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

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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 PRE-REQUISITE 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 4-6) Mtrices (pg 39-4). Surds (pg - ) Sequences nd Series (pg 7-8) Liner Trnsformtions (pg 4-43) 3. Binry Opertions (pg 3) Indices nd Logrithms (pg 8-9) Logic (pg 44) 4. Reltions nd Functions (pg 3-4) Trigonometric Rtios nd Rules (pg 0 - ) Correltion nd Regression (pg 45-47) 5. Polynomil Functions (pg 4-7) Compound nd Multiple Angles (pg - ) Spermn s Rnk Correltion (pg 45-47) 6. Rtionl Functions (pg 7) Trigonometric Functions/Equtions (pg ) Dynmics (pg 48-49) 7. Binomil Theorem (pg 7-8) Differentition (pg 3-4) 8. Inequlities nd Liner Progrmming (pg 8) Appliction of Differentition (pg 4-5) 9. Coordinte Geometry I (pg 9-0) Integrtion (pg 6) 0. Proility I (pg ) Appliction of Integrtion (pg 6-7). Vectors I (pg - 3) Permuttion nd Comintions (pg 8). Proility II (pg 9) 3. Sttistics I (pg 30-3) 4. Appliction of vectors in Geometry (pg 33-35) 5. Sttics (pg 36-38) 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 drill-oriented 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 re-order 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 re-order 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, re-mix, 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 Criterion-Referenced 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 ojective-type 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 SCHOOL-BASED 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 school-sed 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: Mid-term test, Group Exercise, End-of-Term 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.. Mid-Term Test: The mid-term 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. End-of-Tem Test: The end of-term test is summtive ssessment system nd should consist of the knowledge nd skills students hve cquired in the term. The end-of-term 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 End-of-Term 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: 80-00% - Excellent Grde B: 70-79% - Very Good Grde B3: 60-69% - Good Grde C4: 55-59% - Credit Grde C5: 50-54% - Credit Grde C6: 45-49% - Credit Grde D7: 40-44% - Pss Grde D8: 35-39% - 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., 60-69%, 50-54% etc., re the grde cut-off scores. For instnce, the grde cut-off score for B grde is 70-79% in the exmple. When you dopt fixed cut-off score grding system s in this exmple, you re using the criterion-referenced 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 cut-off scores. When you dopt fixed cut-off score grding system you re using the criterion-referenced 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 two-set 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. Three-set prolems Assist students to solve rel life prolems involving three-sets 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 one-to-one Guide students to determine whether given function is one-to-one 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 one-to-one 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 mother-in-lw 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 x-3 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 mid-point 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 mid-point 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) two-point form (ii) grdient-intercept 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 mgnitude-direction forms Discuss the nottion i nd j for the unit vectors write vectors in the stndrd sis form 0 nd long x-xis nd y-xis 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 mgnitude-direction form. Scle drwing nd resolution of vectors Assist students to drw the resultnts of vectors given in mgnitude-direction form y scle drwing Assist students to resolve vectors into components nd vice vers resolve vectors given in mgnitude-direction 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 CO-ORDINATE 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 mid-point 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

29 The student will e le to Let students Tht is log log (ii). log log log c c Unit.4 (cont d) Indices nd Logrithmic Functions.4.4 reduce reltions to liner form using logrithm nd drw the grph. Grphs of exponentil reltions nd their pplictions Assist students to otin the liner form of the reltion, y x s log y log x log nd drw the grph. Guide students to recognize the similrity etween the ove reltions nd the line y mx c drw nd interpret grphs of x the reltions y nd y x Assist students to otin the liner form of the x reltion, y log x y s log log, nd drw the grph. Assist students to estimte the vlues of the constnts nd from their grphs nd determine grphiclly one vlue given the other. Introduce the use of grphic clcultors nd computer in drwing logrithmic grphs 9

30 SENIOR HIGH SCHOOL - YEAR SECTION 3 TRIGONOMETRY Generl Ojectives: The student will:. pply the concept of the trigonometric rtios nd their reciprocls to solve relted prolems. use trigonometric identities to find solutions to trigonometric equtions of compound nd multiple ngles. 3. pply trigonometry to solve relted prolems. UNIT SPECIFIC OBJECTIVE CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION Unit.5 Trigonometr ic rtios nd rules.5. determine trigonometric rtios nd their reciprocls. Bsic trigonometric rtios nd their reciprocls Assist students to revise the three sic trigonometric rtios; sine, cosine nd tngent Guide students to find the sic trigonometry rtios using the qudrnts. Assist students to find the reciprocls of the trigonometric rtios Assist students to relte trigonometric rtios to Crtesin co-ordintes of the point ( x, y) on the circle x y r use sic trigonometric rtios nd reciprocls to prove given trigonometric identities Assist students to derive the trigonometric identities sin e.g. tn ; cos sin ; cos tn sec, etc Assist students to form the concept of negtive ngles nd to estlish the following reltions evlute sine, cosine nd tngent of negtive ngles sin( ) sin(360 ) sin cos( ) cos(360 ) cos 0

