Efficient written methods for calculating addition, subtraction, multiplication and division by the end of year 6

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 = ?

200300500

201030+

729+ = 539

36 + 42 =

Use informal pencil and paper methods to support, record or explain addition

227 + 312 = 227 + 315 =

227 + 315 =

227315500030012542

Using a standard written method

+227315542

+

1

Most significant digits first Least significant digits first

+

Prepares for carrying

227315012030500542

227 + 315 =

Using a standard written method

4567 + 315 =

Using a standard written method; carrying

4567 3154882

+

1

4567 + 315 = 421.36 + 25.7 =

+

1

421.36

25.70

447.06

Using a standard written method; carrying

4567 + 315 = 421.36 + 25.7 =

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

-

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

84 – 56 =84 – 52 =46 – 25 =

Counting up from the smaller to larger number (complementary addition)

84 – 56 =84 – 52 =46 – 25 =

Begin to record calculations in preparation for an efficient standard method; decomposition

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

89 – 57 = 84 – 57 = 284 – 57 =

Record calculations in preparation for an efficient standard method; decomposition

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

754 – 286 =5821 – 764 = 4567 – 893 =

Continue to develop an efficient standard method; decomposition

Develop and use an efficient standard written method; decomposition

6467

2684

3783

-

5 1613

12467

84

12383

163

-

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.

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)

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.

4 x 3 = 6 x 3 =


5 x 6 =

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

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

2700 360 54x9

300 40 6

= 2700

360

54

3114

+

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

346 x 9


Grid method Short multiplication

543 x 7

Develop an efficient standard written method


Long multiplication 72 x 38

72

38

2160

576

2736

x

72 x 30

72 x 8

72 x 38


Long multiplication

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

42 ÷ 321 ÷ 3 54 ÷ 6

Use informal pencil and paper methods to support, record or explain divisions

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

-

-

÷

72 ÷ 3 = 72 ÷ 5 =72 ÷ 4 =

Use informal pencil and paper methods to support, record or explain division


Short division 196 ÷ 6

)6

32 R 4

196

18 (30 x 6 = 180)

16

12 (2 x 6 = 12)

4 (remainder)

196 ÷ 6


Short division

256 ÷ 7


Short division (thousands) 2196 ÷ 6

)6

366

2196

1800 (300 x 6 = 1800)

396

360 (60 x 6 = 360)

36 (6 x 6= 36)


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!

2196 ÷ 6


Short division (thousands)

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.


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.

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?

What is an array?


3 x 5 = 15

5 + 5 + 5 = 15 3

5

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

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!

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



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 divisions.

227 + 300


What is counting on and back on a number line?

227

+ 100

337537

+ 100 + 100

437



227 + 312 = ?

200300500

201030+

729+ = 539



£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?

23 x 3 23 x 3 = (20 x 3) + (3 x 3)

60 9= 69

20 3

3

xTimes tables

Partitioning


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


Understand and use the principle of the associative law.

Extend written methods to short division of HTU by U

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

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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Efficient written methods for calculating addition, subtraction, multiplication and division by the end of year 6