M I N I S T R Y O F E D U C A T I O N


 Arron Bailey
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1 M I N I S T R Y O F E D U C A T I O N Repulic of Ghn TEACHING SYLLABUS FOR SENIOR HIGH SCHOOL ELECTIVE MATHEMATICS Enquiries nd comments on this syllus should e ddressed to: The Director Curriculum Reserch nd Development Division (CRDD) P. O. Box 739 Accr Ghn Tel: Septemer, 00 i
2 RATIONALE FOR TEACHING ELECTIVE MATHEMATICS The ilities to red, nlyze nd clculte re the three fundmentl skills tht re vitl for living nd working. The level of mthemtics one my study depends upon the type of work or profession one my choose in life nd on one s ptitude nd interest. Elective mthemtics dels with resoning y nlogies, mking judgments through discrimintion of vlues, nlysis of dt, nd communiction of one s thoughts through symolic expression nd grphs. Elective Mthemtics t the Senior High School level uilds on the Core Mthemtics of Senior High School. It is requirement s foundtion for those who would wish to emrk on professionl studies in engineering, scientific reserch, nd numer of studies in tertiry nd other institutions of higher lerning. GENERAL AIMS The syllus is designed to help students to: pprecite the use of mthemtics s tool for nlysis, criticl nd effective thinking. discover order, ptterns nd reltions. 3 communicte their thoughts through symolic expressions nd grphs. 4 develop mthemticl ilities useful in commerce, trde nd pulic service. 5 mke competent use of ICT in prolem solving nd investigtion of rel life situtions. SCOPE OF CONTENT Elective mthemtics covers the following content res:. Alger 5. Trigonometry. Coordinte Geometry 6. Clculus 3. Vectors nd Mechnics 7. Mtrices nd Trnsformtion 4. Logic 8. Sttistics nd Proility PREREQUISITE SKILLS AND ALLIED SUBJECTS Success in the study of Elective Mthemtics requires proficiency in English Lnguge nd in Core Mthemtics. Other sujects tht my help the effective study of Elective Mthemtics include Physics nd Technicl Drwing. ii
3 ORGANIZATION OF THE SYLLABUS This syllus hs een structured to cover the three (3) yers of Senior High School. Ech yer s work consists of numer of sections with ech section comprising numer of units. The unit topics for the three yers course re indicted in the tle elow. SCOPE AND SEQUENCE FOR SHS ELECTIVE MATHEMATICS SHS SHS SHS3. Sets (pg ) Coordinte Geometry (pg 46) Mtrices (pg 394). Surds (pg  ) Sequences nd Series (pg 78) Liner Trnsformtions (pg 443) 3. Binry Opertions (pg 3) Indices nd Logrithms (pg 89) Logic (pg 44) 4. Reltions nd Functions (pg 34) Trigonometric Rtios nd Rules (pg 0  ) Correltion nd Regression (pg 4547) 5. Polynomil Functions (pg 47) Compound nd Multiple Angles (pg  ) Spermn s Rnk Correltion (pg 4547) 6. Rtionl Functions (pg 7) Trigonometric Functions/Equtions (pg ) Dynmics (pg 4849) 7. Binomil Theorem (pg 78) Differentition (pg 34) 8. Inequlities nd Liner Progrmming (pg 8) Appliction of Differentition (pg 45) 9. Coordinte Geometry I (pg 90) Integrtion (pg 6) 0. Proility I (pg ) Appliction of Integrtion (pg 67). Vectors I (pg  3) Permuttion nd Comintions (pg 8). Proility II (pg 9) 3. Sttistics I (pg 303) 4. Appliction of vectors in Geometry (pg 3335) 5. Sttics (pg 3638) TIME ALLOCATION Elective Mthemtics is llocted six periods week, ech period consisting of forty (40) minutes. iii
4 SUGGESTIONS FOR TEACHING THE SYLLABUS This syllus hs een plnned to incorporte lmost ll rnches of mthemtics:  Alger, Logic, Trigonometry, Coordinte Geometry, Clculus, Liner Trnsformtion, Vectors, Mechnics, Sttistics nd Proility. In rod frmework of this nture, schools will hve to dopt tem teching pproch for this course. Besides, the techer s ttention is drwn to the use of clcultors nd ICT in teching of Elective mthemtics. The syllus hs een uilt on the core mthemtics syllus. It is therefore necessry for the student to hve sound foundtion in core mthemtics. Techers re dvised to red through the entire syllus in order to pprecite its scope nd demnds. Agin, techers re to link up the core nd elective sylluses when deling especilly with the topics. Generl Ojectives Generl ojectives hve een listed t the eginning of ech section. The generl ojectives re linked to the Generl Aims of this suject nd specify the skills nd ehviours the student should cquire fter lerning the units of section. Section nd Units The syllus hs een plnned on the sis of sections nd units. Ech yer s work is divided into sections. A section consists of numer of units nd specific ojectives. The syllus is structured in five columns: Unit, Specific ojectives, Content, Teching nd Lerning Activities nd Evlution. A description of the contents of ech column is s follows: Column Units The units in column re the mjor topics of the section. The numering of the units is different from the numering dopted in other sylluses. The unit numers consist of two digits. The first digit shows the yer or clss, while the second digit shows the numer of the unit. A unit numer like. is interpreted s Unit of SHS. Similrly unit numer like 3.4 mens Unit 4 of SHS3. This type of unit numering hs een dopted to ensure tht the selected topics nd skills re tught ppropritely in the suggested sequence. The order in which the units re rrnged is just to guide you pln your work. If however, you find t some point tht teching nd lerning in your clss will e more effective if you rnch to nother unit efore coming ck to the unit in the sequence, you re encourged to do so. It is hoped tht no topics will e glossed over for lck of time, ecuse it is not desirle to crete gps in students knowledge. Column Specific Ojectives Column shows the specific ojectives for ech unit. The specific ojectives in this syllus egin with numers such s..3 or 3... These numers re referred to s Syllus Reference Numers SRN. The first digit in this elective mthemtics syllus reference numer refers to the yer of the SHS clss; the second digit refers to the unit, while the third digit refers to the rnk order of the specific ojective. For exmple,..3 mens SHS, unit nd specific ojective 3. In other words..3 refers to specific ojective 3 of unit of SHS. Similrly, the syllus reference numer 3... simply mens syllus ojective Numer of unit t SHS3. Using syllus reference numers provides n esy wy for communiction mong techers nd other eductors. It further provides n esy wy for selecting ojectives for test construction. For instnce, if unit hs five specific ojectives , the techer my wnt to se his/her questions on ojectives.4.3 to.4.5 nd not use the other first two specific ojectives. In this wy iv
