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Heswall Primary School Written Calculation Policy This policy reflects the values and philosophy of Heswall Primary School in relation to the written calculation policy. It has been developed to ensure progression and continuity in the recording of mathematics across all year groups. This will result in a consistent approach throughout the school. This document is intended for use by all teaching staff, and for information to governors, inspectors and the Local Authority advisory service. Each teacher will develop the mental strategies for their children appropriate to the year group with modifications for the more able and less able. Less able children do not have to progress through all levels. Within the school there will be a development of recording mental strategies, which allows the children to make their thinking visible. The work recorded in the children’s books will only reflect a sample of the mathematical learning which they experience, as some work is done practically and through investigative tasks. Children will record their calculations on squared and plain paper, depending on the topic and year group. Children in all year groups may also record on white boards. In Foundation Stage Two the emphasis is placed on children working practically to develop their emerging mathematical knowledge. They will begin to demonstrate their thinking through their own drawings, pictures and marks. Initially children’s learning will focus on counting skills and recognising the size of numbers. E.g. a group of five teddies can be counted and labelled with the figure ‘5’ to represent the total number of the group. In Foundation Two the children will be given opportunities to recognise, write and compare numbers to ten. Children will develop understanding of plus and minus signs by using appropriate contexts, alongside the use of the equals sign. The equals sign will be used accurately: representing ‘the same as’. E.g. 3 + 4 =7 + 8 = 15 is not appropriate. Children will have experience of a range of word sentences in a diversity of forms: eg 3 + 4 + 5 = 12 7=4+3 3+ 4 = 5 + 2 In all calculations, ensure children: Maintain, develop and refine their mental methods. Understand and use an appropriate written method if necessary. Recognise when and why methods are efficient. Always begin by asking ‘Can I do this in my head?’ Record numbers using 1 digit per square with the exception of fractions where the denominator is to be written in one square. The decimal point is to written on the line between squares. Policies: Curriculum – 3/10 ADDITION Children will learn to use a full range of mathematical vocabulary as they progress through the school. Children will use a variety of addition methods, beginning with a pictorial representation of what they have done practically, combining two groups of objects into one group and saying how many altogether by counting all the objects. Children will develop their ability to add two sets together by counting on from a given number, usually the largest number. The children will develop their addition strategies using a number line, initially practically, then using a marked number line, progressing to an empty number line. Empty number lines should not be introduced until children have a lot of experience using numbered number lines and 100 squares to aid calculation. Addition recorded on the empty number lines supports and improves pupils’ mental strategies but does not progress into a standard written form. Numbered number line example (Y1) 8 + 4 = 12 0 1 2 3 4 5 6 7 8 9 10 11 12 The empty number line helps the children to demonstrate their mental calculations. Empty number line example (Y2/Y3) 15 + 7 =22 +5 15 +2 20 22 Children will then move on to ‘ partitioning ’ or splitting numbers up into tens and units, then hundreds, tens and units and ‘recombining’ them to find the total. These calculations can be set out horizontally. Policies: Curriculum – 3/10 2 24 + 23 = 47 +10 24 +10 34 +3 44 +20 47 +3 24 44 47 Or partitioning, eg 24 + 23 40 24 20 4 Tens Units Total + 7 = 47 + + + 23 20 = 3 = 40 7 47 Y3 example 238 + 123 = 361 +10 +100 238 338 +100 238 348 +20 338 +3 +10 358 361 +3 358 361 Or vertically, eg Hundreds Tens Units Total 238 200 30 8 + 147 = + 100 = 300 + 40 = 70 + 7 = 15 385 It is important to keep the digits in the correct place value. Policies: Curriculum – 3/10 3 Expanded Method The next stage is an ‘expanded’ method of calculation. This method will only be taught when the children can: - add and subtract two single digit numbers mentally add and