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Maths Written Calculation Policy

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Maths Written Calculation Policy Powered By Docstoc
					       St Paul’s R.C. Primary School.

Progression from mental to written methods of
calculation …………….
Foundation through to Year 6.


November 2006.




IMPORTANT ASPECTS OF CALULATION:

       There needs to be a systematic teaching of number facts and strategies to
        support pupil’s calculations.

The following needs to be built in at all stages of teaching written methods of calculation.

       1st thought should be, ‘ Can I do it in my head?’
       Children need to be taught the stages of layout for calculation
        i.e. presented horizontally then calculated in a vertical format if
        appropriate.
       Children need to be encouraged to EXPLAIN how they arrived at their answer; this
        is a way of supporting their mathematical thinking. Good use of questioning will
        help them.
       Children need to ESTIMATE the answer to a calculation first.
       Children need to CHECK their answer e.g. Is the answer reasonable?
      Children should aim to use the most EFFICIENT STRATEGY for the calculation
       they are working on.
      When appropriate you need to link mental strategies to the introduction of
       expanded and compact written methods.
      Children need to use personal jottings to support and record mental strategies or
       explain their methods of calculation.
      Errors and misconceptions need to be addressed quickly.
      Effective assessment enables teachers to help the children move forward with
       their learning.
      THERE NEEDS TO BE AN AGREED COMMON APPROACH IN RECORDING AND
       LAYING OUT OF CHILDRENS WORK. [ Poor recording eg. In place value can cause
       careless errors. ]
      If in doubt use your National Numeracy Strategy there are examples for every
       year group, check for guidance in the book ‘The National Numeracy Strategy,
       Teaching written calculations.’ If still unsure ask your Maths Coordinator.




STRUCTURED APPROACH: EXAMPLES FOR + AND –

Mental counting, counting objects, FU to year 6.
Counting in 1’s from and back to zero, using a wide and varied range of approaches.
Counting from different starting points.
Counting in tens: 46,56,66,76,86,96,106….. Then 100’s.
Counting in decimals …. In fractions…
Through zero to negative numbers.

Early stages of mental calculation, learning the facts, years 1,2 and 3.
Understanding and using the correct vocabulary.
Knowing all addition and subtraction facts to 10, then 20.

                                      ,
Empty box: 10= + 3, 1 + 2 + 3 + 1 =  1 + 2 + 3 = + 1.

Knowing and understanding that 6 + 4 = 10 so 10 – 6= 4* the children need to be clear
about the inverse relationship between + and -. [especially useful is empty boxes, eg. +
13 = 17. ]
Knowing that 8 + 8 = 16, so that 8 + 9= 17 and 7 + 8 = 15.
Knowing that 67 – 10 = 57, so 67 – 9 = 58.
Complementary addition is an informal mental method building on the understanding of
number lines from K.S.1.
Children need to identify cases that can be calculated mentally eg.
583 - 499 [by adjustment to 583 – 500 + 1]
361 – 358 [by counting up]
Working with larger numbers and informal jottings, years 2 and 3
Adding and subtracting multiples of 10 and 100
Finding complements in 100, eg. 66 + = 100
56 + 37 = 56 + 30 + 7
_________30__________4____3______
56                     86 90         93




68 + 79 =   60 + 8
             70 + 9
            130 + 17 = 147



 Teachers need to demonstrate a bridge between recording in horizontal and vertical
layouts.
Standard expanded written methods beginning late in year 3 and used in year 4.
Partitioning :
                      HTU                  HTU
                      438                  438
                      275                  275
                      600                    13 Beginning with most
                      100                   100 significant number
                        13                  600 moving to least
                      713                   713 significant number.




  Standard written methods begin in year 4 depending on ability and carried on in year 5.
Column addition and subtraction with ‘carrying’ and adjustment, money and decimals.

Use of calculators beginning in year 4. The children need to be taught to judge when it is
sensible to use mental, written or calculator methods, to choose the appropriate method
or combination of methods, and to apply accordingly. Extending written methods to
complex numbers, decimals in year 6.