31 tn( ) tn(360 ) tn Students to e encourged to use the clcultor to verify these reltions Unit.5 (cont d).5. convert ngles into rdins. Angles in rdins Assist students to otin rdin equivlents of ngles. [ rdins = 80 o ] clculte the rdin equivlents of given ngles Trigonometr ic rtios nd rules.5.3 stte nd use the sine nd cosine rules. Sine nd Cosine rules Assist students to deduce the sine rule: c A sin A sin B sin C, c C B where, nd c re the sides of tringle Guide students to use the sine rule to solve relted prolems Assist students to deduce the cosine rule: tringles. c ccosa, etc nd use it to solve solve tringles using the sine nd cosine rules.5.4 pply the sine nd cosine rules to solve prolems involving erings. Appliction of sine nd cosine rules to erings Assist students to pply the sine nd cosine rules to solve prolems involving erings pply sine nd cosine rules to solve rel life prolems. E.g. An eroplne cn fly t 800kmh - in still ir. The pilot sets course due est when there is wind lowing t 80kmh - from the south-west. Find the mgnitude nd direction of the velocity of the eroplne reltive to the ground.

32 Unit.6 Compound nd Multiple ngles.6. use simple trigonometric identities to find trigonometric rtios for compound ngles. Compound ngles Guide students to derive the compound ngles identities: sin( A B) sin AcosB sin BcosA cos( A tn( A B) B) cosacosb sin Asin B tn A tn B tn Atn B Assist students to use the identities to solve relted prolems prove the compound ngle identities nd pply them to evlute given ngles without using clcultors. E.g. Find (i) sin 75 (ii) sin 5 without using clcultor leving your nswer s surd. Encourge students to use clcultors nd specific vlues to verify these reltions Unit.6 (cont d) Compound nd Multiple ngles.6. use simple trigonometric identities to find trigonometric rtios for multiple ngles. Multiple ngles (up to 3A) Assist students to derive the doule ngle identities for sin A, cosa nd tna nd use them to write identities for sin 3A nd cos3a in terms of sin A nd cos A respectively Encourge students to use the clcultor nd specific vlues to verify these reltions prove multiple ngle identities. E.g. Express sin 3A nd cos 3A in terms of sin A nd cosa respectively Unit.7 Trigonometr ic Functions.7. drw grphs of trigonometric functions. Grphs of trigonometric functions - tn x Revise grphs of sin x nd cos x Assist students to drw the grph of tn x nd compre it with the sine nd cosine grphs. Encourge students to use the clcultor nd computer to investigte the nture of grphs of trigonometric functions drw the grphs of sin x, cos x nd tn x nd use them to estimte given vlues of x.7. solve trigonometric equtions (up to qudrtic). Solving trigonometric equtions (up to qudrtic) Assist students to find solution sets to trigonometric equtions up to qudrtic. E.g. sin x sin x 3 0, 0 x 360 Encourge students to use the clcultor nd computer to drw grphs of trigonometric functions nd find their solutions (grphicl pproch) solve trigonometric equtions

33 .7.3 clculte the mximum nd minimum points of given trigonometric function. Mximum nd minimum points of trigonometric functions Revise grphs of trigonometric functions of the form f ( x) sin x cosx Guide students to express the trigonometric function, f ( x) sin x cosx in the forms R cos(x ) or R sin(x ) for 0 90 Assist students to use the result to clculte the mximum nd minimum points of the function find the mximum nd minimum points of given trigonometric functions 3

34 SENIOR HIGH SCHOOL- YEAR SECTION 4 CALCULUS Generl Ojectives: The student will:. pprecite the principles of differentition nd Integrtion. crry out differentition nd integrtion on polynomils 3. explin ll relevnt terms under differentition nd integrtion 4. pply differentition in relted processes e.g. kinemtics, smll chnges, etc 5. pply differentition to determine grdient of curve t point, the turning points, etc. 6. pply integrtion to determine res under curves, volumes of revolution, etc UNIT SPECIFIC OBJECTIVE CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION Unit.8 Differentition.8. find the limit of function nd relte it to the grdient of curve. Limit of function Assist students to form the concept of limit Assist students to find the limits of functions of the 3 forms: lim x x x, lim, lim, etc x x x x x evlute limits of given functions Let students drw lines through given point, P on curve cutting the curve gin t Q, Q, Q3, Qn with Q pproching P. Let students compre the grdients of the vrious lines with tht of the tngent t P..8. find the limit of simple trigonometric functions. Limit of simple trigonometric functions Assist students to find the limits of sin x nd cos x, s x pproches zero, etc. using tles or clcultors. Guide students to use tles or clcultors to discover sin x tht the lim x 0 x determine the derivtive of simple trigonometric functions 4

