maths 0580 2014

7/23/2019 maths 0580 2014

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Scheme of workCambridge IGCSE® Mathematics0580For examination from 205

Contents

!"er"iew################################################################################################################################################################################################################################################### $

%nit & '(mber########################################################################################################################################################################################################################################## )

%nit 2& *+gebra and gra,hs############################################################################################################################################################################################################ ######### -

%nit $& Geometr.##################################################################################################################################################################################################################################### 2/

%nit & Mens(ration################################################################################################################################################################################################################################# $

%nit 5& Co1ordinate geometr.################################################################################################################################################################################################################## $8

%nit )& rigonometr.######################################################################################################################################################################################################################### ######

%nit -& Matrices and transformations###################################################################################################################################################################################################### 5

%nit 8& 3robabi+it.#################################################################################################################################################################################################################################### /

%nit /& Statistics################################################################################################################################################################################################################################### ### 52

7/23/2019 maths 0580 2014


Overview

his scheme of work ,ro"ides ideas abo(t how to constr(ct and de+i"er a co(rse# he s.++ab(s has been broken down into teaching (nits with s(ggested teachingacti"ities and +earning reso(rces to (se in the c+assroom# his scheme of work4 +ike an. other4 is meant to be a g(ide+ine4 offering ad"ice4 ti,s and ideas# It can ne"er becom,+ete b(t ho,ef(++. ,ro"ides teachers with a basis to ,+an their +essons# It co"ers the minim(m re(ired for the Cambridge IGCSE co(rse b(t a+so addsenhancement and de"e+o,ment ideas on to,ics# It does not take into acco(nt that different schoo+s take different amo(nts of time to co"er the Cambridge IGCSE

co(rse#

Recommended prior knowledgeIt is recommended that candidates ha"e fo++owed the Secondar. Mathematics C(rric(+(m Framework which can be fo(nd at&3TU htt,&66www#cie#org#(k6(a+ifications6academic6+owersec6cambridgesecondar.6reso(rcesU3T or fo++owed co(rses which co"er the materia+ contained in the %7 'ationa+C(rric(+(m for Mathematics at 7e. Stage $ 3TU htt,&66www#ed(cation#go"#(k6schoo+s6teachingand+earning6c(rric(+(m6secondar.6b00//00$6mathematics6ks$U3T#

Outlineho+e c+ass (W)4 gro(, work (G)4 ,air (P) and indi"id(a+ acti"ities (I) are indicated4 where a,,ro,riate4 within this scheme of work# S(ggestions for homework (H) andformati"e assessment (F) are a+so inc+(ded# he acti"ities in the scheme of work are on+. s(ggestions and there are man. other (sef(+ acti"ities to be fo(nd in themateria+s referred to in the +earning reso(rce +ist#

!,,ort(nities for differentiation are indicated as basic and callenging# here is the ,otentia+ for differentiation b. reso(rce4 +ength4 gro(,ing4 ex,ected +e"e+ of

o(tcome4 and degree of s(,,ort b. the teacher4 thro(gho(t the scheme of work# imings for acti"ities and feedback are +eft to the 9(dgment of the teacher4 according tothe +e"e+ of the +earners and si:e of the c+ass# ;ength of time a++ocated to a task is another ,ossib+e area for differentiation#

he (nits within the scheme of work are&

!nit "# $umber!nit %# &lgebra and graps!nit '# Geometr!nit # *ensuration!nit +# Co,ordinate geometr!nit -# .rigonometr!nit /# *atrices and trans0ormations!nit 1# Probabilit

!nit 2# 3tatistics

3llabus contenthe s.++ab(s references (se C for Core and E for Extended c(rric(+(m# In this scheme of work the. are +isted together (sing black te4t to identif. where both the Coreand 54tended curriculum cover te same content and blue text for where the content is onl covered b te 54tended curriculum#

"#0 < Cambridge IGCSE Mathematics =0580> ? from 205 2

http://www.cie.org.uk/qualifications/academic/lowersec/cambridgesecondary1/resources




http://www.education.gov.uk/schools/teachingandlearning/curriculum/secondary/b00199003/mathematics/ks3







7/23/2019 maths 0580 2014


.eacer supporteacher S(,,ort =3TU http://teachers.cie.org.ukU3T> is a sec(re on+ine reso(rce bank and comm(nit. for(m for Cambridge teachers4 where .o( can down+oad s,ecimenand ,ast (estion ,a,ers4 mark schemes and other reso(rces# e a+so offer on+ine and face1to1face training@ detai+s of forthcoming training o,,ort(nities are ,ostedon+ine#

his scheme of work is a"ai+ab+e as 3AF and an editab+e "ersion in Microsoft ord format@ both are a"ai+ab+e on eacher S(,,ort at 3TU http://teachers.cie.org.ukU3T# If.o( are (nab+e to (se Microsoft ord .o( can down+oad !,en !ffice free of charge from 3TU www#o,enoffice#org U3T#

Resource listhe reso(rce +ist for this s.++ab(s4 inc+(ding textbooks endorsed b. Cambridge4 can be fo(nd at U www#cie#org#(k U and eacher S(,,ort U htt,&66teachers#cie#org#(k U#

Endorsed textbooks ha"e been written to be c+ose+. a+igned to the s.++ab(s the. s(,,ort4 and ha"e been thro(gh a detai+ed (a+it. ass(rance ,rocess# *s s(ch4 a++textbooks endorsed b. Cambridge for this s.++ab(s are the idea+ reso(rce to be (sed a+ongside this scheme of work as the. co"er each +earning ob9ecti"e#

.e4tbookshe most common+. (sed textbooks referenced in this scheme of work inc+(de&Barton4 A Essential Mathematics for IGCSE Extended Teacher Resource Kit =!xford %ni"ersit. 3ress4 202>aighton4 D et a+ Core Mathematics for Cambridge IGCSE ( 'e+son hornes4 202>aighton4 D et a+ Extended Mathematics for Cambridge IGCSE ( 'e+son hornes4 202>

Morrison4 7 and amshaw4 ' Cambridge IGCSE Mathematics Core and Extended Coursebook (with C!R"M# ( Cambridge %ni"ersit. 3ress4 202>'.e4 C IGCSE Core Mathematics ( einemann4 200/>'.e4 C IGCSE Extended Mathematics =einemann4 200/>3earce4 C Cambridge IGCSE Maths Student $ook =Co++ins Ed(cationa+4 20>3emberton4 S Essential Mathematics for Cambridge IGCSE Extended (with C!R"M# =!xford %ni"ersit. 3ress4 202>3imenta+4 and a++4 Cambridge IGCSE Mathematics Second Edition u%dated with C =odder Ed(cation4 20>a.ner4 A Core Mathematics for Cambridge IGCSE (with C!R"M# =!xford %ni"ersit. 3ress4 20>a.ner4 A Extended Mathematics for Cambridge IGCSE (with C!R"M > =!xford %ni"ersit. 3ress4 20>Sim,son4 * Core Mathematics for Cambridge IGCSE =Cambridge %ni"ersit. 3ress4 200>Sim,son4 * Extended Mathematics for Cambridge IGCSE =Cambridge %ni"ersit. 3ress4 20>

Websiteshis scheme of work inc+(des website +inks ,ro"iding direct access to internet reso(rces# Cambridge Internationa+ Examinations is not res,onsib+e for the acc(rac. or

content of information contained in these sites# he inc+(sion of a +ink to an externa+ website sho(+d not be (nderstood to be an endorsement of that website or thesites owners =or their ,rod(cts6ser"ices>#

he ,artic(+ar website ,ages in the +earning reso(rce co+(mn of this scheme of work were se+ected when the scheme of work was ,rod(ced# !ther as,ects of the siteswere not checked and on+. the ,artic(+ar reso(rces are recommended#

ebsites in this scheme of work4 and some other (sef(+ websites4 inc+(de&3TU htt,&66nrich#maths#org6front,ageU3T

"#0 < Cambridge IGCSE Mathematics =0580> ? from 205 $

http://teachers.cie.org.uk/







http://www.openoffice.org/




http://www.cie.org.uk/











http://nrich.maths.org/frontpage










7/23/2019 maths 0580 2014


3TU htt,&66www#tes#co#(k6teaching1reso(rces6U3T

3TU htt,&66www#bbc#co#(k6schoo+s6gcsebitesi:e6maths6U3T

3TU htt,&66www#wa+domaths#com6U3T

3TU htt,s&66www#khanacadem.#org6 U3T

3TU htt,&66www#geogebra#org6cms6en6U3T

3TU htt,&66(i:+et#comU3T

3TU htt,&66www#cimt#,+.mo(th#ac#(kU3T

3TU htt,&66www#math#com6U3T

3TU htt,&66www#mm+soft#comU3T

3TU htt,&66www#mrbartonmaths#com69igsaw#htmU3T

3TU htt,&66www#mathsisf(n#com6U3T

3TU htt,&66www#mathsre"ision#net6U3T 3TU htt,&66www#ec+i,secrossword#com6U3T

3TU htt,&66www#basic1mathematics#com6U3T

3TU htt,&66math#abo(t#com6U3T

3TU htt,&66www#.o(t(be#com6U3T

3TU htt,&66reso(rces#wood+ands19(nior#kent#sch#(k6maths6U3T

3TU htt,&66mrbartonmaths#com6ebook#htmU3T

3TU htt,&66i++(minations#nctm#orgU3T

3TU htt,&66www#weatherbase#com6U3T

3TU htt,&66www#co+inbi++ett#word,ress#com6U3T

3TU htt,&66www#bgrademaths#b+ogs,ot#co#(k6U3T 3TU htt,&66www#timeanddate#com6wor+dc+ock6U3T

3TU htt,&66www#s,ringfrog#com6U3T

3TU htt,&66www#timde"ere(x#co#(k6maths6mathsintro#htm+U3T

3TU htt,&66www#+earner#org6U3T

3TU htt,s&66ma,s#goog+e#com6U3T

3TU htt,&66www#regents,re,#org6regents6math6a+gtrig6math1*;GIG#htm U3T

3TU htt,&66www#tesse++ations#org6index#shtm+U3T

3TU htt,&66www#nationa+stemcentre#org#(k6U3T

3TU htt,&66www#on+inenews,a,ers#com6U3T

!nit "# $umber

Recommended prior knowledge;earners sho(+d be ab+e to add4 s(btract4 m(+ti,+. and di"ide confident+. with integers and identif. the correct o,eration from a word ,rob+em# he. sho(+d be fami+iarwith directed n(mbers and ha"e an (nderstanding of a n(mber +ine in"o+"ing ,ositi"e and negati"e "a+(es# he. sho(+d (nderstand how to ro(nd to the nearest who+e

"#0 < Cambridge IGCSE Mathematics =0580> ? from 205

http://www.tes.co.uk/teaching-resources/




http://www.bbc.co.uk/schools/gcsebitesize/maths/




http://www.waldomaths.com/




https://www.khanacademy.org/




http://www.geogebra.org/cms/en/




http://quizlet.com/

http://quizlet.com/

http://quizlet.com/

http://quizlet.com/

http://www.cimt.plymouth.ac.uk/




http://www.math.com/




http://www.mmlsoft.com/




http://www.mrbartonmaths.com/jigsaw.htm




http://www.mathsisfun.com/




http://www.mathsrevision.net/





http://www.eclipsecrossword.com/




http://www.basic-mathematics.com/




http://math.about.com/




http://www.youtube.com/




http://resources.woodlands-junior.kent.sch.uk/maths/




http://mrbartonmaths.com/ebook.htm




http://illuminations.nctm.org/




http://www.weatherbase.com/




http://www.colinbillett.wordpress.com/




http://www.bgrademaths.blogspot.co.uk/




http://www.timeanddate.com/worldclock/




http://www.springfrog.com/




http://www.timdevereux.co.uk/maths/maths_intro.html




http://www.learner.org/




https://maps.google.com/




http://www.regentsprep.org/regents/math/algtrig/math-ALGTRIG.htm




http://www.tessellations.org/index.shtml




http://www.nationalstemcentre.org.uk/




http://www.onlinenewspapers.com/









http://quizlet.com/


























7/23/2019 maths 0580 2014


n(mber4 04 00 and 000 and ha"e some fami+iarit. with decima+ ,+aces# ;earners sho(+d be fami+iar with m(+ti,+.ing and di"iding who+e n(mbers and decima+s b. 0to the ,ower of an. ,ositi"e or negati"e integer and recognise the e(i"a+ence of sim,+e decima+s4 fractions and ,ercentages4 e#g# 0#254 H and 25# ;earners need to(nderstand 2 and 2 ho(r c+ock and be ab+e to con"ert between these# he. sho(+d a+so be confident in working with sim,+e fractions and decima+s4 for exam,+ewriting a fraction in its sim,+est form b. cance++ing common factors@ adding and s(btracting fractions with the same denominator@ adding and s(btracting decima+s withthe same n(mber of decima+ ,+aces# he. sho(+d be aware of the order of o,erations4 inc+(ding brackets and recognise the effects of m(+ti,+.ing and di"iding b.n(mbers bigger than or sma++er than #

Conte4this first (nit re"ises and de"e+o,s mathematica+ conce,ts in n(mber that (nder,in the co(rse# he work is f(ndamenta+ to the st(d. of a++ the other (nits and ,arts of itwi++ need to be re"isited when teaching s(bse(ent (nits# his (nit is a,,ro,riate for a++ +earners4 with the exce,tion of a++ of sections #2 and #- and the indicated,arts of sections #54 #-4 #04 #4 #24 and #) which are on+. for extended +earners# It is antici,ated that +earners st(d.ing the extended s.++ab(s wi++ work thro(ghat a faster ,ace#

OutlineIt is intended for the to,ics in this (nit to be st(died se(entia++.4 a+tho(gh this is not essentia+ as certain to,ics4 for exam,+e ime =#>4 can be st(died ear+ier as it is asim,+er and ,robab+. (ite fami+iar conce,t# *+so .o( ma. want to st(d. (nit #) after #24 as the. are re+ated4 or .o( ma. choose to +ea"e a ga, between them sothat ,ercentages can be re"ised when to,ic #) is st(died# his (nit co"ers a++ as,ects of n(mber from the s.++ab(s4 name+. fractions4 decima+s4 ,ercentages4 ratios4indices4 directed n(mbers4 bo(nds4 time4 mone. and finance# Some teachers ,refer to not teach n(mber a++ in one b+ock4 it is ,ossib+e to +ea"e some sections (nti+ +aterin the co(rse4 for exam,+e (,,er and +ower bo(nds =#0> and ex,onentia+ growth and deca. =#-> co(+d be ta(ght +ater on in the co(rse4 simi+ar+. with other to,ics#

.eacing timeIt is recommended that this (nit sho(+d take a,,roximate+. $0?$5 =Core +earners> 5?20 =Extended +earners> of the o"era++ IGCSE co(rse#


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Re0 3llabus content 3uggested teacing activities 6earning resources

1.1 Identif. and (se nat(ra+ n(mbers4integers =,ositi"e4 negati"e and:ero>4 ,rime n(mbers4 s(aren(mbers4 common factors andcommon m(+ti,+es4 rationa+ and

irrationa+ n(mbers =e#g# J4 2 >4 rea+n(mbers#

Inc+(des ex,ressing n(mbers as a,rod(ct of ,rime factors#

Finding the ;owest CommonM(+ti,+e =;CM> and ighestCommon Factor =CF> of twon(mbers#

A useful starting point would be to revise positive and negative numbersusing a number line and explain the dierence between natural numbersand integers. (W) (Basic)

!earners would "nd it useful to have a de"nition of the terms #e.g. factor$multiple$ s%uare number& which can be found on the maths revision website.(W) (I) (Basic)

A fun activit' would be to allocate a number to each learner in the class andask them to stand up if the' are$ for example$ (a multiple of )*$ (a factor of1+* etc. Use this to show interesting facts such as prime numbers will have ,people standing up #emphasises 1 is not prime&- s%uare numbers will havean odd number of people standing up. ee which are commonfactors/common multiples for pairs of numbers. This could be extended to0 and !02. (W) (Basic)

A followon activit' would be for learners to identif' a number from a

description of its properties. or example$ sa' to the class (which numberless than 45 has 3 and 4 as factors and is a multiple of 67* !earners couldthen make up their own descriptions and test one another. (G) (Basic)

Another interesting task is to look a ermat8s discover' that some primenumbers are the sum of two s%uares$ e.g. ,6 9 ,4 ) 9 4 ;

,; ,;

,;. !earners

could see what primes the' can form in this wa'$ and an' the' can8t form inthis wa'. !earners can look for a rule which tests whether or not a prime canbe made like this. (I) (Challenging)

2ove on to looking at how to write an' integer as a product of primes. <nemethod that can be used is the factor tree approach which can be foundonline or in ;emberton8s =ssential maths 0>. After demonstrating$ or

showing the presentation$ ask learners to practise using the method to writeother numbers as products of primes. Then ask learners to look at "ndingthe product of primes of other numbers$ for example ?5$ )45$ ),$ 314$ butthis time the' can be encouraged to look for alternative methods$ forexample b' researching on the internet. Another useful method is therepeated division method. (I) (H)

!earners would "nd it useful to have a de"nition of the terms rational$

Online3TU htt,&66www#mathsre"ision#net6content6n(mbersU3T

3 TUhtt,&66www#mathsisf(n#com6irrationa+1n(mbers#htm+

3TU htt,&66"imeo#com608$20U3T thefactor tree a,,roach

C7,RO*3emberton# %nit s+ides / and 0

"#0 < Cambridge IGCSE Mathematics =0580> ? from 205 )

http://www.mathsrevision.net/content/numbers





http://www.mathsisfun.com/irrational-numbers.html



http://vimeo.com/101831240









7/23/2019 maths 0580 2014



irrational and real numbers which can be found on the 2aths is un website.<n the website there are %uestions on rationa+ and irrationa+ n(mbers for +earnersto tr.# hese start simple and soon become more challenging. (I) (F)

Extended curriculum only

%se +ang(age4 notation and Kenndiagrams to describe sets andre,resent re+ationshi,s betweensets#Aefinition of setse#g# & L x & x is a nat(ra+ n(mberN $ L = x 4' >& ' L mx O c N C L x & a < x < bN L a4 b4 c 4 PN'otation'(mber of e+ements in set & n= &>QPis an e+ement ofPR @

QPis not an e+ement ofPR Com,+ement of set & &he em,t. set B%ni"ersa+ set & is a s(bset of $ & C $ & is a ,ro,er s(bset of $ & D $ & is not a s(bset of $ & E $ & is not a ,ro,er s(bset of $ & F $%nion of & and $ & G $Intersection of & and $ & T $

It is (sef(+ to start with re"ising sim,+e Kenn diagrams4 for exam,+e with ,eo,+e who

wear g+asses in one circ+e and ,eo,+e with brown hair in another circ+e asking +earnersto identif. the t.,e of ,eo,+e in the o"er+a,,ing region# (W) (8asic)

his can be extended to +ooking at genera+ Kenn diagrams concentrating more on theshading of the regions re,resenting the sets * G B4 * T B4 * G B4 * G B4 * T B4 * T B4 * G B and * T B he+,ing +earners to (nderstand the notation# *n exce++ent acti"e1+earning reso(rce is the Kenn diagrams card sort in Bartons eacher eso(rce 7it,ages )1)# *sk +earners to work in gro(,s to com,+ete this acti"it.# (G) (Callenging)

;earners wo(+d find it (sef(+ to know that =* G B> is the same as * T B and that =* TB> is the same as * G B and to (nderstand the +ang(age associated with sets andKenn diagrams# Morrison and amshaws book ,ages -21-/4 for exam,+e4 (sesKenn diagrams to so+"e ,rob+ems in"o+"ing sets#

;earners need to be ab+e to disting(ish between a s(bset and a ,ro,er s(bset# hework on Kenn diagrams can be extended to +ook at (nions and intersections when thereare three sets# (W) (Callenging)

*sk +earners to tr. the ,ast ,a,er (estion# (H) (F)

.e4tbooks

Barton ,#)1)Morrison ,#-21-/

Past paper 3a,er 4 D(ne 2024 U8

1.3 Ca+c(+ate s(ares4 s(are roots4c(bes and c(be roots of n(mbers#

Using simple examples illustrate s%uares$ s%uare roots$ cubes and cube rootsof integers. (W) (Basic)

=xtend the task b' asking more able learners to s%uare and cube fractionsand decimals without a calculator$ it ma' be worth doing topic 1.+ "rst tohelp with this. (W) (Challenging)

