TEACHER'S EDITION GRADE 4 SAMPLE TOPIC - Pearson ...

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Topics 9 and 10 clusTer overview

Math Background: Focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461A

Math Background: Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461C

Math Background: Rigor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .461E

Math Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .461F

Differentiated Instruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461G

The Language of Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461H

Topic 9Understand addition and sUbtraction of fractions

Planner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461I

Math and Science Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

Review What You Know . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462–464

Vocabulary Cards and Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462–464

Lesson 9-1 Model Addition of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465A

Lesson 9-2 Decompose Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471A

Lesson 9-3 Add Fractions with Like Denominators . . . . . . . . . . . . . . . . . . . . . . 477A

Lesson 9-4 Model Subtraction of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 483A

Lesson 9-5 Subtract Fractions with Like Denominators . . . . . . . . . . . . . . . . . . . 489A

Lesson 9-6 Add and Subtract Fractions with Like Denominators . . . . . . . . . . . . 495A

Lesson 9-7 Estimate Fraction Sums and Differences . . . . . . . . . . . . . . . . . . . . . 501A

Lesson 9-8 Model Addition and Subtraction of Mixed Numbers . . . . . . . . . . . . 507A

Lesson 9-9 Add Mixed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513A

Lesson 9-10 Subtract Mixed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519A

Lesson 9-11 MATH PRACTICES AND PROBLEM SOLVING: Model with Math . . 525A

Fluency Practice Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

Vocabulary Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532

Reteaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

Topic Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535–536

Topic Performance Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537–538

TEACHER'S EDITION: Grade 4 Sample Topic

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Content Focus in (continued)

•Solve Problems Involving Fraction Addition and Subtraction ThroughoutTopic9,studentsusemodelsandequationstorepresentreal-worldproblemsinvolvingadditionandsubtractionoflikefractionsthatrefertothesame-sizewhole.(4.NF.B.3d)

•Multiply with Unit Fractions InLesson10-1,therepeated-additioninterpretationofwhole-numbermultiplicationisusedtolaythegroundworkformultiplyingawholenumberbyaunitfraction.Justas34canberepresentedasthesumof unitfractions,sotoo,can34berepresentedasaproductofawholenumberandaunitfraction.(4.NF.B.4a)

34 = 1

4 + 14 + 1

434 = 3 × 1

4

•Multiply Other Fractions InLessons10-2and10-3,unitfractionsareusedtounderstandmultiplyingnon-unitfractionsandwholenumbers.(4.NF.B.4b)

5 × 23

Thepictureshows5 × 23 = 10 × 1

3 = 103 .

So,5 × 23 = 5 × 2

3 = 103 .

•Solve Problems Involving Fraction Multiplication ThroughoutTopic10,studentssolveavarietyofreal-worldproblemsinvolvingmultiplicationoffractionsandwholenumbers.Theyusebothvisualmodelsandequationstorepresentthesituation.(4.NF.B.4c)

Sadieneeds7identicalpiecesofribbon.Eachpieceis34yardlong.HowmuchribbondoesSadieneed?Ifshecanonlybuyawholenumberofyardsofribbon,howmanyyardsshouldshebuy?

7 × 34 = 7 × 3

4 = 214 or514

Sadieneeds514yardsofribbon.

Sheshouldbuy6yardsofribbon.

PD

Professional Development Videos TopicOverviewVideosandListenandLookForLessonVideospresentimportantinformationaboutthecontent.

MUltIPly A FrActIOn by A whOle nUMber

461B

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MAJOR CLUSTER

SUPPORTING CLUSTER

ADDITIONAL CLUSTER

FOCUS COHERENCE RIGORF C R

TOPICS

9–10Major Cluster 4.NF.BMaTH BaCkGROUNd: FOCUS

TOPIC 9 Understand Addition and Subtraction of Fractions

TOPIC 10 Extend Multiplication Concepts to Fractions

TOPICS 9 AND 10 FOCUS ON

MAJOR CLUSTER 4.NF.B Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

Content Focus in

Topics 9 and 10 focus on the deep understanding of adding and subtracting fractions with like denominators and multiplying fractions by whole numbers called for in the Grade 4 Common Core Standards.

•Joining and Separating Fractions Throughout Topic 9, students extend to fractions their understanding of whole-number addition as joining, whole-number subtraction as separating, and the inverse relationship between addition and subtraction. (4.NF.B.3a)

•decompose Fractions In Lesson 9-2, students learn that a fraction with a numerator greater than 1, such as 45, can be written as the sum of unit fractions: 45 = 1

5 + 15 + 1

5 + 15.

Fractions and mixed numbers can be decomposed in more than one way. (4.NF.B.3b)

225 = 1 + 1 + 2

5 = 55 + 5

5 + 25

•Visual Models for Fraction addition and Subtraction Lessons 9-1, 9-4, 9-6, and 9-8 highlight the use of visual models, such as fraction strips, number lines, and partitioned regions, to show fraction operations. (4.NF.B.3c)

add aNd SUBTRaCT FRaCTIONS

FOCUS ONCOMMON CORE

CLUSTERS

LESSONS

TOPICS

CLUSTERS

1

2

3

4

5

6

78

9

10

11

12

13

14

15

16

4.NBT.A

4.NBT.B

4.OA.A

4.OA.C

4.OA.B4.NF.A

4.NF.B

4.MD.B

4.MD.C

4.G.A

4.MD.A

4.NF.C

461A Topics 9–10 Cluster Overview

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=23

46

= 68 = ×3

434

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Major Cluster 4.nf.bMatH BaCkGROUNd: COHERENCE

tOPICS

9–10

Content Coherence in

Students learn best when ideas are connected in a coherent curriculum. This coherence is achieved through various types of connections including connections within clusters, across clusters, across domains, and across grades.

BIG IdEaS IN GRadES k–6

Big Ideas are the conceptual underpinnings of enVisionmath2.0 and provide conceptual cohesion of the content. Big Ideas connect Essential Understandings throughout the program.

A Big Idea that connects most of the work in this cluster is that complex calculations involving fractions and mixed numbers can be broken into simpler equivalent calculations involving unit fractions. For example,

34 + 24 = (14 + 1

4 + 14 ) + (14 + 1

4 )2 × 4

4 = 2 × (4 × 14 )

For a complete list of Big Ideas, see pages 90–91 in the Teacher’s Edition Program Overview.

How do Topics 9–10 connect to what students learned earlier?

GRadE 3

•Meaning of Fractions In Topic 12, students strengthened their understanding of the meaning of fractions and used various models to represent them. (3.NF.A.1, 3.NF.A.2a, 3.NF.A.2b)

34

•Simple Equivalent Fractions In Topic 13, students learned how to recognize and generate simple equivalent fractions and to express whole numbers as fractions. (3.NF.A.3a, 3.NF.A.3b, 3.NF.A.3c)

EaRLIER IN GRadE 4

•Equivalent Fractions In Topic 8, students recognized and generated equivalent fractions that have a greater variety of denominators. (4.NF.A.1)

LOOk BaCk

461C Topics 9–10 Cluster Overview

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Content Coherence in (continued)

How is content connected within Topics 9–10?

•Fractions and Mixed Numbers InLessons9-1through9-6,studentscometounderstandfractionadditionandsubtraction.(4.NF.B.3a)InLessons9-8through9-10,theyextendtheseunderstandingstoadditionandsubtractionofmixednumbers.(4.NF.B.3c)

•Fraction Addition and Multiplication InLessons9-1,9-2,9-3,and9-6,studentsaddfractionswithlikedenominators.(4.NF.B.3a)InLessons10-1and10-2,thisadditionworkisusedtobuildafoundationforfractionmultiplication.(4.NF.B.4a,4.NF.B.4b)

•Multiplication and Unit Fractions InLesson10-2,studentscometounderstandthattheycanthinkofafractionastheproductofawholenumberandaunitfraction.(4.NF.B.4a)ThishelpstheminLessons10-2and10-3toviewtheproductofawholenumberandafractionasamultipleofawholenumberandaunitfraction.(4.NF.B.4b)

•Word Problems Involving Fractions ThroughoutTopics9and10,studentsapplytheirknowledgeoffractionaddition,subtraction,andmultiplicationtosolveproblems.(4.NF.B.3d,4.NF.B.4c)

•Time Problems Involving Fractions Topic10alsodevelopscontentfromSupportingCluster4.MD.Aonsolvingmeasurementproblemsinvolvingsimplefractions.SeeLesson10-5.(4.MD.A.2)

How will Topics 9–10 connect to what students will learn later?

LATER IN GRADE 4

•Fractions in Data and Measurement InTopics11and13,studentswillsolvedataandmeasurementproblemsinvolvingaddingandsubtractingfractionswithlikedenominatorsandmultiplyingafractionbyawholenumber.(4.MD.A.2,4.MD.B.4)

Thetallestorchidis248inchestallerthantheshortestorchid.

•Decimal Fractions InTopic12,studentswilladdfractionswithdenominatorsof10and100.(4.NF.C.5)

•Fractions and Angle Measure InTopic15,studentswilllearnthatann-degreeangleisananglethatturnsntimesthrough n

360ofacircle.(4.MD.C.5a,4.MD.C.5b)

GRADE 5

•Fraction Computation InTopic7,studentswilladdandsubtractfractionsandmixednumberswithunlikedenominators.(5.NF.A.1)InTopic8,theywillmultiplyfractions.(5.NF.B.4a)InTopic9,theywilldividefractionsandwholenumbers.(5.NF.B.4a,5.NF.B.7a,5NF.B.7b)

ToPICs 9 AND 10 Look AhEAD

46 = 1

6 + 16 + 1

6 + 16

46 = 4 × 1

6 = 4 × 16

1578

- 1338

248

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TOPICS

9–10Major Cluster 4.NF.BMaTH BaCkGROUNd: RIGOR

Content Rigor in

A rigorous curriculum provides a balance between conceptual understanding, procedural skills and fluency, and applications.

•Understand Why the Procedure for adding and Subtracting Fractions Works Early in Topic 9, students represent non-unit fractions, ab, where (a 7 1), as a sum of unit fractions, 1b. This approach illustrates why numerators can be added and subtracted when adding and subtracting fractions with like denominators.

•Using Number Sense to Estimate Fraction Sums and differences In Lesson 9-7, students extend their understanding of fraction addition and subtraction by using number sense to estimate whether fraction sums are greater than or less than one, and fraction differences are greater than or less than one-half.

•Make Sense of Multiplying a Whole Number by a Fraction or Mixed Number In Lesson 10-1, a key understanding developed is that a fraction like 34 can be represented as 3 × 1

4. This understanding is developed throughout Topic 10 using visual models to make sense of multiplying a whole number by a fraction or mixed number.

•Connect addition and Multiplication In Lessons 10-1 and 10-2, multiplication is connected to addition to strengthen understanding of why the procedures with fractions make sense.

•Using Conceptual Understanding to add and Subtract Fractions Early in Topic 9, students decompose non-unit fractions into unit fractions. These parts are then joined or separated. There are no fluency expectations for fraction addition and subtraction in Grade 4.

•Using Conceptual Understanding to Multiply a Whole Number by a Fraction In Lesson 10-3, students use models and symbols to recognize that in general n × (a

b) = n × ab . There are no fluency expectations for fraction

multiplication in 4.NF.B.

•addition and Subtraction Situations Throughout Topic 9, there are addition and subtraction situations involving fractions. These situations include add to, take from, put together, take apart, and comparison.

•Multiplication Situations Throughout Topic 10, there are multiplication situations that involve multiplying a whole number by a fraction. These situations include equal groups and comparison.

CONCEPTUal UNdERSTaNdING PROCEdURal SkIll aNd FlUENCy

aPPlICaTIONS

5 × 24 = 5 × (2 × 1

4 ) = (5 × 2) × 1

4

= 10 × 14

= 104 or 22

4

24 + 2

4 + 24 + 2

4 + 24 = 10

4 or 224

461E Topics 9–10 Cluster Overview

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MP

PearsonRealize.comMajor Cluster 4.nf.bMath Practices

Connecting Math Practices and Content Standards in

Math practices and content standards are connected within all lessons including the lessons that focus on math practices.

•MP.1 Make sense of problems and persevere in solving them.

Students persevere as they try to understand problems involving fractions, plan how to solve them, and determine if their solution makes sense. (e.g., p. 515, Item 2)

•MP.2 Reason abstractly and quantitatively.

Students use reasoning to analyze relationships between quantities in problems involving adding, subtracting, and multiplying fractions. (e.g., p. 476, Item 10)

•MP.3 Construct viable arguments and critique the reasoning of others.

Students critique strategies for computing with fractions, and they construct arguments to justify results. (e.g., p. 491, Item 1)

•MP.4 Model with mathematics.

Students model with math when they use expressions, equations, number lines, fraction strips, and other pictures to represent problems involving fractions. (e.g., p. 470, Item 16)

•MP.5 Use appropriate tools strategically.

Students use tools such as fraction strips to represent fraction operations and solve problems. (e.g., p. 465)

•MP.6 Attend to precision.

Students attend to precision when they use and explain fraction computations and when they choose the correct units in their answer. (e.g., p. 578, Item 7)

•MP.7 Look for and make use of structure.

Students look for structure when they examine relationships in fraction computations. (e.g., p. 556)

•MP.8 Look for and express regularity in repeated reasoning.

Students use repeated reasoning when they generalize about benchmark fractions and fraction operations. (e.g., p. 501)

•Lessons 9-11 and 10-6 These lessons focus on MP.4. Students model with math to solve real-world problems involving fraction addition, subtraction, and multiplication. They use representations, such as equations, number lines, pictures, and bar diagrams, to map the relationships between the important quantities in the problem.

Revisit the information about MP.4 in these other resources:

•Math Practices and Problem Solving Handbook before Topic 1; includes Math Practices Proficiency Rubrics.

•Math Practices Posters to display in your classroom

MP

Math Practices Animations, one for each math practice, available at PearsonRealize.com.

Math Practices within Lessons Lessons that focus on Math Practices

n miles

910

510

610

910 + 510 + 6

10 = n

910 + 510 + 6

10 = 2010

2010 or 2

461F

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Games

Tools

Assessment

Learn Practice Buddy

TOPICS

9 –10major cluster 4.nf.bdIfferenTIaTed InSTruCTIOnI Intervention O On-Level A advanced

Intensive Intervention

as needed, provide more instruction for struggling students that is on or below grade level.

Math diagnosis and Intervention System 2.0•Diagnosis Use the diagnostic tests in the

system. Also, use the item analysis charts given with program assessments at the start of a grade or topic, or at the end of a topic, group of topics, or the year.

•Intervention Lessons These two-page lessons include guided instruction followed by practice. The system includes lessons below, on, and above grade level.

•Teacher Support Teacher Notes provide the support needed to conduct a short lesson. The lesson focuses on vocabulary, concept development, and practice. The Teacher’s Guide contains individual and class record forms and correlations to Student’s Edition lessons.

resources for fluency Success•A variety of print and digital resources

are provided to ensure success on Common Core fluency standards. See Steps to Fluency Success on pages 43K–43N.

33

Strategic Intervention

at the end of the lesson, assess to identify students’ strengths and needs and then provide appropriate support.

Quick CheckIn the Student’s Edition Assess the lesson using 3 items checked in the Teacher’s Edition.

Assessment

Online Quick Check You can also assess the lesson using 5 online, machine-scored, multiple-choice items.

Intervention activity ITeachers work with struggling students.

reteach to Build understanding IThis is a page of guided reteaching.

Technology Center I O A

Tools

Digital Math Tools Activities reinforce the lesson content or previously taught content using a suite of digital math tools.

Games

Online Games provide practice on the lesson content or previously taught content.

Homework and Practice I O AUse the leveled assignment to provide differentiated homework and practice.

Additional resources to support differentiated instruction for on-level and advanced students include:

On-Level and advanced activity Centers O A•Center Games are provided in on-level

and advanced versions.

•Math and Science Activity is related to the topic science theme introduced at the start of the topic.

•Problem-Solving Reading Mat is used with a lesson-specific activity.

22

Ongoing Intervention

during the core lesson, monitor progress, reteach as needed, and extend students’ thinking.

Guiding Questions•In the Teacher’s Edition Guiding

questions are used to monitor understanding during instruction.

Learn

Online Guiding Questions Guiding questions are also in the online Visual Learning Animation Plus.

Prevent MisconceptionsThis feature in the Teacher’s Edition is embedded in the guiding questions.

error Intervention: If... then...This feature in the Teacher’s Edition is provided during Guided Practice. It spotlights common errors and gives suggestions for addressing them.

reteachingReteaching sets are at the end of the topic in the Student’s Edition. They provide additional examples, reminders, and practice. Use these sets as needed before students do the Independent Practice.

Higher Order ThinkingThese problems require students to think more deeply about the rich, conceptual knowledge developed in the lesson.

Practice Buddy Online

Practice Buddy

Online auto-scored practice is provided for each lesson. On-screen learning aids include Help Me Solve This and View an Example.

11

461G Topics 9–10 Cluster Overview

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Glossary Games

major cluster 4.nf.bthe language of math

English Language Learners

Provide ell support through visual learning throughout the program, ELL instruction in every lesson, and additional ideas in an ELL Toolkit.

Visual learningThe visual learning that is infused in enVisionmath2.0 provides support for English language learners. This support includes a Visual Learning Animation Plus and a Visual Learning Bridge for each lesson.

english language learners Instruction Lessons provide instruction for English language learners at Beginning, Intermediate, and Advanced levels of English proficiency.

english language learners toolkitThis resource provides professional development and resources for supporting English language learners.

Math Vocabulary

Build math vocabulary using the vocabulary cards, vocabulary activities, vocabulary review, and glossary plus the online glossary and vocabulary game.

my Word CardsVocabulary cards for a topic are provided in the Student’s Edition. Students use the example on the front of the card to complete the definition on the back.

Vocabulary activitiesThe Teacher’s Edition provides vocabulary activities at the start of topics. These include activities for vocabulary in My Word Cards and/or activities for vocabulary in Review What You Know.

Vocabulary ReviewA page of vocabulary review is provided at the end of each topic. It reviews vocabulary used in the topic.

glossaryA glossary is provided at the back of the Student’s Edition.

animated glossary

Glossary

An online, bilingual, animated glossary uses motion and sound to build understanding of math vocabulary.

online Vocabulary game

Games

An online vocabulary game is available in the Game Center.

In a number, a is a group of three digits, separated by commas, starting from the right.

0000,000,1, 00

onesthousands

millionsbillions

period

Math and Reading

Connect reading and math using a data-filled reading mat for the topic with accompanying activity masters and guide.

Problem-Solving Reading matsThere is a large, beautiful mat for each topic. At the start of the topic, help students become familiar with the mat and the vocabulary used by reading the mat aloud as students follow along. Use the Problem-Solving Reading Activity Guide for suggestions about how to use the mat.

Problem-Solving Reading activityAt the end of some lessons, a Problem-Solving Reading Activity provides a page of math problems to solve by using the data on the mat.

461H

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TOPIC

9Topic plannerUndersTand addITIOn and sUbTraCTIOn Of fraCTIOns

Digital

Print

•Student’sEdition•DailyCommonCoreReview•ReteachtoBuildUnderstanding•CenterGames•MathandScienceActivity•Problem-SolvingReadingMat•Problem-SolvingReadingActivity

Digital

•ListenandLookForPDLessonVideo•Student’sEditioneText•Today’sChallenge•Solve&Share•VisualLearningAnimationPlus

•AnimatedGlossary•MathTools•PracticeBuddyOnlinePractice•QuickCheck•AnotherLookHomeworkVideo•MathGames

Lesson 9-3

ADD FrActions with Like DenominAtors pp.477–482

content standards 4.nf.b.3a, 4.nf.b.3dmathematical Practices MP.1, MP.3, MP.4, MP.7

Objective Solveproblemsinvolvingjoiningpartsofthesamewholebyadding fractions.

essential Understanding Twofractionscanbejoinedoraddedtofindthetotal.Thereisageneralmethodforaddingfractionswithlikedenominators.

Vocabulary None

eLL speaking: Expressideasaloud.

Materials Numberlines(TT12),Fractionstrips(orTT13),Circlefractionmodels(TT14),coloredpencils

On-Level and advanced activity Centers•CenterGames

Lesson 9-2

DecomPose FrActionspp.471–476

content standard 4.nf.b.3bmathematical Practices MP.2, MP.4, MP.5

Objective Decomposeafractionormixednumberintoasumoffractionsinmorethanoneway.

essential Understanding Afractionab,wherea 7 1,canbedecomposedintothesumoftwoormoreunitornon-unitfractionsinoneormorewayswherethesumofthefractionsisequaltotheoriginalfraction.

Vocabulary Decompose,Compose,Mixednumber

eLL strategies: Usepriorknowledgetounderstandmeanings.

Materials Fractionstrips(orTT13),Circlefractionmodels(TT14),coloredpencils

On-Level and advanced activity Centers•MathandScienceActivity

Lesson 9-1

moDeL ADDition oF FrActions pp.465–470

content standard 4.nf.b.3amathematical Practices MP.1, MP.2, MP.3, MP.4, MP.5

Objective Usefractionstripsandnumberlinestoaddfractions.

essential Understanding Themeaningofadditionoffractionsisthesameasthemeaningofadditionofwholenumbers,butthealgorithmsaredifferent.

Vocabulary None

eLL speaking: Speakusinggrade-levelcontentareavocabulary.

Materials Numberlines(TT12),Fractionstrips(orTT13),Circlefractionmodels(TT14),measuringcups

On-Level and advanced activity Centers•CenterGames

LessOn resOUrCes

461I Topic 9

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Digital

PearsonRealize.com

Digital

Print

Start of Topic•MathandScienceProject•Home-SchoolConnection•ReviewWhatYouKnow•MyWordCards

End of Topic •FluencyPracticeActivity•VocabularyReview•Reteaching•TopicAssessment•TopicPerformanceAssessment

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Lesson 9-6

ADD AnD SubTrAcT FrAcTionS WiTH LiKE DEnoMinATorS pp.495–500

content Standard 4.NF.B.3aMathematical Practices MP.2, MP.4, MP.5

Objective Countforwardorbackwardonanumberlinetoaddorsubtract.

Essential Understanding Fractionadditionandsubtractioncanbethoughtaboutasjoiningandseparatingsegmentsonthenumberline.Theycanalsobethoughtaboutascountingforwardorcountingbackwardonthenumberline.

Vocabulary None

ELL reading: Usevisualandcontextual support.

Materials Numberlines(TT12),red,blue,andgreencrayons

On-Level and Advanced Activity Centers•Problem-SolvingReadingMat

Lesson 9-5

SubTrAcT FrAcTionS WiTH LiKE DEnoMinATorS pp.489–494

content Standards 4.NF.B.3a, 4.NF.B.3dMathematical Practices MP.2, MP.3, MP.4

Objective Solveproblemsinvolvingseparatingpartsofthesamewholebysubtractingfractions.

Essential Understanding Thedifferencebetweentwofractionswithlikedenominatorscanbefoundbyseparatingonefractionalamountfromtheother.Thereisageneralmethodforsubtractingfractionswithlikedenominators.

Vocabulary None

ELL Listening: Learnvocabularythatis heard.

Materials Numberlines(TT12)

On-Level and Advanced Activity Centers•CenterGames

Lesson 9-4

MoDEL SubTrAcTion oF FrAcTionS pp.483–488

content Standard 4.NF.B.3aMathematical Practices MP.1, MP.2, MP.4, MP.5, MP.6

Objective Usetoolssuchasfractionstrips,areamodels,andnumberlinestosubtractfractions.

Essential Understanding Similartowholenumbersubtraction,onewaytothinkaboutsubtractionoffractionsistoseparateapartfromawhole.Thealgorithmforsubtractingfractionswithlikedenominatorsisnotthesameasthealgorithmforsubtractingwholenumbers.

Vocabulary None

ELL reading: Usecontextualsupporttodevelopvocabulary.

Materials Numberlines(TT12),Fractionstrips(orTT13),Circlefractionmodels(TT14)

On-Level and Advanced Activity Centers•CenterGames

TOPIC REsOURCEs

461JPearsonRealize.com

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Notes

TOPIC

9Topic plannerUndersTand addITIOn and sUbTraCTIOn Of fraCTIOns

Lesson 9-9

Add Mixed NuMbers pp. 513–518

Content standard 4.nf.b.3cMathematical Practices MP.1, MP.2, MP.3, MP.8

Objective Use equivalent fractions and properties of operations to add mixed numbers with like denominators.

essential Understanding Two procedures for adding mixed numbers both involve changing the calculation to a simpler equivalent calculation.

Vocabulary None

eLL Listening: Learn academic vocabulary.

Materials None

On-Level and advanced activity Centers•Center Games

Lesson 9-8

ModeL AdditioN ANd subtrACtioN of Mixed NuMbers pp. 507–512

Content standard 4.nf.b.3cMathematical Practices MP.2, MP.3, MP.5

Objective Use models and equivalent fractions to add and subtract mixed numbers.

essential Understanding Adding and subtracting mixed numbers is an extension of the ideas and procedures for adding and subtracting fractions.

Vocabulary None

eLL speaking: Express ideas on a variety of topics.

Materials Number lines (TT 12), Fraction strips (or TT 13), markers

On-Level and advanced activity Centers•Center Games

Lesson 9-7

estiMAte frACtioN suMs ANd differeNCes pp. 501–506

Content standard 4.nf.b.3aMathematical Practices MP.1, MP.2, MP.3, MP.4, MP.8

Objective Use number lines and benchmark fractions to estimate fraction sums and differences.

essential Understanding Fraction sums and differences can be estimated by thinking about how each fraction relates to other fractions that are easy to add and subtract mentally.

Vocabulary None

eLL speaking: Express opinions

Materials Number lines (TT 12), Fraction strips (or TT 13), number cubes

On-Level and advanced activity Centers•Math and Science Activity

461K Topic 9

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Lesson 9-11

Math Practices anD ProbleM solving: MoDel with Math pp. 525–530

Mathematical Practices MP.4 Also MP.1, MP.2, MP.5content standards 4.NF.B.3d, 4.NF.B.3a

Objective Use previously learned concepts and skills to represent and solve problems.

Essential Understanding Good math thinkers choose and apply math they know to show and solve problems from everyday life.

Vocabulary None

ELL reading: Derive meaning from print.

Materials None

On-Level and Advanced Activity Centers•Problem-Solving Reading Mat

Lesson 9-10

subtract MixeD nuMbers pp. 519–524

content standard 4.NF.B.3cMathematical Practices MP.1, MP.2, MP.3, MP.8

Objective Use equivalent fractions, properties of operations, and the relationship between addition and subtraction to subtract mixed numbers with like denominators.

Essential Understanding Two procedures for subtracting mixed numbers both involve changing the calculation to a simpler equivalent calculation. These are extensions of the same procedures used for adding mixed numbers with like denominators.

Vocabulary None

ELL learning: Use prior knowledge.

Materials None

On-Level and Advanced Activity Centers•Center Games

461L

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TOPIC

9 Essential Questions: How do you add and subtract fractions and mixed numbers with like denominators? How can fractions be added and subtracted on a number line?

Understand Addition and Subtraction of Fractions

How do you write, “I love math?” using Morse

code? Here is a project about fractions and information.

Do Research Morse code uses patterns to transfer information. Any word can be written using Morse code. Use the Internet or other sources to find how to write fourth, grade, and school using Morse code.

Journal: Write a Report Include what you found. Also in your report:

• Write one in Morse code. Write a fraction that tells what part of the code for one is dashes.

• Write three in Morse code. Write a fraction that tells what part of the code for three is dots.

• Write and solve an equation to find how much greater the fraction for dots is than the fraction for dashes in the word three.

Math and Science Project: Fractions and Information Transfer

A combination of dots and dashes stands for each letter, each

number, and even some whole words.

Morse code uses a special machine to

transfer information using a series

of tones.

Topic 9 461

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Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. 4

NameHome-School Connection

Topic 9Understand Addition and Subtraction of FractionsTopic 9 Standards 4.NF.B.3a, 4.NF.B.3b, 4.NF.B.3c, 4.NF.B.3d See the front of the Student’s Edition for complete standards.

Dear Family,

In this topic, your child will learn to add and subtract fractions with like denominators, or denominators that are the same. To add fractions with like denominators, add the numerators and write the sum over the like denominator. For example, 3

12 + 812 = 11

12.Your child will also learn to use fraction strips and number lines to represent the

addition and subtraction of fractions with like denominators. Here are some activities that you can do to help your child understand adding

and subtracting fractions with like denominators.

Fraction Writing

Materials: Paper and pencil

Step 1 Write 14, 12, 24, 34, 18, and 112 on a piece of paper.

Step 2 Have your child name the fractions that have a common denominator and explain how to add those fractions.

Fraction Toss-Off

Materials: Number cube labeled 1–6

Step 1 Toss a number cube once to generate a numerator and once or twice to generate a one-digit or two-digit denominator. Repeat to create several fractions.

Step 2 Have your child decompose each fraction in two or more ways.

Observe Your ChildFocus on Mathematical Practice 4 Model with Mathematics

Help your child become proficient with Mathematical Practice 4. Ask your child to represent the addition in the Fraction Writing activity with bar diagrams and equations.

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ONE dashes

THREE dots

4_6

_810

TOPIC

9TOPIC OPENERUNDERSTAND ADDITION AND SUBTRACTION OF FRACTIONS

TOPIC ESSENTIAL QUESTIONS

MATH AND SCIENCE PROJECT STEM

How do you add and subtract fractions and mixed numbers with like denominators? How can fractions be added and subtracted on a number line?Revisit the Topic Essential Questions throughout the topic, and see a note about answering the questions in the Teacher’s Edition for the Topic Assessment.

Science Theme The science theme for this project is Information Transfer. This theme will be revisited in the Math and Science Activities in Lessons 9-2 and 9-7 and in some lesson exercises.

See if students can identify the machine that uses Morse code shown in the photo. Ask why people would want to use a machine like this.

Explain that the photo shows a telegraph machine and that it was used even before telephones were invented.

Project-Based Learning Have students work on the Math and Science Project over the course of several days.

EXTENSIONHave students write a message in Morse code.

Sample Student Work for Math and Science Project

Home-School ConnectionSend this page home at the start of Topic 9 to give families an overview of the content in the topic.

461 Topic 9

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© Pearson Education, Inc. 4

Name

Equivalent Fractions

Write the missing values to show pairs of equivalent fractions.

3. 23 =□6 4. □4 = 3

12 5. 65 =□10

6. 12 = 50

7. 15 =□10

8. 3

□ = 30100

Benchmark Fractions

Use the number line to find a benchmark fraction or whole number for each given fraction.

0 114

12

34

9. 78 is close to . 10. 8

12 is close to . 11. 26 is close to .

Problem Solving12. Adult admission to the dog show is $16. Children’s admission is $9. How much

would it cost 3 adults and 2 children to enter the dog show?

13. Meg saved coins she found for a year. She found a total of 95 pennies, 13 nickels, 41 dimes, and 11 quarters. She would like to evenly divide the coins into 4 piggy banks. How many coins will go in each piggy bank?

A-ZA-Z Vocabulary

Choose the best term from the box. Write it on the blank.

1. In 23, 2 is the of

the fraction and 3 is the of the fraction.

2. Fractions that name the same region, part of a set, or part of a segment are called .

• benchmark fractions

• denominator

• equivalent fractions

• numeratornumerator

equivalent fractions

denominator

40 coins

4

2

1

$66

12

10100

1

34

14

462 Topic 9 Review What You Know

MTH16_SE04_CC2_T09_TO.indd 462 17/05/14 8:58 PM

✂My Word Cards


Complete each definition. Extend learning by writing your own definitions.

means to break into parts.

A number that has a whole number part and a fraction part is a

.

means to combine parts.Compose

mixed number

Decompose

464 Topic 9 My Word Cards

MTH16_SE04_CC2_T09_MWC.indd 464 12/05/14 12:23 PM

✂

Glossary

My Word Cards

decompose

mixed number

compose

Use the examples for each word on the front of the card to help complete the definitions on the back.

45 = 1

5 + 25 + 1

514 + 1

4 + 14 = 3

4

1

13, 4

12, 6

58

463Topic 9 My Word Cards

MTH16_SE04_CC2_T09_MWC.indd 463 24/04/14 9:39 AM

Use the Topic 13 activity on p. 673–674 with the Topic 9 words at the right.

Topic 9 Vocabulary Words Activity

Item Analysis for Diagnosis and Intervention

Item Standard MDIS

1–2 4.NF.A.1 H1

3–8 4.NF.A.1 H16

9–11 4.NF.A.2 H18

12–13 4.OA.A.3 J2, J4

462–464

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Name Daily Common Core Review

9-1

1. Airport security guards choose some travelers for an extra safety check. So far, the guards have chosen the 6th, 12th, 18th, and 24th travelers in line. Which traveler will most likely be chosen next for the extra safety check?

� The 25th traveler in line




2. Shannon says, “My apartment number cannot be found using a factor of 3.” Select all the possible numbers for Shannon’s apartment.

15

27

31

42

73

3. Jake said he ate 34 of his dinner. Which fraction is equivalent to 34?

� 26

� 48

� 912

� 1012

4. Which fraction is greater than 23?

� 45

� 46

� 12

� 13

5. Kendra made 111 pastries for a bake sale. How many bags can she make if she puts 3 pastries in each bag? How many pastries are left over?

