~ t.. JL(jJ,.JyS <j? ~?-?,J-? flpsl..i.? .Too-. " ~I- ~ -c:\ w ~ a.'1 ~ ~ O~ 0'&0. ~ 1>") \). '1 s., /' om. l~,; ~ I'..Is cr-i." ft. ~ 'S ~+--..e (.

Transcription

1 C'l According o her SeinglPhysics Core Curriculudens ::: Jf:.A-Jl.- I' S -..Jc.f;. "c,. 'S' ;.,,f,j{ ecr :.. JL(jJ,.JyS <j??-?,j-? flpsl..i.? -J rj"! r-oa 5S ;, /ll 'I5"/;> ;JT'" 0-,. III sus-?,.;,.r <;i.k... RELAT.REPRESENTATIONS IN <INEMATICSI/ This aricle focuses on he imporance, deelopmen, srucure, and deliery of an ssmen ool ha I creaed called Relaing Represenaions in Kinemaics (RRK). -ea<:hing physics, we consanly remind he sudens ha here are generally four en ways o represen moon-erbally (V), mahemaically (M), graphically (G), and dagrammalcallygrams,. D). Th,:...:. sudens on heir abiliy o differeniae beween esenalons and(jy o ransfer informaion from Cj ne represenaion o anoher. This formaie assessmen ool can also be used as an. aciiy o help sudens pracice,i. relaing MIweer. A"" inerpreai 4'moion. L. LL-,., Inroducion: - required o be able o disinguish, relae and inerpre symbolic, erbal, and mahemaical are ---. informaion (Sandard I, Key Idea I). Research also shows ha sudens mus know how ----=""'::: o relae one inerpreaion of moion o anoher in order o maser heir undersanding of kinemaics. "The hinking inoled in making he ranslaions o and from graphs help regiser he conceps [OfkinemaiCS],'6:;:>APparenly correcly ranslae from one rcinemaics graph o anoh he bes ou { l1e I- /1 " fj undersanding of kinemaics graphs," sudens who could This assessmen ool is only 4 quesions long, bu each quesion incl udes a.too-. " I- -c:\ w a.'1 O 0'&0. 1>") \). '1 s., /' om. l,; I'..Is cr-i." f. 'S +--..e (. 1 ')1\.10"'"'?, e"," I.Y" "k'" L y... A/_. s,f3-k.. I/-if 7 C8e., Jl-6'3rf'.., T/'l;" ""' J, Co...1,; 1Iq/''''J quesion. They mus hen ake he graphical informaion and explain he moion erbally. The nex sep is o use he erbal par of he quesion o answer a mahemaical problem. Finally, hey are asked o draw do diagrams o represen he sarpe moion. Since 'U'!:J sudens mus graph somehing 3 imes, here are 5 problems per quesion. So here are acually 20 problems for his assessmen.? / Mahemaically, here should be 24 differen ways o arrange hese 4 7 'J AAN'1. ec,' represenaions, howeer, mos of he combinaions repea isel? f-for example: rr _. - Fi. a) G, V, M, D J b) G, V, D, M Alhough he combinaion is differen, sudens are relaing a erbal represenaion from he graphical for boh quesions. ThiS roue WIll yield a 120 problem assessmen which IS ery excessie. Therefore, he forma for he assessmen IS GVMD )\,,, S.e7 ";).I.JL. V, D, G, M D,M,V,G M, G, D; rh, c.l I J 'I J. l c. Se"1 j.. "'CJb (.j) I 1-) f This esures ha he sudens will be assessed on eery possible way o relae he represenaions wih he minimum amoun of quesions so ha i is possible o use his assessmen as a full period exam. combinaion of relaing inerpreaions. For example, quesion 1 has he forma "G, V, M, D." Sudens mus analyze a elociy s. ime graph and qualiaiely skech he acceleraion and posiion s. ime graphs o complee he graphing secion of he Srucure of he RRK: Quesion I: GVMD rj

