Download St Augustine`s Catholic Primary School Parent Numeracy Workshop

St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
March 2013
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
One of the key features of written calculations is that children are only expected to
move on when they have fully grasped the concepts behind what it is they are doing.
We set a rough guide of year groups for different methods as often this is where they
are introduced, but it is the progression of stages which today attempts to make clear.
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Practical and Mental Methods
Bead strings
Numicon
Number lines
Number bonds
Adding multiples of 10
Partitioning
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Partitioning
276
200
+
70
+
6
Repartitioning
100
+
170
+
6
Addition
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Addition
Prerequisite Skills
Partitioning
(Splitting up a number by place value e.g. 126 = 100 + 20 + 6)
adding multiples of 10
addition facts to 20
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Addition
Stage 1
Mental method, using partitioning:
Initially carried out with jottings and then mentally.
47 + 76
47 + 76 =
(40 + 70)
= 110 + 13
= 123
+ (7 + 6)
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Addition
Stage 2
Vertical layout, expanded working, moving from adding the most significant digit
first to adding the least significant digit first:
The next step is to show children the vertical format (units under units, tens under tens, etc.) and link it to the
mental method. They first practise this method with calculations they can do mentally, and then extend to
three-digit numbers, which provides justification for developing a written method.
An important point is that children should be able to describe what they are doing by referring to
the actual values of the digits in the columns i.e. ‘20 + 50’ or ‘2 tens + 5 tens’, never ‘2 + 5’.
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Addition
Stage 2
47 + 76
47 + 76
Adding the least significant digit first
Adding the most significant digit first
Leading to:
47
76
+
110
(40 + 70)
13
123
(7 + 6)
47
76
13
(7 + 6)
110
123
(40 + 70)
+
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Addition
Stage 3
Vertical layout, contracting the working to a compact, efficient form:
A conventional compact, efficient method, with carrying, is introduced.
If, with a little practise, they cannot use the compact method without making errors, they should return to the
expanded format.
Again, when describing each step, children should refer to the actual values of the digits.
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Addition
Stage 3
H TU
13
110
123
Becomes
+
47
76
47
76
1 2 3
1 1
Place value becomes really important, as
does presentation
+
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Addition
Stage 4
Extend to bigger numbers and decimals:
When returning to written calculations at a later stage, e.g. to revise or to extend to decimals or numbers with
more digits, it is a good idea to start again with informal, expanded methods. This helps children to retain
their understanding of the link between different methods, and makes it easier for them to resort to an
expanded method if they need to do so.
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Addition
Stage 4
13.2 + 12.6
TU. t
13.2
12.6
2 5. 8
+
Subtraction 1
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Subtraction – Complementary Addition
Counting Up
Prerequisite Skills
Rapid recall of all pairs of numbers with a total of 10.
Complements to 100.
Counting on in steps of 10/100 etc
Rounding to the next nearest multiple of 10.
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Subtraction – Complementary Addition
Counting Up
Stage 1
Show the children the vertical layout for a calculation they can do mentally.
Link the steps to those on an empty number line.
74 - 27
74
27
3 ( 30)
4 0 ( 70)
4 ( 74)
47
+3
27 30
+40
+4
70 74
-
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Subtraction – Complementary Addition
Counting Up
Stage 2
Show how this form of recording can help organise the steps involve in subtracting a three-digit number from
another three-digit number.
326 - 178
326
178
2 ( 180)
2 0 ( 200)
1 0 0 ( 300)
2 6 ( 326)
148
-
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Subtraction – Complementary Addition
Counting Up
Stage 2
You can reduce the number of stages further, by using knowledge of pairs of numbers that total 100.
326 - 178
326
178
2 2 ( 200)
1 2 6 ( 326)
148
-
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Subtraction – Complementary Addition
Counting Up
Stage 3
Extend to bigger numbers and decimals.
22.4 - 17.8
22.4
17.8
0 . 2 ( 18)
4 . 0 ( 22)
0 . 4 ( 22.4)
4.6
-
Subtraction 2
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Subtraction – Decomposition
Prerequisite Skills
Children need to have a good understanding of how to partition and repartition numbers.
e.g. 63 = 60 + 3
=50 + 13 (by making the tens 10 less and the units 10 more)
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Subtraction – Decomposition
Stage 1
Expanded vertical method
The expanded vertical method of decomposition involves partitioning and repartitioning. The subtraction
calculation is carried out in columns, starting with the least significant digits (the units).
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Subtraction – Decomposition
Stage 1
263 – 141
342 – 126
30
200 + 60 + 3
100 + 40 + 1
-
100 + 20 + 2 = 122
Leading To Repartitioning
300 + 40 + 2
100 + 20 + 6
12
200 + 10 + 6 = 216
-
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Subtraction – Decomposition
Stage 2
Standard compact method of decomposition
A conventional compact, efficient method, with carrying below the line, is introduced by linking it directly to
the expanded method, starting with the units. Children should be able to explain the link and appreciate that
the compact method saves time recording their working. If, with a little practice, they cannot use the compact
method without making errors, they should return to the expanded format or to counting up on a number
line.
*Again, when describing each step, children should refer to the actual values of the digits.
