Unit 7:Fractions and Their Uses; Chance and Probabilityellis2020.org/iTLG/iTLG Grade...

$: Unit 7:Fractions and Their Uses; Chance and Probabilityellis2020.org/iTLG/iTLG Grade 4/U7.pdfOverview Unit 7 begins with a review of fraction ideas previously introduced, and extends$
OverviewOverviewUnit 7 begins with a review of fraction ideas previously introduced, and extends knowledge by developing a good understanding ofequivalent fractions. Unit 7 also provides informal activities related to chance and probability. Unit 7 has four main areas of focus:

◆ To review fractions as parts of a whole (ONE), fractions on number lines, and uses of fractions,

◆ To guide students as they order fractions and find fractional parts of sets and regions,

◆ To provide practice identifying equivalent fractions, and

◆ To review basic ideas of probability, comparing predicted and actual results, and guiding the application of fractions to chance experiments.

554 Unit 7 Fractions and Their Uses; Chance and Probability

Unit Organizer 555

Lesson Objective Page

Contents

7◆1 Review of Basic Fraction Concepts 570To review fractions as parts of a whole (ONE), fractions on number lines, and uses of fractions.

7◆2 Fractions of Sets 576To provide practice finding fractional parts of sets.

7◆3 Probabilities When Outcomes Are Equally Likely 581To review basic vocabulary and concepts of probability; and to introduce finding probabilities for events when all the possible outcomes are equally likely.

7◆4 Pattern-Block Fractions 587To guide students as they find fractional parts of polygonal regions.

7◆5 Fraction Addition and Subtraction 592To guide students in the use of pattern blocks to add and subtract fractions.

7◆6 Many Names for Fractions 598To provide practice identifying equivalent fractions.

7◆7 Equivalent Fractions 603To guide the development and use of a rule for generating equivalent fractions.

7◆8 Fractions and Decimals 609To provide experience renaming fractions as decimals and decimals as fractions; and to develop an understanding of the relationship between fractions and division.

7◆9 Comparing Fractions 615To provide practice ordering sets of fractions.

7◆10 The ONE for Fractions 621To guide students as they find the whole, or the ONE, for given fractions.

7◆11 Probability, Fractions, and Spinners 626To review basic ideas of probability, including fairness and expected results; and to guide the application of fractions to spinners.

7◆12 A Cube-Drop Experiment 632To guide students in comparing predicted and actual results from an experiment with equally likely outcomes.

7◆13 Progress Check 7 638To assess students’ progress on mathematical content through the end of Unit 7.


To review fractions as parts of a whole(ONE), fractions on number lines, anduses of fractions.

To provide practice finding fractionalparts of sets.

To review basic vocabulary andconcepts of probability; and to introducefinding probabilities for events when allthe possible outcomes are equally likely.

To guide students as they find fractionalparts of polygonal regions.

To guide students in the use of patternblocks to add and subtract fractions.

To provide practice identifyingequivalent fractions.

To guide the development and use of arule for generating equivalent fractions.

To provide experience renamingfractions as decimals and decimals asfractions; and to develop anunderstanding of the relationshipbetween fractions and division.

To provide practice ordering sets offractions.

To guide students as they find thewhole, or the ONE, for given fractions.

To review basic ideas of probability,including fairness and expected results;and to guide the application of fractionsto spinners.

To guide students in comparingpredicted and actual results from anexperiment with equally likelyoutcomes.

Lesson Objectives Links to the Past Links to the Future

Learning In Perspective

Grades 1–3: Name parts of a whole as fractions.

Grade 3: Name numbers of fractional parts ofcollections and regions as fractions and mixednumbers.

Grade 3: Introduce the vocabulary of chanceevents. Conduct probability experiments: predictoutcomes; test predictions; make frequency tablesand bar graphs.

Grade 3: Name numbers of fractional parts ofcollections and regions as fractions and mixednumbers.

Grade 3: Review basic fraction concepts andnotation. Make a number-line poster for fractions.

Grade 2: Identify equivalent fractions with fractioncards. Play Equivalent Fractions.Grade 3: Create name-collection boxes forfractions.

Grade 2: Identify equivalent fractions with fractioncards. Play Equivalent Fractions.Grade 3: Create name-collection boxes forfractions.

Grades 1 and 2: Model decimals through hundredthswith base-10 blocks, 10-by-10 grids, and money.Grade 3: Use place-value tools to display decimalsthrough thousandths.

