Rational Numbers as Fractions

Transcription

1 Rtionl Numbers s Frctions 6 CHAPTER Preliminry Problem Brndy bought horse for $0 nd immeditely strted pying for his keep. She sold the horse for $0. Considering the cost of his keep, she found tht she hd lost n mount equl to hlf of wht she pid for the horse, plus one-fourth of the cost of his keep. How much did Brndy lose on the horse? 0 A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc. ISBN

2 Section 6- The Set of Rtionl Numbers Integers such s - were invented to solve equtions like x + = 0. Similrly, new type of number is needed to solve n eqution like x =. Becuse there is no integer tht stisfies this eqution, number tht does ws invented. The new number must be such tht it is between 0 nd. One reson is tht 0 6 6, which upon substituting x for becomes # 0 6 x 6 #, or 0 6 x 6. There re no integers between 0 nd tht meet these conditions. Therefore, similr to the development of new nottion for negtive integers, we need nottion for this new number. If multipliction is to work with this new number s with whole numbers, then x = x + x, so x + x =. In other words, the number creted must be dded to itself to get. The number invented to solve the eqution is one-hlf, denoted It is n element of the set of numbers of the form where. b, b Z 0 nd nd b re integers. Moreover, numbers of the form re solutions to equtions of b the form bx =. This set, denoted Q, is the set of rtionl numbers nd is defined s follows: Q = e b ƒ nd b re integers nd b Z 0 f Reserch Note Children frequently think of frction s two numbers seprted by line, rther thn s single number (Bn, Frrell, nd McIntosh 99). Q is subset of nother set of numbers clled frctions. Frctions re of the form where b, b Z 0 but nd b re not necessrily integers. For exmple, is frction but not rtionl number. (In this text, we restrict ourselves to frctions where nd b re rel numbers, but tht restriction is not necessry.) The fct tht b Z 0 is lwys necessry, becuse division by 0 is undefined. As indicted in the Principles nd Stndrds excerpt below, children should be introduced to frctions through concrete ctivities. Beyond understnding whole numbers, young children cn be encourged to understnd nd represent commonly used frctions in context, such s / of cookie or /8 of pizz, nd to see frctions s prt of unit whole or of collection. Techers should help students develop n understnding of frctions s division of numbers. And in the middle grdes, in prt s bsis for their work with proportionlity, students need to solidify their understnding of frctions s numbers. (p. ) As is evident from the reserch note, it it importnt to emphsize tht frction such s represents single number. Rtionl Numbers s Frctions Expecttions The Principles nd Stndrds expecttions of students tht re covered in this chpter include the following: ISBN In grdes K, children should understnd nd represent commonly used frctions such s /, /, nd /. (p. 9) In grdes, children should develop n understnding of frctions to include them s prts of unit wholes, s prts of collection, s loctions on number lines, nd s divisions of whole numbers. Children should use models, benchmrks, nd equivlent forms to judge the size of frctions. They should be ble to develop nd use strtegies to estimte computtions involving frctions. (p. 9) A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

3 Rtionl Numbers s Frctions In grdes 6 8, children work flexibly with frctions to solve problems, compre nd order frctions, nd find their loctions on number line. They understnd nd use rtios nd proportions to represent quntittive reltionships. Children understnd the menings nd effects of rithmetic opertions with frctions. They select pproprite methods nd tools for computing with frctions from mong mentl computtion, estimtion, clcultors, or pper nd pencil nd pply the selected method. (p. 9) 6- The Set of Rtionl Numbers In the rtionl number is the numertor nd b is the denomintor. The rtionl number b, my lso be represented s >b or s, b. The word frction is derived from the Ltin word b frctus, mening broken. The word numertor comes from Ltin word mening numberer, nd denomintor comes from Ltin word mening nmer. Tble 6- shows severl wys in which we use rtionl numbers. Tble 6- Uses of Rtionl Numbers Use Exmple Division problem or solution to The solution to x = is. multipliction problem Prtition, or prt, of whole Joe received of Mry s slry ech month for limony. Rtio The rtio of Republicns to Democrts on Sente committee is three to five. Probbility When you toss fir coin, the probbility of getting heds is. Historicl Note / The erly Egyptin numertion system hd symbols for frctions with numertors of. Most frctions with numertors other thn were expressed s sum of different frctions with numertors of, for exmple, = +. Frctions with denomintor 60 or powers of 60 were common in ncient Bbylon bout 000 BCE, where, ment + The method ws lter dopted by the Greek 60. stronomer Ptolemy (pproximtely BCE ). The sme method ws lso used in Islmic nd Europen countries nd is presently used in the mesurements of ngles, where 9 mens + 9 degrees The modern nottion for frctions br between numertor nd denomintor is of Hindu origin. It cme into generl use in Europe in sixteenth-century books. ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

4 Section 6- The Set of Rtionl Numbers Reserch Note When students construct n understnding of the concept of frction, the re model is preferred over the set model becuse the totl re is more flexible, visible ttribute. The re model llows students to encode lmost ny frction, wheres the set model (for exmple, group of colored chips) hs distinct limittions, especilly for prt/whole interprettion. For exmple, try to represent using either four cookies or piece of pper (English nd Hlford 99; Hope nd Owens, 98). ISBN () Are model 0 (b) Number-line model (c) Set model Figure 6- Figure 6- illustrtes the use of rtionl numbers s prt of whole nd s prt of given set. For exmple, in the re model in Figure 6-(), one prt out of three congruent prts, or of the lrgest rectngle, is shded. In Figure 6-(b), two prts out of three congruent prts, or of the unit segment, re shded. In Figure 6-(c), three circles out of five congruent circles, or of the circles, re shded. Notice tht s indicted in the Reserch Note, the re model is the preferred model. Our erly exposure to frctions, or rtionl numbers, usully tkes the form of orl description rther thn mthemticl nottion. We her phrses such s one-hlf of pizz, one-third of cke, or three-fourths of pie. We encounter such questions s If three identicl fruit brs re distributed eqully mong four friends, how much does ech get? The nswer is tht ech receives of br. The English words used for rtionl numbers re the sme words we use to tell order, for exmple, the fourth person in line nd three-fourths for This cuses confusion for. students lerning bout frctions. In contrst, in Chinese is red out of four prts, (tke) three. The Chinese model enforces the ide of prtitioning quntities into equl prts nd choosing some number of these prts. The concept of shring quntities nd compring sizes of shres cn provide n entry point to introduce students to rtionl numbers. For rtionl numbers, shring cn ply the role tht counting does for whole numbers. When rtionl numbers re introduced s frctions tht represent prt of whole, we must py ttention to the whole from which rtionl number is derived. For exmple, if we tlk bout of pizz, then the mount of pizz is determined both by the frctionl prt, nd the size of the pizz. Three-fourths of lrge pizz is certinly more thn, three-fourths of smll pizz. Attention must be pid to the context nd the size of the whole being considered. To understnd the mening of ny frction, using the prts-to-whole model, we must b, consider ech of the following:. The whole being considered.. The number b of equl-size prts into which the whole is divided.. The number of prts of the whole tht we re selecting. A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

5 Rtionl Numbers s Frctions The crtoon shows tht the concept of frction is not esy for some children. NOW TRY THIS 6-. How would you nswer Billy s question in the crtoon? b. Jim clims tht becuse in Figure 6- the shded portion for is lrger thn the shded portion for How would you help him?. () (b) Figure 6- Rtionl numbers cn be represented on number line. Once the integers 0 nd re ssigned to points on line, the unit segment is defined nd every other rtionl number is ssigned to specific point. For exmple, to represent on the number line, we divide the segment from 0 to into segments of equl length nd mrk the line ccordingly. Then, strting from 0, we count of these segments nd stop t the mrk corresponding to the right endpoint of the third segment to obtin the point ssigned to the rtionl number Figure 6- - shows the points tht correspond to -, nd., - -,, 0,,,., A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc. ISBN

6 Section 6- The Set of Rtionl Numbers 0 Figure 6- In the grde Focl Points, we find tht students develop n understnding of the mening nd uses of frctions to represent prts of whole, prts of set, or points or distnces on number line (p. ). NOW TRY THIS 6-. If represents drw rectngles for nd,,,,,. b. In Figure 6-, locte points tht correspond to nd -, -,,. A frction where 0 6 b, is proper frction. For exmple, is proper b, 9 frction, but nd re not; is n improper frction. In generl is n improper,, b frction if Ú b 0. Typiclly, in erly grdes students re introduced only to positive frctions. An exmple of this is shown on the following student pge. NOW TRY THIS 6- Answer the three Tlk About It questions on the student pge (pge 6). REMARK Notice tht every integer n cn be represented s rtionl number becuse where is ny non-zero integer. Thus if then 0 = 0 # k n = nk k k Z 0 = 0 k, k k. NOW TRY THIS 6- Drw Venn digrm to show the reltionship mong nturl numbers, whole numbers, integers, nd rtionl numbers. ISBN Equivlent or Equl Frctions Frctions cn be introduced in the clssroom through concrete ctivity such s pper folding. In Figure 6-(), one of congruent prts, or is shded. In this cse, the whole, is the rectngle. In Figure 6-(b), ech of the thirds hs been folded in hlf so tht now we hve 6 sections, nd of 6 congruent prts, or re shded. Thus, both nd represent exctly the sme shded portion. Although the symbols nd do not look like, 6, 6 6 they represent the sme rtionl number nd re equivlent frctions or equl frctions. A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

7 6 Rtionl Numbers s Frctions School Book Pge NAMING FRACTIONAL PARTS Source: Mthemtics, Dimond Edition, Grde, Scott Foresmn-Addison Wesley, 008 ( p. 0). ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

8 Section 6- The Set of Rtionl Numbers E-Mnipultives Activity For prctice with the concept of frctions, see the Nming Frctions nd Visulizing Frctions modules on the E-Mnipultives disk. Equivlent frctions re numbers tht represent the sme point on number line. Becuse they represent equl mounts, we write nd sy tht equls = () (b) (c) Figure 6- Figure 6-(c) shows the rectngle with ech of the originl thirds folded into equl prts with of the prts now shded. Thus, is equl to becuse the sme portion of the model is shded. Similrly, we could illustrte tht re equl., 6, 9,,, Á Frction strips cn lso be used for generting equivlent frctions, s seen on the student pge. Prt () on the student pge shows tht Wht equivlent frctions re = 6 = 6. modeled in prt (b)? Also on the student pge on pge 8, in Exmple A we see nother wy to find equivlent frctions. This technique mkes use of the Fundmentl Lw of Frctions, which cn be stted s follows: The vlue of frction does not chnge if its numertor nd denomintor re multiplied by the sme nonzero integer. Under certin ssumptions this Lw of Frctions cn be proved nd hence it is stted s theorem. Theorem 6 : Let b Fundmentl Lw of Frctions be ny frction nd n nonzero integer. Then, b = n bn. REMARK. In Theorem 6 it cn be shown tht n cn be ny nonzero number. This version of the Fundmentl Lw of Frctions is used lter in this book.. Theorem 6 implies tht if d is common fctor of nd b then. b =, d b, d. At this point, some students justify the Fundmentl Lw of Frctions s follows: n bn = # n b # = # n n b n = # = b b This is correct pproch; however in our development of properties of frctions, we hve not yet discussed multipliction of frctions. ISBN NOW TRY THIS 6- Explin why in the Fundmentl Lw of Frctions n must be nonzero. A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

9 8 Rtionl Numbers s Frctions School Book Pge EQUIVALENT FRACTIONS Source: Scott Foresmn-Addison Wesley Mthemtics, Grde 6, 008 ( p. 6). ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

10 Section 6- The Set of Rtionl Numbers 9 - From the Fundmentl Lw of Frctions, - = becuse - - Similrly, - = The form where b is positive number, is usully preferred. b, b b. - = # - - # - = -. Exmple 6- Find vlue for x such tht = x 0. Solution Becuse 0, =, we use the Fundmentl Lw of Frctions to obtin x Hence, nd x = = 60 = # # = 60 0, 0. Alterntive pproch: Therefore x = 60. = # 6 # = 6 = # 0 # = Simplifying Frctions The Fundmentl Lw of Frctions justifies the process of simplifying frctions. Consider the following: Also, 60 0 = 6 # 0 # = = # # = 60 We cn simplify becuse the numertor nd denomintor hve common fctor of We cn simplify becuse 6 nd hve common fctor of. However, we cnnot simplify becuse nd hve no positive common fctor other thn. Notice tht we could lso simplify in one step by: is the simplest form of 0 = # 0 # = becuse both 60 nd 0 hve been divided by their gretest common divisor, 0. To write frction in simplest form; tht is, in lowest terms, we divide both nd b by the GCD(, b). b Definition of Simplest Form A rtionl number is in simplest form if b 0 nd GCD, b = ; tht is, if nd b hve no b common fctor greter thn, nd b 0. ISBN We cn use scientific/frction clcultors to simplify frctions. For exmple, to simplify 6 we enter 6 / nd press SIMP =, nd ppers on the screen. At this point,, 6 A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

11 0 Rtionl Numbers s Frctions n indictor tells us tht this is not in simplest form, so we press SIMP = gin to obtin At ny time, we cn view the fctor tht ws removed by pressing the x y key.. The Fundmentl Lw of Frctions cn be used to simplify lgebric expressions, s seen in the following exmple. Exmple 6- Write ech of the following frctions in simplest form if they re not lredy so:. 8b + b x + x + x b. c. d. b + b x + x e. + x - b + b f. g. x - b + b Solution. b. 8 b b = b b = + b + b + b = = + b, where + b Z 0 + b + b c. x + x xx + xx + = =, where x Z - x + x + x + = x = x d. + x cnnot be simplified becuse + x nd x hve no fctors in common except. x e. + x x = + x x = + x x f. Recll tht in Chpter we used the distributive properties to show tht - b + b = - b. Thus, - b - b + b = = + b = + b, where Z b - b - b g. The frction is lredy in simplest form becuse + b does not hve + b s fctor. Notice tht + b Z + b. REMARK. When we write n lgebric expression tht is frction, we must ssume tht the denomintor is not 0. Thus, even fter the frction is reduced, this restriction hs to x + x be mintined. For exmple, in prt (c) of Exmple 6-, if x Z -, nd x + = x in prt (f ) the result holds if - b Z 0; tht is, if Z b. + x. In prt (d) we pointed out tht cnnot be simplified. However, for some prticulr vlues of x, the corresponding frction cn be simplified. This is the cse if x is x + x multiple of ; for exmple, if then the vlue of is + 9 # = # x =, 9 # = x 9 9. ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

12 Section 6- The Set of Rtionl Numbers. Some students, thinking bout the Fundmentl Lw of Frctions s cncelltion 6 + # + # property, often simplify n expression like by thinking of it s nd cnceling equl numbers in the products to obtin + s the nswer. Emphsizing the fctor pproch tht neither nor is fctor of 6 + my help to void such mistkes. ISBN E-Mnipultive Activity Additionl prctice with ordering frctions is vilble in the Rnking Frctions module on the E-Mnipultive disk. Equlity of Frctions 0 We cn use three methods to show tht two frctions, such s nd re equl.,. Simplify both frctions to the sme simplest form: Thus,. Rewrite both frctions with the sme lest common denomintor. Since LCM, = 0, then Thus,. Rewrite both frctions with common denomintor (not necessrily the lest). A common multiple of nd my be found by finding the product #, or 0. Now, Hence, = # # # = = 60 0 = 0 0 nd 0 = # # = = 0 nd 0 = 60 0 = 0 nd 0 = 0 0 = 0 c The third method suggests generl lgorithm for determining if two frctions nd re b d equl. Rewrite both frctions with common denomintor bd. Tht is, b = d bd nd c d = bc bd d Becuse the denomintors re the sme, if, nd only if, d = bc. For exmple, bd = bc bd becuse # 9 = 6 = 6 # 6. In generl, the following theorem results. 6 = 6 9 A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

13 Rtionl Numbers s Frctions Theorem 6 c Two frctions nd re equl if, nd only if, d = bc. b d Using clcultor, we cn determine if two frctions re equl by using the preceding property. Since both * 9 6 = nd * = yield disply of 9, we see tht = Ordering Rtionl Numbers As discussed in Principles nd Stndrds (p. 9), students in lower grdes should experience compring frctions between 0 nd in reltion to such benchmrks s 0, nd. In the,, middle grdes, the comprison of frctions becomes more difficult. We first consider the comprison of frctions with like denomintors. Children know tht becuse if pizz is divided into 8 prts of equl size, then 8 8 prts of pizz is more thn prts. Similrly, Thus, given two frctions with common positive denomintors, the one with the greter numertor is the greter frction. 6. This cn be stted s follows. Theorem 6 If, b, nd c re integers nd b 0, then if, nd only if, c. b c b Compring the size of frctions is shown on the following student pge. NOW TRY THIS 6-6 Determine if Theorem 6 is true if b 6 0. To compre frctions with unlike denomintors, some students my incorrectly reson 6 tht becuse 8 is greter thn. In other cses, they might flsely believe tht is 8 equl to becuse in both cses the difference between the numertor nd the denomintor is. Compring frctions with unlike denomintors my be ided by using 8 frction ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

14 Section 6- The Set of Rtionl Numbers School Book Pge ISBN Source: Mthemtics, Dimond Edition, Grde Three, Scott Foresmn-Addison Wesley, 008 ( p. 0). A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

15 Rtionl Numbers s Frctions strips to compre the frctions visully. For exmple, consider the frctions nd shown in Figure 6-. From Figure 6-, students see tht ech frction is one piece less thn the sme-size whole unit. However, they see tht the missing piece for is smller thn the missing piece for so must be greter thn,. Compring frctions with unlike denomintors cn be ccomplished by rewriting the frctions with the sme positive denomintor. Then we cn compre them using previously c lerned technique. Using the common denomintor bd, we cn write the frctions nd s b d d bc nd Becuse b 0 nd d 0, then bd 0; nd we cn pply Theorem 6 s follows: bd bd. b c d if, nd only if, d bd bc bd Therefore, we hve the following theorem. Figure 6- nd d bd bc bd if, nd only if, d bc Theorem 6 If, b, c, nd d re integers with b 0 nd d 0, then if, nd only if, d bc. b c d 9 NOW TRY THIS 6- Order the frctions nd from lest to gretest., 6, 8, Reserch Note Students tught the common denomintor method for compring two frctions tend to ignore it nd focus on rules ssocited with ordering whole numbers. Students who correctly compre numertors if the denomintors re equl often compre denomintors if the numertors re equl (Behr et l. 98). ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

16 Section 6- The Set of Rtionl Numbers As suggested in the Reserch Note, if students compre denomintors of frctions if the numertors re the sme, then this cn be good strtegy if hndled properly. For exmple, consider nd If the whole is the sme for both frctions, this mens tht we hve three 0. nd three Becuse is greter thn then three of the lrger prts is greter thn 0. 0, three of the smller prts, so 0. NOW TRY THIS 6-8 Generlize the pproch bove for compring frctions whose numertors nd denomintors re positive integers. Denseness of Rtionl Numbers The set of rtionl numbers hs property tht the set of whole numbers nd the set of integers do not hve. Consider nd To find rtionl number between nd we., first rewrite the frctions with common denomintor, s nd Becuse there is no 6 6. whole number between the numertors nd, we next find two frctions equl, respectively, to nd with greter denomintors. For exmple, nd nd is = 8 = 6, 6 8 between the two frctions nd So is between nd This property is generlized s follows nd stted s.. theorem. Theorem 6 : Denseness property for rtionl numbers c Given two different rtionl numbers nd there is nother rtionl number between these b d, two numbers. NOW TRY THIS 6-9 Explin why there re infinitely mny rtionl numbers between ny two rtionl numbers. ISBN Exmple 6-. Find two frctions between nd 8. b. Show tht the sequence is n incresing sequence, tht is, tht ech term,,,, Á strting from the second term is greter thn the preceding term. 8 9 Solution. Becuse we see tht or is between nd To find = # 9 # = 9 8, 9, , nother frction between the given frctions, we find two frctions equl to A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

17 6 Rtionl Numbers s Frctions 8 nd 9 8, respectively, but with greter denomintors; for exmple, 9 nd 8 = We now see tht nd re ll between nd nd thus 6, 6 6, between nd 8. n n + b. Becuse the nth term of the sequence is the next term is n +, n + +, n + or We need to show tht for ll positive integers n, n +. n + n + 8 = 6 n n +. By Theorem 6, the inequlity will be true if, nd only if, n + n + nn +. This inequlity is equivlent to n + n + n + n, which is true for ll n since the left side is greter thn the right side. + c Some students incorrectly dd s Although the technique does not produce b + c d b + d. the correct sum, s will be seen in Section 6-, it provides wy to find number between ny two rtionl numbers. In Exmple 6-, to find number between nd we could 8 + dd the numertors nd dd the denomintors to produce We see tht 8 + = becuse 0 6. Also, becuse We stte the generl property in the following theorem, whose proof is explored in Now Try This Theorem 6 6 c Let nd be ny rtionl numbers with positive denomintors, where Then, b 6 c b d d. b 6 + c b + d 6 c d. c NOW TRY THIS 6-0 Prove Theorem 6 6; tht is, if nd re ny rtionl numbers with positive b d denomintors, where, then. Hint: If, then d 6 bc. Use this to prove b 6 c b 6 + c b + d 6 c b 6 c d d d tht nd + c b + d 6 c b 6 + c b + d d.b REMARK Notice tht the proof of Theorem 6 6 suggested in Now Try This 6-0 lso proves Theorem 6. ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