5 techer would smple the ojectives within units nd within the yer to e le to develop test tht ccurtely reflects the importnce of the vrious concepts nd skills tught in clss. You will note lso tht specific ojectives hve een stted in terms of the student,  i.e., wht the student will e le to do during nd fter instruction nd lerning in the unit. Ech specific ojective hence strts with the following The student will e le to This in effect mens tht the techer hs to ddress the lerning prolems of ech individul student. It mens individulizing your instruction s much s possile such tht the mjority of students will e le to mster the ojectives of ech unit of the syllus. Column 3 Content: The content in the third column of the syllus presents selected ody of informtion tht you will need to use in teching the prticulr unit. In some cses, the content presented is quite exhustive. In other cses, you could dd more informtion to the content presented. Column 4 Teching nd Lerning Activities (T/LA): T/LA ctivities tht will ensure mximum student prticiption in the lessons re presented in column 4. Avoid instrumentl lerning nd drilloriented methods nd rther emphsize prticiptory teching nd lerning, nd lso emphsize the cognitive, ffective nd psychomotor domins of knowledge in your instructionl system wherever pproprite. You re encourged to reorder the suggested teching nd lerning ctivities nd lso dd to them where necessry in order to chieve optimum student lerning. A suggestion tht will help your students cquire the hit of nlyticl thinking nd e le to pply their knowledge to prolems is to egin ech lesson with rel life prolem. Select rel life or prcticl prolem for ech lesson. The selection must e mde such tht students cn extend the knowledge gined in the previous lesson nd other generic skills to new situtions not specificlly tught in clss. This is to enle students see the relevnce of mthemtics to rel life sitution. At the eginning of lesson, stte the prolem, or write the prolem on the ord. Let students pply (George Poly s) prolem solving techniques, nlyze the prolem, suggest solutions, etc., criticize solutions offered, justify solutions nd evlute the worth of possile solutions. There my e numer of units where you need to reorder specific ojectives to chieve required lerning effects. Column 5 Evlution: Suggestions nd exercises for evluting the lessons of ech unit re indicted in Column 5. Evlution exercises cn e in the form of orl questions, quizzes, clss ssignments, structured questions, project work, etc. Try to sk questions nd set tsks nd ssignments tht will chllenge your students to pply their knowledge to issues nd prolems nd engge them in developing solutions nd developing positive ttitudes towrds the suject s result of hving undergone instruction in this suject. The suggested evlution tsks re not exhustive. You re encourged to develop other cretive evlution tsks to ensure tht students hve mstered the instruction nd ehviour implied in the specific ojectives of ech unit. Lstly, er in mind tht the syllus cnnot e tken s sustitute for lesson plns. It is, therefore, necessry tht you develop scheme of work nd lesson plns for teching the units of this syllus. DEFINITION OF PROFILE DIMENSIONS A centrl spect of this syllus is the concept of profile dimensions tht should e the sis for instruction nd ssessment. A dimension is psychologicl unit for descriing prticulr lerning ehviour. More thn one dimension constitute profile of dimensions. A specific ojective such s follows: The student will e le to descrie etc., contins n ction ver descrie tht indictes wht the student will e le to do fter teching hs tken plce. Being le to descrie something fter the instruction hs een completed mens tht the student hs cquired knowledge. Being le to explin, summrize, give exmples, etc. mens tht the student hs understood the lesson tught. Similrly, eing le to develop, pln, construct etc, mens tht the student hs lernt to crete, innovte or synthesize knowledge. Ech of the specific ojectives in this syllus contins n ction ver tht descries the ehviour the student will e le to demonstrte fter the instruction. Knowledge, Appliction, etc. re dimensions tht should e the prime focus of teching nd lerning in schools. Instruction in most cses hs tended to stress knowledge cquisition to the detriment of other higher level ehviours such s ppliction, nlysis, etc. The v
6 importnce of lerning is to help students to e le to pply their knowledge, develop nlyticl thinking skills, synthesize informtion, nd use their knowledge in vriety of wys to del with lerning prolems nd issues in life. Ech ction ver indictes the underlying profile dimension of ech prticulr specific ojective. Red ech ojective crefully to know the profile dimension towrd which you hve to tech. Profile dimensions descrie the underlying ehviours for teching, lerning nd ssessment. In Elective Mthemtics, the two profile dimensions tht hve een specified for teching, lerning nd testing re: Knowledge nd Understnding 30% Appliction of Knowledge 70% Ech of the dimensions hs een given percentge weight tht should e reflected in teching, lerning nd testing. The weights, indicted on the right of the dimensions, show the reltive emphsis tht the techer should give in the teching, lerning nd testing processes. The focus of this syllus is to get students not only to cquire knowledge ut lso to understnd wht they hve lernt nd pply them in prcticl situtions. The explntion nd key words involved in ech of the dimensions re s follows: Knowledge nd Understnding (KU) Knowledge The ility to: Rememer informtion, recognize, retrieve, locte, find, do ullet pointing, highlight, ookmrk, network socilly, ookmrk socilly, serch, google, fvourite, recll, identify, define, descrie, list, nme, mtch, stte principles, fcts nd concepts. Knowledge is simply the ility to rememer or recll mteril lredy lerned nd constitutes the lowest level of lerning. Understnding The ility to: Interpret, explin, infer, compre, explin, exemplify, do dvnced