Structure and calculation

Transcription

1 Strn/sustrn N1 R Bsi ountion ontnt Orr positiv n ngtiv intgrs, imls n rtions; us th symols =,, <, >,, Apply th our oprtions (+, ) inluing orml writtn mthos, to intgrs, imls n simpl rtions (propr n impropr), n mix numrs ll oth positiv n ngtiv; N2 pply th our oprtions (, ), inluing orml writtn mthos, to intgrs, imls n simpl rtions (propr n impropr), n mix numrs ll oth positiv n ngtiv; g unrstn n us pl vlu (.g. whn working with vry lrg or vry smll numrs, mr Strutur n lultion N3 N4 N5 h g h i n whn lulting with imls). Rognis n us rltionships twn oprtions, inluing invrs oprtions (.g. nlltion to simpliy lultions n xprssions); us onvntionl nottion or priority o oprtions, inluing rkts, powrs, roots n riprols. Us th onpts n voulry o prim numrs, tors (ivisors), multipls, ommon tors, ommon multipls highst ommon tor, lowst ommon multipl, prim toristion, inluing using prout nottion n th uniqu toristion thorm. Apply systmti listing strtgis Us positiv intgr powrs

2 Num N6 n ssoit rl roots (squr, u n highr), rognis powrs o 2, 3, 4, 5; N7 N8 Clult xtly with rtions, Frtions, imls n prntgs N9 N10 N11 N12 N13 N14 N15 Clult with n intrprt stnr orm A x 10 n, whr 1 A < 10 n n is n intgr. Work intrhngly with trminting imls n thir orrsponing rtions (suh s 3.5 n 7/2 or or 3/8 ). Intiy n work with rtions in rtio prolms. Intrprt rtions s oprtors n prntgs s oprtors. Us stnr units o mss, lngth, tim, mony n othr msurs (inluing stnr ompoun msurs) using iml quntitis whr pproprit. Estimt nswrs; hk lultions using pproximtion n stimtion, inluing nswrs otin using thnology roun numrs n msurs to n pproprit gr o ury (.g. to spii numr o iml pls or signiint igurs) N16 Us n intrprt lgri nottion, inluing: in pl o

3 A1 3y in pl o y + y + y n 3 y 2 in pl o, 3 in pl o, 2 in pl o Nottion, voulry n mnipultion A2 A3 A4 A5 g g h i j k l m n o / in pl o oiints writtn s rtions rthr thn s imls rkts. Sustitut numril vlus into ormul n xprssions, inluing sintii ormul. Unrstn n us th onpts n voulry o xprssions, qutions, ormul, inqulitis, trms, n tors. Simpliy n mnipult lgri xprssions ollting lik trms multiplying singl trm ovr rkt tking out ommon tors simpliying xprssions involving sums, prouts n powrs, inluing th lws o inis. Unrstn n us stnr mthmtil ormul; rrrng ormul to hng th sujt. A6

4 Whr pproprit, intrprt simpl xprssions s untions with inputs n outputs; A7 A8 Work with oorints in ll our qurnts. Plot grphs o qutions tht orrspon to strightlin grphs in th oorint pln; A9 Intiy n intrprt grints o linr untions Algr Grphs A10 A11 A12 grphilly n lgrilly; n intrpts o linr untions grphilly n lgrilly. Rognis, skth n intrprt grphs o linr untions, qurti untions, A13 A14 Plot n intrprt grphs n grphs o non-stnr untions in rl ontxts, to in pproximt solutions to prolms

5 suh s simpl kinmti prolms involving istn, sp n lrtion. A15 A16 Solv linr qutions in on unknown lgrilly; A17 Solving qutions n inqulitis A18 A19 A20 A21 A22 A23 in pproximt solutions using grph. Gnrt trms o squn rom ithr trm-totrm or position-to-trm rul. Rognis n us squns o tringulr, squr

