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St Catherine’s R C Primary School
Calculations Policy
Rationale
This policy is designed to outline our approach to mental and written calculations in a progressive and consistent format. We recognise the
importance of mental strategies being introduced and taught systematically from reception classes onwards and pupils being given regular
opportunities to develop and practise these skills. We aim to ensure that mental and written calculations are there to complement each
other. Every written method has elements of mental operations whole written recording helps pupils clarify their thinking and supports the
development of more fluent mental strategies. It is important that children do not abandon jotting and empty number line approaches once
paper and pencil methods have been introduced.
Pupils of all ages and abilities should be encouraged to:
 To examine the calculation or problems and the numbers involved as the size of or complexity of the numbers involved often influences the
chosen methods of calculation.
 Decide upon the best method to use. This may be mental calculation with or without jottings, an empty number line, expanded written
method, compact written method, using a calculator or another chosen method that the child is comfortable and accurate in using.
 Be aware of the importance of approximating and estimating an answer before carrying out the calculation and then complete th e calculation
and check the appropriateness of their answer.
When faced with a calculation, pupils are able to select an efficient method of their choice that is appropriate for a given task. They should always
ask themselves.
 Have I estimated my answer?
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



Can I do this in my head?
Do I need jottings such as an empty number line?
Do I need a pencil and paper method?
Should I use a calculator?
At whatever stage in their learning, and whatever method being used, it must still be underpinned by a secure and appropriate knowledge of number
facts, along with those mental skills that are needed to carry out the process and judge if it was successful.
Our overall aim is that when our children leave primary school they:
 have a secure knowledge of number facts and a good understanding of the four operations;
 are able to use this knowledge to carry out calculations mentally and to apply general strategies involving bigger numbers;
 make use of diagrams and jottings to help record steps and part answers when using mental methods that generate more information that
they can keep in their heads;
 have an efficient and reliable written method of calculation for each operation that children can apply with confidence ;
 use a calculator effectively, using their mental skills to monitor the process, check the steps involved and decide if the numbers displayed
make sense.
Teaching children to calculate mentally – (DFE publication 2010)
The four chapters of the booklet cover:
1. Progression in mental calculation skills
This describes the progression in the number facts that children should derive and recall, the calculations that they are expected to do mentally
and the range of calculation strategies or methods that they can draw on.
2. Principles of teaching mental calculation
This promotes a broad interpretation of mental calculation and identifies principles that underpin teaching: for example, encouraging children to
share their mental methods, to choose efficient strategies and to use informal jottings to keep track of the information they need when
calculating. It also looks at the role of tests and questioning.
3. Addition and subtraction strategies
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This sets out the main strategies for adding and subtracting mentally. It describes activities to support teaching of these strategies and typical
problems.
4. Multiplication and division strategies
This sets out the main strategies for multiplying and dividing mentally. Again, it describes activities to support teaching of these strategies and
typical problems.
Written methods for addition of whole numbers
The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written
method accurately and with confidence..
To add successfully, children need to be able to:
•
recall all addition pairs to 9 + 9 and complements in 10;
•
add mentally a series of one-digit numbers, such as 5 + 8 + 4;
•
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;
•
partition two-digit and three-digit numbers into multiples of 100, 10 and 1 in different ways.
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.
Written methods for subtraction of whole numbers
The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written
method accurately and with confidence
These notes show the stages in building up to using an efficient method for subtraction of two-digit and three-digit whole numbers by the end of
Year 4.
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).