Grades 2 and 3: Sort fractions by size (relative to �12

�).Play Fraction Top-It to compare fractions.

Grades 1–3: Name parts of a whole asfractions.



Grade 5: Find the whole, given a fraction or percent of the whole.

Grade 5: Solve parts and whole problems. Review concept of whole or ONE as applied to sets.

Grade 5: Perform experiments to estimate theprobability of a chance event; record probabilities on a Probability Meter Poster.

Grade 5: Solve parts and whole problems. Review concept of whole or ONE as applied to sets.

Grade 5: Use common denominators to � and �

fractions with unlike denominators. Add andsubtract mixed numbers. Grade 6: � , �, �,and � fractions and mixed numbers with like orunlike denominators.

Grade 5: Rename fractions and mixed numbers in simplest form.Grade 6: Applications and maintenance.

Grade 5: Formulate a division rule for findingequivalent fractions.

Grade 5: Rename fractions as decimals. Grade 6:Rename numbers expressed by fractions, mixednumbers, decimals, and percents.

Grade 5: Compare fractions by renaming them with a common denominator. Grade 6: Comparefractions by renaming them as decimals. Compareratios by renaming them as n-to-1 ratios.

Grade 5: Find the whole, given a fraction orpercent of the whole.



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Identify fractions as equal parts of a whole or the ONE and solve problems involving Number and Numeration Goal 2fractional parts of regions.Identify equivalent fractions and mixed numbers. Number and Numeration Goal 5Identify a triangle, hexagon, trapezoid, and rhombus. Geometry Goal 2Find fractions and mixed numbers on number lines. Patterns, Functions, and Algebra Goal 1

Solve problems involving fractional parts of collections. Number and Numeration Goal 2 Identify the whole or the ONE when given the “fraction-of.” Number and Numeration Goal 2Identify equivalent fractions. Number and Numeration Goal 5Use an equal-sharing division strategy. Operations and Computation Goal 4

Add fractions with like denominators. Operations and Computation Goal 5Use basic probability terms to describe and compare the likelihood of an event; explain Data and Chance Goal 3the choice of term.Predict the outcomes of an experiment. Data and Chance Goal 4Express the probability of an event as a fraction. Data and Chance Goal 4

Identify the whole or the ONE. Number and Numeration Goal 2Find fractional parts of polygonal regions. Number and Numeration Goal 2Identify equivalent fractions. Number and Numeration Goal 5Model fraction addition with pattern blocks. Operations and Computation Goal 5Identify a triangle, hexagon, trapezoid, and rhombus. Geometry Goal 2

Identify the whole or the ONE. Number and Numeration Goal 2Represent fractions with pattern blocks. Number and Numeration Goal 2Identify equivalent fractions. Number and Numeration Goal 5Model fraction addition and subtraction with pattern blocks. Operations and Computation Goal 5Identify a triangle, hexagon, trapezoid, and rhombus. Geometry Goal 2

Identify fractional parts of regions. Number and Numeration Goal 2Name equivalent fractions. Number and Numeration Goal 5Use patterns in a table to find equivalent fractions. Patterns, Functions, and Algebra Goal 1

Identify fractional parts of regions. Number and Numeration Goal 2Name equivalent fractions. Number and Numeration Goal 5Use a rule for generating equivalent fractions. Number and Numeration Goal 5Develop a rule for generating equivalent fractions. Patterns, Functions, and Algebra Goal 1

Read and write decimals through hundredths. Number and Numeration Goal 1Represent a shaded region as a fraction and a decimal. Number and Numeration Goal 2Rename fractions with 10 and 100 in the denominator as decimals. Number and Numeration Goal 5Use fraction notation and equal sharing to solve division problems. Operations and Computation Goal 4

Compare fractions. Number and Numeration Goal 6Order fractions. Number and Numeration Goal 6Explain strategies used to compare and order fractions. Number and Numeration Goal 6Use patterns to compare and order fractions. Patterns, Functions, and Algebra Goal 1

Given a fractional part of a region, name the ONE. Number and Numeration Goal 2Given a fractional part of a collection, name the ONE. Number and Numeration Goal 2Identify a hexagon, trapezoid, and rhombus. Geometry Goal 2

Name fractional parts of regions. Number and Numeration Goal 2Use equivalent fractions to design spinners. Number and Numeration Goal 5Use probability language to describe the likelihood of events. Data and Chance Goal 3Conduct experiments and calculate expected probability. Data and Chance Goal 4

Rename fractions as percents. Number and Numeration Goal 5Use basic probability terms to describe the likelihood of events. Data and Chance Goal 3Conduct a cube-drop experiment. Data and Chance Goal 4Use fractions and percents to predict the outcomes of an experiment. Data and Chance Goal 4Compare predicted outcomes and actual results. Data and Chance Goal 4

Key Concepts and Skills Grade 4 Goals*

* For a detailed listing of all Grade 4 Goals, see the Appendix.

