Download Progression in Calculations 2015

÷
Introduction
Written methods of calculations are based on mental
strategies. Each of the four operations builds on mental
skills which provide the foundation for jottings and
informal written methods of recording. Skills need to be
taught, practised and reviewed constantly. These skills
lead on to more formal written methods of calculation.
Strategies for calculation need to be supported by
familiar models and images to reinforce understanding.
When teaching a new strategy it is important to start
with numbers the child can easily manipulate so that
they can understand the concept.
The transition between stages should not be hurried as
not all children will be ready to move on to the next
stage at the same time, therefore the progression in this
document is outlined in stages. Previous stages may
need to be revisited to consolidate understanding when
introducing a new strategy.
A sound understanding of the number system is
essential for children to carry out calculations efficiently
and accurately.
KS1
Progression in Teaching Addition
Mental Skills
• Recognise the size and position of numbers
• Count on in units and tens
• Know number bonds to 10, 20 and 100
• Add multiples of 10 to any number
• Partition and recombine numbers
• Bridge through 10
Models and Images
• Place value apparatus
• Place value cards
• Number tracks
• Numbered number lines
• Marked but unnumbered number lines
• Empty number lines
• Hundred square
• Counting stick
• Bead string
• Models and Images charts
• Numicon
Key Vocabulary
• add
• addition
• plus
• and
• count on
• more
• sum
• total
• altogether
• increase
40
8
Y1
Recognise, count, write and order
numbers 0 to 100
Count reliably up to 10
Count 2, 5, 10 backwards and
Find one more and one less than a
number
One more than
three is four
Begin to relate addition to combining
two groups of objects
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
6 + 5 = 11
Read and write numbers from 1 to 20 in numerals
and words.
6 + 4 = 10
Manipulate numbers based on facts that you
already know. This helps to develop
understanding of the equal sign balancing an
equation.
Know doubles of numbers
Know by heart all pairs of numbers with a total of
10 and 20.
3 7
Addition of
more than
two numbers
to spot bonds
3 + 2 + 7 = 12
Know that addition can be
done in any order
8 + 7 = 15
+2
8
Add one and two digit numbers
that bridge 10 up to 20
+5
10
Solve missing number problems
involving addition
15
Y1
Y2
Know that addition of numbers can be done in
any order (commutative)
+10
Continue to use numberlines to develop
understanding of:
+2
23
Counting on in tens and ones
23 + 12 = 23 + 10 + 2
= 33 + 2
= 35
35
33
Partitioning and bridging through 10.
The steps in addition often bridge through a
multiple of 10
e.g. Children should be able to partition the 7 to
relate adding the 2 and then the 5.
8 + 7 = 15
Adding 9 or 11 by adding 10 and adjusting by 1
e.g. Add 9 by adding 10 and adjusting by 1
35 + 9 = 44
Towards a Written Method
Partitioning in different ways and recombine
Leading to exchanging:
72
47+25
47
25
60 + 12
Towards a Written Method
Introduce expanded column addition
modelled with place value counters (Dienes
could be used for those who need a less
abstract representation)
H
T
U
247 + 125 = 372
Progression in Teaching Addition
Mental Skills
• Add numbers mentally including:
–
three digit number and units
–
three digit number and tens
–
three digit number and hundreds
Models and Images
• Place value apparatus
• Place value cards
•Empty number lines
• Hundred square
• Counting stick
• Bead string
• Models and Images charts
• Teaching packs 1,2 and 3 (on server)
Key Vocabulary
• add
• addition
• plus
• and
• count on
• more
• sum
• total
• altogether
• increase
40
8
KS2
H
T
U
Expanded Method
Pupils should be able to add numbers with
up to three digits (this can include
decimals) using the column method
Compact written method
Extend to numbers with at least three digits.
Children should be able to make the
choice of reverting to expanded
methods if experiencing any difficulty.
Extend to up to two places of decimals
(same number of decimals places) and
adding several numbers (with different
numbers of digits).
72.8
+ 54.6
127.4
1 1
End of Y6 expectation
2016 Sample Paper Arithmetic
LJ
End of Y6 expectation
2016 Sample Paper Reasoning, paper
3
Y1
Y2
Y4
Y3
Y5
Y6
KS1
Progression in Teaching Subtraction
Mental Skills
• Recognise the size and position of numbers
• Count back in units 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
•
•
•
•
•
•
•
•
•
•
•
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
Numicon
Key Vocabulary
Subtract
Take away
Minus
Count back
Less
Fewer
Difference between
40
8
Begin to count backwards in
familiar contexts such as
number rhymes or stories
Five fat sausages
frying in a pan …
Y1
Ten green bottles
hanging on the wall …
Continue the count back in units
from any given number
Begin to relate subtraction to ‘
taking away ’
Three teddies take away
two teddies leaves one
teddy
Use concrete objects and pictorial representations. If
appropriate, progress from using number lines with
every number shown to number lines with significant
numbers shown.
If I take away four shells
there are six left
Missing number problems
e.g. 7 = □ - 9; 20 - □ = 9; 15 – 9 = □; □ - □ = 11;
16 – 0 = □
Count back in tens
Y1
Begin to use the – and = signs to
record mental calculations in a
number sentence
Maria had six sweets and
she ate four. How many
did
she have left?
6-4=2
Know by heart subtraction facts for
numbers up to 10 and 20
15 - 7 = 8
Subtract single digit numbers
often bridging through 10
Begin to find the
difference by
counting up from
the smallest
number
Mastery
questions for Y1
Use mental images
such as a 100
square, numberline
or Numicon to
subtract 1 from a
two-digit number
45 - 1
Use mental images, such as a
100 square, numberline or
Numicon to subtract 10 from a
two-digit number
20
25
45
74 - 27 = 47
Now where’s
the answer?
Recording addition and subtraction in expanded
columns can support understanding of the quantity
aspect of place value and prepare for efficient
written methods with larger numbers. The numbers
may be represented with Dienes apparatus. E.g. 75
– 42
30
Use mental images, such
as a 100 square,
numberline or Numicon to
subtract multiples of 10
from any number
Counting on to find the
difference
Towards written methods
Y2
Written methods (progressing to 3-digits)
Introduce expanded column subtraction with no
decomposition, modelled with place value counters
(Dienes could be used for those who need a less
abstract representation)
KS2
For some children this will lead to exchanging,
modelled using place value counters or
dienes.
Standard written method
The previous stages reinforce what happens to numbers
when they are subtracted using more formal written
methods. It is important that the children have a good
understanding of place value and partitioning.
3
4 31
- 2 7
1 6
Written methods
Continue using the written method for subtraction including decimals. Throughout the year groups,
pupils will progress to larger numbers, aiming for both conceptual understanding and procedural fluency
with decomposition to be secured.
End of Y6 expectation
2016 Sample Paper Reasoning, paper
3
Y3
Y1
Y2
Y4
Y5
Y6
KS1
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
Multiplication can be done in any order (commutative law)
Models and Images
•
•
•
•
•
•
•
•
•
•
•
•
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
Numicon
Vocabulary
•
•
•
•
•
•
•
•
•
•
Lots of
Groups of
Times
Multiply
Multiplication
Doubling
Once, twice, three times
Array, row, column
Double
Repeated addition
40
8
Y1
Count in tens
from zero
0
20
30
40
50
Count in twos
from zero
0
2
4
6
8
10
15
Count in fives
from zero
0
20
25
Know doubles and
corresponding halves
Use arrays to understand multiplication can
be done in any order (commutative
)
30
Y2
Expressing multiplication as a
number sentence using times
symbol (x).
Using understanding of the inverse
and practical resources to solve
missing number problems.
7x2=
7 x  = 14
 x 2 = 14
=2x7
14 =  x 7
14 = 2 x 
Begin to develop understanding of
multiplication as scaling (2 times bigger/taller)
Doubling numbers up to 10 + 10 and link with understanding scaling
Using known doubles to work out double 2 digit numbers (double 15 = double 10 +
double 5)
Towards written methods
Use jottings to develop an
understanding of doubling two digit
numbers.
Progression in Teaching Multiplication
Mental Skills
•
•
•
•
•
•
•
•
Square numbers up to 12 x 12
Cube numbers
Quick recall of multiplication facts
By the end of Y4 children need to know upto 12x12
Recognise multiplication as repeated addition
Use known facts to derive associated facts
Multiplying by 10, 100, 1000 and understanding the effect
Multiplying by multiples of 10
Models and Images
•
•
•
•
•
•
•
•
•
•
•
•
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
Numicon
Vocabulary
•
•
•
•
•
•
•
•
•
•
Lots of
Groups of
Times
Multiply
Multiplication
Multiple
Product
Once, twice, three times
Double
Repeated addition
40
8
KS2
10
Use place value apparatus to support the
multiplication of U x TU
4 x 13 = 52
4
4
KS2
3
4
10
3
40
12
10
3
40
12
Use place value apparatus to support the
multiplication of U x TU alongside the grid
method
4 x 13 = 52
40 + 12 = 52
10
10
3
4
Use place value apparatus to
represent the multiplication of U
x TU alongside the grid method
10
4
4 x 23 = 92
4
Always start with the Units column.