18 Section 6- The Set of Rtionl Numbers LABORATORY ACTIVITY Obtin tngrm pieces or build them nd cut them out s shown in Figure 6-6. Answer ech of the following.. If the re of the entire squre is squre unit, find the re of ech tngrm piece. b. If the re of piece is squre unit, find the re of ech tngrm piece. b c d e f g Figure 6-6 Assessment 6-A. Write sentence tht illustrtes the use of in ech of 8 the following wys:. As division problem b. As prt of whole c. As rtio. For ech of the following, write frction to represent the shded portion:. For ech of the following four squres, write frction to represent the shded portion. Wht property of frctions does the digrm illustrte?. b. c. d.. b.. Bsed on your observtions could the shded portions in the following figures represent the indicted frctions? Tell why. ISBN c. d.. b. c. A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

19 8 Rtionl Numbers s Frctions. In ech cse, subdivide the whole shown on the right to show the equivlent frction.. b. c Referring to the figure, represent ech of the following s frction: 9. Determine if the following pirs re equl by chnging both to the sme denomintor: nd b. nd Drw n re model to show tht = If frction is equl to nd the sum of the numertor nd denomintor is 8, wht is the frction?. Mr. Gomez filled his cr s 6 gl gs tnk. He took short trip nd used 6 gl of gs. Drw n rrow in the following figure to show wht his gs guge looked like fter the trip:. The dots in the interior of the circle s prt of ll the dots b. The dots in the interior of the rectngle s prt of ll the dots c. The dots in the intersection of the interiors of the rectngle nd the circle s prt of ll the dots d. The dots outside the circulr region but inside the rectngulr region s prt of ll the dots. For ech of the following, write three frctions equl to the given frction: -. b. 9 0 c. d. 8. Find the simplest form for ech of the following frctions: 6-6. b. c For ech of the following, choose the expression in prentheses tht equls or describes best the given frction: 0., undefined, 0 0 b. undefined,, c. undefined,, 0 + d.,, cnnot be simplified + x e. + x,, cnnot be simplifiedb x x 0. Find the simplest form for ech of the following: -b x y. b. + b 6xy. Determine if the following pirs re equl: 8. nd b. nd Solve for x in ech of the following: -. b. = x 6 = x. For ech of the following pirs of frctions, replce the comm with the correct symbol 6, =, to mke true sttement: - -. b. c. 8, 6, 6 8, 8. Arrnge ech of the following in decresing order:., 6, b., 6, 0 9. Show tht the sequence (in which ech,,, 6,, 6 8, Á successive term is obtined from the previous term by dding to the numertor nd the denomintor), is n incresing sequence; tht is, show tht ech term in the sequence is greter thn the preceding one. 0. For ech of the following, find two rtionl numbers between the given frctions:. nd b. nd 9 9. A scle on mp is mi to the inch. Wht is the irline milege between two cities tht re 8 in. prt on the mp?.. Six oz is wht prt of pound? A ton? b. A dime is wht frction of dollr? c. min is wht frction of n hour? d. 8 hr is wht frction of dy? ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

20 Section 6- The Set of Rtionl Numbers 9 Assessment 6-B. Write sentence tht illustrtes the use of in ech of 0 the following wys:. As division problem b. As prt of whole c. As rtio. For ech of the following, write frction to represent the shded portion:.. Could the shded portions in the following figures represent the indicted frctions? Tell why. 8. b. c.. If ech of the following models represents the given frction, drw model tht represents the whole. Shde your nswer.. b. 8 b. c. c. d. 6. Bsed on your visul observtion write frction to represent the shded portion.. Complete ech of the following figures so tht it shows : 0. b.. Referring to the figure, represent ech of the following s frction: ISBN c. d. e. f.. The dots outside the circulr region s prt of ll the dots b. The dots outside the rectngulr region s prt of ll the dots c. The dots in the union of the rectngulr nd the circulr regions s prt of ll the dots d. The dots inside the circulr region but outside the rectngulr region s prt of ll the dots 8. Find the simplest form for ech of the following frctions: b. c Mr. Gonzles nd Ms. Price gve the sme test to their fifth-grde clsses. In Mr. Gonzles s clss, 0 out of students pssed the test, nd in Ms. Price s clss, out A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

21 60 Rtionl Numbers s Frctions of 0 students pssed the test. One of Ms. Price s students herd bout the results of the tests nd climed tht the clsses did eqully well. Is the student right? Explin. 0. For ech of the following, choose the expression in prentheses tht equls or describes best the given frction: 6 + x. + x,, cnnot be simplifiedb x x 6 + b. +,, cnnot be simplifiedb c ,, too lrge to simplifyb. Find the simplest form for ech of the following: + b. b. + b + b. Determine if the following pirs re equl:,0. nd b. nd 8 0,000. Determine if the following pirs re equl by chnging both to the sme denomintor: nd b. nd 0. A bord is needed tht is exctly wide to fill hole. in. Cn bord tht is be shved down to fit the hole? If 8 in. so, how much must be shved from the bord?. The following two prking meters re next to ech other with the times left s shown. Which meter hs more time left on it? How much more? 8. Solve for x in ech of the following:. = x 8 b. x = x x 9.. If wht must be true? c = b c, b. If wht must be true? b = c, 0. In Amy s lgebr clss, 6 of the students received As on test. The sme test ws given to Bren s clss nd of the students received As. Which clss hd the higher rte of As?. For ech of the following pirs of frctions, replce the comm with the correct symbol 6, =, to mke true sttement: 0. b. c., 0, -, c d. If b nd c 0, d 0, compre the size of c with bd.. Show tht the sequence is decresing,,,, 6, 6 sequence; tht is, show tht ech term in the sequence is less thn the preceding one.. For ech of the following, find two rtionl numbers between the given frctions: 8 -. nd b. nd Consider the following number grid. The circled numbers form rhombus (tht is, ll sides re the sme length) hr 0 hr b. c. d. D Meter A C B A Meter B 6. Red ech mesurement s shown on the following ruler: 0 Inches. Determine if ech of the following is lwys correct. If not find when it is true. Explin. b + c + b. = + c b. + c = b b c b + c c. d. + = b + c = + c c If A is the sum of the four circled numbers nd B is A the sum of the four interior numbers, find B. b. Form rhombus by circling the numbers 6, 8,, A nd. If A nd B re defined s in (), find B. c. How do the nswers in () nd (b) compre? Why does this hppen? 6. A scle on mp is 0 mi to the inch. Wht is the irline milege between two cities tht re in. prt on the mp?.. oz is wht prt of pound? b. A nickel is wht frction of dollr? c. min is wht frction of n hour? d. 6 hr is wht frction of hr dy? ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

22 Section 6- The Set of Rtionl Numbers 6 Mthemticl Connections 6- ISBN Communiction. Jne hs recipe tht clls for c of flour. She wnts to mke of the recipe. Insted of determining directly how mny cups re needed for the new recipe, she fills of cup times. Explin why Jne s method works.. Ann clims tht she cn t show of the following fces becuse some re big nd some re smll. Wht do you tell her?. A student clims tht becuse if you dd to both = 6 the top nd the bottom of frction, the frction does not chnge. How do you respond?. In ech of two different fourth-grde clsses, of the members re girls. Does ech clss hve the sme number of girls? Explin.. Consider the set of ll frctions equl to If you tke. ny 0 of those frctions, dd their numertors to obtin the numertor of new frction nd dd their denomintors to obtin the denomintor of new frction, how? does the new frction relte to Generlize wht you found nd explin. 6. Should frctions lwys be reduced to their simplest form? Why or why not?. How would you respond to ech of the following students?. Iris clims tht if we hve two positive rtionl numbers, the one with the greter numertor is the greter. b. Shirley clims tht if we hve two positive rtionl numbers, the one with the greter denomintor is the lesser. 8. If we tke the set of frctions equivlent to nd grph them s points on coordinte system so tht the numertor becomes the x-coordinte nd the denomintor becomes the y-coordinte for ech point, explin wht type of grph we would get. 9. Write n explntion of how to convert inches to yrds nd vice vers. Open-Ended 0. List three types of mesure tht require rtionl numbers s the pproprite number of units in the mesurements.. Some people hve rgued tht the system of integers is more understndble thn the system of positive rtionl numbers. If you could decide which should be tught first in school, which would you choose nd why?. Mke three sttements bout yourself or your environment nd use frctions in ech. Explin why your sttements re true (for exmple, your prents hve three children, two of whom live t home; hence of their children live t home). Coopertive Lerning. Assume the tllest person in your group is unit tll nd do the following:. Find rtionl numbers to pproximtely represent the heights of other members of the group. b. Mke number line nd plot the rtionl number for ech person ordered ccording to height. Questions from the Clssroom 0. A student sks if is in its simplest form. How do you 6 respond?. A student writes becuse # 6 #. Another 6 student writes Where is the fllcy? = 6. The numertors nd the denomintors of the following frctions form rithmetic sequences. A student clims tht the frctions form n rithmetic sequence. How would you respond?,,,, 6, 6, 8, Á. A student clims tht there re no numbers between 999 nd becuse they re so close together. Wht is 000 your response? m + n m 8. A student simplified the frction to How p + n p. would you help this student? 9. Without thinking, student rgued tht pizz cut into pieces ws more thn pizz cut into 6 pieces. How would you respond? 0. A student sks if dding the sme very lrge number to both the numertor nd the denomintor of frction yields quotient of. How do you respond? d. Joe reported tht if then c - d = b b = c d Z, - b. Jon sid she didn t believe it. How do you respond? A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

