Addition – key objectives Step 1 Begin to relate addition to combining two groups of objects and subtraction to ‘taking away’ In practical activities and discussion begin to use the vocabulary involved in adding and subtracting Step 2 Step 3 Relate addition to counting on; recognize that addition can be done in any order; use practical and informal written methods to support the addition of a one-digit number or a multiple of 10 to a one digit or two-digit number. Use vocabulary related to addition and subtraction and symbols to describe and record addition and subtraction number sentences. Use number songs and stories, counting books etc… Practical recording eg. • • + Add or subtract mentally a onedigit number or a multiple of 10 to or from any two-digit number; use practical and informal written methods to add and subtract twodigit numbers. Understand that subtraction is the inverse of addition and vice versa; use this to derive and record related addition and subtraction number sentences. Step 4 Step 5 Add or subtract mentally combinations of one-digit and twodigit numbers. Add or subtract mentally pairs of two-digit whole numbers (eg. 47 + 58, 91-35). Develop and use written methods to record, support or explain addition and subtraction of twodigit and three-digit numbers. Refine and use efficient written methods to add and subtract twodigit and three-digit whole numbers and £.p Use a calculator to carry out onestep and two-step calculations involving all four operations; recognize negative numbers in the display, correct mistaken entries and interpret the display correctly in the context of money. Use the symbols +, -, x, ÷ and = to record and interpret number sentences involving all four operations; calculate the value of an unknown in a number sentence (eg. ÷2=6, 30- =24) ENL (bead strings) • • “Hit the tens” method Counting in ones on bead string /frames • Number bonds to 10 (Hearts in love etc) • ENL jumping to the next ten. • +3 +10 +6 -1 37 66 = 37 40 50 13 +2 1+2=3 8 60 37 + 29 = 66 66 37 + 29 = 66 +3 10 13 8 + 5 = 13 • • • • Dice games Number tracks Introduce bead strings Counting in ones from numbers to ten. • “Tens jumping” (preferred method) • 66 +10 +10 37 47 +3 57 37 + 29 = 66 +6 60 66 • 66 +30 66 +10 “Over jumping” Partitioning 37 = 30 + 7 29 = 20 + 9 30+ 20 = 50 7 + 9 = 16 So 37+29 = 66 67 “Expanded method” 37 +29 50 (30+20) 16 (7+9) 66 Addition – key objectives Step 6 Step 7 Use efficient written methods to add and subtract whole numbers and decimals with up to two places. Use efficient written methods to add and subtract integers and decimals. Extend mental methods for whole number calculations. Use a calculator to solve problems involving multi-step calculations. • “Compact method” 37 +29 66 1 Calculate mentally with integers and decimals: U.t ± U.t, TUxU, TU÷U, U.txU, U.t÷U • “Compact method” 126.4 +115.3 241.7 1 Step 8 Extend carrying method to numbers with at least four digits, including decimals in a range of contexts. Eg. Money and measurement. • Adding decimals with a mixed number of decimal places. 3.59 + 0.7 + 5 3.59 0.70 +5.00 9.29 1 Step 9 Step 10 Extend understanding of addition to add fractions with the same denominator and fractions with denominators of the same number. • Add fractions with equal denominators. 2+2=4 5 5 5 4+5=9= 6 6 6 Extend understanding of addition and equivalent fractions to add fractions with different denominators and mixed numbers. • Add fractions with unequal denominators. 1+1=4+1=5 2 8 8 8 8 1 3 6 2 + 3 = 8 + 9 = 17 = 3 4 12 12 12 1125 Subtraction – key objectives Step 1 Step 2 Begin to relate addition to combining two groups of objects and subtraction to ‘taking away’ In practical activities and discussion begin to use the vocabulary involved in adding and subtracting Step 3 Understand subtraction as “take away” and find a “difference” by counting up; use practical and informal written methods to support the subtraction of a onedigit number from a one digit or two-digit number and a multiple of ten from a two-digit number. Use vocabulary related to addition and subtraction and symbols to describe and record addition and subtraction