Fractions, Decimals, and Percentages Activity Set...

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Fractions, Decimals, and Percentages

Activity Set 9

Trainer Guide

fractions, decimals, and percentages—activity set 9 Int_RaN_09_TGCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development

fractions, decimals, and percentages—activity set 9 Int_RaN_09_TG

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

FraCtions, DeCiMals, anD PerCentaGesaCtivity set #9

NGSSS 5.A.6.5 NGSSS 6.A.1.1 NGSSS 6.A.1.2 NGSSS 6.A.1.3 NGSSS 7.A.3.2 NGSSS 7.A.5.2

Divide and Conquer

In this activity, participants divide fractions.

materials

• Transparency/Page:DivideandConquer• Transparency/Page:InvertandMultiply• Transparency/Page:FractionDivisionProblems• Transparency/Page:PracticalDivision• Transparency/Page:FractionDivisionProblems

AnswerKey• Transparency/Page:PracticalDivisionAnswerKey• overheadpatternblocks• patternblocks(1setperpairofparticipants)• 2-colorcounters• blanktransparencies

Vocabulary

• reciprocal• MultiplicativeInverseProperty

time: 35minutes

introDuCe

•Place12countersina3x4gridonablanktransparency.

•Writebelowthegridtheproblem12÷4=.

•Askparticipants,“Whatistheproblemasking?” (Howmanygroupsof4arein12?–3)

•Pullgroupsoffouroutofthegrid,countingthemoffasyougo.

•Place4yellowpatternblocks(hexagons)onablanktransparencyandwritetheproblem4÷1=.




•Askparticipantswhattheproblemisasking.(Howmanygroupsof1arein4?—4)

•Removetheunitblocksonebyone,countingthemoffasyougo.

•DisplayTransparency:DivideandConquerwiththebottomhalfcovered.

•Place3bluepatternblocks(rhombuses)inthefirsthexagonfieldabovetheequation1÷3=.

•Askparticipantswhattheproblemisasking.(Howmanygroupsof3arein1?)

•Moveeachofthebluepatternblockstothethreehexagonfieldsatthebottomofthetransparency—onetoeachfield.

•Pointouttoparticipantsthat1cannotbedividedintofullgroupsof3,butonlypartialgroups.

•Askwhatvalueofeachhexagonfitsintothefirstunitwhole.( 13)

•Replacethe3bluepatternblocksinthefirsthexa-gon.

•Write 13 aftertheequalsigntofinishtheequation.

•Tellparticipantsthattheexpression ab is another

waytorepresenta÷b.

•Uncoverthebottomhalfofthetransparency.

•Place1yellow,1red,and1bluepatternblocksonthe shapes.

•Askparticipantswhattheequation1 12÷ 13=isask-

ing.(Howmanygroupsof 13arein1and 1

2?)

•Replacetheyellowhexagonwith3bluerhombus-es,andtheredtrapezoidwith1bluerhombusand1greentriangle.

•Askparticipantshowmany 13shapes(rhombuses)arerepresented.(4completeand1half)

1 12

13

1 ÷ 3 =

÷ =

Divide and conquer

McGraw-Hill Professional Development fractions, decimals, and percentages B/tr41

Transparency: Divide and Conquer




•Write4 12aftertheequalssignintheequation.

•Explaintoparticipantsthattheyhavejustseenhowthedivisionoffractionsisconceptuallynodifferentthanthedivisionofwholenumbersandintegers.

•TellparticipantsthattheywillnowlearnhowtousetheMultiplicativeInversePropertytodividewithfractions.

•Place4orangesquaressidebysideonablank transparency.

•Askparticipantswhatfractionofthewholeeachsquarerepresents.( 14)

•Tellparticipantsyouwanttoreducethissquareby 14,

andaskthemhowtodoso.(Remove1square)

•Remove1squareandaskparticipantswhatsizeofthewholeisleft.( 34)

•Write 34belowtheshapeonthetransparency.

•Askparticipantswhatfractionofthereducedwholeeachsquarerepresentsnow.( 1

3)

•Counteachsquarealoudsaying,“One-third,two-thirds,andthree-thirdsmakesawhole.”

•Tellparticipantsyouwanttoenlargethisshapesoitistheexactsizeastheoriginal,andaskthemhowtodoso.(Add1square)

•Add1squareandaskparticipantswhatsizeofthereducedwholeisrepresentednow.( 43)

•Write 43totherightof 3

4 on the transparency.

•Askparticipantswhattheyobserveaboutthesetwofractions.(Thenumeratorsanddenominatorsarereversed.)

•Tellparticipantsthatthesetwonumbersare reciprocals of each other.




•Addmultiplicationandequalssigntothefractionstomaketheequation 3

4• 43=.

•Askparticipantswhattheanswertothisequationis. (1212,whichreducesto1)

•Pointoutthatthe1representstheoriginalshapeof4orangesquares,whichhasbeenreducedto 3

4 of its sizeandthenenlargedto 4

3.

•Takeoutanotherblanktransparencyandplace9greentrianglessidebysidesotheyformalong trapezoid.

•Repeattheabovestepsbytakingaway2trianglestoreduceitto 79oftheoriginalandadding2trianglestoenlargeitto 9

7ofthereducedshape,showingthat 79 •9

7 =6363(or1).

•Pointoutthatmultiplyingafractionbyitsreciprocalalwaysresultsintheanswer1.

•DisplayTransparency:InvertandMultiply and have participantstakeouttheirmatchingpages.

•ReadthedefinitionoftheMultiplicativeInverseProperty.

•TellparticipantsthattheMultiplicativeInversePropertyisneededtodividewithfractions.

•Walkparticipantsthroughthestepsofdividingbyfractions as presented on the transparency.

•PointouttoparticipantsthatbecauseapplyingtheMultiplicativeInversePropertyalwayschangesthevalueofthedenominatorto1,theonlystepstheyhavetoperformwhendividingwithfractionsisfindtheproductofthenumeratorandthereciprocalofthedenominator—invertandmultiply.

•DisplayTransparency:FractionDivisionProblems and haveparticipantstakeouttheirmatchingpages.

•Havetheparticipantsmoveintopairs,anddistributeasetofpatternblockstoeachpair.

invert and multiply

McGraw-Hill Professional Development fractions, decimals, and percentages B/41

34

14

14

14

14

13

Whole = 1 Reduced to

13

13

13

13

43

Enlarged to

34

43

1212

• = = 1

Multiplicative Inverse Property: The product of a number and its reciprocal always equals 1.

12

ab

÷ =

Rewrite problem from the form a ÷ b to the form :

Use the Multiplicative Inverse Property to change the value of the denominator to 1:

1213

3131

•

•=

32

1= 3

2Multiply by its reciprocal1

3

Dividing by a fraction is the same as multiplying by its reciprocal.

1213

=÷ 13

12

Transparency: Invert and Multiply

Fraction Division Problems


Use the Multiplicative Inverse Property to divide the following equations. Use pattern blocks to check your answers.

16

1 ÷ =

61

1 • = 6

13

÷ =

31

• =

56

56

52

156

=

32

÷ =52

16

÷ =

61

• =

23

23

4123

=

14

÷ =78

21

• =43

83

23

• =52

53

106

= 41

• =78

72

288

=

12

÷ =43

Transparency: Fraction Division Problems




•Givethepairs5-10minutestosolvetheproblemsonthepage.

•Getthegroup’sattention.

•Haveparticipantvolunteerscomeupandsharetheiranswers,includinghowtheyusedthepatternblockstochecktheirwork.

•RefertoTransparency:FractionDivisionProblemsAnswerKeytoresolveanyquestions.

•Suggesttoparticipantsthattheyarereadytoapplywhattheyhavelearnedaboutfractiondivision.

•TellparticipantstheywillremainintheirpairsfortherestofDivideandConquer.