An interesting activit' is to look at "nding the s%uare root of an integer b'repeated subtraction of consecutive odd numbers until 'ou reach Hero. orexample$ for ,4 subtract in turn 1$ 3$ 4$ I$ and then 6 to get to 5. ive odd

Past paper 3a,er $24 D(ne 2024 U$

"#0 < Cambridge IGCSE Mathematics =0580> ? from 205 -

7/23/2019 maths 0580 2014



numbers have been subtracted so the s%uare root of ,4 is 4. Ask learners toinvestigate this method for other$ larger$ s%uare numbers. (I) (H)

Another interesting challenge is to look at the palindromic s%uare number1,1. #;alindromic means when the digits are reversed it is the samenumber&. Ask learners to "nd all the palindromic s%uare numbers less than1555. (I) (H)

*sk +earners to tr. the ,ast ,a,er (estion# (H) (F)

1.) %se directed n(mbers in ,ractica+sit(ations#

*n effecti"e start for this to,ic is to draw a n(mber +ine from 120 to O204 then ,oint to"ario(s n(mbers =both ,ositi"e and negati"e> asking +earners4 for exam,+e4 Qwhat is 5more than this n(mberVR4 Qhat is ) +ess than this n(mberVR (W) <o( can kee, it sim,+e b. (sing on+. integers (8asic) or extend the task b. (singdecima+s or fractions# (Callenging)

*n interesting extension to this is to then +ook at directed n(mbers in the context of,ractica+ sit(ations# For exam,+e4 temperature changes$ Jood levels$ bank credits

and debits. ;earners can see weather statistics for o"er 2/000 cities on+ine atweatherbase#com4 which can be (sed for them to in"estigate a "ariet. of tem,erat(rechanges in"o+"ing ,ositi"e and negati"e tem,erat(res9 (G) (8asic)

Online3TU htt,&66www#weatherbase#com6U3T

1.4 %se the +ang(age and notation ofsim,+e "(+gar and decima+ fractionsand ,ercentages in a,,ro,riatecontexts#

ecognise e(i"a+ence and con"ertbetween these forms#

!earners would "nd it useful to have a de"nition of the terms =e#g# numeratordenominator e)ui*alent fractions sim%lif' *ulgar fraction im%ro%er fraction mixednumber decimal fraction4 and %ercentage># * f(n acti"it. wo(+d be to ask +earners to,rod(ce a crossword with the terms defined# *sk them to add an. other terms that the.can think of to do with fractions4 decima+s and ,ercentages# Crosswords can be easi+.created (sing the exce++ent on+ine software at ec+i,secrossword#com# (I) (H)

* (sef(+ acti"it. for +earners wo(+d be (sing c+ear exam,+es and (estions to tr.co"ering con"erting between fractions4 decima+s and ,ercentages4 s(ch as Metca+f

,#81/0# ;earners sho(+d (nderstand how to (se ,+ace "a+(e =(nits4 tenths4 h(ndredths4etc#> to change a sim,+e decima+ into a fraction# For exam,+e 0#$ has $ in the tenths

co+(mn so it is 0

$

# (W) (8asic)

Online3TU htt,&66www#ec+i,secrossword#com6U3T

.e4tbookMetca+f ,#81/0


Inc+(des the con"ersion of rec(rringdecima+s to fractions#

* (sef(+ acti"it. for +earners is to +ook at the on+ine +esson at basic1mathematics#com to+earn how to con"ert rec(rring decima+s to fractions# It (ses the method& x L 0#5555P#00 x L 5#5555P s(btract these to get

Online3TU htt,&66www#basic1mathematics#com6con"erting1re,eating1decima+s1to1fractions#htm+U3T











http://www.basic-mathematics.com/converting-repeating-decimals-to-fractions.html











7/23/2019 maths 0580 2014



// x L 5 so x L15

99=

5

33 (W) (Callenging)

1.? !rder (antities b. magnit(de anddemonstrate fami+iarit. with the

s.mbo+sL4 W4 X4 Y4 Z 4 [

* good acti"e +earning a,,roach to this to,ic is to gi"e +earners a set of cards with thes.mbo+s L4 W4 X4 Y4 Z4 [# *sk them to choose which card sho(+d go in between ,airs of

(antities that .o( gi"e them# For exam,+e4 00m and 000 cm@ 20 and 0#2@ 18 and104 etc# (W) (8asic)

Extend this b. asking +earners to consider when4 or if4 more than one card can be (sed=e#g# W can be (sed in ,+ace of X or Y># (W)

Mo"e on to gi"ing +earners a +ist of fractions4 decima+s and ,ercentages# *sk them toorder these b. magnit(de (sing the ine(a+it. signs# (G) (8asic)

o check their (nderstanding4 +earners can then tr. the ,ast ,a,er (estion# (H) (F)

Past paper 3a,er 24 D(ne 20$4 U-

1.I %nderstand the meaning and r(+esof indices#E"a+(ate 2;

5;4 5;

?2;4 00;

0

ork o(t 2;

?$; \ 2;

%se the standard form & \ 0;

n;

where n is a ,ositi"e or negati"einteger4 and [ & Y 0#Con"ert n(mbers into and o(t ofstandard form#Ca+c(+ate with "a+(es in standardform#


5;

];

L √ 5 4 E"a+(ate 00;

];

4 8;

?26$

* good starting ,oint is to begin working on+. with ,ositi"e indices b. re"ising themeaning of these and the basic r(+es of indices s(ch as 2 ;

$;\ 2;

5; L 2;

8; 4 5;

;^ 5;

$; L 5;

; L 5

etc# Gi"e sim,+e exam,+es to re"ise writing an integer as a ,rod(ct of ,rimes inc+(dingwriting answers (sing index notation# (W) (8asic)

*n interesting cha++enge for +earners is the ,(::+e Q3ower Cra:.R on the nrich#maths#orgwebsite# *sk +earners to work in gro(,s to com,+ete the cha++enge# (G)

Extend this b. working with negati"e and :ero indices4 and for extended +earners4

fractiona+ indices# %sef(+ exam,+es are 2;

1; L 2;

=21$>; L

22

23=

1

2 and

2;

0; L 2;

=$1$>; L

23

23

L # <o( can mo"e on to introd(cing fractiona+ indices b. re+ating

them to roots =of ,ositi"e integers>4 for exam,+e4

12 \

412 L ;

; L so

412 L

√ 4 L 2# he r(+es of indices can be (sed to show how "a+(es s(ch as16

3

4 can

be sim,+ified# ;earners sho(+d tr. +ots of exam,+es and (estions4 s(ch as in 3earce4St(dent book ,#2)212-$# (I) (Callenging)

he next ste, is to gi"e +earners a range of exam,+es showing how to write n(mbers in

Online3TU htt,&66nrich#maths#org68-U3T

3TU htt,&66www#tes#co#(k6teaching1reso(rce6Standard1Form1orksheet1)/$2/06U3T

.e4tbook3earce4 St(dent book ,#2)212-$

"#0 < Cambridge IGCSE Mathematics =0580> ? from 205 /

http://nrich.maths.org/847




http://www.tes.co.uk/teaching-resource/Standard-Form-Worksheet-6193290/










7/23/2019 maths 0580 2014



standard form and "ice1"ersa# Em,hasise to +earners that different ca+c(+ators dis,+a.standard form in different wa.s and check that the. know how to in,(t n(mbers instandard form into their ,artic(+ar ca+c(+ator# (W) (8asic)

Extend this b. (sing the fo(r r(+es of ca+c(+ation with n(mbers in standard form4 both

with and witho(t a ca+c(+ator# It wo(+d be (sef(+ for +earners if .o( can em,hasisecommon errors# For exam,+e4 if +earners are asked to work o(t the answer to 2# \ 0 ;

;

? 2 \ 0;

; in standard form it is common to see an answer of 0# \ 0 ;

;# 3oint o(t that

a+tho(gh 2# \ 0 ;

; ? 2 \ 0;

;L 0# \ 0;

; the answer is not in standard form4 since 0# is

+ess than # (W) (8asic)

* (sef(+ acti"it. wo(+d be to ask +earners to tr. the standard form worksheet from thetes#co#(k website# (H)#

1.+ %se the fo(r r(+es for ca+c(+ationswith who+e n(mbers4 decima+s and"(+gar =and mixed> fractions4inc+(ding correct ordering of

o,erations and (se of brackets#

* good starting acti"it. is to ask +earners to work in gro(,s to (se fo(r s and the fo(rr(+es for ca+c(+ations to obtain a++ the who+e n(mbers from to 204 e#g# O \ ? L)# (G) (8asic)

he next ste, is to +ook at +ong m(+ti,+ication and short and +ong di"ision# <o( can seethe traditiona+ and ch(nking =re,eated s(btraction> exam,+es on the bbc#co#(k website#his sho(+d be re"ision for most b(t is worth s,ending a bit of time on to ens(re+earners are confident in the methods# (W) (8asic)

Extend this b. c+arif.ing the order of o,erations4 inc+(ding the (se of brackets ,ointingo(t common errors4 for exam,+e when +earners do ca+c(+ations working from +eft to rightinstead of (sing the order of o,erations r(+e4 BIAM*S# =Brackets Indices Ai"isionM(+ti,+ication *ddition and S(btraction># Gi"e +earners some exam,+es i++(strating ther(+es for m(+ti,+.ing and di"iding with negati"e n(mbers# (W) (8asic)

Extend this to (sing the fo(r r(+es with fractions =inc+(ding mixed n(mbers> anddecima+s# It is im,ortant that +earners can do these ca+c(+ations both with and witho(t

the (se of a ca+c(+ator as the. ma. be ex,ected to show working# * (sef(+ book withgood exam,+es and exercises is 3embertons Essentia+ Maths ,# 21 and 01$# *sk+earners to tr. the (estions from the exercises# (I) (Callenging)

Online3TU htt,&66www#bbc#co#(k6schoo+s6gcsebitesi:e6maths6n(mber6m(+ti,+icationdi"isionre"2#shtm+U3T

.e4tbook3emberton4 Essentia+ maths4 ,#21401$

#/ Make estimates of n(mbers4(antities and +engths4 gi"ea,,roximations to s,ecifiedn(mbers of significant fig(res and

* sim,+e starting ,oint is to re"ise ro(nding n(mbers to the nearest 04 004 0004 etc#4or to a set n(mber of decima+ ,+aces# <o( can show +earners how to ro(nd a n(mber toa gi"en n(mber of significant fig(res ex,+aining the difference and simi+arities betweensignificant fig(res and decima+ ,+aces# (W) (8asic)

Online3TU htt,&66www#math#com6schoo+6s(b9ect6+essons6S%;$G;#htm+U3T


http://www.bbc.co.uk/schools/gcsebitesize/maths/number/multiplicationdivisionrev2.shtml






http://www.math.com/school/subject1/lessons/S1U1L3GL.html










7/23/2019 maths 0580 2014



decima+ ,+aces and ro(nd offanswers to reasonab+e acc(rac. inthe context of a gi"en ,rob+em#

It is (sef(+ for +earners to ex,+ain common misconce,tions s(ch as $#/8 to d, is #0not # Em,hasise that on this s.++ab(s non1exact answers are re(ired to threesignificant fig(res (n+ess the (estion sa.s otherwise# e"ision of estimating andro(nding can be fo(nd at the website math#com# (W) (8asic)

#0 Gi"e a,,ro,riate (,,er and +owerbo(nds for data gi"en to a s,ecifiedacc(rac.4 e#g# meas(red +engths#

<o( wi++ ,robab+. want to start this to,ic with exam,+es to determine (,,er and +owerbo(nds for data# Start with sim,+e exam,+es and then ,rogressi"e+. harder ones4 forexam,+e4 a +ength4 l 4 meas(red as $ cm to the nearest mi++imetre has +ower bo(nd 2#/5cm and (,,er bo(nd $#05 cm# Em,hasise that the bo(nds4 in this case4 are not 2#5 and$#5 which wo(+d be a common misconce,tion# Show +earners how this information canbe written (sing ine(a+it. signs e#g# 2#/5 cm [ + Y $#05 cm# (W) (8asic)

*n interesting a+ternati"e to an exercise is the (,,er and +ower bo(nds re"ision whee+in Bartons eacher eso(rce 7it ,#/51/-# (G)

.e4tbookBarton ,#/51/-


!btain a,,ro,riate (,,er and +owerbo(nds to so+(tions of sim,+e,rob+ems gi"en data to a s,ecifiedacc(rac.4 e#g# the ca+c(+ation of the,erimeter or the area of a rectang+e#

For extended +earners mo"e on to +ooking at (,,er and +ower bo(nds for (antitiesca+c(+ated from gi"en form(+ae# * (sef(+ exercise can be fo(nd in 3earces St(dentbook ,#0-10/# (I) (Callenging)9

o check their (nderstanding4 +earners can then tr. the ,ast ,a,er (estion# (H) (F)

.e4tbook3earce4 St(dent book ,#0-10/

Past papers3a,er 24 D(ne 20$4 U/3a,er 224 D(ne 20$4 U8

# Aemonstrate an (nderstanding ofratio and ,ro,ortion#

Ai"ide a (antit. in a gi"en ratio#

Airect and in"erse ,ro,ortion#

%se common meas(res of rate#

Ca+c(+ate a"erage s,eed#

%se sca+es in ,ractica+ sit(ations#

;earners wi++ find it (sef(+ to ha"e a definition of ratio with a ,ractica+ demonstration4 forexam,+e the ratio of different co+o(red beads on a neck+ace# (W) (8asic)

he next ste, is to +ook at exam,+es i++(strating how a (antit. can be di"ided into an(mber of (ne(a+ ,arts4 for exam,+e share _$)0 in the ratio 2 & $ & -# <o( wi++ thenwant to mo"e on to writing ratios in an e(i"a+ent form4 for exam,+e ) & 8 can be writtenas $ & 4 +eading on to the form & n# (W) (8asic)

*n interesting a+ternati"e to an exercise is the acti"e1+earning ratio 9igsaw atwww#tes#co#(k which +earners can work in gro(,s to com,+ete# (G) (8asic)#

* f(n homework task wo(+d be to ask +earners to ,rod(ce their own 9igsaw on ratiosimi+ar to this one# (I) (H) (8asic)

;earners co(+d ,rod(ce their own 9igsaw (sing b+ank e(i+atera+ triang+es and ,a,er#owe"er4 if the. ,refer to do this task e+ectronica++. then arsia software =9igsaw making

Online3TU htt,&66www#tes#co#(k6teaching1reso(rce6arsia1atio1genera+1)0-0U3T

3TU htt,&66www#mm+soft#com6index#,h,Vo,tionLcomcontent`taskL"iew`idL/`ItemidL0U3T

3TU htt,&66www#co+inbi++ett#word,ress#com6U3T

.e4tbook3emberton ,#28122


http://www.tes.co.uk/teaching-resource/Tarsia-Ratio-general-6107044






http://www.mmlsoft.com/index.php?option=com_content&task=view&id=9&Itemid=10



















7/23/2019 maths 0580 2014



software> is a"ai+ab+e for down+oad at mm+soft#com# (P) (I) (H) (8asic)

he next ste, is to +ook at ratio ,rob+ems where .o( are not gi"en the tota+4 for exam,+etwo +engths are in the ratio & - if the shorter +ength is 8 cm how +ong is the +onger+engthV (W) (8asic)

Extend this to exam,+es where .o( are gi"en the difference# For exam,+e4 the mass oftwo ob9ects are in the ratio 2 & 54 one ob9ect is $) g hea"ier than the other what is themass of each ob9ectV (W) (Callenging)

<o( wi++ then want to +ook at drawing gra,hs to determine whether two (antities are indirect ,ro,ortion# *sk +earners to so+"e a "ariet. of ,rob+ems in"o+"ing direct ,ro,ortionb. either the ratio method or the (nitar. method# ;ook at (antities in in"erse,ro,ortion4 for exam,+e the n(mber of da.s to ,erform a 9ob and the n(mber of ,eo,+eworking on the 9ob# <o( wi++ be ab+e to +ink ,ro,ortion to meas(res of rate and sca+es4for exam,+e exchange rates4 a"erage s,eed4 densit.4 ma, sca+es and other ,ractica+exam,+es# For some ideas read the on+ine b+og4 it started with a ma%4 'o"ember 202at co+inbi++ett#word,ress#com# <o( can a+so find some good (estions and exam,+es in3embertons Essentia+ Maths book ,ages 28122# (G) (P) (Callenging)


Increase and decrease a (antit. b.a gi"en ratio#

For extended +earners ,ro"ide some good exam,+es and (estions on increasing anddecreasing a (antit. b. a gi"en ratio4 e#g# in 3embertons Essentia+ Maths book ,#222122$# (P) (I)

.e4tbook3emberton ,#222122$

#2 Ca+c(+ate a gi"en ,ercentage of a(antit.#

Ex,ress one (antit. as a,ercentage of another#

Ca+c(+ate ,ercentage increase ordecrease#

he best starting ,oint here is to re"ise con"erting between ,ercentages and decima+s#<o( can (se exam,+es to find ,ercentages of (antities4 for exam,+e to find 5 of _2do 0#5 \ 2 L $#) so _$#)04 =remind +earners that in mone. ca+c(+ations it iscon"entiona+ to (se 2 d, for do++ar answers># <o( sho(+d enco(rage +earners to ,racticementa+ arithmetic methods too4 for exam,+e4 di"ide b. 0 to find 04 ha+"e this to find5 and add these res(+ts to find 5# (W) (8asic)

he next ste, is to (se exam,+es to show how to ex,ress one (antit. as a ,ercentageof another inc+(ding where there is a mixt(re of (nits# (W) (8asic)

Extend the work on finding ,ercentages of (antities to +ooking at how to ca+c(+ate,ercentage increases and decreases# For exam,+e to increase something b. 5m(+ti,+. b. #54 to decrease something b. 5 m(+ti,+. b. 0#85# 3ro"ide ,racticeexam,+es4 either write them .o(rse+f or from a textbook4 e#g# 3imente+ ,#)/1-$# (I)

.e4tbooks3imente+ ,#)/1-$Barton ,#012


7/23/2019 maths 0580 2014



(8asic)

It wo(+d be (sef(+ to +earners if .o( e+iminate the misconce,tion that increasing a(antit. b. 50 then decreasing the res(+ting (antit. b. 50 +eads back to theorigina+ "a+(e# * good wa. to do this is b. (sing the acti"e1+earning card sorting acti"it.

in A Bartons4 eacher eso(rce 7it ,ages 012# (G) (Callenging)


Carr. o(t ca+c(+ations in"o+"ingre"erse ,ercentages4 e#g# findingthe cost ,rice gi"en the se++ing ,riceand the ,ercentage ,rofit#

For extended +earners .o( wi++ need to mo"e on to ca+c(+ations in"o+"ing re"erse,ercentages# here are two good "ideos on+ine at .o(t(be#com and atbgrademaths#b+ocgs,ot#co#(k ex,+aining two different a,,roaches for re"erse,ercentages (estions# *sk +earners to com,are these methods and to decide whichmethod the. think is easier# (G) (H)

Online3TU htt,&66www#.o(t(be#com6watchV"L!U/10%,)IU3T

3TU htt,&66www#bgrademaths#b+ogs,ot#co#(k6200/60)6re"erse1,ercentages#htm+U3T

#$ %se a ca+c(+ator efficient+.#

*,,+. a,,ro,riate checks ofacc(rac.#

<o( might want to start this to,ic (sing exam,+es to show how to estimate the answerto a ca+c(+ation b. ro(nding each fig(re in the ca+c(+ation to sf# ;earners can thencheck their estimates b. doing the origina+ ca+c(+ation (sing a ca+c(+ator# Some goodexam,+es and ,ractice on this can be fo(nd in 3emberton ,#)# (P) (I) (8asic)

*n interesting extension acti"it.4 +inking this work to ,ercentages4 wo(+d be toin"estigate the ,ercentage error ,rod(ced b. ro(nding in ca+c(+ations (singaddition6s(btraction and m(+ti,+ication6di"ision# =<o( wo(+d need to ex,+ain ,ercentageerror beforehand># (G) (Callenging)

.e4tbook3emberton ,#)

# Ca+c(+ate times in terms of the 21ho(r and 21ho(r c+ock#

ead c+ocks4 dia+s and timetab+es#

* basic starting ,oint wo(+d be to re"ise the (nits (sed for meas(ring time4 withexam,+es showing how to con"ert between ho(rs4 min(tes and seconds# It is (sef(+ to(se te+e"ision sched(+es and b(s6train timetab+es to he+, with ca+c(+ations of timeinter"a+s and con"ersions between 21ho(r and 21ho(r c+ock formats# (W) (8asic)

*sk +earners to work in ,airs or sma++ gro(,s to create a timetab+e for a b(ses or trainsr(nning between two +oca+ towns# o extend a to,ic that is re+ati"e+. eas. for more ab+e

+earners4 there is an interesting case st(d. on+ine ca++ed sched(+ing an aircraft4 whichcan be fo(nd at cimt#,+.mo(th#ac#(k# (G) (P) (8asic) (Callenging)

It is (sef(+ for +earners to +ook at wor+d time differences and the different time :ones#<o( co(+d ask them to research and annotate a wor+d ma, with times in "ario(s citiesass(ming it is noon where .o( +i"e# imes can be fo(nd on+ine at timeanddate#com# (I)(H)

Online3TU htt,&66www#cimt#,+.mo(th#ac#(k6reso(rces6res6schedair#,df U3T

3 TUhtt,&66www#timeanddate#com6wor+dc+ock6

3TU htt,&66www#s,ringfrog#com6con"erter 6decima+1time#htmU3T

Past paper 3a,er $$4 D(ne 2024 U=b>=c>


http://www.youtube.com/watch?v=OQ9T1-0Up6I





http://www.bgrademaths.blogspot.co.uk/2009/06/reverse-percentages.html







http://www.cimt.plymouth.ac.uk/resources/res1/schedair.pdf








http://www.springfrog.com/converter/decimal-time.htm
















7/23/2019 maths 0580 2014



*n im,ortant ,oint for +earners to consider is common misconce,tions associated withtime ca+c(+ations# hen +earners do time ca+c(+ations on a ca+c(+ator and ha"e adecima+ answer for exam,+e4 5#$ em,hasise that this means 5 ho(rs 8 min(tes not 5ho(rs $ min(tes or 5 ho(rs $0 min(tes# <o( can i++(strate this we++ (sing the on+inedecima+ time con"erter at s,ringfrog#com# (Callenging)

o check their (nderstanding4 +earners can tr. the ,ast ,a,er (estion# (H) (F)

#5 Ca+c(+ate (sing mone. and con"ertfrom one c(rrenc. to another#

<o( can (se exam,+es showing how to so+"e straightforward ,rob+ems in"o+"ingexchange rates# * good reso(rce for this is Morrison and amshaws Co(rsebook,#2)12)5# It is (sef(+ for +earners if .o( +ink this work to s.++ab(s section 2#/ =(singcon"ersion gra,hs># %,1to1date exchange rates can be fo(nd from a dai+. news,a,er or on+ine at cnnfn#cnn#com# (W) (8asic)

Online3TU htt,&66cnnfn#cnn#com6markets6c(rrencies6U3T

.e4tbookMorrison ,#2)12)5

#) %se gi"en data to so+"e ,rob+emson ,ersona+ and ho(seho+d financein"o+"ing earnings4 sim,+e interestand com,o(nd interest#

Inc+(des disco(nt4 ,rofit and +oss#

Extract data from tab+es and charts#

* good ,+ace to start is to +ook at sim,+e ,rob+ems on ,ersona+ and ho(seho+d finance4(sing ,ractica+ exam,+es where ,ossib+e# For exam,+e4 taking information from,(b+ished tab+es or ad"ertisements# It wo(+d be (sef(+ for +earners if .o( introd(ce arange of sim,+e words and conce,ts here to describe different as,ects of finance4 forexam,+e tax4 ,ercentage ,rofit4 de,osit4 +oan4 etc# (W) (8asic)

he next ste, is to introd(ce the form(+a I L 3 =I L interest earned4 3 is thein"estment4 is the ,ercentage rate and is the time> to so+"e a "ariet. of ,rob+emsin"o+"ing sim,+e interest4 inc+(ding those re(iring +earners to (se rearranged "ersionsof the form(+a# Core +earners sho(+d a+so ha"e an (nderstanding of how to work o(tcom,o(nd interest4 idea++. in a sing+e ca+c(+ation4 for exam,+e the com,o(nd interestearned on an in"estment of _500 o"er .ears at a rate of $ interest is 500 \ #0$ ;

;#

(W) (8asic)

*n interesting homework task is to ask +earners to research the cost of borrowingmone. from different banks =or mone. +enders># (I) (H)

o check their (nderstanding4 +earners can tr. the ,ast ,a,er (estion# (H) (F)

Past paper 3a,er 24 'o" 2024 U)


7now+edge of com,o(nd interestform(+a is re(ired#Ka+(e of in"estment L +( Or 600>;

n;

where + is the amo(nt in"ested4 r is

Em,hasise to extended +earners that the. sho(+d know and be ab+e to (se the form(+aP( 1 r /155& ;

n; for compound interest. ome examples and %uestions can be

found in 2etcalf8s =xtended 0ourse book p.,65,63 (I)

.e4tbookMetca+f ,#2/012/$


http://cnnfn.cnn.com/markets/currencies/







7/23/2019 maths 0580 2014



the ,ercentage rate of interest andn is the n(mber of .ears ofcom,o(nd interest#

#- Extended curriculum only

%se ex,onentia+ growth and deca.in re+ation to ,o,(+ation and finance4e#g# de,reciation4 bacteria growth#

o introd(ce the to,ic of ex,onentia+ growth and deca. .o( wi++ find some good

exam,+es from the website khanacadem.#org which (ses the a,,roach nL ak ;

t ;

where n L n(mber at time t 4 a is the initia+ "a+(e and k is the rate# *sk +earners to com,are thesimi+arities between this ex,onentia+ growth form(+a and the com,o(nd interestform(+a# (W) (Callenging)

Online3TU

htt,&66www#khanacadem.#org6math6trigonometr.6ex,onentia+and+ogarithmicf(nc6ex,growthdeca.6"6wor d1,rob+em1so+"ing11ex,onentia+1growth1and1deca.U3T

!nit %# &lgebra and graps

Recommended prior knowledge

;earners sho(+d be ab+e to (nderstand the conce,t of (sing +etters to re,resent to re,resent (nknown n(mbers or "ariab+es@ know the meanings of the words termex%ression4 e)uation4 formula and function# he. sho(+d be confident with the work on directed n(mbers4 r(+es of indices and order of o,erations from %nit and knowthat a+gebraic o,erations fo++ow the same order as arithmetic o,erations# ;earners sho(+d be ab+e to generate terms of a sim,+e integer se(ence and find a term gi"enits ,osition in the se(ence@ find sim,+e term1to1term r(+es# he. sho(+d be ab+e to sim,+if. or transform +inear ex,ressions with integer coefficients co++ect +ike terms@m(+ti,+. a sing+e term o"er a bracket# he. sho(+d a+so know how to ex,ress sim,+e f(nctions a+gebraica++. and re,resent them in ma,,ings4 s(bstit(te ,ositi"e andnegati"e integers into form(+ae4 +inear ex,ressions and ex,ressions in"o+"ing sma++ ,owers4 e#g# $ x ;2; O or 2 x ;$;# ;earners sho(+d (nderstand the work from %nit 5 onCartesian co1ordinates in two dimensions and finding the gradient of a +ine and be ab+e to draw hori:onta+4 "ertica+ and diagona+ +ines from e(ations e#g# ' L $4 x L 124 ' L 5 x ? and x O $' L 2# ;earners sho(+d know how to work o(t areas of triang+es4 rectang+es and tra,e:i(ms#

Conte4this (nit re"ises and de"e+o,s mathematica+ conce,ts in a+gebra that are im,ortant in other ,arts of the co(rse# *,,roximate+. ha+f of the (nit is for core and extended+earners and the other ha+f is for extended +earners on+. which is ref+ected in the different teaching time recommendations# Sections which are for extended +earnerson+. are 2#$4 2#)4 2#84 2# and 2#24 a++ of the other sections ha"e ,arts for extended +earners on+. and these are indicated as s(ch thro(gho(t# It is antici,ated that+earners st(d.ing the extended s.++ab(s wi++ work thro(gh at a faster ,ace and sho(+d ha"e more ,rior know+edge of as,ects of the core s.++ab(s#

OutlineIt is intended for the to,ics in this (nit to be st(died se(entia++. as man. +ater to,ics re(ire know+edge from ear+ier ones4 for exam,+e finding the in"erse of a f(nctionis m(ch easier ha"ing st(died how to transform e(ations first# owe"er4 some ear+ier extended to,ics re(ire the know+edge of +ater core to,ics4 e#g# to transform aform(+a where the s(b9ect a,,ears more than once =extended section 2#> re(ires the ski++ of factorising =core section 2#2>4 conse(ent+.4 for extended +earners4 itmight be (sef(+ to co"er a++ the core work thoro(gh+. and se(entia++. before starting the extended work# he (nit co"ers a++ as,ects of a+gebra from the s.++ab(s4name+. constr(cting and rearranging e(ations@ ex,anding and factorising@ mani,(+ating a "ariet. of a+gebraic fractions@ working with indices@ so+"ing e(ations 1


http://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/word-problem-solving--exponential-growth-and-decay












7/23/2019 maths 0580 2014


inc+(ding +inear4 (adratic and sim(+taneo(s e(ations@ ine(a+ities@ direct and in"erse "ariation@ rates of change 1 inc+(ding tra"e+ gra,hs@ so+"ing e(ations gra,hica++.@estimating gradients of c(r"es and working with f(nctions# Some teachers ,refer to not teach a+gebra a++ in one b+ock4 it is ,ossib+e to +ea"e some sections (nti+ +ater inthe co(rse for exam,+e +inear ,rogramming =2#)> and direct and in"erse "ariation =2#8> co(+d be ta(ght +ater on in the co(rse4 simi+ar+. with man. other to,ics#

.eacing timeIt is recommended that this (nit sho(+d take a,,roximate+. 20?25 =Core +earners> $5?0 =Extended +earners> of the o"era++ IGCSE co(rse#


7/23/2019 maths 0580 2014



2# %se +etters to ex,ress genera+isedn(mbers and ex,ress basicarithmetic ,rocesses a+gebraica++.#

S(bstit(te n(mbers for words and

+etters in form(+ae#

ransform sim,+e form(+ae#

Constr(ct sim,+e ex,ressions andset (, sim,+e e(ations#

*n effecti"e start to this to,ic is re"ising basic a+gebraic notation4 for exam,+e4 a O a L2a4 b \ c L bc =em,hasising that cb is the same as bc b(t that the con"ention is to write+etters in a+,habetica+ order># *+so +ook at sim,+e exam,+es with indices d \ d L d;

2; and e

\ e \ e \ e L e;

;# Ex,+ain to +earners how to s(bstit(te n(mbers into a form(+a4 inc+(ding

form(+ae that contain brackets# (W) (8asic)

*n interesting in"estigation is to ask +earners to work in gro(,s to +ook at the differencebetween sim,+e a+gebraic ex,ressions which are often conf(sed# For exam,+e4 find thedifference between 2 x 4 2 O x and x ;2; for different "a+(es of x # *sk +earners Qis there an(mber that makes them a++ e(a+VR (G) (8asic)

!nce the basics are sec(re mo"e on to transforming sim,+e form(+ae4 for exam,+erearranging ' L ax O b to make x the s(b9ect# ;earners need to (nderstand how toconstr(ct sim,+e ex,ressions and e(ations from word ,rob+ems# (W) (8asic)

*n exce++ent extension acti"it. is the ,(::+e4 ,erimeter ex,ressions4 at nrich#maths#org(I) (H) (Callenging)

Online3TU htt,&66nrich#maths#org6-28$U3T


S(bstit(te n(mbers for words and+etters in com,+icated form(+ae#

Constr(ct and transformcom,+icated form(+ae ande(ations4 e#g# transform form(+aewhere the s(b9ect a,,ears twice#

For extended +earners .o( wi++ need to b(i+d on a++ of the work from the core ,art ofto,ic 2## Mo"ing on to more com,+icated form(+ae when s(bstit(ting4 for exam,+ethose with man. orders of o,erations to consider# <o( can +ink the work on transformingform(+ae to the work on so+"ing e(ations4 asking +earners to think abo(t the ba+ancemethod (sed in both# (W) (Callenging)

* (sef(+ assessment too+ is the ,ast ,a,er4 D(ne 20$ U5# (I) (H) (F)

Exam,+es of constr(cting more com,+icated e(ations and ex,ressions can be fo(nd inthe ,ast ,a,er4 D(ne 20$ U5abc# (F)

he fina+ ste, is to ex,+ain to +earners how to transform com,+ex form(+ae4 for exam,+e4

x ;2 PO ' ;2 PL r ;2P4 s L ut O ]at ;2P4 ex,ressions in"o+"ing s(are roots4 etc# <o( can (se

a series of exam,+es to i++(strate how to transform form(+ae containing a+gebraic

fractions4 =with ,ossib+e +inks to the work in to,ic 2#$> for exam,+e1

f =

1

u+

1

v (W)

(Callenging)

*sk +earners to tr. the ,ast ,a,er4 D(ne 20$ U20# (F)

Past papers3a,er 24 D(ne 20$4 U53a,er 24 D(ne 20$4 U5=a>=b>=c>3a,er 224 D(ne 20$4 U203a,er 24 'o" 2024 U)







7/23/2019 maths 0580 2014



he most cha++enging form(+a to transform4 which deser"es time s,ending on it4 iswhere the s(b9ect a,,ears twice# %sing exam,+es showing the factorising and di"idingthro(gh methods4 .o( can disc(ss the benefits of each# <o( can +ink this work to to,ic2#2# (W) (Callenging)

* good ,ast ,a,er (estion on this to,ic is 'o" 202 U)# (F)

2#2 Mani,(+ate directed n(mbers#

%se brackets and extract commonfactors#

e#g# ex,and$ x =2 x ? ' >4 = x O >= x ? ->e#g# factorise / x ;2; O 5 x'

*n im,ortant starting ,oint is to re"ise a++ as,ects of directed n(mbers with a++ fo(ro,erations and +ink this to ,ositi"e and negati"e a+gebraic terms with the fo(ro,erations# he inabi+it. to dea+ with negati"e n(mbers can otherwise ca(se an(nnecessar. st(mb+ing b+ock in a+gebraic work# (W) (8asic)

<o( wi++ need to (se exam,+es4 with both ,ositi"e and negati"e n(mbers4 to i++(strateex,anding brackets# Start sim,+. with a sing+e term being m(+ti,+ied o"er a bracketcontaining two or more terms# Extend this techni(e to m(+ti,+.ing two sim,+e +inearbrackets together for exam,+e = x ? $>= x O -># ;earners ma. find it (sef(+ to see a 2\2a+gebraic m(+ti,+ication grid to he+, with their (nderstanding# (W) (8asic)

he next ste, is to (se exam,+es4 with both ,ositi"e and negati"e n(mbers4 to i++(stratefactorising sim,+e ex,ressions with one bracket# Ex,+ain that factorising is the re"erseof ex,anding# * good so(rce of exam,+es and (estions can be fo(nd in 3imente+ ,#/)1/8 and 0102# (I) (8asic)

*sk +earners to tr. the ,ast ,a,er D(ne 20$ U)# (F)

.e4tbook3imente+ ,#/)1/84 0102

Past paper 3a,er 24 D(ne 20$4 U)


Ex,and ,rod(cts of a+gebraicex,ressions#Factorise where ,ossib+eex,ressions of the form&ax O bx O ka' O kb' a;

2; x ;2; ? b;

2;' ;2

a;

2; O 2ab O b;

2

ax ;2; O bx O c

For extended +earners mo"e on to exam,+es where the. wi++ need to find the ,rod(cts of a+gebraic ex,ressions4 for exam,+e = x ;2; O $ x ? >= x ? 5># (W) (Callenging)

B(i+ding on the ear+ier factorising work (se exam,+es to show +earners how to factorisethree term (adratic e(ations initia++. where the coefficient of x ;2; is # Inc+(de sim,+edifference of two s(are exam,+es s(ch as x ;

2; ? ) em,hasising that these can be

so+"ed (sing the same method as three term (adratics bearing in mind that thecoefficient of the x term is 0# <o( can gi"e +earners some (estions ,racticingfactorising sim,+e (adratics from the ,ower ,oint ,resentation4 s+ide 54 on+ine attes#co#(k# (I) (H)

he next ste, is +ooking at factorising b. gro(,ing# *n exam,+e of the kind of (estion+earners might see can be fo(nd in the ,ast ,a,er D(ne 20$ U0# (F)

Online3TU htt,&66www#tes#co#(k6teaching1reso(rce6Factorising1U(adratic1Ex,ressions1)$20226U3T

5,8ook3TU htt,&66mrbartonmaths#com6ebook#htmU3T

Past paper 3a,er 24 D(ne 20$4 U0


http://www.tes.co.uk/teaching-resource/Factorising-Quadratic-Expressions-6320242/

















7/23/2019 maths 0580 2014



* rea++. cha++enging to,ic is that of factorising (adratics where the coefficient of x ;2; isnot # It is worth s,ending a considerab+e amo(nt of time on this to,ic inc+(dingre"isiting it thro(gho(t the co(rse to ens(re methods are not forgotten# * higher orderthinking ski++ is to ask +earners to com,are different methods for tack+ing a (estion#his is ,artic(+ar+. (sef(+ for more ab+e +earners# *sk them to com,are the two different

methods for factorising (adratics of the form ax ;2; O bx c4 where a ≠ # he firstmethod can be fo(nd in the ,ower ,oint ,resentation4 s+ide )4 on+ine at tes#co#(k=which (ses s,+itting the x term into two terms and then factorising b. gro(,ing># hesecond method can be fo(nd in Mr Bartons E1book4 the section more factorising)uadratics ,ages - and 8 =which (ses a tria+ and error a,,roach># (W) (Callenging)

Fina++. gi"e +earners exam,+es how to factorise harder difference of two s(ares,rob+ems4 for exam,+e# ) x ;2; ? 25' ;2;# It is a+so worth mentioning two1term (adraticfactorising exam,+es s(ch as 8 x ;2; ? 2 x # Em,hasise that these are often ,oor+.answered# 3oint o(t that beca(se the. are (adratics +earners often tr. to (se two setsof brackets instead of 9(st the one set of brackets re(ired# (W) (Callenging)

2#$ Extended curriculum only

Mani,(+ate a+gebraic fractions#

Factorise and sim,+if. rationa+ex,ressions#

B(i+ding on the work on factorising in to,ic 2#2 show +earners how to factorise and

sim,+if. rationa+ ex,ressions s(ch as for exam,+e x2−2 x

x2−5 x+6

(W) (Callenging)

3ro"ide +earners with ,+ent. of exam,+es and (estions =e#g# '.e ,#0210$># It is worth+inking this work on sim,+if.ing rationa+ ex,ressions to the work on (sing the fo(r r(+eswith a+gebraic fractions so that +earners a+wa.s gi"e the most sim,+ified answer# (P) (I)(Callenging)

o assess +earners (nderstanding of this to,ic ask them to com,+ete the ,ast ,a,erD(ne 20$ U8# (F)

hen .o( wi++ need to s,end time re"ising adding and s(btracting sim,+e fractions with

+earners4 for exam,+e 25+ 3

8 # Ex,+aining the ,rocess of finding a common

denominator b.4 in this case4 m(+ti,+.ing the two denominators# *sk +earners to disc(sswhen the +owest common denominator doesnt need to be the ,rod(ct of the two

denominators4 if .o( need to .o( can gi"e the exam,+e3

10+

5

8# (W) (8asic)

.e4tbook

'.e ,#0210)

Past paper 3a,er 24 D(ne 20$4 U83a,er 24 D(ne 20$4 U22


7/23/2019 maths 0580 2014



he next ste, is to mo"e on to a+gebraic fractions starting with n(merica+ denominators4

for exam,+e x

3 O

x−4

24

2 x

3 1

3 ( x−5)2

then extending this to a+gebraic

denominators4 for exam,+e1

x−2

O2

x−3

# <o( wi++ need to em,hasise common

errors that occ(r when s(btracting a+gebraic fractions# For exam,+e in the (estion3

x−5−

4

x+2 ex,+ain that it is common to see sign errors on the n(merator when x

? 5 is m(+ti,+ied b. 1# (W) (Callenging)

*fter this mo"e on to exam,+es demonstrating m(+ti,+.ing and di"iding with n(merica+fractions4 reminding +earners that instead of di"iding b. a fraction .o( m(+ti,+. b. itsreci,roca+# (W) (8asic)

Extend this b. +ooking at a+gebraic fractions for exam,+e 3a

4 \

9a

104

3a

4 ^

9a

10# (W) (Callenging)

3ro"ide exam,+e (estions for +earners to ,ractice =from textbooks4 e#g# '.e ,#010)>and the ,ast ,a,er D(ne 20$ U22 is a+so worth doing# (I) (F)

2# %se and inter,ret ,ositi"e4 negati"eand :ero indices#%se the r(+es of indices#

* good starting ,oint is to gi"e +earners exam,+es re"ising the r(+es of indices work from%nit to,ic #-# Extend this to (sing and inter,reting ,ositi"e4 negati"e and :ero indicesand (sing the r(+es of indices with a+gebraic terms4 for exam,+e4 sim,+if.& $ x ;; \ 5 x 4 0 x ;$;

^ 2 x ;2;4 = x ;);>;

2;# (W) (8asic)


%se and inter,ret fractiona+ indices#%se the r(+es of indices#

For extended +earners4 mo"e on to +ooking at fractiona+ indices# For exam,+e4 sim,+if.&

3 x

−4

×

2

3 x

1

2

4

2

5 x

1

2

÷ 2 x

−2

4

(2 x

5

3 )3

and so+"ing ex,onentia+ sim,+e

e(ations4 e#g# so+"e $2 ;

x

; L 2# (W) (Callenging)

For an exce++ent acti"e +earning reso(rce4 that .o( can (se with +earners working ingro(,s4 see the indices card sorting acti"it. in Bartons eacher eso(rce 7it ,ages )51))# (G)

.e4tbookBarton ,#)51))

Past paper 3a,er 24 D(ne 20$4 U


7/23/2019 maths 0580 2014



o assess +earners (nderstanding of the to,ic ask them to com,+ete the ,ast ,a,erD(ne 20$ U# (F)

2#5 So+"e sim,+e +inear e(ations in one(nknown#

So+"e sim(+taneo(s +inear e(ationsin two (nknowns#

Begin this work with re"ising how to so+"e sim,+e +inear e(ations4 inc+(ding those withnegati"es4 for exam,+e $ x O 2 L 1# <o( wi++ a+so want to inc+(de exam,+es showing how

to so+"e +inear e(ations with brackets4 for exam,+e 5= x O > L $= x O 0># (W) (8asic)

For a f(n acti"e +earning reso(rce ask +earners to work in gro(,s to com,+ete the sim,+ee(ations 9igsaw acti"it. from tes#co#(k# Man. more 9igsaws are a"ai+ab+e atmrbartonmaths#com which a+so contains the +ink for down+oading the arsia software to"iew the 9igsaws# (G) (8asic)

* good introd(ction to sim(+taneo(s e(ations is a non1a+gebraic a,,roach4 forexam,+e $ coffees and 2 teas cost _)#50 and 5 coffees and 2 teas cost _/#50# Showing+earners how the sim(+taneo(s e(ation from these statements can be formed andem,hasising that the cost of tea and coffee does not change# (W) (8asic)

Extend this b. +ooking at exam,+es to i++(strating how to so+"e sim(+taneo(s +inear

e(ations with two (nknowns b. e+imination4 s(bstit(tion and finding a,,roximateso+(tions (sing gra,hica+ methods =+inking to to,ic 2#0># (W)

* sam,+e (estion can be fo(nd in the ,ast ,a,er D(ne 20$ U0# (I) (F)

Online3TU htt,&66www#tes#co#(k6eso(rceAetai

+#as,xVstor.CodeL)08-58U3T

3TU htt,&66www#mrbartonmaths#com69igsaw#htmU3T



So+"e (adratic e(ations b.factorisation4 com,+eting the s(areor b. (se of the form(+a#

So+"e sim,+e +inear ine(a+ities#

Extended +earners wi++ then need to ex,+ore a++ the different methods for so+"ing(adratic e(ations4 name+. b. factorisation4 (sing the (adratic form(+a andcom,+eting the s(are =for rea+ so+(tions on+.># he best starting ,oint is (singexam,+es of the form ax ;2; O bx O c L 0 then extend this b. +ooking at e(ations re(iringrearranging into this form first4 e#g# a.ner ,#-/18)# (W) (G) (P) (I)

* more cha++enging acti"it. in"o+"es +earners needing to constr(ct their own e(ationsfrom information gi"en and then so+"e them to find the (nknown (antit. or (antities#his co(+d in"o+"e the so+(tion of +inear e(ations4 sim(+taneo(s e(ations or (adratice(ations# (W) (Callenging)

o introd(ce the to,ic of so+"ing +inear ine(a+ities it is a good idea starting with 9(stn(mbers4 for exam,+e - X 54 showing that that m(+ti,+.ing or di"iding an ine(a+it. b. anegati"e n(mber re"erses the ine(a+it. sign4 i#e# 1- Y 15# <o( can (se exam,+es toi++(strate how to so+"e sim,+e +inear ine(a+ities inc+(ding re,resenting the ine(a+it. ona n(mber +ine4 e#g# from textbooks s(ch as 3emberton CA %nit 2# (W) (Callenging)

.e4tbooka.ner ,#-/18)

C7,Rom3emberton %nit 2 ? a+gebra4 so+"ing+inear ine(a+ities



http://www.tes.co.uk/ResourceDetail.aspx?storyCode=6108758














7/23/2019 maths 0580 2014



he most cha++enging ine(a+ities for +earners to so+"e are those where the ine(a+it.needs to be s,+it into two ,arts and each ,art so+"ed se,arate+. =see the fina+ s+ide ofthe ,ower ,oint ,resentation># he D(ne 20$ ,ast ,a,er U8 is a good exam,+es# (W)(F) (Callenging)

2#) Extended curriculum only

e,resent ine(a+ities gra,hica++.and (se this re,resentation in theso+(tion of sim,+e +inear,rogramming ,rob+ems#

'ote& he con"entions of (singbroken +ines for strict ine(a+itiesand shading (nwanted regions wi++be ex,ected#

* good starting ,oint is to begin b. asking +earners to draw a n(mber of straight +ines ona set of axes4 ,ossib+. on mini white boards4 for exam,+e ' L 24 x L 154 ' L $ x and x O 2' L 0# *sk +earners to consider a ,oint on one side of each of these +ines4 theorigin if ,ossib+e4 and (se s(bstit(tion to see if the ine(a+ities ' Y 24 x X 1 54 ' Y $ x and x O 2' X 0 are tr(e for that ,artic(+ar ,oint# *sk +earners to work in gro(,s to do theirown exam,+es simi+ar to those a+read. s(ggested# (G) (8asic)

Extend this work b. asking +earners to +ook at exam,+es i++(strating how to so+"e +inear,rogramming ,rob+ems b. gra,hica+ means4 high+ighting the convention of usingbroken lines for strict ine%ualities K and L and solid lines for the ine%ualitiesM and N# (W) (Callenging)

Fina++.4 +earners wi++ need to (nderstand how to constr(ct ine(a+ities from constraintsgi"en4 showing that a n(mber of ,ossib+e so+(tions to a ,rob+em exist4 indicated b. the(nshaded region on a gra,h# Create .o(r own exam,+es and (estions or (se those intextbooks4 e#g# 3emberton ,#281$# (W) (Callenging)

.e4tbook3emberton ,#281$

Past paper 3a,er $4 D(ne 20$4 U$

2#- Contin(e a gi"en n(mber se(ence#

ecognise ,atterns in se(encesand re+ationshi,s between differentse(ences#

Find the nth term of +inearse(ences4 sim,+e (adratic andc(bic se(ences#

;earners wi++ find it (sef(+ to ha"e the definition of a se(ence of n(mbers# Begin b.asking +earners to work in gro(,s to in"estigate some sim,+e se(ences4 for exam,+efinding the next two n(mbers in a se(ence of e"en4 odd4 s(are4 triang+e or Fibonaccin(mbers4 etc# (G) (8asic)

Extend this to +ooking at finding the term1to1term r(+e for a se(ence4 for exam,+e these(ence $4 /4 54 24 2-4 ###4 has a term1to1term r(+e of O)@ the se(ence 04 204 04 542#54 P4 has a term1to1term r(+e of ^2# ;earners wi++ need to ha"e some a,,reciation ofthe +imitations of a term1to1term r(+e4 i#e# that the. are not "er. (sef(+ for finding termsthat are a +ong wa. down the se(ence# his +eads on nice+. to finding the ,osition1to1term r(+e for a se(ence b. examining the common difference4 for exam,+e the nth term in the se(ence $4 /4 54 24 2-4 P4 is )n 1 $# (W) (8asic)

*sk +earners to tr. the ,ast ,a,er D(ne 20$ U$# (F)

*n interesting in"estigation is to +ook at s(are tab+es ,+aced in a row so that ,eo,+e

Past paper 3a,er 224 D(ne 20$4 U$


7/23/2019 maths 0580 2014



can sit aro(nd one tab+e4 ) ,eo,+e can sit aro(nd 2 tab+es 9oined4 8 ,eo,+e can sitaro(nd $ tab+es 9oined4 and so on# *sk +earners to work o(t how man. ,eo,+e can sitaro(nd n tab+es# o add extra cha++enge ask +earners to in"estigate tab+es of differentsha,es and si:es and to tr. to re+ate the nth term form(+a to the ,ractica+ sit(ationex,+aining how the n(mbers in the form(+a re+ate to the arrangements of the tab+es# (P)

(I) (H)

ith more ab+e +earners .o( co(+d +ook at deri"ing the form(+a for a +inear se(ence nthterm L a O =n 1 >d where a is the first term and d is the common difference4 this form(+ais not essentia+ know+edge# (P) (I) (Callenging)#

*nother a,,roach is +ooking at ,atterns and re+ationshi,s between different se(ences#For exam,+e4 the se(ence 24 54 04 -4 2)4 P4 is the s(are n(mbers O # <o( cangi"e +earners se"era+ exam,+es of these asking them to find the nth term4 (sing 9(stsim,+e (adratic and c(bic se(ences i#e# of the form an;

2; c or an;

$; c # (P) (I) (H)


Find the nth term (adratic andc(bic se(ences4 ex,onentia+se(ences and sim,+e combinationsof these#

For extended +earners extend the core work b. +ooking at exam,+es of finding the nthterm of harder (adratic se(ences# * (sef(+ reso(rce is the "ideo and in ,artic(+ar the

a,,+et which can be fo(nd on+ine at wa+domaths#com# ;earners can work in gro(,s(sing the a,,+et to in"estigate finding the nth term of harder (adratic se(ences# (G) (Callenging)

For e"en more cha++enge .o( can extend this f(rther sti++ b. in"estigating c(bicse(ences (sing the "ideo and a,,+ets again# (G) (Callenging)

!ther methods for finding nth terms are ,ossib+e# *sk +earners to search on+ine fora+ternati"e methods for finding nth terms# (P) (I) (H) (Callenging)

o assess +earners (nderstanding of this to,ic .o( can (se the ,ast ,a,er D(ne 20$U0# (F)

Fina++.4 +earners wi++ need to +ook at ex,onentia+ se(ences with a common m(+ti,+ier =or ratio> instead of a common difference# ith more ab+e +earners deri"e the form(+a forthe nth term L ar ;=n1>

; where a is the first term and r is the common ratio4 this form(+a isnot essentia+ know+edge# (Callenging)

OnlineKideo (adratic se(ences&3TU

htt,&66www#wa+domaths#com6"ideo6U(adSe06U(adSe0#9s,U3T

*,,+et (adratic se(ences&3TU htt,&66www#wa+domaths#com6U(adSe2;#9s,U3T

Kideo c(bic se(ences&3TU htt,&66www#wa+domaths#com6"ideo6C(bSe06C(bSe0#9s,U3T

*,,+et c(bic se(ences&3TU htt,&66www#wa+domaths#com6C(bSe

;#9s,U3T


2#8 Extended curriculum only

Ex,ress direct and in"erse "ariation

;earners wi++ need to be ab+e to so+"e a "ariet. of ,rob+ems in"o+"ing direct or in"erse"ariation# Efficient notation to be enco(raged is mo"ing from the (estion4 for exam,+e4' "aries direct+. with x =or ' is direct+. ,ro,ortiona+ to x > to each of these ste,s in t(rn '

.e4tbooksMorrison ,#)$1)-Barton ,#8/1/

"#0 < Cambridge IGCSE Mathematics =0580> ? from 205 2$

http://www.waldomaths.com/video/QuadSeq01/QuadSeq01.jsp





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7/23/2019 maths 0580 2014



in a+gebraic terms and (se this formof ex,ression to find (nknown(antities#

O x P ' L kx # *nother (sef(+ exam,+e is t "aries in"erse+. as the s(are root of * P t O1

v2

P t Lk

v2

4 where k is a constant# Em,hasise the common error of re"ersing

direct and in"erse "ariation# !nce the form(+a has been estab+ished ask +earners to (se

gi"en "a+(es to work o(t the "a+(e of the constant4 k4 and then (se the form(+ae withthe e"a+(ated k# Exam,+es and (estions for +earners to ,ractise this can be fo(nd intextbooks4 e#g# Morrison ,#)$1)- and Barton ,#8/1/# (W) (8asic)

o assess (nderstanding4 ask +earners to tr. the ,ast ,a,er D(ne 20$ U/# (I) (H) (F)

Past paper 3a,er 24 D(ne 20$4 U/

2#/ Inter,ret and (se gra,hs in ,ractica+sit(ations inc+(ding tra"e+ gra,hsand con"ersion gra,hs#

Araw gra,hs from gi"en data#

* good starting ,oint is to draw and (se straight +ine gra,hs to con"ert betweendifferent (nits4 for exam,+e between metric and im,eria+ (nits or between differentc(rrencies# Exchange rates can be fo(nd at cnnfn#cnn#com which can be (sef(+ forsetting (estions# ;earners need to be confident in so+"ing ,rob+ems (sing com,o(ndmeas(res# It wi++ be (sef(+ to +earners to +ink this work to the work from to,ic # and#5 of the s.++ab(s# (W) (8asic)

It is im,ortant for +earners to be ab+e to draw a "ariet. of gra,hs from gi"en data4 forexam,+e to determine whether two (antities =' and x or for more ab+e +earners ' and x ;2;4 etc#> are in ,ro,ortion# <o( wi++ be ab+e to +ink this to the work in to,ic 2#8 on directand in"erse "ariation =for extended +earners># 3ro"ide +earners with exam,+es and(estions4 either ,re,ared .o(rse+f or from textbooks4 e#g# Metca+f ,#2-$12-# (W)(8asic)

Online3TU htt,&66cnnfn#cnn#com6markets6c(rrencies6U3T

.e4tbooksMetca+f ,#2-$12-/Barton ,#-$1-5


*,,+. the idea of rate of change toeas. kinematics in"o+"ing distance1time and s,eed1time gra,hs4acce+eration and dece+eration#

Ca+c(+ate distance tra"e++ed as area(nder a +inear s,eed1time gra,h#

For extended +earners .o( wi++ want to ,ro"ide exam,+es of how to draw and (sedistance1time gra,hs to ca+c(+ate a"erage s,eed =+inking this to the ca+c(+atinggradients work in to,ic 5#2># ;earners sho(+d be ab+e to inter,ret the information shownin tra"e+ gra,hs and to be ab+e to draw tra"e+ gra,hs from gi"en data# *sk +earners todraw a tra"e+ gra,h for an imaginar. 9o(rne. and to write a set of (estions abo(t this 9o(rne.# For exam,+e Qwhat was the a"erage s,eedVR (I) (H)

hen +earners ha"e drawn their gra,hs and written their (estion the. can then gi"ethese to other members of a gro(, to answer# (G) (P)

<o( wi++ need to ens(re that +earners ha"e st(died to,ic #2 and that the. canconfident+. ca+c(+ate areas of rectang+es4 triang+es4 tra,e:i(ms and com,o(nd sha,esderi"ed from these# (W) (8asic)

Extend this work b. +ooking at exam,+es of s,eed1time gra,hs being (sed to find










7/23/2019 maths 0580 2014



acce+eration and dece+eration and to ca+c(+ate distance tra"e++ed as area (nder a +inears,eed1time gra,h# %sef(+ exam,+es and (estions can be fo(nd in Metca+fs ExtendedCo(rse book ,#2-512-/# (W)

Cha++enge can be ,ro"ided b. +ooking at exam,+es where +earners are re(ired to

con"ert between different (nits# For exam,+e4 where different (nits are being (sed inthe (estion and in the gra,h# (W) (Callenging)

An interesting group activit' is to ask learners to look at the distancetimeand speedtime card sorting activit' from Qarton8s Teacher Resource Sitp.I3I4. (G)

o assess their (nderstanding of this work ask +earners to tr. the ,ast ,a,er (estion#(I) (H) (F)

2#0 Constr(ct tab+es of "a+(es forf(nctions of the formax O b4 x ;2 ;O ax O b4 a,x (x W -#

where a and b are integra+constants#

Araw and inter,ret s(ch gra,hs#So+"e +inear and (adratice(ations a,,roximate+. b.gra,hica+ methods#

Begin this to,ic b. drawing a series of +ines with x L constant and . L constant# *sk+earners to identif. the e(ations of the +ines that .o( ha"e drawn# Em,hasise theim,ortance of (sing a r(+er and a shar, ,enci+ in mathematica+ diagrams thro(gho(t

this to,ic# (W) (8asic)

Mo"e on to exam,+es of drawing diagona+ straight +ine gra,hs from a tab+e of "a+(eswhere the gradient and interce,t are integers# <o( can +ink this to the work on gradientin to,ic 5#2# (W) (8asic)

*s an extension acti"it. .o( co(+d ask +earners to find o(t how to (se the gradient andinterce,t to draw a +ine4 an exam,+e can be fo(nd in the on+ine +esson at math#com#;earners can work as a gro(, to ex,+ore and com,are the methods for drawing +inesfrom e(ations# (G) (Callenging)

Extend this to +ooking at drawing (adratic f(nctions of the form x ;2 ;O ax O b4 andsim,+e reci,roca+ f(nctions s(ch as a,x (x W -#. ;earners sho(+d be ab+e to draw a

"ariet. of these gra,hs confident+. and acc(rate+. from a tab+e of "a+(es4 introd(ce theterms ,arabo+a and h.,erbo+a =a+tho(gh these are not re(ired># <o( can then disc(sswith +earners the s.mmetr. ,ro,erties of a (adratic gra,h and how this is (sef(+# (W)

he next ste, is to show how the so+(tions to a (adratic e(ation ma. bea,,roximated (sing a gra,h# Extending this work to show how the so+(tion=s> to ,airs of e(ations =for exam,+e ' L x ;2; 1 2 x 1 $ and ' L x > can be estimated (sing a gra,h# hiswork can be +inked to the work on sim(+taneo(s e(ations from to,ic 2#5# (W)

Online3TU htt,&66www#math#com6schoo+6s(b9ect26+essons6S2%;$G;#htm+U3T









7/23/2019 maths 0580 2014



(Callenging)


Constr(ct tab+es of "a+(es and drawgra,hs for f(nctions of the form ax ;n;4where a is a rationa+ constant4 andn L ?24 ?4 04 4 24 $4 and sim,+es(ms of not more than three ofthese and for f(nctions of the forma;

x;4 where a is a ,ositi"e integer#

So+"e associated e(ationsa,,roximate+. b. gra,hica+methods#

Araw and inter,ret gra,hsre,resenting ex,onentia+ growth

and deca. ,rob+ems#

Software drawing ,ackages s(ch as Geogebra are (sef(+ here for +earners to (se toin"estigate different feat(res of gra,hs# Geogebra is free to down+oad4 on+ine fromgeogebra#org# <o( wi++ ,robab+. want to start b. asking +earners to draw f(nctions of theform a6 x ;2;@ a6 x @ ax ;$;@ a;

x ;@ where a is a constant4 (sing a gra,h drawing ,ackage +ike

Geogebra# *sk +earners to work in gro(,s to (se the software to gain an awareness ofwhat each of the different t.,es of gra,hs +ook +ike# ;earners sho(+d be in a ,osition torecognise common t.,es of f(nctions from their gra,hs4 for exam,+e from the ,arabo+a4h.,erbo+a4 (adratic4 c(bic and ex,onentia+ gra,hs# (G) (8asic)

hen mo"e on to asking +earners to draw the gra,hs from tab+es of "a+(es# * (sef(+"ideo exam,+e +esson can be fo(nd on+ine at khanacadem.#org# Extend the work toinc+(de sim,+e s(ms of not more than three f(nctions in the form ax ;n;4 where a is arationa+ constant4 and n L ?24 ?4 04 4 24 $# *sk +earners to so+"e associated e(ationsa,,roximate+. (sing these gra,hs# (W) (Callenging)

