Making Faction Division Concete: A Ne Way to Undestand the Invet and Multily Algoithm Intoduction us is not to eason hy, just invet and multily. This is a hyme that I emembe leaning ay back in the fifth gade. Fifteen yeas late, at the beginning of my caee as a mathematics teache, I found that I anted to give my students moe than instuction on hat to do to get the ight anse. I anted them to undestand hy the mathematics made sense. Fast foad the clock and tenty-five yeas afte fifth gade I began an eanest quest to discove hy the invet and multily algoithm fo factions made sense. h, I kne about the algebaic maniulations and the use of comlex factions to ove that a c a d b d actually does equal, but that asn t hat I as b c looking fo. I anted a stategy that I could use to develo the algoithm using concete mateials and mathematical concets that ee accessible to an ue elementay o junio high student. My quest aid off and my objective fo this aticle is to shae the logic and easoning involved in faction division and the ste-by-ste constuction of a concete model using Cuisenaie Rods that the exeienced teache can use as a guide hen develoing lessons fo naïve leanes. While othe excellent faction division activities using Cuisenaie Rods ae available, this aticle goes futhe than most of those activities. It also: Constucts faction division as an extension of hole numbe division and faction multilication; Illuminates not just that the algoithm oks, but also hy the algoithm makes sense; and Facilitates the develoment of stong estimation stategies and skills. Pat 1 of this aticle evisits the language and meaning of division ith hole numbes as a singboad fo constucting the meaning of division hen factions ae involved. Pat 2 esents a sequence of faction division examles that can be modeled ith the Cuisenaie Rods. These examles ae designed to illuminate the meaning of faction division, set u stategies fo estimating the quotients, and finally, to establish a atten that ill be analyzed in Pat 3 and then linked to the logical constuction of the invet and multily algoithm in Pat 4. A Bief Wod About Readiness Befoe I launch into constucting the logic of the faction division algoithm, it is imotant to take a bief moment to list the concets that a student needs to kno and undestand in ode to be eady to lean hy the invet and multily algoithm makes sense. The folloing eadiness concets ovide the foundation uon hich the faction division concets develoed in this aticle can be constucted: The language and meaning of factions and faction notation (vocabulay, the infomation communicated by the numeato and denominato, etc). The language and meaning of multilication ith hole numbes and factions (vocabulay, the infomation communicated by the factos and the oduct, multilication as a eesentation fo multile 1 gous of a fixed quantity, etc). Why the taditional faction multilication algoithm (numeato times numeato ove denominato times denominato) makes sense. None of these concets ae develoed in this aticle, hoeve a comanion aticle, entitled Faction Multilication (Rusch, Texas Mathematics Teache, Sing 2005), develos all of these eadiness concets ith an eye toad setting the stage fo develoing the faction division concets addessed hee. A Note About the Stategies Used to Develo the Concets Thee ae to stategies I use to develo the faction division model hen oking ith my on students. 2 Since my students have found these stategies to be helful, I ill also use them in this aticle. The 1 In the case of faction multilication, multile gous may include a factional at. Fo examle: 5 8 of a gou of 1 3 o 3 1 2 gous of 3 4. 2 Pe-sevice teaches in both mathematics content and mathematics methods couseok and exeienced in-sevice teaches aticiating in ofessional develoment couses and okshos. age 6 / Fall 2005 Texas Mathematics Teache
fist stategy is to begin ith vey simle examles and then methodically incease the comlexity of those examles hile emhasizing the logic that links one to the next. The second is to emhasize the use of fomal language as a tool to illuminate the connections fom one examle to the next. The simlicity that may be evidenced in the examles and the exlanations ae in no ay intended to insult the eade. Instead, they ae designed to highlight the logical connections beteen concets that ae comfotable and the ne concets that ae being constucted. In the atitioning context ictued above, the dividend communicates the quantity you begin ith, the diviso communicates the numbe of gous, and the quotient communicates the quantity in each gou: 12 is the dividend. 