Southwestern Advantage Sample

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Make better grades, save time, and prep for your future!

Math 1&2

Science

Social Studies & Language

Honors & DVD

Topic Source & DVD

MATH Math’s New Methods

Saxon, Chicago/Everyday, Singapore

Foundations of Math Numbers and Operations Arithmetic, Fractions, Decimals, Ratio & Percent

Measurement Geometry Trigonometry Formulas & Equations Statistics and Probability Algebra Advanced Algebra Calculus SOCIAL STUDIES & LANGUAGE US History Canadian History Government Economics World History Geography Grammar Writing and Research Vocabulary Reading Speaking Literature Foreign Languages

SCIENCE Scientific Methods Biology Chemistry Earth Science Ecology Physics Space Science TOPIC SOURCE (including DVD) People Places Wars Animals History Science Writing Research Papers HONORS (including AP Tests DVD) English Language US History English Literature Calculus AB US Gov’t & Politics Biology, Chemistry, Calculus AB, Calculus BC, English Language, Psychology, English Literature, Statistics, US Government & Politics, US History, Writing the AP Essay

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4PVUIXFTUFSO "EWBOUBHF Southwestern Advantage, Math © 2011 Southwestern Nashville, Tennessee ISBN 978-0-87197-558-4 3FWJTFE © Page 1198 constitutes an extension of this copyright page. Dustin Hillis $IJFG &YFDVUJWF 0ffiDFS 4PVUIXFTUFSO 'BNJMZ PG $PNQBOJFT Dan Moore 1SFTJEFOU 4PVUIXFTUFSO "EWBOUBHF 4BMFT Iskander Ahmet Timo Aleste Dave Causer Vasil Chakardakov Ryan Groom Edgar Ibarra-Uzeta Ranol Kasevali Karel Koiv Erki Kukk Andres Martin Priit Martin Mark Metsla Madis Pajo

Maret Pajo Amy Pigozzi Groom Mark Rau, .BOBHJOH %JSFDUPS 6 4 4BMFT

Erin Reynolds Veiko Roos Matt Ross Virgie Sandford AJ Skalsky Annika Tell Veiko Tell Emil Tsvetkov

All rights reserved. Printed in the Republic of Korea (ROK) 'SPOU DPWFS iNPOPHSBNw BSU BOE i4PVUIXFTUFSO "EWBOUBHF MPHPw BSF CPUI QSPQSJFUBSZ USBEFNBSLT PG 4PVUIXFTUFSO (SFBU "NFSJDBO *OD

&YFDVUJWF &EJUPS BOE 1SFTJEFOU Dan Moore $VSSJDVMVN %JSFDUPS Janet D. Sweet 4FDUJPO &EJUPST Genevieve Brand Carrie Davisson Deborah Ward Ketner Margaret Kriegbaum Susan Longley George Pasles "SU %JSFDUPS Travis Rader

1SFGBDF Welcome to 4PVUIXFTUFSO "EWBOUBHF .BUI. We are pleased to bring you these unique, user-friendly reference books. Designed in such a way that students can spend “more time learning, less time looking,� the pages are open and inviting, and critical information is summarized in boxes, lists, and other easily usable and understandable pieces. Problems are shown worked out step-by-step. Where a problem can be worked by more than one method, those methods are also shown. Additionally, simply by keying in the page number, you can access step-by-step videos of each problem from that page at 48BEWBOUBHF DPN. (PU UP ,OPX boxes summarize the most essential information; cross-references in the /FFE .PSF )FMQ boxes direct you to pages where you can find additional information or review material. 5SZ *U 5IJT 8BZ suggests alternative ways people with various learning styles can use to more effectively approach, work, or visualize problems and concepts. 8BUDI 0VU boxes alert students to things that might be easily confused or that might give students difficulty. We hope you will find this book both useful and enjoyable. Every effort has been made to ensure that the information in this book is as accurate as possible. If errors should be found, however, we would appreciate hearing from you. Please share your comments or suggestions with us at 1-888-551-5901 or send them to Editor c/o Southwestern Advantage, P.O. Box 305142, Nashville, TN 37230.

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$POUSJCVUPST BOE "EWJTPST We gratefully acknowledge the invaluable contributions of these educators and writers to the development and production of this book. Their academic and professional awards are testament to their breadth of knowledge and excellence in the classroom, and the accomplishments listed here are merely highlights from their careers.

+FBO "SNTUSPOH . 4 $VSSJDVMVN BOE 5FBDIJOH 5FBDIFST $PMMFHF $PMVNCJB 6OJWFSTJUZ # 4 ;PPMPHZ .FNQIJT 4UBUF 6OJWFSTJUZ Excellence in Publishing Award; Holt, Rinehart & Winston

5FBDIJOH &YQFSJFODF Montessori pre-K and transitional K to 1 Grade 6, traditional classroom Grades 2–4, elementary science specialist Middle school science and math tutor +BOF #PPLT + % $PMVNCJB 4DIPPM PG -BX # " .BUIFNBUJDT .BOIBUUBOWJMMF $PMMFHF ,BUIZ $BSUFS # 4 .BUIFNBUJDT 5SJOJUZ $PMMFHF President’s Club Award at Pearson Prentice Hall (once for innovation, once for best product)

5FBDIJOH FYQFSJFODF High school math, grades 9–12 Alternative education for both adults and youth High school math tutor

/PSBMJF $PY . " 5 .BUIFNBUJDT #SJEHFXBUFS 4UBUF $PMMFHF " # .BUIFNBUJDBM #JPMPHZ #SPXO 6OJWFSTJUZ

5FBDIJOH FYQFSJFODF High school math, grades 9–12 %BOJFMMF 1PUWJO $VSSBO . " "QQMJFE .BUIFNBUJDT BOE $VSSJDVMVN %FWFMPQNFOU #PTUPO $PMMFHF Marketing Manager of the Year (Houghton Mifflin) Strategic Achievement Reward (STAR) (McDougal Littell) University of Michigan Student Recognition Award John F. Kennedy Award for Service and Leadership

5FBDIJOH FYQFSJFODF Curriculum specialist, 6–12+ Teacher Department chair $PODFUUB . %VWBM &E % .BUIFNBUJDT $VSSJDVMVN BOE "ENJOJTUSBUJPO 6OJWFSTJUZ PG 3PDIFTUFS Director of Mathematics and Science, Brighton School District, Rochester, New York Other titles after leaving public education include Director of Mathematics, Wasatch Education Systems, Jostens Learning Corp. Senior Software Designer, Riverdeep (now HMH) Educational Consultant, Duval & Associates .JDIBFM (SFFO 1I % .BUIFNBUJDT 6OJWFSTJUZ PG 8JTDPOTJO . " 5 :BMF 6OJWFSTJUZ # " .BUIFNBUJDT "NIFSTU $PMMFHF

5FBDIJOH FYQFSJFODF High school math College instructor

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5FBDIJOH FYQFSJFODF High school math, grades 9–12 Math department chair University professor, Algebra and Statistics Curriculum writer for secondary and post-secondary courses Author of university courses, on-ground and online Senior editor: Grade level/content/lead editor, across multiple grades, multiple companies, multiple projects Extra Mile Award presented by Oral Roberts University to secondary teachers -BVSB + 0TUFSCSPDL . " .BUIFNBUJDT 6OJWFSTJUZ PG .JOOFTPUB # " .BUIFNBUJDT $BSMFUPO $PMMFHF

5FBDIJOH FYQFSJFODF Math, grades 8–12 Math coach and tutor, grades K, 3, and 6

Project SMART; high school math Coauthor of, among other books, .BUIFNBUJDT GPS &MFNFOUBSZ 5FBDIFST " $POUFNQPSBSZ "QQSPBDI UI &EJUJPO author of encyclopedia articles; frequent contributor to such scholarly publications as +PVSOBM GPS 3FTFBSDI JO .BUIFNBUJDT &EVDBUJPO and +PVSOBM PG .BUIFNBUJDT 5FBDIFS &EVDBUJPO +PTFQI $ 1PXFS . " .BUIFNBUJDT 6OJWFSTJUZ PG .BSZMBOE

5FBDIJOH FYQFSJFODF Junior high school math High school math Junior college instructor University instructor Author of "MHFCSB &TTFOUJBMT BOE "QQMJDBUJPOT, Holt, Rinehart and Winston 5IBEEFVT 5 8FSU . &E 7BOEFSCJMU 6OJWFSTJUZ # 4 &OHJOFFSJOH 7BOEFSCJMU 6OJWFSTJUZ Awarded the Ellen Bowers Hofstead Chair in Mathematics and Science

5FBDIJOH FYQFSJFODF Math grades 9–12 Department chair Curriculum developer (MFOO 8PSUINBO . 4 .BOBHFNFOU BOE 1PMJDZ 4UPOZ #SPPL 6OJWFSTJUZ # 4 "QQMJFE .BUIFNBUJDT BOE 4UBUJTUJDT 4UPOZ #SPPL 6OJWFSTJUZ

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$POUSJCVUPST BOE "EWJTPST We gratefully acknowledge the invaluable contributions of these educators and writers to the development and production of this book. Their academic awards are testament to their breadth of knowledge and excellence in the classroom, and the accomplishments listed here are merely highlights from their careers.

+PBO #SVNNPOE . " -FTMFZ $PMMFHF # " 6OJWFSTJUZ PG 8ZPNJOH 26 years as a classroom teacher, K–Grade 3; Wyoming Teacher of the Year; developed new programs in guided reading; coached and consulted for a migrant workers education program; developed a beforeschool early-bird library program; tutors middle school students; coaches early literacy teachers ,FOU $SJQQFO 1I % . &E BOE # 4 6OJWFSTJUZ PG /FCSBTLB Associate professor, Curriculum and Instruction, UNLV; coauthor of “Computer Uses in Chemical Education� in 5IF /FX $IFN4PVSDF; associate editor of the +PVSOBM PG 4DJFODF &EVDBUJPO BOE 5FDIOPMPHZ; associate director, Center for Mathematics and Science Education, University of Nevada, Las Vegas; his research involves the design and implementation of Web-based learning systems to support selfregulated learning %FOJTF $SPLFS . &E 1FBCPEZ $PMMFHF PG 7BOEFSCJMU 6OJWFSTJUZ # " 6OJWFSTJUZ PG ,BOTBT more than 20 years as a classroom teacher, in English and Journalism; advisor to award-winning student newspaper and news site; frequent contributor to such scholarly publications as the &OHMJTI +PVSOBM, which gave her their Paul and Kate Farmer writing award; State Media Adviser of the Year; named a Dow Jones Newspaper Fund Special Recognition Adviser; frequent speaker at regional and national conferences

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"SUIVS 3 &DIFSE +S 1I % . " BOE # " 6OJWFSTJUZ PG /PSUI $BSPMJOB 25 years as a classroom teacher, in European History, AP Comparative Government, AP U.S. Government, World Religions; Tennessee Humanities Council Outstanding Teacher Award; Presidential Scholar’s Inspirational Teacher Award; past holder of the Ellen Bowers Hofstead Chair in the Humanities +FTVT (BSDJB &E % BOE . " 6OJWFSTJUZ PG $BMJGPSOJB #FSLFMFZ # " 4BO 'SBODJTDP 4UBUF 6OJWFSTJUZ Professor, Curriculum and Instruction, UNLV; coauthor of 'JFME &YQFSJFODF 4USBUFHJFT GPS &YQMPSJOH %JWFSTJUZ JO 4DIPPM 4PDJBM 4UVEJFT GPS $IJMESFO " (VJEF UP #BTJD *OTUSVDUJPO and $POUFYUT PG 5FBDIJOH .FUIPET GPS .JEEMF BOE )JHI 4DIPPM *OTUSVDUJPO; past president, National Council for the Social Studies 4IFSSJ (PVME . &E BOE # 4 6OJWFSTJUZ PG .BJOF 28 years as a classroom teacher, in English and Literacy; department chair; Maine Teacher of the Year; UMPI Alumni Educator of the Year; frequent speaker at conferences and workshops; former secretary and former vice president of National State Teachers of the Year Association; supervisor and trainer of preservice teachers 1BU (SBff # 4 0LMBIPNB 4UBUF 6OJWFSTJUZ 29 years as a classroom teacher, in Journalism, Humanities, and Social Studies; New Mexico Teacher of the Year; Governor’s Award for Outstanding New Mexico Woman; Distinguished Service Award, National Council of Teachers of English; Medal of Merit, Journalism Education Association %BMF " (SPUF 1I % 6OJWFSTJUZ PG 8JTDPOTJO . " 6OJWFSTJUZ PG *PXB # 4 $PSOFMM 6OJWFSTJUZ Associate professor, Classics, University of North Carolina; author of " $PNQSFIFOTJWF (VJEF UP 8IFFMPDL T -BUJO; president of the North Carolina Classical

Association; frequent speaker at education conferences; conducts study tours to Greece and Rome #BSSZ )FSU[ . &E 6OJWFSTJUZ PG "MCFSUB . 4 BOE # 4 4PVUI %BLPUB 4UBUF 6OJWFSTJUZ 21 years as a classroom teacher, in Biology, Chemistry, Physics, and IB (International Baccalaureate) Biology; Prime Minister’s Award for Teaching Excellence ,FJM )JMFNBO . 4 BOE # 4 6OJWFSTJUZ PG ,BOTBT more than 20 years as a classroom teacher, Social Studies/Museum Studies; Kansas Teacher of the Year; 2004 National Teacher of the Year finalist; creator of the Museum Connections class, which has amassed more than 20,000 teaching artifacts; teaches museum courses for other social studies teachers 3PMMJF + .ZFST 1I % 6OJWFSTJUZ PG $BMJGPSOJB . 4 BOE # 4 $BMJGPSOJB *OTUJUVUF PG 5FDIOPMPHZ Emeritus professor, Chemistry, University of California, Berkeley; Guggenheim Fellow; ACS International Award Fellow; former visiting professor, Harvard University; former faculty senior scientist, Lawrence Berkeley National Laboratory; author of, among other books, 6OJWFSTJUZ $IFNJTUSZ and .PMFDVMBS .BHOFUJTN BOE .BHOFUJD 3FTPOBODF 4QFDUSPTDPQZ +BNFT " 3PF 1I % 6OJWFSTJUZ PG $BMJGPSOJB #FSLFMFZ # 4 8JMMJBNT $PMMFHF Associate professor, Chemistry and Biochemistry, Loyola Marymount University; contributor to such scholarly publications as +PVSOBM PG #JPMPHJDBM $IFNJTUSZ and 'SFF 3BEJDBM #JPMPHZ BOE .FEJDJOF &EOB 3PHFST . 4 BOE # 4 6OJWFSTJUZ PG 5FOOFTTFF 33 years as a classroom teacher, in Pre-K, Grade 2, and Grade 5; Tennessee Teacher of the Year; National Teachers Hall of Fame; Tennessee Educators Association Friend of Education Award; Presidential

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Award for Excellence in Science and Mathematics; Governor’s Outstanding Tennessean Award; on the boards of, among others, the Dollywood Foundation and Berkshire Education Scholarship Foundation; director of Dolly Parton’s Chasing Rainbows Award &SOFTU 4DIJMMFS 1I % BOE . 4 6OJWFSTJUZ PG *PXB # 4 *PXB 4UBUF 6OJWFSTJUZ 33 years as a classroom teacher, in Biology and Advanced Biology; Iowa Teacher of the Year; Outstanding Young Iowan Educator Award; Excellence in Teaching Science Award; Presidential Award in Secondary Science; Christa McAuliffe Award .JDIBFM 4FJEFM 1I % . " BOE # " 6OJWFSTJUZ PG $BMJGPSOJB -PT "OHFMFT Emeritus Jesse and George Siegel Professor in the Humanities, Columbia University; Department Chair; associate editor of $PMVNCJB )JTUPSZ PG #SJUJTI 'JDUJPO $PMVNCJB 8PSME PG 2VPUBUJPOT, and 5IF 8PSLT PG %BOJFM %FGPF; author of, among other books, &QJD (FPHSBQIZ +BNFT +PZDF T 6MZTTFT and 4USFBL +PF %J.BHHJP BOE UIF 4VNNFS PG ; frequent contributor to such scholarly publications as &JHIUFFOUI $FOUVSZ 'JDUJPO and +BNFT +PZDF 2VBSUFSMZ .BSZ &MJ[BCFUI 4QBMEJOH 1I % *OEJBOB 6OJWFSTJUZ . " BOE # " 8FTU 7JSHJOJB 6OJWFSTJUZ Editor, +PVSOBM PG 5FBDIFS &EVDBUJPO; frequent contributor to such scholarly publications as &EVDBUJPOBM 'PSVN &OHMJTI &EVDBUJPO &OHMJTI +PVSOBM and 5FBDIJOH BOE 5FBDIFS &EVDBUJPO member, executive committee, Conference on English Education of the National Council of Teachers of English; her research interests include performance and portfolio assessment, learning communities in teacher education, and secondary English teaching and teachers

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&TUJNBUJOH XJUI 8IPMF /VNCFST &TUJNBUJOH 4VNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &TUJNBUJOH %JGGFSFODFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &TUJNBUJOH 1SPEVDUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &TUJNBUJOH 2VPUJFOUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

%BUB BOE (SBQIT 5BMMZ $IBSUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1JDUPHSBQIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . #BS (SBQIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -JOF (SBQIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $JSDMF (SBQIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .FBO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .FEJBO .PEF BOE 3BOHF . . . . . . . . . . . . . . . . . . . . . . . . . .

"OBMZ[JOH 4PMJE 'JHVSFT 1SJTNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ZMJOEFST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1ZSBNJET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $POFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4QIFSFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1MBUPOJD 4PMJET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5BCMFT GPS 'PVOEBUJPOT PG .BUIFNBUJDT. . . . . . .

www.swadvantage.com

7

$POUFOUT /VNCFST BOE 0QFSBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decimals 1MBDF 7BMVF UP 5IPVTBOEUIT . . . . . . . . . . . . . . . . . . . . . . . . . 3FBEJOH BOE 8SJUJOH %FDJNBMT. . . . . . . . . . . . . . . . . . . . . . . $PNQBSJOH BOE 0SEFSJOH %FDJNBMT . . . . . . . . . . . . . . . . . . 3PVOEJOH %FDJNBMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "EEJOH %FDJNBMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4VCUSBDUJOH %FDJNBMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .VMUJQMZJOH %FDJNBMT CZ 8IPMF /VNCFST . . . . . . . . . . . . .VMUJQMZJOH %FDJNBMT CZ %FDJNBMT . . . . . . . . . . . . . . . . . . . %JWJEJOH %FDJNBMT CZ 8IPMF /VNCFST . . . . . . . . . . . . . . . %JWJEJOH 8IPMF /VNCFST CZ %FDJNBMT . . . . . . . . . . . . . . . %JWJEJOH %FDJNBMT CZ %FDJNBMT . . . . . . . . . . . . . . . . . . . . . . 8PSE 1SPCMFNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . /VNCFS 5IFPSy 3BUJPOBM /VNCFST BOE /VNCFS 4FUT . . . . . . . . . . . . . . . . . %FDJNBMT 'SBDUJPOT BOE 1FSDFOUT . . . . . . . . . . . . . . . . . . . . 3BUJPOBM /VNCFST PO B /VNCFS -JOF . . . . . . . . . . . . . . . . . $PNQBSJOH BOE 0SEFSJOH 3BUJPOBM /VNCFST . . . . . . . . . . 1SJNF BOE $PNQPTJUF /VNCFST . . . . . . . . . . . . . . . . . . . . . 'BDUPST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .VMUJQMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . %JWJTJCJMJUZ 3VMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &YQPOFOUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1SJNF 'BDUPSJ[BUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

'SBDUJPOT BOE .JYFE Numbers 'SBDUJPOT 1BSU PG B 8IPMF . . . . . . . . . . . . . . . . . . . . . . . . . . 'SBDUJPOT 1BSU PG B 4FU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *NQSPQFS 'SBDUJPOT BOE .JYFE /VNCFST . . . . . . . . . . . . . &RVJWBMFOU 'SBDUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $PNQBSJOH BOE 0SEFSJOH 'SBDUJPOT BOE .JYFE /VNCFST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3PVOEJOH 'SBDUJPOT BOE .JYFE /VNCFST . . . . . . . . . . . . . "EEJOH 'SBDUJPOT XJUI -JLF %FOPNJOBUPST . . . . . . . . . . . . "EEJOH 'SBDUJPOT XJUI 6OMJLF %FOPNJOBUPST . . . . . . . . . 4VCUSBDUJOH 'SBDUJPOT XJUI -JLF %FOPNJOBUPST . . . . . . . 4VCUSBDUJOH 'SBDUJPOT XJUI 6OMJLF %FOPNJOBUPST . . . . . .VMUJQMZJOH 'SBDUJPOT CZ 'SBDUJPOT . . . . . . . . . . . . . . . . . . . .VMUJQMZJOH 8IPMF /VNCFST BOE 'SBDUJPOT . . . . . . . . . . . .VMUJQMZJOH .JYFE /VNCFST . . . . . . . . . . . . . . . . . . . . . . . . . %JWJEJOH 'SBDUJPOT CZ 'SBDUJPOT . . . . . . . . . . . . . . . . . . . . . . %JWJEJOH 'SBDUJPOT BOE 8IPMF /VNCFST . . . . . . . . . . . . . . %JWJEJOH .JYFE /VNCFST . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8PSE 1SPCMFNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3BUJP BOE 1SPQPSUJPO 8SJUJOH BOE 4JNQMJGZJOH 3BUJPT . . . . . . . . . . . . . . . . . . . . . . 8SJUJOH 3BUFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &RVJWBMFOU 3BUJPT BOE 1SPQPSUJPOT . . . . . . . . . . . . . . . . . . . 4PMWJOH 1SPQPSUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4DBMF %SBXJOHT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8PSE 1SPCMFNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1BUUFSOT 3FQFBUJOH 1BUUFSOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5SJBOHVMBS BOE 4RVBSF /VNCFS 1BUUFSOT . . . . . . . . . . . . . (SPXJOH 1BUUFSOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4ISJOLJOH 1BUUFSOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1BUUFSOT XJUI 5XP 0QFSBUJPOT . . . . . . . . . . . . . . . . . . . . . . . 8PSE 1SPCMFNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1FSDFOU .FBOJOH PG 1FSDFOUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 'JOEJOH UIF 1FSDFOU PG B 8IPMF /VNCFS . . . . . . . . . . . . . . 'JOEJOH UIF 1FSDFOU 0OF /VNCFS *T PG "OPUIFS /VNCFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 'JOEJOH B /VNCFS 8IFO (JWFO B 1FSDFOU PG UIF /VNCFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1FSDFOUT < BOE > . . . . . . . . . . . . . . . . . . . . . . . . . 1FSDFOU *ODSFBTF BOE %FDSFBTF . . . . . . . . . . . . . . . . . . . . . . 4JNQMF *OUFSFTU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $PNQPVOE *OUFSFTU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8PSE 1SPCMFNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1SPCMFN 4PMWJOH 'PVS 4UFQ 1SPDFTT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4USBUFHZ 8SJUF B /VNCFS 4FOUFODF . . . . . . . . . . . . . . . . . . 4USBUFHZ %SBX B %JBHSBN . . . . . . . . . . . . . . . . . . . . . . . . . . . 4USBUFHZ .BLF B 5BCMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4USBUFHZ -PPL GPS B 1BUUFSO . . . . . . . . . . . . . . . . . . . . . . . . . 4USBUFHZ (VFTT $IFDL BOE 3FWJTF . . . . . . . . . . . . . . . . . . 4USBUFHZ 6TF -PHJDBM 3FBTPOJOH . . . . . . . . . . . . . . . . . . . . . 4USBUFHZ 8PSL #BDLXBSET . . . . . . . . . . . . . . . . . . . . . . . . . . . 4USBUFHZ 4PMWF B 4JNQMFS 1SPCMFN . . . . . . . . . . . . . . . . . . . 4USBUFHZ .BLF BO 0SHBOJ[FE -JTU . . . . . . . . . . . . . . . . . . . . 4USBUFHZ .BLF PS 6TF B (SBQI . . . . . . . . . . . . . . . . . . . . . . .

&TUJNBUF XJUI %FDJNBMT BOE 'SBDUJPOT &TUJNBUJOH UP 1SFEJDU PS $IFDL . . . . . . . . . . . . . . . . . . . . . . . &TUJNBUJOH %FDJNBM 4VNT BOE %JGGFSFODFT . . . . . . . . . . . . &TUJNBUJOH %FDJNBM 1SPEVDUT BOE 2VPUJFOUT . . . . . . . . . . &TUJNBUJOH 'SBDUJPO 4VNT BOE %JGGFSFODFT . . . . . . . . . . . . &TUJNBUJOH 'SBDUJPO 1SPEVDUT BOE 2VPUJFOUT . . . . . . . . . . 8PSE 1SPCMFNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5BCMFT GPS /VNCFST BOE 0QFSBUJPOT . . . . . . . . . .

9

$POUFOUT .FBTVSFNFOU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 $VTUPNBSZ 4ZTUFN .FBTVSJOH 6 4 $VTUPNBSZ -FOHUI .FBTVSJOH "SFB JO UIF 6 4 $VTUPNBSZ 4ZTUFN .FBTVSJOH $BQBDJUZ JO UIF 6 4 $VTUPNBSZ 4ZTUFN .FBTVSJOH 8FJHIU JO UIF 6 4 $VTUPNBSZ 4ZTUFN %JNFOTJPOBM "OBMZTJT 5FNQFSBUVSF .FUSJD 4ZTUFN .FBTVSJOH -FOHUI JO UIF .FUSJD 4ZTUFN .FBTVSJOH "SFB JO UIF .FUSJD 4ZTUFN .FBTVSJOH $BQBDJUZ JO UIF .FUSJD 4ZTUFN .FBTVSJOH .BTT JO UIF .FUSJD 4ZTUFN $POWFSUJOH 6 4 $VTUPNBSZ BOE .FUSJD 5FNQFSBUVSF 5FNQFSBUVSF $POWFSTJPO 'PSNVMBT 1FSJNFUFS BOE "SFB 1FSJNFUFS &TUJNBUJOH "SFB "SFBT PG 3FDUBOHMFT BOE 1BSBMMFMPHSBNT "SFBT PG 5SJBOHMFT "SFBT PG 5SBQF[PJET $JSDVNGFSFODF "SFBT PG $JSDMFT $PNQPTJUF 'JHVSFT

10

4VSGBDF "SFB BOE 7PMVNF 4VSGBDF "SFB PG 1SJTNT 4VSGBDF "SFB PG $ZMJOEFST 4VSGBDF "SFB PG 1ZSBNJET BOE $POFT 7PMVNFT PG 1SJTNT BOE $ZMJOEFST 7PMVNFT PG 1ZSBNJET BOE $POFT 4VSGBDF "SFB BOE 7PMVNFT PG 4QIFSFT 5ISFF %JNFOTJPOBM $PNQPTJUF 'JHVSFT 4DJFOUJGJD /PUBUJPO 8SJUJOH /VNCFST JO 4DJFOUJGJD /PUBUJPO $POWFSUJOH UP 4UBOEBSE /PUBUJPO .VMUJQMZJOH XJUI 4DJFOUJGJD /PUBUJPO %JWJEJOH XJUI 4DJFOUJGJD /PUBUJPO 1SFDJTJPO BOE &SSPS

5BCMFT GPS .FBTVSFNFOU

(FPNFUSZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3FBTPOJOH BOE 1SPPG $POEJUJPOBMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . #JDPOEJUJPOBMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . %FEVDUJWF 3FBTPOJOH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *OEVDUJWF 3FBTPOJOH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1SPQFSUJFT PG $POHSVFODF . . . . . . . . . . . . . . . . . . . . . . . . . . . 1SPWJOH "OHMFT $POHSVFOU . . . . . . . . . . . . . . . . . . . . . . . . . . 5IFPSFNT "CPVU 1BSBMMFM -JOFT . . . . . . . . . . . . . . . . . . . . . .

$POHSVFODF $POHSVFOU 'JHVSFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6TJOH 444 BOE 4"4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6TJOH "4" BOE ""4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6TJOH $1$5$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $POHSVFODF JO 3JHIU 5SJBOHMFT . . . . . . . . . . . . . . . . . . . . . . . *TPTDFMFT BOE &RVJMBUFSBM 5SJBOHMFT . . . . . . . . . . . . . . . . . . .

2VBESJMBUFSBMT 1SPQFSUJFT PG 1BSBMMFMPHSBNT . . . . . . . . . . . . . . . . . . . . . . . . 1SPWJOH B 2VBESJMBUFSBM *T B 1BSBMMFMPHSBN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1SPQFSUJFT PG 4QFDJBM 1BSBMMFMPHSBNT . . . . . . . . . . . . . . . . . 1SPQFSUJFT PG 5SBQF[PJET BOE ,JUFT . . . . . . . . . . . . . . . . . . .

4JNJMBSJUZ 4JNJMBS 1PMZHPOT 1SPWJOH 5SJBOHMFT 4JNJMBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4JNJMBSJUZ JO 3JHIU 5SJBOHMFT . . . . . . . . . . . . . . . . . . . . . . . . . *OEJSFDU .FBTVSFNFOU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1SPQPSUJPOT JO 5SJBOHMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 'SBDUBMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5SJBOHMFT BOE 1PMZHPOT 5SJBOHMF *OFRVBMJUJFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5SJBOHMF "OHMF 4VN 5IFPSFN . . . . . . . . . . . . . . . . . . . . . . . "OHMFT PG B 1PMZHPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1ZUIBHPSFBO 5IFPSFN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4QFDJBM 3JHIU 5SJBOHMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "SFB PG B 3FHVMBS 1PMZHPO . . . . . . . . . . . . . . . . . . . . . . . . . . .

$PPSEJOBUF (FPNFUSZ 4MPQFT PG 1BSBMMFM BOE 1FSQFOEJDVMBS -JOFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . %JTUBODF 'PSNVMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .JEQPJOU 'PSNVMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &RVBUJPOT PG $JSDMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1SPPGT 6TJOH $PPSEJOBUF (FPNFUSZ . . . . . . . . . . . . . . . . . .

4QBUJBM 3FBTPOJOH /FUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *TPNFUSJD %SBXJOHT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0SUIPHSBQIJD %SBXJOHT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . #BTF %FTJHOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -BUFSBM "SFBT PG 1ZSBNJET BOE $POFT . . . . . . . . . . . . . . . . . 4PMJET PG 3FWPMVUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $SPTT 4FDUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -PDVT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &VMFS T 'PSNVMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5SBOTGPSNBUJPOT 5SBOTMBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3FGMFDUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3PUBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4ZNNFUSZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . %JMBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $PNQPTJUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5FTTFMMBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

$JSDMFT "SDT BOE 4FDUPST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $IPSET BOE "SDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1SPQFSUJFT PG 5BOHFOUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *OTDSJCFE "OHMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5BOHFOUT $IPSET BOE "OHMFT . . . . . . . . . . . . . . . . . . . . . . . 'JOEJOH 0UIFS "OHMF .FBTVSFT . . . . . . . . . . . . . . . . . . . . . . 'JOEJOH 4FHNFOU -FOHUIT . . . . . . . . . . . . . . . . . . . . . . . . . . .

$POTUSVDUJPOT $POHSVFOU 4FHNFOUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $POHSVFOU "OHMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1FSQFOEJDVMBS #JTFDUPS PG B 4FHNFOU . . . . . . . . . . . . . . . . . "OHMF #JTFDUPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1FSQFOEJDVMBS -JOFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1BSBMMFM -JOFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $JSDVNTDSJCF B $JSDMF "CPVU B 5SJBOHMF . . . . . . . . . . . . . . . *OTDSJCF B $JSDMF JO B 5SJBOHMF . . . . . . . . . . . . . . . . . . . . . . . .

