Scheme of Work
Cambridge International AS & A Level Mathematics 9709/03 Pure Mathematics 3 P3! "or e#amination $rom %07 Contents Introduction''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' Introduction'''''''''''''''''''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' 3 ' Algebra'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' Algebra''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' ''''''''''' ( %' Logarithmic and e#)onential $unctions''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' $unctions'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' 9 3' *rigonometr+''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' *rigonometr+''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' % ,' -i$$erentiation''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' -i$$erentiation'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' ( (' Integration'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' Integration''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''' 9 .' umerical solution o$ euations''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' euations'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' %3 7' 1ectors''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' 1ectors''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' ''''''''''''''' %. 2' -i$$erential -i$$erential euations''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' euations'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '' %9 9' Com)le# numbers'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' numbers''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' 3%
Cambridge International AS & A Level Mathematics 9709! $rom %07
Scheme o$ 4or5
Introduction *his scheme o$ 6or5 )rovides ideas about ho6 to construct and deliver a course' It has been bro5en do6n into di$$erent units o$ the three subect areas o$ Pure Mathematics units P8 P% and P3!8 Mechanics units M and M%! and Probabilit+ & Statistics units S and S%!' "or each unit there are suggested teaching activities and learning resources to use in the classroom $or all o$ the s+llabus learning obectives' *his scheme o$ 6or58 li5e an+ other8 is meant to be a guideline8 o$$ering advice8 ti)s and ideas' It can never be com)lete but ho)e$ull+ )rovides teachers 6ith a basis to )lan their lessons' It covers the minimum reuired $or the Cambridge International AS & A Level course but also adds enhancement and develo)ment ideas' It does not ta5e into account that di$$erent schools ta5e di$$erent amounts o$ time to cover the Cambridge International AS & A Level course' *he mathematical content o$ Pure Mathematics 3 in the s+llabus is detailed in the tables belo6' *he order in 6hich to)ics are listed is not intended to im)l+ an+thing about the order in 6hich the+ might be taught' Recommended prior knowledge no6ledge o$ the content o$ unit P is assumed8 and candidates ma+ be reuired to demonstrate such 5no6ledge in ans6ering uestions' Candidates 6ill be e#)ected to be $amiliar 6ith scienti$ic notation $or the e#)ression o$ com)ound units8 e'g' ( m s $or ( metres )er second' As 6ell as demonstrating s5ill s5ill in the a))ro)riate techniues8 techniues8 candidates 6ill be e#)ected e#)ected to a))l+ their 5no6ledge 5no6ledge in the solution o$ )roblems' )roblems' Individual uestions set ma+ involve ideas and methods $rom more than one section o$ the relevant content list' Outline Suggestions $or inde)endent stud+ (I) and (I) and $ormative assessment (F) are (F) are indicated8 6here a))ro)riate8 6ithin this scheme o$ 6or5' *he activities in the scheme o$ 6or5 are onl+ suggestions and there are man+ other use$ul activities to be $ound in the materials re$erred to in the learning resource list' :))ortunities $or di$$erentiation are indicated as basic/consolidation and basic/consolidation and challenging/extension challenging/extension'' *here is the )otential $or di$$erentiation b+ resource8 length8 grou)ing8 e#)ected level o$ outcome8 and degree o$ su))ort b+ the teacher8 throughout the scheme o$ 6or5' *imings $or activities and $eedbac5 are le$t to the udgment o$ the teacher8 teacher8 according to the level level o$ the learners and si;e si;e o$ the class' Length o$ time time allocated to a tas5 is another another )ossible area $or di$$erentiation' di$$erentiation' Teacher support *eacher Su))ort http://teachers.cie.org.uk! is a secure online resource ban5 and communit+ $orum $or Cambridge teachers8 6here +ou can do6nload s)ecimen and )ast uestion )a)ers8 mar5 schemes and other resources' 4e also o$$er online and $ace
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Cambridge International AS & A Level Mathematics 9709! $rom %07
Scheme o$ 4or5
Resources *he u)
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Cambridge International AS & A Level Mathematics 9709! $rom %07
Scheme o$ 4or5
"# $lgebra %earning ob&ecti!es
'uggested teaching acti!ities
x
Understand the meaning of relations such as
x − a
a = b
and use
⇔ a = b and 2
2
< b ⇔a – b < x < a + b in the
course of solving equations and inequalities.
*o introduce the notation8 start 6ith a numerical value8 e'g' (8 and discuss the meaning o$ learners to deduce the results
a = b
⇔ a = b and 2
2
x − a
−5
' >ou could hel)
< b ⇔a – b < x < a + b as part of a class
discussion. *his lin5 leads to $our $iles 6hich are e#tremel+ use$ul' Clic5 on htt)s//666'tes'co'u5/teaching
ou could demonstrate some initiall+ to learners using a gra)h )lotter' (I) BAlternative Methods $or Solving Modulus ?uations is a 6or5sheet 6hich hel)s learners to e#)lore the di$$erent 6a+s o$ solving this t+)e o$ euation' (I) *he lin5 belo6 demonstrates the gra)hs o$ various modulus $unctions' htt)//666'mathsmutt'co'u5/$iles/mod'htm ast papers (I)(F) Dune %0, )a)er 3%8 uestion Dune %0, )a)er 3%8 uestion ovember %0, )a)er 338 uestion Dune %03 )a)er 38 uestion , involves logarithms! Dune %03 )a)er 3%8 uestion
ivide a pol!nomial" of degree not e#ceeding $" %! a linear or quadratic pol!nomial" and identif! the quotient and remainder &'hich ma! %e (ero).
*here are several di$$erent methods o$ )ol+nomial division including ins)ection8 the table method8 and long division' *his Po6erPoint )resentation introduces all three methods $or $actorising cubics' >ou can use the methods $or an+ )ol+nomial and also $or division that results in a remainder htt)//666'$urthermaths'org'u5/$iles/sam)le/$iles/ed#/"actorisingEcubics'))t 4hen teaching an+ o$ the methods8 start 6ith a numerical e#am)le to remind learners o$ the thought )rocess the+
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Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
Scheme o$ 4or5
'uggested teaching acti!ities
÷ 8 leads to a uotient o$ + 4 x + 1) ÷ ( x + 2) 6hich leads to a
need8 and use this to introduce the terms Buotient and Bremainder ' "or e#am)le 54763 .2,( and a remainder o$ 3' Continue 6ith a sim)le algebraic e#am)le
( x
+
2
uotient o$ x 2 and a remainder o$ −3 ' >ou 6ill )robabl+ need to sho6 learners $urther e#am)les involving more com)le# )ol+nomials be$ore the+ )ractise on their o6n'
*he lin5s belo6 )rovide ideas on )ossible a))roaches +ou can ta5e $or long division htt)s//666'5hanacadem+'org/math/algebra%/)ol+nomialEandErational/dividingE)ol+nomials/v/dividing<)ol+nomials< 6ithou 6ill $ind man+ use$ul uestions in te#tboo5s $or learners to )ractise'
Use the factor theorem and the remainder theorem" e.g. to *nd factors" solve pol!nomial equations or evaluate unkno'n coecients.
Summarise the 6or5 alread+ done on )ol+nomial division to sho6 that p x ! F divisor × uotient! G remainder' Sho6 that algebraic division can o$ten be avoided in uestions b+ substituting into p x ! the value o$ x that ma5es the divisor ;ero e'g' substituting 3 i$ the divisor is x 3 and calculating )3! to $ind the remainder!' Sho6 that the $actor theorem is a s)ecial case o$ the remainder theorem 6hen the remainder is ;ero' *he lin5 belo6 gives a good a))roach o$ this t+)e 6hich +ou could use 6ith a 6hole class' htt)s//666'mathsis$un'com/algebra/)ol+nomialsou could sho6 e#am)les involving $inding $actors8 solving )ol+nomial euations and evaluating un5no6n coe$$icients to the 6hole class8 uestioning learners individuall+ throughout' Hemind learners that the+ should sho6 all their 6or5ing as the use o$ a calculator $or $inding solutions to )ol+nomial euations 6ill not be acce)ted in an e#am' ere is a use$ul 6or5sheet 6hich covers basic use o$ the remainder theorem and evaluating un5no6n coe$$icients
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Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
Scheme o$ 4or5
'uggested teaching acti!ities log in $or $ree do6nload! htt)s//666'tes'co'u5/teaching
,ecall an appropriate form for e#pressing rational functions in partial fractions" and carr! out the decomposition" in cases 'here the denominator is no more complicated than: - ax G b!cx G d !ex G f ! -
ax G b!cx G d)2 ax G b! x 2 G c 2 !
?#am)les o$ the three main t+)es o$ )artial $raction are here log in $or $ree do6nload! htt)s//666'tes'com/teaching
and 'here the degree of the numerator does not e#ceed that of the denominator.
In man+ uestions8 the $irst )art 6ill involve brea5ing do6n rational $unctions into )artial $ractions and later )arts 6ill use )artial $ractions 6ith another mathematical techniue such as binomial e#)ansion8 integration or solving di$$erential euations' >ou can set learners uestions involving these to)ics 6hen the+ have covered them'
Use the e#pansion of & + x )n" 'here
Learners have alread+ met the binomial e#)ansion in unit P so8 to chec5 their understanding8 +ou could set them
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ast papers (I)(F) Dune %0, )a)er 38 uestion 9 includes a binomial e#)ansion! Dune %0, )a)er 338 uestion 2 includes integration! ovember %0, )a)er 38 uestion 9 includes a binomial e#)ansion! ovember %0, )a)er 3%8 uestion 9 includes a binomial e#)ansion! Dune %03 )a)er 38 uestion 3 Dune %03 )a)er 3%8 uestion 2 includes di$$erential euations!'
.
Cambridge International AS & A Level Mathematics 9709! $rom %07
Scheme o$ 4or5
%earning ob&ecti!es
'uggested teaching acti!ities
n is a rational num%er and x < &*nding a general term is not included" %ut adapting the standard
some )re)arator+ uestions on basic binomial e#)ansions using the $ormula (I)
( a + b)
n
8 6here n is a )ositive integer'
−1
2 − 1 x 2 series to e#pand e.g. included).
is
As5 learners to 6or5 out the $irst $e6 terms o$ the e#)ansion o$ n
( 1 + x ) = 1 + nx + obtain and $ractional )o6ers'
n ( n − 1) 2!
x2 +
n ( n − 1) ( n − 2) 3!
( 1 + x )
n
$rom the $ormula $or e#)anding
( a + b)
n
8 to
x3... *his is no6 in a use$ul $orm $or introducing negative
x < 1
*he tutorial at this lin5 sho6s that +ou need the condition $or negative )o6ers because the+ generate an in$inite series' *he $irst $e6 terms are onl+ a good a))ro#imation i$ the values o$ x meet this condition and the series converges' htt)//666'e#amsolutions'net/maths
*his lin5 uses an e#am)le 6ith n F 2 and has an interesting gra)hical dis)la+ o$ the a))ro#imation' htt)//666'intmath'com/series
2 − 1 x 2 >ou can demonstrate to learners ho6 to re<6rite e#am)les o$ the t+)e
1
x
as 2
4
1 −
÷
−1
so that the!
can go on to e#pand them. ast papers (I)(F) Dune %0, )a)er 38 uestion 9 includes )artial $ractions! Dune %0, )a)er 338 uestion % ovember %0, )a)er 38 uestion 9 includes )artial $ractions!
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Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
Scheme o$ 4or5
'uggested teaching acti!ities Dune %03 )a)er 38 uestion %
*# %ogarithmic and exponential +unctions %earning ob&ecti!es
'uggested teaching acti!ities
Understand the relationship %et'een logarithms and indices" and use the la's of logarithms &e#cluding change of %ase).
Start b+ de$ining the terms Blogarithm and Be#)onential8 lin5ing to the conce)t o$ indices' *o hel) learners
log x
=b
a understand a statement such as 8 +ou could describe it to them in 6ords such as J4hat )o6er o$ a is x K Ans6er b *his lin5 gives an introduction 6ith animation sho6ing the relationshi) bet6een logarithms and e#)onentials htt)//666')ur)lemath'com/modules/logs'htm' Learners should )ractise converting e#)ressions $rom logarithmic to e#)onential $orm and $rom e#)onential $orm to logarithmic' Most te#tboo5s 6ill have )lent+ o$ e#am)les o$ this t+)e'
A use$ul 6or5sheet is here includes the la6s o$ logarithms!' htt)//maths'm'edu'au/numerac+/6ebEmums/module%/4or5sheet%7/module%')d$ (I)
= b and loga y = c ' se targeted uestioning to ab +c = xy 8 encourage learners to 6rite the e#)onential $orms o$ these statements and reach the conclusion that log a xy = loga x + log a y. re6riting this in logarithmic $orm to obtain >ou could as5 learners to obtain the other t6o *o introduce the la6s o$ logarithms8 start 6ith statements
log a x
la6s in a similar 6a+' Learners 6ill then need to )ractise a))l+ing these la6s' *he lin5 belo6 )rovides eight $iles o$ notes8 6or5sheets and revision log in $or $ree do6nload!' htt)s//666'tes'co'u5/teaching
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Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
Scheme o$ 4or5
'uggested teaching acti!ities ast papers (I)(F) Dune %0, )a)er 38 uestion . ovember %0, )a)er 38 uestion
Understand the de*nition and properties of e x and 0n x " including their relationship as inverse functions and their graphs.
x
>ou could introduce the e#)onential $unction e in various 6a+s' :ne a))roach 6ould be using a gra)h )lotter to sho6 learners the gra)hs o$ various e#)onential $unctions
y = 2 x , y = 3 x , y = 5 x ' -evelo) the idea o$ a )articular e#)onential $unction 6hich lies bet6een x x y = 2 and y = 3 8 such that its gradient $unction is the same as itsel$' 4ith a suitable gra)h )lotter +ou could
e'g'
demonstrate that the gradient $unction o$
y = e x is e x '
*here are other8 $ormal8 a))roaches that +ou could use 6ith more ca)able learners' "or e#am)le +ou could consider
1 + 1 n÷ com)ound interest and the limit o$ the series
n
as sho6n at this lin5 htt)//666'mathsis$un'com/numbers/eou could encourage learners to obtain the logarithmic $orm o$ the statement e = a and so introduce them to natural logarithms' Nuilding on the 6or5 done on $unctions in unit P8 +ou could develo) this into the inverse x
x
relationshi) bet6een e and ln x and demonstrate the inverses on a gra)h )lotter' *he lin5 belo6 leads to an interactive e#ercise covering this relationshi) htt)//hotmath'com/hel)/gt/genericalg%/sectionE2E('html(I)
Use logarithms to solve equations of the form a x 1 b" and similar inequalities.
As a 6hole class e#ercise8 +ou could 6or5 through some e#am)les o$ increasing di$$icult+8 using care$ull+ directed uestioning to 6or5 through the solutions' *e#tboo5s 6ill include man+ e#am)les o$ this t +)e o$ uestion and the interactive e#ercise at the lin5 above includes some too' >ou could demonstrate e#am)les using ineualities8 6ith learners $inding critical values $irst and then deducing the
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Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
Scheme o$ 4or5
'uggested teaching acti!ities set o$ solutions' It is hel)$ul to highlight to learners the sign o$ ln x $or 0 the ineualit+ reverses'
< x ≤ 1 8 )erha)s through an e#am)le 6here
ast papers (I)(F) Dune %0, )a)er 3%8 uestion % Dune %0, Pa)er 338 uestion
Use logarithms to transform a given relationship to linear form" and hence determine unkno'n constants %! considering the gradient and/or intercept.