2 wl.y r'? The RRK begins wih a elociy s. ime graph of a ball in moion. The ball is slowing down while raeling in he posiie direcion and speeding up while raeling in he negaie direcion. This is a classical case of a ball being hrown up in he air or,.,... Pzc.. 'i,.. C-ol'" " \. ey wan, and he grader mus accep muliple answers ThiS also helps reinsae he Idea (Y'. a an objec ha slows down does no rael backward unil i sops Sudens frequenly! j-. dralons. ime graph as an "Inerse" cure when hey are asked o graphically i 1e i? ieswn 'he moion of an objec decreasmg IS speed k CjJ Sudens are hen asked o wne a erbal descnplon of he moion "Sudens VI. should be led o ranslae graphs ino erbal descripion. They should also ranslae erbal -:J>.;:C J ) w- j.l-. 1.' Ii Jr. J I. S b"; I,ll.. (s- i I,J ) 1 {,.. sudens are o use heir do diagram o answer he nex sep, no he erbal Jip?-rJJ (":;> '" > J... presenaion. This ensures 2 hings: I) Sudens are following he inended order and descripion ino graphs," (Arons, 35). Sudens should include enough informaion as o,? 2) sudens can go back o check if all heir seps make sense when he seps are proide hem wih a sufficien amoun of giens o sole he nex par of he quesion, A which is o calculae he acceleraion. Using heir calculaed acceleraion, sudens muaw do diagr7 f 9 r he 1" IJ... moion of he objec. Since heir firs do will represen =O, hey should ha 9. A h his sep, The sudens mus no only figure ou he separaion ofhe dos bu also e r.. idea ha acceleraion means he change in elociy for eery do and no elociy iself. J)f "If asked o describe, in simple, eeryday words, wha "acceleraion" means, many sudens respond 'how fas i goes,' wih no ery clear aneceden forhe pronoun' i.' If ('Ifi '. ) hen asked o describe wha 'elociy means, hey gie he same response," (Arons, 32). compared o each oher. A his poin, sudens can quaniaiely skech he elociy and ( ejl.lj!.<.. A) i,,-s As furher pracice and pro concep of acceleraion, sudens ;:: hol '" Ouesion 2' VOGM firs figure ou how many dos hey mus use in order o correcly draw heir do diagram based on he gien erbal informaion. They are no plugging numbers ino an equaion bu raher hinking concepually abou how long i will ake for an objec o reach ha speed wih he gien acceleraion. (O_J...} If a suden makes a misake on he do diagram, hen he graph should be graded based on he do diagram and no he erbal descripion, The quesion clearly saes ha acceleraion V5. ime graphs bu only qualiaiely skech he posiion s. ime graph. The nex sep is a good way o check if he sudens are following insrucions. The quesion clearly includes "Using your graphical descripion." If he sudens ry. calculae displacemen by using one of he kinemaics equaions, he is -= rolled up a ramp. I sar wih his quesion o reinforce he imporance of consan accel ion whi slowing down. A similar qusionbeen menioned in he June 9 ''/ VA,. ((; 20071NYS P cs Regens Exam (Quesion rja -.L--e.. -<-.""",---. Sudens are hen asked o draw he posiion s. ime and acceleraion s. ime C... raphs accordingly. I is imporan o menion ha hey mus sar a he origin for he elf posiion graph for ease ofgrading. Oherwise, sudens will slarl a whaeer posiion o-- ;J c... +j..i (J}"r.i:?1)SS:U.. P/J:i:. direcion and size of he elociy and acceleraion ecors for each momen of ime. correcly. Sudens mus calculae he area under he elociy s. ime cure in order o Sinceealculaed he acceleraion for he sep before, hey ge exra pracice wih he <;,

3 properly arrie o heir answer. "Sudens do no recognize he meaning of areas under kinemaics graph cures," (Beichner, 755). r Quesion 3: DMVG The do diagram represens a jogger ru;nwa consan elociy. Sudens are o use heir general consan elociy equaion (d=) in order o sole he problem. Any M inclusion of acceleraion in a suden's answer for his problem is a clear indicaion of his lack of concepual undersanding of acceleraion. For he erbal represenaion, sudens mus menion eiher consan elociy or consan acceleraion. The grader should no accep consan speed unless i is accompanied by consan direcion. The do diagram clearly shows ha he objec is raeling in a sraigh line and he elociy ecors are no changing direcions. Sudens mus now make graphs ou of heir erbal represenaion. "Sudens should be asked o ranslae froll moion eens o kinemaics graphs and back again," (Beichner, 755). They should be able o figure ou ha he acceleraion has a alue of zero mfs/s and he elociy has a alue of4 m/s for he enire ime ofmoion. The posiion will increase wih a consan slope unil i reaches a alue of20 m on he 5'h f raeling eas, he acceleraion ecor is poining wes. Sudens are asked o calculae he ime bu hey mus firs calculae he acceleraion which should yield a posiie number. This reinforces he idea ha a posiie acceleraion does no always mean speeding up. Once hey deermine he acceleraion, hey can easily figure ou he ime of moion. Once he ime of moion is deermine, sudens hae enough quaniies o skech all hree graphs. The posiion s. ime graph for his problem will be he hardes o skech. Sudens mus firs realize ha he objec is moing in he negaie direcion; herefore he graph should be below he horizonal axis. Nex, hey mus draw a cure / wih a seep negaie slope iniially and a flaer, non-zero negaie slope a he end. They usually hae a ery difficul ime figuring ou his soluion. Sudens will now draw do diagrams based on heir graph. "Unless hey are.explicily led o do so, sudens do no consciously connec he graphs wih acual or / isualized moions; hey rea hem as un inerpreed absracions," (Arons, 28). Agf,,'.r for ease ofgrading, I included he direcion for wha would be considered he posiie direcion for his problem. Obiously here is no definiie posiie or negaie dire f n, bu by including he sign of he direcion for his problem, here will only be one correc direcion. Sudens mus now draw dos from righ o lef decrease he spaces beween / V f (f.."'\. second. Quesion 4: MGDV The final quesion of he RRK sars wih a rain ha is slowing down in he negaie direcion. Sudens mus undersand ha an objec ha slows down is acceleraing in he epposie direcion. For example, if a car is slowing down while P-..::>,.-().. he dos, and hae acceleraion ecors poining o he righ while he elociy ecors, which should decrease in size sequenially, will poin o he lef. Finally, sudens should be able o exrapolae informaion from he do diagram o make a erbal descripion of he moion. This seems like a ery easy ask, bu i's as difficul as he one before. Jus as saed, sudens mus use heir informaion from he./.' do diagram in order o answer he erbal par. If hey hae an incorrec dodiar-..