6. Luis has $20. He buys 4 cans of tennis balls and gets $8 back as change. How much did one can of tennis balls cost?

7. Which is greater 23 or 38? Explain how to compare using the benchmark fraction, 12.

8. Write 21,407 in expanded form and using number names.

9. Round 16,049 to the nearest ten, hundred, and thousand.

$3

37 bags; 0 left over

23 +

12 and 38 *

12,

so 23 +38.

4.OA.B.4 4.NBT.B.6

4.OA.A.3

4.NF.A.2

4.NBT.A.2

4.NBT.A.3

4.OA.B.4

4.NF.A.1

4.NF.A.2

20,000 + 1,000 + 400 +

7; twenty-one thousand,

four hundred seven

16,050; 16,000;16,000

D 9•1

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Daily Common Core Review

LESSON 9-1MODEL ADDITION OF FRACTIONS

Today’s ChallengeUse the Topic 9 problems any time during this topic.

FOCUSDomain 4.NF Number and Operations—FractionsCluster 4.NF.B Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.Content Standard 4.NF.B.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.Mathematical Practices MP.1, MP.2, MP.3, MP.4, MP.5Objective Use fraction strips and number lines to add fractions.Essential Understanding The meaning of addition of fractions is the same as the meaning of addition of whole numbers, but the algorithms are different. Materials Number lines (Teaching Tool 12), Fraction strips (or Teaching Tool 13), Circle fraction models (Teaching Tool 14)

COHERENCEPrior to this lesson students have developed an understanding of the meaning of addition of whole numbers as joining and they have developed an understanding of the meaning of a fraction ab as a number of unit fractions 1b. These two ideas are brought together in this lesson where representations are used to show how fractions with the same denominator can be joined.

RIGORThis lesson emphasizes conceptual understanding. It connects to addition of whole numbers by showing that addition of fractions also involves joining quantities to find a total. Also, the lesson uses students’ understanding that a fraction like 5

10 can be represented as 5 of the unit fraction 1

10. So, finding the sum of 5

10 and 210 means joining

5 of the unit fraction 110 with 2 of the unit

fraction 110 to get 7 of the unit fraction 1

10 or 7

10.

PD

Watch the Listen and Look For Lesson Video.

ENGLISH LANGUAGE LEARNERS

Speaking Speak using grade-level content area vocabulary.

Use with the Visual Learning Bridge on Student’s Edition p. 466.

Remind students that some words have more than one meaning. Ask students to describe what the word like means. Explain that in math, like means the same. For example, 48 and 58 have like denominators because both of the denominators are the same.

Beginning Write these three fractions: “25,

35, 45.” Point to the denominators. Say: These fractions have like denominators. Write other fractions with like denominators. Ask students to point to and say the denominators and complete the sentence stem: “These fractions have .”Intermediate Write these six fractions: “25, 4

12, 712, 35, 8

12, 45.” Ask students to identify which fractions have like denominators. Ask: How are these fractions alike?

Advanced Ask students to write six fractions, some with like and some with unlike denominators. Have students trade papers and identify the fractions that have like denominators. Ask students to explain like denominators to a partner.Summarize What are like denominators?

Think

LESSON OVERVIEW MATH ANYTIMEFOCUS • COHERENCE • RIGORF C R

465A Topic 9

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10–15 min

Solve

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Name

I can …

Lesson 9-1Model Addition of Fractions

Look Back! MP.5 Use Appropriate Tools Kyle says 18 + 1

8 + 18 = 3

8. Jillian says 18 + 18 + 1

8 = 324. Use fraction strips

to decide who is correct.

You can use appropriate tools. You can

use drawings, area models, or fraction strips to solve this problem. Show your work in

the space below!

Content Standard 4.NF.B.3a Mathematical Practices MP.1, MP.2, MP.3, MP.4, MP.5

use tools such as fraction strips or area models to add fractions.

Kyle and Jillian are working on a sports banner. They painted 38 of the banner green and 48 purple. How much of the banner have they painted? Solve this problem any way you choose.

Sample answer: Using fraction strips, 18 + 18 + 1

8 = 38.

Kyle is correct.

See margin for sample student work.

18

18

118

465Topic 9 Lesson 9-1

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Jack’s Work Conner’s Work

STEP

1 PearsonRealize.com

Solve

COHERENCE: Engage learners by connecting prior knowledge to new ideas.Students model addition of fractions with like denominators by using appropriate tools and models, including fraction strips, drawings, and number lines.

DEVELOP: PROBLEM-BASED LEARNING

BEFORE

1. Pose the Solve-and-Share ProblemMP.5 Use Appropriate Tools Strategically Listen and look for students who connect the fraction strips (or Teaching Tool 13) to combining the parts of the banner.

2. Build UnderstandingWhat information are you given? [Kyle and Jillian painted 38 of the banner green and 48 of the banner purple.] What are you asked to do? [Determine how much of the banner Kyle and Jillian have painted.]

DURING

3. Ask Guiding Questions As NeededHow can you represent the whole banner that Kyle and Jillian are painting? [I can use the 1 fraction strip, or 1 whole.] Do you need to combine or separate the parts of the banner? [Combine] What operation should you use to find the amount of the banner that Kyle and Jillian painted so far? [Addition]

AFTER

4. Share and Discuss SolutionsStart with students’ solutions. If needed, project Jack’s work to discuss how to use fraction strips and an equation.

5. Transition to the Visual Learning BridgeFraction strips can be used to model how to find the sum of two fractions with like denominators by combining parts of the whole.When adding fractions with like denominators, the numerators are added without changing the denominator.

6. Extension for Early FinishersWhat fraction of a dollar is a penny? [ 1

100] A dime? [ 110]

A quarter? [14]

Analyze Student Work

Jack uses fraction strips and writes an equation. Conner draws an area model to represent the addition.

Solve

Whole Class

Whole Class

Small Group

465

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Visual Learning

A

Learn Glossary

B C

0 1

510=

110+

110

210=

110+

110

+ 110+

110+

110

Convince Me! MP.1 Make Sense and Persevere What two fractions would you add to find the fraction of the canoes that are either green or brown? What is the sum? How do you know your sum is correct?

You can use tools such as fraction strips to add two or more fractions.

Ten canoeing teams are racing downriver. Five teams have silver canoes and two teams have brown canoes. What fraction of the canoes are either silver or brown?

How Can You Use Tools to Add Fractions?

110

110 1

10110

110

110

110

1

Find 510 + 2

10. Use five 110 fraction strips

to show 510 and two 1

10 strips to show 210.

Five 110 strips joined with two 1

10 strips are

seven 110 strips.

Add the numerators. Then write the sum over the like denominator.

510 + 2

10 = 710

Find 510 + 2

10. Mark five 110 segments to show

510 and two 1

10 segments to show 210.

Adding 510 and 2

10 means joining five 1

10 segments and two 110 segments.

710 of the canoes are either silver

or brown.


310 and 2

10; 310 + 2

10 = 510; Sample answer: Three 1

10 parts

joined with two 110 parts are five 1

10 parts or 510.

466 Topic 9 Lesson 9-1

MTH16_SE04_CC2_T09_L01_VLB.indd 466 13/05/14 5:12 PM

DEVELOP: VISUAL LEARNINGSTEP

2

Learn Glossary

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The Visual Learning Bridge connects students’ thinking in the Solve & Share to important math ideas in the lesson. Use the Visual Learning Bridge to make these ideas explicit. Also available as a Visual Learning Animation Plus at PearsonRealize.com

MP.1 Make Sense and PersevereWhat information are you given in this problem? [5 out of 10 canoes are silver and 2 out of 10 canoes are brown.] What operation will you use to solve the problem? [Addition] Why? [I need to combine the parts of the canoes that are each color.]

In 510, what does the

numerator represent? [5 silver canoes] What do 5

10 and 210

have in common? [The denominator, 10] What does the like denominator represent? [The total number of canoes]

How many equal segments are between 0 and 1 on the number line? [10] Why? [There are 10 canoes and each segment represents a canoe.]Prevent MisconceptionsSome students may want to add the denominators. While working with the models, point out that after combining the 5 segments and the 2 segments, there are still 10 segments or 10 equal parts between 0 and 1.

11

Convince Me! MP.1 Make Sense and Persevere How could you find the sum of the fractions of the canoes that are green and the fraction that are brown? [You could use fraction strips or a number line.]

Coherence To solve the canoe problem, students connect two representations for fractions, fraction strips and a number line. Both show the sum of two fractions can be found by adding the numerators and keeping the same denominator. This focus on conceptual development of adding fractions should help students with the symbolic work later in the topic.

Revisit the essential question. Students can use fraction strips or number lines to show when you combine 5 tenths and 2 tenths, you get 7 tenths. The sum is the same whether combining fraction strips or segments on a number line.

Visual Learning Bridge

466 Topic 9

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20–30 min

18

18

18

1

18

18

18


Common Core Assessment

17. Number Sense Using three different numerators, write an equation in which three fractions, when added, have a sum of 1.

20. What addition problem is shown by the fraction strips below?

19. A bakery sells about 9 dozen bagels per day. About how many bagels does the bakery sell in a typical week? Explain.

18. MP.4 Model with Math A rope is divided into 8 equal parts. Draw a picture to show 18 + 3

8 = 48.

21. Higher Order Thinking Terry ran 110 of

the distance from school to home. He walked 3

10 more of the distance and then

skipped 210 more of the distance. What

fraction of the distance home does Terry still have to go?

22. Jackson said, “I am thinking of two fractions that when added have a sum of one.” Which fractions could Jackson have been thinking about?

𝖠𝖠𝖠𝖠 53 and 53

𝖡𝖡𝖡𝖡 14 and 34

𝖢𝖢𝖢𝖢 25 and 45

𝖣𝖣𝖣𝖣 38 and 48

23. Lindsay has 5 red hats, 2 blue hats, and 3 black hats. Which statement is true?

𝖠𝖠𝖠𝖠 810 of the hats are either red or black.

𝖡𝖡𝖡𝖡 53 of the hats are either red or black.

𝖢𝖢𝖢𝖢 510 of the hats are either red or black.

𝖣𝖣𝖣𝖣 310 of the hats are either red or black.

Look back to see if you answered the question that

was asked.

There are 12 bagels in one dozen.

Sample answer: 16 + 26 + 3

6 = 66

About 756 bagels; Sample answer: 12 × 9 = 108; 108 × 7 = 756

Check students’ drawings.

38 + 3

8 = 68

Sample answer: 410 of the distance


MTH16_SE04_CC2_T09_L01.indd 468 13/05/14 5:16 PM

*Tools AssessmentPractice

Buddy

18

18

18

18

18

115

15

15

110

110

110

110

110

112

112

112

112

112

112

112

112

112

112

112

112

112

16

16

0 1

24 = 1

4 + 14

14

0 126 = 1

6 + 16

36 = 1

6 + 16 + 1

616

0 1 0 1

13

13

Do You Understand? Do You Know How?

Name

*For another example, see Set A on page 533.

1. MP.2 Reasoning In the problem on the previous page, why aren’t the purple 1

10 strips the same length as the red strip?

2. What two fractions are being added below? What is the sum?

For 3–4, find each sum.

Leveled Practice For 5–16, find each sum. Use fraction strips or other tools.

3. 25 + 1

5

4. 16 + 1

6

5. 312 + 4

12 6. 410 + 1

10 7. 212 + 4

12

14. 14 + 1

4 15. 25 + 2

5 16. 110 + 2

10 + 110

8. 16 + 2

6 + 36 9. 1

4 + 24 10. 1

3 + 13

11. 58 + 1

8 12. 14 + 3

4 13. 712 + 2

12

Sample answer: The red strip represents 10

10, or all of the canoes. The purple 1

10 strips represent the 7 silver and brown canoes or 7

10.

Sample answers given.

28 and 38; 58

26


712

510

612

24

45

66

68

34

44

23

912

410

35


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PearsonRealize.com

ToolsPractice Buddy

Assessment

QUICK CHECKCheck mark indicates items for prescribing differentiation on the next page.Items 14 and 22 are worth 1 point. Item 21 is worth up to 3 points.

Error Intervention: Item 2If students have difficulty writing the fractions,then remind them of their work with fractions in Grade 3. Two copies of 18 is 28. Three copies of 18 is 38.

11 Reteaching Assign Reteaching Set A on p. 533.

Multi-Step Problems Page 468 Items 19 and 21

Item 17 Number Sense The easiest solution is to choose three different numbers for the numerators, and use their sum as the denominator.

Item 18 MP.4 Model with Math Students can represent the situation with a picture of the divided rope or with fraction strips.

Item 20 Students should recognize that 3 of the unit fraction 18 are joining another 3 of the unit fraction 18 to get 6 of the unit fraction 18 or 68.

Item 21 Higher Order Thinking To find the solution to this two-step question, students need to determine which operations to use. What is the hidden question you need to answer to be able to solve the problem? [Find the sum of 1

10 + 310 + 2

10] How do you find the

distance Terry still has to go? [Subtract : 1010 - 6

10 = 410]

467–468

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22


Name Reteach to Build Understanding

9-1

4. What fraction does the area model show?

5. Shade 25 more of the area model.

6. What part is shaded in all?

7. 15 + 2

5 =

8. What fraction does the area model show?

9. Shade 16 more of the area model.

10. What part is shaded in all?

11. 36 + 1

6 =

On the Back!

12. Use fraction strips or draw an area model to find 512 + 6

12.

Vocabulary

1. A fraction represents a part of a whole. An area model is a tool used to model fractions.

Shade 7 parts of the area model to show 710.

2. The denominator of a fraction is the bottom number. It tells how many equal parts make 1 whole.

What is the denominator of 710?

3. The numerator of a fraction is the top number. It tells how many of the equal parts to use.

What is the numerator of 710?

1112; Check students’ drawings.

10

7

35

35

36

46

46

15

R 9•1

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Share your thinking while you work.

PartnerTalk

Clip and Cover

Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. 4Center Game ★

Get Started or

Get 10 squares in one color and 10 in another color, two paper clips, two number cubes, and fraction strips or other tools. Take turns.

At Your Turn

Toss two cubes to find your ovals. EXAMPLE: Choose the 3rd oval on the left and the 5th oval on the right, or choose the 5th oval on the left and the 3rd oval on the right. Mark your ovals with paper clips.

How to Play

Explain how to add the two numbers you chose. Use fraction strips or other tools to model each problem. Find and cover the sum. Lose your turn if the answer is taken.

How to Win The first player or team to cover three connected rectangles in a row or column wins.

If you have more time Play again! Talk about how you know your answer is reasonable.

1112

612

712

212

912

812

912

312

412

712

512

1112

612

1112

412

812

312

612

112

412

312

112

512

212

112

512

312

212

9•1

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PartnerTalk

Clip and Cover

Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. 4Center Game ★★

Get Started or

Get 10 squares in one color and 10 in another color, one paper clip, one number cube, and fraction strips or other tools. Take turns.

At Your Turn

Toss one cube to find your ovals. EXAMPLE: Choose the 3rd oval on the left, or choose the 3rd oval on the right. Mark your oval with a paper clip.

How to Play

The number you chose is a sum. Find two numbers that you can add to get that sum. Find and cover the answer. Lose your turn if the answer is taken.

How to Win The first player or team to cover three connected rectangles in a row or column wins.

If you have more time Play again! Talk about your strategies as you play.

56

+ 16

310

+ 410

38

+ 18

35

+ 15

510

+ 310

712

+ 312

110

+ 610

28

+ 38

25

+ 25

36

+ 26

612

+ 312

48

+ 18

18

+ 28

412

+ 612

14

+ 14

310

+ 310

1012

610

38

912

45

1

56

48

58

810

24

710

9•1


ASSESS AND DIFFERENTIATE

Use the QUICK CHECK on the previous page to prescribe differentiated instruction.

I Intervention O On-Level A Advanced 0–3 points on the Quick Check 4 points on the Quick Check 5 points on the Quick Check

STEP

3

Reteach I

On-Level and Advanced Activity Centers O A

Modeling Addition of FractionsMaterialsMeasuring cups

•Have students work in groups with two measuring cups per group.

•Begin by adding 14 cup of water to one measuring cup. Measure 14 cup of water in the second measuring cup and add it to the first measuring cup. Have students read the total amount. What equation represents this situation? [14 + 1

4 = 24]

•Add 14 cup more water to the measuring cup. Have students use an equation to represent this addition. [24 + 1

4 = 34]

•Have students repeat the exercise using different measurements. Have them write an equation for each model.

Advanced On-LevelCenter GamesStudents work in pairs or small groups to find the sums of fractions with like denominators. Have students record their equations as they play the game.

Intervention Activity I

469A Topic 9


15–30 min

18

18

1918

18

18

18

1811

8

1



13. MP.3 Critique Reasoning When Jared found 15 + 2

5, he wrote the sum 310. Is Jared

correct? Explain.

16. MP.4 Model with Math Sandy made 8 friendship bracelets. She gave 1 bracelet to her best friend and 5 bracelets to her friends on the tennis team. Use the model to find the fraction that represents the total number of bracelets Sandy gave away.

14. Number Sense Leah wrote 2 different fractions with the same denominator. Both fractions were less than 1. Can their sum equal 1? Can their sum be greater than 1? Explain.

17. Higher Order Thinking Julia writes 2 fractions with the same denominator that have numerators 5 and 7. What could the denominator be if the sum is less than 1? Equal to 1? Greater than 1?

15. Sasha has a box of antique letters. She wants to give an equal number of letters to each of her 5 friends. How many antique letters will each friend receive?

18. Billy did 16 of his homework on Friday. He did 16 more on Saturday. Billy still had 46 to finish. How much of his homework did Billy do on Friday and Saturday?

𝖠𝖠𝖠𝖠 26 𝖢𝖢𝖢𝖢 4

6

𝖡𝖡𝖡𝖡 36 𝖣𝖣𝖣𝖣 5

6

19. Roberto shares a bag of almonds with 2 friends. He shares 18 bag with Jeremy and 28 bag with Emily. He eats 38 of the almonds himself. What fraction of the almonds do Roberto and his friends eat?

𝖠𝖠𝖠𝖠 112 𝖢𝖢𝖢𝖢 6

8

𝖡𝖡𝖡𝖡 38 𝖣𝖣𝖣𝖣 7

8

There are 130 antique

letters in the box.

Sample answer: Yes, 38 + 58 = 1;

Yes, 36 + 46 = 7

6, and 76 + 1.

No; Sample answer: Jared added the denominators, 5 + 5, instead of keeping them the same. Fraction strips would show the answer is 35, not 3

10.

26 antique letters

Sample answer: For the sum to be less than 1, the denominator can be any number greater than or equal to 13. The denominator must be 12 if the sum is equal to 1. If the sum is greater than 1, then the denominator can be any number 1 through 11.Sample answer: 68 of the bracelets



GamesToolsHelp Practice Buddy

Another Look!


NameHomework & Practice 9-1Model Addition of Fractions

You can use a circle fraction model to add fractions.

For 1–12, find each sum. Use fraction strips or other tools.

Eight friends went out to lunch. Four of them had pizza. Two had hamburgers and two had soup. What fraction of the group had either pizza or soup?

Divide a circle into eighths to represent each of the 8 people in the group.

Four people had pizza. Shade 4 of the sections to represent 48.

Two people had soup. Shade 2 more sections to represent 28.

Count the number of 18 sections. There are six 18 sections shaded. So, 68 of the group had either pizza or soup.

48 + 2

8 = 68

1. 15 + 1

5 2. 46 + 1

6 3. 58 + 2

8

4. 212 + 2

12 5. 25 + 3

5 6. 210 + 3

10

7. 58 + 3

8 8. 310 + 1

10 9. 34 + 1

4

10. 510 + 4

10 11. 16 + 1

6 + 16 12. 1

12 + 512 + 2

12


25

910

88

412

56

36

410

55

78

812

44

510



TIMINGThe time allocated to Step 3 will depend on the teacher’s instructional decisions and differentiation routines.

PearsonRealize.com

ToolsHelp Practice Buddy

Games


Math Tools and Math GamesA link to a specific math tools activity or math game to use with this lesson is provided at PearsonRealize.com.

Tools Games


Leveled Assignment I Items 1–4, 15–16, 18–19 O Items 5–8, 13, 17–19 A Items 9–12, 14–15, 17–19

469–470

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9-2Name

1. Which fraction is equivalent to 12?

� 26

� 48

� 610

� 912

2. Shaquille bought 12 jumbo packs of trading cards. Each jumbo pack contains 33 cards. How many trading cards did Shaquille buy?

� 99 trading cards




3. Over the summer Hans ran 89 laps of the track and swam 147 laps of the pool. Hans completed twice as many laps as his friend Lee. How many laps did Lee run and swim?

� 118 laps

� 128 laps

� 236 laps

� 472 laps

4. Select all the statements that are true.

13 6 1

4

13 7 1

6

12 6 1

5

14 7 1

10

110 7 1

2

5. Which is greater 54 or 45? Explain.

6. Write 968,472 in expanded form.

7. A school has 158 computers. They want each of the 6 grades to have the same number of computers. How many computers should each grade receive? Explain.

900,000 + 60,000+ 8,000 + 400 +70 + 2

26 computers; Sample

answer: 158 ÷ 6 = 26 R2.

Each grade will have

26 computers, and

2 computers will be left

over.

54

; 54 + 1 and 45 * 1,

so 54 +45.

4.NF.A.1 4.NF.A.2

4.NBT.A.2

4.NBT.B.6

4.NBT.B.5

4.OA.A.2

4.NF.A.2

D 9•2



Think

Today’s Challenge

Learn


Glossary

Animated Glossary

eText


Tools

Math Tools

Games

Math Games

Assessment

Quick Check

PD


Solve

Solve and Share

Help



Practice Buddy


LESSON 9-2DECOMPOSE FRACTIONS


FOCUSDomain 4.NF Number and Operations—FractionsCluster 4.NF.B Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.Content Standard 4.NF.B.3b Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 38 = 1

8 + 18 + 1

8; 38 = 18 + 2

8;

218 = 1 + 1 + 1

8 = 88 + 8

8 + 18.

Mathematical Practices MP.2, MP.4, MP.5Objective Decompose a fraction or mixed number into a sum of fractions in more than one way.Essential Understanding A fraction ab, where a 7 1, can be decomposed into the sum of two or more unit or non-unit fractions in one or more ways where the sum of the fractions is equal to the original fraction.Vocabulary Decompose, Compose, Mixed number

Materials Fraction strips (or Teaching Tool 13)

COHERENCEPrior to this topic students learned that a fraction ab can be decomposed into the sum of a unit fractions, 1b. They used this idea in the previous lesson to think about addition of fractions ab and c

b as the joining of unit fractions. In this lesson they learn that a fraction ab, where a 7 1, can be broken into the sum of unit and non-unit fractions in one or more ways. These ideas are brought together in the following lessons on addition and subtraction of fractions.

RIGORThis lesson emphasizes conceptual understanding. Understanding addition and subtraction of fractions both build on the idea of decomposing fractions. In addition, fractions can be decomposed into unit fractions and then joined to the find the total. In subtraction, a total is decomposed into parts, unit or non-unit fractions, and then one part is separated from the other.

PD



Strategies Use prior knowledge to understand meanings.


Remind students that word parts can be helpful in understanding unfamiliar words. Explain that composing means “putting something together.” The prefix de- means “the opposite of.” Explain that decompose means “take apart.”

Beginning Show a set of three red counters and two yellow counters. Write 55. Say: 3 of the 5 counters, or 35, are red. 2 of the 5 counters, or 25, are yellow. Have students complete the sentence stem: “One way to decompose 55 is and .”

Intermediate Show a set of three red counters and two yellow counters. Say: This set of counters can be decomposed into red and yellow counters. 35 of the counters

are red. 25 of the counters are yellow. Have students discuss and model other ways to decompose the set of counters.

Advanced Give student partners a set of 5 counters. Ask partners to discuss and model different ways to decompose the whole set into two or more smaller sets.

Summarize How can you use your understanding of decomposing when solving fraction problems?

Think


471A Topic 9


10–15 min

Solve


Name

I can …

Lesson 9-2Decompose Fractions

Look Back! MP.5 Use Appropriate Tools Use a drawing or fraction strips to help write equivalent fractions for the amount of chili in one of the bowls.

How can you model the amount of chili Karyn puts in each bowl? Show your

work in the space below! Content Standard 4.NF.B.3b Mathematical Practices MP.2, MP.4, MP.5

use fraction strips, area models, or drawings to decompose fractions.

Karyn has 118 pounds of chili to put into

three bowls. The amount of chili in each bowl does not have to be the same. How much chili could Karyn put into each bowl? Solve this problem any way you choose.

Sample answers: 48 and 12; 48 and 24.



MTH16_SE04_CC2_T09_L02_VLB.indd 471 21/04/14 10:46 AM

Isaiah’s Work Estela’s Work

STEP


Solve

COHERENCE: Engage learners by connecting prior knowledge to new ideas.Students use their knowledge of decomposing whole numbers, as in making 10, to decompose a fraction.


BEFORE

1. Pose the Solve-and-Share ProblemYou may wish to provide fraction strips (or Teaching Tool 13).MP.4 Model with Math In this problem, look for students who represent 3 bowls of chili using drawings or fraction strips.

2. Build UnderstandingWhat information are you given? [Karyn has 11

8 pounds of chili to put into three bowls. The amount of chili does not have to be the same in each bowl.] What are you asked to do? [Find how much chili Karyn could put into each bowl.]

DURING

3. Ask Guiding Questions As NeededWhat tools can you use to help solve the problem? [Sample answer: fraction strips] How many eighths is 11

8 ? [11]

AFTER

4. Share and Discuss SolutionsStart with students’ solutions. If needed, project Isaiah’s work to discuss how to decompose a fraction.

5. Transition to the Visual Learning BridgeA fraction can be broken into parts, or decomposed, in more than one way.

6. Extension for Early FinishersUse fraction strips to find another way to represent 11

8 as a sum of fractions. [Sample answer: 38 + 3

8 + 38 + 2

8]


Isaiah decomposes the original fraction into 3 fractions and writes his answer as an addition equation.

Estela decomposes incorrectly because her 3 fractions do not total the original amount.

Solve

Whole Class

Whole Class

Small Group

471

MTH16_TE04_CC2_T09_L02.indd Page 471 10/07/14 8:14 AM sw-102 /113/PE01513_PS/MATH_2016/NA/ANCILLARY/Product_Sampler/Product_Sampler_3_6/Layout ...

Visual Learning

A

Learn Glossary

B COne Way Another Way

Convince Me! MP.5 Use Appropriate Tools Draw pictures or use fraction strips to show why these equations are true.

56 = 3

6 + 26

56 = 1

6 + 26 + 2

6

Decompose means to break into parts. Compose means to combine parts. The fraction of the garden that Charlene will

plant can be decomposed in more than one way.

Charlene wants to leave 16 of her garden empty. What are some different ways Charlene can plant the rest of her garden?

Charlene could plant four 16 sections with blue flowers and one 16 section with red peppers.

56 is 46 and 16.

56 = 4

6 + 16

Charlene could plant one 16 section with green beans, one 16 section with yellow squash, one 16 section with red peppers, and two 16 sections with blue flowers.

56 is 16 and 16 and 16 and 26.

56 = 1

6 + 16 + 1

6 + 26

How Can You Represent a Fraction in a Variety of Ways?

16 empty

56 planted


Sample pictures shown.




2

Learn Glossary

PearsonRealize.com


MP.1 Make Sense and PersevereInto how many equal parts is the garden divided? [6] What is the name of each part? [One-sixth] How much of the garden will be planted? [56]

MP.2 Reason QuantitativelyDoes 46 + 1

6 equal 16 + 16 +

16 + 2

6? [Yes] How

do you know? [Both expressions are equal to 56: 46 + 1

6 = 16 + 1

6 + 16 + 2

6]

Prevent MisconceptionsSome students may not understand there are several ways to decompose 56. Remind them of making ten in many different ways. Explain to students that another way to decompose 56 is to use what they know about unit fractions, 56 = 1

6 + 16 + 1

6 +16 + 1

6.

11

Convince Me! MP.5 Use Appropriate Tools Strategically Fraction strips provide a useful tool for decomposing fractions into sums. When the fraction is greater than one, the process is the same.

Revisit the essential question. Point out that, just as whole numbers can be decomposed into a sum in more than one way (e.g., 5 = 3 + 2; 5 = 1 + 2 + 2), fractional amounts can

be decomposed into a sum of fractions with the same denominators in more than one way (e.g., 56 = 3

6 + 26; 56 = 1

6 + 26 + 2

6).


MP.4 Model with Math What does 46 represent? [The fraction of the garden that is planted with blue flowers] What does 16 represent? [The fraction of the garden that is planted with red peppers] What does the sum 56 represent? [The fraction of the entire garden that is planted] How can you show this situation in an addition equation? [56 = 4

6 + 16]

472 Topic 9


20–30 min

123 inches



11. Jackie ate 15 of a bag of popcorn. She shared the rest with Enrique. List three ways they could have shared the remaining popcorn.

14. There were 45 girls and 67 boys at a sold-out performance. Each ticket to the performance costs $9. How much were all the tickets to the performance?

12. MP.4 Model with Math Draw an area model to show 4

10 + 310 + 2

10 = 910.

15. Higher Order Thinking Jason wrote 11

3 as the sum of three fractions. None of the fractions had a denominator of 3. What fractions might Jason have used?

13. In a class of 12 students, 8 students are boys. Write two equivalent fractions that tell which part of the class is boys.

16. A teacher distributes a stack of paper to 3 groups. Each group receives a different amount of paper. Select all the ways the teacher can distribute the paper by decomposing 12

3 inches. Use fraction strips if needed.

1 + 13 + 1

3

23 + 1

3 + 13

23 + 2

3 + 13

13 + 1

3 + 13 + 1

3 + 13

1 + 23

1101

101

101

10

1101

101

101

101

10

The area model shows 12 sections. Each section

is 112 of the class.

Jackie eats 15, Enrique eats 35.



Sample answer:

Sample answer: 812, 46

$1,008 Sample answer: 36 + 26 + 3

6



Tools AssessmentPractice Buddy

*


Name


318 is 1 whole + 1 whole + 1 whole + 1

8.

Each whole can also be shown as eight equal parts.

318 = 1 + 1 + 1 + 1

8

318 = 8

8 + 88 + 8

8 + 18

Another Example!

1. MP.4 Model with Math Draw a model to show one way to decompose 78.

2. Paul said the sum of 110 + 7

10 + 410 is

the same as the sum of 510 + 5

10 + 210. Is

Paul correct? Explain.3. 3

5 =□□ +□□ 35 =□□ +□□ +□□

4. 1

34 =□□ +□□ 1

34 =□□ +□□

For 3–4, decompose each fraction or mixed number in two different ways. Use drawings or fraction strips as needed.

Leveled Practice For 5–10, decompose each fraction or mixed number in two different ways. Use drawings or fraction strips as needed.

5. 46 =□□ +□□ 46 =□□ +□□ +□□ 6. 7

8 = □□ +□□ 78 = □□ +□□ +□□

7. 135 =□□ +□□ 13

5 =□□ +□□ +□□ 8. 212 =□□ +□□ 21

2 =□□ +□□ +□□

A mixed number has a whole number part and a

fraction part.

How can you decompose 318?

9. 912 = □□ +□□ 9

12 = □□ +□□ +□□ 10. 113 = □□ +□□ 11

3 = □□ +□□ +□□

Check students’ drawings; Sample answer: 78 = 3

8 + 48

Yes; Sample answer: 110 + 7

10 + 410 = 12

10 and 510 + 5

10 + 210 = 12

10.



25

15

15

15

15

44

54

24

34

16

16

16

26

36

55

212

32

33

58

25

612

22

23

28

35

212

22

13

28

35

112

12

13

38

35

712

22

13

28



PearsonRealize.com

ToolsPractice Buddy

Assessment


Another Example! What are you asked to find? [One way to decompose 31

8] Why do you replace the whole number with three groups of 88? [ 88 is another way to show 1, and three groups of 88 equal 3.]

Error Intervention: Item 4If students have trouble decomposing a mixed number,then ask: How many fourths are in one whole? [One whole is 44.] How many fourths are in 13

4? [4 fourths plus 3 more fourths is 7 fourths.]


Multi-Step Problems Page 474 Items 11 and 14; Page 476 Items 9 and 12

Item 11 Students will need to answer a hidden question before they can solve the problem. What are you asked to find? [The different ways Jackie could have shared the remainder of the bag of popcorn with Enrique] What part of the bag of popcorn did they share? [45]

Item 12 MP.4 Model with Math Area models should show one shape (rectangle, circle, etc.) divided equally into 10 pieces. Students can label or shade 4 pieces to show 4

10, 3 pieces to show 310, and 2

pieces to show 210.

Item 14 What do you need to find before you can answer the question? [How many tickets were sold.] How will that help you answer the question? [I can multiply the number of tickets sold by the cost per ticket to find the answer.]

473–474


22

15 + 1

5 + 15 + 1

5 = 45



9-2

3. There are different ways to decompose a fraction. This area model shows another way to decompose 45. Write the fractions to complete the equation.

4. There are several ways to decompose 45. This equation uses 3 addends to decompose 45. Shade the area model to show the equation.

5. You can decompose a mixed number. The area models show two ways to decompose 23

4. Complete the equations.

On the Back!

6. Use fraction strips or draw an area model to show two different ways to decompose 78. Write an equation for each model.

Vocabulary

1. To decompose a fraction means to break it into the sum of two or more parts.

The equation at the right shows one way to

decompose .