35 Unit.8 (cont d) Differentiti on.8.3 differentite polynomils nd simple trigonometric functions from first principles. Differentition from first principles Assist students to find the derivtive of polynomils ( n 3) from first principles using the ide of smll increment, x nd y or the definition dy f ( x h) f ( x) lim dx h 0 h find the derivtives of given functions from first principles Limit differentition from first principles to only polynomils.8.4 recognise nd use the rules of differentition. Rules of differentition Assist students to differentite polynomils using the power rule, i.e. (i) use the vrious rules of differentition to find the derivtives of given functions d dx ( x n ) nx n, (ii) Differentition of sums nd differences d dx (x n x m ) d dx x n d dx x m Assist students to differentite functions which re products of polynomils [e.g. f ( x) g( x) ] using the product rule Assist students to differentite rtionl expressions of the form H ( x) f ( x) g( x), where g(x) 0 using the quotient rule Assist students to use the chin rule to differentite n m function of function of the form ( x ) or G ( x) f ( x) n 5

36 .8.5 differentite implicit functions. Differentition of implicit functions Assist students to differentite implicit functions find the derivtives of given implicit functions. E.g. Find the derivtive of 3 3 x y xy with respect to Unit.9 Applictions of Differentiti on.9. find the equtions of tngent nd norml to curve t point. Equtions of tngent nd norml to curve Assist students to pply differentition to find the equtions of tngents nd normls to given curves pply derivtives to find the equtions of tngents nd normls to given curves t given points of contct Unit.9 (cont d).9. find the rtes of chnge nd smll chnges. Rtes of chnges Assist students to pply the chin rule to find the rtes of chnge nd smll chnges solve rel life prolems involving rtes of chnge Applictions of Differentiti on Assist students to clculte the percentge chnges.9.3 find the mxim nd minim vlues nd points. Mxim nd minim Guide students to find the turning points (sttionry points) of curves nd determine their nture Assist students to find the second derivtive of functions nd use it to determine the nture of turning points Assist students to pply derivtives to solve mxim nd minim prolems pply derivtives to solve mxim nd minim prolems. E.g. A frmer wishes to construct rectngulr got pen using n existing wll s one of its sides with 60 metres of fencing. Clculte the dimensions of the pen tht would llow for mximum grzing re..9.4 sketch curves up to cuic functions. Curve sketching Assist students to (i) use the ide of clculus to otin the turning points of functions, (ii) find the intercepts nd use these points to sketch curves pply derivtives to sketch given curves 6

37 Unit.0 Integrtion.9.5 solve simple prolems involving liner kinemtics Liner kinemtics Assist students to solve prolems involving liner kinemtics displcements, velocity nd ccelertion.0.. integrte polynomils. Indefinite integrl Assist students to recognise integrtion s the reverse of differentition Assist students to integrte monomils. E.g. x n dx x n n C, n nd C is constnt Assist students to integrte polynomils pply derivtives to solve prolems on liner kinemtics E.g. A ll is thrown verticlly upwrds nd its height (h metres) ove the ground t time t seconds is given y h 6 t 4 t. Find the velocity nd ccelertion fter one second nd the mximum height reched. integrte given functions. E.g. ( x x) dx Unit.0 (cont d) Integrtion.0.. find definite integrls. Definite Integrls Assist students to find vlues for definite integrls for given limits of the form f ( x) dx evlute given definite integrl. E.g. Evlute 3 ( 4x 3x ) dx find definite integrls y the sustitution method. Integrtion y sustitution Assist students to find integrls y the sustitution method e-g x x dx integrte given functions y method of sustitution where we let x u so tht du xdx.0.4. integrte simple trigonometric functions. Integrtion of simple trigonometric functions Assist students to integrte simple trigonometric functions e.g. sin x dx find the integrl of simple trigonometric functions 7

38 Unit. Appliction of integrtion.. solve simple prolems involving liner kinemtics. Kinemtics Assist students to pply integrtion to solve simple prolems involving kinemtics - displcements, velocity nd ccelertion solve rel life prolems involving kinemtics. E.g. A prticle moves long stright line OP such tht its velocity fter t seconds is given y v t t. 8 (i) Find the time it tkes the prticle to rech the point P; (ii) How fr is the prticle from O when it chnges direction.. find the re under curve. Ares under curves Assist students to sketch given curves nd identify the specific res Assist students to pply definite integrl to find the indicted re under the curve pply integrtion to clculte res under given curves. E.g Find the re under the prol y x x ove the x xis, etween x 0 x x nd..3 find volumes generted when curves re rotted out the x- nd y- xes. Volumes of revolution Assist students to find the volumes of solids of revolution out (i) the x-xis nd (ii) the y-xis using the rules V y dx nd V x dy respectively pply integrtion to clculte the volume of solids of revolution. E.g. Find the volume of the solid formed y revolving the region under the curve y x, from x 0 to x out the x-xis Unit. (cont d) Appliction of integrtion..4 clculte n pproximte vlue of definite integrl using the Trpezium Rule. The Trpezium Rule Assist students to recognize tht the re under curve is pproximtely equl to tht in the trpezium formed y the limits. Assist students to estlish the trpezium rule nd pply it to find re under curves. 8 use the trpezium rule to pproximte the vlue of given definite integrl. E.g. Using the trpezium rule with 5 ordintes, clculte dx x

39 SENIOR HIGH SCHOOL - YEAR SECTION 5 PERMUTATION, COMBINATION & PROBABILITY Generl Ojectives: The student will:. differentite etween permuttion (deling with rrngement) nd comintion (deling with selection). recognize proility s rtio of eqully likely events 3. distinguish etween mutully exclusive nd independent events 4. find proility using the inomil proility rule UNIT SPECIFIC OBJECTIVE CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION Unit. Permuttion nd Comintion.. rrnge ojects or things in row... rrnge n ojects in circulr form. Permuttion ordered choices Circulr Arrngement Assist student to distinguish etween ordered nd unordered choices Led students to use the formul, n n! P r ( n r)! to solve prolems involving permuttion. Students to e encourged to check their nswers using the clcultor. Assist students to rrnge given numer of ojects in circulr form Assist students to discover nd use the rule (n )! to find the numer of wys of rrnging n ojects in circulr form solve rel life prolems on permuttion solve rel life prolems on circulr rrngements. E.g. How mny distinct wys cn 5 people sit round circulr tle?..3 drw ojects from collection with nd without replcement. Comintion Drwing with replcement Drwing without replcement Assist students to find the numer of wys of drwing ojects from collection with replcement. Assist students to find the numer of wys of drwing ojects from collection without replcement. Assist students to use the formul n n! c r r!( n r)! to find the numer of wys of selecting r ojects from n given ojects solve rel life prolems on comintion. E.g. How mny wys cn 4 red nd lue lls e drwn (without replcement) from g contining 6 red nd 4 lue lls? 9

40 Assist students to find the numer of wys of selecting r ojects from collection with given condition. Students should e encourged to check their nswers using the clcultor Unit.3 Proility II.3. use ide of comintion to clculte simple proilities..3. use inomil distriution to clculte simple proilities. Appliction of comintion Binomil proility Assist students to clculte simple proilities using comintion. e.g. Proility of drwing 3 green nd white mrles from ox contining 5 green nd 6 white mrles is: P(3green nd white) = 5 Introduce the concept of inomil distriution Discuss the fetures of inomil distriution, e.g. There re n independent trils; the proility of n event remins constnt from tril to tril; nd the proility of success is p nd of filure is q p Assist students to clculte simple proilities using the inomil distriution given y, n r n r P ( x r) Cr p q n is the numer of trilsp is the proility of success, q is the proility of filure q p,nd r is the numer of successes.. C 3 6 C C 5 pply comintion to solve rel life prolems on proility. E.g. Wht is the proility of forming committee of 3 oys nd girls from clss of 5 oys nd 0 girls? pply inomil distriution to clculte proilities 30

41 SENIOR HIGH SCHOOL- YEAR SECTION 6 STATISTICS I Generl Ojectives: The student will:. represent dt grphiclly nd interpret the grph.. clculte the mesures of loction nd spred nd interpret them. UNIT SPECIFIC OBJECTIVE CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION Unit.4 Sttistics.4. drw histogrm from grouped dt. Histogrm Revise histogrm with equl intervls with students Assist students to construct frequency tle with unequl intervls nd guide them to clculte the frequency densities Assist students to drw histogrm for dt with unequl intervls nd use it to estimte the medin drw histogrm with unequl intervls nd use it to estimte the medin.4. drw cumultive frequency curve nd use it to estimte the qurtile nd inter-qurtile rnges. Cumultive Frequency Curve (Ogive) Revise drwing of cumultive frequency curves nd estimtion of medins, deciles nd percentiles from the grph Assist students to estimte the qurtiles from the grph nd use them to clculte the inter-qurtile nd semiinter-qurtile rnges drw cumultive frequency curves nd use them to estimte semi inter-qurtile rnges.4.3 clculte the men, mode nd medin from grouped frequency tle. Mesures of centrl tendency Assist students to use the ssumed men method to clculte the men of grouped dt using the formul fd x A, f clculte the men of set of given dt using the ssumed men method where A the Assumed men, nd d x A = devition from the ssumed men 3

42 Unit.4 (cont d) Sttistics Assist students to clculte the mode using the formul where, Mode L L = the lower clss oundry of the modl clss = excess frequency of modl clss over frequency of the next lower clss = excess frequency of modl clss over frequency of the next higher clss C C = the size of the modl clss clculte the mode nd medin from grouped dt. Assist students to clculte the medin using the formul Medin L N F m F C where L = the lower clss oundry of the medin clss N = totl frequency F = sum of frequencies of ll clsses lower thn the medin clss F m = frequency of the medin clss C = the size of the medin clss 3

43 .4.4 clculte the vrince nd stndrd devition of set of dt. Mesures of dispersion Assist students to clculte the rnge for given set of dt Assist students to clculte the men devition using the formul Men Devition (MD) clculte the men devition, vrince nd stndrd devition of set of grouped dt nd interpret the result x n x MD for simple dt without frequencies; nd f i xi x MD for frequency distriutions f i Unit.4 (cont d) Sttistics Assist students to clculte the vrince nd stndrd devition from ungrouped dt using the true men Given popultion of N dt vlues, the vrince nd stndrd devition re given y ( x ) ( x ) nd N N where is the popultion men Discuss the interprettion of stndrd devition with students Assist students to clculte the vrince nd stndrd devition from grouped dt using the true men y f ( x x ) Assist students to use the ssumed men to clculte the stndrd devition, i.e. fd fd sd. f f f 33

44 SENIOR HIGH SCHOOL - YEAR SECTION 7 VECTORS II Generl Ojectives: The student will:. pply vectors nd dot product of vectors to solve prolems. use vector pproch to derive trigonometric identities 3. find the vector in the direction of nother given vector UNIT SPECIFIC OBJECTIVE CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION Unit.5 Appliction of Vectors to Geometry.5. pply vectors to solve simple geometric prolems. Appliction of vectors to geometry. Revise equlity, ddition, sutrction, nd sclr multipliction of vectors in the form i j nd Assist students to deduce the position vector of point tht divides line segment internlly in rtio ( : ) shown in the digrm s OA OB OC find the position vector of point tht divides line segment oth internlly nd externlly in given rtio B C A O Assist students to find the position vector of point tht divides line segment externlly in given rtio 34

45 Unit.5 (cont d) Appliction of Vectors Assist students to prove, for exmple, tht if M nd N re the mid-points of sides AC nd BC respectively of tringle ABC shown, then MN AB C.5. use the ide of sclr product to find ngle etween two vectors. Concept of Sclr (Dot) product for finding ngle etween two vectors M N A B Assist students to find the dot product of two given vectors. E.g. The dot product of the vectors u i nd v i dj c is given y u v c d Assist students to estlish the fct tht ngle etween two vectors is the ngle etween the directions of the vectors Assist students to use the concept of sclr product to find the ngle etween ny two vectors i.e cos where θ is the ngle etween the vectors Led students to estlish the fct tht if 0, then nd re perpendiculr j find the dot product of two given vectors pply sclr (dot) products to clculte ngle etween two vectors. E.g. Given tht p p 5cm, q cm q 9, find nd (i) the ngle etween the vectors p nd q. (ii) (ii) the sclr (dot) product of p nd q.5.3 use vectors to estlish sine nd cosine rules. The sine nd cosine rules Assist students to use vector pproch to derive the reltions: For ny tringle with sides,, nd c, use vector pproch to derive the sine nd cosine rules (i) Sine rule: sin A sin B c sin C (ii) Cosine rules: c ccos A c ccos B c cosc 35

46 Unit.5 (cont d) Appliction of Vectors.5.4 use vector pproch to derive trigonometric identities. Compound ngles for cosine nd sine. Guide students to use vector pproch to derive compound ngle identities: Cos( A B) cos AcosB sin Asin B Sin( A B) sin AcosB cosasin B use vector pproch to derive the compound ngle identities.5.5 find the projection of one vector on nother. Direction vectors Assist students to pply ˆ where â is the unit vector of to estlish tht the projection of vector on is A find the vector in the direction of nother given vector cos O θ K B Note tht the projection of OA on OB is cos 36

47 SENIOR HIGH SCHOOL- YEAR SECTION 8 Generl Ojectives: MECHANICS I The student will:. recognize forces s vectors. distinguish etween sttic nd dynmics 3. resolve forces nd find their resultnts 4. determine coefficient of friction 5. find moments out point 6. pply ides in mechnics to solve prcticl prolems UNIT SPECIFIC OBJECTIVE CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION Unit.6 Sttics.6. distinguish etween sclr nd vector quntities. Differences etween sclr nd vector quntities Guide students to recognize mss, distnce nd speed s sclr quntities nd velocity, ccelertion, force nd momentum s vector quntities. Discuss the definitions of vrious quntities with students clssify given quntities under sclrs nd vectors Assist students to use pproprite units - e.g. kg, m, s, ms Assist students to relize tht, for exmple, speed of 5 ms in the direction 060 o is the velocity vector o ( - - 5ms, 060 ). Similrly, (0ms, 030 ) is n ccelertion vector nd ( 5N, 040 ) is force vector.6.3 resolve forces cting t point. Coplnr forces cting t point Assist students to resolve forces cting t point nd find their resultnt. e.g. the resolution of the force (5N, 030 o ) is 5cos60 5sin 60 or 5sin 30 5 cos30 resolve given forces nd find the mgnitude nd direction of the resultnt 37

48 Unit.6 (cont d) Sttics.6.4 find the resultnt of forces nd consider forces in equilirium. Equilirium of prticles Resultnt of forces Lmi s theorem Assist students to estlish the fct tht prticle cted upon y forces F, F, F 3,, F n is sid to e in equilirium if F F F... F 3 n Assist students to determine tensions in strings tht re suspending prticles Discuss with students Lmi s theorem in reltion to T sin T sin T3 sin where T, T, nd T 3 re three forces cting on the ody nd, nd re the ngles etween them 0 determine the force tht will keep prticle under given forces in equilirium. E.g. A oject C of weight 0N is suspended y two light strings AC nd BC from points A nd B on the sme horizontl level ove C. AB AC cm nd CB 8 cm. Find the tensions in AC nd BC T T 3 T α γ β T T T3 Guide students to pply Lmi s Theorem to solve simple prolems on equilirium system of forces nd led students to pprecite the Lmi s theorem is equilirium to the sine rule. pply Lmi s theorem to solve relted prolems 38

49 Unit.6 (cont d) Sttics.6.5 find moments of forces. Moments of forces Assist students to determine the moment of force s.6.6 determine coefficient of friction etween odies. Friction Moment of force (m) is defined s the product of the mgnitude of force (F) nd the perpendiculr distnce (d) of line of force from the xis; i.e. P d Guide students to use the principle of moments to solve simple prolems Assist students to distinguish etween smooth nd rough plnes nd explin the mening of friction nd coefficient of friction Guide students to determine coefficient of friction etween ody nd rough plne F Line of ction of F stte nd use the principle of moments to solve relted prolems. E.g. Two pupils, Aku nd A re sitting on non-uniform plnk AB of mss 4 kg nd length m. the plnk is pivoted t M, the midpoint of AB. C is the centre of mss AB nd AC 0. 8m. If Aku s mss is 4 kg nd she sits t A while the mss of A is 30 kg, find where A cn sit for the plnk to e horizontl clculte the coefficient of friction etween odies in contct. E.g. A prticle of weight 30 N rests in equilirium on rough horizontl surfce. A string is ttched to the prticle mking n ngle of 30 with the horizontl. If the tension in the string is 4 N, find the mgnitude of the frictionl force cting on the prticle 39

50 SENIOR HIGH SCHOOL - YEAR 3 SECTION MATRICES AND LINEAR TRANSFORMATIONS Generl Ojectives: The student will:. use liner trnsformtions to find imges of points nd oject points. pply liner trnsformtion in finding reflections nd rottions of points nd plne figures 3. recognize the use of mtrices in liner trnsformtions UNIT SPECIFIC OBJECTIVE CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION Unit 3. Mtrices 3.. recognise mtrix nd stte its order Ide of mtrix Assist students to use everydy situtions to explin the ide of mtrices, e.g. Legue tles, seting rrngements in clss. stte the order of given mtrix nd indicte the type Guide students to stte the order of given mtrices Assist students to identify the types of mtrices unit, zero, digonl, squre nd rectngulr mtrices 3.. pply equl mtrices in relted prolems Equl mtrices Assist students to recognize tht if two mtrices re equl, their corresponding elements re equl.,,, pply equlity of mtrices to find missing entries of given mtrices Guide students to use the ide of equl mtrices to find missing entries in given mtrix dd nd sutrct mtrices. Addition nd sutrction of mtrices Assist students to dd nd sutrct mtrices (up to 3 3 ) y dding or sutrcting corresponding elements 40 find the sum nd difference of given mtrices. E.g. A shopkeeper hs two seprte shops which she opens on Mondy,

51 Unit 3. (cont d) Mtrices (Use rel life situtions) Assist students demonstrte tht the zero mtrix is the identity mtrix for ddition i.e O A A O A Wednesdy nd Fridy. In prticulr week she mde the following sles in the two shops. Represent the totl sles she mde in the week in mtrix form Shop A Fnt Mlt Energ y Mon Wedn Fri 9 Shop B Fnt Mlt Energ y Mon Wedn Fri multiply mtrix y sclr nd mtrix y mtrix Multipliction of mtrices Multipliction of mtrix y sclr. Multipliction of mtrices (up to 3 3 mtrices) Assist students to recognize tht multipliction of mtrix y sclr k involves multiplying ech element of A y k k Assist students to multiply n mtrix, e.g. c d x y k k k k m n mtrix y n n x cx y dy find the product of mtrix y sclr nd mtrix y mtrix E.g. Given tht A B nd find AB nd BA nd comment. Guide students to multiply two mtrices: 4

52 p q p r q s c d r s cp dr cq ds Assist students to multiply two 3 3 mtrices Unit 3. (cont d) NB. We multiply mtrices tht re conformle. Mtrix multipliction is not commuttive. Mtrices 3..5 find the inverse of mtrix. Inverse of mtrix Assist students to find the determinnts of mtrices. Tht is, if Determinnt of mtrix A c d then det A A d c Assist students to find the djoint mtrix s d c find the inverses of given mtrices Guide students to write the inverse of the mtrix A s A d c d c where d c 0 4

53 Unit 3. Liner trnsformti ons 3.. use liner trnsformtion to clculte imge nd oject points. 3.. stte the mtrix representing liner trnsformtion. Imges of points nd oject points under liner trnsformtion Mtrix of liner trnsformtion [Restrict to mtrices] Assist students to find imges of vrious points under liner trnsformtions of the form ( x, y) ( x, y ), where x x y nd y cx dy Assist students to write down the mtrix of liner trnsformtion, e.g. the mtrix of the liner trnsformtion x 4x 3y nd y 5 x y is A find imges of points under given liner trnsformtions determine the mtrices of given liner trnsformtions Unit 3. (cont d) Liner trnsformti ons 3..3 find the inverse of liner trnsformtion. Inverse of liner trnsformtion Assist students to find the inverse of liner trnsformtion y (i) finding the mtrix of trnsformtion A (ii) (iii) finding the inverse of the mtrix using the inverse mtrix to write the inverse of the liner trnsformtion; i.e. A A x y find the inverses of given liner trnsformtions 3..4 find the composition of liner trnsformtions. Composition of liner trnsformtions Assist students to find the composition of liner trnsformtions y (i) writing the mtrices A nd B of the given trnsformtions (ii) finding the mtrix product AB (iii) writing the liner trnsformtion for AB s the composition of the liner trnsformtions i.e find single liner trnsformtion tht represents the composition of two given liner trnsformtions Guide students to interpret s the trnsformtion for the mtrix B followed y trnsformtion for the mtrix A. 43

