Interpretation of written signs

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CARTERHATCH JUNIOR SCHOOL
CALCULATION POLICY
September 2013
Introduction: The aim is for all children to have one written method for
each of the four operations which is reliable and efficient

Each page refers to a different operation, i.e +, -, x and ÷. Stages refer to the guidance paper on written calculation
strategies in the renewed framework.

There is progression in calculation from N to Y6.

In order for progression to run smoothly vocabulary, mental calculation strategies and rapid recall facts should be introduced at
the appropriate stage and reinforced.

Throughout KS1 and KS2 it is assumed that mathematical vocabulary will be introduced and used in context. Refer to the
electronic Renewed framework

Numerical examples have been included to illustrate progression from year to year as well as development within the year.

The different approaches to mental and written calculation should be made explicit.

It is important to notice consistency between methods of written calculations (+ and – starting with units; when dividing in Y5 and
Y6, be consistent with subtraction method needed for children to divide by "chunking", i.e. repeated subtraction), therefore when
adding or subtracting in earlier years, begin with the least significant digits.

Use the Framework for Teaching Mathematics to ensure the appropriate numbers are used for appropriate year groups or levels
of attainment.

Some methods rely heavily on:
A firm understanding of place value
Interpretation of written signs
Familiarity with a range of vocabulary / Efficient mental calculation
Addition
Progression examples:
To add successfully, children need to be able to:
23
+42
315
+624

recall all addition pairs to 9 + 9 and complements in 10;
1. No 'carrying'

add mentally a series of one-digit numbers, such as
5 + 8 + 4;
2. Extra digit in answer
94
+73
561
+718
add multiples of 10 (such as 60 + 70) or of 100 (such as
600 + 700) using the related addition fact, 6 + 7, and their
knowledge of place value;
3. Carrying U (units/ones) to T (tens)
47
+25
237
+516
partition two-digit and three-digit numbers into multiples
of 100, 10 and 1 in different ways.
4. Carrying T to H
371
+485
293
+541
5. Carrying U to T and T to H
376
+485
295
+547
6. More than two numbers to be added
463
921
+759
7. Different numbers of digits
24
375
+ 48
4756
20375
+ 752
£4.21
+£3.87
24.90kg
+ 7.25kg


Note: It is important that children’s mental methods of calculation
are practised and secured alongside their learning and use of an
efficient written method for addition.
8. Real life problems involving money or
measures
9. Numbers with decimals
2.13
+5.62
Addition
Pre-stage
Stage 2
Partitioning
(by the end of year 3)
 Counting on fingers
Steps in addition can be recorded on a Record steps in addition using
 Count on one digit from a number line. The steps often bridge
partitioning:
two digit number
through a multiple of 10.
47 + 76 = 47 + 70 + 6 = 117 + 6
 Counting forwards and
8 + 7 = 15
= 123
backwards on a number
47 + 76 = 40 + 70 + 7 + 6 = 110
line
+ 13 = 123
 Partition into multiples of 10
Partitioned numbers are then
and 1 and recombine
48 + 36 = 84
written under one another:
 Derive fact families using
47  40  7
addition and subtraction
 76
70  6
20 – 7 = 13
20 – 13 = 7
110
 13  123
or:
13 + 7 = 20
7 + 13 = 20
Say how a set of objects can
be separated into 2 groups
8 is 5 and 3
Stage 1
The Empty Number Line
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.
Stage 3
Expanded method in columns
Practical apparatus to be used
when learning carrying and
exchanging
Write the numbers in columns.
Adding the tens first:
47
 76
110
13
123
Adding the ones first:
47
 76
13
110
123
Discuss how adding the ones first
gives the same answer as adding
the tens first. Refine over time to
adding the ones digits first
consistently.
Stage 4
Column Method
(by the end of Year 4)
47
258
366
 76
 87
 458
824
123
345
11
11
11
Column addition remains efficient
when used with larger whole
numbers and decimals. Once
learned, the method is quick and
reliable.
Carry digits are recorded below the
line, using the words ‘carry ten’ or
‘carry one hundred’, not ‘carry one’.
Addition
N


R
Through practical activities in
meaningful contexts.
Counting on fingers in a consistent way
 Partitioning
say how a set of
objects can be
separated into 2
groups
8 is 5 and 3
 counting on using
fingers
Y1
Adding multiples of
10.
Y4
Stage 4 by the end
of year 4
67 + 24
10 + 5 = 15
Stage 3




Deliberately count on the wrong number.
Ask the children how to put it right.
Y3
Stage 2
70 + 30 = 100
42 + 30 = 72
45 + 13
30 + 20 = 50
Adding multiples of 10 = 45 + 3 + 10
= 58
and units.

Illustrate number stories with number
sentences
 Model and encourage use of
mathematical language
count on
altogether
one more, etc.
 Model interpretation of and sign using
appropriate language
Combine numbers of
objects – count all the
objects
Y2

+3 +10
45
48
58
combining sets to
make a total
(units first)
steps along a
number line
 partition into tens
(counting on)
and ones (units
record using +
first)
and = signs
Add a one digit
number to a one 45 + 13
of two digit
40 + 5 + 10 + 3
number
Stage 1 –
8 + 50 = 58
number line
Derive fact families
Using add / subtract
eg
20-7=13
20-13=7
13 + 7 = 20
7 + 13 = 20
367
+ 85
12
140
300
452
= 60 + 7 + 20 + 4
= 11 + 80 = 91
+
67
24
11
80
91
Y5
7587
675
12
150
1100
7000
8262
(units first)
+
add
mentally
from
top to
bottom
Y6
7648
1486
14
120
1000
8000
9134
(units first)
+
(units first)
(7 + 4)
(60 +20)
(units first)
"carrying"
Extend to 3 digit
numbers
Relies heavily on firm
understanding of
place value
+
3587
675
4262
1 1 1
+
7648
1486
9134
1 1 1
Real life money problems
decimals
£4.21
72.5 km
124.90 kg
£3.87
+
54.6 km
+
7.25 kg
0.08
127.1 km
132.15 kg
1 1
1 1
1.00
7.00
Numbers with any amount of digits, e.g.
£8.08
72
3 numbers less than
567
1000
26 +
+
Addition
367
569
826
+
22
140
1600
Leading to ‘carrying’
367
85 +
452
11
Units first. Call each
digit by its value, e.g.
“60 plus 80 is 150”
* links to Y2 partitioning
Subtraction
Progression examples:
To subtract successfully, children need to be able to:
 recall all addition and subtraction facts to 20;
 subtract multiples of 10 (such as 160 – 70) using the
related subtraction fact,16 – 7, and their knowledge of
place value;
 partition two-digit and three-digit numbers into multiples of
one hundred, ten and one in different ways (e.g. partition
74 into 70 + 4 or 60 + 14).
Note: It is important that children’s mental methods of calculation
are practised and secured alongside their learning and use of an
efficient written method for subtraction.
1. No adjustment
47
-23
864
-621
2. Adjustment T to U
51
-36
432
-217
3. Adjustment H to T
437
-182
618
-217
4. Adjustment H to T and T to U
432
-187
5. Noughts
470
-142
700
-485
604
-347
Subtraction
Pre-stage
 Counting back from a larger
number
 One less, two less
 Counting on from a smaller
number
 Find the difference by counting
up – important to use visual
images
Stage 1
The Empty Number Line
Steps in subtraction can be
recorded on a number line. The
steps often bridge through a
multiple of 10.
15 – 7 = 8
74 – 27 = 47 worked by counting
back:
The Counting Up Method
74
 27
3
40
4
47
 30
 70
 74
or:
______
_______________
The steps may be recorded in a
different order:
or combined:
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.
74
 27
3
44
47
 30
 74
Stage 2
Partitioning
Subtraction can be recorded
using partitioning:
74 – 27 = 74 – 20 – 7 = 54 –
7 = 47
74 – 27 = 70 + 4 – 20 –
7 = 60 + 14 – 20 – 7 = 40 + 7
This requires children to subtract
a single-digit number or a
multiple of 10 from a two-digit
number mentally. The method of
recording links to counting back
on the number line.
Stage 3
Expanded method in columns
Practical apparatus to be used
when learning carrying and
exchanging
Partitioned numbers are then
written under one another:
Example: 74 − 27
70  4
 20  7
60
6 14
14
70  4
 20  7
40  7
7 4
27
4 7
Example: 741 − 367
700  40  1
 300  60  7
600
130
11
700  40  1
 300  60  7
300  70  4
6 13 11
7 41
 3 67
374
The expanded method leads
children to the more compact
method so that they understand
its structure and efficiency. The
amount of time that should be
spent teaching and practising the
expanded method will depend on
how secure the children are in
their recall of number facts and in
their understanding of place
value.
Subtraction
N
R
Y1
Y2
Y3
Y4
counting back by partitioning
counting back
Through practical activities in meaningful
contexts
"How many are left?"
"Take away"
"The difference between
Taking away and
count how many
are left.
1
2
3
4
5
counting back from
the larger number
one less, etc.
-10
5
-8
15
-50
23
5
15
+10