4 heir erbal descripion should be a coninuance of heir diagram and no he graph, which mos sudens will be inclined o do. Therefore, sudens may no ge full credi een if hey correcly describe he moion erbally if hey incorrecly described i diagrammaically Grading: Alhough he es is consruced in a way ha is relaiely easy o grade, here are sill some unaoidable issues ha mayor may no be soled depending on ihe grader. For sarers, he RRK follows he rule ha he approach for a problem wihin a quesion mus sequenially follow he problem before i. For he VDGM quesion, sudens mus use he graph in order o sole he problem mahemaically. The problem his causes is similar o problems we face when grading he NYS Physics Regens exam. If a suden ges he iniial problem incorrec, she can sill receie full credi for he nex problem if she carries along he preiously incorrec answer. In his case, a suden may no hae any idea how o draw he do diagram for Quesion 4 c...5' --Ii) ;'L/: c;. credi ifshe saes ha he objec is slowing down in he negaie direcion ifher do (o I 7:).J diagram does no display his. The purpose of his exercise/assessmen is o ensure ha i "" cl,e-:; ) here is a leas one quesion where he sudens mus ranslae a do diagram ino words. ::!:.; "",. Deliery of RRK: For mos formaie assessmens, I feel ha i is bes o adminiser he assessmen hree imes--once before he inroducion o he uni, once direcly afer, and once a he end of he year. The firs ime should be used o help eachers and sudens analyze where heir srenghs and weaknesses are regarding he conceps of he opic. The second exam should show wherelif improemens were made. Unlike mos aricles, a hird ex is equally imporan Sudens could someimes memorize a few ideas wihou acually nalizing he concep in order o pass an exam. Those memorized ideas will lea? " re"$eir sem in a maer of ime and will be harder o recall for a laer exam unless hey...a= compleely undersood he maerial. Ci For ha reason, I adminiser he RRK righ Defore he subopic of graphical d,1 D I... dl'j. and will draw 5 eenly spaced dos wih no acceleraion and equally sized ecors for all 5 dos. They will clearly be marked incorrec for he incorrec do diagram. Bu for he 1 erbal descripion, a suden could wrie "he objec is raeling a a consan elociy of1 50 m/s for 5 seconds." Technically, she will be marked correc because her erbal descripion correcly relaes o her diagrammaical represenaion. I hae ye o figure ou a way o eliminae his shorcoming. The suden was able o ranslae a do diagram ino words, bu she clearly ook he easy way ou. On he oher hand, I would no gie her. moion, direcly afer ha opic, and once again in June when hey are reiewing for he NYS Regens exam. Sudens need o know imporan words like posiion, iniial elociy, final elociy, ec. in order o answer any of he quesions, so I wai unil we a leas inroduce he general conceps and mahemaical equaions of kinemaics. An imporan aspec of any assessmen ool is he mehod of deliery. o Assessmen daa is mos accurae when i is properly adminisered. For exampl firs handed ou he Force Concep Inenory o my sudens, I made Ihe enible misake of saying, "his will no be graded." A er all, hey canno be graded on somehing

