8Algebraic UNCORRECTED SAMPLE PAGES. techniques. What you will learn. Australian curriculum. Chapter 8A 8B 8C 8D 8E 8F
|
|
- Zoe Shields
- 6 years ago
- Views:
Transcription
1 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K Chptr Wht you will lrn 8Algri thniqus Epning inomil prouts Prt squrs n irn o prt squrs Ftorising lgri prssions Ftorising th irn o two squrs Ftoristion y grouping Ftorising qurti trinomils (Etning) Ftorising trinomils o th orm + + (Etning) Simpliying lgri rtions: multiplition n ivision Simpliying lgri rtions: ition n sutrtion Furthr simpliition o lgri rtions (Etning) Equtions with lgri rtions (Etning) Austrlin urriulum NUMBER AND ALGEBRA Pttrns n lgr Apply th istriutiv lw to th pnsion o lgri prssions, inluing inomils, n ollt lik trms whr pproprit Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
2 Fr lling Th istn ( units) o n ojt rom th top o uiling tr it hs n ropp (whr ir rsistn is ngligil) n oun using th ormul ut + _ 1 t whr u is th initil vloity o th ojt, t th tim sin th ojt hs n ropp n th lrtion u to grvity, whih is pproimtly qul to 9.8 m/s. Whn n ojt is ropp it hs n initil vloity o 0 m/s, so th istn th ojt hs lln oms.9t. Using lgr, th istn rom th uiling tr t sons n oun or th tim tkn to rh groun lvl oul lult. I th ojt is inst ropp rom hot ir lloon sning t 10 m/s, th ojt i rst trvls in n upwr irtion. Its istn ( mtrs) ov or low th hight o th lloon rom whn th ojt is ropp n oun using 10t.9t. Knowing th tim tkn or th ojt to rh th groun, w oul gin us lgr to i n tors, suh s th hight o th lloon, th grtst hight rh y th ojt n th tim tkn or th ojt to rturn to th hight rom whih it ws rls. Onlin rsours Chptr pr-tst Vios o ll work mpls Intrtiv wigts Intrtiv wlkthroughs Downlol HOTshts Ass to HOTmths Austrlin Curriulum ourss Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
3 88 Chptr 8 Algri thniqus 8A Ky is Epning inomil prouts A inomil is n prssion with two trms suh s + or +. You will rll rom Chptr tht w look t th prout o singl trm with inomil prssion,.g. ( ) or ( 1). Th prout o two inomil prssions n lso pn using th istriutiv lw. This involvs multiplying vry trm in on prssion y vry trm in th othr prssion. Lt s strt: Rtngulr pnsions Epning th prout o two inomil prssions n ppli to prolms involving th pnsion o rtngulr rs suh s rmr s pok. I ( + 1) n ( + ) r th si lngths o rtngl s shown, th totl r n oun s n prssion in two irnt wys. Writ n prssion or th totl r o th rtngl using lngth ( + ) n with ( + 1). Now in th r o h o th our prts o th rtngl n omin to giv n prssion or th totl r. Compr your two prssions ov n omplt this qution: ( + )( ) + +. Cn you plin mtho or pning th lt-hn si to giv th right-hn si? Epning inomil prouts uss th istriutiv lw. ( + )( + ) ( + ) + ( + ) Digrmmtilly ( + )( + ) For mpl: ( + 1)( + ) Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
4 Numr n Algr 89 Empl 1 Epning inomil prouts Epn th ollowing. ( + )( + ) ( )( + 7) ( 1)( 6) ( )( + 7) SOLUTION ( + )( + ) ( )( + 7) ( 1)( 6) ( )( + 7) Eris 8A Th givn igrm shows th r ( + )( + ). + + EXPLANATION Us th istriutiv lw to pn th rkts n thn ollt th lik trms n. Atr pning to gt th our trms, ollt th lik trms 7 n. Rmmr n 1 ( 6) 6. Rll 1. Writ own n prssion or th r o h o th our rgions insi th rtngl. Copy n omplt: ( + )( + ) Th givn igrm shows th r ( + )( + 1). 1 (½) Writ own n prssion or th r o h o th our rgions insi th rtngl. Copy n omplt: ( + )( ) UNDERSTANDING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
5 90 Chptr 8 Algri thniqus 8A Empl 1 Empl 1,, Copy n omplt ths pnsions. ( + 1)( + ) ( )( + ) ( )(7 + ) + 6 ( 1)( ) + + (½) (½) Epn th ollowing. ( + )( + ) ( + )( + ) (t + 8)(t + 7) (p + 6)(p + 6) ( + 9)( + 6) ( + 1)( + ) g ( + 1)( + 7) h (y + 10)(y + ) i (m + )(m + 1) Epn th ollowing. ( + )( ) ( + )( ) ( + )( 8) ( 6)( + ) ( 1)( + 10) ( 7)( + 9) g ( )( + 7) h ( 1)( ) i ( )( ) j ( + )( + ) k ( + )( + 1) l ( + 1)( + ) m ( )( + ) n (8 )( + ) o ( )( + 1) p ( + )( 7) q ( + )( ) r ( + 1)( ) s ( )(6 ) t ( )( 1) u (7 )( ) 6 Epn ths inomil prouts. ( + )( + ) ( )( + ) ( )( + ) ( y)(y z) (y )(z y) (1 )(1 + y) g ( + y)( y) h ( + )( ) i ( y)( + y) j ( )( + ) k ( y)( y) l (y yz)(z + ) 7 A room in hous with imnsions m y m is to tn. Both th lngth n with r to inrs y m. Fin n pn prssion or th r o th nw room. I : i in th r o th nw room ii y how muh hs th r inrs? 7 7, 8 6(½) 8, 9 UNDERSTANDING FLUENCY PROBLEM-SOLVING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
6 Numr n Algr 91 8 A pitur rm m wi hs lngth whih is twi th with m. Fin n prssion or th totl r o th rm n pitur. Fin n prssion in pn orm or th r o th pitur only. Pitur m m 9 Th outsi g o pth roun rtngulr swimming pool is 1 m long n 10 m wi. Th pth is mtrs wi. m Fin n prssion or th r o th pool in pn orm. Pool Fin th r o th pool i. 1 m 10(½) 10(½), Writ th missing trms in ths pnsions. ( + )( + ) ( + )( + ) ( + 1)( + ) ( + )( + 9) ( + )( ) + ( )( + ) g ( + 1)( + ) + + h ( )( 1) 9 + i ( + )( + ) j ( )( 1) Consir th inomil prout ( + )( + ). Fin th possil intgr vlus o n i: ( + )( + ) ( + )( + ) + 6 ( + )( + ) + 6 ( + )( + ) 6 Trinomil pnsions 1 Using th istriutiv lw ( + )( + + ) Us this knowlg to pn n simpliy ths prouts. Not:. ( + 1)( + + 1) ( )( + ) ( 1)( + ) ( + 1)( + ) ( + )( ) ( + 7)( 7) g ( + )( + ) h ( )( ) i ( + )( + ) j ( )( + + ) 1 Now try to pn ( + 1)( + )( + ). 10(½), 11 m 10 m 1, 1 PROBLEM-SOLVING REASONING ENRICHMENT 8A Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
7 9 Chptr 8 Algri thniqus 8B Prt squrs n irn o prt squrs W know tht, 1, n ( + ) r ll mpls o prt squrs. To pn ( + ) w multiply ( + ) y ( + ) n us th istriutiv lw: ( + ) ( + )( + ) A similr rsult is otin or th squr o ( ): ( ) ( )( ) + + Anothr typ o pnsion involvs th s tht ls with th prout o th sum n irn o th sm two trms. Th rsult is th irn o two prt squrs: ( + )( ) + Binomil prouts n us to lult th most iint wy to ut th shps rquir or rition out o mtl sht. (sin, th two mil trms nl h othr out.) Lt s strt: Sing th pttrn Using ( + )( + ) + + +, pn n simpliy th inomil prouts in th two sts low. St A ( + 1)( + 1) ( + )( + ) ( )( ) St B ( + 1)( 1) + 1 ( )( + ) ( )( + ) Dsri wht pttrns you s in oth sts o pnsions ov. Gnrlis your osrvtions y omplting th ollowing pnsions. A ( + )( + ) B ( + )( ) + ( )( ) + + Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
8 Numr n Algr 9 9,, (y), ( 1) n ( y) r ll mpls o prt squrs. Epning prt squrs ( + ) ( + )( + ) ( ) ( )( ) + + Dirn o prt squrs (DOPS) ( + )( ) + ( )( + ) lso pns to Th rsult is irn o two prt squrs. Empl Epning prt squrs Epn h o th ollowing. ( ) ( + ) SOLUTION ( ) ( )( ) + + Altrntiv solution: ( ) + + ( + ) ( + )( + ) Altrntiv solution: ( + ) () EXPLANATION Writ in pn orm. Us th istriutiv lw. Collt lik trms. Epn using ( ) + whr n. Writ in pn orm. Us th istriutiv lw. Collt lik trms. Epn using ( + ) + + whr n. Rll (). Ky is Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
9 9 Chptr 8 Algri thniqus Empl Forming irn o prt squrs Epn n simpliy th ollowing. ( + )( ) ( y)( + y) SOLUTION ( + )( ) + Altrntiv solution: ( + )( ) () () EXPLANATION Epn using th istriutiv lw ( + )( ). Hr n. ( y)( + y) 9 + 6y 6y y Epn using th istriutiv lw. 9 y 6y 6y 0. Altrntiv solution: ( y)( + y) () (y) Eris 8B ( + )( ) with n y 9 y hr. 1 Complt ths pnsions. ( + )( + ) ( + )( + ) ( )( ) + ( 7)( 7) 7 + Sustitut th givn vlu o into + + n simpliy. i ii 11 iii 1 Sustitut th givn vlu o into + n simpliy. i ii 9 iii 0 Complt ths pnsions. ( + )( ) + ( 10)( + 10) + 10 ( 1)( + 1) + ( + )( ) (½), (½) UNDERSTANDING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
10 Numr n Algr 9 Empl Empl Empl Empl 7(½) 8(½) 8(½) Epn h o th ollowing prt squrs. ( + 1) ( + ) ( + ) ( + ) ( + ) ( + 9) g ( + 7) h ( + 10) i ( ) j ( 6) k ( 1) l ( ) m ( 9) n ( 7) o ( ) p ( 1) Epn h o th ollowing prt squrs. ( + 1) ( + ) ( + ) ( + 1) ( + ) ( + ) g (7 + ) h ( + ) i ( ) j ( 1) k ( ) l ( 9) m ( + y) n ( + y) o (7 + y) p (6 + y) q ( 9y) r ( 7y) s ( 10y) t ( 6y) u (9 y) 6 Epn h o th ollowing prt squrs. ( ) ( ) (1 ) (6 ) (11 ) ( ) g (7 ) h (1 ) i (8 ) j ( ) k (9 ) l (10 ) 7 8 Epn n simpliy th ollowing to orm irn o prt squrs. ( + 1)( 1) ( + )( ) ( + 8)( 8) ( + )( ) ( + 1)( 1) ( + 11)( 11) g ( 9)( + 9) h ( )( + ) i ( 6)( + 6) j ( )( + ) k ( )( + ) l (7 )(7 + ) Epn n simpliy th ollowing. ( )( + ) ( )( + ) ( )( + ) (7 y)(7 + y) (9 y)(9 + y) (11 y)(11 + y) g (8 + y)(8 y) h (10 9y)(10 + 9y) i (7 y)(7 + y) j (6 11y)(6 + 11y) k (8 y)(8 + y) l (9 y)(9 + y) 9 Lr is yrs ol n hr two st rins r ( ) n ( + ) yrs ol. 9 9, 10 Writ n prssion or: i th squr o Lr s g ii th prout o th gs o Lr s st rins (in pn orm). Ar th nswrs rom prts i n ii qul? I not, y how muh o thy ir? 9, 10 FLUENCY PROBLEM-SOLVING 8B Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
