Download º º º º º º º º º º º º º º º º º º º - St. George`s Junior School, Shrewsbury

St George’s Junior School
January 2013
Calculation Policy
Introduction
This booklet has been designed to explain how the different methods of
calculating are taught at St George’s.
The stages are progressive and the children will be working at a stage with which
they feel confident. When they have secured this stage they will be challenged
to move on to the next stage, but will always have a method to fall back on.
All calculations should be written horizontally at first; when the children are
ready, they will move on to vertical written methods.
Children will be encouraged to carry out a quick mental estimation prior to
calculating a problem, so that they can check that their answer is a ‘sensible’
answer.
Quick recall of number bonds and times tables are essential in enabling children
to move forward with calculation.
1of 19
Key Stage 2
Addition
Stage 1
Single digits/units
Horizontal 7 + 6 [using counters, number line, Dienes apparatus]
Using number line+ 1
+1
+1
+1
+1
+1
+1
7
13
Stage 2
Then number line alongside vertical method.
+6
7
7
+6
13
13
Stage 3
Tens and Units
Horizontal 23 + 13
[using number line]
+ 10
23
+3
33
36
Always start with the larger number at the beginning of the number line.
Then add the tens and then the units.
Then number line, alongside vertical method.
+ 10
23
+3
33
36
23
+13
36
Stage 4
Adding larger numbers—increase the jump in tens
45 + 34
+ 30
45
+4
75
2of 19
79
45
+34
79
Stage 4
Using a number line becomes very slow with larger numbers, so more efficient to move on to vertical method.
Vertical method—no carrying
46 + 33
Always start with the unit first, then the tens.
Say ‘5 add 4 equals 9, write it down in the units column. Then 4 tens add 3
tens equals 7 tens, write it down in the tens column, my total is 79.’
45 + 34
45
+34
79
[with Dienes to start
with, then without]
Vertical method—with carrying
56 + 37
Always start with the unit first, then the tens.
Say ‘6 add 7 equals 13 [1 ten, 3 units], write 3 down in the units column, carry the ten to the tens column, jot down underneath. Then 5 tens add 3 tens,
equals 8 tens, add in the 1 ten you have carried, equals 9 tens, write it down
in the tens column, my total is 93.’ Can be demonstrated using whiteboard
software.
56 + 37
56
+ 37
93
[with Dienes to start
with, then without]
1
Stage 5
Continue with vertical method, moving on to hundreds. Without carrying
initially and then with when secure.
345 + 123
345
+123
468
357 + 367
357
+367
724
[with Dienes
to start with,
then without]
1 1
Stage 6
Continue with vertical addition through progressively larger numbers.
[Dienes should not be needed by this stage, as the children should be
secure with the method used]
3of 19
Stage 7
Addition of money will be introduced when children have a sound understanding of the method involved.
Step 1
£5 · 5 6
£3 · 3 2
£8 · 8 8
Step 2
£3 6 · 7 8
£4 3 · 2 1
£7 9 · 9 9
Step 3
£3 · 4 5
£2 · 7 8
£6 · 2 3
Step 4
£5 6 · 6 4
£3 6 · 5 9
£9 3 · 2 3
1
1
1
1
1
Stage 8
Addition of decimals will be introduced when the children have an
understanding of the place value of tenths and hundredths etc.
Step 1
6·5
3·4
9·9
Step 2
34·65
52·34
86·99
Step 3
6·6
2·5
9·1
Step 4
56·67
36·56
93·23
1
1
4of 19
1
1
Stage 9
Addition of negative numbers
Step 1 Using a number line
When adding positive numbers ‘jump’ to the right.
When adding negative numbers ’jump’ to the left.
-5 + +7 = +2
-5 -4 -3 -2 -1 0 +1
Step 2 Adding a negative number
+2 + -7 = -5
+2 +3
+4
+5 +6
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6
Children should be able to work without number line when confident.
Step 3 Two signs rule
When adding or subtracting negative numbers, remember that when two signs
appear next to each other and are different, then you subtract. When two
signs are next to each other and they are the same, you add:
+7 + - 5 =
+7 — + 5 =
+7 + + 5 =
+7 — - 5 =
Stage 10
Addition of fractions
Step 1 same denominators
1 + 3 = 4 =
4
4
4
1
3 + 3 = 6 = 1 2 =
4
4
4
4
Step 2 different denominators
1 + 1
4
2
1 + 2 =
8 3
= 1 + 2 = 3
4
4
4
3 +
24
16 = 19
24
24
+2
+2
+12
+12
11
2
3 + 7 = 10 = 1 2 = 1 1
8
8
8
8
4
Find common denominator.
