67 um to Ifiity - a Ope-eded Ivestigatio Duca amso* Rhodes Uiversity Grahamstow *With the participatio of Nyasha Chiuraise, Christopher Gobae, olomo Johso, Phidile Madiyasi, Mzikayise Mai, Xoliswa Mbelai, Irma Moller, May Moya, Washigto Mushwaa, Cheriyaparambil Raghava, Thembile adi, Maud iaw, Beauty olai ad Mzwadile Zigela. FRF Mathematics Educatio Chair Ope-eded ivestigatios are a woderful way to access a diverse rage of mathematical topics i a meaigful ad egagig maer. Not oly do such topics ofte arise uexpectedly or seredipitously, but ope-eded ivestigatios ca provide a ideal cotext for urturig such importat dispositios as curiosity, creativity ad self-cofidece, alog with feeligs of persoal relevace ad a desire to egage dyamically i a process of geuie mathematical discovery. I preseted Figure to a group of secodary school mathematics teachers i a erichmet programme ru by the FRF Mathematics Educatio Chair at Rhodes Uiversity. What follows is a sythesis of some of the observatios that arose from the sessio, alog with further ideas ad possible aveues for additioal exploratio. FRACTION UM 6 etc. 6 6 Ivestigate Figure. A ope-eded ivestigatio. Table was completed i order to scaffold the ivestigatio, ad this was foud to be a effective meas of ecouragig iitial egagemet with the task. tadard calculators geerally show a maximum of oly te decimal places, but the Widows calculator set to scietific mode displays up to thirty decimal places. Table. ums of fractios expressed i differet formats. um showig idividual terms um as a decimal 0. 6 0. 6 6 0. 8 6 6 6 0. 0 6 6 6 0 0. 0078 6 6 6 0 096 0.9 6 6 6 0 096 68 0.9888 um as a sigle fractio 6 6 8 6 0 6 096 6 68 Learig ad Teachig Mathematics, 8, 67-7
68 The teachers sat i groups comprisig three or four members ad were challeged to fid as may patters or iterestig observatios as they could i Table. Before readig o, take twety miutes to try this task yourself. How may of the followig did you otice? The sums ca be expressed as. The sum approaches 0. as more terms are added. The last digit of the deomiators of the idividual terms alterate betwee ad 6. The umerator of the sum expressed as a sigle fractio is always more tha the sum of the deomiators of all but the last of the idividual terms. By way of example, i the fractio 6 / 096, 6 6 6 6 0, while i the fractio 8 / 6, 8 6 6. The sum expressed as a sigle fractio ever simplifies, i.e. the umerator ad deomiator ever share ay commo factors. The deomiator of the sum expressed as a sigle fractio is always more tha times the umerator. The sum of the umerator ad deomiator of ay fractio i the third colum gives the value of the umerator i the followig cell. By way of example, for the fractio / 6, 6 ad is the umerator of the followig cell, / 6. This observatio leads to the followig recursive formula for the sum of the first terms: Numerator of Deomiator of Expadig the last two colums of Table yields a umber of iterestig cyclig patters. ice we have idetified a umber of patters relatig to the sum expressed as a sigle fractio, we ca use these patters to expad the fial colum maually ad use the Widows calculator i scietific mode to express the fractio as a decimal. Table. ums of fractios expressed as decimals ad sigle fractios 0. 0. um as a decimal 0.8 0.0 0.0078 0.9 6 0.9888 7 0.87070 8 0.0676778 9 um as a sigle fractio 6 6 8 6 0 6 096 6 68 8 66 878 6 Learig ad Teachig Mathematics, 8,67-7
69 0.089 0 0.860768 0.68080 0.86679600078 0.096990069 0.08978788 0.7878070 6 0.907796908678 7 9 0876 980 90 90 67776 696 670886 89788 686 799 0778 676 996796 76606 7798698 The sum expressed as a decimal always eds with. Other tha 0., all other decimal sums ed with. The digit positioed fourth from the ed of the decimal sums alterates betwee ad 8, i.e. a - digit repeatig patter. The digit positioed fifth from the ed of the decimal sums cycles through a -digit repeatig patter:,0,7,. The digit positioed sixth from the ed of the decimal sums cycles through a 8-digit repeatig patter:,,0,9,8,7,,. We could possibly cojecture that the digit positioed seveth from the ed cycles through a 6-digit repeatig patter. The last digit of the umerator of the sum expressed as a sigle fractio alterates betwee ad, a -digit repeatig patter. The first digit of the umerator of the sum expressed as a sigle fractio cycles