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Partners for Mathematics Learning

PARTNERS for Mathematics Learning. Grade 8 Module 4. Partners for Mathematics Learning. 2. Module 4  Proportional Reasoning. Partners for Mathematics Learning. 3. Which Is a Better Buy?  12 tickets for $15.00 or 20 tickets for $23.00?. Partners for Mathematics Learning. 4.

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Partners for Mathematics Learning

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  1. PARTNERS forMathematicsLearning Grade8 Module4 Partners forMathematicsLearning

  2. 2 Module4 ProportionalReasoning Partners forMathematicsLearning

  3. 3 WhichIsaBetterBuy? 12ticketsfor$15.00or20ticketsfor $23.00? Partners forMathematicsLearning

  4. 4 WhichisaBetterBuy? Whatisyouranswer? Howdidyouobtainyouranswer? Whataresomestrategiesthatyour studentsmightuse? Partners forMathematicsLearning

  5. 5 WhichIsaBetterBuy? Ifastudentvalueseachticketasworth $1.00,whatmightthestudentsayabout eachdealusing… Additivereasoning Proportionalreasoning Partners forMathematicsLearning

  6. 6 ProportionalThinking Asdifferentwaystothinkaboutproportions areconsideredanddiscussed,teachers shouldhelpstudentsrecognizewhenand howvariouswaysofreasoningabout proportionsmightbeappropriatetosolve problems PSSM,2000 Partners forMathematicsLearning

  7. 7 CapstoneoftheCurriculum! “Proportionalreasoninghasbeenreferred toasthecapstonefortheelementary curriculumandthecornerstoneofalgebra andbeyond.” VandeWalle,J.A(2004).ElementaryandMiddleSchool Mathematics:TeachingDevelopmentally.PearsonLearningInc. Partners forMathematicsLearning

  8. 8 Amazing–Isn’tIt? “Itisestimatedthatmorethanhalfofthe adultpopulationcannotbeviewedas proportionalthinkers.Thatmeansthatwe donotacquirethehabitsandskillsof proportionalreasoningsimplybygetting older.” VandeWalle,J.A(2004).ElementaryandMiddleSchool Mathematics:TeachingDevelopmentally.PearsonLearningInc. Partners forMathematicsLearning

  9. 9 ResearchSays… “Researchindicatesthatinstructioncanhavean effect,especiallyifrulesandalgorithmsfor fractioncomputation,forcomparingratios,and forsolvingproportionsaredelayed.Premature useofrulesencouragesstudentstoapplyrules withoutthinkingand,thus,theabilitytoreason proportionallyoftendoesnotdevelop.” VandeWalle,J.A(2004).ElementaryandMiddleSchool Mathematics:TeachingDevelopmentally.PearsonLearningInc. Partners forMathematicsLearning

  10. 10 TeachingEffectively Instructioninsolvingproportionsshouldinclude methodsthathaveastrongintuitivebasis PSSM,2000 Inagroupofstudentswhocansuccessfully applyanalgorithm,howcanyoudistinguish betweenthosewhocanreasonproportionally andthosewhocannot? Partners forMathematicsLearning

  11. 11 “OneInchTall”byShelSilverstein Ifyouwereonlyoneinchtall,you’dridea wormtoschool Theteardropofacryingant wouldbeyourswimmingpool. Partners forMathematicsLearning

  12. 12 WouldThisBeTrue? Ifyouwereonlyoneinchtall,youcouldwear athimbleonyourhead Partners forMathematicsLearning

  13. 13 HowAboutThis? Ifyouwereonlyoneinchtall,itwouldtake aboutamonthtogetdowntothestore Partners forMathematicsLearning

  14. 14 Let’sInvestigate! Ifyouwereonlyoneinchtall… Couldyourideawormtoschool? Couldyouwearathimbleonyourhead? Wouldittakeamonthtogettothestore? Partners forMathematicsLearning

  15. 15 JustTheFacts,Ma’am! Whatinformationwillyouneedtosolve theseproblems? Partners forMathematicsLearning

  16. 16 TimeToInvestigate Partners forMathematicsLearning

  17. 17 Let’sTalk Whatareourconclusions? Couldyourideawormtoschool? Couldyouwearathimbleonyourhead? Wouldittakeamonthtogettothestore? Partners forMathematicsLearning

  18. 18 IncreasedStudentUnderstanding Problemsthatinvolveconstructingor interpretingscaledrawingsofferstudents opportunitiestouseandincreasetheir knowledgeofsimilarity,ratio,and proportionality PSSM Partners forMathematicsLearning

  19. 19 TangramTime! Partners forMathematicsLearning

  20. 20 TangramTime! Partners forMathematicsLearning

  21. 21 InvestigationandExploration “Studentswholearnedthrough investigationandexplorationwerenot onlymoresuccessfulatgivingcorrect responsestoproportionalreasoning tasksbutalsobetterabletojustify thoseanswers.” Fey,J.T.,Miller,J.L.(2000).ProportionalReasoning.Mathematics TeachingintheMiddleSchool.5(5),312 Partners forMathematicsLearning

  22. 22 ContinuingOurInvestigations… Partners forMathematicsLearning

  23. 23 TheSierpinskiTriangleActivity Partners forMathematicsLearning

  24. 24 TheSierpinskiTriangle STAGE0 Usingdotpaper,constructanequilateral triangleofsidelength16 Whatistheareaofthistriangle? Partners forMathematicsLearning

  25. 25 TheSierpinskiTriangle STAGE1 Markthemidpointofeachsideofthe triangle Jointhemidpointsto form4smallertriangles Partners forMathematicsLearning

  26. 26 TheSierpinskiTriangle STAGE1 Determinesomerelationshipsbetweenthenew trianglesandtheoriginaltriangle Similarity? Congruence? Whatistheareaofeachnewtriangle? Whatfractionoftheoriginaltriangledoeseach newtrianglerepresent? Partners forMathematicsLearning

  27. 27 TheSierpinskiTriangle STAGE1 Remove(shade)thecentertriangle Whatfractionoftheoriginal areaisnotshaded? Updatethechart Partners forMathematicsLearning

  28. 28 TheSierpinskiTriangle STAGE2 Bisecteachsideofthe“new”(unshaded) triangles Jointhemidpointsineachtoformatotal of12smallertriangles(16ifyoudivided thelargershadedtriangle) Partners forMathematicsLearning

  29. 29 TheSierpinskiTriangle STAGE2 Determinesomerelationships betweenthenewtriangles andtheoriginaltriangle Remove(shade)thecentertriangles Whatistheareaofthenewtriangle? Whatfractionoftheoriginalareaisnot shaded? Partners forMathematicsLearning

  30. 30 TheSierpinskiTriangle STAGE3 Bisecteachsideofthe“new”(unshaded) triangles Jointhemidpointsineachtoformatotal of_?_smallertriangles Partners forMathematicsLearning

  31. 31 TheSierpinskiTriangle STAGE3 Remove(shade)thecentertriangles Whatfractionoftheoriginalareaisnot shaded? Updatethechart Partners forMathematicsLearning

  32. 32 TheSierpinskiTriangle Isanexampleofafractal(aself-similar object) Ingeneral,afractalisageometricobject whosepartsarereduced-sizedcopiesof thewhole Givesomereal-lifeexamplesoffractals Partners forMathematicsLearning

  33. 33 TheSierpinskiTriangle Determineamathematicalrelationshipfor Thenumberoftrianglesateachstage Theareaofeachnewtriangle Thefractionoftheoriginalareathateachnew trianglerepresents Thefractionoftheoriginalareathatisnot shaded Partners forMathematicsLearning

