Aims Introduction Progression of calculation strategies - Addition Mental Skills Models and Images Vocabulary Expectations end of FS2

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1 CALCULATION POLICY Date for adoption: April 2014 Growing and Learning Through Christian Values This policy reflects the values and philosophy of Alderbury and West Grimstead CEVA Primary School. It sets out a framework within which all staff work.

2 Aims The national curriculum for mathematics aims to ensure that all pupils: Become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately. Reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language. Can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions. Introduction Each of the four operations build on mental skills which provide the foundations for jottings and informal written methods of recording. Skills need to be taught, practised and reviewed constantly. These skills lead on to more compact written methods of calculation. Children need to build up a repertoire of increasing complexity. Progression of calculation strategies - Addition Mental Skills Recognise the size and position of numbers. Count in different steps and derive facts. Know number bonds to 10 and 20. Partition and recombine numbers. Bridge through 10, 100 etc. Models and Images Counting apparatus. Place value apparatus. Number lines. Empty number lines. Counting stick. Bead String. ITPs Number facts, ordering. Numbers, number grid, counting on and back in ones and tens. Numicon. Vocabulary Add, addition, plus, and, count on, more than, sum, total, altogether, increase one more, ten more, equal, most, estimation. Expectations end of FS2 Count to 20 and beyond. Combine two sets of objects practically, using cubes/counters etc. Draw objects. Read and write numerals 0-9 accurately. Number sentence with objects drawn. Partition numbers to 10. Record: 5= =5

3 Progression of calculation strategies - Subtraction Mental Skills Recognize the size and position of numbers Count back in different steps and derive facts Know number facts for all numbers to 20 Subtract multiples of 10 from any number Models and Images Counting apparatus Place value apparatus Number lines Empty number lines. Counting stick Bead strings ITPs Number Facts, Counting on and back in ones and tens, Difference Numicon. Vocabulary Take away, How many left? How many fewer than? How many more to make count back, less than, fewer, difference between, subtract, minus, subtraction, decrease, estimation. Early Experiences Begin to count backwards Practical taking away from a set of objects Draw objects and cross out number taken away. Number sentence with objects drawn. Group of cubes on a tray, part of them covered and need children visualise the calculation. Compare lengths of objects ready to find the difference

4 Expecations by end of Year 1 Expectations by end of Year 1 + Number Range Visual/ Number lines No number line Vertical Method Facts Addition as counting on. Put the larger number in your head and count on (know addition can be done in any order) Include 0 Number track/number line jumps of 1 3+2= 5 Bead strings = 23 Number line efficient jumps = = 7 50 = 7 = 57 Missing number 7 + = 10 Know by heart all pairs of numbers with a total of 10 and 20. Represent and use number facts within or 10 more than a number = 13 8 in my head then count on 9,10,11,12,13 TU+U missing number problems _ Number Range Visual/ Number lines - taking away Number lines counting on Vertical Method Decomposition at Step 3 and above Facts Subtraction as taking away and finding the difference, by counting on. Include 0 U U Take away jumps of 1 Using bead strings 6 3 = Find a small difference by counting up from the smallest number 14 11= 3 Subtraction Facts to and 10 less. TU U bridging 10 Solve 1 step problems using manipulatives missing number problems Missing number 10 - = 8

5 Number number and place value Count to and across 100, forwards and backwards, beginning with 0 or 1, or from any given number. Count, read and write numbers to 100 in numerals; count in multiples of twos, fives and tens. Given a number, identify one more and one less. Identify and represent numbers using objects and pictorial representations including the number line, and use the language of: equal to, more than, less than (fewer), most least. Read and write numbers from 1 to 20 in numerals and words. Number addition and subtraction Read, write and interpret mathematical statements involving addition (+), subtraction (-) and equals (=) signs. Represent and use number bonds and related subtraction facts within 20 Add and subtract one-digit and two-digit numbers to 20, including zero Solve one-step problems that involve addition and subtraction, using concrete objects and pictorial representations, and missing number problems such as 7 = - 9.