number sentences. • • • • • Use number songs and stories, counting books etc… Dice games Number tracks Introduce bead strings. Practical recording eg. • • • • • • - Number bonds to 10 (Hearts in love etc) Introduce vocabulary of subtraction. Bead strings to assist mental calculation. Use +/-/= to record calculation. Use number line for counting back. Use addition/ counting on to solve subtraction, find the difference. Understand that subtraction is the inverse of addition and vice versa; use this to derive and record related addition and subtraction number sentences. • • • ENL (bead strings) Counting back Counting on / find the difference. 3 = +1 2 3-1=2 Step 4 Add or subtract mentally a one-digit number or a multiple of 10 to or from any two-digit number; use practical and informal written methods to add and subtract two-digit numbers. +1 3 5-2=3 +10 +2 5 Develop and use written methods to record, support or explain addition and subtraction of two-digit and three-digit numbers. • • Consolidate ENL methods used in Y2. Use number complements to make larger jumps. 51 15 +1 4 Step 5 Add or subtract mentally combinations of one-digit and twodigit numbers. 18 20 +1 +3 30 33 - 18 = 15 +50 33 39 40 90 - 39 = 51 90 Add or subtract mentally pairs of twodigit whole numbers (eg. 47 + 58, 91-35). Refine and use efficient written methods to add and subtract two-digit and three-digit whole numbers and £.p Use a calculator to carry out one-step and two-step calculations involving all four operations; recognize negative numbers in the display, correct mistaken entries and interpret the display correctly in the context of money. • Convert ENL into column methods of recording. • Many pupils may continue to use number line if more comfortable withy this. 130 - 79 21 (100) 30 (130) 51 Subtraction – key objectives Step 6 Step 7 Use efficient written methods to add and subtract whole numbers and decimals with up to two places. Extend mental methods for whole number calculations for example to subtract one near multiple of a thousand from another (eg. 60704097) • Extend column subtraction with larger numbers and decimals. 675 -136 4 (140) 60 (200) 400 (600) 75 (675) 539 Step 9 Step 8 Use efficient written methods to add and subtract integers and decimals, to multiply and divide integers and decimals by a onedigit integer, and to multiply two digit and three digit integers by a two digit integer. Use understanding of subtraction to subtract fractions with the same denominator. Step 10 Apply understanding of subtraction and equivalent fractions to subtract fractions with different denominators and mixed numbers. Use a calculator to solve problems involving multi-step calculations. • 600 Introduce expanded decomposition. 140 700 40 9 - 200 50 2 400 90 7 Calculate mentally with integers and decimals: U.t ± U.t, TUxU, TU÷U, U.txU, U.t÷U • Move towards Compact decomposition including decimals. 2 • Subtract fractions. • 4–2=2 5 5 5 1–1=4–1 =3 2 8 8 8 8 1 36.57 - 17.46 19.11 Subtract fractions with unequal denominators. 2 1 – 3 = 10 – 3 2 4 4 4 =7= 4 1 43 Multiplication – key objectives Step 1 Step 2 Count repeated groups of the same size. Solve practical problems that involve combining groups of 2, 5 or 10, or sharing into equal groups. Know doubles of all numbers to ten. Step 3 Step 4 Step 5 Represent repeated addition and arrays as multiplication, and sharing and repeated subtraction (grouping) as division; use practical and informal written methods and related vocabulary to support multiplication and division, including calculations with remainders. Derive and recall multiplication facts for the 2, 3, 4, 5, 6 and 10 times-tables and the corresponding division facts; recognize multiples of 2, 5 or 10 up to 1000. Multiply one-digit and two-digit numbers by 10 or 100, and describe the effect. Know by heart all multiplication facts for times table up to 10 x 10. Use the symbols +, -, x, ÷ and = to record and interpret number sentences involving all four operations; calculate the value of an unknown in a number sentence (eg ÷ 2 = 6, 30 - = 24) • Count objects in sets, be able to say whether sets are the same, larger