             *******************************

Structured Approach : examples for multiplication and division.

Mental counting and counting objects, years 2 and 3
Counting in 2’s or 5’s ……
Breaking off ‘sticks’ of cubes: how many 2’s make 20.
Counting on from and back to zero in 3’s and 4’s

Early stages of mental calculation, learning facts, years 2 and 3.
Knowing doubles of small numbers and corresponding halves.
Knowing that 10 + 10 + 10 = 30
Knowing that 10 x 3 = 30 and 30/3 = 10
= 10 x 3 10 x = 30

Working with larger numbers and informal jottings, years 2,3,4.
Double 17 is double 10 plus double 7, 20 + 14
37/5 = 7r
Multiplying, then dividing by 10, 100, 1000

Non- standard expanded written methods, beginning in year 2 for top group




   [ Year 2 ] 15 x 2



                            ____10_________5___



                       2       20        10    = 30
                           ___________________




14 x 6
                 ___10___________4__
               6      60        24             = 84      10 x 6 = 60
                 ___________________                      4 x 6 = 24
                                                         14 x 6 = 84

Standard written methods, years 5 and 6.
Short multiplication, long multiplication, short division
Multiplication and division of decimals by whole numbers
Use of calculators beginning in year 4

Division …… Chunking

Progression in objectives:
Year 2: begin to understand division as grouping [ repeated subtraction ] or sharing
Contexts: division as grouping
          Bakers rolls/buns in packets of 4
          Eggs packed 6 to a box
          Children into groups of 7
          Apples in bags of 6

Year 3: understand division as grouping [ repeated subtraction ] or sharing.
Begin to model repeated subtraction as an informal written method eg. A grocer has a
box of 34 apples. He wants to sell them in bags of 5. How many bags can he fill?
           34/5       34
                       -5
                       29
                       -5
                       24
                       -5
                     ETC       answer 6 r4

Year 4: Develop and refine written methods for TU/U
Initially repeatedly subtract the divisor as in year 3, but using larger numbers eg. 72/5.
Children will find this time consuming and will be more than ready to spot short cuts eg.
Subtracting a multiple of 10 of the divisor
            72/5           72
                           50   10 groups of 5
                           22
                           20    4 groups of 5
                            2

                         answer 14r2

Begin to introduce bracket notation for division eg. 96/6

                        6 96
                          60       [ 10 x 6 ]
                          36
                          36       [6x6]
                           0     answer 16




Year 5: Extend written methods to short division HTU by U [ with integer remainder ] As
year 4 but beginning to subtract greater multiples of 10 of the divisor eg.
                    256/7
        256
        -70 [ 10 x7 ]
         185
             -140 [ 20 x 7 ]
               46
              -42 [ 6 x7 ]
                4
           answer 36 r4

Moving forwards subtracting the largest multiple of 10 of the divisor as possible

Year 6: extend written methods to….
Short division of TU or HTU by U [ mixed number answers ]
Division of HTU by TU [ long division, whole number answers ]
Short division of numbers involving decimals.
Knowing when to round up and down i.e. boxes of eggs.




                        Y1                              Y2                            Y3                               Y4
            Foundation –                     Signs and missing            Framework – Section 5,          Framework –Section 6 p.48          Fra
            Verbal counting on and back      numbers. S5 P.24             p.43                                                               p.49
            to 20                            Continue using a range of    Autumn and Spring terms-
            Repeated addition.               equations as in Year 1 but   develop Partitioning Nos.       Begin Informal Written             Intro
            Relates addition to              with appropriate, larger     and Empty No.Line with          Method. Then Expanded              Met
            combining two groups.            numbers.                     increasingly difficult nos.     Written Method with most           mas
            Counting real objects.           Extend to:                   Summer term – Children to       significant digit HTU first in     Met
            Counting in 2’s to 20            14 + 5 = 15 + 5 -1           present calculations            order to lead on to least          but
            Counting in 10’s to 100          And adding three numbers     vertically, but only record     significant digit first. (ensure   app
            Combining groups                 32 +     +     = 100         most significant digit first,   largest number to goes at top      (If a
            Begin to count on to 100 by                                   some then use both, units       of sum)                            to u
            summer term depending on         And brackets 12 + 23 = (10   digit first and tens using      231 + 323=                         add
            group ability.                   +2) + (20 + 3)               Informal Written Method.                                           first
            Use of simple number lines.                                                                      323         323                 to e
            Use of practical materials for                                45 + 53 =                       + 231        + 231                   66
            counting & problem solving.      Partitioning into tens and                                     300            8                 + 4
            Drawing adding pictures.         ones and recombine             45           45                   50          50                   60
            Language relating to size         12 + 23 = 10 + 2 + 20 +3      +53         +53                    8         300                   10
            Year 1, S5, P24                  (loop partitioning)             90          8                 554           554
            + = signs and missing            = 30 + 5                         8          90
Addition




            numbers                          = 35                           98          98                Leading to: 323 + 237                70
            3+4=                             Refine to 23 + 12 =                                                                             Enc
                = 3 + 4 Inverse              23 + 10 + 2                                                    323       323                    zero
            Partitioning 5 + 8 =             = 33 + 2                                                     + 237     + 237
            3+2+2+1                          = 35                                                           500       560                    Lea
            Word calculations:                                                                                50       1                     diff
            4 plus 8 = or                         +10         +2                                              10                             mon
            Add 30 to 60                                                                                    560
            Bead & number lines              23            33 35                                                                               6.
            Counting on keeping biggest      By the end of the year                                                                          + 4.
            number in head.                  moving on to:                                                                                    10
            Use of number squares and        Add 9 or 11 by adding 10
            fans to see.                     and adjusting by 1                                           AIM – Most children use
            Drawing number pictures.          35 + 9 = 44                 AIM –Most to add TU + TU        expanded written method             10
            Number lines 100                    +10           -1          using the Informal Written      when appropriate ie
            Using materials to add up                                     Method least sig. Some          HTU + HTU least and most           AIM
            and number fans.                                              children HTU + HTU.             sig or several numbers.            use
            Double 4, half of 6              35           34      45      Most & least Significant        Some using least sig               met
            Simple partitioning T & U i.e    Start with larger number     informal methods side by        informal and standard              Num
            13 = 10 + 3 or 25 = 20 + 5       when adding.                 side.                           together.                          dec
            Range of equations.                                                                                                              num
            Recognise + and = signs          Secure knowledge of Place
            Secure knowledge of              Value & Partitioning. Some
            number bonds.                    simple vertical addition.
Approaches to written methods – expectations for each year
(Subtraction)

                            Y1                                 Y2                              Y3                          Y4
                Foundation –                      Continue to use a range of      Framework – Section 5,        Framework – Section 6,         Fram
                                                  equations                       p45                           p.50                           p51
                Finding one less than, up to      Use mathematical language,      In Summer Term:               (Do not do Complimentary       (Do
                5,10.                             subtract, take away:            Teach children to write the   addition method as a written   com
                Using the number line to take     3 less than 7?                  Informal Written method by    method)                        a wr
                away.                             How many less than 7 is         partitioning number for                                      Tea
                (numbers up to 10)                18?                             subtracting (Ensure                                          Writ
                Taking one away from a group                                      numbers can not be            Teach children to use          ThH
                of objects.                                                       subtracted mentally)          Expanded Written Method        calc
                Picture/story representations     Subtraction is the inverse of                                 for HTU – HTU, initially       exch
                of sums.                          addition.                                                     where calculation has only     exp
                Relate less than to take away.    But 4 - 2 =                     Partitioning                  one exchange (T U), but       dec
                Use of stimulating /appropriate       2 – 4 =?                                                  moving on to 2 exchanges.
                objects and materials.                                            87 – 23 =
                Vocabulary - sharing,             Addition doubles related to                                                                  644
                difference.                       halving:                          80 + 7                      453 – 247 =
                Year 1                            13 + 13 = 26                    - 20 + 3                                     13
  Subtraction




                - = signs and missing             Doubles of multiples from 5       60 + 4 = 64                         40 10 + 3               600
                numbers.                          + 5 to 50 + 50                                                  400 + 50 + 3                 -500
                                                                                                                - 200 + 40 + 7                  100
                7–3=                               25 – 8 = ?                     634 - 23 =
                                                                                                                200 + 0 + 6 = 206              = 11
                                                   10 =    +8                       600 + 30 + 4
                                                                                  -       20 + 3                 (Exchange T to U)              (E
                Use of appropriate number         Partitioning numbers             600 + 10 + 1 = 611                                          Som
                lines/100 squares to count        70 – 11 = 59                                                                                 met
                back.                             70 – 10 = 60 – 1                                                                             exp
                Counting back in head from a                                                                    AIM: Most children to use
                given number.                       47 – 23=                                                    the Expanded Written
                Use fingers to take some                                          AIM: Most children to use     Method
                away.                               40 + 7                        the Informal Written          HTU – HTU and
                Number rhymes with props.         - 20 + 3                        Method for TU – TU.           HTU – TU. Use Informal
                Reciting number bonds.              20 + 4 = 24                   Some HTU – TU Children        Written Method for
                Add or subtract 10 or 20 from                                     able to record 3 digit        subtraction of money.
                29 etc witn materials.            AIM: Some subtracting           numbers in partitioned        Some adjusting T to U.         AIM
                Find the difference using         using the number line.          form.                                                        Stan
                cubes/materials/toys etc.         Most start subtracting                                                                       for
                Phrase questions in different     using partitioning.                                                                          the
                ways = subtract 2 from 10                                                                                                      Met
                Understand the language of                                                                                                     dec
                subtraction difference, take                                                                                                   num
                away, how many more.

        Approaches to written methods – expectations for each
                        year (Multiplication)
                              Y1                                Y2                               Y3                              Y4
                 Foundation –                      Understand and use the           No formal method. But             Framework, Section 6           Fr
                                                   language: double, times,         children should understand        p.66                           p.6
                 Investigate natural multiples     multiply and X                   multiplication as repeated        Teach Grid method with a
                 by grouping, arranging and                                         addition and as describing        range of 2/3 digit numbers.    Sh
                 sorting. E.g. Pairs of socks in   Understand multiplication as     an array.                         (see Framework)                Te
                 2’s                               repeated addition:                                                                                Me
                 Counting in 2’s and 10’s.         5 + 5 + 5 = 3 lots of 5          They need to know their 2x,                                      rec
                 Related addition to combining     or 3 x 5                         5x and10x table and begin to      Using      numbers   which     wo
                 two groups.                                                        know their 3x and 4x table.       children     can    multiply   Re
                                                                     2x3=6                                            mentally      using      the   Y4
                                                                                    In order to provide the           partitioning method from
                                                      3                             children with some of the pre-    year 3, begin to teach the     Lo
                                                   3x2=6                            requisite skills for Y4 written   grid method.                   be
                                                   Doubling / halving               approaches the objective                                         me
                                                   11 x 2 = 22                      ‘Use knowledge of number          23 x 8                         xT
                 Year 1                            Inverse                          facts and place value to
                                                   22 divided by 2 =11              multiply or divide mentally.’                                    7.2
Multiplication




                 Sorting objects/material into                                      (Framework Section 5 p. 57)        x          20        3
                 groups to count.                  6 x 2 =2x6                       is useful.
                 Investigating natural multiples                                                                       8         20        4
                 by grouping, arranging and                                         Begin to develop informal         = 184                           X
                 sorting e.g.                                                       ways of calculating and           Leading on to TU x TU
                 Eggs in a box, corners on a            X       =                   recording eg 17 x 5 by
                 square, fingers, gloves.                                           partitioning and recombining.                                     3
                                                                                    For example:                       x          70        2
                 Counting in 2’s, 5’s and 10’s.    Using number squares                             17 x 5                                            0
                 Grouping numbers, corners of      Responding to oral questions.                                       30         2100      60
                 shapes.                           6 times 2
                 Repeated addition.                5 multiplied by 2                     10x5=50                       8          560       16
                                                                                          7x5=35                                                     =
                                                   12 x 2                                                              = 2736
                                                   (10 x 2) + (2 x 2) = 20 + 4           17x5=85
                                                   =24
                                                                                     x          10         7
                                                    x           10        2                                           AIM: Most children are
                                                                                     5        50         35           able to use the grid           AIM
                                                    2           20        4         = 85                              method with confidence         ab
                                                                                    Most securing partitioning,       to multiply TU x U. Some       for
                                                    Most children to be confident   some moving on to grid            children using TU x TU         dig
                                                   with doubling and halving        method TU xU use both                                            sim
                                                   numbers some children to         methods at same time.                                            Gr
                                                   begin partitioning and grid                                                                       mu
                                                   method with simple numbers.
                                                   TU x U


Approaches to written methods –expectations for each
year (Division)
                       Y1                              Y2                               Y3                            Y4
           Foundation –                    Divisions = signs and missing   written approaches are:-        Framework, Section 6          Fr
                                           numbers.                        - objectives on p49 Section 5   p.68                          p.6
           Sharing out objects, toys and                                   (develop concept of division    Autumn and Spring terms -     Co
           materials into groups. (In      6 –divided by 2 =               as grouping, inverse of         teach ‘chunking’ –            eff
           context where possible)         .. /.. /..                     multiplication. As well as      repeated subtraction,         Fra
           Recognise differences in        Understand division as          sharing).                       initially of the divisor ,    pro
           quantity when comparing         sharing and grouping            - p.51 Section 5                moving on to 10 x divisor     su
           sets of objects. ½ of a cake.   6 divided by 2 can be           (remainders)                    and multiples of divisor.     1–
           Chunks of a number line.        modelled as:                    - table facts, see Framework                                  ch
                                                                                                                                         ap
                                           Grouping – There are 6                                          Recorded like this:           thi
                                           sweets. How many people         Also Make sure children               72 ÷ 5 = 14 r 2
                                           can have 2 each?                become competent at
                                                                           subtracting multiples of 10              72
                                           (How many 2’s make 6?)          from any number eg 117 –              - 50 (10 x 5)
                                                                           20/30/ 40 etc                            22
                                                                                                                  - 20 (4 x 5)
                                            1     2    4 6                                                            2
Division




                                           Saring – 6 sweets are shared             27 ÷ 6 = 4 r 3         For 2,3,4,5,10 as divisor.
                                           between 2 people. How many
                                           do they have each?                      27                      For other divisors subtract
                                                                                  - 6 (1 group of 6)       10 x and the chunks of the
           Year 1                                                       21                      divisor
                                                                                - 12 (2 groups of 6)
           Sharing out objects, toys and   12 divided by 2=                          9                                                   19
           materials into groups. (In                                               6 (1 groups of 6)                                    (C
           context where possible)              12
           Sharing pictures.                   - 4 2 groups of 2                   3
           Solving sharing problems               8
           with number pictures.             - 4 2 groups of 2
           Using hoops to share                   4                        For 2,3,4,5,10 as divisor.
           objects, and sharing equally         - 4 2 groups of 2
           into groups.                            0
           Encourage the vocabulary of                   =6                                                AIM – most children           AIM
           division, divide, share and                                     No formal written method. But   using ‘chunking’ method       ab
           group.                          Clearly identify the number     some of the pre-requisite       by subtracting chunks of      eff
                                           you divide by.                  skills that must be developed   10x the divisor (as           Me
                                                                           for the children to be ready    Method B in Framework).       div
                                                                           for Y4                          Are able to divide 2/3        Bu
                                                                           Some children may be ready      digit number by single        ca
                                                                           for repeated subtraction.       digit.                        rel
                                                                                                                                         les

				
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