SOUTH PACIFIC BOARD FOR EDUCATIONAL ASSESSMENT

Transcription

1 SOUTH PACIFIC BOARD FOR EDUCATIONAL ASSESSMENT Pacific Senir Secndary Certificate Mathematics Prescriptin Frm 6 Effective frm January 008

2 Pacific Senir Secndary Certificate Mathematics Cntents Page Ratinale 1 General Aims Pre-requisite Time Allcatin Curse Cntent & Learning Outcmes 3 Tpic 1: Algebra 4 Tpic : Crdinate Gemetry 4 Tpic 3: Sequences and Series 6 Tpic 4: Prbability 8 Tpic 5: Statistics 9 Tpic 6: Graphs f Functins 10 Tpic 7: Calculus 11 Tpic 8: Trignmetry 1 Assessment 15 External Assessment 16 Internal Assessment 17 Advisry Sectin 19 Texts and Reference Bks 19 Appendices 0

3 Ratinale The imprtance and usefulness f Mathematics in influencing ur daily lives are undeniable. We simply can nt d away with the applicatin f Mathematics, frm simple cunting things at hme t writing sphisticated reprts at wrk, the skills and knwledge f Mathematics play central rles. Mathematics has been taught at schls fr many years, and in many schls, Mathematics has becme a cmpulsry subject at all levels. Many students leave frmal schling at the end f their Frm 6 year, while thers mve n t higher levels and eventually t university and ther higher educatin institutins. It is vital that curses f studies and prescriptins at this level are designed in such a way that the needs f thse wh leave schl at this pint and thse wh cntinue n t higher educatin are bth catered fr. Recent develpments in science and technlgy have been rapid and their impacts n educatin and ur lives are mre than nticeable. It is therefre crucial that we respnd t thse develpmental frces by cnstantly upgrading and imprving ur educatinal fferings s that ur students are adequately equipped t cpe with the demands f mdern living. The review exercise was carried ut with the fllwing cnsideratins in mind: 1. Retain the standard f the mathematical skills and knwledge taught at Frm 6 level in cmparisn with the ther cuntries.. Equip students with the required mathematical skills and knwledge t cntinue n t further and higher educatin. 3. Adequately prepare students wh leave schl at the end f Frm 6 s that they may cntribute cnstructively t sciety. 4. Keep up with the new develpment in Science and Technlgy and prmte the use f technlgy in real life applicatins. 5. Encurage students t use their mathematical skills as a tl t explre and slve mathematical prblems in real life situatins. 1

4 General Aims The PSSC Mathematics curse intends t: 1. Develp enquiring minds in students s that they may take genuine interests in Mathematics and actively engage in research and research-related activities.. Prvide students with mathematical knwledge and skills t enable them t deal successfully with the demands f their careers as well as the intricacies f real life. 3. Encurage students t use the new develpments in Science and Technlgy in their academic wrks and everyday living. 4. Equip students with mathematical skills s that they can make well-judged deductins and reasnings and draw lgical and meaningful cnclusins when facing new r unfamiliar real life situatins. Prerequisite Students undertaking this PSSC curse will be expected t have successfully cmpleted the Frm 5 Mathematics curse. Time Allcatin It is recmmended that schls allw a minimum f five hurs per week cntact time in the classrm with the teacher, with a minimum f thirty prductive weeks per year.

5 Curse Cntent and Learning Outcmes TOPIC 1: ALGEBRA Slve a linear equatin written in any frm Identify linear equatins Simplify linear equatins. (e.g expand brackets, eliminate fractinal terms, cllect like terms etc.) Apply algebraic peratins crrectly. (i.e additin, subtractin, multiplicatin and divisin) Check the accuracy f answers using algebraic methds such as back substitutin when required. Slve tw linear equatins simultaneusly 1..1 Slve tw equatins simultaneusly using eliminatin r substitutin methd. 1.. Operate any ther apprpriate methd t slve the tw linear simultaneus equatins Slve linear inequatins f the frm ax + b cx + d, a, b, c, d R Slve the inequatins by appling algebraic peratins crrectly 1.3. Evaluate the crrect slutin set. Slve linear inequatins f the frm: ax + b cx + d ex + f, a, b, c, d, e, f R Slve the linear inequatins using any apprpriate methd 1.4. Evaluate the crrect slutin set Change the subject f frmulae Identify the subject and cllect like terms 1.5. Rearrange the frmula t btain the crrect subject Slve quadratic equatins Identify quadratic equatins 1.6. Factrise quadratic equatins Expand and simplify factrs f a quadratic equatin Slve quadratic equatins by factrisatin Slve quadratic equatins using quadratic frmula Find the pint r pints f intersectin between a quadratic functin and a straight line. Slve cubic equatins r functin Identify cubic functins 1.7. Expand and simplify factrs f a cubic equatins Factrise a cubic functin using the factr therem (n prf required) Slve cubic functins by factrisatin nly 3