5BCMFT GPS (FPNFUSZ 11

$POUFOUT 5SJHPOPNFUSZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3JHIU 5SJBOHMF 5SJHPOPNFUSZ "OHMFT BOE %FHSFF .FBTVSF . . . . . . . . . . . . . . . . . . . . . . . . $PNQVUJOH XJUI "OHMF .FBTVSFT . . . . . . . . . . . . . . . . . . . . 4QFDJBM 5SJBOHMFT BOE "OHMFT . . . . . . . . . . . . . . . . . . . . . . . . 5SJHPOPNFUSJD 3BUJPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5SJHPOPNFUSJD 3BUJPT PG "DVUF "OHMFT . . . . . . . . . . . . . . . . 5SJHPOPNFUSJD 3BUJPT PG 4QFDJBM "OHMFT . . . . . . . . . . . . . . . 'JOEJOH "OHMF .FBTVSFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4PMWJOH 3JHIU 5SJBOHMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "QQMJDBUJPOT PG 3JHIU 5SJBOHMFT . . . . . . . . . . . . . . . . . . . . . . 3BEJBO .FBTVSF BOE UIF 5SJHPOPNFUSJD 'VODUJPOT

.FBTVSFT PG 3PUBUJPOT PG "OHMFT . . . . . . . . . . . . . . . . . . . . . 'VODUJPOT PG /PO "DVUF "OHMFT . . . . . . . . . . . . . . . . . . . . . . 3BEJBO .FBTVSF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "QQMJDBUJPOT PG 3BEJBO .FBTVSF . . . . . . . . . . . . . . . . . . . . . -JOFBS BOE "OHVMBS 7FMPDJUZ . . . . . . . . . . . . . . . . . . . . . . . . . 5IF 6OJU $JSDMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(SBQIT PG UIF 5SJHPOPNFUSJD 'VODUJPOT 1FSJPEJD 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (SBQIT PG UIF 4JOF BOE $PTJOF 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6TJOH B (SBQIJOH $BMDVMBUPS . . . . . . . . . . . . . . . . . . . . . . . . . 5SBOTGPSNBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (SBQIT PG UIF 5BOHFOU BOE $PUBOHFOU 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (SBQIT PG UIF 4FDBOU BOE $PTFDBOU 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0UIFS 5SBOTGPSNBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $PNCJOJOH 5SJHPOPNFUSJD 'VODUJPOT . . . . . . . . . . . . . . . . . )BSNPOJD .PUJPO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .PEFMJOH XJUI 5SJHPOPNFUSJD 'VODUJPOT. . . . . . . . . . . . . . 5SJHPOPNFUSJD *EFOUJUJFT 'VOEBNFOUBM *EFOUJUJFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7FSJGZJOH 5SJHPOPNFUSJD *EFOUJUJFT . . . . . . . . . . . . . . . . . . 4VN BOE %JGGFSFODF *EFOUJUJFT * . . . . . . . . . . . . . . . . . . . . . 4VN BOE %JGGFSFODF *EFOUJUJFT ** . . . . . . . . . . . . . . . . . . . . %PVCMF "OHMF *EFOUJUJFT . . . . . . . . . . . . . . . . . . . . . . . . . . . )BMG "OHMF *EFOUJUJFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1SPEVDU UP 4VN BOE 4VN UP 1SPEVDU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

*OWFSTF 5SJHPOPNFUSJD 'VODUJPOT BOE 5SJHPOPNFUSJD &RVBUJPOT

*OWFSTF 4JOF 'VODUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *OWFSTF $PTJOF 'VODUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . *OWFSTF 5BOHFOU 'VODUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . 4PMWJOH 5SJHPOPNFUSJD &RVBUJPOT * . . . . . . . . . . . . . . . . . . 4PMWJOH 5SJHPOPNFUSJD &RVBUJPOT ** . . . . . . . . . . . . . . . . . . 4PMWJOH 5SJHPOPNFUSJD *OFRVBMJUJFT . . . . . . . . . . . . . . . . . . *OWFSTF 5SJHPOPNFUSJD 'VODUJPOT . . . . . . . . . . . . . . . . . . . .

0CMJRVF 5SJBOHMFT BOE 7FDUPST 5IF -BX PG 4JOFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5IF "NCJHVPVT $BTF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5IF -BX PG $PTJOFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "SFB PG B 5SJBOHMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7FDUPST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $PNQPOFOUT PG B 7FDUPS . . . . . . . . . . . . . . . . . . . . . . . . . . . "EEJUJPO BOE 4VCUSBDUJPO PG 7FDUPST . . . . . . . . . . . . . . . . .VMUJQMJDBUJPO CZ B 4DBMBS . . . . . . . . . . . . . . . . . . . . . . . . . . 5IF %PU 1SPEVDU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "QQMJDBUJPOT PG 7FDUPST . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1PMBS $PPSEJOBUFT BOE UIF $PNQMFY 1MBOF 1PMBS $PPSEJOBUFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1PMBS BOE 3FDUBOHVMBS $PPSEJOBUFT . . . . . . . . . . . . . . . . . (SBQIT PG 1PMBS &RVBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . 1BSBNFUSJD &RVBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $PNQMFY 3PPUT PG &RVBUJPOT . . . . . . . . . . . . . . . . . . . . . . . 1PMBS 'PSN PG $PNQMFY /VNCFST . . . . . . . . . . . . . . . . . . . 1SPEVDU BOE 2VPUJFOU 5IFPSFNT . . . . . . . . . . . . . . . . . . . . %F.PJWSF T 5IFPSFN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3PPUT PG $PNQMFY /VNCFST . . . . . . . . . . . . . . . . . . . . . . . .

5BCMFT GPS 5SJHPOPNFUSZ . . . . . . . . . . . . . . . . . . .

4QFDJBM 'FBUVSFT .BUI T /FX .FUIPET 8IBU 1BSFOUT /FFE UP ,OPX . . . . . . . . . . . . .

*O UIF 0UIFS #PPL 4UBUJTUJDT BOE 1SPCBCJMJUZ. . .

5BCMFT GPS 'PVOEBUJPOT PG .BUIFNBUJDT . . . . . . . . . . . . . . . . . . . . . . .

"MHFCSB. . . . . . . . . . . . . . . . . .

5BCMFT GPS /VNCFST BOE 0QFSBUJPOT . . . . .

$BMDVMVT . . . . . . . . . . . . . . . . .

5BCMFT GPS .FBTVSFNFOU . . . . . . . . . . . . . . .

*OEFY. . . . . . . . . . . . . . . . . . . .

"EWBODFE "MHFCSB. . . . . . . . .

5BCMFT GPS (FPNFUSZ . . . . . . . . . . . . . . . . . . . 5BCMFT GPS 5SJHPOPNFUSZ . . . . . . . . . . . . . . 5BCMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (MPTTBSZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

$POUFOUT *O UIF 0UIFS #PPL .BUI T /FX .FUIPET 8IBU 1BSFOUT /FFE UP ,OPX

/VNCFST BOE 0QFSBUJPOT 'PVOEBUJPOT PG .BUIFNBUJDT

/VNCFST BOE 0QFSBUJPOT

.FBTVSFNFOU

(FPNFUSZ

5SJHPOPNFUSZ

*O 5IJT #PPL 4UBUJTUJDT BOE 1SPCBCJMJUZ

"MHFCSB

"EWBODFE "MHFCSB

$BMDVMVT

U X X X 48BEWB 7JTJU VT POMJOF B

1206

OUBHF DPN

4UBUJTUJDT BOE 1SPCBCJMJUZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (BUIFSJOH %BUB 4BNQMFT BOE 1PQVMBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . 4VSWFZT BOE #JBT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3BOEPN 4BNQMJOH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0UIFS 4BNQMJOH .FUIPET . . . . . . . . . . . . . . . . . . . . . . . . . . 5IFPSFUJDBM BOE &YQFSJNFOUBM 1SPCBCJMJUZ 5IFPSFUJDBM 1SPCBCJMJUZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &YQFSJNFOUBM 1SPCBCJMJUZ . . . . . . . . . . . . . . . . . . . . . . . . . . . 5IF $PVOUJOH 1SJODJQMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1FSNVUBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $PNCJOBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0EET BOE $PNQMFNFOUT . . . . . . . . . . . . . . . . . . . . . . . . . . . %JTQMBZJOH %BUB 'SFRVFODZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4DBUUFS 1MPUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . )JTUPHSBNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $IPPTJOH BO "QQSPQSJBUF (SBQI . . . . . . . . . . . . . . . . . . . . 4UFN BOE -FBG 1MPUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . #PY BOE 8IJTLFS 1MPUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . %JTQMBZJOH 5XP %BUB 4FUT . . . . . . . . . . . . . . . . . . . . . . . . . . 7FOO %JBHSBNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . %FTDSJCJOH %BUB 5IF i$FOUFSw PG B %BUB 4FU . . . . . . . . . . . . . . . . . . . . . . . . . . 0VUMJFST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &GGFDUT PG $IBOHJOH %BUB 7BMVFT . . . . . . . . . . . . . . . . . . . . .JTMFBEJOH 4UBUJTUJDT .JTMFBEJOH (SBQIT . . . . . . . . . . . .

www.swadvantage.com

"OBMZ[JOH BOE 6TJOH %BUB $PSSFMBUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *OUFSQPMBUJOH XJUIJO B %BUB 4FU . . . . . . . . . . . . . . . . . . . . . &YUSBQPMBUJOH GSPN B %BUB 4FU . . . . . . . . . . . . . . . . . . . . . . -JOF 1MPUT BOE %BUB %JTUSJCVUJPO . . . . . . . . . . . . . . . . . . . . 4UBOEBSE %FWJBUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2VBSUJMFT BOE 1FSDFOUJMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . -JOFBS 3FHSFTTJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2VBESBUJD 3FHSFTTJPO BOE &YQPOFOUJBM 3FHSFTTJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . /PSNBM %JTUSJCVUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1BTDBM T 5SJBOHMF BOE #JOPNJBM %JTUSJCVUJPO . . . . . . . . . . 4BNQMF 4J[F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &YQFSJNFOUBM %FTJHO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $PNQPVOE &WFOUT BOE $POEJUJPOBM 1SPCBCJMJUZ

*OEFQFOEFOU BOE %FQFOEFOU &WFOUT . . . . . . . . . . . . . . . . 1SPCBCJMJUZ BOE (FPNFUSZ . . . . . . . . . . . . . . . . . . . . . . . . . . .VUVBMMZ &YDMVTJWF PS 0WFSMBQQJOH &WFOUT . . . . . . . . . . . 1SPCBCJMJUZ BOE 7FOO %JBHSBNT . . . . . . . . . . . . . . . . . . . . . 4JNVMBUJPOT BOE 5FDIOPMPHZ . . . . . . . . . . . . . . . . . . . . . . . 4USFBLT BOE 0VUDPNFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5BCMFT GPS 4UBUJTUJDT BOE 1SPCBCJMJUZ . . . . . . . . .

1207

$POUFOUT "MHFCSB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . /VNCFST 7BSJBCMFT BOE &YQSFTTJPOT *OUFHFST BOE "CTPMVUF 7BMVF . . . . . . . . . . . . . . . . . . . . . . . . "EEJOH *OUFHFST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4VCUSBDUJOH *OUFHFST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .VMUJQMZJOH BOE %JWJEJOH *OUFHFST . . . . . . . . . . . . . . . . . . &YQPOFOUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4RVBSF 3PPUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5IF 3FBM /VNCFS 4ZTUFN . . . . . . . . . . . . . . . . . . . . . . . . . . $PNNVUBUJWF BOE "TTPDJBUJWF 1SPQFSUJFT . . . . . . . . . . . . %JTUSJCVUJWF 1SPQFSUZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5IF *EFOUJUZ BOE *OWFSTF 1SPQFSUJFT . . . . . . . . . . . . . . . . . 0SEFS PG 0QFSBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7BSJBCMFT BOE "MHFCSBJD &YQSFTTJPOT . . . . . . . . . . . . . . . . &WBMVBUJOH 7BSJBCMF &YQSFTTJPOT . . . . . . . . . . . . . . . . . . . . 5SBOTMBUJOH &OHMJTI JOUP "MHFCSB . . . . . . . . . . . . . . . . . . . . (SBQIJOH 'VODUJPOT 3FMBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5IF $PPSEJOBUF 1MBOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *OUFSQSFUJOH -JOF (SBQIT . . . . . . . . . . . . . . . . . . . . . . . . . . . -JOFBS 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5IF 7FSUJDBM -JOF 5FTU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0OF 7BSJBCMF -JOFBS &RVBUJPOT &RVBUJPO $PODFQUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "EEJUJPO BOE 4VCUSBDUJPO &RVBUJPOT . . . . . . . . . . . . . . . . .VMUJQMJDBUJPO BOE %JWJTJPO &RVBUJPOT . . . . . . . . . . . . . . . 5XP 4UFQ &RVBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .VMUJ 4UFQ &RVBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &RVBUJPOT XJUI 'SBDUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . &RVBUJPOT XJUI %FDJNBMT . . . . . . . . . . . . . . . . . . . . . . . . . . . "CTPMVUF 7BMVF &RVBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . /P 4PMVUJPO PS .BOZ 4PMVUJPOT . . . . . . . . . . . . . . . . . . . . . 4PMWJOH 8PSE 1SPCMFNT XJUI &RVBUJPOT . . . . . . . . . . . . .

1208

0OF 7BSJBCMF -JOFBS *OFRVBMJUJFT (SBQIJOH BO *OFRVBMJUZ BT BO *OUFSWBM . . . . . . . . . . . . . . . 0OF 4UFQ "EEJUJPO BOE 4VCUSBDUJPO *OFRVBMJUJFT . . . . . 0OF 4UFQ .VMUJQMJDBUJPO BOE %JWJTJPO *OFRVBMJUJFT . . . .VMUJ 4UFQ *OFRVBMJUJFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *OFRVBMJUJFT XJUI 7BSJBCMFT PO #PUI 4JEFT . . . . . . . . . . . . $PNQPVOE *OFRVBMJUJFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . "CTPMVUF 7BMVF *OFRVBMJUJFT . . . . . . . . . . . . . . . . . . . . . . . . 5XP 7BSJBCMF -JOFBS &RVBUJPOT BOE *OFRVBMJUJFT

'PVS 2VBESBOUT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (SBQIJOH GSPN B 5BCMF PG 7BMVFT . . . . . . . . . . . . . . . . . . . . (SBQIJOH GSPN BO &RVBUJPO . . . . . . . . . . . . . . . . . . . . . . . . 5IF Z *OUFSDFQU BOE UIF Y *OUFSDFQU . . . . . . . . . . . . . . . . . )PSJ[POUBM BOE 7FSUJDBM -JOFT . . . . . . . . . . . . . . . . . . . . . . . 3BUF PG $IBOHF BOE 4MPQF . . . . . . . . . . . . . . . . . . . . . . . . . . 5IF 4MPQF *OUFSDFQU 'PSN PG B -JOFBS &RVBUJPO . . . . . . 5IF 1PJOU 4MPQF 'PSN PG B -JOFBS &RVBUJPO . . . . . . . . . . 5IF 4UBOEBSE 'PSN PG B -JOFBS &RVBUJPO . . . . . . . . . . . . . $POWFSUJOH 'PSNT PG B -JOFBS &RVBUJPO . . . . . . . . . . . . . . 8SJUJOH BO &RVBUJPO PG B -JOF . . . . . . . . . . . . . . . . . . . . . . . (SBQIJOH B -JOFBS *OFRVBMJUZ . . . . . . . . . . . . . . . . . . . . . . .

4ZTUFNT PG -JOFBS &RVBUJPOT BOE *OFRVBMJUJFT

4PMWJOH B 4ZTUFN CZ (SBQIJOH . . . . . . . . . . . . . . . . . . . . . . 4PMWJOH B 4ZTUFN CZ 4VCTUJUVUJPO . . . . . . . . . . . . . . . . . . . 4PMWJOH B 4ZTUFN CZ "EEJUJPO PS 4VCUSBDUJPO . . . . . . . . 4PMWJOH B 4ZTUFN CZ &MJNJOBUJPO . . . . . . . . . . . . . . . . . . . . 4ZTUFNT XJUI ;FSP 0OF PS .BOZ 4PMVUJPOT . . . . . . . . . . 4ZTUFNT PG *OFRVBMJUJFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . $POTUSBJOUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 'FBTJCMF 3FHJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0CKFDUJWF 'VODUJPOT BOE -JOFBS 1SPHSBNNJOH . . . . . . .

0QFSBUJPOT XJUI 1PMZOPNJBMT .POPNJBMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . #JOPNJBMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .VMUJQMZJOH 1PMZOPNJBMT . . . . . . . . . . . . . . . . . . . . . . . . . . . 4QFDJBM 1SPEVDUT PG #JOPNJBMT . . . . . . . . . . . . . . . . . . . . . . 1PMZOPNJBM -POH %JWJTJPO . . . . . . . . . . . . . . . . . . . . . . . . . . 1PMZOPNJBM 'VODUJPOT BOE &RVBUJPOT (SBQIJOH 1PMZOPNJBM 'VODUJPOT . . . . . . . . . . . . . . . . . . . . ;FSPT BOE Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4PMVUJPOT PG B 1PMZOPNJBM &RVBUJPO . . . . . . . . . . . . . . . . . 3BUJPOBM ;FSPT PG B 1PMZOPNJBM 'VODUJPO . . . . . . . . . . . . 3PPUT BOE $PNQMFY $POKVHBUFT . . . . . . . . . . . . . . . . . . . . 5IF 'VOEBNFOUBM 5IFPSFN PG "MHFCSB . . . . . . . . . . . . . . 5IF 3FNBJOEFS 5IFPSFN BOE UIF 'BDUPS 5IFPSFN . . . . 3FBM BOE *NBHJOBSZ ;FSPT . . . . . . . . . . . . . . . . . . . . . . . . . . "QQSPYJNBUJOH 3FBM ;FSPT . . . . . . . . . . . . . . . . . . . . . . . . . . #FIBWJPS PG B 1PMZOPNJBM 'VODUJPO . . . . . . . . . . . . . . . . . . 'BDUPSJOH 1PMZOPNJBMT 'BDUPSJOH #JOPNJBMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 'BDUPSJOH x + CY + D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 'BDUPSJOH BY + CY + D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 'BDUPSJOH CZ (SPVQJOH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 'BDUPSJOH 4QFDJBM 1PMZOPNJBMT . . . . . . . . . . . . . . . . . . . . . . 4ZOUIFUJD %JWJTJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5IF 3BUJPOBM 3PPUT 5IFPSFN . . . . . . . . . . . . . . . . . . . . . . .

2VBESBUJD 'VODUJPOT BOE &RVBUJPOT 2VBESBUJD 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1SPQFSUJFT PG 2VBESBUJD 'VODUJPOT . . . . . . . . . . . . . . . . . . 5SBOTGPSNJOH 2VBESBUJDT . . . . . . . . . . . . . . . . . . . . . . . . . . . 4PMWJOH ax = C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5IF ;FSP 1SPEVDU 1SPQFSUZ . . . . . . . . . . . . . . . . . . . . . . . . . 'BDUPSJOH UP 4PMWF 2VBESBUJDT . . . . . . . . . . . . . . . . . . . . . . $PNQMFUJOH UIF 4RVBSF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5IF 2VBESBUJD 'PSNVMB 6TJOH UIF %JTDSJNJOBOU 4VN BOE 1SPEVDU PG UIF 3PPUT . . . . . . . . . . . . . . . . . . . . . . 4PMWJOH )JHIFS 0SEFS 1PMZOPNJBM &RVBUJPOT . . . . . . . . . 1SPKFDUJMF .PUJPO 1SPCMFNT . . . . . . . . . . . . . . . . . . . . . . . . 4ZTUFNT PG 2VBESBUJDT -JOFBS 2VBESBUJD 4ZTUFNT . . . . . . . . . . . . . . . . . . . . . . . . . . 2VBESBUJD 2VBESBUJD 4ZTUFNT . . . . . . . . . . . . . . . . . . . . . . 2VBESBUJD *OFRVBMJUJFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . /POMJOFBS 4ZTUFNT PG *OFRVBMJUJFT . . . . . . . . . . . . . . . . . . (SBQIJOH $BMDVMBUPST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5BCMFT GPS "MHFCSB . . . . . . . . . . . . . . . . . . . . . . . . .

1209

$POUFOUT "EWBODFE "MHFCSB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $PPSEJOBUF 4ZTUFN JO 5ISFF %JNFOTJPOT 1MPUUJOH 1PJOUT JO 5ISFF %JNFOTJPOT . . . . . . . . . . . . . . . . (SBQIJOH JO 5ISFF %JNFOTJPOT . . . . . . . . . . . . . . . . . . . . . 4PMWJOH 4ZTUFNT CZ 4VCTUJUVUJPO . . . . . . . . . . . . . . . . . . . . 4PMWJOH 4ZTUFNT CZ &MJNJOBUJPO . . . . . . . . . . . . . . . . . . . . . .BUSJDFT 3PXT BOE $PMVNOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "EEJOH BOE 4VCUSBDUJOH .BUSJDFT . . . . . . . . . . . . . . . . . . . 4DBMBS .VMUJQMJDBUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .VMUJQMZJOH CZ .BUSJDFT . . . . . . . . . . . . . . . . . . . . . . . . .BUSJY .VMUJQMJDBUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5IF *EFOUJUZ .BUSJY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . %FUFSNJOBOUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5IF *OWFSTF PG B .BUSJY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4PMWJOH .BUSJY &RVBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . "VHNFOUFE .BUSJDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6TJOH .BUSJDFT GPS *OWFOUPSZ . . . . . . . . . . . . . . . . . . . . . . . 6TJOH .BUSJDFT JO (FPNFUSZ . . . . . . . . . . . . . . . . . . . . . . . . $SBNFS T 3VMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1SPCMFN 4PMWJOH BOE 7BSJBUJPO .PEFMJOH 8PSE 1SPCMFNT . . . . . . . . . . . . . . . . . . . . . . . . . . /VNCFS BOE *OUFHFS 1SPCMFNT . . . . . . . . . . . . . . . . . . . . . "HF 1SPCMFNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $PJO 1SPCMFNT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . %JTUBODF 1SPCMFNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .JYUVSF 1SPCMFNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *OUFSFTU 1SPCMFNT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 'PSNVMBT BOE -JUFSBM &RVBUJPOT . . . . . . . . . . . . . . . . . . . . . %JSFDU 7BSJBUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *OWFSTF 7BSJBUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +PJOU 7BSJBUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .JYFE 7BSJBUJPO 1SPCMFNT . . . . . . . . . . . . . . . . . . . . . . . . . .

1210

4FRVFODFT BOE 4FSJFT "SJUINFUJD 4FRVFODFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "SJUINFUJD 4FSJFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (FPNFUSJD 4FRVFODFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (FPNFUSJD 4FSJFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *OGJOJUF (FPNFUSJD 4FSJFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 5IF 'JCPOBDDJ 4FRVFODF . . . . . . . . . . . . . . . . . . . . . . . . . . . 3FDVSTJWF 3VMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &YQMJDJU BOE 3FDVSTJWF 3VMFT . . . . . . . . . . . . . . . . . . . . . . . . .BUIFNBUJDBM *OEVDUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . $PNQMFY /VNCFST 5IF 4ZNCPM J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "EEJOH BOE 4VCUSBDUJOH $PNQMFY /VNCFST . . . . . . . . . .VMUJQMZJOH $PNQMFY /VNCFST . . . . . . . . . . . . . . . . . . . . . %JWJTJPO BOE $PNQMFY $POKVHBUFT . . . . . . . . . . . . . . . . . . "CTPMVUF 7BMVF PG B $PNQMFY /VNCFS . . . . . . . . . . . . . . . $POJD 4FDUJPOT (SBQIJOH 1BSBCPMBT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &RVBUJPOT GPS 1BSBCPMBT . . . . . . . . . . . . . . . . . . . . . . . . . . . . (SBQIJOH $JSDMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &RVBUJPOT GPS $JSDMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (SBQIJOH &MMJQTFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &RVBUJPOT GPS &MMJQTFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (SBQIJOH )ZQFSCPMBT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &RVBUJPOT GPS )ZQFSCPMBT . . . . . . . . . . . . . . . . . . . . . . . . . . (SBQIJOH $POJD *OFRVBMJUJFT . . . . . . . . . . . . . . . . . . . . . . . . &DDFOUSJDJUZ BOE $POJD 4FDUJPOT . . . . . . . . . . . . . . . . . . . . . 5SBOTMBUFE $POJDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3BEJDBMT &YQPOFOUT -PHBSJUINT BOE 3BUJPOBM &YQSFTTJPOT

4RVBSF 3PPUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0UIFS 3PPUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .VMUJQMZJOH 3BEJDBM &YQSFTTJPOT . . . . . . . . . . . . . . . . . . . . "EEJOH BOE 4VCUSBDUJOH 3BEJDBM &YQSFTTJPOT . . . . . . . . %JWJEJOH 3BEJDBM &YQSFTTJPOT . . . . . . . . . . . . . . . . . . . . . . . *OUFHFS BOE 3BUJPOBM &YQPOFOUT . . . . . . . . . . . . . . . . . . . . 1SPQFSUJFT PG &YQPOFOUT . . . . . . . . . . . . . . . . . . . . . . . . . . . -PHBSJUINT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $PNNPO -PHBSJUINT BOE /BUVSBM -PHBSJUINT . . . . . . . .VMUJQMZJOH BOE %JWJEJOH 3BUJPOBM &YQSFTTJPOT . . . . . . "EEJOH BOE 4VCUSBDUJOH 3BUJPOBM &YQSFTTJPOT . . . . . . . $PNQMFY 'SBDUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

&YQPOFOUJBM BOE -PHBSJUINJD 'VODUJPOT &YQPOFOUJBM (SPXUI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &YQPOFOUJBM %FDBZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . #BTF F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -PHBSJUINJD 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &YQPOFOUJBM BOE -PHBSJUINJD &RVBUJPOT . . . . . . . . . . . . . &YQPOFOUJBM BOE -PHBSJUINJD *OFRVBMJUJFT . . . . . . . . . . .

8SJUJOH .PEFMT GPS 'VODUJPOT $POTUBOU %JGGFSFODFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4FDPOE 0SEFS $POTUBOU %JGGFSFODFT . . . . . . . . . . . . . . . . . 5IJSE 0SEFS $POTUBOU %JGGFSFODFT . . . . . . . . . . . . . . . . . . . .PEFMT GPS &YQPOFOUJBM 'VODUJPOT . . . . . . . . . . . . . . . . . . .PEFMT GPS 1PXFS 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . "OBMZ[JOH 'VODUJPOT $PNQPTJOH 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "EEJOH BOE 4VCUSBDUJOH 'VODUJPOT . . . . . . . . . . . . . . . . . .VMUJQMZJOH 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . %JWJEJOH 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1JFDFXJTF 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5SBOTMBUJOH 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4USFUDIJOH BOE $PNQSFTTJOH 'VODUJPOT . . . . . . . . . . . . . 3FGMFDUJOH 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *OWFSTFT PG -JOFBS 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . *OWFSTFT PG /PO -JOFBS 'VODUJPOT . . . . . . . . . . . . . . . . . . . 1BSFOU 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5BCMFT GPS "EWBODFE "MHFCSB . . . . . . . . . . . . . . .

3BUJPOBM BOE 3BEJDBM 'VODUJPOT 3BUJPOBM 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "TZNQUPUFT PG 3BUJPOBM 'VODUJPOT . . . . . . . . . . . . . . . . . . 3BUJPOBM &RVBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3BUJPOBM *OFRVBMJUJFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3BEJDBM 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3BEJDBM &RVBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3BEJDBM *OFRVBMJUJFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1211

$POUFOUT $BMDVMVT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -JNJUT BOE $POUJOVJUZ 8IBU *T B -JNJU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6TJOH 5BCMFT BOE (SBQIT UP 'JOE -JNJUT . . . . . . . . . . . . . . $BMDVMBUJOH -JNJUT "MHFCSBJDBMMZ . . . . . . . . . . . . . . . . . . . . -JNJUT PG *OEFUFSNJOBUF 'PSNT . . . . . . . . . . . . . . . . . . . . . . 4QFDJBM -JNJUT *OWPMWJOH 5SJHPOPNFUSJD 'VODUJPOT . . . . -FGU BOE 3JHIU )BOE -JNJUT . . . . . . . . . . . . . . . . . . . . . . . . *OGJOJUF -JNJUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -JNJU BOE 7FSUJDBM "TZNQUPUFT . . . . . . . . . . . . . . . . . . . . . . -JNJUT BOE &OE #FIBWJPS )PSJ[POUBM "TZNQUPUFT . . . . $POUJOVJUZ BU B 1PJOU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $POUJOVJUZ PO BO *OUFSWBM . . . . . . . . . . . . . . . . . . . . . . . . . . . *OUFSNFEJBUF 7BMVF 5IFPSFN . . . . . . . . . . . . . . . . . . . . . . . %FSJWBUJWFT %FGJOJUJPO PG B %FSJWBUJWF . . . . . . . . . . . . . . . . . . . . . . . . . . . "QQSPYJNBUF 3BUF PG $IBOHF . . . . . . . . . . . . . . . . . . . . . . . %JGGFSFOUJBCJMJUZ BOE $POUJOVJUZ . . . . . . . . . . . . . . . . . . . . . -PDBM -JOFBSJUZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $POTUBOU 4VN BOE %JGGFSFODF 3VMFT . . . . . . . . . . . . . . . . 1PXFS 1SPEVDU BOE 2VPUJFOU 3VMFT . . . . . . . . . . . . . . . . . $IBJO 3VMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . %FSJWBUJWFT PG 5SJHPOPNFUSJD 'VODUJPOT . . . . . . . . . . . . . )JHIFS 0SEFS %FSJWBUJWFT . . . . . . . . . . . . . . . . . . . . . . . . . . . %FSJWBUJWFT PG *OWFSTF 'VODUJPOT . . . . . . . . . . . . . . . . . . . . "QQMJDBUJPOT PG %FSJWBUJWFT 3FMBUJPOTIJQT #FUXFFO G f ′ f ″ . . . . . . . . . . . . . . . . . . . . . . . 1PTJUJPO 7FMPDJUZ BOE "DDFMFSBUJPO . . . . . . . . . . . . . . . . . $VSWF 4LFUDIJOH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *NQMJDJU %JGGFSFOUJBUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5BOHFOU BOE /PSNBM -JOFT . . . . . . . . . . . . . . . . . . . . . . . . . 3FMBUFE 3BUFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0QUJNJ[BUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .FBO 7BMVF 5IFPSFN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &YUSFNF 7BMVF 5IFPSFN . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1212

*OUFHSBUJPO #BTJD "OUJEJGGFSFOUJBUJPO 5FDIOJRVFT . . . . . . . . . . . . . . . . 3JFNBOO 4VNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 'VOEBNFOUBM 5IFPSFN PG $BMDVMVT . . . . . . . . . . . . . . . . . . 5SJHPOPNFUSJD 3VMFT GPS *OUFHSBUJPO . . . . . . . . . . . . . . . . . V 4VCTUJUVUJPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *OUFHSBUJPO CZ 1BSUT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "QQMJDBUJPOT PG *OUFHSBMT "SFB #FUXFFO 5XP $VSWFT . . . . . . . . . . . . . . . . . . . . . . . . . . 4FDPOE 'VOEBNFOUBM 5IFPSFN PG $BMDVMVT . . . . . . . . . . .FBO 7BMVF 5IFPSFN GPS *OUFHSBMT . . . . . . . . . . . . . . . . . . "WFSBHF 7BMVF PG B 'VODUJPO . . . . . . . . . . . . . . . . . . . . . . . . .PUJPO "MPOH B -JOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7PMVNF PG 4PMJET %JTL .FUIPE . . . . . . . . . . . . . . . . . . . . . 7PMVNF PG 4PMJET 8BTIFS BOE 4IFMM .FUIPE . . . . . . . . . 7PMVNF PG 4PMJET 3FWPMVUJPO "CPVU B -JOF . . . . . . . . . . 4PMJET XJUI ,OPXO $SPTT 4FDUJPOT . . . . . . . . . . . . . . . . . . %JGGFSFOUJBM &RVBUJPOT %JGGFSFOUJBM &RVBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4MPQF 'JFMET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4FQBSBUJPO PG 7BSJBCMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . (SPXUI BOE %FDBZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -PHBSJUINJD &YQPOFOUJBM BOE 5SBOTDFOEFOUBM 'VODUPOT

%FSJWBUJWFT &YQPOFOUJBM 3VMF . . . . . . . . . . . . . . . . . . . . . . . %FSJWBUJWFT -PHBSJUINJD 3VMF . . . . . . . . . . . . . . . . . . . . . . &YQPOFOUJBM *OUFHSBMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -PHBSJUINJD *OUFHSBMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *OWFSTF 5SJHPOPNFUSJD 'VODUJPOT . . . . . . . . . . . . . . . . . . . .

5BCMFT GPS $BMDVMVT. . . . . . . . . . . . . . . . . . . . . . . . .

4QFDJBM 'FBUVSFT 5BCMFT GPS 4UBUJTUJDT BOE 1SPCBCJMJUZ . . . .

*O UIF 0UIFS #PPL .BUI T /FX .FUIPET

5BCMFT GPS "MHFCSB . . . . . . . . . . . . . . . . . . . .

8IBU 1BSFOUT /FFE UP ,OPX . . . . .

5BCMFT GPS "EWBODFE "MHFCSB . . . . . . . . . .

'PVOEBUJPOT PG .BUIFNBUJDT . . . . . . . . . . . . . . . .

5BCMFT GPS $BMDVMVT. . . . . . . . . . . . . . . . . . . . 5BCMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (MPTTBSZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *OEFY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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/VNCFST BOE 0QFSBUJPOT . . . .FBTVSFNFOU . . . . . . . . . . . . . . (FPNFUSZ . . . . . . . . . . . . . . . . 5SJHPOPNFUSZ . . . . . . . . . . . . .

1213

.BUI T /FX .FUIPET 8I BU 1B S FO U 4 O FFE U P ,O P 8

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HELP

"O BMHPSJUIN JT B TUFQ CZ TUFQ QSPDFEVSF GPS QFSGPSNJOH B NBUI DPNQVUBUJPO

14

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&WFSZEBZ .BUIFNBUJDT

4JOHBQPSF .BUI

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XXX 48BEWBOUBHF DPN

15

"EEJUJPO 1BSUJBM 4VNT Need More

HELP

5IF iQBSUJBM TVNTw NFUIPE JT TPNFUJNFT DBMMFE UIF MFGU UP SJHIU BEEJUJPO NFUIPE CFDBVTF ZPV XPSL GSPN UIF MBSHFTU UP UIF TNBMMFTU QMBDF WBMVF

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1

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+

+

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45&1

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This Way

POFT + POFT = POFT PS

5P TFF BOPUIFS NFUIPE GPS BEEJOH + HP UP &YBNQMF PO page 18.

45&1

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5IF TVN PG BOE JT

16

+

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.BU I T O FX .FU I P E T X I BU 1B S FO U T O FFE U P ,O P X

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+ +

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&Y "NQM &

Need More

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5P SFWJFX CBTJD BEEJUJPO GBDUT HP UP "EEJUJPO 'BDUT JO 'PVOEBUJPOT PG .BUIFNBUJDT Q

3

BEE + "EE UIF IVOESFET + "EE UIF UFOT + "EE UIF POFT + "EE UIF QBSUJBM TVNT 5IF TVN PG BOE JT

XXX 48BEWBOUBHF DPN

+ +

5S Z *U This Way 5P TFF UXP PUIFS NFUIPET GPS BEEJOH + HP UP Example 5 on page 18 BOE &YBNQMF PO page 19.

