Transcript kellingtonCalculation_progression_for_parents_1

Kellington Primary School
Maths Parents’ Workshop
10th December 2012
Aims
To explain how we teach your children
+, -, x and ÷.
To give you ideas of how you can help
your children at home.
Addition
Laying the
foundations……
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Number lines
Practical equipment
Numicon
Multilink cubes
Real life contexts
Number bonds
Patterns
Partitioning……..
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Arrow cards
Place value
Partitioning
Recombining
Use of a number line
6+5=
1
2
3
4
5
6
7
8
9
10
11
12
13
Use of a 100 square
34+12=
Beginning to use column
addition, step 1…..
• Continue to use partitioning
• 364+ 34
=300+60+4+30+4
=300+90+8
• Then we recombine it all, to be left
with the answer, 398.
Using column addition,
step 2……
• Continue to use partitioning.
• 364+54=
364
+ 54
300
110
+ 8
418
Column addition….
• The final step, when the children
have a sound grasp of place value &
of the whole process…
364
+ 54
418
1
Subtraction
Subtraction
Taking away practically.
3-2=
Use of a number line/100
square
12-6=6
1
2
3
4
5
6
7
8
9
10
11
12
13
Written methods for
Subtraction
Stage 1: The empty number line
The empty number line helps to record the steps in
mental subtraction. There are several ways to do this:
• Counting Back - a calculation like 74 - 27 can be
recorded by counting back 27 from 74 to reach 47.
or
• 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 2: Partitioning
Subtraction can be recorded using
partitioning to write equivalent
calculations that are easier to carry out
mentally. For 74 - 27 this involves
partitioning the 27 into 20 and 7, then
subtracting 20 and 7 in turn.
74 – 27 is the same as 74 – 20 – 7
74 – 20 = 54
54 – 7 = 47
Written methods for
Subtraction
Stage 3: Expanded column method
The partitioning stage should be followed by the
expanded column method, where tens and units are
placed under each other. This is where the concept
of ‘borrowing’ is introduced
Example: 74 - 27
Written methods for
Subtraction
Stage 3: Expanded column method
It can also be applied to three and four digit
numbers.
Example: 741 - 367
Written methods for
Subtraction
Stage 3: Expanded column method
Depending on the numbers it can get quite
complicated and this stage may need a lot of
time and perseverance!
Written methods for
Subtraction
Stage 4: Column method
The expanded method is eventually reduced
to:
Multiplication
Multiplication- repeated
addition
3x5=
(3 groups of 5)
xx
xx
x
5
xx
xx
x
+ 5
xx
xx
x
+
5= 15
Times tables
• By end of Year 2 children should
know x2,x5,x10 Plus ?????
• Practise counting in 2’s, 3’s, 4’s, 5’s,
10’s
• Matching pairs (question on one card,
answer on another)
2013 By the end of year 4 – all times
tables?
Arrays
Children should be able to model a multiplication
calculation using an array.
This knowledge will support with the development of the
grid method.
3x5
5x3
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......
14  3  (10  4)  3
 (10  3)  (4  3)  30  12  42
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 left-hand 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
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. Some children should be able to use this expanded method
for TU × U by the end of Year 5.
30  8
 7
210
56
266
30  7  210
8  7  56
38
 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
350
42
1512
1
50  20  1000
6  20  120
50  7  350
6  7  42
56
 27
1120
392
1512
1
56  20
56  7
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.
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
38
 7
266
5
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 some children to
use this long multiplication
method for TU × TU by the end
of Year 6.
Written methods for
Multiplication
In Year 6, children apply the same steps to
multiply HTU x TU
286
 29
4000
1600
120
1800
720
54
8294
Start with the
grid method,
asking the
children to
estimate their
answer first.
200  20  4000
80  20  1600
6  20  120
200  9  1800
80  9  720
6  9  54
1
This expanded method
is cumbersome, so there
is plenty of incentive to
move on to a more
efficient method.
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.
Division
÷
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
Written methods for
Division
Stage 1: Mental division using
partitioning
One way to work out TU ÷ U mentally is
to partition TU into smaller multiples of
the divisor, then divide each part
separately.
Informal recording in Year 4 for 84 ÷ 7
might be:
In this example, using knowledge of multiples,
the 84 is partitioned into 70 (most children
will be secure with a multiple of 10) plus 14
Written methods for
Division
Stage 1: Mental division using partitioning
or……
and with a remainder
WDIK
10 x 4
20 x 4
30 x 4
etc
Written methods for
Division
Stage 2: Short division of TU ÷ U
'Short' division of TU ÷ U can be introduced
as a more compact recording of the mental
method of partitioning, to children who are
confident with multiplication and division
facts and whose understanding of partitioning
and place value is sound. For most children
this will be during Year 5.
Written methods for
Division
Stage 2: '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
Stage 3: 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.
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.
Thank you and Goodbye!