Download NORTHBROOK PRIMARY SCHOOL

June 2014
PROGRESSION THROUGH CALCULATIONS FOR MULTIPLICATION
Policy
Stage 1
Mental Multiplication using counting in steps and using number facts
Children will experience equal groups of objects and will count in 2s and 10s and begin to count in 5s.
Mental Multiplication using doubling and halving
Children to be able to find doubles from 1 to 10 using their fingers.
Mental Multiplication using grouping
They will work on practical problem solving activities involving equal sets or groups.
Stage 2
Mental Multiplication using counting in steps and using number facts
Children are able to count in 2s, 10s and 5s and will begin to count in 3s.
e.g. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24
This will progress to learning 2, 3, 10 and 5 times tables (up to 12x) and linking multiplication facts to
counting in steps.
E.g. 5 x 10 = 50 or 5 steps of 10 = 10, 20, 30, 40, 50
Children should know their doubles to double 20.
1
E.g. double 8 is 16.
Mental Multiplication using doubling and halving
Children will begin to know doubles of multiples of 5 to 100.
E.g. double 35 is 70.
Children will begin to double 2 digit numbers less than 50 with one's digits of 1, 2,3,4,5.
Mental Multiplication using grouping
Children will develop their understanding of multiplication and use jottings to support calculation:
Children know that multiplication is repeated addition.
3 times 5 is 5 + 5 + 5 = 15 or 3 lots of 5 or 5 x 3
Repeated addition can be shown easily on a number line:
E.g. 5 x 3 = 5 + 5 + 5
5
0
1
2
5
3
4
5
6
7
5
8
9
10 11 12 13 14 15
Number lines show repeated groupings (repeated addition of sets of the same size).
And on a bead bar:
5x3=5+5+5
5
5
5

Children develop their understanding of multiplication as repeated grouping (repeated addition of sets of the
same size) using practical apparatus and diagrams.
These may also be viewed as 3 x 4 (i.e. 3 lots of 4)
2
They will use visual arrays or sets of objects to find "lots of".
e.g. 3 lots of 4
3 x 4 = 12
4 x 3 = 12
Children should be able to model a multiplication calculation using an array. This knowledge will support with
the development of the grid method.
5 x 3 = 15
3 x 5 = 15
Stage 3
Mental Multiplication using counting in steps and using number facts
Children must be able to count in 2s, 3s, 4s, 5s, 8s and 10s.
Children must be able to recall the 2, 3, 4 ,5 , 8 and 10 times tables up to 12x. They must recall multiplication
and division facts for these times tables at speed. Children will be able to solve inverse operation problems
for these times tables.
e.g. □ x 5 = 20
Mental Multiplication using doubling and halving
Children to be able to find doubles of numbers to 50 using partitioning.
E.g. double 32 = double 30 + double 2 = 60 + 4 = 64
Children must recognise doubling as a strategy for multiplying by 2.
E.g. 24 x 2 is double 24 = 48
Children must know their doubles to double 20.
E.g. double 16 is 32
Know doubles of multiples of 5 to 100.
E.g. double 75 is 150
Children will also develop an understanding of scaling.
e.g. Find a ribbon that is 4 times as long as the blue ribbon.
5 cm
3
20 cm
Mental Multiplication using grouping
Children should know that 3 x 5 has the same answer as 5 x 3. This is known as commutativity and can also
be shown on the number line.
5
0
1
2
3
5
3
4
5
6
7
5
8
9
10 11 12 13 14 15
3
3
3
3
Arrays and other practical apparatus can be used to illustrate commutativity (that multiplication calculations
can be carried out in any order) e.g. 2 x 5 arrives at the same product as 5 x 2.
Children will continue to use repeated addition for multiplication.
4 times 6 is 6 + 6 + 6 + 6 = 24 or 4 lots of 6 or 6 x 4
Children should use number lines or bead bars to support their understanding.
6
0
6
6
6
6
12
6
6
18
6
24
6
Note: The beads are grouped in lots of 6. The red and white beads are also organised into lots of 10. Th
Children should be able to model a multiplication calculation using an array. This knowledge will support with
the development of the grid method.
