Developing Written Calculations Multiplication and Division

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Developing Written Calculations Multiplication
and Division

• A workshop for all at
• Endon Hall Primary School
• Learning Together And Having Fun
• March 2007

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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 (now the Primary
  Framework for Mathematics) 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 know multiplication and division facts?
• 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, subtraction, multiplication and
  division 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 multiplication and division.
• 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. 6 x 7 42 60 x 70 4200
• 70 x 5, 70 x 50, 700 x 5, 700 x 50

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Establishing Place Value

• Understanding and using relationships between
  addition, subtraction, multiplication and
  division to find answers and check results e.g. 6
  x 7, 42 6, 42 7, 7 x 6
• To know that multiplication is repeated addition
  and that division is repeated subtraction
• 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

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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 facts 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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Multiplication and division skills and strategies

• Knowing multiplication and division facts to 10
  x 10
• Multiplying and dividing by multiples of 10

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Multiplication and division skills and strategies

• Multiplying and dividing by single-digit numbers
  and multiplying by two-digit numbers
• Doubling and halving

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Multiplication and division skills and strategies

• Partitioning and recombining

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Examples for x and

• Mental counting and counting objects, Counting in
  2s, 5s, 10s, Breaking off 'sticks' of cubes
  How many twos make 20?Counting on from and back
  to zero in 1s, 2s, 5s, 10s, then moving on to
  3s, 4s.
• Early stages of mental calculation, learning
  facts, Knowing doubles of small numbers, and
  corresponding halvesKnowing that 10 10 10
  30Knowing that 10 x 3 30, and 30 3 10

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Examples for x and

• Working with larger numbers and informal
  jottings. Double 17 is double 10 plus double 7,
  or 20 1437 5 7 R 2Multiplying, then
  dividing, by 10, 100, 1000
• Non-standard expanded written methods, Standard
  written methods.

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Examples for x

• Working with larger numbers and informal jottings

4 x 8
32
1x4
1x4
1x4
1x4
1x4
1x4
1x4
1x4
4
0
8
12
16
20
24
28
32
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Recording mental multiplication using partitioning

• 14 x 3
• Partition 14 into 10 4
• 10 x 3 30 4 x 3 12 30 12 42
• 43 x 6
• Partition 43 into 40 3
• 40 x 6 240 3 x 6 18 240 18 258

x
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Examples for x
Progression

• Grid method of multiplication

Using Arrays
Written method
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Examples for x

• Grid method of multiplication

13
3
10
60 18 78
60
18
6
6
13 x 6
78
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Examples for x
Progression

• Grid method of multiplication

Using Arrays
Written method
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Examples for x

• Grid method of multiplication

• Have a go at

47 x 35

• Remember to draw
• your grid and record
• your answers.

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Expanded short multiplication
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• 56  27 is approximately 60  30  1800.

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Examples for

• Division by Chunking

Grouping
Written method
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Examples for
Progression
14
r2
86 6
10 x 6
4 x 6
1x6
1x6
1x6
1x6
0
26
20
14
8
2
86
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Short Division

• 81 divided by 3
• Partition 81 into 60 21
• 60 divided by 3 20
• 21 divided by 3 7
• 81 divided by 3
• 20 7 27

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• Have a go at

Progression
86 6
14
r2
847 17
6) 86

• Remember to decide
• on the size of chunks you
• can jump in.

60
10 x 6
26
24
4 x 6
2
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What can you do?

• Be positive about maths, even if you dont feel
  confident about it yourself.
• Talk and listen to your child about their work in
  maths. It will help your child if they have to
  explain to you.
• Share maths activities they take home.
• Play number games
• Help with learning number facts

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How do you feel now?