Unit Organizer 557

Key Concepts and Skills

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Ongoing Learning and Practice

Math BoxesMath Boxes are paired across lessons as shown in the brackets below.This makes them useful as assessment tools. Math Boxes also preview content of the next unit.

Ongoing Learning and Practice

7◆1 Product Pile-Up Developing automaticity with multiplication facts Operations and Computation Goal 3

7◆2, 7◆3 Fraction Of Identifying fractions of collections Number and Numeration Goal 2

7◆3, 7◆6 Grab Bag Calculating the probability of an eventData and Chance Goal 4

7◆5 Angle Tangle Measuring and estimating the measures of anglesMeasurement and Reference Frames Goal 1

7◆6, 7◆7 Fraction Match Identifying equivalent fractionsNumber and Numeration Goal 5

7◆9 Over and Up Squares Plotting ordered number pairsMeasurement and Reference Frames Goal 4

7◆9, 7◆10 Fraction Top-It Comparing and ordering fractionsNumber and Numeration Goal 6

7◆10 Getting to One Applying proportional reasoning skillsNumber and Numeration Goal 2

7◆11 Chances Are Using basic probability termsData and Chance Goal 3

Lesson Game Skill Practiced

See the Differentiation Handbook for ways to adapt games to meet students’ needs.

▲

Home Communication Study Links provide homework and home communication.

Home Connection Handbook provides more ideas to communicateeffectively with parents.

Unit 7 Family Letter provides families with an overview, Do AnytimeActivities, Building Skills Through Games, and a list of vocabulary.

Mixed practice [7◆1, 7◆3], [7◆2, 7◆4], [7◆5, 7◆7], [7◆6, 7◆8], [7◆9, 7◆11], [7◆10, 7◆12]

Mixed practice with multiple choice 7◆2, 7◆3, 7◆6, 7◆7, 7◆9, 7◆10

Mixed practice with writing/reasoning opportunity 7◆1, 7◆2, 7◆4, 7◆7, 7◆8, 7◆9, 7◆10, 7◆12

Practice through Games Games are an essential component of practice in the Everyday Mathematicsprogram. Games offer skills practice and promote strategic thinking.

132

4


Encourage students to use a variety of strategies to solve problems and toexplain those strategies. Strategies that students might use in this unit:

◆ Acting out the problem ◆ Using and making a table◆ Using and drawing a picture ◆ Using estimation◆ Using computation

Lesson Activity

See Chapter 18 in the Teacher’s Reference Manual for more information about problem solving.

7◆1 Identify fractional parts of number lines.

7◆2 Use pennies to model "fraction-of" problems.

7◆4 Explore fractional parts of regions with pattern blocks.

7◆5 Model fraction sums and differences with pattern blocks.

7◆6 Collect fraction names.

7◆10 Use pattern blocks to find the ONE.

7◆11 Design spinners.

7◆12 Make predictions for a cube drop experiment.

Unit Organizer 559

Problem SolvingProblem Solving

Unit 7Lessons

NCTMStandards

7 ◆1 7 ◆2 7 ◆3 7 ◆4 7 ◆5 7 ◆6 7 ◆7 7 ◆8 7 ◆9 7 ◆10 7 ◆11 7 ◆12 7 ◆13

1, 3,6–10

Content Standards: 1 Number and Operations, 2 Algebra, 3 Geometry, 4 Measurement, 5 Data Analysis and ProbabilityProcess Standards: 6 Problem Solving, 7 Reasoning and Proof, 8 Communication, 9 Connections, 10 Representation

1, 6–10 1, 5, 6–8

1, 3,6–10

1–3, 6–8 1, 6–8 1, 6–9 1, 6–8 1, 2,

6–81, 2,6–10 1, 5–10 1, 5–10 6–10

Planning Tips

Lessons thatteach throughproblem solving,not just aboutproblem solving

PacingPacing depends on a number of factors, such as students’ individual needsand how long your school has been using Everyday Mathematics. At thebeginning of Unit 7, review your Content by Strand Poster to help you seta monthly pace.

NCTM Standards

MOST CLASSROOMS

J A N U A R Y F E B R U A R Y M A R C H

http://standards.nctm.org


Balanced Assessment

7◆1 Describe fractions as equal parts of a whole. [Number and Numeration Goal 2]

7◆2 Solve "fraction-of" problems.[Number and Numeration Goal 2]

7◆3 Use basic probability terms to indicate the likelihood of an event. [Data and Chance Goal 3]

7◆4 Describe the relationship between the whole and its fractional parts. [Number and Numeration Goal 2]

7◆5 Use pattern blocks to solve fraction addition problems. [Operations and Computation Goal 5]

7◆6 Estimate the measure of an angle. [Measurement and Reference Frames Goal 1]

7◆7 Describe a method for determining fraction equivalency.[Number and Numeration Goal 5]

7◆8 Rename tenths and hundredths as decimals with the assistance of a visual model.[Number and Numeration Goal 5]

7◆9 Compare fractions and explain strategies. [Number and Numeration Goal 6]

7◆10 Compare fractions and write a number model to illustrate the comparison. [Number and Numeration Goal 6]

7◆11 Express the probability of an event as a fraction. [Data and Chance Goal 4]

7◆12 Predict the outcomes of an experiment and test the predictions using manipulatives.[Data and Chance Goal 4]

Lesson Content Assessed

Use the Assessment

Management System

to collect and analyze dataabout students’ progressthroughout the year.

Ongoing Assessment

Recognizing Student AchievementOpportunities to assess students’ progress toward Grade 4 Goals:

Informing InstructionTo anticipate common student errors and to highlight problem-solvingstrategies:

Lesson 7◆2 Use equivalent fractions to solve “fraction-of” problems

Lesson 7◆4 Emphasize that the ONE or the whole can change

Lesson 7◆6 Understand what the numerator and denominator in afraction represent

Lesson 7◆7 Find equivalent names for fractions

Lesson 7◆9 Understand that the denominator represents the number of pieces in the whole

Lesson 7◆11 Understand the term expect

Lesson 7◆12 Convert fractions to percents

Unit Organizer 561

Portfolio OpportunitiesOpportunities to gather samples of students’ mathematical writings, drawings, and creations to add balance to the assessment process:

◆ Naming fractional parts of a region, Lesson 7◆1◆ Describing relationship between fractions and minutes, Lesson 7◆2◆ Explaining why the same pattern blocks take on different fractional values for

different-sized shapes, Lesson 7◆4◆ Drawing a picture to show understanding of equivalent fractions, Lesson 7◆7◆ Explaining how to find a fraction of a set, Lesson 7◆7◆ Explaining why an answer to a fraction subtraction problem is incorrect, Lesson 7◆9◆ Determining how a candy bar is divided, Lesson 7◆10◆ Determining which fraction is greater, Lesson 7◆12◆ Comparing actual to predicted results, Lesson 7◆12◆ Using fractions to divide land, Lesson 7◆13

Assessment HandbookUnit 7 Assessment Support

◆ Grade 4 Goals, pp. 37–50 ◆ Unit 7 Open Response◆ Unit 7 Assessment Overview, pp. 102–109 • Detailed rubric, p. 106

• Sample student responses, pp. 107–109

Solve problems involving fractional parts of regions andcollections; identify the ONE.[Number and Numeration Goal 2]

Rename tenths and hundredths as decimals. [Number and Numeration Goal 5]

Find equivalent fractions.[Number and Numeration Goal 5]

Compare and order fractions.[Number and Numeration Goal 6]

Solve multidigit multiplication and division problems.[Operations and Computation Goal 4]

Add and subtract fractions.[Operations and Computation Goal 5]

Use ordered number pairs to locate points on acoordinate grid. [Measurement and Reference Frames Goal 4]

Use basic probability terms. [Data and Chance Goal 3]

Calculate expected probability. [Data and Chance Goal 4]

CONTENT ASSESSED Self Oral/Slate Written Open Response

ASSESSMENT ITEMS

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Periodic Assessment7◆13 Progress Check 7

