Transcript Progression In Calculation – EYFS to Year 6

Progression In Calculation – EYFS
to Year 6
Aims
• To have an overview of the skills children need
to calculate.
• To understand how to support your child with
maths.
• To be more aware of the models, images and
resources used to support the teaching and
learning of maths.
• Think about the progression from mental
towards written methods.
Session 1
Addition
Beginning to add
Practical, counting objects and relating addition
to combining two groups of objects
Beginning to use a number track
Use of the number track- hopping and recording.
(a)
2 and 3 makes 5
0 1 2 3 4 5 6 7 8 9 10
Mental Strategies for Addition
Secure mental addition requires the ability to:
•recall key number facts instantly (number pairs to 10, 20 & 100,
doubles etc) and to apply these to similar calculations
•recognise that addition can be done in any order and use this to
add mentally different combinations of one and two digit
numbers
• partition two-digit numbers in different ways, including adding
the tens and units separately before recombining
•understand the language of addition including more than, sum,
plus, greater than, total, altogether etc)
Written methods for Addition
Stage 1: The empty number line
The empty number line helps to record the steps on
the way to calculating the total. The steps often
bridge through a multiple of 10.
8 + 7 = 15
48 + 36 = 84
or:
Over to you!
Use a number line to find answers to these sums.
53 + 24
86 + 17
149 + 38
Written methods for Addition
Stage 2: Partitioning
The next stage is to record mental methods using
partitioning. Partitioning both numbers into tens and
ones mirrors the column method where ones are
placed under ones and tens under tens. This also links
to mental methods.
Eg: 47 + 76 = 47 + 70 + 6 = 117 + 6 = 123
or 47 + 76 = 40 + 70 + 7 + 6 = 110 + 13 = 123
Partitioned numbers are then written under one
another:
Over to you!
Use partitioning to find answers to these sums.
65 + 38
71 + 26
94 + 45
Written methods for Addition
Stage 3: Expanded method in columns
Children can now move on to a layout showing the
addition of the tens to the tens and the ones to the
ones separately. Children should start by adding the
ones digits first.
NB The addition of the tens in the
calculation 47 + 76 is described as
40 + 70 = 110 as opposed to
4 + 7 = 11.
Written methods for Addition
Stage 3: Expanded method in columns
The expanded method leads children to the more
compact method so that they understand its
structure and efficiency. The amount of time that
should be spent teaching and practising the
expanded method will depend on how secure the
children are in their recall of number facts and in
their understanding of place value.
Over to you!
Use the expanded column method to find answers to
these sums.
65 + 38
123 + 59
315 + 172
Written methods for Addition
Stage 4: Column method
In this method, recording is reduced further. Carry
digits are recorded below the line, using the words
'carry 10' or 'carry 100', not 'carry 1'.
Later, extend to adding three two-digit numbers,
two three-digit numbers, numbers with different
numbers of digits and decimals.
Session2
Subtraction
and take away
Introducing ‘take away’
Begins with practical demonstrations of subtraction relating to ‘take away’. Use of
number tracks, pictures and songs (10 green bottles, 5 little speckled frogs).
Beginning to take away
• Number tracks leading to number lines
introduced for recording ‘jumps’ back.
• 8-3=5
1
2
3
4
5
6
7
8
Mental Strategies for Subtraction
Secure mental subtraction requires the ability to:
• recall key subtraction facts instantly (inverse of number pairs
to 10, 20 & 100, halves etc) and to apply these to similar
calculations
• mentally subtract combinations of one and two digit
numbers
• understand that subtraction is the inverse of addition and
recognise that subtraction can’t be done in any order (it has to
start with the larger number)
• understand the language of subtraction including less,
minus, take away, difference between etc)
The problem with subtraction
Typical Questions
• Sam has saved 57p. Her sister has saved 83p
How much more money does Sam have than his sister?
• Samir is running a 50 metre potato race. He drops out after
18 metres
How much further does he have to go?
• Nisha and Charlie weigh fruit. Nisha’s weighs 38g. Charlies
weighs 50g.
How much heavier is Charlies fruit than Nishas?
• One sunflower is now 38cm high. Another is 83cm high.
What is the difference between the heights of the sunflowers?
Progression In Subtraction - Difference
Written methods for Subtraction
Stage 1: The empty number line
The empty number line helps to record the steps in mental
subtraction.
• Counting Up - the steps can also be recorded by counting up
from the smaller number to find the difference
or
Written methods for Subtraction
Stage 1: The empty number line
With practice, children will need to record less information and decide
whether to count back or forward. It is useful to ask children whether
counting up or back is the more efficient for calculations such as 57 - 12,
86 - 77 or 43 - 28.
