PROGRESSION THROUGH CALCULATIONS
FOR
MULTIPLICATION AND DIVISION
MENTAL CALCULATIONS (ongoing)
This is a selection of mental calculation strategies:
Doubling and halving
Applying the knowledge of doubles and halves to known facts.
e.g. 8 x 4 is double 4 x 4
Using multiplication facts
Tables should be taught every day from Y2 onwards, either as part of the mental oral
starter or other times as appropriate within the day. This should include ‘how/what do
we know?’ questions:
How do we know a number is in the five times table (is a multiple of 5 in KS2)
What can we say if at least one of the numbers (multiples) is even / if all numbers are odd.
How can we use X2 to find X4
Children should be encouraged to learn tables at home using the three-week-method.
Learning tables should not be seen as a race. A table should be fully embedded before
moving on to the next. A table should not be considered learnt until children can give
related division facts.
Equipment and images should be used extensively in KS1. It can also be valuable in KS2,
especially when introducing new concepts, such as working with groups of decimals.
Once table stars have been achieved
Using and applying division facts
Children should be able to utilise their tables knowledge to derive other facts.
e.g. If I know 3 x 7 = 21, what else do I know?
30 x 7 = 210, 300 x 7 = 2100, 3000 x 7 = 21 000, 0.3 x 7 = 2.1 etc
210 ÷ 7 = 30
2.1 ÷ 7 = 0.3
Use closely related facts already known
13 x 11 = (13 x 10) + (13 x 1)
= 130 + 13
= 143
Multiplying and dividing by 10 or 100
Knowing that the effect of multiplying by 10 is a shift in the digits one place value to the
left.
Knowing that the effect of multiplying by 100 is a shift in the digits two place values to the
left.
Knowing that the effect of dividing by 10 is a shift in the digits one place value to the right.
Knowing that the effect of dividing by 100 is a shift in the digits two place values to the
right.
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Partitioning
23 x 4 = (20 x 4) + (3 x 4)
= 80 + 12
= 102
Use of factors
8 x 12 = 8 x 4 x 3
MANY MENTAL CALCULATION STRATEGIES WILL CONTINUE TO BE USED. THEY ARE
NOT REPLACED BY WRITTEN METHODS.
MENTAL METHODS CAN INCLUDED SIMPLE INFORMAL JOTTINGS E.G. WHEN
PARTITIONING JOTTING THE ANSWERS TO EACH SECTION.
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Key
Stage
One
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Written Multiplication and Division
Year 1
With support, solve one step problems involving multiplication using concrete objects,
pictorial representations and arrays.
Count in twos, fives and tens, making connections to arrays
Year 2
Recall and use multiplication and division facts for 2, 5 and 10. Make statements related
to the tables (eg, times 10s have 0 in ones column).
Solve multiplication problems in context by using arrays, number lines, bead strings,
pictorial representations etc. to develop a clear understanding of multiplication as
repeated addition and commutative.
x
7
7 X 5 = 35
5 X 7 = 35
5
35
35 ÷ 5 = 7
35 ÷ 7 = 5
+5
0
+5
5
+5
10
+5
+5
15
20
+5
25
+5
30
35
Solve division problems in context by using arrays, number lines, bead strings, pictorial
representations etc. to develop a clear understanding of division as repeated
subtraction, but is not commutative.
-5
0
-5
5
-5
10
-5
-5
15
20
-5
25
-5
30
35
Calculate mathematical statements using X, ÷ and = signs.
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Years
Three
and
Four
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Year 3
Recall and use multiplication and division facts for 3, 4 and 8. Make statements
related to the tables (eg. 3 X odd = odd, 3 X even = even; Dividing a number by 8 gives a
quotient that is half of dividing the same number by 4)
Solve problems using mental strategies and jottings to multiply and divide 2
digits by 1 digit.
64 X 3 = 192
60 X 3 =180 (6 X 3 X 10 = 180)
4 X 3 = 12
180+12=192
63 ÷ 3 = 21
60 ÷3 =20 (6 ÷ 3 X 10 = 20)
3÷3=1
20+1=21
Introduce scaling problems (eg. eight times as high; 4 times less)
Year 4
Recall and use multiplication facts up to 12 X 12. Make statements related to the
tables
(eg. odd X odd = odd, any integer X even = even: If 6 X 7 = 42 then 42 ÷ 6 = 7 and 42 ÷ 7 = 6)
Use known facts to multiply mentally including 3 numbers (eg. 3X4X6)
Begin to use their knowledge of tables to recognise factors of number.
Extend mental methods to three digit numbers, explaining the connections to
tables (eg. 600 ÷ 3 = 200).
Children should be encouraged to approximate their answers before calculating.
Children should be encouraged to consider if a mental calculation would be
appropriate before using written methods.
Begin to use more formalised written methods
Grid method
TU x U
(Short multiplication – multiplication by a single digit)
23 x 8
Children will approximate first
23 x 8 is approximately 25 x 8 = 200
Please note the grid is drawn vertically as opposed to horizontally to encourage the use
of place value and make a clearer link to other written methods.
x
20
3
8
1 60
2 4
18 4
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Once comfortable with grids, children should be introduced to expanded columns
for multiplying TO X O and HTO X O (after the method has been introduced for
addition)
23
X 8
160
24
184
When comfortable, they can then move on to the traditional algorithm.
23
X 8
184
2
341
X 6
2046
2
Children should continue to use mental methods and jottings to solve division
problems. They should be able to make division statements related to their
written multiplications.
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Years
Five
and
Six
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Children should be encouraged to approximate their answers before calculating.
Children should be encouraged to consider if a mental calculation would be
appropriate (efficient and accurate) before using written methods.
Year 5
Multiplication
Children should continue to us expanded methods for numbers up to 4 digits.
7346
X
9
54
360
2700
63000
65114
Once fully confident, children can move on to formal traditional methods for 4
digits by 1 digit.
2346
X
9
21114
3 4 5
Using similar methods, they will be able to multiply decimals with one decimal
place by a single digit number and two decimal places in context, approximating
first. They should know that the decimal points line up under each other.
They begin to use long multiplication for up to 4 digits by 2 digits.
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Division
Children will use written methods to solve short division TU ÷ U.
Initially, children should be encouraged to use facts boxes in order to encourage
efficient jumps and develop understanding of connection to tables.
Short division HTO ÷ O
Facts box
6
12
18
24
30
36
42
48
54
60
195 ÷ 6 = 32 r 3
6)
195
- 180
15
- 12
3
30x
2x
Answer : 32 remainder 3 or 32 r 3
Children should understand that remainders can be shown as fractions and
decimals.
1
32 r 3 = 32 36 = 32 2 = 32.5
Short division ThHTO ÷ O
3195 ÷ 6 = 532 r 3
6 ) 3195
3000
195
- 180
15
- 12
3
Facts box
6
12
18
24
30
500X
30x
2x
Answer : 532 remainder 3 = 532 r 3 = 532
3
6=
532
1
2=
532.5
As children become more confident, facts boxes can be simplified or not used.
Children need to be able to decide what to do after division and round up or down
accordingly. They should make sensible decisions about rounding up or down
after division. For example 3195 ÷ 6 is 532 remainder 3, but whether the answer
should be rounded up to 533, rounded down to 532 or expressed as a mixed
number depends on the context.
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Children should then move on to the traditional short division.
They should be able to express what they are doing in terms of place value. For
instance, in Example 1, 98 is divided by 70 then 28 is divided by 7.
Year 6
Develop calculations requiring the use of formal methods of long multiplication and
division for 4 digits by two digits.
Multiply simple pairs of proper fractions, writing the answer in its simplest form
1
4
For example
X
1
2
=
1
8
Divide proper fractions by whole numbers
For example
1
3
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÷ 2 =
1
6
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