or smaller. • • • Learn double to twenty Count on and back in 2s, 5s and 10s. Using bead frames look at the pattern of five. • • • Multiplication as repeated addition 4x3=4+4+4 Use number lines to solve problems with missing numbers eg. Use practical and informal written methods to multiply and divide two-digit numbers (eg. 13 x 3, 50 ÷ 4); round remainders up or down, depending on the context. Understand that division is the inverse of multiplication and vice versa; use this to derive and record related multiplication and division number sentences. • Understand multiplication is the inverse of division. • Partitioning. 0 2x2 Develop and use written methods to record, support and explain multiplication and division of twodigit numbers by a one-digit number, including division with remainders (eg. 15 x 9, 98 ÷ 6) • Grid method of multiplication, then develop into more formal recording. 24 x 6 x 20 x2=6 1x2 Multiply and divide numbers to 1000 by 10 and then 100 (wholenumber answers), understanding the effect; relate to scaling up or down. 4 6 120 24 3x2 144 6 • • Learn multiplication facts for 2s, 5s and 10s by heart. Introduce formal recording and x sign. 13 x 3 = 10 x 3 = 30 3x3= 9 39 • Work with place value charts x10 and x100 H T U 6 2 x 10 6 2 0 Multiplication – key objectives Step 6 Step 7 Step 8 Extend mental methods for wholenumber calculations, for example to multiply a two-digit by a onedigit number (eg. 12 x 9) to multiply by 25 (eg. 16 x 25). Use knowledge of place value and multiplication facts to 10 x 10 to derive related multiplication and division facts involving decimals (eg. 0.8 x 7, 4.8 ÷ 6) Use understanding of place value to multiply and divide whole numbers and decimals by 10, 100 or 1000. Use knowledge of multiplication facts to derive quickly squares of numbers to 12 x 12 and the corresponding squares of multiples of 10. Refine and use efficient written methods to multiply and divide HTU xU, TU x TU, U.t x U • Extend grid method 34 x 56 x 30 4 50 1500 200 1700 6 180 24 204 Use efficient written methods to multiply integers and decimals by a one-digit integer, and to multiply two-digit and three-digit integers by a two-digit number. Further extend grid • method. • Develop into expanded column method. 56 x 34 1500 (30x50) 180 (30 x 6) 200 (50 x 4) 24 (6 x 4) 1904 Extend the compact method to ThHTUxU, ThHTUxTU and decimals up to 3 decimal places. • Short multiplication 342 X 7 2394 • Long multiplication 124 X 26 2480 744 3224 • 3 30 6000 1200 90 7290 6 1200 240 18 1458 Apply understanding in the use of formal written methods for short and long multiplication and to multiply pairs of proper fractions. Short multiplication 4.38 X 7 30.66 21 x 200 40 Step 10 Apply their knowledge and understanding of multiplication and use materials and diagrams to multiply proper fractions and mixed numbers by whole numbers. 243 x 36 8748 1904 Step 9 2 5 • Long multiplication 16.741 X 35 502.23 83.705 585.935 • Multiply proper fractions by whole numbers. 1 1 • Compact column method 56 x 34 1680 (56x30) 224 (56 x 4) 1904 • Short multiplication 2741 X 6 16446 4 2 • Long multiplication 3124 X 26 62480 18744 81224 111 2 x 3 3= 2 2 12 x 3 = 7 21 • Multiply pairs of proper fractions. 2 x 4= 8 3 5 15 Division – key objectives Step 1 Share objects into equal groups and count how many in each group. Step 2 Step 3 Step 4 Solve practical problems that involve combining groups of 2, 5 or 10 or sharing into equal groups. Represent repeated addition and arrays as multiplication, and sharing and repeated subtraction (grouping) as division; use practical and informal written methods and related vocabulary to support multiplication and division, including calculations with remainders. Use practical and informal written methods to multiply and divide two-digit numbers (eg 13x3, 50÷4); round remainders up or down, depending on the context. Multiply and divide numbers to 1000 by 10 and then 100 (wholenumber answers), understanding the effect; relate to scaling up or down. Understand that division is the inverse of multiplication and vice versa; use this to derive and record related multiplication and division