3 is the diviso. you begin ith. of gous. 12 3 = 4 4 is the quotient. in each gou. PART 1: Setting the Foundations Revisiting the Language And Meaning of Division ne of the key factos to constucting an undestanding of faction division and the invet and multily algoithm is having a clea and ecise undestanding of the infomation being communicated by the dividend, diviso and quotient, and then using that language vey consistently thoughout the develoment of the ne concets and connections. The language and meaning of division is a bit moe comlicated than the language and meaning of multilication because the infomation communicated by the diviso and the quotient ae deendent on the context of the oblem. Fo examle: Context #1: I have 12 floes and 3 vases. I ant to ut the same numbe of floes in each vase. Ho many floes ill each vase get? A. B. C. D. E. 12 3 = 4 To illustate the division ocess fo this context (atitive division), I begin ith thee vases and distibute the 12 floes equally among those vases: Context #2: I have 12 floes and some vases. I ant to ut 3 floes in each vase. Ho many vases ill I need? 12 3 = 4 To illustate the division ocess fo this context (measuement division), I begin ith some vases and distibute the 12 floes so that thee ae thee floes in each vase: A. B. C. D. E. In the measuement context ictued above, the dividend again communicates the quantity you begin ith, but this time the diviso communicates the numbe in each gou, and the quotient communicates the numbe of gous.: 12 is the dividend. 3 is the diviso. you begin ith. in each gou. 12 3 = 4 4 is the quotient. of gous. htt://.tctmonline.net Fall 2005 / age
The faction division algoithm is much moe easily develoed using the second context, so fom this oint on all of the examles ill model the second (o the measuement ) context. In all of the folloing examles, I ill emhasize statements that define the dividend, diviso and quotient as follos: Dividend The amount you begin ith. age / Fall 2005 Diviso The size of each gou. a b = c Quotient The numbe of gous of the diviso in the dividend. and the division statement ill alays ask: Ho many gous of the diviso can I get fom the dividend? PART 2: An Intoduction to Faction Division Using Cuisenaie Rods A Concete / Pictoial Develoment of Faction Division This section ovides a seies of examles that ae designed to constuct a fameok fo easoning that ill lead to an undestanding of the meaning of faction division, stategies fo estimating the quotients, and ultimately shed light on the logic undeinning the invet and multily algoithm. These examles begin ith vey simle tasks, and then the comlexity of the tasks gadually inceases. When I m oking ith my students I ask them to bea ith me though the simle tasks thee is moe challenge coming and the simle tasks ill seve a uose late on don the oad. The standad symbols fo Cuisenaie Rod colos ae in the folloing table. Symbol lg y b b o Colo hite ed light geen ule yello dak geen black bon blue oange An imotant at of the Cuisenaie Rod exloation tasks is data collection. While the exloations seve to illuminate the logic and meaning of faction division and ovide an ootunity to develo estimation stategies, the data collected and ecoded duing the exloations ill late be used to discen the attens that ill hel illuminate the logic of the faction division algoithm. As I ok though the exloation tasks hee I ll model a helful stategy fo ecoding the data. I ll exlain ho to use the data in the table in Pat 3, afte all the data has been collected and ecoded. Although I ovide ictues fo the examles esented, I think it might be vey helful (and cetainly moe fun!) to ull out a set of Cuisenaie Rods and ok though the tasks concetely as I esent the ictoial and abstact eesentations. When I ok though the questions in each examle ith my students I atch them caefully and ait until they have had enough time to find the ods they need and they begin to egiste the concete connections befoe I oceed to the next question o examle. This is esecially imotant in the initial examles hile the students get used to the ods and ho to eesent a faction of the unit using anothe colo od. Fo the uoses of efficiency in this esentation hoeve, I have listed all the questions fo each examle as a gou and I have shon only a ictue of the model that the students ae likely to constuct in ode to anse the division question asked. In this section, the dividend is alays one and evey division question asks Ho many gous of ae in one hole? The eason fo this stategy ill become clea in Pat 3. Examle #1: Let the oange od eesent one hole. What colo od ould eesent 1 10 of the hole? Ho many gous of 1 10 ae in one hole? To hel emhasize thinking that ill suot using easoning to detemine the quotient, I intoduce the division question using symbolic tems, but then estate the oblem using visual tems. Fo Texas Mathematics Teache