23 6 Rtionl Numbers s Frctions. Steve clims tht the shded portions cn t represent since there re circles shded nd is less thn. How do you respond? Third Interntionl Mthemtics nd Science Study (TIMSS) Questions Which shows of the squre shded? A B C D E. Dryl sys tht ech piece of the pie shown represents of the pie. How do you respond? TIMSS 00, Grde In the figure, how mny MORE smll squres need to be shded so tht of the smll squres re shded?. b. c. d. e... Cssidy noticed tht, but In generl 6. she thinks tht if nd b re positive integers, nd b, then She would like to know if this is 6 b. lwys true. How do you respond? b. Cssidy would like to know if her discovery in prt () is true when nd b re negtive integers, nd if so, why or why not. How do you respond?. Sven would like to know if the following sttement is true, nd why or why not. How do you respond? - - If then 6 x 6 6 x Crl sys tht becuse nd 8. How 8 would you help Crl? TIMSS 00, Grde 8 Ntionl Assessment of Eductionl Progress (NAEP) Question Wht frction of the figure is shded? NAEP, Grde, 00 In which of the following re the three frctions rrnged from lest to gretest?. b. c., 9,,,,, 9 9 d. e. 9,, 9,, NAEP, Grde 8, 00 BRAIN TEASER In n old Sm Loyd puzzle, wtch is described s hving stopped when the minute nd hour hnds formed stright line nd the second hnd ws not on. At wht times cn this hppen? ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

24 Section 6- Addition, Subtrction, nd Estimtion with Rtionl Numbers 6 6- Addition, Subtrction, nd Estimtion with Rtionl Numbers Addition nd subtrction of rtionl numbers is very much like ddition nd subtrction of whole numbers nd integers. We first demonstrte the ddition of two rtionl numbers with like denomintors, using n re model in Figure 6-() nd number-line +, model in Figure 6-(b). () + = (b) ISBN Reserch Note Students lerning computtionl lgorithms involving frctions hve difficulty connecting their models involving mnipultives with symbolic procedures. This personl competence with rote procedure my outstrip their conceptul understnding of frctions; one result cn be tht students unble to judge the resonbleness of the nswer cn check their work only by repeting the rote procedure. (Werne nd Hiebert 988). 0 Figure 6- Why does the re model in Figure 6-() mke sense? Suppose tht someone gives you of pie to strt with nd then gives you nother of the pie. In Figure 6-(), is represented by pieces when the pie is cut into equl-size pieces nd is represented by piece of the equl-size pieces. So ll together you hve + = pieces of the equlsize pieces, or of the totl (whole) pie. The number-line model in Figure 6-(b) works the sme s the number-line model for whole numbers. The ides illustrted in Figure 6- cn be pplied to the sum of two rtionl numbers with like denomintors nd re summrized in the following definition. Definition of Addition of Rtionl Numbers with Like Denomintors c If nd re rtionl numbers, then b + c b = + c. b b b Next we consider the ddition of two rtionl numbers with unlike denomintors, using Poly s four-step, problem-solving process. A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

25 6 Rtionl Numbers s Frctions Problem Solving Adding Rtionl Numbers Problem Determine how to dd the rtionl numbers Understnding the Problem We cn model nd s prts of whole, s seen in Figure 6-8, but we need wy to combine the two drwings to find the sum. nd. Figure Figure 6-9 Devising Pln We use the strtegy of solving relted problem: dding rtionl numbers with the sme denomintors. We cn find the sum by writing ech frction with common denomintor nd then completing the computtion. Crrying Out the Pln From erlier work in this chpter, we know tht hs infinitely mny representtions, including nd so on. Also hs infinitely mny representtions, including nd so on. By compring the two sets of rtionl numbers, we see 6, 6 9, 8, 8,, 6, 8 tht nd hve the sme denomintor. One is 8 prts of equl prts, while the other is prts of equl prts. Consequently, the sum is Figure = 8 + =. illustrtes the ddition. Looking Bck To dd two rtionl numbers with unlike denomintors, we considered equl rtionl numbers with like denomintors. The common denomintor for nd is. This is lso the lest common denomintor, or the LCM of nd. To dd two frctions with unequl denomintors such s nd we could find equl frctions with the denomintor 8, LCM(, 8), or 6. However, ny common denomintor will work s well, for exmple, or even # 8. By considering the sum we cn generlize to + = # # + # # = 8 + =, finding the sum of two rtionl numbers with unlike denomintors, s in the following theorem. ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

26 Section 6- Addition, Subtrction nd Estimtion with Rtionl Numbers 6 Theorem 6 c If nd re ny two rtionl numbers, then b + c b d d d + bc =. bd Reserch Note The most frequent error students mke when dding frctions is dding the numertors nd dding the denomintors (Bn, Frrell, nd McIntosh 99). As pointed out in the Reserch Note, students don t lwys dd frctions correctly. In grde Curriculum Focl Points, students re expected to pply their understnding of frction models to represent the ddition nd subtrction of frctions with unlike denomintors (p. ). REMARK From the definition of ddition of rtionl numbers with like denomintors we hve The sme result would be obtined if we used Theorem 6 : b + c b = + c. b b + c b + cb + cb = b b # = b b # = + c. b b Exmple 6- Find ech of the following sums:. b c. + b + 6 d. e. b + x + y b LCM, = # #, Solution. Becuse then = = + - ) b. c. hence, + = # - ) + # # = 9 0 ; 9 # # 6 or 0 # = , d. x + y = y xy + x y + x =. xy xy e. LCM b, b = b ; b + b = + = # # + # # = = = - = = + b + 6 = = b b # b + # b = b + b. -. ISBN Mixed Numbers In everydy life, we often use mixed numbers; tht is, numbers tht re mde up of n integer nd proper frction. For exmple, Figure 6-0 shows tht the nil is long. The mixed number mens + It is sometimes inferred tht mens in.. A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

27 66 Rtionl Numbers s Frctions times since xy mens but this is not correct. Also, the number - x # y, mens, or - not - + -,. - b, 0 Inches Figure 6-0 REMARK In Ntionl Assessment of Eductionl Progress (NAEP) test, students were given the following problem: is the sme s: () + (b) - (c) * (d), Only % of the seventh grders chose the correct response, (), nd only % of the eleventh grders chose the correct response. A mixed number is rtionl number, nd therefore it cn lwys be written in the form For exmple, b. = + = + = # + # # = 8 + = Exmple 6- Chnge ech of the following mixed numbers to the form. b. - b, where nd b re integers: Solution. = + = + = # + # # = + = b. - = - + b = - + b = - # + # # b = - b NOW TRY THIS 6- Use the ides in Exmple 6- to write s mixed number. + 8 ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

28 Section 6- Addition, Subtrction nd Estimtion with Rtionl Numbers 6 Exmple 6-6 Chnge 9 to mixed number. Solution We divide 9 by nd use the division lgorithm s follows: 9 = # + = # + = + = REMARK In elementry schools, problems like Exmple 6-6 re usully computed using division, s follows: 9 Hence, 9 = + =. We cn use scientific/frction clcultors to chnge improper frctions to mixed numbers. For exmple, if we enter 9 / nd press Ab/c, then > ppers, which mens. We cn lso use scientific/frction clcultors to dd mixed numbers. For exmple, to dd we enter + 6, Unit / + Unit =, nd the disply reds 9 9>0. We then press Ab/c to obtin 6 9>0, which mens 6 0. Becuse mixed numbers re rtionl numbers, the methods of dding rtionls cn be extended to include mixed numbers. The student pge (pge 68) shows method for computing sums of mixed numbers. Work the three problems on the bottom of the student pge. Notice tht ll the frctions on the student pge re nonnegtive. This is typicl in grdes 6. / 6 Properties of Addition for Rtionl Numbers Rtionl numbers hve the following properties for ddition: closure, commuttive, ssocitive, dditive identity, nd dditive inverse. To emphsize the dditive inverse property of rtionl numbers, we stte it explicitly, s follows. ISBN Theorem 6 8: Additive Inverse Property of Rtionl Numbers For ny rtionl number there exists unique rtionl number - the dditive inverse of b, b, b, such tht b + - b b = 0 = - b b + b A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

29 68 Rtionl Numbers s Frctions School Book Pge ADDING MIXED NUMBERS Source: Mthemtics, Dimond Edition, Grde Five, Scott Foresmn-Addison Wesley, 008 ( p. 6). ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

30 Section 6- Addition, Subtrction nd Estimtion with Rtionl Numbers 69 Another form of - cn be found by considering the sum Becuse b + - b b. b + - b = + - b = 0 b = 0 - it follows tht - nd re both dditive inverses of so - b = - b b b, b. Exmple 6- Find the dditive inverses for ech of the following: -. b. c. - Solution. or b c. - or b = =, - 9 Properties of the dditive inverse for rtionl numbers re nlogous to those of the dditive inverse for integers, s shown in Tble 6-. As with the set of integers, the set of rtionl numbers lso hs the ddition property of equlity, which sys tht you cn dd the sme number to both sides of n eqution. Tble 6- Integers. -- =. Rtionl Numbers - - b b = b.. - b + c b = - + -b d b = b + - c d Theorem 6 9: Addition Property of Equlity c e If nd re ny rtionl numbers such tht nd if is ny rtionl number, then b = c b d d, f b + e f = c d + e f. REMARK Theorem 6 9 cn be stted s follows: If x = y then x + z = y + z where x, y, z re ny rtionl numbers. ISBN Subtrction of Rtionl Numbers In elementry school, subtrction of rtionl numbers is usully introduced by using tkewy model. If we hve of pizz nd of the originl pizz is tken wy, of the pizz 6 A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

31 0 Rtionl Numbers s Frctions 6 remins; tht is, In generl, subtrction of rtionl numbers with like - = 6 - =. denomintors is determined s follows: b - c b = - c b As with integers, number line cn be used to model subtrction. If line is mrked off in units of length then is equl to - c units of length which implies tht When the denomintors re not the sme we cn perform the subtrction b - c b = - c b - c b, b b,. b by finding common denomintor. For exmple, - = # # - # # = 9-8 = 9-8 = Subtrction of rtionl numbers, like subtrction of integers, cn be defined in terms of ddition s follows. Definition of Subtrction of Rtionl Numbers in Terms of Addition c e If nd re ny rtionl numbers, then is the unique rtionl number such tht b - c b d d f b = c d + e f. As with integers, we cn see tht subtrction of rtionl numbers cn be performed by dding the dditive inverses. The following theorem sttes this. Theorem 6 0 c If nd re ny rtionl numbers, then b - c d = - c b d b + d. Now, using Theorem 6 0, we obtin the following: (Theorem 6 0) b - c d = b + - c d = d + b- c (Theorem 6 ) bd = d + - bc (Multipliction with integers) bd d - bc = bd ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

