Unit 5: Big Numbers, Estimation, and Computationellis2020.org/iTLG/iTLG Grade 4/U5.pdf · 5 8 Big...

OverviewUnit 5 begins with two lessons that focus on extending multiplicationskills, in preparation for the introduction of the partial-productsalgorithm for multiplication. The lattice algorithm is also introduced asan alternative to the partial-products algorithm. Unit 5 also emphasizesreading, writing, and using large numbers, including the use of powersof 10 to represent large numbers. Unit 5 has four main areas of focus:

◆ To extend basic multiplication facts and to review the basic principles of multiplication of multidigit numbers,

◆ To provide practice estimating and deciding when estimation is appropriate,

◆ To review and provide practice with the partial-products algorithmand the lattice method for multiplication, and

◆ To provide practice reading, writing, and comparing large numbersusing patterns in the base-ten place-value system.

298 Unit 5 Big Numbers, Estimation, and Computation

Unit Organizer 299

Lesson Objective Page

Contents

Unit Organizer 299

Lesson Objective Page

Contents

5◆1 Extended Multiplication Facts 314To extend basic multiplication facts to products of ones and tens and products of tens and tens.

5◆2 Multiplication Wrestling 320To provide practice with extended multiplication facts; and to introduce the basic principles of multiplication with multidigit numbers.

5◆3 Estimating Sums 325To provide practice deciding whether estimation is appropriate in a given situation; and to provide practice estimating sums.

5◆4 Estimating Products 331To provide practice estimating whether a product is in the tens, hundreds, thousands, or more.

5◆5 Partial-Products Multiplication (Part 1) 337To review and provide practice with the partial-products algorithm for 1-digit multipliers.

5◆6 Partial-Products Multiplication (Part 2) 343To introduce and provide practice with the partial-products algorithm for 2-digit multipliers.

5◆7 Lattice Multiplication 349To review and provide practice with the lattice method for multiplication.

5◆8 Big Numbers 355To provide practice reading, writing, and comparing large numbers using patterns in the base-ten place-value system.

5◆9 Powers of 10 361To introduce exponential notation for powers of 10 as a way of naming the values of places in our base-ten system.

5◆10 Rounding and Reporting Large Numbers 367To discuss sensible ways of reporting a count when a large numberof items has been counted.

5◆11 Comparing Data 373To guide students as they look up and compare numerical data, including geographical measurements.

5◆12 Progress Check 5 378To assess students’ progress on mathematical content through the end of Unit 5.


To extend basic multiplication factsto products of ones and tens andproducts of tens and tens.

To provide practice with extendedmultiplication facts; and to introducethe basic principles of multiplicationwith multidigit numbers.

To provide practice deciding whetherestimation is appropriate in a givensituation; and to provide practiceestimating sums.

To provide practice estimatingwhether a product is in the tens,hundreds, thousands, or more.

To review and provide practice withthe partial-products algorithm for 1-digit multipliers.

To introduce and provide practicewith the partial-products algorithmfor 2-digit multipliers.

To review and provide practice withthe lattice method for multiplication.

To provide practice reading, writing,and comparing large numbers usingpatterns in the base-ten place-valuesystem.

To introduce exponential notation forpowers of 10 as a way of naming thevalues of places in our base-tensystem.

To discuss sensible ways of reportinga count when a large number ofitems has been counted.

To guide students as they look upand compare numerical data,including geographicalmeasurements.

Lesson Objectives Links to the Past Links to the Future

Learning In Perspective

Grade 3: Use mental strategies to calculateproducts of 1-digit and multidigit numbers, and of 2-digit numbers and powers of 10.

Grade 3: Use mental strategies to calculateproducts of 1-digit and multidigit numbers, andof 2-digit numbers and powers of 10.

Grades 1-3: Use estimates to check thereasonableness of answers. Grades 2 and 3:Use terms such as about, a little more than,almost, and in between.