17

"EEJUJPO

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5

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18

"EE UIF EJHJUT JO FBDI QMBDF

)VOESFET +

5SBEF POFT GPS UFO

5SBEF UFOT GPS IVOESFE

5IF TVN PG BOE JT

5FOT

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6

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5S Z *U This Way 5P TFF UXP PUIFS NFUIPET GPS BEEJOH + HP UP Example 3 on page 17 BOE &YBNQMF PO page 18.

.FU I 0E

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XXX 48BEWBOUBHF DPN

19

4VCUSBDUJPO 5SBEF 'JSTU 4VCUSBDUJPO Need More

HELP

" TVCUSBDUJPO QSPCMFN DPOTJTUT PG UIFTF UISFF QBSUT minuend − TVCUSBIFOE EJGGFSFODF

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1

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20

.BU I T O FX .FU I P E T X I BU 1B S FO U T O FFE U P ,O P X

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−

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5P SFWJFX CBTJD TVCUSBDUJPO GBDUT HP UP 4VCUSBDUJPO 'BDUT JO 'PVOEBUJPOT PG .BUIFNBUJDT Q

t 5IPVTBOET DPMVNO :PV DBO TVCUSBDU − 4U FQ

4VCUSBDU #FDBVTF ZPV IBWF BMSFBEZ EPOF UIF USBEJOH ZPV DBO TVCUSBDU GSPN SJHIU UP MFGU or GSPN MFGU UP SJHIU − = − =

5S Z *U

− = − =

−

5IF EJGGFSFODF CFUXFFO BOE JT

&Y "NQM &

4&"3$)

5SBEF GSPN SJHIU UP MFGU

:PV DBOOPU TVCUSBDU − CVU UIFSF BSF OP Uens UP USBEF. 'JSTU USBEF IVOESFE GPS UFOT 5IFO USBEF UFO GPS POFT. /PX ZPV IBWF IVOESFET UFOT BOE POFT 4U FQ

5P TFF PUIFS NFUIPET GPS TVCUSBDUJOH − HP UP &YBNQMF PO QBHF BOE &YBNQMF PO QBHF

3

TVCUSBDU − 4U FQ

This Way

−

5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

4VCUSBDU − =

− =

− =

5IF EJGGFSFODF CFUXFFO BOE JT XXX 48BEWBOUBHF DPN

21

4VCUSBDUJPO

$PVOUJOH 6Q 4VCUSBDUJPO Need More

HELP

5P SFWJFX CBTJD BEEJUJPO GBDUT HP UP "EEJUJPO 'BDUT JO 'PVOEBUJPOT PG .BUIFNBUJDT Q

8IFO VTJOH UIF DPVOUJOH VQ TVCUSBDUJPO NFUIPE GJSTU XSJUF UIF TVCUSBIFOE UIF OVNCFS CFJOH TVCUSBDUFE 5IFO DPVOU VQ UP HFU UIF OFYU QMBDF WBMVF $JSDMF FBDI OVNCFS VTFE UP DPVOU VQ 5IFO BEE UIF DJSDMFE OVNCFST UP GJOE UIF EJGGFSFODF FY B NQM F

4

4VCUSBDU − 8SJUF UIF TVCUSBIFOE

$PVOU VQ UP UIF OFBSFTU UFO

+

$PVOU VQ UP UIF UFOT QMBDF JO UIF NJOVFOE

+ +

$PVOU VQ UP UIF NJOVFOE "EE UIF DJSDMFE OVNCFST

+ + =

5IF EJGGFSFODF CFUXFFO BOE JT

FY B NQM F

5

4VCUSBDU − 8SJUF UIF TVCUSBIFOE

+

$PVOU VQ UP UIF OFBSFTU UFO

+

$PVOU VQ UP POF IVOESFE $PVOU VQ UP UIF IVOESFET QMBDF JO UIF NJOVFOE

+

$PVOU VQ UP UIF NJOVFOE

+

"EE UIF DJSDMFE OVNCFST

+ + + =

5IF EJGGFSFODF CFUXFFO BOE JT

22

.BU I T O FX .FU I P E T X I BU 1B S FO U T O FFE U P ,O P X

-FGU UP 3JHIU 4VCUSBDUJPO 8IFO VTJOH UIF MFGU UP SJHIU TVCUSBDUJPO NFUIPd GJSTU TVCUSBDU UIF MFGUNPTU EJHJUT UIF EJHJUT XJUI UIF HSFBUFTU QMBDF WBMVF $POUJOVF TVCUSBDUJOH XPSLJOH UPXBSE UIF SJHIU VOUJM UIF POFT IBWF CFFO TVCUSBDUFE :PV DBO XPSL IPSJ[POUBMMZ PS WFSUJDBMMZ &Y "NQM &

6

TVCUSBDU −

4&"3$)

5IF TVCUSBIFOE JT UIF TBNF BT + + .FU I P E

4VCUSBDU UIF IVOESFET 4VCUSBDU UIF UFOT 4VCUSBDU UIF POFT

− = − = − =

5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

5IF EJGGFSFODF CFUXFFO BOE JT .FU I P E

4VCUSBDU UIF IVOESFET 4VCUSBDU UIF UFOT 4VCUSBDU UIF POFT

− − −

5IF EJGGFSFODF CFUXFFO BOE JT

&Y "NQM &

7

TVCUSBDU − 5IF TVCUSBIFOE JT UIF TBNF BT + + + 4VCUSBDU UIF UIPVTBOET

− =

4VCUSBDU UIF IVOESFET

− =

4VCUSBDU UIF UFOT

− =

4VCUSBDU UIF POFT

− =

5S Z *U This Way 5P TFF PUIFS NFUIPET GPS TVCUSBDUJOH − HP UP &YBNQMF PO QBHF BOE &YBNQMF PO QBHF

5IF EJGGFSFODF CFUXFFO BOE JT

XXX 48BEWBOUBHF DPN

23

4VCUSBDUJPO

1BSUJBM %JGGFSFODFT 5IF QBSUJBM EJGGFSFODFT TVCUSBDUJPO NFUIPd JT TJNJMBS UP UIF QBSUJBM TVNT NFUIPE GPS BEEJUJPO 4UBSUJOH PO UIF MFGU XJUI UIF HSFBUFTU QMBDF WBMVF TVCUSBDU GSPN MFGU UP SJHIU VOUJM ZPV IBWF TVCUSBDUFE UIF POFT 5IFO DPNCJOF UIF QBSUJBM EJGGFSFODFT UP GJOE UIF UPUBM EJGGFSFODF &Y "NQM & 4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

8

4VCUSBDU − 493 − 4VCUSBDU UIF IVOESFET − 300 4VCUSBDU UIF UFOT −

4VCUSBDU UIF POFT − "EE UIF QBSUJBM EJGGFSFODFT

+ + =

− =

&Y "NQM &

9

4VCUSBDU − 6,378 − 4VCUSBDU UIF UIPVTBOET − 1,000 4VCUSBDU UIF IVOESFET − 200 4VCUSBDU UIF UFOT − 4VCUSBDU UIF POFT − "EE UIF QBSUJBM EJGGFSFODFT − =

24

+ + + =

.BU I T O FX .FU I P E T X I BU 1B S FO U T O FFE U P ,O P X

4BNF $IBOHF 4VCUSBDUJPO 8IFO VTJOH UIF TBNF DIBOHF TVCUSBDUJPO NFUIPd UIF TBNF OVNCFS JT BEEFE UP PS TVCUSBDUFE GSPN CPUI UIF NJOVFOE BOE UIF TVCUSBIFOE 5IFO UIF SFTVMUJOH OVNCFST BSF TVCUSBDUFE 5IF JEFB JT UP DIBOHF UIF OVNCFST TP UIF TVCUSBIFOE FOET JO [FSP PS JO B TFSJFT PG [FSPT *G ZPV DIPPTF UIF OVNCFS BEEFE PS TVCUSBDUFE DBSFGVMMZ ZPV NBZ CF BCMF UP HFU UXP [FSPT JO POF TUFQ BT TIPXO JO &YBNQMF FY B NQM E

10

TVCUSBDU − 5IJOL )PX DBO * HFU UIF TVCUSBIFOE UP FOE JO [FSPT 4U FQ

4VCUSBDU UP DIBOHF UIF SJHIU NPTU JO UP 4VCUSBDU GSPN CPUI OVNCFST

4U FQ

− − −

−

"EE UP DIBOHF UIF JO UP "EE UP CPUI OVNCFST 4VCUSBDU UIF SFTVMUJOH OVNCFST

+ − +

− −

− =

FY B NQM E

11

TVCUSBDU −

5S Z *U

5IJOL JT BMNPTU 4U FQ

"EE UP UP HFU "EE UP CPUI OVNCFST

4U FQ

This Way + − +

−

4VCUSBDU UIF SFTVMUJOH OVNCFST

−

5P TFF PUIFS NFUIPET GPS TVCUSBDUJOH − HP UP Example 2 on page 21 BOE &YBNQMF PO page 23.

− =

XXX 48BEWBOUBHF DPN

25

.VMUJQMJDBUJPO 1BSUJBM 1SPEVDUT Need More

HELP

5P SFWJFX CBTJD NVMUJQMJDBUJPO GBDUT HP UP .VMUJQMJDBUJPO 'BDUT JO 'PVOEBUJPOT PG .BUIFNBUJDT Q

8IFO VTJOH UIF QBSUJBM QSPEVDUT NVMUJQMJDBUJPO NFUIPd GJSTU CSFBL EPXO FBDI GBDUPS UP TIPX UIF WBMVF PG JUT EJHJUT 5IFO NVMUJQMZ FBDI QBSU PG POF GBDUPS CZ FBDI QBSU PG UIF PUIFS GBDUPS UP HFU UIF QBSUJBM QSPEVDU 'JOBMMZ BEE BMM PG UIF QBSUJBM QSPEVDUT FY B NQM E

1

.VMUJQMZ × 5IJOL + × +

.VMUJQMZ UFOT BOE UFO

× =

.VMUJQMZ UFOT BOE POFT

× =

.VMUJQMZ POFT BOE UFO

× =

.VMUJQMZ POFT BOE POFT

× =

"EE UIF QBSUJBM QSPEVDUT

+ + + =

5IF QSPEVDU PG BOE JT

FY B NQM E

2

.VMUJQMZ × 5IJOL + + × +

.VMUJQMZ IVOESFE BOE UFOT

× =

.VMUJQMZ IVOESFE BOE POFT

× =

.VMUJQMZ UFOT BOE UFOT

× =

.VMUJQMZ UFOT BOE POFT

× =

.VMUJQMZ POFT BOE UFOT

× =

.VMUJQMZ POFT BOE POFT

× =

"EE UIF QBSUJBM QSPEVDUT

+ + + + + =

5IF QSPEVDU PG BOE JT

26

.BU I T O FX .FU I P E T X I BU 1B S FO U T O FFE U P ,O P X

:PV DBO BMTP VTF B WFSUJDBM GPSNBU GPS UIF QBSUJBM QSPEVDUT NFUIPE BT TIPXO JO &YBNQMF BOE &YBNQMF CFMPX &Y "NQM &

3

.VMUJQMZ × 4U FQ 4U FQ

5IJOL + × +

8SJUF UIF QSPCMFN JO WFSUJDBM GPSNBU 5IFO NVMUJQMZ FBDI QBSU PG POF GBDUPS CZ FBDI QBSU PG UIF PUIFS GBDUPS

× = × = 420 × = × = 14 4U FQ

4&"3$)

→ → → →

62 ×

5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

"EE UIF QBSUJBM QSPEVDUT + + + =

5IF QSPEVDU PG BOE JT

&Y "NQM &

4

.VMUJQMZ × 4U FQ 4U FQ

5IJOL + + × +

8SJUF UIF QSPCMFN JO WFSUJDBM GPSNBU 5IFO NVMUJQMZ FBDI QBSU PG POF GBDUPS CZ FBDI QBSU PG UIF PUIFS GBDUPS

× = × = 800 × = × = 120 × = 350 × = 20 4U FQ

→ → → → → →

235 ×

5S Z *U This Way 5P TFF BOPUIFS NFUIPE GPS NVMUJQMZJOH × HP UP &YBNQMF PO QBHF

"EE UIF QBSUJBM QSPEVDUT + + + + + =

5IF QSPEVDU PG BOE JT

XXX 48BEWBOUBHF DPN

27

.VMUJQMJDBUJPO

&HZQUJBO .VMUJQMJDBUJPO Need More

HELP

&HZQUJBO .VMUJQMJDBUJPO XBT EFWFMPQFE CZ UIF BODJFOU &HZQUJBOT BMUIPVHI UIFZ VTFE IJFSPHMZQIT SBUIFS UIBO OVNFSBMT *U JT WFSZ TJNJMBS UP BOPUIFS NFUIPE DBMMFE 3VTTJBO 1FBTBOU .VMUJQMJDBUJPO

8IFO VTJOH UIF &HZQUJBO NVMUJQMJDBUJPO NFUIPd XSJUF UIF QSPCMFN IPSJ[POUBMMZ XJUI B DPMVNO PG OVNCFST CFMPX FBDI GBDUPS t 5IF MFGU DPMVNO BMXBZT TIPXT QPXFST PG UIF OVNCFS JT EPVCMFE %PVCMF VOUJM ZPV SFBDI B OVNCFS UIBU JT HSFBUFS UIBO UIF GBDUPS BCPWF UIF DPMVNO t 5IF SJHIU DPMVNO TUBSUT XJUI UIF GBDUPS BCPWF JU %PVCMF UIJT GBDUPS VOUJM UIFSF BSF UIF TBNF OVNCFS PG FOUSJFT BT JO UIF DPMVNO PO UIF MFGU 'JOE UXP PS NPSF OVNCFST JO UIF MFGU DPMVNO UIBU FRVBM UIF GBDUPS PO UIF MFGU 5IFO BEE UIF DPSSFTQPOEJOH OVNCFST JO UIF SJHIU DPMVNO UP GJOE UIF QSPEVDU FY B NQM E

5S Z *U This Way

.VMUJQMZ × 45&1

:PV DBO XSJUF UIF GBDUPST JO BOZ PSEFS *O &YBNQMF ZPV DPVME XSJUF UIF QSPCMFN BT × BOE EPVCMF 'PS BOZ QSPCMFN MPPL BU UIF GBDUPST DBSFGVMMZ BOE EFDJEF XIJDI XJMM CF FBTJFTU UP EPVCMF 1MBDF UIBU GBDUPS PO UIF SJHIU

5

8SJUF UIF NVMUJQMJDBUJPO IPSJ[POUBMMZ #FMPX UIF GBDUPS PO UIF MFGU EPVCMF UIF OVNCFS VOUJM ZPV SFBDI B OVNCFS UIBU JT HSFBUFS UIBO UIF GBDUPS 4JODF > TUPQ XJUI 'PS UIJT NVMUJQMJDBUJPO UIFSF BSF TJY FOUSJFT JO UIF MFGU DPMVNO

×

×

45&1

#FMPX UIF GBDUPS PO UIF SJHIU XSJUF UIF GBDUPS 5IFO EPVCMF UIF GBDUPS VOUJM UIFSF BSF TJY FOUSJFT JO UIF DPMVNO PO UIF SJHIU

1 2 4 8 16 32

45&1

'JOE UXP OVNCFST JO UIF DPMVNO PO UIF MFGU XIPTF TVN JT 36. + = 'JOE UIF DPSSFTQPOEJOH OVNCFST JO UIF DPMVNO PO UIF SJHIU BOE

× 1 2 → 8 16 →

45&1

"EE UIF UXP OVNCFST JO UIF DPMVNO PO UIF SJHIU

+ =

5IF QSPEVDU PG BOE JT

28

36

.BU I T O FX .FU I P E T X I BU 1B S FO U T O FFE U P ,O P X

-BUUJDF .VMUJQMJDBUJPO 5IF MBUUJDF NVMUJQMJDBUJPO NFUIPE VTFT B HSJE PG TRVBSFT XJUI FBDI TRVBSF EJWJEFE CZ B EJBHPOBM 5IF GBDUPST BSF QMBDFE BT GPMMPXT t 0OF GBDUPS JT XSJUUFO BMPOH UIF UPQ MFGU UP SJHIU XJUI B EJHJU BCPWF FBDI TRVBSF t 5IF PUIFS GBDUPS JT XSJUUFO BMPOH UIF SJHIU TJEF PG UIF HSJE GSPN UPQ UP CPUUPN XJUI B EJHJU UP UIF SJHIU PG FBDI TRVBSF .VMUJQMZ FBDI EJHJU JO POF GBDUPS CZ FBDI EJHJU JO UIF PUIFS GBDUPS 'PS FBDI QSPEVDU QMBDF UIF UFOT EJHJU BCPWF UIF EJBHPOBM BOE UIF POFT EJHJU CFMPX JU 5IFO BEE UIF EJHJUT BMPOH FBDI EJBHPOBM TUBSUJOH BU UIF CPUUPN SJHIU DPSOFS 1MBDF UIF TVN BU UIF CPUUPN PG FBDI EJBHPOBM PVUTJEF PG UIF HSJE *G OFDFTTBSZ DBSSZ UIF UFOT EJHJU UP UIF OFYU EJBHPOBM 3FBE UIF BOTXFS EPXO UIF MFGU TJEF BOE UIFO GSPN MFGU UP SJHIU BDSPTT UIF CPUUPN &Y "NQM &

6

.VMUJQMZ × 4U FQ

5S Z *U

1MBDF UIF NVMUJQMJDBUJPO PO B HSJE 'JSTU NVMUJQMZ × = 1MBDF UIF BCPWF UIF EJBHPOBM BOE UIF CFMPX UIF EJBHPOBM $POUJOVF NVMUJQMZJOH FBDI UJNF XSJUJOH UIF QBSUJBM QSPEVDU PO UIF HSJE

4U FQ

× =

× =

× =

× =

This Way

× = "EE UIF QBSUJBM QSPEVDUT TUBSUJOH BU UIF CPUUPN SJHIU DPSOFS BOE NPWJOH MFGU *G ZPV FYUFOE UIF EJBHPOBM MJOFT ZPV DBO TFF UIF EJBHPOBMT CFUUFS

TU EJBHPOBM OE EJBHPOBM + + = SE EJBHPOBM + + + = 8SJUF JO UIF EJBHPOBM BOE QMBDF UIF UFO JO UIF OFYU EJBHPOBM UI EJBHPOBM + + + =

4&"3$)

5P TFF BOPUIFS NFUIPE GPS NVMUJQMZJOH × HP UP &YBNQMF PO QBHF

5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

UI EJBHPOBM 4U FQ

'JOE UIF QSPEVDU 4UBSU XJUI UIF UPQ MFGU EJHJU PVUTJEF PG UIF HSJE .PWF EPXO BOE UIFO BDSPTT UIF CPUUPN PG UIF HSJE 8SJUF UIF EJHJUT JO PSEFS

5IF QSPEVDU PG BOE JT

XXX 48BEWBOUBHF DPN

29

.VMUJQMJDBUJPO

-BUUJDF .FUIPE BOE (SJET 4PNFUJNFT MBUUJDF NVMUJQMJDBUJPO JT TIPXO VTJOH B HSJE GPS BMM PG UIF OVNCFST BT TIPXO JO &YBNQMF BOE &YBNQMF 5IF DFMMT PO UIF HSJE UIBU TIPX UIF QSPEVDUT NBZ IBWF B IFBWJFS CPSEFS PS UIFZ NBZ CF UJOUFE "MUIPVHI MBUUJDF TUZMFT NBZ EJGGFS TPNFXIBU UIF TUFQT GPS GJOEJOH UIF QSPEVDU SFNBJO UIF TBNF FY B NQM E

7

.VMUJQMZ × 4U FQ

1MBDF UIF NVMUJQMJDBUJPO PO B HSJE 'JSTU NVMUJQMZ × = 5IFSF BSF OP UFOT TP QMBDF B BCPWF UIF EJBHPOBM BOE UIF CFMPX UIF EJBHPOBM

$POUJOVF NVMUJQMZJOH FBDI UJNF XSJUJOH UIF QBSUJBM QSPEVDU PO UIF HSJE × =9

× =

× = 12

× =

× = 4U FQ

"EE UIF QBSUJBM QSPEVDUT TUBSUJOH BU UIF CPUUPN SJHIU DPSOFS BOE NPWJOH MFGU TU EJBHPOBM OE EJBHPOBM + + = 8SJUF JO UIF CPY BOE QMBDF UIF UFO JO UIF OFYU EJBHPOBM SE EJBHPOBM + + + = 8SJUF JO UIF CPY BOE QMBDF UIF UFO JO UIF OFYU EJBHPOBM

UI EJBHPOBM + + + = UI EJBHPOBM 4U FQ

'JOE UIF QSPEVDU 4UBSU XJUI UIF UPQ MFGU EJHJU PVUTJEF PG UIF HSJE .PWF EPXO BOE UIFO BDSPTT UIF CPUUPN PG UIF HSJE 8SJUF UIF EJHJUT JO PSEFS

5IF QSPEVDU PG BOE JT

30

.BU I T O FX .FU I P E T X I BU 1B S FO U T O FFE U P ,O P X &Y "NQM &

8

.VMUJQMZ × 4U FQ

4&"3$)

1MBDF UIF NVMUJQMJDBUJPO PO B HSJE

5IFO NVMUJQMZ 3FNFNCFS UP QMBDF UIF UFOT EJHJU BCPWF UIF EJBHPOBM BOE UIF POFT EJHJU CFMPX JU

'JSTU NVMUJQMZ FBDI EJHJU JO UIF GBDUPS CZ

/FYU NVMUJQMZ FBDI EJHJU CZ

5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

'JOBMMZ NVMUJQMZ FBDI EJHJU CZ

4U FQ

"EE UIF QBSUJBM QSPEVDUT TUBSUJOH BU UIF CPUUPN SJHIU DPSOFS BOE NPWJOH MFGU TU EJBHPOBM SE EJBHPOBM + + = 8SJUF JO UIF CPY BOE QMBDF UIF UFO JO UIF OFYU EJBHPOBM UI EJBHPOBM + + + + = 8SJUF JO UIF CPY BOE QMBDF UIF UFO JO UIF OFYU EJBHPOBM UI EJBHPOBM + + + = 8SJUF JO UIF CPY BOE QMBDF UIF UFO JO UIF OFYU EJBHPOBM

OE EJBHPOBM + + =

UI EJBHPOBM + + + = UI EJBHPOBM 4U FQ

'JOE UIF QSPEVDU 4UBSU XJUI UIF UPQ MFGU EJHJU PVUTJEF PG UIF HSJE .PWF EPXO BOE UIFO BDSPTT UIF CPUUPN PG UIF HSJE 8SJUF UIF EJHJUT JO PSEFS

5IF QSPEVDU PG BOE JT

XXX 48BEWBOUBHF DPN

31

%JWJTJPO 1BSUJBM 2VPUJFOUT Need More

HELP

" EJWJTJPO QSPCMFN DPOTJTUT PG UIFTF UISFF QBSUT RVPUJFOU EJWJTPS EJWJEFOE

5IF QBSUJBM RVPUJFOUT EJWJTJPO NFUIPd JOWPMWFT NBLJOH B TFSJFT PG FTUJNBUFT CZ NVMUJQMZJOH UIF EJWJTPS CZ B TNBMM OVNCFS BOE TVCUSBDUJOH $POUJOVF VOUJM OP NPSF EJWJTJPOT BSF QPTTJCMF "EE UIF QBSUJBM RVPUJFOUT *ODMVEF BOZ SFNBJOEFS JO UIF GJOBM RVPUJFOU FY B NQM E

1

%JWJEF ÷ 5IJOL 5IF EJWJTPS JT 4JODF × = UIFSF BSF BU MFBTU UXP hT JO

5IFSF JT B SFNBJOEFS PG TP UIFSF JT BOPUIFS JO

*O UIF OVNCFS UIFSF BSF HSPVQT PG XJUI B SFNBJOEFS PG

8SJUF × 5IFO TVCUSBDU − =

8SJUF × 5IFO TVCUSBDU − =

8SJUF UIF RVPUJFOU 3

× −

× − × −

3 × − × −

5IF RVPUJFOU ÷ JT 3

FY B NQM E

5S Z *U This Way 5P TFF BOPUIFS NFUIPE GPS EJWJEJOH ÷ HP UP &YBNQMF PO QBHF

2

%JWJEF ÷ &TUJNBUF UIF OVNCFS PG UJNFT UIF EJWJTPS XJMM HP JOUP UIF EJWJEFOE 5IJOL 4JODF × = JU XJMM HP BU MFBTU UJNFT 8SJUF × 4VCUSBDU − = 5IFSF JT B SFNBJOEFS PG &TUJNBUF BHBJO

3 × − × −

5IJOL 4JODF × = JU XJMM HP UJNFT 8SJUF × = 4VCUSBDU − = 5IFSF BSF OP NPSF HSPVQT PG TP JT UIF SFNBJOEFS 5IFSF BSF HSPVQT PG XJUI MFGU PWFS TP UIF RVPUJFOU ÷ JT 3

32

.BU I T O FX .FU I P E T X I BU 1B S FO U T O FFE U P ,O P X

$PMVNO %JWJTJPO 5IF DPMVNO EJWJTJPO NFUIPd JT B WBSJBUJPO PG UIF TUBOEBSE MPOH EJWJTJPO BMHPSJUIN 7FSUJDBM MJOFT XIJDI GPSN DPMVNOT BSF JOTFSUFE UP TFQBSBUF UIF EJHJUT JO UIF EJWJEFOE 3FNBJOEFST BSF BMXBZT SFQPTJUJPOFE CFGPSF QFSGPSNJOH UIF OFYU EJWJTJPO &YBNQMF TIPXT B WBSJBUJPO PG UIF BMHPSJUIN GPS UIJT NFUIPE &Y "NQM &

3

EJWJEF ÷ 45&1

'JSTU %JWJTJPO • %FUFSNJOF IPX NBOZ UJNFT XJMM HP JOUP 4JODF > JU XJMM OPU HP • 8SJUF BCPWF UIF JO UIF EJWJEFOE 5IFO TVCUSBDU − = • #SJOH EPXO UIF 3FQPTJUJPO UIF SFNBJOEFS UP UIF MFGU PG UIF UP TIPX

45&1

4FDPOE %JWJTJPO

Need More

HELP

5P SFWJFX CBTJD EJWJTJPO GBDUT HP UP %JWJTJPO 'BDUT JO 'PVOEBUJPOT PG .BUIFNBUJDT Q

3 − − −

• %FUFSNJOF IPX NBOZ UJNFT XJMM HP JOUP 4JODF × = JU XJMM HP UJNF • 8SJUF BCPWF UIF JO UIF EJWJEFOE 5IFO TVCUSBDU − = • #SJOH EPXO UIF 3FQPTJUJPO UIF SFNBJOEFS UP UIF MFGU PG UIF UP TIPX 45&1

5IJSE %JWJTJPO • %FUFSNJOF IPX NBOZ UJNFT XJMM HP JOUP 4JODF × = JU XJMM HP UJNFT • 8SJUF BCPWF UIF JO UIF EJWJEFOE 5IFSF BSF OP NPSF EJHJUT JO UIF EJWJEFOE UP CSJOH EPXO 4VCUSBDU − =

5IF RVPUJFOU ÷ JT 3

&Y "NQM &

5S Z *U This Way 5P TFF BOPUIFS NFUIPE GPS EJWJEJOH ÷ HP UP &YBNQMF PO QBHF

4

EJWJEF ÷ %FUFSNJOF IPX NBOZ UJNFT XJMM HP JOUP 9. • 4JODF > XSJUF JO UIF EJWJEFOE 5IFO TVCUSBDU − = • 1MBDF B OFYU UP UIF OFYU OVNCFS JO UIF EJWJEFOE UP HFU • $POUJOVF EJWJEJOH JO UIJT XBZ BT TIPXO PO UIF SJHIU

3 −0 85

4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

5IF RVPUJFOU ÷ JT 3

XXX 48BEWBOUBHF DPN

33

8PSE 1SPCMFNT #PY .PEFMT Need More

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'PS NPSF XPSE QSPCMFNT JOWPMWJOH XIPMF OVNCFST HP UP 8PSE 1SPCMFNT JO 'PVOEBUJPOT PG .BUIFNBUJDT Q

*O B CPY NPEFl UIF iCPYw JT B SFDUBOHVMBS CBS UIBU TIPXT UIF SFMBUJPOTIJQ CFUXFFO B LOPXO BOE BO VOLOPXO RVBOUJUZ 5IJT UZQF PG NPEFM DBO IFMQ ZPV WJTVBMJ[F XIBU PQFSBUJPO JT OFFEFE UP TPMWF B QSPCMFN 'PS NPSF DPNQMFY QSPCMFNT UXP PS NPSF CBST NBZ CF VTFE UP SFQSFTFOU UIF TJUVBUJPO FY B NQM E

1

)FDUPS IBT GJTI JO IJT GJTI UBOL )F CVZT NPSF BU UIF QFU TUPSF )PX NBOZ GJTI EPFT IF IBWF OPX %SBX B CBS BOE EJWJEF JU JOUP UXP QBSUT -FU POF QBSU SFQSFTFOU GJTI -FU UIF PUIFS QBSU SFQSFTFOU NPSF GJTI

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FY B NQM E

2

%BOJFM DPMMFDUFE SPDLT 4POZB GPVOE GFXFS SPDLT UIBO %BOJFM )PX NBOZ SPDLT EPFT 4POZB IBWF SPDLT

%BOJFM 4POZB

SPDLT SPDLT

5IF XIPMF JT SPDLT UIF OVNCFS %BOJFM DPMMFDUFE " LOPXO QBSU JT SPDLT 4VCUSBDU UP GJOE UIF VOLOPXO QBSU − = 4POZB IBT SPDLT

34

.BU I T O FX .FU I P E T X I BU 1B S FO U T O FFE U P ,O P X

&Y "NQM &

3

U IF TVN PG UXP OVNCFST JT P OF PG UIF OVNCFST JT UISFF UJNFT HSFBUFS UIBO UIF PUIFS OVNCFS X IBU BSF UIF UXP OVNCFST 3FQSFTFOU UIF UXP OVNCFST BT SFDUBOHMFT 5IF SFDUBOHMF GPS UIF MBSHFS OVNCFS JT UISFF UJNFT UIF MFOHUI PG UIF SFDUBOHMF GPS UIF TNBMMFS OVNCFS

4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

5IFSF BSF FRVBM TFDUJPOT TP UIF TNBMMFS OVNCFS JT ÷ = 5IF MBSHFS OVNCFS is 24 × = 4JODF + = UIFTF WBMVFT NVTU CF DPSSFDU 5IF OVNCFST BSF BOE

&Y "NQM &

4

-FF IBE USBEJOH DBSET EFCPSB IBE GFXFS USBEJOH DBSET -FF HJWFT EFCPSB TPNF PG IJT DBSET OPX IF IBT UJNFT BT NBOZ DBSET BT EFCPSB B BU GJSTU IPX NBOZ USBEJOH DBSET EJE EFCPSB IBWF -FF %FCPSB

GFXFS

%FCPSB − =

U GJSTU %FCPSB IBE " USBEJOH DBSET

C IPX NBOZ USBEJOH DBSET EP UIFZ CPUI IBWF OPX 5IF UPUBM OVNCFS PG DBSET JT + = 5IF DBSET BSF TIBSFE CFUXFFO -FF BOE %FCPSB "GUFS HJWJOH %FCPSB TPNF DBSET -FF IBT × BT NBOZ DBSET BT TIF IBT -FF DBSET %FCPSB

%FCPSB ÷ =

/PX -FF IBT DBSET BOE %FCPSB IBT DBSET

XXX 48BEWBOUBHF DPN

-FF = × =

5PUBM + =

35

'PVOEBUJPOT PG Foundations of Mathematics .BUIFNBUJDT

" UJMF TFUUFS OFFET UP VOEFSTUBOE UXP EJNFOTJPOBM HFPNFUSJD TIBQFT UP GPMMPX B EFTJHO QBUUFSO BOE TFU UIFÂ UJMFT DPSSFDUMZ

36

1MBDF 7BMVF 8IPMF /VNCFST &TUJNBUJOH XJUI 8IPMF /VNCFST 1PJOUT -JOFT 1MBOFT BOE "OHMFT "OBMZ[JOH 4IBQFT JO B 1MBOF "OBMZ[JOH 4PMJE 'JHVSFT %BUB BOE (SBQIT

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37

.VMUJQMJDBUJPO 'BDUT 8IBU *T .VMUJQMJDBUJPO 7ATCH /UT

!

