Eurpean Schls Office f the Secretary-General Pedaggical Develpment Unit Ref. : 011-01-D-8-en- Orig. : EN MATHEMATICS SYLLABUS SECONDARY 5th YEAR 6 perid/week curse APPROVED BY THE JOINT TEACHING COMMITTEE ON 9, 10 AND 11 FEBRUARY 011 IN BRUSSELS Entry int frce in September 011 011-01-D-8-en- 1/14
ALGEBRA (fr guidance: 80 perids) TOPIC KNOWLEDGE & SKILLS USE OF TECHNOLOGY Abslute Value use the rules cncerning the sum, difference, prduct and qutient f abslute values slve equatins and inequalities f the kind ax b c 0, ax b c slve equatins and inequalities invlving the abslute value use graphs t slve equatins and inequalities Pwers and rts express the n th rts f a real number (and the pwers f these rts) as pwers with ratinal indices understand that finding the n th rt and raising t the n th pwer are inverse peratins understand that the rules f calculatin fr pwers where the index is a whle number extends t pwers where the index is a ratinal number use these prperties in prblems invlving expnential grwth and decay simplify algebraic expressins like: 1 1 16 1ab 3 9,,, 3a b 18ab 4 7ab explre values f x in simple expnential equatins such as 4 x x 1 t find pssible slutins use the tl t calculate expressins and verify slutins use a spreadsheet and a scatter plt t slve prblems invlving expnential grwth and decay slve and verify slutins f simple expnential equatins such as x 4 x1 011-01-D-8-en- /14
Simultaneus Equatins f the type: ax by cz d ex fy gz h ix jy kz l slve these systems algebraically frm and slve systems frm simple wrded prblems frm and slve a system f equatins t find the cefficients f the functin f a parabla passing thrugh three given pints fr simple cases slve these systems use the tl t fit a parabla given three pints use the tl t verify the equatin f the parabla Plynmials divide Px by Qx (where Qx is f first r secnd degree nly) understand and apply the remainder therem factrise a plynmial (easy cases, degree 4) apply the identities belw (and knw them) a 3 b 3 a ba ab b a 3 b 3 a ba ab b a b a 3a b 3ab b 3 3 3 find the zers f plynmials find the signs f plynmials simplify, add, subtract, multiply and divide ratinal fractins study the variatins in sign f a ratinal fractin P x Q x, where the degree f r equal t P x and Q x is less simplify expressins invlving divisins f plynmials factrise a plynmial find the zers f plynmials find the signs f plynmials simplify, add, subtract, multiply and divide ratinal P x fractins Q x 011-01-D-8-en- 3/14
Quadratic equatins and inequalities understand and apply the relatin between the cefficients f a quadratic equatin and its slutins: ax bx c 0 which can be expressed as b c xx 0 where and a a slve quadratic inequalities slve equatins which can be reduced t a quadratic equatin (e.g. a biquadratic equatin) [simple cases nly] slve wrded prblems which lead t equatins f the abve types slve quadratic inequalities algebraically and graphically slve equatins which can be reduced t a quadratic equatin Real functins use the idea f the dmain f definitin fr the fllwing functins plynmial functins f rder 3 r less f x ax b f x sin f x cs f x tan f x x x x ax b cx d fr functins f the type y ax b draw and recgnise their graphs find the zer and, where apprpriate, the y intercept verify that graphs have been successfully drawn find the zers and the y intercept graphically deduce frm a graph the equatins f the asympttes f a hyperbla draw the graphs f a large variety f the real functins 011-01-D-8-en- 4/14
fr functins f the type y sin x, y cs x and y tan x draw and recgnise their graphs understand the idea f the perid f these functins a fr rectangular hyperblas f the type y and x b ax b y cx d draw and recgnise their graphs determine that ax b A B cx d cx d give the equatin f the vertical and hrizntal asympttes find the zer and the y intercept, when apprpriate knw the behaviur f the graph at psitive and negative infinity knw the behaviur f the graph clse t the vertical asymptte determine the centre f symmetry draw and recgnise the graphs fllwing functins: g x ax b g x ax bx c g x ax b cx d find, where apprpriate, fr y g x fr the y g x the equatins f the asympttes the centre f symmetry 011-01-D-8-en- 5/14
the zer the y intercept sketch the graphs f different piecewise functins cmpsed f functins f the types g x ax b g x ax bx c g x ax b cx d 3x x 1 e.g. f( x) x² x 1 011-01-D-8-en- 6/14
PROBABILTY and STATISTICS (fr guidance: 5 perids) TOPIC KNOWLEDGE & SKILLS USE OF TECHNOLOGY Elementary ideas in prbability determine the universal set f pssible utcmes f a randm trial define an event A as sub-set f (cntaining ne r mre elements) shw the universal set in the frm f a Venn diagram Algebra f events express the events A B and A B in wrds and in set frm knw that, fr exclusive events, A B find a cmplementary event A Prbability understand fr the prbability f an event A that: 0P A 1 calculate the prbability f an event A understand the idea f prbability leading n frm relative frequency use the randm number generatr create by using a spreadsheet, the relative frequency f a virtual generated experiment and cmpare with the theretical prbabilities (e.g. dice simulatin) calculate prbability values as a fractin and as a decimal 011-01-D-8-en- 7/14
Cmbined events use fr independent events (sampling with replacement): ttal pssibility space diagrams Venn diagrams tree diagrams use fr cnditinal events (sampling withut replacement), tree diagrams (limit t up t three sets f branches) slve prblems which require the fllwing: P A P A 1 fr cmplementary events B +P P A 0 fr mutually exclusive events P A B P A P B P A B P A B P A B fr mutually exclusive events B P A 1 fr exhausitive events P A B P B P A B (use nly in tree diagrams) P A B P B PB ( A) P A B P A P B fr independent events use the tl in prbability prblems 011-01-D-8-en- 8/14