Unit 7 Assessment Masters◆ Unit 7 Self Assessment, p. 184 ◆ Exit Slip, p. 311◆ Unit 7 Written Assessment, pp. 185–187 ◆ Math Logs, pp. 306–308◆ Unit 7 Open Response, p. 188 ◆ Other Student Assessment Forms, pp. 304, ◆ Unit 7 Class Checklist, pp. 272, 273, and 303 305, 309, and 310◆ Unit 7 Individual Profile of Progress,

pp. 270, 271, and 302

Daily Lesson Support

Differentiated Instruction

ENGLISH LANGUAGE LEARNERS

READINESS ENRICHMENT

EXTRA PRACTICE7◆1 Building a Math Word Bank7◆3 Discussing the likelihood of events in

everyday life7◆11 Building a Math Word Bank7◆12 Building a Math Word Bank

7◆2 Playing Fraction Of7◆3 Playing Grab Bag7◆5 Solving Frames-and-Arrows problems7◆6 Playing Fraction Match7◆7 Completing name-collection boxes7◆9 Playing Fraction Top-It

5-Minute Math 7◆7 Finding equilvalentfractions; 7◆12 Offering more experiencewith probability


Adjusting the Activity7◆1 Modeling with pattern blocks ELL7◆1 Using the Fraction Number-Line Poster

ELL7◆2 Representing equal groups ELL7◆3 Referring to playing card information7◆3 Highlighting words and phrases7◆3 Rewording a formula7◆3 Focusing on unit fractions7◆4 Associating names with shapes ELL7◆4 Writing number models ELL7◆5 Using colored chalk ELL7◆6 Drawing pictures to model problems ELL

7◆7 Presenting a more abstract rationale 7◆7 Using a table of equivalent fractions 7◆8 Modeling decimal numbers ELL7◆9 Ordering fractions ELL7◆9 Using a calculator to compare fractions7◆10 Creating a visual representation ELL7◆10 Using pattern blocks and counters ELL7◆10 Playing Fraction Top-It ELL7◆11 Displaying probability terms ELL7◆11 Expressing probability as a fraction ELL7◆12 Describing similarities between

spinning a spinner and dropping a cube

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

Cross-Curricular LinksArt LiteratureLesson 7◆1 Students draw and color a design. Lesson 7◆5 Students read Gator Pie.

Social Studies Lesson 7◆11 Students conduct experiments found in DoLesson 7◆2 Students follow the World Tour Routine. You Wanna Bet?: Your Chance to Find Out About Probability.

TechnologyLesson 7◆11 Students visit a Web site to create their own spinners.

Using the ProjectsUse Project 3, A Carnival Game, after Unit 7. See the Differentiation Handbook formodifications to Project 3.

7◆1 Constructing an equilateral triangle7◆1 Naming fractional parts of designs 7◆2 Solving "fraction-of" problems7◆3 Conducting an experiment7◆4 Determining values of tangram pieces7◆6 Modeling equivalencies of fractions7◆7 Investigating fraction representations 7◆8 Representing fractions7◆9 Using digits to create specified fractions7◆10 Playing Getting to One7◆10 Using clues to find the ONE7◆11 Conducting probability experiments 7◆12 Comparing data with expected results

7◆1 Creating a Fraction Number-Line Poster7◆2 Acting out a fraction situation7◆3 Examining a deck of playing cards7◆4 Building rectangles7◆5 Writing number models for fraction

addition7◆6 Finding equivalent fractions 7◆7 Using a Fraction Number-Line Poster 7◆8 Creating base-10 block designs7◆9 Exploring relative sizes of fractions7◆11 Exploring fractional parts of regions7◆12 Modeling fractions and their percent

equivalents

Unit Organizer 563

Unit 7 Vocabularydenominatorequal chanceequally (more, less) likelyequivalent fractionsEquivalent Fractions Ruleeventexpectfair die favorable outcomemixed numbernumeratoroutcomeprobability"whole" boxwhole (or ONE or unit)

Grade 3

Grade 4

Grade 5

8◆1,8◆3

8◆1 8◆5 8◆6 8◆6 11◆4 1◆3,11◆3

7◆1 7◆2 7◆3 7◆4 7◆5 7◆6 7◆7 7◆8 7◆9 7◆10 7◆11 7◆12

5◆1 6◆8 6◆9 6◆10 5◆4 5◆5–5◆7 5◆3

Language SupportEveryday Mathematics provides lesson-specific suggestions to help allstudents, including non-native English speakers, to acquire, process, andexpress mathematical ideas.

Connecting Math and LiteracyLesson 7◆11 Do You Wanna Bet?: Your Chance to Find Out AboutProbability, by Jean Cushman, Clarion, 1991Count Your Way through Brazil, by Jim Haskins and Kathleen Benson,Carolrhoda Books, 1996

Student Reference Bookpp. 43, 46, 230, 236, 237, 243–245, 247–249, 257, and 259

Multiage Classroom ◆ Companion LessonsCompanion Lessons from Grades 3 and 5 can help you meet instructionalneeds of a multiage classroom. The full Scope and Sequence can be foundin the Appendix.

Professional Development

Teacher’s Reference Manual LinksTopicLesson Section

See 7◆6

Fraction and Decimal Notation

Fraction/Decimal/Percent Conversion

Numeric Relations

Fraction Multiplication

Fraction Division

The Language of Chance

Making Predictions

Data and Chance Tools and Techniques

Collecting and Recording Data

Organizing and Displaying Data

Making Predictions

7◆7

7◆8

7◆9

7◆10

7◆11

7◆12

9.3.1

11.4.3

9.7

11.3.4

11.3.5

12.1.2

12.1.3

12.4

12.2.2

12.2.3

12.1.3

Lesson

7◆1

7◆2

7◆3

7◆4

7◆5

7◆6

Topic

Number Grids/Number Lines

Uses of Fractions


See 7◆1

Why Study Probability

The Language of Chance

Uses of Fractions


Fraction Addition and Subtraction

Calculators

Entering Fractions and Mixed Numbers

Common Denominators

Section

1.3.4

9.3.2

9.3.1

12.1.1

12.1.2

9.3.2

9.3.1

11.3.2

3.1.1

11.4.1

11.3.1


Materials

Lesson Masters Manipulative Kit Items Other Items

* Denotes optional materials

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7◆9

7◆4

7◆13

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7◆12

7◆10

Technology Assessment Management System, Unit 7iTLG, Unit 7

Study Link Master, p. 203 per group: 8 each of number cards straightedge; scissors; tape; crayons or Teaching Masters, pp. 204–206 1–10; pattern blocks; calculator; colored pencils

slate; Geometry Template; compass

Study Link 7◆1; Teaching Aid Masters, straws*; slate 20 pennies or other counters; pp. 388 or 389 and 419–421*; Study Link quarter-sheets of paper*Master, p. 207; Teaching Master, p. 208;Game Masters, pp. 477–480

Study Link 7◆2; Teaching Masters, pp. 210 per group: 3 six-sided dice; slate per group: 1 deck of regular playing cards; and 211; Study Link Master, p. 209; 1 large grocery bag; chart paper*; scissorsGame Masters, pp. 477–480 and 483–485

Study Link 7◆3; Teaching Masters, pattern blocks; slate overhead pattern blocks*; red pencil or pp. 212, 214 and 215; Teaching Aid marker*; straightedge; chart paper*; red, Masters, pp. 388 or 389 and 441; blue, green crayons; scissors; red, blue, transparency of Math Masters, p. 212*; green color tiles*Study Link Master, p. 213

Study Link 7◆4; Study Link Master, pattern blocks; calculator; slate overhead pattern blocks*; colored chalk*; p. 216; Game Master, p. 457; Teaching protractor; straightedge; Gator PieMaster, p. 217; Teaching Aid Master p. 393

Study Link 7◆5; Study Link Master, p. 218; Fraction Cards (Activity Sheets 5 and 6); Game Masters, pp. 473–476 and 483–485; scissors; glueTeaching Masters, pp. 219–222

Study Link 7◆6; Teaching Masters, calculator colored chalk; straightedgepp. 224 and 225; Study Link Master,p. 223; Game Masters, pp. 473–476; Teaching Aid Masters, pp. 388 or 389 and 397; Math Masters, pp. 204 and 205

Study Link 7◆7; transparency of Math base-10 blocks; calculator; slate overhead base-10 blocks*; pen or Masters, p. 426; Study Link Master, colored pencilp. 226; Teaching Aid Masters, pp. 412, 414, 416, 426*, and 442; Teaching Master, p. 227

Study Link 7◆8; Study Link Master, per group: 2 six-sided dice; Fraction Cards (Activity Sheets 5 and 6); p. 228; Game Masters, pp. 494 and 506; calculator*; slate colored pencils; scissors; tapeTeaching Masters, pp. 229 and 230

