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Luckwell Primary School Calculation Strategy Models and Images and Known Facts Concrete-symbols-abstract Gemma Fricker 2014 Array- a regular arrangement of objects 4 x 3 or 3 x 4 Base 10 Base 10- cubes used to give a visual representation of 1, 10, 100 and 1000 Bridging (through 10/100)-to use knowledge of number bonds to count on or back to the nearest/best 10 as an efficient strategy to get to the final number e.g. 13 + 8 as 13 +7 = 20 + 1 = 21 Cardinality-the size (value) of a number compared to another number Commutativity-that addition or multiplication can be done in any order e.g. 2X3 is the same as 3X2 Compensation- to use known number facts to calculate a number that is near to the one known e.g. 27 + 9 as 27 + 10 then adjust (compensate) by subtracting 1 Cuisenaire-different coloured rods that can be used to represent any number or value Cuisenaire Denominator-the bottom number in a fraction representing the number of parts that make up the whole Distributivity-a law used in algebra e.g. 3(2a+4c) = 6a +12 can be understood as 3 lots of 2 times the first number added to 4 times another number is the same as 6 lots of the first number add 12 Factor-a whole number that multiplies with another number to make a third number e.g. 2 and 5 are factors of 10 Integers-a complete number, not a fraction or decimal e.g. 3, -3, 103 Inverse- opposite/reverse operations e.g. 4 + 6= 10 10 – 4= 6 Numerator-the top number in a fraction representing the number of equal parts out of the whole Numicon- shapes that give a visual representation of numbers from 1-10 and can be combined to make larger numbers Numicon Ordinal number-1st, 2nd, 3rd etc Proportionality-the relationship between two variables that remains constant e.g. in enlarging a piece of A4 paper the long side will always be 2 times the length of the short side For more definitions visit: http://www.amathsdictionaryforkids.com/dictionary.html Guidance This strategy has calculation methods for all 4 operations and is designed to be used in a linear progression. The year groups attached are guidelines in line with the Nation Curriculum 2012, however children should use the strategy appropriate to their development as assessed by the class teacher. Secure knowledge of number facts and in particular place value are key to children’s success in mastering written methods and should be a priority By the end of year 6, children will have a range of calculation methods, mental and written. Selection will depend upon the numbers involved. Children should not move onto the next stage if they are not ready or they are not confident and show a lack of understanding of the method. Children should be encouraged to approximate their answers before calculating. Children should be encouraged to consider if a mental calculation would be appropriate before using written methods. M/O Facts R e c Addition Recognise numbers Children are encouraged to develop a 0-5, then 0-10, to 20 mental picture of the number system in their heads to use for calculation. Count with and order numbers to 20 Children should count with and be able to order nos to 20 Find 1 more and 1 less than numbers to 5, then 10, to 20 Some number bonds Record marks they can interpret and explain Recognise odd and even numbers using apparatus Doubles and halves in context and using apparatus The cardinality of numbers must be made explicit. e.g. The ‘eightness’ of 8 Subtraction Children are encouraged to develop a mental picture of the number system in their heads to use for calculation. They develop ways of recording calculations using pictures, objects etc. Use Numicon covers to ‘hide’ parts e.g. Using quantities and objects, they add two single-digit numbers and count on to find the answer. They develop ways of recording calculations using images, pictures, objects etc. Multiplication 6-2=4 Say the number that is 1 less than a number to 20 In practical activities begin to use the vocabulary involved in subtraction + = Say the number that is 1 more than a number to 20 In practical activities begin to use the vocabulary involved in addition Teachers model the use of the number line Bead strings or bead bars can be used to illustrate subtraction. 6-2=4 . Children will experience equal groups of objects. In songs and rhymes, they will count in 2s and 10s and begin to count in 5s. They solve problems, including doubling Overlaying Numicon They will work on practical problem solving activities involving equal sets or groups. Division Children will understand equal groups and share items out in play and problem solving. They solve problems, including halving and sharing. Overlaying Numicon Y 1 Count to and across Use numbers to 20 100 forwards and Add and subtract 1-digit numbers to back, from 1, 0 or 20, inc 0 any number, inc in ones, twos, fives and Record their work using pictures and tens images Read and write nos to 20 in numerals and words Identify and represent numbers using objects and pictorial representations inc the number line and use language equal to, more than, less than, fewer, most and least Represent and use number bonds within 20 and related subtraction facts 4+6 Doubling and halving Recognise odd and even numbers Use numbers to 20 Record their work using pictures and images 6-2=4 Solve one-step problems involving multiplication and division using concrete objects, pictorial representations and arrays with the support of the teacher Use Numicon covers to ‘hide’ parts + 4x3=12 9÷3=3 = Begin to use symbols = - + < and > to record L1 calculations Begin to use symbols = - + < and > to record L1 calculations Partition numbers in different ways e.g. 7=3+4, 7=2+5, 7-3=4 Solve one-step problems involving addition and subtraction using concrete objects, pictorial representations , and missing number problems e.g 7 = - 9 Memorise and reason with number bonds in different forms e.g. 3+7=10, 10=3+7, 10-3=7 Bead strings or bead bars can be used to illustrate subtraction including modelling how to bridge through ten by counting back 3 then counting back 2. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Children begin to use numbered number lines to support their own calculations counting on in ones Bead strings or bead bars can be used to illustrate addition inc modelling how to bridge through 10 8+5 (8+2+3)=10 Solve one-step problems involving multiplication and division using concrete objects, pictorial representations and arrays with the support of the teacher 13-5=8 Children use numbered number lines to support their own calculations counting back in ones. The number line should also be used to show that 6 - 3 means the ‘difference between 6 and 3’ or ‘the difference between 3 and 6’ and how many jumps they are apart by counting on. Using measures as a context supports this notion e.g. difference between lengths They make connections between arrays, number patterns and counting in 2s, 5s and 10s Through grouping and sharing small quantities pupils begin to understand multiplication...doubling numbers and quantities This should be in the context of measures as well as numbers e.g. 4 cups of rice is equal to 1 jug of rice, 1 apple is equal in mass to 3 satsumas ie. 3x Satsuma=apple 12÷3=4 Recognise, find and name a half as one of two equal parts of an object or a shape or quantity Recognise, find and name a quarter as one of four equal parts of an object or a shape or quantity Through grouping and sharing small quantities pupils begin to understand division...finding simple fractions of objects and quantities This should be in the context of measures as well as numbers e.g. 4 cups of rice is equal to 1 jug of rice, 1 apple is equal in mass to 3 satsumas Mental/Oral Facts Y 2 Count in steps of 2,3, 5, 0 and 10 from any number forwards and back Recall and use addition facts to 20; Using pictorial representations, inc numbers, quantities and measures to derive facts e.g. subtraction facts to 20 20/100] Addition Subtraction Add and subtract numbers using concrete objects, pictorial representations and mentally, inc. a 2-digit number and ones; a 2-digit number and tens; 2 2-digit numbers e.g. + = Children will begin to use empty number lines to support calculations. = 0r Y 2 Count, read and write nos to 100 in numerals and words Use PV and number facts to solve problems + Division Children will develop their understanding of multiplication and use jottings to support calculation: Repeated addition 3 times 5 is 5 + 5 + 5 = 15 or 3 lots of 5 or 5 x3 Repeated addition can be shown easily on a number line: Children will develop their understanding of division and use jottings to support calculation Sharing equally 6 sweets shared between 2 people, how many do they each get? Grouping or repeated subtraction There are 6 sweets, how many people can have 2 sweets each? and on a bead bar: Counting back: First counting back in tens and ones. Or with Numicon: Compare and order numbers from 0-100 use < > and = signs Recognise the place value of each digit in a 2digit number Add and subtract numbers using concrete objects, pictorial representations and mentally, inc. a 2-digit number and ones; a 2-digit number and tens; 2 2-digit numbers Multiplication = Using known facts: Then helping children to become more efficient by subtracting the units in one jump (by using the known fact 7 – 3 = 4). Subtracting the tens in one jump and the units in one jump. 3X3=9 Commutativity: Children should know that 3 x 5 has the same answer as 5 x 3. Numicon shapes illustrate this well It can also be shown on the number line. Repeated subtraction using a number line or bead bar 12 ÷ 3 = 4 Or Numicon: 9÷3=3 Children will begin to use ‘empty number lines’ themselves starting with the larger number and counting on. PTO First counting on in tens and ones. Bridging through ten can help children become more efficient. Arrays Children should be able to model a Using symbols to stand for unknown numbers to complete equations using inverse operations 3X5=15 so 15÷ =3 must be 5 Use x and ÷ signs to write multiplication and division statements Recall and use multiplication and division facts for the 2, 5 and 10 times tables Recognise odd and even numbers Then helping children to become more efficient by adding the units in one jump (by using the known fact 4 + 3 = 7). Y 2 Followed by adding the tens in one jump and the units in one jump. Bridging through ten can help children become more efficient. Partitioning numbers in different ways e.g. 17=13+4, 17=2+15, 17-3=14 Partitioning to add using base 10 materials as support Adding ‘tens’ and ‘ones’ e.g. 34+23 30 + 4 20 + 3 50 + 7 Commutativity Addition can be done in any order 2+3=5 and 3+2=5 Recording in different ways 5=3+2 0r 1+4=2+3; 3+4>5 multiplication calculation using an array. This knowledge will support with the development of the grid method. Counting on: The number line should still show 0 so children can cross out the section from 0 to the smallest number. They then associate this method with ‘taking away’. Children can be given a ‘rule’ for when to count on to ‘find the difference.’ Measures are a useful context e.g. long jumps, lengths of rope Partitioning and decomposition Using base 10 materials e.g. 89-57=32 80 + 9 - 50 + 7 30 + 2 Children would not be expected to record this as column subtraction in Year 2 Recognise and use the inverse relationship between addition and subtraction and use this to check calculations and solve missing number problems 5+7=12 so 12- =5 must be 7 Using symbols to stand for unknown numbers to complete equations using inverse operations Use x and ÷ signs to write multiplication and division statements 4 x5=20 and 20÷4=5 Solve problems involving multiplication and division using, material,arrays, repeated addition, mental methods and multiplication and division facts, including problems in context 4 x5=20 and 20÷4=5 Using symbols to stand for unknown numbers to complete equations using inverse operations ÷ 2 = 4 20 ÷ = 4 ÷=4 Solve problems involving multiplication and division using, material,arrays, repeated addition, mental methods and multiplication and division facts, including problems in context M/O facts Read, write Y3 (numbers and words) order and compare and order numbers up to 1000 Using a variety of representations, including measures, count in ones, tens and hundreds, so that they become fluent in the order and place value of numbers to 1000 Count on from zero in multiples of 4, 8 50 and 100 Find 10 or 100 more or less than a given number Addition Children will continue to use empty number lines with increasingly large numbers, including compensation where appropriate. Count on from the largest number irrespective of the order of the calculation. Compensation Adding ‘tens’ then ‘ones’. Children will begin to use informal pencil and paper methods (jottings) to support, record and explain partial mental methods building on existing mental strategies (until secure in strategy). Base 10 should be used alongside written jottings. Adding the least significant digits first Subtraction Children will continue to use empty number lines with increasingly large numbers. Partitioning and decomposition Partitioning – demonstrated using arrow cards Decomposition - Base 10 materials NOTE When solving the calculation 89 – 57, children should know that 57 does NOT EXIST AS AN AMOUNT it is what you are subtracting from the other number. Therefore, when using base 10 materials, children would need to count out only the 89. Begin to exchange.(‘steal’ not ‘borrow)’ Multiplication Division Using multiplication tables that they know: Children will continue to use: Repeated addition 4 times 6 is 6 + 6 + 6 + 6 = 24 or 4 lots of 6 or 6 x 4 Children should use number lines or bead bars to support their understanding. (Teaching bead bars are available.) Cuisenaire rods can be used to support multiplicative relation ships ie. If this (rod) is 3 then what is this (2x or 3x larger rod)? Commutativity Children should know that 3 x 5 has the same answer as 5 x 3. Numicon pieces can illustrate this clearly. This can also be shown on the number line. Arrays Children should be able to model a multiplication calculation using an array. This knowledge will support with the development of the grid method. Then add numbers up to 3-digits Recognise the using formal written methods of value of each column addition. digit in a 3-digit number Estimate answers to calculations e.g. use inverse operations to check answers Solve problems including missing number problems using number facts, place value and more complex addition and subtraction Finding the difference Where the numbers are involved in the calculation are close together or Scaling near to multiples of 10, 100 etc e.g. Find a ribbon that is 4 times counting on using a number line should as long as the blue ribbon be used. PTO You could also use capacity or Cuisenaire rods to develop this Using multiplication tables that they know: Ensure that the emphasis in Y3 is on grouping rather than sharing. Children will continue to use: Repeated subtraction using a number line Cuisenaire rods can be used to support multiplicative relation ships ie. If this (rod) is 8 then what is this (2x or 3x smaller rod)? Children should also move onto calculations involving remainders. Using symbols to stand for unknown numbers to complete equations using inverse operations 26 ÷ 2 = 24 ÷ = 12 ÷ 10 = 8 Scaling Use bar method Scaling should be used to solve problems in context e.g. Find …4 times shorter than this… Or James has saved £8. Sarah has saved twice as much. How much have they saved altogether? a Fractions Recognise, find and write fractions of discrete sets of objects: unit fractions and non-unit fractions with small denominators. This should go beyond 0, 1 Y3 Add and subtract mentally including a 3digit number and ones, a 3-digit number and tens, , a 3-digit number and hundreds Multiplication facts for (2,5,10) 3,4,6 and 8 x tables up to 12X… Division facts for times tables 2,5,10,3,4,6 e.g.30 ÷ 6= 5; 6 = 30÷5 Count up and down in tenths…by dividing an object and in dividing 1-digit numbers or quantities by 10 Working within the context of measures supports this notion. Can support this using Cuisenaire (bar method) Scaling should be used to solve problems in context e.g. Find …4 times as long as this… Using symbols to stand for unknown numbers to complete equations using inverse operations x 5 = 20 3 x = 18 x = 32 Partitioning Jottings should be used to support this. Some children will require concrete apparatus at this point. 