Mathematical language...Short multiplication Short multiplication Long multiplication Teaching points 1) Calculation layout with place value 2) Multiply from least significant digit

1

Progression in Maths

Mathematical language

Abbey Primary School

2

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.

3

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

4

Addition Progression methods

Vertica

l co

lumn

addition

Column addition

Column addition with

carrying


carrying and different

number of digits

5

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



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

6

Addition Language

Column addition

with decim

als


3 hundredths + 6 hundredths = 9 hundredths 4 tenths + 2 tenths = 6 tenths 8 Ones + 1 One = 9 Ones 1 Ten + 2 Tens = 3 Tens


decim

als and

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.



number of digits


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

7

Subtraction Progression

Number line

Partitioning

Column subtraction

8

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

9

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


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


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

10

Subtraction

Language

Partitioning with

exch

anging and decim

als


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


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


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

11

Multiplication Whole number methods

Grid Method

Grid Method

Expanded

Expanded

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.

12

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.

13

Multiplication Language

Expanded


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



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


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)

14

Multiplication Language

Expanded


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



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

15

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

16

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


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)


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)

Documents

Mathematical language...Short multiplication Short multiplication Long multiplication Teaching points 1) Calculation layout with place value 2) Multiply from least significant digit