11 96 Chptr 8 Algri thniqus 8B 10 A squr pi o tin o si lngth 0 m hs our squrs o si lngth m rmov rom h ornr. Th sis r ol up to orm try. Th ntr squr orms th try s. Writ n prssion or th si lngth o th s o th try. Writ n prssion or th s o th try. Epn your nswr. Fin th r o th try s i. Fin th volum o th try i. m m Try s 0 m 0 m 11 Four tnnis ourts r rrng s shown with squr storg r in th ntr. Eh ourt r hs th sm imnsions. Writ n prssion or th si lngth o th totl r. Writ n prssion or th totl r. Writ n prssion or th si lngth o th insi storg r. Writ n prssion or th r o th insi storg r. Sutrt your nswr to prt rom your nswr to prt to in th r o th our ourts. Fin th r o on ourt. Dos your nswr onirm tht your nswr to prt is orrt? 1 A squr o si lngth units hs on si ru y 1 unit n th othr inrs y 1 unit. 11 1, 1 Fin n pn prssion or th r o th rsulting rtngl. Is th r o th originl squr th sm s th r o th rsulting rtngl? Eplin why/why not? 1, 1 PROBLEM-SOLVING REASONING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
12 Numr n Algr 97 1 A squr o si lngth is rmov rom squr o si lngth. Using sutrtion writ own n prssion or th rmining r. Writ prssions or th r o th rgions: i A ii B iii C A ll th prssions rom prt to s i you gt your nswr rom prt. Etn pnsions 1 Epn n simpliy ths prssions. ( + ) ( 1) ( + )( ) ( + 1) ( + 1)( 1) ( + 1) ( 1) g ( )( + ) ( + ) h ( 1) ( + 1)( 1) i ( + y) ( y) + ( + y)( y) j ( ) + ( + ) k ( ) ( + ) l ( ) + ( ) m ( ) ( )( + ) n ( + y) ( y) C A B Th logil skills o lgr hv pplitions in omputr progrmming. 1 REASONING ENRICHMENT 8B Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
13 98 Chptr 8 Algri thniqus 8C Ky is Ftorising lgri prssions Th pross o toristion is ky stp in th simpliition o mny lgri prssions n in th solution o qutions. It is th rvrs pross o pnsion n involvs writing n prssion s prout o its tors. pning ( ) 6 torising Lt s strt: Whih toris orm? Th prout ( + 8) whn pn givs + 8. Ftorising is ky mthmtil skill rquir in mny ivrs ouptions, suh s in usinss, sin, thnology n nginring. Writ own thr othr prouts tht whn pn giv + 8. (Do not us rtions.) Whih o your prouts uss th highst ommon tor o n 8? Wht is this highst ommon tor? Whn torising prssions with ommon tors, tk out th highst ommon tor (HCF). Th HCF oul : numr For mpl: + 10 ( + ) pronumrl (or vril) For mpl: + ( + ) th prout o numrs n pronumrls For mpl: + 10 ( + ) A toris prssion n hk y using pnsion. For mpl: ( + ) Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
14 Numr n Algr 99 Empl Fining th HCF Dtrmin th HCF o th ollowing. 6 n 8 SOLUTION n 6y EXPLANATION HCF o 6 n 8 is. HCF o n is. HCF o n 6 is. Empl Ftorising prssions Ftoris th ollowing SOLUTION HCF o n y is. 8 1 EXPLANATION ( ) Th HCF o 0 n 16 is 8. Pl 8 in ront o th rkts n ivi h trm y ( + ) Th HCF o th trms is, inluing th ommon ngtiv. Pl th tor in ront o th rkts n ivi h trm y. Empl 6 Tking out inomil tor Ftoris th ollowing. ( + y) + ( + y) (7 ) (7 ) SOLUTION ( + y) + ( + y) ( + y)( + ) (7 ) (7 ) 1(7 ) (7 ) (7 )(1 ) EXPLANATION HCF ( + y). Th son pir o rkts ontins wht rmins whn ( + y) n ( + y) r ivi y ( + y). Insrt 1 in ront o th irst rkt. HCF (7 ). Th son rkt must ontin 1 tr iviing (7 ) n (7 ) y (7 ). Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
15 00 Chptr 8 Algri thniqus Eris 8C 1 (½), Empl Empl Empl Empl 6 1 Writ own th highst ommon tor (HCF) o ths pirs o numrs. 8, 1 10, 0, 60, 0, 100, 7 g 16, h 6, 7 Writ own th missing tor y 6y 1 6 g 6 h 0 i 7y 1 y Writ own th missing tor in h prt. i ( + ) iii ( + ) ii ( + ) Whih qution ov uss th HCF o 6 n 1? By looking t th trms lt in th rkts, how o you know you hv tkn out th HCF? 7(½) 8(½) Dtrmin th HCF o th ollowing. 6 n 1y 1 n 18 10m n 1y n 8 1t n 6s 1 n p g 9 n y h 6n n 1mn i 10y n y j 8 n 1 k y n 18y l n 1 Ftoris th ollowing y 10 + g 9 h 6 i 1 + j 6m + 6n k 10 8y l 0 m + n o y 7y p q p + p r 8 8 s + 1 t 6y 10y u 1 1 v 9m + 18m w 16y Ftoris th ollowing y toring out th ngtiv sign s prt o th HCF y y 8 g 10 1y h m 0n i 18 j 8 1 k 16y 6y l 10 m 6 0 n 6p 1p o 16 8 p 9 7 Ftoris th ollowing whih involv inomil ommon tor. ( + ) + ( + ) ( + 1) + ( + 1) 7(m ) + m(m ) ( 7) + ( 7) 8( + ) ( + ) ( + 1) ( + 1) g y(y + ) (y + ) h ( + ) ( + ) i t(t + ) + (t + ) j m(m ) + (m ) k y(y 1) (y 1) l (7 ) + (7 ) 8(½) UNDERSTANDING FLUENCY Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
16 Numr n Algr 01 8 Ftoris ths mi prssions y + 9y g y y + y h i m(m + ) + (m + ) j ( + ) ( + ) k ( ) + ( ) l ( + 1) ( + 1) m y( y) ( y) n ( + ) + ( + ) o (y + 1) (y + 1) 9 Writ own th primtr o ths shps in toris orm Th prssion or th r o rtngl is ( + 8) squr units. Fin n prssion or its with i th lngth is ( + ) units Th hight, in mtrs, o ll thrown in th ir is givn y t t, whr t is th tim in sons. Writ n prssion or th ll s hight in toris orm. Fin th ll s hight t ths tims: i t 0 ii t iii t How long os it tk or th ll s hight to rturn to 0 mtrs? Us tril n rror i rquir. 9 9, (½), 10, 11 FLUENCY PROBLEM-SOLVING 8C Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
17 0 Chptr 8 Algri thniqus 8C n vlut y irstly torising to 7(9 + ). This givs Us similr thniqu to vlut th ollowing Common tors n lso rmov rom prssions with mor thn two trms. For mpl: y ( + + y) Ftoris ths prssions y tking out th HCF z 10z + zy y + y y + 6 1y 8yz 0yz Somtims w n hoos to tor out ngtiv or positiv HCF. Both toristions r orrt. For mpl: ( ) (HCF is 1) OR ( + ) (HCF is 1) 1( ) 1 1, 1 Ftoris in two irnt wys: th irst y toring out ngtiv n th son y positiv HCF n m + m g + h y + y i 8n + 1n j 8y + 0 k 1mn + 10 l 1 + Ftoring out ngtiv 1 Using th t tht ( ) you n toris ( ) ( ) y ollowing ths stps. ( ) ( ) ( ) + ( ) ( )( + ) Us this i to toris ths prssions. ( ) + ( ) ( ) ( ) ( ) ( ) ( ) + ( ) ( ) + ( ) ( ) + ( ) g ( ) ( ) h ( ) + (10 ) i ( ) + (6 ) 1, 1 1 REASONING ENRICHMENT Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
18 Numr n Algr 0 8D Ftorising th irn o two squrs Rll tht irn o two prt squrs is orm whn pning th prout o th sum n irn o two trms. For mpl, ( + )( ). Rvrsing this pross mns tht irn o two prt squrs n toris into two inomil prssions o th orm ( + ) n ( ). Lt s strt: Epning to unrstn torising Complt th stps in ths pnsions thn writ th onlusion. ( + )( ) + ( )( + ) ( + )( ) ( + )( ) + ( + )( ) ( + )( ) Ftorising th irn o prt squrs (DOPS) uss th rul ( + )( ). 16 ( + )( ) () 10 ( + 10)( 10) y (y) ( + y)( y) First tk out ommon tors whr possil. 18 ( 9) ( + )( ) Empl 7 Ftorising DOPS Ftoris h o th ollowing y ( + 1) Ky is SOLUTION ( + )( ) EXPLANATION Writ s DOPS ( is th sm s ). Writ in toris orm ( + )( ). Hr n. Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
19 0 Chptr 8 Algri thniqus 9 () ( + )( ) Writ s DOPS. 9 is th sm s (). Writ in toris orm. Empl 7 81 y (9) y (9 + y)(9 y) ( 16) ( ) ( + )( ) ( + 1) ( + 1) Eris 8D ( )( + 1 ) ( + )( 1) 81 (9) Us ( + )( ) First, tor out th ommon tor o. Writ s DOPS n thn toris. Writ s DOPS. In hr, is th prssion + 1 n. Writ in toris orm n simpliy. 1 Epn ths inomil prouts to orm irn o prt squrs. ( + )( ) ( 7)( + 7) ( 1)( + 1) ( + y)( y) ( y)( + y) ( + )( ) Writ th missing trm. Assum it is positiv numr. ( ) 9 ( ) 11 ( ) 81 ( ) 00 ( ) ( ) 9 g ( ) h ( ) 9y Complt ths toristions ( ) ( + )( ) ( + 1)( ) 16 1 ( ) ( ) 9 ( ) ( ) ( + )( 1) 1 (½) (½) ( + )( ) 6(½) 7(½) Ftoris h o th ollowing. 9 y y g 81 h y i j 16 k l 1 m 6 y n 11 o 00 p 900 y 7(½) UNDERSTANDING FLUENCY Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
20 Numr n Algr 0 Empl 7, Empl 7 Empl 7 Ftoris h o th ollowing. 9 9 m y 9 81 g 1 h 6 i 16 9y j 6 y k y l 6 9 m p q n 81m n o 9 p Ftoris h o th ollowing y irst tking out th ommon tor y m g y h y i 6 7 Ftoris h o th ollowing. ( + ) 9 ( + ) ( + 10) 16 ( ) ( 7) 1 ( ) 6 g 9 ( + ) h ( + ) i 81 ( + 8) 8 Th hight ov groun (in mtrs) o n ojt thrown o th top o uiling is givn y 6 t whr t is in sons. Ftoris th prssion or th hight o th ojt y irstly tking out th ommon tor. Fin th hight o th ojt: i initilly (t 0) ii t sons (t ). How long os it tk or th ojt to hit th groun? Us tril n rror i you wish. 9 This multisiz squr pitur rm hs si lngth 0 m n n hol squr pitur with ny si lngth lss thn 6 m. 8 8, 9 I th si lngth o th pitur is m, writ n prssion or: i th r o th pitur ii th r o th rm (in toris orm). Us your rsult rom prt ii to in th r o th rm i: i 0 ii th r o th pitur is m. m 0 m 8, 9 FLUENCY PROBLEM-SOLVING 8D Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
21 06 Chptr 8 Algri thniqus 8D 10 Initilly it my not ppr tht n prssion suh s + 9 is irn o prt squrs. Howvr, swpping th position o th two trms mks + 9 9, whih n toris to ( + )( ). Us this i to toris ths irn o prt squrs y g 16 + y z h Olivi toriss 16 to gt ( + )( ) ut th nswr sys ( + 1)( 1). Wht shoul Olivi o to gt rom hr nswr to th tul nswr? Wht shoul Olivi hv on initilly to voi this issu? 1 Fin n plin th rror in this working n orrt it. 9 ( 1) ( + 1)( 1) ( + )( ) Ftorising with rtions n powrs o 10 10, 11 1 Som prssions with rtions or powrs o n toris in similr wy. Ftoris ths g i y j k 1 1 h l y REASONING ENRICHMENT Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