6 Expectations by end of Year 2 Expectations by end of Year 2 + Number Range Visual/ Number lines No number line Vertical Method Facts TU + U TU + TU bridging 10s/100 Add 3 or more single digit numbers = 13 Add together multiples of = in my head 60, 70, 80 Partition two digit numbers using tens and ones Number line efficient jumps Put the largest number first = = = 82 No bridging Dienes (TU grid) Know bonds to 10 and pairs to 20 and use related facts to 100. Pairs to 100, using multiples of 10. Use _ Number Range Visual/ Number lines - taking away Number lines counting on Vertical Method Decomposition Facts Subtraction as the inverse of addition. Mentally TU U TU - TU TU TU bridging 10 Subtraction is not communitive = = = Dienes (TU grid) Subtraction facts to at least 10. Use inverse to check calculations.

7 Number number and place value Count in steps of 2, 3 and 5 from 0, and in tens from any number, forward and backward. Recognise the place value of each digit in a two-digit number (tens, ones) Identify, represent and estimate numbers using different representations, including the number line. Compare and order numbers from 0 up to 100; use, and signs. Read and write numbers to at least 100 in numerals and in words. Use place value and number facts to solve problems. Number addition and subtraction Solve problems with addition and subtraction: o Using concrete objects and pictorial representations, including those involving numbers, quantities and measures o Applying their increasing knowledge of mental and written methods. Recall and use addition and subtraction facts to 20 fluently, and derive and use related facts up to 100. Add and subtact numbers using concrete objects, pictorial representations, and mentally, including: o A two-digit number and ones o A two-digit number and tens o Two two-digit numbers o Adding three one-digit numbers. Show that addition of two numbers can be done in any order (commutative) and subtraction of one number from another cannot. Recognise and use the inverse relationship between addition and subtraction and use this to check calculations and solve missing number problems.

8 Expectations by end of Year 3 Expectations by end of Year 3 + Number Range Visual/ TU + TU bridging 100 HTU + TU not bridging 1000 HTU + HTU not bridging 1000 HTU + U (mentally) Include money Estimate answers = = 142 Number lines No number line Columnar Formal written methods = Largest number first = = = = = = 142 H T U Expanded vertical = = = 142 H T U progress to formal Facts Bonds to 20 and 100 Multiples of 50 that total 1000 _ Number Range Visual/ TU TU HTU TU HTU HTU Mentally HTU U HTU TU HTU HTU Estimating answers and using inverse operations to check answers. Solve problems include missing number = = = = 148 Number lines - taking away Number lines counting on Small difference Small difference Vertical Method Decomposition Facts = 207 Subtraction facts to 20. Differences of multiples of 10. Compact method

9 Number number and place value Count from 0 in multiples of 4, 8, 50 and 100; find 10 or 100 more or less than a given number. Recognise the place value of each digit in a three-digit number (hundreds, tens, ones). Compare and order numbers up to Identify, represent and estimate numbers using different representations. Read and write numbers up to 1000 in numerals and in words. Solve number problems and practical problems involving these ideas. Number addition and subtraction Add and subtract numbers mentally, including: o A three-digit number and ones o A three-digit number and tens o A three-digit number and hundreds Add and subtract numbers with up to three digits, using formal written methods of columnar addition and subtraction. Estimate the answer to a calculation and use inverse operations to check answers. Solve problems, including missing number problems, using number facts, place value, and more complex addition and subtraction.

10 Expectations by end of Year 4 Expectations by end of Year 4 + Number Range Visual/ Number lines No number line Vertical Method Facts HTU + HTU including bridging 1000 Decimals, money Estimate and use inverse operation to check answers. Solve +2 step problems money = = 8 = = = 622 Expanded vertical upto 4 digits Compact Vertical Bonds to 100 Derive sums of pairs of multiples of 10, 100, Pairs of fractions to make _ Number Range Visual/ Number lines - taking away Number lines counting on Vertical Method Decomposition Facts Up to 4 digit numbers. Decimals, money step problems and deciding which operation and method to use = = = = Subtraction facts to 50.