serches, ctegorize, comment, twitter, tg, nnotte, suscrie, summrize, trnslte, rewrite, prphrse, give exmples, generlize, estimte or predict consequences sed upon trend. Understnding is generlly the ility to grsp the mening of some mteril tht my e verl, pictoril, or symolic Appliction of Knowledge (AK) The ility to use knowledge or pply knowledge, s implied in this syllus, hs numer of lerning/ehviour levels. These levels include ppliction, nlysis, innovtion or cretivity, nd evlution. These my e considered nd tught seprtely, pying ttention to reflect ech of them eqully in your teching. The dimension Applying Knowledge is summry dimension for ll four lerning levels. Detils of ech of the four su levels re s follows: Appliction Anlysis The ility to: Apply rules, methods, principles, theories, etc. to concrete situtions tht re new nd unfmilir. It lso involves the ility to produce, solve, operte, demonstrte, discover, implement, crry out, use, execute, run, lod, ply, hck, uplod, shre, edit etc. The ility to: Brek down piece of mteril into its component prts, to differentite, compre, deconstruct, ttriute, outline, find, structure, integrte, msh, link, vlidte, crck, distinguish, seprte, identify significnt points etc., recognize unstted ssumptions nd logicl fllcies, recognize inferences from fcts etc. vi
7 Innovtion/Cretivity Innovtion or cretivity involves the ility to: Put prts together to form novel, coherent whole or mke n originl product. It involves the ility to comine, compile, compose, devise, construct, pln, produce, invent, devise, mke, progrm, film, nimte, mix, remix, pulish, video cst, podcst, direct, rodcst, suggest n ide or possile wys, revise, design, orgnize, crete, nd generte new ides nd solutions. The ility to innovte or crete is the highest form of lerning. The world ecomes more comfortle ecuse some people, sed on their lerning, generte new ides nd solutions, design nd crete new things. Evlution  The ility to pprise, compre fetures of different things nd mke comments or judgments, contrst, criticize, justify, hypothesize, experiment, test, detect, monitor, review, post, moderte, collorte, network, refrctor, support, discuss, conclude, mke recommendtions etc. Evlution refers to the ility to judge the worth or vlue of some mteril sed on some criteri nd stndrds. Evlution is constnt decision mking ctivity. We generlly compre, pprise nd select throughout the dy. Every decision we mke involves evlution. Evlution is high level ility just s ppliction, nlysis nd innovtion or cretivity since it goes eyond simple knowledge cquisition nd understnding. The ction vers provided under the vrious profile dimensions nd in the specific ojectives of the syllus should help you to structure your teching such s to chieve the effects needed. Select from the ction vers provided for your teching, in evluting lerning efore, during nd fter the instruction. Use the ction vers lso in writing your test questions. FORM OF ASSESSMENT It must e emphsized gin tht it is importnt tht oth instruction nd ssessment e sed on the profile dimensions of the suject. In developing ssessment procedures, select specific ojectives in such wy tht you will e le to ssess representtive smple of the syllus ojectives. Ech specific ojective in the syllus is considered criterion to e chieved y the student. When you develop test tht consists of items or questions tht re sed on representtive smple of the specific ojectives tught, the test is referred to s CriterionReferenced Test. In mny cses, techer cnnot test ll the ojectives tught in term, in yer, etc. The ssessment procedure you use i.e. clss tests, home work, projects, etc. must e developed in such wy tht it will consist of smple of the importnt ojectives tught over period. The exmple elow shows n exmintion consisting of two ppers, Pper nd Pper. School Bsed ssessment hs een dded to the structure. Pper will usully e n ojectivetype pper; Pper will consist of structured questions demnding higher order thinking, nd school sed ssessment will consist numer of tsks. The distriution of mrks for the questions in the two ppers should e in line with the weights of the profile dimensions lredy indicted nd s shown in the lst column of the tle elow. The West Africn Exmintions Council (WAEC) generlly sets out 40 ojective test items t the WASSCE. Use this s guide to develop n ojective test pper (Pper ) tht consists of 40 items. Pper could consist of some structured questions nd more demnding questions put in sections. In generl, let students nswer the compulsory section nd t lest four questions from ten questions put in three prts in Pper. In the smple ssessment structure presented elow, Pper is mrked out of 00; Pper is mrked out of 00 nd school sed ssessment is mrked out of 60, giving totl of 60 mrks. Depending upon the school s exmintion nd mrking systems, you could use totl mrk convenient to the techer nd the school. Ber in mind of course, tht using different totl mrk will chnge the mrk lloctions for the test ppers, etc. vii
8 The lst row shows the weight of the mrks llocted to ech of the four test components. The two ppers nd SBA re weighted differently. Pper, the ojective test pper is weighted 0%. Pper is more intellectully demnding pper nd is therefore weighted more thn the ojective test pper. Pper is designed to test minly ppliction of knowledge. Ppers nd the SBA re weighted 30%, nd 50% respectively. Dimensions Pper Pper Distriution of Exmintion Pper Weights nd Mrks School Bsed Assessment Totl Mrks % Weight of Dimension Knowledge nd Understnding Appliction of Knowledge Totl Mrks % Contriution of Pper You will note tht Pper hs contriution of 0% to the totl mrks; Pper hs contriution of 30% to the totl mrks nd SBA hs contriution of 50% to the totl mrks. The numers in the cells indicte the mrks to e llocted to the items/questions tht test ech of the dimensions within the respective ppers. The lst ut one column shows the totl mrks llocted to ech of the dimensions. The numers in this column re dditions of the numers in the cells nd they gree with the profile dimension weights indicted in the lst column. Of the totl mrks of 0, the 78 mrks for Knowledge nd Understnding is equivlent to 30%. The 8 mrks for Appliction is equivlent to 70% of the totl mrks. Item Bnk: Oviously the structure of