38 x 5 = (30 x 5) + (8 x 5) = 150 + 40 = 190 Counters numbered in 1s, 10s, or 100s could be used Use measures as context e.g. ½ of 2 metres; ¼ of 2 metres Recognise and use fractions as numbers: unit fractions and non-unit fractions with small denominators E.g Fractions of shapes Or Bar method e.g. NNS ITP fractions Recognise and show with diagrams equivalent fractions with small denominators Children should then, when ready, move to reliable written methods Grid method: Add and subtract fractions with the same denominators within 1 whole e.g. 5/7 + 1/7 = 6/7 The grid method is important as it leads into the partitioning model (below) for mental multiplication and division of 2 digit by 1 digit numbers 2-digit by 1-digit numbers Compare and order unit fractions and fractions with the same denominator e.g. Using measures 1 ½ litres, ¼ litre, 1 1/3 litre, ¾ Recall multiplication and Y4 division facts for multiplication tables to 12x12 Count on from zero in multiples of 6, 7, 9, 25 and 1000 Count back through zero to negative numbers Add numbers with up to 4 digits using column addition where appropriate * Subtract numbers with up to 4 digits using column subtraction where appropriate* *Children should be taught to recognise where metal strategies would be more efficient *Children should be taught to recognise where metal strategies would be more efficient Carry below the line. Partitioning and decomposition Adjusting from ‘tens’ to ‘ones’ and ‘hundreds’ to tens’ Base 10 or place value counters should be used alongside written algorithm to support children Base 10 or place value counters should be used alongside written algorithms to support children Use place value and known and derived facts to multiply and divide mentally inc. multiplying by 1 and 1 dividing by 1 and multiplying together 3 numbers e.g. 200 × 3 = 600 into 600 ÷ 3 = 200, to become fluent. Children will continue to use arrays where appropriate leading into the grid method of multiplication. Modelling the two side-by-side can secure understanding. Use place value and known and derived facts to multiply and divide mentally inc. multiplying by 1 and 1 dividing by 1 e.g. 600 ÷ 3 = 200 can be derived from 2X3=6 Children will develop their use of repeated subtraction to be able to subtract multiples of the divisor. Initially, these should be multiples of 10s, 5s, 2s and 1s – numbers with which the children are more familiar. Recognise the value of each digit in a 4-digit number Order and compare numbers beyond 1000 Identify, represent and estimate numbers using different representations (inc measures) Round any number to the nearest 10, 100 or 1000(inc measures) Children should continue to practice both mental and written strategies with increasingly large numbers as well as mixed numbers of digits to aid fluency Children should know that the decimal points should line up under each other, particularly when adding or subtracting mixed amounts, e.g. £3.59 + 78p. Children must be taught to Decomposition Children should be able to subtract numbers with different numbers of digits and know that decimal points should line up under each other. Grid method Multiply 2-digit and 3-digit numbers by a 1-digit number using formal written methods TU x O (to HTO x TO) (Short multiplication – multiplication by a single digit) 23 x 8 Children will approximate first 23 x 8 is approximately 25 x 8 = 200 Then onto the vertical method: Short division TU ÷ O Showing working alongside. Leading to subtraction of other multiples. (PTO) Any remainders should be shown Y4 Read Roman numerals to 100 and know that over time the number system changes to include the concept of zero Count up or down in hundredths, recognise that hundredths arise when dividing and object by 100 and dividing tenths by ten Count in simple fractions estimate and use inverse operations to check a calculation Children must be taught to estimate and use inverse operations to check a calculation HTO x O (Short multiplication – multiplication by a single digit) 346 x 9 Children will approximate first 346 x 9 is approximately 350 x 10 = 3500 as integers, i.e. 14 remainder 2 or 14 r 2. Children need to be able to decide what to do after division and round up or down accordingly. They should make sensible decisions about rounding up or down after division. Complete equations using inverse operations Complete equations using inverse operations Integer scaling problems e.g. scaling up shapes, models, linear measures. Correspondence problems such as n objects are connected to m objects Eg. Three cakes shared equally between 10 children Fractions Recognise and show using diagrams, families of equivalent fractions Use bar and linear model Cuisenaire rods can support this model number diagrams to support as well as going beyond 0-1 eg. 0-2 number lines Add and subtract fractions with the same denominator Recognise and write decimal equivalents of any number of tenths or hundredths Measures are a good context here. Recognise and write decimal equivalents to ¼, ½ ¾ Use liquids to aid visualisation Y4 Find the effect of dividing a 1 or 2-digit number by 10 and 100 Aid memoir e.g. ‘Zero the Super Hero’. Round decimals with 1 decimal place to the nearest whole number Compare numbers with the same number of decimal places to the nearest two decimal place