11 Number number and place value Count in multiples of 6, 7, 9, 25 and Find 1000 more or less than a given number. Count backwards through zero to include negative numbers. Recognise the place value of each digit in a four-digit number (thousands, hundreds, tens and ones). Order and compare numbers beyond Identify, represent and estimate numbers using different representations. Round any number to the nearest 10, 100 or Solve number and practical problems that involve all of the above and with increasingly large positive numbers. Read Roman numerals to 100 (I to C) and know that over time, the numeral system changed to include the concept of zero and place value. Number addition and subtraction Add and subtract numbers with up to 4 digits using the formal written methods of columnar addition and subtraction where appropriate. Estimate and use inverse operations to check answers to a calculation. Solve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and why.

12 Expectations by end of Year 5 Expectations by end of Year 5 + Number Range Visual/ Number lines No number line Vertical Method Facts Decimals up to 2 decimal places. More than 4 digits and with increasingly large numbers. Solve multistep problems. Rounding to check answers = = = = = = = 2434 Expanded vertical Derive bonds to up to 2 decimal places. Derive bonds to 10 up to 1 decimal place. Compact vertical _ Number Range Visual/ More than 4 digits Decimals up to 2 places. Subtract mentally increasingly large numbers. Use rounding to check multistep subtraction problems = = = 26.8 Number lines - taking away Number lines counting on = Vertical Method Decomposition = Compact method Facts Subtraction facts to 100. Use number facts for mental subtractions. 9-2 = = = 0.07

13 Number number and place value Read, write, order and compare numbers to at least 1,000,000 and determine the value of each digit. Count forwards or backwards in steps of powers of 10 for any given number up to 1,000,000. Interpret negative numbers in context, count forwards and backwards with positive and negative whole numbers, including through zero. Round any number up to 1,000,000 to the nearest 10, 100, 1000, 10,000 and 100,000. Solve number problems and practical problems that involve all of the above. Read Roman numerals to 1000 (M) and recognise years written in Roman numerals. Number addition and subtraction Add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and subtraction). Add and subtract numbers mentally with increasingly large numbers. Use rounding to check answers to calculations and determine, in the context of a problem, levels of accuracy. Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why.

14 Expectations by end of Year 6 Expectations by end of Year 6 + Number Range Visual/ Number lines No number line Vertical Method Facts Extend Step 5 - three numbers. Decimals up to 3 decimal places relate to measures km km = = 0.2 0, = = km km 3.243km km = = = = Expanded vertical Compact vertical Common factors, common multiples, common prime number. BODMAS Expectation _ Number Range Visual/ Number lines - taking away Number lines counting on Vertical Method Decomposition Facts Consolidate and extend Step 5. Use estimation. Decimals up to 3 places relate this to measure. BODMAS Recognise when one written method from Step 5 is more efficient for particular calculations. Multistep problems in context.

15 Number number and place value Read, write, order and compare numbers up to 10,000,000 and determine the value of each digit. Round any whole number to a required degree of accuracy. Use negative numbers in context, and calculate intervals across zero. Solve number and practical problems that involve all of the above. Number addition and subtraction Perform mental calculations, including with mixed operations and large numbers. Identify common factors, common multiples and prime numbers. Use their knowledge of the order of operations to carry out calculations involving the four operations BODMAS. Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why.

16 Progression of calculation strategies Multiplication Mental Skills Recognise the size and position of numbers Count on in different steps 2s, 5s, 10s Double numbers up to 10 Recognize multiplication as repeated addition Quick recall of multiplication facts Multiplying by 10, 100, 1000 and understanding the effect Multiplying by multiples of 10 Resources, Models and Images Counting apparatus Place value apparatus Arrays e.g. egg boxes, windows drawer units. Multiplication squares Number lines Counting stick Bead strings ITPs Multiplication grid, Number Dials, Multiplication Facts, Array Creator Squared paper lino or wallpaper with squares on- for grid method. Numicon. Vocabulary lots of, groups of, multiply, multiplication, multiple, product, once, twice, three times, array, row, column, double, repeated addition, times, 10 times as big/long/high etc Solve problems including doubling, halving and sharing (put in % as well). Counting in 2s, 5s and 10s Practical grouping of objects