number sentences. Develop and use written methods to record, support and explain multiplication and division of twodigit numbers by a one-digit number, including division with remainders (eg 15x9, 98÷6) Use the symbols +, -, x, ÷ and = to record and interpret number sentences involving all four operations. Step 5 Find unit fractions of numbers and quantities (eg. ½, ¼ and ⅛ of 16 litres) Find fractions of numbers, quantities or shapes (eg ⅛ of 24 plums, ⅜ of a 6 by 4 rectangle) Use a calculator to carry out onestep and two-step calculations involving all four operations; correct mistaken entries and interpret the display correctly. • • Link to doubling and halving Know halves of even numbers to 20 • • Introduce division symbol ÷ Division as sharing … 14÷3 = 4 r 2 • Inverse operation 4x3=12 • So … • Use knowledge of times tables facts to divide on ENL 12÷3=4 Counting on/back on ENL 12 10x6 -3 Sharing • Division as grouping (this is more useful as an introduction to ENL) 14÷3 = 4 r 2 0 -3 -3 0 -3 12 2x6 60 72 Division – key objectives Step 6 Step 7 Use understanding of place value to multiply and divide whole numbers and decimals by 10, 100 or 1000. Refine and use efficient written methods to multiply and divide HTU x U, TUxTU, U.txU and HTU ÷U Find fractions using division (eg 1/100 of 5kg) and percentages of numbers and quantities (eg 10%, 5% and 15% of £80) Use a calculator to solve problems, including those involving decimals or fractions (eg find ¾ of 150g); interpret the display correctly. • Lead into more formal recording of “Target method” 56 56÷4=14 14x4 56 10x4 40 4x4 16 2x4 8 1x4 4 0 Continue to develop “Target method”, refine to make steps more efficient. • 182 182÷7=26 26x7 182 25x7 20x7 10x7 5x7 175 140 70 35 1x7 7 0 Step 8 Step 9 Step 10 Calculate mentally with integers and decimals: U.t±U.t, TUxU, TU÷U, U.txU, U.t ÷U Use efficient written methods to add and subtract integers and decimals, to multiply and divide integers and decimals by a onedigit integer, and to multiply twodigit and three-digit integers by a two-digit integer. Relate fractions to multiplication and division (eg 6÷2= ½of6 = 6 x ½); express a quotient as a fraction or decimal (eg 67÷4 = 16.75 or 16¾); find fractions and percentages of whole-number quantities (eg ⅝ of 96, 65% of £260) Use a calculator to solve problems involving multi-step calculations. • Lead into more formal recording. 12 6 72 60 (10 x 6) 12 (2 x6) • Continue to develop compact methods. 12 6 72 • Include decimals 2.14 7 14.98 Division – key objectives Step 11 Children will use their understanding of the relationship between multiplication and division to apply short division for whole numbers up to ThHTU÷U and interpret remainders in context. • Continue to develop compact methods. Step 12 Children should extend understanding to convert remainders to fractional OR decimals as requested. • Include decimals 432 449 2 1 5 8 3456 11 12 5391 449 r 3 5 3 12 11 12 5391 449.25 5 11 3 6 12 5391.00 Step 13 Step 14 Step 15 Children use their understanding of division to use a formal written method to divide numbers up to 4 digits by 2 digits. Any remainders should be shown as fractions when requested. Children should also apply their knowledge and understanding of division and use materials and diagrams to divide proper fractions by whole numbers . • Include decimals 28 r 12 15 432 300 (20x15) 132 120 (8x15) 12 • Include decimals 28 15 432 300 (20x15) 132 120 (8x15) 12 4 5 12 = 4 15 5 • Include decimals 28.8 15432.0 30 132 120 12.0 12.0 0 Calculation Strategy Lantern Community Primary School 2014 Foundation Stage Key Stage 1 Key Stage 2 Foundation to Year 6 Pencil and paper procedures This policy contains the key pencil and paper procedures that will be taught at the Lantern. It has been written to ensure consistency and progression throughout the school. Although the focus of the policy is on pencil and paper procedures it is important to recognise that the ability to calculate mentally lies at the heart of the Numeracy Framework. The mental methods in the Teaching children to calculate mentally document will be taught systematically from Reception onwards