For assessment ,(r,oses the ,ast ,a,er (estion is "er. (sef(+ b(t it does re(ire

to,ic 2# to ha"e a+so been st(died# (F)

he fina+ ste, is to +ook at exam,+es of how to draw and inter,ret gra,hs re,resentingex,onentia+ growth and deca. ,rob+ems# It wi++ be (sef(+ to +earners to +ink this to thework from to,ic #-# (W) (Callenging)

Online3TU htt,&66www#geogebra#org6cms6en6U3T

3TU htt,&66www#khanacadem.#org6math6trigonometr.6ex,onentia+and+ogarithmicf(nc6ex,growthdeca.6"6gra,hing1ex,onentia+1f(nctionsU3T


2# Extended curriculum only

Estimate gradients of c(r"es b.drawing tangents#

Ens(re +earners ha"e st(died to,ic 5#2 =finding the gradient of a straight +ine> beforebeginning this to,ic# he. sho(+d a+read. be ab+e to confident+. find the gradient of astraight +ine# ;earners wi++ find it (sef(+ to ha"e a definition of the term tangent# (W)(8asic)

Mo"e on to +ooking at exam,+es showing how to find the gradient at a ,oint on a c(r"eb. drawing a tangent at that ,oint# (W) (Callenging)

;earners can tr. the ,ast ,a,er (estion4 bearing in mind this (estion a+so re(iresto,ic 2#0 to ha"e been st(died# (F)



%se f(nction notation4e#g# f= x > L $ x ? 54 f& x $ x ? 54 todescribe sim,+e f(nctions#

* (sef(+ staring ,oint is to gi"e +earners a definition of a f(nction4 f=x> i#e# that it is a r(+ea,,+ied to "a+(es of x# ;ook at e"a+(ating sim,+e f(nctions4 for exam,+e +inear f(nctions4for s,ecific "a+(es4 describing the f(nctions (sing f=x> notation and ma,,ing notation#(W) (8asic)

.e4tbook3earce ,#2/1$0$

Online3TU htt,&66www#khanacadem.#org6math6

"#0 < Cambridge IGCSE Mathematics =0580> ? from 205 2)






http://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/graphing-exponential-functions








http://www.khanacademy.org/math/algebra/algebra-functions/function_inverses/v/function-inverse-example-1








7/23/2019 maths 0580 2014



Find in"erse f(nctions f ; ?; = x >#

Form com,osite f(nctions asdefined b. gf= x > L g=f= x >>#

he next ste, is to introd(ce the in"erse f(nction as an o,eration which (ndoes theeffect of a f(nction# Aemonstrate how +earners can e"a+(ate sim,+e in"erse f(nctions for s,ecific "a+(es4 describing the f(nctions (sing the f ;1;=x> notation and ma,,ing notation#It wi++ be (sef(+ to +earners to +ink this to the work on transforming form(+ae from to,ic2## Ex,+ain to +earners that to find the in"erse of the f(nction f= x > L 2 x ? 54 a (sef(+

method is to rewrite this as ' L 2 x ? 54 then to interchange the x and ' to get x L 2' ? 54then to make . the s(b9ect ' L = x O 5>62 and fina++. to re1write (sing the in"erse f(nctionnotation asf ;1;= x > L = x O 5>62# (W) (Callenging)

%sing +inear and6or (adratic f(nctions4 f= x > and g= x >4 show +earners how to formcom,osite f(nctions s(ch as gf= x >4 and how to e"a+(ate them for s,ecific "a+(es of x # Itwi++ be (sef(+ for +earners to in"estigate4 for a "ariet. of different f(nctions gf= x > and fg= x >in order to see that these are not "er. often the same# Em,hasise that it is im,ortantthat +earners know the correct order to a,,+. the f(nctions# (W) (Callenging)

3ro"ide +earners with exam,+es and (estions4 either ,re,ared .o(rse+f or fromtextbooks4 e#g# 3earce4 or the "ideo at khanacadem.#org which a+so ta+ks abo(t what

the gra,h of an in"erse f(nction +ooks +ike# 7nowing that the gra,h of an in"ersef(nction is a ref+ection in the +ine ' L x is not re(ired know+edge b(t is a (sef(+extension for the more ab+e +earners# (W)

*sk +earners to tr. the ,ast ,a,er (estion to assess their (nderstanding on this to,ic# (F)

a+gebra6a+gebra1f(nctions6f(nctionin"erses6"6f(nction1in"erse1exam,+e1U3T

Past paper

3a,er 224 D(ne 20$4 U2

!nit '# Geometr

Recommended prior knowledge;earners sho(+d be ab+e to confident+. (se a r(+er4 ,rotractor and ,air of com,asses4 the first two as meas(ring too+s as we++ as for drawing# ;earners sho(+d a+so knowthe names of common 2A and $A sha,es s(ch as (adri+atera+4 rectang+e4 triang+e =e(i+atera+4 isosce+es4 right ang+ed and sca+ene>4 ,ara++e+ogram4 tra,e:i(m4 kite4circ+e4 s,here4 c.+inder4 ,rism4 ,.ramid4 c(be4 c(boid and cone# he. sho(+d a+so know the names of the ,arts of a circ+e& radi(s4 diameter4 circ(mference and chord4(nderstand the terms ,er,endic(+ar and ,ara++e+ and be fami+iar with the (se of these in re+ation to s,ecia+ (adri+atera+s sides and diagona+s#

Conte4t

"#0 < Cambridge IGCSE Mathematics =0580> ? from 205 2-








7/23/2019 maths 0580 2014


his (nit re"ises and de"e+o,s mathematica+ conce,ts in geometr.# It is a,,ro,riate for a++ +earners4 with the exce,tion of the indicated ,arts of sections $#4 $#5 and $#)which are on+. for extended +earners# It is antici,ated that +earners st(d.ing the extended s.++ab(s wi++ work thro(gh at a faster ,ace#

OutlineIt is intended for the to,ics in this (nit to be st(died se(entia++.@ howe"er it is ,ossib+e to st(d. sections as stand1a+one to,ics# he (nit co"ers a++ as,ects of geometr.from the s.++ab(s4 name+. working with ang+es4 constr(ctions4 simi+ar fig(res4 s.mmetr. and +oci# Some teachers ,refer to not teach geometr. a++ in one b+ock4 it is,ossib+e to +ea"e some sections (nti+ +ater in the co(rse4 for exam,+e +oci co(+d be ta(ght +ater on in the co(rse4 simi+ar+. with other to,ics#

.eacing timeIt is recommended that this (nit sho(+d take a,,roximate+. - of the o"era++ IGCSE co(rse#


7/23/2019 maths 0580 2014



$# Use and interpret thegeometrical terms: point$ line$parallel$ bearing$ right angle$acute$ obtuse and reJex angles$perpendicular$ similarit' and

congruence.

Use and interpret vocabular' of triangles$ %uadrilaterals$ circles$pol'gons and simple solid"gures including nets.

%se f+ashcards at (i:+et#com to +ook at the geometrica+ termino+og. or the Geometr.crossword in Bartons eacher eso(rce 7it ,#-01-2# (W) (8asic)

Introd(ce the termino+og. for bearings4 simi+arit. and congr(ence brief+.4 =simi+arsha,es and three fig(re bearings wi++ be st(died in more detai+ in to,ics $# and )#>#

I++(strate common so+ids4 e#g# c(be4 c(boid4 tetrahedron4 c.+inder4 cone4 s,here4 ,rism4,.ramid4 etc# Aefine the terms "ertex4 edge and face# Ex,+ore some geometric so+idsand their ,ro,erties at i++(minations#nctm#org# (I) (H)

Online3TU htt,&66(i:+et#com62)-0226f+ashcardsU3T

3TU htt,&66i++(minations#nctm#org6*cti"it.

Aetai+#as,xVIAL-0U3T

3TU htt,&66i++(minations#nctm#org6;essons6imath6GeoSo+ids6GeoSo+ids1 *S#,df U3T

.e4tbookBarton ,#-01-2

$#2 2easure lines and angles.

0onstruct a triangle given thethree sides using ruler and pair

of compasses onl'.

0onstruct other simplegeometrical "gures from givendata using ruler and protractoras necessar'.

0onstruct angle bisectors andperpendicular bisectors usingstraight edge and pair ofcompasses onl'.

einforce acc(rate meas(rement of +ines and ang+es thro(gh "ario(s exercises# Forexam,+e4 each +earner draws two +ines that intersect# Meas(re the +ength of each +ine tothe nearest mi++imetre and one of the ang+es to the nearest degree# Each +earner sho(+dthen meas(re another +earners drawing and com,are answers# *+so4 draw an. triang+e4

meas(re the three ang+es and check that the. add (, to 80# (G) (8asic)

Show how to&: constr(ct a triang+e (sing a r(+er and com,asses on+.4 gi"en the +engths of a++ three

sides@: bisect an ang+e (sing a straight edge and com,asses on+.@: constr(ct a ,er,endic(+ar bisector (sing a straight edge and com,asses on+.#: Constr(ct a range of sim,+e geometrica+ fig(res from gi"en data4 e#g# constr(ct a

circ+e ,assing thro(gh three gi"en ,oints#See "ario(s constr(ctions on+ine at mathsisf(n#com# (W)

3ro"ide exam,+es and ,ractice (estions on a++ of the abo"e4 either ,re,ared b. .o( orfrom a textbook# (I)

Gi"e +earners a written descri,tion of a com,o(nd sha,e =with no side +engths> and askthem to recreate the sha,e# For exam,+e4 draw a rectang+e4 constr(ct an e(i+atera+triang+e (sing the to, edge of the rectang+e as one of the sides4 etc# (I) (H)

%se the ,ast ,a,er (estion# (I) (H) (F)

Online3TU htt,&66www#mathsisf(n#com6geometr .6constr(ct1r(+er1com,ass1#htm+U3T

3TU

htt,&66www#mathsisf(n#com6geometr .6constr(ctions#htm+U3T

.e4tbookMetca+f ,#2-2128

Past paper 3a,er 4 D(ne 2024 U2

"#0 < Cambridge IGCSE Mathematics =0580> ? from 205 2/

http://quizlet.com/26701221/flashcards





http://illuminations.nctm.org/ActivityDetail.aspx?ID=70





http://illuminations.nctm.org/Lessons/imath/GeoSolids/GeoSolids-AS1.pdf






http://www.mathsisfun.com/geometry/construct-ruler-compass-1.html





http://www.mathsisfun.com/geometry/constructions.html
















7/23/2019 maths 0580 2014



$#$ Read and make scale drawings. %se an exam,+e to re"ise the to,ic of sca+e drawing# Show how to ca+c(+ate the sca+e of a drawing gi"en a +ength on the drawing and the corres,onding rea+ +ength# 3oint o(tthat meas(rements do not need to be inc+(ded on a sca+e drawing and that man. sca+edrawings (s(a++. ha"e a sca+e written in the form & n# (W) (8asic)

%se the on+ine reso(rces at tes#co#(k for some exam,+es and (estions# (G)

Araw "ario(s sit(ations to sca+e and inter,ret res(+ts# For exam,+e4 ask +earners todraw a ,+an of a room in their ho(se to sca+e and (se it to determine the area of car,etneeded to co"er the f+oor4 ,+an an orienteering co(rse4 etc# (P) (I) (H)

Online3TU htt,&66www#tes#co#(k6eso(rceAetai+#as,xVstor.CodeL)280/8U3T

$# 0alculate lengths of similar"gures.

Extended curriculum only:Use the relationships betweenareas of similar triangles$ withcorresponding results for

similar "gures and extension tovolumes and surface areas ofsimilar solids.

Aisc(ss the conditions for congr(ent triang+es# 3oint o(t that in naming triang+es whichare congr(ent it is (s(a+ to state +etters in corres,onding order4 i#e# &$C is congr(entto E/G im,+ies that the ang+e at & is the same as the ang+e at E # (W) (8asic)

Extend the work on congr(ent sha,es to introd(ce simi+ar triang+es6sha,es# %se thefact that corres,onding sides are in the same ratio to ca+c(+ate the +ength of an(nknown side# ;ink this work to work on transformations since rotation4 ref+ection and

trans+ation +ea"e sha,es congr(ent and en+argements form simi+ar sha,es# (W) (8asic)

For extended +earners4 ex,and on the work on ca+c(+ating +engths of simi+ar fig(res to(sing the re+ationshi,s between areas4 s(rface areas and "o+(mes of simi+ar sha,esand so+ids# (W) (Callenging)

%se the ,ast ,a,er (estions# (F)

.e4tbookMorrison ,#2)122

Past papers3a,er 4 D(ne 20$4 U8=d>3a,er 24 D(ne 20$4 U

$#5 Recognise rotational and lines'mmetr' #including order ofrotational s'mmetr'& in twodimensions.

ote: ncludes properties oftriangles$ %uadrilaterals andcircles directl' related to theirs'mmetries.

Extended curriculum only:

Recognise s'mmetr' properties

Aefine the terms +ine of s.mmetr. and order of rotationa+ s.mmetr. for two dimensiona+sha,es# e"ise the s.mmetries of triang+es =e(i+atera+4 isosce+es> and (adri+atera+s=s(are4 rectang+e4 rhomb(s4 ,ara++e+ogram4 tra,e:i(m4 kite> inc+(ding consideringdiagona+ ,ro,erties# Aisc(ss the infinite s.mmetr. ,ro,erties of a circ+e# (W) (8asic)

For extended +earners4 define the terms ,+ane of s.mmetr. and order of rotationa+

s.mmetr. for three dimensiona+ sha,es# %se diagrams to i++(strate the s.mmetries ofc(boids =inc+(ding a c(be>4 ,risms =inc+(ding a c.+inder>4 ,.ramids =inc+(ding a cone>#;ook at diagrams for the s.mmetr. ,ro,erties of a circ+es ,a.ing attention to chordsand tangents# (W) (Callenging)

C7,RO*3emberton4 Essentia+ maths forCambridge IGCSE CA# %nit 4sha,e and s,ace@ s.mmetr.

.e4tbook

'.e ,#$251$28

"#0 < Cambridge IGCSE Mathematics =0580> ? from 205 $0








7/23/2019 maths 0580 2014



of theprism #including c'linder& andthe p'ramid #including cone&.

Use the following s'mmetr'

properties of circles:• e%ual chords are e%uidistant

from the centre• the perpendicular bisector

of a chord passes throughthe centre

• tangents from an externalpoint are e%ual in length.

$#) Ca+c(+ate (nknown ang+es (sing thefo++owing geometrica+ ,ro,erties&• ang+es at a ,oint

• ang+es at a ,oint on a straight+ine and intersecting straight+ines

• ang+es formed within ,ara++e++ines

• ang+e ,ro,erties of triang+es and(adri+atera+s

• ang+e ,ro,erties of reg(+ar,o+.gons

• ang+e in a semi1circ+e

• ang+e between tangent andradi(s of a circ+e#

Extended curriculum only:• ang+e ,ro,erties of irreg(+ar

,o+.gons• ang+e at the centre of a circ+e is

twice the ang+e at thecirc(mference

• ang+es in the same segment are

e"ise basic ang+e ,ro,erties b. drawing sim,+e diagrams which i++(strate ang+es at a,oint@ ang+es on a straight +ine and intersecting +ines@ ang+es formed within ,ara++e+ +inesand ang+e ,ro,erties of triang+es and (adri+atera+s# (W) (8asic)

Aefine the terms irreg(+ar ,o+.gon4 reg(+ar ,o+.gon4 conca"e and con"ex# %seexam,+es that inc+(de& triang+es4 (adri+atera+s4 ,entagons4 hexagons and octagons#Show that each exterior ang+e of a reg(+ar ,o+.gon is $)06n4 where n is the n(mber ofsides4 and that the interior ang+e is 80 min(s the exterior ang+e# So+"e a "ariet. of,rob+ems that (se these form(+ae# Araw a tab+e of information for reg(+ar ,o+.gons# %seas headings& n(mber of sides4 name4 exterior ang+e4 s(m of interior ang+es4 interiorang+e# (I) (H)#

%se diagrams to show the ang+e in a semi1circ+e and the ang+e between tangent andradi(s of a circ+e are /0# %se the d.namic ,ages on timde"ere(x#co#(k to see thecirc+e theorems come to +ife# (W)

3ro"ide the so+(tion to an examination st.+e (estion on the to,ic of ang+es that

contains a mistake in the working# *sk +earners to identif. the mistake# (G)

For extended +earners mo"e on to +ook at ang+e ,ro,erties of irreg(+ar ,o+.gons# B.di"iding an n1sided ,o+.gon into a n(mber of triang+es show that the s(m of the interiorang+es is 80=n 1 2> degrees and that the interior and exterior ang+es s(m to 80# (W)

Ex,+ain the theor. that ang+es in o,,osite segments are s(,,+ementar.# In"estigate

8ookBarton ,#)-1)/

Online3TU htt,&66www#timde"ere(x#co#(k6maths6geom,ages68theorem#,h,U3T

3TU htt,&66www#tes#co#(k6teaching1reso(rce6Circ+e1heorems1GCSE1igher17S1with1answers1)8/0/6U3T

Past papers3a,er 24 'o" 2024 U54 203a,er 24 'o" 2024 U)


http://www.timdevereux.co.uk/maths/geompages/8theorem.php





http://www.tes.co.uk/teaching-resource/Circle-Theorems-GCSE-Higher-KS4-with-answers-6181909/











7/23/2019 maths 0580 2014



e(a+• ang+es in o,,osite segments

are s(,,+ementar.@ c.c+ic(adri+atera+s#

'ote& Candidates wi++ be ex,ectedto (se the correct geometrica+termino+og. when gi"ing reasons for answers.

c.c+ic (adri+atera+s# For exam,+e4 ex,+ain wh. a++ rectang+es are c.c+ic (adri+atera+s#hat other (adri+atera+ is a+wa.s c.c+icV Is it ,ossib+e to draw a ,ara++e+ogram that isc.c+icV etc# %se exam,+es to show that the ang+e at the centre of a circ+e is twice theang+e at the circ(mference and that ang+es in the same segment are e(a+# *gain (sethe d.namic ,ages on timde"ere(x#co#(k to see the circ+e theorems come to +ife# (W)

(Callenging)

So+"e a "ariet. of ,rob+ems (sing a++ the circ+e theorems making s(re that +earnersknow the correct +ang(age for describing the reasoning for their answers# r. theworksheet on+ine at tes#co#(k with a +arge n(mber of (estions in examination st.+e oncirc+e theorems# *+so (se the ,ast ,a,er (estions# (I) (H) (F)

$#- Use the following loci and themethod of intersecting loci forsets of points in two dimensionswhich are:• at a given distance from a

given point

• at a given distance from agiven straight line

• e%uidistant from two givenpoints

• e%uidistant from two givenintersecting straight lines.

o introd(ce the conce,t of +oci ask +earners in the c+ass to stand (, if the. f(+fi+ certaincriteria4 e#g# if the. are exact+. 2 m from the door4 +ess than 2 m from the board4 etc# *2 m +ong ,iece of string can he+, if +earners are not confident with estimating +engths#Mo"e on to drawing sim,+e diagrams to i++(strate the fo(r different +oci# %se thecon"ention of a broken +ine to re,resent a bo(ndar. that is not inc+(ded in the +oc(s of,oints# (W) (8asic)

;ook at the on+ine exam,+e of a circ+e ro++ing aro(nd a s(are at nrich#maths#org# %se asimi+ar idea of an exam,+e of a rectang(+ar card being ro++ed a+ong a f+at s(rface# orko(t the +oc(s of one of the "ertices of the rectang+e as it mo"es# (I) (H)

Extend the work to +ook at o"er+a,,ing regions of intersecting loci$ use the pastpaper %uestions. (F)

.e4tbook3imente+ ,#/1/-

Online3TU htt,&66nrich#maths#org625/U3T

Past paper 3a,er $4 D(ne 20$4 U2=a>

!nit # *ensuration

Recommended prior knowledge;earners sho(+d be ab+e to choose s(itab+e (nits of meas(rement to estimate4 meas(re4 and so+"e ,rob+ems in a range of sim,+e contexts4 inc+(ding (nits of mass4+ength4 area4 "o+(me or ca,acit.# he. sho(+d know abbre"iations for4 and some re+ationshi,s between4 metric (nits& ki+ometres =km>4 metres =m>4 centimetres =cm>4mi++imetres =mm>@ tonnes =t>4 ki+ograms =kg> and grams =g>@ +itres =+> and mi++i+itres =m+># he. sho(+d be ab+e to read the sca+es on a range of ana+og(e and digita+meas(ring instr(ments and be fami+iar with the terms ,erimeter4 area and "o+(me# he. sho(+d a+so know the names of common 2A and $A sha,es s(ch as rectang+e4triang+e =e(i+atera+4 isosce+es4 right ang+ed and sca+ene>4 ,ara++e+ogram4 tra,e:i(m4 kite4 circ+e4 s,here4 c.+inder4 ,rism4 ,.ramid4 c(be4 c(boid and cone# ;earners







7/23/2019 maths 0580 2014


sho(+d know the names of the ,arts of a circ+e& radi(s4 diameter and circ(mference and (nderstand the terms ,er,endic(+ar and ,ara++e+ and be fami+iar with the (se ofthese in re+ation to s,ecia+ (adri+atera+s sides and diagona+s# It is s(ggested that %nit $ is st(died before %nit 4 in ,artic(+ar s.++ab(s reference $##

Conte4this (nit re"ises and de"e+o,s mathematica+ conce,ts in mens(ration# It is a,,ro,riate for a++ +earners4 with the exce,tion of the indicated ,arts of sections #$ and #which are on+. for extended +earners# It is antici,ated that +earners st(d.ing the extended s.++ab(s wi++ work thro(gh at a faster ,ace# ;earners sho(+d (se ca+c(+atorswhere a,,ro,riate@ howe"er4 it is recommended that reg(+ar non1ca+c(+ator work is com,+eted to strengthen +earners menta+ arithmetic#

OutlineIt is intended for the to,ics in this (nit to be st(died se(entia++.4 as +atter to,ics re(ire the know+edge from ear+ier ones4 for exam,+e a +earner wi++ not be ab+e to worko(t the "o+(me of a c.+inder witho(t ha"ing first +earned how to ca+c(+ate the area of a circ+e# he (nit co"ers a++ as,ects of mens(ration from the s.++ab(s4 name+. mass4+ength4 ,erimeter4 area4 "o+(me and ca,acit.# Some teachers ,refer to not teach mens(ration a++ in one b+ock4 it is ,ossib+e to +ea"e some sections (nti+ +ater in theco(rse for exam,+e areas and "o+(mes of com,o(nd sha,es co(+d be ta(ght +ater on in the co(rse4 simi+ar+. with other to,ics#


"#0 < Cambridge IGCSE Mathematics =0580> ? from 205 $$

7/23/2019 maths 0580 2014



# Use current units of mass$length$ area$ volume andcapacit' in practical situationsand express %uantities in termsof larger or smaller units.

0onvert between unitsincluding units of area andvolume.

* good starting ,oint is to (se ,ractica+ exam,+es to i++(strate how to con"ert between&mi++imetres4 centimetres4 metres and ki+ometres@ grams4 ki+ograms and tonnes@mi++i+itres4 centi+itres and +itres# For exam,+e +ooking at "ario(s meas(ring sca+es# (W)(8asic)

Extend this work to +ooking at con"erting between (nits of area mm;

2;

4 cm;

2;

and m;

2;

and"o+(me mm;

$;4 cm;

$; and m;

$;# * sim,+e +esson ex,+aining this can be fo(nd in the on+ine

"ideo +esson on .o(t(be#com# (W) (Callenging)

More ab+e +earners wi++ ,robab+. find it interesting to ex,+ore the +ink between the workon con"erting between area (nits to the work on ratio and simi+ar sha,es and can +ookat (sing sca+es on ma,s to work with areas# (G) (Callenging)

o assess +earners (nderstanding4 ask them to tr. the ,ast ,a,er (estion (I) (H) (F)

Online3TU htt,&66www#.o(t(be#com6watchV"LdnSgd2.''IU3T

Past paper

3a,er 2$4 'o" 2024 U5

#2 0arr' out calculations involvingthe perimeter and area of arectangle$ triangle$

parallelogram and trapeHiumand compound shapes derivedfrom these.

<o( might want to begin this to,ic b. reminding +earners how to ca+c(+ate the ,erimeterand area of a rectang+e4 s(are and a triang+e# (W) (8asic)

his can then be extended b. +ooking at how to ca+c(+ate the area of a ,ara++e+ogramand a tra,e:i(m and a "ariet. of com,o(nd sha,es# (W) (8asic)

*n interesting in"estigation is to +ook at (sing isometric dot ,a,er to find the area ofsha,es that ha"e a ,erimeter of 54 )4 -4 ### 4 (nits# (G)

*sk +earners to find o(t what sha,e (adri+atera+ has the +argest area when the,erimeter is4 for exam,+e 2 cm# (H)

C7,RO*3emberton4 Essentia+ maths forCambridge IGCSE CA# %nit 24

sha,e and s,ace@ ,erimeter andarea

#$ 0arr' out calculations involvingthe circumference and area of acircle.

* (sef(+ starting ,oint is re"ising4 (sing straightforward exam,+es4 how to ca+c(+ate thecirc(mference and area of a circ+e# ;earners are ex,ected to know the form(+ae# (W)(8asic)

Extend this b. +ooking at how to find com,o(nd areas in"o+"ing circ+es4 for exam,+e4 a

circ+e with the radi(s of 5#$ cm is drawn to(ching the sides of a s(are# *sk+earners Qhat area of the s(are is not co"ered b. the circ+eVR the (estioncan be extended to consider the area of waste materia+ when c(tting se"era+ circ+es ofthis si:e o(t of an * sheet of ,a,er# (I) (H)


http://www.youtube.com/watch?v=dnSgdT2yNNI







7/23/2019 maths 0580 2014




olve problems involving thearc length and sector area asfractions of the circumference

and area of a circle.

For extended +earners the next ste, is to (se exam,+es to i++(strate how to ca+c(+ate thearc +ength and the sector area b. (sing fractions of f(++ circ+es# (W) (Callenging)

;earners wi++ need to combine their work on sector area with area of a triang+e work=s.++ab(s reference )#$> to find segment areas# For an exam,+e of this see the ,ast

,a,er (estion# (F) (Callenging)

.e4tbook'.e4 Extended Mathematics ,#)-1)/

Past paper

3a,er 2$4 D(ne 20$4 U8# 0arr' out calculations involving

the volume of a cuboid$ prismand c'linder and the surfacearea of a cuboid and a c'linder.

Starting with sim,+e exam,+es draw the nets of a "ariet. of so+ids asking +earners if the.are ab+e to identif. the so+id from the net# It is (sef(+ for +earners to (nderstand thatthere are man. different right and wrong wa.s to draw the net of a c(be# ;ess ab+e+earners might a,,reciate the o,,ort(nit. to work in gro(,s to draw nets on card and to(se these to make "ario(s geometrica+ sha,es# (G) (8asic)

he next ste, is to demonstrate a ,(r,ose and (se for drawing nets# For exam,+e in the,ackaging ind(str. there are man. different interesting nets (sed to create boxes4,artic(+ar+. those that re(ire +itt+e or no g+(e# *n interesting homework acti"it. wo(+d beto ask +earners to co++ect +ots of different ,ackaging boxes to in"estigate the nets (sedto create them# (H) (8asic)

<o( can then ask +earners to +ook at how to ca+c(+ate the s(rface area of a c(boid and ac.+inder4 (sing the nets to he+,# Extend this to i++(strating how to ca+c(+ate the "o+(me of a c(boid and a "ariet. of ,risms4 inc+(ding c.+inders# ;earners wi++ find it (sef(+ to knowthe form(+a "o+(me of ,rism L cross1sectiona+ area x +ength# * (sef(+ reso(rce on thisto,ic is the on+ine reso(rce4 s(rface area and "o+(me4 at www#+earner#org# (W) (8asic)

*sk +earners to tr. the ,ast ,a,er (estion n (F)9

Online3TU htt,&66www#+earner#org6interacti"es6geometr.6area#htm+U3T

3TU htt,&66math#abo(t#com6od6form(+as6ss6s(rfacearea"o+#htmU3T

.e4tbooka.ner4 Extended Mathematics,#0-12$

Past papers3a,er 4 D(ne 20$4 U3a,er 24 D(ne 20$4 U5


0arr' out calculations involvingthe surface area and volume ofa sphere$ p'ramid and cone.ote: ormulae will be given forthe surface area and volume ofthe sphere$ p'ramid and cone.

For extended +earners mo"e on to (sing nets to i++(strate how to ca+c(+ate the s(rfacearea of a triang(+ar ,rism4 a ,.ramid and a cone# It wi++ be (sef(+ for +earners to(nderstand how to obtain the form(+a πr =r O s> for the s(rface area of a cone =where s Ls+ant +ength># <o( wi++ a+so want to ex,+ain how to ca+c(+ate the s(rface area of a s,here(sing the form(+a πr ;2;# (W) (8asic)

he next ste, is to (se exam,+es to i++(strate how to ca+c(+ate the "o+(me of a ,.ramid

=inc+(ding a cone> (sing the form(+a1

3 \ area of base \ ,er,endic(+ar height# *+so

+ook at how to ca+c(+ate the "o+(me of a s,here (sing the form(+a4

3 πr ;$;# Aiagrams

Online3TU htt,&66math#abo(t#com6od6form(+as6ss6s(rfacearea"o+#htmU3T

.e4tbook

a.ner4 Extended Mathematics,#0-12$



http://www.learner.org/interactives/geometry/area.html





http://math.about.com/od/formulas/ss/surfaceareavol.htm

















7/23/2019 maths 0580 2014



and form(+ae can be fo(nd at www#math#abo(t#com# Em,hasise to +earners that the.sho(+d know which form(+ae to +earn and which wi++ be gi"en# hi+st some form(+ae wi++be gi"en4 .o( co(+d cha++enge those with good memories to +earn the gi"en form(+aetoo# (W) (Callenging)

Exam,+es and ,ractice (estions on this to,ic can be fo(nd in textbooks# (P) (I)

;earners sho(+d be ,re,ared to (se form(+ae to find +engths too# Show them the ,ast,a,er (estion 5 as an exam,+e# (P) (I) (H) (F)

*n interesting task is to show +earners a sheet of * ,a,er# Ex,+ain to them that thiscan be ro++ed into a c.+inder in two wa.s# *sk +earners Qhich gi"es the biggest"o+(meVR o extend this task ask +earners to in"estigate what width and +ength gi"es themaxim(m c.+inder "o+(me if the area of ,a,er remains constant b(t the +ength andwidth can "ar. (I) (H)

#5 0arr' out calculations involvingthe areas and volumes ofcompound shapes.

he fina+ section is a++ abo(t extending a++ the work from sections # to # to find thes(rface area and "o+(me of a wide "ariet. of com,osite sha,es# here are a n(mber of,ast ,a,er (estions for +earners to tr. demonstrating the kind of (estions that the.ma. see# (P) (I) (H) (F)

Past papers3a,er $4 D(ne 20$4 U3a,er $4 D(ne 2024 U53a,er 24 D(ne 2024 U

!nit +# Co,ordinate geometr

Recommended prior knowledge;earners sho(+d be ab+e to (nderstand the conce,t of (sing +etters to re,resent (nknown n(mbers or "ariab+es@ know the meanings of the words term and e)uation#he. sho(+d be confident with the work on directed n(mbers from %nit and be ab+e to sim,+if. or transform +inear ex,ressions with integer coefficients and s(bstit(te,ositi"e and negati"e integers into +inear ex,ressions# ;earners sho(+d a+so (nderstand the work from %nit ) on 3.thagoras theorem =extended +earners on+.> and ha"ean awareness of hori:onta+4 "ertica+ and diagona+ +ine e(ations4 e#g# ' L $4 x L 124 ' L 5 x ? #

Conte4this (nit re"ises and de"e+o,s mathematica+ conce,ts in co1ordinate geometr. that are im,ortant in other ,arts of the co(rse# It is a,,ro,riate for a++ +earners4 with theexce,tion of a++ of sections 5#$ and 5#) and the indicated ,art of section 5#2 which are on+. for extended +earners# It is antici,ated that +earners st(d.ing the extendeds.++ab(s wi++ work thro(gh at a faster ,ace#

Outline

"#0 < Cambridge IGCSE Mathematics =0580> ? from 205 $)

7/23/2019 maths 0580 2014


It is intended for the to,ics in this (nit to be st(died se(entia++.4 as man. +ater to,ics re(ire know+edge from ear+ier ones4 for exam,+e inter,reting the e(ation of astraight +ine gra,h re(ires know+edge of the gradient# he (nit co"ers a++ as,ects of co1ordinate geometr. from the s.++ab(s4 name+. Cartesian co1ordinates4 gradient of a straight +ine4 +engths of +ine segments4 mid,oint of a +ine4 e(ations of +ines inc+(ding ,ara++e+ and ,er,endic(+ar +ines# Some teachers ,refer to not teach co1ordinategeometr. a++ in one b+ock4 it is ,ossib+e to +ea"e some sections (nti+ +ater in the co(rse for exam,+e finding the gradient of ,ara++e+ and ,er,endic(+ar +ines =5#)> co(+d beta(ght +ater on in the co(rse4 simi+ar+. with other to,ics#

.eacing time

It is recommended that this (nit sho(+d take a,,roximate+. - of the o"era++ IGCSE co(rse#

"#0 < Cambridge IGCSE Mathematics =0580> ? from 205 $-

7/23/2019 maths 0580 2014



5# 5#

Aemonstrate fami+iarit. withCartesian co1ordinates in twodimensions#

e"ise co1ordinates in two dimensions see reso(rces#wood+ands19(nior#kent#sch#(k foron+ine games to do with co1ordinates# (W) (8asic)

Araw a ,ict(re b. 9oining dots on a s(are grid# Araw x and ' axes on the grid and writedown the coordinates of each dot# (I) (H)

*sk other +earners to draw these ,ict(res from a +ist of coordinates on+.# (G)

Online3TUhtt,&66reso(rces#wood+ands1 9(nior#kent#sch#(k6maths6sha,es6coordinates#htm+

5#2 5#2

Find the gradient of a straight +ine#

Core& 'ote& 3rob+ems wi++ in"o+"efinding the gradient where the gra,his gi"en#


Ca+c(+ate the gradient of a straight+ine from the co1ordinates of two,oints on it#

Aefine4 with diagrams4 a +ine with a ,ositi"e gradient as one s+o,ing (, and a +ine with anegati"e gradient as one s+o,ing down# %se sim,+e exam,+es to show how to ca+c(+atethe gradient =,ositi"e4 negati"e or :ero> of a straight +ine from a gra,h (sing "ertica+distance di"ided b. hori:onta+ distance in a right ang+ed triang+e# (W) (8asic)

Extend this to consider the gradient of the +ine x L constant# (W)

%se exam,+es to show how to ca+c(+ate the gradient of a straight +ine from the co1ordinates of two ,oints on it4 first+. b. drawing the +ine and then witho(t drawing the+ine# %se gradient L change in ' coordinates o"er change in x coordinates# Ex,+ain thecommon error of s(btracting the coordinates the o,,osite wa. ro(nd on the n(meratorto the denominator ca(sing the sign to be incorrect# (W) (Callenging)

.e4tbook3imente++4 Cambridge IGCSE Maths,#$/10

C7,RO*3emberton4 Essentia+ maths forCambridge IGCSE CA4 %nit 4a+gebra@ gradients and straight +inegra,hs4 s+ides 1

5#$ 5#$


Ca+c(+ate the +ength and the co1ordinates of the mid,oint of astraight +ine from the co1ordinates of its end ,oints#

e"ise 3.thagoras theorem from %nit )# %se exam,+es to show how to ca+c(+ate the+ength of a straight +ine segment from the co1ordinates of its end ,oints# (W)

Ao this first+. b. (sing a sketch (8asic) and then to extend the +earners (se the form(+a

√ ( x1− x

2)2+( y

1− y

2)2

# 'ote that know+edge of this form(+a is not essentia+>

(Callenging)# %se the ,ast ,a,er (estion# (F)

%se exam,+es to show how to f ind the co1ordinates of the mid,oint of a straight +inefrom the co1ordinates of its end ,oints# Inc+(de exam,+es working backwards4 e#g# when

an end ,oint and a mid,oint are known find the other end ,oint# (W) (I)

.e4tbookMorrison Co(rsebook ,#/)1/8

Past Paper 3a,er 224 D(ne 2024 U-a

5# 5#

Inter,ret and obtain the e(ation ofa straight +ine gra,h in the form ' L m x O c.

e"ise drawing a gra,h of ' L m x O c from a tab+e of "a+(es# Inter,ret the meaning of mand c from the e(ation (sing the terms gradient and interce,t# Starting with a straight+ine gra,h show how its e(ation =' L m x O c> can be obtained# (W) (I) (8asic)9

o inter,ret the meaning of an e(ation4 ex,+ain how an e(ation sim,+. gi"es there+ationshi, between the x and ' co1ordinates on the +ine4 e#g# for the e(ation ' L 2 x

.e4tbookMetca+f4 Core Co(rse book ,#$01$02


http://resources.woodlands-junior.kent.sch.uk/maths/shapes/coordinates.html







7/23/2019 maths 0580 2014



Core& 'ote& 3rob+ems wi++ in"o+"efinding the e(ation where thegra,h is gi"en#

this means the ' ordinate is a+wa.s do(b+e the x ordinate# %se this to identif. if a ,oint+ies on the +ine4 e#g# which of these ,oints& =24 8>4 =14 8>4 =-4 >4 =204 0>4 =04 0> +ie onthe +ine ' L 2 x V *sk +earners to come (, simi+ar (estions# (I) (H)

hen gi"e these (estions to others in a gro(, to identif. which ,oints do not +ie on a

gi"en +ine# (G) (P)5#5 5#5

Aetermine the e(ation of a straight+ine ,ara++e+ to a gi"en +ine#

e#g# find the e(ation of a +ine,ara++e+ to ' L x ? that ,assesthro(gh =04 ?$>#

%se exam,+es to show how to f ind the e(ation of a straight +ine ,ara++e+ to a gi"en +ine4e#g# find the e(ation of a +ine ,ara++e+ to ' L x ? that ,asses thro(gh =04 ?$># (W)

%se the ,ast ,a,er (estion =this a+so co"ers work on gradient># (F)

.e4tbook3emberton4 Essentia+ Maths book,#)81-0

Past paper 3a,er 4 'o"ember 2024 U8

5#) 5#)Extended curriculum only

Find the gradient of ,ara++e+ and,er,endic(+ar +ines#

%se exam,+es to show that ,ara++e+ +ines ha"e the same gradient# Inc+(de exam,+eswhere the e(ation is gi"en im,+icit+.4 e#g# which of these +ines are ,ara++e+V ' L 2 x 4 ' O2 x L 04 ' ? 2 x O $4 2' L 2 x O -4 etc# (W)

%se an odd1one1o(t acti"it.4 e#g# which is the odd one o(t =beca(se the +ine is not,ara++e+ to the others>4 gi"ing three or more exam,+esV *sk +earners to come (, withtheir own set of odd one o(t exam,+es# (G)

Find the gradient of ,er,endic(+ar +ines b. (sing the fact that if two +ines are,er,endic(+ar the ,rod(ct of their gradients is 14 e.g. "nd the gradient of a lineperpendicular to y 9 3 x 1. (W)

Use a variet' of examples linking earlier topics from this unit$ e.g. "nd thee%uation of a line perpendicular to one passing through the coordinates #1$3& and #V,$ V6&. (W) (Challenging)

Use the online examples and %uestions at bbc.co.uk. (I) (H)