6OMJLF BEEJUJPO BOE TVCUSBDUJPO UIF OVNCFST JO B NVMUJQMJDBUJPO QSPCMFN EP OPU NFBO UXP EJGGFSFOU BNPVOUT PG UIF TBNF DBUFHPSZ

.VMUJQMJDBUJPO JT B NBUIFNBUJDBM PQFSBUJPO JO XIJDI B QBSUJDVMBS OVNCFS PG FRVBM TJ[FE HSPVQT BSF DPNCJOFE 5IF SFTVMU JT DBMMFE UIF QSPEVDU *O UIF NVMUJQMJDBUJPO CFMPX UIFSF BSF HSPVQT PG SFDUBOHMFT 8IFO UIF HSPVQT BSF DPNCJOFE UIFSF BSF SFDUBOHMFT /VNCFS PG *UFNT QFS (SPVQ

×

0OF OVNCFS UFMMT UIF OVNCFS PG FRVBM HSPVQT 5IF PUIFS OVNCFS UFMMT UIF TJ[F PG FBDI FRVBM HSPVQ

/VNCFS PG &RVBM (SPVQT

=

5PUBM "NPVOU QSPEVDU

JUFNT QFS HSPVQ × HSPVQT =

× NFBOT UIF TBNF UIJOH BT + + = &Y "NQM &

1

,JNJ +PF BOE #PC FBDI IBWF UXP CPPLT )PX NBOZ CPPLT EP UIFZ IBWF JO BMM

4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

CPPLT QFS QFSTPO

×

QFPQMF

=

CPPLT

× = 5IFSF BSF CPPLT JO BMM

o tP t U oU GP (

!

w oX nP KO ,

.VMUJQMJDBUJPO JT B NBUIFNBUJDBM PQFSBUJPO 5IF QSPEVDU HJWFT UIF TBNF SFTVMU BT UIF TVN PG BEEJOH UIF OVNCFS PG JUFNT JO FBDI HSPVQ :PV DBO DIBOHF BOZ NVMUJQMJDBUJPO QSPCMFN JOUP BO BEEJUJPO QSPCMFN .VMUJQMJDBUJPO "EEJUJPO × = + + = JUFNT QFS HSPVQ × HSPVQT = JUFNT QFS HSPVQ + JUFNT QFS HSPVQ + JUFNT QFS HSPVQ =

68

Whole Numbers '06/%"5*0/4 0' ."5)&."5*$4

.PEFMT GPS .VMUJQMJDBUJPO 5ISFF XBZT UP NPEFM NVMUJQMJDBUJPO BSF VTF DPVOUFST NBLF BO BSSBZ BOE TLJQ DPVOU PO B OVNCFS MJOF "O BSSBy JT BO BSSBOHFNFOU PG PCKFDUT JO SPXT BOE DPMVNOT &BDI SPX IBT UIF TBNF OVNCFS PG PCKFDUT HSPVQT PG DPVOUFST

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KVNQT PG TQBDFT

7ATCH /UT

!

*O NVMUJQMJDBUJPO BMM PG UIF HSPVQT NVTU CF UIF TBNF TJ[F + + + + = ×

× =

FY B NQM E

+ + + + = × QMVT

2

+BWJFS T DMBTTSPPN IBT SPXT PG EFTLT 5IFSF BSF EFTLT JO FBDI SPX )PX NBOZ EFTLT JO BMM 5IFSF BSF UIF TBNF OVNCFS PG EFTLT JO FBDI SPX 5IJT JT BO FRVBM HSPVQT QSPCMFN TP ZPV DBO NVMUJQMZ .PEFM UIF QSPCMFN VTJOH BO BSSBZ ×

EFTLT QFS SPX

SPXT

× = 5IFSF BSF EFTLT JO BMM

FY B NQM E

3

6TF B OVNCFS MJOF UP GJOE × = KVNQT PG TQBDFT

× =

XXX 48BEWBOUBHF DPN

69

/VNCFST BOE Numbers and Operations 0QFSBUJPOT

$POTVNFST VTF VOJU SBUFT UP DPNQBSF QSJDFT CFUXFFO JUFNT BOE UP EFUFSNJOF XIJDI RVBOUJUZ JT B CFUUFS CVZ

212

%FDJNBMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . /VNCFS 5IFPSZ . . . . . . . . . . . . . . . . . . . . . . . . . . 'SBDUJPOT BOE .JYFE /VNCFST . . . . . . . . . . . . . 3BUJP BOE 1SPQPSUJPO . . . . . . . . . . . . . . . . . . . . . 1FSDFOU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &TUJNBUF XJUI %FDJNBMT BOE 'SBDUJPOT . . . . . . 1BUUFSOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1SPCMFN 4PMWJOH . . . . . . . . . . . . . . . . . . . . . . . . . .

www.swadvantage.com

213

"EEJOH %FDJNBMT .PEFMJOH %FDJNBM "EEJUJPO :PV DBO VTF UFO CZ UFO HSJET UP NPEFM UIF BEEJUJPO PG EFDJNBM OVNCFST 5IF WBMVF PG FBDI TNBMM TRVBSF JO UIF HSJE JT POF IVOESFEUI &Y "NQM & 4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

1

6TF B HSJE UP NPEFM + 4IBEF IVOESFEUIT PG UIF HSJE

4IBEF NPSF IVOESFEUIT

5IFSF BSF IVOESFEUIT TIBEFE

+ =

&Y "NQM &

2

6TF B HSJE UP NPEFM + 4IBEF XIPMF HSJE BOE IVOESFEUIT PG BOPUIFS HSJE

4IBEF NPSF IVOESFEUIT

5IFSF BSF IVOESFEUIT TIBEFE + = 226

Decimals

/V NCFS T B /E 0QFS BU J0/T

"EEJOH %FDJNBMT PO B 1MBDF 7BMVF $IBSU :PV DBO BMTP BEE EFDJNBMT JO B WFSUJDBM GPSNBU PO B QMBDF WBMVF DIBSU -JOF VQ UIF EFDJNBM QPJOUT JO UIF OVNCFST 5IF EFDJNBM QPJOU JO UIF TVN XJMM CF EJSFDUMZ CFMPX UIF EFDJNBM QPJOUT JO UIF BEEFOET FY B NQM E

3

BEE +

5S Z *U

+

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PO

UFO

T

FT

8SJUF UIF QSPCMFN WFSUJDBMMZ PO B QMBDF WBMVF DIBSU

:PV DBO VTF NFOtal maUI UP add 11.5 + 'JSTU, add the whole number parUT 5IJOL + JT

"EE GSPN SJHIU UP MFGU BT ZPV BEE XIPMF OVNCFST

+

UFO UIT

PO

UFO

T

FT

#SJOH UIF EFDJNBM QPJOU TUSBJHIU EPXO JOUP UIF BOTXFS

5IFO BEE UIF decimal parUT 5IJOL + JT $PNCJOF UIF TVNT

+ =

G'Oot Tto

K+N.O/W 7Ã "EEJOH %FDJNBMT 8IFO BEEJOH EFDJNBMT WFSUJDBMMZ ZPV NVTU BMJHO MJOF VQ UIF EFDJNBM QPJOUT 5IJT LFFQT FBDI EJHJU JO JUT QSPQFS QMBDF 5IF EFDJNBM QPJOU JO UIF TVN JT EJSFDUMZ CFMPX UIF EFDJNBM QPJOUT JO UIF BEEFOET

XXX 48BEWBOUBHF DPN

+ 227

"EEJOH %FDJNBMT

"EEJOH %FDJNBMT XJUI 3FHSPVQJOH

4

"EE + 8SJUF UIF QSPCMFN JO WFSUJDBM GPSNBU

"EE UIF XIPMF OVNCFST BOE #SJOH UIF EFDJNBM QPJOU TUSBJHIU EPXO JOUP UIF BOTXFS

OE IV

FT

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SFE

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&9".1-&

T

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PO

HELP

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UFO

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+

+ =

&9".1-&

8SJUF UIF QSPCMFN JO WFSUJDBM GPSNBU

/05& $ EFOPUFT $BOBEJBO EPMMBST

+

SFE OE IV

PO

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T

4&"3$)

5

.BSMB CPVHIU B %7% GPS $ BOE B CPUUMF PG XBUFS GPS $ )PX NVDI EJE TIF TQFOE JO BMM

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OE SF IV

T

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PO F

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#SJOH UIF EFDJNBM QPJOU TUSBJHIU EPXO JOUP UIF BOTXFS

Decimals

/V NCFS T B /E 0QFS BU J0/T

"EEJOH %FDJNBMT XJUI .PSF 5IBO 5XP "EEFOET 6TF B QMBDF WBMVF DIBSU 8SJUF UIF QSPCMFN JO WFSUJDBM GPSN CFJOH DBSFGVM UP MJOF VQ UIF EFDJNBM QPJOUT FY B NQM E

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6

'PS BOPUIFS WJTVBM SFQSFTFOUBUJPO PG UIF OVNCFST BEE QMBDFIPMEFS [FSPT so each number has UXP XIPMF OVNCFS QMBDFT BOE UISFF EFDJNBM QMBDFT

BEE + + 8SJUF UIF QSPCMFN JO WFSUJDBM GPSNBU

+

IV OE SFE UIT UIP VT BO EUI T

FT PO

UFO

T

#SJOH UIF EFDJNBM QPJOU TUSBJHIU EPXO JOUP UIF BOTXFS

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"EE GSPN SJHIU UP MFGU

5IJT BMJHOT UIF EFDJNBM QPJOUT CVU EPFT OPU DIBOHF UIF WBMVF PG UIF OVNCFST

+ + =

FY B NQM E

5S Z *U

7

BEE + + 8SJUF UIF QSPCMFN JO WFSUJDBM GPSNBU

5

"EE GSPN SJHIU UP MFGU BOE SFHSPVQ BT OFFEFE 3FOBNF UIF UIPVTBOEUIT BT IVOESFEUI BOE [FSP UIPVTBOEUIT

+

0

5

)

5

3FOBNF UIF IVOESFEUIT BT UFOUI BOE IVOESFEUIT

Need More

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$POUJOVF BEEJOH 3FOBNF UIF UFOUIT BT POF BOE UFOUIT

5

"EE UIF XIPMF OVNCFST

XXX 48BEWBOUBHF DPN

)

5

#SJOH UIF EFDJNBM QPJOU TUSBJHIU EPXO JOUP UIF BOTXFS

+ + =

5

+

0

*O &YBNQMF ZPV EP OPU OFFE UP JODMVEF UIF [FSP JO UIF UIPVTBOEUIT QMBDF XIFO ZPV XSJUF UIF GJOBM TVN )PXFWFS JG UIJT XFSF B TDJFOUJGJD NFBTVSFNFOU ZPV XPVME JODMVEF UIF [FSP UP NBLF UIF OVNCFS NPSF QSFDJTF 5IF OVNCFS JT QSFDJTF UP UIPVTBOEUIT XIJMF JT QSFDJTF Uo hundrFEUIT

229

%JWJEJOH %FDJNBMT CZ 8IPMF /VNCFST .PEFMJOH %FDJNBM %JWJTJPO Need More

HELP

5IF EJWJEFOE JT UIF OVNCFS UP CF EJWJEFE 5IF EJWJTPS JT UIF OVNCFS Zou divide bZ 5IF RVPUJFOU JT UIF BOTXFS

:PV DBO VTF B UFO CZ UFO HSJE UP NPEFM UIF EJWJTJPO PG B EFDJNBM OVNCFS CZ B XIPMF OVNCFS -PPL BU &YBNQMF BOE &YBNQMF /PUJDF UIF GPMMPXJOH t 5IF RVPUJFOU IBT UIF TBNF OVNCFS PG EFDJNBM QMBDFT BT UIF EJWJEFOE t 5IF RVPUJFOU JT MFTT UIBO UIF EJWJEFOE FY B NQM E

1

%JWJEF CZ 4IBEF IVOESFEUIT

%JWJEF IVOESFEUIT JOUP FRVBM HSPVQT

&BDI HSPVQ DPOUBJOT IVOESFEUIT ÷ =

FY B NQM E

2

%JWJEF ÷

To G'Oot to

K+N.O/W 7!à

= 4IBEF IVOESFEUIT

%JWJEF IVOESFEUIT JOUP FRVBM HSPVQT

%JWJEJOH B %FDJNBM CZ B 8IPMF /VNCFS 8IFO B EFDJNBM OVNCFS JT EJWJEFE CZ B XIPMF OVNCFS UIF RVPUJFOU JT BMXBZT MFTT UIBO UIF EJWJEFOE ÷ = IVOESFEUIT JT MFTT UIBO IVOESFEUIT

246

&BDI HSPVQ DPOUBJOT IVOESFEUIT ÷ =

Decimals

/6.#&34 "/% 01&3"5*0/4

%JWJEJOH %FDJNBMT CZ 8IPMF /VNCFST 5P EJWJEF B EFDJNBM CZ B XIPMF OVNCFS XSJUF UIF QSPCMFN JO MPOH EJWJTJPO GPSN BOE EJWJEF BT XJUI XIPMF OVNCFST 1MBDF UIF EFDJNBM QPJOU JO UIF RVPUJFOU EJSFDUMZ BCPWF UIF EFDJNBM QPJOU JO UIF EJWJEFOE FY B NQM E

3

%JWJEF รท 8SJUF UIF QSPCMFN JO MPOH EJWJTJPO GPSN

%JWJEF BT ZPV EP GPS XIPMF OVNCFST .VMUJQMZ TVCUSBDU BOE CSJOH EPXO UIF OFYU EJHJU JO FBDI TUFQ

1MBDF UIF EFDJNBM QPJOU JO UIF RVPUJFOU EJSFDUMZ BCPWF UIF EFDJNBM QPJOU JO UIF EJWJEFOE

Need More

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'PS IFMQ XJUI EJWJEJOH XIPMF OVNCFST HP UP %JWJEF 8IPMF /VNCFST JO 'PVOEBUJPOT PG .BUIFNBUJDT Q

รท =

FY B NQM E

4

.S -FF CPVHIU XJOEPX TIBEFT 5IF UPUBM DPTU XBT )PX NVDI EJE FBDI XJOEPX TIBEF DPTU %JWJEF UIF UPUBM DPTU CZ UIF OVNCFS PG XJOEPX TIBEFT

รท

8SJUF UIF QSPCMFN JO MPOH EJWJTJPO GPSN

%JWJEF BT ZPV EP GPS XIPMF OVNCFST .VMUJQMZ TVCUSBDU BOE CSJOH EPXO UIF OFYU EJHJU JO FBDI TUFQ

1MBDF UIF EFDJNBM QPJOU JO UIF RVPUJFOU EJSFDUMZ BCPWF UIF EFDJNBM QPJOU JO UIF EJWJEFOE

4FB 3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

&BDI XJOEPX TIBEF DPTU

XXX 48BEWBOUBHF DPN

247

8PSE 1SPCMFNT 4PMWJOH %FDJNBM 8PSE 1SPCMFNT 0GUFO UIFSF JT NPSF UIBO POF XBZ UP TPMWF B XPSE QSPCMFN 'JOE UIF NFUIPE UIBU XPSLT CFTU GPS ZPV :PV NBZ GJOE JU IFMQGVM UP GPMMPX UIF TUFQT TIPXO JO (PU 5P ,OPX CFMPX 5IF FYBNQMFT TIPX BO BQQMJDBUJPO PG UIF TUFQT FY B NQM E

Need More

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'PS IFMQ TPMWJOH XPSE QSPCMFNT HP UP 1SPCMFN 4PMWJOH PO QBHFT o :PV XJMM GJOE B GPVS TUFQ QSPCMFN TPMWJOH QSPDFTT BOE TUSBUFHJFT GPS QSPCMFN TPMWJOH 5IF $IFDL JO &YBNQMFÂ VTFT UIF TUSBUFHZ 6TF -PHJDBM 3FBTPOJOH PO QBHF

1

4BSBI IBT B XBUFSNFMPO UIBU IBT B NBTT PG LJMPHSBNT 8BUFS NBLFT VQ 6.716 kJMPHSBNT PG UIF NBTT )PX NBOZ LJMPHSBNT PG UIF XBUFSNFMPO T NBTT BSe not wBUFS 3&"%

3FBE UIF QSPCMFN DBSFGVMMZ *EFOUJGZ XIBU ZPV LOPX BOE XIBU ZPV OFFE UP GJOE PVU or need UP LOPX. ,OPX 5PUBM NBTT JT LH BOE LH PG UIF NBTT JT XBUFS /FFE UP ,OPX UIF QBSU PG UIF XBUFSNFMPO T NBTT UIBU JT OPU NBEF VQ PG XBUFS

1M "O

5IJOL BCPVU IPX ZPV DBO GJOE UIF JOGPSNBUJPO ZPV OFFE UP LOPX :PV DPVME TVCUSBDU UIF NBTT UIBU JT XBUFS GSPN UIF FOUJSF NBTT PG UIF XBUFSNFMPO

4P M W &

8SJUF UIF QSPCMFN JO WFSUJDBM GPSN "EE [FSPT GPS QMBDFIPMEFST BOE TVCUSBDU Lg − Lg Lg

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− LH JT OPU XBUFS

%JE * BOTXFS UIF RVFTUJPO *T NZ BOTXFS SFBTPOBCMF LH + LH = LH XIJDI JT HSFBUFS UIBO UIF UPUBM NBTT PG UIF XBUFSNFMPO 5IFSFGPSF UIF NBTT PG UIF XBUFSNFMPO UIBU JT OPU XBUFS NVTU CF MFTT UIBO LH 5IF BOTXFS PG LH JT B SFBTPOBCMF BOTXFS

LH PG UIF XBUFSNFMPO T NBTT JT OPU XBUFS

G'Oot Tto

K+N.O/W 7!Ã

*G ZPV TUBUF ZPVS BOTXFS JO UFSNT PG UIF RVFTUJPO ZPV XJMM LOPX ZPV BOTXFSFE UIF DPSSFDU RVFTUJPO 5IF RVFTUJPO JO &YBNQMF BTLT i)PX NBOZ LJMPHSBNT PG UIF XBUFSNFMPO T NBTT BSF OPU XBUFS w 5IF BOTXFS TUBUFT i LH PG UIF XBUFSNFMPO T NBTT JT OPU XBUFS w

1SPCMFN 4PMWJOH 1SPDFTT 5P TPMWF B XPSE QSPCMFN ZPV DBO VTF UIJT QSPDFTT 3FBE UIF QSPCMFN DBSFGVMMZ *EFOUJGZ XIBU ZPV LOPX BOE XIBU ZPV OFFE UP GJOE PVU 1MBO IPX UP GJOE UIF TPMVUJPO 4PMWF UIF QSPCMFN VTJOH ZPVS QMBO $IFDL ZPVS XPSL BOE BTL JG ZPVS BOTXFS NBLFT TFOTF

Decimals &Y "NQM &

/6.#&34 "/% 01&3"5*0/4

2

+BNBM CPVHIU UXP 5 TIJSUT BOE B %7% 5IF 5 TIJSUT DPTU FBDI BOE UIF %7% Dost $14.99. HoX NVDI EJE +BNBM TQFOE 3FB E

,OPX 5 TIJSUT DPTU FBDI %7% DPTU /FFE UP ,OPX 5PUBM DPTU PG UIF JUFNT

1M B O

5IFSF BSF UXP XBZT UP TPMWF UIJT QSPCMFN NFU I P E

NFU I P E

.VMUJQMZ UP GJOE UIF UPUBM PG UIF 5 TIJSUT 5IFO BEE UIF DPTU PG UIF %7% UP UIF QSPEVDU

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NFU I P E

NFU I P E

4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

4P M W F

× $31.00 $I F$L

+

+

:PV DBO FTUJNBUF FBDI QSJDF BOE UIFO GJOE UIF TVN JT BCPVU . JT BCPVU . JT BCPVU 6TF NFOUBM NBUI × = 5IF DBMDVMBUFE BOTXFS PG JT SFBTPOBCMF

+BNBM TQFOU PO UIF JUFNT

&Y "NQM &

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'PS IFMQ XJUI FTUJNBUJOH UP DIFDL HP UP &TUJNBUJOH UP 1SFEJDU PS $IFDL PO QBHF

3

$BSPMJOB CPVHIU QPVOET PG HSBQFT &BDI QPVOE PG HSBQFT DPTUT $ )PX NVDI EJE $BSPMJOB TQFOE 3FB E

,OPX MC BU $ QFS lb /FFE UP ,OPX 5PUBM DPTU

1M B O

4P M W F

3.62 QPVOET PG HSBQFT × 1.49 GPS QPVOE 5.3938 SPVOET UP $

$I F$L

.VMUJQMZ UP GJOE IPX NVDI $BSPMJOB TQFOU × = BOE × = > > $ JT B SFBTPOBCMF BOTXFS

$BSPMJOB TQFOU $ GPS UIF HSBQFT

XXX 48BEWBOUBHF DPN

259

8PSE 1SPCMFNT

4PMWJOH %FDJNBM 8PSE 1SPCMFNT &Y "NQM & 4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

4

.BSMB JT NBLJOH TBOEXJDIFT 4IF IBT QPVOET PG UVSLFZ &BDI TBOEXJDI VTFT 0.25Â pounds of turkFZ 8IBU JT UIF HSFBUFTU OVNCFS PG TBOEXJDIFT .BSMB DBO NBLF 3FB %

,OPX MC PG UVSLFZ MC QFS sandwich /FFE UP ,OPX /VNCFS PG sandwiches MBSMB DBO NBLF

1M B O

%JWJEF UIF OVNCFS PG QPVOET PG UVSLFZ CZ UIF BNPVOU OFFEFE GPS TBOEXJDI

4P M W F

%JWJEF .PWF UIF EFDJNBM QPJOU UP the righU UXP QMBDFT

$I F$L

6TF MPHJDBM SFBTPOJOH BOE NFOUBM NBUI 5IJOL ÷ = *G TIF DBO NBLF TBOEXJDIFT QFS MC UVSLFZ UIFO TIF DBO NBLF TBOEXJDIFT XJUI MC PG UVSLFZ MC = MC MC ÷ = —

− −

5IF BOTXFS JT SFBTPOBCMF

5IF HSFBUFTU OVNCFS PG TBOEXJDIFT .BSMB DBO NBLF JT

&Y "NQM &

5

4BN CPVHIU UISFF MFNPOT GPS FBDI BOE GPVS PSBOHFT GPS FBDI )F HBWF UIF DMFSL B CJMM )PX NVDI DIBOHF EJE IF SFDFJWF 3FB %

,OPX MFNPOT BU FBDI PSBOHFT BU FBDI HBWF DMFSL /FFE UP ,OPX "NPVOU PG DIBOHF SFDFJWFE

1M B O

.VMUJQMZ UXJDF BOE UIFO BEE UP GJOE UPUBM DPTU PG UIF JUFNT 4VCUSBDU UP GJOE UIF DIBOHF

4P M W F

1FSGPSN UIF PQFSBUJPOT JO UIF QMBO $PTU PG lemons $PTU PG PSanges $0.20 QFS MFNPO $0.65 QFS PSBOHF × 3 MFNPOT × 4 PSBOHFT

5PUBM DPTU +

'JOE UIF DIBOHF $ − $ $ $I F$L

3FWJFX ZPVS NVMUJQMJDBUJPO 5P FTUJNBUF UIF DIBOHF UIJOL − = TP JO DIBOHF JT SFBTPOBCMF

4BN SFDFJWFE JO DIBOHF

260

Decimals FY B NQM E

/6.#&34 "/% 01&3"5*0/4

6

'FOH JT CVJMEJOH B GFODF BSPVOE IJT HBSEFO )F OFFET B UPUBM PG NFUFST PG GFODJOH )F IBT POF QJFDF PG GFODJOH UIBU JT NFUFST JO MFOHUI BOE BOPUIFS QJFDF UIBU JT NFUFST JO MFOHUI %PFT 'FOH IBWF FOPVHI GFODJOH GPS IJT HBSEFO 3FB E

,OPX 'FOH IBT N BOE N PG GFODJOH /FFE UP ,OPX *T UIJT MFOHUI FRVBM UP PS HSFBUFS UIBO N

1M B O

"EE UP GJOE UIF UPUBM BNPVOU PG GFODJOH 'FOH IBT $PNQBSF UIF TVN UP XIBU IF OFFET

4P M W F

1FSGPSN UIF PQFSBUJPOT JO UIF QMBO "EE. $PNQBSF N > N IBT > OFFET +

$I F$L

46.40 5IF BNPVOU PG GFODJOH 'FOH IBT JT HSFBUFS UIBO UIF BNPVOU IF OFFET 'FOH IBT NFUFST PG GFODJOH

7ATCH /UT

3PVOEJOH TIPXT UIBU + = BOE + = 5IJT UFMMT ZPV UIBU ZPVS BOTXFS JT SFBTPOBCMF CVU GPS UIJT QSPCMFN ZPV OFFE B QSFDJTF DPNQBSJTPO 6TF B DBMDVMBUPS UP DIFDL ZPVS BEEJUJPO PS TVCUSBDU − UP TFF JG UIF EJGGFSFODF JT

8IFO B QSPCMFN BTLT JG TPNF RVBOUJUZ JT FOPVHI SPVOEJOH JT OPU UIF CFTU XBZ UP DIFDL UIF BOTXFS

'FOH IBT FOPVHI GFODJOH GPS IJT HBSEFO

FY B NQM E

!

7

"O PSBOHF KVJDF DBSUPO JT MBCFMFE MJUFST TFSWJOHT )PX NBOZ NJMMJMJUFST PG KVJDF BSF JO B TFSWJOH 3FB E

,OPX - PG KVJDF XJMM NBLF TFSWJOHT /FFE UP ,OPX /VNCFS PG N- JO FBDI TFSWJOH

1M B O

%JWJEF UIF BNPVOU PG KVJDF CZ UIF OVNCFS PG TFSWJOHT $POWFSU UIF BOTXFS UP N-

4P M W F

1FSGPSN UIF PQFSBUJPOT JO UIF QMBO %JWJEF. $POWFSU MJUFST UP NJMMJMJUFST MJUFS = NJMMJMJUFST .VMUJQMZ CZ CZ NPWJOH UIF EFDJNBM − QPJOU UISFF QMBDFT UP UIF SJHIU - = N− 5IFSF BSF MJUFST QFS TFSWJOH

$I F$L

6TF DPNQBUJCMF OVNCFST %JWJEF ÷ = - QFS TFSWJOH &YQSFTT - BT N- N- JT DMPTF UP N- 5IF BOTXFS JT SFBTPOBCMF

5IFSF BSF NJMMJMJUFST PG KVJDF JO FBDI TFSWJOH

XXX 48BEWBOUBHF DPN

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'PS IFMQ XJUI DPNQBUJCMF OVNCFST HP UP &TUJNBUJOH %FDJNBM 4VNT BOE %JGGFSFODFT PO page 414.

261

*NQSPQFS 'SBDUJPOT BOE .JYFE /VNCFST 6OEFSTUBOEJOH .JYFE /VNCFST BOE *NQSPQFS 'SBDUJPOT

&YBNQMFT — — —

" QSPQFS GSBDUJPn IBT B OVNFSBUPS UIBU JT MFTT UIBO JUT EFOPNJOBUPS

&YBNQMFT — — —

"O JNQSPQFS GSBDUJPn IBT B OVNFSBUPS UIBU JT HSFBUFS UIBO PS FRVBM UP JUT EFOPNJOBUPS

&YBNQMFT — — —

" NJYFE OVNCFr DPOUBJOT B XIPMF OVNCFS BOE B QSPQFS GSBDUJPO .JYFE OVNCFST OBNF UIF TVN PG B XIPMF BOE B QBSU PG B XIPMF &Y "NQM & 4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

1

-JBN BOE IJT GSJFOET BUF UISFF BOE POF IBMG QJ[[BT 8SJUF B NJYFE OVNCFS UP SFQSFTFOU UIF OVNCFS PG QJ[[BT FBUFO 6TF B NPEFM UP IFMQ ZPV 5ISFF XIPMF QJ[[BT BOE POF IBMG PG BOPUIFS QJ[[B BSF TIBEFE -JBN BOE IJT GSJFOET BUF — QJ[[BT

&YQSFTTJOH .JYFE /VNCFST BT *NQSPQFS 'SBDUJPOT 4PNFUJNFT JU JT OFDFTTBSZ UP FYQSFTT B NJYFE OVNCFS BT BO JNQSPQFS GSBDUJPO &Y "NQM &

2

8SJUF BT BO JNQSPQFS GSBDUJPO .&U I P E

6TF B EJBHSBN UP NPEFM —

.&U I P E

5IJOL 5IFSF BSF IBMWFT JO QJ[[B TP UIFSF BSF × = IBMWFT JO QJ[[BT —

.VMUJQMZ UIF XIPMF OVNCFS CZ UIF EFOPNJOBUPS

5IFSF BSF TFWFO IBMG QJ[[BT TIPXO JO UIF EJBHSBN

"EE UIJT GSBDUJPO UP POF IBMG

— + — = —

= — — = —

XSJUUFO BT BO JNQSPQFS GSBDUJPO JT — 5IF NJYFE OVNCFS —

306

.&U I P E

5IFO BEE UIF OVNFSBUPS × +

Fractions and Mixed Numbers

/6.#&34 "/% 01&3"5*0/4

&YQSFTTJOH *NQSPQFS 'SBDUJPOT BT .JYFE /VNCFST 5P DIBOHF BO JNQSPQFS GSBDUJPO UP B NJYFE OVNCFS GPMMPX UIFTF TUFQT

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HELP

t %JWJEF UIF OVNFSBUPS CZ UIF EFOPNJOBUPS t 6TF UIF RVPUJFOU BT UIF XIPMF OVNCFS QBSU PG UIF NJYFE OVNCFS

'PS IFMQ XJUI EJWJTJPO HP UP %JWJEJOH 8IPMF /VNCFST JO 'PVOEBUJPOT PG .BUIFNBUJDT Q

t 6TF UIF SFNBJOEFS BT UIF OVNFSBUPS PG UIF GSBDUJPO JO UIF NJYFE OVNCFS t ,FFQ UIF PSJHJOBM EFOPNJOBUPS UIBU ZPV VTFE BT UIF EJWJTPS FY B NQM E

3

&YQSFTT FBDI JNQSPQFS GSBDUJPO BT B NJYFE OVNCFS

B � 5IF RVPUJFOU JT UIF XIPMF OVNCFS QBSU PG UIF NJYFE OVNCFS. 5IF SFNBJOEFS JT UIF OVNFSBUPS PG UIF QSPQFS GSBDUJPO 5IF JNQSPQFS GSBDUJPO — XSJUUFO BT B NJYFE OVNCFS JT —

−

C �

5IF RVPUJFOU JT UIF XIPMF OVNCFS QBSU PG UIF NJYFE OVNCFS. 5IF SFNBJOEFS JT UIF OVNFSBUPS PG UIF QSPQFS GSBDUJPO 5IF JNQSPQFS GSBDUJPO — XSJUUFO BT B NJYFE OVNCFS JT —

−

5S Z *U —

—

5IF RVPUJFOU JT UIF XIPMF OVNCFS QBSU PG UIF NJYFE OVNCFS. 5IF SFNBJOEFS JT UIF OVNFSBUPS PG UIF QSPQFS GSBDUJPO

5IF JNQSPQFS GSBDUJPO — XSJUUFO BT B NJYFE OVNCFS JT —

− −

.VMUJQMZ UIF XIPMF OVNCFS CZ UIF EFOPNJOBUPS

t

5IFO BEE UIF OVNFSBUPS

XXX 48BEWBOUBHF DPN

t t t

UIBU SFQSFTFOU — PG the whole

—

3FOBNJOH .JYFE /VNCFST BOE *NQSPQFS 'SBDUJPOT *NQSPQFS GSBDUJPO .JYFE OVNCFS *NQSPQFS GSBDUJPO

t

'PS &YBNQMF B UBLF DPVOUFST BOE EJWJEF UIFN JOUP HSPVQT PG :PV DBO NBLF 1Â gSPVQ PG 5IFSF BSF 2Â cPVOUFST MFGU PWFS

5SZ UIJT NFUIPE GPS &YBNQMF C TUBSUJOH XJUI DPVOUFST BOE NBLJOH HSPVQT PG

D �

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G'Oot Tto

K+N.O/W 7Ã .JYFE OVNCFS

%JWJEF UIF OVNFSBUPS CZ UIF EFOPNJOBUPS 6TF UIF RVPUJFOU BT UIF XIPMF OVNCFS 6TF UIF SFNBJOEFS BT UIF OVNFSBUPS

307

&RVJWBMFOU 'SBDUJPOT .PEFMJOH &RVJWBMFOU 'SBDUJPOT &RVJWBMFOU GSBDUJPOT BSF GSBDUJPOT UIBU OBNF UIF TBNF WBMVF PS BNPVOU :PV DBO NPEFM FRVJWBMFOU GSBDUJPOT VTJOH GSBDUJPO TUSJQT PS B OVNCFS MJOF 5IF NPEFMT CFMPX VTF FRVJWBMFOU GSBDUJPOT UP OBNF UIF TBNF RVBOUJUZ — = — = —

@

&Y "NQM & 4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

@

@

1

6TF GSBDUJPO TUSJQT UP XSJUF UXP GSBDUJPOT UIBU BSF FRVJWBMFOU UP

4U &Q

.PEFM — VTJOH GPVSUIT TUSJQT.

4U &Q

*U UBLFT FJHIUIT UP NBLF B TUSJQ PG FRVBM MFOHUI

4U &Q

*U UBLFT UXFMGUIT UP NBLF a TUSJQ PG FRVBM MFOHUI

—

—

—

—

—

—

—

—

—

— — — — — — — — —

5IF GSBDUJPOT — — BOE — BSF FRVJWBMFOU GSBDUJPOT

&Y "NQM &

2

6TF B OVNCFS MJOF UP XSJUF B GSBDUJPO FRVJWBMFOU UP

%JWJEF UIF EJTUBODF CFUXFFO BOE JOUP UIJSET (SBQI — PO B OVNCFS MJOF.

—

%JWJEF UIF EJTUBODF CFUXFFO BOE JOUP TJYUIT (SBQI — PO B OVNCFS MJOF.