22 Numr n Algr 07 8E Ftoristion y grouping Whn n prssion ontins our trms, suh s +, it my possil to toris it into prout o two inomil trms lik ( 1)( + ). In suh situtions th mtho o grouping is otn us. Lt s strt: Two mthos sm rsult Th our-trm prssion + is writtn on th or. Ftorising y grouping is it lik rrnging sttr ojts into som sort o orr. Tommy hooss to rrrng th trms to giv + thn toriss y grouping. Shron hooss to rrrng th trms to giv + thn lso toriss y grouping. Complt Tommy n Shron s toristion working. Tommy + ( ) + 1( ) ( )( ) Shron + ( ) ( ) Disuss th irns in th mthos. Is thr ny irn in thir nswrs? Whos mtho o you prr? ( + 1)( ) Ftoristion y grouping is mtho whih is otn us to toris our-trm prssion. Trms r group into pirs n toris sprtly. + 6 Th ommon inomil tor is thn tkn out to omplt th toristion. ( + ) ( + ) Trms n rrrng to ssist in th srh o ommon tor. ( + )( ) Empl 8 Ftorising y grouping Us th mtho o grouping to toris ths prssions SOLUTION ( + ) + ( + 6) ( + ) + ( + ) ( + )( + ) EXPLANATION Group th irst n son pir o trms. Ftoris h group. Tk th ommon tor ( + ) out o oth groups. Ky is + 1 ( + ) + ( 1) ( + ) ( + ) ( + )( ) Group th irst n son pir o trms. Ftoris h group. Tk out th ommon tor ( + ). Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
23 08 Chptr 8 Algri thniqus Empl 9 Rrrnging n prssion to toris y grouping Ftoris using grouping. Empl 8 SOLUTION Altrntivly: ( + 1) 9( + 1) ( + 1)( 9) Eris 8E ( 9) + 1( 9) ( 9)( + 1) EXPLANATION Rrrng so tht h group hs ommon tor. Ftoris h group thn tk out ( + 1). Altrntivly, you n group in nothr orr whr h group hs ommn tor. Thn toris. Th nswr will th sm. 1 (½) 1 Epn h prssion. ( 1) ( + ) (1 ) ( ) ( + ) ( ) g ( ) h y( y) i ( + 1) + ( + 1) j ( ) + ( ) k ( ) ( ) l (1 ) (1 ) Copy n thn ill in th missing inormtion. ( + 1) + ( + 1) ( + 1)( ) ( + ) ( + ) ( + )( ) ( + ) ( + ) ( + )( ) ( + 7) + ( + 7) ( + 7)( ) ( ) + ( ) ( )( ) ( + ) ( + ) ( + )( ) g ( ) ( ) ( )( ) h ( ) + ( ) ( )( ) Tk out th ommon inomil trm to toris h prssion. ( ) ( ) ( + ) + ( + ) ( 7) + ( 7) ( + 1) ( + 1) ( ) ( ) ( + ) ( + ) g ( ) + ( ) h ( + 1) ( + 1) i ( ) + ( ) (½) Us th mtho o grouping to toris ths prssions g h + 1 i j + k + 9 l + 1 Us th mtho o grouping to toris ths prssions. Th HCF or h pir inlus pronumrl y 8z + wy 1wz rs 10r + st t + 1y 9y + (½) 6(½) 6(½) UNDERSTANDING FLUENCY Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
24 Numr n Algr 09 Empl 9 6 Ftoris ths prssions. Rmmr to us tor o 1 whr nssry, or mpl, + ( ) + 1( ) , 8 Ftoris ths prssions y irst rrrnging th trms y + 8y g 6m n + mn h 1p 8r pr + i 16 y 8y Wht pn prssion toriss to th ollowing? ( )( + ) ( )( ) ( + y)( z) ( 1)( + ) ( )( ) ( y)(y + z) g ( + )( + ) h (m )(y + z) 9 Not tht whih n toris y grouping. Us similr mtho to toris th ollowing oul rrrng in two irnt wys or torising. Mtho ( + 7) ( ) Mtho ( ) + 7( ) Copy n omplt oth mthos or th ov prssion. Us irnt rrngmnts o th our trms to omplt th toristion o th ollowing in two wys. Show working using oth mthos. i 6 + ii y 8 + y iii m 1n + 6m 10mn iv m + n mn 6 v vi Mk up t lst thr o your own our-trm prssions tht toris to inomil prout. Dsri th mtho tht you us to mk up h our-trm prssion. Grouping with mor thn our trms 1 Ftoris y grouping. ( ) ( ) ( ) ( + 1) + ( + 1) ( + 1) ( + 1) ( + 1) ( ) ( ) + (1 ) + + ( ) + + g h 6y z + 10yz + y y i 8z y + + y 1 z j + y + + 6y Using CAS lultor 8E: Epning n torising This tivity is in th intrtiv ttook in th orm o printl PDF. 7, 9 10, 11 1 PROBLEM-SOLVING FLUENCY REASONING ENRICHMENT 8E Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
25 10 Chptr 8 Algri thniqus 8F Ftorising qurti trinomils EXTENDING Ky is An prssion tht tks th orm + +, whr n r onstnts, is n mpl o moni qurti trinomil whih hs th oiint o qul to 1. To toris qurti prssion, w n to us th istriutiv lw in rvrs. Consir th pnsion shown t right: I w min th pnsion ov w n s how h trm o th prout is orm. Prout o n is ( + )( ) 8 Prout o n is 8 ( ( ) 8, th onstnt trm) Lt s strt: So mny hois ( ) pning ( + )( ) 8 toris orm pn orm torising ( + )( ) 8 Mi sys tht sin 6 thn + 6 must qul ( )( + ). A n to giv th mil trm, ( +, th oiint o ) Epn ( )( + ) to s i Mi is orrt. Wht othr pirs o numrs multiply to giv 6? Whih pir o numrs shoul Mi hoos to orrtly toris + 6? Wht vi n you giv Mi whn trying to toris ths typs o trinomils? To toris qurti trinomil o th orm + +, in two numrs whih: multiply to giv n to giv. For mpl: 10 ( )( + ) hoos n + sin 10 n + Chk toristion stps y pning. hk: ( )( + ) Writ th tors in ny orr. writ ( )( + ) or ( + )( ) Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
26 Numr n Algr 11 Empl 10 Ftorising qurti trinomils Ftoris h o th ollowing qurti prssions SOLUTION EXPLANATION ( + )( + ) Ftors o 10 inlu: (10, 1) n (, ). Th pir tht s to 7 is (, ). + 8 ( + )( ) Ftors o 8 r ( 8, 1) or (8, 1) or (, ) or (, ) n + ( ) so hoos (, ) ( )( ) Ftors o 10 r: (10, 1) or ( 10, 1) or (, ) or (, ). Empl 11 Ftorising with ommon tor Ftoris th qurti prssion 1. SOLUTION 1 ( 6) Eris 8F ( )( + ) To to ngtiv ( 7), oth tors must thn ngtiv: + ( ) 7 so hoos (, ). EXPLANATION First tk out ommon tor o. Ftors o 6 r: ( 6, 1) or (6, 1) or (, ) or (, ). + 1 so hoos (, ). 1 (½) 1 Epn ths inomil prouts. ( + 1)( + ) ( + )( + 7) ( )( + 11) ( )( + 6) ( + 1)( ) ( + 1)( ) g ( )( 6) h ( 0)( 11) i ( 9)( 1) Di wht two numrs multiply to giv th irst numr n to giv th son numr. 6, 10, 7 1, 1 0, 9, 7, 6 g 1, h 0, 1 i 6, j 18, 11 k 0, 1 l 100, (½) UNDERSTANDING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
27 1 Chptr 8 Algri thniqus 8F Empl 10 Empl 10 Empl 10 Empl 11 Ftoris h o th ollowing qurti prssions g h i j k l Ftoris h o th ollowing qurti prssions g + 18 h i + 1 Ftoris h o th ollowing qurti prssions g h i Ftoris h o th ollowing qurti prssions g 1 h 11 1 i 1 7 6(½) 7(½) Ftoris h o th ollowing qurti prssions y irst tking out ommon tor g + 1 h i j + 90 k 6 0 l 6 7(½) 8 Fin th missing trm in ths trinomils i thy r to toris using intgrs. For mpl: th missing trm in oul 7 us toriss to ( + )( + ) n n r intgrs. Thr my mor thn on nswr in h s g 16 h + 9 A kyr, rtngulr in r, hs lngth mtrs mor thn its with ( mtrs). Insi th rtngl r thr squr pv rs h o r m s shown. Th rmining r is lwn. Fin n prssion or: i th totl kyr r ii th r o lwn in pn orm iii th r o lwn in toris orm. Fin th r o lwn i: i 10 ii (½), 9 ( + ) mtrs 8(½), 9 mtrs FLUENCY PROBLEM-SOLVING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
28 Numr n Algr 1 10 Th prssion toriss to ( )( ) ( ), whih is prt squr. Ftoris ths prt squrs g + + h 0 + i (½), Somtims it is not possil to toris qurti trinomils using intgrs. Di whih o th ollowing nnot toris using intgrs Complting th squr 10(½) 1 It is usul to l to writ simpl qurti trinomil in th orm ( + ) +. This involvs ing (n sutrting) spil numr to orm th irst prt squr. This prour is ll omplting th squr. Hr is n mpl. 6 ( ) ( )( ) 17 ( ) 17 10(½) Complt th squr or ths trinomils REASONING ENRICHMENT 8F Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
29 1 Chptr 8 Algri thniqus 8G Ftorising trinomils o th orm + + EXTENDING Ky is So r w hv toris qurti trinomils whr th oiint o is 1, suh s 0. Ths r ll moni trinomils. W will now onsir non-moni trinomils whr th oiint o is not qul to 1 n is lso not ommon tor to ll thr trms, suh s in Th mtho us in this stion uss grouping whih ws isuss in stion 8E. Lt s strt: How th grouping mtho works Consir th trinomil First writ thn toris y grouping. Not tht 9 ws split to giv + n th prout o n 10 is 0. Dsri th link twn th pir o numrs {, } n th pir o numrs {, 10}. Why ws 9 split to giv + n not, sy, + 6? Dsri how th 1 shoul split in so it n toris y grouping. Now try your mtho or 7 1. To toris trinomil o th orm + + y grouping, in two numrs whih sum to giv n multiply to giv. For mpl: ( 6) 0 so th two numrs r n sin 1 + ( ) 1 n 1 ( ) 0. ( + ) ( + ) ( + )( ) Mntlly hk your tors y pning your nswr. 6 ( + )( ) Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
30 Numr n Algr 1 Empl 1 Ftorising trinomils o th orm + + Ftoris SOLUTION ( + 1) + ( + 1) ( + 1)( + ) EXPLANATION Empl 1 Ftorising trinomils with ngtiv numrs Ftoris th qurti trinomils SOLUTION ( + ) ( + ) ( + )( ) Eris 8G ( ) ( ) ( )( ) 6 thn sk wht tors o this numr (6) to 7. Th nswr is 1 n 6, so split Thn toris y grouping. EXPLANATION 10 ( 9) 90 so sk wht tors o 90 to giv 9. Choos 1 n 6. Thn omplt th toristion y grouping so sk wht tors o 7 to giv 17. Choos 9 n 8. Complt mntl hk. ( )( ) , (½) (½) 1 List th two numrs whih stisy h prt. Multiply to giv 6 n to giv Multiply to giv 1 n to giv 8 Multiply to giv 10 n to giv Multiply to giv n to giv Multiply to giv 18 n to giv 9 Multiply to giv n to giv 1 g Multiply to giv 0 n to giv 7 h Multiply to giv 8 n to giv UNDERSTANDING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
31 16 Chptr 8 Algri thniqus 8G Empl 1 Empl 1 Copy n omplt ( ) + ( ) ( )( ) ( ) + ( ) ( )( ) ( ) ( ) ( )( ) ( ) 1( ) ( )( ) ( + ) + ( ) ( )( ) (½) ( ) + 1( ) ( )( ) (½) Ftoris ths qurti trinomils g h i Ftoris ths qurti trinomils g h i j 1 1 k l m 9 + n o 1 Ftoris ths qurti trinomils g h i 9 j k l Ftoris y irstly tking out ommon tor Ftoris ths trinomils (½) 6(½) (½) 6(½), 7 UNDERSTANDING FLUENCY PROBLEM-SOLVING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