17 Progression of calculation strategies - Division Mental Skills Recognize the size and position of numbers Count back in different steps 2s, 5s, 10s Halve numbers to 20 Recognize division as repeated subtraction Quick recall of division facts Use known multiplication facts to derive associated division facts Divide by 10, 100, 1000 and understand the effect Divide by multiples of 10 Models and Images Counting apparatus Arrays Number lines Empty number lines. Multiplication squares ITPs Multiplication grid, Number Dials, Grouping, Remainders Numicon. Vocabulary lots of, equal groups of, shared equally, group, halve, half, divide, division, divided into, divided by, remainder, factor, quotient, divisible by, inverse Practical sharing of objects with vocabulary of sharing not grouping. This may include a leftover. 6 cakes shared between 2. = 3 each

18 Expectations by end of Year 1 Expectations by end of Year 1 X Number Range Pictures and symbols Number tracks and number lines Arrays Grid method Facts Solve (practical) problems that involve combining groups of 2, 5 or 10. Count on or back in 1s, 2s, 5s and 10s. Repeated addition. There are 3 sweets in one bag. How many sweets in 5 bags X 4 4 X Doubles to 20. Odd and even numbers. Count in 2s, 5s, 10s. Solve 1 step problems. Manipulatives. Arrays with teacher support. Number Range Pictures and symbols Number tracks and number lines Grouping vertical layout. Facts Solve practical problems that involve sharing into equal groups. How many apples in each bowl if I share 12 apples between 3 bowls. 1 step % problems use as in x. 6 cakes shared between = =

19 Number multiplication and division Solve one-step problems involving multiplication and division, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher.

20 Expectations by end of Year 2 Expectations by end of Year 2 X Number Range Pictures and symbols Number tracks and number lines Arrays Grid method Facts Multiplication as repeated addition and arrays. Derive multiples of 2, 5 and 10. Relate to X facts and derive related facts. There are 4 apples in each box. How many apples in 6 boxes? Repeated addition can be shown easily on a number line: 5 x 3 = x 6 or 6 x 3 Commutativity of multiplication. Recall and use Know 2, 5 and 10 times tables. Doubles of numbers to 20. Solving problems. Record x number sentences. Number Range Pictures and symbols Number tracks and number lines Grouping vertical layout. Facts Division as sharing and grouping, including remainders. TU U Where divisor is 2, 5, or 10. Solve problems as x. Four eggs fit in a box. How many boxes would you needs to pack 20 eggs? Arrays 20 2 = = 4 Derive and recall division facts for 2, 5 and 10 times tables. Derive and recall halves of even numbers to 40. Not commutative.

21 Number multiplication and division Recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers. Calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication (x), division ( ) and equals (=) signs. Show that multiplication of two numbers can be done in any order (commutative) and division of one number by another cannot. Solve problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and division facts, including problems in contexts.

22 Expectations by end of Year 3 X Number Range Pictures and symbols Number tracks and number lines Arrays Grid method Facts TU X U U up to 5 13 X 4 U / TU X 10 U / TU X 100 Develop efficient mental methods. Solving simple problems in context eg 4 times a high, possible use of number line to record steps. 13 X 4 = X 4 = 40 3 X 4 = X 6 10 X 6 = 60 5 X 6 = = 90 Expanded grid 13 X 4 = 52 4 X 10 = 40 4 X 3 = 12 Compact grid 13 X 4 = 10 X 4 3 X 4 Derive and recall 2, 3, 4, 5, 8 and 10 times tables. Derive related division facts. Recognize multiples of 2, 5 and 10 up to Doubles of TU and HTU Progressing to formal written 24 x 4

23 Expectations at end of Year 3 Number Range Pictures and symbols Number tracks and number lines Grouping vertical layout. Facts TU U where divisor is 2, 3, 4, 5, 8 or 10. HTU = 6 r Partition multiples of the divisor = 12 r2 10 x 4 = 40 2 X 4 = 8 (48) Progress to formal written method Derive and recall division facts for2, 3, 4, 5, 8 and 10 times tables. With practical apparatus.

24 Number multiplication and division Recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables. Write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental and progressing to formal written methods. Solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems and correspondence problems in which n objects are connected to m objects.

