6- FRACTIONS sics MATH 0 F Frctions rise to express PART of UNIT Wht prt of n HOUR is thirty minutes? Fifteen minutes? tw elve minutes? (The UNIT here is HOUR.) Wht frction of the children re hppy? (The UNIT here is the entire GROUP or set of children.) In this exmple, w e my lso perceive the frction s rtio. One person might sy hlf or four-eighths the children re hppy. Another might sy four (re hppy), of eight ltogether, or one of every two. It s miles to Nn s house. We hve driven 9 miles. Wht prt of the drive hve we covered? (The UNIT here is the distnce to Nn s house.) We compre 9 to. 4 Four children must divide three pies mong them. How much of PIE does ech one get? (The UNIT here is PIE.) Most generlly, frction is used to express prt of w hole. The denomintor tells us into how mny equl prts the whole ws divided. The numertor tells us how mny of those prts w e hve. numertor ) 4 denomintor Definition: The set of rtionl numers, denoted Q, is {/ Z nd Z nd 0}. The numertor of / is defined to e, the denomintor of / is. The concept of equivlent frctions is encoded in the c FRACTION STRIPS: FUNDAMENTAL LAW OF FRACTIONS: if c 0, ) )) c In grde school this is clled RENAMING frctions. For exmple, / ( 4)/( 4) 4/, illustrted t right: / is the simplest or reduced form of the frction. A rtionl frction is sid to e in SIM PLEST or reduced form w hen the numertor nd denomintor re reltively prime. (Specil cse: if the denomintor is, we usully omit 4 )) )) )) writing the denomintor.) 4 After w e define multipliction! 7 9 9 7 9 9 Reduce: 960 40 COMPARING FRACTIONS: Are nd 4 equivlent? nd? 7 nd 40? 0 nd 4? 7 7 4 7 0 9 6 How we know tht / nd c/d re equivlent: Reduce, express with common denomintor. Wht out when they re NOT equivlent? How do we put them in order? () use common denomintor (then compre numertors) () compre to esily recognized frctions () if the numertors re the sme, compre denomintors. (Lrger denomintor smller frction!) (Would you rther hve ½ or c of $million?)
6-A More Frction Bsics MATH 0 f Illustrting frctions w ith re models. To illustrte one-hlf, it is importnt tht the UNIT e divided into two EQUAL prts. These re good illustrtions of one-hlf: These re not: In different discussions of frctions the unit my vry, ut one common unit should e used throughout prticulr discussion. Digrms used to illustrte ddition, sutrction, multipliction nd division, s w ell s frction equivlence nd comprison should follow suit. For instnce, to show equivlence, use ONE unit nd sudivide: 4 6 6 use: do NOT use: Rectngles ren t glmorous, ut they re esy to divide tw o w ys! To model the ddition of ½ + ½ ½ + ¼ ½ + ¼ ½ + do not use: do not use: use: use: + Not sme unit! NSU! Sme unit! Mesurement models re very similr to re models, ut re not necessrily -dimensionl. Set models re rrely used for comprisons or illustrting rithmetic opertions. (Usully find of N ) Mny frctions do not exist in W, the set of w hole numers. They elong to the set of rtionl numers, w hich includes the w hole numers, nd ll these new rtios. To sve spce, w e w ill refer to the set of frctions s Q. Getting redy for ddition & sutrction of frctions: We strt w ith common denomintors: ) + ) ) ) ) + ) ) + ) ) + ) 4 4 We step up to denomintors tht re comptile, s one divides the other: ) + ) ) ) ) ) 4 4 6 Until w e re redy for the generl cse: ) + ) ) ) ) + ) Then we will go eyond tht, to: 4 ) + ) ) ) ) ) ) + ) 6 9 6 9 6 9 6