through a -digit repeatig patter:,,,8,. The first digit of the deomiator of the sum expressed as a sigle fractio also cycles through a -digit repeatig patter:,,6,,. May of these patters are fasciatig i their ow right, but the mere observatio of regularity is of little cosequece if it caot be developed ito a potetial learig experiece. This is perhaps the critical challege of ay teacher wishig to use a ivestigative approach i the classroom. However, a simple observatio ofte has the potetial to ope up a diverse rage of additioal topics. The challege lies i idetifyig such momets ad egagig with them appropriately. By way of example, take the observatio made by oe group of teachers that the deomiator of the sum expressed as a sigle fractio is always more tha times the umerator. Expressed differetly, the umerator is always a third of less tha the deomiator. ice the th sum has deomiator, this meas that the umerator ca be expressed as ( ) /. Learig ad Teachig Mathematics, 8, 67-7
70 Thus:, which ca be simplified to, ad fially to... Now, as approaches ifiity, approaches zero, ad thus approaches, which proves oe of. the iitial observatios that as more terms are added the sum approaches 0.. What is pleasig about this outcome is that it ca be arrived at from first priciples, without ay recourse to stadard formulae for geometric series or the sum to ifiity. Havig arrived at the formula, it is iterestig to ote that it ca be rewritte as. x p, which of course is i the stadard format for a expoetial graph, y a. b q. Thus, a graphical represetatio of would yield the basic expoetial graph y reflected about the x-axis ad traslated vertically through of a uit. ice this expoetial graph has y as a horizotal asymptote, this leds visual support to the observatio that as more terms are added the sum approaches 0.. ice ca oly take o discrete values (atural umbers), this also has the potetial to ope up a discussio o discrete versus cotiuous variables. Let us ow cosider the observatio made by aother group that the umerator of the sum expressed as a sigle fractio is always more tha the sum of the deomiators of all but the last of the idividual terms. This ca be represeted symbolically as follows: 0 0 6 i 0 6 i I geeral, 6 0 8 6 6 6 etc. ( ) Usig the formula for a geometric progressio yields, which is the same formula as arrived at previously. It is perhaps also iterestig to ote that the series 6 6 6 forms part of a class of covergig series i which the first term ad commo ratio are ot oly the same, but are of the form p. For such series it ca be readily show that. p Although there are o doubt may more aveues to explore with this ivestigatio, we have already touched o a diverse rage of topics: ratioal umbers, geometric series, sigma otatio, covergig series ad sum to ifiity, cojecturig, attemptig to prove cojectures, expoetial fuctios, reflectios, Learig ad Teachig Mathematics, 8,67-7
traslatios, discrete versus cotiuous variables, ad explicit versus recursive formulae. What is particularly meaigful is that this diverse rage of topics could be accessed ot oly through a sigle ivestigatio, but through a process of geuie mathematical discovery cetred o ecouragig a spirit of dyamic egagemet. Below are two further ivestigatios cetred o fractios. It is strogly recommeded that before usig them i a classroom settig, teachers first sped time thoroughly explorig ad persoally egagig with each task. This will provide critical isights ito the challeges ad appropriateess of each ivestigatio. Furthermore, it will assist i idetifyig those magical momets which have the potetial to lead to meaigful learig experieces which are the hallmark of a ivestigative approach. 7 DENOMINATOR FRACTION 6 6 7 Oe third ca be writte as a sum of two uit fractios. Ivestigate I this sequece, each term is obtaied from the precedig term by the rule: x x y y x y Ivestigate TECHNO TIP If you do t like havig to use the mouse to isert expoets, fractios ad the like i Equatio Editor, here are some useful cotrol keys that will speed up your typig. CTRL + h produces a superscript for expoets CTRL + l produces subscript CTRL + 9 produces ( ) CTRL + f produces a fractio CTRL + g allows you to type the ext letter as a Greek symbol (e.g. q becomes theta, θ) CTRL + r produces square root symbol ( ) Learig ad Teachig Mathematics, 8, 67-7