|
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Can use concrete objects, pictorial representations and arrays
Can count in 2s
Year 1 multiplication
Can count in 5s
Can count in 10s
?
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Can use concrete objects, pictorial representations and arrays
Year 1 division
10
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Year 2 |
Multiplication and Division |
|
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0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0
5
10
15
20
25
30
35
40
0
10
20
30
40
50
60
70
80
Can make connections between arrays, number patterns, and counting in 2s, 5s and 10s
Can recall and use multiplication facts for the 2, 5 and 10 times tables
Year 2 multiplication
How many twos equal 16?
How many fives equal 40?
How many tens equal 80?
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Can calculate mathematical statements for multiplication using the 2, 5 and 10 times tables
0
5
10
15
5 x 3 = 5 + 5 + 5
Year 2 multiplication
Show that multiplication can be done in any order (commutative)
3 x 5 = 15
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
5 x 3 = 15
0
5
10
15
5 x 3 = 15
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0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0
5
10
15
20
25
30
35
40
0
10
20
30
40
50
60
70
80
Can recall and use division facts for the 2, 5 and 10 times tables
Year 2 division
Can calculate mathematical statements for division using the 2, 5 and 10 times tables
15 ÷ 5 = 3
0
5
10
15
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Year 3 |
Multiplication and Division |
|
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6 x 4 = 24
4 x 6 = 24
Understanding commutativity
Year 3 multiplication
Can multiply 2-digit numbers by 1-digit
14 x 3
14 x 3 = 10 x 3 + 4 x 3
14 x 3 = 30 + 12 = 42
x 3
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Can write mathematical statements for known multiplication and division facts using x, ÷ and =
7 x 3 = 21
3 x 7 = 21
Year 3 multiplication and division
21 ÷ 7 = 3
21 ÷ 3 = 7
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Year 4 |
Multiplication and Division |
|
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Can multiply 3 numbers mentally (associative law)
Year 4 multiplication
2 x 3 x 4
Can use the distributive law to multiply 2-digit numbers by a 1-digit number
6 x 32
30
2
6
6 x 30
6 x 2
6 x 32 = 6 x 30 + 6 x 2
6 x 32 = 180 + 12
6 x 32 = 192
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Can multiply a 2-digit number by 1-digit number (no regrouping)
Can multiply a 2-digit number by 1-digit number (regroup)
Can multiply a 3-digit number by 1-digit number (regroup)
Year 4 multiplication
32 x 3
54 x 3
138 x 4
30 × 3
2 × 3
4 × 3
50 × 3
×
+
8 x 4
30 x 4
100 x 4
×
+
+
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Year 4 division
Can divide a 2-digit number
by 1-digit (no regrouping)
Can divide a 2-digit number by 1-digit (with
regrouping)
69 ÷ 3
tens | ones |
| |
| |
| |
tens | ones |
| |
| |
| |
72 ÷ 3
tens | ones |
| |
| |
| |
Can divide a 3-digit number by 1-digit
476 ÷ 7
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Year 5 |
Multiplication and Division |
|
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Can multiply a 4-digit number by 1-digit number using short multiplication
Can multiply a 4-digit number by a 2-digit number using long multiplication
Can multiply a 2-digit number by a 2-digit number using long multiplication
Can multiply a 3-digit number by a 2-digit number using long multiplication
Year 5 multiplication
2,169 x 8
23 x 46
254 x 37
1,836 x 24
1
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Can divide a 4-digit number by a 1-digit number
Short division (no remainder)
Can divide a 4-digit number by a 1-digit number
Short division (with remainder)
Year 5 division
8,034 ÷ 6
6,287 ÷ 5
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Year 6 |
Multiplication and Division |
|
Year 6 |
Fractions (including decimals and percentages) |
|
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Year 6 multiplication
Can multiply up to 4 digits by a two-digit whole number using the formal written method of long multiplication
4,132 x 24
Can multiply a one-digit number with up to two decimal places by whole numbers
2.14 x 6
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Can divide numbers up to 4 digits by a two-digit whole number using long division
Year 6 division
Can divide numbers up to 4 digits by a one-digit whole number using long division
465 ÷ 15
496 ÷ 8
I know
8 x 6 = 48
so 8 x 60 = 480
I know
15 x 3 = 45
so 15 x 30 = 450
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Year 6 division
Can 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 or fractions
Can use written division methods in cases where the answer has up to two decimal places
7.62 ÷ 6
372 ÷ 16
372 ÷ 16 = 23 r 4
or
372 ÷ 16 = 23
372 ÷ 16 = 23
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