a. How might the Egyptians have expressed the number? What about?

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1 A-APR Egytia Fractios II Aligmets to Cotet Stadards: A-APR.D.6 Task Aciet Egytias used uit fractios, such as ad, to rereset all other fractios. For examle, they might exress the umber as +. The Egytias did ot use a 4 4 give uit fractio more tha oce so they would ot have writte a. How might the Egytias have exressed the umber? What about? b. We will see how we ca use idetities betwee ratioal exressios to hel i our uderstadig of Egytia fractios. Verify the followig idetity for ay : x > 0 x x + x(x + ) c. Show that each uit fractio, with, ca be writte as a sum of two or more differet uit fractios. d. Describe a rocedure for writig ay ositive ratioal umber, say, with, as a Egytia fractio., > 0 IM Commetary The urose of this task is for studets rewrite a simle ratioal exressio ad study the arithmetic of these exressios. Egytia fractios rovide a iterestig cotext, both historically ad mathematically, for studets to use ratioal exressios. Aalogous to the stadard A-APR.4, i which studets use olyomial idetities to

2 describe umerical relatioshis, this task has studets rewrite ratioal exressios (A- APR.D) i order to deduce iformatio about ratioal umbers. Secifically, studets will work to rewrite fractios as a sum of distict uit fractios, for which the idetities of ratioal fuctios rovided will be uite useful. I order to solve the roblem, studets will rewrite a simle ratioal exressio ad will eed to study carefully the arithmetic of these uit fractios with a variable i the deomiator. Deedig o imlemetatio (e.g., levels of scaffoldig), the task oses reasoably high levels of cogitive demad ad gives studets the oortuity to model may of the Stadards for Mathematical Practice. Coseuetly, teachers should be aware that a fully oe-eded imlemetatio of this task will reuire amle time ad guidace. The remaider of this commetary describes some discussio of imlemetatio. Teachers may fid wide divergece i aroaches take by studets, e.g., comig u with the rovided algebraic idetities o their ow (if they are iitially withheld), or the largely euivalet "greedy algorithm" demostrated i the solutio, i which studets simly choose the largest uit fractio uder the give fractio (e.g., uder ). As 4 such, the teacher may wish to ecourage strategy-sharig i grou- or whole-class discussio, ad/or give more examles for them to comlete exlicitly before movig o to arts (c) ad (d). Though studets are ecouraged to come u with multile solutio techiues, the solutio targeted by the series of romts i the task is to start with the biggest uit fractio less tha, amely. This leads to the exressio + a algebraic idetity of ratioal exressios. Teachers lookig to icrease the cogitive comlexity of the task might withhold the give idetity, ad have studets search for some o their ow. Alteratives solutio techiues aboud: Aother good method is to use the idetity to deduce (uo dividig by ) that, + + +, A solutio of the task usig this idetity istead is addressed i a secod solutio.

3 As otioal scaffoldig, the istructor may wish to ecourage studets to use art (c) to hel with art (d), as is doe i the solutios. Studets workig o this roblem will egage i several of the stadards for mathematical ractice: MP: Make Sese of Problems ad Persevere i Solvig Them. Studets will eed to uderstad what a Egytia fractio is ad how to covert a fractio to this form. This will reuire atiece ad exerimetatio as we are ot used to exressig fractios this way. MP: Reaso Abstractly ad Quatitatively. Studets will eed to make fractio coversios both i a cocrete ad abstract situatio. MP: Look For ad Make Use of Structure. I order to fid a egytia fractio reresetatio for a geeral fractio the studets will eed to idetify a atter ad also exlai how to imlemet their strategy which will reuire reeated use of a commo argumet. MP8: Look For ad Exress Regularity i Reeated Reasoig. Part c of the first solutio ad the secod solutio both aly a iterative rocess to write a fractio as a Egytia fractio. Some additioal uestios the teacher may wish to address or which may come u i workig o this task are: a. Is there a algorithm for fidig a Egytia fractio for a give umber? For umbers like ad it is ossible to fid a egytia fractio reresetatio through trial ad error. For a umber like this will be much more challegig. b. Ca you determie the smallest umber of uit fractios eeded i a Egytia fractio reresetatio of a give umber? c. Oe remarkable coseuece of art (d) of this roblem is that as grows, the sum evetually becomes larger tha ay give umber. For examle, usig art (b) we ca write,000,000 or,000,000,000 or ay other umber as a Egytia fractio so whe m m

