OverviewOverviewUnit 4 begins the more formal study of decimals through thousandths.Metric units of length, including conversions, are also studied. Unit 4 has four main areas of focus:

◆ To extend the base-ten place-value system to decimals,

◆ To review and extend basic concepts, notation, and applications for decimals,

◆ To extend whole-number methods of addition and subtraction to decimals, and

◆ To review relationships among metric units of length, and guidestudents as they use them.

224 Unit 4 Decimals and Their Uses

Unit Organizer 225

Lesson Objective Page

Contents

4◆1 Decimal Place Value 238To extend the base-ten place-value system to decimals.

4◆2 Review of Basic Decimal Concepts 244To review basic concepts and notation for decimals through hundredths.

4◆3 Comparing and Ordering Decimals 250To guide students as they compare and order decimals in tenths and hundredths.

4◆4 Estimating with Decimals 255To explain why decimals are useful; and to guide estimation of sums and differences of decimals.

4◆5 Decimal Addition and Subtraction 260To extend methods for whole-number addition and subtraction to decimals.

4◆6 Decimals in Money 266To provide practice adding and subtracting decimals to compute balances in a savings account.

4◆7 Thousandths 271To extend basic concepts and notation for decimals through thousandths.

4◆8 Metric Units of Length 277To review the relationships among metric units of length; and to guide students as they work with metric measurements.

4◆9 Personal References for Metric Length 283To assist students as they establish personal references for metric units of length.

4◆10 Measuring in Millimeters 289To guide students as they measure lengths to the nearest millimeter; and to provide practice converting measurements between millimeters and centimeters.

4◆11 Progress Check 4 294To assess students’ progress on mathematical content through the end of Unit 4.

226 Unit 4 Decimals and Their Uses

Learning In Perspective

To extend the base-ten place-valuesystem to decimals.

To review basic concepts and notationfor decimals through hundredths.

To guide students as they compareand order decimals in tenths andhundredths.

To explain why decimals are useful;and to guide estimation of sums anddifferences of decimals.

To extend methods for whole-numberaddition and subtraction to decimals.

To provide practice adding andsubtracting decimals to computebalances in a savings account.

To extend basic concepts and notationfor decimals through thousandths.

To review the relationships amongmetric units of length; and to guidestudents as they work with metricmeasurements.

To assist students as they establishpersonal references for metric units oflength.

To guide students as they measurelengths to the nearest millimeter; andto provide practice convertingmeasurements between millimetersand centimeters.

Lesson Objectives Links to the Past Links to the FutureGrades 1–3: Use dollars and cents notation. Grade 3: Model decimals with base-10 blocks.

Grade 3: Read and write 1-digit and 2-digit decimals.

Grade 3: Compare metric measurements indecimal notation. Compare decimals throughhundredths directly or by coloring 10-by-10 grids.

Grade 3: Relate decimals to metric measures.

Grades 1–3: Invent, share, and discuss paper-and-pencil methods for adding andsubtracting multidigit whole numbers. Use moneyto model decimal addition and subtraction.

Grades 1–3: Use dollars-and-cents notation torepresent money amounts; act out purchasing andchange-making situations to practice adding andsubtracting money amounts.

Grades 1 and 2: Model decimals throughhundredths. Grade 3: Display decimals throughthousandths.

Grades K–2: Introduce and use linear metricmeasures. Use decimals for money, with rulers,and on number lines. Grade 3: Express metricmeasurements in decimal notation.

Grades 1 and 2: Introduce nonstandard linearmeasures, such as hand and arm spans and foot lengths; introduce linear metric measures.Grade 3: Use personal references.

Grades 1–3: Choose and use appropriatemeasuring tools; refer to tables of equivalent unitsof metric measure.

Grades 4–6: Applications and maintenance.

Grade 5: Extend decimal notation throughthousandths.

Grades 5 and 6: Compare fractions by renamingthem as equivalent decimals.

Grades 4–6: Applications and maintenance.

Grades 5 and 6: Extend addition and subtraction methods to decimal places beyond the hundredths.

Grades 4–6: Applications and maintenance; in particular, money amounts are continually used to exemplify and practice algorithms for operations with decimals.

Grades 5 and 6: Read, write, compare, andcalculate with decimals to thousandths; practiceplace-value skills to thousandths.

Grades 5 and 6: Applications and maintenance.

Grades 5 and 6: Applications and maintenance.

Grades 4–6: Applications and maintenance,including repeated opportunities to measure inmetric units.

4◆1

4◆2

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4◆7

4◆8

4◆9

4◆10

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4◆10

Identify the values of digits in decimals. Number and Numeration Goal 1Read and write decimals through thousandths. Number and Numeration Goal 1Order decimals through thousandths on a number line. Number and Numeration Goal 6Use extended division facts to expand the place-value chart to decimals. Operations and Computation Goal 3Describe numeric patterns in number lines. Patterns, Functions, and Algebra Goal 1

Read and write decimals through hundredths. Number and Numeration Goal 1Model decimals through hundredths with base-10 blocks. Number and Numeration Goal 1Name the fractional part of the ONE represented by a base-10 block. Number and Numeration Goal 2Rename fractions with 10 and 100 in the denominator as decimals. Number and Numeration Goal 5

Model decimals through hundredths with base-10 blocks. Number and Numeration Goal 1Read and write decimals through hundredths. Number and Numeration Goal 1Rename fractions with 100 in the denominator as decimals. Number and Numeration Goal 5Compare and order decimals through hundredths. Number and Numeration Goal 6

Read and interpret decimals through tenths. Number and Numeration Goal 1Compare whole numbers and decimals. Number and Numeration Goal 6Estimate sums and differences of decimals; explain the strategies used. Operations and Computation Goal 6Use a table of data to answer questions. Data and Chance Goal 2

Model decimals through hundredths with base-10 blocks. Number and Numeration Goal 1Express the values of digits in decimals. Number and Numeration Goal 1Add and subtract decimals to the hundredths place. Operations and Computation Goal 2Judge the reasonableness of solutions to decimal addition and subtraction problems. Operations and Computation Goal 6

Read and write decimals through hundredths in the context of money. Number and Numeration Goal 1Add and subtract decimals through hundredths in the context of money. Operations and Computation Goal 2Estimate reasonable solutions for decimal addition and subtraction problems. Operations and Computation Goal 6Complete a table of deposits and withdrawals. Data and Chance Goal 1

Read, write, and model (with base-10 blocks) decimals through thousandths. Number and Numeration Goal 1Name the fractional part of the ONE represented by each base-10 block. Number and Numeration Goal 2Rename fractions with 10, 100, and 1,000 in the denominator as decimals. Number and Numeration Goal 5Compare and order decimals through thousandths. Number and Numeration Goal 6

Read and write decimals through hundredths. Number and Numeration Goal 1Use extended multiplication facts to convert between metric measurements. Operations and Computation Goal 3Measure objects or distances to the nearest centimeter. Measurement and Reference Frames Goal 1Describe relationships among metric units of length. Measurement and Reference Frames Goal 3

Identify personal references for metric units of length. Measurement and Reference Frames Goal 1Estimate, without tools, the length of objects or distances in centimeters, decimeters, Measurement and Reference Frames Goal 1

and meters.Measure the length of objects or distances in centimeters, decimeters, and meters. Measurement and Reference Frames Goal 1

Use extended multiplication facts to convert between metric measurements. Operations and Computation Goal 3Use a scale to determine actual size. Operations and Computation Goal 7Measure lengths to the nearest millimeter. Measurement and Reference Frames Goal 1Describe the relationship among metric units of length. Measurement and Reference Frames Goal 3

Key Concepts and Skills Grade 4 Goals*

* See the Appendix for a complete list of Grade 4 Goals.

Unit Organizer 227

Key Concepts and Skills

228 Unit 4 Decimals and Their Uses

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

▲

Home Communication Study Links provide homework and home communication.

Home Connection Handbook provides more ideas to communicateeffectively with parents.

Unit 4 Family Letter provides families with an overview, Do-AnytimeActivities, Building Skills Through Games, and a list of vocabulary.

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

Mixed practice with multiple choice 4◆1, 4◆ 4 , 4◆ 6

Mixed practice with writing/reasoning opportunity 4◆2, 4◆3, 4◆ 5, 4◆8, 4◆9

4◆1 Polygon Pair-Up Identifying properties of polygonsGeometry Goal 2

4◆2 Baseball Multiplication Developing automaticity with multiplicationfacts Operations and Computation Goal 3

4◆2, 4◆7 Base-10 Exchange Exploring relationships in decimal place valuesNumber and Numeration Goal 1

4◆3 Product Pile-Up Developing automaticity with multiplicationfacts Operations and Computation Goal 3

4◆3 Coin Top-It Comparing decimals in a money contextOperations and Computation Goal 6

4◆4, 4◆9 Number Top-It (Decimals) Comparing and ordering sets of decimalsNumber and Numeration Goal 6

4◆6 Name That Number Representing numbers in different waysNumber and Numeration Goal 4

4◆8 Fishing for Digits Expressing values of digits in whole numbers Numbers and Numeration Goal 1

Lesson Game Skill Practiced

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

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:

◆ Using a graph ◆ Using logical reasoning◆ Using a number line ◆ Trying and checking◆ Using data in a table ◆ Using estimation◆ Using a table ◆ Using computation

Lesson Activity

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

4◆ 3 Solve riddles by ordering decimals and using place value.

