Exploring Proportional Reasoning in Grade 8 Mathematics Learning

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.

• Proportional Reasoning
• Grade 8 Mathematics
• Decision Making
• Curriculum Development
• Instructional Strategies

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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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