* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Maths Calculation Policy - The Batt C of E Primary School
Survey
Document related concepts
Mechanical calculator wikipedia , lookup
History of mathematics wikipedia , lookup
Infinitesimal wikipedia , lookup
Ethnomathematics wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
History of logarithms wikipedia , lookup
Approximations of π wikipedia , lookup
Large numbers wikipedia , lookup
Real number wikipedia , lookup
Positional notation wikipedia , lookup
Elementary mathematics wikipedia , lookup
Elementary arithmetic wikipedia , lookup
Transcript
Calculation Objectives (Year 1 – 6) Year 1 Addition and subtraction Pupils should be taught to: read, interpret and practise writing mathematical statements involving addition (+), subtraction (-) and equals (=) signs accurately. add and subtract 1-digit and 2-digit numbers to 20 (9 + 9, 18 - 9), including zero. add three 1-digit numbers. recall and use number bonds and related subtraction facts within 20. solve simple word problems that involve addition and subtraction. Multiplication and division Pupils should be taught to: recognise and write the multiplication symbol (x) and the division symbol (÷) in mathematical statements, calculating the answer with the teacher using concrete objects. solve word problems involving simple multiplication and division, with teacher support. Year 2 Addition and subtraction Pupils should be taught to: rapidly recall and use addition and subtraction facts to 20. add and subtract numbers with up to two 2-digits including using column addition without carrying and column subtraction without borrowing. add and subtract numbers mentally including: - a 2-digit number and ones - a 2-digit number and tens - two 2-digit numbers use subtraction in ‘take away’ and ‘find the difference’ problems. recognise and show that addition can be done in any order (commutative) and subtraction cannot. recognise and use addition and subtraction as inverse operations including to check calculations. solve word problems with addition and subtraction of numbers with up to 2-digits. The Batt School Calculation Policy 2015 1 Multiplication and division Pupils should be taught to: recall multiplication and division facts for the 2, 5 and 10 multiplication tables. use the multiplication (x), division (÷) and equals (=) signs to read and write mathematical statements. write and calculate mathematical statements for multiplication and division within the multiplication tables. recognise and use the inverse relationship between multiplication and division to check calculations. ensure pupils can recognise and show that multiplication can be done in any order (commutative) and division cannot. solve word problems involving multiplication and division. Year 3 Addition and subtraction Pupils should be taught to: add and subtract numbers with up to 3 digits, including using columnar addition and subtraction. accurately add and subtract numbers mentally including: pairs of one- and 2-digit numbers; 3digit numbers and ones; 3-digit numbers and tens; 3-digit numbers and hundreds. solve word problems including missing number problems, using number facts, place value, and more complex addition and subtraction. Multiplication and division Pupils should be taught to: recall and use multiplication and division facts for the 2, 3, 4, 5, 8 and 10 multiplication tables. write and calculate mathematical statements for multiplication and division within the multiplication tables; and for 2-digit numbers x 1-digit numbers, using mental and written methods. solve word problems involving the four operations, including missing number problems. Year 4 Addition and subtraction Pupils should be taught to: add and subtract numbers using formal written methods with up to 4 digits. accurately add and subtract numbers mentally including two 2-digit numbers. estimate, within a range, the answer to a calculation and use inverse operations to check answers. Multiplication and division The Batt School Calculation Policy 2015 2 Pupils should be taught to: recall multiplication and division facts for multiplication tables up to 12 x 12. mentally perform multiplication and division calculations quickly and accurately, including multiplying by 0 and dividing by 1. multiply or divide 2-digit and 3-digit numbers by a 1-digit number using formal written methods; interpret remainders appropriately as integers. recognise and use factor pairs within 144. solve word problems involving the four operations. find the effect of dividing a 2-digit number by 10 and 100, identifying the value of the digits in the answer as units, tenths and hundredths. Year 5 Addition and subtraction Pupils should be taught to: add and subtract whole numbers with up to 5 digits, including using formal written methods. add and subtract numbers mentally with increasingly large numbers. add and subtract numbers with up to three decimal places. Multiplication and division Pupils should be taught to: identify multiples including common multiples, and factors including common