6 Squns A24 g n u numrs, simpl rithmti progrssions A25 R1 h i Du xprssions to lult th n th trm o linr squns Chng rly twn rlt stnr units (.g. tim, lngth, r, volum/pity, mss) in numril ontxts n ompoun units (.g. sp, rts o py, pris) in numril ontxts s o hng g R2 R3 R4 R5 R6 R7 R8 Us sl tors, sl igrms n mps. Exprss on quntity s rtion o nothr, whr th rtion is lss thn 1 or grtr thn 1. Us rtio nottion inluing rution to simplst orm. Divi givn quntity into two prts in givn prt:prt or prt:whol rtio; xprss th ivision o quntity into two prts s rtio; pply rtio to rl ontxts n prolms (suh s thos involving onvrsion, omprison, sling, mixing, onntrtions). Exprss multiplitiv rltionship twn two quntitis s rtio or rtion. Unrstn n us proportion s qulity o rtios. Rlt rtios to rtions n to linr untions. Din prntg s numr o prts pr hunr ; intrprt prntgs s rtion

7 Rtio, proportion n rt Rtio, proportion n rts o hng R9 R10 R11 R12 R13 g or iml, intrprt prntg hngs s rtion or iml n intrprt ths multiplitivly; xprss on quntity s prntg o nothr h ompr two quntitis using prntgs; i work with prntgs grtr thn 100%; j solv prolms involving prntg hng, k inluing prntg inrs/rs l n originl vlu prolms, n simpl intrst inluing in innil m mthmtis. Solv prolms involving irt n invrs proportion, inluing grphil n lgri rprsnttions. Us ompoun units suh s sp, rts o py, unit priing Compr lngths, rs n volums using rtio nottion n sl tors. R14 R15 g

8 R16 G1 Us onvntionl trms n nottions: points, lins, vrtis, gs, plns, prlll lins, prpniulr lins, right ngls, polygons, rgulr polygons n polygons with rltion n/or rottion symmtris; us th stnr onvntions or llling n rrring to th sis n ngls o tringls; rw igrms rom writtn sription. G2 Apply th proprtis o ngls t point, ngls t point on stright lin, vrtilly opposit ngls; Proprtis n onstrutions G3 G4 G5 G6 unrstn n us ltrnt n orrsponing ngls on prlll lins; riv n us th sum o ngls in tringl us th sum o ngls in tringl (.g. to u n us th ngl sum in ny polygon, n to riv proprtis o rgulr polygons). Driv n pply th proprtis n initions o: spil typs o quriltrls, inluing squr, rtngl, prlllogrm, trpzium, kit n rhomus;n tringls n othr pln igurs using pproprit lngug.

9 G6 g Gomtry n msurs G7 G8 G9 G10 G11 G12 G13 G14 G15 Intiy, sri n onstrut ongrunt n similr shps, inluing on oorint xs, y onsiring rottion, rltion, trnsltion n nlrgmnt Intiy n pply irl initions n proprtis, inluing: ntr, rius, hor, imtr, irumrn, Solv gomtril prolms on oorint xs. Intiy proprtis o th s, surs, gs n vrtis o: us, uois, prisms, ylinrs, pyrmis, ons n sphrs. intrprt plns n lvtions o 3D shps. Us stnr units o msur n rlt onpts (lngth, r, volum/pity, mss, tim, mony, t.) Msur lin sgmnts n ngls in gomtri igurs, inluing intrprting mps n sl rwings G16 n us o rings. Know n pply ormul to lult: r o tringls, prlllogrms, trpzi; volum o uois n othr right prisms (inluing ylinrs). Know th ormul: irumrn o irl = 2πr = π, r o irl = πr 2 ;

10 Mnsurtion n lultion G17 G18 G19 lult: primtrs o 2D shps, inluing irls; rs o irls n omposit shps; G20 g G21 G22 G23 Vtors G24 Dsri trnsltions s 2D vtors. G25 P1 Ror sri n nlys th rquny o outoms o proility xprimnts using tls n rquny trs.