Illustrate number stories with number
sentences
Model interpretation of – sign using
appropriate language
Comparing numbers of objects
How many are left – count back
How many have been removed –
count on
How many more are needed – count
on
84
23
Extend to 3 digit
numbers
-30
40
78
+8
70 – 30 = 40
multiples of 10
50 – 20 = 30
-6
70
- 50
20
11
7
4
754 – 86
600 140
80
600
- 200
400
14
6
Record using –
and = signs
5
4
6
140 14
80 6
60 8
= 468
leading to
6
70
(units first)
754 – 286
700
50
- 200
80
81 – 57 = 24
28
14 – 12 = 2
15 – 8 = 7
21 – 10 = 11
Y6
(units first)
84 – 56 = 28
Count back from
the larger number.

Stage 3
(units first)
23 – 18
23 – 8 – 10
Counting on from
smaller number.
0
Stage 2
(units first)
(units first)
5–2=3
Y5
Stage 3 by the end
of stage 4
decomposition
754 – 86
600 140
80
600
60
14
6
8
-
7
2
4
14
5
8
6
1
4
6
8
= 668
extend to decimals using chosen method
Using an empty number line
Find the
difference by
counting up
Understand
inverse is addition
20-13 = 7
13 = 7 = 20
Fact families
Call each digit by its
value.
£6.28
- £3.15
-
6
2
3
13
4
6
7
1
6
8
8
7
4
3
Multiplication
Progression examples:
To multiply successfully, children need to be able to:
 recall all multiplication facts to 10 × 10;
 partition number into multiples of one hundred, ten and one;
 work out products such as 70 × 5, 70 × 50, 700 × 5 or 700 × 50
using the related fact 7 × 5 and their knowledge of place value;
 add two or more single-digit numbers mentally;
 add multiples of 10 (such as 60 + 70) or of 100 (such as 600 +
700) using the related addition fact, 6 + 7, and their knowledge
of place value;
 add combinations of whole numbers using the column method
(see above).
Note: It is important that children’s mental methods of calculation are
practised and secured alongside their learning and use of an efficient
written method for multiplication.
1. No 'carrying'
2. Extra digit
3. 'Carrying' but keeping in same decade
4. 'Carrying' and going into next decade
5. Noughts
6. Multiplying by multiples of 10
7. 'Long' multiplication
32
44
x3
x2
32
51
x4
x4
83
34
x4
x7
78
68
x7
x8
202
430
x4
x6
87
416
x10
x 60
47
832
x23
x74
Multiplication
Stage 1
Mental multiplication using partitioning
Informal recording in Year 4 might be:
Also record mental multiplication using
partitioning:
Stage 2
The Grid Method
(introduced in year 4)
38 × 7 = (30 × 7) + (8 × 7)
= 210 + 56 = 266
x
30
8
7
210
56
Stage 3
Expanded short multiplication
(by end of year 4)
30  8
 7
210
56
266