examle, I ould initially ose Examle #1 ith the symbolic statement Ho many gous of 1 10 ae in one hole? but then almost immediately estate the question using visual tems: Ho many gous that ae the size of the hite od can I make fom the oange od? It is not at all unusual fo me to need to estate the oblem in visual tems multile times. nce the students have detemined the quotient in visual tems (i.e., that thee ae ten gous that ae the size of the hite od in the oange od), I estate and anse the question in symbolic tems, So thee ae ten gous of 1 10 in one hole to emhasize the connection beteen the concete/visual and the symbolic eesentations. I have the students ite out the vebal statement of the oblem to einfoce the meaning of faction division and facilitate develoment of estimation skills. Examle #2: Let the light geen od eesent one hole. What colo od ould eesent 1 3 of the hole? Ho many gous of 1 3 ae in one hole? lg The hite ods illustate that one can get thee gous of 1 3 fom the light geen od that eesents one hole. The next examles of data should be similaly modeled. Examle #3: Let the bon od eesent one hole. What colo od ould eesent 1 4 of the hole? Ho many gous of 1 4 ae in one hole? Examle 4: Let the oange od eesent one hole. What colo od ould eesent 1 5 of the hole? Ho many gous of 1 ae in one hole? 5 Examle #5 belo begins to incease the comlexity of the easoning by intoducing divisos that ae no longe unit factions. As the examles ae esented, I have found that it is imotant to eeatedly emhasize that the quotient tells us ho many gous of the diviso ae in the dividend. In this case, e ant to kno ho many gous of 2/3 can e get out of one hole? Examle #5: Let the blue od eesent one hole. What colo od ould eesent 1 3 of the hole? What colo od ould eesent 2 3 of the hole? Ho many gous of 2 3 ae in one hole? b lg This examle ovides a good lace to intoduce the thinking that ill lead to easonable estimation skills. Questions like the folloing begin to constuct those estimation skills: Befoe you detemine the quotient, let s fist ty to detemine a easonable estimate by asking some questions: Do you think thee is moe than one gou of 2 3 o less than one gou of 2 3 in ou unit? Why? Moe than to gous of 2 3? hy? It is faily easy fo my students to decide that thee ill be moe than one gou of 2 3 in the hole but less than to gous of 2 3. What is not so easy is subsequently aticulating that thee ae exactly 1 1 2 gous of 2 3 in the hole, even hen they ae looking at the folloing model: b lg one comlete 1 one comlete gou of 2 only only 1 2 3 of of the second gou of 2 3 It is not unusual fo students to incoectly conclude that the quotient is 1 1 3. It takes some mental gymnastics to a themselves aound the logic that suggests the coect quotient is 1 1 2. When the students ecod the data fom this examle in thei table, I ask them to ecod the infomation using mixed numbes. I secifically tell them not to convet the mixed numbes to imoe factions. I also tell them not to simlify any factions. The ationale fo not simlifying ill become clea in Pat 3. Examles #6 though #19 ovide simila tasks that develo a visual image as ell as, the meaning of a faction division statement. Asking the students to geneate a easonable estimate befoe detemining htt://.tctmonline.net Fall 2005 / age
the exact quotient hels to develo thei estimation skills, and it also hels to efine the easoning of hy the quotient makes sense ithin the context of the oblem. Examles #6 though #8 ae of simila comlexity. Examle #9 intoduces contexts in hich thee ae moe than to gous of the diviso in the dividend. Examle #6: Let the bon od eesent one hole. What colo od ould eesent 1 4 of the hole? What colo od ould eesent 3 4 of the hole? Ho many gous of 3 4 ae in one hole? b Examle #7: Let the dak geen od eesent one hole. What colo od ould eesent 1 6 of the hole? What colo od ould eesent 5 6 of the hole? Ho many gous of 5 6 ae in one hole? y Examles #8 :Let the oange od eesent one hole. Ho many gous of 3 5 ae in one hole? Ho many gous of 2 5 ae in one hole? o Examles #9 : Let the black od eesent one hole. Ho many gous of 4 7 ae in one hole? Ho many gous of 3 7 ae in one hole? Ho many gous of 2 7 ae in one hole? Finally, examles #10 though #13 atchet u the comlexity just one moe notch by intoducing a diviso that is geate than one. It is esecially useful to have students geneate a easonable estimate befoe detemining the quotient on these examles. Fist I ll ose Examle #10, and then I ll go though the estimation easoning befoe osing Examles #11 though #13. Examle #10: Let the ule