—

5IF GSBDUJPOT — BOE — OBNF UIF TBNF QPJOU PO B OVNCFS MJOF 5IFZ BSF FRVJWBMFOU

GSBDUJPOT

308

Fractions and Mixed Numbers /6.#&34 "/% 01&3"5*0/4

/BNJOH &RVJWBMFOU 'SBDUJPOT FY B NQM E

3

8SJUF UIF FRVJWBMFOU GSBDUJPOT TIPXO JO UIF NPEFM B =

5IF DJSDMF PO UIF MFGU JT EJWJEFE JOUP UIJSET 5XP UIJSET BSF TIBEFE 5IF DJSDMF PO UIF SJHIU JT EJWJEFE JOUP OJOUIT 4JY OJOUIT BSF TIBEFE 4P — = —

C =

5S Z *U

5IF SFDUBOHMF PO UIF MFGU JT EJWJEFE JOUP GJGUIT 'PVS GJGUIT BSF TIBEFE 5IF SFDUBOHMF PO UIF SJHIU JT EJWJEFE JOUP UFOUIT &JHIU UFOUIT BSF TIBEFE 4P — = —

8SJUJOH &RVJWBMFOU 'SBDUJPOT *G ZPV NVMUJQMZ PS EJWJEF UIF OVNFSBUPS BOE UIF EFOPNJOBUPS PG B GSBDUJPO CZ UIF TBNF OPO [FSP OVNCFS UIF SFTVMU JT BO FRVJWBMFOU GSBDUJPO FY B NQM E

4

8SJUF GPVS GSBDUJPOT UIBU BSF FRVJWBMFOU UP

× × × × =— =— =— =— — — — — × × × × 5IF GSBDUJPOT — — — BOE — BSF FRVJWBMFOU UP — 5IFZ BSF FRVJWBMFOU GSBDUJPOT

This Way *O &YBNQMF UIF OVNFSBUPST BSF NVMUJQMFT PG BOE UIF EFOPNJOBUPST BSe multiples PG :PV NBZ GJOE JU FBTJFS UP TLJQ DPVOU UP GJOE FRVJWBMFOU GSBDUJPOT

— — — — —

&BDI QBJS PG NVMUJQMFT GPSNT B GSBDUJPO FRVJWBMFOU UP UXP GJGUIT

G'Oot Tto

K+N.O/W 7Ã

.VMUJQMZJOH UP 'JOE &RVJWBMFOU 'SBDUJPOT .VMUJQMJDBUJPO CZ EPFT OPU DIBOHF UIF WBMVF PG B OVNCFS .VMUJQMZJOH UIF OVNFSBUPS BOE EFOPNJOBUPS PG B GSBDUJPO CZ UIF TBNF OVNCFS JT UIF TBNF BT NVMUJQMZJOH UIF GSBDUJPO CZ XXX 48BEWBOUBHF DPN

&RVJWBMFOU 'SBDUJPOT

8SJUJOH &RVJWBMFOU 'SBDUJPOT FY B NQM E

5

8SJUF UXP FRVJWBMFOU GSBDUJPOT GPS FBDI HJWFO GSBDUJPO

5S Z *U

B �

8IFO GJOEJOH B NJTTJOH OVNFSBUPS PS EFOPNJOBUPS BT JO &YBNQMF TFUUJOH VQ UIF QSPCMFN GJSTU XJMM IFMQ ZPV WJTVBMJ[F JU 4JODF > ZPV NVTU NVMUJQMZ

×

×

=—=— — ×

.VMUJQMZ UIF OVNFSBUPS BOE EFOPNJOBUPS CZ UIF TBNF OVNCFS

This Way

=—=— — ×

5IF GSBDUJPOT — BOE — BSF FRVJWBMFOU UP —

C �

×

÷

.VMUJQMZ UIF OVNFSBUPS BOE EFOPNJOBUPS CZ UIF TBNF OVNCFS

=—=— — ×

*O UIF QSPCMFN CFMPX < TP ZPV NVTU EJWJEF

%JWJEF UIF OVNFSBUPS BOE EFOPNJOBUPS CZ UIF TBNF OVNCFS

=—=— — ÷

=— — ÷

5IF GSBDUJPOT — BOE — BSF FRVJWBMFOU UP —

× =— — ×

÷

FY B NQM E

6

8IBU EFOPNJOBUPS XPVME NBLF UIFTF GSBDUJPOT FRVJWBMFOU � = � 5IJOL 8IBU OVNCFS NVMUJQMJFE CZ JT FRVBM UP × = 50

×

.VMUJQMZ UIF EFOPNJOBUPS CZ UIF TBNF OVNCFS.

4P — = — 5IF EFOPNJOBUPS NBLFT UIF GSBDUJPOT FRVJWBMFOU

G'Oot Tto

K+N.O/W 7!Ã

%JWJEJOH UP 'JOE &RVJWBMFOU 'SBDUJPOT *G ZPV DBOOPU EJWJEF UIF OVNFSBUPS BOE %JWJEJOH UIF OVNFSBUPS BOE EFOPNJOBUPS EFOPNJOBUPS PG B GSBDUJPO CZ UIF TBNF PG B GSBDUJPO CZ UIF TBNF OVNCFS JT UIF OVNCFS UIF GSBDUJPO JT JO TJNQMFTU GPSN TBNF BT EJWJEJOH CZ

÷

=—=— — ÷

310

=—=— — ×

BOE IBWF OP DPNNPO GBDUPST —

Fractions and Mixed Numbers /6.#&34 "/% 01&3"5*0/4

4JNQMFTU 'PSN PG B 'SBDUJPO " GSBDUJPO JT JO TJNQMFTU GPSm XIFO UIF OVNFSBUPS BOE EFOPNJOBUPS IBWF OP DPNNPO GBDUPS PUIFS UIBO POF 5P XSJUF B GSBDUJPO JO TJNQMFTU GPSN EJWJEF UIF OVNFSBUPS BOE EFOPNJOBUPS SFQFBUFEMZ CZ UIFJS DPNNPO GBDUPST PS PODF CZ UIFJS HSFBUFTU DPNNPO GBDUPS &Y "NQM &

7

8SJUF JO TJNQMFTU GPSN NFU I P E

'JOE UIF ($' PG BOE 24.

-JTU GBDUPST PG BOE 24.

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5IF ($' JT %JWJEF UIF OVNFSBUPS BOE EFOPNJOBUPS CZ 6.

'PS IFMQ XJUI GJOEJOH UIF ($' HP UP 'BDUPST PO QBHF

÷ =—=— — ÷

5IF OVNCFST BOE IBWF OP DPNNPO GBDUPST PUIFS UIBO 5IF GSBDUJPO JT JO TJNQMFTU GPSN XSJUUFO JO TJNQMFTU GPSN JT — 5IF GSBDUJPO —

NFU I P E

'JOE B DPNNPO GBDUPS PG BOE #PUI OVNCFST BSF EJWJTJCMF CZ

÷

÷

=—=— — ÷

%JWJEF UIF OVNFSBUPS BOE EFOPNJOBUPS CZ 5IF OVNCFST BOE TIBSF B DPNNPO GBDUPS %JWJEF UIF OVNFSBUPS BOE EFOPNJOBUPS CZ 3.

=—=— — ÷

5IF OVNCFST BOE IBWF OP DPNNPO GBDUPST PUIFS UIBO 5IF GSBDUJPO JT JO TJNQMFTU GPSN XSJUUFO JO TJNQMFTU GPSN JT — 5IF GSBDUJPO —

&Y "NQM &

8

8SJUF JO TJNQMFTU GPSN

'JOE UIF TJNQMFTU GPSN PG UIF GSBDUJPO QBSU PG UIF NJYFE OVNCFS ÷ =— =— 5IF ($' PG BOE JT TP —

÷ 5IF NJYFE OVNCFS — XSJUUFO JO TJNQMFTU GPSN JT —

XXX 48BEWBOUBHF DPN

4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

311

"EEJOH 'SBDUJPOT XJUI -JLF %FOPNJOBUPST .PEFMJOH "EEJOH 'SBDUJPOT XJUI -JLF %FOPNJOBUPST :PV DBO NPEFM BEEJOH GSBDUJPOT XJUI MJLF EFOPNJOBUPST VTJOH GSBDUJPO TUSJQT PS EJBHSBNT

1

&Y "NQM & 4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

6TF GSBDUJPO TUSJQT UP NPEFM +

—

—

QMVT — .PEFM —

—

$PVOU UIF OVNCFS PG GJGUIT 5IFSF BSF GJGUIT

—

—

—

= — + — —

2

&Y "NQM &

—

6TF GSBDUJPO TUSJQT UP NPEFM +

4U FQ

.PEFM — QMVT —

—

$PVOU UIF OVNCFS PG FJHIUIT 5IFSF BSF FJHIUIT

—

— + — = —

Need More

HELP

'PS IFMQ XJUI TJNQMJGZJOH GSBDUJPOT HP UP 4JNQMFTU 'orm of a 'SBDUJPO on page 311.

4U FQ

—

—

—

—

—

—

—

—

4JNQMJGZ

—

÷ = — — ÷

—

—

—

—

—

—

—

+ — = — —

&Y "NQM &

3

6TF GSBDUJPO TUSJQT UP NPEFM +

.PEFM — QMVT —

—

—

5IFSF BSF GPVSUIT

+ — PS — 4JNQMJGZ — = —

= — — + —

318

—

—

—

— =

—

—

—

Fractions and Mixed Numbers /6.#&34 "/% 01&3"5*0/4 FY B NQM F

4

" SPVOE QJ[[B JT EJWJEFE JOUP UFO FRVBM TMJDFT #FO FBUT TMJDFT PS PG UIF QJ[[B 4POJB FBUT TMJDFT PS PG UIF QJ[[B 8IBU GSBDUJPO SFQSFTFOUT UIF QBSU PG UIF QJ[[B UIBU XBT FBUFO 4U FQ

%SBX B EJBHSBN UP SFQSFTFOU UIF TJUVBUJPO 4IBEF UIF QBSU PG UIF QJ[[B UIBU XBT FBUFO

4U FQ

$PVOU UIF OVNCFS PG TMJDFT FBUFO

#FO 4POJB

+—=— — 4U FQ

8SJUF UIF TVN JO TJNQMFTU GPSN ÷ %JWJEF UIF OVNFSBUPS BOE EFOPNJOBUPS CZ — = — ÷

#FO BOE 4POJB BUF POF IBMG PG UIF QJ[[B

"EEJOH 'SBDUJPOT -JLF GSBDUJPOT BSF GSBDUJPOT XJUI UIF TBNF EFOPNJOBUPS 5P BEE MJLF GSBDUJPOT BEE UIF OVNFSBUPST BOE QVU UIF TVN PWFS UIF DPNNPO EFOPNJOBUPS FY B NQM F

5

"EE +

5IF GSBDUJPOT BSF MJLF GSBDUJPOT "EE UIF OVNFSBUPST + =

+—=— —

G'OoTt TtOo

K+N.O/W 7Ã "EEJOH 'SBDUJPOT XJUI -JLF %FOPNJOBUPST t "EE UIF OVNFSBUPST

XXX 48BEWBOUBHF DPN

t

8SJUF UIF TVN PWFS UIFJS DPNNPO EFOPNJOBUPS

t

4JNQMJGZ JG QPTTJCMF

+— — ÷ =— ÷ =—

=—

319

"EEJOH 'SBDUJPOT XJUI -JLF %FOPNJOBUPST

"EEJOH 'SBDUJPOT FY B NQM E

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3FNFNCFS UIF OVNFSBUPS JT BCPWF UIF GSBDUJPO CBS *U JT UIF OVNCFS PG QBSUT CFJOH DPOTJEFSFE 5IF EFOPNJOBUPS JT CFMPX UIF GSBDUJPO CBS *U JT the tPUBM OVNCFS PG FRVBM QBSUT

6

"EE + +

"EE UIF OVNFSBUPST PG UIF MJLF GSBDUJPOT

+ + =

1VU UIF TVN PWFS UIF EFOPNJOBUPS

—

+ — = — + — — FY B NQM E

7

"EE +

"EE UIF OVNFSBUPST PG UIF MJLF GSBDUJPOT

+ =

1VU UIF TVN PWFS UIF EFOPNJOBUPS

—

5IF TVN JT BO JNQSPQFS GSBDUJPO 8SJUF UIF JNQSPQFS GSBDUJPO BT B NJYFE OVNCFS

= — = — + —

4JNQMJGZ UIF GSBDUJPO %JWJEF UIF OVNFSBUPS BOE EFOPNJOBUPS CZ UIFJS ($'

= — = — — ÷

÷

= — + — —

FY B NQM E

8

"EE + +

"EE UIF OVNFSBUPST PG UIF MJLF GSBDUJPOT

+ + =

1VU UIF TVN PWFS UIF EFOPNJOBUPS

—

5IF TVN JT BO JNQSPQFS GSBDUJPO 8SJUF UIF JNQSPQFS GSBDUJPO BT B NJYFE OVNCFS NFU I P E

= —

— = — + —

NFU I P E

%JWJEF UIF OVNFSBUPS CZ UIF EFOPNJOBUPS

320

3 = —

Fractions and Mixed Numbers /6.#&34 "/% 01&3"5*0/4

"EEJOH .JYFE /VNCFST 5P BEE NJYFE OVNCFST BEE UIF XIPMF OVNCFS QBSUT BOE UIF GSBDUJPO QBSUT TFQBSBUFMZ &Y "NQM &

9

"EE � + �

8SJUF UIF QSPCMFN JO WFSUJDBM GPSN "EE UIF XIPMF OVNCFS QBSUT "EE UIF OVNFSBUPST PG UIF MJLF GSBDUJPOT

— + —

—

4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

— + — = — &Y "NQM &

10

QPVOE PG "NFSJDBO DIFFTF BOE 'BSSBI CPVHIU � QPVOET PG 4XJTT DIFFTF �

� QPVOET PG QSPWPMPOF DIFFTF )PX NVDI DIFFTF EJE TIF CVZ JO BMM

"EE UIF BNPVOUT PG DIFFTF 8SJUF UIF QSPCMFN JO WFSUJDBM GPSN "EE UIF XIPMF OVNCFS QBSUT "EE UIF OVNFSBUPST PG UIF MJLF GSBDUJPOT 4JNQMJGZ UIF JNQSPQFS GSBDUJPO JO UIF TVN

'BSSBI CPVHIU — QPVOET PG DIFFTF JO BMM

—

— + — — = + — + — = + + — = —

G'Oot Tto

K+N.O/W 7Ã "EEJOH .JYFE /VNCFST t "EE UIF XIPMF OVNCFS QBSUT t "EE UIF GSBDUJPO QBSUT t 4JNQMJGZ UIF TVN JG OFDFTTBSZ

XXX 48BEWBOUBHF DPN

321

"EEJOH 'SBDUJPOT XJUI 6OMJLF %FOPNJOBUPST .PEFMJOH "EEJUJPO XJUI 6OMJLF 'SBDUJPOT 4PNFUJNFT ZPV OFFE UP BEE GSBDUJPOT XJUI EJGGFSFOU EFOPNJOBUPST PS VOMJLF GSBDUJPOT 5IFTF GSBDUJPOT EP OPU IBWF B DPNNPO EFOPNJOBUPS TP XF TBZ UIFZ IBWF VOMJLF EFOPNJOBUPST :PV DBO VTF GSBDUJPO TUSJQT UP NPEFM BEEJOH GSBDUJPOT XJUI VOMJLF EFOPNJOBUPST &Y "NQM &

7ATCH /UT

!

8IFO ZPV MJOF VQ UIF FRVJWBMFOU GSBDUJPO TUSJQT NBLF DFSUBJO UIBU UIFZ BMJHO XJUI UIF MFGU FEHF PG UIF TUSJQT BCPWF UIFN

1

6TF GSBDUJPO TUSJQT UP NPEFM +

.FU I P E

.PEFM UIF BEEJUJPO XJUI GSBDUJPO TUSJQT 6TF FJHIUIT GSBDUJPO TUSJQT UP SFOBNF — +—=— $PVOU UIF OVNCFS PG FJHIUIT. —

—

—

—

+

—

—

+

—

—

—

—

—

—

—

—

+

=

.FU I P E

.PEFM UIF BEEJUJPO XJUIPVU GPMEJOH GSBDUJPO TUSJQT .PEFM UIF TVN VTJOH FJHIUIT TUSJQT $PVOU UIF OVNCFS PG FJHIUIT

—

—

—

—

—

—

—

—

—

—

—

—

—

—

=

—

—

—

+—=— —

&Y "NQM & 4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

2

6TF GSBDUJPO TUSJQT UP NPEFM + .PEFM UIF BEEJUJPO XJUI GSBDUJPO TUSJQT 6TF UXFMGUIT GSBDUJPO TUSJQT UP SFOBNF CPUI GSBDUJPOT

—

—

—

—

+

—

—

—

—

—

—

— — — — — — — — — — — — — — — — — — — +

—

322

—

=—

—

—

—

Fractions and Mixed Numbers /6.#&34 "/% 01&3"5*0/4 4JNQMJGZ UIF JNQSPQFS GSBDUJPO

—

8SJUF UIF TVN BT B NJYFE OVNCFS

— — — — — — — — — — — — — — — — — — — +

—

= —

= — + — —

—

+

= —

6TJOH &RVJWBMFOU 'SBDUJPOT UP "EE 0UIFS XBZT UP BEE VOMJLF GSBDUJPOT JODMVEF GJOEJOH UIF MFBTU DPNNPO EFOPNJOBUPS -$% PG UIF GSBDUJPOT BOE ESBXJOH B EJBHSBN FY B NQM E

3

"EE +

Need More

N&U I P E

8SJUF FRVJWBMFOU GSBDUJPOT GPS — BOE — VTJOH UIFJS MFBTU DPNNPO EFOPNJOBUPS -$%

5IF -$% PG UXP GSBDUJPOT JT JUT -$. MFBTU DPNNPO NVMUJQMF

.VMUJQMFT PG y .VMUJQMFT PG y

'JOE UIF -$. PG BOE

-$. =

3FOBNF — VTJOH UIF EFOPNJOBUPS

HELP

'PS IFMQ XJUI MFBTU DPNNPO NVMUJQMFT BOE MFBTU DPNNPO EFOPNJOBUPST HP UP .VMUJQMFT PO QBHF

= — × = — — ×

/PX UIF GSBDUJPOT BSF MJLF "EE UIF OVNFSBUPST BOE QVU UIF TVN PWFS UIF EFOPNJOBUPS

+

= — + — = — —

= — + — — N&U I P E

6TF B EJBHSBN 4U &Q

%SBX B EJBHSBN UP NPEFM UIF QSPCMFN

4U &Q

$IBOHF UIF NPEFM GPS — JOUP UFOUIT

— +

+

—

4U &Q

=— —

—

$PVOU UIF TFDUJPOT

+ — = — —

XXX 48BEWBOUBHF DPN

323

8PSE 1SPCMFNT 4PMWJOH 'SBDUJPO BOE .JYFE /VNCFS 8PSE 1SPCMFNT Need More

HELP

'PS IFMQ TPMWJOH XPSE QSPCMFNT HP UP 1SPCMFN 4PMWJOH TUBSUJOH PO Q :PV XJMM GJOE B GPVS TUFQ QSPCMFN TPMWJOH QSPDFTT BOE TUSBUFHJFT GPS QSPCMFN TPMWJOH 5IF $IFDL JO Example 1 VTFT UIF TUSBUFHZ 6TF -PHJDBM 3FBTPOJOH GPVOE PO QBHF

5IFSF JT NPSF UIBO POF XBZ UP TPMWF NBOZ XPSE QSPCMFNT 'JOE UIF NFUIPE UIBU XPSLT CFTU GPS ZPV :PV NBZ GJOE JU IFMQGVM UP GPMMPX UIF TUFQT JO UIF (PU UP ,OPX CPY PO UIF GBDJOH QBHF &YBNQMFT BOE TIPX B GPSNBM BQQMJDBUJPO PG UIF TUFQT &Y "NQM &

1

4JNPO DIFDLFE PVU B CPPL GSPN UIF MJCSBSZ )F SFBE PG UIF CPPL PO 5VFTEBZ PG

UIF CPPL PO 8FEOFTEBZ BOE PG UIF CPPL PO 5IVSTEBZ )PX NVDI PG UIF CPPL IBT

4JNPO SFBE JO BMM 3&"%

3FBE UIF QSPCMFN DBSFGVMMZ *EFOUJGZ XIBU ZPV LOPX BOE XIBU ZPV OFFE UP GJOE PVU ,OPX )F SFBE — — BOE — PG UIF CPPL /FFE UP ,OPX 5IF GSBDUJPO PG UIF CPPL IF SFBE BMUPHFUIFS

1-"/

5IJOL BCPVU IPX ZPV DBO GJOE UIF JOGPSNBUJPO ZPV OFFE UP LOPX 5IF QISBTF iJO BMMw VTVBMMZ NFBOT BEEJUJPO :PV DPVME BEE UIF UISFF GSBDUJPOT

40-7&

8SJUF UIF QSPCMFN JO WFSUJDBM GPSN 'JOE UIF DPNNPO EFOPNJOBUPS BOE BEE

=— —

=— —

8SJUF FRVJWBMFOU GSBDUJPOT VTJOH UIF -$%

=— — =— +—

"EE UIF OVNFSBUPST 4JNQMJGZ UIF TVN $)&$,

5IF -$% PG BOE JT 30.

%JE * BOTXFS UIF RVFTUJPO *T NZ BOTXFS JT SFBTPOBCMF 4JNPO IBT OPU SFBE UIF FOUJSF CPPL TP UIF BOTXFS NVTU CF MFTT UIBO 5IF GSBDUJPOT —

4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

358

BOE — BSF CPUI B MJUUMF MFTT UIBO — TP UIFZ SFQSFTFOU OPU RVJUF — PG UIF CPPL 5IF

GSBDUJPO — JT B MJUUMF HSFBUFS UIBO — 4P 4JNPO IBT SFBE B MJUUMF PWFS IBMG PG UIF CPPL

4JNPO IBT SFBE — PG UIF CPPL

4JODF — = — UIFO — JT B SFBTPOBCMF BOTXFS

Fractions and Mixed Numbers /6.#&34 "/% 01&3"5*0/4 FY B NQM E

2

"MFKBOESP BOE )FBUIFS NBEF DBSECPBSE EFDPSBUJPOT "MFKBOESP NBEF B EFDPSBUJPO

JODIFT MPOH )FBUIFS NBEF B EFDPSBUJPO JODIFT MPOH )PX NVDI MPOHFS XBT

"MFKBOESP T EFDPSBUJPO UIBO )FBUIFS T 3&"%

1-"/

,OPX "MFKBOESP T EFDPSBUJPO XBT — JODIFT )FBUIFS T XBT — JODIFT /FFE UP ,OPX 5IF EJGGFSFODF JO UIF MFOHUIT PG UIF EFDPSBUJPOT

5IFSF BSF UXP XBZT UP TPMWF UIF QSPCMFN N&U I P %

N&U I P %

4VCUSBDU UIF XIPMF OVNCFS QBSUT of UIF NJYFE OVNCFST. 5IFO TVCUSBDU UIF GSBDUJPOT.

8SJUF UIF NJYFE OVNCFST BT JNQSPQFS GSBDUJPOT 5IFO TVCUSBDU

N&U I P %

N&U I P %

— − — — = —

− — = — = — = — —

4P -7&

$I &$,

— = —

&TUJNBUF FBDI MFOHUI 5IFO GJOE BO FTUJNBUFE EJGGFSFODF

— JT BCPVU BOE — JT BCPVU 4VCUSBDU − = 4JODF — JT BDUVBMMZ B MJUUMF NPSF UIBO UIF BOTXFS XJMM CF MFTT UIBO TP — JT B SFBTPOBCMF BOTXFS

"MFKBOESP T EFDPSBUJPO XBT — JODIFT MPOHFS UIBO )FBUIFS T

Need More

HELP

'PS IFMQ XJUI FTUJNBUJOH HP UP &TUJNBUJOH UP 1SFEJDU PS $IFDL PO QBHF

G'Oot Tto

K+N.O/W 7Ã 1SPCMFN 4PMWJOH 1SPDFTT 5P TPMWF B XPSE QSPCMFN ZPV DBO VTF UIJT QSPDFTT 3FBE UIF QSPCMFN DBSFGVMMZ *EFOUJGZ XIBU ZPV LOPX BOE XIBU ZPV OFFE UP GJOE PVU 1MBO IPX UP GJOE UIF TPMVUJPO 4PMWF UIF QSPCMFN VTJOH ZPVS QMBO $IFDL ZPVS XPSL BOE BTL JG ZPVS BOTXFS NBLFT TFOTF

XXX 48BEWBOUBHF DPN

359

8PSE 1SPCMFNT

4PMWJOH 'SBDUJPO BOE .JYFE /VNCFS 8PSE 1SPCMFNT FY B NQM E

7ATCH /UT

!

*O &YBNQMF UIF QSPCMFN BTLT IPX NBOZ BQQMFT BSF SFE TP ZPVS BOTXFS TIPVME CF B XIPMF OVNCFS 'PS UIF DIFDL ZPV DPVME DPOGJSN UIBU JT UXP UIJSET PG

÷

3

" CPXM PG BQQMFT DPOUBJOT BQQMFT 5XP UIJSET PG UIF BQQMFT BSF SFE )PX NBOZ BQQMFT BSF SFE 3FB E

,OPX BQQMFT JO BMM BOE — BSF SFE /FFE UP ,OPX )PX NBOZ BSF SFE

1M B /

.VMUJQMZ UP GJOE — PG BQQMFT

4P M W F

.VMUJQMZ

× = — × — —

=—=— — ÷

×— =—

= D I FD L

6TF MPHJDBM SFBTPOJOH BOE NFOUBM NBUI *G ZPV BSSBOHFE BQQMFT JOUP FRVBM HSPVQT UIFSF XPVME CF JO FBDI HSPVQ 5XP UIJSET PG UIF BQQMFT XPVME CF × = 5IF BOTXFS JT SFBTPOBCMF

5FO PG UIF BQQMFT BSF SFE

FY B NQM E

4

" QJ[[B NBLFS IBT DVQT PG DIFFTF UP EJWJEF FRVBMMZ BNPOH QJ[[BT )PX NBOZ

DVQT PG DIFFTF XJMM CF PO FBDI QJ[[B 3FB E

,OPX — DVQT PG DIFFTF QJ[[BT TBNF BNPVOU PG DIFFTF PO FBDI QJ[[B

/FFE UP ,OPX "NPVOU PG DIFFTF PO FBDI QJ[[B 1M B /

%JWJEF UP GJOE UIF BNPVOU PG DIFFTF PO FBDI QJ[[B

4P M W F

%JWJEF

D I FD L

— ÷ =— ÷—

=—×—

=— = —

5IFSF BSF — DVQT PG DIFFTF PO FBDI QJ[[B

360

"EE UP DIFDL ZPVS BOTXFS — + — + —

= — = + —

= — UIF PSJHJOBM BNPVOU PG DIFFTF

Fractions and Mixed Numbers /6.#&34 "/% 01&3"5*0/4 &Y "NQM &

5

" CPY DPOUBJOT NVGGJOT 0OF TJYUI PG UIF NVGGJOT BSF CBOBOB NVGGJOT 0OF IBMG PG UIF CBOBOB NVGGJOT DPOUBJO OVUT )PX NBOZ NVGGJOT BSF CBOBOB NVGGJOT XJUI OVUT 3FB E

QM B O

,OPX NVGGJOT — BSF CBOBOB — PG UIF CBOBOB NVGGJOT DPOUBJO OVUT /FFE UP ,OPX )PX NBOZ BSF CBOBOB NVGGJOT XJUI OVUT 4U FQ

.VMUJQMZ UP GJOE UIF OVNCFS PG CBOBOB NVGGJOT

4U FQ

.VMUJQMZ UIF OVNCFS PG CBOBOB NVGGJOT CZ —

4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

4U FQ — × — = $)F$, 6TF BOPUIFS NFUIPE UP DIFDL .VMUJQMZ — × — = — UP GJOE UIF GSBDUJPO PG UIF × = NVGGJOT BSF CBOBOB NVGGJOT UIBU BSF CBOBOB OVU NVGGJOT 5IFO NVMUJQMZ —

4P M W F 4U FQ — × — = 4

with nuts ✔ 5XP PG UIF NVGGJOT BSF CBOBOB NVGGJOT XJUI OVUT

&Y "NQM &

6

3BKBU XBMLFE NJMFT .BSJBO XBMLFE BT GBS BT 3BKBU )PX NBOZ NPSF NJMFT EJE

3BKBU XBML UIBO .BSJBO 3FB E

,OPX 3BKBU XBMLFE — NJMFT .BSJBO XBMLFE — PG UIF EJTUBODF 3BKBU XBMLFE /FFE UP ,OPX )PX NVDI GBSUIFS 3BKBU XBMLFE

QM B O 4U FQ

.VMUJQMZ UP GJOE UIF EJTUBODF .BSJBO XBMLFE

4P M W F

4U FQ — PG — = — × — =— NJMFT = —

$)F$,

4U FQ

4U FQ

4VCUSBDU UIJT EJTUBODF GSPN UIF EJTUBODF 3BKBU XBMLFE

— − — =—−— =— NJMFT = —

=—−—

3PVOE BOE FTUJNBUF UP DIFDL 3BKBU XBMLFE BCPVU NJMFT 5XP GJGUIT JT BCPVU POF IBMG 4P .BSJBO XBMLFE BCPVU POF IBMG PG PS NJMFT 4JODF − = 3BKBU XBMLFE BCPVU NJMFT NPSF UIBO .BSJBO 5IF BOTXFS JT SFBTPOBCMF

NJMFT NPSF UIBO .BSJBO 3BKBU XBMLFE —

XXX 48BEWBOUBHF DPN

361

.FBTVSFNFOU Measurement

"SDIJUFDUT DPOWFSU QFSJNFUFS BOE BSFB NFBTVSFNFOUT UP NBLF TDBMF ESBXJOHT BOE NPEFMT PG UIF CVJMEJOHT UIBU UIFZ BSF EFTJHOJOH

474

6 4 $VTUPNBSZ 4ZTUFN .FUSJD 4ZTUFN 1FSJNFUFS BOE "SFB 4VSGBDF "SFB BOE 7PMVNF 4DJFOUJGJD /PUBUJPO

www.swadvantage.com

475

$POWFSUJOH 6 4 $VTUPNBSZ BOE .FUSJD $POWFSUJOH #FUXFFO 6 4 $VTUPNBSZ BOE .FUSJD .FBTVSFT Need More

HELP

3FNFNCFS XIFO XSJUJOH B DPOWFSTJPO GBDUPS VTF UIF VOJU UIBt is CFJOH DIBOHFE BT UIF EFOPNJOBUPS 'PS NPSF IFMQ XJUI DPOWFSTJPO GBDUPST HP UP %JNFOTJPOBM "OBMZTJT PO QBHF

.FBTVSFNFOUT DBO CF FYQSFTTFE JO UIF 6 4 DVTUPNBSZ TZTUFN PS JO UIF NFUSJD TZTUFN :PV OFFE UP VTF B DPOWFSTJPO GBDUPS UP DPOWFSU CFUXFFO UIF UXP TZTUFNT *O UIFTF DPOWFSTJPOT JU EPFT OPU NBUUFS JG ZPV BSF HPJOH GSPN TNBMM VOJUT UP MBSHF PS MBSHF VOJUT UP TNBMM 5IF DPOWFSTJPO GBDUPS BMMPXT GPS UIF EJGGFSFODF JO VOJU TJ[F )PXFWFS EVF UP EJGGFSFODFT JO UIF UXP TZTUFNT BMM DPOWFSTJPOT CFUXFFO UIF 6.S. cVTUPNBSZ BOE NFUSJD NFBTVSFT HJWF BQQSPYJNBUF NFBTVSFT 0OF NFUFS JT BQQSPYJNBUFMZ FRVBM UP ≈ ZBSET 5IF DPOWFSTJPO GBDUPS ZE

GPr cIBOHJOH NFUFST UP ZBSET JT — N

8IFO ZPV NVMUJQMZ CZ UIF DPOWFSTJPO GBDUPS UIF VOJU UIBU ZPV BSF DPOWFSUJOH GSPm cBODFMT PVU 5IF VOJU ZPV BSF DPOWFSUJOH UP SFNBJOT ZE

N × — ≈ × ZE = ZE N N ≈ ZE FY B NQM E

1

NJMFT � LJMPNFUFST 45&1

8SJUF UIF DPOWFSTJPO GBDUPS 6TF NJMF ≈ LJMPNFUFST LN NJ

$POWFSTJPO GBDUPS — 45&1

6TF UIF DPOWFSTJPO GBDUPS UP DPOWFSU NJMFT UP LJMPNFUFST NJ

LN

× — ≈ LN — NJ

G'Oot Tto

NJMFT ≈ LJMPNFUFST

+KN.O/W 7!à 6 4 $VTUPNBSZ BOE .FUSJD &RVJWBMFOUT‰-FOHUI 6 4 $VTUPNBSZ UP .FUSJD

.FUSJD UP 6 4 $VTUPNBSZ

JO ≈ DN

NN ≈ JO

GU ≈ DN

DN ≈ JO

ZE ≈ DN

N ≈ GU ≈ ZE

NJ ≈ LN

LN ≈ NJ

514

Metric System

.FB T V S F.FO U

$POWFSUJOH $BQBDJUZ 5P DPOWFSU VOJUT PG DBQBDJUZ GJSTU XSJUF B DPOWFSTJPO GBDUPS 5IFO NVMUJQMZ UIF VOJUT ZPV XBOU UP DIBOHF CZ UIF DPOWFSTJPO GBDUPS &Y "NQM &

MJUFST = 45&1

2 GMVJE PVODFT

4&"3$)

8SJUF UIF DPOWFSTJPO GBDUPS 6TF MJUFS ≈ GMVJE PVODFT GM P[ $POWFSTJPO GBDUPS — -

45&1

6TF UIF DPOWFSTJPO GBDUPS UP DPOWFSU MJUFST UP GMVJE PVODFT -

GM P[

× — ≈ × GM P[ ≈ GM P[ — -

5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

MJUFST ≈ GMVJE PVODFT

&Y "NQM &

DVQT = 45&1

3

7ATCH /UT

MJUFST

8SJUF UIF DPOWFSTJPO GBDUPS 6TF DVQ ≈ N- N D

$POWFSTJPO GBDUPS — 45&1

6TF UIF DPOWFSTJPO GBDUPS UP DPOWFSU DVQT UP NJMMJMJUFST D

N-

×— ≈ × ≈ N— D 45&1

!

*O &YBNQMF 4Uep 3, ZPV EP OPU OFFE UP GJOE B DPOWFSTJPO GBDUPS CFDBVTF JT B QPXFS PG %JWJEF CZ CZ NPWJOH UIF EFDJNBM QPJOU UISFF QMBDFT UP UIF MFGU

$POWFSU NJMMJMJUFST UP MJUFST 6TF - ≈ N- N- ÷ N- QFS - ≈ -

DVQT ≈ MJUFST

G'Oot Tto

K+N.O/W 7à 6 4 $VTUPNBSZ BOE .FUSJD &RVJWBMFOUT‰$BQBDJUZ 6 4 $VTUPNBSZ UP .FUSJD

.FUSJD UP 6 4 $VTUPNBSZ

GM P[ ≈ N-

N- ≈ GM P[

DVQ ≈ N-

D- GM P[

QU ≈ N-

- ≈ GM P[ ≈ RU

RU ≈ -

L- ≈ HBM

HBM ≈ -

XXX 48BEWBOUBHF DPN

$POWFSUJOH 6 4 $VTUPNBSZ BOE .FUSJD

$POWFSUJOH .BTT BOE 8FJHIU 5P DPOWFSU VOJUT PG XFJHIU BOE NBTT GJSTU XSJUF B DPOWFSTJPO GBDUPS 5IFO NVMUJQMZ UIF VOJUT ZPV XBOU UP DIBOHF CZ UIF DPOWFSTJPO GBDUPS FY B NQM E

4

" EPDUPS OFFET UP DBMDVMBUF IPX NBOZ NJMMJHSBNT PG B DFSUBJO NFEJDJOF UP HJWF Uo a paUJFOU 5IF EPTBHF JT DBMDVMBUFE CBTFE PO UIF NBTT PG UIF QFSTPO UBLJOH JU *G UIF QBUJFOU XFJHIT MC XIBU JT UIF QBUJFOU T NBTT JO LJMPHSBNT 45&1

8SJUF UIF DPOWFSTJPO GBDUPS 6TF QPVOE ≈ LJMPHSBNT LH MC

$POWFSTJPO GBDUPS — 45&1

6TF UIF DPOWFSTJPO GBDUPS UP DPOWFSU QPVOET UP LJMPHSBNT LH

MC

× — ≈ × LH ≈ LH — MC 5IF QBUJFOU T NBTT JT BQQSPYJNBUFMZ FRVBM UP LJMPHSBNT

FY B NQM E

5

" CPVMEFS IBT B NBTT PG NFUSJD UPOT )PX NBOZ QPVOET EPFT UIF CPVMEFS XFJHI 45&1

8SJUF UIF DPOWFSTJPO GBDUPS 6TF NFUSJD UPO ≈ QPVOET MC U

$POWFSTJPO GBDUPS — 45&1

6TF UIF DPOWFSTJPO GBDUPS UP DPOWFSU NFUSJD UPOT UP QPVOET U

MC

× — ≈ × MC ≈ MC — U

'GoOt Tto

+KN.O/W 7!Ã

5IF CPVMEFS XFJHIT BQQSPYJNBUFMZ QPVOET

6 4 $VTUPNBSZ BOE .FUSJD &RVJWBMFOUT‰8FJHIU BOE .BTT 6 4 $VTUPNBSZ UP .FUSJD

.FUSJD UP 6 4 $VTUPNBSZ

P[ ≈ H

NH ≈ P[

MC ≈ LH

H ≈ P[

TIPSU UPO ≈ LH ≈ NFUSJD UPO

LH ≈ P[ ≈ C NFUSJD UPO ≈ MC ≈ TIPSU UPO

516

Metric System

.FB T V S F.FO U

4PMWJOH 1SPCMFNT XJUI 6OJU $POWFSTJPOT 3FNFNCFS XIFO TPMWJOH QSPCMFNT UIBU SFRVJSF B DPOWFSTJPO CFUXFFO 6 4 DVTUPNBSZ BOE NFUSJD TZTUFNT UIF BOTXFS JT BMNPTU BMXBZT BO BQQSPYJNBUF NFBTVSF :PV NBZ OFFE UP SPVOE UIF FTUJNBUF VQ PS EPXO UP BOTXFS UIF RVFTUJPO &Y "NQM &

6

BOJUB OFFET UP CVZ FOPVHI FOFSHZ ESJOL UP GJMM B HBMMPO DPPMFS DPNQMFUFMZ U he brBOE PG FOFSHZ ESJOL UIBU TIF CVZT JT TPME POMZ JO MJUFS CPUUMFT )PX NBOZ CPUUMFT PG FOFSHZ ESJOL TIPVME TIF CVZ #FGPSF ZPV DBO BOTXFS UIJT RVFTUJPO ZPV OFFE UP LOPX UIF OVNCFS PG MJUFST JO HBMMPOT 45&1

8SJUF B DPOWFSTJPO GBDUPS GPS HBMMPOT UP MJUFST 6TF HBMMPO ≈ MJUFST -

$POWFSTJPO GBDUPS — HBM 45&1

4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

6TF UIF DPOWFSTJPO GBDUPS UP DPOWFSU HBMMPOT UP MJUFST HBM

-

× — ≈ × - ≈ — HBM HBM ≈ 45&1

$PNQBSF UIF RVBOUJUJFT - CPUUMFT < - < - CPUUMFT

4JODF UIF DBQBDJUZ PG UIF DPPMFS JT - "OJUB TIPVME CVZ CPUUMFT JG TIF XBOUT UP GJMM UIF DPPMFS DPNQMFUFMZ

&Y "NQM &

7

)FDUPS IBT NFUFST PG SPQF )F OFFET GFFU PG SPQF UP NBSL PGG B TFDUJPO of a parkJOH MPU %PFT IF IBWF FOPVHI SPQF 45&1

8SJUF B DPOWFSTJPO GBDUPS GPS NFUFST UP GFFU 6TF NFUFS ≈ GFFU GU

$POWFSTJPO GBDUPS — N 45&1

6TF UIF DPOWFSTJPO GBDUPS UP DPOWFSU NFUFST UP GFFU N

GU

× — ≈ GU — N 5IF MFOHUI PG )FDUPS T SPQF JT GFFU 4JODF GFFU < GFFU )FDUPS EPFT OPU IBWF FOPVHI SPQF

XXX 48BEWBOUBHF DPN

517

(FPNFUSZ Geometry

-BOETDBQF BSUJTUT VTF DPOHSVFOU GJHVSFT XIFO EFTJHOJOH HBSEFOT UP NBLF UIFN NPSF WJTVBMMZ BQQFBMJOH

608

3FBTPOJOH BOE 1SPPG . . . . . . . . . . . . . . . . . . . . . . 2VBESJMBUFSBMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5SJBOHMFT BOE 1PMZHPOT . . . . . . . . . . . . . . . . . . . 4QBUJBM 3FBTPOJOH . . . . . . . . . . . . . . . . . . . . . . . . $JSDMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $POHSVFODF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4JNJMBSJUZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $PPSEJOBUF (FPNFUSZ . . . . . . . . . . . . . . . . . . . . . 5SBOTGPSNBUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . $POTUSVDUJPOT . . . . . . . . . . . . . . . . . . . . . . . . . . . .

www.swadvantage.com

609

1SPWJOH "OHMFT $POHSVFOU $MBTTJGZJOH "OHMFT :PV DBO DMBTTJGZ BOHMFT CZ UIFJS NFBTVSFT 5IF NFBTVSF PG B SJHIU BOHMe JT ¡ 5IF NFBTVSF PG B TUSBJHIU BOHMe JT ¡ 5IF NFBTVSF PG BO BDVUF BOHMe JT HSFBUFS UIBO ¡ BOE MFTT UIBO ¡ 5IF NFBTVSF PG BO PCUVTF BOHMe JT HSFBUFS UIBO ¡ BOE MFTT UIBO ¡ FY B NQM F

1

4UBUF XIFUIFS FBDI BOHMF BQQFBST UP CF BDVUF SJHIU PCUVTF PS TUSBJHIU

"

#

B ∠'&# -PPL GPS UIF SJHIU BOHMF TZNCPM ∠'&# JT B SJHIU BOHMF

C

8BZT UP

C ∠'

'

∠' BQQFBST UP CF BDVUF

REMEMBER

&

%

D ∠"

5P SFNFNCFS XIBU BO BDVUF BOHMF MPPLT MJLF UIJOL PG UIF DBQJUBM MFUUFS " BT JO "DVUF 5IF BOHMF BU UIF UPQ PG UIF MFUUFS " JT BDVUF

∠" BQQFBST UP CF PCUVTF E ∠'&% 1PJOUT % & BOE ' BSF DPMMJOFBS TP ∠'&% JT B TUSBJHIU BOHMF

"OHMF "EEJUJPO 1PTUVMBUF " QPTUVMBUe JT B TUBUFNFOU UIBU JT BDDFQUFE BT USVF XJUIPVU CFJOH QSPWFO 1PTUVMBUFT BSF POF PG UIF CBTJD UPPMT VTFE JO QSPPGT )FSF JT B CBTJD QPTUVMBUF BCPVU BOHMFT "OHMF "EEJUJPO 1PTUVMBUF *G QPJOU . JT JO UIF JOUFSJPS PG ∠+,- UIFO N∠+,. + N∠.,= N∠+,-

+ .

,

(05 50

,/08

5ZQFT PG "OHMFT

3JHIU

624

"DVUF

0CUVTF

4USBJHIU

-

Reasoning and Proof &Y "NQM &

(&0.&53:

2

'JOE FBDI BOHMF NFBTVSF

4&"3$)

0

B N∠-.0

/

1

¡ ¡

5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

¡

-

6TF UIF "OHMF "EEJUJPO 1PTUVMBUF

.

N∠-.0 = N∠-./ + N∠/.0 = ¡ + ¡

4VCTUJUVUF UIF HJWFO BOHMF NFBTVSFT

= ¡ N∠-.0 = ¡ C N∠0.1 6TF UIF "OHMF "EEJUJPO 1PTUVMBUF

N∠/.1 = N∠/.0 + N∠0.1

4VCTUJUVUF UIF HJWFO BOHMF NFBTVSFT

¡ = ¡ + N∠0.1

4VCUSBDU GSPN CPUI TJEFT

¡ = N∠0.1

N∠0.1 = ¡

*EFOUJGZJOH "OHMF 1BJST

7ATCH /UT

5XP BOHMFT BSF DPNQMFNFOUBSy JG UIF TVN PG UIFJS NFBTVSFT JT ¡ 5XP BOHMFT BSF TVQQMFNFOUBSy JG UIF TVN PG UIFJS NFBTVSFT JT ¡ 5XP BOHMFT UIBU IBWF UIF TBNF WFSUFY BOE TIBSF B TJEF CVU IBWF OP DPNNPO JOUFSJPS QPJOUT BSF DBMMFE BEKBDFOU BOHMFs *G UXP BOHMFT BSF BEKBDFOU BOE UIFJS FYUFSJPS TJEFT MJF PO B TUSBJHIU MJOF UIFO UIF BOHMFT GPSN B MJOFBS QBir &Y "NQM &

∠BOE ∠GPSN B MJOFBS QBJS

!

$PNQMFNFOUBSZ BOHMFT BOE TVQQMFNFOUBSZ BOHMFT EP OPU IBWF UP CF BEKBDFOU BOHMFT

3

6TF UIF EJBHSBN UP OBNF UIF BOHMF QBJST

9

B DPNQMFNFOUBSZ BOHMFT

:

∠97: BOE ∠:78 C TVQQMFNFOUBSZ BOHMFT

6

7

8

∠679 BOE ∠978 ∠67: BOE ∠:78 D BEKBDFOU BOHMFT ∠679 BOE ∠978 ∠67: BOE ∠:78 ∠679 BOE ∠97: E MJOFBS QBJST ∠679 BOE ∠978 ∠67: BOE ∠:78

XXX 48BEWBOUBHF DPN

625

1SPWJOH "OHMFT $POHSVFOU

5IFPSFNT BCPVU "OHMF 1BJST " UIFPSFm JT B TUBUFNFOU UIBU DBO CF QSPWFO UISPVHI EFEVDUJWF SFBTPOJOH 5IJT UIFPSFN TUBUFT UIBU B MJOFBS QBJS PG BOHMFT JT BMXBZT TVQQMFNFOUBSZ -JOFBS 1BJS 5IFPSFN *G UXP BOHMFT GPSN B MJOFBS QBJS UIFO UIFZ BSF TVQQMFNFOUBSZ

8IFO ZPV QSPWF B UIFPSFN ZPV NVTU KVTUJGZ FWFSZ TUBUFNFOU XJUI B EFGJOJUJPO QSPQFSUZ QPTUVMBUF PS QSFWJPVTMZ QSPWFE UIFPSFN " QBSBHSBQI QSPPf VTFT DPNQMFUF TFOUFODFT UP FYQMBJO BOE KVTUJGZ FBDI TUFQ FWFOUVBMMZ SFBDIJOH UIF EFTJSFE DPODMVTJPO &Y "NQM & 4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

4

8SJUF B QBSBHSBQI QSPPG (JWFO ∠JT TVQQMFNFOUBSZ UP ∠∠JT TVQQMFNFOUBSZ UP ∠1SPWF ∠âˆ

∠BOE ∠BSF TVQQMFNFOUBSZ #Z EFGJOJUJPO N∠+ N∠= ¡ 4JNJMBSMZ N∠+ N∠= ¡ 4VCTUJUVUJOH ZPV HFU N∠+ N∠= N∠+ N∠5IF 4VCUSBDUJPO 1SPQFSUZ PG &RVBMJUZ MFUT ZPV TVCUSBDU N∠GSPN CPUI TJEFT 4P N∠= N∠BOE ∠âˆ

5IJT QSPWFT UIF $POHSVFOU 4VQQMFNFOUT 5IFPSFN 5IF QSPPG PG UIF $POHSVFOU $PNQMFNFOUT 5IFPSFN JT TJNJMBS

(05 50

,/08

$POHSVFOU 4VQQMFNFOUT 5IFPSFN *G UXP BOHMFT BSF TVQQMFNFOUBSZ UP UIF TBNF BOHMF PS UP DPOHSVFOU BOHMFT UIFO UIF BOHMFT BSF DPOHSVFOU $POHSVFOU $PNQMFNFOUT 5IFPSFN *G UXP BOHMFT BSF DPNQMFNFOUBSZ UP UIF TBNF BOHMF PS UP DPOHSVFOU BOHMFT UIFO UIF BOHMFT BSF DPOHSVFOU

626

Reasoning and Proof

FY B NQM E

(&0.&53:

5

∠$#% BOE ∠&'( BSF TVQQMFNFOUBSZ 'JOE UIF WBMVF PG Y

' $ &

¡ "

#

(

Y + ¡

%

6TF UIF -JOFBS 1BJS 5IFPSFN BOE UIF $POHSVFOU 4VQQMFNFOUT 5IFPSFN ∠"#$ BOE ∠$#% GPSN B MJOFBS QBJS

N∠"#$ + N∠$#% =

4VCTUJUVUF GPS N∠$#%

N∠"#$ + =

N∠"#$ = ¡

4VQQMFNFOUT PG UIF TBNF BOHMF BSF DPOHSVFOU

N∠"#$ = N∠&'(

4VCTUJUVUF

= Y +

4PMWF UIF FRVBUJPO GPS Y

= Y

Y =

7FSUJDBM "OHMFT 7FSUJDBM BOHMFT BSF UIF QBJST PG OPO BEKBDFOU BOHMFT GPSNFE XIFO UXP MJOFT JOUFSTFDU

∠BOE ∠BSF WFSUJDBM BOHMFT

∠BOE ∠BSF WFSUJDBM BOHMFT 7FSUJDBM BOHMFT IBWF B WFSZ TQFDJBM QSPQFSUZ 7FSUJDBM "OHMFT 5IFPSFN 7FSUJDBM BOHMFT BSF DPOHSVFOU ∠∠BOE ∠âˆ

7ATCH /UT

!

7FSUJDBM BOHMFT BSF OPU BMXBZT JO B WFSUJDBM QPTJUJPO SFMBUJWF UP FBDI PUIFS 5IFZ BSF BMXBZT PQQPTJUF FBDI PUIFS BDSPTT UIFJS DPNNPO WFSUFY

Proof of the Vertical Angle Theorem Given: ∠1 and ∠3 are vertical angles. Prove: ∠1 ∠3

∠BOE ∠GPSN B MJOFBS QBJS #Z UIF -JOFBS 1BJS 5IFPSFN ∠BOE ∠BSF TVQQMFNFOUBSZ 4JNJMBSMZ ∠BOE ∠BSF TVQQMFNFOUBSZ 4JODF ∠BOE ∠BSF CPUI TVQQMFNFOUBSZ UP ∠∠1 ∠CZ UIF $POHSVFOU 4VQQMFNFOUT 5IFPSFN

XXX 48BEWBOUBHF DPN

627

1SPWJOH "OHMFT $POHSVFOU

6TJOH UIF 7FSUJDBM "OHMFT 5IFPSFN &Y "NQM & 4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

6

'JOE UIF WBMVF PG Z. ¡

7FSUJDBM BOHMFT BSF DPOHSVFOU %JWJEF CPUI TJEFT CZ

&Y "NQM &

Z¡

Z = Z =

7

'JOE UIF WBMVFT PG Y BOE Z.

-

0

7FSUJDBM BOHMFT BSF DPOHSVFOU 4VCTUJUVUF %JWJEF CPUI TJEFT CZ -JOFBS QBJST BSF TVQQMFNFOUBSZ 4VCTUJUVUF 4JNQMJGZ 4VCUSBDU GSPN CPUI TJEFT %JWJEF CPUI TJEFT CZ

Z + ¡ Y¡ ¡ /

.

1

N∠-/0 = N∠./1 Y = Y = N∠-/. + N∠./1 = ¡ Z + + = Z + = Z = Z =

4P Y = BOE Z =

3JHIU "OHMFT :PV BMSFBEZ LOPX UIBU UIF NFBTVSF PG B SJHIU BOHMF JT ¡ CZ EFGJOJUJPO )FSF BSF UXP UIFPSFNT UIBU UFMM ZPV NPSF BCPVU SJHIU BOHMFT 3JHIU "OHMF $POHSVFODF 5IFPSFN "MM SJHIU BOHMFT BSF DPOHSVFOU

5IFPSFN *G UXP BOHMFT BSF DPOHSVFOU BOE TVQQMFNFOUBSZ UIFO FBDI BOHMF JT B SJHIU BOHMF

628

Reasoning and Proof

FY B NQM E

(&0.&53:

8

1SPWF UIF 3JHIU "OHMF $POHSVFODF 5IFPSFN

(JWFO ∠BOE ∠BSF SJHIU BOHMFT 1SPWF ∠âˆ

#Z EFGJOJUJPO PG B SJHIU BOHMF N∠= ¡ BOE N∠= ¡ #Z UIF 5SBOTJUJWF 1SPQFSUZ PG &RVBMJUZ N∠1 = N∠PS ∠âˆ

FY B NQM E

9

(JWFO ∠BOE ∠BSF DPOHSVFOU BOE TVQQMFNFOUBSZ 1SPWF ∠BOE ∠BSF SJHIU BOHMFT

#Z EFGJOJUJPO PG TVQQMFNFOUBSZ BOHMFT N∠+ N∠= ¡ #Z EFGJOJUJPO PG DPOHSVFOU BOHMFT N∠= N∠6TJOH TVCTUJUVUJPO BOE BMHFCSB N∠+ N∠= ¡

N∠= ¡ N∠= ¡

5IFO CZ EFGJOJUJPO PG DPOHSVFOU BOHMFT N∠= ¡ TJODF ∠∠4P ∠BOE ∠BSF CPUI SJHIU BOHMFT CZ EFGJOJUJPO PG B SJHIU BOHMF

"O BOHMF CJTFDUPr JT B SBZ UIBU EJWJEFT BO BOHMF JOUP UXP DPOHSVFOU DPQMBOBS BOHMFT @@l

BP CJTFDUT ∠"#$ ∠"#1 ∠$#1 N∠"#1 = N∠$#1 = @ N∠"#$ FY B NQM E

1 " #

$

10

@@l EG CJTFDUT ∠%&' BOE N∠%&' = ¡ 'JOE UIF WBMVFT PG Y BOE Z

(

8SJUF BOE TPMWF UXP FRVBUJPOT JO UXP WBSJBCMFT 6TF UIF "OHMF "EEJUJPO 1PTUVMBUF

N∠%&( + N∠(&' = N∠%&'

4VCTUJUVUF

Y − Z + Y + Z =

4JNQMJGZ %JWJEF CPUI TJEFT CZ 6TF UIF EFGJOJUJPO PG BO BOHMF CJTFDUPS 4VCTUJUVUF 4VCTUJUVUF GPS Y 4JNQMJGZ 4PMWF GPS Z

Y = Y = N∠%&( = N∠(&'

' Y + Z ¡

Y – Z ¡ %

&

Need More

HELP

'PS B SFWJFX PO TPMWJOH TZTUFNT PG FRVBUJPOT HP UP 4PMWJOH B 4ZTUFN CZ 4VCTUJUVUJPO JO "MHFCSB #PPL Q

Y − Z = Y + Z − Z = + Z − Z = + Z = Z Z =

4P Y = BOE Z =

XXX 48BEWBOUBHF DPN

629

5IFPSFNT "CPVU 1BSBMMFM -JOFT 1SPQFSUJFT PG 1BSBMMFM -JOFT 7ATCH /UT

!

-JOFT UIBU EP OPU JOUFSTFDU BSF FJUIFS QBSBMMFM PS TLFX $PQMBOBS MJOFT UIBU EP OPU JOUFSTFDU BSF QBSBMMFl /PO DPQMBOBS MJOFT UIBU EP OPU JOUFSTFDU BSF TLFw %JBHSBNT VTF TPMJE SFE BSSPXIFBET UP EFOPUF MJOFT UIBU BSF QBSBMMFM Q

5XP TFHNFOUT UIBU EP OPU JOUFSTFDU BSF OPU OFDFTTBSJMZ QBSBMMFM 5o be parBMMFM UIFZ NVTU CF QBSUT PG QBSBMMFM MJOFT

R B

C

-JOF B JT QBSBMMFM UP MJOF C B ]] C BOE MJOF Q JT QBSBMMFM UP MJOF R " USBOTWFSTBl JT B MJOF UIBU JOUFSTFDUT UXP PS NPSF DPQMBOBS MJOFT BU EJGGFSFOU QPJOUT 8IFO B USBOTWFSTBM JOUFSTFDUT QBSBMMFM MJOFT UIF QBJST PG BOHMFT GPSNFE IBWF TQFDJBM OBNFT

$PSSFTQPOEJOH BOHMFs MJF JO DPSSFTQPOEJOH QPTJUJPOT PO UIF TBNF TJEF PG B USBOTWFSTBM

∠BOE âˆ

∠BOE âˆ

∠BOE âˆ

∠BOE âˆ

"MUFSOBUF JOUFSJPS BOHMFs MJF CFUXFFO UIF QBSBMMFM MJOFT PO PQQPTJUF TJEFT PG B USBOTWFSTBM ∠BOE âˆ

∠BOE âˆ

4BNF TJEF JOUFSJPS BOHMFs MJF CFUXFFO UIF QBSBMMFM MJOFT PO UIF TBNF TJEF PG B USBOTWFSTBM ∠BOE âˆ

∠BOE âˆ

"MUFSOBUF FYUFSJPS BOHMFs MJF PVUTJEF UIF QBSBMMFM MJOFT PO PQQPTJUF TJEFT PG B USBOTWFSTBM

∠BOE âˆ

∠BOE âˆ

Corresponding Angles Postulate *G UXP QBSBMMFM MJOFT BSF DVU CZ B USBOTWFSTBM UIFO DPSSFTQPOEJOH BOHMFT BSF DPOHSVFOU

(05 50

,/08

FY B NQM F

1

'JOE UIF WBMVF PG Y JG N∠= Y + ¡ BOE N∠= Y − ¡

1BSBMMFM 1PTUVMBUF

5IF EJBHSBN TIPXT UIBU B ]] C

(JWFO B MJOF BOE B QPJOU OPU PO UIF MJOF UIFSF JT FYBDUMZ POF MJOF UISPVHI UIF QPJOU QBSBMMFM Uo the line

$PSSFTQPOEJOH BOHMFT BSF DPOHSVFOU

630

4VCUSBDU Y GSPN CPUI TJEFT "EE UP CPUI TJEFT %JWJEF CPUI TJEFT CZ

Y − = Y + Y − = Y = Y =

B C

Reasoning and Proof

(&0.&53:

:PV DBO VTF UIF $PSSFTQPOEJOH "OHMFT 1PTUVMBUF UP QSPWF PUIFS UIFPSFNT BCPVU QBJST PG BOHMFT GPSNFE XIFO B USBOTWFSTBM DVUT QBSBMMFM MJOFT Alternate Interior Angles Theorem *G UXP QBSBMMFM MJOFT BSF DVU CZ B USBOTWFSTBM UIFO BMUFSOBUF JOUFSJPS BOHMFT BSF DPOHSVFOU

Alternate Exterior Angles Theorem *G UXP QBSBMMFM MJOFT BSF DVU CZ B USBOTWFSTBM UIFO BMUFSOBUF FYUFSJPS BOHMFT BSF DPOHSVFOU

*O B UXP DPMVNO QSPPf ZPV PSHBOJ[F UIF TUFQT PG B QSPPG BOE UIFJS KVTUJGJDBUJPOT JO TFQBSBUF DPMVNOT 5IF GJSTU TUFQ HFOFSBMMZ TUBUFT UIF HJWFO JOGPSNBUJPO )FSF JT B UXP DPMVNO QSPPG PG UIF "MUFSOBUF &YUFSJPS "OHMFT UIFPSFN &Y "NQM &

‹__›

2

‹__›

Given: AB || CD Prove: ∠1 ∠3

"

$

4&"3$)

#

5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

%

Statements

k@@l k@@l ]] CD AB

Reasons (JWFO

∠âˆ

*G UXP ]] MJOFT BSF DVU CZ B USBOTWFSTBM UIFO DPSS ∠T BSF

∠âˆ

7FSUJDBM BOHMFT BSF

∠âˆ

5SBOTJUJWF 1SPQFSUZ PG

5IF QSPPG PG UIF "MUFSOBUF *OUFSJPS "OHMFT UIFPSFN JT WFSZ TJNJMBS &Y "NQM &

3

'JOE N∠(%&

"MUFSOBUF JOUFSJPS ∠T BSF "EE UP CPUI TJEFT 4VCUSBDU Y GSPN CPUI TJEFT 4VCTUJUVUF GPS Y

Y − = Y +

Y – ¡

Y = Y + Y = N∠(%& = Y − ¡ = − =

&

%

'

Y + ¡ (

N∠(%& = ¡

XXX 48BEWBOUBHF DPN

631

5IFPSFNT "CPVU 1BSBMMFM -JOFT

4BNF 4JEF *OUFSJPS "OHMFT &WFSZ QBJS PG BOHMFT GPSNFE CZ QBSBMMFM MJOFT BOE B USBOTWFSTBM JT FJUIFS DPOHSVFOU PS TVQQMFNFOUBSZ 5IF OFYU UIFPSFN JOWPMWFT QBJST PG BOHMFT UIBU MJF PO UIF TBNF TJEF PG B USBOTWFSTBM BOE CFUXFFO UXP QBSBMMFM MJOFT 5IF QSPPG PG UIJT UIFPSFN JT JO &YBNQMF

7ATCH /UT

!

8IFO B USBOTWFSTBM JOUFSTFDUT UXP QBSBMMFM MJOFT BMM UIF BDVUF BOHMFT BSF DPOHSVFOU BOE BMM UIF PCUVTF BOHMFT BSF DPOHSVFOU &WFSZ BDVUF BOHMF JT TVQQMFNFOUBSZ UP FWFSZ PCUVTF BOHMF

Same-Side Interior Angles Theorem *G UXP QBSBMMFM MJOFT BSF DVU CZ B USBOTWFSTBM UIFO TBNF TJEF JOUFSJPS BOHMFT BSF TVQQMFNFOUBSZ

FY B NQM E

4

1SPWF UIF 4BNF 4JEF *OUFSJPS "OHMFT 5IFPSFN (JWFO ℓ ]] N 1SPWF ∠BOE ∠BSF TVQQMFNFOUBSZ

4UBUFNFOUT

3FBTPOT (JWFO

N∠+ N∠= ¡

*G UXP BOHMFT GPSN B MJOFBS QBJS UIFO UIFZ BSF TVQQMFNFOUBSZ

N∠= Nâˆ

*G UXP ]] MJOFT BSF DVU CZ B USBOTWFSTBM UIFO DPSS ∠T BSF

N∠+ N∠= ¡

4VCTUJUVUJPO

∠BOE ∠BSF TVQQMFNFOUBSZ

%FGJOJUJPO PG TVQQMFNFOUBSZ BOHMFT

5

'JOE UIF WBMVFT PG B BOE C

¡

B + C + = B + C =

B + C ¡ B – C ¡

B − C + = B − C =

HELP

'PS IFMQ XJUI TPMWJOH TZTUFNT PG FRVBUJPOT HP UP 4PMWJOH B 4ZTUFN CZ 4VCTUJUVUJPO BOE 4PMWJOH B 4ZTUFN CZ "EEJUJPO PS 4VCUSBDUJPO JO "MHFCSB #PPL QQ

632

N

â„“ ]] N

FY B NQM E

Need More

â„“

¡

NFU I P E

NFU I P E

"EE &RVBUJPOT BOE

4PMWF &RVBUJPO GPS B UIFO TVCTUJUVUF JOUP &RVBUJPO

B + C = B − C = B = TP B = 4VCTUJUVUF JOUP &RVBUJPO

B = + C

+ C + C =

+ C =

C = TP C =

C = TP C = B + C =

B + = TP B =

Reasoning and Proof

(&0.&53:

4PNFUJNFT UIF DPOWFSTF PG B DPOEJUJPOBM JT USVF *O QBSUJDVMBS UIF DPOWFSTF PG UIF $PSSFTQPOEJOH "OHMFT 1PTUVMBUF JT USVF

7ATCH /UT

Converse of the Corresponding Angles Postulate

!

*T UIJT UIFPSFN TUJMM USVF JG ZPV SFNPWF UIF XPSET iJO B QMBOFw /P UIF MJOFT DPVME CF QFSQFOEJDVMBS CVU UIFZ DPVME BMTP CF TLFX

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681

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c = a + b

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c > a + b

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D

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683

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890

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45&1

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2

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Right Triangle Trigonometry

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897

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1214

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1215

5IFPSFUJDBM 1SPCBCJMJUZ 8IBU *T 5IFPSFUJDBM 1SPCBCJMJUZ 1SPCBCJMJUy JT B XBZ UP EFTDSJCF UIF MJLFMJIPPE UIBU B HJWFO FWFOU XJMM UBLF QMBDF 8IFO QSPCBCJMJUZ JT DBMDVMBUFE CZ BOBMZTJT JU JT DBMMFE UIFPSFUJDBM QSPCBCJMJUy 5IFPSFUJDBM QSPCBCJMJUZ UFMMT XIBU ZPV DBO FYQFDU UP IBQQFO OPU XIBU BDUVBMMZ IBQQFOT &BDI QPTTJCMF SFTVMU JT BO PVUDPme "O FWFOt JT BOZ PVUDPNF PS HSPVQ PG PVUDPNFT 5P EFUFSNJOF UIFPSFUJDBM QSPCBCJMJUZ VTF UIF GPSNVMB CFMPX XIFSF GBWPSBCMF PVUDPNFT BSF PVUDPNFT ZPV XBOU BOE UPUBM OVNCFS PG PVUDPNFT JT UIF OVNCFS PG BMM UIF QPTTJCMF PVUDPNFT Need More

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t 5IF QPTTJCMF PVUDPNFT JO UIF TBNQMF TQBDF BSF OPU BMXBZT FRVBMMZ MJLFMZ 5IF TFDUJPOT PG UIF TQJOOFS BU UIF SJHIU BCPWF BSF OPU BMM UIF TBNF TJ[F *G ZPV TQJO UIJs sQJOOFS UIF BSSPX JT NPSF MJLFMZ UP MBOE PO UIBO PO PS

1232

Theoretical and Experimental Probability

45"5*45*$4 "/% 130#"#*-*5:

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1

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'PS IFMQ DPOWFSUJOH GSBDUJPOT UP EFDJNBMT BOE QFSDFOUT TFF %FDJNBMT 'SBDUJPOT BOEÂ 1FSDFOUT JO /VNCFST BOE 0QFSBUJPOT #PPL Q

C 'JOE UIF QSPCBCJMJUZ PG SPMMJOH BO PEE OVNCFS 5IFSF BSF QPTTJCMF PVUDPNFT 5IFSF BSF PEE OVNCFST‰ BOE 1 PEE OVNCFS = @ = =

D 'JOE UIF QSPCBCJMJUZ PG SPMMJOH B OVNCFS MFTT UIBO 5IFSF BSF QPTTJCMF PVUDPNFT 5IFSF BSF OVNCFST MFTT UIBO ‰ BOE 1 OVNCFS MFTT UIBO = @ =@ ≈ =

E 'JOE UIF QSPCBCJMJUZ PG SPMMJOH B OVNCFS HSFBUFS UIBO

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1233

5IFPSFUJDBM 1SPCBCJMJUZ

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2

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D 'JOE UIF QSPCBCJMJUZ PG TQJOOJOH B OVNCFS HSFBUFS UIBO BOE MFTT UIBO 0VU PG UIF QPTTJCMF PVUDPNFT PVUDPNFT BSF CFUXFFO BOE ‰ BOE 1 OVNCFS HSFBUFS UIBO BOE MFTT UIBO = @ =@ = =

FY B NQM E

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3FNFNCFS UIF TVN PG BMM UIF QPTTJCMF PVUDPNFT JT FRVBM UP @ +@ +@ =

3

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@ @ @ + + = @ =

"EE UP GJOE UIF UPUBM OVNCFS PG QPTTJCMF PVUDPNFT + + = 'JOE FBDI QSPCBCJMJUZ OVNCFS PG CMBDL TPDLT 1(black) = @@ = @ = @ UPUBM OVNCFS PG TPDLT OVNCFS PG CMVF TPDLT 1(blue) = @@ = @ = @ UPUBM OVNCFS PG TPDLT OVNCFS PG XIJUF TPDLT @@ @ @ = = 1 XIJUe) = UPUBM OVNCFS PG TPDLT

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1234

Theoretical and Experimental Probability &Y "NQM &

45"5*45*$4 "/% 130#"#*-*5:

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*O B TFU PG OVNCFS DBSET POF OVNCFS DBSd is selecUFE BU SBOEPN

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$PVOU UIF DBSET XIPTF OVNCFST BSF OPU 4FU VQ B QSPCBCJMJUZ SBUJP OVNCFS PG DBSET UIBU BSF OPU 1 OPU = @@@ = @ = @ = = UPUBM OVNCFS PG DBSET

N&5I P E

'JOE UIF QSPCBCJMJUZ PG TFMFDUJOH B DBSE XIPTF OVNCFS JT BOE TVCUSBDU UIF QSPCBCJMJUZ GSPN OVNCFS PG DBSET UIBU BSF =@ 1 = @@ = @

UPUBM OVNCFS PG DBSET @ @ 1 OPU = – = = =

&Y "NQM &

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5

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'PS IFMQ XJUI JOFRVBMJUJFT TFF (SBQIJOH BO *OFRVBMJUZ BT BO *OUFSWBM JO "MHFCSB Q

XXX 48BEWBOUBHF DPN

1235

4DBUUFS 1MPUT 8IBU *T B 4DBUUFS 1MPU

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/P PG 1FBST

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'PS IFMQ XJUI plotting poinUT HP UP 5IF $PPSEJOBUF 1MBOF in "MHFCSB Q

1

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1264

Displaying Data

&Y "NQM &

4U BU J4U JD 4 B O E 1S P CB CJM JU Z

2

B CPPUI BU B TUSFFU GBJS IBT HJGU JUFNT UIBU DPTU MFTT UIBO U IF TDBUUFS QMPU TIPXT UIF QSJDFT PG UIF JUFNT BOE UIF OVNCFS PG JUFNT PGGFSFE BU FBDI QSJDF 'PS FYBNQMF UIFSF BSF UFO JUFNT UIBU DPTU

t 5IF GJSTU EBUB QPJOU JT BU

5IFSF BSF JUFNT UIBU DPTU

t 5IF TFDPOE EBUB QPJOU JT BU

VOUJM BMM UIF EBUB QPJOUT PO UIF HSBQI BSF TIPXO PO UIF UBCMF

&Y "NQM &

5IF GJSTU OVNCFS JO BO PSEFSFE QBJS SFQSFTFOUT B EBUB QPJOU T QPTJUJPO BMPOH UIF IPSJ[POUBM BYJT PS Y BYJT 5IF TFDPOE OVNCFS SFQSFTFOUT JUT QPTJUJPO BMPOH UIF WFSUJDBM BYJT PS Z BYJT

t $POUJOVF BEEJOH DPPSEJOBUFT UP UIF UBCMF

O P PG JUFNT

REMEMBER

5IFSF BSF JUFNT UIBU DPTU

1SJDF

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5IF Y BYJT TIPXT UIF QSJDFT PG UIF JUFNT 5IF Z BYJT TIPXT UIF OVNCFS PG JUFNT BWBJMBCMF BU FBDI QSJDF 'JOE UIF DPPSEJOBUFT GPS FBDI EBUB QPJOU TUBSUJOH XJUI UIF POF PO UIF MFGU

C

6TF UIF TDBUUFS QMPU UP DPNQMFUF UIF EBUB UBCMF CFMPX

1SJDF

3

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U PUBM 4DPSF

5P SFDBMM UIJT PSEFS SFNFNCFS UIBU I DPNFT CFGPSF W JO UIF BMQIBCFU 5P SFDBMM UIF EJSFDUJPOT PG UIF BYFT UIJOL iZ QPJOUT VQ UP UIF TLZ w

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P OF PG UIF QMBZFST PO UIF UFBN EJTQMBZFE UIF EBUB PO B TDBUUFS QMPU CVU NBEF B NJTUBLF JEFOUJGZ UIF EBUB QPJOU PO UIF TDBUUFS QMPU UIBU JT QMPUUFE JODPSSFDUMZ $VNVMBUJWF 4DPSF

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XXX 48BEWBOUBHF DPN

1265

"MHFCSB Algebra

8FMEFST VTF MJOFBS JOFRVBMJUJFT UP BDDPVOU GPS UIF FYQBOTJPO BOE DPOUSBDUJPO PG NFUBM XIFO JU JT IFBUFE Pr cPPMFE

1382

/VNCFST 7BSJBCMFT BOE &YQSFTTJPOT . . . . . . (SBQIJOH 'VODUJPOT . . . . . . . . . . . . . . . . . . . . . . 0OF 7BSJBCMF -JOFBS &RVBUJPOT . . . . . . . . . . . . 0OF 7BSJBCMF -JOFBS *OFRVBMJUJFT . . . . . . . . . . 5XP 7BSJBCMF -JOFBS &RVBUJPOT BOE *OFRVBMJUJFT . . . . . . . . . . . . . . . . . . . . . . . . . 4ZTUFNT PG -JOFBS &RVBUJPOT BOE *OFRVBMJUJFT . . . . . . . . . . . . . . . . . . . . . . . . . 0QFSBUJPOT XJUI 1PMZOPNJBMT . . . . . . . . . . . . . 1PMZOPNJBM 'VODUJPOT BOE &RVBUJPOT . . . . . . 'BDUPSJOH 1PMZOPNJBMT . . . . . . . . . . . . . . . . . . . 2VBESBUJD 'VODUJPOT BOE &RVBUJPOT . . . . . . . 4ZTUFNT PG 2VBESBUJDT . . . . . . . . . . . . . . . . . . .

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1383

.VMUJQMZJOH BOE %JWJEJOH *OUFHFST To G'Oot to

K+N.O/W 7!à .VMUJQMJDBUJWF 1SPQFSUZ PG 'PS BOZ SFBM OVNCFS B B · = BOE · B =

.VMUJQMZJOH *OUFHFST -PPL BU UIF QBUUFSOT CFMPX

× = × = × = × − = − × − = −

− × = − − × = − − × = − × − = − × − =

8IFO ZPV NVMUJQMZ JOUFHFST ZPV NVMUJQMZ BCTPMVUF WBMVFT 5IF TJHOT PG UIF GBDUPST EFUFSNJOF UIF TJHO PG UIF QSPEVDU

.VMUJQMJDBUJWF 1SPQFSUZ PG −

Multiplying Integers with the Same Sign

'PS BOZ SFBM OVNCFS B B · − = −B BOE − · B = −B

*G UXP JOUFHFST IBWF UIF TBNF TJHO UIFJS QSPEVDU JT QPTJUJWF

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1

'JOE FBDI QSPEVDU B

] ] =

4P =

] ] =

] − ] =

=

C − −

] − ] =

4P − − =

C

8BZT UP

Multiplying Integers with Different Signs

REMEMBER

6TF UIJT TFOUFODF UP SFNFNCFS IPX UP EFUFSNJOF UIF TJHO PG B QSPEVDU PS B RVPUJFOU PG UXP JOUFHFST 4MFFQJOH 1FPQMF %P /PUIJOH

*G UXP JOUFHFST IBWF EJGGFSFOU TJHOT UIFJS QSPEVDU JT OFHBUJWF

FY B NQM E

2

'JOE FBDI QSPEVDU B −

*G CPUI TJHOT BSF UIF 4BNF UIF SFTVMU JT 1PTJUJWF

] ]

4P − = −

*G UIF TJHOT BSF %JGGFSFOU UIF SFTVMU is /FHBUJWF

C −

1394

] − ] =

4P − = −

] − ] =

] ]

Numbers, Variables, and Expressions

&Y "NQM &

"-(&#3"

3

*O POF GBOUBTZ GPPUCBMM MFBHVF B UFBN TDPSFT − QPJOUT FWFSZ UJNF JUT QMBZFST BSF DIBSHFE XJUI B UVSOPWFS B GVNCMF PS BO JOUFSDFQUJPO *G B UFBN T QMBZFST BSF DIBSHFE XJUI UVSOPWFST IPX NBOZ QPJOUT BSF TDPSFE 5P GJOE UIF UPUBM OVNCFS PG QPJOUT NVMUJQMZ UIF OVNCFS PG UVSOPWFST CZ UIF OVNCFS PG QPJOUT TDPSFE GPS FBDI UVSOPWFS UVSOPWFS

UJNFT ×

QPJOUT GPS FBDI UVSOPWFS −

FRVBMT =

UPUBM QPJOUT −

4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

5IF OVNCFS PG QPJOUT TDPSFE JT −

%JWJEJOH *OUFHFST 6TF UIF TBNF SVMFT GPS UIF TJHOT XIFO ZPV EJWJEF JOUFHFST *G UXP JOUFHFST IBWF UIF TBNF TJHO UIFJS RVPUJFOU JT QPTJUJWF *G UXP JOUFHFST IBWF EJGGFSFOU TJHOT UIFJS RVPUJFOU JT OFHBUJWF &Y "NQM &

4

'JOE FBDI RVPUJFOU − −

B

] ] =

] − ] =

= —

5IF TJHOT BSF UIF TBNF 4P — = −

−

C − ÷

] − ] =

] ] =

5IF TJHOT BSF EJGGFSFOU 4P − ÷ = −

÷ =

G'Oot Tto

K+N.O/W 7Ã

.VMUJQMZJOH PS %JWJEJOH *OUFHFST XJUI UIF 4BNF 4JHO *G UXP OVNCFST IBWF UIF TBNF TJHO UIFJS QSPEVDU PS RVPUJFOU JT QPTJUJWF − − = + + + = + − ÷ − = + + ÷ + = +

.VMUJQMZJOH PS %JWJEJOH *OUFHFST XJUI %JGGFSFOU 4JHOT *G UXP OVNCFST IBWF EJGGFSFOU TJHOT UIFJS QSPEVDU PS RVPUJFOU JT OFHBUJWF − + = − + − = − − ÷ + = − + ÷ − = −

XXX 48BEWBOUBHF DPN

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BC D = B CD

1409

%JTUSJCVUJWF 1SPQFSUZ 6TJOH UIF %JTUSJCVUJWF 1SPQFSUZ UP 4JNQMJGZ 5IF OVNCFS JT BDUVBMMZ UIF TVN PG BOE 8IFO ZPV NVMUJQMZ B OVNCFS CZ ZPV BSF BDUVBMMZ GJSTU NVMUJQMZJOH UIF OVNCFS CZ /FYU ZPV NVMUJQMZ UIF OVNCFS CZ BOE UIFO ZPV BEE UIFTF UXP QSPEVDUT UIF QSPEVDU PG BOE UIF QSPEVDU PG BOE UIF TVN PG UIFTF QSPEVDUT

× +

%JTUSJCVUJOH POF GBDUPS PWFS UXP BEEFOET PG BOPUIFS GBDUPS JT DBMMFE UIF %JTUSJCVUJWF 1SPQFSUZ 5IJT QSPQFSUZ JT TPNFUJNFT DBMMFE UIF %JTUSJCVUJWF 1SPQFSUZ PG .VMUJQMJDBUJPO PWFS "EEJUJPO 5IF %JTUSJCVUJWF 1SPQFSUZ TUBUFT UIBU GPS BOZ SFBM OVNCFST B C BOE D

B C + D = BC + BD 5IF %JTUSJCVUJWF 1SPQFSUZ DBO IFMQ ZPV EP NFOUBM NBUI &Y "NQM & 4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

1

6TF UIF %JTUSJCVUJWF 1SPQFSUZ UP SFXSJUF 5IFO TJNQMJGZ 3FXSJUF BT +

= +

6TF UIF %JTUSJCVUJWF 1SPQFSUZ

= +

.VMUJQMZ

= +

"EE

=

5IF %JTUSJCVUJWF 1SPQFSUZ BMTP XPSLT GPS NVMUJQMJDBUJPO PWFS TVCUSBDUJPO 'PS BOZ SFBM OVNCFST B C BOE D B C − D = BC − BD &Y "NQM &

2

6TF UIF %JTUSJCVUJWF 1SPQFSUZ UP SFXSJUF 5IFO TJNQMJGZ 3FXSJUF BT –

1410

= −

6TF UIF %JTUSJCVUJWF 1SPQFSUZ

= −

.VMUJQMZ

= −

4VCUSBDU

=

Numbers, Variables, and Expressions

"-(&#3"

6TF UIF %JTUSJCVUJWF 1SPQFSUZ JG ZPV TFF BO FBTJFS DBMDVMBUJPO UIBU JT FRVJWBMFOU FY B NQM E

3

4JNQMJGZ + 5IF OVNCFST BOE BSF DPNQBUJCMF GPS BEEJUJPO 6TF UIF %JTUSJCVUJWF 1SPQFSUZ UP SFXSJUF UIF FYQSFTTJPO

+ = +

"EE

=

.VMUJQMZ

=

5IF %JTUSJCVUJWF 1SPQFSUZ DBO CF FYUFOEFE UP BQQMZ UP NPSF UIBO UXP UFSNT FY B NQM E

4

5IF TVSGBDF BSFB PG B SFDUBOHVMBS QSJTN JT HJWFO CZ 4" = XI + â„“X + â„“I XIFSF X JT UIF XJEUI â„“ JT UIF MFOHUI BOE I JT UIF IFJHIU 8IBU JT UIF TVSGBDF BSFB PG UIJT SFDUBOHVMBS QSJTN

DN

DN

DN

NFU I P E

NFU I P E

4VCTUJUVUF JOUP UIF GPSNVMB

&BDI UFSN IBT B GBDUPS PG 6TF UIF %JTUSJCVUJWF 1SPQFSUZ UP SFXSJUF UIF GPSNVMB

4" = XI + ℓX + ℓI = ∙ ∙ + ∙ ∙ + ∙ ∙

= + + = DN

4" = XI + â„“X + â„“I

= ∙ + ∙ + ∙

= + +

=

= DN

G'Oot Tto

K+N.O/W 7Ã

U IF EJTUSJCVUJWF 1SPQFSUZ 5IF %JTUSJCVUJWF 1SPQFSUZ PG "EEJUJPO PWFS .VMUJQMJDBUJPO 'PS BOZ SFBM OVNCFST B C BOE D B C + D = BC + BD C + D B = CB + DB

+ = +

+ = +

5IF %JTUSJCVUJWF 1SPQFSUZ PG 4VCUSBDUJPO PWFS .VMUJQMJDBUJPO 'PS BOZ SFBM OVNCFST B C BOE D B C − D = BC − BD C − D B = CB − DB

XXX 48BEWBOUBHF DPN

– = −

− = −

%JTUSJCVUJWF 1SPQFSUZ

7BSJBCMF &YQSFTTJPOT 0OF JNQPSUBOU VTF PG UIF %JTUSJCVUJWF 1SPQFSUZ JT JO XSJUJOH FRVJWBMFOU FYQSFTTJPOT 8IFO UIF %JTUSJCVUJWF 1SPQFSUZ JT VTFE UP GBDUPS PVU UIF DPNNPO GBDUPST PG UXP PS NPSF UFSNT UIF SFTVMUJOH FYQSFTTJPO JT FRVJWBMFOU UP UIF PSJHJOBM FYQSFTTJPO 5IF UFSNs PG BO FYQSFTTJPO BSF UIF QBSUT UIBU BSF BEEFE PS TVCUSBDUFE

C

8BZT UP

REMEMBER

0OF XBZ UP SFNFNCFS UIF EJTUSJCVUJWF QSPQFSUZ JT UP UIJOL PG EJTUSJCVUF XIJDI NFBOT UP HJWF TPNFUIJOH PVU UP NFNCFST PG B HSPVQ

Y + Z − UFSNT *O UIF FYQSFTTJPO BCPWF XIJDI UFSNT IBWF B DPNNPO GBDUPS 5IF UFSNT Y BOE IBWF BT B DPNNPO GBDUPS 6TF UIF $PNNVUBUJWF 1SPQFSUZ UP DIBOHF UIF PSEFS PG UIF UFSNT

Y − + Z

3FXSJUF BT

Y − + Z

6TF UIF %JTUSJCVUJWF 1SPQFSUZ UP XSJUF BO FRVJWBMFOU FYQSFTTJPO

Y − + Z

FY B NQM E

5

6TF UIF %JTUSJCVUJWF 1SPQFSUZ UP XSJUF BO FRVJWBMFOU FYQSFTTJPO B − Z −

6TF UIF %JTUSJCVUJWF 1SPQFSUZ

− Z + − −

.VMUJQMZ

− Z +

5IF FYQSFTTJPOT − Z − BOE − Z + BSF FRVJWBMFOU C S + 8SJUF BT

S +

6TF UIF %JTUSJCVUJWF 1SPQFSUZ

S +

5IF FYQSFTTJPOT S + BOE S + BSF FRVJWBMFOU

5IF %JTUSJCVUJWF 1SPQFSUZ DBO BMTP CF VTFE UP TJNQMJGZ FYQSFTTJPOT DPOUBJOJOH MJLF UFSNT -JLF UFSNs DPOUBJO UIF TBNF WBSJBCMFT SBJTFE UP UIF TBNF QPXFST 6OMJLF UFSNs EJGGFS JO FJUIFS B WBSJBCMF PS UIF QPXFS PG B WBSJBCMF 5IF GPMMPXJOH UBCMF QSPWJEFT TFWFSBM FYBNQMFT PG MJLF UFSNT BOE VOMJLF UFSNT *EFOUJGZJOH -JLF 5FSNT 5FSNT

7BSJBCMF T

1PXFS T

-JLF UFSNT

N BOE N

N

:FT

Y BOE Z

Y BOE Z

/P

B BOE B

B

BOE

/P

1412

Numbers, Variables, and Expressions &Y "NQM &

"-(&#3"

6

6TF UIF %JTUSJCVUJWF 1SPQFSUZ UP TJNQMJGZ JG QPTTJCMF B Z � Z

Need More

5IFTF UFSNT BSF MJLF UFSNT Z JT FRVJWBMFOU UP Z 6TF UIF %JTUSJCVUJWF 1SPQFSUZ UP GBDUPS PVU Z

Z − Z = Z −

"EE

= Z

6TF UIF $PNNVUBUJWF 1SPQFSUZ

= Z

HELP

3FNFNCFS UIBU UIF WBSJBCMF Z IBT B DPFGGJDJFOU PG BOE DBO CF XSJUUFO BT Z

C Y � Z 5IFTF UFSNT BSF OPU MJLF UFSNT 5IF WBSJBCMFT BSF EJGGFSFOU 5IJT FYQSFTTJPO DBOOPU CF TJNQMJGJFE D O � n 5IFTF UFSNT BSF OPU MJLF UFSNT 5IF WBSJBCMFT BSF UIF TBNF CVU UIF QPXFST EJGGFS 5IJT FYQSFTTJPO DBOOPU CF TJNQMJGJFE

&Y "NQM &

7

4JNQMJGZ B N � � N 6TF UIF %JTUSJCVUJWF 1SPQFSUZ UP NVMUJQMZ

N – + N = N – + N

6TF UIF $PNNVUBUJWF 1SPQFSUZ

= N + N −

6TF UIF %JTUSJCVUJWF 1SPQFSUZ BHBJO

= N + −

4VCUSBDU

= N −

6TF UIF $PNNVUBUJWF 1SPQFSUZ BHBJO

= N −

C � L �

6TF UIF %JTUSJCVUJWF 1SPQFSUZ UP NVMUJQMZ

4&"3$)

− L − 2) = − L − −

.VMUJQMZ

= − L +

6TF UIF $PNNVUBUJWF 1SPQFSUZ

= + − L

"EE

= − L

XXX 48BEWBOUBHF DPN

5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

1413

0SEFS PG 0QFSBUJPOT 0SEFS PG 0QFSBUJPOT 5IF FYQSFTTJPO – × DPOUBJOT OP HSPVQJOH TZNCPMT 8JUIPVU B TQFDJGJD PSEFS GPS QFSGPSNJOH UIF PQFSBUJPOT UIF FYQSFTTJPO DPVME CF TJNQMJGJFE JO NPSF UIBO POF XBZ )FSF BSF UXP XBZT TPNFPOF NJHIU TJNQMJGZ JU – × ×

− × −

5P QSFWFOU DPOGVTJPO XIFO TJNQMJGZJOH FYQSFTTJPOT XJUI BEEJUJPO TVCUSBDUJPO NVMUJQMJDBUJPO BOE EJWJTJPO BOE OP HSPVQJOH TZNCPMT BMXBZT VTF UIF GPMMPXJOH PSEFS t .VMUJQMZ BOE EJWJEF GSPN MFGU UP SJHIU t "EE BOE TVCUSBDU GSPN MFGU UP SJHIU :PV EPO U OFFE UP EP BMM NVMUJQMJDBUJPOT CFGPSF BMM EJWJTJPOT PS BMM BEEJUJPOT CFGPSF TVCUSBDUJPOT :PV DBO NVMUJQMZ BOE EJWJEF BT ZPV HFU UP FBDI PQFSBUJPO t %JWJEF BOE NVMUJQMZ GSPN MFGU UP SJHIU t 4VCUSBDU BOE BEE GSPN MFGU UP SJHIU &Y "NQM & 4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

1

4JNQMJGZ FBDI FYQSFTTJPO a. 17 − 3 × 4 STEP 1

.VMUJQMZ

STEP 2

4VCUSBDU

− × = − × = − =

C + ÷ − × STEP 1

+ ÷ – ×

%JWJEF BOE NVMUJQMZ GSPN MFGU UP SJHIU.

= + STEP 2

"EE BOE TVCUSBDU GSPN MFGU UP SJHIU.

&Y "NQM &

5

–

=

2

-BTIPOEB VTFT UIF DBMDVMBUPS BQQMJDBUJPO PO IFS DPNQVUFS UP FWBMVBUF UIF FYQSFTTJPO TIPXO 8BT UIF FYQSFTTJPO DBMDVMBUFE DPSSFDUMZ − ÷ + × = 6TF PSEFS PG PQFSBUJPOT UP DIFDL.

− ÷ + ×

%JWJEF BOE NVMUJQMZ GSPN MFGU UP SJHIU GJSTU

= −

4VCUSBDU BOE BEE GSPN MFGU UP SJHIU

=

3

+

/P UIF DPSSFDU BOTXFS JT 5IF DBMDVMBUPS FWBMVBUFE BMM PQFSBUJPOT GSPN MFGU UP SJHIU 1418

Numbers, Variables, and Expressions

"-(&#3"

0SEFS PG 0QFSBUJPOT XJUI &YQPOFOUT &YQPOFOUT SFQSFTFOU SFQFBUFE NVMUJQMJDBUJPO *G BO FYQSFTTJPO DPOUBJOT FYQPOFOUT TJNQMJGZ BOZ QPXFST CFGPSF QFSGPSNJOH BOZ PG UIF GPVS CBTJD PQFSBUJPOT t 4JNQMJGZ QPXFST GSPN MFGU UP SJHIU t .VMUJQMZ BOE EJWJEF GSPN MFGU UP SJHIU t "EE BOE TVCUSBDU GSPN MFGU UP SJHIU

3

FY B NQM E

4JNQMJGZ FBDI FYQSFTTJPO

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HELP

a. 54 − 3 × 2 3

4JNQMJGZ UIF QPXFS GJSTU

− × = − ×

'PS IFMQ XJUI FYQPOFOUT HP UP pages 1396−1399.

= − × .VMUJQMZ.

= −

4VCUSBDU

=

C − + ÷ − 4JNQMJGZ UIF QPXFST GSPN MFGU UP SJHIU GJSTU

− + ÷ − = − + ÷ −

%JWJEF

= − +

"EE BOE TVCUSBDU GSPN MFGU UP SJHIU

=

−

0SEFS PG 0QFSBUJPOT XJUI 1BSFOUIFTFT BOE #SBDLFUT (SPVQJOH TZNCPMT TVDI BT QBSFOUIFTFT BOE CSBDLFUT BSF VTFE UP NBLF TVSF UIBU DBMDVMBUJPOT BSF EPOF JO B TQFDJGJD PSEFS 4JNQMJGZ FYQSFTTJPOT JOTJEF HSPVQJOH TZNCPMT GJSTU t 1FSGPSN PQFSBUJPOT JOTJEF QBSFOUIFTFT t 4JNQMJGZ QPXFST t .VMUJQMZ BOE EJWJEF GSPN MFGU UP SJHIU t "EE BOE TVCUSBDU GSPN MFGU UP SJHIU 5IJT JT UIF FYQSFTTJPO GSPN UIF CFHJOOJOH PG UIF MFTTPO XJUI QBSFOUIFTFT BEEFE – × ×

XXX 48BEWBOUBHF DPN

1419

0SEFS PG 0QFSBUJPOT

&Y "NQM & 4&"3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

4

4JNQMJGZ FBDI FYQSFTTJPO a. 164 − (10 + 5) × 2 3 − + ×

1FSGPSN UIF PQFSBUJPO JO QBSFOUIFTFT GJSTU

= −

×

4JNQMJGZ UIF QPXFS

= −

×

.VMUJQMZ

= −

4VCUSBDU

=

C + − ÷ ÷ + – ÷ ÷

1FSGPSN UIF PQFSBUJPOT JO QBSFOUIFTFT GJSTU 4JNQMJGZ UIF QPXFST

= + ÷

= + ÷

%JWJEF

= +

"EE

= —

—

c. 22 + 3(1+3) − (3 × 2 2 ) 2 1FSGPSN UIF PQFSBUJPOT JO QBSFOUIFTFT GJSTU 4JNQMJGZ UIF QPXFST XJUIJO QBSFOUIFTFT 4JNQMJGZ XJUIJO UIF QBSFOUIFTFT 4JNQMJGZ UIF QPXFST "EE BOE TVCUSBDU

&Y "NQM &

Need More

HELP

8IFO BO FYQSFTTJPO IBT CPUI QBSFOUIFTFT BOE CSBDLFUT PS IBT OFTUFE QBSFOUIFTFT BMXBZT TUBSU XJUI UIF JOOFSNPTU QBJS GJSTU

5

'FMJDJB IBE 4IF CPVHIU TIPFT GPS KFBOT GPS GPVS QBJST PG TPDLT BU B QBJS BOE UXP 5 TIJSUT BU FBDI 5IFO TIF HBWF IBMG PG XIBU TIF IBE MFGU UP IFS NPN &WBMVBUF UIF FYQSFTTJPO CFMPX UP GJOE UIF BNPVOU TIF HBWF IFS NPN < − + + · + · > ÷ < − + + + > ÷

STEP 1

.VMUJQMZ JOTJEF QBSFOUIFTFT.

4U FQ

"EE JOTJEF QBSFOUIFTFT.

= < − > ÷

4U FQ

:PV DBO SFNPWF UIF QBSFOUIFTFT.

= < − > ÷

4U FQ

4VCUSBDU.

= < > ÷

4U FQ

%JWJEF.

=

'FMJDJB HBWF IFS NPN

1420

+ + − × = + + − × = + − = + − = −

Numbers, Variables, and Expressions

"-(&#3"

0SEFS PG 0QFSBUJPOT XJUI 0UIFS (SPVQJOH 4ZNCPMT %JWJTJPO CBST BOE TRVBSF SPPU TZNCPMT BSF BMTP HSPVQJOH TZNCPMT @@@@@@@@@

OVNFSBUPS —

ȼ FYQSFTTJPO

EFOPNJOBUPS

FY B NQM E

6

4JNQMJGZ FBDI FYQSFTTJPO

8BZT UP

1FSGPSN UIF PQFSBUJPOT JO UIF OVNFSBUPS BOE JO UIF EFOPNJOBUPS GJSTU 4JNQMJGZ UIF QPXFST .VMUJQMZ

+

REMEMBER

+

— = — − − + =—

"EE BOE TVCUSBDU %JWJEF

*O UIF 6 4 VTF UIF TFOUFODF 1MFBTF &YDVTF .Z %FBS "VOU 4BMMZ UP SFNFNCFS UIF DPSSFDU PSEFS PG PQFSBUJPOT

− = —

=

1 1BSFOUIFTFT & &YQPOFOUT

b. É„ + ·

. .VMUJQMZ

1FSGPSN UIF PQFSBUJPOT VOEFS UIF SBEJDBM TZNCPM GJSTU .VMUJQMZ VOEFS UIF SBEJDBM

C

+ −

B

@@@@@@@@

% %JWJEF @@@@@

" "EE

@@@

4 4VCUSBDU

É„ + · = É„ +

"EE VOEFS UIF SBEJDBM

= É„

4JNQMJGZ UIF TRVBSF SPPU

=

4JNQMJGZ UIF QPXFS

=

.VMUJQMZ

=

*O $BOBEB CSBDLFUT OPU QBSFOUIFTFT BSF VTFE " IFMQGVM TFOUFODF UP SFNFNCFS JT #JH &MFQIBOUT %FTUSPZ .JDF "OE 4OBJMT

G'Oot Tto

K+N.O/W 7Ã 0SEFS PG 0QFSBUJPOT 1 1FSGPSN PQFSBUJPOT JOTJEF 1BSFOUIFTFT & 4JNQMJGZ &YQPOFOUT .% .VMUJQMZ BOE %JWJEF GSPN MFGU UP SJHIU "4

XXX 48BEWBOUBHF DPN

"EE BOE 4VCUSBDU GSPN MFGU UP SJHIU

1421

-JOFBS 'VODUJPOT 'VODUJPO /PUBUJPO 'VODUJPOT IBWF UIFJS PXO OPUBUJPO *O GVODUJPO OPUBUJPn UIF PVUQVU WBMVF GPS UIF JOQVU Y JT XSJUUFO BT f x) BOE SFBE BT iG PG Y w 5IF OPUBUJPO f Y = NFBOT iUIF SVMF BQQMJFE UP Y JT w *OQVU

Y

3VMF

0VUQVU

G Y

&RVBUJPO Z = Y +

.FBOJOH 5IF WBMVF PG Z FRVBMT UIF WBMVF PG Y +

'VODUJPO G Y = Y +

.FBOJOH 5IF JOQVU Y JO G Y HJWFT UIF PVUQVU Y +

&WBMVBUJOH 'VODUJPOT :PV DBO FWBMVBUF GVODUJPOT JO UIF TBNF XBZ UIBU ZPV FWBMVBUF FYQSFTTJPOT FY B NQM E

1

&WBMVBUF G Y = Y − GPS FBDI WBMVF PG Y G Y = Y –

a. Y =

To G'Oot to

K+N.O/W 7!à 'VODUJPO 3VMFT 'VODUJPOT BSF PGUFO XSJUUFO BT FRVBUJPOT 5IF TUBUFNFOU Z = G Y NFBOT UIBU Z JT B GVODUJPO PG Y 4P UIF GVODUJPO G Y = Y – DBO BMTP CF XSJUUFO BT Z = Y –

1450

C. Y =

G Y = Y –

G = –

G = –

= –

= –

=

=

*OQVU 0VUQVU 5BCMFT 0OF XBZ UP EFTDSJCF B GVODUJPO JT UP NBLF BO JOQVU PVUQVU UBCMF FY B NQM E

2

.BLF BO JOQVU PVUQVU UBCMF GPS UIF GVODUJPO G Y = Y −

*OQVU Y

3VMF Y −

0VUQVU G Y

−

−

−

−

−

= –

−

= −

−

4FMFDU WBMVFT GPS Y 5IFO FWBMVBUF G Y 'PMMPX UIF SVMF .VMUJQMZ UIF JOQVU CZ BOE UIFO TVCUSBDU 8IFO Y = G ) = −

Graphing Functions "M H FCS "

(SBQIJOH 'VODUJPOT 0GUFO B GVODUJPO IBT B EPNBJO UIBU JT DPOUJOVPVT 5IJT UZQF PG GVODUJPO DBOOPU CF EFTDSJCFE CZ B TFU PG PSEFSFE QBJST CFDBVTF UIF EPNBJO JT JOGJOJUF " HSBQI GPS UIJT UZQF PG GVODUJPO JT VTVBMMZ B MJOF PS DVSWF 5P HSBQI B DPOUJOVPVT GVODUJPO QMPU UIF PSEFSFE QBJST 5IFO DPOOFDU UIF QPJOUT CZ B MJOF PS TNPPUI DVSWF

3

FY B NQM E

4FB 3$)

HSBQI UIF GVODUJPO Z = Y GPS Y ≥ 4U FQ

4U FQ

.BLF BO JOQVU PVUQVU UBCMF 8IFO Y = Z = = 8IFO Y = Z = = 8IFO Y = Z = = 1MPU UIF QPJOUT BT PSEFSFE QBJST.

Y

Z

4U FQ

Z

0

Y

Z = Y

0

Z

$POOFDU UIF QPJOUT XJUI B TNPPUI DVSWF

5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

Y

-JOFBS 'VODUJPOT " MJOFBS GVODUJPO JT BOZ GVODUJPO XJUI B HSBQI UIBU JT B TUSBJHIU MJOF FY B NQM E

4

HSBQI UIF GVODUJPO Z = Y GPS Y ≥ *T UIF GVODUJPO MJOFBS 4U FQ

4U FQ

.BLF BO JOQVU PVUQVU UBCMF.

1MPU UIF QPJOUT BT PSEFSFE QBJST.

Z

4U FQ

Z

$POOFDU UIF QPJOUT

0

Y

Y

Z

0

Z = Y Y

5IF HSBQI JT B TUSBJHIU MJOF TP UIF GVODUJPO JT MJOFBS

XXX 48BEWBOUBHF DPN

1451

-JOFBS 'VODUJPOT

6TJOH 1BUUFSOT UP (SBQI 4PNFUJNFT B QSPCMFN SFMBUFT UXP TFUT PG OVNCFST UIBU GPMMPX B QBUUFSO :PV DBO HSBQI UIFTF SFMBUJPOTIJQT UP EFUFSNJOF XIFUIFS UIFZ BSF MJOFBS GVODUJPOT FY B NQM E 4FB 3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

5

5IF EJBHSBN TIPXT UIF OVNCFS PG DIBJST UIBU GJU BSPVOE B SPX PG Y UBCMFT (SBQI UIF OVNCFS PG DIBJST Z BT B GVODUJPO PG UIF OVNCFS PG UBCMFT Y *T UIF GVODUJPO MJOFBS

NFU I P E

.BLF BO *OQVU 0VUQVU 5BCMF 4U F1

.BLF BO JOQVU PVUQVU UBCMF XJUI UIF OVNCFS PG UBCMFT BT UIF JOQVU Y BOE UIF OVNCFS PG DIBJST BT UIF PVUQVU Z 5BCMFT 8IFO Y = Z = Y 8IFO Y = Z = $IBJST 8IFO Y = Z = Z 8IFO Y = Z = 10.

4U F1

1MPU UIF QPJOUT BT PSEFSFE QBJST.

Z

$POOFDU UIF QPJOUT

0

4U F1

Y

Z

0

Y

5IF HSBQI PG UIF GVODUJPO JT B TUSBJHIU MJOF TP UIF QBUUFSO JT MJOFBS #Z FYUFOEJOH UIF MJOF ZPV DBO TFF IPX NBOZ DIBJST BSF OFFEFE GPS FBDI BEEJUJPOBM UBCMF

1452

Graphing Functions "M H FCS " NFU I P E

M PPL GPS B 1BUUFSO

4U FQ

4U FQ

/PUJDF UIBU UIF GJSTU UBCMF JT TVSSPVOEFE CZ DIBJST 1MPU UIF QPJOt (1,

4U FQ

8IFO FBDI BEEJUJPOBM UBCMF JT BEEFE NPSF DIBJST BSF BEEFE .PWF UP UIF SJHIU PO UIF HSBQI VOJU BOE NPWF VQ VOJUT 1MPU BOPUIFS QPJOU BU

4U FQ

Z

VQ SJHIU Y

$POOFDU UIF QPJOUT 5IF HSBQI PG UIF GVODUJPO JT B TUSBJHIU MJOF TP UIF QBUUFSO JT MJOFBS.

'PS IFMQ XJUI HSBQIJOH HP UP (SBQIJOH GSPN BO &RVBUJPO JO "MHFCSB Q

HELP

Z

Need More

0

$POUJOVF NPWJOH SJHIU BOE VQ GSPN FBDI QPJOU BOE QMPU BOPUIFS QPJOU

0

Y

To G'Oot to

K+N.O/W 7!Ã

*EFOUJGZJOH -JOFBS 'VODUJPOT -JOFBS GVODUJPOT BSF JO UIF GPSN Z = BY + C XIFSF B BOE C BSF SFBM OVNCFST 5IFTF BSF MJOFBS GVODUJPOT Z = Y B = BOE C = Z = − Y + B = − BOE C = FY B NQM E

5IFTF BSF OPU MJOFBS GVODUJPOT @@ Z = Y Z = ȼx Z = Y +

6

*T UIF GVODUJPO Z = MJOFBS :FT UIF GVODUJPO JT JO UIF GPSN Z = BY + C XIFSF B = BOE C = 5IF HSBQI PG UIJT GVODUJPO JT B IPSJ[POUBM MJOF BU Z =

-JOFBS 'VODUJPOT 5P HSBQI B GVODUJPO NBLF BO JOQVU PVUQVU UBCMF UP GJOE PSEFSFE QBJST 6TF BU MFBTU UXP JOQVU WBMVFT " MJOFBS GVODUJPO JT BOZ GVODUJPO XJUI B HSBQI UIBU JT B TUSBJHIU MJOF

Z Z =

Y 0

XXX 48BEWBOUBHF DPN

-JOFBS GVODUJPOT DBO CF XSJUUFO JO UIF GPSN Z = BY + C XIFSF B BOE C BSF SFBM OVNCFST

1453

Advanced Algebra "EWBODFE "MHFCSB

5IF TFFET PG B TVOGMPXFS HSPX JO TQJSBMT JO PQQPTJUF EJSFDUJPOT 5IF OVNCFST PG TQJSBMT JO FBDI EJSFDUJPO BSF DPOTFDVUJWF OVNCFST JO UIF 'JCPOBDDJ TFRVFODF

1750

$PPSEJOBUF 4ZTUFN JO 5ISFF %JNFOTJPOT .BUSJDFT 1SPCMFN 4PMWJOH BOE 7BSJBUJPO 4FRVFODFT BOE 4FSJFT $PNQMFY /VNCFST $POJD 4FDUJPOT 3BEJDBMT &YQPOFOUT -PHBSJUINT BOE 3BUJPOBM &YQSFTTJPOT &YQPOFOUJBM BOE -PHBSJUINJD 'VODUJPOT 3BUJPOBM BOE 3BEJDBM 'VODUJPOT 8SJUJOH .PEFMT GPS 'VODUJPOT "OBMZ[JOH 'VODUJPOT

www.swadvantage.com

1751

4PMWJOH 4ZTUFNT CZ 4VCTUJUVUJPO 5IF .FUIPE PG 4VCTUJUVUJPO Need More

HELP

:PV DBO VTF NBOZ PG UIF TBNF TUSBUFHJFT UP TPMWF TZTUFNT PG UISFF FRVBUJPOT XJUI UISFF WBSJBCMFT UIBU ZPV VTFE UP TPMWF TZTUFNT PG UXP FRVBUJPOT XJUI UXP WBSJBCMFT 5P SFWJFX UIFTF TUSBUFHJFT HP UP 4PMWJOH B 4ZTUFN CZ 4VCTUJUVUJPO JO "MHFCSB Q

" TZTUFN PG UISFF FRVBUJPOT JOWPMWFT UISFF WBSJBCMFT 5IF WBMVF PG FBDI WBSJBCMF DBO Ce dFUFSNJOFE CZ VTJOH POF TUSBUFHZ PS B DPNCJOBUJPO PG TFWFSBM TUSBUFHJFT 5IF TPMVUJPO UP B TZTUFN PG UISFF FRVBUJPOT JO UISFF WBSJBCMFT JT BO PSEFSFE USJQMF 5Ie oSEFSFE USJQMF IBT UP TBUJTGZ BMM UISFF FRVBUJPOT 5IF NFUIPE PG TVCTUJUVUJPO JT POF TUSBUFHZ UIBU DBO CF VTFE UP TPMWF B TZTUFN PG UISFe eRVBUJPOT t *O POF PG UIF UISFF FRVBUJPOT B WBSJBCMF JT TPMWFE GPS PS JTPMBUFE t 5IF WBMVF PG UIBU WBSJBCMF JT UIFO TVCTUJUVUFE JOUP UIF PUIFS UXP FRVBUJPOT 5IBU QSPDFTT DPOWFSUT UIF PSJHJOBM TZTUFN PG UISFF FRVBUJPOT JO UISFF WBSJBCMFT JOUP B OFX TZTUFN PG FRVBUJPOT 5IF OFX TZTUFN IBT UXP FRVBUJPOT JO UXP WBSJBCMFT t 5IF QSPDFTT JT UIFO SFQFBUFE GPS UIF TZTUFN PG UXP FRVBUJPOT JO UXP WBSJBCMFT *O UIF FYBNQMFT CFMPX UIF PSJHJOBM FRVBUJPOT PG B TZTUFN BSF MBCFMFE B B BOE B 8IFO BO FRVBUJPO TVDI BT B JT DIBOHFE UIF OFX FRVBUJPO JT MBCFMFE C FY B NQM E

1

6TF UIF NFUIPE PG TVCTUJUVUJPO UP GJOE UIF TPMVUJPO Y Z [ GPS UIF TZTUFN CFMPX /PUF UIBU POF PG UIF FRVBUJPOT IBT KVTU POF WBSJBCMF B Y + Z − [ = B Y − Z + [ = B Y = 45&1

&RVBUJPO B JT BMSFBEZ TPMWFE GPS Y 4VCTUJUVUF UIBU WBMVF PG Y JOUP FRVBUJPOT B BOE B (1a) Y + Z − [ = 5 (2a) Y − Z + [ = + Z − [ = 5 − Z + [ = (1b) Z − [ = −1 (2b) −Z + [ =

45&1

&RVBUJPOT C BOE C GPSN B TZTUFN PG UXP FRVBUJPOT JO UXP WBSJBCMFT 0OF XBZ UP TPMWF UIJT TZTUFN JT UP VTF TVCTUJUVUJPO 4PMWF POF PG UIF FRVBUJPOT GPS POF PG UIF WBSJBCMFT 5IFO TVCTUJUVUF UIBU WBMVF JOUP UIF PUIFS FRVBUJPO 'PS FRVBUJPOT C BOE C JU JT FBTJFS UP TPMWe equaUJPO C GPS Z (2b) −Z + [ = 8 (1b) Z − [ = − Z = [ − [ − − [ = − [ − − [ = − [ = [= *U JT OPX LOPXO UIBU Y = BOE [ = 4VCTUJUVUF UIPTF WBMVFT JOUP FJUIFS FRVBUJPO B PS FRVBUJPO B 5IF BOTXFS XJMM CF UIF TBNF XIJDIFWFS FRVBUJPO ZPV DIPPTF (1a) Y + Z − [ = + Z − = + Z − = Z = Z= 5IF TPMVUJPO UP UIF TZTUFN JT

45&1

45&1

1762

$IFDL UIF TPMVUJPO JO BMM UISFF PSJHJOBM FRVBUJPOT (1a) Y + Z − [ = (2a) Y − Z + [ = 16 + − 5 − + 16 + − = ✔ − + = ✔

(3a) Y = = ✔

Coordinate System in Three Dimensions FY B NQM E

"%7"/$&% "-(&#3"

2

6TF UIF NFUIPE PG TVCTUJUVUJPO UP GJOE UIF TPMVUJPO Y Z [ GPS UIF TZTUFN CFMPX /PUF UIBU POF PG UIF FRVBUJPOT IBT KVTU UXP WBSJBCMFT B Y − Z + [ = B Y + Z + [ = B Z + [ = − 45&1

4PMWF POF PG UIF FRVBUJPOT GPS POF PG UIF WBSJBCMFT *O UIJT FYBNQMF MFU T TPMWF FRVBUJPO B GPS [ (3a) Z + [ = − [ = − Z −

45&1

5IFO TVCTUJUVUF UIBU WBMVF JOUP UIF PUIFS UXP FRVBUJPOT UP GPSN B TZTUFN XJUI UXP OFX FRVBUJPOT JO UXP WBSJBCMFT (1a) Y − Z + [ = 2 (2a) Y + Z + [ = Y − Z + − Z − = 2 Y + Z + − Z − = Y − Z − Z − = 2 Y + Z − Z − = (1b) Y − Z = 10 (2b) Y − Z =

45&1

5IF OFX TZTUFN PG C BOE C IBT UXP FRVBUJPOT BOE UXP WBSJBCMFT 4PMWF GPS FJUIFS WBSJBCMF JO FJUIFS FRVBUJPO -FU T TPMWF GPS Y JO C (1b) Y − Z = Y = + Z Y = + Z

45&1

4VCTUJUVUF UIJT WBMVF JOUP UIF PUIFS UXP WBSJBCMF FRVBUJPO BOE TPMWF GPS UIF WBSJBCMF (2b) Y − Z = + Z − Z = + Z − Z = Z = Z=

45&1

45&1

45&1

4FB 3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

4VCTUJUVUF Z = JOUP FJUIFS FRVBUJPO C PS C 5IFO TPMWF GPS Y (1b) Y − Z = Y − = Y = Y= 4VCTUJUVUF Z = BOE Y = JOUP BOZ PG UIF PSJHJOBM FRVBUJPOT 4PMWF GPS [ (1a) Y − Z + [ = − + [ = [ = − [ = − 5IF TPMVUJPO JT − $IFDL UIF TPMVUJPO − JO BMM UISFF PSJHJOBM FRVBUJPOT (1a) Y − Z + [ = 2 (2a) Y + Z + [ = 9 (3a) Z + [ = − − + − 2 + + − 9 + − − − = − ✔ − − = ✔ + − = ✔

XXX 48BEWBOUBHF DPN

1763

4PMWJOH 4ZTUFNT CZ 4VCTUJUVUJPO

5ISFF &RVBUJPOT XJUI 5ISFF 7BSJBCMFT FY B NQM E 4FB 3$) 5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

7ATCH /UT

1764

B Y + Z − [ = B Y − Z + [ = B Y + Z + [ = 45&1

4PMWF POF PG UIF FRVBUJPOT GPS POF PG UIF WBSJBCMFT *O UIJT FYBNQMF MFU T TPMWF FRVBUJPO B GPS [ B Y + Z − [ = [ = Y + Z −

45&1

5IFO TVCTUJUVUF UIF FYQSFTTJPO GPS [ JOUP UIF PUIFS UXP FRVBUJPOT 5IF SFTVMU JT B TZTUFN XJUI UXP FRVBUJPOT JO UXP WBSJBCMFT (2a) Y − Z + [ = 2 (3a) Y + Z + [ = Y − Z + Y + Z − = 2 Y + Z + Y + Z − = Y − Z + Y + Z − = 2 Y + Z + Y + Z − = Y + Z − = 2 Y + Z − = Y + Z = (3b) Y + Z = (2b) Y+Z=

45&1

5IF OFX TZTUFN PG C BOE C IBT UXP FRVBUJPOT BOE UXP WBSJBCMFT 4PMWF GPS FJUIFS WBSJBCMF JO FJUIFS FRVBUJPO :PV NJHIU DIPPTF UP TPMWF FRVBUJPO C GPS Y (2b) Y + Z = Y = −Z +

45&1

4VCTUJUVUF UIJT WBMVF JOUP UIF PUIFS UXP WBSJBCMF FRVBUJPO BOE TPMWF GPS UIF WBSJBCMF (3b) Y + Z = −Z + + Z = − Z + + Z = Z = Z=

45&1

4VCTUJUVUF Z JOUP FJUIFS FRVBUJPO C PS C BOE TPMWF GPS Y (2b) Y + Z = Y + = Y = −

!

*U JT JNQPSUBOU XIFO DIFDLJOH ZPVS PSEFSFE USJQMF UIBU ZPV TVCTUJUVUF UIF TPMVUJPO JOUP UIF PSJHJOBM TZTUFN PG UISFF FRVBUJPOT JOTUFBE PG UIF OFX TZTUFN PG UXP FRVBUJPOT 6TJOH UIF PSJHJOBM FRVBUJPOT XJMM FOBCMF ZPV UP GJOE BOZ NJTUBLFT ZPV NJHIU IBWF NBEF JO DBMDVMBUJOH UIF OFX TZTUFN PG UXP FRVBUJPOT

3

6TF UIF NFUIPE PG TVCTUJUVUJPO UP GJOE UIF TPMVUJPO Y Z [ GPS UIF TZTUFN TIPXO CFMPX /PUe thaU BMM UISFF FRVBUJPOT IBWF UISFF WBSJBCMFT

45&1

45&1

4VCTUJUVUF Y = – BOE Z = JOUP POF PG UIF PSJHJOBM FRVBUJPOT 4PMWF GPS [ (1a) Y + Z − [ = − + − [ = − + − [ = [= 5IF TPMVUJPO JT − $IFDL UIF TPMVUJPO − JO BMM UISFF PSJHJOBM FRVBUJPOT (1a) Y + Z − [ = 3 (2a) Y − Z + [ = 2 (3a) Y + Z + [ = − + − − − + − + + − + − = ✔ − − + = ✔ − + + = ✔

Coordinate System in Three Dimensions

"%7"/$&% "-(&#3"

&YDFQUJPOT UP BO 0SEFSFE 5SJQMF BT B 4PMVUJPO *O UIF GJSTU UISFF FYBNQMFT UIF TPMVUJPO PG UIF TZTUFNT PG FRVBUJPOT JT BO PSEFSFE USJQMF )PXFWFS TPNFUJNFT UIFSF JT OP TPMVUJPO UP B TZTUFN PG FRVBUJPOT "OE TPNFUJNFT UIFSF JT BO JOGJOJUF OVNCFS PG TPMVUJPOT FY B NQM E

4

6TF UIF NFUIPE PG TVCTUJUVUJPO UP GJOE UIF TPMVUJPO Y Z [ GPS UIF TZTUFN TIPXO CFMPX B Y + Z − [ = − B Y − Z + [ = B Y − Z + [ = 45&1

4PMWF POF PG UIF FRVBUJPOT GPS POF PG UIF WBSJBCMFT *O UIJT FYBNQMF TPMWF FRVBUJPO B GPS Y B Y − Z + [ = Y = Z − [ +

45&1

5IFO TVCTUJUVUF UIF FYQSFTTJPO GPS [ JOUP UIF PUIFS UXP FRVBUJPOT UP PCUBJO B TZTUFN XJUI UXP FRVBUJPOT JO UXP WBSJBCMFT (1a) Y + Z – [ = −1 (2a) Y − Z + [ = Z − [ + + Z – [ = −1 Z − [ + − Z + [ = Z – [ = −2 Z − [ + − Z + [ = â‰

5IF TVCTUJUVUJPO JOUP POF PG UIF FRVBUJPOT B SFTVMUFE JO B GBMTF TUBUFNFOU TP UIFSF BSF OP TPMVUJPOT UP UIJT TZTUFN PG FRVBUJPOT

FY B NQM E

5

6TF UIF NFUIPE PG TVCTUJUVUJPO UP GJOE UIF TPMVUJPO Y Z [ GPS UIF TZTUFN TIPXO CFMPX B Y + Z − [ = − B Y + Z − [ = − B Y + Z − [ = − 4PMWF POF PG UIF FRVBUJPOT GPS POF PG UIF WBSJBCMFT *O UIJT FYBNQMF TPMWF FRVBUJPO B GPS Y (1a) Y + Z − [ = − TP Y = − Z + [ − 8IFO UIJT FYQSFTTJPO JT TVCTUJUVUFE GPS Y JO UIF PUIFS UXP FRVBUJPOT UP PCUBJO B TZTUFN XJUI UXP FRVBUJPOT JO UXP WBSJBCMFT UIF SFTVMU JT = (2a)

Y + Z − [ = −4

(3a)

Y + Z − [ = −

− Z + [ − + Z − [ = −4

− Z + [ − + Z − [ = −

− Z + [ − + Z − [ = −4

− Z + [ − + Z − [ = −

− Z + Z + [ − [ − = −4

− Z + Z + [ − [ − = −

− = −4

− = −

=0

=

5IJT SFTVMU NFBOT UIBU GPS BOZ WBMVF PG Y UIFSF BSF WBMVFT PG Z BOE [ UIBU BSF TPMVUJPOT UP UIF TZTUFN 5IFSF BSF BO JOGJOJUF OVNCFS PG TPMVUJPOT UP UIJT TZTUFN

XXX 48BEWBOUBHF DPN

(05 50

,/08

6TJOH 4VCTUJUVUJPO UP 4PMWF B 4ZTUFN PG 5ISFF &RVBUJPOT JO 5ISFF 7BSJBCMFT 4PMWF POF PG UIF FRVBUJPOT GPS POF WBSJBCMF 4VCTUJUVUF GPS UIBU WBSJBCMF JO UIF PUIFS UXP FRVBUJPOT 5IF SFTVMU JT B TZTUFN PG UXP FRVBUJPOT JO UXP WBSJBCMFT 4PMWF UIF UXP WBSJBCMF TZTUFN GPS BOPUIFS WBSJBCMF 4VCTUJUVUF UIF UXP WBSJBCMFT ZPV LOPX JOUP BO PSJHJOBM FRVBUJPO BOE TPMWF GPS UIF UIJSE WBSJBCMF $IFDL UIF PSEFSFE USJQMF JO BMM UISFF PSJHJOBM FRVBUJPOT

1765

3PXT BOE $PMVNOT 5IF %JNFOTJPOT PG B .BUSJY " NBUSix JT B SFDUBOHVMBS BSSBOHFNFOU PG OVNCFST PS WBMVFT &BDI OVNCFS PS WBMVF JO B NBUSJY JT DBMMFE BO FMFNFOt PG UIF NBUSJY 5IF FMFNFOUT BSF BSSBOHFE JO IPSJ[POUBM SPXT BOE WFSUJDBM DPMVNOT " NBUSJY JT VTVBMMZ JOEJDBUFE VTJOH TRVBSF CSBDLFUT 5IF QMVSBM PG NBUSJY JT NBUSJDFT 5IF EJNFOTJPOs PG B NBUSJY SFGFS UP UIF OVNCFS PG SPXT BOE DPMVNOT UIBU JU IBT 5IF EJNFOTJPOT PG B NBUSJY BSF VTVBMMZ XSJUUFO N × O SFBE iN CZ O w XIFSF UIF GJSTU MFUUFS JOEJDBUFT UIF OVNCFS PG SPXT BOE UIF TFDPOE OVNCFS JOEJDBUFT UIF OVNCFS PG DPMVNOT − − &Y "NQM &

5P TFF TUFQ CZ TUFQ WJEFPT PG UIFTF QSPCMFNT FOUFS UIF QBHF OVNCFS JOUP UIF 48BEWBOUBHF DPN 4FBSDI #BS

45&1

−

−

−— −—

—

−

− − −—

1

45&1

$PVOU UIF OVNCFS PG SPXT BOE DPMVNOT -FGU NBUSJY SPXT BOE DPMVNOT .JEEMF NBUSJY SPXT BOE DPMVNOT 3JHIU NBUSJY SPX BOE DPMVNOT 6TF UIF N × O GPSNBU -FGU NBUSJY × .JEEMF NBUSJY × 3JHIU NBUSJY ×

&Y "NQM &

!

8IFO B NBUSJY JT EFTDSJCFE BT IBWJOH EJNFOTJPOT N × O UIF GJSTU WBMVF BMXBZT SFGFST UP UIF OVNCFS PG SPXT BOE UIF TFDPOE WBMVF BMXBZT SFGFST UP UIF OVNCFS PG DPMVNOT

2

8SJUF B × NBUSJY XIFSF UIF FMFNFOUT JO SPX N BSF N N BOE N 45&1

45&1

4FU VQ B × NBUSJY " × NBUSJY IBT SPXT BOE DPMVNOT 1SFQBSF BO PVUMJOF GPS B × NBUSJY VTJOH N N BOE N UP SFQSFTFOU UIF FMFNFOUT JO UIF NBUSJY BT TIPXO JO UIF NBUSJY UP UIF SJHIU.

N N N N N

N N N N N

N N N N N

8SJUF UIF NBUSJY 5IF QISBTF iSPX Nw NFBOT UIBU N = GPS SPX N = GPS SPX BOE TP PO 6TF UIF WBMVFT PG N UP FWBMVBUF FBDI FMFNFOU JO UIF NBUSJY

1772

5FMM UIF EJNFOTJPOT PG UIF UISFF NBUSJDFT TIPXO BCPWF

4&"3$)

7ATCH /UT

Matrices

"E W"O D FE "M H FCS "

&RVBM .BUSJDFT 5XP NBUSJDFT BSF FRVBM NBUSJDFs JG UIFZ NFFU UIFTF DPOEJUJPOT t 5IF NBUSJDFT IBWF UIF TBNF EJNFOTJPOT t $PSSFTQPOEJOH FMFNFOUT JO UIF UXP NBUSJDFT IBWF UIF TBNF WBMVF FY B NQM F

3

B 5IF UXP NBUSJDFT CFMPX BSF FRVBM 6TF UIF EFGJOJUJPO PG FRVBM NBUSJDFT UP GJOE UIF WBMVFT of Y Z [ BOE X Y −

− X + —

Z −

− [

=

45&1

$PSSFTQPOEJOH FMFNFOUT BSF FRVBM TP TFU VQ GPVS FRVBUJPOT X + Y − = − = Z − = − [ = —

45&1

4PMWF FBDI PG UIF GPVS FRVBUJPOT Y − = − = Z −

− = Z Z = −

Y = Y =

C 6TF UIF SFTVMUT GSPN QBSU B UP XSJUF BMM UIF FMFNFOUT JO UIF GPMMPXJOH NBUSJY BT OVNCFST XJUI OP WBSJBCMFT 45&1

X +

X + = X = Y ÷ Y ÷

Z − Y + Z + [ + X

[ X

3FXSJUF UIF NBUSJY VTJOH UIF WBMVFT PG Y Z [ BOE X − + −

45&1

X − Y + Z

!

8IFO ZPV JEFOUJGZ DPSSFTQPOEJOH FMFNFOUT JO UXP NBUSJDFT CF TVSF the elemenUT BSF JO UIF TBNF SPX BOE in the same cPMVNO in each maUSJY

= —

= − [ − = − [ [ =

7ATCH /UT

÷ ÷

− − + − + +

&WBMVBUF FBDI OVNFSJDBM FYQSFTTJPO JO UIF NBUSJY

− −

− −

(05 50

,/08 8SJUJOH B .BUSJY

SPX

DPMVNO

EJNFOTJPOT SPXT × DPMVNOT

5IF FMFNFOUT JO B NBUSJY DBO CF QPTJUJWF PS OFHBUJWF OVNCFST CF [FSP PS DPOUBJO WBSJBCMFT XXX 48BEWBOUBHF DPN

FMFNFOU

"EEJOH PS 4VCUSBDUJOH .BUSJDFT 7ATCH /UT

4VNT BOE %JGGFSFODFT PG $PSSFTQPOEJOH &MFNFOUT

!

8IFO TVCUSBDUJOH NBUSJDFT CF TVSF UP BMXBZT TVCUSBDU UIF FMFNFOU GSPN UIF TFDPOE NBUSJY GSPN UIF DPSSFTQPOEJOH FMFNFOU JO UIF GJSTU NBUSJY OPU UIF PUIFS XBZ BSPVOE 'PS FYBNQMF GPS " − # FWFSZ FMFNFOU JO UIF SFTVMUJOH NBUSJY JT GPVOE CZ FMFNFOU " − DPSSFTQPOEJOH FMFNFOU #

*G UXP NBUSJDFT IBWF UIF TBNF EJNFOTJPOT ZPV DBO add UIF UXP NBUSJDFT CZ GJOEJOH UIF TVNs oG QBJST PG DPSSFTQPOEJOH FMFNFOUT :PV DBO subtract UIFN CZ GJOEJOH UIF EJGGFSFODFT PG DPSSFTQPOEJOH FMFNFOUT

'PS FYBNQMF JG .BUSJY " = + + −

UIFO " + # = BOE " − # = FY B NQM F

− − −

−

− + + − − −

BOE .BUSJY # = −

+ = + −

−

− − = − −

−

−

− −

1

6TJOH UIF NBUSJDFT CFMPX GJOE 9 + ; : − 9 BOE 9 − ; + : − 9 =

− : = − −

; = − −

−

8SJUF FBDI FYQSFTTJPO TVCTUJUVUF UIF NBUSJDFT BOE FWBMVBUF − 9 + ; =

− + − −

− + = + + −

−

+ − + = − + −

: − 9 = −

− − −

− −

− = − − − −

− − − = − − − −

−

− 9 − ; + : =

(05 50

,/08

− =

(

− − − − − − − −

+ − − − = −

)

−

− − −

*EFOUJGZJOH $PSSFTQPOEJOH &MFNFOUT JO 5XP .BUSJDFT . BLF TVSF UIBU UIF UXP NBUSJDFT IBWF UIF TBNF EJNFOTJPOT $ IFDL UIBU GPS FBDI QBJS PG DPSSFTQPOEJOH FMFNFOUT UIF SPX OVNCFS BOE DPMVNO OVNCFS FYBDUMZ NBUDI

"EEJOH BOE 4VCUSBDUJOH .BUSJDFT " BOE # 0QFSBUJPO

1SPDFEVSF

"EEJUJPO " + #

"EE FBDI FMFNFOU JO " UP JUT DPSSFTQPOEJOH FMFNFOU JO #

4VCUSBDUJPO " − #

4VCUSBDU FBDI FMFNFOU JO # GSPN JUT DPSSFTQPOEJOH FMFNFOU JO "

Matrices

"E W"O D FE "M H FCS "

"MXBZT DIFDL UIBU UXP NBUSJDFT IBWF UIF TBNF EJNFOTJPOT CFGPSF ZPV USZ UP BEE PS TVCUSBDt tIFN &Y "NQM &

2

)FSF BSF TFWFSBM NBUSJDFT " ä

ź # ä

$ ä ź

ź % ä & ä ź

7ATCH /UT

ź

B EFUFSNJOF XIFUIFS JU JT QPTTJCMF UP BEE PS TVCUSBDU NBUSJDFT # BOE & EYQMBJO *G JU JT QPTTJCMF GJOE UIF NBUSJY GPS # á & BOE GJOE UIF NBUSJY GPS # ź &

:FT NBUSJDFT # BOE & BSF CPUI ñ NBUSJDFT 5IFZ IBWF UIF TBNF EJNFOTJPOT TP ZPV DBO GJOE UIF TVN PS EJGGFSFODF GPS FBDI QBJS PG DPSSFTQPOEJOH FMFNFOUT Ź à Ź Ź

# à & â

Ź â Ź

Ź Ź Ź Ź

# o & â Ź

Ź â

Ź Ź

C *T JU QPTTJCMF UP BEE PS TVCUSBDU NBUSJDFT " BOE % EYQMBJO *G JU JT QPTTJCMF GJOE UIF NBUSJY GPS " á % BOE GJOE UIF NBUSJY GPS " ź %

/ P " JT B ñ NBUSJY BOE % JT B ñ NBUSJY 5IFZ EP /05 IBWF UIF TBNF EJNFOTJPOT TP ZPV DBOOPU JEFOUJGZ QBJST PG DPSSFTQPOEJOH FMFNFOUT UP BEE PS TVCUSBDU

D *T JU QPTTJCMF UP BEE PS TVCUSBDU NBUSJDFT " BOE C EYQMBJO *G JU JT QPTTJCMF GJOE UIF NBUSJY GPS " á $ BOE GJOE UIF NBUSJY GPS " ź $

: FT NBUSJDFT " BOE $ BSF CPUI ñ NBUSJDFT 5IFZ IBWF UIF TBNF EJNFOTJPOT TP ZPV DBO GJOE UIF TVN PS EJGGFSFODF PG DPSSFTQPOEJOH FMFNFOUT " à $ â

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â

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" Ź $ â

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8SJUF BO FRVBUJPO 6TF i w TZNCPMT GPS UIF VOLOPXO FMFNFOUT JO NBUSJY 9 " à 9 â $

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2107

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HELP

FY B NQM F

'JOE MJN

1

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Y→

MJN =

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Y→

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= +

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=

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Z

'JOE MJN Y − Y + Y −

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‒ ( − ) ‒ ‒ ‒

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XXX 48BEWBOUBHF DPN

MJN x − x + Y − = − + − Y→

= −

2119

-JNJUT PG *OEFUFSNJOBUF 'PSNT

$POKVHBUF 5FDIOJRVF Need More

HELP

'PS IFMQ XJUIÂ VTJOH DPOKVHBUFT UP SBUJPOBMJ[F B GVODUJPO TFF %JWJEJOH 3BEJDBM &YQSFTTJPOT JO "EWBODFE "MHFCSB Q

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G (Y)=

ˆ Y − Y Y

‒

‒

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0 ‒ ‒

2128

Limits and Continuity

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ɄY − − MJN &WBMVBUF Y→

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[

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]

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$BODFM MJLF GBDUPST

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=—

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XXX 48BEWBOUBHF DPN

2129

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1130

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7รท1= 14 รท 2 = 21 รท 3 = 28 รท 4 = 35 รท 5 = 42 รท 6 = 49 รท 7 = 56 รท 8 = 63 รท 9 = 70 รท 10 =

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10 รท 1 = 20 รท 2 = 30 รท 3 = 40 รท 4 = 50 รท 5 = 60 รท 6 = 70 รท 7 = 80 รท 8 = 90 รท 9 = 100 รท 10 =

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1138

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1139

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1141

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1145

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