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Progression in Maths
Mathematical language
Abbey Primary School
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The ‘Progression in Maths’ improved the teaching of the key operations across the school and allowed for easier transition
between year groups for children as you could build on prior knowledge.
Presentation. Children must be writing with one number in each square to ensure
work in columns (all operations) is clear. Children will also have a separate box for the decimal point to highlight the importance in the identification, position and number value surrounding the
decimal point (specifically when adding numbers with differing numbers of digits).
Place value.
Every operation relies on a solid understanding of place value that not only comes from direct questioning (e.g. partition 348) but
also in verbal modelling (I am adding 4 tens/40 to 7 tens/70 which makes 11 tens/110). Always remain aware of the place value
in modelling so that children understand the value of the numbers they are working with.
Children will also have to be confident multiplying and dividing by 10, 100 and 1000 with the correct language (For 6x70 I know that 6x7=42 but because I am multiplying by 70 I need to make my answer ten times bigger. When I multiply by 10 all digits move one place to the left giving me the answer of 420)
Times tables. Children need confidence knowing their times tables and subsequent
division facts for multiplication and division calculations as this is the foundation of mathematical understanding.
Terminology. Teachers need to ensure they are using the term ‘number’ when
talking about total value e.g. 348 and ‘digits’ when referring to any specific digit e.g. 4 tens. Whenever and wherever possible, use
correct column name when referring to digits to consolidate learning and understanding.
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Contents
Page 4 Page 5 Page 6
Page 7
Page 8 Page 9 Page 10
Page 11 Page 12
Page 13 Page 14
Page 15 Page 16
Addition progression methods. Adding integers language Adding decimals language Subtraction progression methods
Exchanging in subtraction Subtracting integers language Subtracting decimals language Multiplication progression methods Multiplying and dividing by 10, 100 and 1000
Multiplying integers language Multiplying decimals language Division progression methods Division language
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Addition Progression methods
Vertica
l co
lumn
addition
Column addition
Column addition with
carrying
Column addition with
carrying and different
number of digits
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Addition Language
Vertica
l co
lumn
addition
Ensure the place value of each digit is mentioned in modelling.
7 Ones + 1 One = 8 Ones Thirty + Thirty = Sixty 1 Hundred + 2 Hundred = 3 Hundred
Becomes
7 Ones + 1 One = 8 Ones 3 Tens + 3 Tens = 6 Tens 1 Hundred + 2 Hundred = 3 Hundred
Column addition
Ensure the place value of each digit
is mentioned in modelling.
3 Ones + 6 Ones = 9 Ones 4 Tens + 1 Ten = 5 Tens 2 Hundred + 6 Hundred = 8 Hundred 7 Thousand + zero = 7 Thousand
Column addition
with ca
rrying
Ensure the place value of each digit is mentioned when carrying.
5 Ones + 9 Ones = 14 Ones / 1 Ten and 4 Ones 6 Tens + 8 Tens + 1 carried Ten = 15 Tens / 1 Hundred and 5 Tens 2 Hundred + 2 Hundred + 1 carried
Column addition with
carrying and different
number of digits
Hundred = 5 Ensure the place value of each digit is mentioned when
carrying. 7 Ones + 6 Ones = 13 Ones / 1 Ten and 3 Ones 5 Tens + 9 Tens + 1 carried Ten = 15 Tens / 1 Hundred and 5 Tens 5 Hundred + 3 Hundred + 1 carried Hundred = 8 Hundred 2 Thousand + zero = 2 Thousand
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Addition Language
Column addition
with decim
als
Ensure the place value of each digit is mentioned in modelling.
3 hundredths + 6 hundredths = 9 hundredths 4 tenths + 2 tenths = 6 tenths 8 Ones + 1 One = 9 Ones 1 Ten + 2 Tens = 3 Tens
Column addition with
decim
als and
carrying
Ensure the place value of each digit is mentioned when carrying.
5 hundredths + 7 hundredths = 12 hundredths / 1 tenth and 2 hundredths 8 tenths + 3 tenths + 1 carried tenth = 12 tenths / 1 One and 2 tenths 4 Ones + 2 Ones + 1 carried One = 7 Ones
3 Tens + 2 Tens = 5 Tens
When adding decimals with a different number of digits, model adding 0 place value holders to ensure numbers are still
written in columns correctly and to aids addition.
Column addition with
carrying and different
number of digits
Ensure the place value of each digit is mentioned when carrying.
0 hundredths + 0 hundredths + 5 hundredths = 5 hundredths 8 tenths + 4 tenths + 9 tenths = 11 tenths / 1 One and 1 tenth 2 Ones + 5 Ones + 3 Ones + 1 carried One = 12 Ones / 1 Ten and 2 Ones
1 Ten + 1 carried Ten = 2 Tens
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Subtraction Progression
Number line
Partitioning
Column subtraction
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Subtraction Exchanging
Previously, this method has
been referred to as ‘borrowing’,
which is now frowned upon as if you were to borrow something, you would
generally give something back.
We refer to this method as ‘exchanging’, where we develop the children’s place value understanding and
ability to play with numbers to a secure level. We aim to teach the children methods to exchange one value for another so that the actual value of the number never changes. E.g. - I am going to exchange 1 Ten for 10 Ones
- I am going to exchange 1 Hundred for 10 Tens - I am going to exchange 1 Thousand for 10 Hundreds
In the above example, the top number is 358 300 + 50 + 8 = 358
When we exchange 1 Hundred for 10 Tens we create
200 + 150 + 8 = 358
Verbal modelling: - Correct calculation layout with place value headings - 8 Ones subtract 7 Ones = 1 One - 5 Tens subtract 9 Ones I can not do so…
I am going to exchange 1 Hundred for 10 Tens and
create 15 Tens - 15 Tens subtract 9 Tens = 6 Tens - 2 Hundreds – 1 Hundred = 1 Hundred
`Please note, the value of the original digit in the Hundreds column is never mentioned to avoid confusion
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Subtraction
Language
Number line Maintain place value language,
introducing the ‘adding 30’ is the same as adding 3 Tens.
Partitioning
Ensure the place value of each digit is
mentioned in modelling.
I partition 87 into 8 Tens and 7 Ones I partition 34 into 3 Tens and 4 Ones 7 Ones - 4 Ones = 3 Ones 80 - 30 = 50 When linking to column subtraction, this will become 8 Tens – 3 Tens = 5 Tens
Partitioning with
exch
anging
Ensure the place value of each digit is mentioned in modelling.
I partition 72 into 7 Tens and 2 Ones I partition 46 into 4 Tens and 6 Ones 2 Ones – 6 Ones I can not do so…
I exchange 1 Ten for 10 Ones. 12 Ones – 6 Ones = 6 Ones
60 – 40 = 20 When linking to column subtraction, this will become 6 Tens – 4 Tens = 2 Tens
Column
subtraction
Ensure the place value of each digit is mentioned in modelling.
8 Ones – 7 Ones = 1 One 5 Tens – 9 Tens I can not do so…
I exchange 1 Hundred for 10 Tens. 15 Tens- 9 Tens = 6 Tens 2 Hundreds – 1 Hundred = 1 Hundred
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Subtraction
Language
Partitioning with
exch
anging and decim
als
Ensure the place value of each digit is mentioned in modelling.
I partition 8.4 into 8 Ones and 4 tenths I partition 6.9 into 6 Ones and 9 tenths 4 tenths – 9 tenths I can not do so…
I exchange 1 One for 10 tenths.
14 tenths – 9 tenths = 5 tenths 7 ones – 6 Ones = 1 One
Column subtraction with
decim
als
Ensure the place value of each digit is mentioned in modelling.
4 hundredths – 6 hundredths I can’t do
I exchange 1 tenth for 10 hundredths. 14 hundredths – 6 hundredths = 8 hundredths
1 tenths – 7 tenths I can not do so…
I exchange 1 One for 10 tenths. 11 tenths – 7 tenths = 4 tenths
8 Ones – 3 Ones = 5 Ones
Children will also need to exchange across a zero place value holder, ensuring correct place value language is still
used.
Column subtraction with
a 0 place value holder
Ensure the place value of each digit is mentioned in modelling.
3 hundredths – 7 hundredths I can’t do
I exchange 1 tenth for 10 hundredths but there are no tenths so…
I exchange 1 One for 10 tenths I exchange 1 tenth for 10 hundredths
13 hundredths – 7 hundredths = 6 hundredths
9 tenths – 4 tenths = 5 tenths
7 Ones – 3 Ones = 4 Ones
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Multiplication Whole number methods
Grid Method
Grid Method
Expanded
Expanded
Short multiplication
Short multiplication
Long multiplication
Teaching points 1) Calculation layout with place value 2) Multiply from least significant digit 3) Show carried numbers 4) New line when multiplying tens 5) Keep numbers in columns 5) Column addition to find answer.
Long multiplication is where you
multiply by 2 or more digits.
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Multiplication Multiplying and dividing by 10, 100 and 1000
To securely multiply using any method children have to confidently multiply and divide by 10, 100 and 1000.
A dated method of multiplying by 10 is to say, “If you multiply by 10 you add a zero on…”
but this is not always the case.
e.g. following this rule means 2.6 x 10 = 2.60 Instead, we use their understanding of place value and use the rules that show a change in place value of digits.
x 10 = each digit moves 1 place value place to the left x 100 = each digit moves 2 place value place to the left x 1000 = each digit moves 3 place value place to the left
e.g This needs to be directly
taught to the children until it becomes a mental skill and referred to during modelling
whenever used. ÷ 10 = each digit moves 1 place value place to the right ÷ 100 = each digit moves 2 place value place to the right
÷ 1000 = each digit moves 3 place value place to the right e.g
This needs to be directly taught to the children until it becomes a mental skill and referred to during modelling
whenever used.
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Multiplication Language
Expanded
Ensure the place value of each digit is mentioned in modelling.
Layout calculation with place value layout. 6 x 8 = 48
6 x 40 = 240 6 x 200 = 1200
becomes
6 x 8 Ones = 48 Ones / 48 6 x 4 Tens = 24 Tens / 240
6 x 2 Hundreds = 12 Hundreds / 1200
Short multiplication
Ensure the place value of each digit is mentioned in modelling.
Layout calculation with place value layout.
6x8 Ones = 48 Ones / 4 Tens and 8 Ones
6x4 Tens = 24 Tens + 4 carried Tens = 28 Tens / 2 Hundreds and 8 Tens
6x2 Hundreds = 12 Hundreds + 2 Hundreds = 14 Hundreds / 1 Thousand and 4 Hundreds
Long multiplication
Ensure the place value of each digit is mentioned in modelling.
Layout calculation with place value layout. (First step is identical to the above calculation)
2x9 Ones = 18 Ones / 1 Ten and 8 Ones
2x8 Tens = 16 Tens + 1 carried Ten = 17 Tens / 1 Hundreds and 7 Tens
2x2 Hundreds = 4 Hundreds + 1 Hundreds = 5 Hundreds
CT then models multiplying 289 by 40, using understanding of multiplying by a multiple of 10.
As I am multiplying by 40, each digit will have to move 1 place to the left as 40 is a multiple of 10. Therefore I am going to add in my 0 place value holder which
will move each digit one place to the left. My calculation is 40x9 Ones but as I already have a 0 place value holder it is
4 x9 Ones = 36 Ones (actually 360)
My calculation is 40x8 Tens but as I already have a 0 place value holder it is 4
x8 Tens = 32 Tens + 3 carried Tens = 35 Tens (actually 3500) My calculation is 40x2 Hundreds but as I already have a 0 place value holder it
is 4 x2 Hundreds = 8 Hundreds + 3 carried Hundreds = 11 Hundreds (actually 11000)
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Multiplication Language
Expanded
Ensure the place value of each digit is mentioned in modelling.
Layout calculation with place value layout. 8 x 0.7 = 5.6 8 x 4 = 32 becomes
8 x 7 tenths = 56 tenths / 5.6 8 x 4 Ones = 32 Ones / 32
Short multiplication
Ensure the place value of each digit is mentioned in modelling.
Layout calculation with place value layout.
8x7 tenths = 56 tenths / 5 Ones and 6 tenths
8x4 Ones = 32 Ones + 5 carried Ones = 37 Ones / 3 Tens and 7 Ones
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Division
Progression
Chunking Teaching points
1) Identify dividend, divisor and quotient 2) Complete a jotting box relevant to the question.
3) Layout number line. 4) Use jotting box to jump in appropriate sizes.
5) Circle number you multiply divisor by.
6) Informally jot addition sums to ensure
correctness. 7) Count circled numbers to find answer.
Long division Teaching points
1) Identify dividend, divisor and quotient. 2) Complete a jotting box relevant to the question.
3) Separate the dividend from the divisor. 4) Use jotting box to subtract chunks of your
divisor from dividend. 5) Use column subtraction to find remaining
amount 6) Repeat until indivisible amount remains.
7) Count circled numbers to find answer.
Short division 1) Separate your divisor and dividend.
2) How many Hundred lots of your divisor goes
into your dividend? 3) Any additional value is passed down to the
following column. 4) How many Ten lots of your divisor goes into
your dividend? 5) Any additional value is passed down to the
following column. 6) Repeat until question is answered.
Short division
Large numbers
As above
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Division
Language
Chunking
Ensure the place value of each digit is mentioned in modelling to allow for easier
progress through additional methods.
Long division
Ensure the place value of each digit is mentioned in modelling to allow for easier
progress through additional methods.
Short division
Ensure the place value of each digit is mentioned in modelling.
How many Hundred lots of 9 go into 2
Hundred (0) so my 2 Hundred become 20 Tens
How many Ten lots of 9 go into 21 Tens (2)
leaving me 3 Tens remaining that become 30 Ones
How many 9s go into 36 Ones (4)
Short division (Large numbers)
Ensure the place value of each digit is mentioned in modelling.
How many Thousand lots of 8 go into 1 Thousand (0) so my 1 Thousand becomes 10
Hundred
How many Hundred lots of 8 go into 12
Hundred (1) leaving me 4 Hundreds that become 40 Tens
How many Ten lots of 8 go into 49 Tens (6) leaving me 1 Tens remaining that become 10
Ones How many 8s go into 16 Ones (2)