32 Numr n Algr Whn splitting th in + 0, you oul writ: A or B Complt th toristion using A. Complt th toristion using B. Dos th orr mttr whn you split th? Ftoris ths trinomils twi h. Ftoris on y grouping thn rpt ut rvrs th orr o th two mil trms in th irst lin o working. i + 1 ii 1 iii Mk up iv non-moni trinomils with th oiint o not qul to 1 whih toris using th ov mtho. Eplin your mtho in ining ths trinomils. Th ross mtho 10 Th ross mtho is nothr wy to toris trinomils o th orm + +. It involvs ining tors o n tors o thn hoosing pirs o ths tors tht to. For mpl: Ftoris 6 1. Ftors o 6 inlu (, 6) n (, ). Ftors o 1 inlu (1, 1), ( 1, 1), (, ) n (, ). W rrng hosn pir o tors vrtilly thn ross-multiply n to gt ( ) ( 1) ( ) + 6 ( ) , 9 ( + ) You will n to ontinu until prtiulr omintion works. Th thir ross-prout givs sum o 1 so hoos th tors ( + ) n ( ) so: 6 1 ( + )( ) Try this mtho on th trinomils rom Qustions n. 10 REASONING ENRICHMENT 8G Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
33 18 Chptr 8 Algri thniqus Progrss quiz 8pt 8A 8pt 8B 8pt 8B 8pt 8C 8pt 8D 8pt 8E 8pt 8E 8pt 8F Et 8pt 8G Et Epn th ollowing. ( + )( + ) ( )( + 8) ( )( + 6) ( )( ) Epn h o th ollowing. (y + ) ( ) ( ) (7k + m) Epn n simpliy th ollowing. ( + )( ) (11 9y)(11 + 9y) Ftoris th ollowing ( + ) + ( + ) 7(8 + ) (8 + ) k(k ) (k ) Ftoris h o th ollowing y 0 1 y 1 (h + ) 6 Us th mtho o grouping to toris ths prssions Us grouping to toris ths prssions y irst rrrnging p 10 + p 1 Ftoris h o th ollowing qurti prssions m 11m + 0 k + k Ftoris ths qurti trinomils. k + 7k m 19m + 6 h + h Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
34 Numr n Algr 19 8H Simpliying lgri rtions: multiplition n ivision With numril rtion suh s 6, th highst ommon tor o 6 n 9 is, whih n 9 nll For lgri rtions th pross is th sm. I prssions r in toris orm, ommon tors n sily intii n nll. Lt s strt: Corrt nlling Consir this nlling ttmpt: Sustitut 6 into th lt-hn si to vlut Sustitut 6 into th right-hn si to vlut + 1. Wht o you noti out th two nswrs to th ov? How n you plin this? Di how you might orrtly nl th prssion on th lt-hn si. Show your stps n hk y sustituting vlu or. Simpliy lgri rtions y torising n nlling only ommon tors. Inorrt To multiply lgri rtions: toris prssions whr possil nl i possil multiply th numrtors n th nomintors. Corrt + 1 ( + ) 1 + To ivi lgri rtions: multiply y th riprol o th rtion ollowing th ivision sign ollow th ruls or multiplition tr onvrting to th riprol - Th riprol o is. Ky is Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
35 0 Chptr 8 Algri thniqus Empl 1 Simpliying lgri rtions Simpliy th ollowing y nlling. ( + )( ) 6( ) SOLUTION 1 ( + ) ( ) 1 6 ( ) ( ) 1 ( ) ( + )( ) ( + ) EXPLANATION Empl 1 Multiplying n iviing lgri rtions Simpliy th ollowing. ( 1) ( + ) ( + ) 9( 1)( 7) SOLUTION 1 ( 1) 1 ( ( + ) + 1 ) 1 9 ( 1) 1 ( 7) 1 1 ( 7) ( 7) ( )( + ) ( + ) ( + 7) + 7 ( ) ( + ) 1 ( + 7) ( + 7) 1 1 ( + ) 1 ( )( + ) ( + ) ( + 7) Cnl th ommon tors ( ) n. Ftoris th numrtor n nomintor thn nl ommon tor o ( ). Ftoris th irn o squrs in th numrtor thn nl th ommon tor. EXPLANATION + Et First, nl ny tors in th numrtors with ommon tor in th nomintors. Thn multiply th numrtors n th nomintors. Multiply y th riprol o th rtion tr th ivision sign. Cnl ommon tors n multiply rmining numrtors n nomintors. Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
36 Numr n Algr 1 Empl 1 Empl ( )( + ) 1 ( + 1) 1 1 ( ) ( + 1) 1 + Eris 8H 1 Simpliy ths rtions y nlling Ftoris ths y tking out ommon tors Copy n omplt. ( ) ( 1) ( + )( + ) Simpliy th ollowing y nlling. ( + ) ( + ) ( + )( ) ( + ) ( ) ( ) g 6( 1)( + ) 9( + ) First toris ll th lgri prssions. Not tht is irn o prt squrs. Thn nl s norml ( + 1) 6 7 Simpliy th ollowing y torising n thn nlling ( ) ( ) 8(½) 6 6 9(½) h 0( + 7) ( + 7) 1 ( ) ( ) ( )( + ) 1 6 9(½) UNDERSTANDING FLUENCY + 10 g + y + y h 8y 6y Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
37 Chptr 8 Algri thniqus 8H Empl 1 Empl 1 Empl 1 Et Empl 1 Et Simpliy th ollowing. Ths prssions involv irn o prt squrs ( 0) 00 Simpliy th ollowing y nlling. ( ) ( + 1) ( + 1) ( + )( + ) + + Simpliy th ollowing y nlling. ( + 1) ( + )( + 1) ( + ) ( 9)( + ) 9( + )( 9) ( + 6) ( + )( + 6) Simpliy y irstly torising ( 6) ( + )( ) + ( + )( + ) ( + 1) ( + 1)( ) ( + ) ( ) + 8 ( ) ( + )( ) ( + 7) Ths prssions involv omintion o trinomils, irn o prt squrs n simpl ommon tors. Simpliy y irstly torising whr possil (½) FLUENCY PROBLEM-SOLVING g h Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
38 Numr n Algr Et Et 11 Th prssion 1( + ) n writtn in th orm nll to 1. Us this i to simpliy ths lgri rtions ( ), whih n Just lik n nll to ( + ) +, nls to. Us this i to nl ths ( + ) rtions. ( + 1) ( + 1) ( 1)( + ) 18( 1)( + ) All in togthr ( ) ( ) Et 11(½) 11 7( + 7) 1( + 7) 11 + Us your knowlg o toristion n th is in Qustions 11 n 1 ov to simpliy ths lgri rtions. g i ( + ) (1 ) h j ( ) Et 11, 1 1 REASONING ENRICHMENT 8H Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
39 Chptr 8 Algri thniqus 8I Ky is Simpliying lgri rtions: ition n sutrtion Th pross rquir or ing or sutrting lgri rtions is similr to tht us or rtions without pronumrls. To simpliy +, or mpl, you woul in th lowst ommon multipl o th nomintors (1) thn prss h rtion using this nomintor. Aing th numrtors omplts th tsk. Lt s strt: Compr th working Hr is th working or th simpliition o th sum o pir o numril rtions n th sum o pir o lgri rtions Although lgri rtions, sm strt, prorming oprtions on thm n simpliying thm is ssntil to mny lultions in rl-li mthmtil prolms Wht typ o stps wr tkn to simpliy th lgri rtions tht r th sm s or th numril rtions? Writ own th stps rquir to (or sutrt) lgri rtions. To or sutrt lgri rtions: trmin th lowst ommon nomintor (LCD) prss h rtion using th LCD or sutrt th numrtors. Empl 16 Aing n sutrting with numrls in th nomintors Simpliy: SOLUTION EXPLANATION + + Dtrmin th LCD o n, i.. 0. Eprss h rtion s n quivlnt rtion with nomintor o Thn sutrt numrtors. Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
40 Numr n Algr ( + ) ( ) Not th LCD o n 6 is 6 not Simpliy 1 6 to in th inl stp. Th LCD o n is 10, writ s quivlnt rtions with nomintor 10. Epn th rkts n simpliy th numrtor y ing n ollting lik trms. Empl 17 Aing n sutrting with lgri trms in th nomintors Simpliy: SOLUTION Eris 8I + EXPLANATION Th LCD o n is, so rwrit th irst rtion in n quivlnt orm with nomintor lso o. Th LCD o n is so hng th irst rtion so its nomintor is lso, thn numrtors. 1 Fin th lowst ommon multipl o ths pirs o numrs. (6, 8) (, ) (11, 1) (1, 18) Writ quivlnt rtions y stting th missing prssion ( + 1) 1 UNDERSTANDING + 11 ( + ) ( + 1) Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
41 6 Chptr 8 Algri thniqus 8I Empl 16, Empl 16 Empl 17 Empl 17 Copy n omplt ths simpliitions ( + 1) + + Writ own th LCD or ths pirs o rtions , Simpliy: i m 7 + y 7 y 8 m 6 + m 7 9 Simpliy: g j y + + y t 1 + t 8 16 Simpliy: + Simpliy: + 8 7, j n y 8 + y h k , g k o m + m g 7 + +, 6 6(½) i l 7 10, 7(½) h l p 9 + m m 6 p 9 p y + y m 1 + m 6 h + 9 g 7 h + 7(½) 8(½) 8 9(½) 8 10(½) UNDERSTANDING FLUENCY PROBLEM-SOLVING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
42 Numr n Algr 7 9 Simpliy ths mi lgri rtions g + 10 Fin th missing lgri rtion. Th rtion shoul in simplst orm h Fin n sri th rror in h st o working. Thn in th orrt nswr , A stunt thinks tht th LCD to us whn simpliying is 8. 11, 1 Complt th simpliition using ommon nomintor o 8. Now omplt th simpliition using th tul LCD o. How os your working or prts n ompr? Whih mtho is prrl n why? Mor thn two rtions! 1 Simpliy y ining th LCD g + 1 j + m + h k n i l o PROBLEM-SOLVING REASONING ENRICHMENT 8I Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
43 8 Chptr 8 Algri thniqus 8J Furthr simpliition o lgri rtions EXTENDING Ky is Mor ompl ition n sutrtion o lgri rtions involvs prssions lik: 1 + n ( ) In suh mpls, r ns to tkn t h stp in th working to voi ommon rrors. Lt s strt: Thr ritil rrors Th ollowing simpliition o lgri rtions hs thr ritil rrors. Cn you in thm? ( + ) 6 6 Th orrt nswr is Fi th solution to prou th orrt nswr. Whn omining lgri rtions whih involv sutrtion signs, rll tht: th prout o two numrs o opposit sign is ngtiv numr th prout o two ngtiv numrs is positiv numr. For mpl: n ( 1) ( + ) (1 ) ( 1) A ommon nomintor n prout o two inomil trms. For mpl: ( 1) ( + )( 1) + ( + ) ( + )( 1) ( + )( 1) + 7 ( + )( 1) Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
44 Numr n Algr 9 Empl 18 Simpliying with mor ompl numrtors Simpliy: 1 + SOLUTION 1 + ( 1) ( + ) ( ) 6( ) EXPLANATION Empl 19 Simpliying with mor ompl nomintors Simpliy: SOLUTION ( ) ( + 1)( ) + ( + 1) ( + 1)( ) ( + 1)( ) 7 ( + 1)( ) ( 1) 1 ( 1) ( 1) ( 1) + ( 1) ( 1) Th LCD o n is 1. Insrt rkts roun h numrtor whn multiplying. Not: ( + ) 1 not + 1. Dtrmin th LCD n prss s quivlnt rtions. Insrt rkts. Epn th rkts, rll 6 ( ) 6 n thn simpliy th numrtor. ( 1) 1 EXPLANATION Th lowst ommon multipl o ( + 1) n ( ) is ( + 1)( ). Rwrit h rtion s n quivlnt rtion with this nomintor thn numrtors. Just lik th LCD o n is, th LCD o ( 1) n 1 is ( 1). Rmmr tht ( 1) +. Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