ssessment recommended in this syllus will need lot of work on the prt of the techer. In preprtion for setting exmintion ppers, try to develop n item nk. The term item nk is generl term for pool of ojective items, nd essy questions. As you tech the suject, try to write ojective test items, essy questions nd structured essy questions To fit selected specific ojectives which you consider importnt to e tested. If you proceed diligently, you will relize you hve written more thn 00 ojective test items, nd more thn 30 essy questions in spce of one yer. Rndomly select from the item nk to compose the test ppers. Select with replcement. This mens, s items/questions re selected for testing, new ones hve to e written to replce those items/questions lredy used in exmintions. Items nd questions tht hve een used in exmintions my lso e modified nd stored in the item nk. An importnt issue in the preprtion for mjor exmintion such s the WASSCE, is the issue of test wiseness. To e test wise mens tht the student knows the mechnics for tking test. These mechnics include writing the index numer nd other prticulrs ccurtely nd quickly on the nswer pper; reding ll questions efore selecting the est questions to nswer; pportioning equl time to ech question or spending more time on questions tht crry more mrks; mking notes on ech question ttempted efore writing the nswer; leving extr time to red over one s work; finlly checking to see tht the personl prticulrs supplied on the nswer sheet re ccurte. Some good students sometimes fil to do well in mjor exmintions ecuse of wekness in the mechnics of test tking; ecuse they re not test wise. Tke your finl yer students through these necessry mechnics so tht their performnce in mjor exmintions my not e flwed y the slightest wekness in test tking. viii
9 GUIDELINES FOR SCHOOLBASED ASSESSMENT (SBA) A new School Bsed Assessment system (SBA) will e introduced into the school system in 0. The new SBA system is designed to provide schools with n internl ssessment system tht will help schools to chieve the following purposes: o o o o o o o Stndrdize the prctice of internl schoolsed ssessment in ll Senior High Schools in the country Provide reduced ssessment tsks for sujects studied t SHS Provide techers with guidelines for constructing ssessment items/questions nd other ssessment tsks Introduce stndrds of chievement in ech suject nd in ech SHS clss Provide guidnce in mrking nd grding of test items/questions nd other ssessment tsks Introduce system of modertion tht will ensure ccurcy nd reliility of techers mrks Provide techers with dvice on how to conduct remedil instruction on difficult res of the syllus to improve clss performnce. SBA my e conducted in schools using the following: Midterm test, Group Exercise, EndofTerm Test nd Project. Project: This will consist of selected topic to e crried out y groups of students for yer. Segments of the project will e crried out ech term towrd the finl project completion t the end of the yer. The projects my include the following: i) experiment ii) investigtive study (including cse study)\ iii) prcticl work ssignment A report must e written for ech project undertken.. MidTerm Test: The midterm test following prescried SBA formt 3. Group Exercise: This will consist of written ssignments or prcticl work on topic(s) considered importnt or complicted in the term s syllus 4. EndofTem Test: The end ofterm test is summtive ssessment system nd should consist of the knowledge nd skills students hve cquired in the term. The endofterm test for Term 3 for exmple, should e composed of items/questions sed on the specific ojectives studied over the three terms, using different weighting system such s to reflect the importnce of the work done in ech term in pproprite proportions. For exmple, techer my uild n EndofTerm 3 test in such wy tht it would consist of the 0% of the ojectives studied in Term, 0% of ojectives studied in Term nd 60% of the ojectives studied in Term 3. ix
10 GRADING PROCEDURE To improve ssessment nd grding nd lso introduce uniformity in schools, it is recommended tht schools dopt the following WASSCE grde structure for ssigning grdes on students test results. The WASSCE structure is s follows: Grde A: 8000%  Excellent Grde B: 7079%  Very Good Grde B3: 6069%  Good Grde C4: 5559%  Credit Grde C5: 5054%  Credit Grde C6: 4549%  Credit Grde D7: 4044%  Pss Grde D8: 3539%  Pss Grde F9: 34% nd elow  Fil In ssigning grdes to students test results, you re encourged to pply the ove grde oundries nd the descriptors which indicte the mening of ech grde. The grde oundries i.e., 6069%, 5054% etc., re the grde cutoff scores. For instnce, the grde cutoff score for B grde is 7079% in the exmple. When you dopt fixed cutoff score grding system s in this exmple, you re using the criterionreferenced grding system. By this system student must mke specified score to e wrded the requisite grde. This system of grding chllenges students to study hrder to ern etter grdes. It is hence very useful system for grding chievement tests. Alwys rememer to develop nd use mrking scheme for mrking your clss exmintion scripts. A mrking scheme consists of the points for the est nswer you expect for ech question, nd the mrks llocted for ech point rised y the student s well s the totl mrks for the question. For instnce, if question crries 0 mrks nd you expect 6 points in the est nswer, you could llocte 3 mrks or prt of it (depending upon the qulity of the points rised y the student) to ech point, hence totling 8 mrks, nd then give the remining mrks or prt of it for orgniztion of nswer. For ojective test ppers you my develop n nswer key to speed up the mrking. In ssigning grdes to students test results, you my pply the ove grde oundries nd the descriptions, which indicte the mening of ech grde. The grde oundries re lso referred to s grde cutoff scores. When you dopt fixed cutoff score grding system you re using the criterionreferenced grding system. By this system student must mke specified score to e wrded the requisite grde. This system of grding chllenges students to study hrder to ern etter grdes. It is hence very useful system for grding chievement tests. x
