WPC  ΎWtA@D(+FsObבrS]{Ɖ*Q0k[]1WsfQ[{<+O_pQRWa9a6<Qxrc^`KY=TqmHj76ʃ+ϟ%n?ύn5Ef:y!W]Q]b14nZ~&魑JU˽HT r<ɬP6 TF:ՏZ[C #MnLj;Lt1I۸婆e/hieΑkZ!BkyRE'?J6b_{V{̃?m.^ψӫc8׮RRBBΒ׳bOw8 lU^|=Ylpɽ6hnCov1Lb=4^SFY~l8 %XT"[R͋^f#UNU % 0( UH ^  w% 4) = L mN 1e 1u 0`ca 1 0wK 0 0L 0 0 0g 0B 11 1 1? 1 1Mfa 72 1 72 1 72Z 1 72 1E 72b 0D` 0D 72 1 72 0Nv!,O B 0D< A D- 0K A CANON BJC-2010 (COPY 2)M/,,/M0(  Z6Times New Roman RegularX($,w 0Times New Roman Boldd4|+ QLl+,܃ H3|xU(5hCEKQW]cioAutoList11.1.1.1.1.1.1.1.3#37=CIQYag1.a.i.(1)(a)(i)1)a)(;3$2#  0  .3  0  hECEIMQUY]aAutoList2""""""""i)1)a)(O;$0  2#  a  .3  0` (#(#(b$0  0` (#(#2#   .3  0 ` (#` (#(xir$0  0` (#(#0 ` (#` (#2#(  0  )3  0 (# (#($0  0` (#(#0 ` (#` (#0 (# (#2#(  a  )3  0h(#(#(F$0  0` (#(#0 ` (#` (#0 (# (#0h(#(#2#(   )3  0h(#h(#($0  0` (#(#0 ` (#` (#0 (# (#0h(#(#0h(#h(#2#  0  )3  0(#(#({$0  0` (#(#0 ` (#` (#0 (# (#0h(#(#0h(#h(#0(#(#2#  a  )3  0p(#(#`UCEIMQUY]aAutoList3""""""""i)1)a)heCEIMQUY]aAutoList4""""""""i)1)a)huCEIMQUY]aAutoList5""""""""i)1)a)pCEIMQUY]aAutoList6""""""""i)1)a)hCEIMQUY]aAutoList7""""""""i)1)a)TABLE A i)(hCEKQW]cioAutoList81.1.1.1.1.1.1.1.1.(hCEKQW]cioAutoList91.1.1.1.1.1.1.1.!1.(գEGMSY_ekqAutoList101.1.1.1.1.1.1.1.#1.(EGMSY_ekqAutoList111.1.1.1.1.1.1.1.%1.http://www.mathgoodies.com/lessons/toc_vol2.shtm(3$ !  (F$0  0` (#(#0 ` (#` (#0 (# (#0h(#(#0h(#h(#0(#(#0p(#(#2#     )3  0p(#p(# )hYEGKOSW[_cAutoList12"""""""")1)a)+)"(O$http://turnbull.mcs.st-and.ac.uk/~history/Java/Circle.htmlhttp://www.merlot.orgLevel 1Level 2Level 3Level 4Level 5Level 1Level 2Level 3Level 4Level 5Level 1Level 2Level 3Level 4Level 5Level 1Level 2Level 3Level 4Level 5 d(3$ !  ݁Level 1Level 2Level 3Level 4Level 5^\ba I&mage <=8C HKKKK)!dxdx( $ Figure  1  ^\ba&OLE 2.0 Box <=8C HKKKK !  __________@ X@XXXXX@TheCircumference/DiameterRatio  __XðX Content: 0 ` PreAlgebraJ` (#` (#  Topic: 0 ` _Determinetherelationship_Ԁbetweenthecircumferenceanddiameterof_acircle ' usingbothrealworldandsimulateddata__. ` (#` (# _ Objectives:    Thestudentwillbeableto:   Day1 g   (5h83 "3"    325h  1  .3  0 `   constructacircleusingGeometers_Sketchpad_Ԁsoftware34݌T ` (#` (# Ќ  "3"    3)25h  2  .3  0 `   measurethecircumferenceanddiameterofcircles3)b݌A ` (#` (# Ќ  "3"    3/25h  3  .3  0 `   createatableofvaluesbasedonthedatageneratedbyGeometers_Sketchpad_3/h݌.~ ` (#` (# Ќ    Day2 X "3"    325h  4  .3  0 `   measurethecircumferenceanddiameterofvariouscircularitems3݌E` (#` (# Ќ  "3"    325h  5  .3  0 `   createatableofvaluesbasedondata3݌2` (#` (# Ќ  "3"    3 25h  6  .3  0 `   identifythedependentvariableandtheindependentvariable3 ݌` (#` (# Ќ  Day3  "3"    3 25h  7  .3  0 `   plotbothsetsofdatausingGeometers_Sketchpad_Ԁsoftware3  ݌` (#` (# Ќ  "3"    3 25h  8  .3  0 `   analyzethe_data(bothcomputergeneratedandrealworld)_3 A ݌` (#` (# Ќ  "3"    36 25h  9  .3  0 `   discoverthe