2. A mixed number, such as 234, has a whole

number part and a fraction part.

In 234, is the whole number part

and is the fraction part.

15 + 2

5 + 15 = 4

5

45

234

44

1

34

34

25 + =

25

45

Sample answer: 68 + 18 = 7

8; 38 + 48 = 7

8;Check students’ drawings.

44 + + = 23

4

1 + + = 234

R 9•2



Name Math and Science Activity9-2

Decomposing Data

Did You Know? For centuries, cultures have used “talking drums” to communicate by mimicking the patterns of human language. These drums can be used to tell news, relay stories, or create music.

The table below shows an anthropologist’s observations of 100 drum messages. Use this information to answer the questions.

Topics Fraction of Messages

Poetry 410

Announcements 7100

Celebrations 23100

Warnings 210

Commands 10100

➊ Consider the fraction of messages that are announcements. Decompose the fraction in 2 different ways.

➋ Consider the fraction of messages that are celebrations. Decompose the fraction in 2 different ways.

➌ Write two ways to decompose the fraction of messages that are commands.

➍ Decompose the fraction of messages that are warnings.

➎ Extension Decompose the fraction of messages that are poetry in as many ways as you can.4

10 = 110 + 1

10 + 110 + 1

10; 410 = 1

10 + 110 + 2

10;4

10 = 110 + 3

10

Sample answer: 7100 = 3

100 + 4100; 7

100 = 3100 + 3

100 + 1100


100 + 3100; 23

100 = 14100 + 9

100

210 = 1

10 + 110


100 + 5100; 10

100 = 2100 + 3

100 + 5100

9•2Math and Science Activity

MTH16_ANC04_CC2_T09_L02_SC.indd Page 287 02/06/14 2:29 PM taptitr2 /147/PE01513_ENG_TRM/MATH/NA/TRM/2013/G4/XXXXXXXXXX/Layout/Interior_Files/Topic_0 ...

Math and Science Activity STEM

This activity revisits the science theme, Information Transfer, introduced on page 461 in the Student’s Edition.

Sample Student Work

4.




STEP

3

Intervention Activity I Reteach I


Decomposing FractionsMaterialsCircle fraction models (Teaching Tool 14); colored pencils

•Have students work in pairs with fraction circles.

•Have one student color in the fraction circle to model a fraction. Then have the other student use two or three different colors to color in the same number of parts in their fraction circle. Both students should write how the fraction was decomposed.

•Have students switch roles and repeat the activity.

•You can extend the activity by having students decompose the fraction in two ways. Have students color two fraction circles to show different decompositions of the same fraction.

475A Topic 9


15–30 min

18

18

18

18

18

18

18

18

Trail Mix

Trail Mix

Trail Mix

Trail Mix



9. Yvonne ran 38 of the race before stopping for water. She wants to stop for water one more time before finishing the race. List two ways Yvonne can do this.

10. MP.2 Reasoning A teacher noticed 58

of the students were wearing either blue shorts or white shorts. Write two different ways this could be done.

11. Connie made 1

13 pounds of trail mix for a

hike. Is there a way Connie can break up the trail mix into four bags? Explain.

12. Jo’s Donut Express earned $4,378 at a festival by selling chocolate or vanilla donuts for $2 each. If they sold 978 chocolate donuts, how many vanilla donuts did they sell?

13. Higher Order Thinking Mark said he can decompose 56 into three fractions with three different numerators and the same denominator. Is this possible? Explain. Remember, you can use equivalent fractions.

14. Mrs. Evans asked the class to decompose 13

4. Which of the following are NOT ways to decompose 13

4? Select all that apply. Use area models if needed.

1 + 34

14 + 1

4 + 14 + 1

4 + 14 + 1

4 + 14

44 + 3

4

34 + 5

4

1 + 12 + 2

2

15. Ms. Anderson showed her class how to decompose 13

8. Select all the ways Ms. Anderson could have decomposed 13

8. Use fraction strips if needed.

38 + 3

8 + 38 + 2

8

1 + 38

88 + 3

8

58 + 5

8 + 18

1 + 14 + 2

4

Yes; Sample answer: She can put 13 pound of trail mix into each bag.

Yes; 56 = 1012; 10

12 = 212 + 3

12 + 512

Sample answer: 38 wear blue shorts

and 28 wear white shorts; 48 wear

blue shorts and 18 wear white shorts.Sample answer: Run 28, stop, and then

run 38; Run 48, stop, and then run 18.

1,211 vanilla donuts




18

18

18

18

18

1

Another Look!


NameHomework & Practice 9-2Decompose Fractions

There are more than two solutions to this problem.

For 1–8, decompose each fraction or mixed number in two different ways. Use drawings or fraction strips as needed.

Shannon wants to use 58 of her garden space to plant petunias and marigolds. How can Shannon use the available space?

Write 58 as the sum of fractions in two different ways.

58 = 1

8 + 48 5

8 = 28 + 3

8

Shannon could use 18 of the space for petunias and 48 for marigolds, or she could use 28 of the space for petunias and 38 for marigolds.

1. 48 = □□ +□□

48 = □□ +□□ +□□

2. 710 =□□ +□□

710 =□□ +□□ +□□

5. 114 =

114 =

6. 223 =

223 =

3. 45 =

45 =

4. 310 =3

10 =

7. 1

35 =

1

35 =

8. 1

12 =

1

12 =

Challenge yourself! Include ways that break a fraction or mixed number into more

than two parts.


15 + 3

525 + 1

5 + 15

55 + 3

555 + 2

5 + 15

22 + 1

212 + 1

2 + 12

210 + 1

10110 + 1

10 + 110

44 + 1

414 + 2

4 + 24

43 + 4

333 + 3

3 + 23

18

410

310

18

18

28

38

410

210

110




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Games



Tools Games


Leveled Assignment I Items 1–4, 9, 12, 14–15 O Items 2–5, 11, 13–15 A Items 6–8, 10–11, 13–15

475–476


Think

Today’s Challenge

Learn


Glossary

Animated Glossary

eText


Tools

Math Tools

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Math Games

Assessment

Quick Check

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Solve

Solve and Share

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9-3

1. There are 10 campers at Camp Davis. Three campers are swimming and 2 campers are hiking. Which fraction of the campers are swimming or hiking?

� 310

� 410

� 510

� 710

2. A pencil is 410 inch wide. Which

fraction is equivalent to 410?

� 210

� 25

� 35

� 810

3. Select all of the numbers which are a multiple of 9.

45

65

154

162

225

4. Which of the following is one way to decompose 12

3?

� 23 + 1

3 + 13

� 23 + 2

3 + 13

� 13 + 1

3 + 13

� 13 + 2

3 + 33

5. What is 7 times as much as 25?

6. Phoebe has 72 roses that she is putting into 5 vases. If she puts the same number of roses in each vase, how many roses are left over?

7. Gary’s soccer team plays 12 games in a season. Each game is 45 minutes long. His older brother, Hector, plays the same number of games, but each of his games lasts twice as long. How many minutes does Hector’s team play each season? Draw bar diagrams and write and solve equations to solve.

4.OA.A.2

4.OA.A.3

4.NBT.B.6

4.NF.B.3a

4.NF.A.1

4.OA.B.4

4.NF.B.3b

175

2 roses

1,080 minutes each

season; g = 12 × 45;

g = 540; h = 2 × 540,

h = 1,080 Check

students’ bar diagrams.

D 9•3




LESSON 9-3ADD FRACtions with Like DenominAtoRs

today’s ChallengeUse the Topic 9 problems any time during this topic.

FoCUsDomain 4.nF Number and Operations—FractionsCluster 4.nF.B Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.Content Standard 4.nF.B.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Also 4.nF.B.3d.Mathematical Practices mP.1, mP.3, mP.4, mP.7objective Solve problems involving joining parts of the same whole by adding fractions. essential Understanding Two fractions can be joined or added to find the total. There is a general method for adding fractions with like denominators. materials Number Lines (Teaching Tool 12), Fraction strips (or Teaching Tool 13)

CoheRenCeLesson 9-1 connected the meaning of fraction addition to whole number addition. Lesson 9-2 showed that a fraction ab, where a 7 1, can be decomposed in different ways. This lesson combines these ideas where fractions with like denominators are first decomposed into unit fractions. Then the unit fractions are joined to find the total. They also see patterns in the calculations to build a general method for adding fractions with like denominators.

RiGoRThis lesson blends conceptual understanding with procedural skill. The idea of joining unit fractions is used with the number line to show why the sum of two fractions with like denominators can be found by adding the numerators and writing that sum over the like denominator.

PD



speaking Express ideas aloud.

Use before the Visual Learning Bridge on Student’s Edition p. 478.

Have students describe like denominators. Remind students that like denominators means the two denominators are alike, or the same number. Write: “1

4, 23, 13.” Explain to students that 23 and 13 have like denominators: both of the denominators are the same, 3.

Beginning Make flash cards with fractions on them (no mixed numbers). Say: I’m going

to show you two fractions. If they have like denominators, say ‘like denominators.’ If they don’t have like denominators, say ‘different denominators.’

intermediate Ask students to explain and give examples of the how like and unlike denominators are different.

Advanced Ask students to work with a partner to discuss examples of things that are divided into equal parts, such as a pie with eight slices, or an orange with ten pieces. Ask them to think of ways to describe the

whole item by using 3 or more fractions with like denominators, such as “I ate 3

10 of the orange, Frank ate 3

10 of the orange, and Bonita ate 4

10 of the orange.”

summarize How are like denominators used in math to describe parts of a set or group?

Think


477A topic 9


10–15 min

Solve

115

15

15


Name

I can …

Lesson 9-3Add Fractions with Like Denominators

Look Back! MP.7 Look for Relationships What do you notice about the denominators in your equation?

You can model with math. What equation can you write to

represent this problem? Content Standards 4.NF.B.3a, 4.NF.B.3d Mathematical Practices MP.1, MP.3, MP.4, MP.7

use my understanding of addition as joining parts of the same whole to add fractions with like denominators.

Jonas is making nachos and tacos for a family party. He uses 25 bag of shredded cheese for the nachos and 15 bag for the tacos. How much of the bag of shredded cheese does Jonas use? Solve this problem any way you choose.

Sample answer: The denominators are all the same: 5.




Camila’s Work Kevin’s Work

STEP


Solve

COHERENCE: Engage learners by connecting prior knowledge to new ideas.Students formalize what they learned about adding fractions with models in the previous two lessons to add without models.


BEFORE

1. Pose the Solve-and-Share ProblemYou may wish to provide fraction strips (or Teaching Tool 13).MP.4 Model with Math Look for students who use fraction strips and an equation to represent and solve the problem.

2. Build UnderstandingWhat information are you given? [Jonas uses 25 of the bag of shredded cheese for nachos and 15 of the bag of shredded cheese for tacos.] What are you asked to do? [Find how much of the bag Jonas used.]

DURING

3. Ask Guiding Questions As NeededWhat tools can you use to help solve the problem? [Sample answer: fraction strips] What expression can you write to represent this problem? [25 + 1

5]

AFTER

4. Share and Discuss SolutionsStart with students’ solutions. If needed, project Camila’s work to discuss how to use fraction strips to help solve this problem.

5. Transition to the Visual Learning BridgeProblems involving the addition of fractions with like denominators can be solved by using models or by writing an equation.When adding fractions with like denominators, add the numerators without changing the denominator.

6. Extension for Early FinishersSuppose Jonas uses 58 of a tomato for the nachos and 38 of a tomato for the tacos. What fractional part of the tomato did Jonas use? [88 tomato or 1 whole tomato]


Camila uses fraction strips to represent the problem and writes an equation to solve the problem.

Kevin uses fraction strips and writes an answer.

Solve

Whole Class

Whole Class

Small Group

477


Visual Learning

A

Learn Glossary

B C

0 1

212= 1

12+ 112

412= 1

12+ 112 + 1

12 + 112

DA

TA

Favorite Pet

512

412

212

112


Convince Me! MP.3 Critique Reasoning Frank solved the problem above and found 2

12 + 412 = 6

24. What error did Frank make? Explain.

How Can You Add Fractions with Like Denominators?

Find 212 + 4

12 using a model.

Find 212 + 4

12 by joining parts.

Add the numerators. Write the sum over the like denominator.

212 + 4

12 = 612

2 4

212 + 4

12 = 112 + 1

12 + 112 + 1

12 + 112 + 1

12 = 612 6

12 is equivalent to 12. One half of the club members chose a hamster or a dog as

their favorite pet.

The table shows the results of a fourth-grade Pets Club survey. What fraction of the club members chose a hamster or a dog as their favorite pet?

Add the fractions for hamsters and dogs

to find the result.

Sample answer: Frank added the denominators instead of writing the sum of the numerators over the like denominator; 212 + 4

12 = 612.




2

Learn Glossary

PearsonRealize.com


MP.1 Make Sense and Persevere What fraction of the club chose a hamster as their favorite pet? [ 2

12] What fraction of the club chose a dog as their favorite pet? [ 4

12] What operation will you use to solve the problem? Why? [Addition; amounts are joined]

MP.3 Construct Arguments Why do you write the sum of the numerators over the like denominator? Use the number line model in Box B to explain. [When you add 2

12 and 412, you add

2 segments plus 4 segments on the number line. The result is 6 segments. Since the space between 0 and 1 on the number line is divided into 12 equal segments, the 6 segments are 6

12 of the whole.]

Convince Me! MP.3 Critique Reasoning If students have difficulty finding Frank’s mistake, refer back to the number line in Box B. Ask students if the space between 0 and 1 is divided into 12 or 24 equal parts. Point out that when parts of the whole are combined, the number of equal parts does not change.

Coherence It is important that students apply the generalization about adding fractions which was developed with models in this and earlier lessons. A similar generalization will apply to subtracting fractions in the following lessons. Then, the generalizations will be used to add and subtract mixed numbers later in the topic.

Revisit the essential question. Remind students that when adding fractions with like denominators, the parts of the whole are the same size, so the numerators can be added without changing the denominators to find the sum.


What does the sum 6

12 represent? [Out of 12 students, the fraction who chose either a hamster or a dog as their favorite pet]

478 Topic 9


20–30 min

DA

TA

Shapes in the Set

210

410

110

310



20. Which two shapes make up 710 of the shapes

in the set?

21. Which two shapes make up half of the set? Find two possible answers.

19. What fraction of the set is either triangles or rectangles?

22. MP.1 Make Sense and Persevere There are 64 crayons in each box. A school bought 25 boxes of crayons for the art classes. If the crayons are shared equally among 5 classes, how many crayons will each class receive? Explain.

23. Higher Order Thinking Three-tenths of Ken’s buttons are blue, 4

10 are green, and the rest are black. What fraction of Ken’s buttons are black?

For 19–21, use the table at the right.

24. Draw lines connecting each fraction or mixed number on the left with the correct expression on the right.

34 38 + 2

8 + 328

458 2

6 + 46

76 24 + 1

4

1 66 + 1

6

25. Draw lines connecting each fraction or mixed number on the left with the correct expression on the right.

1 110 1

4 + 24 + 8

4

2 110 + 0

10 + 1010

612 3

12 + 312

234 5

4 + 34

Sample answer: 610 of the set

rectangles and circles

triangles and circles; rectangles and hexagons

310

320 crayons; Sample answer: 64 ×25 = 1,600; 1,600 ÷ 5 = 320




*


Name


For 3–6, find each sum. Use drawings or fraction strips as needed.

Another Example!Find 45 + 3

5.

34

45 + 3

5 = 15 + 1

5 + 15 + 1

5 + 15 + 1

5 + 15

Write the fraction as a mixed number.

= 75

75 = 5

5 + 25 = 12

5

1. Using the survey on the previous page, what fraction of the club members chose either a bird or a cat?

2. MP.3 Critique Reasoning Greg found 13 + 2

3 = 36. What error did

Greg make?

3. 24 + 1

4 4. 13 + 2

3

5. 212 + 11

12 6. 110 + 4

10


7. 28 + 1

8 8. 36 + 2

6 9. 18 + 4

8

10. 310 + 2

10 11. 310 + 5

10 12. 512 + 4

12

13. 45 + 3

5 + 25 14. 3

10 + 210 + 6

10 15. 26 + 5

6

16. 36 + 9

6 17. 1110 + 11

10 18. 78 + 1

8

You can write the sum as a fraction or a mixed number.

Sample answer: 612


Greg added the denominators instead of using the like denominator.

34

33 or 1

1312 or 1 1

12510

38

510

56

95 or 14

51110 or 1 1

10

2210 or 2 2

10

76 or 11

6

810

912

58

126 or 2 8

8 or 1




PearsonRealize.com

ToolsPractice Buddy

Assessment


Another Example In order to change 75 to a mixed number, why do you decompose 75 to 55 + 2

5 instead of some other sum? [55 = 1, so 55 + 2

5 = 1 + 25 = 12

5]

Error Intervention: Items 3–6If students are adding the denominators as well as the numerators,then use fraction strips or drawings to show why the denominator does not change in the sum. How will you find the numerator of the fraction in the sum? [Add the numerators of the addends.] How will you find the denominator of the fraction in the sum? [Use the same denominator as the denominator of the addends.]


Multi-Step Problems Page 480 Items 22 and 23; Page 482 Item 22

Item 19 Students should recognize that the question is asking them to find a part of a whole. Why is addition the needed operation? [The fractional amounts are joined.]

Item 24 If students are having difficulty matching each fraction or mixed number on the left to the correct expression on the right, have students decompose the fraction on the left into unit fractions and then find the correct combination of addends.

479–480


22



9-3

3. To find the sum of 310 + 2

10, decompose, or take apart, one or both fractions.

310 can be written as + + .

210 can be written as + .

So, 310 + 2

10 can be written as

+ + + + .

310 + 2

10 =

4. To find the sum of 28 + 58, decompose, or take apart,

one or both fractions.

28 can be written as + .

58 can be written as + + + + .

So, 28 + 58 can be written as

+ + + + + + .

28 + 5

8 =

On the Back!

5. Find the sum of 18 + 58 by decomposing the fractions.

Vocabulary

1. Fractions have a like denominator when the bottom number of both fractions is the same.

The like denominator for 310 and 2

10 is .

2. Two fractions that represent the same part of the whole are equivalent fractions.

Divide to find a fraction equivalent to 510. 5

10 = 510 ,□□ = □□

5 15 2

18 + 1

8 + 18 + 1

8 + 18 + 1

8 = 68

10

110

110

1818

18

18

18

78

18

18

18

18

18

18

18

18

18

110

110

110

110

1105

10

110

110

110

R 9•3



PartnerTalk

Toss and Talk


Play again!

If you have more time

How to Win

You win if you are the first to get four connected rectangles, like:

Toss Explain how to compute the sum. Find the answer.

2 510 + 2

10

3 58 + 3

8

4 412 + 3

12

5 210 + 4

10

6 26 + 5

6

116

912 12

51

58

710

1112

912

712

610

1012

810

116

1112

58

710

7 212 + 4

12 + 312

8 38 + 2

8

9 312 + 8

12

10 45 + 3

5

11 510 + 3

10

12 212 + 3

12 + 512

Get Started or

Get 10 squares in one color and 10 in another color. Get two number cubes. Take turns with another player or team. Talk about math as you play!

At Your Turn

Toss two number cubes. Add the dots. Find your toss below. Follow the directions. Explain your thinking. Cover the answer. If the answer is taken, lose your turn. Have fun!

9•3

MTH16_ANC04_CC2_T09_L03_CG.indd Page 290 02/06/14 2:28 PM taptitr2 /147/PE01513_ENG_TRM/MATH/NA/TRM/2013/G4/XXXXXXXXXX/Layout/Interior_Files/Topic_0 ...


PartnerTalk

Toss and Talk


Play again!


How to Win


Toss Explain how to compute the sum. Find the answer.

2 310 + 4

10 + 310

3 13 + 2

3 + 43

4 28 + 2

8 + 38

5 24 + 2

4 + 34

6 45 + 3

5 + 35

3 1 112 21

6

3 225

78 13

4

213 11

2 156 21

6

134

2 225 2 2

10

7 12 + 1

2 + 12

8 36 + 5

6 + 56

9 35 + 6

5 + 35

10 610 + 8

10 + 810

11 128 + 10

8 + 28

12 46 + 4

6 + 36

Get Started or


At Your Turn


9•3


Advanced On-LevelCenter GamesStudents work in pairs or small groups to find the sums of fractions with like denominators and explain their thinking. Have students record their equations as they play the game.




STEP

3



Adding Fractions with Like DenominatorsMaterialsCircle fraction models (Teaching Tool 14); colored pencils

•Have students work in pairs with circle fraction models. One student chooses two fractions with the same denominator. That student should color in the circle fraction model, using a different color for each fraction.

•The other student then writes an addition sentence and finds the sum of the fractions.

•Have students switch roles and repeat the activity if time allows.

481A Topic 9


15–30 min

Think about how you can rewrite fractions so they have like denominators.

DA

TA


24. In Martha’s pet store, 66 of the hamsters are brown, 36 of the mice are white, 26 of the fish are blue, and 56 of the birds are yellow. Draw lines connecting each fraction with the correct expression.


19. What fraction of the students voted for fruit juice or soda?

23. Higher Order Thinking How can you add 3

10 and 25? Explain.22. MP.1 Make Sense and Persevere

A bus traveled 336 miles in 7 hours. It traveled the same number of miles each hour. If the bus continues at the same number of miles per hour, how many miles will the bus travel in 15 hours? Explain.

20. Which two beverages have a sum of 58 of the student votes?

21. What combination of beverages makes up 68 of the student votes?


Favorite Beverage

Fraction of Student Votes

Iced Tea 38

Fruit Juice 28

Water 18

Soda 28

512 0

12 + 112

212 1

12 + 212 + 1

12

412 3

12 + 212

112 1

12 + 112

66 26 + 0

6

36 36 + 1

6 + 16

26 26 + 1

6

56 16 + 1

6 + 16 + 3

6

25. In Fred’s catering order, 512 of the lunches

are sandwiches, 212 are salads, 4

12 are pastas, and 1

12 are soups. Draw lines connecting each fraction with the correct expression.

Sample answer: I can write 25 as the equivalent fraction 4

10. Then the two fractions will have like denominators and they can be added.

Sample answer: 48

iced tea and fruit juice or iced tea and soda

iced tea, fruit juice, water or iced tea, water, soda

720 miles; Sample answer: 336 ÷ 7 = 48; 48 × 15 = 720




0 1

28 = 1

8 + 18

48 = 1

8 + 18 + 1

8 + 18

Another Look!


Name


Find 48 + 28.

48 + 2

8 = 68

Homework & Practice 9-3Add Fractions with Like Denominators

1. 13 + 1

3 2. 310 + 6

10 3. 512 + 2

12

4. 312 + 7

12 5. 510 + 3

10 6. 28 + 4

8

7. 710 + 3

10 8. 18 + 6

8 9. 110 + 5

10

13. 15 + 2

5 + 45 14. 2

8 + 18 + 12

8 15. 26 + 10

6

10. 45 + 1

5 11. 28 + 6

8 12. 610 + 0

16. 20100 + 25

100 + 25100 17. 2

10 + 610 + 1

10 18. 1010 + 10

10 + 1010

When you add fractions with like denominators, add the numerators and keep the

denominators the same.


23

712

810

1010 or 1 6

10

610

1012

70100

68

78

75 or 12

5158 or 17

8

55 or 1 8

8 or 1

126 or 2

3010 or 3

910

910




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Tools Games


Leveled Assignment I Items 1–6, 19–20, 24–25 O Items 7–12, 20, 22–25 A Items 13–18, 21–25

481–482

MTH16_TE04_CC2_T09_L03.indd Page 481 19/06/14 1:38 PM sw-102 /151/PE01513_TE_1of1/ENVISION_MATH_ENGLISH/NA/TE/2013/G4/XXXXXXXXXX/Layout/Interi ...



9-4Name

1. Karl has 10 shirts. Four shirts are green and 1 is blue. What fraction of Karl’s shirts are green or blue?

𝖠𝖠𝖠𝖠𝖠 110

𝖡𝖡𝖡𝖡𝖠 410

𝖢𝖢𝖢𝖢𝖠 510

𝖣𝖣𝖣𝖣𝖠 610

2. Jack wrote this equation on the board, but Kim erased part of it.

312 + □

12 = 712

Which is the value of the missing numerator?

𝖠𝖠𝖠𝖠𝖠 4

𝖡𝖡𝖡𝖡𝖠 5

𝖢𝖢𝖢𝖢𝖠 10

𝖣𝖣𝖣𝖣𝖠 12

3. The outline of the shape of a unique city park is shown below. Each side is the same length. What is the length of a fence that encloses the entire park?

40 feet

𝖠𝖠𝖠𝖠𝖠 160 feet

𝖡𝖡𝖡𝖡𝖠 200 feet

𝖢𝖢𝖢𝖢𝖠 240 feet

𝖣𝖣𝖣𝖣𝖠 280 feet

4. What is the sum of 28 + 18 + 3

8?

5. Tyrone completed 38 of a report on Wednesday. Then he completed 12 of the report on Thursday. Did Tyrone complete more of his report on Wednesday or Thursday? Explain.

6. Josie read 100 pages of her book last week and 146 pages this week. Her older brother says he read about 3 to 4 times as many total pages as Josie. Explain why 2,500 is NOT a reasonable estimate for the number of pages that Josie’s brother read.

Sample answer: 68

Sample answer: 2,500 is

too great of an estimate.

100 + 146 = 246 which

can be rounded to 250.

250 × 3 = 750

250 × 4 = 1,000

Thursday; 12 = 48 and

48 +

38.

4.NF.B.3a

4.NF.A.2

4.OA.A.3

4.NF.B.3a

4.NF.B.3a

4.NBT.B.5

D 9•4



Think

Today’s Challenge

Learn


Glossary

Animated Glossary

eText


Tools

Math Tools

Games

Math Games

Assessment

Quick Check

PD


Solve

Solve and Share

Help



Practice Buddy


LESSON 9-4MoDel SubtRaCtion of fRaCtionS


foCuSDomain 4.nf Number and Operations—FractionsCluster 4.nf.b Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.Content Standard 4.nf.b.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.Mathematical Practices MP.1, MP.2, MP.4, MP.5, MP.6objective Use tools such as fraction strips, area models, and number lines to subtract fractions.essential understanding Similar to whole number subtraction, one way to think about subtraction of fractions is to separate a part from a whole. The algorithm for subtracting fractions with like denominators is not the same as the algorithm for subtracting whole numbers.

Materials Number lines (Teaching Tool 12), Fraction strips (or Teaching Tool 13), Circle fraction models (Teaching Tool 14)

CoHeRenCeLesson 9-2 developed the idea that a fraction ab, where a 7 1, can be decomposed into unit fractions, 1b. In this lesson subtraction of fractions ab - c

b, where a 7 1, is given meaning by first decomposing ab into a unit fractions, 1b, and then separating c unit fractions from the total. The number of unit fractions remaining is the difference between the two fractions.

RiGoRThis lesson emphasizes conceptual understanding. It brings together students’ understanding of decomposing a fraction into unit fractions and their understanding of subtraction of whole numbers as separating a part from a whole.

PD



Reading Use contextual support to develop vocabulary.

Use with the Solve & Share on Student’s Edition p. 483.

Write expression and have students read the word. Explain that in math, an expression is a way of saying something with numbers and operation symbols. It does not include an equal sign or comparison symbol. For example, 45 - 2

5 is an expression.

beginning Use an area model to demonstrate subtraction. Draw a rectangle

divided into 6 equal parts. Ask a student to shade 4 parts. Ask: What part of the area model is not shaded? Write: “6

6 46.” Ask:

Which symbol should I use to show taking away? [ - ] How is that symbol read? Ask students to read the expression.

intermediate Draw a rectangle divided into 6 equal parts. Ask: What fraction represents the whole rectangle? Ask a student to shade 4 parts to show taking away. Ask: What fraction represents the part taken away? Guide students as they write expressions and read them to a partner.

advanced Have student partners take turns using fractions to write and read addition and subtraction expressions. One partner reads the expression, and the other partner uses drawings to represent it. Partners discuss whether the drawing accurately reflects the expression.

Summarize What information do you get when you read a math expression?

Think


483A topic 9


10–15 min

Solve


Name

I can …

Lesson 9-4Model Subtraction of Fractions

Look Back! MP.6 Be Precise Explain why 46 is subtracted from 66 to find how much of the plywood is left.

You can select tools such as fraction strips, drawings,

or area models to solve this problem. Show your work in

the space below!


use tools such as fractions strips or area models to subtract fractions with like denominators.

Mr. Yetkin uses 46 of a sheet of plywood to board up a window. How much of the plywood is left? Solve this problem any way you choose.

Sample answer: 66 is one whole. Mr. Yetkin uses 46 of one whole sheet of plywood. To find the amount of plywood left, subtract 46 from 66.




STEP


Solve

COHERENCE: Engage learners by connecting prior knowledge to new ideas.Students use tools to model subtraction as separating parts of the same whole as they used the tools to model addition as combining parts of the same whole.


BEFORE

1. Pose the Solve-and-Share ProblemYou may wish to provide fraction strips (or Teaching Tool 13).MP.5 Use Appropriate Tools Strategically In this problem, look for students who select and use an appropriate tool to show how much of the plywood is left.

2. Build UnderstandingWhat are you asked to do? [Find how much of the sheet of plywood is left after 46 of the sheet is used to board up a window.] What tools can you use? [Sample answer: A fraction strip divided into sixths]

DURING

3. Ask Guiding Questions As NeededWhat is another name for the whole? [66] How can you use fraction strips to show subtraction? [Sample answer: Using a fraction strip divided into sixths, I can cross off 46 of the strip to represent the part of the sheet of plywood that was used. The part left is the difference.]

AFTER

4. Share and Discuss SolutionsStart with students’ solutions. If needed, project Marco’s work to discuss how to model and write the equation.

5. Transition to the Visual Learning BridgeYou can use what you know about adding fractions with like denominators to subtract fractions with like denominators.When subtracting fractions with like denominators, the numerators can be subtracted without changing the denominator.

6. Extension for Early FinishersOn the board, write (1 - 1

5) - 25. Use fraction strips to find the

answer to this subtraction problem. [25]


Solve

Whole Class

Whole Class

Small Group

Marco’s Work Sarah’s Work

Marco draws fraction strips to model and help write the equation. Sarah uses an area model to help her solve the problem.483


Visual Learning

A

Learn Glossary

B C

18

18

18

18

18

18

1

18

18

Another Way


Convince Me! MP.5 Use Appropriate Tools In the problem above, suppose six sections of the garden are used for yellow roses and two other sections are used for petunias. How much more of the garden is used for yellow roses than is used for petunias? Use fraction strips or another tool to help. Write your answer as a fraction.

A flower garden is divided into eighths. If 28 of the garden is used to grow yellow roses, what fraction is left to grow other flowers?

Six eighths of the garden is left to grow other flowers.

Find 88 - 28.

Separating the 28 from 88 leaves 68.

88 - 2

8 = 18 + 1

8 + 18 + 1

8 + 18

+ 18 + 1

8 + 18 = 6

8

Find 88 - 28.

88 = 1

8 + 18 + 1

8 + 18 + 1

8 + 18 + 1

8 + 18

28 = 1

8 + 18

Separating the 28 from 88 leaves 68.

88 - 2

8 = 18 + 1

8 + 18 + 1

8 + 18 + 1

8 + 18 + 1

8 = 68

How Can You Use Tools to Subtract Fractions?

0 1

88

18

18= + 1

8+ 18+

28

18

18= +

18+ 1

8+ 18+ 1

8+

One Way

You can use tools such as fraction strips to

represent subtraction.

68 − 2

8 = 48; 48 more of the garden is used for yellow roses than is

used for petunias.




2

Learn Glossary

PearsonRealize.com

The Visual Learning Bridge connects students’ thinking in Solve & Share to important math ideas in the lesson. Use the Visual Learning Bridge to make these ideas explicit. Also available as a Visual Learning Animation Plus at PearsonRealize.com

MP.1 Make Sense and PersevereWhat fraction can you use to represent the whole garden? [88] What fraction represents the part of the whole garden used for yellow roses? [28]

MP.2 Reason QuantitativelyWhy do you take away two of the 18 strips? [To find the fraction of the garden left over for other flowers]

MP.5 Use Appropriate Tools StrategicallyHow is using a number line like using fraction strips? [With both tools, you start with 88, take away 28, and see that you have 68 left.]Prevent MisconceptionsAsk students if the number of equal parts changes after taking away 28. Pointing out that the result is 68 and the denominator is still eighths will help students make the transition to subtracting fractions symbolically in later lessons without subtracting the denominators.

11

Convince Me! MP.5 Use Appropriate Tools Strategically Remind students of the subtraction concepts they learned in kindergarten and first grade. The problem in the Visual Learning Bridge involves taking away or separating parts of the whole. The problem in the Convince Me! involves comparison subtraction. Have students use fraction strips to show the two types of problems.

Revisit the essential question. Point out that when subtracting fractions with like denominators (such as subtracting 28 from 68), they are subtracting pieces or portions of the same size.


484 Topic 9


20–30 min

18

18

18

18

1

18

18

18



15. MP.4 Model with Math What subtraction problem did Miles show using the fraction strips below?

17. In Kayla’s class, some of the students are wearing blue shirts. 68 of the students are NOT wearing blue shirts. What fraction of the students are wearing blue shirts? Show your work.

19. Rick shared his bag of grapes with friends. He gave 2

10 of the bag to Melissa and 4

10 of the bag to Ryan. What fraction of the bag of grapes does Rick have left? Show your work.

18. MP.2 Reasoning In Exercise 17, what number represents the whole class? How do you know what fraction to use to represent this number?

20. Higher Order Thinking Teresa gave away 8 baseball cards and has 4 baseball cards left. Write a subtraction problem to show the fraction of the baseball cards Teresa has left.

16. Using only odd numbers for numerators, write two different subtraction problems that have a difference of 12. Remember, you can find equivalent fractions for 12.

21. Audry wrote a subtraction problem that has a difference of 13. Which problem did Audry write?

𝖠𝖠𝖠𝖠 22 - 1

2

𝖡𝖡𝖡𝖡 53 - 3

3

𝖢𝖢𝖢𝖢 43 - 3

3

𝖣𝖣𝖣𝖣 53 - 1

3

22. Kinsey wrote a subtraction problem that has a difference of 10

8 . What problem did Kinsey write?

𝖠𝖠𝖠𝖠 208 - 10

8

𝖡𝖡𝖡𝖡 810 + 2

10

𝖢𝖢𝖢𝖢 108 - 4

8

𝖣𝖣𝖣𝖣 68 - 1

4

Sample answer: 88 − 6

8 = 28 of the students

Sample answer: 210 + 4

10 = 610;

1010 − 6

10 = 410 of the bag of grapes

1 represents the whole class. Since the fraction of the class not wearing blue shirts is 68, in order to have a like denominator, I used 88 to represent 1 whole.


12 = 412 of the cards left

Sample answer: 78 − 38 = 4

8


4 = 24 = 1

2; 118 − 7

8 = 48 = 1

2




*

113

13

1112

112

112

112

112

112

112

112

112

112

112

0 1

22

12

12= +

12


Name

*For another example, see Set B on page 533.

Find 118 - 2

8.

Use eleven 18-fraction strips to show 11

8 . Take 2 strips away.

118 - 2

8 = 98

98 = 8

8 + 18 = 11

8

Another Example!

1 118

18

18

18

18

18

18

18

18

18

18

18

1. MP.2 Reasoning In the problem at the top of the previous page, suppose one other 18 section was used to grow peonies. What fraction of the garden is now available for flowers?

For 2–5, use fraction strips or other tools to subtract.

2. 13 - 1

3 3. 55 - 2

5

4. 712 - 3

12 5. 78 - 1

8

Leveled Practice For 6–14, find each difference. Use fraction strips or other tools as needed.

6. 1112 - 5

12 7. 22 - 1

2 8. 23 - 1

3

12. 96 - 1

6 13. 2110 - 1

10 14. 15 - 1

5

9. 45 - 2

5 10. 1710 - 3

10 11. 86 - 2

6

You can write the difference as a fraction or

a mixed number.

68 − 1

8 = 58 of the garden

Sample answers given.0 3

5

68

412


612

12

13

25

1410 or 1 4

1066 or 1

86 or 12

62010 or 2 0



PearsonRealize.com

ToolsPractice Buddy

Assessment

QUICK CHECKCheck mark indicates exercises for prescribing differentiation on the next page. Items 13 and 21 are worth 1 point. Item 20 is worth up to 3 points.

Another Example How is this problem different than the one on the previous page? [You start with more than one whole.] How is it the same? [You start with a number of eighths, take some eighths away, and see how many eighths are left.]

Error Intervention: Items 2–5If students try to subtract symbolically and make mistakes such as subtracting the denominators,then encourage them to use fractions strips, draw an area model, or draw a number line.

11 Reteaching Assign Reteaching Set B on p. 533.

Multi-Step Problems Page 486 Item 19; Page 488 Items 13–14 and 16

Item 17 If students are not sure how to start the problem, ask: What fraction would represent the whole of Kayla’s class? [88] What operation is needed to solve this problem? Why? [Subtraction; the total and one part are known.]

Item 19 Coherence Subtracting fractions with like denominators uses the concepts of subtracting whole numbers and of adding fractions with like denominators.

Item 20 Higher Order Thinking If students have difficulty writing a subtraction problem, have them write a number sentence about the total number of baseball cards Teresa has rather than the fraction. Ask: If Teresa gave away 8 cards and has 4 left, how many did she have to start? [12] What subtraction sentence with whole numbers can you write to represent this situation? [12 - 8 = 4] What subtraction sentence can you write to represent what fraction of the cards Teresa has left? [12

12 - 812 = 4

12]

485–486


22



9-4

3. An area model can be used to model the subtraction of fractions.

Find 1112 - 7

12.

Step 1: Shade the area model to show the greater fraction.

Step 2: Cross out the number of parts equal to the lesser fraction.

Step 3: Count the remaining shaded parts to find the difference.

4. Use the area model to find 58 - 38.

58 - 3

8 =

5. Use the area model to find 64 - 34.

64 - 3

4 =

6. Use the area model to find 73 - 23. You can write

the difference as a fraction or a mixed number.

73 - 2

3 = or

On the Back!

7. Use fraction strips or draw an area model to find 710 - 3

10.

Vocabulary1. In a fraction, the numerator can be

less than, equal to, or greater than the denominator. Write an example of each.

2. A mixed number has a whole number part and a fraction part. Write three examples of mixed numbers.

less equal greater

1112 - 7

12 =



23

123

123

53

34

28

33

245

43

1058

410; Check students’ drawings.

412

R 9•4

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PartnerTalk

Clip and Cover


Get Started or

Get 10 squares in one color and 10 in another color, two paper clips, two number cubes, and fraction strips or other tools. Take turns.

At Your Turn

Toss two cubes to find your ovals. EXAMPLE: Choose the 3rd oval on the left and the 5th oval on the right, or choose the 5th oval on the left and the 3rd oval on the right. Mark your ovals with paper clips.

How to Play

Explain how to find each difference. Use fraction strips or other tools to model each problem. Find and cover the difference. Lose your turn if the answer is taken.

How to Win The first player or team to get any three connected rectangles in a row or column wins.


212

812

712

112

512

112

712

312

312

512

212

412

212

612

312

1112

512

912

112

412

512

912

612

1212

712

812

1212

712

9•4



PartnerTalk

Clip and Cover


Get Started or

Get 10 squares in one color and 10 in another color, one paper clip, one number cube, and models to help solve. Take turns.

At Your Turn

Toss one cube to find your ovals. EXAMPLE: Choose the 2nd oval on the left, or choose the 2nd oval on the right. Mark your oval with a paper clip.

How to Play

The number you chose is a difference. Find two fractions that you can subtract to get that difference. Cover your answer. Lose your turn if the answer is taken.

How to Win The first player or team to get any three connected rectangles in a row or column wins.


712 — 3

1258 — 1

856 — 2

6710 — 4

10

310 — 0

1034 — 2

455 — 2

51112 — 2

12

910 — 6

1078 — 1

81012 — 8

12910 — 5

10

78 — 4

878 — 3

824 — 1

468 — 3

8

35

14

412

410

68

14

310

912

48

38

212

36

9•4





STEP

3



Modeling Subtraction of FractionsMaterialsFraction strips (or Teaching Tool 13)

•Have students use fraction strips to model 78 - 3

8. What is the difference? [48]

•Continue modeling subtraction with other fractions. Have students draw pictures of their models.

Advanced On-LevelCenter GamesStudents work in pairs or small groups to subtract fractions with like denominators. Have students record their equations as they play the game.

487A Topic 9


15–30 min

110

110

110

1

110

110

110

110

110

110

110

0 1



You can use fraction strips to help solve

the problem.

13. Eddie noticed that out of 10 students, one student was wearing brown shoes, and seven students were wearing black shoes. What fraction of students were NOT wearing brown or black shoes?

14. MP.1 Make Sense and Persevere A marathon is a race that covers about 26 miles. Cindy ran 5 miles before taking her first water break. Then she ran another 7 miles to get to her next water break. After 6 more miles, she took her last water break. About how much farther does Cindy have until she reaches the finish line?

15. Algebra Jeffrey has already run 38 of the race. What fraction of the race does Jeffrey have left? Write and solve an equation.

16. Higher Order Thinking Rob’s tablet is fully charged. He uses 1

12 of the charge playing games, 5

12 of the charge reading, and 3

12 completing homework. What fraction of the charge remains on Rob’s tablet?

17. Roger found he had 25 of his quarters left to use at the arcade. Which of the following subtraction problems could NOT be used to find the fraction of quarters Roger had left?

𝖠𝖠𝖠𝖠 45 - 2

5

𝖡𝖡𝖡𝖡 36 - 1

2

𝖢𝖢𝖢𝖢 35 - 1

5

𝖣𝖣𝖣𝖣 55 - 3

5

18. Krys has 23 of her homework finished. Which of the following does NOT have a difference of 23?

𝖠𝖠𝖠𝖠 73 - 3

3

𝖡𝖡𝖡𝖡 43 - 2

3

𝖢𝖢𝖢𝖢 33 - 1

3

𝖣𝖣𝖣𝖣 93 - 7

3

About 8 miles

Sample answer: 210 of the students

88 − 3

8 = r ; r = 58 of the race Sample answer: 3

12


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Another Look!

Step 1 Step 2 Step 3 Step 1 Step 2 Step 3


NameHomework & Practice 9-4Model Subtraction of Fractions

For 1–12, find each difference. Use fraction strips or other tools as needed.

Kimberly cut a pizza into 10 equal slices. She ate two of the slices. What fraction of the pizza is left? Remember, 10

10 = 1 whole pizza.

Divide a circle into tenths to show the pizza cut into 10 slices.

Take away the 2 slices or 2

10 of the pizza that Kimberly ate.

Count the remaining slices and write the subtraction.

1010 - 2

10 = 810

810 of the pizza is left.

1. 35 - 2

5 2. 710 - 3

10 3. 44 - 2

4

4. 810 - 5

10 5. 66 - 3

6 6. 1112 - 7

12

7. 56 - 2

6 8. 48 - 2

8 9. 1112 - 8

12

10. 98 - 2

8 11. 244 - 18

4 12. 3010 - 20

10

0 1

35

15

15= + 1

5+

25

15

15= +

101

101

101

101

101

101

101

1


15

36

310

78

410

64 or 12

4

28

36

24

1010 or 1

312

412


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PearsonRealize.com


Games



Tools Games


Leveled Assignment I Items 1–6, 14–15, 17–18 O Items 1–6, 13, 16–18 A Items 7–12, 14, 16–18

487–488


Think

Today’s Challenge

Learn


Glossary

Animated Glossary

eText


Tools

Math Tools

Games

Math Games

Assessment

Quick Check

PD


Solve

Solve and Share

Help



Practice Buddy



9-5Name

The diagram shows David’s campsite, the park ranger’s cabin, and Badger Lake at Hundred Pines State Park.

18

mile

58

mile

David’sCampsite

Ranger’sCabin

BadgerLake

1. Select all the true statements about the distances.

18 6 5

8

58 7 1

8

58 6 4

8

18 7 4

8

48 7 1

8

2. David hikes from his campsite to the ranger’s cabin and then back to his campsite. How far does David hike?

� 48 mile

� 38 mile

� 28 mile

� 18 mile

3. How far is it from the ranger’s cabin to Badger Lake?

� 18 mile

� 28 mile

� 38 mile

� 48 mile

4. Find the sum.1

12 + 412

5. What is 85 + 45? Write your answer as

a fraction and as a mixed number.

6. In a survey, 410 of the students voted

for less homework, while 210 of the

students voted for fewer tests. What fraction of the students voted either for less homework or fewer tests?

7. Draw fraction strips to show how to add 26 + 3

6. Explain how your drawing shows the sum.

8. James won 7 out of 10 games. What fraction of games did James lose?

512

310

Check students’ drawings;16 + 1

6 = 26

16 + 1

6 + 16 = 3

616 + 1

6 + 16 + 1

6 + 16 = 5

6

Sample answer: 610

125 ; 22

54.NF.A.2

4.NF.B.3d

4.NF.B.3d

4.NF.B.3a

4.NF.B.3a

4.NF.B.3d

4.NF.B.3a

4.NF.B.3d

D 9•5

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LESSON 9-5SubtRaCt FRaCtionS with Like DenominatoRS


FoCuSDomain 4.nF Number and Operations—FractionsCluster 4.nF.b Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.Content Standard 4.nF.b.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Also 4.nF.b.3d.Mathematical Practices mP.2, mP.3, mP.4objective Solve problems involving separating parts of the same whole by subtracting fractions.essential understanding The difference between two fractions with like denominators can be found by separating one fractional amount from the other. There is a general method for subtracting fractions with like denominators.materials Number lines (Teaching Tool 12)

CoheRenCeIn the previous lesson students subtracted fractions with like denominators by decomposing the minuend (the first number) into unit fractions and then separating from that the subtrahend (the second number). In this lesson students use the relationship between addition and subtraction to find the difference of two fractions with like denominators. They also see patterns in the calculations to build a general method for subtracting fractions with like denominators.

RiGoRThis lesson blends conceptual understanding with procedural skill. The idea of decomposing the minuend into the sum of two fractions is used to find fraction differences. This work together with the work in the previous lesson leads students to understand why the difference of two fractions with like denominators can be found by subtracting the numerators and writing that difference over the like denominator.

PD



Listening Learn vocabulary that is heard.Use before the Visual Learning Bridge on Student’s Edition p. 490.Write “difference,” and ask students to listen as it is read aloud. Have students read it. Explain that in math, the word difference means the answer to a subtraction problem. Write: “7

8 - 48 = 3

8.” The solution, 38, is the difference of 78 and 48. beginning Students write 18 through 88,- , and = on 10 cards. Students listen as you

say: 58 is the difference of 68 and 18. Students use the cards to model the equation they hear. Monitor students and discuss any misunderstandings. Repeat using other fractions.

intermediate Students listen as you say: 58 is the difference of 68 and 18. Students write the equation they hear. Monitor students and discuss any misunderstandings. Repeat using other fractions.

advanced Partners write 18 through 88, - , and = on 10 cards. One partner uses the cards to make an equation. He or she reads the equation as the other partner writes what he or she hears. Partners compare their equations and discuss any differences.

Summarize What does the word difference mean in math?

Think


489A topic 9


10–15 min

Solve


Name

I can …

Lesson 9-5Subtract Fractions with Like Denominators

Look Back! MP.3. Critique Reasoning Sarah wrote the expression 8

10 - 510 to solve the problem. Jared wrote the expressions 10

10 - 810,

1010 - 5

10, and 510 - 2

10 to solve the problem. Who is correct? Explain.

You can model with math. What expression can you use to

represent this problem? Content Standards 4.NF.B.3a, 4.NF.B.3d Mathematical Practices MP.2, MP.3, MP.4

use my understanding of subtraction as separating parts of the same whole to subtract fractions with like denominators.

Leah and Josh live the same direction from school and on the same side of Forest Road. Leah’s house is 8

10 mile from school. Josh’s house is 5

10 mile from school. How much farther does Leah have to walk home when she reaches Josh’s house? Solve this problem any way you choose.

Sample answer: Both are correct. Sarah finds the distance between Leah and Josh’s houses. Jared finds the distance each student walks home, then subtracts to find the difference between the distances.




STEP


Solve

COHERENCE: Engage learners by connecting prior knowledge to new ideas.Students formalize what they learned about subtracting fractions with models in the previous lesson to subtract without models.


BEFORE

1. Pose the Solve-and-Share ProblemYou may wish to provide number lines (Teaching Tool 12) or fraction strips (or Teaching Tool 13).MP.4 Model with Math In this problem, look for students who subtract fractions to find how much distance is remaining.

2. Build UnderstandingWhat information are you given? [Leah’s house is 8

10 mile from school. Josh’s house is 5

10 mile from school.] What are you asked to do? [Find how much farther Leah has to walk home when she reaches Josh’s house.]

DURING

3. Ask Guiding Questions As NeededWhat operation will you use to find how much farther Leah has to walk? Why? [Subtraction; I need to find the difference of the total and the part.] What tools can you use? [Sample answers: Fraction strips, number lines, paper and pencil]

AFTER

4. Share and Discuss SolutionsStart with students’ solutions. If needed, project Ashley’s work to discuss how to write an equation.

5. Transition to the Visual Learning BridgeSubtracting fractions with like denominators is similar to adding fractions with like denominators.When subtracting fractions with like denominators, subtract the numerators without changing the denominator.

6. Extension for Early FinishersJoy buys a large carton of juice. She drinks 18 carton each day. How much juice will Joy have left after three days? [58]


Solve

Whole Class

Whole Class

Small Group

Ashley’s Work Tyler’s Work

Ashley uses fraction strips to represent the subtraction equation she writes.

Tyler subtracts the numerators, but incorrectly subtracts the denominators.

489


Visual Learning

A

Learn Glossary

B C

1 cup

12

14

34 5

8

38

Another WayOne Way


Convince Me! MP.2 Reasoning In the problem above, suppose Tania decided to double the amount of lemonade she wants to make. Then how much more lemon juice would Tania need to squeeze?

Tania is squeezing lemons to make lemonade. The recipe calls for 58 cup of lemon juice. The amount Tania has squeezed is shown at the right. What fraction of a cup of lemon juice does Tania still need to squeeze?

How Can You Subtract Fractions with Like Denominators?

Subtract the fractions to find the difference.

28 is equivalent to

14.

Tania needs to squeeze 14 cup more lemon juice.

Find 58 - 38 using a general method.

58 - 3

8 = n

Subtract the numerators. Write the difference over the like denominator.

58 - 3

8 = 5 - 38 = 2

8

Find 58 - 38 using the relationship

between addition and subtraction.

Break 58 apart. Write a related addition equation: 58 = 2

8 + 38

Write a related subtraction equation: 58 - 3

8 = 28

38 cup

cup

n38

58

0 138

18

18= + 1

8+

28

18

18= +

58 + 5

8 = 108 ; 10

8 − 38 = 7

8Tania would need to squeeze 78 cup more lemon juice.




2

Learn Glossary

PearsonRealize.com

The Visual Learning Bridge connects students’ thinking in Solve & Share to important math ideas in the lesson. Use the Visual Learning Bridge to make these ideas explicit. Also available as a Visual Learning Animation Plus at PearsonRealize.com

MP.2 Reason QuantitativelyHow do you know that you are being asked to subtract to solve this problem? [Since I know how much lemon juice is called for in the recipe and how much Tania has so far, I have to find how much more she needs, so I subtract.]

MP.7 Use StructureWould it help to decompose 58 as 18 + 4

8? Why or why not? [No; Since Tania has squeezed 38 cup of lemon juice, you need to decompose 58 so one of the addends is 38.]

MP.4 Model with MathWhat does the bar diagram show? [The whole, 58 cup; the known part, 38 cup; and the unknown part, how much more is needed]

Prevent MisconceptionsEmphasize the similarity between the generalization of how to subtract fractions with like denominators to the generalization for how to add fractions with like denominators. The denominator stays the same in both. This can help prevent students from subtracting the denominators.

11

Convince Me! MP.2 Reason Quantitatively Another way this problem can be solved is by using the answer given in Box C above. If Tania wants to double the amount, she needs to squeeze 58 cup more of juice. She needed 28 more, so she needs 28 + 5

8 = 78 more juice.

Coherence Continue to emphasize that adding and subtracting fractions involve combining or separating parts of the same whole. The number of parts of the whole is the same after combining or separating them, so the denominator stays the same for the sum or difference.

Revisit the essential question. Point out that the algorithm for subtracting fractions with like denominators is similar to the algorithm for adding fractions with like denominators. To

subtract fractions with like denominators, subtract the numerators and write the difference as the new numerator over the like denominator.


490 Topic 9


20–30 min

Denmark

Finland

Iceland

Norway

Sweden



19. Joey ran 14 mile in the morning and 14 mile farther than in the morning in the afternoon. If he wants to run a full mile, how much more does Joey have to run? Write equations to explain.

20. MP.2 Reasoning Write to explain how subtracting 45 - 3

5 is similar to subtracting 4 - 3.

21. Higher Order Thinking The flags of all 5 Nordic countries are displayed. What fraction describes how many more of the flags displayed are 2-color flags than are 3-color flags?

22. A piece of chalk is 910 centimeter long.

Brian breaks off a piece 610 centimeter

long. How long is the piece of chalk that is left? Write and solve an equation.

23. Marietta baked a chicken pot pie. She serves 23 of the pie at dinner. How much of the pie remains? Write and solve an equation.

n

centimeter910

610

First find how many flags in all, then find how many 2-color and

3-color flags.

14 + (14 + 1

4) = 34;

1 − 34 = 4

4 − 34 = 1

4 mile more

910 − 6

10 = 310 centimeter 3

3 − 23 = 1

3 of the pie

15 more 2-color flags

Sample answer: To subtract fractions with like denominators, subtract the numerators and write the difference over the like denominator. Subtracting whole numbers is the same as subtracting fractions that have a denominator of 1.

23

33

n




Buddy


Name

*For another example, see Set B on page 533.

1. MP.3 Critique Reasoning Jesse has a bottle that contains 7

10 liter of water. He drinks 2

10 liter. Jesse says he has 12 liter left. Is he correct? Explain.

2. Subtract 410 from 9

10. What addition sentence can you use to check your answer?

For 3–10, subtract the fractions.

Leveled Practice For 11–18, subtract the fractions.

3. 23 - 1

3 4. 34 - 2

4

5. 56 - 2

6 6. 912 - 3

12

7. 98 - 3

8 8. 1710 - 9

10

11. 56 - 1

6 12. 8100 - 3

100

16

56

n 3100

8100

n

13. 34 - 1

4 14. 68 - 4

8

15. 56 - 4

6 16. 4010 - 20

10

9. 48 - 1

810. 1

2 - 12

17. 80100 - 40

100 18. 1910 - 8

10

510; 4

10 + 510 = 9

10

Yes; 710 − 2

10 = 510 liter, and 5

10 is equivalent to 12.



13

14

612

36

810

0

68

38

46 5

100

24

40100

28

2010 or 2

1110 or 1 1

10

16

14

34

n 48

68

n



PearsonRealize.com

ToolsPractice Buddy

Assessment

QUICK CHECKCheck mark indicates exercises for prescribing differentiation on the next page.Items 15 and 22 are worth 1 point. Item 21 is worth up to 3 points.

Error Intervention: Item 2If students have difficulty determining an addition sentence to check their answer,then have students look at the numerators. What is 9 - 4? [5] What is the related addition sentence? [5 + 4 = 9]

11 Reteaching Assign Reteaching Set B on p. 533.

Multi-Step Problems Page 492 Item 19; Page 494 Items 12, 14, and 17

Item 19 Students can add the distance run in the morning and the distance run in the afternoon before subtracting from 1. They can also subtract twice to solve the problem.

Item 22 How can the bar diagram help you write an equation? [The bar diagram shows that n + 6

10 = 910 or 9

10 - 610 = n.

Item 23 Coherence Students have used bar diagrams to add and subtract whole numbers. Students can use bar diagrams to add and subtract fractions in the same way.

491–492


22

1112

112

112

112

112

112

112

112



9-5

To subtract fractions, subtract the numerators. Write the difference over the like denominator.

3. Find 812 - 5

12.

Step 1: Subtract the numerators. - =

Step 2: Write the difference of the numerators over the like denominator.

□12

4. Find 56 - 26.

Subtract the numerators. 5 - 2 =

Write the difference over the like denominator. □

6

5. Find 510 - 4

10.

Subtract the numerators. 5 - 4 =

Write the difference over the like denominator. □10

6. 35 - 2

5 =

7. 22 - 1

2 =

On the Back!

8. Find the difference between 58 and 28.

Vocabulary

1. The denominator of a fraction is the bottom number. It tells how many equal parts make 1 whole.

What is the denominator of 812?

2. The numerator of a fraction is the top number. It tells how many equal parts are shaded.

What is the numerator of 812?

12

8

1512

38

8 5 3

3

1

3

3

1

R 9•5



PartnerTalk

Toss and Talk


Play again!


How to Win


Toss Explain how to compute the difference. Find the answer.

2 910 — 2

10

3 78 — 1

8

4 912 — 2

12

5 56 — 1

6

6 812 — 5

12

38

210

312

46

36

710

35

212

712

68 0 3

8

210

46

312

36

7 56 — 1

6 — 16

8 68 — 2

8 — 18

9 810 — 6

10

10 1012 — 8

12

11 810 — 8

10

12 45 — 1

5

Get Started or


At Your Turn


9•5



PartnerTalk

Toss and Talk


Play again!


How to Win


Toss Explain how to compute the difference. Find the answer.

2 78 — 2

8 — 18

3 56 — 1

6 — 26

4 25100 — 10

100 — 5100

5 2012 — 4

12 — 112

6 1710 — 5

10 — 610

512

16

38

610

26

10100

5100

610

5100

48

1 16

1 610

1512

312

7 56 — 3

6 — 16

8 80100 — 65

100 — 10100

9 163 — 5

3 — 83

10 1112 — 3

12 — 512

11 910 — 2

10 — 110

12 68 — 1

8 — 28

Get Started or


At Your Turn


9•5





STEP

3



Subtracting Fractions with Like Denominators•Write 45 - 2

5 and have students read the subtraction sentence aloud.

•What does the denominator represent? [The number of equal parts in the whole, 5]

•How many parts are there? [4] How many parts are being subtracted? [2]

•Ask students: What is 45 - 25? [25]

•How do you know? [Subtract the numerators, 4 - 2 = 2, write the 2 over the like denominator of 5 to get 25.]

•Repeat the activity using different subtraction sentences, such as 7

10 - 410.

Advanced On-LevelCenter GamesStudents work in pairs or small groups to subtract fractions with like denominators and explain their thinking. Have students record their equations as they play the game.

493A Topic 9


15–30 min

n

centimeter910

710

A D D



11. MP.4 Model with Math An engineer was supposed to draw a line exactly 7

10 centimeter long. An error was made, and he drew the line 9

10 centimeter long. How much longer than needed was the line the engineer drew? Write an equation.

12. A mosaic wall is divided into 100 equal sections. If 30 sections are reserved for orange tiles and 40 sections are reserved for blue tiles, what fraction of the mosaic wall is left for other colors?

13. Number Sense Jonah is thinking of a 2-digit number. It is a multiple of 6 and 12. It is a factor of 108. The sum of its digits is 9. What number is Jonah thinking of?

15. Math and Science Morse code is a way to transmit text using a series of dots or dashes. The Morse code for “Add” is shown. What fraction of the shapes are dots in the Morse code for “Add?”

16. Higher Order Thinking Diego compared the differences for 10

10 - 110 and

100100 - 10

100. He said the differences both

equal 910. Is Diego correct? Explain.

14. In a bag of 100 balloons, 12 are red and 13 are green. What fraction of the balloons in the bag are NOT red or green?

17. 58 of Marie’s marbles are red and 28 are blue. The rest of the marbles are white. Draw a model to represent Marie’s marbles. Write and solve equations to find the fraction of the marbles that are white. 5

8 + 28 = 7

8; 88 − 78 = 1

8; 18 of Marie’s marbles are white.

Sample answer: 75100 of the balloons

Sample answer: 58 of the shapes are dots

Sample answer: 30100 of the mosaic wall


10 = 210 centimeter

36

Yes; Sample answer:1010 − 1

10 = 910;

100100 − 10

100 = 90100;

910 = 90

100

88

58

28 n




38

68

n16

46

n

Another Look!


NameHomework & Practice 9-5Subtract Fractions with Like Denominators

For 1–10, subtract the fractions.

Flora needs an additional 28 cup flour to make

her dough. The dough recipe calls for 68 cup flour.

How many cups of flour does Flora have?

28

68

n

Subtract the numerators. Write the difference over the like denominator.

68 - 2

8 = 48

Flora has 48 cup flour.

3. 45 - 3

5 4. 36 - 1

6

5. 97100 - 40

100 6. 58 - 1

8

7. 1010 - 9

10 8. 1712 - 5

12

9. 33100 - 4

100 10. 50100 - 10

100

1. 68 - 3

8 2. 46 - 1

6

35

45

n

Bar diagrams can help you represent

the problem.

15

57100

29100

110

26

48

40100

1212 or 1

38

36


16

36

n




PearsonRealize.com


Games



Tools Games


Leveled Assignment I Items 1–4, 12–13, 15, 17 O Items 3–6, 11, 14, 16–17 A Items 7–12, 16–17

493–494




9-6

1. Shira has 10 books on her bookshelf. Of those, 5 are about horses. The rest are about nature. What fraction of Shira’s books are about nature?

� 510

� 410

� 310

� 210

2. The table shows the amount of time four people spent exercising.

Exercise Log

Name Time

Bill 12 hour

Carly 34 hour

Dimitri 23 hour

Emma 28 hour

Which person exercised for the least amount of time?

� Bill

� Carly

� Dimitri

� Emma

3. Which fraction is represented by point M on the number line below?

M

08

88

� 18

� 28

� 68

� 88

4. What is the difference of 56 - 26 ?

5. Rachel made 48 cookies. She wants to divide them equally among 3 of her friends’ families. How many cookies should she give to each family?

6. Junior has 6 baseball cards and 4 basketball cards. What fraction of Junior’s cards are basketball cards? Write two equivalent fractions.

7. Write the missing number. What is true about the numbers in the list?

3 6 9 15 18 21

8. Shelly added 13, 23, 43, and 13. She determined the sum of the fractions to be 83. What is 83 written as a mixed number?

Sample answer: 36

223

16 cookies

Sample answer: 410; 25

Sample answer: The numbers are all multiples of 3.

12

4.NF.B.3a

4.NF.A.2

4.NF.A.1

4.NF.B.3a

4.NBT.B.6

4.NF.A.1

4.OA.B.4

4.NF.B.3a

D 9•6



Think

Today’s Challenge

Learn


Glossary

Animated Glossary

eText


Tools

Math Tools

Games

Math Games

Assessment

Quick Check

PD


Solve

Solve and Share

Help



Practice Buddy


LESSON 9-6ADD AnD SubtRACt FRACtionS with Like DenominAtoRS


FoCuSDomain 4.nF Number and Operations—FractionsCluster 4.nF.b Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.Content Standard 4.nF.b.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.Mathematical Practices mP.2, mP.4, mP.5objective Count forward or backward on a number line to add or subtract.essential understanding Fraction addition and subtraction can be thought about as joining and separating segments on the number line. They can also be thought about as counting forward or counting backward on the number line.materials Number lines (Teaching Tool 12)

CoheRenCeIn previous lessons, students added and subtracted fractions with like denominators by joining segments or by separating segments on the number line. In this lesson, the meanings of adding and subtracting fractions is extended to counting forward on the number line for addition and counting backward on the number line for subtraction.

RiGoRThis lesson blends conceptual understanding with procedural skill. It is conceptual in that it extends students’ understanding of addition and subtraction from joining and separating parts to the idea of counting forward and backward. It is procedural because the general methods learned for adding and subtracting fractions in previous lessons can be practiced.

PD



Reading Use visual and contextual support.


Draw a number line for fourths. Label 0, 14, 24, 34, and 1. Have students read the fractions. Discuss how the fractions decrease in value to the left and increase in value to the right. Relate this to whole numbers on the number line.

beginning Draw a number line for sixths. Label with 0, 16, 26, 36, 46, 56, and 1. Ask students to take turns pointing to the tick marks and reading the fraction labels.

intermediate Draw a number line for sixths. Label the tick marks for 0, 16, and 1. Ask students to take turns filling in the missing fractions and reading them aloud. Do the fractions increase or decrease as you move to the right on the number line?

Advanced Draw a number line for sixths. Label the tick marks for 0, 16, and 1. Ask students to take turns filling in the missing fractions and reading them aloud. Ask

students to discuss the relationship between the values on the number line. Do the fractions increase or decrease as you move to the right on the number line? What fraction is two equal segments to the right of 36?

Summarize How can reading a number line help you solve math problems?

Think


495A topic 9


10–15 min

Solve


Name

I can …

Lesson 9-6Add and Subtract Fractions with Like Denominators

Look Back! MP.2 Reasoning Write a fraction that is equivalent to the amount of a full charge that Sebastian used when playing the game.

You can use appropriate tools such as a number line to

show this problem. Content Standard 4.NF.B.3a Mathematical Practices MP.2, MP.4, MP.5

use a number line to add and subtract fractions when the fractions refer to the same whole.

Sebastian has 68 of the full charge left on his phone. He uses 28 of the full charge playing a game. What fraction of the full charge does Sebastian have left? Solve this problem any way you choose.

Sample answer: 14




STEP


Solve

COHERENCE: Engage learners by connecting prior knowledge to new ideas.Students extend their understanding of adding and subtracting fractions with like denominators by showing these operations on a number line.


BEFORE

1. Pose the Solve-and-Share ProblemMP.5 Use Appropriate Tools Strategically In this problem, look for students who use a number line to find the fraction of the charge left on a phone and explain their work.

2. Build UnderstandingWhat information are you given? [Sebastian has 68 of the full charge left on his phone and uses 28 of the full charge playing a game.] What are you asked to do? [Find what fraction of the full charge Sebastian has left.]

DURING

3. Ask Guiding Questions As NeededWhat operation will you use to find the amount of charge Sebastian’s phone has left? [Subtraction] How can you draw a number line for this problem? [Draw a number line from 0 to 1 that is divided into 8 equal parts. Label each tick mark as 18, 28, 38, and so on.]

AFTER

4. Share and Discuss SolutionsStart with students’ solutions. If needed, project Grace’s work to discuss how to subtract fractions using a number line.