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Subtraction – Decomposition
Stage 2
725 – 367
6 11 1
725
367
3 58
-
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Subtraction – Decomposition
Stage 3
Extend to bigger numbers and decimals
When returning to written calculations at a later stage, e.g. to revise or to extend to decimals or numbers with
more digits, it is a good idea to start again with informal, expanded methods. This helps children to retain
their understanding of the link between different methods, and makes it easier for them to resort to an
expanded method if they need to do so.
Multiplication
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Multiplication
Prerequisite Skills
Partitioning
Doubling and halving
Multiplying by 10
Multiplication facts/calculation strategies
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Multiplication
Stage 1
Mental method, using partitioning:
38 x 7
(30 x 7)
= 210
= 266
+
+
(8 x 7)
56
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Multiplication
Stage 2
Grid method, expanded working
The mental method from which written methods are developed involves partitioning, and then multiplying
the tens and ones separately. It is common to start with the tens when working mentally. A useful way of
recording intermediate steps is the ‘grid’ method.
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Multiplication
Stage 2
38 x 7
x
30
7
210
(30 x 7)
8
+
56
(8 x 7)
266
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Multiplication
Stage 3: Extended to bigger numbers:
56 x 27
x
20
7
50
1000
6
+
(50 x 20)
350
(50 x 7)
120
1120
(6 x 20)
+
42
(6 x 7)
392
1512
1
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Multiplication
Stage 4: Extended to decimals
23.5 x 12
x
10
2
20
200
3
+
(20 x 10)
40
(20 x 2)
30
0.5
+
(3 x 10)
+
6
(3 x 2)
5
235
(0.5 x 10)
+
1
(0.5 x 3)
47
282
1
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Multiplication
Stage 5
Vertical format, expanded working
Eventually, children may be introduced to a vertical format. They should first practise this with calculations
they can do mentally. They can also investigate starting with the ones first rather than the tens. Children
should describe what they do by referring to the actual values of the digits in the columns. The method is
then extended to multiplying by two-digit numbers.
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Multiplication
Stage 5
56 x 27
38
7
x
2 1 0 (30 x 7 = 210)
5 6 (8 x 7 = 56)
266
56
27
x
38 x 7
1 0 0 0 (50 x 20 = 1000)
1 2 0 (6 x 20 = 120)
3 5 0 (50 x 7 = 350)
4 2 (6 x 7 = 42)
1512
1
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Multiplication
Stage 6
Vertical format, compact method
Eventually, children may be introduced to a vertical format. They should first practise this with calculations
they can do mentally. They can also investigate starting with the ones first rather than the tens. Children
should describe what they do by referring to the actual values of the digits in the columns. The method is
then extended to multiplying by two-digit numbers.
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Multiplication
Stage 6
56 x 27
38
7
x
26
6
5
56
27
x
38 x 7
1 1 2 0 (56 x 20)
3 9 2 (56 x 7)
1512
1
Division
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Division
Stage 1
Understanding division as:
Sharing equally occurs when a quantity is shared out equally into a given number of portions, and we can
work out how many are in each portion.
e.g. 6 ÷ 2 (share 6 sweets between 2 children)
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Division
Stage 1
Understanding division as:
Grouping (or repeated subtraction) occurs when we are asked to find how many groups of a given size are
equivalent to the original quantity. For example how many groups of 2 marbles are in a set of 6 marbles, the
calculation
6 ÷ 2 (how many 2s in 6?)
-2
Shown/calculated on a number line:
0
-2
2 4 6
-2
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Division
Prerequisite Skills
Counting back in steps of equal size
Halving
Multiplication and division facts/strategies
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Division
Stage 2
Informal written methods – subtracting multiples of the divisor (‘Chunking’)
Division by subtracting multiples of the divisor…
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Division
Key facts:
10  7 = 70
5  7 = 35
2  7 = 14
17=7
Stage 2
100 ÷ 7
2 lots 2 lots
-14
0
2
10 lots
-14
16
-70
30
100 ÷ 7 = 14 r2
100
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Division
Stage 3
Progressing to an expanded vertical method:
234 ÷ 13
234
130
104
65
39
26
13
13
0
(10 x 13)
(5 x 13)
(2 x 13)
(1 x 13)
Key facts:
10  13 = 130
5  13 = 65
2  13 = 26
1  13 = 13
234 ÷ 13 = 18
St Augustine’s Catholic Primary School
Parent Numeracy Workshop
Progression in Calculation
Division
Stage 4
Standard written methods:
196 ÷ 6
6 196
180
16
12
(30)
(2)
4
196 ÷ 6 = 32 r4
or
196 ÷ 6 = 32 4/6
Useful Sites
National curriculum
http://www.education.gov.uk/schools/teachingandlearning
Information and services for parents
https://www.education.gov.uk/publications/standard/publicationDetail/Page1/DCSF-00267-2010
www.parentchannel.tv
www.parentlineplus.org.uk
www.direct.gov.uk/en/EducationAndLearning
Homework and enrichment
www.bbc.co.uk/schools
www.bbc.co.uk/learning
www.channel4learning.com/apps/homeworkhigh
www.gridclub.com
www.schoolzone.co.uk
www.nrich.maths.org
www.sumdog.com (Children have been signed up to this site)
http://www.educationcity.com