11 SENIOR HIGH SCHOOL  YEAR SECTION : ALGEBRA Generl Ojectives: The student will:. pply set theory in solving prolems. recognize the difference etween reltion nd function 3. use the principles relted to polynomil functions 4. pply the concept of inry opertions to ny given sitution 5. resolve rtionl function into prtil frctions, 6. find domin, rnge, zero of rtionl function nd stte when it is undefined. 7 use the inomil theorem in pproximtions. 8. pply solutions of simultneous liner inequlities to liner progrmming. UNIT SPECIFIC OBJECTIVE CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION UNIT. Sets.. estlish set identities nd pply them. Alger of sets De Morgn s lws Revise twoset prolems with students Assist students to estlish set identities: (A B C) A B C ; (A B C) A B C ; A ( B C) ( A B) ( A C) ; use De Morgn s lws to solve relted prolems A ( B C) ( A B) ( A C).. find solutions to threeset prolems. Threeset prolems Assist students to solve rel life prolems involving threesets nd the use of Venn digrms use lger of sets to solve rel life prolems involving three sets Unit. Surds.. crry out the four opertions on surds. Surds of the form n, where n is not perfect squre, Assist students to dd, sutrct nd multiply surds find the sum nd products of surds nd simplify them Unit. (cont d) Surds.. rtionlize surds with inomil denomintors. Rtionlizing surds with inomil denomintors Assist students to write the conjugte form of given surd rtionlize surds with inomil denomintors
12 Guide students to rtionlize surds, e.g. E.g. Simplify c d n m c d n m c c d d m m 3 3 Unit.3 Binry Opertions.3. determine the commuttive, ssocitive nd distriutive properties of inry opertions. Properties of inry opertions Revise the ide of inry opertions Assist students to investigte the commuttive nd ssocitive properties of inry opertions Assist students to determine whether or not for ny two given inry opertions, one distriutes over the other use the properties of inry opertions to solve relted prolems.3. determine the property of closure. Closure Assist students to determine whether or not given set is closed under given inry opertion E.g. x y xy where x, y R determine whether or not the result of inry opertion is memer of the given set..3.3 find the identity element nd use it to find the inverse of given element. Identity nd inverse elements Assist students to find the identity element (e) of inry opertion stisfying Guide students to find the inverse of n element,, for given inry opertion stisfying e e find the identity element of given inry opertion nd determine the inverse of ny element in the given set * * e where is the inverse of
13 Unit.4 Reltions nd Functions.4. represent functions grphiclly. Grphicl representtion of function Revise the ide of reltions, functions nd types of functions Assist students to represent functions grphiclly use mpping digrms to estlish vrious types of reltions Introduce the use of grphic clcultors nd computers to investigte nture of grphs of functions determine whether or not given function is onetoone Guide students to determine whether given function is onetoone Unit.4 (cont d).4. determine the domin nd rnge of function. Domin nd rnge of function Assist students to stte the domin of function s the set of vlues of x tht mkes the function defined find the domin nd rnge of given function Reltions nd Functions Guide students to stte the rnge of function s the set of vlues of y (dependent vrile) tht cn result from the sustitutions for the independent vrile, x.4.3 determine the inverse of onetoone function. Inverse of function Assist students to find the inverse of function nd stte the domin e.g. if f : x 3x 4 then the inverse reltion is find the inverse of given function.4.4 determine the composite of two given functions. Composite functions f : x 4 3 x 3, x R Assist students to find the composite of two functions defined s fog ( x) f [ g( x)]; gof ( x) g[ f ( x)] Use rel life situtions to introduce composite functions. E.g., Suppose h(x) husnd of x nd m(x) mother of x. Then, hom (x) the husnd of the mother of x which is step fther or fther of x, nd moh (x) mother of the husnd of x, which is the motherinlw of x. find fog ; gof ; f og of given rel life functions nd interpret the results Include composites of inverses of functions 3
14 Unit.5 Polynomil functions.5. recognise liner nd qudrtic functions. Liner nd qudrtic functions Assist students to ssocite liner function with the stright line grph. Assist students to ssocite qudrtic function with the prol. find the mximum nd minimum vlues/points of given qudrtic functions Encourge the use of grphic clcultors nd computers to investigte the shpes of grphs s coefficients of the vriles nd the constnts chnge. Unit.5 (cont d) Polynomil functions Guide students to express the qudrtic function f ( x) x x c in the form f ( x) ( x d) k, where k is the mximum or the minimum vlue.5. sketch the curve of qudrtic function. Sketching curves of qudrtic functions Assist students to sketch curves of qudrtic functions nd indicte their shpes Guide students to identify the vertex, the xis of symmetry, mximum nd minimum points, incresing nd decresing prts of prol sketch given qudrtic curves nd identify the xes of symmetry, turning points, etc.5.3 solve qudrtic equtions y method of completing the squres nd y formul. Solving qudrtic equtions Completing the squre Assist students to solve qudrtic equtions using the method of completing the squres Assist students to deduce the generl formul of qudrtic eqution tht is, solve qudrtic equtions using (i) method of completing the squres (ii) qudrtic formul Qudrtic formul x 4c nd use it to solve qudrtic equtions.5.4 use the discriminnt to descrie the nture of roots of qudrtic equtions. Roots of qudrtic equtions Assist students to use the discriminnt 4 d 4c to descrie the nture of the roots of qudrtic equtions s stte the nture of roots of given qudrtic equtions. E.g. Find the vlues of k tht mke the eqution x ( k ) x k 0