5. Transition to the Visual Learning BridgeYou can count forward or backward on a number line to add or subtract.

6. Extension for Early FinishersOn the board, write (56 - 2

6) - 16. Use a number line to simplify

this expression. [26 or 13]


Solve

Whole Class

Whole Class

Grace’s Work Gabrielle’s Work

Grace uses a number line and writes an equation to help explain her work.

Gabrielle draws a number line but does not show her work.

Small Group

495


Visual Learning

A

Learn Glossary

B C

Mary’shouse

Marcy’shouse Soccer

210

mile 510

mile

0 138

58

78

0 ? 1210

510


Convince Me! MP.5 Use Appropriate Tools Use the number line below to

find 58 + 28. Can you also use the number line to find 58 - 2

8? Explain.

Mary rides her bike 210 mile to pick up her friend

Marcy for soccer practice. Together, they ride 5

10 mile to the soccer field. What is the distance from Mary’s house to the soccer field?

The distance from Mary’s house to the

soccer field is 710 mile.

How Do You Add and Subtract Fractions on a Number Line?

You can use jumps on the number line to add or subtract

fractions.

Write the addition equation.

Add the numerators. Write the sum over the like denominator.

210 + 5

10 = 2 + 510 = 7

10

Use a number line to show 210 + 5

10.

Draw a number line for tenths. Locate 210 on the

number line.

To add, move 510 to the right.

When you add, you move to the right on the number line.

58 + 2

8 = 78; Yes, I used the number line to subtract by moving to the left: 58 − 2

8 = 38.




2

Learn Glossary

PearsonRealize.com


MP.1 Make Sense and PersevereWhat information are you given in this problem? [Mary rides her bike 2

10 mile to Marcy’s house; then she rides her bike 5

10 mile to the soccer field.] What must you find? [The total distance Mary rides her bike to get to soccer practice] What operation can you use to find the distance? [Addition]

MP.5 Use Appropriate Tools StrategicallyWhat does each segment on the number line represent? [ 1

10] How do you use counting and the number line to add fractions? [Show the first addend. Then count or jump forward the amount for the second addend to find the total.]

MP.3 Construct ArgumentsHow can using a number line help you write an equation and solve the problem? [Sample answer: It helps me find the total distance to the soccer field, the distance Mary rides to Marcy’s house, and the distance Mary and Marcy ride their bikes to the soccer field.]

Prevent MisconceptionsSome students may still add the denominators. Ask them how many equal segments the number line is divided into and if that changes after they add. Point out that the number of equal segments on the number line stays the same when parts of the same whole are combined or separated.

11

Convince Me! MP.5 Use Appropriate Tools Strategically Both addition and subtraction can be shown on a number line. For both, the number line is divided into equal segments as determined by the like denominator.

Coherence Students have added and subtracted fractions with like denominators, that are parts of the same whole, with fraction strips, number lines, and symbolically without any tools. Next, they will use what they have learned to add and subtract mixed numbers.

Point out that fractions can be added or subtracted by locating a fraction on the number line and then moving to the right to add (as for 58 + 2

8) or to the left to subtract (as for 58 - 28).


496 Topic 9


20–30 min

LookoutTower

Picnic Area

•3__4

mile

2__4 m

ile



13. Higher Order Thinking Sarah and Jenny are running an hour long endurance race. Sarah ran 26 hour before passing the baton to Jenny. Jenny ran 36 hour, then passed the baton back to Sarah. What fraction of the hour does Sarah still need to run to complete the race?

14. Choose numbers from the box to fill in the missing numbers in each equation. Use each number once.

a. □

4 + 2

□ = 34

b. 812 -□12 = 2

□c. □

8 + 2

□ = 58

15. Choose numbers from the box to fill in the missing numbers in each equation. Use each number once.

a. 310 +

□10 = 9

□b. 9

12 - 6

□ =□12

c. 14 +□4 = 3

□

9. Number Sense How do you know the quotient of 639 , 6 is greater than 100 before you actually divide?

11. Isaac started his bike ride at the trailhead. He reached the picnic area and continued to the lookout tower. If Isaac rode his bike for a total of 10

4 miles, how much farther did he ride beyond the lookout tower?

10. MP.2 Reasoning Maria saved 14 of her allowance. Tomas saved 16 of his allowance. Who saved a greater part of his or her allowance? Explain your reasoning.

12. MP.4 Model with Math Ricky completely filled a bucket to wash his car. After he finished washing the car, 58 of the water remained in the bucket. Write and solve an equation to show the fraction of the water Ricky used.

1 3

4 6

8 12

2 3

4 6

10 12

Sample answer: 6 × 100 = 600. 639 + 600, so the quotient is greater than 100.

Sample answer: If Maria and Tomas each get the same amount for allowance, then Maria saved more. If they get different amounts, then the answer is unknown.

24 + 3

4 = 54; 10

4 − 54 = 5

4 or 114 miles

38 of the water; Sample answer: 88 − 5

8 = n; n = 38

16 of the hour

1

612

4

38

6

3

4212

10


MTH16_SE04_CC2_T09_L06.indd Page 498 19/06/14 11:53 PM s-w-100 /132/PE01513_SE/ENVISION_MATH_ENGLISH/NA/SE/2013/G4/XXXXXXXXXX/Layout/Interior_Fi ...


*

0

Start

116

26

36

46

56

0

Start

115

25

35

45

0

Start

1110

210

310

410

510

610

710

810

910

0

Start

115

25

35

45

0

Start

1? 68

48

Do You Know How?Do You Understand?

Name

*For another example, see Set C on page 533.

1. In the example above, how is the denominator illustrated on the number line?

For 3–4, write the equation shown by each number line.


Find 68 - 48.

Start at 68. To subtract, move 48 to the left. The ending point is 28.

So, 68 - 48 = 2

8.

2. MP.4 Model with Math Draw a number line to represent 3

12 + 512.

Another Example!

3.

4.

5. 6.

7.

0 114

24

34

Start

8.

0

Start

116

26

36

46

56

The denominator is the number of equal-size segments that are between 0 and 1 on the number line. 1

5 + 25 = 3

5

56 − 2

6 = 36

210 + 4

10 = 610

24 + 1

4 = 34

45 − 3

5 = 15

46 − 2

6 = 26

0 1812

312



PearsonRealize.com

ToolsPractice Buddy

Assessment


Another Example To subtract on the number line, do you move right or left? [Left] Do you start at zero? [No] Where do you start? [You start at the point representing the first number in the subtraction equation.]

Error Intervention: Items 3–4If students have difficulty,then ask: Where does the arrow start? How many 15s does the arrow move to the right? [2] Where does the arrow end? [35] Write the equation. [15 + 2

5 = 35] Does the arrow go right or left?

[Right] Does that mean you add or subtract? [Add]

11 Reteaching Assign Reteaching Set C on p. 533.

Multi-Step Problems Page 498 Items 11 and 13; Page 500 Items 15 and 17

Item 12 MP.4 Model with Math What fraction should you use to represent the amount of water in the completely filled bucket? [88] Why? [Since the fraction of water remaining in the bucket is written in eighths, the amount of water Ricky started with should also be written in eighths, 88 = 1.]

Items 14–15 Coherence Make sure students know how to add, subtract, and decompose fractions with like denominators representing parts of the same whole before they move on from this lesson. Students will use these skills when they add and subtract mixed numbers in later lessons.

497–498


22

0 114

24

34

0 114

24

34

0 116

26

36

46

56

0 115

25

35

45

0 115

25

35

45



9-6

2. You can use a number line to add 14 + 24.

Locate 14 on the number line. Move 2 segments to the right to show adding two 14 parts.

The fraction that names the ending point is .

So, 14 + 24 = .

3. You can use a number line to subtract 56 - 36.

Start at 56. Move three segments to the left to show subtracting three 16 parts.

The fraction that names the ending point is .

So, 56 - 36 = .

4. Use the number line to add 15 + 35.

15 + 3

5 =

5. Use the number line to subtract 35 - 25.

35 - 2

5 =

On the Back!

6. Draw a number line to show how to find each solution. 38 + 3

8 68 - 4

8

Vocabulary

1. A number line can be used to add and subtract fractions.

This number line is divided into equal segments.

Each segment is of 1 whole unit.

414

34

262

6

45

15

68

28

34

Check students’ drawings.

R 9•6





STEP

3


Adding and Subtracting on the Number LineMaterialsRed, blue, and green crayons•Ask students to draw and label number

lines divided into sixths.

•Have students mark 56 with a red dot.

•Have students draw a green arc from 56 to 26 and write 36 above the arc.

•Students should use a blue dot to mark the location of the answer. [26]

•Have students write the equation modeled by the number line. [56 - 3

6 = 26]

•Repeat the process by drawing a number line divided into eighths and showing 28 + 3

8.

Problem-Solving Reading MatHave students read the Problem-Solving Reading Mat for Topic 9 and then complete Problem-Solving Reading Activity 9-6.

See the Problem-Solving Reading Activity Guide for other suggestions on how to use this mat.


499A Topic 9


15–30 min

0 145

15

Amount of Juice

1

0Sue Jon Sam Aliya

Orange juiceApple juice

14

24

34

Cu

ps

0

Start

1110

210

310

410

510

610

810

910

710

210



310 + 4

10 = 710; 10

10 − 710 = 3

10; Sofia

spent 310 of her money on milk.

14. Robbie drew the number line below to find 45 - 1

5. Explain why Robbie is incorrect.

15. MP.2 Reasoning Kayla used 410 of her

allowance to buy yogurt and 510 to go

skating. What fraction of her allowance does Kayla have left? Explain.

16. Which child drank the most juice? How much juice did that child drink?

17. Higher Order Thinking Sofia bought bananas, cereal, and milk at the store. She spent all of her money. She spent 3

10 of her money on bananas and 410 on

cereal. What fraction of her money did Sofia spend on milk? Write and solve equations.

18. Val’s construction team was supposed to build a frame 7

10 meter long. They ended up building the frame 2

10 meter too long. How long was the frame Val’s team built? Use each fraction from the box once to fill in the missing numbers on the number line.

210 7

10 910

26 3

6 56

19. Corinne ran 56 mile from the start of the trail, turned around and ran 36 mile back. How far is Corinne from the start of the trail? Use each fraction from the box once to fill in the missing numbers on the number line.

0

Start

116

36

46

26

36

56

Sample answer: The arrow goes to the right, which shows addition. It should go to the left to show subtraction.

Kayla has 110 of her allowance left.

Her total allowance is 1010. She spent

410 + 5

10 = 910; 10

10 − 910 = 1

10.

Jon drank the most; 54 or 114 cups




0 158

68

78

38

48

18

28

Start0 15

868

78

38

48

18

28

Start

0

Start

116

26

36

46

56

0

Start

2115

25

35

45

65

75

85

95

0 178

38

Another Look!

What You Show What You Write


NameHomework & Practice 9-6Add and Subtract Fractions with Like Denominators


For 5–13, add or subtract the fractions. Use a number line if needed.

There were 7 slices remaining of an apple pie divided into eighths. Katie and her 3 friends each ate a slice of the remaining pie. Calculate 78 - 4

8 to find how much of the apple pie is now left.

78 - 4

8 = 38

38 of the pie is left.

1.

3. 4.

2.

5. 26 + 1

6 6. 712 - 2

12 7. 18 + 5

8

8. 14 + 3

4 9. 910 - 3

10 10. 23 + 3

3

11. 45 + 3

5 12. 98 - 6

8 13. 13 + 5

3

Subtract to find how much of

the pie is left.

58 − 3

8 = 28 2

8 + 48 = 6

8

46 − 3

6 = 16

45 + 2

5 = 65 or 11

5

36

44 or 1

75 or 12

5

68

53 or 12

3

63 or 2

512

610

38





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Tools Games


Leveled Assignment I Items 1–2, 5–7, 14, 16, 18–19 O Items 3–4, 8–10, 14, 17–19 A Items 3–4, 11–13, 15, 17–19

499–500


Think

Today’s Challenge

Learn


Glossary

Animated Glossary

eText


Tools

Math Tools

Games

Math Games

Assessment

Quick Check

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Solve

Solve and Share

Help



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9-7

1. It takes 4 pounds of grapes to make 1 pound of raisins. How many pounds of grapes would you need to make 3,000 pounds of raisins?

� 1,000 pounds

� 6,000 pounds

� 9,000 pounds

� 12,000 pounds

2. Which number does n represent in the bar diagram below?

5,600

n n n n n n n n

� 7

� 80

� 700

� 800

3. Which equation is modeled by this number line?

78

28 10

� 78 - 5

8 = 28

� 710 - 5

10 = 210

� 28 + 5

8 = 78

� 210 + 5

10 = 710

4. What is the sum of 910 and 1

10?

5. Bob mixed 58 cup of water and 18 cup of lemon juice in a cup. The rest of the mixture was lime juice. How much lime juice did Bob add if the cup was full?

6. Jen needs to save $180 for a new camping tent. She is able to save $9 each week. How many weeks will Jen need to save to reach her goal? Write an equation to solve the problem. Explain how you found your answer.

1010 or 1

Sample answer: 28 cup

20 weeks; Sample

answer: Divide $180 by

the amount Jen

can save each week,

180 ÷ 9 = x; x = 20

4.NBT.B.5

4.NBT.B.6

4.NF.B.3a

4.NBT.B.6

4.NF.B.3a

4.NF.B.3a

D 9•7




LESSON 9-7EstimatE FRaCtion sums anD DiFFEREnCEs


FoCusDomain 4.nF Number and Operations—FractionsCluster 4.nF.B Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.Content Standard 4.nF.B.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Mathematical Practices mP.1, mP.2, mP.3, mP.4, mP.8objective Use number lines and benchmark fractions to estimate fraction sums and differences. Essential understanding Fraction sums and differences can be estimated by thinking about how each fraction relates to other fractions that are easy to add and subtract mentally. materials Number lines (Teaching Tool 12)

CoHEREnCEIn previous lessons, students added and subtracted fractions with like denominators using different strategies and found exact answers. In this lesson, they estimate fraction sums and differences by replacing fractions with benchmark fractions that are close and easy to add and subtract mentally. These estimation skills are then used in subsequent lessons with mixed number operations.

RiGoRThis lesson emphasizes conceptual undertanding: number sense. By analyzing the relative size of the numerator and denominator, students can determine whether a given fraction is close to benchmark numbers 0, 14, 12, 34, and 1. Using these benchmarks, they can then use mental math strategies to decide if sums and differences are greater than or less than 1, close to 0, and so forth.

PD



speaking Express opinions.


Explain that the benchmark fractions 14, 12, and 34 are useful when estimating parts of a whole. Remind students that estimating means finding an answer that is close but not exact.

Beginning Draw a number line and label 0, 14, 12, 34, and 1. Draw a line from 0 to

almost 14. Ask students to complete the sentence stem: “The line is about of a whole.” Draw other lines just short or just past a benchmark fraction.

intermediate Draw a number line and label 0, 14, 12, 34, and 1. Draw a line from 0 to almost 14. Ask students to estimate the length of the line using complete sentences that include the word about.

advanced Draw two number lines and label each with 0, 14, 12, 34, and 1. Draw a line from 0 to almost 14 on one number line. Draw a line from 0 to just past 12 on the other. Ask students to estimate the sum of the lengths of the lines and explain their reasoning using the word about.

summarize How can benchmark fractions be used to estimate sums or differences?

Think


501A topic 9


10–15 min

Solve

0 1110

210

310

410

510

610

710

810

910

0 158

68

78

38

48

18

28

0 115

25

35

45


Name

I can …

Lesson 9-7Estimate Fraction Sums and Differences

Look Back! MP.8 Generalize Which fraction in the exercises above

is close to 12? How can you use whole numbers and benchmark fractions

such as 12 to estimate fraction sums and differences?

You can use reasoning to think about how fractions relate to

0, 12, and 1.


use number lines and benchmark fractions to estimate fraction sums and differences.

For 1–2, use the number lines to decide if each sum is greater than or less than 1. Tell how you decided.

For 3–4, use the number lines to decide if each difference is less than 12 or greater than 12. Tell how you decided.

58 + 8

10 18 + 1

5 + 110

78 - 1

10 910 - 5

83.

1.

4.

2.

58; Sample answer: Round fractions to whole numbers or benchmark fractions, then add or subtract. Compare the estimated sum or difference to the actual sum or difference to check if the answer is reasonable.




STEP


Solve

CoHEREnCE: Engage learners by connecting prior knowledge to new ideas.Students extend their understanding of fractions by using a number line to connect fractions in addition and subtraction equations to benchmark fractions and to explain why fractions from different-sized wholes cannot be compared, added, or subtracted.

DEVELoP: PROBLEM-BASED LEARNING

BEFORE

1. Pose the solve-and-share ProblemmP.2 Reason Quantitatively In this problem, students compare sums to 1 and differences to 12.

2. Build understandingAre you asked to find the exact sums or differences in these problems? [No, I must compare sums to 1 and differences to 12.]

DURING

3. ask Guiding Questions as neededHow can you tell just by looking if a fraction is less than 12? [The denominator is more than twice the numerator.] How can you tell if a fraction is close to 1? [The numerator is close to the denominator.]

AFTER

4. share and Discuss solutionsStart with students’ solutions. If necessary, project Thai’s work to show how to write good explanations.

5. transition to the Visual Learning BridgeComparing fractions to 0, 12, or 1 can help with estimating sums or differences.

6. Extension for Early FinishersName two fractions with different denominators that have a sum greater than 1 and a difference less than 12. [Sample answer: 58 and 12]

analyze student Work

Thai correctly explains how the fractions are related to benchmark fractions and constructs good arguments.

Estela has correct answers for the sums but may not understand how to estimate sums and differences.

Solve

Whole Class

Whole Class

Small Group

Thai’s Work Estela’s Work

501


Visual Learning

A

Learn Glossary

B C

lb

Chucklb

Fana

23 lb

Randylb

Martha15

910

14

0 1

23

34

14

12

34

Close to

0 114

12

34

910

Close to 1

0 1

15

14

12

34

14

Close to

0 114

12

34

14


Convince Me! MP.3 Critique Reasoning Cammy said, “Each of these pies is about 12. So if I put them together, I have about 1 whole pie.” Do you agree with this reasoning? Explain.

Are the following statements reasonable?

Together, Randy and Martha bought about 1 pound of nuts. Fana bought about 14 pound less than Chuck.

You can use whole numbers and benchmark fractions such as 14 and

12 to

decide if a fraction sum or difference is reasonable.

How Can You Decide if a Fraction Sum or Difference Is Reasonable?

Is 23 + 15 about 1?

Think: 23 is close to but less than 34.


34 + 1

4 = 44 or 1

So, 23 + 15 is close to but less than 1.

The statement is reasonable.

Is 910 - 1

4 about 14?


Think: 14 is a benchmark fraction.

44 - 1

4 = 34

So, 910 - 1

4 is close to but less than 34. The statement is NOT reasonable.

No; Sample answer: Fractions must refer to the same whole before they can be compared, added, or subtracted. The pies are not the same size so you cannot say 12 of one pie plus 12 of a different-size pie is 1 whole.




2

Learn Glossary

PearsonRealize.com


MP.1 Make Sense and Persevere Do you need to find exact answers to solve the problems? [No, I can use estimation and number sense to decide whether each statement is reasonable.]

MP.3 Construct Arguments How would the answer change if 9

10 were replaced by 34? [I would

have 34 - 14 = 2

4. Since 910

is closer to 1, the actual difference would be greater than 24.]Prevent MisconceptionsRemind students to use equivalent fractions to subtract 1- 1

4.

11

Convince Me! MP.3 Critique Reasoning Cammy is trying to add parts of two pies that are different sizes. If the pies were the same size, the fractions could be considered parts of the same whole.

Revisit the essential question. Benchmark fractions can be used to estimate sums and differences, but each fraction must refer to the same whole to join, separate, or compare fractions.

Visual Learning BridgeMP.5 Use Appropriate Tools StrategicallyWhat tool can you use to decide if 23 is close to 34 and 15 is close to 14? [I can locate the fractions on the number lines and compare them to the benchmark fractions.]

MP.8 GeneralizeWhy do you think 14, 12, and 34 are called benchmark fractions? [They are easy to compute with mentally. I can use whichever benchmark fraction is closest as an estimate.]

502 Topic 9


20–30 min

0 114

24

34

0 114

24

34

0 114

24

34

28

110

15

0 114

12

34

15

0 114

12

34

610

0 114

12

34

15

0 114

12

34

610



13. Lucy ate 28 of a watermelon, Lily ate 110,

and Madelyn ate 15. Estimate how much of the watermelon they ate. Explain.

14. Gavin, Olivia, and Michael are writing a report together. Gavin has written 25 of the report, Olivia has written 18, and Michael has written 2

10. Use number lines or benchmark fractions to estimate if they have more or less than 12 of their report left to write.

15. MP.1 Make Sense and Persevere Last year the Levitz family sold 16 boxes of nuts for $6 a box. This year, they only have 8 boxes to sell. How much should they charge per box to have the same income selling nuts as last year? Explain.

Use benchmark fractions. You can draw number

lines to help.

16. Higher Order Thinking Choose two fractions from the list that meet each condition.

58

110

34

15

a. Their sum is greater than 1.b. Their difference is close to 0. c. Their sum is between 12 and 1.

17. Harry filled 15 of a pitcher with water. Then he filled another 6

10 of the pitcher with water. Estimate what fraction of the pitcher is filled with water. Use the number lines to explain.

Sample answer: The pitcher is close to 34 filled. Using the number lines, I can see 15 is close to 14, and 6

10 is close to 12 or 24; 14 + 2

4 = 34.

Sample answer: They ate about 24 of

the watermelon. 28 is equivalent to 14, 110 is close to 0, and 15 is close to 14. 14 + 0 + 1

4 = 24

Check students’ work; Less than 12

$12; 16 × $6 = $96, $96 ÷ 8 = $12

Sample answers:

a. 58 and 34

b. 15 and 1

10

c. 58 and 1

10




Buddy

0 114

12

34

0 114

12

34

110

35

0 114

12

34

0 114

12

34

25

710

0 2114

12

34

54

32

74

0 2114

12

34

54

32

74


Name

Draw number lines to identify benchmark fractions as needed.

*For another example, see Set D on page 534.

1. MP.3 Construct Arguments Use benchmark fractions to estimate 4

10 + 38. Explain.

2. MP.3 Critique Reasoning Charlie said 8

10 - 15 is about 14. Do you agree?

Explain.

For 3–6, use < or > to complete each equation. Use the number lines as needed.

For 7–12, estimate whether each sum or difference is reasonable. If it is NOT reasonable, estimate the sum or difference.

3. 13 + 1

2 ○1 4. 54 + 2

4 ○ 2

5. 34 - 1

8 ○ 12 6. 3

2 - 14 ○ 1

2

9. 12 + 2

3 7 1 10. 710 + 2

3 7 1

11. 910 - 1

8 6 12 12. 4

5 - 23 6 1

2

7. 110 + 3

5 is about 1. 8. 710 - 2

5 is about 14.

No; Sample answer: 810 is close to 34; 15 is close to 14; 810 − 1

5 is about 24 or 12.

Sample answer: 410 is close to 12; 38

is close to 12; 410 + 3

8 is about 1.

*

+

*

+

Not reasonable; about 34 Reasonable

Reasonable

Not reasonable; about 34

Reasonable

Reasonable

Sample estimates given.



PearsonRealize.com

ToolsPractice Buddy

Assessment


Error Intervention: Item 2If students have difficulty determining which benchmark fractions to use,then have students draw number lines from 0 and 1. They should divide the number lines into parts based on the denominators in the problem. How should the number line be divided? [Sample answer: Into fourths] Which benchmark fraction will you use to decide if the difference is reasonable? [Sample answer: 34 and 14.]

11 Reteaching Assign Reteaching Set D on p. 534.


Item 13 What fraction on the number line is equivalent to 28? [14]

Item 15 MP.1 Make Sense and Persevere This item reminds students to look for hidden questions that must be answered before the final answer can be found. Students must understand that the item requires them to find how much the Levitz family made last year before they can determine how much they should sell the boxes for this year. Furthermore, students are required to make sense of the information they are given in order to choose an operation for each step in solving the problem.

Item 16 Higher Order Thinking Point out to students that they do not need to actually add or subtract the fractions to find the fraction pairs that meet each condition. Ask students to explain how they used estimation to choose the fraction pairs.

503–504


22

Vocabulary

0 1

0 115

25

35

45

14

12

34

0 1

0 115

25

35

45

0 118

28

38

48

58

68

78

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310

410

510

610

710

810

910



9-7

1. Benchmark fractions are fractions that are commonly used for estimation, such as 14, 12, and 34.

Write these benchmark fractions on the number line at the right.

2. An estimate is a number or an answer that tells about how many.

Use the number lines to name a benchmark

fraction close to 15.

3. Use the number lines to estimate 15 + 14.

15 + 1

4 is about + 14 = .

4. Use the number lines to estimate 58 - 310.

58 is about 12, or 24. 3

10 is about .

- =

So, 58 - 310 is about .

5. Use the number lines to estimate 12 + 13.

6. Use the number lines to estimate 910 - 5

8.

On the Back!

7. Use number lines and benchmark fractions to estimate each sum or difference. Explain.

Estimate 25 + 510. Estimate 8

10 - 35.

Sample answer: 25 is about 12;5

10 is equal to 12; 12 + 12 = 1;

25 + 5

10 is about 1.

14

14

14

24

24

Sample answer: 810 is about 1;

35 is about 34; 1 is equal to 44; 44 − 3

4 = 14; 8

10 − 35 is about 14.

14

14

14

Sample answer: 34

Sample answer: 14

R 9•7

MTH16_ANC04_CC2_T09_L07_BU.indd 4 6/11/14 10:53 PM user /142/PE00875_SE_G8_R2/PEARSON_LITERATURE_15/NA/SE/PHLIT1_5/G8/XXXXXXXXXX/Layout/I ...


Name Math and Science Activity9-7

Analyzing Data

Did You Know? Speech recognition software uses speech pattern algorithms to recognize sound waves and translate them into text. The longer a person uses the software, the better the software becomes at recognizing the person’s unique speech patterns.

Dina is developing a new speech-to-text application. She programmed the application to recognize 150 words. The table shows the fraction for each word type of the 150 words Dina programmed.

Word Type Fraction of Words

Names 25

Adjectives 16

Verbs 13

Pronouns 110

➊ Show how to estimate the fraction of words Dina programmed that are names and pronouns.

➋ Choose two word types whose sum is greater than 12. Explain.

➌ About what fraction of the words are NOT names. Use benchmark fractions to estimate.

➍ Extension Draw a number line and plot each fraction of words. Show which benchmark fraction each word type is closest to.

25 is close to 12; 1

10 is close to 0; 12 + 0 = 12


Check students’ work.

24; Sample answer: 16 is close to 14; 13 is close to 14; 1

10 is

close to 0; 14 + 14 + 0 = 2

4

Verbs and Names; Sample answer: Both fractions are close to 12, so the sum of the two groups must be greater than 12.

9•7Math and Science Activity

MTH16_ANC04_CC2_T09_L07_SC.indd Page 305 6/11/14 10:55 PM user /142/PE00875_SE_G8_R2/PEARSON_LITERATURE_15/NA/SE/PHLIT1_5/G8/XXXXXXXXXX/Layout/I ...

Math and Science Activity STEM

This activity revisits the science theme, Information Transfer, introduced on page 461 in the Student’s Edition.

Sample Student Work

4.




STEP

3



Estimating Fraction Sums and DifferencesMaterials Number cubes, Fraction strips (or Teaching Tool 13)•Have students work in pairs. Each

student tosses two number cubes and uses the digits to create a fraction. Each student uses fraction strips to determine which benchmark fraction is closest to his or her fraction.

•Using both fractions, one student estimates the sum of the two fractions. The other student estimates the difference.

•Students then check each other’s work.

•Students repeat the activity, switching roles to find the sum or difference.

•Have students record their tosses, estimates, and equations.

505A Topic 9

MTH16_TE04_CC2_T09_L07.indd Page 505 28/06/14 10:04 AM s-w-100 /113/PE01513_PS/MATH_2016/NA/ANCILLARY/Product_Sampler/Product_Sampler_3_6/Layout ...

15–30 min

0 114

28

24

34

0 114

15

24

34



9. MP.4 Model with Math Elena ate 28 of a

pizza and Dylan ate 15. Use the number lines and benchmark fractions to estimate how much of the pizza they ate. Explain.

10. Kara’s grandmother is knitting a baby cap. She knitted 15 of the cap yesterday morning. By evening, she had knitted 7

10 of the cap. Estimate how much of the cap she knitted during the afternoon.

11. Math and Science Drums can be used to communicate. If a drummer beats his drum 240 times in a message, how many drum beats will be made if he plays the message twice.

12. In a beaded necklace, 312 of the beads

are blue and 13 are green. Use benchmark fractions to estimate about what fraction of the beads are blue or green. Explain.

13. Higher Order Thinking Jonathan spends 28 of his money on food, 15 of his money on fuel, and 2

10 of his money on clothes. Estimate what fraction of his money Jonathan has left. Explain.

14. Margaret’s bottle of shampoo is 78 full. She uses 13 of the shampoo in the bottle to wash the dog. Estimate what fraction of the shampoo is left. Use whole numbers and benchmark fractions to explain.

You can draw a number line to find benchmark

fractions.

Sample answer: 28 = 14. 15 is close to 14.

14 + 1

4 = 24. So, Elena and Dylan ate

about 12 of the pizza.

480 drumbeatsAbout 12 of the cap

Sample answer: About 12 of the beads

are blue or green. 312 = 1

4. 13 is close

to 14. 14 + 14 = 2

4 or 12.

Sample answer: 28 = 14, 15 is close to 14,

and 210 is close to 14. So, Jonathan

spends almost 34 of his money

(14 + 14 + 1

4). He has about 14 of his money left.

Sample answer: About 12 of the

shampoo is left. 78 is close to 1 (or 22) and 13 is close to 12. 22 − 1

2 = 12.


MTH16_SE04_CC2_T09_L07.indd 506 15/05/14 12:23 AM


0 114

910

12

34

0 114

25

12

34

0 114

12

34

110

0 114

26

12

34

0 114

15

12

34

0 114

58

12

34

Another Look!


NameHomework & Practice 9-7Estimate Fraction Sums and Differences

3. 15 + 1

2 7 1 4. 25 + 2

3 7 1 5. 33 + 4

3 6 2

6. 710 - 2

5 6 12 7. 9

10 - 13 6 1

2 8. 76 - 1

2 7 1

For 1–8, estimate whether each sum or difference is reasonable. If it is NOT reasonable, estimate the sum or difference.

110 + 2

6 is close to but greater than 0 + 14 = 1

4.

The whole distance to school is 44 or 1.

44 - 1

4 = 34

Jake still needs to travel close to, but less than 34 of the distance to get to school.

Think: 110 is close to but greater than 0.

26 is close to but greater than 14.

Jake ran 110 of the distance to the school

and walked 26 of the distance. Estimate what fraction of the distance Jake still needs to travel to get to school. Use the benchmark

fractions on the number lines to help you.

1. 15 + 5

8 is about 34. 2. 910 - 2

5 is about 14.

Reasonable

Reasonable

Reasonable Not reasonable; about 21

4





Sample estimates given.




PearsonRealize.com


Games



Tools Games


Leveled Assignment I Items 1–3, 9–11, 14 O Items 2, 4–5, 10, 13–14 A Items 6–8, 12–14

505–506




9-8Name

1. Pete is packing toys into boxes. Each box holds 30 toys. There are 94 toys and 3 boxes. How many extra toys are there?

� 4 toys

� 3 toys

� 2 toys

� 1 toy

2. Samantha drives 95 miles each day. How many miles does she drive in 40 days?

� 2,800 miles

� 3,400 miles

� 3,800 miles

� 4,800 miles

3. Julie bikes 30 weekends each year. On the weekends when she bikes, she bikes 9 kilometers. How many kilometers does she bike in four years?

� 270 kilometers

� 1,000 kilometers



4. Rose is planning a surprise party for her father. She is inviting 94 people besides her father and herself. She can seat 8 people at each table. How many tables will Rose need?

� 10 tables

� 11 tables

� 12 tables

� 13 tables

5. Joe does 25 sit-ups each day. How many sit-ups does he do in 3 weeks? Remember, one week has 7 days.

6. Estimate the quotient 522 , 9. Explain how you estimate.

7. Jenna needs 3 cups of flour to make 24 of her favorite cookies. How many cups of flour does Jenna need to make 120 of her favorite cookies? Write equations and use them to explain how to solve.

8. Find the unknown value in the equation.16 + n

6 = 56

525 sit-ups

60; Sample answer:

540 ÷ 9 = 60

n = 4

4.OA.A.3

4.NBT.B.5

4.NBT.B.5

4.OA.A.3

4.NBT.B.5

4.NBT.B.6

4.OA.A.3

4.NF.B.3a

15 cups; Sample answer:

24 ÷ 3 = c, c = 8 cookies

with each cup;

120 ÷ 8 = f , f = 15

D 9•8



Think

Today’s Challenge

Learn


Glossary

Animated Glossary

eText


Tools

Math Tools

Games

Math Games

Assessment

Quick Check

PD


Solve

Solve and Share

Help



Practice Buddy


LESSON 9-8MODEL ADDITION AND SUBTRACTION OF MIXED NUMBERS


FOCUSDomain 4.NF Number and Operations—FractionsCluster 4.NF.B Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.Content Standard 4.NF.B.3c Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.Mathematical Practices MP.2, MP.3, MP.5Objective Use models and equivalent fractions to add and subtract mixed numbers.Essential Understanding Adding and subtracting mixed numbers is an extension of the ideas and procedures for adding and subtracting fractions. Materials Number lines (Teaching Tool 12) Fraction strips (or Teaching Tool 13)

COHERENCEIn previous lessons, students added and subtracted fractions with like denominators using different strategies. They used fraction strips and number lines to make sense of the procedures. In this lesson, they extend their understanding of addition and subtraction of fractions with like denominators to addition and subtraction of mixed numbers with like denominators. The same representations are used and extended to make sense of procedures.

RIGORThis lesson emphasizes conceptual understanding. Fraction strips and number lines are used to extend work with fractions to include mixed numbers. Both addition and subtraction can be shown using each representation.

PD



Speaking Express ideas on a variety of topics.


Read the Solve & Share problem. Explain that students could draw pictures or use fraction strips to solve the problem. Ask students to think about how they would solve this problem then share their ideas with a partner.

Beginning Read the Solve & Share problem to the students. First draw a picture representation of the problem, then use fraction strips to show students how to solve

the problem. Ask students to think about how they would solve the problem. Students will turn and talk to partners using the following sentence stem: I would use a ________ to solve the problem.

Intermediate Read the Solve & Share problem with students. Work with students to draw a picture representation of the problem, then use fraction strips to show students how to solve the problem. Ask students to talk to partners to explain how the problem could best be solved.

Advanced Have pairs read the Solve & Share problem. Ask students to work with their partners to draw a representation of the problem, then use fraction strips to demonstrate how to solve the problem. Students will discuss their ideas for solving the problem.

Summarize How does drawing pictures or using fraction strips help to solve problems with mixed numbers?

Think


507A Topic 9


10–15 min

Solve


Name

I can …

Lesson 9-8Model Addition and Subtraction of Mixed Numbers

Look Back! MP.2 Reasoning How can you estimate the sum above?

You can select tools such as fraction strips or number lines to

add mixed numbers.

Content Standard 4.NF.B.3c Mathematical Practices MP.2, MP.3, MP.5

use models and equivalent fractions to help add and subtract mixed numbers.

Tory is cutting loaves of bread into fourths. She needs to wrap 33

4 loaves to take to a luncheon and 12

4 loaves for a bake sale. How many loaves does Tory need to wrap for the luncheon and the bake sale? Solve this problem any way you choose.

Sample answer: I can estimate by rounding to the next whole number; 33

4 is about 4; 124 is about 2; 4 + 2 = 6; About 6 loaves.




STEP


Solve

COHERENCE: Engage learners by connecting prior knowledge to new ideas.Students have learned how to add and subtract fractions with like denominators. They extend this knowledge to model addition and subtraction of mixed numbers.


BEFORE

1. Pose the Solve-and-Share ProblemMP.5 Use Appropriate Tools Strategically In this problem, look for students who use fraction strips (or Teaching Tool 13) to show 33

4 + 124.

2. Build UnderstandingWhat information are you given? [Tory needs to wrap 33

4 loaves for a luncheon and 12

4 loaves for a bake sale.]

DURING

3. Ask Guiding Questions As NeededWhich fraction strips will you need to model the problem? [The fourths] What do you know about adding fractions that can help you solve the problem? [I can find the sum by first combining the fourths and the wholes separately.] How can you use fraction strips to help rename 4 54? Ask a volunteer to demonstrate how four of the fourths strips can be composed for a one-whole, and explain why 45

4 = 514.

AFTER

4. Share and Discuss SolutionsStart with students’ solutions. Then project Brian’s and Margaret’s work as needed.

5. Transition to the Visual Learning BridgeYou can show the addition of mixed numbers with like denominators using fraction strips.

6. Extension for Early FinishersMegan measures 12

3 cups of brown sugar and 213 cups of white

sugar. How much sugar does she have in all? How would you show the addition? [4 cups; Check students’ work.]


Brian correctly adds 334 and 12

4. He draws a picture of fraction strips to show how 54 can be renamed as 11

4.Margaret correctly adds 33

4 and 124 using fraction strips.

Solve

Whole Class

Whole Class

Small Group

Brian’s Work Margaret’s Work

507


Visual Learning

A

Learn Glossary

B C

2 512 feet

11112 feet

0 1 2 3

1112

1

512

2612


Convince Me! MP.5 Use Appropriate Tools Suppose Bill’s boards were

21112 feet and 1 5

12 feet. What would be the total length of the two boards?

How much longer is one board than the other? Use fraction strips or draw number lines to show your work.

Bill has 2 boards to use to make picture frames. What is the total length of the two boards? How much longer is one board than the other?

You can use addition to find the total length

of the two boards.You can use subtraction

to find how much longer one board is than the other.

Use fraction strips to show .

Add the fractional parts:

Rename 1612 as 1 4

12.

Add the whole number parts: 2 + 1 = 3

Then add the sum of the whole number parts to the sum of the fractional parts.

3 + 1 412 = 4 4

12

So, 2 512 + 111

12 = 4 412 feet.

Use a number line to show .

Mark the number you are subtracting from, 2 5

12.

To subtract, move 11112 to the left on the

number line.

Write the difference as a fraction: 612

So, 2 512 - 111

12 = 612 foot.

How Can You Add or Subtract Mixed Numbers?

2 512 + 111

12 2 512 - 11112

512 + 11

12 = 1612

1

112

112

112

112

112

112

112

112

112

112

112

112

112

112

112

112

112

112

112

112

Sample answer: 4 412 feet; 1 6

12 or 112 feet longer. Check students’ work.




2

Learn Glossary

PearsonRealize.com

The Visual Learning Bridge connects students’ thinking in the Solve & Share to important math ideas in the lesson. Use the Visual Learning Bridge to make these ideas explicit. Also available as a Visual Learning Animation Plus at PearsonRealize.com.

MP.1 Make Sense and Persevere What type of numbers are you adding and subtracting? [Mixed numbers]

Convince Me! MP.5 Use Appropriate Tools Strategically Using fraction strips, help students see that the sum of 211

12 and 1 512 is the

same as the sum of 2 512 and 111

12. In both problems, they are adding (2 + 1) + (11

12 + 512). Also, help them see that regrouping is not needed

on the subtraction because 1112 is greater than 5

12.

Coherence To add mixed numbers, students use what they know about adding fractions and adding whole numbers. They add the fractions and they add the whole numbers (in either order). Then both sums are combined. To subtract mixed numbers, sometimes students need to regroup before using what they know about subtracting fractions.

Revisit the essential question. To model addition of mixed numbers, students can combine fraction strips and count the number of parts in the sum. Remind students that when

subtracting mixed numbers, they can trade one whole for fraction strips that make up an equivalent fraction, then use what they know about regrouping to subtract.

Visual Learning BridgeMP.5 Use Appropriate Tools StrategicallyYou may want to have a volunteer show both addends with fraction strips and then rearrange the strips to show 316

12. Why is 16

12 renamed as 1 4

12? [So that a whole number part and a fraction part can be used to help add the other two numbers]

Prevent MisconceptionsSome students will recognize that 6

12 and 12 are equivalent fractions. Remind students that, because they are equivalent, neither is more or less mathematically precise.

11

508 Topic 9


20–30 min



18. MP.5 Use Appropriate Tools Kit said, “On summer vacation, I spent 11

2 weeks with my grandma and one week more with my aunt than with my grandma.” How many weeks did she spend with her grandmother and her aunt? Use fraction strips or number lines to find the sum.

20. Hannah used 158 gallons of paint for

the ceiling and some gallons of paint for the walls. Hannah used 6 gallons of paint in all. How much paint did Hannah use for the walls?

22. Higher Order Thinking A recipe calls for 12

3 cups of brown sugar for the granola bars and 11

3 cups of brown sugar for the topping. Dara has 31

4 cups of brown sugar. Does she have enough brown sugar to make the granola bars and the topping? Explain.

19. MP.5 Use Appropriate Tools If Kit spent 31

2 weeks in swimming lessons, how much more time did Kit spend visiting family than in swimming lessons? Use fraction strips or number lines to find the difference.

21. A furlong is a unit of length still used today in racing and agriculture. A race that is 8 furlongs is 1 mile. A furlong is 660 feet. How many furlongs are in 1 mile?

23. Megan is knitting a scarf. She has

knitted 2 712 feet so far. She needs to knit

another 21112 feet. Which of the following

expressions can Megan use to find the

length of the completed scarf?

𝖠𝖠𝖠𝖠 2 712 + 211

12

𝖡𝖡𝖡𝖡 2 512 + 2 7

12

𝖢𝖢𝖢𝖢 7 112 + 11 1

12

𝖣𝖣𝖣𝖣 4 + 1112

24. Megan finishes the scarf. It is 5 612 feet in

length. She finds a mistake in her knitting and unravels 2 4

12 feet to correct the mistake. How long is the scarf now?

𝖠𝖠𝖠𝖠 81012 feet

𝖡𝖡𝖡𝖡 5 412 feet

𝖢𝖢𝖢𝖢 3 212 feet

𝖣𝖣𝖣𝖣 1 412 feet

You can use fraction strips or a number line to

compare amounts.

112 + 21

2 = 4 weeks

438 gallons

Sample answer: 123 + 11

3 = 3; 31

4 + 3; Dara has enough brown sugar.

5,280 feet

4 − 312 = 1

2 week longer




Buddy

1414

14

14

114


Name

*For another example, see Set E on page 534.

1. MP.3 Construct Arguments When adding mixed numbers, is it always necessary to rename the fractional sum? Explain.

2. 125 + 24

5 3. 114 + 23

4

For 2–5, use fraction strips or number lines to find each sum or difference.

For 6–9, use each model to find the sum or difference.


6. 214 - 13

4 7. 123 + 22

3

8. 234 - 13

4 9. 136 + 13

6

10. 235 + 13

5 11. 4 512 + 1 7

12 12. 4 910 + 3 7

10 13. 534 + 23

4

14. 1238 - 95

8 15. 813 - 72

3 16. 1378 - 107

8 17. 314 - 23

4

0 54321 23

1

23

2

4. 423 - 21

3 5. 414 - 33

4

415

213

24

4No; Sample answer: If the numerator is less than the denominator, it does not need to be renamed.



24

1 3

413

415

268

23

8 610

3

824

24

6


0 321

341

342

116

16

16

16

16

16

1


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ToolsPractice Buddy

Assessment


Error Intervention: Items 2–5If students do not know what fraction strips to use,then have students identify the whole number and the fractional part of each mixed number. Ask: What fraction strip represents 1? [The whole strip] What kinds of fraction strips match the denominator of the fractional part? [Answers will vary.]

11 Reteaching Assign Reteaching Set E on p. 534.

Multi-Step Problems Page 510 Item 22; Page 512 Items 13–15

Items 18–19 MP.5 Use Appropriate Tools Strategically Draw two number lines from 0 to 4 divided into increments of 12. Have students show how to solve using the number lines. For item 18, they should start at 11

2 and move right 212. For item 19, they should start

at 4 and move left 312.

Item 20 Have students demonstrate how they used fraction strips to find the solution.

Item 22 Higher Order Thinking Have students describe what step they did first to solve the problem. [Add the brown sugar measurements.]

509–510


22

Vocabulary

11

16

16

116

16

16

16

16

111

16

16

16

16

16

16

16



9-8

1. Whole numbers are the numbers 0, 1, 2, 3, 4, and so on. Each mixed number has a whole number part and a fraction part.

Look at the mixed numbers below. Circle the whole number in each mixed number. Draw a rectangle around the fraction part.

1015 12

3 1258 12 7

12

2. Use the model to add 226 + 15

6.

3. Use the model to subtract 225 - 14

5.

Model the number you are subtracting

from, 225.

Rename 225 as 17

5. Cross out one whole

and 45 to show subtracting 145.

1

1

1

15

15

15

15

15

15

15

15

15

On the Back!

4. Use fraction strips to find the sum of 258 + 43

8 and the difference of 3 3

10 - 1 610.

Model each number using fraction strips.

Add the whole numbers. Add the fractions.

Write the sum.

Write the difference.

226

156+

376 or 41

6

7; 1 710

35

R 9•8



PartnerTalk

Teamwork


Get Started or

Get paper and a pencil. Put in a bag. Take turns.

Repeat for Each Round

Pick a number tile. Read the expression next to that number. Find the picture below that matches your expression. Explain why you chose that picture. Decide on the correct sum or difference. Use the picture to explain why your answer is correct.

4321


Create an expression with mixed numbers that your team can solve by drawing a picture of fraction strips.

235 +11

51 21

3 —123

2 238 —15

83 23

4 +134

4

a

b

d

c

9•8



PartnerTalk

Teamwork


Get Started or

Get paper and a pencil. Put in a bag. Take turns.


Pick a number tile. Read the expression next to that number. Find the picture below that matches your expression. Explain why you chose that picture. Decide on the correct sum or difference. Use the picture to explain why your answer is correct.

4321


Create an expression with mixed numbers that your team can solve by drawing a picture of fraction strips.

148 + 27

81 22

4 —114

2 258 —16

83 11

6 + 226

4

a

b

d

c

9•8


Advanced On-LevelCenter GamesStudents work in pairs or small groups to use drawings to find sums and differences of mixed numbers. Have students record the completed equations as they play the game.




STEP

3

Reteach I


Modeling Addition and Subtraction of Mixed NumbersMaterialsFraction strips (or Teaching Tool 13), board, markers•Write these problems on the board.

323 + 12

3 313 - 12

3

•Work together to show each mixed number in the first problem using fraction strips. Then discuss how to combine the fractional parts of the two numbers: 23 and 23. Ask: What is the sum of these two fraction parts? [43] How can you use fraction strips to rename 43 as a mixed number? [Trade 33 for a 1 strip.]

•What is the total of the fractional parts as a mixed number? [11

3] Continue by

adding the whole number parts [3 + 1 = 4], and then combining the two sums [51

3].

•Have students complete the subtraction problem using fraction strips.

1

1

13_ 1

3_ 1

3_

13_

3 13_

1 23_-

2 43_

1 23_

1 23_

-=

=

Intervention Activity I

510A Topic 9


15–30 min

DA

TA

Beetles by Length

Beetle Length in Inches

Hercules beetle 634

Ladybug 14

Stag beetle 118 to 2 48

7 b book reports

Points to pass

Points for eachbook report



13. Stan needs 90 points to get a passing grade in class. He already has 6 points. If each book report is worth 7 points, what is the fewest number of book reports Stan can do and still pass the class?

14. Higher Order Thinking Nicole, Tasha, Maria, and Joan each walk to school from home. Nicole walks 111

12 miles. Tasha walks 2 1

12 miles. Maria walks 1 712 miles.

Joan walks 2 212 miles. How can you find

how much farther Joan walks to school than Maria?

15. Alyssa used 123 gallons of white paint for

the ceiling of her kitchen and 123 gallons

of white paint for her bedroom. She used 32

3 gallons of green paint for the walls of her kitchen and 12

3 gallons of yellow paint for the walls of her bedroom. How much more green paint did Alyssa use than white paint?

𝖠𝖠𝖠𝖠 13 gallon

𝖡𝖡𝖡𝖡 23 gallon

𝖢𝖢𝖢𝖢 1 gallon

𝖣𝖣𝖣𝖣 2 gallons


10. How many inches longer is a Hercules beetle than a ladybug?

11. What is the difference between the largest and the smallest stag beetles?

12. How long are a Hercules beetle and a ladybug combined?

16. Jerome’s rain gauge showed 13 910

centimeters of rain fell last month. This month, the rain gauge measured 15 3

10 centimeters. How many more centimeters of rain fell this month than last month?

𝖠𝖠𝖠𝖠 2915 centimeters

𝖡𝖡𝖡𝖡 15 310 centimeters

𝖢𝖢𝖢𝖢 225 centimeters

𝖣𝖣𝖣𝖣 1 410 centimeters

Sample answer: I used fraction strips to show Joan’s distance, 2 2

12 miles, and regrouped 1 whole to subtract Maria’s distance 2 2

12 − 1 712 = 7

12 mile.12 book reports

Sample answer: 624 inches

138 inches

7 inches


MTH16_SE04_CC2_T09_L08.indd Page 512 19/06/14 7:47 AM s-w-100 /132/PE01513_SE/ENVISION_MATH_ENGLISH/NA/SE/2013/G4/XXXXXXXXXX/Layout/Interior_Fi ...


0 54321 78

1

38

2

11515

15

15

15

15

11

15

1

1

1

14

14

14

0 54321 12

3

12

1

0 4321

34

1

34

1

Another Look!


NameHomework & Practice 9-8Model Addition and Subtraction of Mixed Numbers


1. 312 + 11

2 2. 334 - 21

4 3. 134 + 13

4

7. 2 512 + 4 3

12 8. 1213 - 52

3 9. 224 + 63

4

4. 345 - 12

5 5. 526 + 35

6 6. 1028 - 75

8

Use a number line to find 178 + 23

8.

Use a number line for eighths. Start at 178.

To add, move 238 to the right.

Write the sum as a fraction or a mixed number.

So, 178 + 23

8 = 428.

Use fraction strips to find 215 − 12

5.

Model the number you are subtracting from, 21

5.

Rename 215 as 16

5. Cross out one whole and 25 to show subtracting 12

5.

Write the difference as a fraction.

So, 215 - 12

5 = 45.

You can use fraction strips or number lines to show the addition and

subtraction of mixed numbers.


5

6 812

225

124

623

916

324

914

258




PearsonRealize.com


Games



Tools Games


Leveled Assignment I Items 1–4, 10–12, 15–16 O Items 2, 4–6, 13–16 A Items 6–9, 13–16

511–512


Think

Today’s Challenge

Learn


Glossary

Animated Glossary

eText


Tools

Math Tools

Games

Math Games

Assessment

Quick Check

PD


Solve

Solve and Share

Help



Practice Buddy



9-9

1. Which is the difference? Remember, you can use equivalent fractions. Select all that apply.4

12 - 112

112

212

312

14

13

2. After Ronaldo’s Fourth of July party, 46 of his cake is left. How much cake will be left after his cousin Max eats another 16?

𝖠𝖠𝖠𝖠 56

𝖡𝖡𝖡𝖡 36

𝖢𝖢𝖢𝖢 13

𝖣𝖣𝖣𝖣 312

3. Noreen bought two shirts for $13 each and two pairs of shoes for $18 a pair. How much did Noreen pay in all?

𝖠𝖠𝖠𝖠 $31

𝖡𝖡𝖡𝖡 $44

𝖢𝖢𝖢𝖢 $49

𝖣𝖣𝖣𝖣 $62

4. A day pass at a theme park costs $16 for a child and $24 for an adult. How much would it cost to get day passes for 5 adults and a class of 26 children?

5. Is 78 - 35 greater than or less than 12?

Explain.

6. Use division to write two equivalent fractions for 6

12.

7. Find the sum.

12,479+ 32,612

4.NF.B.3a

4.NF.B.3d

4.NF.B.3a

4.NF.A.1

4.NBT.B.4

4.OA.A.3

4.OA.A.3

$536

Less; Sample answer:78 is a little less than 1

and 35 is a little more

than 12, so 78 − 35

is a little less than

1 − 12 = 1

2.

Sample answer:6

12 ÷ 22 = 3

6;6

12 ÷ 33 = 2

4

45,091

D 9•9




LESSON 9-9ADD MixeD NuMbeRs


FOCusDomain 4.NF Number and Operations—FractionsCluster 4.NF.b Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.Content Standard 4.NF.b.3c Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.Mathematical Practices MP.1, MP.2, MP.3, MP.8Objective Use equivalent fractions and properties of operations to add mixed numbers with like denominators.essential understanding Two procedures for adding mixed numbers both involve changing the calculation to a simpler equivalent calculation.

COHeReNCeIn the previous lesson, students added mixed numbers with like denominators using fraction strips and number lines. In this lesson, two procedures for adding mixed numbers are developed.

RiGORThis lesson blends conceptual understanding and procedural skill. Underlying both procedures is changing the calculation to simpler equivalent calculations, as previously learned. The first procedure breaks the calculation into adding the fractional parts and then the whole number parts. The second procedure changes the mixed numbers to equivalent fractions, then uses the procedure for adding fractions with like denominators.

PD



Listening Learn academic vocabulary.Use with the Visual Learning Bridge on Student’s Edition p. 514.Activity Read and write mixed numbers and define the term. For example, 27

8 is a mixed number. Two is the whole number and 78 is the fraction. Write 27

8 + 113 and read the

addition sentence aloud. Explain to students that when adding mixed numbers, they will first add the fractions and then add the whole numbers.

beginning Write 224 + 41

4 and read the addition sentence aloud. Point to 24 and 14.

Say: To add mixed numbers, add the fractions first. Point to 2 and 4. Say: Then add the whole numbers. Write 33

8 + 468.

Have student pairs use the sentence stem: To add [mixed numbers], add

[fractions] first. Then add [whole numbers].

intermediate Write 224 + 41

4. Say: To add mixed numbers, add the fractions first. Point to 2 and 4. Say: Then add the whole numbers. Write 33

8 + 468. Have student pairs

explain to each other how to add using the term mixed numbers.

Advanced Ask students to write an addition sentence with mixed numbers. Have students read each other’s addition sentence. Ask student pairs to discuss how to solve addition sentences with mixed numbers.summarize How are addition sentences with mixed numbers solved?

Think


513A Topic 9


10–15 min

Solve

136 cups apple juice 14

6 cups orange juice


Name

I can …

Lesson 9-9Add Mixed Numbers

Look Back! MP.2 Reasoning Without solving the problem, use reasoning to estimate the sum. Will the sum be more or less than 3? How much more or less? Explain.

Generalize. You can use what you know about adding fractions

to solve this problem.

Content Standard 4.NF.B.3c Mathematical Practices MP.1, MP.2, MP.3, MP.8

use equivalent fractions and properties of operations to add mixed numbers with like denominators.

Joaquin used 136 cups of apple juice and

146 cups of orange juice in a recipe for punch. How

much juice did Joaquin use? Solve this problem any way you choose.

Sample answer: 136 is equal to11

2. 146 is a little more than11

2. 11

2 + 112 = 3, so the sum will be a little more than 3.




STEP


Solve

COHERENCE: Engage learners by connecting prior knowledge to new ideas.Students extend their understanding of how to add mixed numbers using models to add them symbolically. They also use their understanding of the Commutative and Associative Properties of addition and of decomposing fractions.


BEFORE

1. Pose the Solve-and-Share ProblemMP.8 Generalize Listen and look for students who generalize what they have learned about adding fractions, writing mixed numbers as fractions, or properties of addition to add the mixed numbers without using models.

2. Build UnderstandingWhat information is given in the problem? [Joaquin used 13

6 cups of apple juice and 14

6 cups of orange juice.] What do you need to find? [How much juice Joaquin used]

DURING

3. Ask Guiding Questions As NeededHow could you decompose 13

6 and 146? [Sample answer:

136 = 6

6 + 36 and 14

6 = 66 + 4

6] How could you use the decomposed fractions to add the two mixed numbers? [Sample answer: Use the Associative Property to add (66 + 3

6) + (66 + 46) = 9

6 + 106 = 19

6 . Then change 19

6 to a mixed number: 196 = 31

6.]

AFTER

4. Share and Discuss SolutionsStart with students’ solutions. If necessary, project and discuss Benny‘s and Ginny’s work to show two different ways to add without using models.

5. Transition to the Visual Learning BridgeOne way to add mixed numbers is to add the fraction parts and then add the whole number parts. Sometimes the sum of the fractions needs to be renamed. Another way is to add equivalent fractions.

6. Extension for Early FinishersMarty has three lengths of wood molding. They are 14

8 in., 22

8 in., and 138 in. long. What is the total length of all three

pieces of molding? [518 in.]


Solve

Whole Class

Whole Class

Small Group

Benny’s Work Ginny’s Work

Benny used reasoning and properties of addition to add the fraction parts, add the whole number parts, and then add the two sums together.

Ginny changed the mixed numbers to fractions before adding and then changed the fraction to a mixed number after adding.

1 36_

1So, 36_

1 36_

3 16_

1 36_ 6

6_ 3

6_+ += = 9

6_=

1 46_

1 46_

1 46_ 6

6_ 4

6_+

1 46_

+

66_ 6

6_+

+

66_+ 1

6_+

= = 106_=

96_+ += 19

6_

196_

106_ =

=

3 16_

=

=

Benny’s Work

First, add fraction parts, then add the whole number parts.

Add the whole number part to the

fraction part 2 + 11__6

= 3 1__6 cups.

+ 4__

67__

6 = 1 1__

6

1+1 2

3__

6

513


Visual Learning

A

Learn Glossary

B C

138 cups sand


Find 278 + 13

8 by adding equivalent fractions.

Write 348 as a mixed number.

348 = 8

8 + 88 + 8

8 + 88 + 2

8 = 428

Convince Me! MP.2 Reasoning How is adding mixed numbers like adding fractions and whole numbers?

Brenda mixes sand with 278 cups of potting mixture to

prepare soil for her plant. After mixing them together, how many cups of soil does Brenda have?

Find 278 + 13

8 by breaking up mixed numbers.

278 + 13

8 = (2 + 1) + (78 + 3

8) Add the Then add fractions. whole numbers.

278

+ 138

3108

Write the fraction as a mixed number.

3108 = 3 + 8

8 + 28 = 42

8

How Can You Add Mixed Numbers?

You can use properties of operations to add mixed

numbers. When you break apart a mixed number to add, you are using the Commutative and the

Associative Properties.

Brenda has 428 cups

of soil.

278

+ 138

108

278 = 2 + 78 = 16

8 + 78 = 23

8

138 = 1 + 38 = 8

8 + 38 = 11

8238 + 11

8 = 348

Sample answer: They are similar because you add the fractions, renaming them if necessary, and then add the whole numbers.




2

Learn Glossary

PearsonRealize.com


MP.1 Make Sense and PersevereWhat information is in the problem? [Brenda mixed 27

8 cups of potting mixture and 13

8 cups of sand.] What do you need to find? [How much soil Brenda had after mixing them together]

How are the Commutative and Associative Properties used? [The Commutative Property allows you to change the order of the addends and the Associative Property allows you to change the way addends are grouped. Together, the properties let you change (2 + 7

8) + (1 + 38) to

(2 + 1) + (78 + 38).]

Prevent Misconceptions

Some students may have difficulty understanding 310

8 and 428 are equivalent.

Show the equivalence with a number line or fraction strips.

11

Convince Me! MP.2 Reason Quantitatively After students use the Commutative and Associative Properties to rearrange the addends, the computation involves adding fractions and adding whole numbers. If they change the mixed numbers to fractions, they then use what they learned about adding fractions.

Coherence Adding mixed numbers uses several concepts students have learned previously. They use what they have learned about decomposing fractions and about adding fractions with like denominators. They also use properties of addition learned in earlier grades. They formalize what they learned in the previous lesson about adding mixed numbers using models to add symbolically.

Revisit the essential question. One way to add mixed numbers is to use properties to add the fraction parts, add the whole number parts, and then combine the sums. Another way is to change both mixed numbers to equivalent fractions and then add.


What operation should you use to find how much soil Brenda has? [Addition]

MP.7 Use StructureWhy does 27

8 + 138 = 23

8 + 118 ? [27

8 is equivalent to 238 and 13

8 is equivalent to 11

8 , so their sums are equivalent] Is there another way to change 34

8 to 428?

[Sample answer: Change 348 to 32

8 + 28 = 42

8]

514 Topic 9


20–30 min



23. a. Find the distance from the start of the trail to the end of the trail.

26. Higher Order Thinking A male Parson’s chameleon can be up to 233

4 inches long. It can extend its tongue up to 351

4 inches. What are 3 possible lengths for the chameleon when its tongue is extended?

b. Linda walked from the start of the trail to the bird lookout and back. Did Linda walk more or less than if she had walked from the start of the trail to the end?

27. How long an extension cord can Julie make by attaching a 223

8 foot and a 2668 foot cord

together? Select all the possible sums.

2238 + 266

8 = 3938

2238 + 266

8 = 49 912

2238 + 266

8 = 4918

2238 + 266

8 = 4898

2238 + 266

8 = 48

28. Mary skips 2213 yards down a trail, then

hops another 1523 yards. How far is Mary

down the trail? Select all the possible sums.

2213 + 152

3 = 37

2213 + 152

3 = 3733

2213 + 152

3 = 38

2213 - 152

3 = 623

2213 + 152

3 = 1143

For 23, use the map at the right.

24. Joe biked 1 912 miles from home to the

lake, then went some miles around the lake, and then back home. Joe biked a total of 4 9

12 miles. How many miles did Joe bike around the lake?

25. MP.2 Reasoning The bus took 435 hours

to get from the station to Portland and 34

5 hours to get from Portland to Seattle. How long did the bus take to get from the station to Seattle?

Tongue can extend up to 351

4 inches.

378 miles

278 miles

Sample answer: 668 miles

More; 768 miles + 66

8 miles

Sample answer: 1 312 miles

Sample answer: 2768 inches;

3134 inches; 59 inches

825 hours


MTH16_SE04_CC2_T09_L09.indd Page 516 19/06/14 7:54 AM s-w-100 /132/PE01513_SE/ENVISION_MATH_ENGLISH/NA/SE/2013/G4/XXXXXXXXXX/Layout/Interior_Fi ...


Buddy


Name


1. Brenda adds 118 cups of peat moss

to her soil in the problem on the previous page. How much soil does Brenda now have? Explain.

3. 178

+ 128

4. 2 410

+ 5 510

2. MP.1 Make Sense and Persevere Use another strategy to find the sum of 42

8 + 118.

For 3–8, find each sum.

Leveled Practice For 9–22, find each sum by adding mixed numbers or by adding equivalent fractions.

11. 256

+ 546

12. 11 710

+ 10 910

13. 978

+ 758

14. 578

+ 818

9. a. Add the fractions.

b. Add the whole numbers.

c. Write the fraction as a mixed number.

10. a. Write the mixed numbers as fractions.

b. Add the fractions.


15. 4 110 + 6 5

10 16. 9 712 + 4 9

12 17. 5 + 318 18. 83

4 + 734

19. 245 + 73

5 20. 326 + 85

6 21. 1 712 + 210

12 22. 368 + 93

8

5. 423 + 12

3 6. 6 512 + 411

12

7. 213 + 21

3 8. 8 912 + 5 5

12

136

+ 246

214 =

94

+ 324 = + 14

4

Sample answer: Using equivalent

fractions, 348 + 9

8 = 438 = 53

8



318 7 9

10

538 cups; Sample answer:

428 + 11

8 = 538

613

423

836

818

4 512

10 610

1025 121

6

1748

1624

1318

22 610

14

14 412

11 412

14 212

234 = 53

437

6 = 416



PearsonRealize.com

ToolsPractice Buddy

Assessment


Error Intervention: Item 8If students make computational errors changing mixed numbers with larger whole number parts to fractions,then encourage them to add the fractions, add the whole numbers, and then combine the sums.



Item 23b Explain that students can use number sense to solve this problem. Since 37

8 7 278, 37

8 + 378 7 37

8 + 278.

Item 24 Students can use a bar diagram to decide what computations are needed to solve the problem. They also may want to write equations.

Item 26 Higher Order Thinking Students should name fractions greater than the length of the chameleon, 233

4 inches, and less than or equal to the combined length of the chameleon and the fully extended tongue, 59 inches.

Items 27–28 Make sure students realize there is more than one way to represent a sum.

515–516


22



9-9

3. Use properties to add 146 + 25

6.

Add the fraction part. Add the whole numbers.

1

46

+ 256

14

6

+ 256

4. Use equivalent fractions to add 146 + 25

6.

146 = 6

6 + 46 =

256 = 6

6 + + 56 =

146 + 25

6 = +

= or

5. Find 218 + 32

8 using properties.

6. Find 112 + 31

2 using equivalent fractions.

On the Back!

7. Use properties or equivalent fractions to find 125 + 31

5.

538

Vocabulary

1. According to the Commutative Property of Addition, you can add numbers in any order.46 + 2 = 2 +

2. According to the Associative Property of Addition, you can regroup addends.

(1 + 46) + 2 = 1 + ( + 2)

5

46

396 or 43

696

66

46

106

106

176

276 43

6

176

435; Check students’ work.

R 9•9



PartnerTalk

Display the Digits


1 36 + 3 2

6 = 4 ab

9 38

+ 5 48

14 cd

12710

+ 9 810

22 5e f

414

+ 4 34

g

15 312 + 5 0

12 = 20 h1 i

11 310

+ 10 110

21 j10

a

b

c

d

e

f

g

h

i

j


Make up another addition puzzle with mixed numbers.Ask your partner to display the answers with 0–9 tiles.

Get Started or

Find each sum by adding mixed numbers. Use mental math. Display each 0–9 tile exactly once. If you have a partner, take turns.

9•9



PartnerTalk

Display the Digits



Make up another addition puzzle with mixed numbers.Ask your partner to display the answers with 0–9 tiles.

5 a8

+ 21 08

2 b 1 c

d 23

+ 5 e3

13 1f

1 81 g

+ 32 4h 0

3 i 210

136

+ 36j

a

b

c

d

e

f

g

h

i

j

Get Started or

Find each sum by adding mixed numbers. Use mental math. Display each 0–9 tile exactly once. If you have a partner, take turns.

