Number, Place Value, Approximation & Estimation / Rounding .
StrandYear 1Number, place value, approximation & estimation / rounding1N1a Count to & across 100, forward &backwards, beginning with 0 or 1, or fromN1 Counting (inany given numbermultiples)1N1b Count in multiples of twos, fives &tens1N2a Count, read & write numbers to 100 innumeralsN2 Read, write,order & compare1N2b Given a number, identify one more &numbersone less1N2c Read & write numbers from 1 to 20 innumerals & wordsYear 22N1 Count in steps of 2, 3, & 5 from 0, & intens from any number, forward or backward2N2a Read & write numbers to at least 100in numerals & in words2N2b Compare & order numbers from 0 upto 100; use , & signs2N3 Recognise the place value of each digitin a two-digit number (tens, ones)N3 Place value;RomannumeralsN4 Identify,represent &estimate;rounding1N4 Identify & represent numbers usingobjects & pictorial representations includingnumber lines, & use the language of: equalto, more than, less than (fewer), most, leastNational Curriculum Strand Tracker: taken from Key stage 1 & 2: mathematics test frameworkYear 3Year 42N4 Identify, represent & estimate numbersusing different representations, including thenumber line3N1b Count from 0 in multiples of 4, 8, 50 &1003N2a Compare & order numbers up to 1000;Read & write numbers to 1000 in numerals &in words3N2b Find 10 or 100 more or less than agiven number3N3 Recognise the place value of each digitin a three-digit number (hundreds, tens,ones)3N4 Identify, represent & estimate numbersusing different representationsN5 NegativenumbersN6 NumberproblemsAddition, subtraction, multiplication & division (calculations)1C1 Represent & use number bonds &related subtraction facts within 20C1 Add /subtractmentally1C2a Add & subtract one-digit & two-digitnumbers to 20, including zeroC2 Add /subtract usingwritten methods2N6 Use place value & number facts to solveproblems3N6 Solve number problems & practicalproblems involving 3N1–3N52C1a Recall & use addition & subtractionfacts to 20 fluently, & derive & use relatedfacts up to 1003C1 Add & subtract numbers mentally,including: a three-digit number & ones a three-digit number & tens a three-digit number & hundreds2C1b Add & subtract numbers mentally, i.e: a two-digit number & ones a two-digit number & tens two two-digit numbers adding three one-digit numbers2C2 Add & subtract numbers using concreteobjects & pictorial representations, including: a two-digit number & ones a two-digit number & tens two two-digit numbers adding three one-digit numbersYear 5Year 64N1 Count in multiples of 6, 7, 9, 25 & 10005N1 Count forwards or backwards in steps ofpowers of 10 for any given number up to 1000 0004N2a Order & compare numbers beyond10005N2 Read, write, order & compare numbersto at least 1 000 0006N2 Read, write, order & compare numbersup to 10 000 0004N3a Recognise the place value of eachdigit in a four-digit number (thousands,hundreds, tens & ones)4N3b Read Roman numerals to 100 (I to C)& know that over time, the numeral systemchanged to include the concept of zero &place value4N4a Identify, represent & estimate numbersusing different representations5N3a Determine the value of each digit innumbers up to 1 000 0006N3 Determine the value of each digit innumbers up to 10 000 0004N4b Round any number to the nearest 10,100 or 10004N5 Count backwards through zero toinclude negative numbers5N4 Round any number up to 1 000 000 tothe nearest 10, 100, 1000, 10 000 & 100 0005N5 Interpret negative numbers in context,count forwards & backwards with positive &negative whole numbers, including across zero6N4 Round any whole number to a requireddegree of accuracy6N5 Use negative numbers in context, &calculate intervals across zero4N6 Solve number & practical problems thatinvolve 4N1–4N5 & with increasingly largepositive numbers5N6 Solve number problems & practicalproblems that involve 5N1–5N56N6 Solve number problems & practicalproblems that involve 6N2–6N54N2b Find 1000 more or less than a givennumber5N3b Read Roman numerals to 1000 (M) &recognise years written in Roman numeral5C1 Add & subtract numbers mentally withincreasingly large numbers3C2 Add & subtract numbers with up to threedigits, using formal written methods ofcolumnar addition & subtraction4C2 Add & subtract numbers with up to 4digits using the formal written methods ofcolumnar addition & subtraction whereappropriate5C2 Add & subtract whole numbers withmore than 4 digits, including using formalwritten methods (columnar addition &subtraction3C3 Estimate the answer toa calculation & use inverse operations