subtract multiples of ten mentally partition numbers into tens and units mentally add and subtract two, two digit numbers mentally and record the method informally. The following expanded form of algorithm will be used for addition. This serves as an introductory stage to more condensed formats. Children can use brackets to show the expanded format for each number. Y4 example H 2 2 + T 3 2 5 0 5 4 4 U 2 5 7 0 0 7 ( ( 2 ( 3 0 2 0 0 + + + 5 2 0 0 0 2 ) ) ) Progressing to: + H 2 2 T 3 2 4 4 5 0 5 U 2 5 7 0 0 7 Children should only be working on numbers this large when they are able to add multiples of 10 and 100 with confidence. Standard Compact Method The standard compact method will only be introduced to children who are already confident using the expanded method. It is not expected that all children will be using this method and they may choose a method of calculation in which they are confident and accurate. Some children will continue to calculate using a number line throughout the school. + H 2 1 3 T 3 4 8 U 8 7 5 1 Policies: Curriculum – 3/10 Th 7 8 H 6 3 0 T 5 4 0 1 1 1 + U 8 6 4 4 Explicit language must be used in standard compact method. + H 2 3 6 T 4 9 4 1 1 U 7 5 2 7 plus 5 equals 12, 12 is one ten and two units. Put the two in the units column and carry the ten in the tens column [underneath]. Forty plus ninety equals 130, plus the extra ten equals 140. Put the 40 in the tens column and carry the 100 in the hundreds column [underneath]. 200 plus 300 plus the extra 100 equals 600. Put the 600 in the hundreds column. The total equals 642. Policies: Curriculum – 3/10 5 SUBTRACTION Children will begin with a pictorial representation of what they have done practically, taking away a number of objects. 8–2=6 Pop! Pop! Empty Number Lines [from Year 2] Empty number lines should not be introduced until children have had a lot of experience using numbered number lines to aid calculation. The empty number line will be used to find the ‘difference’ between two numbers. The difference can be found by counting on. Count On Method The term ‘finding the difference’ is introduced in Year 2: Eg find the difference between 7 and 12 Finding the difference between 7 and 12 is 5 +3 7 +2 10 12 Counting on can be done in several ways. Y3 example: counting on in tens or multiples of ten. Policies: Curriculum – 3/10 6 Children will add numbers from the top of the number lines in their heads or with simple jottings. 81 - 37 = 44 +10 37 +10 47 57 +10 67 +40 +3 37 +10 +3 77 +1 80 81 +1 40 80 81 Y4 example – Pupils continue to record on the number line by ‘counting on’. The calculations should be extended to bigger numbers. 146 - 78 = 68 +20 +2 78 80 +40 100 +6 140 146 Children can record their jottings in this way: 40 6 + + 20 2 Policies: Curriculum – 3/10 = = 60 8 68 7 314 - 186 = 128 +10 +4 186 190 100 10 4 + + 10 4 +100 0 200 = = +10 300 +4 310 314 100 20 8 128 NB Some children may not progress beyond this stage – this is ok! Standard Compact Method Explicit language must be used in Standard Compact Method. - H T 7 12 8 5 2 U 3 8 4 1 4 6 8 4 minus 6 you can’t do without going into negative numbers, so I will rearrange the numbers by borrowing a ten from 30 to make it 20. 14 minus 6 is 8. Put the 8 in the units column of the answer. 20 minus 80 you can’t do without going into negative numbers, so I will rearrange the numbers again by borrowing a hundred from 800 to make it 700. 120 minus 80 is 40. Put the 40 in the tens column of the answer. 700 minus 500 is 200. Put the 200 in the hundreds column of the answer. The answer is 248. Check that I still have 834 as the first number – 700 + 120 + 14 is 834. There is no need to teach decomposition at any stage. Many pupils will continue to calculate using a number line to the end of Y6. Policies: Curriculum – 3/10 8 MULTIPLICATION The multiplication and division signs will be introduced in Year 2. Children will use a pictorial representation of what they have done practically, drawing pictures to record a number of sets with a number of objects in: eg 3 sets of 3 Children will then start to use repeated addition and some multiplication facts, to solve multiplication problems: 3 + 3 + 3 + 3 = 12 (4 x 3 = 12) Number Lines Number lines will be used to demonstrate repeated addition +2 0 +2 2 +2 4 6 Arrays Arrays will be used to link mental strategies to written calculation and to demonstrate the distributive and commutative law Y2 Example Arrays 3x4= 4x3= 4 0000 3 0000 0000 Y3 Example 3 x 4 = 4 x 3 (commutative law) Distributive law ie partitioning to aid mental calculation (3 x 4) + (4 x 4) = 7 x 4 16 x 3 = (10 x 