9•9





STEP

3



Adding Mixed Numbers•Have students fold a piece of paper

into three sections and write one of the following steps in each section.

•Add the fractions and write the fraction as a mixed number if possible.

•Add the whole numbers.

•Add the sums.

•Ask them to write the problem 4 8

10 + 5 510 in the first section.

•Add the fractions in the first section. [ 810 + 5

10 = 1310] Can the sum be written

as a mixed number? How? [Yes; 1310 = 10

10 + 310 = 1 3

10]

•Add the whole numbers in the second section. [4 + 5 = 9]

•Add the sums in the third section. [9 + 1 3

10 = 10 310]

•Repeat with other pieces of paper and other problems.

Advanced On-LevelCenter GamesStudents work in pairs or small groups to add mixed numbers. Have students record the sums as they play the game.

517A Topic 9


15–30 min



13. A-ZA-Z Vocabulary Use the vocabulary words mixed number and fractions to complete the sentence.

When adding mixed numbers, you first add the , then add the whole numbers. Finally, you write the

.

14. MP.3 Critique Reasoning Alan used 9 as an estimate for 3 7

10 + 5 410. He added

and got 9 110 for the actual sum. Is Alan’s

answer reasonable?

15. Ruth needs 214 cups of flour for one cake

recipe and 234 cups of flour for another

cake recipe. If she makes both cakes, how much flour will Ruth need altogether?

17. Higher Order Thinking Tirzah wants to put a fence around her garden. She has 22 yards of fence material. Does Tirzah have enough to go all the way around the garden?

16. A “stone” is an old unit of weight used in Ireland and England to measure potatoes. A stone is 14 pounds and 80 stones make up half of a “long ton.” How many pounds is half of a long ton?

18. Pookie weighs 1278 pounds. Rascal weighs

1338 pounds. What is the total weight of

both cats? Select all the possible sums.

2628 pounds

2614 pounds

2108 pounds

1338 pounds

1278 pounds

19. Rex weighs 3014 pounds. Buckey weighs

5024 pounds. What is the total weight of

both dogs? Select all possible sums.

2014 pounds

80 pounds

8034 pounds

3234 pounds

81 pounds

4 812 yards

6 912 yards

No; Sample answer: Tirzah needs 2210

12 yards to go around the garden.

Yes; Sample answer: Since the sum, 9 1

10, is close to the estimate 9, Alan’s answer is reasonable.

1,120 pounds

fractions

mixed number

5 cups




Another Look!


NameHomework & Practice 9-9Add Mixed Numbers

You can add mixed numbers with like denominators using properties of operations.

For 1–12, find each sum by adding mixed numbers or adding equivalent fractions.

1.

210

12

+ 3 312

2.

13

8

+ 668

3.

5 4

10

+ 4 210

4.

102

6

+ 36

5. 3 312 + 6 8

12 6. 125 + 31

5 7. 21012 + 3 9

12 8. 726 + 85

6

9. 434 + 22

4 10. 11 910 + 3 2

10 11. 5 812 + 3 5

12 12. 211112 + 17 5

12

Randy played basketball for 256 hours on Saturday.

He played for 136 hours on Sunday. How many hours

did Randy play basketball on the weekend?

Add Mixed Numbers

a. Add the fractions.

b. Add the whole numbers.


Randy played basketball for 426 hours on

the weekend.

Add Fractions

a. Write the mixed numbers as fractions.

b. Add the fractions.


386 = 42

6

256

+ 136

= 426

256 = 17

6

+ 136 = + 9

6 266


6 112

91112

714

435

15 110

6 712

9 112

1616

39 412

818 9 6

10 1056




PearsonRealize.com


Games



Tools Games


Leveled Assignment I Items 1–6, 13, 15–19 O Items 5–10, 13–14, 16–19 A Items 8–14, 16–19

517–518



Name Daily Common Core Review9-10

1. Select all the ways to decompose 712.

112 + 3

12 + 212

212 + 4

12 + 112

512 + 1

12 + 212

412 + 3

12

112 + 1

12 + 112 + 1

12 + 112

2. Neil spends 128 hours washing the car

and 258 hours mowing and weeding

the yard. How many total hours does Neil spend on his chores?

� 338 hours

� 358 hours

� 368 hours

� 378 hours

3. Which of the fractions is equivalent to 23?

� 812

� 610

� 48

� 25

4. Find the difference.

2,612 – 1,009

� 3,621

� 1,611

� 1,603

� 1,207

5. Mr. DeWitt carved a wooden boat. He began with a piece of wood that was 2,037 millimeters long. He used 1,674 millimeters of wood for the boat. How many millimeters of the wood did Mr. DeWitt not use?

6. The distance between Miami and Naples is 107 miles. The distance between Miami and Jacksonville is about three times this distance. Landon estimates the distance between Miami and Jacksonville is about 400 miles. Is Landon’s estimate reasonable? Why or why not?

7. Use multiplication to write two equivalent fractions for 34.

4.NF.B.3b

4.NF.B.3c

4.OA.A.3

4.NF.A.1

4.NF.A.1

4.NBT.B.4

4.NBT.B.4

363 millimeters

No; Sample answer:

The distance is about

100 × 3 = 300 miles.

400 miles is too great

a number.

Sample answer:34 × 2

2 = 68;

34 × 3

3 = 912

D 9•10



Think

Today’s Challenge

Learn


Glossary

Animated Glossary

eText


Tools

Math Tools

Games

Math Games

Assessment

Quick Check

PD


Solve

Solve and Share

Help



Practice Buddy


LESSON 9-10SubtRaCt MixeD NuMbeRS


FOCuSDomain 4.NF Number and Operations—FractionsCluster 4.NF.b Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.Content Standard 4.NF.b.3c Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.Mathematical Practices MP.1, MP.2, MP.3, MP.8Objective Use equivalent fractions, properties of operations, and the relationship between addition and subtraction to subtract mixed numbers with like denominators.essential understanding Two procedures for subtracting mixed numbers both involve changing the calculation to a simpler equivalent calculation. These are extensions of the same procedures used for adding mixed numbers with like denominators.

COHeReNCeIn Lesson 9-8 students subtracted mixed numbers with like denominators using fraction strips and number lines. In the previous lesson, two procedures for adding mixed numbers with like denominators were developed. In this lesson, those same procedures are extended to subtracting mixed numbers with like denominators.

RiGORSimilar to Lesson 9-9, this lesson blends conceptual understanding and procedural skill. Both procedures require students to change the calculation to simpler (previously-learned) equivalent calculations. The first breaks the calculation into subtracting the fractional parts and then whole number parts. The second changes the mixed numbers to equivalent fractions; then the procedure for subtracting fractions with like denominators is used.

PD



Learning Use prior knowledge.


Write 234 + 12

4. Ask students what they remember about adding mixed numbers. Write 23

6 - 146. Explain that when subtracting

mixed numbers, first determine if the second fraction can be subtracted from the first fraction without renaming fractions. Remind students how to change a mixed number by renaming the whole number.

beginning Point to 236, then 14

6. Ask: Can you subtract 14

6 from 236? [Yes, if you

rename 236.] Write 23

6 = 196. Explain that

in order to complete the subtraction, the 236

needs to be renamed. Write 413 - 22

3. Ask students to rename the first mixed number.

intermediate Ask students to explain what mixed numbers are. Point to 23

6, then 146. Ask

students to identify the mixed number that will need to be renamed. Ask students to rewrite

the problem by renaming the first mixed number [19

6 - 146].

advanced Ask students to explain to partners what mixed numbers are. Ask students to rewrite and solve 23

6 - 146

by renaming the first mixed number [19

6 - 146 = 5

6]. Ask students to create mixed number problems for their partners to solve.

Summarize How are mixed number subtraction problems solved?

Think


519A topic 9


10–15 min

Solve

m

Home Aunt’s House

mile68

miles182


Name

I can …

Lesson 9-10Subtract Mixed Numbers

Look Back! MP.3 Critique Reasoning Sarah found Evan

has 278 miles left to walk. Is Sarah’s answer reasonable? Use estimation

to explain.

Generalize. You can use what you know about subtracting fractions to

solve this problem. Content Standard 4.NF.B.3c Mathematical Practices MP.1, MP.2, MP.3, MP.8

use equivalent fractions, properties of operations, and the relationship between addition and subtraction to subtract mixed numbers with like denominators.

Evan is walking 218 miles to his aunt’s house.

He has already walked 68 mile. How much farther does Evan have to go? Solve this problem any way you choose.

No; Sample answer: Evan has to walk about 2 miles. He has already walked more than a half mile. 2 − 1

2 = 112. Evan has

about 112 miles left to walk.




0

=

= =-

1 2 3

1

88_

38_

1 38_

2 18_

2 18_

Mark’s Work

118_

168_

68_

178_

178_

178_

118_

STEP


Solve

COHERENCE: Engage learners by connecting prior knowledge to new ideas.Students extend what they have learned about subtracting fractions with like denominators, subtracting mixed numbers with models, decomposing fractions, and the relationship between addition and subtraction to subtract mixed numbers.


BEFORE

1. Pose the Solve-and-Share ProblemMP.8 Generalize Listen and look for students who generalize what they have learned about subtracting fractions, writing mixed numbers as a fraction, or the relationship between addition and subtraction to add mixed numbers without using representations.

2. Build UnderstandingWhat do you know? [Evan needs to walk 21

8 miles in all; He has already walked 68 mile] What do you need to find? [How much farther Evan needs to walk] What operation can you use? [Subtraction]

DURING

3. Ask Guiding Questions As NeededWhich is greater 18 or 68? [68] How can you write an equivalent fraction for 21

8? [Sample answers: 218 = 1 + 8

8 + 18 = 19

8 or 21

8 = 88 + 8

8 + 18 = 17

8 ]

AFTER

4. Share and Discuss SolutionsStart with students’ solutions. If needed, project and discuss Carly‘s work to show a way to subtract.

5. Transition to the Visual Learning BridgeAs with adding, you can subtract mixed numbers by subtracting the fraction parts and the whole number parts. Sometimes, you need to regroup one whole before subtracting the fractions. You can also change both mixed numbers to equivalent fractions before subtracting.

6. Extension for Early FinishersCreate a word problem involving subtracting mixed numbers. Trade with a partner to solve.


Carly regrouped 218 as 19

8. Then she subtracted the fraction parts and the whole number parts.

Mark used a number line to represent the problem and renamed 21

8 as 178 . Then he subtracted 68 from 17

8 and used the number line to change his answer, 11

8 , to 138.

Solve

Whole Class

Whole Class

Small Group

Carly’s Work Mark’s Work

2 1 __ 8 = 1 + 1 + 1 __ 8

= 1 + + 1 __ 8 = 1 9 __ 8

2 1 __ 8 = 1 9 __ 8

— 6 __ 8 = — 6 __ 8

1 3 __ 8 miles to go

88

519


Visual Learning

A

Learn Glossary

B C

2 inches36

236 = 19

6

- 146 = 14

656

236 = 2 + 36 = 12

6 + 36 = 15

6

146 = 1 + 46 = 6

6 + 46 = 10

6156 - 10

6 = 56


Convince Me! MP.2 Reasoning Explain why you rename 414 to

find 414 - 3

4.

A golf ball measures about 146 inches across the

center. What is the difference between the distances across the centers of a tennis ball and a golf ball?

Find 236 - 14

6 by subtracting mixed numbers.

The tennis ball is 56 inch wider than the golf ball.

Find 236 - 14

6 by subtracting equivalent fractions.

How Can You Subtract Mixed Numbers?

You can use properties of operations and the relationship

between addition and subtraction to help subtract mixed numbers.

To subtract 46 from 36, rename 236.

Remember, 1 = 66.

236 = 2 + 36 = 1 + 6

6 + 36 = 196

You can count up to check your work! 146 + 26 = 2 and 2 + 36 = 23

6 26 + 36 = 56

Sample answer: Since 14 *34 you must decompose 41

4 and rename as 35

4.




2

Learn Glossary

PearsonRealize.com


MP.1 Make Sense and Persevere What do you know? [A golf ball measures 14

6 inches across the center and a tennis ball measures 23

6 inches.] What do you need to find? [The difference between these lengths.]

Convince Me! MP.2 Reason Quantitatively Tell students to rename the mixed number whenever the fraction in the number they are subtracting from is less than the fraction in the number they are subtracting. Have students use fraction strips to help them understand this concept.

Coherence There are three ways to subtract mixed numbers. All 3 build on concepts and skills developed earlier. Students can use what they learned about counting up mentally to subtract whole numbers and use this method with mixed numbers. They use what they have learned about subtracting fractions and about decomposing with the other two methods.

Revisit the essential question. There are three ways to subtract mixed numbers. One way is to subtract the whole number and the fraction parts separately, which may require renaming the

minuend. Another way is to change to equivalent fractions. The third way is to count up.

Visual Learning BridgeWhy do you need to rename 23

6 before you subtract? [36 6 4

6] Is the answer 56 reasonable? How do you know? [Yes; 23

6 - 146 is about

212 - 11

2 = 1. Since 56 is close to 1, it is a reasonable answer.]Prevent MisconceptionsSome students may not subtract the whole numbers

and get an answer of 156.

Ask them what 1 - 1 is and

have them write 056 = 5

6.

11

MP.3 Construct ArgumentsWhich method is easier to use, subtracting fractions and whole numbers separately or changing to equivalent fractions? Explain. [Answers will vary; Check that students can justify their choice.] When counting up to check your answer, why do you add 26 + 3

6? [First add 26 to 14

6 to get 2, then add another 36. Add 26 and 36 to see how much was added in all.]

520 Topic 9


20–30 min



31. Last week, the office used 7 112 boxes

of paper. This week, they used 3 512 boxes

of paper. How many more boxes did they use last week than this week? Use equivalent fractions to solve.

𝖠𝖠𝖠𝖠 1012 boxes

𝖡𝖡𝖡𝖡 423 boxes

𝖢𝖢𝖢𝖢 413 boxes

𝖣𝖣𝖣𝖣 3 812 boxes

32. A store sold 615 cases of juice on Friday and

445 cases of juice on Saturday. How many

more cases of juice did the store sell on Friday than on Saturday?

𝖠𝖠𝖠𝖠 11 cases

𝖡𝖡𝖡𝖡 315 cases

𝖢𝖢𝖢𝖢 225 cases

𝖣𝖣𝖣𝖣 125 cases

25. The average weight of a basketball is 211

8 ounces. The average weight of a baseball is 52

8 ounces. How many more ounces does the basketball weigh?

29. Jack made 514 dozen cookies for the

bake sale, and his sister made 334 dozen

cookies. How many more dozen cookies did Jack make than his sister?

27. Two of the smallest mammals on Earth are the Bumblebee Bat and the Etruscan Pygmy Shrew. How much shorter is the bat than the shrew?

28. MP.1 Make Sense and Persevere The average length of an adult female hand is about 63

5 inches. About how much longer is the hand than the lengths of the bat and shrew combined?

26. What is the value of the 4 in 284,612?

30. Higher Order Thinking Jenna has a spool that contains 53

4 meters of ribbon. She uses 32

4 meters for a school project and 11

4 meters for a bow. How much ribbon remains on the spool?

Bumblebee BatLength 11

5 inches

Etruscan Pygmy Shrew Length 12

5 inches

1578 ounces

1 meter

4,000

15 inch

About 4 inches

Sample answer: 124 dozen




Buddy


Name


1. A hole at the golf course is 336 inches

wide. How much wider is the hole than the golf ball?

2. MP.3 Construct Arguments Why might you need to rename some whole numbers when subtracting?

For 3–8, find each difference.

For 9–24, find each difference by subtracting mixed numbers or subtracting equivalent fractions.

3. 758

- 248

518

4. 5

- 234

214

5. 6 310 - 1 8

10 6. 9 412 - 4 9

12

7. 456 - 21

6 8. 1 912 - 10

12

13. 613 - 52

3 14. 924 - 63

4 15. 838 - 35

8 16. 7 - 312

17. 1516 - 45

6 18. 13 112 - 8 3

12 19. 625 - 23

5 20. 10 510 - 4 7

10

21. 12 912 - 10 7

12 22. 2514 - 20 23. 17 - 21

8 24. 2635 - 134

5

9. 878

- 248

10. 4 510

- 1 910

11. 418

- 148

12. 6

- 245

Sample answer: When subtracting larger fractions from smaller fractions you need to rename whole numbers with equivalent fractions.

156 inches

4 510

246

4 712

1112



468

345

1245

1478

23

1026

2 212

312

5 810

234

41012

514

638 2 6

10 258

315



PearsonRealize.com

ToolsPractice Buddy

Assessment


Error Intervention: Item 6If students have difficulty with subtracting mixed numbers,then encourage them to draw fraction strips or a number line as needed. For a problem like item 6, encourage students to subtract the fraction and whole number parts separately or to count up. Changing to equivalent fractions can lead to computations with greater numbers and errors.



Item 25 Encourage students to think about the best way to solve the problem before starting. Since 211

8 is relatively large, renaming as 209

8 may be the best approach.

Item 28 MP.1 Make Sense and Persevere Students can use bar diagrams to help them decide what computations are needed to solve this two-step problem. They may also want to write equations. Ask: What should you do first to solve this problem? [Find 11

5 + 125] What

operation will you use to find how much longer an average female hand is than the combined lengths of the bat and shrew? [Subtraction]

Coherence Subtracting mixed numbers uses many concepts and skills students have already learned, such as subtracting fractions and decomposing. In the next lesson, students apply the computational skills they have developed throughout the topic to model and solve problems.

521–522


22



9-10

2. You can subtract 338 - 15

8 by decomposing and writing 338 as an

equivalent mixed number. Subtract the fraction part, and then subtract the whole numbers.

3. You can also use equivalent fractions to subtract 338 - 15

8.

338 = 8

8 + + + 38 =

158 = + 5

8 =

338 - 15

8 = -

= or

On the Back!

4. Decompose or use equivalent fractions to find 4 310 - 2 9

10.

Vocabulary

1. To decompose a mixed number means to break the number into the sum of two or more parts.

338 = 3 + 3

8

You can decompose a mixed number to find an equivalent mixed number or two mixed numbers that represent the same amount.

338 = 3 + 3

8

Decompose the 3.

338 = (2 + 8

8) + 38

Use the Associative Property to change the grouping.

338 = 2 + ( + 3

8)= 2 +

=

338 =

- 158 = 15

8

88

88

88

88

278

138138

148 16

8

278

118

2118

168

2118

1 410; Check students’ work.

R 9•10



PartnerTalk

Teamwork


Get Started or

Get paper and a pencil. Put in a bag.


Choose A, B, C, D, E, or f. Pick a tile. Pick two tiles if your group has only two students. Do the jobs listed below in order. To find your job, find the number that matches the tile you chose.

321 4


Make up a subtraction problem with mixed numbers. Ask your team to complete steps 1–4 for your problem.

1 Read the subtraction problem to your team.

2 Explain how to find the difference by subtracting the mixed numbers.

3 Work with your team to compute the difference. Make sure everyone has the same answer.

4 Explain why the answer is reasonable.

834

— 214

1225

— 415

738 — 12

8

536 — 21

69 7

10— 3 3

10

1146

— 126

a

d

b

e

c

f

9•10



PartnerTalk

Teamwork


Get Started or

Get paper and a pencil. Put in a bag.


Choose A, B, C, D, E, or f. Pick a tile. Pick two tiles if your group has only two students. Do the jobs listed below in order. To find your job, find the number that matches the tile you chose.


Make up a subtraction problem with whole numbers and mixed numbers. Ask your team to complete steps 1–4 for your problem.

1 Read the subtraction problem to your team.

2 Decide whether or not you will need to find an equivalent fraction for the first number. Tell your group how you know.

3 Work with your team to subtract the numbers. Make sure everyone has the same answer.

4 Explain why the answer is reasonable.

5 — 123

4 — 13

4

2 — 11

2

8 — 21

66 — 31

37

— 216

321 4

a

d

b

e

c

f

9•10


Advanced On-LevelCenter GamesStudents work in pairs or small groups to subtract mixed numbers. Have students record the differences as they play the game.

Rename ifnecessary

Subtractfractions

Subtract thewhole numbersand write thedifference




STEP

3



Subtracting Mixed Numbers•Have students fold a piece of paper

into three sections and write one of the following steps in each section.

•Rename if necessary

•Subtract fractions

•Subtract the whole numbers and write the difference

•Ask them to write 1216 - 53

6 in the first section.

•Do you need to rename 1216? [Yes]

Show how to rename it in the first section of the paper. [121

6 = 11 + 66 +

16 = 117

6]

•Subtract the fractions in the second section. [76 - 3

6 = 46]

•Subtract the whole numbers in the third section and write the difference. [11 - 5 = 6; 64

6]

•Repeat with other pieces of paper and other problems.

523A Topic 9


15–30 min

Giant Walking Stick Length 53 3

10 cmRhinoceros Beetle

16 710 cm

Giant Weta Beetle 8 5

10 cm


17. A-ZA-Z Vocabulary Use a vocabulary word to complete the sentence.

A number that has a whole number part and a fraction part is a called a(n)

.

18. Some of the world’s smallest horses include Thumbelina who stands 171

4 inches tall, Black Beauty who stands 182

4 inches tall, and Einstein who stands 14 inches tall.

a. How much taller is Black Beauty than Thumbelina?

b. How much taller is Thumbelina than Einstein?

19. MP.2 Reasoning If Carol hangs a picture using 38 yard of a wire that is 11

8 yards long, how much wire will Carol have left?

20. Write 6,219 in expanded form.

21. Higher Order Thinking Some of the largest insects in the world include the Rhinoceros Beetle, the Giant Walking Stick, and the Giant Weta Beetle. How much longer is the Giant Walking Stick than the Rhinoceros Beetle and the Giant Weta Beetle combined?

Common Core Assessment22. Jessie needs a board 7 9

12 feet long. She has a board 9 1

12 feet long. How much of the length does Jessie need to cut from the board? Use equivalent fractions to solve.

𝖠𝖠𝖠𝖠 113 feet

𝖡𝖡𝖡𝖡 2 812 feet

𝖢𝖢𝖢𝖢 223 feet

𝖣𝖣𝖣𝖣 161012 feet

23. Robyn ran 534 miles last week. She ran

414 miles this week. How many more miles

did Robyn run last week? Use equivalent fractions to solve.

𝖠𝖠𝖠𝖠 114 miles

𝖡𝖡𝖡𝖡 112 miles

𝖢𝖢𝖢𝖢 134 miles

𝖣𝖣𝖣𝖣 10 miles

6,000 + 200 + 10 + 9

Sample answer: 68 yard

mixed number

114 inches

314 inches

28 110 cm




Another Look!


NameHomework & Practice 9-10Subtract Mixed Numbers

You can subtract mixed numbers with like denominators using properties of operations.

Janet grew a pumpkin that weighs 1334 pounds

and a melon that weighs 824 pounds. How much

heavier is the pumpkin than the melon?

The pumpkin is 514 pounds

heavier than the melon.

For 1–16, find each difference by subtracting mixed numbers or subtracting equivalent fractions.

9. 814 - 73

4 10. 2 910 - 2 5

10 11. 656 - 54

6 12. 3 - 134

1. 1034

- 714

2. 746

- 236

3. 3

- 223

4. 17 812

- 12 312


Subtract Mixed Numbers

a. Subtract the fractions. Rename whole numbers as fractions as needed.

b. Subtract the whole numbers.

Subtract Fractions

a. Write the mixed numbers as fractions.

b. Subtract the fractions.


1334 = 55

4

- 824 = - 34

4214 = 51

4

13. 11 - 212 14. 42 6

10 - 10 15. 1815 - 22

5 16. 2726 - 121

6

1334

- 824

514

5. 926 - 65

6 6. 415 - 23

5 7. 6 312 - 3 4

12 8. 528 - 37

8236

24

812

135

410

32 610

21112

116

1545

138

114

1516

324 51

6

13 5 5

12




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Tools Games


Leveled Assignment I Items 1–8, 17–18, 20, 22–23 O Items 1–4, 9–12, 17–18, 21–23 A Items 9–17, 19, 21–23

523–524



Name Daily Common Core Review9-11

1. Kevin drank 314 pints of water and

134 pints of juice. How much more

water than juice did Kevin drink?

𝖠𝖠𝖠𝖠 34 pint

𝖡𝖡𝖡𝖡 114 pints

𝖢𝖢𝖢𝖢 124 pints

𝖣𝖣𝖣𝖣 134 pints

2. Which fraction is the sum of 28 + 48?

𝖠𝖠𝖠𝖠 28

𝖡𝖡𝖡𝖡 48

𝖢𝖢𝖢𝖢 68

𝖣𝖣𝖣𝖣 88

3. Which fraction makes the number sentence true?35 7 □

𝖠𝖠𝖠𝖠 36

𝖡𝖡𝖡𝖡 46

𝖢𝖢𝖢𝖢 56

𝖣𝖣𝖣𝖣 66

4. Round 12,218. Select all that apply.

10,000

12,000

12,200

12,220

12,500

5. The table below shows the populations of 4 cities.

City Population

Happy Valley 49,604

Lakeside 50,104

Stoneyville 49,984

Rutherton 50,673

Write a comparison using 6 or 7 for the city with the greatest population and the city with the least population.

6. Is the sum of 710 + 1

5 closer to 0, 12, or 1?

7. Kelly ran 38 of a race before resting. Then she ran another 38 of the race before resting again. How much of the race does Kelly have left to run? Write and solve equations.

50,673 + 49,604 or

49,604 * 50,673

Closer to 1

28; Sample answer:38 + 3

8 = r; r = 68;

88 − 6

8 = n; n = 28

4.NF.B.3c

4.NF.B.3a

4.NF.A.2

4.NBT.A.3

4.NBT.A.2

4.NF.B.3a

4.NF.B.3d

D 9•11


Think

Today’s Challenge

Learn


Glossary

Animated Glossary

eText


Tools

Math Tools

Games

Math Games

Assessment

Quick Check

PD


Solve

Solve and Share

Help



Practice Buddy

MP

Math Practices Animations


MATH PRACTICES AND PROBLEM SOLVING | LESSON 9-11

MODEL WITH MATH


FOCUSMathematical Practices MP.4 Model with Mathematics. Also MP.1, MP.2, MP.5

Domain 4.NF Number and Operations—FractionsCluster 4.NF.B Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.Content Standard 4.NF.B.3d Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. Also 4.NF.B.3a.Objective Use previously learned concepts and skills to represent and solve problems.

Essential Understanding Good math thinkers choose and apply math they know to show and solve problems from everyday life.

COHERENCEStudents have used this Mathematical Practice throughout the program prior to this lesson. This lesson is an opportunity to stop and focus on the Thinking Habits good problem solvers use when they model with mathematics. Although the content used in this lesson was developed in this topic, instruction should focus on the use and management of MP.4.

RIGORThis lesson emphasizes applications. Rigorous mathematics instruction calls for the selection, use, and management of multiple mathematical practices. All of the problems in this lesson elicit the use of multiple mathematical practices. For example, making sense of problems and persevering, MP.1, is required to solve all problems. The classroom conversation should focus on the meaning and use of the Thinking Habits shown on the Solve & Share task for MP.4.

PD



Reading Derive meaning from print.


Read the Thinking Habits to students. Ask: How are these habits useful when solving problems? Ask students to copy the Thinking Habits in their notebooks. As students are writing the Thinking Habits, write them on a poster for classroom display. Tell students they will refer to the habits to solve problems.

Reread the Thinking Habits with students. Ask students which of the habits would help them solve the problem.

Beginning Ask students to draw symbols or pictures next to each habit in their notebooks to help remember each habit. Students will work as a group to solve the problem.

Intermediate Ask students to read the Thinking Habits aloud, and explain what each habit means in their own words.

Advanced Students will solve the problem using one of the Thinking Habits. They will explain to a partner how the Thinking Habits helped solve the problem.

Summarize How can students use the Thinking Habits to help solve problems?

Think



525A Topic 9

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Solve

DA

TA

Day of the Week

Time Jamie Studied

Tuesday 134 hours

Wednesday 34 hour

Thursday 24 hour

Math Practices and Problem Solving

Name


I can …

Look Back! MP.4 Model with Math What representations can you use to help solve this problem?

Mathematical Practices MP.4 Also MP.1, MP.2, MP.5 Content Standards 4.NF.B.3d, 4.NF.B.3a

use math I know to represent and solve problems.

Lesson 9-11Model with Math

The table shows how long Jamie studied for a math test over 3 days. How much more time did Jamie spend studying on Tuesday and Wednesday than on Thursday?

Thinking HabitsBe a good thinker!

These questions can help you.

• How can I use math I know to help solve this problem?

• How can I use pictures, objects, or an equation to represent the problem?

• How can I use numbers, words, and symbols to solve the problem?

Sample answer: I can use bar diagrams or drawings to help solve the problem.



MTH16_SE04_CC2_T09_L11_MP_VLB.indd 525 12/05/14 10:06 AM

10–15 min

DEVELOP: PROBLEM-BASED LEARNINGSTEP


Solve

COHERENCE: Engage learners by connecting prior knowledge to new ideas. Students extend their understanding of solving problems involving addition and subtraction of whole numbers to solving problems with fractions and mixed numbers. They use what they learned about adding and subtracting fractions and mixed numbers with like denominators.

BEFORE

1. Pose the Solve-and-Share ProblemMP.4 Model with Math In this problem, students draw bar diagrams to analyze given information, and write and solve equations to solve a two-step problem about time.

2. Build Understanding What information are you given? [The amount of time Jamie studied over 3 days.] What are you asked to do? [Find how much more time Jamie studied on Tuesday and Wednesday than on Thursday.]

DURING

3. Ask Guiding Questions As NeededWhat operation will you use to find how many hours Jamie studied on Tuesday and Wednesday? [Addition] What operation will you use to find how much more time Jamie studied on Thursday than on the other two days combined? [Subtraction] How can bar diagrams help you analyze information? [Sample answer: A bar diagram shows how information is related.]

AFTER

4. Share and Discuss SolutionsStart with students’ solutions. If needed, project Estela’s work to discuss how to write equations using bar diagrams.

5. Transition to the Visual Learning BridgeMaking a bar diagram can help you understand how quantities in a problem are related.Analyzing relationships can help you write an equation to solve a problem.

6. Extension for Early FinishersHow many hours did Jamie study in all? Draw a picture and write an equation to solve. [Check students’ drawings; 13

4 + 34 + 2

4 = x; x = 18

4 or 3]


Solve

Whole Class

Whole Class

Small Group

Estela’s Work Aiden’s Work

Estela draws bar diagrams and writes equations to model and solve the problem.

Aiden solves the wrong problem.

y

24_hr2

24_hr

+34_

1 or64_ 2 2

4_

34_

x =

= x

y = 2

2_ 24_- = y 2

4

Jamie spent 2 hours more studying on Tuesday and Wednesday than on Thursday.

x

34_hr1

1

34_hr

525

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Visual Learning

B

A

Learn Glossary

C

510

9101

410

2 miles on Saturday

Convince Me! MP.4 Model with Math How do the bar diagrams help you decide if your answer makes sense?

Here’s my thinking.

Find 2 410 - 6

10.

Use a bar diagram and write an equation to solve.

2 410 - 6

10 = d d = 1 810

Brad and his father hiked 1 810 miles

farther on Saturday than on Sunday.

I can

• use previously learned concepts and skills.

• use bar diagrams and equations to represent and solve this problem.

• decide if my results make sense.

How can I model with math?

What do you need to find?

I need to find how far Brad and his father hiked on Saturday and much farther they hiked on Saturday than on Sunday.

Brad and his father hiked the Gadsen Trail and the Rosebriar Trail on Saturday. They hiked the Eureka Trail on Sunday. How much farther did they hike on Saturday than on Sunday?

How Can You Use Math to Model Problems?

Gadsen Trail 1 9

10 mile

Eureka Trail 6

10 mile

Rosebriar Trail5

10 mile

610

410

2 miles

d

Sample answer: The first bar diagram represents the sum of 1 910 + 5

10.

1 910 is close to 2 and 5

10 is equal to 12. 2 + 12 = 21

2 which is close to 2 410.

The second bar diagram represents the difference. 2 410 − 6

10 is a little

more than 212 − 1

2. The answer 1 810 is a little more than 11

2, so the

difference is reasonable.

526 Topic 9 Lesson 9-11 © Pearson Education, Inc. 4

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STEP

2

Learn Glossary

PearsonRealize.comDEVELOP: VISUAL LEARNING

PURPOSE: The Visual Learning Bridge connects students’ thinking in the Solve & Share to important math ideas in the lesson. Use the Visual Learning Bridge to make these ideas explicit. Also available as a Visual Learning Animation Plus at PearsonRealize.com.

What kind of diagram would help you solve this problem? [A bar diagram]

MP.4 Model with MathWhat previously learned concepts and skills can you use to solve this problem? [Adding and subtracting fractions and mixed numbers] What equation can you write to go with the first step in the problem? [1 9

10 + 510 = h]

Convince Me! MP.4 Model with Math Modeling with math involves translating a problem situation into mathematics, such as an equation. Bar diagrams can help with this translation. Then, the equation is solved and translated back to the problem situation. It also involves the important, but often overlooked step, of making sure the answer makes sense in terms of the mathematics and the situation.

Coherence Students have learned to draw bar diagrams and write equations to model and solve problems involving whole numbers with all 4 operations. In this lesson, they extend that understanding for problems involving adding and subtracting fractions and mixed numbers. They also use the computation skills developed throughout the chapter.