With three-digit numbers the number of steps can again be reduced,
provided that children are able to work out answers to calculations such
as 178 + ? = 200 and 200 + ? = 326 mentally.
or
Over to you!
When would you use a numberline?
59 - 11
86 – 68
142 – 35
92-9
Written methods for Subtraction
Expanded column method
It can also be applied to three and four digit
numbers.
Example: 741 - 367
Written methods for Subtraction
Expanded column method
Depending on the numbers it can get quite
complicated and this stage may need a lot of time
and perseverance!
Over to you!
Use the expanded column method to find answers to
these sums.
73 - 39
123 - 58
315 - 177
Written methods for Subtraction
Stage 4: Column method
The expanded method is eventually reduced to:
Over to you!
Use the compact column method to find answers to
these sums.
83 - 58
166 - 47
402 - 175
Session 3
Multiplication
Beginning to multiply
When we begin to multiply we start by counting in steps of 2, 3 across a numberline.
Mental Strategies for Multiplication
To multiply successfully, children need to be able to:
• recall all multiplication facts to 10 × 10
• apply times tables facts to similar calculations such as
7x5
70 × 5, 70 × 50, 700 × 5 or 700 × 50 using their
knowledge of place value;
• partition numbers into multiples of Hundreds, Tens and Units
• add two or more single-digit numbers, multiples of 10 and 100
and combinations of whole numbers using the column method.
• understand the language of multiplication including lots of,
groups of, times, multiply, product
Written methods for Multiplication
Initially multiplication is introduced as ‘repeated
addition’ using vocabulary such as ‘lots of’ or ‘groups
of’ and real objects or pictures.
3 lots of 3 = 9
leading to
3x3=9
Solving Multiplication Calculations –
before written methods
Use of arrays to solve simple problems
••••
•••• 4x2=8
2x4=8
Use a numberline to multiply
5x6=30
0
5
10
15
20
25
30
Written methods for Multiplication
In KS2 the aim is that children develop rapid recall of
all times tables to 12 x 12 and can use an efficient
written method for
• two-digit by one-digit multiplication by the end of
Year 4 (TU x U)
•two-digit by two-digit multiplication by the end of
Year 5 (TU x TU)
•three-digit by two-digit multiplication by the end of
Year 6 (HTU x TU)
Written methods for Multiplication
Stage 1: Mental multiplication using
partitioning
This allows the tens and ones to be multiplied separately to form partial
products. These are then added to find the total product. Either the tens
or the ones can be multiplied first but it is more common to start with the
tens. This can look like......
1 4  3  (1 0  4 )  3
 (1 0  3 )  ( 4  3 )  3 0  1 2  4 2
Written methods for Multiplication
Stage 2: The Grid Method
This links directly to the mental method. It is an alternative way of recording
the same steps. It is better to place the number with the most digits in the lefthand column of the grid so that it is easier to add the partial products. For TU x
TU, the partial products in each row are added, and then the two sums at the
end of each row are added to find the total product
Written methods for Multiplication
The next step is to move the number being multiplied (38 in the example
shown) to an extra row at the top. Presenting the grid this way helps
children to set out the addition of the partial products in preparation for
the standard method.
Over to you!
Have a go at solving these multiplications using the
grid method.
65 x 8
74 x 45
92 x 53
Written methods for Multiplication
Stage 3: Expanded short multiplication
The next step is to represent the method in a column format, but showing the
working. Attention should be drawn to the links with the grid method above.
Children should describe what they do by referring to the actual values of the
digits in the columns. For example, the first step in 38 × 7 is ‘thirty multiplied by
seven’, not ‘three times seven’, although the relationship 3 × 7 should be
stressed. Most children should be able to use this expanded method for TU × U
by the end of Year 4.
30  8
 7
210
56
266
38

30  7  210
8  7  56
7
210
56
266
Written methods for Multiplication
Stage 3: Expanded short multiplication
The same steps can be used when introducing TU x TU.
56
27

1000
120
50  20  1000
6  20  120
350
50  7 
350
42
1512
67 
42
1
56
 27
1120
392
1512
1
56  20
56  7
Over to you!
Have a go at solving these multiplications using the
expanded short method.
32 x 8
56 x 15
78 x 37
Written methods for Multiplication
Stage 4: Short multiplication
The expanded method is eventually reduced to the standard method for
short multiplication. The recording is reduced further, with carry digits
recorded below the line. If, after practice, children cannot use the
compact method without making errors, they should return to the
expanded format of stage 3.