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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
Progression in Teaching Written Methods for Multiplication and Division
It is important to teach both of the above operations together to ensure that pupils understand the relationship between them and they can use the
inverse to check their calculations.
We also need to be aware that when introducing the next stage the links between methods are made explicit by showing methods side by side.
It is also necessary to reinforce the importance of estimating and approximating so children can identify if their answers make sense.
Written methods for multiplication of whole numbers
The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written
method accurately and with confidence.
These notes show the stages in building up to using an efficient method for two-digit by one-digit multiplication by the end of Year 4, two-digit by
two-digit multiplication by the end of Year 5, and three-digit by two-digit multiplication by the end of Year 6.
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
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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.
Written methods for division of whole numbers
The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written
method accurately and with confidence..
These notes show the stages in building up to long division through Years 4 to 6 – first long division TU ÷ U, extending to HTU ÷ U, then HTU ÷ TU,
and then short division HTU ÷ U.
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.
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.
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.
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As teachers we may need to consider how we present calculations to pupils. Calculations that are presented horizontally are more likely to prompt a
pupil to ask him or herself the above questions.
As pupils gain confidence at using expanded written methods of calculation we can encourage them to move towards a more compact form of
recording. If errors appear at this stage teachers are advised to return to the expanded method stage until full understanding is consolidated.
Pupils needs to be given regular opportunities to use and apply calculation methods efficiently to solve a range of problems including word problems
and investigations.
Addition
Early
stage
s
Subtraction
Children are encouraged to develop a mental picture of the
number system in their heads to use for calculation.
They develop ways of recording calculations using pictures
Multiplication
Children are encouraged to develop a mental
picture of the number system in their heads to
use for calculation.
They develop ways of recording calculations using
pictures.
Children will experience equal groups of objects.
They will count in 1s, 2s and 10s and begin to count
in 5s.
Division
Children will understand equal groups and share
items out in play and problem solving. They will
count in1s, 2s and 10s and later in 5s.
They will work on practical problem solving activities
involving equal sets or groups.
Bead strings or bead bars can be used to illustrate addition
8+2=10
They use numberlines and practical resources to support
calculation and teachers demonstrate the use of the numberline.
Bead strings or bead bars can be used to
illustrate subtraction including bridging through
ten by counting back 3 then counting back 2.
6-2=4
They use numberlines and practical resources to
support calculation. Teachers demonstrate the
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Addition
Subtraction
Multiplication
Division
use of the numberline.
KS1
Continue using pictures to record
Continue using pictures to record
Children will experience equal groups of objects.
Bead strings or bead bars can be used to illustrate addition
including bridging through ten by counting on 2 then counting on
3.
Bead strings or bead bars can be used to
illustrate subtraction including bridging through
ten by counting back 3 then counting back 2.
They will count in 2s and 10s and begin to count in 5s.
Children will understand equal groups and share
items out in play and problem solving. They will
count in 1s, 2s and 10s and later in 5s.
They will work on practical problem solving activities
involving equal sets or groups.
13-5=8
They use numberlines and practical resources to support
calculation and teachers demonstrate the use of the numberline.
Children then begin to use numbered lines to support their own
calculations using a numbered line to count on in ones
Children then begin to use numbered lines to
support their own calculations - using a numbered
line to count back in ones.
The numberline should also be used to show that 6
- 3 means the ‘difference between
6 and 3’ or ‘the difference between 3 and 6’ and
how many jumps they are apart.
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Addition
Children will begin to use ‘empty number lines’ themselves
starting with the larger number and counting on.

