Sectio 1 of Uit 0 (Pure Mathematics ) Algebra Recommeded Prior Kowledge Studets should have studied the algebraic techiques i Pure Mathematics 1. Cotet This Sectio should be studied early i the course sice the work o epoetial ad logarithmic fuctios, ad o partial fractios, is eeded for Sectio. www.xtremepapers.com Outlie The Sectio solves equatios ad iequalities ivolvig the modulus fuctio ad eamies the use of the factor ad remaider theorems i solvig cubic ad quartic equatios. Work o the biomial epasio ( 1 + ), is eteded to cosider all values of. The Sectio deals with the decompositio of ratioal fuctios ito partial fractios (Topic 1). The epoetial ad logarithmic fuctios are itroduced. Logarithms are used to solve equatios of the form a = b ad to reduce equatios to liear form (Topic ). The Sectio cocludes by dealig with the umerical solutio of equatios by use of a iterative formula (Topic 6).
Topic Learig Outcomes Suggested Teachig Activities Resources O-Lie Resources 1 Algebra Uderstad the meaig of ad use relatios such as a = b a = b ad a b a b a + b i the course of solvig equatios ad iequalities. Geeral discussio o the modulus fuctio. Studets must appreciate that 4 = 4 = 4. They should be ecouraged to draw graphs of fuctios such as y =, recogisig the V shape of the graph ad the poit (1.5,0) o the - ais. They should be ecouraged to sketch graphs of the modulus of other fuctios such as simple trigoometric graphs ad quadratics with statioary poits below the - ais. Studets should be appreciative of the fact that a = b a = b ad be able to use these as appropriate for solvig equatios such as = 7. The graphical solutio of this equatio should also be eamied, as should the alterative method of equatig (-) to ± 7. Studets should realise that the solutio of the iequality < 7 ca be obtaied either from ( ) < 49 or from 7 < < 7. OHP with graphs of such fuctios as y = si, y = 4 ad y = 5. www.bbc.co.uk/ educatio/asgur u/maths pure fuctios modulus
Divide a polyomial, of degree ot eceedig 4, by a liear or quadratic polyomial, ad idetify the quotiet ad remaider (which may be zero). Use the factor theorem ad the remaider theorem, e.g. to fid factors, solve polyomial equatios or evaluate ukow coefficiets. The process of dividig a polyomial (cubic or quartic) by either a liear or quadratic polyomial should be discussed (a OHP is particularly useful). Practice eercises should iclude cases i which a polyomial cotais a coefficiet of zero (for eample the divisio of a quartic that has o coefficiet i ). The alterative method of avoidig the log divisio by comparig coefficiets should also be eamied, especially with more able groups. Studets should be aware that, i the case of divisio of f() by, say, ( ), the remaider is f().this aswer should be verified by the divisio process. This leads aturally to the factor theorem i.e. if the remaider is zero whe dividig by (- ), the (-) is a factor of f() ad that = is a solutio of f()=0. Studets should be able to factorise polyomial epressios ad to solve polyomial equatios by first usig trial ad error to fid a solutio or factor ad the completig the process by divisio. OHP showig i detail the process of log divisio of a cubic polyomial with a liear polyomial. Recall a appropriate form for epressig ratioal fuctios i partial fractios, ad carry out the decompositio, i cases where the deomiator is o more complicated tha ( a + b)( c + d)( e + d) ( a + b)( c + d) ( a + b)( + c ) ad where the degree of the umerator does ot eceed that of the umerator. As a itroductio to partial fractios, it is a worthwhile eercise revisig some of the basic algebra from O Level or IGCSE by askig the studets to epress the followig as sigle fractios:- 4 + 5 + + + ( + ) 1 + 4 OHP showig the process eeded at O Level or at IGCSE for the additio of two or more partial fractios to form a sigle fractio.
