Transcript Addition Year 7 Year 8 Year 9

Pencil and Paper Methods – Part II
LEA Recommendations for
Key Stages 1 – 3 & 4..?
Leicestershire Numeracy Team 2004
The approach to calculation
When faced with a calculation children should ask
themselves;
• Can I do this in my head?
• Can I do this in my head using drawings or
jottings?
• Do I need to use a pencil or paper procedure?
• Do I need a calculator?
This policy looks at the progression in developing
consistent pencil and paper procedures across the
school.
Key messages
• The policy shows the route MOST children
should be able to follow successfully and with
understanding. You might want to introduce
other methods to more able pupils so that
they can have the opportunity to explore and
use them too.
• The methods build on the mental strategies
children have/ are developing.
Structure of policy…
Addition
Year 7
Year 8
Year 9
Addition objectives
Addition objectives
82-85 Understand addition, as it applies to
whole numbers and decimals; know how to
use the laws of arithmetic and inverse
operations.
82–5 Understand addition of fractions and integers; use the laws of
arithmetic and inverse operations.
86–7 Use the order of operations, including brackets.
92-101 Consolidate and extend mental
methods of calculation, to include
decimals, fractions and percentages
accompanied where appropriate by suitable
jottings.
Addition – key questions
• Which methods for addition have the feeder
high schools taught?
•Do we have enough emphasis on the use of
number lines throughout the school? Are they
used to model all four operations?
•Are calculations presented with the equals
sign and empty numbers in a variety of
places?
• ‘Partitioning and recombining’, ‘empty
number lines’, ‘vertical methods’ – which
method do we teach?.
Addition
Year 7
Year 8
Mental methods
Year 9
Use compensation by adding too much, and then
compensating
Use jottings such as an empty number line to support
or explain methods for adding mentally e.g.
Partition and deal with the most significant digits first
Pencil and paper procedures (Written methods)
Extend to decimals with up to 2 decimal
places, including:
 sums with different numbers of digits;
 totals of more than two numbers.
Consolidate methods learned and
used in Year 6 and extend to harder
examples of sums with different
numbers of digits and differing
numbers of decimals.
e.g.76.56 + 312.2 + 5.07 = 398.83
e.g.5.05 + 3.9 + 8 + 0.97 = 17.92
590.005 + 0.0045 = 590.0095
Add and subtract fractions – use
diagrams to illustrate.
Subtraction – key questions
Do we use a range of vocabulary for
subtraction or do students only associate it with
take away?
Do we place enough emphasis on difference
(how many more/how many less) throughout
the department?
Do we agree to teach complementary addition
instead of decomposition?
Do we want to show how complementary
addition can be written vertically or do we just
want to teach it using an empty number line?
Subtraction
Year 7
Year 8
Year 9
Mental methods
Use jottings such as an empty number line to support or explain methods for adding mentally.
Use compensation by subtracting too much, and then compensating
Pencil and paper procedures (Written methods)
Extend to decimals with up to 2 decimal
places, including:
• differences with different numbers of
digits
• totals of more than two numbers.
Consolidate methods learned and used in Subtract more complex fractions
previous years and extend to harder
For example:
examples of differences with different
numbers of digits.
Counting back
Complementary addition
Multiplication – key questions
Which method for multiplication is taught in the
department/across the school?
Is the grid method of multiplication taught in the
school?
Do students need another method for
multiplication
(e.g. long multiplication)?
Multiplication
Year 7
Year 8
Year 9
Mental methods
Partition either part of the product e.g.7.3 x 11 = (7.3 x 10) + 7.3 = 80.3
13 x 1.4 = (10 x 1.4) + (3 x 1.4) = 18.2
OR
Use the grid method of multiplication (as below).
Pencil and paper procedures (Written methods)
Use written methods to support, record or explain
multiplication of:
3 digit by a 2 digit number
a decimal with one or two decimal places by a
single digit
decimals with up to two decimal places
.
Pencil and paper procedures (Written methods)
The grid method can be used to develop the understanding that
algebra is a way of generalising from arithmetic, for example, when
expanding the product of two linear expressions.
Multiply a fraction by a fraction
Key questions - division
Is chunking used for division in the High
Schools?
Have empty number lines been used to
demonstrate the idea of ‘repeated subtraction’?
Has enough emphasis been placed on division
by grouping (or do pupils only know how to
share)?
Division
Year 7
Year 8
Year 9
Mental methods
Use mental or informal written methods to calculate e.g.
Use inverses to check results e.g.
703 ÷ 19 = 37 appears to be about right, because 36 x 20 = 720
Pencil and paper procedures (Written
methods)
Use written methods to support, record or explain
division of:
§a three-digit number by a two-digit number
§a decimal with one or two decimal places by a
single digit.
Refine methods to improve efficiency while maintaining
accuracy and understanding.
109.6 ÷ 8 is approximately 110 ÷ 10 = 11.
109.6
- 80
29.6
- 24
5.6
5.6
0.0
(10 groups of 8)
Pencil and paper procedures (Written methods)
Continue to use the same method as in Year 7 and Year 8. Adjust the dividend and
divisor by a common factor before the division so that no further adjustment is needed
after the calculation
e.g. 361.6 ÷ 0.8 is equivalent to 3616 ÷ 8
Use the inverse rule to divide fractions, first converting mixed numbers to improper
fractions.
Look at one half of a shape.
How many sixths of the shape can
You see? (six)
So, how many sixths in one half? (three)
So
(3)
( 0.7 )
Answer: 13.7
6785 ÷ 25 = 6785 ÷ 5 ÷ 5
½ ÷ 1/6
= ½ x 6/1
= 6/2
= 3
Additional Questions for Heads of Maths to
consider…
• Are pupils encouraged to estimate the size of an
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answer and use it to check that their answer to
a written calculation is reasonable and sensible?
Are pupils given regular opportunities to use and
apply written calculation methods efficiently to
solve a range of problems, including word
problems?
Do you know the approaches to written
calculation used by your feeder schools? If not,
how can you find out?
Do you emphasise the same approaches to
calculation within your school, to those used in
your feeder schools, particularly for those
working below expected levels of attainment?
Additional Questions for Heads of Maths to
consider continued…
• Do you have an agreed statement of practice for written
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•
calculation within your department and across your
school?
Does emphasis continue to be placed on the
development of mental strategies which help pupils in
their recording of written methods?
Do you use pupil’s errors and misconceptions in written
methods as part of a ‘cognitive conflict’ teaching
approach to help improve understanding and accuracy?
And finally …
Is there whole school agreement on the methods in the
policy? If not are there alternatives, which all staff
agree on? How will consistency and progression be
maintained?