Make equivalent fractions
with new denominator
Add the numerators.
5of 19
Subtraction
Stage 1
Take-away
Single digits
Horizontal 6 – 3 [using counters, number line, bead strings, Dienes]
Stage 2
Find the difference — using number line count back in ones, total the number of jumps. Show that this is the same as adding on from 6 to 12.
-1
12 - 6 = 6
Same as
6 + 6 = 12
-1
-1
-1
-1
-1
6
12
+1
+1
+1
+1
+1
+1
Stage 3
Find the difference—using number line count on in bigger jumps.
Start at 23, count on in tens, final jump of 9 at the end. Total up all the
jumps mentally.
62 – 23 = 39
+ 10
23
+ 10
33
+ 10
43
+9
53
= 39
62
Stage 4
When confident, same as above, but with bigger jumps.
62 - 23 = 39
+ 30
23
+9
53
6of 19
62
= 39
Stage 5
Any numbers larger than previous stage, become unwieldy and therefore
it becomes more time efficient to move on to vertical layout, using Dienes
apparatus initially. No decomposition to start with.
Start always with units column. Subtract 4 from 5, jot answer in units
column. Then subtract 3 tens from 4 tens, jot answer in tens column.
45 - 34 =
45
- 34
11
When confident and able to subtract without Dienes apparatus, move on
to H T U, again using Dienes if needed. The without Dienes when confident.
467 - 245
467
-245
222
Again when confident move on to Th, H T U etc.
Stage 6
By now the children should be ready to move on to decomposition. Start
with Dienes apparatus [can be demonstrated using whiteboard program].
They will need lots of repetition and practice.
55 - 28
Again always start with the unit column. Say ‘subtract 8 from 5, can I do
it? No, I need to exchange a ten for 10 units. I now only have 4 tens,
cross out my 5 tens, change it to 4. Now I have exchanged my ten for ten
units I need to record it in the units column, jot it down above the unit.
Now I have a total of 15 units, now I can subtract my 8 units. My answer
is 7, jot it down in the units column. Now I can subtract 2 tens from 4
tens, my answer is 2, jot it down in the tens column. My answer is 27.’
Step 1
4
55
-28
4
Step 2
1
55
-28
7
7of 19
Step 3
4
1
55
-28
2 7
Stage 7
Once confident and secure in the method, and can do without Dienes
apparatus, move on to H T U. Use same method as before, exchanging tens
for units, if necessary and them a hundred for 10 tens if needed. Lots of
practice will be required again.
5 1
568 - 249
568
-249
3 19
538 - 249 =
1
2
step 1
1
538
-249
9
4
1
2 1
4
step 2 5 3 8
-249
89
2
1
step 3 5 3 8
-249
289
9
Further step involves exchanging
when subtracting from a hundred.
1
1
500
-249
251
4
Stage 8
Using same methods as above, subtraction with decimals will be introduced,
initially with money.
9
9
8
Step 1
£6 · 7 5
- £3 · 4 5
£3 · 3 0
Step 2
1
£4 9 · 5 0
- £3 4 · 6 0
£1 4 · 9 0
4
1
1
1
Step 3 £5 0 · 0 0
£3 4 · 5 7
£15·43
As with addition, subtraction of decimals will be introduced when children
are secure with the place value of tenths and hundredths etc.
Step 1
7·5
- 5·2
2·3
Step 2
1
7·4
- 4·5
2·9
6
Step 3 6 2 · 4 5
-31·43
31·02
8of 19
4
1
2
1
Step 4 6 5 · 3 6
-32·68
32·68
Stage 9
Subtraction of negative numbers
Link to addition - use of number lines initially
Remember the two sign rule.
Understanding that
Then
- 7 — + 7 = [- 7 — 7] = -14
- 7 — - 7 = [- 7 + 7] = 0
Stage 10
Subtraction of fractions
Same denominator
Step 1. The denominators numbers are already the same. Go straight to
step 2.
Step 2. Subtract the numerators and put the answer over the same
denominator.
Step 3. Simplify the fraction.
e.g
3 - 1 =
4
4
3—1 = 2= 1
4
4 2
Stage 11
Different denominator
Step 1. The denominators are different, these need to be changed to the
same before you can go further.
Step 2. Now you can subtract the numerators and put the answer over the
same denominator.
Step 3. Simplify the fraction.
e.g. 1 — 1 = 3 - 1 = 2 = 1
2
6
6
6
6
3
9of 19
Multiplication
Stage 1
Initially all linked with times tables work. Children can use apparatus or draw.