In Grades 1–3, students solve number-storyproblems using estimation and use estimates to check the reasonableness of answers. InGrades 2 and 3, students list situations that call for estimates; use terms such as about,a little more than, almost, and in between.

Grade 3: Use arrays and base-10 blocks tomodel a partial-products method for findingproducts of 2-digit numbers.

Grade 3: Use arrays and base-10 blocks tomodel a partial-products method for findingproducts of 2-digit numbers.

Grade 3: Introduce the lattice method formultidigit multiplication.

Grades 1 and 2: Explore place value through base-10 blocks, place-value books, and games;read and write numbers less than 100,000. Grade 3: Extend place value to millions.

Grades 2 and 3: Model square numbers witharrays.

Grade 3: Estimate to solve number-storyproblems and to check the reasonableness of answers; use a slate routine to practicerounding to multiples of 10.

Grades 2 and 3: Use relation symbols (=, >, <)to compare numbers.

Grades 4–6: Applications and maintenance,including extensions to solve related division facts.

Grades 4–6: Applications and maintenance,including extensions to solve related division facts.

Grades 4–6: Round numbers and then usemagnitude estimates to locate the decimal point when multiplying and dividing decimals.

In Grades 4–6, students round numbers and then use magnitude estimates to locate the decimal point when multiplying and dividing decimals.

Grade 4: Extend the partial-products method to multiplication of decimals; use multiplication and division to calculate rates and unit rates (Units 6, 9, 12). Grades 5 and6: Application and maintenance.

Grade 4: Extend the partial-products method to multiplication of decimals; use multiplication and division to calculate rates and unit rates (Units 6, 9, 12).

Grade 4: Extend the lattice method tomultiplication of decimals; use multiplication and division to calculate rates and unit rates(Units 9, 12).

Grade 5: Extend place-value facility to trillionsand to thousandths; use exponential notationand scientific notation to represent both largeand small numbers.

Grades 5 and 6: Compare numbers written in exponential and scientific notation.

Grades 5 and 6: Applications and extensions,including rounding to decimal place values.

Grades 4-6: Applications and maintenance.

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Solve basic multiplication facts. Operations and Computation Goal 3Use basic multiplication facts to compute fact extensions. Operations and Computation Goal 3Use repeated addition and arrays to model multiplication. Operations and Computation Goal 7Describe rules to solve problems involving products of ones and tens and products Patterns, Functions, and Algebra Goal 1

of tens and tens.

Write numbers in expanded notation. Number and Numeration Goal 4Add multidigit numbers. Operations and Computation Goal 2Use basic facts to compute extended facts. Operations and Computation Goal 3Solve multidigit multiplication problems. Operations and Computation Goal 4Evaluate numeric expressions containing parentheses. Patterns, Functions, and Algebra Goal 3Use the Distributive Property of Multiplication over Addition. Patterns, Functions, and Algebra Goal 4

Estimate sums. Operations and Computation Goal 6Compare appropriate situations for the use of exact answers and estimates. Operations and Computation Goal 6Use a travel map to find driving distance and driving time. Data and Chance Goal 2

Solve problems involving products where factors are multiples of 10, 100, 1,000, and so on. Operations and Computation Goal 3Estimate whether a product is in the tens, hundreds, thousands, or more. Operations and Computation Goal 6Explore meanings of average. Data and Chance Goal 2

Write numbers in expanded notation. Number and Numeration Goal 4Add multidigit numbers. Operations and Computation Goal 2Use basic facts to compute extended facts. Operations and Computation Goal 3Use the partial-products algorithm to solve multiplication problems with 1-digit multipliers. Operations and Computation Goal 4Estimate whether a product is in the tens, hundreds, thousands, or more. Operations and Computation Goal 6Apply the Distributive Property of Multiplication over Addition. Patterns, Functions, and Algebra Goal 4

Write numbers in expanded notation. Number and Numeration Goal 4Use the partial-products algorithm to solve multiplication problems with 2-digit multipliers. Operations and Computation Goal 4Estimate whether a product is in the tens, hundreds, thousands, or more. Operations and Computation Goal 6Apply the Distributive Property of Multiplication over Addition. Patterns, Functions, and Algebra Goal 4