First counting on in tens and ones.
Subtraction
Children will begin to use empty number lines to
support calculations.
Counting back:

First counting back in tens and ones.
Multiplication
Division
Children will develop their understanding of
multiplication and use jottings to support calculation:

Repeated addition
3 times 5 is 5 + 5 + 5 = 15 or 3 lots of 5 or
5 x3
Children will develop their understanding of division
and use jottings to support calculation

Sharing equally
6 sweets shared between 2 people, how many do
they each get?
Repeated addition can be shown easily on a number
line:



Then helping children to become more efficient by
adding the units in one jump (by using the known fact 4 +
3 = 7).
Followed by adding the tens in one jump and the units in
one jump.
Bridging through ten can help children become more
efficient.

Then helping children to become more
efficient by subtracting the units in one
jump (by using the known fact 7 – 3 = 4).

Subtracting the tens in one jump and the
units in one jump.

Bridging through ten can help children
become more efficient.
and on a bead bar:

Grouping or repeated subtraction
There are 6 sweets, how many people can have 2
sweets each?

Commutativity
Children should know that 3 x 5 has the same answer
as 5 x 3. This can also be shown on the number line.
Repeated subtraction using a number line
or bead bar
12 ÷ 3 = 4


Arrays
Children should be able to model a multiplication
calculation using an array. This knowledge will
support with the development of the grid method.
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Addition
Subtraction
Multiplication
Division

Using symbols to stand for unknown
numbers to complete equations using
inverse operations
÷2=4
20 ÷  = 4
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÷=4
10
Addition
1. Empty number lines
KS
2 Children will continue to use empty number lines
with increasingly large numbers, including
compensation where appropriate.


Count on from the largest number
irrespective of the order of the
calculation
Compensation
Subtraction
Children will continue to use empty number lines with
increasingly large numbers.
Children will begin to use informal pencil and paper
methods (jottings).
Partitioning
74 -27
= 74 – 20 – 7
= 54 -7
= 47
Expanded method

Partitioning and decomposition
 Partitioning – demonstrated using arrow cards
 Decomposition - base 10 materials
NOTE When solving the calculation 89 – 57, children
should know that 57 does NOT EXIST AS AN
AMOUNT it is what you are subtracting from the
other number. Therefore, when using base 10
materials, children would need to count out only the
89.
Multiplication
Children will continue to use:

Repeated addition
4 times 6 is 6 + 6 + 6 + 6 = 24 or 4 lots of 6
or 6 x 4
Children should use number lines or bead bars to
support their understanding.
Ensure that the emphasis in Y3 is on grouping
rather than sharing.
Children will continue to use:


Repeated subtraction using a number line
Arrays
Children should be able to model a multiplication
calculation using an array. This knowledge will support
with the development of the grid method.

Scaling
e.g. Find a ribbon that is 4 times as long as the blue
ribbon
Children will begin to use informal pencil and paper
methods (jottings) to support, record and explain
partial mental methods building on existing mental
strategies.
Division
Using symbols to stand for unknown
numbers to complete equations using
inverse operations
 x 5 = 20
3 x  = 18
x=
32
Children should also move onto calculations involving
remainders.

Using symbols to stand for unknown
numbers to complete equations using
inverse operations

The + sign may be taken out but leave a clear space
between numbers

26 ÷ 2 = 
24 ÷  = 12
 ÷ 10 = 8
Partitioning
38 x 5 = (30 x 5) + (8 x 5)
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Addition
2. Partitioning
Partitioning both numbers into tens and units
mirrors the column method
Using partitioning
43 + 65
= 40 + 60 + 3 + 5
= 100 + 8
=108
Subtraction

Multiplication
= 150 + 40
= 190
Division
Begin to exchange.
1–6
Partitioned numbers are then written under on
another
40
3
60
5
100
8 = 108
Where the numbers are involved in the calculation are
close together or near to multiples of 10, 100 etc
counting on using a number line should be used.
Expanded method in columns
Write the numbers in columns adding units first
Brackets can be removed when more children
more confident
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Addition
Column Method
Subtraction

Partitioning and decomposition
Multiplication
Children will continue to use arrays where appropriate
leading into the grid method of multiplication.
In this method the recording is reduced further

Carry below the line under the correct
column
Using similar methods, children will:

add several numbers with different
numbers of digits;

begin to add two or more three-digit sums
of money, with or without adjustment from
the pence to the pounds;

know that the decimal points should line up
under each other, particularly when adding
or subtracting mixed amounts, e.g. £3.59
+ 78p.

Division
Children will develop their use of repeated
subtraction to be able to subtract multiples of the
divisor. Initially, these should be multiples of 10s,
5s, 2s and 1s – numbers with which the children are
more familiar.
Grid method
TU x U
(Short multiplication – multiplication by a single digit)
23 x 8
Children will approximate first
23 x 8 is approximately 25 x 8 = 200

Decomposition
Then onto the vertical method:
Short division TU ÷ U
27
3 8 21
The carry digit ‘2’ represents the 2 tens that have
been exchanged for 20 ones. In the first recording
above it is written in front of the 1 to show that 21
is to be divided by 3.
The 27 written above the line represents the
answer: 20 + 7, or 2 tens and 7 ones.
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Addition
Subtraction
Multiplication
Division
Chunking method
This method is based on subtracting multiples
Children should:

be able to subtract numbers with different
numbers of digits;

using this method, children should also begin
to find the difference between two threedigit sums of money, with or without
‘adjustment’ from the pence to the pounds;

know that decimal points should line up under
each other.
6 196
 60 6  10
136
 60 6  10
76
 60 6  10
16
 12 6  2
4
32
Answer:
32 R 4
Any remainders should be shown as integers, i.e. 14
remainder 2 or 14 r 2.
Children need to be able to decide what to do after
division and round up or down accordingly. They
should make sensible decisions about rounding up or
down after division to solve a word problem
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Addition
Children should extend the carrying method to
numbers with at least four digits.
Subtraction
Partitioning and decomposition
3 digit - 3 digit
Multiplication
Grid method
HTU x U
(Short multiplication – multiplication by a single
digit)
346 x 9
Children will approximate first
346 x 9 is approximately 350 x 10 = 3500
Division
Children will continue to use written methods
to solve short division TU ÷ U.
Children can start to subtract larger
multiples of the divisor, e.g. 30x
Short division HTU ÷ U
6 196
Using similar methods, children will:

add several numbers with different numbers
of digits;

begin to add two or more decimal fractions
with up to three digits and the same number
of decimal places;

know that decimal points should line up under
each other, particularly when adding or
subtracting mixed amounts, e.g. 3.2 m – 280
cm.
Decomposition
Children should:

be able to subtract numbers with different
numbers of digits;

begin to find the difference between two
decimal fractions with up to three digits and
the same number of decimal places;
know that decimal points should line up under each
other
Where the numbers are involved in the calculation are
close together or near to multiples of 10, 100 etc
counting on using a number line should be used.
TU x TU
(Long multiplication – multiplication by more than a
single digit)
72 x 38
Children will approximate first
72 x 38 is approximately 70 x 40 = 2800
 180 6  30
16
 12 6  2
4
32
Answer:
32 R 4
Using similar methods, they will be able to multiply
decimals with one decimal place by a single digit
number, approximating first. They should know
that the decimal points line up under each other.
e.g. 4.9 x 3
Children will approximate first
4.9 x 3 is approximately 5 x 3 = 15
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Addition
Subtraction
Multiplication
Division
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Addition
Children should extend the carrying method to number
with any number of digits.
Subtraction
Decomposition
4 digit – 4 digit
Multiplication
ThHTU x U
(Short multiplication – multiplication by a single
digit)
4346 x 8
Children will approximate first
4346 x 8 is approximately 4346 x 10 = 43460
Division
Children will continue to use written methods
to solve short division TU ÷ U and HTU ÷ U.
Long division HTU ÷ TU
23
560
 480
80
72
8
Answer: 23 R 8
24
Using similar methods, children will

add several numbers with different numbers
of digits;

begin to add two or more decimal fractions
with up to four digits and either one or two
decimal places;

know that decimal points should line up under
each other, particularly when adding or
subtracting mixed amounts, e.g. 401.2 + 26.85
+ 0.71.
Children should:

be able to subtract numbers with different
numbers of digits;

be able to subtract two or more decimal
fractions with up to three digits and either
one or two decimal places;

know that decimal points should line up under
each other.
Where the numbers are involved in the calculation are
close together or near to multiples of 10, 100 etc
counting on using a number line should be used.
HTU x TU
(Long multiplication – multiplication by more than a
single digit)
372 x 24
Children will approximate first
372 x 24 is approximately 400 x 25 = 10000
Extend to decimals with up to two decimal
places. Children should know that decimal
points line up under each other.
87.5 ÷ 7
1 2 . 5
7
Using similar methods, they will be able to multiply
decimals with up to two decimal places by a single
digit number and then two digit numbers,
approximating first. They should know that the
decimal points line up under each other.
For example:
8
- 7
1
- 1
7 . 5
0. 0
7. 5
4. 0
3. 5
3.5
7 x10
7x2
7 x 0.5
0
4.92 x 3
Children will approximate first
4.92 x 3 is approximately 5 x 3 = 15
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Addition
Subtraction
Multiplication
Division
Children who are already secure with multiplication
should have little difficulty in using more compact
method
286 x 29
X
286
29
2574
5720
286 x 9
286 x20
8294
1
1)
2)
By the end of year 6, children will have a range of calculation methods, mental and written. Selection will depend upon the numbers involved.
Children should not be made to go onto the next stage if:
they are not ready.
they are not confident.
Children should be encouraged to approximate their answers before calculating.
Children should be encouraged to consider if a mental calculation would be appropriate before using written methods.
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Objectives
The objectives in the revised Framework show the progression in children’s use of written methods of calculation in the strands ‘Using and
applying mathematics’ and ‘Calculating’.
Using and applying mathematics
Calculating
Year 1
• Solve problems involving counting, adding,
subtracting, doubling or halving in the
context of numbers, measures or money, for
example to ‘pay’ and ‘give change’
• Describe a puzzle or problem using
numbers, practical materials and diagrams;
use these to solve the problem and set the
solution in the original context
Year 1
• 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 ‘take away’ and
find a ‘difference’ by counting up; use
practical and informal written methods to
support the subtraction of a one-digit number
from a one-digit or two-digit number and a
multiple of 10 from a two-digit number
• Use the vocabulary related to addition and
subtraction and symbols to describe and
record addition and subtraction number
sentences
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Using and applying mathematics
Calculating
Year 2
• Solve problems involving addition,
subtraction, multiplication or division in
contexts of numbers, measures or pounds
and pence
• Identify and record the information or
calculation needed to solve a puzzle or
problem; carry out the steps or calculations
and check the solution in the context of the
problem
Year 2
• 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
• Use the symbols +, –, ×, ÷ and = to record
and interpret number sentences involving all
four operations; calculate the value of an
unknown in a number sentence (e.g.
 ÷ 2 = 6, 30 –  = 24)
Year 3
Year 3
• Solve one-step and two-step problems
involving numbers, money or measures,
including time, choosing and carrying out
appropriate calculations
• Represent the information in a puzzle or
problem using numbers, images or
diagrams; use these to find a solution and
present it in context, where appropriate using
£.p notation or units of measure
• Develop and use written methods to record,
support or explain addition and subtraction of
two-digit and three-digit numbers
• Use practical and informal written methods to
multiply and divide two-digit numbers (e.g.
13 × 3, 50 ÷ 4); round remainders up or
down, depending on the context
• Understand that division is the inverse of
multiplication and vice versa; use this to
derive and record related multiplication and
division number sentences
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Using and applying mathematics
Calculating
Year 4
• Solve one-step and two-step problems
involving numbers, money or measures,
including time; choose and carry out
appropriate calculations, using calculator
methods where appropriate
• Represent a puzzle or problem using number
sentences, statements or diagrams; use
these to solve the problem; present and
interpret the solution in the context of the
problem
Year 4
• Refine and use efficient written methods to
add and subtract two-digit and three-digit
whole numbers and £.p
• Develop and use written methods to record,
support and explain multiplication and
division of two-digit numbers by a one-digit
number, including division with remainders
(e.g. 15 × 9, 98 ÷ 6)
Year 5
Year 5
• Solve one-step and two-step problems
involving whole numbers and decimals and
all four operations, choosing and using
appropriate calculation strategies, including
calculator use
• Represent a puzzle or problem by identifying
and recording the information or calculations
needed to solve it; find possible solutions
and confirm them in the context of the
problem
• Use efficient written methods to add and
subtract whole numbers and decimals with
up to two places
• 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 and divide HTU × U, TU × TU,
U.t × U and HTU ÷ U
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Using and applying mathematics
Calculating
Year 6
• Solve multi-step problems, and problems
involving fractions, decimals and
percentages; choose and use appropriate
calculation strategies at each stage,
including calculator use
• Represent and interpret sequences, patterns
and relationships involving numbers and
shapes; suggest and test hypotheses;
construct and use simple expressions and
formulae in words then symbols (e.g. the
cost of c pens at 15 pence each is 15c
pence)
Year 6
• Use efficient written methods to add and
subtract integers and decimals, to multiply
and divide integers and decimals by a onedigit integer, and to multiply two-digit and
three-digit integers by a two-digit integer
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