Revisit the essential question. Bar diagrams can be used to show how the information in a problem is related. They can also show how to write and solve an equation.

How does the bar diagram help with writing the equation? [The bar diagram compares the greater quantity to the lesser quantity. The bar diagram shows the relationships in the problem and can be represented by the equation 2 4

10 - 610 = d.]

526 Topic 9


20–30 min

Start Finish

mile38 mile4

8 mile28

miles581

Common Core Performance AssessmentOn Safari Sandra and Ron traveled in a safari car while they were in Tanzania. The diagram shows the distances in miles they traveled from start to finish. How far did Sandra and Ron travel from the leopards to the elephants?

7. MP.2 Reasoning What quantities are given in the problem and what do they mean?

8. MP.1 Make Sense and Persevere What is a good plan for solving the problem?

9. MP.4 Model with Math Draw pictures and write and solve equations to find how far Sandra and Ron travel from the leopards to the elephants.

Zebras Leopards

Elephants

Sample answer: 38 mile is the distance from the start to the zebras, 48 mile from the zebras to the leopards, 15

8 miles from the start to the elephants, and 28 mile from the elephants to the finish.

Sample answer: Add the distances from the start to the zebras and the zebras to the leopards. Subtract that sum from 15

8 mi to find the distance from the leopards to the elephants.

When you model with math, you use a picture, which shows how the quantities in the

problem are related.

38 + 4

8 = m; m = 78 mile

m mile

38

48

78

58 miles1

f

158 − 7

8 = f ; f = 68 mile


MTH16_SE04_CC2_T09_L11_MP.indd 528 14/05/14 7:14 PM


Buddy

Name

MP.4 Model with Math

Alisa hiked a trail that was 910 mile and Joseph

hiked a trail that was 510 mile. How much farther

did Alisa hike than Joseph?

1. Draw a bar diagram to represent the problem and show the relationships among the quantities.

The smallest female spider measures about 35 millimeter in length. The smallest male spider measures about 15 millimeter in length. How much longer is the smallest female spider than the smallest male spider? Use Exercises 4–6 to answer the question.

4. Draw a picture and write an equation to represent the problem.

2. What equation can you write to represent the problem?

5. What previously learned math can you use to solve the problem?

MP.4 Model with Math When you model with math, you use math to represent and solve

a problem.

3. How much farther did Alisa hike than Joseph?

6. How much longer is the smallest female spider than the smallest male spider?

*For another example, see Set F on page 534.

Sample answer: I know how to subtract fractions that have the same denominator.

Sample answer: n = 35 − 1

5

Sample answer: d = 910 − 5

10

410 mile

25 mm

Sample drawing given.

d510

910

n15

35 mm


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PearsonRealize.com

ToolsPractice Buddy

Assessment

QUICK CHECKCheck mark indicates items for prescribing differentiation on the next page. Items 4 and 6 are worth 1 point. Items 7–9 are worth up to 3 points.

MP.4 Model with Math Listen and look for these behaviors as evidence that students are exhibiting proficiency with MP.4.

•Identifies the correct prior knowledge that needs to be applied to solve a problem

•Identifies the hidden question(s) in multiple-step problems•Uses numbers, symbols, and words to solve problems•Identifies the operation(s) needed to solve a problem•Uses estimation as appropriate

Item 4 MP.4 Model with Math If students have difficulty drawing a picture and writing an equation, have them identify the quantities given in the problem and what the numbers mean.

11 Reteaching Assign Reteaching Set F on p. 534.

Item 7 MP.2 Reason Quantitatively This item has 4 given quantities and one unknown quantity; students should use reasoning to identify each.

Item 8 MP.1 Make Sense and Persevere Students must decide which quantities are needed to solve the problem, which are not, and which quantity they are asked to find.

Item 9 MP.4 Model with Math Students will apply their skills with adding fractions and subtracting fractions and mixed numbers.

527–528


22

512

1112

n

610

910

mile from school

distance to Margo’s house

x

910

610

- = x



9-11

3. Complete the bar diagram and the equation for the following problem. Then solve.

Janice lives 910 mile from school. Each day, she walks

610 mile to Margo’s house. Then they walk the rest of the way to school together. What fraction of a mile does Margo live from school?

Margo lives mile from school.

4. Draw a bar diagram and write an equation to solve the problem.

Jeff has a full tub of popcorn. He eats 56 of the popcorn. What fraction of the popcorn remains in the tub?

On the Back!

5. Draw a bar diagram and write an equation to solve the problem.

Jared fills 58 of his photo album. Then he fills another 28 of the album. What fraction of the album is filled?

Vocabulary

1. You can model with math by drawing a bar diagram and writing an equation.

A variable is used to show the unknown information in a bar diagram.

What is the variable in the bar diagram shown?

2. An equation is a number sentence that uses an equal sign to show that two expressions have the same value. Write an equation to represent the situation shown in the bar diagram.

512 + 11

12 = n

n

78; Sample answer: 58 + 2

8 = x ; x = 78 ;


310

16; Sample answer: 66 − 5

6 = p; p = 16;


R 9•11





STEP

3



Math Practices and Problem Solving: Model with Math•Write the following on the board:

Shawn had 78 pound of dried fruit. He shared 58 pound with his friends. How much fruit does Shawn have left?

•Have students model the problem with a bar diagram. Which operation will you use to solve the problem? [Subtraction] What is the initial amount? [78] Which part will you take away? [58] What do you need to find? [The amount or part that is left]

•Have students write and solve an equation for their diagrams. [78 - 5

8 = 28 pound]

•Repeat the process with an addition problem. For example, Sophia walked 36 mile to the library. Then she walked 26 mile to the park. How far did Sophia walk in all?

36 + 2

6 = 56 mile

Problem-Solving Reading MatHave students read the Problem-Solving Reading Mat for Topic 9 and then complete Problem-Solving Reading Activity 9-11.

See the Problem-Solving Reading Activity Guide for other suggestions on how to use this mat.

529A Topic 9

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15–30 min

Trail Mix Ingredients34 cup almonds14 cup pumpkin seeds24 cup coconut34 cup dried cranberries

124 cup walnuts

1 cup granola24 cup dried bananas

Common Core Performance Assessment

When you model with math, you represent the relationships in the problem.

Ian and Rachel each made a trail mix. The amounts of ingredients they have are shown. Ian used all of the coconut, dried cranberries, and dried bananas to make his trail mix. Rachel made 2 cups of trail mix containing all of the almonds, pumpkin seeds, and granola. How much trail mix did Ian make? How much more trail mix did Rachel make than Ian?

4. MP.1 Make Sense and Persevere What do you know, and what do you need to find?

5. MP.5 Use Appropriate Tools What tools could you use to help solve this problem?

7. MP.1 Make Sense and Persevere Write and solve an equation to find how much trail mix Ian made.

6. MP.2 Reasoning How can you use a bar diagram to show how the quantities are related?

8. MP.2 Reasoning Explain how you were able to calculate how much more trail mix Rachel made than Ian.

Sample answer: The amounts of the ingredients are given; I need to determine how much trail mix Ian made and how much more Rachel made than Ian.

Sample answer: Fraction strips, paper and pencil, or bar diagrams

Sample answer: 2 − 134 = m; m = 1

4 cup

24 + 3

4 + 24 = c; c = 7

4 or 134 cups

Sample answer: Use the bar diagram to show the sum of the numbers of cups of ingredients Ian used to make his trail mix. Another bar diagram is used to show the difference between the sum of the number of cups Ian’s ingredients and the 2 cups of Rachel’s trail mix.

24

24

34

c cups 2 cups

m341




58

38 n

Another Look!


Name

MP.4 Model with Math On Nick’s playlist, 5

12 of the songs are pop. What fraction of the songs are NOT pop? Use Exercises 1–3 to answer the question.

2. What previously learned math can you use to solve the problem?

1. How can you draw a picture and write an equation to represent the problem?

Homework & Practice 9-11Model with Math

When you model with math, you use previously learned math to solve

a problem.

Tina built 18 of a model airplane on Saturday and 48 on Sunday. She built 38 more on Monday. How much more of the model airplane did she build on the weekend than on the weekday?

Tell how you can use math to model the problem.

• I can use previously learned concepts and skills.

• I can use bar diagrams and equations to represent and solve this problem.

• I can decide if my results make sense.

Draw a bar diagram and write and solve equations.

18 + 4

8 = 58 on the weekend

58 - 38 = n

n = 28

Tina built 28 more of the model airplane on the weekend than on the weekday.

3. What fraction of the songs on Nick’s playlist are NOT pop?712 of the songs

Sample answer: I know how to subtract fractions with the same denominator. I also know 12

12 = 1.

Sample answer: n = 1212 − 5

12

n

1212

512



PearsonRealize.com


Games



Tools Games


Leveled Assignment I Items 1–8 O Items 1–8 A Items 1–8

529–530


531 Topic 9

PearsonRealize.com

Practice Buddy

Games

Name Fluency Practice Activity

Clues The sum is exactly 1,000.A The difference is exactly 437.E

The sum is exactly 1,001.B The difference is between 150 and 160.

F

The difference is exactly 371.C The sum is between 995 and 1,000.

G

The difference is between 40 and 45.

D The sum is exactly 1,899.H

409− 252

900− 529

909+ 990

506+ 494

580+ 417

560− 123

601− 560

309+ 692

Work with a partner. Point to a clue. Read the clue.

Look below the clues to find a match. Write the clue letter in the box next to the match.

Find a match for every clue. Content Standard 4.NBT.B.4

I can …add and subtract multi-digit whole numbers.

TOPIC

9

F C H A

G E D B

157 371

997

1,899

437 41 1,001

1,000

531Topic 9 Fluency Practice Activity

MTH16_SE04_CC2_T09_FPA.indd 531 21/04/14 7:35 AM

fluency practice activityTOPIC

9

Students practice fluently adding and subtracting whole numbers and estimating sums and differences during a partner activity that reinforces mathematical practices.

Common Core StandardsContent Standard 4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.Mathematical Practices MP.2, MP.6

Getting Started Ask students to work with a partner. Tell them to record their matches on their own page. Go over the directions.

As Students Do the Activity Remind students that each clue can be matched with only one problem.

Encourage students to use estimation to help choose the matches efficiently. Some students may find all of the answers first and then match the clues. Allow this strategy as it provides the same fluency practice.

Another Activity Have students work together to write a new set of clues for the problems on the page. Ask them to record the new clues on a separate sheet of paper.

Extra Challenge Create your own Find a Match activity. Use the same clues on your page. Write a new problem for each clue. Then trade your activity with your partner and complete your partner’s Find a Match activity.

Games

Online Game The Game Center at PearsonRealize.com provides opportunities for fluency practice.

Practice Buddy

Steps to Fluency Success To ensure all students achieve fluency, see pages 43K–43N for additional resources including practice/assessment masters and online practice/assessment on fluency subskills. You can also use the ExamView® CD-ROM to generate worksheets with multiple-choice or free-response items on fluency subskills.

fluenCy PraCtiCe aCtivity

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Glossary Games

PearsonRealize.com

GlossaryVocabulary Review


TOPIC

9 Word List

• decompose

• denominator

• equivalentfractions

• fraction

• likedenominators

• mixednumber

• numerator

• wholenumber

Understand Vocabulary

1. Circlethelabelthatbestdescribes12.

fraction mixednumber wholenumber





4. Drawalinefromeachtermtoitsexample.

Use Vocabulary in Writing

5. Find113 + 22

3.Useatleast3termsfromtheWordListtodescribehowtofindthesum.

decompose 12 = 5

10

denominator 23 = 1

3 + 13

equivalentfractions 56

likedenominators 13 + 2

3 = 33

numerator 78

4; Sample answer: To add mixed numbers with like denominators, you can change the mixed numbers to equivalent fractions and add. You add the fractions, regroup, and then add the whole numbers. 11

3 + 223 = 4

532 Topic 9 VocabularyReview

MTH16_SE04_CC2_T09_VR.indd 532 24/04/14 11:59 AM

vocabulary review

Students review vocabulary words used in the topic.

Oral Language Before students complete the page, you might reinforce oral language through a class discussion involving one or more of the following activities.

• Have students define the terms in their own words.

• Have students say math sentences or math questions that use the words.

• Play a “What’s My Word?” guessing game in which you or a student thinks about one of the words and says a clue that others listen to before they guess the word.

• Play a “Right or Wrong?” game in which you or a student says a sentence that uses one of the words correctly or incorrectly. Then, others say “right” or “wrong.”

Writing in Math After students complete the page, you might further reinforce writing in math by doing one or more of the following activities.

• Tell students to close their books. Then, say the words and have students write them. Students trade papers to check if the words are spelled correctly.

• Have students work with a partner. Each partner writes a math question that uses one of the words. Then students trade papers and give a written answer that uses the word.

Games

Online Game The Game Center at PearsonRealize.com includes a vocabulary game that students can access anytime.

vocabulary review

532

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0 124

14

34

Reteaching

Set A

Set B

Set C

Name TOPIC

9pages 465–482

Find 58 + 28.

Add the numerators. Keep the like denominator.

pages 483–494

pages 495–500

Remember you can decompose fractions and use properties of operations to help add.

1. 25 + 2

5 2. 24 + 1

4 + 14

3. 38 + 4

8 4. 410 + 2

10 + 310

5. 410 + 3

106. 7

12 + 212

Find 58 - 28.

Subtract the numerators. Keep the like denominator.

Find the sum or difference shown on each number line.

Remember you can use fraction strips and number lines to show how to subtract fractions.

1. 33 - 1

3 2. 56 - 2

6

3. 68 - 3

8 4. 410 - 3

10

5. 55 - 3

56. 4

6 - 26

0 1

28 = 1

8 + 18

58 = 1

8 + 18 + 1

8 + 18 + 1

8

58 + 2

8 = 78

58 - 2

8 = 38

18

18

18

1

18

18

?210

0

Start

1

78

0

Start

? 1

210 + 4

10 = 610

78 - 3

8 = 48

0 126

46

16

36

56

2.

56 − 1

6 = 46

Remember that when adding or subtracting fractions with like denominators on a number line, the denominator does not change.

Write and solve the equation shown by each number line.

1.

45 4

4

78 9

10

710 9

12

23 3

6

38 1

10

25 2

6




14 + 2

4 = 34

533Topic 9 Reteaching

MTH16_SE04_CC2_T09_RT.indd 533 12/05/14 1:29 PM

TOPIC

9RETEACHINGUNDERSTAND ADDITION AND SUBTRACTION OF FRACTIONS


Reteaching Sets Standard MDIS

Set A 4.NF.B.3a, 4.NF.B.3b, 4.NF.B.3d H38

Set B 4.NF.B.3a, 4.NF.B.3d H39

Set C 4.NF.B.3a H41

Set D 4.NF.B.3a H85

Set E 4.NF.B.3c H45, H46

Set F MP.1, MP.2, MP.4, MP.5 J11, J12

533 Topic 9

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Set D

Set E

Set F


Remember to use benchmark fractions to estimate.

1. Is 25 + 12 about 1? Explain.

2. Is 78 - 25 about 1? Explain.

3. Is 34 + 13 about 1? Explain.

Find 515 - 33

5. Find 178 + 23

8.

515 = 46

5

- 335 = 33

5

135

178

+ 238

3108 = 42

8

Remember you can use fraction strips and number lines to help add and subtract mixed numbers.

1. 548 + 21

8 2. 336 + 15

6

3. 5 710 + 4 4

10 4. 9 - 338

Remember to draw a bar diagram to help write an equation.

Bonnie ran 14 mile, Olga ran 34 mile, Gracie ran 54 miles, and Maria ran 24 mile.

1. How much farther did Olga run than Bonnie?

pages 501–506

Estimate 15 + 710.

15 is close to but 7

10 is close to but less than 14. less than 34.

Since 14 + 34 = 1, 15 + 7

10 is close to but less than 1.

pages 507–524

pages 525–530

Think about these questions to help you model with math.

Thinking Habits• How can I use math I know to

help solve this problem?

• How can I use pictures, objects, or an equation to represent the problem?

• How can I use numbers, words, and symbols to solve the problem?

0 114

15

12

34

0 114

12

34

710

2. How far did Bonnie, Olga, and Maria run altogether?

758 52

6

10 110 55

8


Yes; 13 is close to 14; 34 + 14 = 1.


Check students’ diagrams; 34 − 1

4 = 24 mile farther

Check students’ diagrams; 14 + 3

4 + 24 = 6

4 or 124 miles

Yes; 25 is close to 12; 12 + 1

2 = 1.

No; 78 is close to 1 and 25 is close to 12;

1 − 12 = 1

2.

534 Topic 9 Reteaching

MTH16_SE04_CC2_T09_RT.indd 534 27/09/14 1:59 AM

Response to Intervention

Ongoing Intervention• Lessonswithguiding

questionstoassessunderstanding

• Supporttopreventmisconceptionsandtoreteach

Strategic Intervention• Targetedtosmallgroups

whoneedmoresupport• Easytoimplement

Intensive Intervention• Instructiontoaccelerate

progress• Instructionfocusedon

foundationalskills

11

22

33

534

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0 1610

810


7. Roger and Sulee each decomposed 116.

Roger wrote 16 + 16 + 2

6 + 36. Sulee wrote

36 + 4

6. Who was correct? Explain.

8. It is 810 mile from Liz’s house to the market.

Liz walked 610 mile, stopped for a break,

and walked the rest of the way to the market. Which equation represents Liz’s walk?

𝖠𝖠𝖠𝖠 0 + 610 = 8

10 𝖢𝖢𝖢𝖢 810 - 6

10 = 210

𝖡𝖡𝖡𝖡 610 + 2

10 = 810 𝖣𝖣𝖣𝖣 10

10 - 810 = 2

10

9. Ryan kayaks 178 miles before lunch and

238 miles after lunch. Select all of the

equations you would use to find how far Ryan kayacked.

178 + 23

8 = 414 miles

158 + 19

8 = 348 miles

158 + 19

8 = 48 miles

178 + 23

8 = 428 miles

178 + 23

8 = 3218 miles

DA

TA

Driving Time

Day Hours Driving

Monday 534

Tuesday 434

Wednesday 214

Thursday 634

n hours

345 3

44

118

18

18

18

18

18

18

11

18

18

18

10. The Jacobys kept track of the time they spent driving on their trip.

Part A

Find how many hours the Jacobys drove on Monday and Tuesday. Draw a bar diagram to represent the problem.

Part B

Find how many hours the Jacobys drove in all. Explain your work.

Sample answer: Roger and Sulee are both correct because 16 + 1

6 + 26 + 3

6 = 116

and 36 + 46 = 11

6.

2 points

1 point

1 point 2 points

2 points

Sample answer: I added the whole numbers and the fractions. 5 + 4 + 2 + 6 = 17 and 34 + 3

4 + 14 + 3

4 = 104 = 22

4;

17 + 224 = 192

4 hours

Sample answer:

534 + 43

4 = 1024 hours

536 Topic 9 Assessment

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Assessment

Name

1. Draw lines to match each expression on the left to an equivalent expression on the right.

2. On Monday, 312 of the students went on

a field trip. What fraction of the students did NOT go on the field trip?

1112

112

112

112

112

112

112

112

112

112

112

112

3. Riley planted flowers in some of her garden. Then, she planted vegetables in 28 of her garden. Now, 78 of Riley’s garden is planted. What fraction of Riley’s garden is planted with flowers?

𝖠𝖠𝖠𝖠 28 of her garden

𝖡𝖡𝖡𝖡 38 of her garden

𝖢𝖢𝖢𝖢 48 of her garden

𝖣𝖣𝖣𝖣 58 of her garden

4. Select all the expressions that show a way to decompose 78.

38 + 48

18 + 18 + 58

34 + 44

18 + 3

8 + 38

18 + 28 + 3

8 + 18

5. For questions 5a–5d, choose Yes or No to tell if 4

12 will make each equation true.

5a. 312 +□ = 7

12 ○ Yes ○ No

5b. 1612 -□ = 1 ○ Yes ○ No

5c. 1 112 + □ = 5 1

12 ○ Yes ○ No

5d. 1 512 - □ = 1 1

12 ○ Yes ○ No

6. Use benchmark fractions to estimate sums and differences less than or greater than 1. Write each expression in the correct answer space.

TOPIC

9

78 + 5

10 158 - 5

6 1010 - 2

3

12 + 2

3 512 + 1

4 116 + 7

8

Less Than 1 Greater Than 1

110 + 1

10 + 110 5

10 + 410

410 + 5

10 210 + ( 3

10 + 610)

( 210 + 3

10) + 610 2

10 + 110

1110 + 4

10 1610 - 1

10

Sample answer: 9

12 of the students

1 point

1 point

1 point

1 point

1 point

1 point

1010 ∙ 2

3 78 ∙ 5

10512 ∙ 1

4 12 ∙ 2

315

8 ∙ 56 11

6 ∙ 78

535Topic 9 Assessment

MTH16_SE04_CC2_T09_AS.indd 535 23/04/14 12:23 PM

TOPIC

9Topic ASSESSMENTUNDERSTAND ADDITION AND SUBTRACTION OF FRACTIONS

Restate the Topic Essential Questions from the Topic Opener or project them from the Student’s Edition eText.

Ask students to answer the Essential Questions (verbally or in writing) and give examples that support their answers. The following are key elements of the answers to the Essential Questions. Be sure these are made explicit when discussing students’ answers.

How do you add and subtract fractions and mixed numbers with like denominators?• The meanings of addition and subtraction are the same when

working with fractions, mixed numbers, and whole numbers.• Fractions can be decomposed and then joined or separated to find

sums and differences. This shows why the standard procedures for adding and subtracting fractions make sense.Example:

28 + 3

8 = 18 + 1

8 + 18 + 1

8 + 18 = 2 + 3

8 = 58

} }2 3

• Mixed numbers can be added and subtracted by decomposing the mixed number into a whole number and a fraction. Whole numbers may need to be renamed to subtract fractions. Another way to add or subtract mixed numbers is to change each to an equivalent fraction first. A third way to subtract mixed numbers is to count up.

How can fractions be added and subtracted on a number line?• Segments on a number line can be joined to add fractions and

separated to subtract fractions. Also, jumps on a number line can show addition or subtraction.Example: To find 56 - 1

6, find the point for 56 on the number line. Jump to the left a length of 16 landing at the point 46.

0 126

46

16

36

56

ANSWERING THE TOPIC ESSENTIAL QUESTION

535–536 Topic 9

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PearsonRealize.com

Assessment

Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. 4 Assessment

Topic 9Assessment

Name

1. Draw lines to match each expression on the left to an equivalent expression on the right.

112 + 1

12 + 112 5

12 + 412

412 + 5

12 212 + ( 3

12 + 612)

( 212 + 3

12) + 612 2

12 + 112

1112 + 4

12 1612 - 1

12

2. On Friday, 15 of the students in class were absent. What fraction of the students were NOT absent?

115

15

15

15

15

3. Cole spent some time working on his history homework. Then, he spent 5

12 hour working on his Spanish homework. Cole spent 1 hour on homework. What fraction of an hour did Cole spend on history?

𝖠𝖠𝖠𝖠 212 hour

𝖡𝖡𝖡𝖡 512 hour

𝖢𝖢𝖢𝖢 712 hour

𝖣𝖣𝖣𝖣 1212 hour

4. Select all the expressions that show a way to decompose 5

10.

310 + 2

10

110 + 1

10 + 310

34 + 2

6

110 + 2

10 + 210

110 + 1

10 + 110 + 1

10 + 110

5. For questions 5a–5d, choose Yes or No to tell if 4

10 will make each equation true.

5a. 310 +□ = 7

10 ○ Yes ○ No

5b. 1610 -□ = 1 ○ Yes ○ No

5c. 1 110 +□ = 5 1

10 ○ Yes ○ No

5d. 1 510 -□ = 1 1

10 ○ Yes ○ No

6. Use benchmark fractions to estimate sums and differences less than or greater than 1. Write each expression in the correct answer space.

710 + 1

2 148 - 8

8 1212 - 2

3

12 + 5

6

412 + 1

4 118 + 7

10

45 of the students

1 point

1 point

1 point

1 point

1 point

1 point

Less Than 1 Greater Than 1

1212 ∙ 2

34

12 ∙ 14

148 ∙ 8

8

710 ∙ 1

212 ∙ 5

6

118 ∙ 7

10

1 of 2

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Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. 4 Assessment

7. Tammi and Orlando each decomposed 13

4. Tammi wrote 24 + 2

4 + 34. Orlando wrote 44 + 3

4. Who was correct? Explain.

8. Tim’s house is 1 mile from school. Tim biked 4

10 mile, stopped for a rest, and biked some more. Which equation represents Tim’s bike ride to school?

0 1410

710

𝖠𝖠𝖠𝖠 0 + 410 = 4

10

𝖡𝖡𝖡𝖡 410 + 3

10 = 710

𝖢𝖢𝖢𝖢 710 - 4

10 = 310

𝖣𝖣𝖣𝖣 1010 - 7

10 = 310

9. Jean and Ricky used fraction strips to add. What is the sum of 25

6 + 126?

Select all that apply.

256 + 12

6 = 416

176 + 8

6 = 256

56 + 2

6 = 76

256 + 12

6 = 376

256 + 12

6 = 3

10. Grandma Meyer uses the recipe to make a soup.

Part A

Draw a bar diagram to find how much vegetable stock and cream are needed.

Part B

Find how many cups of soup will be made with all the ingredients. Explain your work.

Sample answer: Tammi and Orlando are both correct because24 + 2

4 + 34 = 13

4 and44 + 3

4 = 134.

Sample answer: n cups

342 1

41

234 + 11

4 = 4 cups

Soup Recipe

Ingredient Quantity

Chicken broth 234 cups

Water 124 cups

Cream 114 cups

Vegetable stock 234 cups

814; Sample answer: Add

the whole numbers and the fractions; 2 + 1 + 1 + 2 = 6 and 34 + 2

4 +14 + 3

4 = 94 = 21

4; 6 + 214

= 814 cups

1

1

1

16

16

16

16

16

16

16

2 points

1 point

1 point

2 points

2 points

2 of 2

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Topic Assessment Masters

➤

Scoring Guide

Item Points Topic Assessment (Student’s Edition and Masters)

1 1 All matches correct.

2 1 Correct answer.

3 1 Correct choice selected.

4 1 All correct choices selected.


6 1 All expressions in the correct place.

7 21

Correct answer and explanation.Correct answer or explanation.

8 1 Correct choice selected.


10A 21

Correct bar diagram and correct answer.Correct bar diagram or correct answer.

10B 21

Correct answer and explanation.Correct answer or explanation.

Item Standard DOK MDIS

1 4.NF.B.3b 1 H38

2 4.NF.B.3a 1 H39

3 4.NF.B.3a 1 H38, H39, H40

4 4.NF.B.3b 1 H38

5 4.NF.B.3a, 4.NF.B.3c

1 H38, H39, H40

6 4.NF.B.3a, 4.NF.B.3c

2 H85

7 4.NF.B.3b 2 H38

8 4.NF.B.3a, 4.NF.B.3d

1 H41

9 4.NF.B.3c 1 H45

10A 4.NF.B.3c 2 H45

10B 4.OA.C.3 3 H45

The Topic Assessment Masters assess the same content item for item as the Topic Assessment in the Student’s Edition.


ONLINE TOPIC ASSESSMENTAn auto-scored Topic Assessment is provided at PearsonRealize.com.

EXAMVIEW® TEST GENERATORExamView can be used to create a blackline-master Topic Assessment with multiple-choice and free-response items.

536A

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Part C

How many cups of water did Team 3 carry? Use the number line to show the sum.

Part D

Which team carried the most water?

2. Team 1 wanted to know how they did compared to Team 2.

Part A

Draw bar diagrams and write equations to show how to solve the problem.

Part B

How much more water did Team 2 carry than Team 1? Explain how to solve the problem using your equations from Part A. Show your work.

Team 2

Sample answer: c = 68 + 7

8

Team 2 carried 78 cup more; c = 68 + 7

8;

c = 158 cups

m = 248 − 15

8; m = 78 cup

1 point

2 points

2 points

2 points

248 = 112

8

− 158 = 1

5878

Sample answer: m = 248 − 15

8

0 321 681 3

82

58

238 cups

c cups Team 1 carried

68 cups 7

8 cups 58 cups1

48 cups2

m

c cups Team 1 carried

68 cups 7

8 cups 58 cups1

48 cups2

m

538 Topic 9 Performance Assessment

MTH16_SE04_CC2_T09_PA.indd 538 12/05/14 10:57 AM

DA

TA

Water Race Teams

Team Members

1 Jay and Victor

2 Abbie and Shawn

3 Suki and Kira

DA

TA

Water Race Results

Student Cups of Water

Abbie 58

Jay 68

Kira 58

Shawn 178

Suki 168

Victor 78

Name

Performance Assessment

TOPIC

9Water RaceIn one of the games at the class picnic, students balanced containers filled with water on their heads. The goal was to carry the most water to the finish line. The teams are listed in the Water Race Teams table. The amount of water each student carried is listed in the Water-Race Results table.

1. Mia will hand out the prize to the winning team.

Part A

Did Team 1 carry more or less than 2 cups of water? Tell how you estimated.

Part B

How many cups of water did Team 2 carry? Use fraction strips to show the sum.

248 cups

18

18

18

18

18

18

18

118

18

18

18

18

Less; Sample answer: 68 + 78

is less than 2 since 68 * 1 and 78 * 1.

2 points

2 points

58

1128 = 24

8

+ 178

537Topic 9 Performance Assessment

MTH16_SE04_CC2_T09_PA.indd 537 12/05/14 12:16 PM

TOPIC

9

Scoring Guide

Item Points Topic Performance Assessment in the Student’s Edition

1A 2 1

Correct estimate and explanationCorrect estimate or explanation

1B 2 1

Correct fraction strips model and answerCorrect fraction strips model or answer

1C 2 1

Correct number line and answerCorrect number line or answer

1D 1 Correct answer

2A 2 1

Correct bar diagrams and correct equationsPartially correct bar diagrams and equations

2B 2 1

Correct answer and explanationCorrect answer or explanation

Topic performance aSSeSSmenTUNDERSTAND ADDITION AND SUBTRACTION OF FRACTIONS



1A 4.NF.B.3a 2 H38

1B 4.NF.B.3c, 4.NF.B.3d, MP.4

3 H45

1C 4.NF.B.3c, 4.NF.B.3d, MP.4

3 H45

1D 4.NF.A.2, MP.2 1 H21

2A 4.NF.B.3d, MP.4 4 H38, H46

2B 4.NF.B.3a, 4.NF.B.3c, MP.4

4 H38, H46

537–538 Topic 9

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Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. 4 Performance Assessment

Topic 9Performance

Assessment

Name

Puppy GrowthGretchen’s dog had a litter of 4 puppies. The Puppy Growth table shows the mass of each puppy over two months.

1. Examine each puppy’s growth in Parts A–D.

Part A

Did Buddy gain more or less than 12 kilogram from month 1 to month 2? Use benchmark fractions to estimate the difference.

Part B

How much mass did Teddy gain from month 1 to month 2? Draw fraction strips and write an equation to find the difference.

Part C

How much more mass did Socko have in month 2 than in month 1? Use the number line to show the difference.

More; 1 210 − 4

10 is more than 12,

because 1 210 + 1 and 4

10 *12.

Puppy Growth

PuppyKilograms Month 1

Kilograms Month 2

Buddy 410 1 2

10

Teddy 610 1 7

10

Socko 310

910

Peanut 510 1 3

10

1 710 − 6

10 = 1 110 kilogram

2 points

2 points

2 points

610 kilogram; Check students’ drawings.

610

910

310

0 1

1 11

101

101

101

101

101

101

10

1 of 2

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Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. 4 Performance Assessment

Part D

How much more mass did Peanut have in month 2 than in month 1? Show your work.

2. In the second month, how much more combined mass did Buddy and Teddy have than Socko and Peanut? Draw bar diagrams and write equations to explain.

2 points

3 points

810 kilogram

1 310 = 13

10− 5

10 = 5108

10

210 kg

kg

kg

1

Di�erence in mass:

2101 t7

1011 5

t 91025

9102 d2

1022 5

710d 5

Buddy and Teddy: Socko and Peanut:

910 kg2

710 kg1

t

910 kg 3

10 kg

kg

1

s

d210 kg2

910

31011 5 s

s 12101 2

102or5

2 of 2

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Topic Performance Assessment Masters

➤

Scoring Guide

Item Points Topic Performance Assessment Masters

1A 2 1

Correct explanation and estimateCorrect explanation or estimate

1B 2 1

Correct fraction strips model and answerCorrect fraction strips model or answer

1C 2 1

Correct number line and answerCorrect number line or answer

1D 2 1

Correct answer and algorithms shownCorrect answer or algorithms shown

2 3

2

1

Correct answer, appropriate equations, and appropriate diagramsCorrect answer, appropriate equations, and appropriate diagrams, except for minor errorsPartially correct answer but numerous errors



1A 4.NF.B.3a, MP.2 2 H85

1B 4.NF.B.3c, 4.NF.B.3d, MP.4

2 H46

1C 4.NF.B.3c, 4.NF.B.3d, MP.4

2 H41

1D 4.NF.A.2, MP.2 2 H46

2 4.NF.B.3a, 4.NF.B.3d, MP.4

3 H45, H46

538A

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NOTES

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