38
 7
266
5
The step here involves adding 210 and 50
mentally with only the 5 in the 50 recorded.
This highlights the need for children to be
able to add a multiple of 10 to a two-digit or
three-digit number mentally before they
reach this stage
Written methods for Multiplication
Stage 5: Long multiplication
This is applied to TU x TU as follows.
56
 27
1120
392
1512
1
56  20
56  7
The carry digits in the partial
products of 56 × 20 = 120 and
56 × 7 = 392 are usually carried
mentally.
The aim is for most children to use
this long multiplication method for
TU × TU by the end of Year 5.
Written methods for Multiplication
In Year 6, children apply the same steps to multiply
HTU x TU
286
 29
4000
1600
120
200  20  4000
80  20  1600
6  20  120
1800
720
200  9  1800
80  9  720
54
8294
6 9 
1
Start with the grid
method, asking the
children to estimate
their answer first.
This expanded method
is cumbersome, so there
is plenty of incentive to
move on to a more
efficient method.
54
286
 29
5720
2574
8294
286  20
286  9
1
Children who are already
secure with multiplication
for TU × U and TU × TU
should have little difficulty
in using the same method
for HTU × TU.
Session 4
Division
Last but not least……
• Many children can partition and multiply with confidence.
But this is not the case for division. One reason for this
may be that mental methods of division, stressing the
correspondence to mental methods of multiplication,
have not in the past been given enough attention.
• The aim is that children use mental methods when
appropriate, but for calculations that they cannot do in
their heads they use an efficient written method
accurately and with confidence.
• The stages building up to long division span Years 4 to 6 first introducing TU ÷ U, then extending to HTU ÷ U, and
finally HTU ÷ TU.
Mental Strategies for Division
To divide successfully, children need to be able to:
• partition two-digit and three-digit numbers into multiples
of 100, 10 and 1
• recall multiplication and division facts to 10 × 10 and
recognise multiples of one-digit numbers
• know how to find a remainder working mentally - for
example, find the remainder when 48 is divided by 5;
• understand and use multiplication and division as inverse
operations.
• understand and use the vocabulary of division - for
example in 18 ÷ 3 = 6,the 18 is the dividend, the 3 is the
divisor and the 6 is the quotient;
Written methods for Division
Initially division is introduced as ‘sharing’ using real
objects or pictures.
Share 10 apples equally between 2 children
which eventually becomes 10 ÷ 2 = 5
Beginning to divide
Solve problems using repeated addition along a number line.
+5
0
+5
5
+5
10
15
+5
+5
+5
20
25
30
30 ÷ 5 = 6
Finding remainders after simple division
Know when to round the remainder up or down, depending on context of problem
+4
+4
+4
rem 2
14 ÷ 4 = 3 r 2
0
4
8
12 14
Written methods for Division
To carry out written methods of division successful, children
also need to be able to:
• understand division as repeated subtraction;
• estimate how many times one number divides into
another - for example, how many 6s there are in 47,
or how many 23s there are in 92;
• multiply a two-digit number by a single-digit
number mentally;
• subtract numbers using the column method.
Written methods for Division
Stage 3: 'Expanded' method for TU
÷ U and HTU ÷ U
This method, often referred to as
'chunking', is based on subtracting
multiples of the divisor, or 'chunks'.
It is useful for reminding children of
the link between division and
repeated subtraction. However,
children need to recognise that
chunking is inefficient if too many
subtractions have to be carried out.
Written methods for Division
Refining the 'Expanded' method for
HTU ÷ U
Initially children subtract several
chunks, but with practice they should
look for the biggest multiples that they
can find to subtract, to reduce the
number of steps.
Once they understand and can apply
the expanded method, children should
try the standard method for short
division. For most children this will be
at the end of Year 5 or the beginning of
Year 6
Over to you!
Have a go at solving these divisions using the short
method for TU ÷ U.
76 ÷ 8
91 ÷ 7
92 ÷ 4
Written methods for Division
Stage 2: Short division of TU ÷ U
For 81 ÷ 3, the dividend of 81 is split into 60, the highest multiple of 3 that is also a multiple 10
and less than 81, to give 60 + 21.Each number is then divided by 3.
leading to
Written methods for Division
Stage 4: Long division for HTU ÷ TU
The next step is to tackle HTU ÷ TU, which for most children will be in
Year 6. The layout on the right, which links to chunking, is in essence the
'long division' method. Conventionally the 20, or 2 tens, and the 3 ones
forming the answer are recorded above the line, as in the second
recording.
Over to you!
Have a go at solving these divisions using ‘chunking’.
86 ÷ 7
156 ÷ 5
178 ÷ 8