30  7  210
8  7  56
38
7
210
56
266
266
14  3  (10  4)  3
 (10  3)  (4  3)  30  12  42
43  6  (40  3)  6
 (40  6)  (3  6)  240  18  258
Note: These methods are based on the
distributive law. Children should be
introduced to the principle of this law (not its
name) in Years 2 and 3, for example when
they use their knowledge of the 2, 5 and 10 Derive and recall ALL
times-tables to work out multiples of 7:
multiplication facts up to
10 x 10, the corresponding
division facts and multiples
7  3  (5  2)  3  (5  3)  (2  3)  15  6  21 of numbers to 10 up to the
tenth multiple.
Children should describe what
they do by referring to the actual
values of the digits in the
columns. For example, the first
step in 38 × 7 is ‘thirty multiplied
by seven’, not ‘three times
seven’, although the relationship
3 × 7 should be stressed.
Stage 4
Short Multiplication

38
7
266
Stage 5
Two-digit by Two-digit products
(by end of year 5)
56 × 27 is approximately
60 × 30 = 1800.
5
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.
If, after practice, children cannot
use the compact method without
making errors, they should return to
the expanded format of stage 3.
56 × 27 is approximately
60 × 30 = 1800.
56
 27
1000
50  20  1000
120
6  20  120
350
50  7  350
42
6  7  42
1512
1
56 × 27 is approximately
60 × 30 = 1800.
56
 27
1120
56  20
392
56  7
1512
1
Multiplication
N
R
Through practical activities in meaningful
contexts
Grouping objects in twos or threes, then
adding groups of the same number
counting in 2s, 10s
Doubles up to 5 + 5
Y1
Y2
Y3
Doubles up to
10 + 10
Counting in 2s,5’s
10s
Y4
Y5
Y6
Stage 4 by the end
of stage 4
double by partitioning and recombining
Doubles to 20+20
Multiples of 5 to 50
Doubles of
multiples of 10 to
100
Counting in 2s, 10s
+ 5s.
Reading
4x2=8
∆∆∆∆
∆∆∆∆
or
2x4=8
∆∆
∆∆
∆∆
∆∆
 Record using x
and = signs
 Understand
3x2 as 2+2+2 Repeated addition
 Understand
15x3 is (10x3)
+ (5x3)
understand
multiplication as
repeated addition
15 x 3
15 added together
3 times
15 + 15 + 15
32 x 3
30 x 3 = 90
2x3=6
90 + 6 = 96

Record + as x
and vice versa.
Multiply 1 or 2
digit nos by 10
or 100
Family facts for x /

