1. South Molton C of E Primary School
Progression
in Calculations - x ÷ +

2. Language Mathematical image/picture
Introduction
The purpose of this policy is to outline and exemplify the strategies and methods that we will teach children to enable them to become highly proficient and skilled in mental and written calculations for the four operations of addition (+), subtraction (-), multiplication (x) and division (÷). It will also show the progression in skills and strategies through the maths curriculum. Written methods of calculations are based on mental strategies and therefore children’s ability to calculate mentally and use a range of known facts is given a high priority.
It is vital that children develop a strong and secure conceptual understanding of the four operations. Therefore we use apparatus such as NUMICON, Base 10, Cuisenaire and other relevant resources to support the development of this understanding. We also use the connective model of maths so that children understand the language, the images and models, the strategies and how to apply them for each operation.
It is important that children are able to apply the most efficient and suitable method for a given calculation and that they think like mathematicians, choosing and applying the best strategies and estimating prior to calculating. They should also check their calculations using the inverse.
Symbols
Language Mathematical image/picture
Context
Figure1.1 The connective model of learning mathematics (adapted by the Devon Primary Maths Team)

3. The Importance of Mental Maths Methods
It is vitally important the all children develop their ability to calculate mentally using all four operations and are able to draw upon their bank of known facts to support them when calculating with larger numbers and decimal numbers.
Learning addition and subtraction facts as well as been able to count on and back in 1s, 10s, 100s, 1000s as well as decimal steps from any start number are crucial in supporting all mental calculations for addition and subtraction. They also support the use of more formal written methods.
Learning times table facts up to 12 x 12 and the associated division facts provides children with the key skills to enable them to multiply and divide both mentally and using written methods. Understanding what happens when a number is multiplied or divided by 10, 100 and 1000, supports their understanding of formal written methods.
Known facts need to be regularly practised so the children can recall them at speed and with accuracy. There is also a strong emphasis on teaching them to utilise these facts to support other calculations.

4. Progression in Teaching Addition
Mental Skills
Recognise the size and position of numbers
Count on in ones and tens
Know number bonds to 10 and 20
Add multiples of 10 to any number
Partition and recombine numbers
Bridge through 10
Models and Images
NUMICON
Cuisenaire
Base 10
Place value apparatus
Place value cards
Number tracks
Numbered number lines
Marked but blank number lines
Empty number lines
Hundred square
Counting stick
Bead string
Models and Images charts
ITPs – Number Facts, Ordering Numbers, Number Grid, Counting on and back in ones and tens
Key Vocabulary
add
addition
plus
and
count on
more
sum
total
altogether
increase + 40 8

5. Recognise numbers 0 to 10 1 2 3 4 5 6 7 8 9 10
1, 2, 3, 4, 5, 6 … there are 6 teddies
Count reliably up to 10 everyday objects
Find one more than a number
One more than three is four
Count in ones and tens
Begin to relate addition to combining two groups of objects
and
makes 5
Count along a number line to add numbers together
3 + 2 = 5
Begin to use the + and = signs to record mental calculations in a number sentence
6 + 4 = 10
Know doubles of numbers
= 10 +

6. Know by heart all pairs of numbers with a total of 10 and 209 2 8 3 7 4 6 5 + 2 = 3 1 1 + 2 = 3
Know that addition can be done in any order
3 + 5
Put the biggest number first and count on
4 + 6 = 1 + 9
Understand ‘equals’ sign = as ‘same as’
4 + 6 = 1 + 9
Begin to partition numbers in order to add

7. = 16
Know which digit changes when adding 1s or 10s to any number 15 16
= 25 15 25
= 35 15 25 35
Adding two two-digit numbers (without bridging)
Counting in tens and ones
Partitioning and recombining +3
+10 15 25 28
= 28
Using place value cards and Base 10 to partition numbers and recombine 40 8 30 6
= 84
40 + 8
30 + 6
= 84

8. Standard written method – Column Addition
Using a number line
This is also a mental method which can be done mentally or written down. It doesn’t matter how smaller number is added once it has been partitioned. I.e. It can be added hundreds first or ones.
= 634
+ 50
+ 200
+ 8
376
576
626
634
Standard written method – Column Addition
The previous stages reinforce what happens to the numbers when they are added together using more formal written methods. This method can be used for any numbers and decimals. 48
+ 36 84
258
+ 376
634 1
1 1

9. Progression in Teaching Subtraction
Mental Skills
Recognise the size and position of numbers
Count back in ones and tens
Know number facts for all numbers to 20
Subtract multiples of 10 from any number
Partition and recombine numbers (only partition the number to be subtracted)
Bridge through 10
Models and Images
NUMICON
Base 10
Cuisenaire
Place value apparatus
Place value cards
Number tracks
Numbered number lines
Marked but unnumbered lines
Hundred square
Empty number lines.
Counting stick
Bead strings
Models and Images Charts
ITPs – Number Facts, Counting on and back in ones and tens, Difference
Key Vocabulary
Subtract
Take away
Minus
Count back
Less
Fewer
Difference between
Decrease - 40 8

10. 7 8 9 10 , ... Ten green bottles hanging on the wall …
Five fat sausages frying in a pan …
Begin to count backwards in familiar contexts such as number rhymes or stories 7 8 9 10 ,
...
Continue the count back in ones from any given number
Three teddies take away two teddies leaves one teddy
Begin to relate subtraction to ‘ taking away ’
Find one less than a number
Count back in tens
If I take away four shells there are six left
Count backwards along a number line to ‘ take away
Count below the number line when counting back 4 5 6 7 8 9 10