The questio of reversig the process ca the be itroduced. Studets should be ecouraged to realise that a 1 ratioal fuctio such as ca be reduced to ( + )( 1) A B two partial fractios of the form +. Studets + 1 should be ecouraged to reach the idetity 1 A( 1) + B( + ) for themselves. They should be show that such a idetity holds for all values of ad that substitutio of ay values of, but particularly = 1 ad = -, will eable the values of A ad B to be foud. They should appreciate why these two particular values of have bee selected. They should also appreciate that equatios for A ad B ca be obtaied by comparig the coefficiets of ad, i this case, the costat o each side of the idetity. Similar processes should eable the studets to epress 6 1 such ratioal fuctios as ad ( 1) ( + ) ( i + 4) partial fractios. Studets should be made aware of the eed to divide first i cases where the degree of the umerator is the same as that of the deomiator. I such cases the divisio ca ofte be doe sythetically more easily that formally, especially with such fractios as. They should be ecouraged to commit to + memory all the possible decompositios withi the syllabus. OHP showig all the possible decompositios withi this syllabus.
Use the epasio of (1 + ), where is a ratioal umber ad p1 (fidig a geeral term is ot icluded, but adaptig the 1 1 stadard series to ( ) is icluded) The work i Topic 7 (Series) o Uit P1 should be revised. Studets should be comfortable i fidig epressios for ( 1 + ) for simple itegral values of. They should be aware that the coefficiet of r i this epasio ca be r obtaied from the recurrece relatio C = C r + 1 r + 1 They should be made aware that this same epasio holds for values of that are o-itegral, providig that <1. Studets should realise that the epasio of ( 1 + ) ca be similarly used with such series as 1 + 6 with beig replaced by ad = ½ ad for (1 ) with beig replaced by ad =. Lots of practice is eeded. Studets the eed to realise that the 1 1 epasio of ( ) ca also be obtaied by rewritig it as 1 1 1 (1 ) ad the usig the basic epasio of ( 1 + ) 4 with replaced by ¼ ad = 1. r
Logarithmic ad Epoetial Fuctios Uderstad the relatioship betwee logarithms ad idices, ad use the laws of logarithms. Geeral discussio o the defiitio of a logarithm. Show studets that the statemet 10 =100 ca be writte as log 10 100 =. If time allows, show studets how multiplicatio ad divisio was carried out i pre-calculator days by usig logarithmic tables. Show studets, agai o the calculator, that log 10 =0.01 is equivalet to 10 0.01 =. Illustrate, o the calculator, why it is that = 6 is equivalet to the additio log 10 +log 10 =log 10 6 ad give similar results for divisio ad for epoetial calculatios. Deduce the laws of logarithms ad give the studets plety of practice i evaluatig such epressios as log 8 etc, writig epressios such as log k a+log k b-log k c as a sigle logarithm ad vice versa OHP with eamples of the logarithmic calculatios used i the time before the calculator appeared i schools. Iclude i this a eplaatio of why the process works, referrig back to powers of 10. www.bbc.co.uk/ educatio/asgur u/maths pure epoetials ad logarithms Uderstad the defiitio ad properties of e ad l, icludig their relatioship as iverse fuctios ad their graphs. Discuss the properties of the graphs of y = e ad y = l. Draw accurate graphs ad use these to deduce that the gradiet of the curve y = e is equal to e ad that the gradiet of the curve y = l is equal to -1 (this makes a ecellet piece of homework). These results ca be held back util the et sectio (Calculus) is covered, but it is worth poitig out that the gradiet of e is always positive ad always icreasig ad that the gradiet of l is always positive ad always decreasig. Studets should be aware that the fuctios e ad l are iverse fuctios (check o the calculator) ad that the graphs of y = e ad y = l are mirror images i the lie y =. Studets should be aware that l(e ) = ad that e l =. It should be poited out that the rage of f : a e + for R is R ad that the rage of f: a l for + R is R. OHP showig the accurate graphs of y=e ad y=l ad illustratig the iverse property (reflectio i the lie y = ) ad showig how the gradiet fuctios of each ca be verified.