od eesent one hole. What colo od ould eesent 1 1 2? Ho many gous of 1 1 2 ae in one hole? Befoe you detemine the quotient, geneate an estimate: The question is asking Ho many gous of the size of the geen od can e get out of the ule od? Is thee enough quantity in the ule od to be able to ceate one comlete gou of the size of the geen od? Ho does this infom ou estimate? It can take students some time to figue out hat to do hen the diviso is lage than the dividend. I usually have a handful of students ho esist exloing the question asked (Ho many gous of the size of the geen od can e make fom the ule od?) and instead, in thei heads, evise the question to ask, Ho many gous of the ule od can e make fom the geen od? It is imotant fo the teache to kee u the intellectual ess so that the students confont the dissonance head on and ok though to a state of claity in thei undestanding. Vey secifically linking the vebal symbolic ith the vebal visual and the ictue can be helful: Ho many gous of 1 1 2 ae in one hole? Ho many gous of the size of the geen od can e get out of the ule od? =?? When the ods ae stacked it becomes easie to see that thee is not enough quantity in the ule od fo us to ceate a gou the size of one geen od. It is ossible to get at of a geen od, but not all of a geen od fom the ule od: The quotient (the numbe of gous of the geen od that e can get fom the ule od) ill be only at (o only a faction) of one geen od. The ed ods age 10 / Fall 2005 Texas Mathematics Teache
ovide some hel hen tying to detemine exactly hat faction of one comlete gou of the geen od can be ganeed fom the ule od: thee is only Thee is only 2 3 of a geen od in the ule od. 1 1 1 2 = 2 3 Colo of the Unit ange Lt. Geen Bon Statement in Wods Ho many gous of 1 10 ae in 1? Ho many gous of 1 3 ae in 1? Ho many gous of 1 4 ae in 1? Statement in Symbols Change all divisos & quotients to mixed numbes 1 1 10 = 10 1 1 10 = 10 1 1 1 3 = 3 1 1 3 = 3 1 1 1 4 = 4 1 1 4 = 4 1 Examle #11: Let the dak geen od eesent one hole. What colo od ould eesent 1 1 3? Ho many gous of 1 1 3 ae in one hole? ange Blue Bon Ho many gous of 1 5 ae in 1? Ho many gous of 2 3 ae in 1? Ho many gous of 3 4 ae in 1? 1 1 5 = 5 1 1 5 = 5 1 1 2 3 = 1 1 2 1 2 3 = 3 2 1 3 4 = 1 1 3 1 3 4 = 4 3 b Examle #12: Let the dak geen od eesent one hole. What colo od ould eesent 1 1 6? Ho many gous of 1 1 6 ae in one hole? Dak Geen ange ange Black Black Black Ho many gous of 5 6 ae in 1? Ho many gous of 3 5 ae in 1? Ho many gous of 2 5 ae in 1? Ho many gous of 4 7 ae in 1? Ho many gous of 3 7 ae in 1? Ho many gous of 2 7 ae in 1? 1 5 6 = 1 1 5 1 5 6 = 6 5 1 3 5 =1 2 3 1 3 5 = 5 3 1 2 5 = 2 1 2 1 2 5 = 5 2 1 4 7 = 1 3 4 1 4 7 = 7 4 1 3 7 = 2 1 3 1 3 7 = 7 3 1 2 7 = 3 1 2 1 2 7 = 7 2 Examle #13: Let the yello od eesent one hole. What colo od ould eesent 1 2 5? Ho many gous of 1 2 5 ae in one hole? y Pule Dak Geen Dak Geen Yello Ho many gous of 1 1 2 ae in 1? Ho many gous of 1 1 3 ae in 1? Ho many gous of 1 1 6 ae in 1? Ho many gous of 1 2 5 ae in 1? 1 1 1 2 = 2 3 1 3 2 = 2 3 1 1 1 3 = 3 4 1 4 3 = 3 4 1 1 1 6 = 6 7 1 7 6 = 6 7 1 1 2 5 = 5 7 1 7 5 = 5 7 PART 3: Making Connections Comleting a Data Table I use a data table ith my students that looks like the one ictued belo. This table hels students make the connection beteen thei concete eesentations and symbolic eesentations hen they comlete the second and thid columns. When they comlete the last column, students should begin to discove that all of the quotients ae the eciocal of the diviso. This atten illustates that heneve the dividend is one, the quotient is the eciocal of the diviso. Dividend Diviso 1 2 3 = 3 2 Quotient An altenate, and vey helful, ay to state this discovey is: The eciocal of the diviso alays eesents the numbe of gous of the diviso in one hole. htt://.tctmonline.net Fall 2005 / age 11
PART 4: Constucting the Invet and Multily Algothm 3 This is it, the last ste in the constuction of the invet and multily algoithm. The ocess to get fom hee to the end ill ely on undestanding and using the folloing concets: In multilication, one facto communicates the size of the gou and the othe facto communicates the numbe of sets of the gou that e ant. Fo examle, 3 x 5 = 15 communicates that e have thee gous of five. If eithe o both of the factos contain a factional at, the same communication alies. Fo examle, 1 1 2 6 communicates that e have one and one half gous of six. We can think of this as having one comlete gou of six, and an additional half a gou of six. This easoning is consistent ith the distibutive oety: 1 6 = ( 1 6) + ( 6). 1 2 1 2 In the model of division that e have used fo this exloation, the dividend communicates the amount e begin ith, the diviso communicates the size of each gou, and the quotient communicates the numbes of gous of the diviso in the dividend. In moe colloquial language, the quotient tells us ho many gous of the diviso e can make fom the quantity e began ith (i.e., the dividend). Fom the exloations, data collection, and eiting of the symbolic statements, e discoveed that the eciocal of the diviso tells us ho many gous of the diviso e can make fom the dividend. Keeing these thee concets in mind, e ll take a look at the fist illuminating examle. Examle #1: Let the oange od eesent one hole. What colo od ould eesent 1 10 of the hole? Ho many gous of 1 ae in one hole? 10 The symbolic statement fo this question, ith the quotient, is 1 1 10 = 10. The eciocal of the 1 diviso, 10/1, tells us that thee ae ten gous of 1 10 in 1. No suose I began ith to units athe than only one. My question ould then be Ho many gous of 1 10 ae in to holes? The concete/ ictoial and the symbolic eesentations ould look like this: 2 1 10 =??? Since I kno that thee ae ten gous of 1/10 in one unit, then it ould make sense to think about to units as to sets of one unit: + 1 1 10 = 10 1 1 10 = 10 This illustates that I have one gou of ten one tenths fom my fist unit, and a second set of ten one tenths fom my second unit. Thinking this ay, I could say that I am able to ceate to gous of ten fom my to units. I can eesent to gous of ten using symbols this ay: 10 + 10 o I can eesent it using multilication this ay: 2 x 10 No I ll ut this all togethe so that it is easie to see the flo of the thinking. The oiginal question asked Ho many gous of 1/10 can I get fom 2 units? The ictue and the symbolic statement ae shon togethe. 2 1 10 =??? The next ste in the thinking is to look at to units as to gous of one unit: 2 1 10 =??? + ( 1 1 10 = 10 ) + ( 1 1 10 = 10 ) 3 The undestanding to be develoed in this section is deendent on the student s io undestanding of both hole numbe and faction multilication. If the student s ability to exlain the logic of the hole numbe and/o the faction multilication algoithms is eak, it is likely that it ill be difficult fo the student to gas the concets constucted belo. age 12 / Fall 2005 Texas Mathematics Teache
The numbe of gous of 1 10 in to units is the same as the numbe of gous of 1 10 in one unit lus the numbe of gous of 1 10 in the othe unit Ste thee is to ecod the numbe of gous of 1/10 that can be made fom the fist unit and the numbe that can be made fom the othe unit. + 1 1 10 = 10 1 1 10 = 10 Thee ae ten gous of 1 10 in the fist unit and ten gous of 1 10 in the othe unit so I can eesent this infomation by iting 10 + 10 Ste fou is to estate the eeated addition statement as a multilication statement: No, I can eite 10 + 10 as a multilication statement by iting 2 10 This easoning ocess enables us to sho that ou oiginal symbolic statement is the same value as the last symbolic statement. 2 1 10 = 2 10 This same easoning can be alied to any faction division statement to illustate the logic that leads to the invet and multily algoithm. To hel sho this, I ll alk though the stes of easoning ith one moe examle: Examle #2: Let the black od eesent one hole. This eesentation illustates that 3 4 7 = 7 4 + 7 4 + 7 4 = 3 x 7 4 And thee e have it, the invet and multily algoithm! Summay Both national and state standads documents, and the moden needs of ou economy, suggest that students need to lean mathematics ith undestanding. Unfotunately, common justification fo the faction division algoithm deends on mathematics that is inaccessible to most students ho ae leaning faction division fo the fist time. Futhemoe, those algebaic justification stategies sho that the invet and multily algoithm oks; that a c does in fact b d equal a d, but they do not exlain hy it ould b c make sense that this ould be so. This aticle has ovided a line of easoning, suoted by concete and visual images, hich teaches can use to hel thei students constuct the tye of undestanding ecommended in both the national and state standads in mathematics. Tacy Rusch, Ph.D., <tacy.usch@ight.edu> Assistant Pofesso, Wight State Univesity, hio Ho many gous of 4 7 ae in thee holes? The ule od eesents 4 7 of the black od. The ictue shos that thee ae 1 3 4 ule ods, o 7 4 ule ods in one black od. Symbolically, I ould ite this as 1 4 7 = 7 4. The question asked is Ho many gous of 4 7 ae thee in thee holes? Since I kno that the eciocal of the diviso tells us the numbe of gous of the diviso in one hole, I could illustate the question being asked using the ictue belo to sho that I have thee sets of 7 4 in thee holes: + + 7 4 + 7 4 + 7 4 htt://.tctmonline.net Fall 2005 / age 13