4 m is big eough, the sum of all of the uit fractios with deomiators at most be made to exceed,000,000 or,000,000,000 or ay other ositive umber. m ca To view this task as art of the rogressio of learig iheret i CCSSM, teachers may wish to observe relatios with aalogous cotet i the th grade, e.g., foud i the task -NF Egytia Fractios. Solutios Edit this solutio Solutio: a. There are may strategies for doig this, ad ideed may ways of writig a fractio as a Egytia fractio. Though we'll ursue oe articular lie of thought i subseuet arts, we'll address a few otios here. For, oe way to write this as a sum of uit fractios would be to ote that < ad it is the biggest uit fractio less 4 tha. Subtractig it from, we fid that 4 ad this is a Egytia fractio reresetatio of. We ca emloy the same method for, which is just larger tha, the largest of the uit fractios. We have We ow have to deal with which lies betwee ad. We have Puttig this together with our revious euatio gives a Egytia fractio reresetatio of :

5 Startig over with, a alterative idea is to write as Now we have to break dow oe of the 's ad we could do this by writig foreshadowig the ucomig idetity = + 8 6, Puttig these two exressios together gives Similarly, for For 4 we could also start by writig this as this is a little less tha a third ad more tha oe fourth. We have 4 For this is a little more tha ad less tha. We fid 8 Puttig these calculatios together gives These umbers get large very uickly ad so a good, uiform method for how to roceed is defiitely eeded. x + = x + b. Oe way to verify the idetity is begiig with the idetity for ay, x(x + ) ad the dividig both sides by (valid sice is ositive, ad so this term is ot zero) ad distributig. This gives x x + x

6 x + x(x + ) x x(x + ) x(x + ) After cacellig commo terms, we're left with the desired idetity. A secod derivatio is oted i the ext art of the solutio. c. To write as a sum of smaller uit fractios, we eed oly aly the revious idetity. Note that, mirrorig art (a), we ca view this idetity as begiig the Egytia fractio reresetatio with the largest uit fractio smaller tha, amely,. We the have + + = + ( + ) =, ( + ) ( + ) which also rovides a alterative derivatio of the idetity i art (b). Rewritig this we get + ( + ) which is a alterative Egytia fractio reresetatio of. d. We ca write + where there are coies of the uit fractio. If the this is a Egytia fractio ad we are doe. If ot, we ca kee the first ad the relace the secod oe with +. If = the we are doe, havig exressed as a sum of three uit + (+) fractios. If > the we ca write the third as + ad the relace each + (+) of these usig the same idea: so could be writte as + ad (+) + ca be relaced with + = ( + )( + ) 6

7 ( + ) ( + ) + ( + )(( + ) + ) These exressios get very comlicated very uickly but the imortat thig to ote is that sice is at least the deomiators of these uit fractios are gettig bigger ad bigger. So if we cotiue to reeat this rocedure, relacig every dulicate uit fractio with two uit fractios with larger deomiators, the we will evetually remove all dulicates ad have a (very comlex) Egytia fractio reresetatio for. Edit this solutio Solutio: For the curious, we reset the solutios to art (c) ad (d) which make use of the remark i the commetary that a secod solutio stems from the idetity Multilyig both sides by gives , 6 so we have rereseted as a sum of three differet uit fractios. For a arbitrary, we begi by writig + where there are terms i the sum. We ca oly have oe i the exressio so we may take the secod ad relace it with We ow move to the third coy of ad relace it with Each of these uit fractios has already bee used so these also eed to be relaced.

8 Sice each relacemet makes the deomiators bigger, we will be able to evetually get all differet uit fractios i our exasio, that is we will evetually get a exressio of as a Egytia fractio. A-APR Egytia Fractios II Tyeset May 4, 06 at 0:9:. Licesed by Illustrative uder a Creative Commos Attributio-NoCommercial-ShareAlike 4.0 Iteratioal Licese. 8