32 Section 6- Addition, Subtrction nd Estimtion with Rtionl Numbers We proved the following theorem: Theorem 6 c If nd re ny rtionl numbers, then b - c b d d d - bc =. bd Exmple 6-8 Find ech difference in the following:. b Solution. One pproch is to find the LCM for the frctions. Becuse LCM8, = 8, we hve 8 - = 8-8 = 8 An lterntive pproch is s follows: b. Two methods of solution re given: = = + = 6 - = - 9 = = # - 8 # 8 # = = - =, or 8 - = 6 - = 6 # - # # = 6 - =, or The following exmples show the use of frctions in lgebr. Exmple 6-9 Add or subtrct ech of the following. Write your nswer in simplest form.. x - x b. 6 - x + - x + x x - 6 c. d. x - + b - - b x ISBN Solution. x + x = x # + x # = x + x 6 = x 6 A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

33 Rtionl Numbers s Frctions b. We first write ech frction in simplest form: d. x - x = x # x # - x x - x 6 - x = - x - x = # - x # - x = if x Z - x - x - x - 6 = x - Thus, if the sum is + - = - x Z,. c. Using Theorem 6 : + b - - b = x x - x = x - x = = = - if x Z = - b - + b + b - b - b - - b + b - b - b + b - b or - b - b Definition of Greter Thn nd Less Thn in Terms of Subtrction For integers nd b, 6 b (or, equivlently, b ) if, nd only if, there exists positive integer k such tht + k = b, from which it followed tht b - = k nd hence b - 0. For rtionl numbers definition of greter thn nd less thn follows: Definition of Less Thn nd Greter Thn for Rtionl Numbers b 6 c d if c d - b 0, c if, nd only if, b 6 c d b d. This definition is useful in justifying certin inequlities. Exmple 6-0 uses the definition of greter thn nd provides prctice with lgebr. Exmple 6-0 () Verify ech of the following: (i) (ii) (iii) ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

34 Section 6- Addition, Subtrction nd Estimtion with Rtionl Numbers (b) Bsed on these exmples, conjecture generl sttement nd justify it. Solution () (i) (ii) (iii) hence = = 6 6 0, hence = = 0, hence = = 0, (b) Conjecture: If nd b re positive integers nd Z b, then b + b. Justifiction: b + b - = + b - b b - b = 0, b since Z b, nd b 0. Estimtion with Rtionl Numbers Estimtion helps us mke prcticl decisions in our everydy lives. For exmple, suppose we need to double recipe tht clls for of cup of flour. Will we need more or less thn 8 cup of flour? Mny of the estimtion nd mentl mth techniques tht we lerned to use with whole numbers lso work with rtionl numbers. The grde Focl Points clls for students to mke resonble estimtes of frction sums nd differences. The student pge (pge ) from grde textbook hs students estimte frctions of bords for building shelves. Estimtion plys n importnt role in judging the resonbleness of computtions. Students do not necessrily hve this skill. For exmple, when sked to estimte only % of -yer-old students on ntionl ssessment + 8, sid the nswer ws close to. Most sid it ws close to, 9, or. These incorrect estimtes suggest common computtionl errors in dding frctions nd lck of understnding of the opertion being crried out. These incorrect estimtes lso suggest lck of number sense. NOW TRY THIS 6- A student dded nd obtined How would you use estimtion to + 6. show this student tht his nswer could not be correct? ISBN Sometimes to obtin n estimte it is desirble to round frctions to convenient frction, such s or. For exmple, if student hd 9 correct nswers out of 80 questions, the student nswered of the questions correctly, which is pproximtely or,,,,,, , We know is greter thn On number line, the greter frction is to the right A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

35 Rtionl Numbers s Frctions School Book Pge ESTIMATING FRACTIONAL AMOUNTS Source: Mthemtics, Dimond Edition, Grde Five, Scott Foresmn-Addison Wesley 008 ( p. 0). ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

36 Section 6- Addition, Subtrction nd Estimtion with Rtionl Numbers 9 of the lesser. The estimte for is high estimte. In similr wy, we cn estimte 80 0 by or In this cse, the estimte of is low estimte ,. Exmple 6- A sixth-grde clss is collecting cns to tke to the recycling center. Becky s group brought the following mounts (in pounds). About how mny pounds does her group hve ll together? 8, 0, 8, 6 0 Solution We cn estimte the mount by using front-end estimtion nd then djusting by using 0, nd s reference points. The front-end estimte is + +, or 9. The, djustment is 0 + or. An djusted estimte would be 9 + or lb. + + b, Exmple 6- Estimte ech of the following: 9. b Solution. Becuse is slightly more thn nd is slightly more thn, n estimte 9 is number close to but more thn. b. We first dd the whole-number prts to obtin +, or. Becuse ech of the 9 frctions, nd is close to but less thn, their sum is close to but 0, 8,, less thn. The pproximte nswer is + or 8. Assessment 6-A ISBN Perform the following dditions or subtrctions:. + b. - - c. x + y - d. x y + xy + x e f Chnge ech of the following frctions to mixed numbers: 6. b A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

37 6 Rtionl Numbers s Frctions. Chnge ech of the following mixed numbers to frctions in the form where nd b re integers nd b Z 0: b,. b Approximte ech of the following situtions with convenient frction. Explin your resoning. Tell whether your estimte is high or low.. Giorgio hd bse hits out of 6 times t bt. b. Ruth mde gols out of shots. c. Lur nswered 6 problems correctly out of 80. d. Jonthn mde 9 bskets out of 9.. Use the informtion in the tble to nswer ech of the following questions:. Which tem won more thn of its gmes, but ws closest to winning of its gmes? b. Which tems won just under of their gmes? c. Which tems won just over of their gmes? 6. Sort the following frction crds into the ovls by estimting in which ovl the frction belongs: Sort these frction crds About 0 About About Tem Gmes Plyed Gmes Won Ducks 0 Bevers 9 0 Tigers 8 9 Bers 8 Lions Wildcts 6 Bdgers 9 8. Approximte ech of the following frctions by 0, or. Tell whether your estimte is high or low. 9. b c. d Without ctully finding the exct nswer, stte which of the numbers given in prentheses in the following is the best pproximtion for the given sum or difference: ,,, b b ,,, b,,, 9. Compute ech of the following mentlly:. b The following ruler hs regions mrked M, A, T, H: M Inch Ruler Use mentl mthemtics nd estimtion to determine which region ech of the following flls into (for exmple, flls into region A): in b. 8 in. 8 in c. d. 6 in. in.. Use clustering to estimte the following sum: A clss consists of freshmen, sophomores, nd 0 juniors; the rest re seniors. Wht frction of the clss is seniors?. A clerk sold three pieces of one type of ribbon to different customers. One piece ws long, nother ws long, nd the third ws yd yd long. Wht ws the totl yd length of tht type of ribbon sold? A. Mrtine bought 8 of fbric. She wnts to mke skirt yd using pnts using nd vest using 8 yd, 8 yd, yd. How much fbric will be left over?. Students from Rttlesnke School formed four tems to collect cns for recycling during the months of April nd My. The students received 0 for ech lb of cns. A record of their efforts follows: Number of Pounds Collected April 8 My. Which tem collected the most for the -month period? How much did they collect? b. Wht ws the difference in the totl pounds collected in April nd the totl pounds collected in My? T Tem Tem Tem Tem H 6 A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc. ISBN

38 Section 6- Addition, Subtrction nd Estimtion with Rtionl Numbers 6. Give n exmple illustrting ech of the following properties of rtionl numbers ddition:. Closure b. Commuttive c. Associtive. Given tht the sequence in prt () is rithmetic nd in prt (b) the sequence of numertors is rithmetic nd so is the sequence of denomintors, nswer the following: (i) Write three more terms of ech sequence. (ii) Is the sequence in prt (b) rithmetic? Justify your nswer..,,,,, Á b.,,,, 6, Á 8. Find the nth term in ech of the sequences in problem. 9. Insert five frctions between the numbers nd so tht the seven numbers (including nd ) constitute n rithmetic sequence. 0. Let f x = x + where the domin is the set of rtionl, numbers.. Find the outputs if the inputs re the following: - (i) 0 (ii) (iii) b. For which inputs will the outputs be the following? (i) (ii) - (iii). Let f x = x + nd let the domin of the function be x - the set of ll integers except. Find the following:. f 0 b. c. f - f - d. f.. Check tht ech of the following is true: (i) = + # (ii) = + # (iii) = 6 + # 6 b. Bsed on the exmples in (), write s sum of two n unit frctions, tht is, s sum of frctions with numertor.. Solve for x in ech of the following:. x + = b. x - = 6. Find ech sum or difference; simplify if possible. x. xy + y x b. xy - b xyz c. - b - - b + b Assessment 6-B ISBN Perform the following dditions or subtrctions: -. b c. x + y - d. x y + xy + x y e. f Chnge ech of the following frctions to mixed numbers:. b Chnge ech of the following mixed numbers to frctions in the form where nd b re integers nd b Z 0. b,. b. -. Plce the numbers,, 6, nd 8 in the following boxes to mke the eqution true: n n + n n = A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

39 8 Rtionl Numbers s Frctions. Use the informtion in the tble to nswer ech of the following questions: Tem Gmes Plyed Gmes Won Ducks 0 Bevers 9 0 Tigers 8 9 Bers 8 Lions Wildcts 6 Bdgers. Which tems won less thn of their gmes? b. Which tems won more thn of their gmes? c. Which tems won just under of their gmes? 6. Sort the following frction crds into the ovls by estimting in which ovl the frction belongs: Sort these frction crds About 0 About About Approximte ech of the following frctions by 0,,,, or. Tell whether your estimte is high or low.. b c. d Without ctully finding the exct nswer, stte which of the numbers given in prentheses in the following is the best pproximtion for the given sum or difference: ,,, b 0 b ,,, b 9. Compute ech of the following mentlly:. b The following ruler hs regions mrked M, A, T, H: Use mentl mthemtics nd estimtion to determine which region ech of the following flls into (for exmple, flls in region A): in b. 8 in. 8 in. 0 c. d. 6 in. in.. A clss consists of freshmen, sophomores, nd 0 juniors; the rest re seniors. Wht frction of the clss is seniors?. The Nturls Compny sells its products in mny countries. The following two circle grphs show the frctions of the compny s ernings for 00 nd 009. Bsed on this informtion, nswer the following questions:. In 00, how much greter ws the frction of sles for Jpn thn for Cnd? b. In 009, how much less ws the frction of sles for Englnd thn for the United Sttes? c. How much greter ws the frction of totl sles for the United Sttes in 009 thn in 00? d. Is it true tht the mount of sles in dollrs in Austrli ws less in 00 thn in 009? Why? Austrli 0 United Sttes United Sttes Austrli Jpn Jpn Frction of Totl Sles, 00 Frction of Totl Sles, Cnd Englnd 6 Cnd Englnd M A T Inch Ruler H. A recipe requires of milk. Rn put in nd emp- c c tied the continer. How much more milk does he need to put in? ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

40 Section 6- Addition, Subtrction nd Estimtion with Rtionl Numbers 9. A in. bord is cut from 8 in. bord. The sw cut tkes in. How much of the 8 in. bord is left fter 8 cutting?. In ech of the following sequences, the numertors form n rithmetic sequence. Write three more terms of the sequence nd determine which of the sequences re rithmetic, nd which re not? Justify your nswers.. b.,, - -,, 8,,, Á,,, Á 6. Find the nth term in ech of the sequences in problem.. Insert four frctions between the numbers nd so tht the six numbers (including nd ) constitute n rithmetic sequence. 8. Let f x = x - where the domin is the set of rtionl, numbers.. Find the outputs if the inputs re the following: - (i) 0 (ii) (iii) b. For which inputs will the outputs be the following? (i) (ii) - (iii) 9. Let f x = x + nd let the domin of the function x - + be the set of ll integers except. Find the following:. f 0 b. f - c. f - d. f 0.. Check tht ech of the following is true: (i) - = # (ii) - = # (iii) - 6 = # 6 b. Bsed on prt (), write s difference of two nn + frctions with numertors. Justify your nswer.. Solve for x in ech of the following:. x - 6 = b. x - # = #. Find ech sum or difference. x. xy + x b. xy - b xyz + x y Mthemticl Connections 6- ISBN Communiction. Suppose lrge pizz is divided into equl-size pieces nd smll pizz is divided into equl-size pieces nd you get piece from ech pizz. Does represent + the mount tht you received? Explin why or why not.. Explin why we choose common denomintor to dd +... When we dd two frctions with unlike denomintors nd convert them to frctions with the sme denomintor, must we use the lest common denomintor? Wht re the dvntges of using the lest common denomintor? b. When the lest common denomintor is used in dding or subtrcting frctions, is the result lwys frction in simplest form?. Explin why we cn do the following to convert to mixed number: # + =. To show the techer drew the following picture. Ken sid this shows picture of not Wht is =,,. Ken thinking nd how should the techer respond? A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

41 80 Rtionl Numbers s Frctions 6. Kr spent of her llownce on Sturdy nd of wht she hd left on Sundy. Cn this sitution be modeled s Explin why or why not. -?. Compute in two different wys nd leve your + nswer s mixed number. Tell which wy you prefer nd why. 8. Slly clims tht it is esier to dd two frctions if she dds the numertors nd then dds the denomintors. How cn you help her? 9. Does ech of the following properties hold for subtrction of rtionl numbers? Justify your nswer.. Closure b. Commuttive c. Associtive d. Identity e. Inverse 0. Explin n error pttern in ech of the following:. =, =, 6 6 = b. + = 6 8, + = 9, 8 + = 8 c. d =, 8 - =, - = # = 6 9, # 6 = 6, # = 8 0. Notice tht the sequence, whose nth term,,,, Á, is is decresing sequence. Explin how to use this fct n, to determine if the following sequences re incresing or decresing:.,,,,, Á, n b. Hint: b = -, = -,,,, Á, n n + c.,,,,, Á, n Open-Ended. Write story problem for -... Write two frctions whose sum is. If one of the frctions is wht is the other? b, b. Write three frctions whose sum is. c. Write two frctions whose difference is very close to but not exctly... With the exception of the Egyptins used only unit, frctions (frctions tht hve numertors of ). Every unit frction cn be expressed s the sum of two unit frctions in more thn one wy, for exmple, nd Find t lest two different = + = + 6. unit frction representtions for ech of the following: (i) (ii) b. Compute nd simplify your nswer. n - n + c. Re-write prt (b) s sum nd then use the sum to nswer the question in prt (). Write s sum of two different unit frctions. Coopertive Lerning. Interview 0 people nd sk them if nd when they dd nd subtrct frctions in their lives. Combine their responses with those of the rest of the clss to get view of how ordinry people must use computtion of rtionl numbers in their dily lives. Questions from the Clssroom d 6. Joe reported tht if then Jon c + d = b b = c d, + b. sid she didn t believe it. How do you respond?. Kendr showed tht by using the following + = figure. How would you help her? 8. Jill clims tht for positive frctions, b + c = b + c becuse the frctions hve common numertor. How do you respond? Review Problems 9. Simplify if possible.. b. ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

42 Section 6- Multipliction nd Division of Rtionl Numbers 8.. If the sme positive number is dded to the numertor c. nd denomintor of positive proper frction, is the + d. + + e. + - b f. - b 0. Determine if the frctions in ech of the following pirs re equl: b. nd b b 8 b. nd c. nd d. nd where Z b b b +,. If we ssume yr = 6 dys, nswer the following:. Wht month of the yer hs the smllest frction of dys of the yer? b. Wht frction of dys of the yer occur before July? new frction greter thn, less thn, or equl to the originl frction? Justify your nswer. b. Wht if the sme positive number, less thn the numertor nd less thn the denomintor, is subtrcted from the numertor nd denomintor of positive proper frction nd the new denomintor is positive? Third Interntionl Mthemtics nd Science Study (TIMSS) Question Jnis, Mij, nd their mother were eting cke. Jnis te of the cke. Mij te of the cke. Their mother te of the cke. How much of the cke is left?. b. c. d. None TIMSS 00, Grde 6- Multipliction nd Division of Rtionl Numbers Multipliction of Rtionl Numbers To motivte the definition of multipliction of rtionl numbers, we use the interprettion of multipliction s repeted ddition. Using repeted ddition, we cn interpret # s b follows: # b = + + = 9 = The re model in Figure 6- is nother wy to clculte this product. ISBN =. = + + = Figure 6-9, or A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

43 8 Rtionl Numbers s Frctions We next consider. How should this product be interpreted? If the commuttive b # () (b) Figure 6- property of multipliction of rtionl numbers is to hold, then. Next, we consider nother interprettion of multipliction. Wht is of? Recll tht of quntity is the mount resulting from dividing the quntity into equl prts nd tking of these prts. To see wht of is consider tking of equl size brs. This cn be done by tking of ech of the brs, tht is, or. Thus of is , or #, which s we hve seen bove is equl to #. Thus we cn interpret # s of nd in generl for non-negtive rtionl numbers # c c cn be represented s of. b d b d If forests once covered bout of Erth s lnd nd only bout of these forests remin, wht frction of Erth is covered with forests tody? We need to find of, nd cn use n re model to find the nswer. Figure 6-() shows rectngle representing the whole seprted into fifths, with shded. To find of, we divide the shded portion of the rectngle in Figure 6-() into two congruent prts nd tke one of those prts. The result would be the green portion of Figure 6-(b). However, the green portion represents prts out of 0, or, of the whole. Thus, 0 # = 0 = # # b # = # b = 9 This discussion leds to the following definition of multipliction for rtionl numbers. Definition of Multipliction of Rtionl Numbers c If nd re ny rtionl numbers, then # c. b d = # c b d b # d REMARK. Notice tht the definition bove follows from the interprettion of # c c s of. b d b d. Also notice tht when multiplying frctions it is possible nd more efficient to cncel common fctors before multiplying out the numertors nd the denomintors. An re model like the one in Figure 6- is used on the student pge tht follows (pge 8). NOW TRY THIS 6- Answer the questions on the student pge. ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

44 Section 6- Multipliction nd Division of Rtionl Numbers 8 School Book Pge MULTIPLYING FRACTIONS ISBN Source: Mthemtics, Dimond Edition, Grde Five, Scott Foresmn-Addison Wesley 008 ( p. 96). A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

45 8 Rtionl Numbers s Frctions Exmple 6- Answer the following: If of the popultion of certin city re college grdutes nd of the city s college grdutes re femle, wht frction of the popultion of tht city is femle college grdutes? 6 Solution The frction should be of, or #. 6 = # # = The frction of the popultion tht is femle college grdutes is. 66 Properties of Multipliction of Rtionl Numbers Multipliction of rtionl numbers hs properties nlogous to the properties of ddition of rtionl numbers. These include the following properties for multipliction: closure, commuttive, ssocitive, multiplictive identity, nd multiplictive inverse. For emphsis, we list the lst two properties. Theorem 6 : Numbers Multiplictive Identity nd Multiplictive Inverse of Rtionl. The number is the unique number such tht for every rtionl number, b # b b = b = b b #. For ny nonzero rtionl number is the unique rtionl number such tht # b. b = = b # b, b b REMARK The multiplictive inverse of is lso clled the reciprocl of. b b Exmple 6- Find the multiplictive inverse, if possible, for ech of the following rtionl numbers: -. b. c. d. 0 e. 6 Solution. - b. -, or c. Becuse =, the multiplictive inverse of is. d. Even though 0 = 0 is undefined; there is no multiplictive inverse of 0., 0 e. Becuse 6, the multiplictive inverse of 6 is. = Multipliction nd ddition re connected through the distributive property of multipliction over ddition. Also there is multipliction property of equlity for rtionl numbers nd multipliction property of zero similr to those for whole numbers nd integers. A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc. ISBN

46 Section 6- Multipliction nd Division of Rtionl Numbers 8 Theorem 6 :. Distributive Property of Multipliction Over Addition for Rtionl Numbers e If, nd re ny rtionl numbers, then b, c d f. Multipliction Property of Equlity for Rtionl Numbers If c e nd re ny rtionl numbers such tht, nd b = c b d d f is ny rtionl number, then. b # e f = c d # e f. Multipliction Property of Inequlity for Rtionl Numbers e (i) If nd then b c d f 0, e (ii) If nd then b c d f 6 0, b c d + e f b = # c b d b + # e b f b # e b f c # e d f. # e b f 6 c # e d f.. Multipliction Property of Zero for Rtionl Numbers If is ny rtionl number, then # 0 = 0 = 0 #. b b b REMARK Theorem 6 cn be proved using the corresponding properties of integers. For exmple, b # 0 = b # 0 = # 0 b # = 0 b = 0. ISBN Exmple 6- A bicycle is on sle t of its originl price. If the sle price is $0, wht ws the originl price? Solution Let x be the originl price. Then of the originl price is. Becuse the sle x price is $0, we hve. Solving for x gives x = 0 # x = # 0 # x = 0 x = 0 Thus, the originl price ws $0. An lterntive pproch, which does not use lgebr, follows. Becuse is of the prts, then of the prts is $0 nd of the prts is $0. If of the prts is $0, then of these is # $0, or $0. A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