I$ +ou can relate this techniue to )ractical situations8 this could hel) learners 6hen the+ need to use it in their
y = Ab
x
y = Ax
b
scienti$ic subects' Common $orms o$ euation are and ' Learners 6ill need to be able to 6rite these euations in logarithmic $orm and hence relate them to the euation o$ a straight line' Sometimes the variables 6ill be letters other than x and y so learners need to s)ot the $orm o$ the euation in order to distinguish the variables $rom the constants'
y = Ax
b
*his lin5 )rovides a use$ul summar+ $or dealing 6ith situations involving htt)//mathbench'umd'edu/modules/miscEscaling/)age'htm >ou could 6or5 through this 6ith learners in class or the+ could stud+ it inde)endentl+'(I) >ou could use a similar
y = Ab x ' 4or5 through such an e#am)le in class8 ma5ing use o$ a gra)h )lotter
a))roach $or euations o$ the t+)e to demonstrate the straight line obtained#
*e#tboo5s 6ill )rovide learners 6ith man+ use$ul )ractice uestions' "or variet+8 tr+ to choose e#am)les 6hich involve variables other than x and y ' :$ten8 learners are as5ed to 6or5 $rom a given gra)h in straight line $orm'
y
ln y
Common errors involve learners considering values rather than values8 so the+ 6ill need to )ractise uestions to avoid such errors' *he P3 )ast e#am )a)ers have e#am)les o$ this t+)e' *o hel) rein$orce this )oint8 +ou could s)lit the learners into grou)s or )airs and as5 each o$ them to )re)are a uestion' *he easiest 6a+ 6ould be $or them to B6or5 bac56ards $rom a logarithmic relationshi) e'g' P = At ' b
?ach grou) could choose values $or A and b 8 6or5 out the coordinates o$ t6o )airs o$ coordinates and dra6 an a))ro)riate straight line gra)h' Learners could circulate their gra)hs around the other grou)s 6ho 6ould then identi$+ the logarithmic euations used to dra6 the gra)hs#
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Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
Scheme o$ 4or5
'uggested teaching acti!ities
ast papers (I)(F) Dune %03 )a)er 3%8 uestion 3
,# Trigonometr%earning ob&ecti!es
Understand the relationship of the secant" cosecant and cotangent functions to cosine" sine and tangent" and use properties and graphs of all si# trigonometric functions for angles of an! magnitude.
'uggested teaching acti!ities >ou could start b+ de$ining the secant8 cosecant and cotangent $unctions' Learners should 5no6 the gra)hs o$ the sine8 cosine and tangent $unctions so8 as a grou) or individual tas58 +ou could as5 them to thin5 6hat the gra)hs o$ the secant8 cosecant and cotangent $unctions 6ould loo5 li5e' "or instance8 +ou could give them the gra)h o$ y F sin x $rom <3.0O to 7%0O! and as5 them to s5etch y F cosec x on the same a#es' *hen the+ could chec5 using a gra)h )lotter' A similar gra)hical a))roach could be used $or y F sec x and y F cot x ' ast papers (I)(F) Dune %0, )a)er 38 uestion 2
Use trigonometrical identities for the simpli*cation and e#act evaluation of e#pressions and" in the course of solving equations" select an identit! or identities appropriate to the conte#t" sho'ing familiarit! in particular 'ith the use of: 2 2 –– sec θ ≡ 1 + tan θ and cosec2θ ≡ 1 + cot 2 θ
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>ou could start 6ith the identit+ sin θ
+ cos2 θ ≡ 1 6hich learners 5no6 alread+! and as5 6hat the+ $ind 6hen a! 2
2
the+ divide each term in this identit+ b+ cos θ and b! the+ divide each term in the original identit+ b+ sin θ . *he lin5 belo6 )rovides t6o $iles8 one o$ 6hich is a matching e#ercise and the other a 6or5sheet $or learners to com)lete as consolidation and )ractice log in $or $ree do6nload! htt)s//666'tes'com/teaching
Cambridge International AS & A Level Mathematics 9709! $rom %07
Scheme o$ 4or5
%earning ob&ecti!es
'uggested teaching acti!ities
–– the e#pansions of sin& A 2 B)" cos& A 2 B) and tan& A 2 B) –– the formulae for sin 3 A" cos 3 A and tan 3 A –– the e#pressions of a sin θ + b cos θ in the forms R sin( θ ± α ) and R cos(θ ± α ) .
the le$t hand side! and mani)ulate it using the identities covered so $ar' *e#tboo5s 6ill include some )ractice uestions' *his lin5 )rovides an e#ercise on sim)li$ication' htt)//6or5sheets'tutorvista'com/)roving
usuall+ uadratic! in terms o$ one trigonometric ratio e'g' 2sec θ
− 3 + tan θ = 0 6ill sim)li$+ to
2 tan θ + tan θ − 1 = 0 6hich $actorises' 2
"or the com)ound angle addition! $ormulae8 it is a good idea to 6or5 through an e#am)le o$ ho6 one $ormula is derived8 )erha)s as a 6hole class e#ercise' A video )roo$ is here htt)s//666'+outube'com/6atchKvFa0Lv$lM#, *he lin5 belo6 covers the )roo$ o$ one $ormula in a similar 6a+' As an e#ercise $or more ca)able learners8 +ou could as5 them to 6or5 out the )roo$s o$ some o$ the other $ormulae' htt)//666'trans,mind'com/)ersonalEdevelo)ment/mathematics/trigonometr+/com)oundAngleProo$s'htmQmo;*ocId .9.0% Alternativel+8 +ou could start b+ giving learners the challenge o$ deriving the com)ound angle $ormulae gra)hicall+ using this interesting investigation htt)s//666'tes'co'u5/teaching
cos θ = 2
1' .>0
1 2
( 1 + cos 2θ )
sin θ = 2
and
1 2
( 1 − cos 2θ ) $or use in other a))lications such as integration'
%
Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
Scheme o$ 4or5
'uggested teaching acti!ities *e#tboo5s include man+ use$ul )ractice e#ercises on solving euations using the com)ound and double angle $ormulae' >ou should ensure that learners are )ro$icient at using radians as 6ell as degrees' (I) *his lin5 gives a clear summar+ o$ ho6 to deal 6ith e#)ressions o$ the t+)e a sin θ htt)//666'intmath'com/anal+tic
+ b cos θ
o + 4 cos θ and sho6 that it ma+ be 6ritten in the $orm 5 sin ( θ + 53.13 ) ' *his
could also be veri$ied b+ use o$ a gra)h )lotter sho6 learners the gra)h o$ y = 3 sin + 4 cos and8 6ith a discussion on trans$ormations8 +ou could encourage learners to 6rite this e#)ression in a di$$erent 6a+' *he+ can chec5 the result b+ )lotting the euivalent e#)ression and seeing that it gives the same gra)h'
θ
θ
As5 learners to $ind the ma#imum and minimum values o$ the e#)ression and the values o$ θ at 6hich the+ occur' >ou should discourage the use o$ calculus $or uestions o$ this t+)e'! *e#tboo5s include man+ e#am)les o$ 6riting euivalent e#)ressions8 solving euations and $ inding ma#imum and minimum values' Learners 6ill need to be )ro$icient at using radians as 6ell as degrees' (I) ast papers (I)(F) Dune %0, )a)er 38 uestion and uestion 2ii! Dune %0, )a)er 3%8 uestion 3 Dune %0, )a)er 338 uestion 3 ovember %0, )a)er 38 uestion 2 ovember %0, )a)er 338 uestion , Dune %03 )a)er 38 uestion 9 also involves integration! Dune %03 )a)er 3%8 uestion 7 Dune %03 )a)er 338 uestion 3 and uestion ,
1' .>0
3
Cambridge International AS & A Level Mathematics 9709! $rom %07
Scheme o$ 4or5
.# i++erentiation %earning ob&ecti!es
'uggested teaching acti!ities
Use the derivatives of e x " 0n x " sin x " cos x " tan x " together 'ith constant multiples" sums" di4erences and composites.