54 3..5 recognise the identity trnsformtions. Identity trnsformtions Assist students to stte the mtrices of trnsformtion corresponding to specil liner trnsformtions cos sin i., the generl mtrix for reflection in sin cos line through the origin mking n ngle θ with the positive x-xis use the identity trnsformtions to reflect nd rotte given points. Unit 3. (cont d) Liner trnsformti ons ii. iii. iv. 0 0 for reflection in x-xis 0 for reflection in y-xis for reflection in the line y = x. cos sin v. sin cos out the origin. for nticlockwise rottion through θ Include enlrgement y scle fctor k, centre the origin 3..6 find the eqution of the imge of line under liner trnsformtion. Liner trnsformtion of line Assist students to find the eqution of the imge of given line under liner trnsformtion. find the eqution of the imge of line under given liner trnsformtion 44

55 SENIOR HIGH SCHOOL - YEAR 3 SECTION LOGIC Generl Ojectives: The student will:. pprecite the concepts of logicl resoning. pply the concepts to determine the vlidity of compound sttements 3. drw vlid conclusions from truth tles 4. pply the rule of logic to deduce vlid conclusions from rguments in generl 5. e le to convince others logiclly on the vlidity of sttements mde UNIT SPECIFIC OBJECTIVE CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION Unit 3.3 Logic 3.3. identify compound sttements. Compound sttements Assist students to form compound sttements from simple sttements using the connectives nd Assist students to form converse sttements nd comment on them use connectives to form compound sttements nd stte their converses 3.3. drw implictions from given sttements nd their converses. Implictions nd their converses Assist students to drw implictions from given sttements using the impliction sign e.g. P Q drw conclusions from given sttements drw the truth tle of compound sttement. The truth tle Assist students to drw the truth tle using the rule of logic drw the truth tle for given compound sttements E.g. P Q P Q T T T T F F F T T F F T Include negtion, conjunction nd disjunction. 45

56 3.3.4 use the truth tle to deduce conclusions of compound sttements. Vlidity of rguments Assist students to use the truth tle in drwing conclusions using the rule of syntx: true or flse sttement, rule of logic pplied to rguments, implictions nd deductions. drw vlid conclusions from compound sttements using the truth tle 46

57 SENIOR HIGH SCHOOL - YEAR 3 SECTION 3 STATISTICS II Generl Ojectives: The student will:. represent ivrite dt grphiclly nd interpret the grph. use the eqution of the line of est fit to mke predictions 3. clculte correltion coefficients etween two sets of dt 4. use correltion coefficient to compre the reltionship etween the two vriles UNIT SPECIFIC OBJECTIVE CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION Unit 3.4. Correltion nd Regression 3.4. drw the sctter digrm for given dt The sctter digrm Assist students to distinguish etween univrite nd ivrite distriutions nd to drw the sctter digrms for ivrite distriutions. Sctter digrm is used to grphiclly disply ivrite distriutions. One vrile is long the x- xis nd the other long the y-xis. E.g. The test scores of seven students in English nd Mthemtics re s shown: English (x) Mthemtics(y) The ordered pirs (, 3), (4, 44), etc re plotted in the x-y plne in () drw sctter digrms for given sets of ivrite dt ssist students to use grphic clcultors nd computers to disply ivrite distriutions to cross 47 A

58 check their result Unit 3.4 (cont d) Correltion nd Regression 3.4. identify nd explin the vrious forms of correltion Forms of correltion in ivrite distriutions Discuss the vrious forms of correltion with students. Assist students to explin positive, negtive nd no correltions. - If the points cluster round stright line with positive slope s shown in (), then there is positive correltion etween the two vriles, tht is, s one increses so does the other. - If the points cluster round stright line with negtive slope s shown in digrm (), then there is negtive correltion etween the two vriles, tht is, s one increses the other decreses. - If the points re rndomly scttered, s shown in digrm (c), then there is no liner correltion (reltionship) etween the two vriles. () (c) distinguish mong vrious types of sctter digrms drw the line of est fit nd use it to predict one vrile given the other. Line of est fit Eqution of line of est fit (regression) Assist students to drw the line of est fit y first computing the: (i) men vlues x, y for ll vlues of x nd y ; (ii) men vlues x nd y respectively for which drw the line of est fit nd find its eqution y grph nd y lest squres method x is less thn x ; (iii) men vlues x nd y respectively for which x is greter thn x 48

