Download Maths Workshop - St Michael`s C of E Primary School

Efficient written methods
for calculating addition,
subtraction, multiplication
and division by the end of
year 6.
Partition into tens and units and recombine
12 +15 + 21 =
12 = 10 + 2
15 = 10 + 5
21 = 20 + 1
Recombine
10 + 10 + 20 =40
2 + 5 + 1 = 8, 40+ 8 = 48
Partition into tens and units and recombine
12 +15 + 21 =
36 + 42 =
Use informal pencil and paper methods to
support, record or explain addition and
subtraction.
227 + 312 = ?
200
20
300
10
500 + 30 +
7
2
9
=
539
Use informal pencil and paper methods to
support, record or explain addition
36 + 42 =
227 + 312 =
227 + 315 =
Using a standard written method
227 + 315 =
Most significant digits first
227
+ 315
500
030
012
542
Least significant digits first
227
+ 315
012
030
500
542
Prepares for
carrying
+
227
315
542
1
Using a standard written method
227 + 315 =
4567 + 315 =
Using a standard written method; carrying
4567 + 315 =
4567
+ 315
4882
1
421.36 + 25.7 =
421.36
+
25.70
447.06
1
Using a standard written method; carrying
4567 + 315 =
421.36 + 25.7 =
Counting up from the smaller to larger
number (complementary addition)
84 – 56 =
56 + 4 + 20 + 4 = 84
+20
+4
84
-
+4
56
4 to 60
20 to 80
60
56
4 to 84
80
84
28
Begin to record calculations in
preparation for an efficient standard
method; decomposition
84 – 56 =
84
-
56
4 to 60
20 to 80
4 to 84
28
Apply
partitioning
skills
-
84 = 80 + 4 = 70 + 14
56
50 + 6 = 50 + 6
20 + 8 = 28
Counting up from the smaller to larger number
(complementary addition)
46 – 25 =
84 – 52 =
84 – 56 =
Begin to record calculations in preparation for an
efficient standard method; decomposition
46 – 25 =
84 – 52 =
84 – 56 =
Record calculations in preparation for
an efficient standard method;
decomposition
No decomposition
89 = 80 + 9
-
57
50 + 7
30 + 2 = 32
Decomposition
81 = 80 + 1 = 70 + 11
-
57
50 + 7 = 50 + 7
20 + 4 = 24
Record calculations in preparation for
an efficient standard method;
decomposition
89 – 57 =
84 – 57 =
284 – 57 =
Continue to develop an efficient
standard method; decomposition
-
754
= 700 + 50 + 4
286
200 + 80 + 6
leading to
1
= 700 + 40 + 14
744
6
- 286
6 14 14
= 600 +140 + 14
11
644
754
200 + 80 +
200 + 80 + 6
400 + 60 + 8
- 286 -286
468
468
Continue to develop an efficient
standard method; decomposition
754 – 286 =
5821 – 764 =
4567 – 893 =
Develop and use an efficient standard
written method; decomposition
5 13 16
3 16
6467
- 2684
3783
12467
-
84
12383
Develop and use an efficient standard
written method; decomposition
6467 – 2684 =
782 175 – 4387 =
324.9 – 7.25 =
Understand multiplication as repeated
addition and as an array.
(rows are always mentioned before columns)
3 rows of 5 = 15
5
3 x 5 = 15
5 + 5 + 5 = 15
5 rows of 3 = 15
5 x 3 = 15
3 + 3 + 3 + 3 + 3 = 15
3
Modelling the number properties
involving multiplication using an
array of objects not only allows
children to represent their thinking
with concrete materials, but it can
also assist the children to form
useful mental pictures to support
memory and reasoning.
Commutative property
The commutative property of multiplication can be neatly
illustrated using an array. For example, the array above
could be read as 2 rows of 6, or as 6 columns of 2. Or
the array could be physically turned around to show that
2 rows of 6 has the same number as 6 rows of 2.
Regardless of the way you look at it, there remain 12
objects. Therefore, the array illustrates that 2 x 6 = 6 x 2,
which is an example of the commutative property for
multiplication. Being able to apply the commutative
property means that the number of multiplication facts
that have to be memorised is halved.
Understand multiplication as repeated
addition and as an array.
4x3=
6x3=
5x6=
Use informal pencil and paper methods to
support, record or explain multiplication.
23 x 3
23 x 3 = (20 x 3) + (3 x 3)
Grid method
Times tables
x
Partitioning
Array
3
20
3
60
9
= 69
Use informal pencil and paper methods to
support, record or explain multiplication
12 x 9 =
26 x 3 =
35 x 16 =
Use informal pencil and paper methods to
support, record or explain multiplication
Grid method
346 x 9
x
9
300
40
2700 360
6
54
2700
=
360
+
54
3114
Continue to develop an efficient
standard written method
Short multiplication
346 x 9
346
x
9
300 x 9
2700
40 x 9
360
6x9
54
3114
leading to
346
9
x
3114
45
Use informal pencil and paper
methods to support, record or
explain multiplication
Grid method
346 x 9
Develop an efficient standard
written method
Short multiplication
543 x 7
Continue to develop an efficient
standard written method
Long multiplication
72
x
38
72 x 30
2160
72 x 8
576
2736
72 x 38
Continue to develop an efficient
standard written method
Long multiplication
72 x 38
Chunking! Division made easy!
Use informal pencil and paper methods to support, record
or explain divisions
42 ÷ 3 = ?
10 groups of 3
4 groups of 3
Chunks
–12
–-30
- 30
- 12
0
0
12
12
10 chunks + 4 chunks = 14
42
Use informal pencil and paper methods to support, record
or explain divisions
21 ÷ 3
42 ÷ 3
54 ÷ 6
Use informal pencil and paper methods to
support, record or explain division
Using multiples of the divisor (CHUNKING)
72
72 ÷ 5
-
50
10 x 5
22
-
20
4x5
2
Answer: 14 remainder 2
Use informal pencil and paper methods to
support, record or explain division
72 ÷ 3 =
72 ÷ 4 =
72 ÷ 5 =
Develop an efficient
standard written method
Short division 196 ÷ 6
32 R 4
6
) 196
18
(30 x 6 = 180)
16
12
4
(2 x 6 = 12)
(remainder)
Develop an efficient standard written method
Short division
196 ÷ 6
256 ÷ 7
Continue to develop an
efficient standard written
method
Short division (thousands)
2196 ÷ 6
366
6
) 2196
1800
(300 x 6 = 1800)
396
360
36
(60 x 6 = 360)
(6 x 6= 36)
Continue to develop an
efficient standard written
method
Short division (Bus Stop Method!)
366
6
233
) 2196
6 into 2 doesn’t go – carry the 2 over.
6 into 21 goes 3 remainder 3 – carry the 3 over.
6 into 39 goes 6 remainder 3 – carry the 3 over.
6 into 36 goes 6 exactly.
2196 divided by 6 = 366!
2196 ÷ 6
Continue to develop an efficient standard
written method
Short division (thousands)
2196 ÷ 6
4321 ÷ 6
8104 ÷ 6
Know what each digit represents and partition three-digit numbers into a
multiple of 100, a multiple of 10, and ones.
Understand multiplication as repeated addition and as an array.
Add three two-digit numbers using apparatus or informal methods.
Partition into tens and units and recombine.
Add or subtract a near multiple of 10 to a two-digit number, by adding or
subtracting the nearest multiple of 10, and adjusting.
Choose appropriate number operations and calculation methods to solve
word problems.
What is partitioning?
Know what each digit represents and
partition three-digit numbers into a multiple
of 100, a multiple of 10, and ones.
243
3
200
40
What is an array?
Understand multiplication as repeated
addition and as an array.
3 x 5 = 15
5 + 5 + 5 = 15
3
5
What does it mean by informal method?
Add three two-digit numbers using apparatus or
informal methods.
12 +15 + 21 =
+ 20
+7
41
21
48
10 + 10 + 20 =40
2 + 5 + 1 = 8, 40+ 8 = 48
Why is partitioning so useful?
Partition into tens and units and recombine.
12 +15 + 21 =
10 + 10 + 20 =40
2 + 5 + 1 = 8, 40+ 8 = 48
It’s one of
those informal
methods!
What does it mean by adjusting
Add or subtract a near multiple of 10 to a two-digit number, by
adding or subtracting the nearest multiple of 10, and adjusting.
39 + 34 = ?
40 + 34 = 74
Round up 39 to 40
40 + 30 = 70 Partitioning
70 + 4 = 74
74 – 1 = 73
Recombine
A clearly
written answer
To add/subtract by counting on or back in repeated steps of 1, 10, 100.
Use informal pencil and paper methods to support, record or explain
addition and subtraction.
Use informal pencil and paper methods to support, record or explain multiplication.
Use informal pencil and paper methods to support, record or explain divisions.
Choose appropriate number operations and calculation methods to solve
word problems.
What is counting on and back on a
number line?
To add/subtract by counting on or back in
repeated steps of 1, 10, 100.
227 + 300
+ 100
+ 100
337
227
+ 100
437
537
What does it mean by informal method?
Use informal pencil and paper methods to
support, record or explain addition and
subtraction.
227 + 312 = ?
200
20
300
10
500 + 30 +
7
2
9
=
539
What does it mean by informal method?
Use informal pencil and paper methods to
support, record or explain addition and
subtraction.
£5.20 - £3.60 = ?
Demonstrate the need to re-partition the numbers as:
£5.20 - £3.60
= £4 + 120p subtract £3 + 60p
= £4 - £3 and 120p – 60p
= £1 and 60p
= £1.60
What another informal method?
Use informal pencil and paper methods to
support, record or explain multiplication.
23 x 3
23 x 3 = (20 x 3) + (3 x 3)
Times tables
x
Partitioning
3
20
3
60
9
= 69
Division made easy!
Use informal pencil and paper methods to support, record
or explain divisions.
42 ÷ 3 = ?
10 groups of 3
4 groups of 3
Chunks
–12
–-30
- 30
- 12
0
0
12
12
4 chunks + 10 chunks = 14
42
Understand and use the principle of the associative law.
Extend written methods to short division of HTU by U
Choose appropriate number operations and calculation methods to solve
word problems.
What is associative law
and do the majority of children care?
Understand and use the principle of the
associative law.
Establish 5 x 16 is the same as 10 x 8 etc.
5 x 10 = 50
GRID!
5 x 6 = 30
= 80
Chunking!
Extend written methods to short division of HTU by U
183 ÷ 5 = ?
183 – 150 (30 x 5) = 33
33 – 30 (6 x 5) = 3
Answer 36 r 3
To add/subtract by counting on or back in repeated steps of 1, 10, 100.
Use informal pencil and paper methods to support, record or explain
addition and subtraction.
Use informal pencil and paper methods to support, record or explain multiplication.
Use informal pencil and paper methods to support, record or explain divisions.
Choose appropriate number operations and calculation methods to solve
word problems.