143
100
40
3
x2
x2
x2
x2
487
400
80
7
Stage 5
x2
x2
x2
x2
Stage 1
200 + 80 + 6
= 286
800 + 160 + 14
= 974
Stage 2
grid method
x
8
Stage 5 grid
method 2 digit by
2 digit
23 x 8
20 3
160 24
3620
3000
600
20
x2
x2
x2
x2
6000+1200+40
= 7240
372 x 24
X
300
70 2
20 6000 1400 40
4 1200
160 + 24 = 184
72 x 38
x
70
2
30 2100 60
8 560 16
280 8
6000+1400+1200+2
80+40+8 = 8928
2100+560+60+16=
2736
Decimals
x
9
346 x 9
300 40
6
4.9 x 3
x
4
3
12
0.9
2.7
12 + 2.7 = 14.7
Multiply by 10 and
100 numbers to a
1000
Stage 3 expanded
Short multiplication
vertically
23 20 + 3
7
x 7
346
9
54
360
2700
x
(6x9)
(40x9)
(300x9)
4.92 x 3
x
4 0.9 0.02
3 12 2.7 0.06
12 + 2.7 + 0.06 =
14.76
4346
8
48
320
2400
x
(6x8)
(40x8)
(300x8)
Multiplication
21
140
161
(3x7)
(20x7)
3114
32000
34768
(4000x8)
leading to:
leading to:
leading to:
23
7
161
2
x
72
38
576
2160
2736
x
(72x8)
(72x30)
352
27
2464
7040
9504
Units first
(relies on mental calculation strategies
x
Division
To divide successfully in their heads, children need to be able to:
 understand and use the vocabulary of division – for example in 18 ÷ 3 =
6, the 18 is the dividend, the 3 is the divisor and the 6 is the quotient;
 partition two-digit and three-digit numbers into multiples of 100, 10 and 1
in different ways;
 recall multiplication and division facts to 10 × 10, recognise multiples of
one-digit numbers and divide multiples of 10 or 100 by a single-digit
number using their knowledge of division facts and place value;
 know how to find a remainder working mentally – for example, find the
remainder when 48 is divided by 5;
 understand and use multiplication and division as inverse operations.
Single-digit division
Examples:
To carry out written methods of division successful, children also need to be able
to:
 understand division as repeated subtraction;
 estimate how many times one number divides into another – for example,
how many sixes there are in 47, or how many 23s there are in 92;
 multiply a two-digit number by a single-digit number mentally;
 subtract numbers using the column method.
Two digit division
Note: It is important that children’s mental methods of calculation are practised
and secured alongside their learning and use of an efficient written method for
division.
1. No remainder, no carrying
3 69
2. Remainder, no carrying
3 68
3. No remainder, carrying
3 45
4. Remainder, carrying
3 47
5. Placing of the quotient
7 287
6. Noughts in quotient
4 816
8 5608
7. No remainder
32 64
31 93
8. Similar but remainder
13 29
31 97
9. Quotient not so apparent
22 56
41 92
10. Placing the quotient
21 126
32 224
11. No remainder
21 483
32 224
12. Remainder
33 718
13. Noughts in quotient
17 6834
14. Divisors like 29, 39, 48
15. Divisors like 45, 37, 24, 56
3 264
Division
Stage 1
Grouping and sharing
Stage 2
Repeated subtraction
Stage 3
Stage 4
Expanded short division leading to compact short division (by end
Long Division
of year 4)
(by end of year 5)
Counting in multiples
Division as Grouping
97 ÷ 9
How many packs of 24 can we make
Use hands:
from 560 biscuits? Start by multiplying
A bag of 6 sweets, how many
9 97
How many groups of 5 in 15?
24 by multiples of 10 to get an estimate.
children can have 2 sweets each.
 90 9  10
How many 5s have been counted?
As 24 × 20 = 480 and 24 × 30 = 720, we
The division sign (÷) ‘divided into How many more 5s do we need to
7
know the answer lies between 20 and 30
groups of’
Answer:
10 R 7
reach 25?
packs. We start by subtracting 480 from
Use number square to
560.
demonstrate counting in multiples.
Division as Sharing
6 196
24 560
Using a number line:
Share equally
 60 6  10
12 ÷ 3 = 4
20  480
24  20
136
Share a bag of 6 sweets between
12 - 3 - 3 - 3 – 3 = 4
80
 60 6  10
2 children – one for you, one for
3
72
24  3
76
me…