11 Proility Proility P2 P3 P4 P5 P6 P7 P8 Apply is o rnomnss, irnss n qully likly vnts to lult xpt outoms o multipl utur xprimnts. Rlt rltiv xpt rqunis to thortil proility, using pproprit lngug n th 0-1 proility sl. Apply th proprty tht th proilitis o n xhustiv st o outoms sum to on; pply th proprty tht th proilitis o n xhustiv st o mutully xlusiv vnts sum to on. Enumrt sts n omintions o sts systmtilly, using tls, gris, Vnn igrms Construt thortil possiility sps or singl xprimnts with qully likly outoms n us ths to lult thortil proilitis n omin xprimnts with qully likly outoms n us ths to lult thortil proilitis. P9 S1 Intrprt n onstrut tls, hrts n igrms, S2 g h inluing rquny tls, r hrts, pi hrts n pitogrms or tgoril t, vrtil lin hrts or ungroup isrt numril t, n know thir pproprit us.

12 Sttistis Sttistis S3 S4 g h Intrprt, nlys n ompr th istriutions o t sts rom univrit mpiril istriutions through: - pproprit grphil rprsnttion involving isrt, ontinuous n group t - pproprit msurs o ntrl tnny (min, i mn, j mo n mol lss) k n spr (rng, l inluing onsirtion o outlirs, m n S5 Apply sttistis to sri popultion. Us n intrprt sttr grphs o ivrit t; S6 rognis orrltion

13 Aitionl ountion ontnt Highr ontnt only inluing us o th prout rul or ounting (i.. i thr r m wys o oing on tsk n or h o ths, thr r n wys o oing nothr tsk, thn th totl numr o wys th two tsks n on is m n wys).

14 Clult with roots, n with intgr inis n multipls o π; stimt powrs n roots o ny givn positiv numr. n rtionl inis. surs, simpliy sur xprssions involving squrs (.g. 12 = 4 3) = 4 3 = 2 3) n rtionlis nomintors. Chng rurring imls into thir orrsponing rtions n vi vrs. us inqulity nottion to spiy simpl rror intrvls u to truntion or rouning. Apply n intrprt limits o ury, inluing uppr n lowr ouns.

15 intitis, (inluing thos involving surs n lgri rtions) y: xpning prouts o two inomils torising qurti xprssions o th orm x 2 + x +, inluing th irn o two squrs; or mor thn two inomils torising qurti xprssions o th orm x 2 + x + Know th irn twn n qution n n intity; rgu mthmtilly to show lgri xprssions r quivlnt, n us lgr to support n onstrut rgumnts

16 n proos. intrprt th rvrs pross s th invrs untion ; intrprt th sussion o two untions s omposit untion (th us o orml untion nottion is xpt). us th orm y = mx + to intiy prlll lins; in th qution o th lin through two givn points, or through on point with givn grint. n prpniulr lins; Intiy n intrprt roots, intrpts, turning points o qurti untions grphilly; u roots lgrilly n turning points y omplting th squr. simpl ui untions, th riprol untion y = 1/x with x 0 (inluing riprol grphs xponntil untions y = k x or positiv vlus o k, n th trigonomtri untions (with rgumnts in grs) y = sin x, y = os x n y = tn x or ngls o ny siz. Skth trnsltions n rltions o givn untion. n xponntil grphs)

17 Clult or stimt grints o grphs (inluing qurti n othr non-linr grphs) n rs unr grphs (inluing qurti n othr non-linr grphs), n intrprt rsults in ss suh s istn-tim grphs, vloity-tim grphs n grphs in innil ontxts. Rognis n us th qution o irl with ntr t th origin; in th qution o tngnt to irl t givn point. (inluing thos with th unknown on oth sis o th qution); Solv qurti qutions lgrilly y torising in pproximt solutions using grph. Solv two simultnous qutions in two vrils (linr/linr) in pproximt solutions using grph. trnslt simpl situtions or prours into lgri xprssions or ormul; riv n qution (or two simultnous qutions), solv th qution(s) n intrprt th solution. (inluing thos tht rquir rrrngmnt); y omplting th squr n y using th qurti ormul; or linr/qurti) lgrilly; Fin pproximt solutions to qutions numrilly using itrtion Solv linr inqulitis in on vril rprsnt th solution st on numr lin, or two vrils; n qurti inqulitis in on vril; using st nottion n on grph.

18 Fioni typ squns, qurti squns, n simpl gomtri progrssions ( r n whr n is n intgr, n r is rtionl numr > 0 or sur) n othr squns. n qurti squns. hng rly twn rlt stnr units (.g. tim, lngth, r, volum/pity, mss) in lgri ontxts n ompoun units (.g. sp, rts o py, pris, nsity, prssur) in lgri ontxts.

19 nsity n prssur. mk links to similrity (inluing trigonomtri rtios) Unrstn tht X is invrsly proportionl to Y is quivlnt to X is proportionl to 1/Y ; intrprt qutions tht sri irt n invrs proportion. Intrprt th grint o stright lin grph s rt o hng; rognis n intrprt grphs tht illustrt irt n invrs proportion. St up, onstrut n Intrprt th grint t point on urv s th instntnous rt o hng; pply th onpts o vrg rt o hng (grints o hors) in numril, lgri n grphil ontxts, n instntnous rt o hng (grints o tngnts) in numril, lgri n grphil ontxts.

20 solv n intrprt th nswrs in growth n y prolms, inluing ompoun intrst n work with gnrl itrtiv prosss. Us th stnr rulr n ompss onstrutions (prpniulr istor o lin sgmnt, onstruting prpniulr to givn lin rom/t givn point, isting givn ngl); us ths to onstrut givn igurs n solv loi prolms; know tht th prpniulr istn rom point to lin is th shortst istn to th lin. Us th si ongrun ritri or tringls (SSS, SAS, ASA, RHS). Apply ngl ts, tringl ongrun, similrity n proprtis o quriltrls to onjtur n riv rsults out ngls n sis,

21 inluing Pythgors Thorm n th t tht th s ngls o n isosls tringl r qul, n us known rsults to otin simpl proos. (inluing rtionl n ngtiv sl tors). Dsri th hngs n invrin hiv y omintions o rottions, rltions n trnsltions. tngnt, r, stor n sgmnt. Apply n prov th stnr irl thorms onrning ngls, rii, tngnts n hors, n us thm to prov rlt rsults. Construt n

22 sur r n volum o sphrs, pyrmis, ons n omposit solis. Clult r lngths, ngls n rs o stors o irls. Apply th onpts o ongrun n similrity, inluing th rltionships twn lngths, Know th ormul or: Pythgors thorm, = 2, n th trigonomtri rtios, sinθ = opposit/hypotnus, osθ = jnt/hypotnus rs n volums in similr igurs. n tnθ = opposit/jnt; pply thm to in ngls n lngths in right-ngl tringls in two imnsionl igurs Know th xt vlus o sinθ n osθ or θ = 0, 30, 45, 60 n 90 ; know th xt vlu o tnθ or θ = 0, 30, 45 n 60. n, whr possil, gnrl tringls in two n thr imnsionl igurs. Know n pply th sin rul, /sin A = /sin B = C /sin C n osin rul, 2 = os A, to in unknown lngths n ngls. Know n pply Ar = 1/2 sin C to lult th r, sis or ngls o ny tringl. Apply ition n sutrtion o vtors, multiplition o vtors y slr, n igrmmti n olumn rprsnttions o vtors. Us vtors to onstrut gomtri rgumnts n proos.

23 Unrstn tht mpiril unis smpls tn towrs thortil proility istriutions, with inrsing smpl siz. n tr igrms. Clult th proility o inpnnt n pnnt omin vnts, inluing using tr igrms n othr rprsnttions, n know th unrlying ssumptions. Inr proprtis o popultions or istriutions rom smpl, whilst knowing th limittions o smpling. Clult n intrprt onitionl proilitis through rprsnttion using xpt rqunis with two-wy tls, tr igrms n Vnn igrms. tls n lin grphs or tim sris t

24 Construt n intrprt igrms or group isrt t n ontinuous t, i.. histogrms with qul n unqul lss intrvls n umultiv rquny grphs, n know thir pproprit us. inluing ox plots qurtils n intr-qurtil rng). n know tht it os not init ustion; rw stimt lins o st it; mk pritions; intrpolt n xtrpolt pprnt trns whilst knowing th ngrs o so oing.