Study Link 7◆9; Study Link Master, pattern blocks; slate; Geometry overhead pattern blocks*; beans, p. 231; Game Master, p. 506; Teaching Template; calculator pennies, or other counters; FractionMaster, p. 232 Cards (Activity Sheets 5 and 6)

Study Link 7◆10; Teaching Masters, per group: 2 six-sided dice*; slate per student: 1 large (2") paper clip and pp. 233 and 237; Study Link Masters, 2 pieces of removable tape; crayons, pp. 234–236; Game Masters, markers, or colored pencils; straightedge; pp. 462–466 data pad or chart paper*; computer with

Internet access*

Study Link 7◆11; Teaching Master, pp. 238 slate; base-10 blocks; 1 cm cube colored pencils, markers, or crayons; shoe and 240–242; Teaching Aid Master, p. 388 box or copier-paper box*or 389; Study Link Master, p. 239

Study Link 7◆12; Assessment Masters, slate; pattern blocks straightedge; colored pencils*pp. 184–188; Study Link Masters,pp. 243–246

Unit Organizer 565

The discussion below highlights the major content ideas presented in Unit 7 and helps establish instructional priorities.

Fraction Concepts, Notation, and Uses (Lessons 7◆1, 7◆2, 7◆4, and 7◆10)In today’s world, people seldom add, subtract, or divide using fractionalnotation. But very often, they do use fractional notation to express and convey information such as the following:

◆ fractions of sets or collections of discrete things (things difficult topartition or break into parts) Examples: half-dozen eggs; �

14� of the cars in the parking lot

◆ fractions as parts of continuous things Examples: A recipe calls for �

23� cup of milk, �

12� cup of sugar, and �

14� pound

of butter.◆ fractions to name points between whole numbers on rulers, other

measurement scales, and number lines Examples: rulers marked in inches and sixteenths of inches; scale on a measuring cup

◆ fractions to express rates Example: 24 miles per gallon, perhaps set up originally as the rate �1

2040

gamlliolenss�

◆ fractions to set up ratio comparisons or express scales on architecturalplans, maps, or pictures Examples: Johannes got �

14� of the vote (�2

5132

vvootteess�); the scale of the

encyclopedia pictures of the aardvark is �112�, or 1 to 12.

◆ fraction notation for division Example: As indicated above, information in rate or ratio comparison is often expressed first as a fraction before being divided.

Mathematical Background

Note

Except for those casesin which fractions areused to expresscomparisons in rates orratios, a fraction will bea fraction of something.Everyday Mathematicsrefers to this “something”as the “whole,” or the“ONE.” It is important toemphasize that fractionscan be meaninglessunless one thinks ofthem in reference to thewhole. Half a glass ofmilk is different fromhalf a quart; and half asecond is very differentfrom half an hour.

Everyday Mathematics uses a device called the “whole” box to remindstudents that a fraction is a part of a whole thing (�

14� of an orange is a

fraction of a whole orange). A box, pictured next to a problem or at the top of a page, contains a word or phrase that describes the whole, such as “quart of milk” or “l hour.”

In Lesson 7-2, students solve a variety of “fraction-of” problems, in whichthe whole is a collection of objects. Pattern blocks are used in Lesson 7-4to partition various 2-dimensional shapes. They also provide students withpractice naming fractional parts of a region. These activities reinforce theidea that fractions should always be viewed in relation to the whole,unless they are used to indicate rates or ratios.

In Lesson 7-10, students use pattern blocks and counters to find the wholefor given fractions.

Section 9.3 of the Teacher’s Reference Manual contains moreinformation about fractions, concepts, notation, and uses.

Continuing the World Tour (Lessons 7◆2)

Students continue the World Tour by flying from Budapest, Hungary, toBrasília, Brazil. They follow the established World Tour routine to updatethe Route Map and begin gathering information about countries in South America.

Whole

1 hour


Detail of roof of Mathias Church, Budapest, Hungary

Modern architecture,Salvador, Brazil

Unit Organizer 567

NoteUnderstanding equivalentfractions is the keyingredient needed tocompare and computewith fractions. By placingspecial emphasis onequivalent fractions, theauthors hope to avoiddifficulties faced by manystudents and even adultswhen working withfractions.

Fraction Addition and Subtraction(Lessons 7◆5)In Lesson 7-5, fractional relationships between pattern blocks are appliedto solve simple fraction addition and subtraction problems, such as �

23� � �

16�

and �56� � �

23�. The focus is not on using common denominators to add and

subtract fractions, but on gaining hands-on experience with these kinds ofproblems. Paper-and-pencil methods for fraction addition and subtractionare treated in detail in Fifth Grade Everyday Mathematics.