11. Maria had six sweets and she ate four. How many did she have left?
Begin to use the – and = signs to record mental calculations in a number sentence
6 - 4 = 2
Know by heart subtraction facts for numbers up to 10 and 20
45 – 1 =
Subtract 1 from a two-digit number -1 43 44 45
Subtract 10 from a two-digit number
-10
45 – 10 = 45 35
45 – 30 =
Subtract multiples of 10 from any number
-10
-10
-10 15 25 35 45

12. Standard written method
Begin to find the difference by counting up from the smallest number. This method is absolutely crucial in developing a good understanding of subtraction. It is also the most efficient mental method. It should be used when numbers are close together.
Base 10 apparatus & HTU grids can be used as additional support if necessary
= 47
Now where’s the answer?
= 176 +3 +3
+70
+100
227
230
300
400
403
Standard written method
It is important that the children have a good understanding of place value and partitioning before using this method. You don’t “borrow” or “exchange”, you TAKE a ten or hundred.
4 1 ¹3 3 10
4 3
1 6 1 3

13. Progression in Teaching Multiplication
Mental Skills
Recognise the size and position of numbers
Count on in different steps 2s, 5s, 10s
Double numbers to 10
Recognise multiplication as repeated addition
Quick recall of multiplication facts
Use known facts to derive associated facts
Multiplying by 10, 100, 1000 and understanding the effect
Multiplying by multiples of 10
Models and Images
NUMICON
Cuisenaire
Place value apparatus
Arrays
100 squares
Number tracks
Numbered number lines
Marked but unnumbered lines
Empty number lines.
Multiplication squares
Counting stick
Bead strings
Models and Images charts
ITPs – Multiplication grid, Number Dials, Multiplication Facts
Vocabulary
Lots of
Groups of
Times
Multiply
Multiplication
Multiple
Product
Once, twice, three times
Array, row, column
Double
Repeated addition 40 8 x

14. 2 2 2 2 + 2 + 2 + 2 + 2 = 8 4 x 2 = 8 2 multiplied by 4 4 lots of 2
Count in tens from zero 10 20 30 40
Count in twos from zero 8 6 4 2
Count in fives from zero 5 10 15 20 25
Know doubles and corresponding halves +
Understand multiplication as repeated addition
= 8
4 x 2 = 8
2 multiplied by 4
4 lots of 2

15. Know multiplication tables to 12X12 by end of Year 4.
Use known facts to work out new ones
2x5=10
3x5=15
6x5=30
8x5=40
10x5=50
Understand that …
24 x = x x 10
24 x = x x 10
Use factors to multiply
2 4 x x
4 2
Understand multiplication as an array
Understand how to represent arrays on a number line 4 4
2 hops of 4 2 2 2 2
4 hops of 2

16. 13
Use place value apparatus to support multiplication 4
4 x 13 =
Multiplying TU x TU partitioning the numbers
Multiplying U x TU
4 x 13 = 10 4 30 3
300
120 12 X 10 3 4 40 12
= 52
Multiplying TU x TU
14 x 33
Most children will use this method. With this method it is vital that children make an estimate of the answer and can multiply by 10 and 100. =
= 132
462

17. Formal Written Methods for Multiplication: Short Multiplication and Long Multiplication

18. Progression in Teaching Division
Mental Skills
Recognise the size and position of numbers
Count back in different steps 2s, 5s, 10s
Halve numbers to 20
Recognise division as repeated subtraction
Quick recall of division facts
Use known facts to derive associated facts
Divide by 10, 100, 1000 and understanding the effect
Divide by multiples of 10
Models and Images
NUMICON
Counting apparatus
Arrays
100 squares
Number tracks
Numbered number lines
Marked but unnumbered lines
Empty number lines.
Multiplication squares
Models and Images charts
ITPs – Multiplication remainders grid, Number Dials, Grouping,
Vocabulary
Lots of
Groups of
Share
Group
Divide
Division
Divided by
Remainder
Factor
Quotient
Divisible ÷ 40 8

19. Count back in tens 10 20 30
Count back in twos 2 4 6 8 10
Count back in fives 5 10 15
Know halves
Half of 6 is 3
½ of 6 = 3
Use known multiplication facts to work out corresponding division facts
If 2 x 10 = 20
then
20  10 = 2
20  2 = 10   X

20. Understand division as sharing USING APPARATUS.
Understand division as grouping USING APPARATUS.
18 divided into groups of 3
18  3 = 6
Represent ‘groups’ for division on a number line using apparatus alongside the line 3 6 9 12 15 18
18  3 = 6
18  6 = 3
8  2 = 4
4 jumps of 2
Understand division as repeated addition using a horizontal line and apparatus to make the links +2 +2 +2 +2 2 4 6 8

21. What facts do I know about the 7 times-table?
Children need to see that as the numbers get larger, large chunk addition is the more efficient method. Multiples of the divisor (large chunks) are added. Multiplication facts are needed to see the size of the ‘chunk’.
Fact Box
1 x 7 = 7
2 x 7 = 14
5 x 7 = 35
10 x 7 = 70
20 X 7 = 140
50 x 7 = 350
100 x 7 = 700
518  7= 74
Estimate answer first
40x7
20x7
10x7
4x7
280
420
490
518
519  7= 74 r 1
40x7
20x7
10x7
4x7
r 1
280
420
490
518
519

22. Formal Written Methods for Division Short Division and Long Division