25 Expectations by end of Year 4 Expectations by end of Year 4 X Number Range Vertical method Grid method Facts TU X U U up to 10 HTU x U Use place value and derive facts to x mentally include x by 0, 1 x 3 numbers 43 X 6 = X 6 = X 6 = 18 Expanded 43 X6 18 3X X6 258 Compact 43 X X X 12= = 516 Derive and recall all facts to and including 12 X 12. Numbers up to 1000 X10 and X100. Doubles of TU and HTU. Doubles of multiples of 10 and 100 Solve problems including distributive law. Number Range Pictures and symbols Number tracks and number lines TU U Numbers to or 100. Use place value and known derived facts to % mentally including % by = X 6 6 X = 16 r 10 X 4 = 40 6 X 4 = 24 (64) Grouping vertical layout (7 x 10) (7 x 3) 5 Answer 13 r 5. Progress to formal written method 1 4 Facts Derive and recall all number facts up to and including 12 x 12. Halves of TU, HTU and multiples of 10 and With practical apparatus.

26 Number multiplication and division Recall multiplication and division facts for multiplication tables up to 12 x 12. Use place value, known and derived facts to multiply and divide mentally, including multiplying by 0 and 1; dividing by 1; multiplying together three numbers. Recognise and use factor pairs and commutativity in mental calculations. Multiply two-digit and three-digit numbers by a one-digit number using formal written layout. Solve problems involving multiplying and adding, including using the distributive law to multiply two digit numbers by one digit, integer scaling problems and harder correspondence problems such as n objects are connected to m objects.

27 Expectations by end of Year 5 Expectations by end of Year 5 X Number Range Vertical method Grid method Facts Refine and use efficient methods. HTU X U TU X TU U.T X U Doubles of U.t and 0.th THHTU x U x TU x whole numbers and those involving decimals, 10, 100, Expanded 237 X X X x Compact 4.7 X x 36 estimate as 50 X 40 = 2000 X Recall quickly up to and including 12X12. Use facts to multiply pairs of multiples of 10 and 100. Use known facts to derive other facts. Identify common multiples of two numbers. Recall prime numbers up to 19 and establish which numbers to 100 are prime. Common factors. x numbers mentally drawing on known facts. Prime factors. Number Range Short Division Grouping vertical layout Refine and use efficient methods. THHTU U + remainders Divide using factors of the divisor, e.g. 8, 4, 2 % whole nos. including decimals by 10, 100, = 35 r2 30 X 7 = x 7 = 35 (245) Shortened to Long Division Facts Chunking Efficient Recall quickly division X X X X 2 4 Answer 32 r (8 X 40) (8 X 3 ) 2 Answer 43 r 2 facts up to and including the 12 times table. Divide numbers by 12, 100 and Halves of U.t and 0.th

28 Number multiplication and division Identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers. 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 prime numbers up to 19. Multiply numbers up to digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers. Multiply and divide numbers mentally drawing upon known facts. Divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context. Multiply and divide whole numbers and those involving decimals by 10, 100 and 1000.

29 Expectations by end of Year 6 X Number Range Vertical method Grid method Facts Use efficient methods. Integer X U 2307 X 8 Decimal X U 45.7 X 7 TU X TU HTU X U. Integer X 1000, 100, 10, 0.1, 0.01 X numbers up to 4 digits X TU and use long multiplication. Expanded x X X X X X Compact 256 X X X This can then be compacted and numbers carried X 9 estimate as 6 X 9 = 54 X Use facts up to 12 X 12 to derive facts involving multiples of 10 and 100. E.g. 8 X 3 80 X 30 8 X X 7 BODMAS. Mental calculation including mixed operations. Solve problems in context. Multistep and large numbers. Estimate to check answers.

30 Expectations by end of Year 6 Number Range Short Division Grouping vertical layout Long Division Facts Use efficient methods. Integer U Integer 1000, 100 or 10 Decimal U HTU TU U.t U = x 7 = X 7 = 1.4 (43.4) Efficient Grouping Derive division facts involving multiples of and 100, also decimals (8 X 3.0) (8 X 0.2) Answer 3.2 Answer 23 r 8 THHTU x TU Using long division and interpret remainders, as fraction or by rounding. Use short where appropriate. Estimate to check answers.

31 Number multiplication and division Multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication. Divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context. Divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context.