Online3TU htt,&66www#bbc#co#(k6schoo+s6gcsebitesi:e6maths6a+gebra6gra,hshire"#

shtm+U3T

!nit -# .rigonometr

"#0 < Cambridge IGCSE Mathematics =0580> ? from 205 $/

http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/graphshirev1.shtml









7/23/2019 maths 0580 2014


Recommended prior knowledge;earners sho(+d be fami+iar with the work on ang+es from %nit $ and ha"e some (nderstanding of how to +abe+ a triang+e4 with the con"ention that "ertices are +abe++edwith ca,ita+ +etters and o,,osite sides are +abe++ed with +ower case corres,onding +etters# ;earners sho(+d a+so (nderstand how an ang+e is named (sing three +ettersand be fami+iar with the term right!angle triangle4 know that the +ongest side in a right ang+ed triang+e is ca++ed the h.,oten(se and that ang+es in a triang+e add (, to80# ;earners sho(+d be comfortab+e with three dimensiona+ diagrams inc+(ding being ab+e to "is(a+ise ,ara++e+ and ,er,endic(+ar sides in them and be ab+e to workwith s(ares and s(are roots and to be ab+e to so+"e e(ations simi+ar to those arising from (sing 3.thagoras theorem# Extended +earners sho(+d know how to worko(t a sector area (sing the work from section #$#

Conte4this (nit de"e+o,s mathematica+ conce,ts in trigonometr.# Section )# and indicated ,arts of section )#2 are a,,ro,riate for a++ +earners# he rest of the (nit is on+. forextended +earners# It is antici,ated that +earners st(d.ing the core s.++ab(s wi++ s,end c+oser to $ of their time on this (nit as the. are on+. com,+eting the first twosections of the s.++ab(s4 extended +earners ma. re(ire c+oser to 5 as there is a higher content for them to st(d.#

OutlineIt is intended for the to,ics in this (nit to be st(died se(entia++. as +ater to,ics4 for exam,+e trigonometrica+ ,rob+ems in three dimensions4 re(ire know+edge from,re"io(s areas# he (nit co"ers a++ as,ects of trigonometr. and re+ated work from the s.++ab(s4 name+. bearings4 3.thagoras theorem4 trigonometrica+ ,rob+ems in twoand three dimensions =inc+(ding ange+s of e+e"ation and de,ression>4 (sing the sine and cosine r(+e and the form(+a for the area of a triang+e# Some teachers ,refer tonot teach trigonometr. a++ in one b+ock4 it is ,ossib+e to +ea"e some sections (nti+ +ater in the co(rse for exam,+e the trigonometrica+ ,rob+ems in three dimensions =)#>co(+d be ta(ght +ater on in the co(rse4 simi+ar+. with other to,ics#

.eacing timeIt is recommended that this (nit sho(+d take a,,roximate+. $?5 of the o"era++ IGCSE co(rse#


7/23/2019 maths 0580 2014



)# Inter,ret and (se three1fig(rebearings#

'ote& Meas(red c+ockwise from the'orth i#e# 000?$)0

Introd(ce three fig(re bearings and (se exam,+es of meas(ring and drawing in"o+"ingbearings# 3earces st(dent book ,#$1$- has (sef(+ exam,+es# <o( ma. want to +inkthis work to that on sca+e drawings in to,ic $#$# (W) (8asic)

%se exam,+es to show how to ca+c(+ate bearings4 e#g# ca+c(+ate the bearing of B from *

if .o( know the bearing of * from B# (W) (8asic)

%se a ma, to determine distance and direction =bearing> between two ,+aces4 e#g#+earners home and schoo+4 etc# Ma,s from aro(nd the wor+d can be fo(nd on+ine atma,s#goog+e#com (I) (H)

.e4tbook3earce4 St(dent book ,#$1$-

Online3TU htt,s&66ma,s#goog+e#com6U3T

)#2 *,,+. 3.thagoras theorem and thesine4 cosine and tangent ratios forac(te ang+es to the ca+c(+ation of aside or of an ang+e of a right1ang+edtriang+e#

'ote& *ng+es wi++ be (oted in4 andanswers re(ired in4 degrees anddecima+s to one decima+ ,+ace#


So+"e trigonometrica+ ,rob+ems intwo dimensions in"o+"ing ang+es ofe+e"ation and de,ression#

Extend sine and cosine "a+(es toang+es between /0 and 80#

e"ise s(ares and s(are roots# %se sim,+e exam,+es in"o+"ing right ang+ed triang+esto i++(strate 3.thagoras theorem# Start with finding the +ength of the h.,oten(se thenmo"e on to finding the +ength of one of the shorter sides# See exam,+es on+ine atmathsisf(n#com# (W) (8asic)

Extend this work to co"er diagrams where the right ang+ed triang+e isnt ex,+icit+. drawnor the ,rob+em is ,resented witho(t a diagram4 e#g# find the diagona+ +ength across arectang(+ar fie+d or the height of a b(i+ding# (W) (8asic)

hen introd(cing trigonometr. s,end some time on +abe++ing the sides of triang+es witha marked ang+e& ad9acent4 h.,oten(se and o,,osite# *sk +earners to work in gro(,s todraw right ang+e triang+es with a $0 ang+e of "ario(s si:es# *sk them to work o(t theratio o,,osite side ^ ad9acent side for a++ the different triang+es to find the. sho(+d a++ bea simi+ar "a+(e# (G)

hen (se exam,+es in"o+"ing the sine4 cosine and tangent ratios to ca+c(+ate the +engthof an (nknown side of a right1ang+ed triang+e gi"en an ang+e and the +ength of one side#%se a mix of exam,+es4 some exam,+es where di"ision is re(ired and some exam,+eswhere m(+ti,+ication is re(ired# For +earners who str(gg+e with rearranging thetrigonometrica+ ratios it is ,ossib+e to (se the form(+a triang+e a,,roach =see the

worksheet on+ine at tes#co#(k># (W) (G) (P)

For more ab+e +earners enco(rage the rearranging a,,roach# Mo"e on to exam,+esin"o+"ing in"erse ratios to ca+c(+ate an (nknown ang+e gi"en the +ength of two sides of aright1ang+ed triang+e# (W) (G) (P)

So+"e a wide "ariet. of ,rob+ems in context (sing 3.thagoras theorem andtrigonometric ratios =inc+(de work with an. sha,e that ma. be ,artitioned into right1

Online3TU htt,&66www#mathsisf(n#com6,.thagoras#htm+U3T

3TU htt,&66www#tes#co#(k6eso(rceAetai+#as,xVstor.CodeL)$255U3T

.e4tbooksBarton4 eacher eso(rce 7it ,#51-

Morrison4 Core and ExtendedCo(rse book ,#$0$1$2

C7,RO*3emberton4 Essentia+ maths forCambridge IGCSE CA# %nit 84sha,e and s,ace@ Sine and cosineratios for ang+es (, to 80

Past Paper 3a,er $4 D(ne 20$4 U-abc






http://www.mathsisfun.com/pythagoras.html
















7/23/2019 maths 0580 2014



ang+ed triang+es># (I) (Callenging)

%se exam,+es to i++(strate how to so+"e ,rob+ems in"o+"ing bearings (sing trigonometr.#%se the ,ast ,a,er (estion# (F)

For extended +earners define ang+es of e+e"ation and de,ression# %se exam,+es toi++(strate how to so+"e ,rob+ems in"o+"ing ang+es of e+e"ation and de,ression (singtrigonometr.# (W)

Araw a sine c(r"e and disc(ss its ,ro,erties# %se the c(r"e to show4 for exam,+e4 sin50 L sin $0# e,eat for the cosine c(r"e# (W) (Callenging)

)#$ Extended curriculum only

So+"e ,rob+ems (sing the sine andcosine r(+es for an. triang+e and the

form(+a area of triang+e L1

2ab

sin C.

earrange the form(+a for the area of a triang+e =]bh> to the form ]absinC =see theon+ine reso(rce at regents,re,#org># I++(strate its (se with a few sim,+e exam,+es#Ex,+ain that the +etters in the form(+a ma. change from ,rob+em to ,rob+em4 so +earnerssho(+d tr. to remember the ,attern of two sides and the sine of the inc+(ded ang+e# (W)

Extend this to see if +earners can (se the form(+a to work o(t other ,rob+ems4 e#g#ca+c(+ate the area of a segment of a circ+e gi"en the radi(s and the sector ang+e =(singtheir know+edge of sector area work from section #$> or ca+c(+ate the area of a,ara++e+ogram gi"en two ad9acent side +engths and an. ang+e# (I) (Callenging)

%se exam,+es to show how to so+"e ,rob+ems (sing the sine r(+e ex,+aining that the

"ersiona

sin A L

b

sinB is ,referab+e for finding a side and the "ersion

sin A

a

LsinB

b is ,referab+e for finding an ang+e# %se exam,+es to show how to so+"e

,rob+ems (sing the cosine r(+e making s(re that +earners either +earn bothrearrangements of the form(+a name+. to find a side a;

2; L b;

2; O c ;2; ? 2bc cos & and to find

an ang+e cos & Lb

2+c2−a2

2bc or can confident+. rearrange from one to the other#

Gi"e +earners a set of (estions where the. can either (se the sine r(+e or the cosiner(+e# *sk them not to work o(t the answers b(t instead to decide which r(+e to (se =seethe "ideo on+ine at .o(t(be#com for an exam,+e acti"it.># (G)

Ex,+ain how +earners can te++ whether the. need the sine r(+e or the cosine r(+e4 i#e# (sethe cosine r(+e when .o( know a++ three sides in a triang+e or an enc+osed ang+e and

.e4tbooka.ner4 Extended Mathematics,#20/12$

Online3TU htt,&66www#regents,re,#org6regents6math6a+gtrig6*$6areatrig+esson#htmU3T

3TU htt,&66www#.o(t(be#com6watchV"LeMS%Fsh8hsU3T

Past Papers3a,er 4 D(ne 20$4 U)3a,er 2$4 D(ne 20$4 U8


http://www.regentsprep.org/regents/math/algtrig/ATT13/areatriglesson.htm






http://www.youtube.com/watch?v=ZeMSUFsh8hs










7/23/2019 maths 0580 2014



two sides otherwise (se the sine r(+e# (W) (Callenging)%se the ,ast ,a,er (estions# (P) (I) (H) (F)

)# Extended curriculum only

So+"e sim,+e trigonometrica+

,rob+ems in three dimensionsinc+(ding ang+e between a +ine anda ,+ane#

Introd(ce ,rob+ems in three dimensions b. finding the +ength of the diagona+ of a c(boidand determining the ang+e it makes with the base# Extend b. (sing more com,+exfig(res4 e#g# a ,.ramid# (W) (Callenging)

%se the ,ast ,a,er (estion# (P) (I) (H) (F)

.e4tbook3imente+4 Cambridge IGCSE Maths,#2012-

Past Paper 3a,er 24 D(ne 20$4 U2$

!nit /# *atrices and trans0ormations

Recommended prior knowledge;earners sho(+d be fami+iar with the work on Cartesian co1ordinates and e(ations of straight +ine gra,hs from %nit 5 and ha"e some (nderstanding of the terms reflect 4rotate4 translate and enlarge# he. sho(+d recognise +ine and rotation s.mmetr. in two1dimensiona+ sha,es and ,atterns@ draw +ines of s.mmetr. and com,+ete ,atterns

with two +ines of s.mmetr.@ identif. the order of rotationa+ s.mmetr. and be fami+iar with the work on s.mmetr. in %nit $# he. sho(+d a+so be fami+iar with the work onsimi+arit. and congr(ence from %nit $4 in ,artic(+ar know that sha,es remain congr(ent after rotation4 ref+ection and trans+ation b(t are simi+ar after an en+argement4 andbe ab+e to (se 3.thagoras theorem from %nit )#

Conte4this (nit re"ises and de"e+o,s mathematica+ conce,ts in transformations and introd(ces "ectors and matrices# Sections -# and -#2 are a,,ro,riate for a++ +earners4 withthe exce,tion of the negati"e sca+e factors ,arts# he rest of the (nit is on+. for extended +earners# It is antici,ated that +earners st(d.ing the core s.++ab(s wi++ s,endc+oser to 5 of their time on this (nit as the. are on+. com,+eting the first two sections4 extended +earners ma. re(ire c+oser to 8 as there is a higher content for themto st(d.#

OutlineIt is intended for the to,ics in this (nit to be st(died se(entia++.4 a+tho(gh this is not essentia+ as certain to,ics for exam,+e matrices =-#> can be st(died ear+ier# he(nit co"ers a++ as,ects of matrices and transformations from the s.++ab(s4 name+. drawing and describing rotations4 ref+ections4 trans+ations and en+argements =for

extended +earners describing transformations (sing matrices>@ "ectors4 and matrices# Some teachers ,refer to not teach matrices and transformations a++ in one b+ock4 itis ,ossib+e to +ea"e some sections (nti+ +ater in the co(rse for exam,+e "ectors =-#$> co(+d be ta(ght +ater on in the co(rse4 simi+ar+. with other to,ics#

.eacing timeIt is recommended that this (nit sho(+d take a,,roximate+. 5?8 of the o"era++ IGCSE co(rse#


7/23/2019 maths 0580 2014



-# Aescribe a trans+ation b. (sing a"ector re,resented b.

e#g# ( x y ) 4 AB 4 or a#

*dd and s(btract "ectors#

M(+ti,+. a "ector b. a sca+ar#

%se the conce,t of trans+ation to ex,+ain a "ector# %se sim,+e diagrams to i++(strateco+(mn "ectors in two dimensions4 ex,+aining the significance of ,ositi"e and negati"en(mbers# Introd(ce the "ario(s forms of "ector notation# Show how to add and s(btract"ectors a+gebraica++. i++(strate b. making (se of a "ector triang+e# Show how to m(+ti,+.a co+(mn "ector b. a sca+ar and i++(strate this with a diagram# %se4 for exam,+e4 the

exam,+es on+ine at m.maths#co#(k =s(bscri,tion re(ired for this website># (W) (8asic)

.e4tbookBarton4 eacher eso(rce 7it ,#0510-

Online3TU

htt,&66www#m.maths#co#(k6tasks6+ibr ar.6+oad;esson#as,Vtit+eL"ectors6"ectors,artU3T


-#2 ef+ect sim,+e ,+ane fig(res inhori:onta+ or "ertica+ +ines#

otate sim,+e ,+ane fig(res abo(tthe origin4 "ertices or mid,oints ofedges of the fig(res4 thro(ghm(+ti,+es of /0#

Constr(ct gi"en trans+ations anden+argements of sim,+e ,+anefig(res#

ecognise and describe ref+ections4rotations4 trans+ations anden+argements#

'ote ? Core#3ositi"e and fractiona+ sca+e factorsfor en+argements on+.#

'ote ? Extended curriculum only:

'egati"e sca+e factors foren+argements#

Araw an arrow sha,e on a s(ared grid# %se this to i++(strate& ref+ection in a +ine =mirror+ine>4 rotation abo(t an. ,oint =centre of rotation> thro(gh m(+ti,+es of /0 =in bothc+ockwise and anti1c+ockwise directions> and trans+ation b. a "ector# Se"era+ differentexam,+es of each transformation sho(+d be shown# %se the word image a,,ro,riate+.#(W) (8asic)

In"estigate how transformations are (sed to make tesse++ations and ,rod(ce anEscher1t.,e drawing# For ins,iration and ste, b. ste, g(ides see tesse++ations#org# (I)(H)

%se a ,re1drawn sha,e on = x 4 ' > coordinate axes to com,+ete a n(mber oftransformations (sing the e(ations of +ines to re,resent mirror +ines and coordinates tore,resent centres of rotation# ork with = x 4 ' > coordinate axes to show how to find& thee(ation of a sim,+e mirror +ine gi"en a sha,e and its =ref+ected> image4 the centre andang+e of rotation gi"en a sha,e and its =rotated> image4 the "ector of a trans+ation#Em,hasi:e a++ the detai+ that is re(ired to describe each of the transformations# (W)(G) (8asic)

Araw a triang+e on a s(ared grid# %se this to i++(strate en+argement b. a ,ositi"einteger sca+e factor abo(t an. ,oint =centre of en+argement># %se both of the methods&co(nting s(ares and drawing ra.s# Show how to find the centre of en+argement gi"ena sha,e and its =en+arged> image# (W)

For extended +earners show how to draw en+argements (sing negati"e and6or fractiona+sca+e factors# (W) (Callenging)%se the ,ast ,a,er (estions# (F)

C7,RO*3emberton4 Essentia+ maths forCambridge IGCSE CA# %nit $4sha,e and s,ace

Online3TU htt,&66www#tesse++ations#org6index#shtm+U3T

Past papers3a,er 24 D(ne 20$4 U2a =i> =ii>3a,er $4 'o"ember 2024 U0

.e4tbookBarton4 eacher eso(rce 7it ,#2/1$


http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=vectors/vectorspart1

















7/23/2019 maths 0580 2014



-#$ Extended curriculum only

Ca+c(+ate the magnit(de of a "ector

( x y ) as √ x2+ y2 #

e,resent "ectors b. directed +inesegments#

%se the s(m and difference of two"ectors to ex,ress gi"en "ectors interms of two co,+anar "ectors#

%se ,osition "ectors#

e"ise the work from section -## %se diagrams to he+, show how to ca+c(+ate themagnit(de of a "ector4 +ink this to the work on 3.thagoras theorem from to,ic )#2#Ex,+ain the notation re(ired i#e# AB or a for "ectors and for their magnit(des

|⃗ AB| or a =with mod(+(s signs># (W) (G) (P)

Aefine a ,osition "ector and so+"e "ario(s ,rob+ems in "ector geometr.# =xplain tolearners that in their answers to %uestions$ the' are expected to indicate a

in some de"nite wa'$ e.g. b' an arrow or b' underlining$ thus AB or a.(W) (G) (P) (Callenging)9

%se the ,ast ,a,er (estion# (F)

.e4tbookMorrison4 Core and ExtendedCo(rse book ,#50015

Past papers

3a,er 24 D(ne 20$4 U203a,er 2$4 D(ne 20$4 U2$

-# Extended curriculum only

Ais,+a. information in the form of a

matrix of an. order#

Ca+c(+ate the s(m and ,rod(ct=where a,,ro,riate> of two matrices#

Ca+c(+ate the ,rod(ct of a matrixand a sca+ar (antit.#

%se the a+gebra of 2 \ 2 matricesinc+(ding the :ero and identit. 2 \ 2matrices#

Ca+c(+ate the determinant & and

in"erse &;

1; of a non1sing(+ar matrix

&#

%se sim,+e exam,+es to i++(strate that information can be stored in a matrix# Forexam,+e4 the n(mber of different t.,es of choco+ate bar so+d b. a sho, each da. for aweek# Aefine the order6si:e of a matrix as the n(mber of rows \ n(mber of co+(mns#

(W) (8asic)

Ex,+ain how to identif. matrices that .o( ma. add6s(btract or m(+ti,+. together withreference to the order of the matrices# %se exam,+es to i++(strate how to add6s(btractand m(+ti,+. matrices together#Aefine the identit. matrix and the :ero matrix for 2 \ 2 matrices# %se sim,+e exam,+esto i++(strate m(+ti,+.ing a matrix b. a sca+ar (antit.#%se exam,+es to i++(strate how to ca+c(+ate the determinant and the in"erse of a non1sing(+ar 2 \ 2 matrix4 ex,+aining the term non1sing(+ar Ex,+ain when a matrix wi++ ha"eno in"erse# (W) (G)

*s an extension for the more ab+e4 in"estigate how to (se matrices to he+, so+"esim(+taneo(s e(ations (Callenging)# %se the ,ast ,a,er (estions (F).

Past papers3a,er 2$4 D(ne 20$4 U-3a,er 24 D(ne 20$4 U-

.e4tbookBarton4 eacher eso(rce 7it ,#$1


7/23/2019 maths 0580 2014



-#5 Extended curriculum only

%se the fo++owing transformations of the ,+ane& ref+ection =M>4 rotation=>4 trans+ation =>4 en+argement =E>4

and their combinations#

Identif. and gi"e ,recisedescri,tions of transformationsconnecting gi"en fig(res#

Aescribe transformations (sing co1ordinates and matrices =sing(+armatrices are exc+(ded>#

'ote& If M=a> L b and =b> L c thenotation M=a> L c wi++ be (sed#In"ariants (nder these

transformations ma. be ass(med#

Starting with a +etter E drawn on = x 4 ' > coordinate axes4 ,erform combinations of thefo++owing transformations& trans+ation4 rotation4 ref+ection and en+argement# (W) (8asic)

%se a (nit s(are and the base "ectors (1

0) and (0

1) to identif. matrices which

re,resent the "ario(s transformations met so far4 e#g# (0 −1

1 0 ) re,resents a rotation

abo(t =04 0> thro(gh /0 anti1c+ockwise# ork with a sim,+e ob9ect drawn on = x 4 ' >coordinate axes to i++(strate how it is transformed b. a "ariet. of gi"en matrices# %seone of these transformations to i++(strate the effect of an in"erse matrix# %se the on+inereso(rce at nationa+stemcentre#org#(k for +ots of interacti"e and d.namic exam,+es tohe+, +earners to describe transformations (sing matrices# (W) (Callenging)

*sk +earners to in"estigate the connection between the area sca+e factor of atransformation and the determinant of the transformation matrix4 e#g# (sing the matrix

(2 0

0 2

) and working with a rectang+e drawn on = x 4 ' > coordinate axes# (G) (P)

%se the ,ast ,a,er (estion# (F).