It is )robabl+ best to teach this section using a 6hole class a))roach and targeted uestioning o$ learners' "or the
y = e 8 learners alread+ 5no6 that the gradient $unction is e x so +ou can build on this b+ di$$erentiating f x y = e mx , y = e ( ) 8 ma5ing use o$ the chain rule 6here a))ro)riate' *o di$$erentiate y = ln x other $unctions such as x
$unction
d x 8 6rite x = e 8 so d y y
= e y d d x
d y and +ou can obtain the result d x
( ln f ( x ) ) =
f ' ( x ) f ( x )
=
1
x ' sing the chain rule8 +ou can generalise to
.
e#)ressions o$ the $orm *e#tboo5s 6ill have e#ercises $or learners to )ractice' (I) *o obtain the derivatives o$ sin x and cos x 8 +ou could consider the gradient o$ a chord $rom the origin to a )oint h8 sin h! on the curve y F sin x ' As5 learners to calculate the gradient sin h / h 6here h is 0' then 0'0 then 0'00! and use this to deduce the gradient at x F 0' *he+ can deduce the gradient at other 5e+ )oints on the gra)h8 $or instance x F 08 π /%8 π, 3π /%8 %π 8 use their values to )lot the gradient $unction on a gra)h o$ y F sin x and name the gra)h obtained' Sho6 them that a similar a))roach 6ill give them the gradient $unction $or y F cos x ' >ou can $ind this method in man+ te#tboo5s' It is also covered at the lin5 belo68 together 6ith di$$erentiation $rom $irst )rinci)les 6hich is suitable as an e#tension $or the more ca)able learner htt)//666'mathcentre'ac'u5/resources/u)loaded/mcou could encourage learners to obtain results $or the derivatives o$ sin mx 8 cos mx 8 sin $ x ! and cos $ x ! during a class discussion8 ma5ing use o$ the chain rule' Leave the di$$erentiation o$ y F tan x until the uotient rule has been covered' Man+ te#tboo5s 6ill have e#ercises $or learners to )ractice' (I)
1' .>0
,
Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
Scheme o$ 4or5
'uggested teaching acti!ities ast papers (I)(F) Dune %0, )a)er 3%8 uestion 2 Dune %0, )a)er 338 uestion 9 Dune 303 )a)er 3%8 uestion . Dune %03 )a)er 338 uestion 9
i4erentiate products and quotients.
It 6ould be an advantage to derive the )roduct and uotient rules as a 6hole class e#ercise so that learners es)eciall+ the more ca)able! can understand the $ormulae more thoroughl+' *here is a )roo$ here using $unction notation htt)//nrich'maths'org/002.' Alternativel+8 +ou can 6rite the )roduct as uv 6here u and v are $unctions o$ x ! then consider increasing the area o$ a rectangle uv to u G δu!v G δv !' ?#)anding the brac5ets8 6riting ever+ term over δx and considering the limit as δx
y = >ou could set learners the tas5 o$ deriving the uotient rule b+ di$$erentiating y uv−1 8 using the )roduct rule' o$ x 8 as a )roduct
u v 8 6here u and v are $unctions
=
As5 learners to di$$erentiate y
= tan x using the uotient rule'
*he lin5 belo6 gives three $iles 6hich include e#am)les/6or5sheets on di$$erentiation o$ uotients log in $or $ree do6nload! htt)s//666'tes'co'u5/teaching
1' .>0
(
Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
Scheme o$ 4or5
'uggested teaching acti!ities Dune %0, )a)er 38 uestion 0 Dune %03 )a)er 38 uestion (
5ind and use the *rst derivative of a function 'hich is de*ned parametricall! or implicitl!.
>ou can introduce the idea o$ )arametric euations to learners b+ as5ing them to imagine t 6o cars moving to6ards each other along di$$erent straight lines on the x ou 5no6 their lines 6ill intersect but ho6 do +ou 5no6 i$ the cars 6ill collide or miss each otherK >ou need to consider a third )arameter e'g' time!8 and e#)ress both x and y in terms o$ this )arameter8 in order to sa+ 6hether or not there 6ill be a collision'
x = 2t , y = 3t 2 + 5
*hen +ou can sho6 learners some sim)le e#am)les e'g' and eliminate t to obtain the Cartesian $orm o$ the curve' A gra)h )lotter ma+ be use$ul' Sho6 that the gradient $unction ma+ be obtained using the
d x
d y
derivatives dt and dt together 6ith the chain rule' ?#tend the 6or5 to include )arametric euations involving
= 3cos2θ ,
= 4sin θ
x trigonometric $unctions e'g' y to hel) learners to consolidate their 5no6ledge o$ trigonometric identities and di$$erentiation o$ trigonometric $unctions' *his lin5 gives a clear and thorough treatment o$ the to)ic 6ith 6or5ed e#am)les' See 7' Cartesian and )arametric euations and 7', Parametric di$$erentiation htt)//666'cimt')l+mouth'ac'u5/)roects/me)res/alevel/)ureEch7')d$ *he lin5 belo6 )rovides a good overvie6 o$ the to)ic second derivatives are not reuired! htt)//666'mathcentre'ac'u5/resources/u)loaded/mc
y
As5 learners to consider e'g'
1' .>0
2
= x 8 re<6rite it as y = x
1 2
d y then di$$erentiate to obtain d x
1
= x 2
−
1 2
' *he+ can re<
.
Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
Scheme o$ 4or5
'uggested teaching acti!ities
d y 6rite this as d x
=
1 2 y leading to the statement
2 y
d y d x
= 1.
He)eat this e#ercise 6ith several similar e#am)les )o6ers o$ y ! so that learners can identi$+ a )attern' *his e#ercise could be done 6ith the 6hole class or 6ith grou)s' Sho6 learners terms o$ various t+)es the+ no6 5no6 ho6 to di$$erentiate )o6ers o$ x or y 6ith res)ect to x ' >ou can introduce the idea o$ a )roduct term b+ as5ing them to di$$erentiate euations such as using the )roduct rule and b+ rearranging them and di$$erentiating y 6ith res)ect to x '
xy = x 8 x 2 y3
= 4 im)licitl+
>ou can no6 as5 learners to 6or5 through an euation $rom le$t to right and di$$erentiate it im)licitl+ 6ithout rearranging it $irst' It is a good idea to give them euations 6hich cannot be rearranged to )revent this'! *he lin5s belo6 )rovide use$ul e#am)les or 6or5sheets htt)s//666'5hanacadem+'org/math/di$$erential
1' .>0
7
Cambridge International AS & A Level Mathematics 9709! $rom %07
Scheme o$ 4or5
0# Integration %earning ob&ecti!es
'uggested teaching acti!ities
6#tend the idea of 7reverse di4erentiation8 to include the 1 ax + b integration of e " ax + b " sin ( ax + b )
Start 6ith a uic5 revie6 o$ integration $rom unit P8 )erha)s as a uestion and ans6er session 6ith learners 6riting on mini 6hiteboards and holding u) their res)onses' *his 6ill enable +ou to assess all learners understanding be$ore moving on to e#am)les in this section'
sec 2 ( ax + b ) " and &kno'ledge of the general method of integration %! su%stitution is not required). cos( ax + b )
>ou could divide learners into grou)s and give them sets o$ e#)ressions to integrate' As5 them to consider 6hat 6ould need to be di$$erentiated to obtain the given e#)ression8 then to 6or5 out some general )rinci)les' *he $ollo6ing lin5 )rovides a good a))roach $or integration involving logarithmic $unctions' Some o$ the e#am)les ma+ be be+ond the range o$ this s+llabus' htt)//666'mathcentre'ac'u5/resources/u)loaded/mc
Use trigonometrical relationships &such as dou%le-angle formulae) to facilitate the integration of functions 2 such as cos x .
2
As5 learners to recall the three $orms o$ the trigonometric identit+ $or cos2 x and then to use them to re6rite cos x 2
and sin x in terms o$ cos2 x '
∫ 2sin x cos x dx ∫ cos
2
2 x dx
∫ tan
2
3 x + 1 dx
Introduce learners to integrals o$ the t+)e 8 and ' A))ro)riate te#tboo5s 6ill have e#am)les o$ these' *r+ to relate them to areas and also to sim)le $irst order
d y di$$erential euations8 $or e#am)le $ind the euation o$ the curve8 6ith gradient $unction d x π π 0 ≤ x < x = 2 8 6hich )asses through the )oint 4 (I)
0ntegrate rational functions %! means of decomposition into partial fractions
1' .>0
= 2 sec2 x + 1 $or
>ou could cover this section 6ith the section on )artial $ractions see ' Algebra! or later8 )erha)s chec5ing learners understanding b+ setting them some )re)arator+ uestions involving linear denominators' (I)
2
Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
&restricted to the t!pes of partial fractions speci*ed in paragraph a%ove).
Scheme o$ 4or5
'uggested teaching acti!ities
∫
a
ax + b N+ considering the integration o$ 6hich deals 6ith non
d x to
ln ( ax + b )
+ c 8 +ou can sho6 learners the lin5 to the ne#t section
Learners 6ill need to )ractise de$inite integrals o$ this t+)e8 using la6s o$ logarithms to sim)li$+ their ans6ers 6hen a))ro)riate' >ou 6ill need to cover the la6s $rom section %' Logarithms and e#)onentials $irst!' *e#tboo5s 6ill have man+ suitable uestions $or learners to )ractise' ast papers (I)(F) Dune %0, )a)er 38 uestion 9 ovember %0, )a)er 38 uestion 9
,ecognise an integrand of the form k f ′( x ) f ( x )
and integrate" for e#ample"
x x + 1 or tan x. 2
As a 6hole class e#ercise8 +ou can e#tend the 6or5 done in the )revious section b+ considering di$$erent e#am)les
k f ′( x ) o$ the $orm f ( x ) 'here f& x ) is non-linear.
9ou 'ill *nd some useful e#amples" some of 'hich relate to ph!sical situations" at this link: htt)//666'intmath'com/methods
,ecognise 'hen an integrand can usefull! %e regarded as a p roduct" and use. integration %! parts to integrate"
>ou could challenge able learners to start 6ith the )roduct rule and see i$ the+ can derive a $ormula $or integrating a )roduct' *he+ ma+ need some hints to rearrange the )roduct rule then integrate all the terms 6ith res)ect to x ' *he resource at this lin5 includes the derivation as 6ell as reasons $or using the $ormula8 and a set o$ uestions
1' .>0
9
Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
'uggested teaching acti!ities
2 x for e#ample" x sin 3 x " x e or 0n x.
htt)//666'mathcentre'ac'u5/resources/u)loaded/mc
Scheme o$ 4or5
ln ( 2 x +1)
' See i$ the+ can
*he lin5 belo6 leads to a 6or5sheet and solutions that ma+ be used $or )ractice or consolidation log in $or $ree do6nload! htt)s//666'tes'co'u5/teaching
Use a given su%stitution to simplif! and evaluate either a de*nite or an inde*nite integral.
>ou ma+ 6ish to start 6ith a sim)le e#am)le 6hich can easil+ be chec5ed b+ other means e'g'
∫ ( 2 x + 5)
3
dx
b+
using the substitution u = 2 x + 5 or b+ e#)anding $irst' *his 6ill give learners con$idence that the right substitution 6ill 6or5' *e#tboo5s contain man+ use$ul e#am)les o$ both inde$inite and de$inite integrals' *here are also e#am)les at this lin5 htt)//666'mathcentre'ac'u5/resources/6or5boo5s/mathcentre/6eb
1' .>0
%0
Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
Scheme o$ 4or5
'uggested teaching acti!ities *he lin5 belo6 )rovides a 6or5sheet 6hich learners can use $or )ractice and consolidation log in $or $ree do6nload! htt)s//666'tes'co'u5/teaching
Use the trape(ium rule to estimate the value of a de*nite integral" and use sketch graphs in simple cases to determine 'hether the trape(ium rule gives an over-estimate or an underestimate.
>ou could start b+ s5etching on the board )art o$ a curve 6ith an un5no6n euation' As5 learners to consider the area under this curve8 enclosed b+ the x
1' .>0
%
Cambridge International AS & A Level Mathematics 9709! $rom %07
Scheme o$ 4or5
$
<
$
$
1# 2umerical solution o+ e3uations
<
%earning ob&ecti!es
'uggested teaching acti!ities
ocate appro#imatel! a root of an equation" %! means of graphical considerations and/or searching for a sign change.
>ou could introduce this to)ic b+ using a gra)h )lotter to demonstrate both sign changes and gra)hical
$
<
considerations e'g' Change o$ sign
y = x2 + 5 x + 1 =
+ 5 x + 1 = 0 in the intervals −5 < x < −4 y and −1 < x < 0. Learners consider the sign o$ either side o$ the )oints o$ intersection o$ the curve 6ith the x < In this 6a+8 +ou can see clearl+ that there are solutions to the euation x
2
a#is i'e' using the boundaries above' -emonstrate also that the same result ma+ be obtained b+ )lotting
1' .>0
y = x2 against y = −5 x − 1 '
%%
Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
Scheme o$ 4or5
'uggested teaching acti!ities
Learners 6ill need to )ractise e#am)les o$ both t+)es' >ou should encourage them to set out their 6or5 clearl+ and accuratel+' "or e#am)le8 to sho6 that the euation x
f ( x ) = x + 5x + 1
2
= −5 x − 1 has a solution in the interval −5 < x < −4 8
2
learners should state BLet the values o$
f ( −5 )
and
f ( −4 )
then 6rite the euation as
f ( x ) = 0
' N+ calculating and 6riting do6n
8 the+ can demonstrate that there is a sign change and state their conclusion e'g' B
*here is a change o$ sign8 so a solution lies in the interval
−5 < x < −4 '
*his lin5 includes a use$ul overvie6 o$ the to)ic8 6ith e#am)les htt)//666'cimt')l+mouth'ac'u5/)roects/me)res/alevel/)ureEch9')d$
Understand the idea of" and use the notation for" a sequence of appro#imations 'hich converges to a root of an equation.
*he second )art o$ this cha)ter deals 6ith convergence to a root o$ an euation' htt)//666'cimt')l+mouth'ac'u5/)roects/me)res/alevel/)ureEch9')d$
Understand ho' a given simple iterative formula of the form x n + 1
*here is a video tutorial here 6hich students could 6atch inde)endentl+ or +ou could use it 6ith a 6hole class htt)s//666'tes'com/teaching
1' .>0
*he $irst )art o$ this lin5 demonstrates a $ormal a))roach to the idea o$ a seuence o$ a))ro#imations converging to a root o$ an euation' >ou could use it 6ith able learners or )erha)s 6ith a 6hole class' It e#)lains ho6 an iterative $ormula generates the seuence= this is the ne#t learning obective' htt)//666
%3
Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
5& x n) relates to the equation %eing solved" and use a given iteration" or an iteration %ased on a given rearrangement of an equation" to determine a root to a prescri%ed degree of accurac! &kno'ledge of the condition for convergence is not included" %ut candidates should understand that an iteration ma! fail to converge).
Scheme o$ 4or5
'uggested teaching acti!ities
Iterative $ormulae are covered in the cha)ter alread+ lin5ed' It includes e#am)les and activities $or learners to tr+ htt)//666'cimt')l+mouth'ac'u5/)roects/me)res/alevel/)ureEch9')d$ It is a good idea $or learners to ma5e $ull use o$ their calculator $or the iteration )rocess' sing the AS ans6er! 5e+ 6ill save them time in $inding a root o$ an euation' ere is an e#am)le and the method used to $ ind a root'
xn +1
= 3−
e'g' sing the iterative $ormula $inal ans6er to 3 decimal )laces'
1
xn
6ith
x0
= 3 8 sho6 successive iterations to ( decimal )laces and a
x
•
Start b+ entering the value o$ 0 into the calculator )ress 3 then BFB or Benter8 de)ending on the calculator!8 so 3 a))ears as an ans6er'
•
e+ in the right hand side o$ the iterative $ormula8 re)lacing
xn
6ith AS or the 5e+ that dis)la+s a )revious
3 − (1 ÷ AN) ' *he calculator 6ill dis)la+ %'........7' 4rite this do6n to ( decimal )laces' ans6er! i'e' •
•
ee) )ressing the BF 5e+ and successive iterations 6ill a))ear' 4rite do6n as man+ as the uestion reuires8 all correct to ( decimal )laces' %'.%(00 %'.90( %'.22 %'.20. %'.20, %'.203 >ou have no6 done enough iterations to sho6 that an ans6er o$ %'.2 is correct to 3 decimal )laces'
Learners 6ill need )ractice at entering the correct $ormula into their calculator8 using brac5ets 6here necessar+' ast papers (I)(F) Man+ o$ these uestions 6ill involve other )arts o$ the s+llabus as 6ell as other )arts o$ this section' Dune %0, )a)er 38 uestion 2 Dune %0, )a)er 3%8 uestion . Dune %0, )a)er 338 uestion ,
1' .>0
%,
Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
Scheme o$ 4or5
'uggested teaching acti!ities ovember %0, )a)er 38 uestion . ovember %0, )a)er 338 uestion 9 Dune %03 )a)er 3%8 uestion % Dune %03 )a)er 338 uestion .
4# 5ectors %earning ob&ecti!es
'uggested teaching acti!ities
Understand the signi*cance of all the s!m%ols used 'hen the equation of a straight line is e#pressed in the form r 1 a + t b.
>ou could start b+ as5ing learners to use )osition vectors to $ind the vector euation o$ a straight line i$ the line )asses through a )oint 6ith )osition vector a and is )arallel to a vector b' *his 6ill give them the idea o$ um)ing $rom the origin to the line8 then moving along it' As5 learners 6hat it means to choose di$$erent values o$ the scalar t and rein$orce the conce)t o$ the line as a set o$ )oints8 each o$ 6hich is described in this $orm r F )osition vector o$ a )oint on the line! G t direction vector o$ the line! 4or5ing in three dimensions ma+ hel) learners to see 6h+ the+ need a vector euation $or a line y F mx G c is not enough and vectors are a )o6er$ul tool' *he lin5s belo6 give use$ul introductor+ e#am)les htt)//666$'im)erial'ac'u5/metric/metricE)ublic/vectors/vectorEcoordinateEgeometr+/vectorEeuationEo$Eline'html htt)//666'cimt')l+mouth'ac'u5/)roects/me)res/alevel/$)ureEch(')d$ *he second lin5 6ill also be use$ul in the $ollo6ing sections' Learners can )ractise using this $orm o$ the euation' *he lin5 belo6 leads to three $iles= the $ile B1ector euation o$ a line )rovides e#am)les o$ this t+)e in t6o and three dimensions log in $or $ree do6nload! htt)s//666'tes'co'u5/teaching
etermine 'hether t'o lines are parallel" intersect or are ske'.
1' .>0
"rom the vector euation o$ the line8 +ou can as5 learners ho6 the+ could determine 6hether lines are )arallel' *r+
%(
Cambridge International AS & A Level Mathematics 9709! $rom %07
%earning ob&ecti!es
Scheme o$ 4or5
'uggested teaching acti!ities
r
giving some e#am)les o$ vector euations in di$$erent $orms8 e'g' the line
r
2 5 = 1÷÷ + λ 6÷÷ 3÷ 7÷
is )arallel to the line
−3 + 10 µ = 12 µ ÷÷ 1 + 14 µ ÷ = this is eas+ to see i$ learners re<6rite the second one in the same $orm'
"or intersecting lines8 there is some value o$ λ and μ that satis$ies all three euations $or the vector com)onents x 8 y and z ' *here is an e#am)le at this lin5 and +ou ma+ $ind Activit+ % use$ul= learners 6ill need to decide 6hich )airs o$ lines intersect htt)//666'cimt')l+mouth'ac'u5/)roects/me)res/alevel/$)ureEch(')d$ Learners can o$ten $ind s5e6 lines di$$icult to visualise8 so +ou could sho6 them an image Uoogle Bs5e6 lines! or illustrate the geometr+ b+ holding u) t6o long rulers' "rom one direction8 the rulers loo5 as though the+ are intersecting in a )lane8 but $rom a )er)endicular direction the+ are clearl+ not' *his 6ill tie in 6ith solving euations +ou can $ind values $or λ and μ $rom t6o o$ the euations a )lane! but the values do not $it the third euation 3rd dimension!'
5ind the angle %et'een t'o lines" and the point of intersection of t'o lines 'hen it e#ists.
Learners 6ill have used the scalar )roduct in unit P' >ou could set them some )re)arator+ uestions to chec5 the+ can use it to $ind angles bet6een vectors and to determine 6hether lines are )er)endicular' (I) *o $ind the coordinates o$ the )oint o$ intersection8 learners ust need to substitute their value o$ λ or μ to $ind the )osition vector and hence coordinates o$ the )oint' *he lin5s belo6 )rovide $urther e#am)les and uestions 6hich +ou ma+ $ind use$ul htt)s//666'tes'co'u5/teaching
1' .>0
%.
Cambridge International AS & A Level Mathematics 9709! $rom %07
Scheme o$ 4or5
%earning ob&ecti!es
'uggested teaching acti!ities
Understand the signi*cance of all the s!m%ols used 'hen the equation of a plane is e#pressed in either of the forms ax + by + cz 1 d or &r – a).n 1 > .
>ou could start b+ as5ing learners 6hat is the minimum in$ormation the+ 6ould need to de$ine a line or a )lane' "or instance8 to de$ine a line t6o )oints on it are needed= to de$ine a )lane three nonou can encourage them to dra6 diagrams or e#)eriment 6ith a )encil vector! and )iece o$ )a)er )lane!8 steering the discussion to6ards considering the vector normal to the )lane' >ou could go on to derive the vector euation o$ the )lane 6ith them' *he lin5 belo6 is use$ul $or relating the di$$erent $orms o$ the euation o$ a )lane and has e#am)les $or learners to tr+' htt)//666'cimt')l+mouth'ac'u5/)roects/me)res/alevel/$)ureEch(')d$
Use equations of lines and planes to >ou 6ill $ind diagrams ver+ use$ul $or hel)ing learners to visualise various geometrical situations' *his lin5 gives solve pro%lems concerning distances" some diagrams sho6ing the angle bet6een )lanes8 )er)endicular and )arallel )lanes htt)//666'emathematics'net/angle)lane)lane')h) angles and intersections" and in *here are man+ more diagrams on the internet' particular: - *nd the equation of a line or a Alternativel+8 6or5ing in )airs or grou)s8 learners could use )ieces o$ )a)er and )encils to simulate )lanes and plane" given sucient normals and to illustrate various geometrical relationshi)s' information - determine 'hether a line lies 4or5 through e#am)les o$ di$$erent geometrical )roblems' Learners 6ill need to )ractise as man+ uestions as in a plane" is parallel to a )ossible so that the+ become )ro$icient at deciding ho6 to tac5le a uestion using the techniues the+ 5no6' plane" or intersects a plane" and *nd the point of intersection of a line and a *he lin5 belo6 )rovides a 6or5sheet 6ith e#am)les involving )lanes log in $or $ree do6nload! plane 'hen it e#ists htt)s//666'tes'co'u5/teaching
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Cambridge International AS & A Level Mathematics 9709! $rom %07
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planes" and the angle %et'een a line and a plane.