45 0 Chptr 8 Algri thniqus Eris 8J 1, (½) Empl 18 Empl 18 Empl 19 Empl 19 1 Epn th ollowing. ( + ) ( + 1) 7( + ) ( 1) 10( ) 16(1 ) Writ th LCD or ths pirs o rtions. 6 1, 9 1, + 1 Simpliy: g Simpliy: g Simpliy: g Simpliy: ( + 1) + 1 g + 8 ( ) 16, 1 8 7, + h h h ( + ) +,, 1, 1 g ( + 1), h + 1 (½) i i i (½) ( ) (½) ( + 1) h ( 6) 6(½) 9 ( + ) + i 6 7(½) ( ) (1 ) 1 6 7(½) UNDERSTANDING FLUENCY PROBLEM-SOLVING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
46 Numr n Algr 1 7 Simpliy: ( 1) + 1 g ( ) h On o th most ommon rrors m whn sutrting lgri rtions is hin in this working shown on th right: 9 Simpliy: Wht is th rror n in whih stp is it m? By orrting th rror how os th nswr hng? 1 ( + )( + ) + ( + )( + ) ( 1)( ) 6 ( 1)( ) + 8 ( )( ) 10 Us th t tht 1( ) to hlp simpliy ths Ftoris irst + i 1 8 8, 9(½) ( ) ( + 1) 7 ( ) ( + 1)( + ) ( + 1)( + ) ( + 1)( ) ( + )( 1) ( + )( + ) Ftorising nomintor or urthr simpliition is usul stp. Simpliy ths y irstly torising th nomintors i possil g i h j ( + 1) , 10(½) PROBLEM-SOLVING REASONING ENRICHMENT 8J Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
47 Chptr 8 Algri thniqus 8K Equtions with lgri rtions EXTENDING Ky is For qutions with mor thn on rtion it is otn st to try to simpliy th qution y ling with ll th nomintors t on. This involvs ining n multiplying oth sis y th lowst ommon nomintor. Lt s strt: Why us th LCD? For this qution ollow h instrution Multiply vry trm in th qution y. Wht t os this hv on th rtions on th lt-hn si? Strting with th originl qution, multiply vry trm in th qution y. Wht t os this hv on th rtions on th lt-hn si? Strting with th originl qution, multiply vry trm in th qution y 1 n simpliy. Whih instrution ov os th st jo in simpliying th lgri rtions? Why? For qutions with mor thn on rtion multiply oth sis y th lowst ommon nomintor (LCD). Multiply vry trm on oth sis, not just th rtions. Simpliy th rtions n solv th qution using th mthos lrnt rlir. Altrntivly, prss h rtion using th sm nomintor thn simpliy y ing or sutrting n solv. Empl 0 Solving qutions involving lgri rtions Solv h o th ollowing qutions SOLUTION EXPLANATION Multiply h trm y th LCD (LCD o n is 6) n nl. Simpliy n solv or. Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
48 Numr n Algr OR ( ) 1 ( 1) ( ) ( 1) ( + 1)( + ) ( + 1) ( + ) ( + 1) ( + ) ( + ) ( + 1) OR ( + ) ( + 1) Altrntivly, writ h trm on th lt-hn si using th LCD 6. Simpliy y ing th numrtors n solv th rmining qution. Multiply h trm on oth sis y 1 (LCD o n is 1) n nl. Epn th rkts n simpliy y omining lik trms. Not: ( 1) + not. (Altrntivly, writ h trm using th LCD 1 thn omin th numrtors n ( ) ( 1) solv. 1) 1 1 LCD o n is 6. Multiply h trm y 6. Cnl n simpliy. Solv or lving th nswr in rtion orm. (Altrntiv solution: ) Multiply h trm y th ommon nomintor ( + 1)( + ). Epn th rkts. Sutrt rom oth sis to gthr trms on on si thn sutrt 1 rom oth sis. Sin h si is singl rtion you n ross-multiply : This givs th sm rsult s ov. Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
49 Chptr 8 Algri thniqus Eris 8K 1 (½) (½) Empl 0 Empl 0 Empl 0 1 Writ own th lowst ommon nomintor o ll th rtions in ths qutions Simpliy th rtions y nlling g ( + ) 6( 7)( 1) 9( 7) h ( )(1 ) 9( ) Solv h o th ollowing qutions g 1 1 h m m 1 8 Solv h o th ollowing qutions g m + m i i ( + ) 7( + 1)( + ) 7( + 1) 8( )( 1) 8( ) y + y n + + n h Solv h o th ollowing qutions y y (½) i 7(½) y 1 y 6 1 n + n 1 m + m + 6 Solv h o th ollowing qutions. 1 1 m m y + 1 y g h 1 i 7 1 7(½) UNDERSTANDING FLUENCY Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
50 Numr n Algr Empl 0 7 Solv h o th ollowing qutions Hl o numr () plus on-thir o twi th sm numr is qul to. Writ n qution sriing th sitution. Solv th qution to in th numr Us your omin knowlg o ll th mthos lrnt rlir to solv ths qutions with lgri rtions g h Molly n Billy h hv th sm numr o omputr gms ( omputr gms h). Hzl tks on-thir o Molly s omputr gms n qurtr o Billy s omputr gms to giv hr totl o 77 omputr gms. Writ n qution sriing th totl numr o omputr gms or Hzl. Solv th qution to in how mny omputr gms Molly n Billy h h. 11 A ommon rror whn solving qutions with lgri rtions is m in this working. Fin th rror n plin how to voi it. 8 8, 9(½) i 1 + 9(½), ( + )( ) 11, 1 11, 1 1( 1) + 1 (LCD 1) ( 1) , 1 FLUENCY PROBLEM-SOLVING REASONING 8K 7 Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
51 6 Chptr 8 Algri thniqus 8K 1 Anothr ommon rror is m in this working. Fin n plin how to voi this rror. 1 1 (LCD 6) 6 6( 1) 6 ( 1) Som qutions with imls n y solv y irstly multiplying y powr o 10. Hr is n mpl Multiply oth sis y 10 to rmov ll imls. Solv ths iml qutions using th sm i. For prts you will n to multiply y Litrl qutions 1 Solv h o th ollowing qutions or in trms o th othr pronumrls. Hint: you my n to us toristion g + 1 h 1 1 j m + + k n i l o REASONING ENRICHMENT Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
52 Numr n Algr 7 Invstigtion Epning qurtis using rs Consir th pnsion o th qurti ( + )( + 6). This n rprsnt y ining th r o th rtngl shown. Totl r A 1 + A + A + A Thror: ( + )( + 6) Epning with positiv signs 6 Drw igrm n lult th r to trmin th pnsion o th ollowing qurtis. i ( + )( + ) ii ( + 7)( + 8) iii ( + ) iv ( + ) Using th sm thniqu stlish th rul or pning ( + ). Epning with ngtiv signs Consir th pnsion o ( )( 7). Ar rquir totl r (A + A + A ) Thror: [(A + A ) + (A + A ) A ] (7 + 8) ( )( 7) A 1 A A 1 A A A A A This r is ount twi whn w 7 +. Drw igrm n lult th r to trmin th pnsion o th ollowing qurtis. i ( )( ) ii ( 6)( ) iii ( ) iv ( ) Using th sm thniqu, stlish th rul or pning ( ). Dirn o prt squrs Using igrm to rprsnt ( )( + ), trmin th pproprit r n stlish rul or th pnsion o ( )( + ). 7 Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
53 8 Chptr 8 Algri thniqus Numril pplitions o prt squrs Th pnsion n toristion o prt squrs n irn o prt squrs n ppli to th mntl lultion o som numril prolms. Evluting prt squr Th prt squr n vlut using ( + ) + +. (0 + ) (Lt 0, ) 0 + (0)() Us th sm thniqu to vlut ths prt squrs. i ii 1 iii iv 1 v 1. vi. vii 6.1 viii 9.01 Similrly, th prt squr 9 n vlut using ( ) +. 9 (0 1) (Lt 0, 1) 0 (0)(1) Us th sm thniqu to vlut ths prt squrs. i 19 ii 9 iii 98 iv 87 v 1.9 vi.7 vii 8.8 viii.96 Evluting th irn o prt squrs Th irn o prt squrs 1 9 n vlut using ( + )( ). 1 9 (1 + 9)(1 9) (Lt 1, 9) 11 Us th sm thniqu to vlut ths irn o prt squrs. i 1 8 ii iii 1 iv 8 8 v vi.9.7 vii viii Th pnsion ( + )( ) n lso us to vlut som prouts. Hr is n mpl: 1 9 (0 + 1)(0 1) (Lt 0, 1) Us th sm thniqu to vlut ths prouts. i 1 19 ii 8 iii 6 7 iv 10 9 v vi vii 0 80 viii Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
54 Numr n Algr 9 Prolms n hllngs 1 Th irn twn th squrs o two onsutiv numrs is 97. Wht r th two numrs? Th irn twn th squrs o two onsutiv o numrs is 16. Wht r th two numrs? Th irn twn th squrs o two onsutiv multipls o is 81. Wht r th two numrs? I + y 6 n ( + y) 6, in th vlu o y. I + y 10 n y, in th vlu o y. Up or hllng? I you gt stuk on qustion, hk out th 'Working with Unmilir Qustions' postr t th n o th ook to hlp you. Fin th vlus o th irnt igits,, n i th our igit numr. Fin th qurti rul tht rlts th with n to th numr o mths in th pttrn low. n 1 n n Drw possil pttrn or ths ruls. i n + ii n(n 1) Ftoris n 1 n us th toris orm to plin why whn n is prim n grtr thn, n 1 is: i ivisil y ii iii ivisil y thus ivisil y 1. 6 Prov tht this prssion is qul to 1. 7 Prov tht or ll In r ovr km Ryn rn t onstnt sp. Sophi, howvr, rn th irst km t sp 1 km/h mor thn Ryn n rn th son km t sp 1 km/h lss thn Ryn. Who won th r? Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
55 0 Chptr 8 Algri thniqus Chptr summry Moni qurti trinomils (Et) Ths r o th orm: + + Rquir two numrs tht multiply to n to.g ( 9)( + ) sin 9 18 n Ftoristion Th rvrs pross o pnsion. Alwys rmov th highst ommon tor irst..g. + 8 HCF ( + ).g. ( + ) + y( + ) HCF + ( + )( + y) By grouping I thr r our trms, w my l to group into two inomil trms..g ( + ) ( + ) ( + )( ) DOPS ( )( + ).g. 16 ( )( + ).g. 9 () ( )( + ) Algri thniqus A/sutrt Must in lowst ommon nomintor (LCD) or pplying oprtion..g ( 1) Trinomils o th orm + + (Et) Cn toris using grouping lso.g , , Us 9 n ( + ) ( + ) sin ( + )( ) 9 ( ) 18 n 9 + ( ) 7 Epnsion Th pross o rmoving rkts..g. ( + ) + 10 ( + )( ) ( ) + ( ) Spil ss DOPS (irn o prt squrs) ( )( + ).g. ( )( + ) + Prt squrs ( + ) + + ( ) +.g. ( + ) ( + )( + ) Algri rtions Ths involv lgri prssions in th numrtor n/or nomintor..g.,, 7 7 ( 1) Multiply/ivi To ivi, multiply y th riprol. Ftoris ll prssions, nl n thn multiply..g ( + ) 1 ( )( + ) 1 1 ( ) Solving qutions with lgri rtions (Et) Fin lowst ommon nomintor (LCD) n multiply vry trm y th LCD..g. LCD o n is 6 LCD o ( 1) n ( + ) is ( 1)( + ) LCD o n is Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph
S i m p l i f y i n g A l g e b r a SIMPLIFYING ALGEBRA.