Consult Section 11.3.2 of the Teacher’s Reference Manual for additionalinformation on fraction addition and subtraction.

Equivalent Fractions (Lesson 7◆6, 7◆7, and 7◆9)By now, students should be very familiar with the idea of equivalentnames for numbers. They have been filling in name-collection boxes,usually with names for whole numbers, since the middle of first grade.And in third grade, students began collecting names for equivalentfractions. However, the application of this idea to fractions will requirecareful teaching, time, and practice.

There is an unlimited choice of names for any fraction. For example, thenumber “one-half” can be written as �

12� or as �

24�, �

36�, �

48�, �1

50�, �1

62�, …, �

12

00

00�, …,

�12

,,00

00

00�, …, and so on. It can also be written in nonfraction notation, such

as 0.5 and 50%. The ability to change the name or form of a number incountless ways is a very powerful tool in mathematics.

In this unit, ideas of equivalent fractions are developed with decks ofFraction Cards, which students cut out from pages in the journal. Thecards show fraction symbols on one side and shaded pictures for thefractions on the other side.

In Lesson 7-6, students use their Fraction Cards to identify equivalentfractions by matching cards that have equal amounts of shading. Theybegin a collection of fraction names in the journal and add names to thistable throughout the school year.

In Lesson 7-7, students learn that they can rename a fraction bymultiplying both its numerator and denominator by the same number.Students will expand their collections of fraction names with the help of this rule.

Fraction Cards play an important role in Lesson 7-9, where students usethem to determine whether a fraction is greater or less than anotherfraction; to order sets of fractions from the smallest fraction to the largestfraction; and to compare fractions to �

12�.

Fraction Cards are used again in Lesson 7-10 to play a game calledFraction Top-It, which is an adaptation of the card game War. Equivalentnames for fractions are used to compare pairs of fractions. Players cancheck who has the larger fraction by comparing the amount shaded.Eventually, students who play this game frequently will not need to relyon visual confirmation as much.

To further investigate equivalent fractions, refer to Section 9.3.1 in the Teacher’s Reference Manual.


Fractions and Decimals (Lesson 7◆8)Equivalent fractions and shaded grid squares are used to rename fractionsas decimals. Students add these decimal versions to their collections offraction names.

Advantages and Disadvantages of Fractions and Decimals

Both fractions and decimals are used to represent numbers that arebetween whole numbers. Each form has its advantages and disadvantages.Relative size is easy to determine with decimals (for example, 0.45 >0.0095), but sometimes difficult with fractions (for example, is �

79� greater

than or less than �56�?). Decimals are much easier to use in most calculations

because the place-value structure and algorithms for decimals are closelylinked to those of whole numbers. Decimals also appear in scientificnotation for very large and very small numbers. Hence, decimals areuniversally used in science and industry. But people often want to refer to a part of something or compare one thing with another, and, for thatreason, they find fractions very useful.

More information on fractions and decimals can be found in the Teacher’s Reference Manual in Section 9.3.

Chance and Probability(Lessons 7◆3, 7◆11, and 7◆12)The authors want students to feel comfortable talking about chanceevents. Therefore, one focus in Lessons 7-3, 7-11, and 7-12 is onvocabulary development. While many expressions are suggested (forexample, chance, unlikely, more likely, probably, certain), they should notbe taught formally. Students will gradually make these words part of theirvocabulary through repeated use. Choose expressions that are meaningfulto the class. Students are familiar with expect and predict, but probabilityis a difficult word which does not need to be used at this time.

Most of the probability activities follow a similarpattern: Students make predictions about thelikelihood of a particular outcome and then checktheir predictions by performing an experiment.The authors want students to become aware ofthe fact that the more often they repeat anexperiment, the more reliable their predictionswill be. In these activities, some individualresults may be very close to the expected results,but others may be far off. However, when theclass data are combined, the result should be veryclose to the predicted results.

See Section 12.1 of the Teacher’s ReferenceManual for more information concerningchance and probability.

Note

The authors believe thatmost students shouldbe exposed to conceptsand skills many timesand in many differentways, often only briefly,before they can masterthem. The probabilityactivities in EverydayMathematics aregood examples.

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Unit 7:Fractions and Their Uses; Chance and Probabilityellis2020.org/iTLG/iTLG Grade...

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