and pupils will be given regular opportunities to practise the necessary skills. However mental calculation is not at the exclusion of written recording and should be seen as complementary to and not as separate from it. In every written method there is an element of mental processing. Sharing written methods with the teacher encourages the children to think about the mental strategies that underpin them to develop new ideas. Therefore written recording both helps children to clarify their thinking and supports and extends the development of more fluent and sophisticated mental strategies. During the time at school children will be encouraged to see mathematics both as a written and spoken language. Teachers will support and guide children through the following important stages: • Developing the use of pictures and a mixture of words and symbols to represent numerical activities; • Using standard symbols and conventions; • Use of jottings to aid mental strategy; • Use of pencil and paper procedures; • Use of a calculator. This policy concentrates on the introduction of standard symbols, the use of the empty number line as a jotting to aid mental calculation and on the introduction of pencil and paper procedures. It is important that children DO NOT abandon jottings and mental methods once pencil and paper procedures are introduced. Therefore children will always be encouraged to look at a calculation/problem and then decide what is the best method to choose: pictures, mental calculation with or without jottings, structured recording or a calculator. Our long-term aim is for children to be able to select an efficient method of their choice that is appropriate for a given task. They will do this by always asking themselves: ‘Can I do this in my head?’ ‘Can I do this in my head using drawings or jottings?’ ‘Do I need to use a pencil and paper procedure?’ ‘Do I need a calculator?’ The following pages aim to provide a quick reference guide for teachers illustrating how these procedures are developed stage by stage. It is important to realise that pupils must not abandon stages until they can demonstrate that they fully understand it and can employ it successfully in a range of contexts. These steps are not a direct substitute for year by year progression. Young mathematicians can have their confidence dashed very easily if they are forced into rigorous recipe methods that they don’t understand, are unclear and have no connection to prior learning. As a result the use of the empty number line can be seen as a firm basis for all four operations in any year as appropriate. Progression in calculation Key skills • • • • • • • • Develop counting and the language of counting Develop understanding of the process of counting and the number system Secure conceptual understanding of place value Use knowledge of known facts Develop confidence in mental calculation skills Understand operations and the relationship between them Foster mathematical thinking Develop skills in recording including jottings Counting development • Enactive – actions, practical, real life context • Iconic – representation, models, images, pictures • Symbolic – recording, numbers, letters Stages • • • • • • Counting all Counting on Counting on from the first number Counting on from the larger number Using a known fact Using knowledge of place value Preparing for place value From Iconic to symbolic: Use concrete objects, practical experiences and pictures to develop conceptual understanding of place value. Progression using models, images and pictures will secure symbolic understanding and secure knowledge of place value. Towards Calculation Confidence in mental calculation skills Conceptual understanding of place value is secure Practise and use the skill of estimation Understand the process being used Compare and evaluate methods Develop a broad range of experiences in recording calculations – Can I do this in my head? Do I need a jotting? Do I need a written method? • Foster mathematical thinking and discussion • Make the maths real – link mathematics across the curriculum and to the outdoors • • • • • •
© Copyright 2026 Paperzz