Past paper 3a,er 24 D(ne 20$4 U2a =iii>

Online3TU htt,&66www#nationa+stemcentre#org#(

k6e+ibrar.6reso(rce65$06aa1f,1matrices1transformationsU3T

!nit 1# Probabilit

Recommended prior knowledge;earners sho(+d be fami+iar with the work from (nit on fractions4 decima+s and ,ercentages inc+(ding the fo(r r(+es with them# he. sho(+d a+so know the +ang(age of,robabi+it. e#g# certain4 im,ossib+e4 e"en chance4 +ike+.4 (n+ike+.4 o(tcomes4 ex,eriment4 biased4 fair and random and know how to extract information from tab+es and

gra,hs# ;earners sho(+d a+so (nderstand that greater than - does not inc+(de - b(t that greater than or e(a+ to - does# Extended +earners sho(+d ha"e st(died theKenn diagrams work from (nit 4 inc+(ding drawing and inter,reting them#

Conte4this (nit re"ises and de"e+o,s mathematica+ conce,ts in ,robabi+it.# It is a,,ro,riate for a++ +earners4 with the exce,tion of ,art 8#5 which is on+. for extended +earners# Itis antici,ated that +earners st(d.ing the extended s.++ab(s wi++ work thro(gh at a faster ,ace# ;earners sho(+d (se ca+c(+ators where a,,ro,riate@ howe"er4 it isrecommended that reg(+ar non1ca+c(+ator work is com,+eted to strengthen +earners menta+ arithmetic#


http://www.nationalstemcentre.org.uk/elibrary/resource/530/aqa-fp1-matrices-transformations










7/23/2019 maths 0580 2014


OutlineIt is intended for the to,ics in this (nit to be st(died se(entia++.4 as +atter to,ics re(ire the know+edge from ear+ier ones4 for exam,+e a +earner wi++ not be ab+e to worko(t the ,robabi+it. of combined e"ents witho(t ha"ing first +earned how to work with ,robabi+it. of sing+e e"ents# he (nit co"ers a++ as,ects of ,robabi+it. from thes.++ab(s4 name+. (nderstanding the ,robabi+it. sca+e4 working with sing+e and combined e"ents4 (nderstanding how to work o(t the ,robabi+it. of an e"en notha,,ening4 (sing re+ati"e fre(enc. as an estimate of ,robabi+it. and (sing tree diagrams# Some teachers ,refer to not teach ,robabi+it. a++ in one b+ock4 it is ,ossib+e to+ea"e some sections (nti+ +ater in the co(rse for exam,+e ca+c(+ating the ,robabi+it. of combined e"ents co(+d be ta(ght +ater on in the co(rse4 simi+ar+. with other to,ics#



7/23/2019 maths 0580 2014



8# 0alculate the probabilit' of asingle event as either afraction$ decimal or percentage.

ote: ;roblems could be set

involving extracting informationfrom tables or graphs.

%se theoretica+ ,robabi+it. to ,redict the +ike+ihood of a sing+e e"ent# For exam,+e4 findthe ,robabi+it. of choosing the +etter M from the +etters of the word M&T0EM&TICS# %sethe form(+a& ,robabi+it. L fa"o(rab+e o(tcomes6,ossib+e o(tcomes# (W) (8asic)

Aisc(ss when fractions4 decima+s or ,ercentages are ,referab+e for re,resenting

,robabi+ities4 e#g# if the ,robabi+it. is 26$ then a fraction is ,referab+e beca(se it is exact#(W) (8asic)

;earners (se exam,+e (estions that .o("e ,re,ared or from textbooks# (P) (I)

%se the ,ast ,a,er (estions# (F)

For extended +earners (se Kenn diagrams for ,robabi+it. (estions4 for exam,+e see3embertons Essentia+ Maths book ,#$2-# (G) (P) (I) (H) (Callenging)

.e4tbooks3earce4 St(dent book ,#5-015-$

3emberton4 Essentia+ Maths ,#$2-

Past papers3a,er $4 'o" 2024 U20b3a,er 24 'o" 2024 U8a

8#2 Understand and use theprobabilit' scale from 5 to 1.

Aisc(ss ,robabi+ities of 0 and 4 +eading to the o(tcome that a ,robabi+it. +ies betweenthese two "a+(es# e"ise the +ang(age of ,robabi+it. associated with the ,robabi+it.sca+e# %se the ,robabi+it. sca+e b. estimating fre(encies of e"ents occ(rring based on,robabi+ities# (W) (8asic)

*sk +earners to ,rod(ce their own ,robabi+it. sca+e with e"ents marked on it4 see anexam,+e at tes#co#(k# (I) (H)

*sk +earners to find o(t the meaning of m(t(a++. exc+(si"e and exha(sti"e# (I) (H)

Online3TU htt,&66www#tes#co#(k6eso(rceAetai+#as,xVstor.CodeL)$2U3T

.e4tbookBarton4 eacher eso(rce 7it ,#$-1$/

8#$ Understand that the probabilit'of an event occurring 9 1 V theprobabilit' of the event notoccurring.

%se exam,+es to show the ,robabi+it. of an e"ent occ(rring L ? the ,robabi+it. of thee"ent not occ(rring4 inc+(ding those where there are on+. two o(tcomes and thosewhen there are more than two o(tcomes# (W) (8asic)

r. the ,ast ,a,er (estion (F)9

Past paper 3a,er 24 D(ne 20$4 U/

8# Understand relative fre%uenc'as an estimate of probabilit'.

Com,are estimated ex,erimenta+ ,robabi+ities4 or re+ati"e fre(enc.4 with theoretica+,robabi+ities# ;earners need to recognise that when ex,eriments are re,eated differento(tcomes ma. res(+t and increasing the n(mber of times an ex,eriment is re,eatedgenera++. +eads to better estimates of ,robabi+it.# (W) (8asic)

Cond(ct a c+ass ex,eriment into ro++ing dice $00 times4 e#g# 5 ,airs of +earners ro++ing adice 20 times each# Co++ect and combine res(+ts from gro(,s to create a +arge sam,+eset4 show how estimates change as more data is added to the set# (W) (G) (P) (8asic)

Online3TU htt,&66www#mathsisf(n#com6acti"it.6b(ffons1need+e#htm+U3T

Past paper 3a,er $$4 D(ne 20$4 U)a







http://www.mathsisfun.com/activity/buffons-needle.html









7/23/2019 maths 0580 2014



e,eat the ex,eriment where the theoretica+ ,robabi+it. is not known4 e#g# the chanceof a drawing ,in +anding ,oint down when thrown in the air# r. B(ffons need+eex,eriment at mathsisf(n#com# (W)

Carr. o(t ex,eriments to sam,+e the n(mber of (nknown co+o(red co(nters in a bag#

*sk +earners to s(ggest how man. of each t.,e of co+o(red co(nter there are in thebag4 gi"en the known tota+# (G)

%se the ,ast ,a,er (estion# (F)


0alculate the probabilit' ofsimple combined events$ usingpossibilit' diagrams and treediagrams where appropriate.

ote: n possibilit' diagrams$outcomes will be representedb' points on a grid$ and in treediagrams$ outcomes will bewritten at the end of branchesand probabilities b' the side ofthe branches.

o++ two different dice or two s,inners and +ist a++ of the o(tcomes# %se sim,+e exam,+esto i++(strate how ,ossibi+it. diagrams and tree diagrams can he+, to organise data# (W)(Callenging)

%se ,ossibi+it. diagrams and tree diagrams to he+, ca+c(+ate ,robabi+ities of sim,+ecombined e"ents4 ,a.ing ,artic(+ar attention to how diagrams are +abe++ed# (W) (P))(Callenging)

So+"e ,rob+ems in"o+"ing inde,endent and de,endent e"ents4 e#g# ,icking co(nters

from a bag with and witho(t re,+acement# (W) (P) (Callenging)

Aisc(ss conditiona+ ,robabi+it.# (W) (Callenging)

r. the QIn a boxR ,robabi+it. ,rob+em on+ine at nrich#maths#org# (I) (H)

C7,RO*3emberton4 Essentia+ maths forCambridge IGCSE CA# %nit )4,robabi+it. and statistics@ ,robabi+it.2

Online3TU htt,&66nrich#maths#org6//U3T

!nit 2# 3tatistics

Recommended prior knowledge

;earners sho(+d be ab+e to decide which data wo(+d be re+e"ant to an in(ir. and co++ect and organise the data4 as we++ as design and (se a data co++ection sheet or(estionnaire for a sim,+e s(r"e.# he. sho(+d ha"e some fami+iarit. with the three a"erages of mean4 median and mode4 and the meas(re of s,read and range4 a++ insim,+e cases# he. sho(+d a+so know how to draw and inter,ret sim,+e bar +ine gra,hs and bar charts4 ,ie charts and ,ictograms and be fami+iar with the terms discrete and continuous data# ;earners sho(+d be ab+e to (se a ca+c(+ator effecti"e+. and be confident working with Cartesian coordinates and choosing a,,ro,riate sca+es onaxes#

Conte4t







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his (nit re"ises and de"e+o,s mathematica+ conce,ts in statistics# It is a,,ro,riate for a++ +earners4 with the exce,tion of a++ of sections /# and /#5 and the work on(ne(a+ inter"a+s in histograms in section /#2 which are on+. for extended +earners# It is antici,ated that +earners st(d.ing the extended s.++ab(s wi++ work thro(gh at afaster ,ace#

OutlineIt is intended for the to,ics in this (nit to be st(died se(entia++.4 a+tho(gh this is not essentia+ as certain to,ics for exam,+e corre+ation and +ines of best fit =/#) and /#->can be st(died ear+ier# he (nit co"ers a++ as,ects of statistics from the s.++ab(s4 name+. tab+es and statistica+ diagrams4 a"erages4 range4 c(m(+ati"e fre(enc.4 scatter

diagrams and corre+ation# Some teachers ,refer to not teach statistics a++ in one b+ock4 it is ,ossib+e to +ea"e some sections (nti+ +ater in the co(rse for exam,+ecorre+ation and +ines of best fit =/#) and /#-> co(+d be ta(ght +ater on in the co(rse4 simi+ar+. with other to,ics#



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/# Co++ect4 c+assif. and tab(+atestatistica+ data#

ead4 inter,ret and draw sim,+einferences from tab+es and

statistica+ diagrams#

%se sim,+e exam,+es to re"ise co++ecting data and ,resenting it in a fre(enc. =ta++.>chart# For exam,+e4 record the different makes of car in a car ,ark4 record the n(mberof words on the first ,age of a series of different books etc# (W) (8asic)

*sk +earners to cond(ct an ex,eriment of this t.,e4 tab(+ating their data# (G) (P)

%se exam,+es to c+assif. data (sing statistica+ termino+og.4 e#g# discrete4 contin(o(s4n(merica+ =(antitati"e> non1n(merica+ =(a+itati"e>4 etc# %se exam,+es to show how todraw sim,+e inferences from statistica+ diagrams4 and tab+es inc+(ding two1wa. tab+es#%se exam,+es from textbooks and the ,ast ,a,er (estion# (W) (P) (I) (H) (F) (8asic)

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/#2 Constr(ct and read bar charts4 ,iecharts4 ,ictograms4 sim,+efre(enc. distrib(tions4 histogramswith e(a+ inter"a+s and scatterdiagrams#


Constr(ct and read histograms with(ne(a+ inter"a+s#

For (ne(a+ inter"a+s onhistograms4 areas are ,ro,ortiona+to fre(encies and the "ertica+ axisis +abe++ed fre(enc. densit.#

%se the data co++ected4 in section /#4 to constr(ct a ,ictogram4 a bar chart and a ,iechart# 3oint o(t that the bars in a bar chart can be drawn a,art# (W) (8asic)

%se an exam,+e to show how discrete data can be gro(,ed into e(a+ c+asses# (W)(8asic)

Araw a histogram to i++(strate the data =i#e# with a contin(o(s sca+e a+ong the hori:onta+axis># 3oint o(t that this information co(+d a+so be dis,+a.ed in a bar chart =i#e# with bars

se,arated> beca(se data is discrete# Ex,+ain how to draw scatter diagrams with sim,+eexam,+es =.o( ma. choose to do this at the same time as to,ic /#)># (W) (8asic)

In"estigate the +ength of words (sed in two different news,a,ers and ,resent thefindings (sing statistica+ diagrams =+inks to news,a,ers can be fo(nd on+ine aton+inenews,a,ers#com># (G) (P) (8asic)

ecord sets of contin(o(s data4 e#g# heights4 masses etc#4 in gro(,ed fre(enc. tab+es#%se exam,+es that i++(strate e(a+ c+ass widths for core +earners and (ne(a+ c+asswidths for extended +earners# Araw the corres,onding histograms# Em,hasi:e the factthat for contin(o(s data bars of a histogram m(st to(ch# %se the bar charts andhistograms section of the e1book4 to i++(strate wh. f re(enc. densit. is a fairer wa. tore,resent data than fre(enc. on the "ertica+ axis# ;abe+ the "ertica+ axis of a histogramas fre(enc. densit. and show that the area of each bar is ,ro,ortiona+ to thefre(enc.# Show how to ca+c(+ate fre(enc. densities from a fre(enc. tab+e withgro(,ed data and how to ca+c(+ate fre(encies from a gi"en histogram# (W)(Callenging)

r. the ,ast ,a,er (estion (I) (H) (F)

.e4tbookMorrison4 Co(rsebook ,#2)

Online3TU htt,&66www#on+inenews,a,ers#com6U3T

3TU htt,&66mrbartonmaths#com6ebook#htmU3T the maths e1book of notes and

exam,+es

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/#$ Ca+c(+ate the mean4 median4 mode From data co++ected4 show how to work o(t the mean4 the median and the mode from a Online














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and range for indi"id(a+ and discretedata and disting(ish between the,(r,oses for which the. are (sed#

+ist of data or f rom a fre(enc. tab+e# Ex,+ain4 that if there are two midd+e "a+(es4 how tofind the ha+f1wa. ,oint for the median as there can on+. be one median b(t that therecan be more than one mode or no mode# %se sim,+e exam,+es to high+ight how thesea"erages ma. be (sed# For exam,+e4 in a disc(ssion abo(t a"erage wages the ownerof a com,an. with a few high+. ,aid managers and a +arge work force ma. wish to

(ote the mean wage rather than the median# See the on+ine exam,+es of which is thebest a"erage to (se for different sit(ations at mathsteacher#com# (W) (8asic)

Inc+(de exam,+es where the mean is gi"en and the n(mber of ,eo,+e4 tota+ or anindi"id(a+ "a+(e need to be fo(nd# (Callenging)

%se the ,ast ,a,er (estion# (I) (H) (F)

Ex,+ain how the mode can be recognised from a fre(enc. diagram# (G)

r. the Bat ings ,rob+em on+ine at nrich#maths#org# (I) (H)

3TU htt,&66www#mathsteacher#com#a(6.ear86ch-stat602mean6mean#htmU3T

3TU htt,&66nrich#maths#org6505U3T

.e4tbookBarton4 eacher eso(rce 7it ,#2122

Past paper 3a,er $4 'o" 2024 U$=a>

/# Extended curriculum only

Ca+c(+ate an estimate of the meanfor gro(,ed and contin(o(s data#

Identif. the moda+ c+ass from agro(,ed fre(enc. distrib(tion#

%se exam,+es to show how to ca+c(+ate an estimate for the mean of data in a gro(,edfre(enc. tab+e (sing the mid1inter"a+ "a+(es# Ex,+ain how the moda+ c+ass can be

fo(nd in a gro(,ed fre(enc. distrib(tion# (W)

;ook at the exam,+es and (estions on+ine at cimt#,+.mo(th#ac#(k# (I) (H)#

Ex,+ain how to find the inter"a+ that contains the median b(t that working o(t themedian is not re(ired at this +e"e+4 for more ab+e +earners .o( co(+d show them theidea of +inear inter,o+ation as extension work# (Callenging)

r. the ,ast ,a,er (estion# (F)

.e4tbook3imente++4 Cambridge IGCSE Maths

,#$81$/

Online3TU htt,&66www#cimt#,+.mo(th#ac#(k6,ro9ects6me,res6book/6bk/i)6bk/)i2#htm+U3T

Past paper 3a,er 224 D(ne 20$4 U20ab

/#5 Extended curriculum only

Constr(ct and (se c(m(+ati"e

fre(enc. diagrams#

Estimate and inter,ret the median4,ercenti+es4 (arti+es and inter1(arti+e range#

Ex,+ain c(m(+ati"e fre(enc. and (se an exam,+e to i++(strate how a c(m(+ati"efre(enc. tab+e is constr(cted# Araw the corres,onding c(m(+ati"e fre(enc. c(r"eem,hasising that ,oints are ,+otted at (,,er c+ass +imits4 the c(r"e m(st a+wa.s be

increasing and high+ight its distincti"e sha,e# Ex,+ain that this can be a,,roximated b.a c(m(+ati"e fre(enc. ,o+.gon# (W) (8asic)

%se a c(m(+ati"e fre(enc. c(r"e to he+, ex,+ain and inter,ret ,ercenti+es# Introd(cethe names gi"en to the 25th4 50th and -5th ,ercenti+es and show how to estimate thesefrom a gra,h# Show how to estimate the inter1(arti+e range from a c(m(+ati"efre(enc. diagram# Ex,+ain how to (se a c(m(+ati"e fre(enc. c(r"e to com,+ete a

.e4tbook3emberton4 Essentia+ maths ,#$)81$-5

Past paper 3a,er 4 D(ne 20$4 U$


http://www.mathsteacher.com.au/year8/ch17_stat/02_mean/mean.htm









http://www.cimt.plymouth.ac.uk/projects/mepres/book9/bk9i16/bk9_16i2.html













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fre(enc. tab+e# (W) (Callenging)

%se the ,ast ,a,er (estion and exercises from a textbook# (W) (P) (I) (H) (F)(Callenging)

/#) %nderstand what is meant b.

,ositi"e4 negati"e and :erocorre+ation with reference to ascatter diagram#

%se sim,+e exam,+es of scatter diagrams to ex,+ain the terms and meanings of

,ositi"e4 negati"e and :ero corre+ation# e"ise drawing scatter diagrams and describethe res(+ting corre+ation# Ex,+ain wh. and where scatter gra,hs are (sef(+4 e#g# inmaking ,redictions# (W) (8asic)

*sk +earners to co++ect some bi"ariate data of their choice and to ,redict the corre+ation4if an.4 that the. ex,ect to find4 e#g# height and arms s,an for members of the c+ass# %seco++ected data to draw a scatter diagram and to then +ook for the ex,ected corre+ation#Aisc(ss res(+ts# (G) (P)

Ex,+ain that if there are too few ,oints on a scatter diagram a corre+ation ma. a,,eara,,arent when in fact there is no rea+ re+ationshi, between the "ariab+es# (W) (G)

;ook at the on+ine Aa"id and Go+iath ,rob+em at nrich#maths#org# (I) (H)

r. the ,ast ,a,er (estion# (I) (H) (F)

.e4tbook

a.ner4 Core Mathematics ,#$812

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Past paper 3a,er $4 D(ne 20$4 U8

/#- Araw a straight +ine of best fit b.e.e#

Ex,+ain4 with diagrams4 that the ,(r,ose of a good +ine of best fit is to ha"e the s(m ofthe "ertica+ distances from each ,oint to the +ine as sma++ as ,ossib+e# In sim,+er termsask +earners to aim for a simi+ar n(mber of ,oints on each side of the +ine and as man.,oints as ,ossib+e on the +ine or as c+ose to it as ,ossib+e# (W) (G) (P)

Araw diagrams showing bad +ines of best f it ex,+aining what is wrong with them# Forexam,+e4 the common error when a +earner draws the +ine thro(gh the origin when thatdoesnt fit with the trend of the data# %se a textbook for exam,+e exercises# (W) (8asic)

r. the ,ast ,a,er (estion# (I) (H) (F)

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