S i m p l i y i n g A l g r SIMPLIFYING ALGEBRA www.mthltis.o.nz Simpliying SIMPLIFYING Algr ALGEBRA Algr is mthmtis with mor thn just numrs. Numrs hv ix vlu, ut lgr introus vrils whos vlus n hng. Ths
More information# 1 ' 10 ' 100. Decimal point = 4 hundred. = 6 tens (or sixty) = 5 ones (or five) = 2 tenths. = 7 hundredths.
How os it work? Pl vlu o imls rprsnt prts o whol numr or ojt # 0 000 Tns o thousns # 000 # 00 Thousns Hunrs Tns Ons # 0 Diml point st iml pl: ' 0 # 0 on tnth n iml pl: ' 0 # 00 on hunrth r iml pl: ' 0
More information1 Introduction to Modulo 7 Arithmetic
1 Introution to Moulo 7 Arithmti Bor w try our hn t solvin som hr Moulr KnKns, lt s tk los look t on moulr rithmti, mo 7 rithmti. You ll s in this sminr tht rithmti moulo prim is quit irnt rom th ons w
More informationBinomials and Pascal s Triangle
Binomils n Psl s Tringl Binomils n Psl s Tringl Curriulum R AC: 0, 0, 08 ACS: 00 www.mthltis.om Binomils n Psl s Tringl Bsis 0. Intif th prts of th polnomil: 8. (i) Th gr. Th gr is. (Sin is th highst
More informationPaths. Connectivity. Euler and Hamilton Paths. Planar graphs.
Pths.. Eulr n Hmilton Pths.. Pth D. A pth rom s to t is squn o gs {x 0, x 1 }, {x 1, x 2 },... {x n 1, x n }, whr x 0 = s, n x n = t. D. Th lngth o pth is th numr o gs in it. {, } {, } {, } {, } {, } {,
More information(2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely
. DETERMINANT.. Dtrminnt. Introution:I you think row vtor o mtrix s oorint o vtors in sp, thn th gomtri mning o th rnk o th mtrix is th imnsion o th prlllppi spnn y thm. But w r not only r out th imnsion,
More informationModule graph.py. 1 Introduction. 2 Graph basics. 3 Module graph.py. 3.1 Objects. CS 231 Naomi Nishimura
Moul grph.py CS 231 Nomi Nishimur 1 Introution Just lik th Python list n th Python itionry provi wys of storing, ssing, n moifying t, grph n viw s wy of storing, ssing, n moifying t. Bus Python os not
More informationECE COMBINATIONAL BUILDING BLOCKS - INVEST 13 DECODERS AND ENCODERS
C 24 - COMBINATIONAL BUILDING BLOCKS - INVST 3 DCODS AND NCODS FALL 23 AP FLZ To o "wll" on this invstition you must not only t th riht nswrs ut must lso o nt, omplt n onis writups tht mk ovious wht h
More informationb. How many ternary words of length 23 with eight 0 s, nine 1 s and six 2 s?
MATH 3012 Finl Exm, My 4, 2006, WTT Stunt Nm n ID Numr 1. All our prts o this prolm r onrn with trnry strings o lngth n, i.., wors o lngth n with lttrs rom th lpht {0, 1, 2}.. How mny trnry wors o lngth
More informationUNCORRECTED SAMPLE PAGES. Length, area, surface 5area and volume. Online resources. What you will learn
Onlin rsours Auto-mrk hptr pr-tst Vio monstrtions o ll work xmpls Intrtiv wigts Intrtiv wlkthroughs Downlol HOTshts Ass to ll HOTmths Austrlin Curriulum ourss Ass to th HOTmths gms lirry Lngth, r, sur
More informationGarnir Polynomial and their Properties
Univrsity of Cliforni, Dvis Dprtmnt of Mthmtis Grnir Polynomil n thir Proprtis Author: Yu Wng Suprvisor: Prof. Gorsky Eugny My 8, 07 Grnir Polynomil n thir Proprtis Yu Wng mil: uywng@uvis.u. In this ppr,
More informationPresent state Next state Q + M N
Qustion 1. An M-N lip-lop works s ollows: I MN=00, th nxt stt o th lip lop is 0. I MN=01, th nxt stt o th lip-lop is th sm s th prsnt stt I MN=10, th nxt stt o th lip-lop is th omplmnt o th prsnt stt I
More informationCSE 373: AVL trees. Warmup: Warmup. Interlude: Exploring the balance invariant. AVL Trees: Invariants. AVL tree invariants review
rmup CSE 7: AVL trs rmup: ht is n invrint? Mihl L Friy, Jn 9, 0 ht r th AVL tr invrints, xtly? Disuss with your nighor. AVL Trs: Invrints Intrlu: Exploring th ln invrint Cor i: xtr invrint to BSTs tht
More informationIntegration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals
Intgrtion Continud Intgrtion y Prts Solving Dinit Intgrls: Ar Undr Curv Impropr Intgrls Intgrtion y Prts Prticulrly usul whn you r trying to tk th intgrl o som unction tht is th product o n lgric prssion
More informationUNCORRECTED SAMPLE PAGES 4-1. Naming fractions KEY IDEAS. 1 Each shape represents ONE whole. a i ii. b i ii
- Nming frtions Chptr Frtions Eh shp rprsnts ONE whol. i ii Wht frtion is shdd? Writ s frtion nd in words. Wht frtion is not shdd? Writ s frtion nd in words. i ii i ii Writ s mny diffrnt frtions s you
More informationUsing the Printable Sticker Function. Using the Edit Screen. Computer. Tablet. ScanNCutCanvas
SnNCutCnvs Using th Printl Stikr Funtion On-o--kin stikrs n sily rt y using your inkjt printr n th Dirt Cut untion o th SnNCut mhin. For inormtion on si oprtions o th SnNCutCnvs, rr to th Hlp. To viw th
More informationThe University of Sydney MATH2969/2069. Graph Theory Tutorial 5 (Week 12) Solutions 2008
Th Univrsity o Syny MATH2969/2069 Grph Thory Tutoril 5 (Wk 12) Solutions 2008 1. (i) Lt G th isonnt plnr grph shown. Drw its ul G, n th ul o th ul (G ). (ii) Show tht i G is isonnt plnr grph, thn G is
More informationEE1000 Project 4 Digital Volt Meter
Ovrviw EE1000 Projt 4 Diitl Volt Mtr In this projt, w mk vi tht n msur volts in th rn o 0 to 4 Volts with on iit o ury. Th input is n nlo volt n th output is sinl 7-smnt iit tht tlls us wht tht input s
More information, each of which is a tree, and whose roots r 1. , respectively, are children of r. Data Structures & File Management
nrl tr T is init st o on or mor nos suh tht thr is on sint no r, ll th root o T, n th rminin nos r prtition into n isjoint susts T, T,, T n, h o whih is tr, n whos roots r, r,, r n, rsptivly, r hilrn o
More informationExam 1 Solution. CS 542 Advanced Data Structures and Algorithms 2/14/2013
CS Avn Dt Struturs n Algorithms Exm Solution Jon Turnr //. ( points) Suppos you r givn grph G=(V,E) with g wights w() n minimum spnning tr T o G. Now, suppos nw g {u,v} is to G. Dsri (in wors) mtho or
More informationOn each of them are the numbers +6, 5, +4, 3, +2, 1. The two dice are rolled. The score is obtained by adding the numbers on the upper faces.
Cmrig Essntils Mthmtis Cor 8 N1.1 Homwork N1.1 Homwork 1 A thr shows hr lss 2 six-si i. On h of thm r th numrs +6, 5, +4, 3, +2, 1. Th two i r roll. Th sor is otin y ing th numrs on th uppr fs. Clult th
More informationAn undirected graph G = (V, E) V a set of vertices E a set of unordered edges (v,w) where v, w in V
Unirt Grphs An unirt grph G = (V, E) V st o vrtis E st o unorr gs (v,w) whr v, w in V USE: to mol symmtri rltionships twn ntitis vrtis v n w r jnt i thr is n g (v,w) [or (w,v)] th g (v,w) is inint upon
More informationMath 61 : Discrete Structures Final Exam Instructor: Ciprian Manolescu. You have 180 minutes.
Nm: UCA ID Numr: Stion lttr: th 61 : Disrt Struturs Finl Exm Instrutor: Ciprin nolsu You hv 180 minuts. No ooks, nots or lultors r llow. Do not us your own srth ppr. 1. (2 points h) Tru/Fls: Cirl th right
More informationModule 2 Motion Instructions
Moul 2 Motion Instrutions CAUTION: Bor you strt this xprimnt, unrstn tht you r xpt to ollow irtions EXPLICITLY! Tk your tim n r th irtions or h stp n or h prt o th xprimnt. You will rquir to ntr t in prtiulr
More informationConstructive Geometric Constraint Solving
Construtiv Gomtri Constrint Solving Antoni Soto i Rir Dprtmnt Llngutgs i Sistms Inormàtis Univrsitt Politèni Ctluny Brlon, Sptmr 2002 CGCS p.1/37 Prliminris CGCS p.2/37 Gomtri onstrint prolm C 2 D L BC
More informationIndices. Indices. Curriculum Ready ACMNA: 209, 210, 212,
Inis Inis Curriulum Ry ACMNA: 09, 0,, 6 www.mtltis.om Inis INDICES Inis is t plurl or inx. An inx is us to writ prouts o numrs or pronumrls sily. For xmpl is tully sortr wy o writin #. T is t inx. Anotr
More informationDUET WITH DIAMONDS COLOR SHIFTING BRACELET By Leslie Rogalski
Dut with Dimons Brlt DUET WITH DIAMONDS COLOR SHIFTING BRACELET By Lsli Roglski Photo y Anrw Wirth Supruo DUETS TM from BSmith rt olor shifting fft tht mks your work tk on lif of its own s you mov! This
More informationQUESTIONS BEGIN HERE!
Points miss: Stunt's Nm: Totl sor: /100 points Est Tnnss Stt Univrsity Dprtmnt o Computr n Inormtion Sins CSCI 2710 (Trno) Disrt Struturs TEST or Sprin Smstr, 2005 R this or strtin! This tst is los ook
More informationCOMPLEXITY OF COUNTING PLANAR TILINGS BY TWO BARS
OMPLXITY O OUNTING PLNR TILINGS Y TWO RS KYL MYR strt. W show tht th prolm o trmining th numr o wys o tiling plnr igur with horizontl n vrtil r is #P-omplt. W uil o o th rsults o uquir, Nivt, Rmil, n Roson
More informationDecimals DECIMALS.
Dimls DECIMALS www.mthltis.o.uk ow os it work? Solutions Dimls P qustions Pl vlu o imls 0 000 00 000 0 000 00 0 000 00 0 000 00 0 000 tnths or 0 thousnths or 000 hunrths or 00 hunrths or 00 0 tn thousnths
More informationSOLVED EXAMPLES. be the foci of an ellipse with eccentricity e. For any point P on the ellipse, prove that. tan
LOCUS 58 SOLVED EXAMPLES Empl Lt F n F th foci of n llips with ccntricit. For n point P on th llips, prov tht tn PF F tn PF F Assum th llips to, n lt P th point (, sin ). P(, sin ) F F F = (-, 0) F = (,
More informationSAMPLE PAGES. Primary. Primary Maths Basics Series THE SUBTRACTION BOOK. A progression of subtraction skills. written by Jillian Cockings
PAGES Primry Primry Mths Bsis Sris THE SUBTRACTION BOOK A prorssion o sutrtion skills writtn y Jillin Cokins INTRODUCTION This ook is intn to hlp sur th mthmtil onpt o sutrtion in hilrn o ll s. Th mstry
More informationPage 1. Question 19.1b Electric Charge II Question 19.2a Conductors I. ConcepTest Clicker Questions Chapter 19. Physics, 4 th Edition James S.
