Developing Written Calculations Addition and Subtraction

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Developing Written Calculations Addition and
Subtraction

• A workshop for all at
• Endon Hall Primary School
• Learning Together And Having Fun
• October 2006

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Mathematics?

• What images and emotions are racing through your
  mind?

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How was it for you?

• What was Maths like for you when you were at
  school?
• Were you good at it?
• How do you use Maths now in your daily life?

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Why are we here?

• I can

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We can
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Programme of Workshops

• Developing Mental Strategies June 2006
• Developing Written Calculations addition and
  subtraction Autumn 2006
• Developing Written Calculations multiplication
  and division Spring 2007

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The Role of Written Calculations

• As calculations become more complex, written
  methods become more important.
• Recording in Maths is an important tool both
  for furthering the understanding of ideas and for
  communicating those ideas to others.
• Warning The introduction of written methods
  too early can undermine childrens fluency with
  number.

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What do we mean by the phrase written calculation?

• Pencil and paper procedure
• Written method
• Formal written method
• Standard written algorithm
• Personal written jottings
• A useful written method is one that helps
  children to carry out a calculation and can be
  understood by others.

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Why Written Calculations?

• To assist in a mental calculation by writing
  down some of the numbers involved
• To clarify a mental procedure for the pupil
• To help to communicate methods and solutions
• To provide a record of work
• To work out calculations which are too difficult
  to be done mentally
• To develop, refine and use a set of rules for
  correct and efficient calculations

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Remember

• The National Numeracy Strategy emphasises the
  teaching of mental calculation throughout the
  Primary phase.
• - To recognise when calculations can
  be done in their heads.
• - To choose effective and efficient
  strategies to work out the answers.

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Why?

• Many children have recurring difficulties with
  written methods of calculation, such as
  forgetting or misapplying a standard method,
  working inefficiently because they do not know
  number bonds or multiplication facts, or using a
  written method when a mental method would be more
  appropriate.

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When Are Children Ready For Written Calculations?

• Do they know addition and subtraction facts to
  20?
• Do they understand place value and can they
  partition numbers?
• Can they add three single digit numbers
  mentally?
• Can they add and subtract any pair of two digit
  numbers mentally?
• Can they explain their mental strategies orally
  and record them using informal jottings?
• Do they have a good feel for number - being
  able to make a good estimation of the answer?

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What do we do in school?

• Jottings, to support a mental calculation
• A record of a mental strategy, written in a way
  that helps someone else to understand
• Extended methods of calculation that prepare the
  way for increasingly efficient written
  calculations that cannot easily be done mentally

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Reception and Key Stage 1

• Make a record in pictures, words or symbols of
  addition and subtraction activities and construct
  number sentences
• Explain to someone else what they have done
• Interpret information that requires practical,
  oral or mental calculation
• Begin to read records made by the teacher
• Help work out steps in a calculation they will
  later do mentally

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Vocabulary

• Fundamental, if the children are to develop the
  concept of addition and subtraction.
• Understanding the words, signs and symbols
  enables the child to try and put into words their
  own developing thought processes.

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Steps In Establishing Place Value

• Remembering number facts and recalling them
  without hesitation e.g. pairs of numbers to make
  10 doubles and halves to 20
• Using known facts to calculate unknown facts
  e.g. 6612 therefore 6713 241034 therefore
  24933

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Establishing Place Value

• Understanding and using relationships between
  addition and subtraction to find answers and
  check results e.g. 14620 therefore 20-614
• Having a repertoire of mental strategies to
  solve calculations e.g. doubles / near doubles
  bridging 10 / bridging 20 adding 9 by 10 -1
  adding 11 by 10 1

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Establishing Place Value

• Making use of informal jottings such as blank
  number lines to assist in calculations with
  larger numbers e.g. 83-18
• 18 20 80
  83
• 65 70 80
  83

60
2
3
-10
-3
-5
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Establishing Place Value

• Solving one-step word problems (either mentally
  or with jottings) by identifying which operation
  to use, drawing upon their knowledge of number
  bonds and explaining their reasoning.
• Beginning to present calculations in a
  horizontal format and explain mental steps using
  numbers, symbols or words.

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Establishing Place Value

• Learn to estimate or approximate first e.g.
  2930 (round up to the nearest 10, the answer
  will be near to 60).
• Developing a wider understanding of numbers in
  context e.g. decimals fractions measurement
  length, weight, capacity, time and money and
  their inter relationships.

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Strategies for addition and subtraction

• Counting forwards and backwards
• Reordering
• Partitioning using multiples of 10 and 100

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Strategies for and -

• Partitioning bridging through multiples of 10
• Partitioning Compensating

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Strategies for and -

• Partitioning using near doubles
• Partitioning bridging through numbers other
  than 10

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Stages in Addition

• signs and missing numbers

3 4
3 7

14 5 10

32 100


35 1 5

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Partitioning

• Partition into tens and ones and recombine

40 50 90
12 23
36 53
53 30 6
10 2 20 3 30 5 35
83 6
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Partitioning
360 70 430

• Partition into hundreds, tens and ones and
  recombine
• Partition into hundreds, tens, ones and decimal
  fractions and recombine

70
3
358 73
358
428
431
358 70 3
35.8 7.3
35.8
42.8
43.1
35.8 7 0.3
0.3
7
36 7 43
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Partitioning as an introduction to the expanded
method in columns

• Add tens first Add ones first

47
40 7
76
70 6
110 13 123
50 80 130
47
47
76
76
13
110
110
13
123
123
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Compact Recording For Addition
630 50 680
625

48
673
370 90 460
367
1
85
452
16.40
17 8 25
1 1
7.68
24.08
1 1
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Stages in Subtraction

• - signs and missing numbers
• Using the empty number line count back

7 3
7 - 3
14 5 20 -
75 30 45
74 27
-20
-4
-3
74
54
50
47
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Using The Empty Number Line

• Subtract 9 or 11 19 or 21 by adjusting.
• The counting on method. (Find the difference)

35 10 25
35 9
-10
1
35
25
26
21 17
1
1
1
1
21
17 18 19 20
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Partitioning
74 30 44

• 74 27
• 74 20 7
• 54 7 47
• 329 115
• 300 100 200
• 20 - 10 10
• 9 - 5 4

330 120 210
214
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The Counting Up Method

74 - 27
75 30 45
3 40 4
30 70 74
47
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Decomposition

2 1 5 1
3167 - 739
3150 -750 3100 100 600
3000 600 2400
2428
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How do you feel now?