6 TOPIC 1: ALGEBRA Operate ratinal algebraic expressins Simplify algebraic expressins 1.8. Add algebraic expressins Subtract algebraic expressins Multiply algebraic expressins Divide algebraic expressins Slve expnential equatins (including thse with negative and fractinal indices) Apply laws f indices t simplify expnential expressins r equatins 1.9. Slve the expnential equatins using any apprpriate methd. Slve lgarithmic equatins Apply laws f lgarithms t simplify lgarithmic expressins r equatins Cnvert t and frm lgarithmic frm and index frm Slve lgarithmic equatins using any apprpriate methd TOPIC : COORDINATE GEOMETRY Calculate the distance between tw pints.1.1 Identify the variables x 1, x, y 1, y.1. Find the distance between tw pints using Pythagras Therem.1.3 Find the distance between tw pints using the Distance Frmula. Calculate the midpint f a line segment..1 Identify the variables x 1, x, y 1, y.. Crrectly substitute values f x and y int frmula..3 Find the midpint using any apprpriate methd Calculate the gradient f a line segment.3.1 Calculate gradient using the crdinates f tw pints.3. Find the gradient using y/ x.3.3 Evaluate the gradient using trignmetric relatinships Calculate the angle between a line segment and the hrizntal.4.1 Determine the size f the angle between a line segment and the hrizntal.4. Calculate the gradients using trignmetric relatinships.4.3 Find the angle between line segments using a knwn gradient Use the gradient t find the equatin f a straight line.5.1 Identify the general frmulae f a straight line.5. Express the frmula in the frm y=mx+c.5.3 Find the value f c. 4

7 TOPIC : COORDINATE GEOMETRY Use tw pints n a line t find the equatin f the line.6.1 Identify the x and y crdinates f tw pints n the line.6. Find the gradient m f the line.6.3 Write the equatin in the frm y=mx+c Sketch the graph f a straight line.7.1 Find the x- and y- intercepts f the line.7. Determine the gradient f the line.7.3 Sketch the line n the x-y plane Find the gradients f parallel and perpendicular lines.8.1 Define parallel and perpendicular lines.8. Find the gradients f parallel lines.8.3 Find the gradients f perpendicular lines.8.4 Slve applicatin prblems invlving parallel and perpendicular lines Find the pint f intersectin f tw lines.9.1 Identify cnditins which give rise t tw lines intersecting at ne pint.9. Slve the tw linear equatins simultaneusly.9.3 Determine the x- and y- crdinates f the pint f intersectin Identify the types f triangles and quadrilaterals using the lengths and gradients f their sides.10.1 Find the lengths f sides f figures using the distance frmula r any apprpriate methd.10. Find the gradients f the sides f figures n the x-y plane.10.3 Identify the type f triangle r quadrilateral by cmparing the lengths and/r gradients f their sides Sketch and label the graph f a circle centred at the rigin and f the frm: x + y = r, r R.11.1 Identify features f a circle centred at the rigin.11. Find the equatin f a circle centred at the rigin.11.3 Sketch a circle centred at the rigin n the x-y plane Sketch and label the graph f a circle NOT centred at the rigin and f the frm ( x a) + ( y b) = r, {a,b,r} R, r f the frm x + mx + y + ny = k, {m,n,k} R.1.1 Identify imprtant features such as radius and centre, f a circle nt centred at the rigin.1. Apply translatin methd t transfrm a circle f the frm x + y = r, r R t the frm ( x a) + ( y b) = r, { a, b, r} R and vice versa.1.3 Evaluate the translatin vectrs in Sketch a circle nt centred at the rigin n the x-y plane 5

8 TOPIC : COORDINATE GEOMETRY.1.5 Find the equatin f a circle nt centred at the rigin..1.6 Cnvert equatins f circles f the frm x + mx + y + ny = k t the frm ( x a) + ( y b) = r and vice versa. Variables a, b, k, m, n and r are Real Numbers.13 Find the equatin f a circle f knwn radius centred n the pint (a, b).13.1 Recgnise the general frm f the equatin f a circle ( x a) + ( y b) = r.13. Substitute the values f a, b and r int the general frm f the equatin t btain the equatin f the circle.13.3 Write ther frms f the circle equatin int ( x a) + ( y b) = r and vice versa eg. ( x 1) + ( y ) = 4 x + y x 4y 11 = 0 TOPIC 3: SEQUENCES AND SERIES Generate a sequence given a general term Identify features f a general term 3.1. Use a general term t generate a sequence Determine the general term frm a given list f terms f an Arithmetic sequence 3..1 Shw that a list f terms is an Arithmetic sequence 3.. Find the first term f an Arithmetic sequence 3..3 Evaluate the cmmn difference f an Arithmetic sequence 3..4 Determine the general term f an Arithmetic sequence Determine the general term frm a given list f terms f a Gemetric sequence Shw that a list f terms is a Gemetric sequence 3.3. Find the first term f a Gemetric sequence Evaluate the cmmn rati f a Gemetric sequence Determine the general term f a Gemetric sequence Find the sum f an Arithmetic series Apply the apprpriate frmula t find the sum f an Arithmetic series. 3.5 Find the sum f an Gemetric series Apply the apprpriate frmula t find the sum f a Gemetric series. 6

9 TOPIC 3: SEQUENCES AND SERIES Apply n th term f an Arithmetic sequence, T n = a + ( n 1) d, and n S n = [ a + ( n 1) d], the sum f an Arithmetic series Evaluate any unknwn variable in T n 3.6. Evaluate any unknwn variable in S n Slve wrd prblems invlving the applicatin f Arithmetic sequence and series including real life applicatins. Apply the n th n 1 term f a Gemetric sequence, T n = ar, and the sum n n a( 1 r ) a( r = 1) f a Gemetric series. Sn = r Sn =. 1 r r Evaluate any unknwn variable in T n 3.7. Evaluate any unknwn variable in S n Slve wrd prblems invlving the applicatin f Gemetric sequence and series including real life applicatins. a Apply the sum t infinity, S = 1 r State the cnditins fr the applicatin f the sum t infinity 3.8. Evaluate any unknwn variable in S Calculate the sum t infinity S Slve wrd prblems invlving the applicatin f the Sum t infinity including real life applicatins. Evaluate sums using sigma ntatin List a series using a sigma ntatin 3.9. Apply the apprpriate methd t find the sum f the series Write series in sigma ntatin Identify features f a sigma ntatin Evaluate limits and equatin f a sigma ntatin fr a given series Find the crrect sigma ntatin fr a given series. 7

10 TOPIC 4 : PROBABILITY List all pssible utcmes f an event Describe the event 4.1. Identify the utcmes List all pssible utcmes f an event Define and use the fllwing terms: event, sample space, element, trial, utcme, equally likely, bias, independent, cnditinal/dependent. Calculate the prbabilities f equally likely events using the nx ( ) frmula p( x) = ns () 4..1 Define equally likely events 4.. Knw the cmpnents f the frmula 4..3 Use the frmula t calculate the prbability Use tree diagram t predict utcmes Identify imprtant features f tree diagrams 4.3. Draw tree diagrams Calculate prbabilities f familiar r simple applicatins using tree diagrams Cnstruct and Use frequency tables t predict utcmes Identify imprtant features f frequency tables 4.4. Cnstruct frequency and cumulative frequency tables Draw graphs frm frequency table data Interpret graphs and data frm frequency tables Use frequency table data t calculate statistical parameters (mean, median, mde, range, standard deviatin etc.) Use cumulative frequency tables t draw cumulative curves and slve simple applicatin prblems Calculate lng run relative frequencies Identify frmula 4.5. Substitute crrect values int frmula Calculate the expected value f a number f trials Define expected value 4.6. Determine the number f trials Identify crrect frmula Substitute crrect values int frmula Calculate the mean and standard deviatin f a given set f data Knw the difference between mean, mde and median 4.7. Knw and use the frmula fr finding the mean and standard deviatin Substitute values int frmula 8

11 TOPIC 4 : PROBABILITY Use Nrmal Distributin Curve t determine prbabilities Recgnise the shape f a nrmal curve 4.8. Recgnise the Nrmal Distributin as a descriptin f a data set Identify and lcate the psitins f the mean and standard deviatin n the Nrmal curve Knw the relatinship between prbabilities and standard deviatin values Assciate standard deviatin values (1sd, sd, 3sd) with qualitative prbability descriptives likely, very likely, almst certainly, and with their respective prbability values. Use the Standard Nrmal Distributin Knw the parameters fr a Nrmal Distributin 4.9. Draw the nrmal curve using the parameters Use the z-scre t calculate prbabilities Define z-scre Knw the significance f a z-scre Identify and use the frmula fr finding z-scres Lcate the psitins f z-scres n the nrmal curve Use the Table t determine the prbability values represented by z-scres Calculate a specific statistics assciated with a given Nrmal Distributin fr a given prbability Draw the nrmal curve using the crrect parameters TOPIC 5 : STATISTICS Differentiate between cntinuus and nn-cntinuus data Knw and recgnise cntinuus data and discrete data 5.1. Categrise data as cntinuus r discrete Understand cmmnly used statistical terminlgies 5..1 Define: sample, survey, census, bias, nrmal distributin Select a randm sample frm a nrmally distributed ppulatin Define the nature f the ppulatin 5.3. Understand varius sampling methds Identify apprpriate sampling methds Select sample using methd identified in Cnduct statistical analysis n sample data Identify and state the purpse and methd f data cllectin 5.4. Clean and rganise data Determine statistical parameters t use in the analysis Calculate statistical parameters 9

12 TOPIC 5 : STATISTICS Use apprpriate ways f data presentatin t display data 5.5 Interpret data and draw cnclusins Recgnise patterns r trends shwn by parameters 5.5. Recgnise patterns r trends shwn by data presentatins Relate patterns r trends t purpse f data cllectin Cmment n the significance f the patterns and trends Draw cnclusins abut the purpse f data cllectin frm the patterns and trends revealed by the data. TOPIC 6 : GRAPHS OF FUNCTIONS Plt and sketch the graph f: 3 y = ax + bx + c y = ax + bx + cx + d y = lg a ax + b 1 x y = y = a y = ax cx + d Use functin t generate table f values Plt graph f functin, shwing all intercepts Determine asympttes if any Describe behaviur f functins fr large values f x Define limit f a functin Find limit(s) f functins fr specific values f x Identify symmetries f dd and even functins Determine dmain and range f functins. Recgnise the cncepts f transfrming functins by sketching the translatins, change f scale, and reflectin in the x-axis f the fllwing basic functins: 1 3 y = x y = x y = x x a y = a y = y = lg a x x 6..1 Sketch the graph f a given functin shwing apprpriate labels. 6.. Determine transfrmatin and any change f scale invlved using any f the basic functins abve Determine transfrmatin and translatin invlving any f the basic functins Determine reflectin n the x-axis using any f the basic functins Find the equatins and sketch the graphs f the inverse functins f: x y = a ax y = ax y = mx + c 1 y = y = lg x Find the equatin f the inverse functin Use the inverse functin t generate a set f values Sketch r plt the graph f the inverse functin shwing all apprpriate labels. a x 10

13 TOPIC 7: CALCULUS Understand and recgnise the relatinship between the rate f change and the gradient Define gradient in terms f rate f change Use apprpriate frmulae t find gradient. n Find the derivative f ax where a and n are ratinal numbers Find the derivative f a functin using the First Principle 7.. Differentiate functins using apprpriate rule. Find the derivative f a plynmial Identify plynmial and use the rule f differentiatin apprpriately. Recgnise the derivative as the gradient f the tangent t a curve at a pint Apply derivative t find differential functin Slve fr x, using y=f(x) given the gradient Determine gradient f tangent line at the pint (x,y) Determine gradient f nrmal line at the pint (x,y) Use apprpriate methd(s) t find the equatins f the tangent and nrmal t the curve at a given pint Sketch the graphs f tangent r nrmal lines. Apply derivative t statinary pints and minima/maxima prblems dy Find statinary pints f a plynmial using = f ( x) = 0 fr x. dx 7.5. Determine the minima r maxima values Determine the nature f a curve using Secnd Derivative Tests Sketch graphs f plynmials using derivative infrmatin. 7.6 Find the anti-derivative f and n 1. n ax where a and n are ratinal numbers Apply anti-derivative rule (fr plynmial) apprpriately Find the anti-derivative f a plynmial Identify plynmial and use apprpriate rule(s) f anti-derivative Find the value f the cnstant. Evaluate definite integrals Evaluate definite integrals using apprpriate anti-derivative rule(s). Find the area between a curve and x-axis Draw graph f the curve, shwing area and limits 7.9. Evaluate area between curve and x-axis 11

14 TOPIC 7: CALCULUS 7.10 Find area between tw curves Draw graphs f the tw curves Write intended area in terms f definite integrals Evaluate definite integrals TOPIC 8: TRIGONOMETRY 8.1 Calculate values f sin x, cs x, tan x fr 0 x Calculate values f sin x fr 0 x Calculate values f cs x fr 0 x Calculate values f tan x fr 0 x Sketch the graph f y = sin x, y = cs x, y = tan x fr x Sketch graph f y = sin x fr 8.. Sketch graph f y = cs x fr 8..3 Sketch graph f y = tan x fr x 360 with apprpriate labels. 0 x 360 with apprpriate labels. x with apprpriate labels. 8.3 Draw graph f any trignmetric functin f the sin frm y = A ( Bx + C) + K cs Define, calculate and shw crrect amplitude Define, calculate and shw crrect perid and frequency Determine hw the graph f y = sinx changes scale, translates r reflects in the x-axis in rder t get y = asinx r y = sinax r y = sin(x + a) Determine hw the graph f y = cs x changes scale, translates r reflects in the x-axis in rder t get y = a cs x r y = csax r y = cs(x + a) Draw graphs shwing apprpriate translatins (shift) with apprpriate labels acrss bth axes Identify graphs f y = asinx, y = sinax r y = sin(x + a) as transfrmatins f the graph f y = sin x Identify graphs f y = acsx, y = csax r y = cs(x + a) as transfrmatins f the graph f y = cs x. 8.4 Slve trignmetric equatins f the type: asin (x + b) = c, acs (x + b) = c, asin bx = c, acs bx = c fr 0 x 360 and where a, b, c R c Slve asin(x + b) = c using x b = sin 1 ( ) a, b, c R. + fr a x and where 1

15 TOPIC 8: TRIGONOMETRY 8.4. c Slve acs(x + b) = c using x b = cs 1 ( ) where a, b, c R c Slve asinbx = c using bx sin 1 ( ) a, b, c R. + fr = fr c Slve acsbx = c using bx cs 1 ( ) a, b, c R. a = fr a a x and x and where x and where 8.5 sin x Prve simple identities usingsin x + cs x = 1, = tan x r ther given cs x csθ simpler identities such assec θ =, cs ec θ =, ct θ = =. csθ sinθ tanθ sinθ Apply any f the given identities t shw prf r equality by manipulating either r bth sides f the equatin Calculate the area f any triangle Identify triangle and apprpriate frmulae t use Equate variables crrectly t knwn and unknwn quantities Apply frmulae A = absinc t find the unknwn variable. Slve any triangle using the Sine and Csine Rules Identify triangle and apprpriate frmulae t use Equate variables (frmulae) t knwn and unknwn quantities Apply frmulae a = b + c bccsa t find an unknwn side b + c a Apply frmulae: CsA = t find an unknwn angle. bc a b c Apply frmulae = = t find either an unknwn side SinA SinB SinC r an unknwn angle. Cnvert t and frm radians and degrees Identify angle needed and use the apprpriate relatins: π = 180 ;π = Cnvert angle size frm radians t degrees and vice versa. Calculate the length f an arc f a circle given the angle subtended at the centre and the radius Identify the arc f the circle tgether with the angle subtended at the centre and the radius Use the frmulae fr the length f an arc f a circle: S = Rθ Calculate the angle subtended at the centre given the length f an arc and the radius f the circle Identify the angle subtended at the centre, the length f the arc and the radius f the circle. 13

16 TOPIC 8: TRIGONOMETRY Identify and apply the frmula fr finding the length f an arc f a S circle: θ =, where θ is measured in radians. R Calculate the area f a sectr f a circle Identify the area f a sectr f a circle, the angle subtended at the centre and the radius f the circle Identify and apply the frmula fr finding the area f a sectr f a circle: A = 1 R θ Calculate the area f a segment f a circle Find area f a segment using the area f a sectr f a circle: A = 1 R θ 8.1. Verify hw the area f the segment f a circle is equal t the area f the sectr minus the area f the triangle: Area f the segment = 1 R ( θ sin θ ) Calculate the area f a segment f a circle. 14

17 ASSESSMENT In rder t be able t assess a wide range f mathematical skills and especially thse that are difficult t be effectively assessed by an external examinatin, a cmpnent f schl based internal assessment is included in this prescriptin. The internal assessment cmpnents are aimed at prviding students with an pprtunity, withut the pressures f examinatin cnditins, and with access t resurces, t demnstrate their ability t apply their mathematical knwledge and skills in real life situatins. This pprtunity, will allw them t demnstrate their ability t cllect and prcess infrmatin, t make apprpriate cnclusins frm their investigatins and justify them, and t cmmunicate their findings in a written and graphical frm. Teachers and students are als given pprtunities t display creativity and innvatin in develping and designing assessment activities as part f the internal assessment tasks. The prgramme will be assessed by an external examinatin at the end f the year, ONE cmmn assessment task (CAT), ONE minr research task and THREE teacher designed tasks (TDT). The weighting rati will be 70% fr the external examinatin, 10% fr the CAT, 10% fr the minr prject and 10% fr the teacher designed tasks (TDTs). This weighting will be applied by SPBEA as part f their standard prcessing prcedures. The CAT will be based n a tpic r parts f a tpic f the prescriptin t be selected by SPBEA and annunced t the schls befre the end f January each year. It is intended that the bjectives assessed by internal assessment methds will nt be directly assessed again by the external examinatin. Hwever, sme verlaps may ccur where sme previusly learned skills are used in ther tpics. In rder fr students t qualify fr full assessment in this curse, it is necessary that they sit the external examinatin and present wrks fr the internally assessed cmpnents. Assessment Schedule Weight (%) External Examinatin 70 Internal Assessments: CAT (10%) Minr Research Task (10%) Teacher Design Tasks (TDT) (10%) 30 TOTAL

18 Fur aspects f mathematical skills will be assessed by this prescriptin: Apprx Aspect Weighting (%) Mathematical knwledge and skills 5 Infrmatin prcessing 15 Cmmunicatin f mathematical ideas 10 Applicatins f mathematics t prblem slving 50 TOTAL External Assessment (70%) The external Assessment will still be an Examinatin with a ttal mark f 10 which is equivalent t 70% f the final mark. The examinatin will ffer a range f questins in which students will be required t demnstrate their: Knwledge f mathematical principles Ability t apply their mathematical knwledge t practical situatins Ability t write crrect mathematical statements Ability t prcess infrmatin Ability t make deductins and draw cnclusins The examinatin paper will cnsist f: Sectin A (0 Multiple Chice questins wrth 1 mark each) Sectin B (10 Lng answer questins wrth 10 marks each] 0 marks 100 marks. The apprximate weight f each tpic in the examinatin paper will be as fllws: WEIGHT TOPIC (%) (Marks) 1 Algebra 14 4 Crdinate Gemetry Sequence and Series Prbability Graphs and Functins Calculus Trignmetry 1 0 TOTAL Questins in the external examinatin will be designed within a meaningful cntext that is apprpriate t the varied and diversified backgrunds f the students. Cntextual questins give students the pprtunity t demnstrate their mathematical skills and abilities. 16

19 . Internal Assessment (30%).1 Task 1 : Minr Research Task (10%) The research task is designed t allw assessment f a student s ability t apply a sectin f the prescriptin t a tpic f their wn chsing withut pressured time restrictin. It is anticipated that students take abut 10 t 1 hurs f class time t cmplete all aspects f the prject. This is equivalent t abut tw weeks if time allcatin fr mathematics class is ne hur per day. It is imprtant that bth teacher and students pay attentin t the fllwing elements while carrying ut the task. (i) (ii) (iii) (iv) (v) (vi) (vii) Structure and design f the research task Chsing the tpic and framing the hypthesis Data cllectin and data preparatin Data display and presentatin Data analysis Data interpretatin Presenting findings and drawing cnclusins Fr (i) and (ii) the teacher is expected t d mst f the wrk and students are expected t carry ut (iii) (vii) n their wn. Guidelines fr carrying ut the prject are presented in Appendix A.. Cmmn Assessment Task (CAT) (10%) This will be based n a tpic r parts f a tpic t be selected by SPBEA. SPBEA will infrm schls abut the CAT tpic befre the end f January each year. Details such as cmpletin date, deadline fr submissin f marks and any ther special requirements will als be cnveyed t schls befre the end f January. Teachers will be required t administer and mark the CAT under examinatin cnditins. Any breach f regulatins by bth students and teachers will be treated in a similar manner t a breach f examinatin regulatins..3 Teacher Design Tasks (10%) These are tasks ther than written paper and pen tests. The teacher is required t cnstruct THREE tasks with a cmbined weight f 10%. The tasks shuld fcus n any aspect f mathematics (cncept, idea, applicatin, thery, histry etc.) 17

20 Pssible tasks culd include: Seminar presentatin Mdel making Pster prductin Speech/verbal presentatin/pwer-pint presentatin Research/bigraphy Assignments Games/cmpetitins Fieldtrip reprt Creative writing (e.g. pem, shrt essay) Skits/actin sngs/dramatisatin/rle play Activities that demnstrate a mathematical cncept r an applicatin f it Other tasks planned by the teacher that culd be submitted fr apprval It is recmmended that teachers rganise students, either individually r in small grups (-3 students) t demnstrate r present their activities n a weekly basis during class time. 18

21 ADVISORY SECTION 1. Texts and References Barret, R. ( ), Frm 6 Mathematics Revisin, ESA Publicatins Bartn, D. and Jhnsn, W. (199), Theta Mathematics, Lngman, Auckland, New Zealand. Bishp & Wallace, ( ), Encunters, Bks A, B, C and D., Lngman Sealy and Agnew, (,) Senir Mathematics, Lngman Turner et al, ( ), Cnnectins, New Huse Publishers 19

22 APPENDICES APPENDIX A 1. GUIDELINES TO MANAGING THE MINOR RESEARCH TASK The intentin f this minr research task is t prvide students with the pprtunity t demnstrate their ability t apply their knwledge and skills in cllecting, presenting, analysing and reprting numerical data. If this assessment methd is t have credibility (that is the marks awarded t students are t reflect the quality f the students wrk, discriminate between different levels f success, and be fair, valid and cnsistent), the mnitring f the prject must be fair, cnsistent and reliable. The teacher is expected t prvide students with guidance thrughut the develpment and the cmpletin f the prject. All develpment shuld be carefully mnitred t ensure that the final prduct is the student s wn wrk. It is recmmended that students shuld nt spend t lng n this prject. Tw weeks, r three weeks at the lngest, shuld be allwed fr this task. AIM Fr each student t prduce a written reprt abut their statistical investigatin f a randmly cllected sample f 30 numerical items frm a ppulatin f apprximately 00. OBJECTIVES Students will: (i) (ii) be respnsible fr ensuring that timelines and deadlines are adhered t, all paper wrk are secured and in gd cnditin, and that the wrk submitted fr assessment is their wn. select a tpic t study, frmulate a hypthesis, and have the teacher apprve bth. An example f a hypthesis is The mean height f all Frm 3 students in Schl X is 1.54 metres. The activities in the study aim at supprting (prving) r rejecting (disprving) the hypthesis. The teacher is expected t d mst f the wrk in designing the task and in frmulating the hyptheses. (iii) by a due date, submit t the teacher a written statement called Chapter 1, which briefly describes the task, defines the questin, and utlines hw the students will g abut selecting their samples. 0

23 (iv) (v) (vi) use a randm numbers prcess t select a sample f 30 numerical items frm a ppulatin f apprximately 00. This shuld be cmpleted within ne week after submitting Chapter 1. immediately after selecting the sample, prepare a secnd written statement called Chapter which utlines the sample selectin prcess. by a due date, submit t the teacher a final written reprt that cntains Chapter 3, Chapter 4 and References with the fllwing details: Chapter 3 Presentatin and display f sample data using apprpriate tables and graphs. Use nt mre than tw types f graphs t display yur data. Analysis f data. The fllwing questins shuld be used t guide the data analysis: What are the measures f central tendency and what d they tell yu abut the data? What are the measures f the spread f the distributin and what d they tell yu abut the data? What are the cnfidence intervals and what are their significance? Chapter 4 A well presented summary f findings Discussin n the Secnd Cnfidence Interval that has been calculated. Cnclusin relating the findings t the hypthesis. References A list f the surces f infrmatin (bks, ther printed materials, interviews etc.) Sign the prject register t shw that the student has handed in the cmpleted prject t the teacher The teacher will: discuss with the students the nature, aims, prcess, reprting, and assessment f the prject. give each student a written descriptin f the prcess fr the management f their prject which will include the way the teacher will manage the prject, all due dates, penalties fr late wrk, and hw the prject will be assessed. apprve and keep a register f each student s chice f tpic and hypthesis; review Chapter 1 and cmment abut it n the mark sheet which will be attached t it by the teacher; 1

24 review Chapters 1 and and cmment n them n the same mark sheet sign and date the prject register t shw that Chapters 1 and have been apprved mark Chapters 3 and 4 accrding t the assessment schedule take part in any in-schl mderatin prcess if it is required adjust the marks f the prjects if necessary, accrding t the mderatin utcmes keep recrds f the marks btained by students submit students marks tgether with a cpy f the signed prject checklist t the PSSC Crdinatr r Principal. select six (6) samples fr mderatin and submit t the PSSC Crdinatr. The samples shuld cnsist f the tw () tp prjects, tw () middle prjects, ne frm between the tp and the middle, and ne frm between the middle and the lwest prject.