factors. know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers. establish whether a number up to 100 is prime and recall the prime numbers up to 19. multiply numbers up to 4-digits by a 1 or 2-digit number using a formal written method, including long multiplication. accurately multiply and divide numbers mentally drawing upon known facts. divide numbers up to 4 digits by a 1-digit number and 10 and interpret remainders appropriately. multiply and divide numbers by 10, 100 and 1000. recognise and use square numbers and square roots and the notation for square (2) and square root (√) solve word problems involving addition and subtraction, multiplication and division. Year 6 Addition, subtraction, multiplication and division Pupils should be taught to: add and subtract negative integers. multiply numbers with at least 4-digits by 2-digits of whole number using long multiplication. The Batt School Calculation Policy 2015 3 divide numbers up to 4-digits by a 2-digit whole number using long division, and interpret remainders as whole number remainders, fractions, decimals or by rounding. perform mental calculations, including with mixed operations and large numbers. use estimation to check answers to calculations and determine in the context of a problem whether an answer should be rounded, or written as a fraction or a decimal. carry out combined operations involving the four operations accurately and state the order of operations. solve word problems involving addition, subtraction, multiplication and division. identify the value of each digit to three decimal places and multiply and divide numbers up to three decimal place by 10, 100 and 1000. multiply and divide numbers with up to two decimal places by 1-digit and 2-digit whole numbers recognise and use division in the context of fractions, percentages and ratio. solve linear missing number problems, including those involving decimals and fractions, and find pairs of number that satisfy number sentences involving two unknowns. The following pages show our written calculation methods for maths. They are organised by number operations – Addition (+), Subtraction (-), Division () and Multiplication (x). Each section shows the progression of written calculation methods for each operation starting from simple methods and progressing to more advanced ones. Children will progress through the stages according to their mathematical knowledge and understanding rather than year group. The Batt School Calculation Policy 2015 4 Addition Calculation Method . Explanation Step 1 Initially, children will work with pictures and objects. 2 + 3 = 5 At a party, I eat 2 cakes and my friend eats 3. How many cakes did we eat altogether? Children draw a picture to help them work out the answer. 7 + 4 = 11 Children use dots or symbols to represent objects (quicker than drawing a picture) 7 people are on the bus. 4 more get on at the next stop. How many people are on the bus now? ……. …. We would then progress to; 7 + …. We would then progress to counting on from 7. = 11 Children use a number line to record jumps made. Using single jumps… 5 + 3 = 8 +1 5 6 +1 +1 7 8 9 10 11 6 + 4 + 3 =13 + 4 6 or larger jumps. + 3 10 The Batt School Calculation Policy 2015 13 5 Step 2 Draw an empty number line. Children will be taught to place the largest number at the left side of the line and jump forward in appropriate steps. 7+9=16 +1 9 +6 10 16 13 + 19= +1 19 +10 20 This method is also used when adding 19, 29, 39 etc +2 30 32 7 + 9 = (7 + 10) – 1= 16 + 10 7 -1 16 17 Children use a sound 17 – 1 = 16 or….. understanding of adding 10 to adapt calculations. Instead of + 10 adding 9, they add 10 then adjust - 1 the calculation by subtracting 1. 7 This is also used for adding other numbers e.g. 15 + 19 (15 + 20, - 1) This is a mental strategy. 127 + 74 = 201 There are 127 boys and 74 girls in a Children will move to partitioning school. How many children are there using a number line. altogether? Children will start their number line at 127. Add a jump of 70 to 197. Either add 3 then 1 or simply +70 +4 a jump of 4 to land on 201. 127 197 201 The Batt School Calculation Policy 2015 6 Step 3 57 + 25 = 50 + 20 = 70 7 + 5 = 12 70 + 12 = 82 Or 57 + 25 = 57 + 20 = 77 77 + 5 = 82 The next stage is to record mental methods partitioning tens and ones separately. This can be done by adding the tens and then the ones and finding the total or by partitioning only the second number. This can also be done working with the ones first to prepare for more formal written methods. Step 4 24 + 35 9 50 59 Children will be introduced to a formal written method. No regrouping is involved at this early stage and the expanded method should be taught first to ensure children understand the place value. It is important that the correct language is used. They are adding 2 tens to 3 tens and 4 ones to 5 ones. 24 + 35 59 Step 5 298 +358 6 1 298 +3 5 8 56 1 1 298 +3 5 8 656 1 1 The Batt School Calculation Policy 2015 When children are confident using the expanded method, this can be shortened into the traditional compact method using regrouping, initially using two digit numbers. 7 Step 6 2786 + 2568 = 2 786 people visited the museum last month. The numbers increased by 2 568 this month. How many people altogether visited this month? Children continue to use the traditional compact method using regrouping to solve problems with larger numbers. 2786 + 2568 5354 111 Step 7 24.566 39 . 7 0 0 0.560 Children will use the compact method to add larger numbers and decimals up to 3 places. 64.826 1 1 1 The Batt School Calculation Policy 2015 8 Subtraction Calculation Method Explanation Step 1 Take away 5 - 2 = 3 Initially, children will work with pictures and objects, removing a quantity and understanding there is a smaller number left. I had five balloons. Two burst. How Drawing a picture helps children to many did I have left? visualise the problem. Using dots or tally marks is quicker Mum baked 7 biscuits. I ate 3. How than drawing a detailed picture. many were left? 7 - 3 = 4 Take away Step 2 Children start counting backwards on a marked number line in order to subtract. The Batt School Calculation Policy 2015 9 The empty number line helps to record or explain the steps in mental subtraction by counting back from the highest number in appropriate steps which enable the children to bridge through a multiple of ten. Counting Back 15 – 7 = 8 74 – 27 = 47 The steps may be recorded in a different order Step 3 Lisa has 7 felt tip pens and Tim has 3. How many more does Lisa have? Find the difference The Batt School Calculation Policy 2015 Once children have a secure understanding of how to subtract by counting backwards, the concept of ‘finding the difference’ by counting on can be taught using pictures or objects and then a number line. 10 Step 4 834 - 378 = The library owns 834 books. 378 are out on loan. How many are on the shelves? -8 456 -70 464 -300 534 834 John has £5.60. If he spends £3.99 on a game, how much will he have left? Once children have a secure understanding of how to subtract by counting backwards and by finding the difference, both are practised using larger numbers. Children should be taught how to choose which method is more efficient for a particular calculation. Children count up (from the smallest number to the biggest) using an empty number line. It is easiest to count up to a multiple of 10 or 100 (a friendly number). Step 5 7 4 – 2 7 = 7 4 – 2 0 = 5 4 5 4 – 7 = 4 7 Children can progress to recording their subtraction using partitioning. For 74 – 27 this involves partitioning the 27 into 20 and 7, and then subtracting from 74, the 20 and the 7 in turn. This can also be done subtracting the ones first. The Batt School Calculation Policy 2015 11 Step 6 Children will be introduced to a formal written method. No regrouping (decomposition) from other columns is introduced until the children have a secure understanding of place value through regrouping in column addition. 8 9 - 5 5 3 4 Step 7 Decomposition 6 1 5 7 2 - 2 4 5 3 2 7 Once the children are secure with place value and regrouping in addition, they can move onto regrouping (decomposition). _____ 2 1 3 3 The children will use the same method to work with; column and decomposition but the value of the digits should not exceed 4 digits. 1 3 5 4 7 - 2 6 2 8 0 9 1 9 Step 8 The children will then progress to decomposition using numbers which contain zero as a place holder. Step 9 2 1 3 1 1 7 1 3 5 4 7.885 - 2 6 2 8.278 0 9 1 0.507 The Batt School Calculation Policy 2015 The children will use the same method to work with larger numbers up to 5 digits and decimals to three decimal places. 12 Division Calculation Method Explanation Step 1 Sharing spots between two parts of a ladybird More pictures! Drawing often gives children a way into solving the problem. 6 ÷ 2 = 3 6 Easter eggs are shared between 2 children. How many eggs do they get each? Sharing between 2 There are 6 Easter eggs. How many children can have two each? Grouping in twos The Batt School Calculation Policy 2015 13 Step 2 12 ÷ 4 = 3 4 apples are packed in a basket. How many baskets can you fill with 12 apples? Grouping in fours Dots or tally marks can either be shared out one at a time or split up into groups. During practical activities, use the language of sharing and grouping. 6 ÷ 3 = 2 6 is shared into 3 groups. 15 ÷ 5 = 3 Step 3 15 ÷ 3 = 5 5 Number lines can be used to show groupings and divisions. 5 5 The above method would progress into a number line. When you 3 2 1 = 3 lots of/sets of 4 subtract ‘lots of 4’ or ‘sets of 4’ -4 -4 -4 from 12 until you reach 0. You then 0 4 8 12 count up how many lots of 4 you have subtracted to get the answer 3. 12 ÷ 4= 3 The Batt School Calculation Policy 2015 14 Children should also be aware that not all numbers are exactly divisible = 3 lots of/sets of 4 and that dividing some numbers will with 3 left over leave a remainder. 15 ÷ 4= 3 r 3 3 -4 3 2 -4 7 1 -4 11 15 Step 4 84 ÷ 6= 14 4 lots of6 0 It would take a long time for the children to jump in sixes from 84 so children can jump back in bigger ‘jumps’. 