47 86 Rtionl Numbers s Frctions Multipliction with Mixed Numbers In the Penuts crtoon, Slly is hving trouble multiplying two mixed numbers. If she used n estimte to check whether her nswer ws resonble, she would notice tht if she multiplies two numbers tht re both less thn, the nswer must be less thn 9. One wy to multiply # is to chnge the mixed numbers to improper frctions nd use the definition of multipliction, s shown here. # = # = We could then chnge to the mixed number 6. Another wy to multiply mixed numbers uses the distributive property of multipliction over ddition. For exmple, # = + b # + b = + b # + + b # = # # + + # + # = = 6 + = 6 Exmple 6-6 Multipliction of frctions enbles us to obtin equivlent frctions, to perform ddition nd subtrction of frctions, s well s to solve equtions in different wy, s shown in the following exmple. Use the definition of multipliction of frctions nd other properties to justify or solve the following:. The Fundmentl Lw of Frctions, if n Z 0. b = n bn b. Addition of frctions using common denomintor. c. Enlrging the denomintor of positive frction by fctor of m is the sme s multiplying the frction by. m A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc. ISBN

48 Section 6- Multipliction nd Division of Rtionl Numbers 8 d. Solve the following equtions: (i) x + x = x (ii) (find x in terms of nd b) + x b = Solution. b. b = # = # n b b n = n bn = d bd + bc bd d + bc = bd c. bm = # b # = # m b m d. There re mny wys to solve these equtions; we show one wy tht uses multipliction of frctions. (i) is equivlent to ech of the following: x + x =, + b x = # x + # x = #, # x = x x = x # = x, #, = # x b + c d = b # d d + c d # b b x =. Alterntively we could solve the eqution by dding the frctions using the lest common denomintor of x: (ii) x + x =, 8 + x x # x = # x = = x, x = x is equivlent to ech of the following: + x b = x # + x # b = x + b b = ISBN x b + b = b x b + b # b b + b = # b + b x # b = + b b x =, nd Z - b + b A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

49 88 Rtionl Numbers s Frctions Division of Rtionl Numbers In the Principles nd Stndrds, we find the following sttement concerning division of rtionl numbers: The division of frctions hs trditionlly been quite vexing for students. Although invert nd multiply hs been stple of conventionl mthemtics instruction nd lthough it seems to be simple wy to remember how to divide frctions, students hve for long time hd difficulty doing so. Some students forget which number is to be inverted, nd others re confused bout when it is pproprite to pply the procedure. A common wy of formlly justifying the invert nd multiply procedure is to use sophisticted rguments involving the mnipultion of lgebric rtionl expressions rguments beyond the rech of mny middle-grdes students. This process cn seem very remote nd mysterious to mny students. Lcking n understnding of the underlying rtionle, mny students re therefore unble to repir their errors nd cler up their confusions bout division of frctions on their own. An lterntive pproch involves helping students understnd the division of frctions by building on wht they know bout the division of whole numbers. (p. 9) We try to follow tht dvice in developing the concept of division of rtionl numbers. Recll tht 6, mens How mny s re there in 6? We found tht 6, = becuse # = 6 nd, in generl, if, b, c W, then, if, nd only if c is the unique whole b = c number such tht bc =. Consider,, which is equivlent to finding how mny b hlves there re in. We see from the re model in Figure 6- tht there re 6 hlf pieces in the whole pieces. We record this s. This is true becuse,. b # 6 = b = = Figure 6- Figure 6- With whole numbers one wy to think bout division ws in terms of repeted subtrction. We found tht 6, = becuse could be subtrcted from 6 three times; tht is, 6 - # With, = 0., we wnt to know how mny hlves cn be subtrcted from. b Becuse - 6 #, we know tht,. b = 0 b = 6 Next, consider. This mens How mny s re in? Figure 6- shows b, 8 b 8 tht there re six s in the shded portion, which represents of the whole. Therefore, 8. This is true becuse. Using repeted subtrction, we see 8 b # 6 = b, 8 b = 6 6 tht nd tht so., 8-6, 8 = 6 8, 8 8 b = 0, 8 = 6 ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

50 Section 6- Multipliction nd Division of Rtionl Numbers 89 Two dditionl models for division of frctions re shown on the student pge on pge 90. The mesurement or number-line model used in Exmple A on the student pge is lso useful. For exmple, consider. First we drw mesurement or number line divided into 8, eighths, s shown in Figure 6-. Next we wnt to know how mny s there re in. 8 The br of length is mde up of 6 equl-size pieces of length. We see tht there is t lest 8 one length of in. If we put nother br of length on the number line, we see there is 8 more of the 6 equl-length segments needed to mke. Therefore, the nswer is, 8 6 or Figure 6- In the previous exmples, we sw reltionship between division nd multipliction of rtionl numbers. We cn define division for rtionl numbers formlly in terms of multipliction in the sme wy tht we defined division for whole numbers. Definition of Division of Rtionl Numbers c e If nd re ny rtionl numbers, then if, nd only if, is the unique rtionl c number such tht # e. d f = b, c d = e b d f f b c c REMARK In the definition of division, becuse if, then the eqution c hs no solution if nd hs infinitely mny solutions if d x = d Z 0 d = 0 b b Z 0 b = 0. ISBN NOW TRY THIS 6- Students often confuse division by nd division by Notice tht., = but, if, nd only if, x =, # x =, x =. =, = x Write rel-life story tht will help students see the difference between division by nd division by. A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

51 90 Rtionl Numbers s Frctions School Book Pge DIVIDING FRACTIONS Source: Mthemtics, Dimond Edition, Grde Six, Scott Foresmn-Addison Wesley 008 ( p. 66). ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

52 Section 6- Multipliction nd Division of Rtionl Numbers 9 Algorithm for Division of Rtionl Numbers As we see in the Penuts crtoon, Peppermint Ptty doesn t understnd why the lgorithm for division of frctions works. She is not lone in her confusion, nd we will explin why to divide frctions we use the reciprocl nd multiply. Does the method in the crtoon, often clled the invert-nd-multiply method, mke sense bsed on wht we know bout rtionl numbers? We know tht one wy to interpret is b, b. We lso know tht. Therefore,, cn be written s. If the, = invert-nd-multiply technique works, then, =. Since, = * = # # =, =, this is consistent with wht we hve done before. To develop the generl lgorithm for division of rtionl numbers, we consider wht such division might men. For exmple,, = x implies = x We multiply both sides of the eqution by, the reciprocl of. Thus, # = # xb = # bx = # x = x Therefore,., = # A trditionl justifiction of the division lgorithm lso follows. The lgorithm for division of frctions is usully justified in the middle grdes by using the Fundmentl Lw of Frctions,, where, b, nd c re ll frctions, or equivlently, the identity property b = c bc of multipliction. For exmple, ISBN Thus,, = = # = # =, = # # = # A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

53 9 Rtionl Numbers s Frctions NOW TRY THIS 6- Use n rgument similr to the preceding one to show tht, in generl, if re rtionl numbers nd, then. b, c d = # d b c c d d Z 0 b c nd We summrize the lgorithm s follows: Theorem 6 : Algorithm for Division of Frctions c c If nd re ny rtionl numbers nd, then b d d Z 0 b, c d = # d b c Alternte Algorithm for Division of Rtionl Numbers An lterntive lgorithm for division of frctions cn be found by first dividing frctions 9 tht hve equl denomintors. For exmple, nd., 0, 0 = 9, =, These exmples suggest tht when two frctions with the sme denomintor re divided, the result cn be obtined by dividing the numertor of the first frction by the numertor of the second. To divide frctions with different denomintors, we renme the frctions so tht the denomintors re equl. Thus, b, c d = d bd, bc bd = d, bc, or d bc NOW TRY THIS 6-6 Show tht b, c d = # d b c nd b, c d =, c b, d re equivlent. The next three exmples illustrte the use of division of rtionl numbers. Exmple 6- A rdio sttion provides 6 min for public service nnouncements for every hr of brodcsting.. Wht prt of the brodcsting dy is llotted to public service nnouncements? b. How mny -min public service nnouncements cn be llowed in the 6 min? Solution. There re 60 min in n hour nd 60 # min in dy. Thus, 6>60 #, or, of the dy is llotted for the nnouncements. 0 b. 6>, or 8, nnouncements re llowed. b = 6 b ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

54 Section 6- Multipliction nd Division of Rtionl Numbers 9 Exmple 6-8 We hve yd of mteril vilble to mke towels. Ech towel requires yd of mteril. 8. How mny towels cn be mde? b. How much mteril will be left over? Solution. We need to find the integer prt of the nswer to. The division follows:, 8, 8 = # 8 = 8 = 9 Thus we cn mke 9 towels. b. Becuse the division in () ws by, the mount of mteril left over is of, 8 8 or #, or yd. This cn lso be nswered by noting tht the in prt () is 8 two-thirds of towel, which requires of yd of mteril. 8 Exmple 6-9 If of jump rope is yd long, wht is the length of the entire rope? Solution If the length of the rope is l yd, then of the rope s length is long. Thus, l l = l = l = # =, or yd Notice tht this problem could be solved without using division, by simply multiplying both sides of the first eqution by. Estimtion nd Mentl Mth with Rtionl Numbers Estimtion nd mentl mth strtegies tht were developed with whole numbers cn lso be used with rtionl numbers. ISBN Exmple 6-0 Use mentl mth to find. b. c. 6 b # # # Solution Ech of the following is possible pproch:. # # = # # b = # = # 0 A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

55 9 Rtionl Numbers s Frctions b. c. 6 # = + 6 b = # + # 0 = # 0b = # = 6 6 # = 60 + = 6 Exmple 6- Estimte ech of the following:. # 8 b., 9 8 Solution. Using rounding, the product will be close to # 8 =. If we use the rnge strtegy, we cn sy the product must be between # = nd # 8 =. b. We cn use comptible numbers nd think of the estimte s, = 6 or, =. Extending the Notion of Exponents Recll tht ws defined for ny integer nd ny nturl number m s the product of m s. We define m for ny rtionl number in similr wy s follows. m Definition of to the mth Power m = # # # Á #, m fctors where is ny rtionl number nd m is ny nturl number.. From the definition, # = # # # # = +. In similr wy, it follows tht where is ny rtionl number nd m nd n re nturl numbers. If () is to be true for ll whole numbers m nd n, then becuse # 0 = +0 =, we must hve 0 =. Hence, it is useful to give mening to 0 when Z 0 s follows.. For ny nonzero number, 0 =. The properties bove re lso true for rtionl number vlues of. For exmple, consider the following: b # b m # n = m+n = # b # # # b = b + = b If is to be extended to ll integer powers of, then how should - m # n = m+n be defined? If () is to be true for ll integers m nd n, then - # = -+ = 0 =. Therefore, - = >. This is true in generl nd we hve the following.. For nonzero number, -n = n. In elementry grdes the definition of is typiclly motivted by looking t ptterns. Notice tht s the following exponents decrese by, the numbers on the right re divided by 0. Thus the pttern might be continued, s shown. - n ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