6- ADDITION AND SUBTRACTION OF RATIONAL NUMBERS F Def'n ADDITION on Q: if,,c,d,d W & D 0 & d 0 + in generl, for rtionl c d c d + c )) + )) ))))) )) + )) )) + )) )))))) D D D numers / & c/d d d d d Def'n SUBTRACTION on Q: if,,c,d,d W & D> D or d> c & D 0 & d 0 in generl, for rtionl c d c d c )) )) ))))) )) )) )) )) )))))) D D D numers / & c/d d d d d E.g. + + 4 / 4 + + Deling with mixed numers Notice if the rtionl ddends re integers, + + + w e otin the ccustomed results. Also notice if d, the ove generl ddition formul yields: c d + dc d( + c) + c...s w e E.g. )) + )) ))))))) ))))))) ))))) ) + ) ))))))) d d d d d d d expect. 7 7 Although the given formul for + "works", it leds to unnecessrily lrge numers plus extr work to reduce! The preferred method is to write equivlent frctions using the lest common denomintor, D (LCD )... nd then ppel to the intuitively cler " /D + /D ( + )/D". In short, rewrite the frctions with the LCD, then dd numertors. (Note the LCD is the ctully the Lest Common Multiple of the denomintors.) E.g. 7 )) + )) )) + )) )) 6 4 4 4 Sketch model to illustrte the entire process of dding / + /: 4 9 ) + ) )) + )) )) 0 0 0 Properties of + on Q: Closure: Given ny rtionls / nd c/d, their sum, (d + c)/d Q. (Since:,d,,c Z gurntees d+ c nd d re oth Z y closure of multipliction nd ddition on Z; nd 0 & d 0 gurntees d 0. Commuttivity: / + c/d (d + c)/d The only difference etw een these is in the numertors. c/d + / (c + d)/d But they re equl, y commuttivity of multipliction nd ddition on the set W[or Z] of w hole-numers[or integers]. Therefore, w e might sy commuttivity of ddition on Q is "inherited" from the commuttivity of ddition nd multipliction on W [or Z ]. Associtivity: Are similrly "inherited" from the set of W, since, fter rew riting s equivlent frctions Identity: w ith LCD' s, w e simply dd the w hole numers in the numertors. Inverses: Properties of on Q: Properties re the sme s for W. Mixed numers. Common error. Counting up. Compenstion.
6-B other detils of ddition nd sutrction MATH 0 f Mixed numers: 7 6+ 6+ / The trditionl method: 7 / 7 / 6 / Sutrct the frction from the frction nd 6 6 / / 4 / the w hole numer from the w hole. 4 / Exchnge unit for the pproprite frction if needed. Alterntive: Use improper frctions. 9 9 7 7 / / 4 (7+ /) ( + /) 4 / / 4 / / / 4 / Plus: voids mixed numers t the point of sutrcting. Minus: Look t those FUN ig numers, nd the extr conversion steps t the eginning & end. Alterntive: Count up + / +4 + / * * * * / 4 6 7 7 / 4 4 So w e see the difference is 4 + / 4 + / 4 / Alterntive: compenstion (Bsed in prt on some perceptions from the ove picture): + / + / 6 7 / / 7/ 4/ shifts oth #s up the Motivtion: It s so much esier to sutrct! # line the sme mount. Addition compenstion: / 6 + / 6 (Boost one, drop the other so the / + / / + / / + totl is unchnged.) 6 6 6 6 Common error: ) + ) ) How silly is tht? > ½ + > ½ <???? This is ll BAD SCIENCE!!! ) + ) )???? ) + ) ) 4 Word prolems Three-fifths of the children in Ms Jones clss prticipte on school tems. If children in Ms Jones clss do not prticipte on school tems, how mny children re in Ms. Jones clss? Fifths prtitions the clss into equl-size groups. Since the Boys mke up of those groups, we see tht ech Girls Boys group must contin / 4 children. So there re 4 0 children ltogether in Ms. Jones clss. Mel is wrpping gifts for the techers. Ech gift needs ½ feet of rion. If there re 0 techers getting gifts, how much rion does Mel need? If she ought spool with 0 yrds of rion does she hve enough? Et c 0 4 6 7 9 0
6- MULTIPLICATION AND DIVISION OF RATIONAL NUMBERS f Introduction to multipliction: Whole numer Frction vi REPEATED ADDITION: We see the w hole # EG ) ) + ) ) ) ) + ) + ) ) 4 4 4 4 4 multiplies the numertor. Frction Whole numer vi THE WORD OF : two units With this & few more exmples w e cn see tht EG ) ) of? this multipliction is ) of is ) commuttive, just like in W Frction Frction One Hlf Hlf of one-hlf is: E.g. ) ) Hlf of one-hlf Consider: ) ) ) ) ) ) Which led to our... Def'n. MULTIPLICATION on Q: c c c ) ) ))))) ))) d d d Mke your ow n Models of multipliction for: of of of of 4 4 4 4 Since w e know ll the fctors in the num & den 4 efore w e multiply, w e know w e cn cncel Multiply: )) )) ))))))) )) BEFORE w e multiply... I prefer w ht s show n. Wht out Mixed Numers (they should cll these Mixed Up ) 0 ) 4 ) ( 0 + ) ) ( 4 + ) ) (FOIL w y) BAD Common Error: multiply wholes, multiply frctions, dd. Improper frctions re NOT d: 9 9 0 ) 4 ) ( 0+ ) ) ( 4 + ) ) )) ) ))) 4 APP: One fourth of the students t WHS re freshmen. Of these, fourths re in geometry clss. Wht prt of the WHS students re freshmen tking geometry? Properties of Multipliction on Q: (ssume /. c/d, e/f re rtionls) Closure: c/d Q since c nd d re in W[Z], nd d 0 ecuse 0 nd d 0. Commuttivity: Associtivity: Identity: semi-inherited from W[Z]: / c/d c/d c/d c/d / ditto: (/ c/d) e/f c/d e/f (c)e/(d)f (ce)/(df) etc! is still (k /): / / / / /. Inverses for ll non-zero rtionls /!: If 0 nd 0, then / x /. (The multiplictive inverse of /, provided 0, is /.) Annihiltor: 0 still tkes ll! 0 w htever 0 Distriutive property of multipliction over ddition: still holds.
6-4 DIVISION OF RATIONAL NUMBERS MATH 0 F Notice tht: ½ of is the sme s ¼ of is Further, consider tht: 4 nd so on. ½,, ¼ 4 ½ 4, ½ 6, nd so on. Division y w hole numer seems to e multipliction y the reciprocl. Division y /n seems to e multipliction y n, the reciprocl. Just s 6...since x 6 Def'n. DIVISION on Q: c d...since c d cd x d c d c cd IF c 0 (&,d 0) x d i.e. d is the mgic numer c c tht solves c )? ) d E.g. c d d since c d d c c d c Other 4 4 view s: 4 0 0 4 4 A mixed numer exmple: c d c d d d d d c c c c ) ) x...x must stisfy: ) x ) (Then solve for x.) d d ) ) d d c ) ) ))) )))) ))) Provided c,,d 0 d c c ) ) d c d d x 0 x "Cncelltion" (?) 7 4 4 ) )) )) )) )))))) ))))))) 4 Properties of division on Q: Almost closure: Closure, except for division y 0. / c/d d/c, rtionl, provided c 0. The ility to divide ny numer y lmost ny numer expnds the prolems for w hich w e cn compute nswers. This property goes hnd in hnd with the property of INVERSES for multipliction. is the multiplictive identity. If we sk, in multipliction, is there wy to get ck to the identity, " using multipliction, the nswer is now yes, just multiply y the reciprocl. )? hs n nswer for every frction except 0. But tht, in turn mens tht ) hs n nsw er except w hen 0.
6- Division Word Prolems Mth 0 F Word prolems put the rithmetic to use. However, you my lso oserve the truth of the rules vi the setting of word prolem. Tke, for instnce, this prolem from the homework: After reding 6 pges, Jennifer hd red / of her ook. How mny pges hs the ook? Solution y scling: 6 is prts (of prts). Ech prt is 6 6. The whole is prts, or 6 0 pges. Look t the lger: Let X totl pges ) of X 6 6? So X 6 / See process ove!! Also importnt: Notice tht the prolem ) of X # results in the division X # / You will e sked to mke up word prolem yielding (for exmple) /. The ove oservtion indictes tht prolem of the form: John used / of his fuel, or gllons. How much fuel did he hve to strt? will hve tht result. (4) (4) (4c) Mke word prolem for using mesurement division. thletes w ere ssigned to prctice in tems of. How mny tems?...ditto, ut this time prtitive division. thletes w ere ssigned eqully to service committees. How mny on ech committee?...mesurement division ¾ It tkes ¾ minutes to grde pge of test, how mny pges cn e grded in minutes? (4d)...Prtitive division for 7 / After 7 hours grding, Mry found she w s / done. How long to grde ll the ppers? Br digrms cn e especilly helpful when Frctions re involved. Consider solving this prolem w ithout r digrms, w ithout lger: st nd (e) A pet shop sold / 4 of its puppies in the week, / 6 of the rest in the week. The pet shop sold puppies ltogether during those two weeks. How mny puppies hd they to strt? Our text emphsizes using ONE r digrm. Tht s fine for this prolem, ut we show here the development of the digrm. st nd week week:: Altogether: 7 4 (ech prt) 4 puppies to strt. Using lger: Let X numer of puppies to strt ( / 4)X + ( / 6) ( / 4)X 7 ( / ) X X ( / 7) There were puppies to strt.
6-6 Frctions AS... Mth 0 F Frction Rules Summry c c (Fundmentl Lw ): ) )) c d c d± c (+ Definition): ) ± ) d )) d ± )) d ))))) d Frction s : ) ( 0 ) c c Multipliction : ) ) d )) d c d Division : ) ) d )) c ( c 0 ) The set of frctions (ll numers tht cn e w ritten s / ) hs the follow ing properties: All the rithmetic rules for the Whole numers pply. In ddition: For every memer x of Q, except if x 0, there is nother memer of Q, clled / x, such tht x ) X X) is clled the reciprocl of x, nd is lso clled the multiplictive inverse of x. For instnce, if x is whole numer,,,, then ) X is ), ), ), (respectively). Other frctions such s ) cn e thought of s ()) w hen it suits. Wht is the multiplictive inverse of )? ) See text for proofs of tht these numers stisfy the rules stted in the summry ove. Complex frctions: ) ) + ) ) )) 9 0 7 ( c) + c ) ) 9 + )) 7 ))))))))))) ) 9 )) 7 ) 4 )))))))) ) + 9 7 4 0 7 0 7 0 7 0 9 7 )) )) )) ))))))))) )) )) )) )))))))))) 7 9 7 9
6-6A Complex frctions work: 9 0 7 ) ) + ) ) )) 7 9 9 0 ) + ) )) ( c) + c ) ( ) c ) + c ) c ) ))) + + c c c c ) c ))) ) )))))) & 7 ) 9 )) 0 ) Order of opertions: prentheses first! Prentheses still first, ut w e cn get redy to do the division, y inverting... Then multiplying Write the divisions in frction form. Redy to multiply nd dd... Add ) + )) ) + )) 9 7 9 7 7 ))))))))))) ))))))))))) ) 9 )) 7 ) 9 )) 7 7 6 + ))))))) 6 Get common denomintor c. The esy w y to clen this up multiply numertor & denomintor y LCM(9,7), w hich is 7. Get: no more frctions w ithin frction! Do the sme thing here multiply numertor & denomintor y. ) 6+ 4 4 )))))))) Get )))))))))) ))) ) + + + Alterntely, comine the w hole numers first, then multiply y : ) + ) + 4 4 )))))))) ))))) )))))) ))) ) + + ) 4 + This is prolem # in St 9 ( 6.6 p 6). Although not ssigned, it illustrtes the prolem creted y disorgnized cncelltion. Here is Kyle s w ork, Here is the sme w ork, orgnized. if deciphered correctly: Anticipting reducing the frction, It s hrd to see w ht s een 0 w s fctored into 0, & into cncelled nd w ht s left. 9, so the s cn e reduced. Etc. Try the cncelltion yourself here: 9 7 4 0 7 0 7 0 7 0 9 7 )) )) )) ))))))))) )) )) )) )))))))))) 7 9 7 9 0