tocheck answers4C3 Estimate & use inverse operations tocheck answers to a calculation5C3 Use rounding to check answers tocalculations & determine, in the context of aproblem, levels of accuracy6C3 Use estimation to check answers tocalculations & determine, in the context of aproblem, an appropriate degree of accuracy3C4 Solve problems, including missingnumber problems, using number facts, placevalue, & more complex addition &subtraction4C4 Solve addition & subtraction two-stepproblems in contexts,deciding which operations & methods to use& why5C4 Solve addition & subtraction multi-stepproblems in contexts, deciding whichoperations & methods to use & why6C4 Solve addition & subtraction multi-stepproblems in contexts, deciding whichoperations & methods to use & why5C5a Identify multiples & factors, includingfinding all factor pairs of a number &common factors of two numbers5C5b Know & use the vocabulary of primenumbers, prime factors & composite(nonprime) numbers5C5c Establish [if] a number up to 100 isprime & recall prime numbers up to 195C5d Recognise & use square numbers &cube numbers, & the notation for squared(2)& cubed (3)6C5 Identify common factors, commonmultiples & prime numbers1C2b Read, write & interpret mathematicalstatements involving addition ( ), subtraction(–) & equals ( ) signsC3 Estimate,use inverses &checkC4 Add /subtract to solveproblemsC5 Properties ofnumber(multiples,factors, primes,squares &cubes)1C4 Solve one-step problems that involveaddition & subtraction, using concreteobjects & pictorial representations, & missingnumber problems such as 7 – 92C3 Recognise & use the inverserelationship between addition & subtraction &use this to check calculations & missingnumber problems2C4 Solve s addition & subtraction problems: using concrete objects & pictorialrepresentations, including those involvingnumbers, quantities & measures applying their increasing knowledge ofmental & written methods
StrandYear 1Year 22C6 Recall & use multiplication & divisionfacts for the 2, 5 & 10 multiplication tables,including recognising odd & even numbersNational Curriculum Strand Tracker: taken from Key stage 1 & 2: mathematics test frameworkYear 3Year 43C6 Recall & use multiplication & division4C6a Recall multiplication & division facts forfacts for the 3, 4 & 8 multiplication tablesmultiplication tables up to 12 124C6b Use place value, known & derivedfacts to multiply & divide mentally, including: multiplying by 0 & 1; dividing by 1; multiplying together three numbers4C6c Recognise & use factor pairs &commutativity in mental calculations4C7 Multiply two-digit & three-digit numbersby a one-digit number using formal writtenlayoutC6 Multiply /divide mentally2C7 Calculate mathematical statements formultiplication & division within themultiplication tables & write them usingmultiplication, division & equals signsC7 Multiply /divide usingwritten methodsC8 Solveproblems(commutative,associative,distributive & allfour operations)1C8 Solve one-step problems involvingmultiplication & division, by calculating theanswer using concrete objects, pictorialrepresentations & arrays with the support ofthe teacherC9 Order ofoperationsFractions, decimals & percentagesF1 Recognise,1F1a Recognise, find & name a half as onefind, write, nameof two equal parts of an object, shape or& countquantityfractions1F1b Recognise, find & name a quarter asone of four equal parts of a object, shape orquantity2C8 Solve problems involving multiplication& division, using materials, arrays, repeatedaddition, mental methods, & multiplication &division facts, including problems in contextsF3 Comparing &orderingfractionsF4 Add /subtractfractionsF5 Multiply /divide fractions4C8 Solve problems involving multiplying &adding, including using the distributive law tomultiply two-digit numbers by one digit,integer scaling problems & hardercorrespondence problems eg. n objects areconnected to m objectsYear 66C6 Perform mental calculations, includingwith mixed operations & large numbers5C6b Multiply & divide whole numbers &those involving decimals by 10, 100 & 10005C7a Multiply numbers up to 4 digits by aone or two-digit number using a formalwritten method, including long multiplicationfor two-digit numbers5C7b Divide numbers up to 4 digits by aone-digit number using the formal writtenmethod of short division & interpretremainders appropriately for the context5C8a Solve problems involvingmultiplication & division including using theirknowledge of factors & multiples, squares &cubes6C7a Multiply multi-digit numbers up to 4digits by a two-digit whole number using