56 Section 6- Multipliction nd Division of Rtionl Numbers 9 0 = 0 # 0 # 0 0 = 0 # 0 0 = = 0 - = 0 = = 0 # 0 = = 0 # 0 = 0 If the pttern is extended, then we would predict tht 0 -n =. Notice tht this is 0 n inductive resoning nd hence is not full mthemticl justifiction. Consider whether the property m # n = m+n cn be extended to include ll powers of, where the exponents re integers. For exmple, is it true tht # - = + - =? The definitions of - nd the properties of nonnegtive exponents ensure this is true, s shown next. # - = # = = # = Also, - # - = = - is true becuse - # - = # = # # = + = = - In generl, with integer exponents, the following theorem holds. Theorem 6 For ny nonzero rtionl number nd ny integers m nd n, m # n = m+n. REMARK If = 0 then Theorem 6 is still vlid s long s m Z 0, n Z 0 nd m + n Z 0. Other properties of exponents cn be developed by using the properties of rtionl numbers. For exmple, With integer exponents, the following theorem holds. Theorem 6 6 # = = = - 8 = # = = - = -8 For ny nonzero rtionl number nd for ny integers m nd = m-n. n, m n ISBN At times, we my encounter n expression like. This expression cn be written s single power of s follows: = # # = ++ = # = A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

57 96 Rtionl Numbers s Frctions In generl, if is ny rtionl number nd m nd n re positive integers, then m n = m # m # m # Á # m = m+m+á+m = nm = mn n fctors Does this theorem hold for negtive-integer exponents? For exmple, does - =? The nswer is yes becuse - =. Also, - = = = - - = - b = # # # = = = -. n terms Theorem 6 For ny rtionl number Z 0 nd ny integers m nd n, m n = mn Using the definitions nd theorems developed, we cn derive dditionl properties. Notice, for exmple, tht b This property cn be generlized s follows. = # # # = # # # # # # = Theorem 6 8 For ny nonzero rtionl number nd ny integer m, b m b b = m b m From the definition of negtive exponents, the preceding theorem, nd division of frctions, we hve - m b b = m = b b m = bm m = b m b b m Theorem 6 9 For ny nonzero rtionl number nd ny integer m, - m. b b = b m b b A property similr to the one in Theorem 6 8 holds for multipliction. For exmple, # - = nd in generl, it is true tht m is n integer. # = # = b # b = - # - # b m = m # b m if nd b re nonzero rtionl numbers nd ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

58 Section 6- Multipliction nd Division of Rtionl Numbers 9 The definitions nd properties of exponents re summrized in the following definition nd theorem. For ny nonzero rtionl numbers nd b nd integers m nd n (except s noted), we hve the following. Definition of to n Integer Power m = # # # Á #,. where m is positive integer nd is ny rtionl number m fctors. 0 =. -m = m Theorem 6 0: Properties of Exponents..... m # n = m+n m n = m - n m n = nm b b m = m b m b m = m b m 6. b b - m = b b m Notice tht properties nd s well s nd 8 re for multipliction nd division. Anlogous properties do not hold for ddition nd subtrction. For exmple, in generl, + b - Z - + b -. To see why, numericl exmple is sufficient, but it is instructive to write ech side with positive exponents: + b - = + b ISBN Exmple 6- nd we know tht in generl, - + b - = + b + b Z + b. In ech of the following, use properties of exponents to justify the equlity or inequlity:. -x - Z -x - b. -x - = -x - c. b - Z b - d. - b - - = b e. - + b - - Z + b Solution. - x - = - x = x - - x = -x - = - x b = - Hence, - x - Z -x -. x A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

59 98 Rtionl Numbers s Frctions b. -x - = -x = - x = - x b - -x - = -x - = x b Hence, -x - = -x -. c. but b - = b- = b - = # Hence, b - Z b -. b = b, b. d. - b - - = - - b - - = - - b - - = b e. - + b - - = + b b - = + b b - b = b + b Z + b Observe tht ll the properties of exponents refer to powers with either the sme bse or the sme exponent. To evlute expressions using exponents where different bses nd powers re used, perform ll the computtions or rewrite the expressions in either the sme bse ( ) or the sme exponent, if possible. For exmple, cn be rewritten 8 8 = ( ) = =. Exmple 6- Write ech of the following in simplest form using positive exponents in the finl nswer:. b. c. d. (x y - ) - (0 - + # # - 0 ) # 0 6 # - 8 0, Solution. 6 # 8 - = ( ) # ( ) - = 8 # - 9 = = - = b. c. 0 # = ( ) # = = (0 - + # # 0 - ) # 0 = 0 - # 0 + # 0 - # 0 + # - 0 # 0 = # # = 0 + # 0 + # 0 0 = d. (x y - ) - = x - y 8 = # x y 8 = y 8 x BRAIN TEASER A cstle in the frwy lnd of Aluossim ws surrounded by four mots. One dy, the cstle ws ttcked nd cptured by fierce tribe from the north. Gurds were sttioned t ech bridge. Prince Jun ws llowed to tke number of bgs of gold from the cstle s he went into exile. However, the gurd t the first bridge demnded hlf the bgs of gold plus one more bg. Jun met this demnd nd proceeded to the next bridge. The gurds t the second, third, nd fourth bridges mde identicl demnds, ll of which the prince met. When Jun finlly crossed ll the bridges, single bg of gold ws left. With how mny bgs did Jun strt? ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

60 Section 6- Multipliction nd Division of Rtionl Numbers 99 Assessment 6-A ISBN In the following figures, unit rectngle is used to illustrte the product of two frctions. Nme the frctions nd their products.. b.. Use rectngulr region to illustrte ech of the following products:. # b. #. Find ech of the following products. Write your nswers in simplest form. 9 xy. # 6 b. # b c. # z 6 98 b z x y. Use the distributive property to find ech product.. # chint: + b + b. d b. #. Find the multiplictive inverse of ech of the following: -. b. x c., if x Z 0 nd y Z 0 d. - y 6. Solve for x in ech of the following:. b., x = x = 6 x c. d. - = x x =. Show tht the following properties do not hold for the division of rtionl numbers:. Commuttive b. Associtive 8. Compute the following mentlly. Find the exct nswers.. # 8 b. # c. d. 8 # 9 # 0 9. Choose the number tht best pproximtes ech of the following from mong the numbers in prentheses:. # 8, 0,, 6 00 b. # 6,,, 0 8 c. 0, 0 0,,, b 0. Estimte the following:. # 0 0 b. # 8. Without ctully doing the computtions, choose the phrse in prentheses tht correctly describes ech:. # (greter thn, less thn ) 9 b. (greter thn, less thn ), 9 c. (greter thn, less thn ), 00. A sewing project requires 6 yd of mteril tht sells for 8 6 per yrd nd yd of mteril tht sells for 8 per yrd. Choose from the best estimte for the cost of the project:. Between $ nd $ b. Between $ nd $6 c. Between $6 nd $8 d. Between $8 nd $0. Five-eighths of the students t Slem Stte College live in dormitories. If 6000 students t the college live in dormitories, how mny students re there in the college?. Alberto owns of the stock in the N.W. Tofu Compny. 9 His sister Rentt owns hlf s much stock s Alberto. Wht prt of the stock is owned by neither Alberto nor Rentt?. A suit is on sle for $80. Wht ws the originl price of the suit if the discount ws of the originl price? 6. John took ll his money out of his svings ccount. He spent $0 on rdio nd of wht remined on presents. Hlf of wht ws left he put bck in his checking ccount, nd the remining $ he donted to chrity. How much money did John originlly hve in his svings ccount? A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.

61 00 Rtionl Numbers s Frctions. Al gives of his mrbles to Bev. Bev gives of these to Crl. Crl gives of these to Dni. If Dni ws given four mrbles, how mny did Al hve originlly? 8. Write ech of the following in simplest form using positive exponents in the finl nswer:. b. # # 6 c. d., -, - e. - - f., where Z 0 9. Write ech of the following in simplest form using positive exponents in the finl nswer:. b. 9 b, 6 b # b b c. d. b, b # 9 b b 0. If nd b re rtionl numbers, with Z 0 nd b Z 0, nd if m nd n re integers, which of the following re true nd which re flse? Justify your nswers.. m # b b. c. m # n = (b) m+n m # b n = (b) mn b m = (b) m d. b 0 = e. + b m = m + b m f. + b - m = m + b m. Solve for the integer n in ech of the following:. n = b. n c. n # = 6 = d. n # = 8. Solve ech of the following inequlities for x, where x is n integer:. x 8 b. x 6 8 c. x d. x. Determine which frction in ech of the following pirs is greter:. or b. or 8 0 b b b b c. or d. or b b b b. Suppose the number of bcteri in certin culture is given s function of time by Q(t) = t, where t b is the time in seconds nd Q(t) is the number of bcteri fter t sec. Find the following:. The initil number of bcteri (tht is, the number of bcteri t t = 0) b. The number of bcteri fter sec. Let S = Á +. Use the distributive property of multipliction over ddition to find n expression for S. b. Show tht S - S = S = - 6. b c. Find simple expression for the sum Á + n 6. If fn = # -n, find the following:. f0, f -, f b. n if fn = # If the nth term of sequence is given by n = # n, nswer the following:. Find the first five terms. b. Show tht the first five terms re in geometric sequence. c. Which terms re less thn? In the following, determine which number is greter:. or b. or Show tht the rithmetic men of two rtionl numbers is between the two numbers; tht is, prove tht 0 6 b 6 b + c d b 6 c d. 6 Assessment 6-B. Use rectngulr region to illustrte ech of the following products:. # b. #. Find ech of the following products of rtionl numbers. Write your nswers in simplest form.. # b. # - c. # d. # x y e. # b f. # z b z x y. Use the distributive property to find ech product of rtionl numbers.. # b. x y + by x - b c. 8 # 00 8 ISBN A Problem Solving Approch to Mthemtics for Elementry School Techers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, nd Johnny W. Lott. Published by Addison-Wesley. Copyright 00 by Person Eduction, Inc.