Circumference_Conjecture_36 o ݌p` (#` (# Ќ  Materials: J   (ex 8 "3 "  32e"3  0    Geometers_Sketchpad_Ԁsoftware3,݌6(#(# Ќ   (Ex x "3 "  3v2E"3  0    Round_objects_3v݌#s(#(# Ќ   (Up x "3 "  32U"3  0    Meterstickormetricsewingtape3݌`(#(# Ќ  "3 "  32U"3  0    Sewingthreadorthinstringtomeasurethecircumferenceofeachround_object3݌M(#(# Ќ  Introduction:  !' Therelationshipofcircumferenceanddiameterwillbeintroducedbygivingstudentgroupsa "  diagramofacircleandaskingthemtomeasurethecircumferenceanddiameterofthecircle #! usingvariousmeasuringdevicessuchasstring,tapemeasure,ruler,oretc.Thestudentshave $" beenintroducedtoGeometers_Sketchpad_Ԁinapreviousactivity. % #  Procedure:  d'"% ! ! (h8p  Day1  P(#& "3"  32h  1  .3  0    Thestudentswillbeassignedtoteamsoftwo.3݌<)$'(#(# Ќ  "3"  32h  2  .3  0    Usingtheattachedactivitysheet,eachteamwillrespondbyusingdataobtainedwiththe )*y%( Geometers_Sketchpad_Ԁsoftware.3݌+f&)(#(# Ќ  "3"  32h  3  .3  0    Eachteamwillprovidetheinstructorwithadiskcontainingtheactivitydata.3>݌,S'*(#(# Ќ  "3"  3)2h  4  .3  0    Theinstructorwillprovidefeedbackonthedisk.3)V݌,@(+(#(# Ќ   --),  Day2   ! ! #(գ88"3"  32գ  1  .3  0    Thestudentswillcontinuewithsameteammemberusinganewactivitysheet.3(݌(#(# Ќ  "3"  32գ  2  .3  0    Usingtheguidelinesprovidedontheactivitysheet,eachteamwillprovidetheinstructor  withdataasindicated.3=݌z(#(# Ќ  "3"  3_2գ  3  .3  0    Theinstructorwillprovidefeedbackonactivitysheet.3_݌g(#(# Ќ  Day3  A  ! ! %(88#EachteamwilluseThink,Pair,Sharetoanalyzetherelationshipofthecircumferencetothe - } diameterbyrespondingtothequestionsontheworksheet.  j  Closure   D  Atotalgroupdiscussionwillfollowtheteaminvestigations.Thestudentswillprovidethe 0  instructorwithteamanswerstoDay3investigationworksheet.    Assessment:     Day1     !(h88%InstructorwillreviewGeometers_Sketchpad_Ԁactivitydiskandprovidefeedbackwhichincludes  anindicationofmasteryornonmasteryoftheactivitymaterial. p  Day2  I Instructorwillcheckforaccuracyofthecircumferencetodiameterratioandprovidefeedback 5 whichincludesindicationofmasteryornonmasteryoftheactivitymaterial.Studentswillneed "r toredotheactivitysheetifnonmasteryisindicated. _  Day3  9 InstructorwillcheckforunderstandingoftheCircumferenceConjecturebyreviewingDay3 % datainvestigationsheetandbythequalityofthelargegroupdiscussion.Instructorwillgive  studentsfeedbackwhichindicatesmasteryornonmasteryofthematerial.   _Websites_ԀforFurtherInvestigation t   Youmaywishtoviewthefollowingsitesforprelessonideasorextendedlessonsforcircles:  M"  4' -O  5  http://www.mathgoodies.com/lessons/toc_vol2.shtm 6 '-O!'  7 O' &$v" 4. -O  5  http://turnbull.mcs.st-and.ac.uk/~history/Java/Circle.html 6'-O(  7 @( %O!