15 (i) equl, if 4c 0 hs equl roots. (ii) rel nd unequl, if 4c 0 Unit.5 (cont d) Polynomil functions (iii) imginry, if 4c 0 Assist students to find the reltion etween the roots nd nd coefficients,, c of x x c 0, 0 : c, nd Assist students to use these reltions to write other qudrtic equtions with given roots use sum nd product of roots in setting up qudrtic equtions E.g. If nd re the roots of x 5x β. 3 0, find +β nd.5.5 define polynomil function. Definition of polynomil functions Assist students to define polynomil function s n n f ( x) n x n x 0, 0 e.g. f ( x) 3x 4 x 3 4x x 3 stte the degree of given polynomil function.5.6 recognise nd drw the grph of cuic function. Cuic functions Assist students to drw grphs of cuic functions for given intervl. NB: The use of grphic clcultor or computer for investigting cuic functions should e encourged drw grphs of cuic functions for given intervls.5.7 perform the sic lgeric opertions on polynomils. Alger of polynomils Addition nd sutrction Multipliction Division Assist students to dd nd sutrct two polynomils Assist students to find the product of two polynomils Guide students to crry out division of one polynomil y nother of lesser degree nd stte the dd,sutrct, multiply nd divide given polynomil functions reminder eg. ( x ) x 3x x 4 3 The use of synthetic division is llowed. 5
16 .5.8 use the reminder nd fctor theorems to find the fctors nd reminders of polynomil of degree not greter thn 4. Reminder nd fctor theorems Assist students to use the reminder theorem to find the reminder R when the polynomil f (x) is divided y ( x ) s f ( ) R Guide students to discover tht if ( x ) is fctor of polynomil f (x) then f ( ) 0 find the reminder/fctor of polynomil of degree not greter thn four when divided y polynomil of lower degree Unit.5 (cont d) Polynomil functions.5.9 find the fctors nd zeros of polynomil function. The fctors nd roots of polynomil functions/ equtions up to n 3 Assist students to use the fctor theorem to find the fctors of given polynomil function nd roots of f (x) 0 solve for the zeros (roots) of given polynomil functions (equtions) Unit.6 Rtionl functions.6. recognise rtionl function nd determine the domin nd rnge. Rtionl functions of the form f ( x) Q ( x), g( x) 0 g( x) Assist students to recognize rtionl function Assist students to otin the domin, zeros nd the rnge of rtionl functions stte the domin nd zeros of rtionl functions.6. crry out the four sic opertions on rtionl functions. Opertions on rtionl functions Assist students to dd, sutrct, multiply nd divide simple rtionl functions perform opertions involving rtionl functions.6.3 resolve rtionl functions into prtil frctions. Prtil frctions Assist students to write rtionl functions s prtil frctions of vrious forms. express given rtionl functions s prtil functions. E.g. Find the prtil frction decomposition of x x x Unit.7 Binomil Theorem.7. write down the Binomil expnsion for positive integrl index. Binomil Theorem Assist students to write down the Binomil expnsion using the Pscl s Tringle Assist student to discover nd write the expnsion using the inomil theorem for positive integrl index. write the expnsion for given inomil expressions Led students to pply the theorem to expnd 6
17 inomil with rtionl index..7. use the comintion method to determine the coefficient nd exponent of given term in n expnsion. Comintion method Assist students to use intuitive method to estlish tht Assist students to use the comintion method to determine the coefficient nd exponents of terms in n expnsion ( ) n r n 0 n r n r r use comintion method to find the indicted coefficients of terms in inomil expnsion. x term nd the E.g. Find the constnt term in the expnsion of (x ) x [Proof not required] Unit.7 (cont d) Binomil Theorem.7.3 use the expnsion for n ( x) to pproximte exponentil vlues. Expnsion nd use of n ( x) to pproximte exponentil vlues Assist students to expnd ( n x) nd use it to find pproximte exponentil vlues. E.g. 4 4 Evlute ( 0.998) y writing ( 0.998) ( ) nd then sustituting x = in the expnsion of ( 4 x ). use inomil expnsion to pproximte exponentil vlues E.g. Use inomil expnsion to 5 evlute ( 0.998) Unit.8 Inequlities nd Liner progrmming.8.. recognise liner inequlity in two vriles nd drw its grph. Liner inequlities in two vriles Assist students to identify liner inequlities in vriles Assist students to drw the grphs of given liner inequlities drw the grphs of given liner inequlities.8.. use grphicl pproch to solve simultneous inequlities nd interpret. Liner progrmming Guide students to find solutions of simultneous liner inequlities using grphicl method Assist students to pply the solutions to liner progrmming solve prcticl prolems on liner inequlities in two vriles (liner progrmming) 7
18 .8.3. recognise nd solve qudrtic inequlities. Qudrtic inequlities Assist students to identify qudrtic inequlities s x x c 0 nd x x c 0, etc, where 0 Assist students to solve qudrtic inequlities using nlytic, deductive tle nd grphicl pproch. solve rel life prolems involving qudrtic inequlities E.g. Find the set of vlues of x for which + x 3x 0 e.g. x 5x 6 0, ( x )(x 3) 0, Tht is x> nd x<3, <x<3 s shown in the tle elow. x 3 x x3 8
19 SENIOR HIGH SCHOOL  YEAR SECTION COORDINATE GEOMETRY I Generl Ojectives: The student will:. pprecite the concept of stright lines nd pply them in relted prolems. relte grdients of prllel, perpendiculr nd intersecting lines UNIT SPECIFIC OBJECTIVE CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION Let students Unit.9 Stright line.9. find the midpoint of line segment joining two given points. Midpoint of line segment Revise distnce etween two given points s n ppliction of the Pythgors theorem Guide students to find the coordintes of the midpoint of line segment joining two given points find the coordintes of the midpoint of line joining two given points.9. find the coordintes of point which divides given line in given rtio. Division of line segment in given rtio Guide students to find the coordintes of point which divides line joining two points internlly in given rtio m : n find the coordintes of point dividing given line in given rtio Guide students to find the coordintes of point which divides line joining two points externlly in given rtio :.9.3 find the eqution of stright line. Eqution of stright line Guide students to find the eqution of stright line in the vrious forms: (i) twopoint form (ii) grdientintercept form (iii) grdient nd one point form (iv) generl (stndrd) form find the equtions of lines with given informtion 9