DisCuss anD Do

•DisplayTransparency:PracticalDivisionwiththe bottom2problemscovered.

•Walkthroughthestepslistedatthetopofthepage,demonstratingtoparticipantshowtouseapictureandequationtosolvetheproblem. (5÷ 14=20batchesofcookies)

•Uncoverthetransparency.

•Haveparticipantstakeouttheirmatchingpages.

•Suggestthattheyusetheirmanipulativesanddrawpicturestosolvetheother2problemsonthepage.

•Giveparticipants5-10minutestocompletetheproblems.

ConCluDe

•Callthegrouptogether.

•Askparticipantstosharetheiranswers.

Fraction Division Problemsanswer Key



16

1 ÷ =

61

1 • = 6

13

÷ =

31

• =

56

56

52

156

=

32

÷ =52

16

÷ =

61

• =

23

23

4123

=

14

÷ =78

21

• =43

83

23

• =52

53

106

= 41

• =78

72

288

=

12

÷ =43

Transparency: Fraction Division Problems Answer Key

Practical Division


1. Model the problem.2. Draw a picture.

3. Write an equation.4. Solve the problem.

Martin has 5 cups of sugar. He needs cup of sugar to make 1 batch of cookies. How many batches of cookies could Martin make with the sugar he has?

51

÷ =14

201

41

=• 20

Martin can make 20 batches of cookies.

Luan made 9 cups of fruit punch for his party. If each serving is cup, how many servings did Luan make?34

91

÷ =34

363

43

=• 12

Luan made 12 servings of punch.

Atlanta is making pillows. It takes yard to make 1 pillow. Atlanta has 2 yards of material. How many pillows can she make?1

2

34

Atlanta can make 3 full pillows.

34

÷ =12

34

52

= • 3÷ 52

= 206

43

= 26

= 3 13

14

Transparency: Practical Division




•Write,onthetransparency,theequationsandtheanswers.

•Write,onthetransparency,thestepsandtheanswerforeachproblemastheyareshared.

◆ 9÷ 34= 9

1 43=36

3—=12servings

◆ 2 12÷ 3

4= 52 4

3=206—=3 1

3=3pillows

•Askparticipantswhytheyshouldnotincludethefractionalportionintheanswertothepillowquestion.(Youcannotmakepartofapillow.)

•Askparticipantsiftheanswertoawholenumberdivisionproblemisgreaterthanorlessthanthedividend.(Iftheydivide24by6,istheanswergreaterthanorlessthan24?Itislessthan—4.)

•Askparticipantsiftheanswertoafractionaldivisionproblemisgreaterthanorlessthanthedividend. (Iftheydivide 9

10by 15,istheanswergreaterthanor

lessthan 910?Itisgreaterthan—4 1

2.)

•Pointoutthatstudentsoftenexpecttheresulttobelessthanfollowingfractiondivisionbecausethatisreflectiveoftheirexperiencewithwholenumbers. Itisimportanttoemphasizethereversalthatoccurswithfractions.Manipulativesareagoodtoolfor thispurpose.

•Haveeachgrouporganizeitsmanipulatives.

•Askavolunteerfromeachgrouptobringupthemanipulativesaftertheyarecountedandcollected.

end of Divide and Conquer

fractions, decimals, and percentages—activity set 9 Int_RaN_09_PMCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development

1 12

13

1 ÷ 3 =

÷ =

Divide and Conquer


Invert and Multiply

34

14

14

14

14

13


13

13

13

13

43

Enlarged to

34

43

1212

• = = 1

Multiplicative Inverse Property: The product of a number and its reciprocal always equals 1.