$ 4/ -O  5  www.merlot.org6(-O )  7 & ;) '(#&  http://_archive.ncsa.uiuc.edu_/Edu/_RSE_/_RSEorange_/_Piactivities.html_ )%(   u,'+  @  GEOMETERS_SKETCHPAD_Ԉ  @ THECIRCUMFERENCE/DIAMETERRATIO  @\\ @@(DAY1ACTIVITYSHEET  Sketch: `  Step10  0` (#(# ConstructAB L ` (#` (# ""  &,0  0` (#(#2h  1  .3  0 ` (#` (#  Clickonthepointiconinthetoolbox.Movethearrowtoapositionon <  thescreenandclick.Placeasecondpointonthescreen.&,S,݌) y (# (# Ќ  ""  -0  0` (#(#2h  2  .3  0 ` (#` (#  Clickonthearrowicon.--݌ f (# (# Ќ  ""  .0  0` (#(#2h  3  .3  0 ` (#` (#  Selectthetwopointsbyholdingdowntheshiftkeyandclickingonboth  S  points...݌ @  (# (# Ќ  ""   00  0` (#(#2h  4  .3  0 ` (#` (#  Constructthelinesegmentbyselecting linesegment inthe Construct  -  menu. 0:0݌  (# (# Ќ  _Step20  0` (#(# Constructthemidpoint_Ԁofthesegment  ` (#` (#  (ux8!"3"     ` 3|22u"3  0   Selectthelinesegmentbyclickingonthesegmentwiththeselection   arrow.3|22݌ (# (# Ќ  "3"     ` 332u"3  0   Select Pointatmidpoint fromthe Construct menu . 333݌q (# (# Ќ     `  _Step30  0` (#(# ConstructcircleCB _ԀN` (#` (#  (xx"3"     ` 352"3  0   Clickonthecircleiconinthetoolbox.MovethearrowtoPointC,hold > downtheleftmousebutton,anddragtocreatethecircle.Besurethe +{ cursorispositioneddirectlyonpointBwhenyoureleasethemouse h button.3576݌U (# (# Ќ  _Step40  0` (#(# Measurethecircumference_Ԁofthecircle /` (#` (# ""  m80  0` (#(#2"3  0 ` (#` (#  Clickonthecirclewiththeselectionarrowandselect circumference   fromthe Measure menu.m88݌ (# (# Ќ  _Step50  0` (#(# MeasureAB_Ԁ(thediameterofthecircle) ` (#` (#  +)(Yxx"3"     ` 3:2Y"3  0   Clickonthelinesegment(diameter)andselect diameter fromthe    Measure menu.3:!;݌|! (# (# Ќ  _Step60  0` (#(#Makethecirclesmall_;thenselect, inorder ,thediametermeasurementandthe Y#! circumferencemeasurement(holddowntheshiftbuttonasyouselectboth I$" measurements)andchoose Tabulate inthe Measure menu.6% #` (#` (# _Step70  0` (#(#Makethecircle_Ԁalittlelarger;thendoubleclickinsidethetabletoaddanother 'c"% entry. (P#&` (#` (#  Step80  0` (#(#Repeatstep7threeorfourmoretimes.)*%(` (#` (# Step90  0` (#(#Save_sketch_Ԁondiskforfurtherinvestigation.Givedisktoinstructoratendofthis +'* activity.,'+` (#` (#  -(, Ї   `  @  THECIRCUMFERENCE/DIAMETERRATIO   @\\  DAY2ACTIVITYSHEET  NamesofTeamMembers򀀀 M   Forthisinvestigation,youwillneedthefollowingmaterials: % u  (xx+)"3"    3A2"3  0 `   Fiveroundobjectsofvarioussizes_3AB݌ O ` (#` (# Ќ  "3"    3B2"3  0 `   Meterstickormetricsewingtape3BB݌ < ` (#` (# Ќ  "3"    3C2"3  0 `   Thinstring3CC݌) ` (#` (# Ќ  __Step_10  0` (#(#With_Ԁthethreadandmeterstick(orsewingtape),measurethecircumferenceand   diameterofeachroundobjecttothenearestmillimeter(tenthofacentimeter). ` (#` (# Step20  0` (#(#Usingthetablebelow,recordthecircumference(C)and_diameter_Ԁ(D) z measurementsforeachroundobject.g` (#` (# *Wddd Xdd Xdd X(#(#,dd ,sdd ,dd ,dd ,dd +  AA Nameofobject               Circumference(C)  &v   &v   &v   &v   &v Diameter(D)            !   " C/D  (x#   (x$   (x%   (x&  (x'  Step30   ` CalculateC/Dandrecordtheanswersinyourtable.2((#(# Step40  0` (#(#CalculatetheaverageofyourC/Dresults_򀀀_ \*` (#` (# _   6, @  THECIRCUMFERENCE/DIAMETERRATIO @@*   @ DAY3INVESTIGATIONWORKSHEET   @  COMPARING______DAY_Ԁ1_AND__ԀDAY_Ԁ2DATA   ___ NamesofTeamMembers򀀀  a _Step10  0` (#(#GotoGeometers_Sketchpad_ԀandopentheSketchfromDay1.: ` (#` (# _Step20  0` (#(#Click_Ԁinsidethetableoncetoselectit;_then_Ԁchoose PlotTableData inthe Graph  d  Menu.Click_ OK _ԀinthePlotPointdialog.Describethepointsthatappearonthe  T  graph.Ifyoudontseeallthepoints,dragthepointat(1,0)toscaleyouraxes. D ` (#` (# _Step30  0` (#(#Constructaray_Ԁfromtheoriginofthegraphtooneofthepointsonthegraph.   (Selectthetwopointsbyholdingdowntheshiftbutton.Clickontheline   segmenticoninthetoolboxand_clickon_Ԁtherayicon._ԀSelect Ray fromthe    Construct Menu.Clickonselectarrowtoleavesegmenticon.)_ԀMeasureand   recordtheslopeofthisray.Whatratiodoesthisslope_represent?` (#` (# ____0  __0` (#(#__Ԁ u Ѐ b Ѐ_O` (#` (# Step_40  0` (#(#Select_,inorder,thediametermeasurementandthecircumferencemeasurement. )y Choose Plotas(_x,y_) inthe Graph Menu.Changethecolor(usetheDisplay f menu)ofthispointsoyoucandistinguishitfromotherpointsonthegraph.Drag V tochangeyourcircleandwatchtheplottedpoint.Howdoesthisdemonstrate C thatthecircumference/diameterratioisthesameforallcircles?_Ԁ򀀀 0 Ѐ  Ѐ   Ѐ_` (#` (# Step50  0` (#(#Longago,mathematiciansnamedthecircumference/diameterratiowiththe   Greekletter_!_Ԁ(pi).Copyandcompletethefollowingconjecture:IfCisthe n! circumferenceandDisthediameterofacircle,thenthereisanumber_!_Ԁsuch ["  thatC=򀀀.BecauseD=2r,whereristheradius,thenC=򀀀__________Ԁ. H#! ThisistheCircumferenceConjecture.5$"` (#` (# _______________   `     h      p   ____ >0+/ ________Now_Ԁthatweseetherelationshipbetweenthecircumferenceandthediameter_fromDay1data,_  wewillplotourdatafromDay2usingGeometers_Sketchpad_Ԁandcomparetheresults.  _Step10  0` (#(#Selectanewsketch__w` (#` (# ____Step2__0  __0` (#(#__Select plotpoints fromthe Graph menuQ ` (#` (# Step3__0  __0` (#(#__EnteryourrealworlddatafromyourDay2tableandclickOK.Ifyoudonotsee . ~ youpoints,dragthepointat(1,0)toscaleyouraxes. k` (#` (# Step4__0  __0` (#(#____Constructarayfromtheoriginofthegraphtooneofthepointsonthegraph.  E  (Selectthetwopointsbyholdingdowntheshiftbutton.Clickontheline 2  segmenticoninthetoolboxandselecttherayicon.Placethecursoronthe   originandclick)Measureandrecordtheslopeofthisray.Whatratiodoesthis   sloperepresent?򀀀   Ѐ   Ѐ__` (#` (# ____Step5__0  __0` (#(#Howdoestheslope(ratio)ofyourDay2_Ԁ_datacomparetoDay1?________ԀArethere ] differencesorsimilarities?Explainwhy?_Ԁ_򀀀 J Ѐ 7 Ѐ $t Ѐ a Ѐ N Ѐ_;` (#` (#    `     h     _Adaptedfrom:KeyCurriculumPress;DiscoveringGeometrywithGeometers_Sketchpad_____ y  _LindaMayGerard @#! GarrettCommunity_College__