20 Unit.9 (cont d) Stright line.9.4 write the eqution of line pssing through n externl point nd prllel to or perpendiculr to given line. Prllel nd perpendiculr lines Assist students to determine the grdient of line tht is prllel to given line Guide students to find the grdient of line tht is perpendiculr to given line Assist students to write the eqution of line pssing through point nd prllel to or perpendiculr to given line Let students find the eqution of line prllel or perpendiculr to given line. E.g. Find the eqution of stright line prllel to 3x 5y 0 0 nd pssing through ( 6, 4) Include the eqution of perpendiculr isector of line joining two given points.9.5 find the perpendiculr distnce from n externl point to line. Perpendiculr distnce from n externl point to line Assist students to find the perpendiculr distnce from n externl point P( x, y) to given line x y c 0 using the formul d x y c clculte the perpendiculr distnce from point to given line E.g. Find the perpendiculr distnce of the point P(8, 3) from the line 3x 4y clculte the cute ngle etween two intersecting lines. Acute ngle etween two intersecting lines Guide students to find grdients m nd m of two intersecting lines nd find the ngle etween them using the pproprite formul: For m m, tn m m m m. clculte the ngle etween two intersecting lines. E.g. Find two possile vlues of p if the lines px y 3 0 nd 3x y 0 intersect t 45 0 But if m m, then the lines re perpendiculr. 0
21 SENIOR HIGH SCHOOL  YEAR SECTION 3 PROBABILITY I Generl Ojectives: The student will:. explin nd use the proility scle from 0 to. list the smple spce of n experiment nd 3. clculte the proilities of independent nd mutully exclusive events 4. use knowledge of conditionl proility to solve rel life prolems. UNIT SPECIFIC OBJECTIVE CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION UNIT. Proility.. define proility s rtio of numer(s) in n event to numer(s) in the smple spce. Eqully likely events Revise smple spce, events nd compound events nd reltive frequency. Assist students to clculte proilities involving compound events clculte proilities involving compound events.. distinguish etween mutully exclusive nd independent events. Mutully exclusive nd independent events Assist students to solve simple prolems using oth the ddition nd multipliction lws of proility. (i) P ( A B) P( A) P( B) P( A B) (ii) If A nd B re mutully exclusive then A B nd P(A B) = 0 P( A B) P( A) P( B) ; If A nd B re independent, then P( A B) P( A) P( B) therefore P(A U B) = P(A) x P(B) pply ddition nd multipliction lws to clculte proilities..3 clculte simple conditionl proilities. Conditionl proility Assist students to clculte simple conditionl proilities Note: Proility of event A, given tht B hd occurred is defined y: P( A B P ( A/ B) P( B) Agin if A nd B re independent then P ( A/ B) P( A) clculte simple conditionl proilities. E.g. Find the proility of scoring numer less thn five in single toss of die if the toss resulted in n even numer
22 SENIOR HIGH SCHOOL  YEAR SECTION 4 VECTORS I Generl Ojectives: The student will:. write vectors in components /Crtesin nd mgnitude direction (ering) forms. find resultnts of vectors UNIT SPECIFIC OBJECTIVE CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION Unit. Vectors in plne.. represent vectors in the form x i yj. Representtion of vectors in stndrd sis form Revise concept of vectors nd representtion of vectors in component nd mgnitudedirection forms Discuss the nottion i nd j for the unit vectors write vectors in the stndrd sis form 0 nd long xxis nd yxis respectively 0 Assist students to represent given vectors in the form x i yj.. find the resultnt of given vectors. Resultnt of vectors Assist students to find the resultnt of given vectors in the stndrd sis form find the resultnt of given vectors in stndrd sis form..3 dd vectors using tringle, prllelogrm nd polygon lws of ddition. Tringle, prllelogrm nd polygon lws of ddition Assist students to find the resultnt OC y completing the prllelogrm with OA nd OB s djcent sides OB AC OC B OA AC C OC OA OB O A Assist students to use the polygon lw to find the resultnt of given vectors forming the sides of polygon pply prllelogrm nd polygon lws to find the resultnt of given vectors. E.g. Find the sum of the following vectors: 3 OA, 6BZ, AO, AB nd 5 OB
23 Unit. (cont d) Vectors in plne..4 use scle drwing to find the resultnt of vectors given in mgnitudedirection form. Scle drwing nd resolution of vectors Assist students to drw the resultnts of vectors given in mgnitudedirection form y scle drwing Assist students to resolve vectors into components nd vice vers resolve vectors given in mgnitudedirection forms into components nd vice vers. E.g. Find the resultnt of the following vectors (0m, 070 o ) nd (5m, 60 o ) y () scle drwing () resolution..5 estlish nd use properties of ddition of vectors nd sclr multipliction of vectors. Properties of ddition of vectors Sclr multipliction of vectors Assist students to estlish the commuttive, ssocitive nd distriutive properties of ddition of vectors Guide students to use the properties to solve prolems on vectors use properties of ddition of vectors to solve prolems..6 use position vectors to find free vectors..7 clculte unit vectors Position vectors Unit vectors Assist students to identify the position vector of point A reltive to point O s vector OA Let students use the ddition of vectors to derive the reltion, AB OB OA for the (position vectors of B nd A Assist students to clculte unit vector of given vector s â use position vectors to find the vector joining two given points in the Crtesin plne. E.g. P(3, ), Q(3, 4), R(3, 6) nd S(x, y) re the vertices of prllelogrm PQRS. Find the vlues of x nd y. 3