4 x 9 = 36
9 x 4 = 36
Once secure, children will begin to derive new facts from known facts
e.g. 3 x 2 = 6 (known fact)
30 x 2 = 60
300 x 2 = 600 etc.
4
Children must be able to multiply multiples of 10 by 1 digit numbers mentally.
e.g.: 30 x 8 = 240 using known fact 3 x 8 is 24
They must also be able to mentally multiply a 2 digit number by a 1 digit number.
e.g. 12 x 4 = 48
Written Multiplication using partitioning
Children must be able to multiply whole numbers by 10 and 100.
E.g. 4 x 10 = 40
E.g. 4 x 100 = 400
Children will begin to use understanding of place value and partitioning to carry out multiplication of two or
three digit by friendly one -digit numbers. A "friendly" one digit number will be a times table they are
familiar with for their year group.
15 x 4
10 5
10 x 4 = 40
5 x 4 = 20
40 + 20 = 60
It will then be presented in a grid format in preparation for grid multiplication.
x
4
10
40
5
20
Grid method of partitioning will be supported using practical apparatus.
E.g. 15 x 4 = 20
5
Stage 4
Mental Multiplication using number facts
Children must be able to count in multiples of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 25, 50, 100 and 1000's.
Children to know their times table up to 12 x 12.
Children should be able to recognise factors up to 12 of 2 digit numbers.
E.g. factors of 24 are 3, 2, 4, 6, 8 and 12
Mental Multiplication using doubling and halving
Children to be able to find doubles to double 100 and beyond using partitioning.
E.g. double 136 = double 100 + double 30 + double 6 = 200 + 60 + 12 = 272
They must also be able to double amounts of money.
E.g. £2.50 doubled is £5
Children must recognise doubling as a strategy for multiplying by 2, 4 and 8.
E.g. 26 x 4 is double 26 (52) doubled again = 104
Use understanding of place value and number facts in mental multiplication.
E.g. 36 x 5 is half of 36 x 10
Mental Multiplication using grouping
Use partitioning to multiply 2 digit numbers by 1 digit number.
E.g. 24 x 5 = 20 x 5 plus 4 x 5 = 100 + 20 = 120
Multiply multiples of 10, 100 and 1000 by 1 digit numbers using times tables facts.
E.g. 400 x 8 can be solved using times table fact 4 x 8 = 32 and is re written as
4 x 8 x 100 = 32 x 100 = 3200
Multiply near multiples using rounding - children to round up or down to the nearest multiple of 10 and
compensate appropriately.
E.g. 24 x 19 round to 24 x 20 less 24 = 456
Written Multiplication
Children to use grid method for the multiplication of up to a 2 digit numbers by 3 digit number.
When the multiplication is by a single digit number, it is known as short multiplication.
e.g. 2 digit number by 1 digit number: TU x U
Children will approximate first
E.g. 23 x 8 is approximately 25 x 8 = 200
x
8
20
160
3
24
e.g. 3 digit number by 1 digit number: HTU x U
6
160
+ 24
184
Children should approximate first
E.g. 346 x 9 is approximately 350 x 10 = 3500
x
9
300
2700
40
360
6
54
2700
+ 360
+ 54
31 1 4
1 1
Some children will use grid multiplication to multiply 2 digit numbers by 2 digit numbers. This is known as long
multiplication TU x TU. It is multiplication by more than a single digit.
Children should approximate first
E.g. 48 x 92 is approximately 50 x 90 = 4500
x
90
2
40
3600
80
8
720
16
3600
+
720
+
80
+
16
44 1 6
1 1
Children will develop the grid method into a vertical written method, initially multiplying a 2 digit number by
a one digit number. Initially children will use expanded recording in columns and then move to formal written
method, using practical apparatus to support as required.
x
T
1
2
4
6
U
leading to
5
4
0 (5 x 4)
0 (10 x 4)
0
x
T
1
6
2
U
5
4
0
Children will extend written approaches to HTU x U, then to ThHTU x U.
Partitioning approaches and grid method will be used to support this method as required.