5 x 4 = 20
[ 5 groups of 4]
• •
• •
4
• •
• •
+
4
• •
• •
+
• •
• •
4
+
4
• •
• •
+
4
4
x 5
20
= 20
Record answer both ways so the children see the link right from the
beginning.
Stage 2
Move on to more formal written method .
Children will need lots of practice, but will get it. Use table square to help
those who do not have recall of tables.
16x4=
Step 1
16
X 4
2 4 [4 x 6]
4 0 [4 x 10]
64
Step 2
16
x 4
64
2
‘Say 6 multiplied by 4 equals
24, write down the 4 units in
the units column, carry the 2
tens, jot down below the line.
Next 1 ten multiplied by 4,
equals 4 tens, add in the
extra 2 tens, equals 6 tens,
write down 6 in tens column,
answer equals 64.’
Stage 3
When children are secure in above method, move on to when the
answer involves a hundred.
67x5
67
x
5
3 319 5
10of
3
3
Stage 4
When secure on multiplying by units, the children are then ready to move on
to multiplying TU by TU.
Start with numbers that require no carrying in the multiplication.
Step 1
21x14
Step 2
56x23
21
x 14
8 4 [4 x 21]
+ 2 1 0 [10 x 21]
294
56
x 23
1
1 6 8 [3 x 56]
+ 1 1 12 0 [20 x 56]
1 288
‘First you multiply by the units, so 1 multiplied
by 4 equals 4, write down the 4 units in the
units column, 2 tens multiplied be 4 equals 8
tens, write down 8 in the tens column.
Next we multiply by the tens, so we first
need to put a zero in the units column . Now
we can say 1 ten multiplied by 1 equals 1 ten,
write it down in the tens column, then 2 tens
multiplied by 1 ten equals 200, write your
answer in the hundreds column. Then add your
two answers together.’
A jotting beside the calculation might help
the children understand the process.
Method same as above, but the carrying
figure needs to be jotted down where
necessary and added in.
Jotting beside the calculation may not be
necessary by this stage, but may be helpful.
Step 3
When confident and secure after lots of practice move on to H T U
multiplied by T U.
234x45
234
x
45
1
2
1 1 7 0 [5 x 234]
1
1
9 3 6 0 [40 x 234]
10530
1 1
11of 19
Stage 5
Next stage is then to multiply HTU by HTU.
453 x 213
453
x 213
1
1 3 5 9 [3 x 453]
4 5 3 0 [10 x 453]
1
9 0 6 0 0 [200 x 453]
96489
Method same as previous stage
but when multiplying by the
hundred 2 zeros needed to be
jotted down first.
Again jotting beside the calculation may not be necessary by
this stage, but may be helpful.
1
Stage 6
Next step is to multiply with decimal point, for example with money.
Step 1
Method same as previous
£3 · 4 0 x 8
£3 · 4 0
stages, children need to be
x
8
aware that decimal point
£2 7 · 2 0
stays in line at all times.
3
Step 2
£4 5 · 2 5 x 2 3
£4 5 · 2 5
x
23
1
1
1 3 5 · 7 5 [3 x 45 · 25]
1
1
9 0 5 · 0 0 [20 x 45 · 25]
£1 0 4 0 · 7 5
1
Brackets stating
actual calculation
might be useful
initially again.
1
Stage 7
Multiplying fractions
Step 1
multiply the numerators
Step 2
multiply the denominators
Step 3
simplify the answer
1
e.g.
1 x 1 = 1
4
5
20
then
1 x2= 1
2 5 5
1
12of 19
[with cross cancelling first]
Division
Stage 1
Initially division is seen as very much the inverse operation of multiplication. Children to use counters or jottings to ‘share’ and to
use tables with which they are familiar.
20 ÷ 5
º º º
º
• •
• •
º
º
• •
• •
4 x 5 =20
º º º º º º º º º
• •
• •
• •
• •
º º º º
• •
• •
20 ÷ 5 = 4
It is necessary at this stage to model sharing into groups of/sets of
e.g. sharing—3 children share 12 sweets, how many will each child get?
grouping—for 12 children how many tents do we need if there are 3
children in each tent
Stage 2
The next step is to move on from numbers that divide exactly to numbers
that have a remainder. To use same method as above or just to use their
tables knowledge if possible. Again the inverse operation is very important.
23 ÷ 4 = 5 r 3
Children need lots of practise at this stage using tables they know.
Numbers divided must be in the tables i.e. ÷ 5 up to 60
Only move on to
when numbers are above tables numbers as this is
when written methods are required—too difficult to skip count large
numbers.
13of 19
Stage 3
Next step is to only use more formal compact method of recording for
numbers above those in tables.