4◆ 4 Solve number stories about a bicycle trip and railroad tunnel lengths.

4◆ 5, 4◆ 6 Solve decimal addition and subtraction number stories involving measures, money, and time.

4◆ 6 Maintain a bank balance.

4◆ 8, 4◆ 10 Measure land invertebrates and convert between metric units.

4◆ 9 Estimate lengths with personal references.

Unit Organizer 229

Problem SolvingProblem Solving

Unit 4Lessons

NCTMStandards

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

1, 2,6–8

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, 2, 5,6–8

1, 2, 7, 8

1, 2, 6–8

1, 2, 6–9 1, 6–10 1, 6–10 1, 4 , 5,

8–10 4, 8–10 4, 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 4, review your Content by Strand Poster to help you seta monthly pace.

NCTM Standards

MOST CLASSROOMS

O C T O B E R N O V E M B E R D E C E M B E R

230 Unit 4 Decimals and Their Uses

Balanced Assessment

4◆ 1 Identify values of digits in whole numbers.[Number and Numeration Goal 1]

4◆ 2 Identify values of decimal digits.[Number and Numeration Goal 1]

4◆ 3 Compare decimals through hundredths.[Number and Numeration Goal 6]

4◆ 4 Estimate sums of decimals and explain strategy.[Operations and Computation Goal 6]

4◆ 5 Identify values of decimal digits.[Number and Numeration Goal 1]

4◆ 6 Identify data landmarks.[Data and Chance Goal 2]

4◆ 7 Write decimals through hundredths.[Number and Numeration Goal 1]

4◆ 8 Measure line segments to the nearest centimeter.[Measurement and Reference Frames Goal 1]

4◆ 9 Compare decimals through thousandths.[Number and Numeration Goal 6]

4◆10 Demonstrate automaticity with multiplication facts.[Operations and Computation Goal 3]

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-solving strategies:

Lesson 4◆2 Use base-10 blocks to model fractions

Lesson 4◆3 Express values of decimal digits

Lesson 4◆4 Use a number line to reinforce concept of decimal value

Lesson 4◆5 Align digits correctly in decimal sums and differences

Lesson 4◆7 Use base-10 blocks to model thousandths

Lesson 4◆8 Convert metric units of length

Unit Organizer 231

Read, write, and represent decimals through thousandths;identify digits and express their values in such numbers.[Number and Numeration Goal 1]

Find multiples of numbers less than 10; find factors ofnumbers. [Number and Numeration Goal 3]

Convert “easy” fractions to decimals.[Number and Numeration Goal 5]

Compare and order decimals through thousandths.[Number and Numeration Goal 6]

Add and subtract decimals to hundredths.[Operations and Computation Goal 2]

Estimate sums and differences of decimals.[Operations and Computation Goal 6]

Measure to the nearest centimeter.[Measurement and Reference Frames Goal 1]

Describe relationships among metric units of length.[Measurement and Reference Frames Goal 3]

Solve open number sentences.[Patterns, Functions, and Algebra Goal 2]

CONTENT ASSESSED Self Oral/Slate Written Open Response

ASSESSMENT ITEMS

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Periodic Assessment4◆11 Progress Check 4

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

◆ Writing and solving place-value puzzles, Lesson 4◆1◆ Expressing values of decimal digits, Lesson 4◆2◆ Converting coin amounts to decimal equivalents, Lesson 4◆2◆ Naming a ray, Lesson 4◆3◆ Writing and solving decimal riddles, Lesson 4◆3◆ Expressing values of decimal digits, Lesson 4◆5◆ Explaining the process for solving an open sentence, Lesson 4◆5◆ Solving decimal computation problems, Lesson 4◆6◆ Explaining the process of ordering decimals, Lesson 4◆9◆ Using metric units of length to create a scavenger hunt, Lesson 4◆9◆ Forming a relay team, Lesson 4◆11

Assessment HandbookUnit 4 Assessment Support

◆ Grade 4 Goals, pp. 37–50 ◆ Unit 4 Open Response◆ Unit 4 Assessment Overview, pp. 76–83 • Detailed rubric, p. 80

• Sample student responses, pp. 81–83Unit 4 Assessment Masters

◆ Unit 4 Self Assessment, p. 169◆ Unit 4 Written Assessment, pp. 170–172◆ Unit 4 Open Response, p. 173◆ Unit 4 Class Checklist, pp. 260, 261, and 303◆ Unit 4 Individual Profile of Progress, pp. 258,

259, and 302

◆ Exit Slips, p. 311◆ Math Logs, pp. 306–308◆ Other Student Assessment Forms, pp. 304,

305, 309, and 310

Daily Lesson Support

Differentiated Instruction

ENGLISH LANGUAGE LEARNERS

READINESS ENRICHMENT

EXTRA PRACTICE4 ◆3 Creating a Decimals All Around Museum4 ◆6 Building a Math Word Bank4 ◆8 Building a Math Word Bank

4 ◆1 Using money to explore decimals4 ◆2 Playing Base-10 Exchange4 ◆3 Playing Coin Top-It4 ◆4 Estimating the cost of a purchase4 ◆5 Investigating a decimal number grid4 ◆6 Finding totals and making change4 ◆7 Counting by thousandths4 ◆8 Exploring metric units4 ◆9 Matching metric units4 ◆10 Exploring standard measurement units

4 ◆1 Writing and solving place-value puzzles4 ◆2 Exploring the ONE4 ◆3 Writing decimal riddles4 ◆4 Solving gasoline mileage problems4 ◆4 Solving a decimal magic square puzzle4 ◆5 Solving hiking trail problems4 ◆6 Solving “goodie bag” problems4 ◆7 Analyzing softball batting averages4 ◆8 Exploring prefix use in metric units4 ◆9 Designing a measurement hunt4 ◆10 Investigating ratios

4 ◆1 Making and using a place-value tool4 ◆7 Playing Base-10 Exchange4 ◆10 Measuring to the nearest millimeter

5-Minute Math4 ◆3 Practicing with decimals 4 ◆9 Practicing metric measurements

Cross-Curricular LinksLiterature Social StudiesLesson 4◆ 3 Students write and solve Lesson 4◆ 7 Students examine and decimal riddles. compare data for African countries.

Lesson 4◆ 10 Students further explore the Scienceconcept of scale by reading If You Hopped Lesson 4◆ 10 Students measure land Like a Frog. invertebrates and convert the metric units.

Differentiation HandbookSee the Differentiation Handbook for materials on Unit 4.

Adjusting the Activity4 ◆1 Exploring names of places 4◆2 Equating tenths and hundredths to

dollars-and-cents notation4 ◆2 Emphasizing the –th in decimal

names ELL4 ◆3 Encouraging the use of tools in

Product Pile-Up ELL4 ◆4 Circling numbers to consider4 ◆4 Playing a 3-place-decimal version of

Number Top-It (Decimals)

4 ◆5 Using manipulatives to solve decimalproblems ELL

4 ◆6 Using manipulatives or number grid4◆7 Practicing reading decimals ELL4 ◆8 Using metersticks4 ◆9 Solving problems that involve more

than one measurement4 ◆9 Discussing the term reference ELL4 ◆10 Exploring the approximate nature

of measurement4 ◆10 Using cm/mm rulers ELL

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

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232 Unit 4 Decimals and Their Uses

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 4◆3 The Everything Kids’ Joke Book: Side-Splitting, Rib-TicklingFun, by Michael Dahl, Adams Media Corporation, 1992Kids’ Funniest Jokes, edited by Shelia Anne Barry, Sterling Publishing Co., 1993Lesson 4◆10 If You Hopped Like a Frog, by David M. Schwartz, ScholasticPress, 1999Count Your Way through Africa, by Jim Haskins, Carolrhoda Books, 1989Millions to Measure, by David M. Schwartz, HarperCollins, 2003Inchworm and a Half, by Elinor J. Pinczes, Houghton Mifflin, 2003

Student Reference Bookpp. 26, 130, 231, 232, 242, 254, 256, 258, 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.

Unit Organizer 233

Unit 4 Vocabularybalancecentimeter (cm)decimaldecimeter (dm)deposithundredthinterestmeter (m)millimeter (mm)ONEpersonal measurement

referencespeedometertenththousandthtrip meterunitwholewithdrawal

Grade 3

Grade 4

Grade 5

5◆11 1◆1, 5◆1,5◆7, 6◆10

1◆5, 5◆2

1◆9,7◆7

2◆1, 2◆2

1◆10,1◆11

5◆5, 5◆8,5 ◆11

3◆1, 3◆3,5◆9

1◆4, 3◆1,10◆1

3◆1, 3◆3,10◆1

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

2 ◆7 5 ◆5 5 ◆ 3,5 ◆6

5 ◆7 2◆2,2◆3

4◆1 6◆2,6◆3

Professional Development

Teacher’s Reference Manual LinksSection

3.2.1

9.9.1

9.3.1

1.2.6

16.1.1

11.2.1

11.2.2

Topic

Numeration and Order Tools

Base-10 Blocks

Fraction and Decimal Notation

Museums

Why Estimate?

Addition Algorithms

Subtraction Algorithms

Lesson

4◆ 1,

4◆ 2,

4◆ 3

4◆ 4

4◆ 5,

4◆ 6

TopicLesson Section

Fraction and Decimal Notation

Metric System

Measuring Tools

Measurement and Estimation

Rulers and Tape Measures

Personal Measures

Converting between Measures

4◆ 7

4◆ 8

4◆ 9

4◆ 10

9.3.1

14.2.2

3.2.5

14.11.1

14.11.3

14.1

14.2.3

234 Unit 4 Decimals and Their Uses

Materials

Lesson Masters Manipulative Kit Items Other Items

* Denotes optional materials

Technology Assessment Management System, Unit 4iTLG, Unit 4

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4◆3

4◆5

4◆6

4◆7

4◆8

4◆9

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4◆11

4◆4

transparencies of Math Masters,pp. 106 and 107*; Study LinkMaster, p. 108; Polygon Pair-Upcards, pp. 496 and 497; TeachingMaster, p. 109; Teaching Aid Master, p. 428; Place-Value FlipBook, pp. 399–402

Study Link 4◆1; Teaching AidMasters, pp. 388 or 389, and 426;transparency of Math Masters, p.426*; Study Link Master, p. 110;Game Masters, pp. 458 and 460;Teaching Master, p. 111

Study Link 4◆2; Study Link Master,p. 112; Game Masters, pp. 467 and 506

Study Link 4◆3; Study Link Master,p. 113; Game Masters, pp. 490,491*, and 506*; Teaching Masters,pp. 114–117; Teaching Aid Master, p. 428

Study Link 4◆4; Teaching Masters,pp. 118, 120, and 121; Teaching AidMasters, pp. 427 and 428; StudyLink Master, p. 119

Study Link 4◆5; Teaching Masters,pp. 114 and 124; Teaching Aid

Masters, pp. 427* and 428*;transparency of Math Masters,p.122; Study Link Master, p. 123;Game Master, p. 489

Study Link 4◆6; transparency ofMath Masters, p. 125*; Study LinkMaster, p. 126; Teaching AidMasters, pp. 419–421*; TeachingMaster, p. 127; Game Master, p. 459

Study Link 4◆7; Study Link Master,p. 128; Game Master, p. 472*;Teaching Masters, pp. 129 and 130

Study Link 4◆8; Study Link Master,p. 131; Game Masters, pp. 491 and506; Teaching Master, p. 132

Study Link 4◆9; Teaching AidMasters, pp. 413, 414, 416, and 429;Study Link Master, p. 133; TeachingMaster, p. 134

Study Link 4◆10; AssessmentMasters, pp. 169–173; Study LinkMasters, pp. 135–138

slatecalculator*dimes and pennies

base-10 blocks (1 flat, 20 longs, 20 cubes); two 6-sided dice

slatecalculator4 pennies

base-10 blocks; number cards 1–10(8 each); slate; coins

number cards 0–9 (4 each)

slatequarters, dimes

base-10 blocks slateNumber Grid Posterquarters*, nickels*, dimes*, pennies*base-10 blocksdeck of number cardscalculator

three 6-sided dice calculatorbase-10 blocksslate

slate; calculator; meterstick; tapemeasure; centimeter ruler; base-10cubes and longs

tape measure; meterstick; number cards 0–9 (4 each)

centimeter rulerslate

slate

overhead calculator*; scissors

� , / Facts Table*

scissors; The Everything Kids’ Joke Book: Side-Splitting, Rib-Tickling Fun; Kids’ Funniest Jokes

money*; � , / Fact Triangles; coins; scissors

chart paper; colored pencils

ruler; � , / Fact Triangles

scissors; highlighter; ruler; If You Hopped Like aFrog

Unit Organizer 235

The discussion below highlights the major content ideas presented in Unit 4 and helps establish instructional priorities.

