Exploring Proportional Reasoning in Grade 8 Mathematics Learning

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Dive into the world of proportional reasoning in Grade 8 mathematics with a focus on identifying better buys, strategies for decision-making, and the importance of developing proportional thinking skills. Discover the significance of proportional reasoning as a key element in the curriculum, its implications on algebraic concepts, and the need for effective instructional practices to foster students' ability to reason proportionally.


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  1. PARTNERS PARTNERS forMathematicsLearning Grade8 Module4 Partners Partners forMathematicsLearning

  2. 2 Module4 ProportionalReasoning Partners Partners forMathematicsLearning

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

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

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

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

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

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

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

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

  12. 12 WouldThisBeTrue? Ifyouwereonlyoneinchtall,youcouldwear athimbleonyourhead Partners Partners forMathematicsLearning

  13. 13 HowAboutThis? Ifyouwereonlyoneinchtall,itwouldtake aboutamonthtogetdowntothestore Partners Partners forMathematicsLearning

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

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

  16. 16 TimeToInvestigate Partners Partners forMathematicsLearning

  17. 17 Let sTalk Whatareourconclusions? Couldyourideawormtoschool? Couldyouwearathimbleonyourhead? Wouldittakeamonthtogettothestore? Partners Partners forMathematicsLearning

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

  19. 19 TangramTime! Partners Partners forMathematicsLearning

  20. 20 TangramTime! Partners 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 Partners forMathematicsLearning

  22. 22 ContinuingOurInvestigations Partners Partners forMathematicsLearning

  23. 23 TheSierpinskiTriangleActivity Partners Partners forMathematicsLearning

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

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

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

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

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

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

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

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

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

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

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

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

  36. 36 TheKochSnowflake Partners Partners forMathematicsLearning

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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