If children are unfamiliar with table, either use table square or get them to
jot table down in ‘working-out’ column
12r1
8 91 7
19r3
4
5 98
Stage 4
The children should now be ready to move on to a compact written method
with larger numbers.
Step 1
Say ‘2 into 3 goes once with 1 left over,
174
record 1 directly above and carry 1 forward,
348 ÷ 2
2 3 14 8
now say 2 into 14, goes 7, none left over,
record above. Finally 2 into 8 goes exactly
twice, record the 2, nothing to carry.
Step 2
Division with remainder.
183r2
2 1
551 ÷ 3
3 551
Use the same method as above, just record
the remainder number. Same method can be
done for Th H T U.
Step 3
Division with answer to 1 decimal place.
130·4
Use the same method as above, instead
2
1
of recording the remainder, add a deci652 ÷ 5
5 652·0
547 ÷ 4
1 36·75
1 2
3 2
4 547·00
mal point and record the carry number
then divide as before. Don’t forget the
decimal point in the answer, directly
above the one in the sum.
Same method as before, but add two
zeros to continue the division.
This method can also be used for division of money [see Stage 6]
It is important that children are aware that they may need to round the
final answers up or down, this is important for problem solving.
14of 19
Stage 5
Division by tens and units
If they don’t know 12 x table, jot it down first in working out column.
335
4 4 6
4020 ÷ 12
124020
12, 24, 36, 48, 60 etc
9923 ÷ 22
or
0451r1
9 11 2
22 9 9 2 3
451r1
22 9 9 2 3
88
1 12
1 10
23
22
1
22, 44, 66, 88, 110 etc
Say 22 into 9, doesn’t go, carry the 9
forward, 22 into 99, goes 4 times, 88,
jot it down underneath and then subtract, 11 left. Next bring down the 2,
now say 22 into 112, goes 5 times, jot
down 110 underneath, subtract, 2 left,
bring down the 3. Next say 22 into 23,
goes once, subtract 22 from 23, 1 remainder. Nothing to bring down. Don’t
forget to jot down your division
factors above at each stage.
Stage 6
Division of money
The same way as above with decimal point and two decimal places.
£35·55 ÷ 15
2· 3 7
5 10
15 3 5 · 5 5
15, 30, 45, 60, 75, 90, 105
3
Answer £2·37
15of 19
Stage 6 cont’d
Division of fractions
Step 1. Turn the second fraction upside-down (it becomes a reciprocal):
Step 2. Multiply the first fraction by that reciprocal:
Step 3. Simplify the fraction:
e.g.
1 ÷ 1 = 1 x 5 = 5
4 5
4
1
4
= 1 1
4
3
1
2
÷ 1 = 1 x 6 = 3 = 3
6
2
1
1
[with cross cancelling first]
1
16of 19
Mathematical Glossary
addition
Addition is finding the total of two or more numbers. The + sign in a calculation
shows that numbers are being added together e.g. 12 + 7 = 19
approximation
bead string
An approximate answer is very close to the right answer, but is not exact.
The sign ≈ is used.
A string of beads, up to a 100, that can help with calculation.
cancel
(a fraction)
To cancel is to simplify a fraction by dividing the numerator and denominator by
the same number. 12/15 dividing numerator and denominator by 3 equals 4/5.
carrying
Where you regroup the 1s into a group of 10 and put a 1 in the 10s column, the
1 now represents a group of ten.
column
A vertical arrangement for a sum e.g list the numbers in a sum going down the
page.
Reducing diagonally opposite digits in fractions to their lowest common factor.
cross
cancelling
decimal
[fraction]
Dienes
apparatus
decimal point
The part of a number to the right of the decimal point is called the decimal
fraction [decimal]. It is a number less than 1.
e.g. 0.12 = 1 tenth and 2 hundredths.
Base ten Dienes blocks are a physical representation of a means of calculation,
and can help children visualise numbers when learning basic addition and
subtraction.
A decimal point is used to show which digits are whole numbers and which are
fractions. Digits to the left are whole, digits to the right are fractions.
decimal place
The position of the digit after the decimal point is its decimal place.
e.g. 0.2 has 1 decimal place, 0.25 has 2 decimal places
decomposition
A method of subtraction where numbers can be exchanged for an equal amount
of another unit e.g exchange 1 ten for 10 units so that a larger number can be
subtracted.
In fractions, the number written below the line, e.g. In the fraction ⅔, the denominator is 3. The number of parts the fraction is divided into.
denominator
digit
Any of the ten numerals, 0 – 9.
difference
The result of a subtraction. The amount by which one number or value is
greater than another.
divide
To share into groups of a specific number e.g. ‚ 3, share into 3 equal groups.
division
An operation on numbers interpreted in a number of ways. Division can
be sharing - the number to be divided is shared equally into the stated
number of parts; or grouping - the number of groups of a given size is
found. Division is the inverse operation to multiplication.
equal
Symbol: =, read as ‘is equal to’ or ‘equals’. Having the same value.
e.g. 7 − 2 = 4 + 1 since both ‘sums’, 7 − 2 and 4 + 1 have the same value 5.
equivalent
fraction
estimate
Fraction with the same value as another e.g. 6/12 = 3/6 = ½ these are equivalent
fractions.
To arrive at a rough or approximate answer by calculating with suitable
approximations or near numbers.
17of 19
exchange
factor
Change a number for another of equal value. The process of exchange is used in
some compact methods of calculation e.g. 'carrying figures' in addition, multiplication or division; and 'decomposition' in subtraction e.g to exchange 10 units for
1 ten, or 10 tens for 1 hundred.
A whole number that divides into another exactly.
find the
difference
horizontal
To find the difference between two numbers by adding on.
inverse
operation
Addition and subtraction are inverse operations e.g. 5 + 6 =11 11 – 6 = 5.
Multiplication and division are inverse operations e.g. 6 × 10 = 60 60 ‚ 10 = 6.
minus
multiplication
The name for the symbol −, representing the operation of subtraction.
The operation of combining two numbers to give a third number, the product.
e.g. 12 x 3 = 36 is a multiplication.
Carry out the process of multiplication.
multiply
To lay ‘sums’ out across the page e. 9 + 5 =14
negative
integer
A whole number less than 0. Examples: -1, -2, -3 etc.
negative
number
number bond
A number less than zero e.g. − 0.25. Can be read aloud as ‘minus one, minus
two’ or as 'negative one, negative two' etc.
A pair of numbers with a particular total.
e.g. number bonds to ten are all pairs of whole numbers with the total 10.
number line
A line where numbers are represented by points upon it, aids calculation.
numerator
In fractions, the number written on the top. In the fraction ⅔ the numerator is 2.
The number of parts of the fraction you have.
partition
To split a number into component parts e.g the two-digit number 38 can be partitioned into 30 + 8 or 19 + 19.
place value
The value of a digit that relates to its position or place in a number.
e.g. in 1482 the digits represent 1 thousand, 4 hundreds, 8 tens and 2 units.
in 12.34 the digits represent 1 ten, 2 units, 3 tenths and 4 hundredths.
plus
The name for the symbol +, representing the operation of addition.
positive
number
A number greater than zero. They are read as 'positive one, positive two.’
product
The result of multiplying one number by another e.g the product of 2 and 3 is 6
since 2 × 3 = 6.
The result of a division e.g. 45 ‚ 3 = 15. The quotient is 15.
The multiplicative inverse of any non-zero number e.g. 1/3 is the reciprocal of 3.
Any number multiplied by its reciprocal gives 1. e.g. 1/3 x 3 = 1
quotient
reciprocal
reduce
(a fraction)
remainder
rounding
Divide the numerator and denominator by a common factor. To cancel a fraction
e.g. divide the numerator and denominator by 5, to reduce 5/15 to 1/3,
its simplest form.
The amount left over if when you divide a number by another and it doesn’t go
exactly e.g 29 divided by 7 = 4 remainder 1 [often written 4 r 1].
To round a number is to change it to the nearest ten, hundred, thousand etc. to
give an approximate answer
e.g. 23 would be rounded down to 20, 27 would be rounded up to 30.
25, 35, 45, etc are always18of
rounded
up to the next ten.
19
share (equally)
short division
simple fraction
simplify (a fraction)
signs
skip count
subtract
subtraction
subtraction by
decomposition
sum
table square
take away
total
unit
vertical
zero
One model for the process of division.
A compact written method of division.
A fraction where it can’t be reduced any lower.
Reduce a fraction to its simplest form. See cancel and reduce (a fraction).
A symbol that shows you which operation to use in a calculation + - x ÷
Skip counting is counting in numbers other then 1 e.g 2s, 5s, 7s.
Carry out the process of subtraction, takeaway e.g 14 – 9 = 5.
The inverse operation to addition, takeaway.
A vertical method of subtraction where a number on the top line can be
exchanged for a different unit of equal value. See exchange.
The result of one or more additions.
A grid that shows the multiples of all the times tables; aids calculation for
children who find recall of times tables difficult.
Subtract.
The total is the result when you add together a group of numbers.
One.
Where sums are set out in a column down the page.
1. Nought or nothing.
2. In a place value system, a place-holder. Example: 105.
19of 19