Download Calculation - Christ Church Moreton CE Primary School

Christ Church C.E. Primary School, Moreton
MATHEMATICS
Calculation Policy
A school community, nurturing achievers, who in Jesus’ sight are:
Respectful
Resilient
Resourceful
Reliable
Reviewed policy agreed by Governing Body on:
Reviewed policy shared with staff on:
Policy to be reviewed again on:
Progression in Addition
EYFS to Year 1
Add and subtract onedigit and two-digit
numbers to 20, including
zero.
Solve one step
problems that involve
addition and
subtraction, using
concrete objects and
pictorial representations
and missing number
problems.
2+5=
Provide context based experiences at each level of development - real life, money, measures…
Use a variety of models and images to support understanding and the development of mathematical language.
2+5=
Add and subtract
numbers using concrete
objects, pictorial
representations and
mentally, including:
a two-digit number and
ones,
Count on from the first
number (cover first
number or display as
numeral).
+
+
adding three one-digit
numbers
+
6 + 18
6 + 58
By counting on from the
largest number
By partitioning the
smaller number through
the multiple of 10.
6 + 8 becomes
8+2+4
5+
Recognise the largest
number in the calculation
and count on from it
mentally or using number
line.
5+2 (without
counters)
Recognise the largest
number in the
calculation and count
on from it (using
objects or fingers for
smaller number if
necessary).
+
TU + TU within
100
37 + 44
Show addition jumps
above the numberline
– ‘rainbow jumps’.
Addition of three
single digits – look
for bonds you
know and doubles
Partition the smaller
number and use the
tens number to bridge
calculation.
5 + 17 becomes
17 + 3 + 2
Special cases:
Adding 9
9 + 33
58 + 2 + 4
58 60
30 + 46
44
64
By counting on in tens
a two-digit number and
tens,
two two-digit numbers
2+5
5+8
4 + 13
11 + 7
Leading to…
Count out each set
then find the total
(using ‘concrete’
objects or pictorial
representations).
2
Year 2
2+5=
22 + 50
46
56
66
76
By counting in groups of
ten and one from largest
number.
50
70
72
74
80 81
or
40 + 30 = 70
7 + 4 = 11
70 + 11 = 81
or
44 + 40 – 3 = 81
Recall of facts to 20
and recall of adding
multiples of 10 will
support this thinking.
6+9+3
6+3=9
Double 9 = 18
+10
33
42 43
-1
Using Doubles
29 + 30 is the same as
30 + 30 – 1
Progression in Addition
Year 3
Add and subtract
numbers mentally,
including:
a three digit number and
ones,
a three digit number and
tens,
a three digit number and
hundreds,
two two-digit numbers
across 100 (nonstatutory guidance).
Partitioning the
numbers for TU +
TU across 100
55 + 78
70 + 50 = 120
8 + 5 = 13
120 + 13 = 133
Provide context based experiences at each level of development - real life, money, measures…
Use a variety of models and images to support understanding and the development of mathematical language.
Special cases
66 + 79
80 + 66 – 1 = 145
Partitioning
Adding ones and
tens to a 3 digit
number
Using doubles
356 + 8
356 + 4 + 4 = 364
76 + 78
55 + 78
78 + 50 = 128
128 + 2 + 3 = 133
Double 70 + double 6 + 2
Double 70 + double 8 – 2
Add and subtract
numbers with up to
three digits, using formal
written methods of
columnar addition and
subtraction.
Recall of number bonds
for all numbers to 20
and by adding multiples
of 10 will support this
thinking.
Recall of number bonds
for all numbers to 20
and by adding multiples
of 10 will support this
thinking.
Year 4
Use mental strategy
where appropriate
Addition of
3 digit + 3 digit;
4 digit + 4 digit
Add and subtract
numbers with up to four
digits using the formal
written methods of
columnar addition and
subtraction where
appropriate.
1460 + 499
1460 + 500 – 1 = 1959
Use mental strategies to
approximate
576
+369
7268
+5179
945
12447
356 + 70
350 + 70 + 6 = 420
356 + 600
300 + 600 + 56 = 956
Addition of numbers
with decimal places
1.5 + 1.5
Double 1 + Double 0.5
1.6 + 1.7
1.7 + 0.3 + 1.3 = 3.3
Addition of 3 digit + 2 digit numbers and 3
digit + 3 digit numbers
268
+179
1 7 (8 + 9)
1 3 0 (60 + 70)
3 0 0 (200 + 100)
447
Then as above but without brackets.
Then…
6 5
+ 7 3
6 8
+ 7 3
5 7 6
+ 3 6 9
1 3 8
141
9 4 5
1
Addition of numbers to 2 decimal places
4.45
+3.55
57.89
+46.67
8.00
104.56
Progression in Addition
Year 5
Add and subtract
numbers mentally, with
increasingly large
numbers e.g. 5 digit – 4
digit multiple of 10.
Using mental
calculation by
counting on
45678 + 3500 = 49178
45678 + 3000 = 48678
42678 = 500 = 49178
Add and subtract whole
numbers with more than
4 digits, including using
formal written methods
(columnar addition
and subtraction).
5.78 + 2.45 = 8.23
5.78 + 2 = 7.78
5.73 + 0.4 = 8.18
5.33 + 0.05 = 8.23
Year 6
Partitioning
Perform mental
calculations, including
with mixed operations
and large numbers.
4.578 + 0.008 = 4.586
Solve addition and
subtraction multi-step
problems in context,
deciding which
operations and methods
to use and why.
Provide context based experiences at each level of development - real life, money, measures…
Use a variety of models and images to support understanding and the development of mathematical language.
Column addition
Mixed decimals
58765
+29648
57.89 + 46.6 + 23.785
88413
23.785
57.890
+46.600
2
6.568 + 0.079 = 6.647
6.568 + 0.07 = 6.638
6.638 + 0.009 = 6.647
128.275
Column addition
with 5 or 6 digits
58765
+29648
88413
include 0
as
placeholder
where
necessary
Using all 4 operations
6 + 7 x 8 = 62
because multiplication first then addition where there are no
brackets.
2780 – 910 + 1220
can be reordered to 2780 + 1220 – 910 = 3090 as long as the
symbol moves with the number.
Misconceptions
•
Models & Images
Linked Vocabulary
Confusion with ‘…teen’ and ‘…ty’
numbers.
When using the number line, children
may count the start number so
answer is out by 1.
Setting out when working in columns –
confusion over the place value.
Reliance on rules and procedures
without conceptual understanding
•
•
•
Bar Models / Thinking Boxes
When solving word problems, children will be
encouraged to draw bar models / thinking boxes.
This will help them to visualise the problem and
decide which operation(s) they will need to use
to solve the problem.
Remember…
•
•
•
•
Addition is commutative.
Children should estimate first to see
if their answer makes sense.
Subtraction is the inverse of
addition – the sooner children
understand this, the better.
Children need to solve missing
number problems:
3+4=?
3+?=7
?+4=7
?+?=7
?=3+4
7=3+?
7=3+?
7=?+4
+
Add
More
Sum
Total
Make
Altogether
Greater
Plus
Addition
Increase
Exchanging / Regrouping
Inverse
Commutativity
Tens boundary
Hundred boundary
Units boundary
Tenths boundary
+
+
?
Part
Part
Progression in Subtraction
EYFS to Year 1
Add and subtract onedigit and two-digit
numbers to 20, including
zero.
Solve one step problems
that involve addition and
subtraction, using
concrete objects and
pictorial representations
and missing number
problems.
Year 2
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
Provide context based experiences at each level of development - real life, money, measures…
Use a variety of models and images to support understanding and the development of mathematical language.
5–2
7–2
Count out 5 and remove
2 to find the answer
Count back on the
numbertrack by saying
‘start on 7, count back
1,2. What number are
you on?’
Draw pictures and cross
out
8–2
15 – 5
Difference
Use tens and ones
when the calculation
doesn’t bridge 10
7 – 6 or find the
difference between 7
and 6
14 – 3
13 – 5
becomes 13 – 3 – 2
7–3
Using a 10 frame –
children may subitise
how many are remaining
without having to count
them all
Show subtraction
jumps below the
number line –
‘smiley faces’.
Subtracting by
counting backwards
in tens and ones
Subtracting in groups
of ten (rather than
counting in tens) or
groups of ones (by
partitioning number being
subtracted through
multiple of 10)
28 – 4
Count backwards mentally
or using a numberline
Numberline – secure
partitioning of the second
number, then counting
back in tens then ones.
Special cases
27 – 14 = 13
28 – 9
28 – 10 + 1
32 – 7
32 – 2 – 5
45 – 20
Use tens and ones
when the calculation
doesn’t bridge 10
Partitioning
28 – 8 = 20
76 – 70 = 6
Difference
When subtracting 9
or 19
23 – 19
+4
-4
27
13 14 15 16 17
65 – 40
Use a numberline or
models and images
Partitioning the number
being subtracted
through the multiple of
10 mentally or using a
numberline.
- 10
-4
+1
18 19
-10
28
When numbers are close
together, count on from the
smallest number through the
multiple of 10 or count back
from the largest to the
smallest through the
multiple of 10.
Progression in Subtraction
Year 3
Add and subtract
numbers mentally,
including:
a three digit number and
ones,
a three digit number and
tens,
a three digit number and
hundreds,
two two-digit numbers
across 100 (nonstatutory guidance).
Use mental strategies
where appropriate e.g.
Partitioning
Subtracting ones and
tens from a 3 digit
number
567 – 60 = 507
745 – 700 = 45
832 – 2 = 830
Provide context based experiences at each level of development - real life, money, measures…
Use a variety of models and images to support understanding and the development of mathematical language.
TU – TU
By counting back in
tens and ones
56
60 61
91
356 – 70
356 – 50 – 20 = 286
93 – 39 as
93 – 40 + 1
£10.00 - £3.59
+1p
+40p
+£6
£3.59 £3.60 £4.00
£10.00
Year 4
Partitioning
3 1
429
- 132
429 - 132
297
+1
53 54
93
-40
1678 – 600 = 1078
2689 – 80 = 2609
6839 – 9 = 6830
7484 – 1100 = 6384
46 – 28
Difference
(see also subtraction up to three digits)
Count on using numberline when working with
money
£6 + 40p + 1p = £6.41
Add and subtract
numbers with up to three
digits, using formal
written methods of
columnar addition and
subtraction.
Use mental strategies
where appropriate
46
- 28
18
Special cases
Add and subtract
numbers with up to
four digits using the
formal written
methods of columnar
addition and
subtraction where
appropriate.
Use models and images to
support understanding of
column subtraction e.g.
Dienes
3 1
91 – 35
91 – 30 – 1 – 4
364 – 8
364 – 4 – 4 = 356
956 – 600
956 – 600 = 356
Subtraction up to three
digits
Using mental
calculation when
appropriate by
counting back
5678 – 2342 =
5678 – 2000 = 3678
3678 – 300 = 3378
3378 – 40 = 3338
3338 – 2 = 3336
See difference too
‘Exchanging’
‘Regrouping’
Subtraction up to four
digits
Subtraction
up to four
digits
Difference
5003 – 3897 = 1106
£50 - £28.25 = £21.75
+75p
£28.25
£29
Column
subtraction
+£20
+£1
£30
£50
N.B. Children tend to make fewer errors
if they count on using a numberline
(rather than using standard algorithm)
when dealing with money.
2 12 11 1
3 3 2 6
+ 2 6 7 8
0 6 4 8
‘Exchanging’
‘Regrouping’
+103
3897 4000
+1003
5003
Progression in Subtraction
Year 5
Partitioning
Add and subtract
numbers mentally, with
increasingly large
numbers e.g. 5 digit – 4
digit multiple of 10.
Mentally
6.76 – 0.06 = 6.7
7.47 – 0.4 = 7.07
Add and subtract whole
numbers with more than
4 digits, including using
formal written methods
(columnar addition
and subtraction).
Using mental
calculation by
counting back
Difference
Use number bonds to
100 to support
45678 – 3500 = 42178
45678 – 3000 = 42678
42678 – 500 = 42178
£10 - £7.71 = £2.29
Use numberline or jottings
Partitioning
Perform mental
calculations, including
with mixed operations
and large numbers.
4.578 – 0.008 = 4.57
6.378 – 0.07 = 6.308
£7.71  £8.00 = 29p
£8  £10 = £2
7 – 2.45 = 4.55
2.45  3 = 0.55
37=4
5.78 – 2.45 = 3.33
5.78 – 0.05 = 5.73
5.73 – 0.4 = 5.33
5.33 – 2 = 3.33
Year 6
Solve addition and
subtraction multi-step
problems in context,
deciding which
operations and methods
to use and why.
Provide context based experiences at each level of development - real life, money, measures…
Use a variety of models and images to support understanding and the development of mathematical language.
Difference using
larger numbers and
number facts
Difference (used
mixed decimals)
6.45 – 1.7 = 4.75
£100 – 67.23 = £32.77
77p
£67.23 £68
1.7 2 = 0.3
2  6.45 = 4.45
£32
£100
Column
subtraction
2
1
5
3 8 7 6 15
- 1 9 2 4 8
1 9 5 1 7
‘Exchanging’
‘Regrouping’
As above with 5
or 6 digits
Misconceptions
•
•
•
•
•
•
•
Confusion with ‘…teen’ and ‘…ty’ numbers.
When using number line – children may count
the start number, so calculation is out by 1.
Setting out when working in columns confusion over the place value
Misunderstanding regarding place value and
concept of exchanging tens for ones,
hundreds for tens etc.
Lack of understanding that when subtracting
from a number that the answer will be smaller
than start number.
Children switch the digits around to be able to
‘do’ the calculation (believe it is commutative
as with +/x).
Reliance on rules and procedures without
conceptual understanding
Models & Images
Whole
Part
When solving word problems, children will be
encouraged to draw bar models / thinking boxes.
This will help them to visualise the problem and
decide which operation(s) they will need to use
to solve the problem.
Remember…
•
•
•
Children should estimate first to see if their
answer makes sense.
Subtraction is the inverse of addition – the
sooner children understand this, the better.
Children need to solve missing number
problems:
12 - 4 = ?
12 - ? = 3
? -4=3
? - ?=3
? = 12 - 4
3 = 12 - ?
3=? - 4
Linked Vocabulary
?
Take
Take away
Leaves
Left
Fewer
Less than
Minus
Subtract
Subtraction
Remove
Decrease
Difference between
Reduced
Inverse
Progression in Multiplication
Year 1
Solve one-step
problems involving
multiplication and
division, by
calculating the answer
using concrete
objects, pictorial
representations and
arrays with the
support of the
teacher.
Count in multiples of
twos, fives and tens.
Year 2
Calculate
mathematical
statements for
multiplication and
division within the
multiplication tables
and write them using
the multiplication
(×), division (÷) and
equals (=) signs.
Recall and use
multiplication and
division facts for the
2, 5 and 10
multiplication tables,
including recognising
odd and even
numbers.
Provide context based experiences at each level of development - real life, money, measures…
Use a variety of models and images to support understanding and development of mathematical language.
There are two apples on one plate.
How many apples on 3 plates?
+
Recall and derive
doubles
+
Recall and derive
doubles
25 × 2
7 + 7 = 14
7 × 2 = 14
Arrays
20 × 2
5 × 4 = 20
4 × 5 = 20
40
5×2
+
25
Repeated addition
40
6 x 2 = 12
1
10
50
2
3
4
5
6
10 = 50
Progression in Multiplication
Year 3
Write and calculate
mathematical statements
for multiplication and
division 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.
Provide context based experiences at each level of development - real life, money, measures…
Use a variety of models and images to support understanding and development of mathematical language.
Use arrays and
numberlines to count in
multiples
Using partitioning to
multiply
Partitioning using the
grid method
4 x 13 = 52
Multiply single digits by
20, 30, 40, 50 and 80.
x
3
20
60
4
12
57 × 2
50 × 2
100
7×2
+
14
= 114
57 × 2
72
0 1 2 3 4 5 6 7 8 9 10 11
100
14
114
Year 4
Use place value, known
and derived facts to
multiply and divide
mentally, including:
multiplying by 0 and 1;
dividing by 1; multiplying
together three numbers.
Multiply and divide twodigit and three-digit
numbers by a one-digit
number using formal
written layout.
Recall multiplication
and division facts for
multiplication tables up
to 12x12 (facts for 6, 7,
9, 11 and 12 are new).
Multiply single digits by
60, 70 and 90.
Mental
Using place value,
multiply by 10 and 100
e.g. 24 x 100
Th
2
H
4
T
U
2
4
0
0
Partitioning
267 × 2
200 × 2 = 400
60 × 2 = 120
7 × 2 = 14
400 + 120 + 14 = 534
Partitioning using the grid
method
x
Expanded method
3
20 60
x
400
6
2400
30
7
180
42
2622
(6x3)
(10x3)
(20x3)
(100x3)
4 12
72
437 x 6 = 2622
(5x3)
Then move from the
expanded method above
to standard algorithm.
1
Progression in Multiplication
Year 5
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 and divide
numbers mentally
drawing upon
known facts.
Multiply and divide
whole numbers
and those
involving decimals
by 10, 100 and
1000.
Provide context based experiences at each level of development - real life, money, measures…
Use a variety of models and images to support understanding and development of mathematical language.
Mental calculation
Partitioning
407 × 4
400 × 4 = 1600
0×4=0
7 × 4 = 28
1600 + 28 = 1628
Rounding and adjusting
£3.99 × 6
£4 × 6 = £24
£24.00 - £0.06 = £23.94
TU × TU by partitioning
24 × 32 =
2 8
× 3 9
×
20
4
30
600
120
2
40
8
640
128
28 × 19
28 × 10 × 2 = 560
560 – 28 = 532
Year 6
Multiply multidigit numbers up
to 4 digits by a
two-digit whole
number using
the formal written
method of long
multiplication.
Perform mental
calculations,
including with
mixed operations
and large
numbers.
Mental calculation
Partitioning
5.7 × 6
5 × 6 = 30
0.7 × 7 = 4.2
30 + 4.2 = 34.2
5.3 × 19
5.3 × 10 × 2 = 106
106 – 5.3 = 100.7
Standard algorithm
124
x
26
2
744
2480
1
Expanded then moving to standard algorithm
1
3224
7
1 8
2 4
6 0
2 (8×9)
0 ( 20 × 9 )
0 ( 8 × 30 )
0 ( 20 × 30 )
10 9 2
768
567 × 86
×
5 6 7
8 6
4
4
3 4 0 2
5 5
4 5 3 6 0
4 8 7 6 2
Misconceptions
Models & Images
• Understanding of multiplying by 10/100
and what happens to place value of
the number
• Rapid recall of multiplication tables is not
secure and is impacting on fluency and
accuracy of calculation
• Interpretation of digits in the T/H columns
as single digits e.g. 4x3 instead of 4x30
•Multiplication does not always increase the
number i.e. when multiplying a positive
number by a fraction which is less than 1.
Repeated addition
Groups of
Lots of
Multiply
Times
Multiplication
Product
Array
Row
Column
Inverse
Double
Multiplied by
Once, twice, three times… as
(big, wide, long and so on)
Scaling
Rate
When solving word problems, children will be
encouraged to draw bar models / thinking
boxes. This will help them to visualise the
problem and decide which operation(s) they
will need to use to solve the problem.
•
•
•
•
Multiplication is commutative.
Children should estimate first to see if their
answer makes sense
Since multiplication and division are
inverse operations (i.e. one is the
mathematical ‘opposite’ of the other) they
should be taught alongside each other
rather than as two separate entities.
Children need to solve missing number
problems:
3x4=?
?=3x4
3 x ? = 12
12 = 3 x ?
? x 4 = 12
12 = ? X 4
? X ? = 12
Linked Vocabulary
?
6 6 6 6 6
Progression in Division
Year 1
Solve one-step problems
involving multiplication and
division, by calculating the
answer using concrete
objects, pictorial
representations and arrays
with the support of the
teacher.
Provide context based experiences at each level of development - real life, money, measures…
Use a variety of models and images to support understanding and the development of mathematical language.
12 cakes shared equally between 3 people
There are 8 oranges. Can you share them equally?
Each cake box holds 3 cakes. If I have 12 cakes, how
many boxes will I need?
How many groups of 3 are there in 12?
How many times can I subtract 3 from 12?
Year 2
Calculate mathematical
statements for
multiplication and division
within the multiplication
tables and write them using
the multiplication (×),
division (÷) and equals (=)
signs.
Recall and use
multiplication and division
facts for the 2, 5 and 10
multiplication tables,
including recognising odd
and even numbers.
Counting
Recall and derive halves
Relate division to counting
and multiplication facts.
Count in 5s to see that
there are four 5s in 20.
How many 10s in 40?
Repeated subtraction
1
2
9÷3=3
3
Look at halves of even
numbers and see halves of
odd numbers as one left
over or ½ .
Division by sharing
10 ÷ 5 = 2
10 footballs shared
between 5 children.
Half of 5
Division by grouping
10 ÷ 5 = 2
10 footballs. 5 in each bag.
How many bags?
Provide context based experiences at each level of development - real life, money, measures…
Use a variety of models and images to support understanding and the development of mathematical language.
Progression in Division
Year 3
Write and calculate
mathematical statements
for multiplication and
division using the
multiplication tables that
they know, including for
two-digit numbers times
one-digit numbers, using
mental and progressing to
formal written methods.
Year 4
Use place value, known
and derived facts to
multiply and divide
mentally, including:
multiplying by 0 and 1;
dividing by 1; multiplying
together three numbers.
Multiply and divide twodigit and three-digit
numbers by a one-digit
number using formal
written layout.
Recall and use
multiplication and
division facts for the
3, 4 and 8
multiplication tables.
Use facts for numbers
up to 10 times the
divisor e.g. 28 ÷ 3
This is between
27 ÷ 3 = 9 and
30 ÷ 3 = 10
So 9 remainder 1
Division facts for
multiplication tables
up to 12x12.
Use facts for numbers
up to 10 times the
divisor e.g. 75 ÷ 9
This is between
72 ÷ 9 = 8 and
81 ÷ 9 = 9
So 8 remainder 3
Counting
Relate division to step
counting and multiplication
facts.
e.g. count in 4s to see that
there are 6 fours in 24.
1
0
2
4
3
8
12
4
16
5
Division as grouping
13 ÷ 3 = 4 r1
10 ×3
+30
6
20
0
1
0
Division as grouping
Combine multiples of the
divisor to support you
Children may first create a
‘bank’ or key if necessary
e.g. 10 × 6 = 60
5 × 6 = 30
2 × 6 = 12
87 ÷ 6 = 14 r3
(How many 6s in 87?)
4×6
+60
0
60
4×3
+12
30
r.1
42 43
24
Array shows
6 groups of 4
so 24÷4=6
10×6
43 ÷ 3 = 14 r1
+24 r.3
84 87
2
3
3
6
4
9
r1
12 13
Children may first create a
‘bank’ or key e.g. 10 × 3 = 30
5 × 3 = 15
2×3=6
Division by grouping, leading to formal division
87 ÷ 6 = 14 r3
(How many 6s in 87?)
6
1
8
6
2
2
4 r3
7
0
7
4
r3
Provide context based experiences at each level of development - real life, money, measures…
Use a variety of models and images to support understanding and the development of mathematical language.
Progression in Division
Year 5
Divide numbers up to 4
digits by a one-digit
number using the formal
written method of short
division and interpret
remainders appropriately
by the context.
Multiply and divide
numbers mentally
drawing upon known
facts.
Divide numbers by 10
and 100.
H
T
U
2
7
1/10
·
·2
Short division
6725 ÷ 7
0 9 6 0 r5
6
4
7 6 7 2 5
1/100
7
Year 6
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.
Use known facts
• Know 378 is a multiple
of 3 because 300, 60
and 18 are all
multiples of 3.
• Know 385 is a multiple
of 7 because 350 and
35 are multiples of 7.
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.
Use place value and
division facts
Short division
432 ÷ 15
Long division
560 ÷ 24 = 23r8
432 ÷ 15 = 28 r12
1.32÷3 = 1/100 of 132÷3
132÷3 = 44
44÷100=0.44
So 1.32÷3=0.4
-
remainder as a
decimal
remainder as a
fraction
Misconceptions
• Understanding of dividing by 10/100
and what happens to place value of
the number
• Rapid recall of multiplication tables is not
secure and impacting on fluency and
accuracy of calculation
•Assuming that division is commutative.
•Writing a remainder which is larger than the
divisor.
•Discarding the remainder and not
understanding its significance.
When solving word problems, children will be
encouraged to draw bar models / thinking
boxes. This will help them to visualise the
problem and decide which operation(s) they
will need to use to solve the problem.
6
•
•
•
?
Children should estimate first to see if their
answer makes sense
Since multiplication and division are
inverse operations (i.e. one is the
mathematical ‘opposite’ of the other) they
should be taught alongside each other
rather than as two separate entities.
Children need to solve missing number
problems:
12 ÷ 3 = ?
? = 12 ÷ 3
12 ÷ ? = 4
4 = 12 ÷ ?
? ÷3=4
4= ? ÷3
?÷?=4
Models & Images
6
24
?
Linked Vocabulary
Groups
Sharing
Fair
Equal
Left over
Repeated subtraction
Divide
Division
Inverse
Quotient
Dividend
Divisor
Remainder
Half
Array
Factor
Divided by
Divisible by
Divided into
Supporting Materials
• NCETM Progression Maps
• Wirral’s ‘Counting into Calculating – A Guide to Progression
Number: Addition and Subtraction
NUMBER BONDS
Year 1
represent and use
number bonds and
related subtraction facts
within 20
Year 2
Year 3
Year 4
Year 5
Year 6
recall and use addition and
subtraction facts to 20
fluently, and derive and use
related facts up to 100
MENTAL CALCULATION
add and subtract onedigit and two-digit
numbers to 20, including
zero
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 onedigit numbers
read, write and interpret
mathematical statements
involving addition (+),
subtraction (-) and equals
(=) signs
show that addition of two
numbers can be done in
any order (commutative)
and subtraction of one
number from another
cannot
(appears also in Written
Methods)
add and subtract
numbers mentally,
including:
*
a three-digit
number and ones
*
a three-digit
number and tens
*
a three-digit
number and
hundreds
add and subtract numbers
mentally with increasingly
large numbers
perform mental
calculations, including with
mixed operations and large
numbers
use their knowledge of the
order of operations to carry
out calculations involving
the four operations
Number: Addition and Subtraction
Year 1
read, write and interpret
mathematical statements
involving addition (+),
subtraction (-) and equals
(=) signs
Year 2
(appears also in Mental
Calculation)
Year 1
solve one-step problems
that involve addition and
subtraction, using concrete
objects and pictorial
representations, and
missing number problems
such as
7=-9
WRITTEN METHODS
Year 3
Year 4
add and subtract
add and subtract numbers
numbers with up to
with up to 4 digits using
three digits, using
the formal written
formal written methods methods of columnar
of columnar addition
addition and subtraction
and subtraction
where appropriate
Year 5
add and subtract whole
numbers with more than 4
digits, including using
formal written methods
(columnar addition and
subtraction)
Year 6
INVERSE OPERATIONS, ESTIMATING AND CHECKING ANSWERS
estimate the answer to a estimate and use inverse
use rounding to check
recognise and use the
calculation and use
operations to check
answers to calculations and
inverse relationship
inverse
operations
to
determine, in the context of
answers
to
a
calculation
between addition and
a problem, levels of
check answers
subtraction and use this to
accuracy
check calculations and solve
missing number problems.
use estimation to check
answers to calculations and
determine, in the context of
a problem, levels of
accuracy.
PROBLEM SOLVING
Year 3
Year 4
solve problems,
solve addition and
including missing
subtraction two-step
number problems, using problems in contexts,
number facts, place
deciding which
value, and more
operations and methods
complex addition and
to use and why
subtraction
Year 6
solve addition and
subtraction multi-step
problems in contexts,
deciding which operations
and methods to use and
why
Year 2
solve problems with
addition and subtraction:
*
using concrete
objects and pictorial
representations,
including those
involving numbers,
quantities and
measures
*
applying their
increasing
knowledge of
mental and written
methods
solve simple problems in a
practical context involving
addition and subtraction of
money of the same unit,
including giving change
(copied from Measurement)
Year 5
solve addition and
subtraction multi-step
problems in contexts,
deciding which operations
and methods to use and
why
Solve problems involving
addition, subtraction,
multiplication and division
Number: Multiplication and Division
MULTIPLICATION & DIVISION FACTS
Year 3
Year 4
Year 1
Year 2
count in multiples of twos,
fives and tens
(copied from Number and
Place Value)
count in steps of 2, 3, and 5
from 0, and in tens from any
number, forward or
backward
(copied from Number and
Place Value)
count from 0 in multiples of 4, 8, 50
and 100
(copied from Number and Place
Value)
count in multiples of 6,
7, 9, 25 and 1 000
(copied from Number
and Place Value)
recall and use
multiplication and
division facts for the 2, 5
and 10 multiplication
tables, including
recognising odd and even
numbers
recall and use multiplication and
division facts for the 3, 4 and 8
multiplication tables
recall multiplication
and division facts for
multiplication tables
up to 12 × 12
show that multiplication
of two numbers can be
done in any order
(commutative) and
division of one number by
another cannot
Year 1
Year 2
calculate mathematical
statements for
multiplication and division
Year 5
MENTAL CALCULATION
write and calculate mathematical use place value,
statements for multiplication and known and derived
division using the multiplication
facts to multiply and
tables that they know, including
divide mentally,
for two-digit numbers times one- including: multiplying
digit numbers, using mental and
by 0 and 1; dividing
progressing to formal written
by 1; multiplying
methods (appears also in Written
together three
Methods)
numbers
recognise and use
factor pairs and
commutativity in
mental calculations
multiply and divide
numbers mentally
drawing upon known
facts
perform mental
calculations, including with
mixed operations and large
numbers
multiply and divide
whole numbers and
those involving decimals
by 10, 100 and 1000
associate a fraction with
division and calculate decimal
fraction equivalents (e.g.
0.375) for a simple fraction
3
(e.g. /8)
(copied from Fractions)
(appears also in
Properties of Numbers)
WRITTEN CALCULATION
Year 3
Year 4
write and calculate
multiply two-digit
mathematical
and three-digit
statements for
numbers by a one-
Year 6
count forwards or backwards
in steps of powers of 10 for
any given number up to
1 000 000
(copied from Number and
Place Value)
Year 5
multiply numbers up
to 4 digits by a one- or
two-digit number
Year 6
multiply multi-digit numbers up to 4
digits by a two-digit whole number
using the formal written method of
Number: Multiplication and Division
Year 1
Year 2
calculate mathematical
statements for multiplication
and division within the
multiplication tables and
write them using the
multiplication (×), division
(÷) and equals (=) signs
WRITTEN CALCULATION
Year 3
Year 4
multiply two-digit
write and calculate
and three-digit
mathematical
numbers by a onestatements for
digit number using
multiplication and
formal written layout
division using the
multiplication tables
that they know,
including for two-digit
numbers times one-digit
numbers, using mental
and progressing to
formal written methods
Year 5
multiply numbers up
to 4 digits by a one- or
two-digit number
using a formal written
method, including long
multiplication for twodigit numbers
Year 6
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 one-digit
number using the
formal written method
of short division and
interpret remainders
appropriately for the
context
divide numbers up to 4-digits by a
two-digit whole number using the
formal written method of short
division where appropriate for the
context 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
(appears also in Mental
Methods)
use written division methods in cases
where the answer has up to two decimal
places (copied from Fractions (including
decimals))
Number: Multiplication and Division
Year 1
PROPERTIES OF NUMBERS: MULTIPLES, FACTORS, PRIMES, SQUARE AND CUBE NUMBERS
Year 2
Year 3
Year 4
Year 5
recognise and use factor
identify multiples and
pairs and commutativity
factors, including finding
in mental calculations
all factor pairs of a
(repeated)
number, and common
factors of two numbers.
know and use the
vocabulary of prime
numbers, prime factors
and composite (nonprime) numbers
establish whether a
number up to 100 is
prime and recall prime
numbers up to 19
recognise and use square
numbers and cube
numbers, and the
2
notation for squared ( )
3
and cubed ( )
Year 6
identify common factors,
common multiples and
prime numbers
use common factors to
simplify fractions; use
common multiples to express
fractions in the same
denomination
(copied from Fractions)
calculate, estimate and
compare volume of cubes
and cuboids using standard
units, including centimetre
3
cubed (cm ) and cubic
3
metres (m ), and extending
to other units such as mm
3
3
and km
(copied from Measures)
Year 1
Year 2
ORDER OF OPERATIONS
Year 3
Year 4
Year 5
Year 6
use their knowledge of
the order of operations to
carry out calculations
involving the four
operations
INVERSE OPERATIONS, ESTIMATING AND CHECKING ANSWERS
estimate the answer to a
calculation and use inverse
operations to check answers
(copied from Addition and
Subtraction)
estimate and use inverse
operations to check answers
to a calculation
(copied from Addition and
Subtraction)
use estimation to check
answers to calculations
and determine, in the
context of a problem,
levels of accuracy
Number: Multiplication and Division
PROBLEM SOLVING
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
solve one-step problems
involving multiplication
and division, by calculating
the answer using concrete
objects, pictorial
representations and arrays
with the support of the
teacher
solve problems involving
multiplication and division,
using materials, arrays,
repeated addition, mental
methods, and
multiplication and division
facts, including problems in
contexts
solve problems, including
missing number problems,
involving multiplication
and division, including
positive integer scaling
problems and
correspondence problems
in which n objects are
connected to m objects
solve problems involving
multiplying and adding,
including using the
distributive law to multiply
two digit numbers by one
digit, integer scaling
problems and harder
correspondence problems
such as n objects are
connected to m objects
solve problems involving
multiplication and division
including using their
knowledge of factors and
multiples, squares and
cubes
solve problems involving
addition, subtraction,
multiplication and division
solve problems involving
addition, subtraction,
multiplication and division
and a combination of
these, including
understanding the
meaning of the equals sign
solve problems involving
multiplication and division,
including scaling by simple
fractions and problems
involving simple rates
solve problems involving
similar shapes where the scale
factor is known or can be
found
(copied from Ratio and
Proportion)