efficient written methods for calculating addition, subtraction, multiplication and division by the...
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Efficient written methods for calculating addition, subtraction, multiplication and division by the end of year 6.
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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
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Partition into tens and units and recombine
12 +15 + 21 = 36 + 42 =
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Use informal pencil and paper methods to support, record or explain addition and subtraction.
227 + 312 = ?
200300500
201030+
729+ = 539
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36 + 42 =
Use informal pencil and paper methods to support, record or explain addition
227 + 312 = 227 + 315 =
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227 + 315 =
227315500030012542
Using a standard written method
+227315542
+
1
Most significant digits first Least significant digits first
+
Prepares for carrying
227315012030500542
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227 + 315 =
Using a standard written method
4567 + 315 =
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Using a standard written method; carrying
4567 3154882
+
1
4567 + 315 = 421.36 + 25.7 =
+
1
421.36
25.70
447.06
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Using a standard written method; carrying
4567 + 315 = 421.36 + 25.7 =
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Counting up from the smaller to larger number (complementary addition)
56
+ 4 +20
60 80
+ 4
84
84 – 56 = 56 + 4 + 20 + 4 = 84 84
56
4 to 60
20 to 80
4 to 84
28
-
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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
-
Begin to record calculations in preparation for an efficient standard method; decomposition
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84 – 56 =84 – 52 =46 – 25 =
Counting up from the smaller to larger number (complementary addition)
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84 – 56 =84 – 52 =46 – 25 =
Begin to record calculations in preparation for an efficient standard method; decomposition
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Record calculations in preparation for an efficient standard method; decomposition
89 = 80 + 9
57 50 + 7
30 + 2 = 32
- -81 = 80 + 1 = 70 + 11
57 50 + 7 = 50 + 7
20 + 4 = 24
No decompositionDecomposition
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89 – 57 = 84 – 57 = 284 – 57 =
Record calculations in preparation for an efficient standard method; decomposition
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Continue to develop an efficient standard method; decomposition
754 = 700 + 50 + 4 leading to
286 200 + 80 + 6
= 700 + 40 + 14 744
200 + 80 + 6 286
= 600 +140 + 14 644 754
200 + 80 + 6 286 286
400 + 60 + 8 468 468
-
-
- -
1
116 14 14
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754 – 286 =5821 – 764 = 4567 – 893 =
Continue to develop an efficient standard method; decomposition
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Develop and use an efficient standard written method; decomposition
6467
2684
3783
-
5 1613
12467
84
12383
163
-
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Develop and use an efficient standard written method; decomposition
6467 – 2684 = 782 175 – 4387 = 324.9 – 7.25 =
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Understand multiplication as repeated addition and as an array.
3
53 rows of 5 = 15
3 x 5 = 15
5 + 5 + 5 = 15
5 rows of 3 = 15
5 x 3 = 15
3 + 3 + 3 + 3 + 3 = 15
(rows are always mentioned before columns)
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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.
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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.
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4 x 3 = 6 x 3 =
Understand multiplication as repeated addition and as an array.
5 x 6 =
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Use informal pencil and paper methods to support, record or explain multiplication.
23 x 3 23 x 3 = (20 x 3) + (3 x 3)
60 9= 69
20 3
3
xTimes tables
Partitioning
Array
Grid method
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12 x 9 = 26 x 3 = 35 x 16 =
Use informal pencil and paper methods to support, record or explain multiplication
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Use informal pencil and paper methods to support, record or explain multiplication
Grid method
346 x 9
2700 360 54x9
300 40 6
= 2700
360
54
3114
+
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Continue to develop an efficient standard written method
Short multiplication 346 x 9
leading to
346
9
2700
360
54
3114
300 x 9
40 x 9
6 x 9
x
346 93114
x
4 5
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346 x 9
Use informal pencil and paper methods to support, record or explain multiplication
Grid method Short multiplication
543 x 7
Develop an efficient standard written method
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Continue to develop an efficient standard written method
Long multiplication 72 x 38
72
38
2160
576
2736
x
72 x 30
72 x 8
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72 x 38
Continue to develop an efficient standard written method
Long multiplication
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Use informal pencil and paper methods to support, record or explain divisions
Chunking! Division made easy!
42 ÷ 3 = ?
–12 –-30
12 0 42
4 groups of 3 10 groups of 3
0- 30- 12
12
Chunks
10 chunks + 4 chunks = 14
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42 ÷ 321 ÷ 3 54 ÷ 6
Use informal pencil and paper methods to support, record or explain divisions
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Use informal pencil and paper methods to support, record or explain division
Using multiples of the divisor (CHUNKING)
72 5 72
50 10 x 5
22
20 4 x 5
2
Answer: 14 remainder 2
-
-
÷
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72 ÷ 3 = 72 ÷ 5 =72 ÷ 4 =
Use informal pencil and paper methods to support, record or explain division
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Develop an efficient standard written method
Short division 196 ÷ 6
)6
32 R 4
196
18 (30 x 6 = 180)
16
12 (2 x 6 = 12)
4 (remainder)
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196 ÷ 6
Develop an efficient standard written method
Short division
256 ÷ 7
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Continue to develop an efficient standard written method
Short division (thousands) 2196 ÷ 6
)6
366
2196
1800 (300 x 6 = 1800)
396
360 (60 x 6 = 360)
36 (6 x 6= 36)
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Continue to develop an efficient standard written method
Short division (Bus Stop Method!) 2196 ÷ 6
)6
366
21963 32
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!
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2196 ÷ 6
Continue to develop an efficient standard written method
Short division (thousands)
4321 ÷ 6 8104 ÷ 6
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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.
Partition into tens and units and recombine.
Add three two-digit numbers using apparatus or informal methods.
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.
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Know what each digit represents and partition three-digit numbers into a multiple of 100, a multiple of 10, and ones.
243
200
40
3
What is partitioning?
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What is an array?
Understand multiplication as repeated addition and as an array.
3 x 5 = 15
5 + 5 + 5 = 15 3
5
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Add three two-digit numbers using apparatus or informal methods.
What does it mean by informal method?
12 +15 + 21 =
21
+ 20 +7
41
48
10 + 10 + 20 =40
2 + 5 + 1 = 8, 40+ 8 = 48
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Partition into tens and units and recombine.
Why is partitioning so useful?
10 + 10 + 20 =40
2 + 5 + 1 = 8, 40+ 8 = 48
12 +15 + 21 =It’s one of
those informal methods!
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Add or subtract a near multiple of 10 to a two-digit number, by adding or subtracting the nearest multiple of 10, and adjusting.
What does it mean by adjusting
39 + 34 = ?
40 + 34 = 74
74 – 1 = 73
Round up 39 to 40
40 + 30 = 70
70 + 4 = 74
Partitioning
Recombine
A clearly written answer
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Choose appropriate number operations and calculation methods to solve word problems.
Use informal pencil and paper methods to support, record or explain addition and subtraction.
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 multiplication.
Use informal pencil and paper methods to support, record or explain divisions.
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227 + 300
To add/subtract by counting on or back in repeated steps of 1, 10, 100.
What is counting on and back on a number line?
227
+ 100
337537
+ 100 + 100
437
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Use informal pencil and paper methods to support, record or explain addition and subtraction.
What does it mean by informal method?
227 + 312 = ?
200300500
201030+
729+ = 539
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Use informal pencil and paper methods to support, record or explain addition and subtraction.
What does it mean by informal method?
£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
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Use informal pencil and paper methods to support, record or explain multiplication.
What another informal method?
23 x 3 23 x 3 = (20 x 3) + (3 x 3)
60 9= 69
20 3
3
xTimes tables
Partitioning
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Use informal pencil and paper methods to support, record or explain divisions.
Division made easy!
42 ÷ 3 = ?
–12 –-30
12 0 42
4 groups of 3 10 groups of 3
0- 30- 12
12
Chunks
4 chunks + 10 chunks = 14
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Choose appropriate number operations and calculation methods to solve word problems.
Understand and use the principle of the associative law.
Extend written methods to short division of HTU by U
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Establish 5 x 16 is the same as 10 x 8 etc.
Understand and use the principle of the associative law.
What is associative law and do the majority of children care?
GRID!5 x 10 = 50
5 x 6 = 30
= 80
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Extend written methods to short division of HTU by U
Chunking!
183 ÷ 5 = ?183 – 150 (30 x 5) = 3333 – 30 (6 x 5) = 3Answer 36 r 3
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Choose appropriate number operations and calculation methods to solve word problems.
Use informal pencil and paper methods to support, record or explain addition and subtraction.
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 multiplication.
Use informal pencil and paper methods to support, record or explain divisions.