theformal written method of long multiplication6C7b Divide numbers up to 4 digits by a twodigit whole number using the formal writtenmethod of long division & interpretremainders as whole number remainders,fractions, or by rounding, as appropriate forthe context6C7c Divide numbers up to 4 digits by atwo-digit number using the formal writtenmethod of short division as appropriate,interpreting remainders according to context6C8 Solve problems involving addition,subtraction, multiplication & division5C8b Solve problems involving addition,subtraction, multiplication & division & acombination of these, including understanding the meaning of the equals sign5C8c Solve problems involving multiplication& division including scaling by simplefractions & problems involving simple rates2C9a Show that addition of two numberscan be done in any order (commutative) &subtraction of one number from anothercannot2C9b Show that multiplication of twonumbers can be done in any order(commutative) & division of one number byanother cannot2F1a Recognise, find, name & write fractions13, 14, 24 & 34 of a length, shape, set ofobjects or quantity2F1b Write simple fractions eg ½ of 6 32F2 Recognise the equivalence of 2/4 & ½F2 Equivalentfractions3C8 Solve problems, including missingnumber problems, involving multiplication &division, including integer scaling problems &correspondence problems in which n objectsare connected to m objectsYear 55C6a Multiply & divide numbers mentallydrawing upon known facts6C9 Use their knowledge of the order ofoperations to carry out calculations involvingthe four operations3F1a Count up & down in tenths; recognisethat tenths arise from dividing an object into10 equal parts & in dividing one-digitnumbers & quantities by 103F1b Recognise, find & write fractions of adiscrete set of objects: unit fractions & nonunit fractions with small denominators3F1c Recognise & use fractions asnumbers: unit fractions & non-unit fractionswith small denominators3F2 Recognise & show, using diagrams,equivalent fractions with small denominators4F1 Count up & down in hundredths;recognise that hundredths arise whendividing an object by a hundred & dividingtenths by 104F2 Recognise & show, using diagrams,families of common equivalent fractions3F3 Compare & order unit fractions &fractions with the same denominators3F4 Add & subtract fractions with the samedenominator within one whole [eg: 5/7 1/7 6/7]4F4 Add & subtract fractions with the samedenominator5F2a Recognise mixed numbers & improperfractions & convert from one form to theother; write mathematical statements 1 as amixed number [eg: 2/5 4/5 6/5 1 1/5]5F2b Identify name & write equivalentfractions of a given fraction, representedvisually, including tenths & hundredths5F3 Compare & order fractions whosedenominators are all multiples of the samenumber5F4 Add & subtract fractions with the samedenominator & denominators that aremultiples of the same number5F5 Multiply proper fractions & mixednumbers by whole numbers, supported bymaterials & diagrams6F2 Use common factors to simplifyfractions; use common multiples to expressfractions in the same denomination6F3 Compare & order fractions, includingfractions 16F4 Add & subtract fractions with differentdenominators & mixed numbers, using theconcept of equivalent fractions6F5a Multiply simple pairs of properfractions, writing the answer in its simplestform [eg: 1/4 1/2 1/8]6F5b Divide proper fractions by wholenumbers [eg: 1/3 2 1/6]
StrandYear 1F6 Fractions /decimalsequivalenceYear 2National Curriculum Strand Tracker: taken from Key stage 1 & 2: mathematics test frameworkYear 3Year 44F6a Recognise & write decimal equivalentsto 1/4, 1/2, 3/44F6b Recognise & write decimal equivalentsof any number of tenths or hundredths4F7 Round decimals with one decimal placeto the nearest whole numberF7 RoundingdecimalsF8 Compare &order decimals4F8 Compare numbers with the samenumber of decimal places up to two dp4F9 Find the effect of dividing a one- or twodigit number by 10 & 100, identifying thevalue of the digits in the answer as ones,tenths & hundredthsF9 Multiply /divide decimals3F10 Solve problems that involve 3F1–3F4F10 Solveproblems withfractions &decimalsF11 Fractions /decimal /percentageequivalenceF12 Solveproblems withpercentages4F10a Solve problems involving increasinglyharder fractions to calculate quantities &fractions to divide quantities, including nonunit fractions where the answer is a wholenumber4F10b Solve simple measure & moneyproblems involving fractions & decimals totwo decimal placesYear 55F6a Read & write decimal numbers asfractions [eg: 0.71 71 100]Year 66F6 Associate a fraction with division tocalculate decimal fraction equivalents (eg:0.375) for a simple fraction eg: 3/85F6b Recognise & use thousandths & relatethem to tenths, hundredths & decimalequivalents5F7 Round decimals with two decimalplaces to the nearest whole number & to onedecimal place5F8 Read, write, order & compare numberswith up to three decimal places5F10 Solve problems involving numbers upto three decimal places6F9a Identify the value of each digit to threedecimal places & multiply & divide numbersby 10, 100 & 1000 giving answers up tothree decimal places6F9b Multiply one-digit numbers with up totwo-decimal places by whole numbers6F9c Use written division methods in caseswhere the answer has up to 2 decimal places6F10 Solve problems which require answersto be rounded to specified degrees ofaccuracy5F11 Recognise the per cent symbol (%) &understand that per cent relates to ‘numberof parts per hundred’; write percentages as afraction with denominator hundred, & as adecimal5F12 Solve problems which require knowingpercentage & decimal equivalents of 1/2,1/4, 1/5, 2/5, 4/5 & those fractions with adenominator of a multiple of 10 or 25Ratio & proportionR1 Relativesizes, similarityR2 Use ofpercentages forcomparisonR3 Scale factorsR4 Unequalsharing &groupingAlgebraA1 Missing number problems expressed in algebraA2 Simple formulae expressed in wordsA3 Generate & describe linear number sequencesA4 Number sentences involving two unknownsA5 Enumerate all possibilities of combinations of twovariables6R1 Solve problems involving the relativesizes of two quantities, where missing valuescan be found by using integer multiplication &division facts6R2 Solve problems involving the calculationof percentages [eg: of measures such as15% of 360] & the use of percentages forcomparison6R3 Solve problem involving similar shapeswhere the scale factor is known or can befound6R4 Solve problems involving unequalsharing & grouping using knowledge offractions & multiples6A1 Express missing number problemsalgebraically6A2 Use simple formulae6A3 Generate & describe linear numbersequences6A4 Find pairs of numbers that satisfy anequation with two unknowns6A5 Enumerate possibilities of combinationsof two variables
StrandYear 1Year 21M1 Compare, describe & solve practicalproblems for: lengths & heights eg: long/short, longer/shorter, tall/short, double/half mass/weight eg: heavy/light, heavierthan, lighter than capacity & volume eg: full/empty, morethan, less than, half, half full, quarter time eg: quicker, slower, earlier, later2M1 Compare & order lengths, mass,volume/capacity & record the results using , & National Curriculum Strand Tracker: taken from Key stage 1 & 2: mathematics test frameworkYear 3Year 4Year 5Year 6MeasurementM1 Compare,describe & ordermeasuresM2 Estimate,measure & readscales1M2 Measure & begin to record thefollowing: lengths & heights mass/weight capacity & volume time (hours, minutes, seconds)2M2 Choose & use appropriate standardunits to estimate & measure length/ height inany direction (m/cm); mass (kg /g);temperature ( C); capacity (litres/ ml) to thenearest appropriate unit using rulers, scales,thermometers & measuring vessels3M1a Compare lengths (m/cm/mm)3M1b Compare mass (kg/g)3M1c Compare volume / capacity (l/ml)3M2a Measure lengths (m/cm/mm)4M1 Compare different measures, includingmoney in pounds & pence4M2 Estimate different measures, includingmoney in pounds & pence3M2b Measure mass (kg/g)3M2c Measure volume / capacity (l/ml)1M3 Recognise & know the value of differentdenominations of coins & notesM3 MoneyM4 Telling time,ordering time,duration & unitsof time1M4a Tell the time to the hour & half past thehour & draw the hands on a clock face toshow these times1M4b Sequence events in chronologicalorder using language [eg: before & after,next, first, today, yesterday, tomorrow,morning, afternoon & evening]1M4c Recognise & use language relating todates, including days of the week, weeks,months & years2M3a Recognise & use symbols for pounds( ) & pence (p); combine amounts to make aparticular value2M3b Find different combinations of coinsthat equal the same amounts of money2M4a Tell & write the time to five minutes,including quarter past/to the hour & draw thehands on a clock face to show these times2M4b Compare & sequence intervals of time2M4c Know the number of minutes in anhour & the number of hours in a day3M4a Tell & write the time from an analogueclock; 12-hour clocks4M4a Read, write & convert time betweenanalogue & digital 12-hour clocks3M4b Tell & write the