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Threshfield Primary School WHOLE SCHOOL PROGRESSION IN CALCULATION STRATEGIES IN MATHEMATICS GUIDANCE FOR PARENTS & FAMILIES Monday 22nd September 2014 1 Stage 1 2 3 4 5 6 Addition Subtraction Multiplication Division Use of practical apparatus to find the total. Recording and developing mental pictures. Use of practical apparatus to take away or find how many more. Recording and developing mental pictures. Progression in the use of a number line. The empty number line as a representation of a mental strategy. Practical equipment using exchange to take away. Use of practical apparatus to find the total of groups of the same size. Recording and developing mental pictures. The bead string, number line and hundred square. Arrays. Use of practical apparatus to share and put into equalsized groups. Recording and developing mental pictures. The bead string, number line and simple multiples. Arrays for division. The grid method. “Short” division Making the link between the practical and columnar subtraction. Compact method. Expanded “short” multiplication. “Long” division Progression in the use of a number line. The empty number line as a representation of a mental strategy. Partitioning into 10s and 1s to lead to a formal written method. Using Diennes / place value counters alongside columnar written method. Compact column method. a) b) c) Short multiplication for TU and HTU multiplied by any number up to 12. Expanded long multiplication. Compact long multiplication. 2 ADDITION 3 Stage 1: Use of practical apparatus to find the total. Recording and developing mental pictures. and 2 + 3 = 5 4 Stage 2: Progression in the use of a number line 8 + 5 = 13 5 Stage 3: The empty number line as a representation of a mental strategy. 34 + 23 = 57 6 Stage 4: Partitioning into tens and ones to lead to a formal written method. 25 + 47 = 72 7 Written partitioning:47 + 25 40 + 20 = 60 60 + 12 = 72 and 7 + 5 = 12 Or 47 + 20 = 67 67 + 5 = 72 8 Stage 5 Using Dienes apparatus / place value counters alongside columnar written method. 25 + 47 = 9 Stage 6: Compact column method 10 SUBTRACTION 11 Stage 1: Use of practical apparatus to take away or find how many more. Recording and developing mental pictures. 12 Stage 2: Progression in the use of a number line. 8–3=5 13 Additional ‘number lines’ - the bead string and hundred square 14 Stage 3: The empty number line as a representation of a mental strategy. Finding an answer by COUNTING BACK – representing taking away. 15 – 7 = 8 15 74 – 27 = 47 “Take away 20” “Take away 7” 16 Finding an answer by COUNTING ON – representing the “difference between”. 17 Stage 4: Practical equipment using exchange to ‘take away’. 72 – 47 72 – 47 = 25 18 Stage 5: Making the link between the practical and columnar subtraction. 72 - 47 19 Stage 6: Compact method 20 MULTIPLICATION 21 Stage 1: Use of practical apparatus to find the total of groups of the same size. Recording and developing mental images. 2 + 2 + 2 + 2 + 2 = 10 5 + 5 + 5 + 5 + 5 + 5 = 30 6 x 5 = 30 22 3 lots of 4 are 12 4 lots of 3 are 12 23 Stage 2: The bead string, number line and hundred square. 24 Stage 3: Arrays 25 26 27 Stage 4: The Grid Method 7 x 8 = 25 + 15 + 10 + 6 = 56 28 4 x13 = 52 This then becomes: x 4 10 40 3 12 40 + 12 = 52 29 18 x 13 100 + 80 = 180 30 + 24 = 54 180 + 54 = 234 30 53 x 16 Estimate 50 x 20 = 1000 x 50 3 10 500 30 6 300 18 Adding the rows is the most efficient calculation: 500 + 300 = 800 30 + 18 = 48 So 800 + 48 = 848 31 Stage 5: Expanded short multiplication 38 x 7 X 30 + 8 7 56 210 266 (7 x 8) (7 x 30) 32 Stage 6: Short multiplication for up to TU x 12 33 53 x 16 53 x 16 18 300 30 500 848 (6 x 3) (6 x 50) (10 x 3) (10 x 50) 34 35 DIVISION 36 Stage 1: Use of practical apparatus to SHARE and put into EQUAL-SIZED GROUPS. Recording and developing mental images 37 Stage 2: Bead strings, number lines and simple multiples. 15 eggs are placed in baskets, with 3 in each basket. How many baskets are needed? 38 Counting on a labelled and then blank number lines. 15 ÷ 3 = 5 39 Stage 3: Arrays for division 56 ÷ 7 Dividend (56) ÷ divisor (7) = Quotient (8) 40 Stage 4: Short division Just as multiplication is repeated addition, so division is repeated subtraction, i.e. taking away the same number repeatedly. 432 ÷ 5 432 200 232 200 32 30 2 (40 x 5) (40 x 5) (6 x 5) “CHUNKING” – initially, this could be done on an open number line. How many 5s have been subtracted in total? Answer = 86, with 2 remaining. 41 42 Stage 5 Long division 43 44 Notes: We teach the children to ask themselves 4 questions – in steps about the calculations they are doing:1. Is it a calculation you can do in your head (mentally)? If yes, then do it mentally. If not, then … 2. Is it a calculation you can do with jottings? If yes, then do it using jottings. If no, then … 3. Is it calculation where you need a more formal written method? If yes, then choose the appropriate method. If no, then … 4. Is it a calculation where you need a calculator? If yes, then use a calculator. 45 There are 7 strands of mathematical learning: Using & applying maths Counting & understanding number Knowing & using number facts Calculating Understanding shape Measuring Handling data Calculating is just one of those strands 46 Using & Applying Maths Calculating Knowing number facts Understanding place value 47