Use logarithms to solve equatios of the form a = b, ad similar iequalities. Use logarithms to trasform a give relatioship to liear form, ad hece determie ukow costats by cosiderig the gradiet ad / or itercept. Studets should cosider the solutio of the equatio a = b ad should be ecouraged to obtai a solutio by trial ad improvemet. They should be show how epressig the equatio i logarithmic form reduces it to a liear equatio. Studets should be able to solve equatios such as +1 =, ad be able to eted the work to iequalities. They eed to be aware that the solutio of such iequalities as (0.6) < 0., results i a solutio > k, rather tha < k. (they should appreciate that log 0.6 is egative ad that divisio by a egative quatity affects the sig of the terms i the iequality). Such questios might be posed i the form fid the smallest value of for which the th term of the geometric progressio with first term ad commo ratio 0.9 is less tha 0.1. Studets should be able to covert: (i) the equatio y = a to logarithmic form recogisig that the resultig equatios is liear, givig a straight lie graph whe logy is plotted agaist log, (ii) the equatio y = A( b ) to logarithmic form recogisig that the resultig equatio is liear, givig a straight lie graph whe logy is plotted agaist. (iii) the equatio y = Ae to the form ly=la+ recogisig that this form is liear, givig a straight lie whe ly is plotted agaist. Studets should have plety of practice i deducig the values of either a ad i (i) or A ad b i (ii) or A ad i (iii) by drawig straight lie graphs from give eperimetal values of ad y. Such eercises should iclude cases i which the variables have symbols other tha ad y (represetig, for eample, particular physical quatities). OHP showig values of (0.6) for differet to illustrate that >k rather tha <k. OHP graphs for the liear forms of the equatios y = a y = A( b ) ad y = Ae.
6 Numerical Solutio of Equatios Locate approimately a root of a equatio, by meas of graphical cosideratios ad / or searchig for a sig chage. Studets will have eperieced the solutio of equatios where methods ca be used to give eact aswers liear equatios, quadratic equatios, trigoometric equatios, equatios of the form = k etc. It is worth recallig all of these techiques. Geeral discussio should follow o the solutio of such equatios as, for eample, (i) + 5 = 100, (ii) = cos ad (iii) e = +. Studets should have used the method of trial ad improvemet for solvig equatios of type (i) ad should realise that a sketch of f ( ) = + 5 100 (or sketches of y = ad y = 100 5), or a search for a sig chage, will lead them to a solutio i the rage = 4 to = 5. I case (ii), sketches of y = ad y = cos should lead to the coclusio that there is oly oe positive root ad that the OHP showig each of the graphs used i eamples (i), (ii) ad (iii). www.bbc.co.uk/ educatio/asgur u/maths pure umerical methods root lies i the rage 0 < < π. Similarly sketches of y = e ad y = + (or y = e ad y = + ) for case (iii) should lead to the coclusio that there is oly oe positive root ad that the root lies i the iterval 0 < < 1. Studets will eed a lot of practice with similar equatios, icludig cases i which some maipulatio may be ecessary to obtai two suitable fuctios e.g. fid the smallest root of e si = 1. Uderstad the idea of, ad use the otatio for, a sequece of approimatios which coverge to a root of a equatio. The idea of a iterative formula will probably have bee used i topic 7 of Uit P1 with arithmetic ad geometric 1 progressios ( = a + ( 1) d ad = ar ). Give studets practice i usig equatios of the form = f ( ) ad of the form = F( ) + 1.
Uderstad how a give simple iterative formula of the form + 1 = F( ) relates to the equatio beig solved, ad use a give iteratio, or a iteratio based o a give rearragemet of a equatio, to determie the root to a prescribed degree of accuracy. (Kowledge of the coditio for covergece is ot icluded, but cadidates should uderstad that a iteratio may fail to coverge). Discuss the meaig of a equatio of the type + 1 = + 10. Show studets that usig a iitial value for 1 of either or will lead to a series of umbers that coverge evetually to.1. This is the ideal place to discuss simple ideas of covergece ad divergece. Studets should be aware of the eed to write dow the result of each iteratio, usually to oe more place of accuracy tha requested for the fial aswer. Studets should be aware that the value of the previous iteratio eed ot be keyed i to the calculator eplicitly at each stage; the as key ca be used istead. Studets should be give plety of practice o this procedure. Discuss with the studets the techique of settig +1 = = L (or other coveiet letter) i the case of covergece, ad usig this to deduce the equatio for which the result of the iteratio is a root. Epressig a equatio of the form ³=+10 as = ca be doe i may ways, each leadig to a iterative formula of the type + 1 = F( ). It is a worthwhile eercise havig several of these differet iterative formulae available o a OHP ad showig studets that some of them coverge, but that i some cases, the series diverges. The studets do ot eed to kow of coditios for covergece. It is also worth demostratig that a particular iterative formula may give differet limitig values for differet values of 0, e.g. the formula = 1 + 1 with 4 0 =0.5 ad with 0 =.75. OHP showig differet iterative formulae for the solutio of =+10.