  34. 34 TheSierpinskiTriangle Whatpatternsemerge? Whatiftheiterationscontinued… Whatwouldbetheareaofoneofthesmallest trianglesinthe4thiteration? Whatfractionoftheoriginaltrianglewouldnot beshadedatthisstage? Whataboutthe100thiteration? Whatishappeningtotheareaoftheun- shadedregionasthenumberofiterations grows? Partners forMathematicsLearning

  35. 35 TheSierpinskiTriangle Whatwillyourstudentsthinkofthis activity? Whatmathematicalconceptsarecovered inthisactivity? Willyougivethisactivityatry? Partners forMathematicsLearning

  36. 36 TheKochSnowflake Partners forMathematicsLearning

  37. 37 TheKochSnowflake STAGE0 Usingdotpaper,constructanequilateral triangleofsidelength9 Whatistheperimeterofthistriangle? Partners forMathematicsLearning

  38. 38 TheKochSnowflake STAGE1 Trisecteachsideofthetriangle Remove(erase)themiddlesegmentof eachside Partners forMathematicsLearning

  39. 39 TheKochSnowflake STAGE1 Createanewequilateraltriangleoneach sideoftheoriginaltrianglebyaddingtwo segmentsofthesamelengthasthe erasedsegmentontotheside Theerasedsegmentwillbethebaseof thenewequilateraltriangle Partners forMathematicsLearning

  40. 40 TheKochSnowflake Thenewshapewillbea six-pointedstar Whatistheperimeter ofthestar? Partners forMathematicsLearning

  41. 41 TheKochSnowflake STAGE2 ReiteratetheprocessdescribedinStage1 Firsttrisectofeachsideofthetriangle Remove(erase)themiddlesegmentof eachside Partners forMathematicsLearning

  42. 42 TheKochSnowflake STAGE2 Createanewequilateraltriangleoneach sideofthe6-pointedstarbyaddingtwo segmentsofthesamelengthasthe erasedsegmentontotheside Theerasedsegmentwillbethebaseof thenewequilateraltriangle Partners forMathematicsLearning

  43. 43 TheKochSnowflake STAGE2 Thenewfigureshouldlooklikea snowflake Whatistheperimeterofthenewfigure? Partners forMathematicsLearning

  44. 44 TheKochSnowflake Lookatthevalueoftheperimeter ateachstage Isthereapatternhere? Theperimeterofeachfigureis__?__ timestheperimeterofthepreviousfigure Partners forMathematicsLearning

  45. 45 TheKochSnowflake Howmanyiterationswouldittaketoobtain aperimeterof100units?(orascloseto 100asyoucanget) Asyouperformmoreandmoreiterations, whathappenstothevalueoftheperimeter andthearea? Partners forMathematicsLearning

  46. 46 TheKochSnowflake Aninfiniteperimeterenclosesafinitearea …nowthat’samazing! Whatwillyourstudentsthinkofthisactivity? Willyougivethisactivityatry? Partners forMathematicsLearning

  47. 47 SummarizingtheWork Whatmathematicalconceptsandskillsare addressedintheseactivities? Understandingofandcomputationwithreal numbers Understandingofanduseofmeasurement concepts Understandofandusepropertiesand relationshipsingeometry Partners forMathematicsLearning

  48. 48 ProportionalReasoningActivities OneInchTall Tangrams SierpinskiTriangle KochSnowflake Whatareyourfavoriteproportional reasoningactivities? Partners forMathematicsLearning

  49. 49 WhatBigIdeasAreAddressed? Fluencywithdifferenttypesofreasoning (quantitative,additive,multiplicative, proportional)isnecessaryformathematical development Fluency(accuracy,efficiency,flexibility) usingoperationswithrationalnumbers becomessolidifiedinthemiddlegrades Partners forMathematicsLearning

  50. 50 …BIGIdeas Twodimensionalfiguresareviewedinthe rectangularcoordinateplaneand transformationsoftwodimensionalfigures withintheplanemayproducefiguresthat aresimilarand/orcongruenttotheoriginal figure Partners forMathematicsLearning

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