Analysis f data understand that the variance and the standard deviatin are a measure f spread calculate, fr a small size sample ( n 6 ), variance and standard deviatin, using ne f the fllwing x x x² ² n n x calculate frm frequency distributins r histgrams the variance and the standard deviatin calculate an estimate f the variance and the standard deviatin fr raw and gruped data x x x² ² n n x a x x ax² ² a a x calculate the variance and the standard deviatin fr raw and gruped data using a spreadsheet Interpretatin and cmparisn f data cmpare and interpret distributins with respect t: the means and variances/standard deviatins their given histgrams use a tl t calculate an estimate f the mean, variance and the standard deviatin cnstruct histgrams fr interpretatin and cmparisn 011-01-D-8-en- 9/14
GEOMETRY (fr guidance: 60 perids) TOPIC KNOWLEDGE & SKILLS USE OF TECHNOLOGY Oriented angles define the unit circle define an riented angle and represent it in the unit circle define radians in relatin t the arc length f a sectr cnvert radians t degrees and vice-versa estimate the size f an angle in radians and degrees cnvert radians t degrees and vice versa use cnstructins and measurements t verify the estimates f an angle Trignmetric Ratis find the trignmetric ratis f an riented angle and f angles assciated with it in bth degrees and radians knw the gemetric meaning f the trignmetric ratis state hw the trignmetric ratis f a signed angle vary state the trignmetric ratis f certain special angles cmpare the trignmetric ratis f angle with thse f its: cmplementary angle: cs sin 90 cs sin supplementary angle: sin sin 180 sin sin use graphs f the trignmetric functins t understand that ther angles will give the same trignmetric ratis as thse f the standard angles shw cmplementary and supplementary prperties dynamically using sliders d calculatins invlving trignmetric frmulae verify slutins t trignmetric equatins slve trignmetric equatins knw and apply frmulae f the type sin cs 1 011-01-D-8-en- 10/14
sin tan cs knw and apply (nly with numerical values f angles) frmulae f the type cs cs cs sin sin sin sin cs cs sin sin sin cs cs cs sin slve, fr 0 and equatins f the types: sin a, cs a and tan a simple equatins invlving these such as: 1 cs 6 and 3cs sin 1 0 Triangles state and prve the fllwing frmulae fr any triangle a b c bc cs A a b c r sin A sinb sinc 1 Area bc sin A apply the frmulae t determine angles and sides f triangles, including real life prblems verify these frmulae using cnstructin and measurements slve prblems invlving these frmulae Vectrs in the plane 011-01-D-8-en- 11/14
recgnise the fllwing: linearly dependent and independent vectrs, basis, crdinate system the dimensin f a vectr space define an rthnrmal basis express a vectr as a linear cmbinatin f tw given vectrs that frm a basis shw the bijectin which exists between the set f vectrs and the set f rdered pairs f real numbers Scalar prduct define and calculate the scalar prduct f tw vectrs define the scalar prduct f a vectr with itself define the magnitude f a vectr define the rthgnality f tw vectrs list and use the prperties f the scalar prduct express the scalar prduct f tw vectrs in terms f their magnitudes and the csine f the angle between them use the scalar prduct t verify rthgnality express a scalar prduct in an rthnrmal basis calculate the distance between tw pints use vectr methds fr gemetric prfs fr instance a b c r sin A sinb sinc 1 Area sin bc A calculate the scalar prduct f tw vectrs shw dynamically the prperties f the scalar prduct using cnstructin and measurements 011-01-D-8-en- 1/14
Lines in the plane give the vectr equatin f a straight line find a parametric equatin f a line find a cartesian equatin f a line determine the relative psitin f tw lines find parallel lines and recgnise parallel lines frm their equatins determine the equatin f a line parallel t a given line, passing thrugh a given pint find perpendicular lines and recgnise perpendicular lines frm their equatins determine the equatin f a line perpendicular t a given line, passing thrugh a given pint find the pint f intersectin between lines calculate the distance between parallel lines calculate the angle between tw intersecting lines determine the relative psitin f a pint and a line calculate the distance frm a pint t a line determine the crdinates f the perpendicular prjectin f a pint t a line transfrm a parametric equatin f a line int a cartesian equatin draw, given a pint and a line, the parallel line passing thrugh this pint draw, given a pint and a line, the perpendicular line passing thrugh this pint find the pint f intersectin between lines measure the distance between parallel lines measure the angle between tw intersecting lines measure the distance frm a pint t a line determine the crdinates f the perpendicular prjectin f a pint t a line Circles in the plane find the equatin f a circle find the equatin f the tangent at a pint n a circle relative psitin f a pint and a circle relative psitin f a circle and a line relative psitin f a circle and a circle calculate the pint(s) f intersectin f a line with a explre by cnstructing, the relative psitin f a circle and a line cnjecture graphically the equatin f the tangent t a circle frm a given pint determine using the dt prduct the equatin f the tangent t a circle frm a given pint slve simultaneus equatins 011-01-D-8-en- 13/14
circle: graphically by slving simultaneus equatins which can be reduced t a first degree equatin and a quadratic equatin 011-01-D-8-en- 14/14