24 SENIOR HIGH SCHOOL  YEAR SECTION COORDINATE GEOMETRY II Generl Ojectives: The student will:. recognise the eqution of the circle nd use it in relted prolems. solve prolems relted to the prol UNIT SPECIFIC OBJECTIVE CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION Unit. The circle.. find the eqution of circle. Eqution of circle Assist students to use the ide of distnce formul to find the eqution of circle, (i) centre, the origin nd given rdius s x y r (ii) given centre (, ) nd rdius s ( x ) ( y ) r Assist students to write the eqution of circle in the generl form s x y gx fy c 0, where ( g, f ) is the centre of the circle Guide students to use the generl form of the eqution to write the eqution of circle pssing through three given points write the eqution of circle in the stndrd form using given informtion. E.g. Find the eqution of circle centre (3, 4) nd which psses through (6, 8). Assist students to find eqution of circle on given dimeter using the midpoint pproch... determine the centre nd the rdius from given eqution of circle. Finding the centre nd rdius given the eqution of circle Assist students to find the centre nd rdius of the circle from given eqution of circle y (i) compring coefficients with the generl (ii) completing the squres form find the centre nd rdius of circle given its eqution. E.g. Find the centre nd rdius of the circle x y 4x 6y 9 0 4
25 Unit. (cont d) The circle..3 find the equtions of tngent nd norml to circle nd find the length of tngent from n externl point. Tngent nd norml to circle Assist students to find the eqution of tngent to given circle Assist students to find the eqution of norml to given circle Guide students to find the length of tngent to given circle from n externl point find the equtions of tngent nd norml to circle t given point find the length of tngent to circle from given externl point. E.g. Find the length of the tngent to the circle x y 8x 0 y 5 0 from the point (5, 7)...4 find the eqution of the locus of vrile point which moves under given condition. Loci Assist students to find the locus of point equidistnt from two fixed points Assist students to find the locus of point P(x, y) such tht PA is perpendiculr to PB where A nd B re fixed points solve loci relted prolems Assist students to find the locus of point moving under other given conditions e.g. AP 3 PB, where A nd B re fixed points, nd PM PN 6, where M nd N re fixed points Assist students to descrie completely such loci. 5
26 Unit. Prol.. find the eqution of prol given the directrix nd focus. Eqution of prol Assist students to recognize the eqution of prol of the forms: y 4x x 4y nd write the eqution of prol given directrix nd focus Guide students to find the eqution of prol given the directrix nd focus Assist students to find the directrix nd focus from the eqution of prol. determine the focus nd directrix from the eqution of prol.. sketch the vrious forms of prol. Sketching prol Assist students to use given directrix nd focus to sketch prol Assist students to determine the eqution of the xis of symmetry of prol sketch the grph of given prol nd find the eqution of the xis of symmetry..3 find the equtions of the tngent nd norml to Prol. Tngent nd norml Guide students to use ICT to investigte the nture of grphs of vrious forms of prol y chnging the directrix nd focus Guide students to find the eqution of tngent to prol y solving the equtions nd y 4x simultneously y mx Assist students to deduce the eqution of norml from the eqution of tngent c find the equtions of the tngent nd norml to prol t given point. E.g. Find the equtions of the tngent nd norml to the prol y x t the points P(,4) nd Q (, 4). At wht point in the Oxy plne will the tngents t P nd Q meet? 6
27 SENIOR HIGH SCHOOL  YEAR SECTION ALGEBRA II Generl Ojectives: The student will:. recognize the importnce of sequences nd series in everydy life situtions. use knowledge of sequences nd series to solve rel life prolems 3. pply lws of logrithms nd logrithmic functions in mthemticl models. UNIT SPECIFIC OBJECTIVE CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION Unit.3 Sequences nd Series The student will e le to.3. pply liner sequence to solve rel life prolems..3. pply exponentil sequence nd series to rel life situtions. Appliction of liner sequence Exponentil Sequence (Geometric Progression GP) Revise concepts of liner sequence nd finding the nth term. Include rithmetic men Use prcticl situtions e.g. slry increment, to illustrte liner sequences Assist students to pply liner sequence to solve rel life prolems Revise concepts of exponentil sequence nd finding the nth term of GP. Assist students to clculte the geometric men of three consecutive terms of GP Use prcticl situtions, e.g. Deprecition nd compound interest, to illustrte exponentil sequences Assist students to find the sum of the first n terms of n exponentil series using the rule: 7 S n u r r n, r, Let students solve rel life prolems involving liner sequence. E.g. A disply in supermrket consists of milk tins piled up in the form of pyrmid. There re 5 tins on the ottom lyer nd ech successive lyer hs fewer tins. How mny tins re displyed? Solve rel life prolems involving exponentil sequence. E.g. The originl vlue of cr is 80m. If the cr deprecites y 5% every yer, find how much it is worth fter i) yers ii) 4 yers.
28 The student will e le to Let students Unit.3 (cont d) Sequences nd Series.3.3 find n explicit formul for the nth term of recurrence sequence. Recurrence sequence u ( r r ) or, S, r n u nd the limit of the sum s S r, r Assist students to pply exponentil sequence to solve rel life prolems Assist students to recognize recurrence sequence nd u find its terms, e.g. n 3 un u, Assist students to find n explicit formul for the generl rule for the recurrence sequence. n After how mny yers is the cr worth 35,496,46.00? generte the terms of recurrence sequence nd find n explicit formul for the sequence Assist students to express recurring decimls s series nd determine their limits, e.g Unit.4 Indices nd Logrithmic Functions.4. use the lws of indices nd logrithms to solve simple prolems. Lws of indices nd logrithms. Assist students to evlute the products, quotients nd powers of numers using the lws of indices nd logrithms evlute products, quotients nd powers using indices nd logrithm. E.g. Given tht log , evlute without using tle or clcultor, log 3 5 log solve equtions involving indices. Equtions involving indices Assist students to solve equtions involving indices x x e.g. 8, 3 x 3 x 3 0 solve equtions involving indices.4.3 solve equtions involving logrithms including chnging the se. Equtions involving logrithm nd chnge of se Assist students to solve simple equtions involving logrithms e.g. Solve: x 7 Guide students to chnge the se of given logrithm nd use it to solve relted prolems. solve equtions involving logrithms nd chnge of se E.g. Solve log x 4log x 5 8
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