time from an analogueclock; 24-hour clocks4M4b Read, write & convert time betweenanalogue & digital 24-hour clocks3M4c Tell & write the time from an analogueclock, including using Roman numerals fromI to XII4M4c Solve problems involving convertingfrom hours to minutes; minutes to seconds;years to months; weeks to days5M4 Solve problems involving convertingbetween units of time4M5 Convert between different units ofmeasurement [eg: kilometre to metre; hour tominute]5M5 Convert between different units ofmetric measure [eg: kilometre & metre;centimetre & metre; centimetre & millimetre;gram & kilogram; litre & millilitre]3M4d Estimate & read time with increasingaccuracy to the nearest minute; record &compare time in terms of seconds, minutes &hours; use vocabulary such as o’clock/a.m./p.m., morning, afternoon, noon & midnight3M4e Know the number of seconds in aminute & the number of days in each month,year & leap year3M4f Compare durations of events, eg:calculate the time taken by particular eventsor tasksM5 Convertbetween metricunitsM6 Convertmetric/imperial3M7 Measure the perimeter of simple 2–DshapesM7 Perimeter,area4M7a Measure & calculate the perimeter ofa rectilinear figure (including squares) incentimetres & metres4M7b Find the area of rectilinear shapes bycounting squares5M6 Understand & use approximateequivalences between metric units &common imperial units such as inches,pounds & pints5M7a Measure & calculate the perimeter ofcomposite rectilinear shapes in centimetres& metres5M7b Calculate & compare the area ofrectangles (including squares), & includingusing standard units, square centimetres(cm2)
2C4 Solve s addition & subtraction problems: using concrete objects & pictorial representations, including those involving numbers, quantities & measures applying their increasing knowledge of mental & written methods 3C4 Solve problems, including missing number problems, using number facts, place value, & more complex addition &
An approximation scheme for problem X is a family of (1 ε)-approximation algorithms Aε (respectively, (1 ε)-approximation algorithms Aε) for problem X over all 0 ε 1. A polynomial time approximation scheme (PTAS) for problem X is an approximation scheme whose time complexity is polynomial in the input size.
Approximation Algorithms Recap An algorithm A is an -approximation algorithmfor problem P if, A runs inpolynomial time A always outputs a solution with value s Here P is an optimisation problem with optimal solution of value Opt If P is a maximisation problem, Opt 6 s 6 Opt within an factor of Opt If P is a minimisation problem, Opt 6 s 6 Opt We have seen: a 3 2-approximation algorithm forBin .
Indeed, the mean- eld approximation in [4] has signi cant gap from the \true" value (from exact DTMC analysis) when the system size is relatively small, e.g., 10-20. To address this gap and improve the approximation accuracy for the discrete-time network in a broader range of settings, we develop a re ned mean- eld approximation based on the
A spreadsheet template for Three Point Estimation is available together with a Worked Example illustrating how the template is used in practice. Estimation Technique 2 - Base and Contingency Estimation Base and Contingency is an alternative estimation technique to Three Point Estimation. It is less
Introduction The EKF has been applied extensively to the field of non-linear estimation. General applicationareasmaybe divided into state-estimation and machine learning. We further di-vide machine learning into parameter estimation and dual estimation. The framework for these areas are briefly re-viewed next. State-estimation
Compare Three-Digit Number Vocabulary Partner B Write ., ,, or 5 to compare the numbers. When I numbers, I always start with the place value. The greatest place value in 367 and 376 is the place. Both numbers have a 3 in the hundreds place. So I look at the tens place. The number 367 has in the tens place. The number 376 has in the tens place.
this increases the running time by a factor O(nd). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-approximation in polynomial time. We also give similar approximation schemes for some other NP-hard Euclidean problems: Mini-mum Steiner Tree, k-TSP, and k-MST.
PACKING [FL81], are known to have polynomial time approximation schemes (PTASs). A PTAS is an algorithm that, for every xed 0, achieves an approximation ratio of 1 in time that is polynomial in the input size (but could grow very fast with 1 , such as O(n1 )). A PTAS thus allows us to trade o approximation accuracy for running time.