Decimal Concepts, Notation, and Applications (Lessons 4◆1–4◆10)Numbers that are used to count are usually expressed as whole numbers.But numbers that are measures frequently fall between two whole numbers.In ancient times people realized this and long ago began inventing ways toexpress those in-between numbers. Some of these ancient inventions arewhat mathematicians now call fractions, and some are what they now calldecimals. The focus of Unit 4 is on decimals; fractions will be covered indetail in Unit 7.

The first six lessons of Unit 4 review basic concepts and notation fortenths and hundredths with the help of base-10 blocks and moneyapplications. These concepts are extended to thousandths, beginning inLesson 4-7, again using base-10 blocks to represent decimals and withmetric linear measure applications.

As your class studies decimals in Unit 4, there are several important ideas to keep in mind.

◆ Decimal notation is a natural and fairly easy extension of “place-value” notation for whole numbers, which by now is very familiar tostudents. The trading rules are exactly the same—each move of oneplace to the left increases the value by a factor of ten, and each moveto the right decreases the value by a factor of one-tenth.

◆ Procedures for all operations with decimals are exactly like thecorresponding procedures with whole numbers, whether studentscompute answers using calculators or paper-and-pencil algorithms. For addition and subtraction, it is necessary to keep track of where the decimal points are so that the digits stay in the correct place-valuecolumns. (For an example, see Lessons 4-5 and 4-6.) For multiplicationand division, the computation is performed as if all numbers werewhole numbers, and then the decimal point is placed by estimating thesize of the answer. (See Lessons 9-8 and 9-9.)

Mathematical Background

1,000s

Thousands

100s

Hundreds

10s

Tens

1s

Ones

0.1s

Tenths

0.01s

Hundredths

0.001s

Thousandths. .

� 10 � 10 � 10 � 10 � 10 � 10

Note

In our modern world,decimals are often usedto express extremelyprecise measures. Twohundred years ago, themost precise timepiecesavailable were thechronometers used bythe British navy. Theseinstruments kept accuratetime merely to seconds.But today, Olympic sportsevents are frequentlytimed to thousandths of asecond. Atomic clocks,used by scientists, areable to measure time totrillionths of a second(picoseconds).

◆ Since first grade, the Everyday Mathematics program has placed astrong emphasis on decimal notation for uses of money, includingcalculations with decimal money notation in response to numberstories. Furthermore, teachers have reported that students often cando fairly difficult calculations—even with 4- and 5-digit numbers—ifthey think of the numbers as money. Additionally, by fourth grade,students have done much work with metric measures, sometimesexpressed in decimal notation.

◆ The natural notation for a calculator world is decimal notation. Allcalculators work in decimal notation, even the special calculators thatalso handle fraction notation.

◆ All of the problems and examples are presented in real-world contexts.This special emphasis on the uses of decimals is consistent with thegeneral intention of Everyday Mathematics to tie mathematics closelyto its uses.

Decimal notation has become indispensable in science and industry. Inparticular, decimal notation linked to “scientific notation” is useful inexpressing both very small and very large numbers. (Scientific notationwill be introduced in Grade 5.)

For additional information about decimal concepts, notation, andapplications, see Section 9.3 of the Teacher’s Reference Manual.

236 Unit 4 Decimals and Their Uses

66

1010

30

5070

90 110 130150

170

190kph

20

3040

50 60 70 8090

100

110

120mph

The odometer above shows 12,963. This tells us that the car has traveled at least12,963 miles but less than 12,964 miles. The trip meter shows 45.6. This means that the car traveled at least 45.6 miles but less than 45.7 miles, after the trip meter was reset to 0.

World Tour Follow-Up (Lesson 4◆7)In Lesson 4-7, the class will divide into small tour groups. Each groupwill select and visit another country in Africa. In Unit 5, the entire classwill travel to another region.

Personal References (Lesson 4◆9)Lesson 4-9 introduces a new routine that focuses on measures. Studentsare asked to find their own personal references for common units ofmeasure. They record the names of familiar objects that happen to begood estimates for different metric lengths. A finger width, for example,may be about a centimeter; the thickness of a dime or paper clip, about a millimeter. Students’ personal references will be used throughout theyear to help estimate lengths of many objects, with the results recordedon Math Journal 1, page 99.

Section 14.1 of the Teacher’s Reference Manual provides additionalinformation about personal references for common units of measure.

Unit Organizer 237