3) + (6 x 3) Policies: Curriculum – 3/10 9 Grid Method When children are secure in their understanding of arrays they will be gradually introduced to the grid method. The grid method involves partitioning numbers to multiply and encourages the children to use known facts. It also provides a visual image of multiplication stemming from arrays. Y4 Example 13 x 7 = 91 x 7 1 7 0 0 3 1 2 70 + 21 = 91 Year 5/6 Examples 137 x 3 = 411 x 3 1 3 0 0 0 0 3 9 0 0 2 7 1 300 + 90 +21 = 411 The grid method can then be expanded to use TU x TU and progress to HTU x TU: When multiplying by more than units, eg TU or HTU then add the answers vertically rather than horizontally in order to maintain place values. 17 x 19 – 323 x 10 9 1 1 1 0 9 9 0 0 0 0 1 7 6 3 7 0 3 3 0 0 0 0 6 1 7 + 1 9 1 3 1 2 2 0 3 2 0 3 3 0 0 3 or 1 + 1 3 9 3 2 0 3 3 1 724 x 35 = 25340 x 30 2 5 2 1 3 4 7 0 5 5 0 0 0 0 2 0 0 0 0 0 0 0 1 1 2 2 4 4 0 0 0 + 2 4 2 5 5 7 1 3 0 0 4 4 0 0 0 0 1 The grid method can also be used to multiply decimals in the same way: 34.5 x 6 = 207 x 6 30 180 4 0.5 24 3 Policies: Curriculum – 3/10 180 + 24 + 3 = 207 10 At each stage the children can use the grid method to multiply 2, then 3, then 4 digit numbers. NB Not all children need to learn to multiply using traditional algorithms. Expanded Method Th H 2 T 6 4 6 1 4 8 0 2 x 1 2 U 5 8 0 0 0 0 ( ( 2 ( 6 0 5 0 0 x x x 8 8 8 ) ) ) Progressing to Th H 2 T 6 4 6 1 4 8 0 2 x 1 2 U 5 8 0 0 0 0 As before, the grid method can help with understanding and can eventually lead to the standard compact method of multiplication: Th x 2 H 2 T 6 1 2 5 4 U 5 8 0 Explicit language must be used in standard compact method. 8 times 5 equal 40, 4 tens and no units; Put the zero in the units column and carry the four in the tens column (underneath); 8 times 60 equals 480, plus the extra 40 equals 520; Put the 2 in the tens column and carry the 5 in the hundreds column (underneath); 8 times 200 equals 1600, plus the extra 500 equals 2100; Put the one in the hundreds column and the 2 in the thousands column; The total equals 2120. It is important that the traditional standard compact method is seen as the ultimate objective. Children will need to use an expanded form for such calculations before steps in the method are meaningful and they can use this efficient method with understanding. Children’s progression and readiness will obviously vary considerably. Many children will leave school using the grid method and this is valued by Maths teachers in Secondary schools. Policies: Curriculum – 3/10 11 DIVISION Children will initially meet division as equal sharing or grouping. Children will begin with pictorial representation of what they have done practically and become familiar with the language of sharing and grouping: Sharing I have 6 sweets to share between 3 children. How many will they each get? 2 Grouping 9 flowers are bundled in bunches of 3. How many bunches can be made? 3 3 3 3 Arrays can be used to show the relationship between multiplication and division, eg How many 3’s in 12? 000 000 000 000 How many 4’s in 12 0000 0000 0000 This can also be demonstrated on a number line by jumping forward in groups. 3 0 3 3 Policies: Curriculum – 3/10 3 3 6 9 12 There are 4 groups of 3 in 12 3+3+3+3=12 12 4 4 0 4 There are 3 groups of 4 in 12 4+4+4=12 4 8 12 Year3/ 4 Continue to use number lines (only to work forwards), eg 36 ÷ 4 = 9 There are nine 4’s in 36 4 4 0 4 4 4 8 4 12 16 4 20 4 24 4 28 4 32 36 Extend to using number lines to show remainders, Eg 17 ÷ 5 = 3 r2 There are three 5's in 17 and 2 left over 5 0 5 5 5 10 1 15 1 16 17 Years 4/5 As the children gain in confidence they will be able to make bigger jumps along the number line using multiples, Eg 74 ÷ 5 = 14 r4 10 x5 0 4 x5 50 1 1 1 1 70 71 72 73 74 = 14 r 4 Number lines can continue to be used to calculate without introducing the standard compact method. Policies: Curriculum – 3/10 13 STANDARD COMPACT METHOD The traditional division algorithm presents a very contracted format for recording. Eg 488 ÷ 4 = 122 4 1 4 2 8 2 8 Progressing to 168 ÷ 4 = 42 4 0 1 4 16 2 8 Progressing to 263÷ 4 = 65 r3 4 0 2 6 5 26 23 r3 Children should be reminded of the term quotient and the written abbreviation r for remainder Eg 263 divided by 4 equals a quotient of 65 with a remainder of 3 NB. Again, not all children will learn to divide using the compact method and may continue to use the number line to the end of Y6. NB. It is more important that children can find answers to calculations by any of the methods outlined, than it is to learn the traditional methods or algorithms. Policies: Curriculum – 3/10 14