S1 Math Notes

CHSY1NotesArrangedaccordingtoCHSOrder of Learning1.Primes,HCF,LCM(TBChapter 1)2.Integers,Rational NumbersandReal Numbers(TBChapter 2)3.ApproximationandEstimation(TBChapter 3)4.Percentage(TBChapter 8)5.RatioandRate(TBChapter 9)6.BasicAlgebraandManipulation(TBChapter 4)7.Linear Equations(TBChapter 5),SimpleInequalities(Sec2TBChapter 2)8.Linear FunctionsandGraphs(TBChapter 6)9.Number Patterns(TBChapter 7)10.BasicGeometry(TBChapter 10)11.PolygonsandGeometrical Constructions(TBChapter 11)12.PythagorasTheorem(Sec2TBChapter 9)13.Perimeter andAreaof PlaneFigures(TBChapter 12)14.VolumeandSurfaceAreaof PrismsandCylinders(TBChapter 13)15.VolumeandSurfaceAreaof Pyramids,conesandSphere(Sec2TBChapter11)16.Statistical DataHandling(TBChapter 14)17.Statistical Diagrams(Sec2TBChapter 13)18.Averagesof Statistical Data(Sec2TBChapter 14) Primes. HCF, LCMPrimeNumber● Number that onlyhas2factorsandisawholenumber greater than1,suchas2, 3, 7, 13UnderstandingPrimeFactorisation● Usedtoexpressacompositenumber asaproduct of itsprimefactors,usingafactor treeor longdivision● WhenPrimeFactorisationisperformedsomefactorsarerepeated,eg8=2x2x2, with thefactor 2appearing3times● Useconcisenotationtorepresent theproduct● Notationwithnumber abovethebelownumber iscalledindexnotation,readas8tothepower of 3● Number locatedat thetopright handcorner calledtheindex,number atthelevel iscalledbase● Indexshowsthenumber of timesthebaseismultipliedbyitself● UseLadder Methodtofindprimefactorizationof anumber,start withsmallest primefactor of thenumber anddivide,keepdividingtill youget 1thentimesall thenumbersSquareroots andCuberoots 1

● 9canbeexpressedasproduct of 2identical numbers, 9=3x3=3 2 ● Since3 2 =9,9isthesquareof 3,wecanalsosay3isthepositivesquarerootof 9andit isdenoted9squareroot=3● 8canbeexpressedasaproduct of 3identical numbersas8=2x2x2=2 3 ,since2 3 =8,8iscalledcubeof 2● Wecanalsosay2isthepositivecuberoot of 8anddenotedcuberoot 8=2● For squarenumbers,power of primefactorsmust bedivisibleby2● For cubenumbers,power of primefactorsmust bedivisibleby3Highest commonfactor andLowest commonmultipleHighest commonfactor● Largest commonfactor of agroupiscalledHCF756expressedasaproduct of itsprimefactorsis2 2 x3 3 x7,360expressedasaproduct of itsprimefactorsis2 3 x3 2 x5,findhcf of 360and756Method1360=2 3 x3 2 x5756=2 2 x3 3 x7HCF=2 2 x3 2 ● Comparebothnumberswhenexpressedasaproduct of itsprimefactorsthenlookfor commonfactors(best squared/cubed)LCM● Findusingprimefactorisation756expressedasaproduct of itsprimefactorsis2 2 x3 3 x7,360expressedasaproduct of itsprimefactorsis2 3 x3 2 x5,findLCMof 360and756360=2 3 x3 2 x5756=2 2 x3³x7LCM=2 2 x3 3 x5x7● Between2 3 and2 2 bringdownbigger number, thenbringdowneverythingelse 2. Integers, Rational NumbersandReal NumbersNegativeNumbersNegativeIntegers● Wholenumbersare1,2,3,4,knownaspositiveintegers● Negativeintegersare-5,-4,-3● Zeroisneither positivenor negativeintegerNumber line● All negativenumberstotheleft of zero,whileAll positivenumberstotheright of zero● NumbersarearrangedinascendingorderHowtodrawanumber line● Drawhorizontal lineandmarkzeropoint 2

● Use1cmruler tomarkpoints1,2,3at equal unit lengthtotheright of 0and-1,-2,-3,onleft of 0● Drawarrowheadsonbothendsof thelinesAdditionandsubtractioninvolvingnegativenumbers (personal methods,idkif insyllabus)AdditionMethod1● Useanumber line● Drawnumber linebeginningat 0● For negativenumbersmovethat manyspacestotheleft● For positivenumbersmovethat manyspacestotheright● Examples-5+4● Beginningat 0,-5negativesomove5spacestotheleft,after that move4spacestotherightMethod2Useabsolutevalue(outsidethesyllabusbut still useful)● Additionof largenumbers● Lookat thesigns● If signsof thenumbersyouareaddingarethesametheyarealike(gointhesamedirection)● Thereforeaddupthosetwonumbersandkeeptheir signExample1+-2=-3(add1and2thenkeepnegativesign)● However if thesignsof thenumbersyouareaddingaredierent subtractabsolutevalueof the2numbers● Whichnumber hasahigher absolutevalue?● Theanswer will havethesamesignthat thisnumber hadat thebeginningSubtractingPositiveandnegativenumbers● SubtractionandAdditionareoppositesof eachother,sowecanchangeasubtractionproblembyusingtheadditiveinverseor oppositeExample5-4● Additiveinverseof 4is-4whichwecanchangetoaadditionproblem,soits5+-4=1Example7-10● Additiveinverseof 10is-107-10=7+(-10)=-3ExampleAbirdisflyingat 42mabovesealevel andafishisswimming12metesbelowsealevel. Howmanymetersapart arethefishandbird?Birdsheight 42mFishheight -12m 3

Subtract, 42-(12)=42+12=54Multiplyinganddividingpositiveandnegativenumbers● Multiplyor dividethenumbers,thencount number of negativenumbers● If thereanoddnumber of negativenumbersanswer isnegative● If thereanevennumber of negativenumbersanswer ispositive 3. ApproximationandEstimationApproximation● Processof roundingoagivennumber togiveapproximatevalueSignificant figures● Roundnumberstoarequirednumber of Significant FiguresFiverules toidentifysignificant digits● All nonzerodigitsaresignificant (eg192has3SF)● All zeroesbetweennonzerodigitsaresignificant (eg32047has5SF0● Inadecimal,all zerosafter anonzeroaresignificant (eg0.10has2SF)● Inadecimal all zeroesbeforeanonzeroarenonsignificant (eg0.010has2SF)● Inwholenumberszeroesat theendmayor maynot besignificant, itdependsonhownumbersareapproximatedExampleApieceof paper weighs0.0004503g,roundit to1dp,2dp,3dp1dp-0.02dp-0.003dp-0.0000.00045034isthe1st SFAndsoon508175.625is1st SF,,0is2ndSF,8is3rdSFRoundingandTruncationerrors● Truncatemeanscut otheend,egsquareroot 162=12.72792206● If weroundif theanswer to3dpits12.728,However if wetruncatetheanswer at 3dp,it is12.727● ThereisnoroundingoExample2divideby3oncalculator hasafewoptions● If calculator shows0.66666666666667answer wasrounded● If calculator shows0.66666666666666answer wastruncatedConclusion● Accuratenumerical valuecannot beachievedwhenthenumber inanexpressionisroundedotooearly 4

● Inpracticeif aproblemrequiresananswer that iscorrectedto3sf weshouldstoretheintermediateworkingvaluesinour calculator or roundthemotomoreSFeg5SF○ Thiswill increasetheaccuracyof our final answerEstimation● Processof guessingvalueof anunknownquantitySummary● 3situationswhenapproximationisused○ Actual valueknownbut not usedfor variousreasonsegactualvaluenot necessary, easier tormbanapproximatedvalue, toomessytowritealongstringof numbers, impossiblefor calculatorstostoreall digitsof nonexact number○ Exact valuecannot beobtained○ Actual valuetootroublesome/impossibletoobtain 4. Percentage● Natural mathextensionof fractions, ratioandproportion● All percentagesareexpressionof relationshipbasedon100● Everyfraction,ratioandproportionexpressedasapercentage● Percentagesalsoexpressedwheredecimalsarerequired,eg66.96%● Calculationof variousratesbywayof percentagesisabackboneof widerangeof mathapplicationsinctaxes, interest, grades, sportsstatistics,etcPercentageChangeandReversePercentage● Changeinthevalueof anitemexpressedasapercentageincreaseordecreaseintheoriginal value● Tocalculatethepercentageincrease,increaseinvalueof aquantityfromitsoriginal valuemust beknown● Increase=NewValue-Original Value● PercentageIncrease=Increase/Original Valuex100%● NewvaluecanbefoundusingNewValue=Original Valuex(100%+percentageincrease)● Tocalculatepercentagedecrease,● Decrease=Original Value-NewValue● PercentageDecrease=Decrease/Original Valuex100%● NewValue=Original Valuex(100%-PercentageDecrease)PercentageandPercentagePoint inpractical situationsProfit andLoss● Goodsproducedat acertaincost,whentheyaresoldat apricehigherthancost priceaprofit ismade● Whengoodsaresoldat apricelower thanthecost pricealossismade● Profit=Sellingprice-Cost price 5

● Loss=Cost price-Sellingprice● Profit or lossusuallyexpressedasapercentageof thecost price● Profit(or loss)/cost pricex100%● Noteinsomecasesprofit or losscanbeexpressedasapercentageof thesellingpriceDiscount● Itemssoldat lower price(saleprice)● Dierencebetweenoriginal sellingprice,or markedpriceandsalepriceiscalleddiscount● Discount=MarkedPrice-SalePrice● Similarlydiscount oftengivenaspercentageo