This is then consistent when moving
onto the formal written methods
10
40
40
3
12
20 ( 2 x 10 )
3
80
12
80 + 12 = 92
Mental methods
KS2
Counting in multiples of 6, 7, 9, 25 and 1000,
and steps of 1/100.
Use place value apparatus to support the
multiplication of 2 digit or 3 digit numbers
by a one digit number using a formal
written method.
4 x 13 = 52
Written methods (progressing to 4digit x 2digit)
Long multiplication using place value counters
Children to explore how the grid method supports an understanding
of long multiplication (for 2d x 2d)
Continue to refine and deepen
understanding of written methods
including fluency for using long
multiplication
KS2
KS2 2016 Sample
Papers
Arithmetic
KS1
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
Recognise odd and even numbers
Models and Images
•
•
•
•
•
•
•
•
•
Counting apparatus
Arrays
100 squares
Number tracks
Numbered number lines
Marked but unnumbered lines
Empty number lines.
Multiplication squares
Models and Images charts
Vocabulary
•
•
•
•
•
•
•
•
Lots of
Groups of
Share
Group
Divide
Division
Divided by
Remainder
40
8
Y1
Count back in tens
0
10
20
30
Count back in twos
?
4
6
8
10
Count back in fives
0
5
10
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
15
Y1
Understand division as
sharing
Understand division as
grouping
Use of arrays as a
pictorial representation
for division. 15 ÷ 3 = 5
There are 5 groups of 3.
15 ÷ 5 = 3 There are 3
groups of 5.
Children should be able to find ½ and ¼ and simple fractions of objects, numbers
and quantities.
6÷2=
6÷=3
÷2=3
÷=3
=6÷2
3=6 ÷
3=÷2
3=÷
÷ = signs and missing
numbers
Grouping using a numberline
Group from zero in jumps of the divisor to find our ‘how many groups of 3 are there in 15?’
15 ÷ 3 = 5
Y2
KS2
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
•
•
•
•
•
•
•
•
•
Counting apparatus
Arrays
100 squares
Number tracks
Numbered number lines
Marked but unnumbered lines
Empty number lines.
Multiplication squares
Models and Images charts
• Vocabulary
•
•
•
•
•
•
•
•
•
•
•
Lots of
Groups of
Share
Group
Divide
Division
Divided by
Remainder
Factor
Quotient
Divisible
40
8
KS2
18 divided into groups of 3
18  3 = 6
Represent ‘groups’
for division on a
number line using
apparatus alongside
the line
0
3
6
9
12
15
18
18  3 = 6
0
18
18  6 = 3
Becoming more efficient using a
numberline
Children need to be able to partition the
dividend in different ways.
Sharing – 49 shared between 4. How many left over? Grouping – How many 4s make 49. How many are left
over?
Place value counters can be used to support children apply their knowledge of grouping.
For example:
60 ÷ 10 = How many groups of 10 in 60?
600 ÷ 100 = How many groups of 100 in 600?
Sharing, Grouping and using a number line
Children will continue to explore division as sharing and grouping, and to represent calculations on a number line until they
have a secure understanding. Children should progress in their use of written division calculations:
• Using tables facts with which they are fluent
• Experiencing a logical progression in the numbers they use, for example:
1. Dividend just over 10x the divisor, e.g. 84 ÷ 7
2. Dividend just over 10x the divisor when the divisor is a teen number, e.g. 173 ÷ 15 (learning sensible
strategies for calculations such as 102 ÷ 17)
3. Dividend over 100x the divisor, e.g. 840 ÷ 7
4. Dividend over 20x the divisor, e.g. 168 ÷ 7
All of the above stages should include calculations with remainders as well as without. Remainders should be interpreted according
to the context. (i.e. rounded up or down to relate to the answer to the problem)
Formal Written Methods
Formal short division
should only be introduced
once children have a good
understanding of division,
its links with
multiplication and the
idea of ‘chunking up’ to
find a target number (see
use of number lines
above)
1 2 3
3
369
369 ÷ 3 =
KS2
Long division E.g. 2364 ÷ 15
KS2