25 APPENDIX B Achievement Based Assessment Criteria fr Marking the Minr Research Task The prject will be marked frm three perspectives: Infrmatin Prcessing: hw the student gathers, manages and analyses the infrmatin btained, Cmmunicatin: hw the student explains r represents the facts, the results, and the significance f the findings, Numerical Skills: the accuracy with which the student perfrms the calculatins. Within each perspective a range f marks are available fr students t earn. Teachers are encuraged t use the full range f marks available. The fllwing prcess fr marking a set f prjects is recmmended: Read all the prjects cmpletely thrugh, making a subjective assessment f the ttal wrth f the prject as yu prceed; On the basis f this preliminary reading, srt the prjects int 4 grups, namely: very gd, abve average, belw average, pr; Mark ne randmly selected prject frm each grup; Repeat this prcess until all the prjects are marked. As yu mark, place the prjects in rder f merit accrding t yur marking; Re-read all prjects in merit rder, making a subjective assessment n the same basis as fr the first reading and cmparing the wrth f the prject relative t thse which came befre. Adjust yur marks t establish relativities between prjects as establishing the rder f merit. Finally, if necessary, interview student(s) whse research tasks r certain aspects f their tasks need t be clarified. This interview shuld nt be used t adjust marks, but t clear up any uncertainties in the teacher s mind abut the task, and t ensure that the wrk handed in is the student s wn wrk. 3

26 Infrmatin Prcessing Gathering and analysing infrmatin r data Marks Achievement Criteria Assessment Schedule 5 Gathers sme relevant infrmatin Presents an incmplete reprt which indicates the use f an apprpriate methd fr gathering Gathers sme relevant infrmatin and assembles it systematically Gathers relevant infrmatin, assembles it systematically and analyses it. Assembles and analyses relevant infrmatin and draws a valid cnclusin Assembles and analyses relevant infrmatin, draws a valid cnclusin r cnclusins and evaluates findings data Presents an incmplete reprt which indicates the use f an apprpriate methd fr gathering data and which displays the data in a systematic and apprpriate manner Presents a cmplete reprt which meets all the deadlines fr the survey, indicates the use f an apprpriate methd fr gathering and displaying data, and in which the apprpriated calculatins are perfrmed. Presents a cmplete reprt which meets all the deadlines fr the survey, and frm the results f the data analysis draws a valid cnclusin in respnse t the questin behind the survey Presents a cmplete reprt which meets all the deadlines fr the survey, draws a valid cnclusin in respnse t the questin, and includes a searching thrugh, and evaluatin f the prcess and its relevance t the riginal questin 4

27 Cmmunicatin Expressing mathematics in written r ral frm, using symbls, graphs, diagrams etc. Recgnitin will be given t presentatin which enhances the visual impressin f the reprt. Marks Achievement Criteria Assessment Schedule 4 Attempts t cmmunicate mathematical ideas Presents a reprt with sme written cmmentary 8 Cmmunicates mathematical ideas Presents a reprt which explains the prcess used t cnduct the 1 Cmmunicates linked mathematical ideas 16 Cmmunicates linked mathematical ideas lgically and clearly 0 Cmmunicates cmplete mathematical arguments lgically and in an apprpriate style survey Presents a reprt which apprpriately cnveys reasned views n the utcme f the survey relative t the hypthesis. Presents a cmpleted reprt which relates each chapter t the questin asked, which supprts the cnclusin by the use f apprpriate graphs and/r diagrams, and which links the cnclusin back t the hypthesis Presents a reprt which leaves the reader in n dubt that the cnclusin is the mst apprpriate ne fr the data cllected. The argument is t be cgent, reasned, supprted by clear and apprpriate graphs and/r diagrams, and the whle is presented as a lgical package linked t the hypthesis. 5

28 Numerical Skills Applying the crrect mathematical technique as required, and calculating utcmes accurately. Marks Achievement Criteria Assessment Schedule 3 Attempts t calculate Sme calculatins are made, but statistics apprpriately nt all the apprpriate frmulae are 6 Calculates sme statistics apprpriately 9 Calculate statistics apprpriately and with a measure f success 1 Crrectly calculates all statistics 15 Crrectly calculates all statistics, and crrectly interprets the results f their calculatins in relatin t the hypthesis used. Sme calculatins are made using mst f the apprpriate frmulae, but nt all calculatins are perfrmed crrectly. All frmulae are applied apprpriately, but nt all are calculated crrectly. All frmulae are applied apprpriately, all calculatins are crrect, but nt all results are related back crrectly t the hypthesis All frmulae are used apprpriately, all calculatins are crrect, and the interpretatins f the results are crrectly related back t the hypthesis. 6

