Download Language in Maths Guidelines

Progression in Maths
Mathematical
language
Abbey Primary School
1
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.
2
Contents
Page 4
Addition progression methods.
Page 5
Adding integers language
Page 6
Adding decimals language
Page 7
Subtraction progression methods
Page 8
Exchanging in subtraction
Page 9
Subtracting integers language
Page 10
Subtracting decimals language
Page 11
Multiplication progression methods
Page 12
Multiplying and dividing by 10, 100 and 1000
Page 13
Multiplying integers language
Page 14
Multiplying decimals language
Page 15
Division progression methods
Page 16
Division language
3
number of digits
carrying and different
Column addition with
carrying
Column addition with
Column addition
addition
Vertical column
Addition
Progression methods
4
Addition
Language
is mentioned in modelling.
addition
Vertical column
Ensure the place value of each digit
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
with carrying
Column addition
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
of each digit is mentioned when
carrying.
number of digits
carrying and different
Column addition with
Hundred = 5 Ensure the place value
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
5
Addition
with decimals
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
Ensure the place value of each digit is
carrying
mentioned when carrying.
decimals and
Column addition with
Column addition
Language
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
Ensure the place value of each digit is
number of digits
carrying and different
Column addition with
addition.
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
6
Column subtraction
Partitioning
Number line
Subtraction
Progression
7
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
8
Subtraction
Language
Number line
Maintain place value language,
introducing the ‘adding 30’ is
the same as adding 3 Tens.
Ensure the place value of each digit is
Partitioning
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
Ensure the place value of each digit is
exchanging
Partitioning with
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
Ensure the place value of each digit is
subtraction
Column
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
9
Subtraction
exchanging and decimals
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
Ensure the place value of each digit is
mentioned in modelling.
4 hundredths – 6 hundredths I can’t do
decimals
Column subtraction with
Partitioning with
Language
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
Ensure the place value of each digit is
a 0 place value holder
Column subtraction with
used.
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
10
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.
11
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.
12
Multiplication
Language
Ensure the place value of each digit is mentioned
Expanded
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)
13
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
14
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)
4)
Separate the dividend from the divisor.
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
Ensure the place value of each digit is
mentioned in modelling.
(Large numbers)
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)
16