10 lots of 6 24 A jump of 10 groups of 6 takes you to 24. Then you need another 4 lots of 6 which is 24 will take you to 0. Altogether this is 14 sixes. 84 142 ÷ 6= 23 r 4 3 lots of6 10 lots of 6 10 lots of6 4 22 82 142 Step 5 142 ÷ 6= 23 r 4 Children should then be able to divide the number by grouping without the use of a number line. 4 10 3 10 10 3 10 10 3 10 10 3 10 This should also be done with numbers that will leave a remainder. 10 3 10 10 3 10 The Batt School Calculation Policy 2015 Initially, the child will recognise that they can divide 60 into 6 groups by placing 10 into each group. This is then repeated, meaning there is now 20 in each group leaving 22 to be divided by 6. Using their knowledge of division facts they should divide 18 by 6 placing an additional 3 in each group. 15 Step 6 Chunking 65 ÷ 5 = 50 ÷ 5 = 10 15 15 ÷ 5 = 3 The children now use their knowledge of the method above to develop a more formal written method without the use of diagrams. 13 142 ÷ 6= 60 ÷ 6= 10 82 60 ÷ 6= 10 22 18 ÷ 6 = 3 4 23 r 4 142 ÷ 6= 120 ÷ 6= 20 22 18 ÷ 6 = 3 4 23 r 4 The Batt School Calculation Policy 2015 As children become more confident with this method they will be able to work with larger groupings. 16 Step 7 Extended Short Division (600 ÷ 3) (60 ÷ 3) Children with a secure knowledge of the Chunking Method will be able to progress to the more formal Extended Short Division which uses the same principle. It is important that the children are aware of the value of each digit E.g. They know they are dividing 600 by 3 and not 6 by 3. (12 ÷ 3) Step 8 Short Division 73 ÷ 3 = Divide 7 tens by 3 which will give an answer of 2 tens and 1 ten left over. Children now use their knowledge of regrouping to progress to standard Short Division. Exhange the 1 ten for 10 ones Divide the 13 ones by 3 which will give an answer of 4 with 1 remaining. The Batt School Calculation Policy 2015 17 Step 9 Translating remainders into decimals Long Division 033r3 15 4 6 8 45 018 15 03 The Batt School Calculation Policy 2015 Once children are confident using the Short Division with regrouping and remainders, they will progress to converting remainders to decimals. Long division requires the children to be competent and confident with all of the above before they can use it as a division strategy. 18 Multiplication Calculation Method Explanation Step 1 4 x 2 = Each child has two feet. How many feet do four children have? 2 + 2 A picture can also be useful for early multiplication. We say “four lots of 2 feet.” “four groups of 2 feet.” “four sets of 2 feet.” Dots or tally marks are often drawn in groups. This shows 3 groups of 5. + 2 + 2 3 x 5 = There are 5 cakes in a pack. How many cakes in 3 packs? 5 + 5 + 5 Step 2 3 x 4 = A chew costs 4p. How much do 3 chews cost? 3 lots of 4 3 sets of 4 3 groups of 4 3 x 4 Drawing an array showing 3 columns of 4, gives children an image of the answer. Children can now start to use a marked number line to record 3 lots of 4 or 4 lots of 3. 4 x 3 The Batt School Calculation Policy 2015 19 Step 3 3 x 4 = A chew costs 4p. How much do 3 chews cost? Drawing an array (4 rows of 3 or 3 columns of 4) gives children an image of the answer. It also helps develop the understanding that 4x3 is equivalent to 3x4. 3 lots of 4 3 sets of 4 4 x 3 = A chew costs 3p. How much do 4 chews cost? 4 lots of 3 4 sets of 3 Step 4 13 x 7 = 91 There are 13 biscuits in a packet. How many biscuits in 7 packets? +70 +21 10x7 0 When numbers get bigger, it is inefficient to do lots of small jumps. Partition 13 (10 and 3). This gives you two jumps (10x7 and 3x7). The answer is the number you land on 91. 3x7 70 Children will partition using dienes apparatus as well as a number line. The Batt School Calculation Policy 2015 20 Step 5 Children use partitioning to multiply mentally and when using the Grid Method. 7 x 13 = Partitioning. 7 x 10 = 70 7 x 3= 21 70 + 21 = 91 Grid multiplication 54 x 7 = X 50 4 7 350 28 = 3 78 The Batt School Calculation Policy 2015 21 Step 6 34 x 6 Children can progress to using use short multiplication to multiply by a single digit number. Some children may require use of the extended method whilst developing their knowledge of regrouping. 4 x 6 = 24 30 x 6 = 180 204 1 Short Multiplication (extended) 100s 10s 1s 3 x 1 2 4 6 2 4 8 0 0 4 It is important that the children understand and say the place value of each digit as they work through the calculation. E.g The ‘3’ in the example would be referred to as ‘3 tens’ rather than ‘3’. Short Multiplication 100s 10s 1s x 3 2 0 2 2 4 6 4 Short Multiplication 1000s 100s 3 10s 1s 2 5 X Children continue to use short multiplication to multiply larger numbers by single digit numbers. 7 2 2 7 1 3 5 The Batt School Calculation Policy 2015 22 Step 7 Children will return to using the Grid Method to develop their understanding of multiplying numbers by more than one digit. Step 8 Long Multiplication (extended) Once secure with Step 7 children will progress to Long Multiplication with some children initially using the extended method. 59 x 26 Th H T U 59 X 26 354 3 5 1180 1 1 1534 1 The Batt School Calculation Policy 2015 23