12

ab

÷ =



1213

3131

•

•=

32

1= 3

2Multiply by its reciprocal1

3


16

1 ÷ =

61

1 • = 6

13

÷ =

31

• =

56

56

52

156

=

32

÷ =52

16

÷ =

61

• =

23

23

4123

=

14

÷ =78

21

• =43

83

23

• =52

53

106

= 41

• =78

72

288

=

12

÷ =43


1213

=÷ 13

12


Fraction Division Problems

34

14

14

14

14

13


13

13

13

13

43

Enlarged to

34

43

1212• = = 1


12

ab

÷ =



1213

3131

•

•=

321

= 32

Multiply by its reciprocal13

16

1 ÷ =

61

1 • = 6

13

÷ =

31

• =

56

56

52

156

=

32

÷ =52

16

÷ =

61

• =

23

23

4123

=

14

÷ =78

21

• =43

83

23

• =52

53

106

= 41

• =78

72

288

=

12

÷ =43


1213

=÷


Practical Division

34

14

14

14

14

13


13

13

13

13

43

Enlarged to

34

43

1212• = = 1


12

ab

÷ =



1213

3131

•

•=

321

= 32


16

1 ÷ =

61

1 • = 6

13

÷ =

31

• =

56

56

52

156

=

32

÷ =52

16

÷ =

61

• =

23

23

4123

=

14

÷ =78

21

• =43

83

23

• =52

53

106

= 41

• =78

72

288

=

12

÷ =43


1213

=÷



51

÷ =14

201

41

=• 20



91

÷ =34

363

43

=• 12



2

34


÷

34

÷ =12

34

52

= • 3÷ 52

= 206

43

= 26

= 3 13

14


Fraction Division ProblemsAnswer Key

34

14

14

14

14

13


13

13

13

13

43

Enlarged to

34

43

1212• = = 1


12

ab

÷ =



1213

3131

•

•=

321

= 32


16

1 ÷ =

61

1 • = 6

13

÷ =

31

• =

56

56

52

156

=

32

÷ =52

16

÷ =

61

• =

23

23

4123

=

14

÷ =78

21

• =43

83

23

• =52

53

106

= 41

• =78

72

288

=

12

÷ =43


1213

=÷


Practical DivisionAnswer Key

34

14

14

14

14

13


13

13

13

13

43

Enlarged to

34

43

1212• = = 1


12

ab

÷ =



1213

3131

•

•=

321

= 32


16

1 ÷ =

61

1 • = 6

13

÷ =

31

• =

56

56

52

156

=

32

÷ =52

16

÷ =

61

• =

23

23

4123

=

14

÷ =78

21

• =43

83

23

• =52

53

106

= 41

• =78

72

288

=

12

÷ =43


1213

=÷



51

÷ =14

201

41

=• 20



91

÷ =34

363

43

=• 12



2

34


÷

34

÷ =12 • 35

2= 20

643

= 26

= 3 13

14

5 =

9 =

2


GlossaryFractions, Decimals, and Percentages

common denominator A common multiple of the denominators of 2 or more fractions (e.g., 20 is a common denominator of the fractions

12 and

110).

common factor A number that divides evenly into 2 or more numbers (e.g., 4 is a factor of both 8 and 12).

decimal number A number with a decimal point.

denominator The bottom number of a fraction; the number of equal-sized parts into which the whole is divided.

equivalent fractions Fractions with the same value but different forms (e.g.,

12 and

24 ).

fraction A number that can be expressed in the form of a

b , where a and b are whole numbers and b ≠ 0.

fraction area model A fractional number shown as part of an area.

fraction linear model A fractional number shown as part of a line.

fraction set model A fractional number shown as a part of a group.

improper fraction A fraction that is equal to or greater than 1 (e.g.,

44 and

65 ).


least common multiple The least number that is a common multiple of 2 or more numbers.

mixed number A number expressed as a whole number and a fraction (e.g., 2 1

2 ).

Multiplicative Property of One Also know as Multiplicative Identity Property; multiplying any number by 1 does not change the value of the number (e.g., 1 • 12 = 12).

numerator The top number of a fraction; the number of parts being used.

percent The ratio of a number to 100; percent means “per hundred.”

percentage A part of a whole expressed in hundredths.

Glossary (Continued)

Fractions, Decimals, and Percentages Activity Set...

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