ConTst Clikr ustions Chtr 19 Physis, 4 th Eition Jms S. Wlkr ustion 19.1 Two hrg blls r rlling h othr s thy hng from th iling. Wht n you sy bout thir hrgs? Eltri Chrg I on is ositiv, th othr is ngtiv both
More informationOpenMx Matrices and Operators
OpnMx Mtris n Oprtors Sr Mln Mtris: t uilin loks Mny typs? Dnots r lmnt mxmtrix( typ= Zro", nrow=, nol=, nm="" ) mxmtrix( typ= Unit", nrow=, nol=, nm="" ) mxmtrix( typ= Int", nrow=, nol=, nm="" ) mxmtrix(
More informationCSC Design and Analysis of Algorithms. Example: Change-Making Problem
CSC 801- Dsign n Anlysis of Algorithms Ltur 11 Gry Thniqu Exmpl: Chng-Mking Prolm Givn unlimit mounts of oins of nomintions 1 > > m, giv hng for mount n with th lst numr of oins Exmpl: 1 = 25, 2 =10, =
More informationNumbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point
GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply
More informationSolutions for HW11. Exercise 34. (a) Use the recurrence relation t(g) = t(g e) + t(g/e) to count the number of spanning trees of v 1
Solutions for HW Exris. () Us th rurrn rltion t(g) = t(g ) + t(g/) to ount th numr of spnning trs of v v v u u u Rmmr to kp multipl gs!! First rrw G so tht non of th gs ross: v u v Rursing on = (v, u ):
More informationCSE 373. Graphs 1: Concepts, Depth/Breadth-First Search reading: Weiss Ch. 9. slides created by Marty Stepp
CSE 373 Grphs 1: Conpts, Dpth/Brth-First Srh ring: Wiss Ch. 9 slis rt y Mrty Stpp http://www.s.wshington.u/373/ Univrsity o Wshington, ll rights rsrv. 1 Wht is grph? 56 Tokyo Sttl Soul 128 16 30 181 140
More information0.1. Exercise 1: the distances between four points in a graph
Mth 707 Spring 2017 (Drij Grinrg): mitrm 3 pg 1 Mth 707 Spring 2017 (Drij Grinrg): mitrm 3 u: W, 3 My 2017, in lss or y mil (grinr@umn.u) or lss S th wsit or rlvnt mtril. Rsults provn in th nots, or in
More informationSeven-Segment Display Driver
7-Smnt Disply Drivr, Ron s in 7-Smnt Disply Drivr, Ron s in Prolm 62. 00 0 0 00 0000 000 00 000 0 000 00 0 00 00 0 0 0 000 00 0 00 BCD Diits in inry Dsin Drivr Loi 4 inputs, 7 outputs 7 mps, h with 6 on
More informationCSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018
CSE 373: Mor on grphs; DFS n BFS Mihl L Wnsy, F 14, 2018 1 Wrmup Wrmup: Disuss with your nighor: Rmin your nighor: wht is simpl grph? Suppos w hv simpl, irt grph with x nos. Wht is th mximum numr of gs
More informationQUESTIONS BEGIN HERE!
Points miss: Stunt's Nm: Totl sor: /100 points Est Tnnss Stt Univrsity Dprtmnt of Computr n Informtion Sins CSCI 710 (Trnoff) Disrt Struturs TEST for Fll Smstr, 00 R this for strtin! This tst is los ook
More informationV={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)}
Introution Computr Sin & Enginring 423/823 Dsign n Anlysis of Algorithms Ltur 03 Elmntry Grph Algorithms (Chptr 22) Stphn Sott (Apt from Vinohnrn N. Vriym) I Grphs r strt t typs tht r pplil to numrous
More informationCOMP108 Algorithmic Foundations
Grdy mthods Prudn Wong http://www.s.liv..uk/~pwong/thing/omp108/01617 Coin Chng Prolm Suppos w hv 3 typs of oins 10p 0p 50p Minimum numr of oins to mk 0.8, 1.0, 1.? Grdy mthod Lrning outoms Undrstnd wht
More informationSection 3: Antiderivatives of Formulas
Chptr Th Intgrl Appli Clculus 96 Sction : Antirivtivs of Formuls Now w cn put th is of rs n ntirivtivs togthr to gt wy of vluting finit intgrls tht is ct n oftn sy. To vlut finit intgrl f(t) t, w cn fin
More informationOutline. 1 Introduction. 2 Min-Cost Spanning Trees. 4 Example
Outlin Computr Sin 33 Computtion o Minimum-Cost Spnnin Trs Prim's Alorithm Introution Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #33 3 Alorithm Gnrl Constrution Mik Joson (Univrsity o Clry)
More informationCycles and Simple Cycles. Paths and Simple Paths. Trees. Problem: There is No Completely Standard Terminology!
Outlin Computr Sin 331, Spnnin, n Surphs Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #30 1 Introution 2 3 Dinition 4 Spnnin 5 6 Mik Joson (Univrsity o Clry) Computr Sin 331 Ltur #30 1 / 20 Mik
More informationTOPIC 5: INTEGRATION
TOPIC 5: INTEGRATION. Th indfinit intgrl In mny rspcts, th oprtion of intgrtion tht w r studying hr is th invrs oprtion of drivtion. Dfinition.. Th function F is n ntidrivtiv (or primitiv) of th function
More information12/3/12. Outline. Part 10. Graphs. Circuits. Euler paths/circuits. Euler s bridge problem (Bridges of Konigsberg Problem)
12/3/12 Outlin Prt 10. Grphs CS 200 Algorithms n Dt Struturs Introution Trminology Implmnting Grphs Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits 1 Ciruits Cyl 2 Eulr
More information5/9/13. Part 10. Graphs. Outline. Circuits. Introduction Terminology Implementing Graphs
Prt 10. Grphs CS 200 Algorithms n Dt Struturs 1 Introution Trminology Implmnting Grphs Outlin Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits 2 Ciruits Cyl A spil yl
More informationNefertiti. Echoes of. Regal components evoke visions of the past MULTIPLE STITCHES. designed by Helena Tang-Lim
MULTIPLE STITCHES Nrtiti Ehos o Rgl omponnts vok visions o th pst sign y Hln Tng-Lim Us vrity o stiths to rt this rgl yt wrl sign. Prt sping llows squr s to mk roun omponnts tht rp utiully. FCT-SC-030617-07
More informationPolygons POLYGONS.
Polgons PLYGNS www.mthltis.o.uk ow os it work? Solutions Polgons Pg qustions Polgons Polgon Not polgon Polgon Not polgon Polgon Not polgon Polgon Not polgon f g h Polgon Not polgon Polgon Not polgon Polgon
More informationV={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)}
s s of s Computr Sin & Enginring 423/823 Dsign n Anlysis of Ltur 03 (Chptr 22) Stphn Sott (Apt from Vinohnrn N. Vriym) s of s s r strt t typs tht r pplil to numrous prolms Cn ptur ntitis, rltionships twn
More informationWhy the Junction Tree Algorithm? The Junction Tree Algorithm. Clique Potential Representation. Overview. Chris Williams 1.
Why th Juntion Tr lgorithm? Th Juntion Tr lgorithm hris Willims 1 Shool of Informtis, Univrsity of Einurgh Otor 2009 Th JT is gnrl-purpos lgorithm for omputing (onitionl) mrginls on grphs. It os this y
More informationUNCORRECTED SAMPLE PAGES
Numrs n surs Aritmti is t stuy o numrs n oprtions on tm. Tis sort ptr rviws wol numrs, intrs, rtionl numrs n rl numrs, wit prtiulr ttntion to t ritmti o surs n tir pproximtions. Most o tis mtril will milir
More informationNumbering Boundary Nodes
Numring Bounry Nos Lh MBri Empori Stt Univrsity August 10, 2001 1 Introution Th purpos of this ppr is to xplor how numring ltril rsistor ntworks ffts thir rspons mtrix, Λ. Morovr, wht n lrn from Λ out
More informationInstructions for Section 1
Instructions for Sction 1 Choos th rspons tht is corrct for th qustion. A corrct nswr scors 1, n incorrct nswr scors 0. Mrks will not b dductd for incorrct nswrs. You should ttmpt vry qustion. No mrks
More informationFundamental Algorithms for System Modeling, Analysis, and Optimization
Fundmntl Algorithms for Sstm Modling, Anlsis, nd Optimiztion Edwrd A. L, Jijt Rohowdhur, Snjit A. Sshi UC Brkl EECS 144/244 Fll 2011 Copright 2010-11, E. A. L, J. Rohowdhur, S. A. Sshi, All rights rsrvd
More informationDesigning A Concrete Arch Bridge
This is th mous Shwnh ri in Switzrln, sin y Rort Millrt in 1933. It spns 37.4 mtrs (122 t) n ws sin usin th sm rphil mths tht will monstrt in this lsson. To pro with this lsson, lik on th Nxt utton hr
More informationProblem solving by search
Prolm solving y srh Tomáš voo Dprtmnt o Cyrntis, Vision or Roots n Autonomous ystms Mrh 5, 208 / 3 Outlin rh prolm. tt sp grphs. rh trs. trtgis, whih tr rnhs to hoos? trtgy/algorithm proprtis? Progrmming
More informationGraph Isomorphism. Graphs - II. Cayley s Formula. Planar Graphs. Outline. Is K 5 planar? The number of labeled trees on n nodes is n n-2
Grt Thortil Is In Computr Sin Vitor Amhik CS 15-251 Ltur 9 Grphs - II Crngi Mllon Univrsity Grph Isomorphism finition. Two simpl grphs G n H r isomorphi G H if thr is vrtx ijtion V H ->V G tht prsrvs jny
More informationPlanar Upward Drawings
C.S. 252 Pro. Rorto Tmssi Computtionl Gomtry Sm. II, 1992 1993 Dt: My 3, 1993 Sri: Shmsi Moussvi Plnr Upwr Drwings 1 Thorm: G is yli i n only i it hs upwr rwing. Proo: 1. An upwr rwing is yli. Follow th
More informationOutline. Computer Science 331. Computation of Min-Cost Spanning Trees. Costs of Spanning Trees in Weighted Graphs
Outlin Computr Sin 33 Computtion o Minimum-Cost Spnnin Trs Prim s Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #34 Introution Min-Cost Spnnin Trs 3 Gnrl Constrution 4 5 Trmintion n Eiiny 6 Aitionl
More informationAlgorithmic and NP-Completeness Aspects of a Total Lict Domination Number of a Graph
Intrntionl J.Mth. Comin. Vol.1(2014), 80-86 Algorithmi n NP-Compltnss Aspts of Totl Lit Domintion Numr of Grph Girish.V.R. (PES Institut of Thnology(South Cmpus), Bnglor, Krntk Stt, Ini) P.Ush (Dprtmnt
More informationa b c cat CAT A B C Aa Bb Cc cat cat Lesson 1 (Part 1) Verbal lesson: Capital Letters Make The Same Sound Lesson 1 (Part 1) continued...
Progrssiv Printing T.M. CPITLS g 4½+ Th sy, fun (n FR!) wy to tch cpitl lttrs. ook : C o - For Kinrgrtn or First Gr (not for pr-school). - Tchs tht cpitl lttrs mk th sm souns s th littl lttrs. - Tchs th
More informationCS 461, Lecture 17. Today s Outline. Example Run
Prim s Algorithm CS 461, Ltur 17 Jr Si Univrsity o Nw Mxio In Prim s lgorithm, th st A mintin y th lgorithm orms singl tr. Th tr strts rom n ritrry root vrtx n grows until it spns ll th vrtis in V At h
More information12. Traffic engineering
lt2.ppt S-38. Introution to Tltrffi Thory Spring 200 2 Topology Pths A tlommunition ntwork onsists of nos n links Lt N not th st of nos in with n Lt J not th st of nos in with j N = {,,,,} J = {,2,3,,2}
More informationINTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x)
Chptr 7 INTEGRALS 7. Ovrviw 7.. Lt d d F () f (). Thn, w writ f ( ) d F () + C. Ths intgrls r clld indfinit intgrls or gnrl intgrls, C is clld constnt of intgrtion. All ths intgrls diffr y constnt. 7..
More informationTangram Fractions Overview: Students will analyze standard and nonstandard
ACTIVITY 1 Mtrils: Stunt opis o tnrm mstrs trnsprnis o tnrm mstrs sissors PROCEDURE Skills: Dsriin n nmin polyons Stuyin onrun Comprin rtions Tnrm Frtions Ovrviw: Stunts will nlyz stnr n nonstnr tnrms
More informationIn which direction do compass needles always align? Why?
AQA Trloy Unt 6.7 Mntsm n Eltromntsm - Hr 1 Complt t p ll: Mnt or s typ o or n t s stronst t t o t mnt. Tr r two typs o mnt pol: n. Wrt wt woul ppn twn t pols n o t mnt ntrtons low: Drw t mnt l lns on
More informationGraphs. Graphs. Graphs: Basic Terminology. Directed Graphs. Dr Papalaskari 1
CSC 00 Disrt Struturs : Introuon to Grph Thory Grphs Grphs CSC 00 Disrt Struturs Villnov Univrsity Grphs r isrt struturs onsisng o vrs n gs tht onnt ths vrs. Grphs n us to mol: omputr systms/ntworks mthml
More informationH SERIES. Decimals. Decimals. Curriculum Ready ACMNA: 103, 128, 129, 130, 152, 154,
Dimls H SERIES Dimls Curriulum Ry ACMNA: 0, 8, 9, 0,,, www.mthltis.om Copyriht 009 P Lrnin. All rihts rsrv. First ition print 009 in Austrli. A tlou ror or this ook is vill rom P Lrnin Lt. ISBN 978--98--9
More informationMAT3707. Tutorial letter 201/1/2017 DISCRETE MATHEMATICS: COMBINATORICS. Semester 1. Department of Mathematical Sciences MAT3707/201/1/2017
MAT3707/201/1/2017 Tutoril lttr 201/1/2017 DISCRETE MATHEMATICS: COMBINATORICS MAT3707 Smstr 1 Dprtmnt o Mtmtil Sins SOLUTIONS TO ASSIGNMENT 01 BARCODE Din tomorrow. univrsity o sout ri SOLUTIONS TO ASSIGNMENT
More informationCS September 2018
Loil los Distriut Systms 06. Loil los Assin squn numrs to msss All ooprtin prosss n r on orr o vnts vs. physil los: rport tim o y Assum no ntrl tim sour Eh systm mintins its own lol lo No totl orrin o
More information5/1/2018. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees
/1/018 W usully no strns y ssnn -lnt os to ll rtrs n t lpt (or mpl, 8-t on n ASCII). Howvr, rnt rtrs our wt rnt rquns, w n sv mmory n ru trnsmttl tm y usn vrl-lnt non. T s to ssn sortr os to rtrs tt our
More informationFactorising FACTORISING.
Ftorising FACTORISING www.mthletis.om.u Ftorising FACTORISING Ftorising is the opposite of expning. It is the proess of putting expressions into rkets rther thn expning them out. In this setion you will
More informationEvans, Lipson, Wallace, Greenwood
Camrig Snior Mathmatial Mthos AC/VCE Units 1& Chaptr Quaratis: Skillsht C 1 Solv ah o th ollowing or x: a (x )(x + 1) = 0 x(5x 1) = 0 x(1 x) = 0 x = 9x Solv ah o th ollowing or x: a x + x 10 = 0 x 8x +
More informationChapter 16. 1) is a particular point on the graph of the function. 1. y, where x y 1
Prctic qustions W now tht th prmtr p is dirctl rltd to th mplitud; thrfor, w cn find tht p. cos d [ sin ] sin sin Not: Evn though ou might not now how to find th prmtr in prt, it is lws dvisl to procd
More informationChem 104A, Fall 2016, Midterm 1 Key
hm 104A, ll 2016, Mitrm 1 Ky 1) onstruct microstt tl for p 4 configurtion. Pls numrt th ms n ml for ch lctron in ch microstt in th tl. (Us th formt ml m s. Tht is spin -½ lctron in n s oritl woul writtn
More informationCONVERTING UNITS. Converting Units PASSPORT
CONVERTING UNITS PASSPORT www.mthltis.om.u This ook shows how to writ th sm vlu using smllr or lrgr units o msurmnt. Mny nint ivilistions msur lngths/istns y rlting thm to rtin oy prts. Invstigt ths trms
More informationCS 241 Analysis of Algorithms
CS 241 Anlysis o Algorithms Prossor Eri Aron Ltur T Th 9:00m Ltur Mting Lotion: OLB 205 Businss HW6 u lry HW7 out tr Thnksgiving Ring: Ch. 22.1-22.3 1 Grphs (S S. B.4) Grphs ommonly rprsnt onntions mong
More informationGraphs. CSC 1300 Discrete Structures Villanova University. Villanova CSC Dr Papalaskari
Grphs CSC 1300 Disrt Struturs Villnov Univrsity Grphs Grphs r isrt struturs onsis?ng of vr?s n gs tht onnt ths vr?s. Grphs n us to mol: omputr systms/ntworks mthm?l rl?ons logi iruit lyout jos/prosss f
More informationOutline. Circuits. Euler paths/circuits 4/25/12. Part 10. Graphs. Euler s bridge problem (Bridges of Konigsberg Problem)
4/25/12 Outlin Prt 10. Grphs CS 200 Algorithms n Dt Struturs Introution Trminology Implmnting Grphs Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits 1 2 Eulr s rig prolm
More informationChem 107: Inorganic Chemistry (40720)
Chm 107: Inorgni Chmistry (40720) Prossor Mtt Lw -mil: lwm@ui.u Oi Hours: W 3:00-4:00p n Thurs 11-noon in NS2 2127 TAs Julit Khosrowi -mil: jkhosrow@ui.u Oi Hours: Tus 2:00-3:00p, 3 r loor tls, Rins Hll
More informationUNCORRECTED PAGE PROOFS
7 Shps 9 Unrstning lngth isovr n ojts Msurmnt n gomtry Wht informtion n w gthr from 2 shps n 3 ojts? 7 7 7 7 7E 7F 7G 7H 7I Tringl proprtis Quriltrl proprtis 2 shps n 3 ojts Isomtri rwings n plns Nts n
More informationA prefix word in each of these sentences is incorrect. Rewrite the prefix words correctly.
Spring Trm 2 Cirl th possssiv pronoun: Th hilrn wr thrill tht th i tht h n hosn or th inl prout ws thirs. Rwrit this sntn with th vril phrs t th ginning. Don t orgt omm! Thr wsn t on pi o pizz lt tr th
More informationWalk Like a Mathematician Learning Task:
Gori Dprtmnt of Euction Wlk Lik Mthmticin Lrnin Tsk: Mtrics llow us to prform mny usful mthmticl tsks which orinrily rquir lr numbr of computtions. Som typs of problms which cn b on fficintly with mtrics
More informationGrade 7/8 Math Circles March 4/5, Graph Theory I- Solutions
ulty o Mtmtis Wtrloo, Ontrio N ntr or ution in Mtmtis n omputin r / Mt irls Mr /, 0 rp Tory - Solutions * inits lln qustion. Tr t ollowin wlks on t rp low. or on, stt wtr it is pt? ow o you know? () n
More informationJonathan Turner Exam 2-10/28/03
CS Algorihm n Progrm Prolm Exm Soluion S Soluion Jonhn Turnr Exm //. ( poin) In h Fioni hp ruur, u wn vrx u n i prn v u ing u v i v h lry lo hil in i l m hil o om ohr vrx. Suppo w hng hi, o h ing u i prorm
More information16.unified Introduction to Computers and Programming. SOLUTIONS to Examination 4/30/04 9:05am - 10:00am
16.unii Introution to Computrs n Prormmin SOLUTIONS to Exmintion /30/0 9:05m - 10:00m Pro. I. Kristin Lunqvist Sprin 00 Grin Stion: Qustion 1 (5) Qustion (15) Qustion 3 (10) Qustion (35) Qustion 5 (10)
More informationN1.1 Homework Answers
Camrig Essntials Mathmatis Cor 8 N. Homwork Answrs N. Homwork Answrs a i 6 ii i 0 ii 3 2 Any pairs of numrs whih satisfy th alulation. For xampl a 4 = 3 3 6 3 = 3 4 6 2 2 8 2 3 3 2 8 5 5 20 30 4 a 5 a
More informationThe University of Sydney MATH 2009
T Unvrsty o Syny MATH 2009 APH THEOY Tutorl 7 Solutons 2004 1. Lt t sonnt plnr rp sown. Drw ts ul, n t ul o t ul ( ). Sow tt s sonnt plnr rp, tn s onnt. Du tt ( ) s not somorp to. ( ) A onnt rp s on n
More informationCSE303 - Introduction to the Theory of Computing Sample Solutions for Exercises on Finite Automata
CSE303 - Introduction to th Thory of Computing Smpl Solutions for Exrciss on Finit Automt Exrcis 2.1.1 A dtrministic finit utomton M ccpts th mpty string (i.., L(M)) if nd only if its initil stt is finl
More informationPractice Test I Bonding and Geometry Name Per
Prti Tst Boning n Gomtry Nm Pr This is prti - Do NOT ht yourslf of fining out wht you r pl of oing. B sur you follow th tsting onitions outlin low. DO NOT USE A CALCULATOR. You my us ONLY th lu prioi tl.
More informationCS200: Graphs. Graphs. Directed Graphs. Graphs/Networks Around Us. What can this represent? Sometimes we want to represent directionality:
CS2: Grphs Prihr Ch. 4 Rosn Ch. Grphs A olltion of nos n gs Wht n this rprsnt? n A omputr ntwork n Astrtion of mp n Soil ntwork CS2 - Hsh Tls 2 Dirt Grphs Grphs/Ntworks Aroun Us A olltion of nos n irt
More informationCS61B Lecture #33. Administrivia: Autograder will run this evening. Today s Readings: Graph Structures: DSIJ, Chapter 12
Aministrivi: CS61B Ltur #33 Autogrr will run this vning. Toy s Rings: Grph Struturs: DSIJ, Chptr 12 Lst moifi: W Nov 8 00:39:28 2017 CS61B: Ltur #33 1 Why Grphs? For xprssing non-hirrhilly rlt itms Exmpls:
More informationBASIC CAGE DETAILS SHOWN 3D MODEL: PSM ASY INNER WALL TABS ARE COINED OVER BASE AND COVER FOR RIGIDITY SPRING FINGERS CLOSED TOP
MO: PSM SY SI TIS SOWN SPRIN INRS OS TOP INNR W TS R OIN OVR S N OVR OR RIIITY. R TURS US WIT OPTION T SINS. R (UNOMPRSS) RR S OPTION (S T ON ST ) IMNSIONS O INNR SIN TO UNTION WIT QU SM ORM-TOR (zqsp+)
More information3 a 15a 6 b 21a 5 c 30a 6 d 12a 9. e 125a 8 f 36a 12 g 90a 13 h 56a a 6a b 5 c 3a 4 d 6a 4. e 10a 4 f 8a 2 g 5a 4 h 12a 2
Answrs Cptr Workin wit surs Eris A Surs 8 6 6 8 6 6 6 + 6 8 6 6 6 6 + 8 + + 8 6 9 6 m ( + m Cptr Simpliin prssions usin t lws o inis Eris A Inis 9 i j 6 k 8 l 6 6 i j k l 6 6 9 8 6 9 6 6 6 8 8 6 9 8 8
More informationStructure and calculation
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),
More informationSurds and Indices. Surds and Indices. Curriculum Ready ACMNA: 233,
Surs n Inies Surs n Inies Curriulum Rey ACMNA:, 6 www.mthletis.om Surs SURDS & & Inies INDICES Inies n surs re very losely relte. A numer uner (squre root sign) is lle sur if the squre root n t e simplifie.
More information