Ratio and Proportional Reasoning Activity Set 4...McGraw-Hill Professional Development RATIO AND PROPORTIONAL REASONING/25 Circle Circumference Diameter C d C ÷d 1 2 3 Transparency:

Ratio and Proportional Reasoning

Activity Set 4

Trainer Guide

RATIO AND PROPORTIONAL REASONING—AcTIvITy SET 4 Int_RPR_04_TGCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development

RATIO AND PROPORTIONAL REASONING—AcTIvITy SET 4 Int_RPR_04_TG

Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development 1

RATIO AND PROPORTIONAL REASONINGACTIvITy SET #4

NGSSS 4.G.5.1 NGSSS 6.G.4.1 A Piece of Pi

In this activity, participants investigate the meaning of pi and use proportions to solve problems involving pi.

Materials

• Transparency/Page:APieceofPi• Transparency/Page:APieceofPiTallySheet• Transparency/Page:PiProblems• Transparency/Page:PiProblemsAnswerKey• blanktransparency• precutcardboardcircles• paperclips• string• pencils• scissors• calculators• ruler• measuringtapes(centimetersandinches)• maskingtape

Vocabulary

• pi• ratio• proportion

tiMe: 30 minutes




teaching tip: Precut random-sized circles in groups of three: small, medium, and large. The range of sizes must be from a minimum of 4" to a maximum of 12" in diameter. You will need one set of circles for each participant group and one set for a model.

Label each group of three circles with the same unit of measure: centimeters, inches, or paper clips. For example, label one group of three “centimeters”; label a second group “paper clips”; label a third group “inches.” Repeat until all the groups of circles are labeled.

INTRODuCE

•Explaintoparticipantsthattheywilldetermine theratiobetweenthecircumferenceofacircleand its diameter.

•Useoneoftheprecutcircles(modelset)totraceacircleontheblanktransparency.

•Usetherulerasastraightedge(notformeasuring) todrawthediameteracrossthecircle.

•Cutapieceofstringtothelengthofthediameter.

•Write“diameter”onapieceofmaskingtapeandattach it to the string.

•Measurethediameterstringusinginches.

•Writethemeasurementonthepieceoftapeandonthetransparencywiththecircle.

•Cutapieceofstringtothelengthofthecircumference of the circle.

•Write“circumference”onapieceofmaskingtapeand attach it to the string.




•Measurethecircumferencestringusinginches.

•Writethemeasurementonthepieceoftapeandonthetransparencywiththecircle.Besurethateachmeasurementisappropriatelylabeledwithincheson the transparency.

•Hangthetwopiecesofstringside-by-sideon thewall.

•Havethegroupestimatethenumberof diameterstringsitwilltaketomakeupthe circumference string.

•Writetheestimateonthetransparency.

•Haveparticipantsusetheircalculatorstodivide thecircumferencebythediametertocheck their estimate.

•Writetheresultonthetransparency.

•DisplayTransparency:APieceofPi and have participantstakeouttheirmatchingpages.

•Explaintoparticipantsthattheywillrepeatthisactivity for three different circles, recording their final measurements on their pages.

Do the following for each circle:

• Cut a string the same length as the diameter of the circle.

• Cut a string the length of the circumference of the circle.

• Measure each string using the assigned unit of measure of your circle.

• Determine the ratio of the length of the diameter to the length of thecircumference.

• Fill in the chart for the circle.

Circle your unit of measure: centimeters inches paper clips

a piece of pi

McGraw-Hill Professional Development RATIO AND PROPORTIONAL REASONING/25

Circle Circumference Diameter Cd C ÷ d

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Transparency: A Piece of Pi




DISCuSS AND DO

•Askparticipantstoworkingroupsof4or5sothatthey can share the precut circles.

•Haveoneparticipantfromeachgrouppickupasetofprecut circles.

•Pointoutthattheunitofmeasurethateachgroupshoulduseiswrittenonitscircles.

teaching tip: Have participants within the groups work in pairs to measure and record their data, with each pair taking one circle and sharing data with the group.

•DisplayTransparency:APieceofPiTallySheet.

•Explaintoparticipantsthatassoonastheyhave completed and recorded their measurements, they should:

◆ send a group representative to the front to record theirinformation(forallthreecircles)onTransparency:APieceofPiTallySheet

◆ hang,onthewall,theircircleswiththerelatedpiecesofstringaswaspreviouslymodeled

•Giveparticipantsapproximately10minutestocompletetheirassignedtasks.

RATIO AND PROPORTIONAL REASONING—AcTIvITy SET 4 TRANS_K6_RA_04Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development


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a piece of pi Tally Sheet

Transparency: A Piece of Pi Tally Sheet




CONCLuDE

•Callthegrouptogether.

•Askparticipantsiftheycanmakeanygeneralizationsabouttherelationshipbetweenthediameterstringsandthecircumferencestrings.(Thediameterstringsare about 1

3 thelengthofthecircumferencestrings.)

•HaveparticipantslookatthelastcolumnonTransparency:APieceofPiTallySheet.

•Pointouttherangeofresponses.

•Pointoutthatall(ormost)ofthenumbersfitinarangethatcloselyapproximatespi(3.1416roundedto4decimalplaces,or3.14roundedto2decimalplaces—orascloseaswecangetusingnonspecificinstrumentsofmeasurement).

•Pointoutanyoutliersanddiscusswithparticipantswhytheymighthaveappeared.(mismeasurements,inaccuratetoolsformeasuring,orcomputationalerrors)

teaching tip: If time permits, tally all the numbers in the final column and compute the mean. (Sum the column and divide the sum by the number of entries.) Point out that the more data points, the more closely the list will approximate pi.

•Write,onablanktransparency,theratio3.141

(theratioof Cd fora1-unitcircle).

•Explainthatwecanuseproportionsandtheratio

shown(3.141 )tohelpsolvecircumferenceand

diameterproblems.Wecansolveforthediameter

whenthecircumferenceisknown,andwecansolve

forthecircumferencewhenthediameterisknown.




•DisplayTransparency:PiProblems and have participantstakeouttheirmatchingpages.

•Walkthroughallthestepstoeachsolutionwith thegroup.(StepsareshownonTransparency:PiProblemsAnswerKey.)

•Askparticipantswhywasthisexercisewascalled APieceofPi.(Theproblemsrelatedtotheratioof C to d—pi.)

•Askparticipantsif,priortothisactivity,theyhadrealizedthatthenumber3.14wasaratio.

End of A Piece of PiJohn bought his mother a round music box as a gift. Ithas a circumference of 12 inches. The only boxes the storehas are square. How wide must the square box be to holdthe music box? (Round your answer to the nearest inch.)

Marilyn swam 4 miles straight across the center ofRound Lake, while Tana decided to walk around the lake.How far did Tana walk?

pi problems Answer Key


d12

1= (Cross multiply to solve.)

Solve for d when C = 12

3.14

4C


Solve for C when d = 4

3.14

12 = 3.14d

=

3.82 = d

The box will be 4 inches on a side.

C = (4)(3.14)

C = 12.56

Tana walked 12.56 miles.

3.14 3.143.14d12

Transparency: Pi Problems Answer Key

John bought his mother a round music box as a gift. It hasa circumference of 12 inches. The only boxes the store hasare square. How wide must the square box be to hold themusic box? (Round your answer to the nearest inch.)

Marilyn swam 4 miles straight across the center of RoundLake, while Tana decided to walk around the lake. How fardid Tana walk?

pi problems


d12



3.14

4C



3.14

Transparency: Pi Problems

RATIO AND PROPORTIONAL REASONING—AcTIvITy SET 4 Int_RPR_04_PMCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development

Do the following for each circle:

• Cutastringthesamelengthasthediameterofthecircle.

• Cutastringthelengthofthecircumferenceofthecircle.

• Measureeachstringusingtheassignedunitofmeasureofyourcircle.

• Determinetheratioofthelengthofthediametertothelengthofthecircumference.

• Fillinthechartforthecircle.

Circle your unit of measure: centimeters inches paperclips

A Piece of Pi


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A Piece of Pi Tally Sheet


Johnboughthismotheraroundmusicboxasagift.Ithasacircumferenceof12inches.Theonlyboxesthestorehasaresquare.Howwidemustthesquareboxbetoholdthemusicbox?(Roundyouranswertothenearestinch.)

Marilynswam4milesstraightacrossthecenterofRoundLake,whileTanadecidedtowalkaroundthelake.HowfardidTanawalk?

Pi Problems

d12



3.14

4C



3.14


Johnboughthismotheraroundmusicboxasagift.Ithasacircumferenceof12inches.Theonlyboxesthestorehasaresquare.Howwidemustthesquareboxbetoholdthemusicbox?(Roundyouranswertothenearestinch.)

Marilynswam4milesstraightacrossthecenterofRoundLake,whileTanadecidedtowalkaroundthelake.HowfardidTanawalk?

Pi Problems Answer Key

d12



3.14

4C



3.14

12 = 3.14d

=

3.82 = d

The box will be 4 inches on a side.

C = (4)(3.14)

C = 12.56

Tana walked 12.56 miles.

3.14 3.143.14d12


GlossaryRatio and Proportional Reasoning

cross multiplication Amethodofverifyingwhether2fractionsareequivalent.

equivalent fractions 2fractionswhich,whenreducedtolowestterms,areequal.

equivalent ratios Ratiosthatcanbeexpressedbyequivalentfactions.

pi Theratioofthecircumferenceofacircletoitsdiameter(approximately3.14).

proportion Astatementthat2ratiosareequivalent.

rate Aratiocomparingtwodifferentunitsofmeasure.

ratio Acomparisonof2quantities.

sample Asubsetofitemstakenatrandomfromacompleteset.

Documents

Ratio and Proportional Reasoning Activity Set 4...McGraw-Hill Professional Development RATIO AND PROPORTIONAL REASONING/25 Circle Circumference Diameter C d C ÷d 1 2 3 Transparency: