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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