5 J J;J hey'e no seen before in my class. The problem is high school suden (r.. s: ::i During he pos es, I ell hem ha i is rpw being graded (since J: s seriously when hey hear hose words. The daa was 1. Arons, A. B. (1997). Teaching 11Iroducory Physics. New York: Wiley. Ch2. eiher chose choice "c" for all quesions or hey lef mas (hem bla The nex y, I f(u Beichner, R.K. (1994, Augus). Tesing suden inerpreaion of kinemaics graphs. used a raher producie mehod, and I'e used i flawless\;:. year af year. I call i he American Journal ofphysics, 62(8) 'lf I 0 c, de/" The Physics Teacher, 30(3) es will deermine he smares suden in he class. I's amazing I>w well sudens work - References Physics IQ Tes" and menion ha i's hands down, documened ani solid proof ha his. 3. Hesenes, D., Wells, M. & Swackhamer, G. (1992, March). Force concep inenory. under hese rondi ions. hey'e seen he maerial in class), and he sudens work equall eery suden and eery school is differen, bu his works impeccably wih he sudens in my class. = Conclusion: Relaing Represenaion in Kinemaics is a wonderful ool ha could be used as a formaie assessmen or a classroom acliiy o help srenghen he conceps of moion 4. Sokoloff, D.R. & Thornon, R.K. (1998, April). Assessing suden learning of Newon's laws: The Foroe and Moion Concepual Ealuaion and he Ealuaion of Acie Learning Laboraory and Lecure Curricula. The Physics Teacher, 66(4) ? 1 and shoren he gaps of misconcepions. The RRK has been shorened o fi ino one or wo periods depending on he class ime and sudens' skill leels. I is also only 4 pages long, which will conenienly no ake much paper or ime while copying. Hopefully, we J/;Arrcan all use he RRK o ulimaely lead he sudens o a beer oerall undersanding of kinemaics. Clear comprehension of kinemaics a he beginning of he year will lead o a srong undersanding of he opics ha follow. "We hae shown ha improing suden undersanding of kinemaics also improes suden learning of dynamics, een if dynamics is augh i1l a radiional manrrer," (SokololT & Thornon, 343) oj<.. 7

6

7 Name Per Dae _ d) Draw a do diagrnm o represen his moion. Assume he dos are aken afer each second. Make sure o include elociy & acceleraion ecors and any quaniies ha are gien in he preious represenaion: O').%r? Relaing Represenaions in Kinemaics (RRK) Quesi. Below is a elociy s. ime graph for a ball in moion: a) Skech (qualiaiely) he posiion s. ime (sar a he origin) and acceleraion s. ime graphs for his moion: 22 m/s 22 m/s s 8s d a Quesion 2: 01:.. 'O""s0 e:-.. A car sars from res and acceleraes a a consan rae of 6 m/s/s unil i reaches a speed of 30 mls. a) Draw a do diagram o represen his moion. Assume he dos are aken afer each second. Make sure o include elociy & acceleraion ecors: b) Wrie a brief sory abou he moion of he ball (wha is he ball doing?). Make sure o include any quaniies ha Me gien in he preious represenaion: b) Using your dol diagram, skech he posiion (sar a he origin), elociy, and acceleraion s. ime graphs for his moion. Make sure o include any quaniies ha are gien in he preious represenaion: d a c) Using your erbal descripion, calculae he acceleraion of he ball hroughou he 8 seconds of moion (Show all work): 4... ol e,l; df 0.J- +- rrj { cjj. Cil,...k.., c) Using your graphical descripion, calculae he displacemen of he car (Show all work): )(;-J... -? f \ { ( i ( & I- ( + -",,

8 (2l, -ru f 'QLA br'7'1: --Yo deff dd d,, S" The following is a do diagram of a person jogging: > I a) =0 r c""" si-e..j- -r 4mis 4ln/s 4mis 4m:s 4m/s 4m i s \'---7 y " Y---7 y " Assuming he dos are aken afer each second, wha is he jogger's displacemen? (Show all work) -i).,j-..l :.)-J: a? Quesion 4: The following numerically represens he moion of a rain: Vi = -50 mls Vf= -25 mls "/} A- a. X:'Srrl d=-150m - r a) Calculae he ime of moion (Show all work): IS (.., u&-\s') ad"" b) Skech he posiion (sar a he origin), elociy, and acceleraion s. ime graphs for his moion. Make sure o include any quaniies ha are gien in he preious represenaion: b) Wrie a brief sory abou he moion of he jogger (wha is she doing?). Make sure o include any quaniies ha are gien in he preious represenaion: d a d c) Using your erbal descripion, skech he posiion (sar a he origin), elociy, and acceleraion s.,ime graphs for his moion. Make sure o include any quaniies ha are gien in he preious represenaion: a c) Using your graphical descripion, draw a do diagram o represen his moion. Assume he dos are aken afer each second. Make sure o include elociy & acceleraion ecors and any quaniies ha are gien in he preious represenaion (righ of he page is he posiie direcion): d) Using your dol diagram, wrie a brief sory abou he moion of he rain (wha is i doing?). Make sure o include any quaniies ha are gien in he preious represenaion: