Download Written Calculations - Hook Norton Primary School

Hook Norton C.E.P. School
A Guide to Written Calculations
2011/2012
Dear Parents
We have put this booklet together to demonstrate the
calculation skills that we teach at Hook Norton School.
They might have changed since we were at school!
This will help you to help your child with their maths
homework if you wish to. The skills are taught carefully in
class and experience has shown that if children are shown a
different method at home then this confuses them and is
counter-productive.
Please don’t worry if you don’t help your child at home – if
your child is struggling please have a word with the class
teacher.
If you would like help understanding the methods for
yourself please speak to Mrs Crouch.
Children should be encouraged to:



Approximate their answers before calculating.
Check their answers after calculation using an
appropriate strategy.
Consider if a mental calculation would be appropriate
before using written methods.
2
Addition
Pupils are taught to understand addition as combining
two sets and counting on.
Stage 1
4+3=
I put 4 pennies in my purse. I add another 3. How many
pennies are in my purse?
4+=7
I need 7 pennies. I already have 4. How many more do I
need?
Pupils work these out in a variety of ways:
 Using actual objects
 Drawing a picture


0

Using tally marks
Using a number track
1
2
3
4
5
6
Using an empty number track
3
7
8
9
10
 Using a number line, numbered and without numbers.
4+3=7
0
1
2
3
4
5
6
7
8
9
10
Stage 2
47 + 25 = 
My sunflower is 47cm tall. It grows another 25cm How tall
is it now?
+ 20
+5
Or
47
67
+20
72
+3
47
67
+2
70
72
Pupils draw an empty number line to record the steps they
take in a calculation. They count up in steps of 100, 10, 5, 2
and 1. This is much more efficient than simply counting in
ones.
4
Stage 3
The empty number line is extended to larger numbers and
use of compensation.
38 + 86 = 124
+30
86
Compensation
49 + 73 = 122
73
+4
116
+4
120
(73 + 50 – 1)
+ 50
124
-1
122
123
Stage 4
Pupils use partitioning (splitting a number into units, tens,
hundreds), then adding the units first, then the tens and
then the hundreds.
487 + 546
There are 487 boys in a school and 546 girls. How many
pupils altogether?
400 + 80 + 7
+ 500 + 40 + 6
900 + 120 + 13 = 1033
5
Stage 5
The method is then contracted.
6
+ 2
1
8
9
7
6
3
0
3
+
4
5
1
1
9
0
8
4
1
2
0
3
7
6
3
0
0
3
We always start by adding the units first, then the tens,
then the hundreds.
Stage 6
The method is contracted further.
+
1
4
5
0
8
4
2
1
1
7
6
3
7 + 6 = 13, so the 3 goes into the units column and the 10 is
saved below the line to be added into the tens column.
6
Stage 7
21.87 + 3.9 + 874 + 5943.95 =
+
+
+
2
5
6
8
9
8
7
4
4
1
3
4
3
3
1
1
1
2
•
•
8
9
7
•
•
9
7
5
2
1
In this case it is important to set the calculation out
correctly and begin adding with the smallest column, so
hundredths are added to hundredths, tenths to tenths,
units to units, tens to tens, hundreds to hundreds,
thousands to thousands.
7
Subtraction
Pupils are taught to understand subtraction as taking
away (counting back), and finding the dfference
(counting up).
Stage 1
5–2=
I have 5 balloons. 2 burst. How many balloons are left?
Take away
A teddy bear cost £5 and a doll cost £2. How much more
does the teddy bear cost?
Find the difference
Pupils work simple subtraction calculations out in a variety
of practical ways, for example:
 Using objects
 Drawing a picture
 Tally marks
 Counting back in ones on numbered and empty number
tracks
 Counting back in ones on numbered and non-numbered
number lines
8
Stage 2
Pupils calculate using empty number lines to record the
steps. They count back in steps of 10, 5, 2 and 1.
I have 75p. I buy a chocolate bar costing 38p. How much
money do I have left?
-3
-5
-30
37
40
45
75
75 – 38 = 37
Stage 3
Pupils use empty number lines to count from the smaller
number up to the larger number – complementary addition.
In a traffic survey 863 vehicles are counted. 546 are cars,
the rest are lorries. How many lorries are counted?
+4
+50
+200
+60
+3
546
550
600
800
860
4 + 50 + 200 + 60 + 3 = 317 therefore 863 – 546 = 317
9
863
Stage 4
Pupils start using vertical methods. We always start by
subtracting the units, then the tens, then the hundreds.
We might have to “borrow” from the next column.
754 – 226 = 328
= 328
642 – 386 = 256
= 256
Stage 5
The method is contracted
10
Stage 6
5003 – 2486 = 2517
In this case we still start with the units, but we cannot do
3 – 6. There are no tens to borrow, or hundreds. We go to
the thousands column, take a thousand out, leaving 4
thousand, and put the thousand we have borrowed into the
hundreds column. We now have 10 hundreds in the hundreds
column. We borrow one of those hundreds and put it in the
tens column, leaving 9 hundreds. We now have 10 tens. We
put one of those tens into the units column, leaving 9 tens.
We now have thirteen in the units column, the subtraction
is now possible.
Stage 7
This method works in the same
way with decimals, we start our
subtraction at the right hand
column.
11
Multiplication
Pupils are taught to understand multiplication as
repeated addition, scaling and arrays.
Stage 1
3 x 4 = 12
3 dogs have 4 legs each. How many legs?
Multiplication is worked out in a variety of practical ways:
 Draw a picture and count the legs
Using an empty and numbered number track

0
1

0
2
4
5
6
7
8
9
10
11
12
Using empty and numbered number lines
+4
+4
+4
1

3
2
3
4
5
6
4
Use tally marks
7
8
8
12
9
10
11
12
12

Drawing an array
o o o o
3x4
o o o o
o o o o
4x3
This shows that 3 x 4 = 4 x 3
Stage 2
Pupils use partitioning
There are 14 cakes in a box. How many in 6 boxes.
14 x 6 = (10 x 6) +( 4 x 6), 14 lots of 6 is the same as 10
lots of six plus 4 lots of 6.
+60
+24
0
60 + 24 = 84
10x6
60
4x6
Stage 3
Pupils start to set their working out in a grid.
14 x 6
or
x
6
x
10
4
10
60
6 60
24
4
24
10x6
4x6
Either way is fine. Then add
Together the answers:
60 + 24 = 84
13
84
Stage 4
The grid method is extended.
e.g. 24 x 79 = 4416
Helpful hint: 20 x 70 = 2 x 10 x 7 x 10
Do the 2x7 first
= 14 x 10 x 10
=140 x 10
= 1400
Stage 5
Extended further
e.g. 248 x 79 = 6992
14
Stage 6
Extend to decimals
e.g. 4.92 x 73 = 359.16
Helpful hint:
70 x 0.02 = 7 x 10 x 2 ÷ 100
Do 7 x 2 first
= 14 x 10 ÷ 100
= 140 ÷ 100
= 1.4
Stage 7
Use a vertical method.
As with all vertical methods
we start with the units. We
do not write down the bits in
brackets, those are just
included here to explain the
method.
15
Stage 8
The method is contracted.
In this case we say: 9 x 8 = 72, write the 2 in the units,
carry the 7 tens on to the tens column, 9 x 40 = 360, add in
the 7 tens = 430, put the 3 tens in the tens column, carry
the 4 hundreds on to the hundreds column, 9 x 200 = 1800,
add in the 4 hundreds = 2200, so a 2 hundreds goes into
the hundreds and a 2 thousands into the thousands.
Then we say: 70 x 8 = 560, so 0 goes into the units and 6
tens into the tens column, the 5 hundreds is carried into
the hundreds column, 70 x 40 = 2800, add in the 5
hundreds = 3300, so 3 hundreds goes into the hundreds
column and 3 thousands are carried on to the thousands, 70
x 200 = 14000, add in the 3 thousands = 17000, so 7
thousands goes into the thousands column and 1 ten
thousand into the ten thousands column.
16
Division
Pupils are taught to understand division as sharing and
grouping.
Stage 1
8÷4=2
8 sweets are shared between 4 children. How many sweets
do they have each?
Sharing between 4
There are 8 lollies. How many children can have 4 each?
How many 4s in 8?
Grouping in 4s
Pupils can draw pictures, dots and tally marks to do the
calculation.
17
Stage 2
42 ÷ 6 = 42
Count up in steps of 6 to find out how many there are in 42.
0
6
12
18
24
30
36
42
Stage 3
91 ÷ 7 = 13
Counting in steps of 7 would take too long, so we start
chunking. A chunk of 10 lots of 7 = 70.
10 x 7
+7
+7
+7
0
70
77
84
91
Stage 4
91 ÷ 7 = 13
10 x 7 = 70
91 – 70 = 21, therefore we still have to find out how many
7s there are in 21.
21 ÷ 7 = 3
So we found there was a chunk of 10 lots of 7 and a chunk
of 3 lots of 7, so 13 lots of 7 altogether.
10 x 7
3x7
0
70
18
91
We can also show this as follows:
91 ÷ 7
10 lots of 7 + 3 lots of 7 is 13 lots of 7 altogether. So
there are 13 lots of 7 in 91.
Stage 5
This method is expressed
vertically, subtracting
chunks of the number as
we go. The chunks are
recorded in the brackets.
We say: 10 lots of 7 is 70,
take this away from the
91, we have 21 left. 3 lots
of 7 is 21, take this away,
we have 0 left so the
calculation is finished. We
count up the number of 7s we took away which is 10 + 3 =13.
We took away 13 lots of 7 altogether, so our answer is 13.
19
Stage 6
Pupils do a fact box to help them. In this case there is a
remainder of 9, so the answer is expressed as 35 r9. The
answer can also be expressed as a fraction: 35 9/12,
(because we are dividing by 12) which can be simplified to
35 3/4. In this case the answer can also be expressed as a
decimal: 35.75. Pupils will be told which type of answer to
give.
Stage 7
Extended to numbers with up to 2 decimal places
Answer 24.05
20