7
Partitioning support
215 x 4
200 10
5
200 x 4 = 800
10 x 4 = 40
5 x 4 = 20
800 + 40 + 20 = 860
Grid method support
Once secure, children will develop expanded recording in columns and then move to formal written method,
using practical apparatus to support as required.
H
2
T
1
x
8
8
2
4
0
6
U
5
4
0 ( 5 x 4)
0 ( 10 x 4)
0 ( 200 x 4)
H
2
T
1
8
6
2
x
U
5
4
0
Stage 5
Mental Multiplication using number facts
Children to use their times tables facts up to 12 x 12 to times multiples of 10 and 100 of the multiplier.
E.g. 4 x 6 = 24 therefore 40 x 6 = 240 and 400 x 6 = 2400
Children will use their knowledge of factors and multiples in mental multiplication problems.
8
E.g. 43 x 6 is double 43 x 3
E.g. 28 x 5 is half 28 x 10
Children to know square numbers and cube numbers.
Mental Multiplication using doubling and halving
Children to be able to double amounts of money using partitioning.
In Katherine Semar we might teach some children to remove the decimal point, do the mental calculation and
then return the decimal afterwards to avoid place value mistakes.
E.g. double £6.73
Children being taught to remove the decimal will understand they have multiplied 6.73 by 100.
Children double £673 is double £600 + double £70 plus double £3 = £1200 + £140 + £6 = £1346
Children who multiplied out the decimal point must remember to divide their answer by 100 to replace the
decimal correctly. That is £1346 / 100 = £13.46
Those children not multiplying out the decimal point will solve by partitioning the decimal number as
double £6 + double 70p + double 3p which is £12 + £1.40 + 6p = £13.46
Children to understand that doubling and halving is a strategy for multiplying by 2, 4, 8, 5 and 10.
E.g. 48 x 5 is half of 48 x 10 (480) = 240
Mental Multiplication using grouping
To be able to multiply whole numbers and decimals by 10, 100, 1000 and 10 000.
E.g. 3.4 x 100 = 340
Use partitioning to multiply 2 and 3 digit numbers by 1 digit numbers mentally.
E.g. 316 x 6 as 300 x 6 (1800) plus 10 x 6 (60) plus 6 x 6 (36) = 1896
Use partitioning to multiply decimal numbers by 1 digit numbers.
E.g. 4.3 x 5 as 4 x 5 (20) plus 0.3 x 5 (1.5) = 21.5
Multiply near multiples by rounding
E.g. 32 x 29 as 32 x 30 less 32 = 928
Written Multiplication
If the multiplier is a 1 digit number, the written multiplication is short multiplication.
Children to be able to complete short multiplication of 2, 3 and 4 digit numbers by 1 digit number.
E.g. 4346 x 8 is approximately 4346 x 10 = 43460
Expanded method
(6 x 8)
9
i)
x
4346
8
48
Compact method
leading to
4346
x
8
8
4
ii) 4 3 4 6
x
8
(6 x 8)
48
(40 x 8)
320
leading to
4346
x
8
68
3
iii) 4 3 4 6
x
8
(6 x 8)
48
(40 x 8)
320
(300 x 8)
2400
leading to
4346
x
8
768
2
iv) 4 3 4 6
x
8
(6 x 8)
48
(40 x 8)
320
(300 x 8)
2400
(400 x 8)
32000
34768
leading to
4
3
4
4346
x
8
34 768
2
3
4
When the multiplier is a 2 or more digit number, the written multiplication method is long multiplication.
This method is used for long multiplication of 2, 3 and 4 digit numbers by teen numbers.
E.g. 315 x 18 estimate to 300 x 20 = 600
3 1 5
x
1 8
2 5 2 0 from 5 x 8 = 40, 10 x 8 = 80 (plus exchanged 40), 300 x 8 = 24 (plus exchanged 100)
1
4
3 1 5 0 from adding a 0 to hold the place, 1 x 5 = 5, 1 x 10 = 10 and 1 x 300 = 300
56 70
Children continue to use grid multiplication to multiply numbers with up to 2 decimal places by 1 digit
numbers.
In Katherine Semar we sometimes teach the children to multiply the decimal number by 10, 100 or 1000 to
remove the decimal and divide the answer by the same amount to return the decimal to the answer.
E.g. £3.46 x 9 when multiplied by 100 becomes 346 x 9
x
9
10
300
2700
40
360
6
54
2700
+ 360
+ 54
31 1 4
The answer 3114 must be divided by 100 to give the correct answer to £3.46 x9 = £31.14
Children should be able to find simple % of amounts i.e. 10%, 5%, 20%, 15%, 50%
Children must be able to multiply fractions by 1 digit numbers.
E.g. 3/4 x 6 = 18/4 = 4 2/4 = 4 1/2
Stage 6
Mental Multiplication using number facts
Use times tables facts up to 12 x 12 in mental multiplication of large numbers.
Use times tables facts up to 12 x 12 in mental multiplication of numbers with up to 2 decimal places.
E.g. 6 x 4 = 24 and 0.06 x 4 = 0.24
Mental Multiplication using doubling and halving
Children must be able to use doubles and half as a mental multiplication strategy to multiply by 2, 4, 8, 5, 20,
50 and 25.
E.g. 28 x 25 is 1/4 of 28 x 100 which is 700
Children must be able to double decimal numbers up to 2 decimal places using partitioning.
E.g. double £36.73
Some children will partition the decimal number and double it.
Double £36.73 by £36 x 2 = £72 plus 73p x 2 = 146p to total £73.46 altogether.
Some children will multiply out the decimal point, double the whole number and then divide their answer to
return the decimal point.
Double £36.73. First multiple by 100 to become 3673p. Then partition the number and double.
Double 3673p becomes double 3000 + double 600 + double 70 + double 3 which totals 7343p
Mental Multiplication using grouping
Children must be able to multiply whole numbers and decimals up to 3 decimal places by 10, 100, 1000
E.g. 234 x 1000 = 234 000
E.g. 0.23 x 1000 = 230
Use partitioning for mental multiplication.
E.g. 3060 x 4 as 3000 x 4 (12000) + 60 x 4 (240) = 12 240
Use factors for mental multiplication
E.g. 421 x 6 calculates as double 421 x 3 x 2 = 1263 x 2 = 2526
Multiply decimal numbers using near multiples by rounding
E.g. 4.3 x 19 as 4.3 x 20 less 4.3 = 81.7
Children must be able to identify common factors, common multiples and prime numbers and use these in
mental multiplications.
Written Multiplication
Children progress to short multiplication of 2, 3 and 4 digit numbers by a 1 digit number.
11
Children progress to long multiplication of 2, 3 and 4 digit numbers by 2 digit numbers.
E.g. 456 x 38 estimate to 500 x 20 = 10 000
4 5 6
x
3 8
36 48
4
4
1 36 80
1
1
1 73 28
1
1
Children are able to use short multiplication of decimal numbers and money - multiplying the decimal by 10,
100 or 1000 to remove the decimal point and then dividing the answer by 10, 100 or 1000 to return the
decimal place.
E.g. 13.72 x 6 becomes (1372 x 6) divided by 100 = 82.32
Children may continue to use grid multiplication to multiply numbers with up to 2 decimal places by 1 digit
numbers or they can use short multiplication, keeping the decimal place in line.
E.g. £13.72 x 6 = £82.32
1 3. 7 2
x
6
8 2. 3 2
2 4
1
Those using grid multiplication of numbers with up to 2 decimal places by a 1 digit number may multiply the
decimal by 10, 100 or 1000 to remove the decimal point and then divide the answer by 10, 100 or 1000 to
return the decimal place. E.g. 6.76 x 4
x
4
600
2400
70
280
6
24
2400
+ 280
+ 24
2704
The answer of 2704 must be divided by 100 to give the correct answer to the question.
6.76 x 4 = 27.04
Children are able to use % for comparisons and are able to calculate simple %.
E.g. 16% of 240 would be calculated as 10% + 5% + 1% = 24 + 12+ 2.4 = 38.4
Multiply simple pairs of fractions
E.g. 1/2 x 1/4 = 1/8
Multiply fractions and mixed numbers by whole numbers.
12