Scheme o$ 4or5
'uggested teaching acti!ities ovember %0, )a)er 338 uestion 7 Dune %03 )a)er 38 uestion . Dune %03 )a)er 3%8 uestion 0 Dune %03 )a)er 338 uestion 0
6# i++erential e3uations %earning ob&ecti!es "ormulate a sim)le statement involving a rate o$ change as a di$$erential euation8 including the introduction i$ necessar+ o$ a constant o$ )ro)ortionalit+'
'uggested teaching acti!ities
d y
y >ou could start b+ as5ing learners to thin5 about the gradient $unction8 d x 8 as the rate o$ change o$ 6ith res)ect d s to x ' sing other variables8 +ou can introduce them to other rates o$ change such as dt 8 the rate o$ change o$ a variable s 6hich could re)resent distance! 6ith res)ect to a variable t 6hich could re)resent time!' *his lin5 )rovides an introduction to $orming di$$erential euations and solving $irst order di$$erential euations8 including some real li$e e#am)les htt)//666'slideshare'net/davidmiles00/core<,ou could use it to chec5 their understanding htt)s//666'tes'co'u5/teaching
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5ind %! integration a general form of solution for a *rst order di4erential equation in 'hich the varia%les are separa%le.
Scheme o$ 4or5
'uggested teaching acti!ities
d y >ou could start 6ith a sim)le di$$erential euation e'g' d x
= x 2 and as5 learners to $ind a solution to this' Introduce
d y the term Bgeneral solution and em)hasise the im)ortance o$ the constant o$ integration' N+ considering d x +ou could lead on to the idea o$ se)arating variables'
= y 2 8
Learners 6ill bene$it $rom )ractice at se)arating variables8 so +ou 6ill need to give them a good variet+ o$ uestions on this be$ore the+ move on to solving euations o$ greater com)le#it+' >ou can use this section to hel) learners re
the la6s' *he+ also need to be a6are that terms such as e
86here c is a constant8 ma+ be 6ritten in the $orm
x
Ae 8 and to )ractise re<6riting solutions in the $orm reuired $or each uestion' At the lin5 belo6 +ou 6ill $ind three $iles8 one o$ 6hich contains notes and e#am)les on general solutions o$ $irst order di$$erential euations log in $or $ree do6nload! htt)s//666'tes'co'u5/teaching
Use an initial condition to *nd a particular solution.
Introduce the idea that a )articular solution relates to s)eci$ic conditions given in the uestion8 and that the conditions lead to $inding a value $or the constant' Learners 6ill consolidate their 6or5 on general solutions 6hen 6or5ing through )roblems reuiring )articular solutions' At the lin5 belo6 +ou 6ill $ind three $iles8 one o$ 6hich contains notes and e#am)les on )articular solutions o$ $irst order di$$erential euations log in $or $ree do6nload! htt)s//666'tes'co'u5/teaching
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Scheme o$ 4or5
'uggested teaching acti!ities *he lin5 belo6 )rovides an interactive e#ercise on )articular solutions o$ $irst order di$$erential euations' htt)//6or5sheets'tutorvista'com/di$$erential
ast papers (I)(F) *hese uestions include all as)ects o$ this section and the )revious sections' Dune %0, )a)er 38 uestion , Dune %0, )a)er 3%8 uestion 9 Dune %0, )a)er 338 uestion ( ovember %0, )a)er 338 uestion 2 Dune %03 )a)er 38 uestion 0 includes numerical solution o$ euations! Dune %03 )a)er 3%8 uestion 2 includes )artial $ractions!
0nterpret the solution of a di4erential equation in the conte#t of a pro%lem %eing modelled %! the equation.
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aving solved a di$$erential euation8 learners o$ten need to inter)ret their solution in conte#t' Sometimes a gra)h can hel) them to deduce 6hat is ha))ening'
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Scheme o$ 4or5
'uggested teaching acti!ities
d P *he gra)h above sho6s the )articular solution to the euation dt
= − ( P − 20 )
6here P re)resents the si;e o$ a
= 0 8 P = 16 ' *his leads to the ' >ou can see $rom the gra)h that8 as t increases8 P → 20 8 so +ou can conclude
)o)ulation8 in 000s8 and t re)resents time in +ears' It is given initiall+ that 6hen t − t
)articular solution P = 20 − 4e that8 over time8 the )o)ulation increases and a))roaches %0 000 but never reaches it' Alternativel+8 +ou can see this algebraicall+ $rom the )articular solution as t increases8 e
− t
→ 0 so P → 20 '
Learners 6ill bene$it $rom )ractising e#am t+)e uestions to build con$idence in this t+)e o$ inter)retation' *e#tboo5s 6ill also )rovide )ractice uestions $or learners to 6or5 through' (I) ast papers (I)(F) *hese include all the above as)ects o$ -i$$erential ?uations' ovember %0, )a)er 38 uestion 7 Dune %03 )a)er 338 uestion 2
7# Complex numbers %earning ob&ecti!es
'uggested teaching acti!ities
? understand the idea of a comple# num%er" recall the meaning of the terms real part" imaginar! part" modulus" argument" con@ugate" and use the fact that t'o comple# num%ers are equal if and onl! if %oth real and imaginar! parts are equal.
>ou could introduce the conce)t o$ com)le# numbers b+ as5ing learners to solve an euation e'g' x + 6 x + 25 = 0 ' sing their 5no6ledge o$ the discriminant8 or b+ using the uadratic $ormula8 learners can deduce that the euation
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has no real roots' As5 them to 6rite do6n the t6o suare roots o$ <., using i =
−1 ' *he+ can give the solutions to x x 3 4i 3 4i = − + = − − the uadratic euation as and ' >ou can introduce the term Bcom)le# number $or these numbers 6ith a real )art and an imaginar+ )art'
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Scheme o$ 4or5
'uggested teaching acti!ities >ou ma+ 6ish to mention that engineers generall+ use
rather than i $or
−1 '
Plotting com)le# numbers on an Argand diagram 6ill hel) learners to visualise them as t6oou can also use the diagram to introduce the terms Bconugate8 Bmodulus and Bargument together 6ith the a))ro)riate notation and conventions $or these' It 6ould be a good idea $or learners to )ractise )lotting basic e#am)les= +ou 6ill $ind man+ in te#tboo5s' *he article here demonstrates a similar a))roach and includes some investigations 6hich learners ma+ $ ind interesting htt)//nrich'maths'org/,03 Learners can move on to consider )roblems 6hich reuire them to euate real and imaginar+ )arts' >ou can dra6 the analog+ 6ith the )rocess o$ euating coe$$icients' "or e#am)le BUiven that the com)le# numbers and
( a + 1) + 2i
2a + (3a − b)i are eual8 $ind the values o$ a and b '
*he lin5 belo6 leads to $our $ iles8 a summar+ o$ the above )oints8 e#am)les and a matching activit+ 6hich +ou could use in grou)s to chec5 learners understanding log in $or $ree do6nload! htt)s//666'tes'co'u5/teaching
? carr! out operations of addition" su%traction" multiplication and division of t'o comple# num%ers e#pressed in cartesian form x + iy.