8
0
3
6
9
12
 60 6  10
Answer: 23 R 8
16
In
effect,
the
recording above is the long
72 ÷ 5
 12 6  2
Can we subtract 10 lots of 5?
division
method,
though conventionally
4
32
How many other lots of 5 can we
the
digits
of
the
answer
are recorded
Answer:
32 R 4
subtract?
above the line as shown below.
To find 196 ÷ 6, we start by multiplying 6 by 10, 20, 30, … to find
10 x 5
2
4x5
23
that 6 × 30 = 180 and 6 × 40 = 240. The multiples of 180 and 240
24 560
trap the number 196. This tells us that the answer to 196 ÷ 6 is
0
2
22
72
480
between 30 and 40.
Turn number line around
80
Start the division by first subtracting 180, leaving 16, and then
72
20
subtracting the largest possible multiple of 6, which is 12, leaving 4.
8
-4
1x4
Answer: 23 R 8
6 196
16
-4
1x4
12
-4
1x4
-4
1x4
8
4
-4
0
1x4
 180 6  30
16
 12 6  2
4
32
Answer:
32 R 4
The quotient 32 (with a remainder of 4) lies between 30 and 40, as
predicted.
Division
N
R
Y1
Y2
Y3
Halving up to 10
understanding
8÷2
as half of 8
Through practical activities in meaningful contexts
Grouping objects equally
10 grouped into 2s
*How many groups?*
Y5
Y6
Halving by partitioning and recombining
Understand division as grouping
(repeated subtraction)
Interpret 8 ÷ 2 as how many 2s make 8?
∆∆/∆∆/∆∆/∆∆
and ‘8 put into groups of 2’
Halving of numbers Use multiplication
up to 20
facts and count
up/back
Halves of multiples
20 ÷ 4 = 5
count up/back in 4s
of 10 to 100
20
Grouping 2,5,10
Represent number
stories using  sign.
Y4
Record using ÷ and
= signs
Know related
division facts for 2 x
5 x 10 x tables
Recognise
relationship
between x and ÷
Include calculations
with remainders
-4
1x4
-4
1x4
-4
1x4
-4
1x4
-4
1x4
16
12
8
4
0
remainders
i.e. 21 ÷ 4 = 5 r 1
21
-4
17
-4
13
72 ÷ 2
70 ÷ 2 = 35
2 ÷2 = 1
35 + 1 = 36
3476
3000
400
70
6
÷2
÷2
÷2
÷2
÷2
150 + 25 + 4
= 179
1500+200+35+3
= 1738
72 ÷ 5
256 ÷ 7
977 ÷ 36
14r2
5 ) 72
- 20
52
- 20
32
- 20
12
- 10
2
36r4
7 ) 256
- 210 (30x7)
46
- 42 (6x7)
4
27r5
36 ) 977
- 720 (20x36)
257
- 180 (5x36)
77
- 72 (2x36)
5
(4x5)
(4x5)
(4x5)
(2x5)
bracketed numbers
represent how
many groups
-4
5
÷2
÷ 2 = 150
÷ 2 = 25
÷2 =4
Concrete ‘acting
out’ of repeated
subtraction
repeated subtraction (chunking)
-4
9
358
300
50
8
-4
1
0
Inverse of x
Remainders rounded depending on context
Decimals
Express remainder
as fraction.
7
) 87.5
Vocabulary
Addition and subtraction
Reception
add, more, and
make, sum, total
altogether
score
double
one more, two more, ten more…
how many more to make… ?
how many more is… than…?
take (away), leave
how many are left/left over?
how many have gone?
one less, two less… ten less…
how many fewer is… than…?
difference between
is the same as
Year 1
double,
how much more
is…?
- , subtract, minus
how much less
is…?
half, halve
Year 3
Year 4
Year 5
addition, one
hundred more,
subtraction, one
hundred less,
tens boundary
ones, units, tens
hundreds
boundary
Increase, decrease,
inverse
units boundary,
tenths boundary
partition
1 digit
2 digit
3 digit
inverse
Thousands
4 digits
decimal point
decimal place
tenths/hundredths
negative
ten thousands
million
lots of, groups of
x, times, multiply,
multiplied by
multiple of
once, twice, three
times… ten
times…
times as (big, long,
wide… and so on)
repeated addition
array
row, column
share equally
one each, two
each, three each…
group in pairs,
threes… tens
equal groups of
÷, divide, divided
by, divided into
multiplication,
product
factor, quotient,
divisible by
inverse
left
left over
remainder
round up/down
grid
row
column
divisor
Year 6
integer
Tens, ones
Count up
Number sentence
digit
double, halve
share, left, left
over
half
Multiplication and division
Year 2
near double
count in
2s,5s,10s
groups of
multiplication
dividend
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