29 Mderatin f Minr Research Task Marks Within-schl mderatin Where a schl has mre than ne class, taught by mre than ne teacher, there will need t be a system in place t ensure that all teachers mark the prjects cnsistently. This prcess shuld be straight frward, but shuld include: a meeting befre the prcess begins t share and bring clser tgether markers views n the interpretatin f the marking schedules, the marking f cmmn prjects (say 3 per 30 students) by all teachers. The marks are then cmpared and teachers adjust their judgements accrdingly fr cnsistency. a check marking prcess where ne teacher is elected t check mark a sample f prjects, say at a rate f 3 per 30 students, a meeting after the prcess t discuss any situatins which may have arisen during the marking and t make any adjustments t the marks. Between-cuntry mderatin Upn cmpletin f the within-schl mderatins, representative samples will be selected by the mderatin leader and sent t SPBEA fr a between-cuntry mderatin by an external mderatr. The external mderatr will advise mderatin leaders if any mark ranges within natinal distributins need adjusting. If any mark adjustments need t be made, schls in that cuntry will be advised by SPBEA f the size f the adjustments t be made. If n mark adjustments are required, schls will be advised t retain the marks apprved during the within-cuntry mderatin. Submissin f IA marks t SPBEA Every schl will send all its IA marks t reach SPBEA by the dates specified frm SPBEA. Every schl will frward the six (6) samples fr mderatin tgether with a cpy f the mark sheet t the PSSC crdinatr. 7

30 Checklist fr the Management f the Minr Research Task T be signed and returned t SPBEA when the raw marks are submitted. SCHOOL: COUNTRY: DATE: Steps taken Instructin sheet utlining prject and including due dates given t all students All students given advice, and each has chsen a suitable tpic Date cmpleted Signature f teacher Chapter 1 mnitring cmpleted Chapter mnitring cmpleted Cmpleted tasks marked accrding t the scheme Within-schl mderatin f marking (if required) carried ut Student interviews cmpleted Student mark verificatin sheets signed All tasks have been carried ut in accrdance with the spirit f the Teachers Manual. SIGNED: POSITION IN SCHOOL: 8

31 STUDENT: TITLE f TASK: Minr Research Task Student Mark Sheet TEACHER: HYPOTHESIS: CHAPTER ONE: The Tpic Due Date: Was this chapter cmpleted n time? Yes/N Is the data t be cllected numeric? Yes/N Is the prcess t cllect the data well described? Yes/N Is the prcess manageable? Yes/N Cmments: CHAPTER TWO: Randm Sampling and data cllectin Due Date: Was this chapter cmpleted n time? Yes/N Descriptin f sampling prcedure? Yes/N Randm number table included? Yes/N Descriptin f the field wrk dne? Yes/N List f data cllected? Yes/N Cmments: CHAPTER THREE: Numerical prcessing f data Was this chapter cmpleted n time? Yes/N Presentatin including cmpleteness Graphs? Yes/N Accuracy f calculatins Explanatins? Yes/N Cmments: Due Date: CHAPTER FOUR: Cnclusins Due Date: Calculatin f the C.I. Explanatin f C.I. as it applies t this prject Accuracy f evaluatins Validity f cnclusins Cmments: MARK ACCOUNTING: Infrmatin prcessing ut f 5 = Cmmunicatin ut f 0 = Numerical skills ut f 15 = Ttal ut f 60 = 9

32 Recrd f Student Minr Research Tasks Entries shuld be by date and signature. An example is shwn in the first line f the table. Student Task Title/Date Ch 1 CH CH 3 & 4 Raw mark/60 Mderated mark All cmpleted Paula Tti 6RU Heights f secndary schl students. Is the mean ver 1.54 m?

33 PSSC MATHEMATICS Task Outline Frm : Minr Research Task Year: Schl: Cuntry: (a) Brief descriptin f aim f task. (b) Brief descriptin f activities: (c) Expected utcmes f activities: (d) List f pssible areas/tpics 31

34 PSSC MATHEMATICS Task Outline Frm : Teacher Design Task Year: Schl: Cuntry: Task N. Title f Task: (a) Brief descriptin f task (aim, duratin, structure etc.) (b) Descriptin f activities (c) Expected utcmes f activities 3

35 Pacific Senir Secndary Certificate MATHEMATICS Internal Assessment Prgramme Summary Frm Cuntry: Schl: TASK Start Date End Date Ttal Mark Ttal Weight A. CAT 0 10% B. MINOR RESEARCH TASK: Chapter 1.Design and Hypthesis Chapter.Sampling and data cllectin Chapter 3. Data analysis 60 10% Chapter 4.Data interpretatin, making inferences and drawing cnclusins C. Teacher Design Tasks (TDT) % 3. Ntes: 1. Unless therwise stated, task utlines fr all tasks must be submitted tgether with this cmpleted IA Summary Frm. Fr Teacher Design Tasks (TDT) select THREE tasks with a cmbined ttal mark f 60 and cmbined ttal weight f 10%. Teachers are expected t devise their wn marking schemes fr these tasks. Teacher s Name: Teacher s Signature: Date: Principal s Name: Principal s Signature: Date: 33

36 PSSC Mathematics IA MARK CAPTURE FORM (Minr Research Task) MAT 1 Schl: Student Identificatin Name SPIN Ttal Teacher Mark (Out f 60) COMMENTS

37 PSSC Mathematics IA MARK CAPTURE FORM (Teacher Design Tasks) MAT Schl: Student Identificatin Name SPIN Ttal Teacher Mark (Out f 60) Cmments