All o$ these methods 6ill be $amiliar to learners $rom other areas o$ mathematics •
addition and subtraction o$ t6o com)le# numbers is similar to adding and subtracting vectors' Learners 6ill $ind it use$ul to deal 6ith this both algebraicall+ and using an Argand diagram'
•
multi)lication o$ t6o com)le# numbers is similar to e#)anding brac5ets'
•
division o$ one com)le# number b+ another is similar to rationalising surds in the denominator'
Although the geometrical inter)retation o$ these o)erations using an Argand diagram a))ears in a later section8 +ou
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Cambridge International AS & A Level Mathematics 9709! $rom %07
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Scheme o$ 4or5
'uggested teaching acti!ities ma+ 6ish to cover it here along 6ith the algebra' It 6ould be hel)$ul to discuss 6ith learners e#am)les o$ each t+)e then set them )lent+ o$ )ractice' (I) *he lin5 belo6 has several 6or5sheets on the to)ics in this section' >ou can do6nload these to )rovide learners 6ith )ractice and consolidation' *here are also interesting investigations and s)ot the error e#ercises' htt)//666'math6or5sheetsgo'com/sheets/algebra<%/com)le#
? use the result that" for a pol!nomial equation 'ith real coecients" an! non-real roots occur in con@ugate pairs ? represent comple# num%ers geometricall! %! means of an Argand diagram.
>ou could start b+ giving learners some basic euations to solve
− 14 z + 53 = 0 2 z − 4 z 2 − 5 z − 3 = 0 ' *he+ ma+ need a hint to $ind one root using the $actor theorem z = 3 !'
e'g' z
2
3
Uive learners more e#am)les o$ this t+)e and as5 them to ma5e a deduction $rom their results' >ou can then give them a more advanced e#am)le e'g' BUiven that 2
− 3i is one o$ the roots o$ the euation
z + 2 z − z + 38 z + 130 = 0 8 solve the euation com)letel+' 4
3
2
A))ro)riate te#tboo5s 6ill have )lent+ o$ uestions $or learners to )ractise' >ou ma+ alread+ have introduced the Argand diagram in the )revious section' *his lin5 6ill hel) learners to thin5 about geometrical relationshi)s bet6een )oints on the Argand diagram htt)//nrich'maths'org/92(9/note ast papers (I)(F) Dune %0, )a)er 3% uestion 7 Dune %03 )a)er 38 uestion 7a! Dune %03 )a)er 3%8 uestion 9a! Dune %03 )a)er 338 uestion 7
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? carr! out operations of multiplication and division of t'o comple# num%ers e#pressed in polar form r &cos θ + isin θ ) B r eiθ .
Scheme o$ 4or5
'uggested teaching acti!ities *his cha)ter )rovides use$ul e#am)les and e#ercises on the )olar $orm o$ com)le# numbers8 ma5ing use o$ the Argand diagram htt)//666'cimt')l+mouth'ac'u5/)roects/me)res/alevel/$)ureEch3')d$ :nce learners are com)etent 6ith the )olar $orm8 +ou could set them e#am)les o$ multi)lication and division and as5 them to deduce 6hat ha))ens to the moduli and arguments' *he+ could inter)ret their results using the Argand diagram too' *his lin5 )rovides interactive uestions $or learners to ans6er and assess their )rogress htt)s//666'5hanacadem+'org/math/)recalculus/imaginar+Ecom)le#E)recalc/e#)onential<$ormou 6ill $ind additional e#am)les in a))ro)riate te#tboo5s' ast papers (I)(F) Dune %0, )a)er 38 uestion (
? *nd the t'o square roots of a comple# num%er.
>ou could start b+ having a class discussion 6ith learners ho6 could the+ $ind the suare roots o$ a com)le#
+ 4i K 4ith care$ul uestioning8 +ou can encourage them to 6rite it in the $orm ( a + bi ) 8 6here a + bi is a suare root o$ 3 + 4i 8 and to euate real and imaginar+ )arts' number such as 3
2
= 3 + 4i
*he lin5 belo6 demonstrates such an a))roach' htt)//666'e#amsolutions'net/maths
? understand in simple terms the geometrical e4ects of con@ugating a comple# num%er and of adding" su%tracting" multipl!ing and dividing
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>ou ma+ alread+ have covered this in earlier sections i$ +ou used Argand diagrams as 6ell as algebra' *here is a use$ul interactive resource here $or visualising multi)lication and division on an Argand diagram' >ou could either use it as a demonstration $or the 6hole class or individual learners could use it to )redict their ans6ers
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Cambridge International AS & A Level Mathematics 9709! $rom %07
Scheme o$ 4or5
%earning ob&ecti!es
'uggested teaching acti!ities
t'o comple# num%ers.
and chec5 them' htt)//666'$urthermaths'org'u5/$iles/sam)le/$iles/Com)le#Multi)lication'html
? illustrate simple equations and inequalities involving comple# num%ers %! means of loci in an
Start b+ as5ing learners 6hat the+ understand b+ the 6ords Blocus and Bloci and to suggest an+ e#am)les' :ne e#am)le is the circle the locus o$ a )oint 6hich moves such that it is al6a+s a constant distance $rom a $i#ed )oint'
Argand diagram" e.g.
z − a
= z −b
z − a
< k "
and arg& z − a ) 1
a.
*he lin5 belo6 )rovides teaching )oints and ideas htt)s//666'ncetm'org'u5/sel$
z − ( 3 + 4i ) = 5
+
sing an e#am)le e'g' 8 as5 learners to )lot the com)le# number 3 4i then to consider the signi$icance o$ the (8 the z and the modulus signs' -ra6 the )arallel 6ith the de$inition o$ a circle above and )lot the circle described b+ this euation' >ou can e#tend this reasoning to the ineualit+ on their diagram'
z − ( 3 + 4i ) < 5
8 as5ing learners to shade the a))ro)riate region
+ − 2i +ou can encourage learners to re<6rite it as z − ( −3 + 2i) then to )lot a circular locus based on the )oint −3 + 2i ' sing an e#am)le such as z 3
x + iy ' *he
>ou could also demonstrate to learners that the+ can $ind the Cartesian euations o$ loci b+ 6riting z as Cartesian $orm could be use$ul $or veri$+ing that8 $or e#am)les o$ the t+)e )er)endicular bisector'
z − a
= z − b " the locus is a
"or e#am)les o$ the t+)e arg& z – a) 1 a" it is im)ortant that learners realise onl+ hal$ lines are needed' *his lin5 )rovides e#am)les and e#ercises that +ou could either use in class or learners could use inde)endentl+ $or revision htt)//666'ilovemaths'com/3argand)lane'as) *he lin5 belo6 )rovides a 6or5sheet on loci in the com)le# )lane that could be used $or )ractice or consolidation' (I) htt)s//666'tes'co'u5/teaching
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Cambridge International AS & A Level Mathematics 9709! $rom %07
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Scheme o$ 4or5
'uggested teaching acti!ities Dune %03 )a)er 38 uestion 7b! Dune %03 )a)er 3%8 uestion 9b! Dune %03 )a)er 338 uestion 7ii!8iii!
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V Cambridge International ?#aminations %0. 1ersion ' )dated 0'03'.
Cambridge International ?#aminations ills Hoad8 Cambridge8 CN %?8 nited ingdom tel G,, %%3 ((3((, $a# G,, %%3 ((3((2 email in$oWcie'org'u5 666'cie'org'u5