59 Unit 3.4. (cont d) Correltion nd Regression NB. The line of est fit must pss through either or x, y x, y. x, y, nd Guide students to use the line of est fit to predict/estimte the vlue of vrile when one is given. Assist students to determine the eqution of the line of est fit from the grph Guide students to find the eqution of line of est fit y lest squres method: y x ; (regression of y on x) where xy y n x nd x x Assist students to use the eqution to predict one vrile given the other. Include regression of x on y. Unit 3.5 Spermn s Rnk Correltion Coefficient 3.5. clculte the correltion coefficient y rnking. Spermn s rnk correltion coefficient (Include dt with ties) Assist students to clculte the correltion coefficient using the Spermn s rnk correltion coefficient formul 6 d r s n n Exmple: The Spermn s rnk correltion for the dt on scores of 7 students in English nd Mthemtics tests in is clculted s follows: x y R x R y R x R y (d) d Totl 49

60 3.5. interpret correltion coefficients. Interprettion of correltion coefficients r s 6 d 7( n n ) Assist students to drw conclusions from the coefficient clculted. r lies etween nd Note: the vlue of s If r s then there is perfect positive correltion etween the two sets of dt If r s, then there is perfect negtive correltion, or complete disgreement etween the two sets of dt interpret correltion coefficients of given sets of dt 50

61 SENIOR HIGH SCHOOL- YEAR 3 SECTION 4 MECHANICS II Generl Ojectives: The student will:. recognize forces s vectors. distinguish etween sttic nd dynmics 3. resolve forces nd find their resultnts 4. determine coefficient of friction 5. find momentum nd impulse 6. pply ides in mechnics to solve prcticl prolems UNIT SPECIFIC OBJECTIVE CONTENT TEACHING AND LEARNING ACTIVITIES EVALUATION Unit 3.6 Dynmics 3.6. explin the concepts of motion, time nd spce. Concepts of motion Displcement, velocity (Reltive velocity) nd ccelertion. Assist students to explin the concept of dynmics nd relte it to sttics Assist students to drw nd interpret distnce time, nd velocity time grphs drw grphs of motion nd interpret them use the equtions of motions to solve rel-life prolems. E.g stte nd use Newton s equtions of motion to solve simple prolems solve simple prolems involving motion under grvity. Equtions of Motion Newton s lws Motion under Grvity Assist students to deduce nd use the following Newton s lws of motion to solve relted prolems: (i) (iii) F m (ii) v u t s ut t (iv) v u s Assist students to solve simple prolems on motion under grvity involving: (i) prticles projected verticlly upwrds (ii) prticles flling freely under grvity (Ignore ir resistnce) A prticle of mss kgt rest is cted upon y three forces F ( 0 N, 090 ), F (8N, 0 ) n d F (8, 330 ). 3 N Clculte the resultnt force nd the ccelertion of the prticle. solve rel-life prolems involving motion under grvity E.g.A oy stnding on top uilding m high throws ll verticlly upwrds with n initil 5

62 velocity of 96 ms () Find the ll s height nd velocity t time t. () When does the ll hit the ground nd wht is the impct velocity? (c) When is the velocity zero? Unit 3.6 (cont d) Dynmics find the force up n inclined plne. Motion long inclined plnes Assist students to resolve force up n inclined plne into Norml nd Frictionl Force N F solve rel life prolems involving motion long inclined plnes E.g. MgSinθ θ Mg Mgcosθ A ody of mss 8kg rests on smooth plne inclined t 30o to the horizontl. Find the lest vlue of the force required to keep it in equilirium nd the resultnt rection of the plne. [Tke g = 0ms - ] Assist students to solve simple relted prolems 5

63 3.6.5 pply the principles of conservtion of liner momentum to solve simple prolems on direct impct. Direct impct Guide students to distinguish etween momentum nd impulse Assist students to pply the principle of conservtion of liner momentum: m (i) u mu mv mv m (ii) u mu ( m m ) v (Exclude coefficient of restitution) stte nd use the principle of conservtion of liner momentum to solve rel life prolems. E.g. Two prticles A nd B with msses 3kgnd 4kgmoving with velocities u i 3j nd u i 4j respectively, collide. After collision prticle A moves with velocity i j. Determine the velocity of prticle B fter collision. 53

64 References. Additionl Mthemtics for West Afric y Tlert, Godmn nd Ogum. Core Mthemtics for Advnced Level y L. Bostock nd S. Chndler 3. Studymte: HSC 3 Unit Mthemtics y Mrgret Grove 4. Additionl Pure Mthemtics, Book y Bckhouse 5. Elective Mthemtics for S.H.S., y P. Aseidu 54

65 YEAR SECTION : ALGEBRA... SECTION COORDINATE GEOMETRY I... 9 SECTION 3 PROBABILITY I... SECTION 4 VECTORS I... YEAR SECTION CO-ORDINATE GEOMETRY II... 4 SECTION ALGEBRA II... 7 SECTION 3 TRIGONOMETRY... 0 SECTION 4 CALCULUS... 4 SECTION 5 PERMUTATION, COMBINATION & PROBABILITY... 9 SECTION 6 STATISTICS I... 3 SECTION 7 VECTORS II SECTION 8 MECHANICS I YEAR 3 SECTION MATRICES AND LINEAR TRANSFORMATIONS SECTION LOGIC SECTION 3 STATISTICS II SECTION 4 MECHANICS II