Add single-digit numbers. Operations and Computation Goal 2Solve basic multiplication facts. Operations and Computation Goal 3Use the lattice method to solve multiplication problems with 1- and 2-digit multipliers. Operations and Computation Goal 4

Read and write whole numbers to hundred billions. Number and Numeration Goal 1Identify digits and their values in whole numbers to hundred billions. Number and Numeration Goal 1Use multiplication to solve a multistep problem. Operations and Computation Goals 3 and 4Make reasonable estimates. Operations and Computation Goal 6

Read and write large numbers; identify the digits and their values. Number and Numeration Goal 1Use exponential notation to represent powers of 10. Number and Numeration Goal 4Use expanded notation to represent powers of 10. Number and Numeration Goal 4Identify and describe patterns in a place-value table. Patterns, Functions, and Algebra Goal 1

Read and write whole numbers; identify digits and their values. Number and Numeration Goal 1Describe differences between estimates and exact counts. Operations and Computation Goal 6Round large numbers to a given place. Operations and Computation Goal 6Use data presented in a table. Data and Chance Goal 2

Read and write large numbers. Number and Numeration Goal 1Develop the meaning of percent as per 100. Number and Numeration Goal 5Compare large numbers. Number and Numeration Goal 6Use a table of information. Data and Chance Goal 2

Key Concepts and Skills Grade 4 Goals*

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

Unit Organizer 301

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Key Concepts and Skills

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

5 ◆ 1 Beat the Calculator Solving extended multiplication factsOperations and Computation Goal 3

5 ◆ 1, 5 ◆ 7 Multiplication Top-It Developing automaticity with multiplicationfacts Operations and Computation Goal 3

5 ◆ 2, 5 ◆ 4 Multiplication Wrestling Calculating and finding the sum of partialproductsOperations and Computation Goal 4; Patterns, Functions, and Algebra Goal 4

5 ◆ 3 Product Pile-Up Developing automaticity with multiplication facts Operations and Computation Goal 3

5 ◆ 6 Name That Number Representing numbers in different waysNumber and Numeration Goal 4

5 ◆ 8, 5 ◆ 11 High-Number Toss Identifying values of digits and comparing large numbersNumber and Numeration Goals 1 and 6

5 ◆ 9 Polygon Pair-Up Identifying properties of polygonsGeometry Goal 2

5 ◆ 11 Number Top-It Comparing large numbersNumber and Numeration Goal 6

Lesson Game Skill Practiced

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

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Home Communication Study Links provide homework and home communication.

Home Connection Handbook provides more ideas to communicate effectively with parents.

Unit 5 Family Letter provides families with an overview, Do-AnytimeActivities, Building Skills Through Games, and a list of vocabulary.

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

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

Mixed practice with multiple choice 5 ◆3, 5 ◆4, 5 ◆5, 5 ◆8, 5 ◆9, 5 ◆10

Mixed practice with writing/reasoning opportunity 5 ◆3, 5 ◆4, 5 ◆5, 5◆6, 5◆8, 5◆9


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

◆ Using computation ◆ Using data in a table◆ Using estimation ◆ Using logical reasoning◆ Using information in pictures and maps ◆ Writing a number sentence

Lesson Activity

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

5◆1, 5◆5 Write and solve number stories involving multiplication.5◆6

5◆3 Plan a driving trip to four cities by using a travel map to estimate distance and time.

5◆4 Compare a student’s weekly and yearly consumption of food and beverages to that of the “average” American.

5◆4 Solve number stories involving questions like “About how many ...?” and “Is that more/less than ...?”

5◆8 Find how many dots are in a 50 by 40 array on one page, then on multiple pages and on multiple reams.

5◆9 Find the largest number in the World Tour section of the Student Reference Book.

5◆11 Identify minimum/maximum values and compare data in population, area, and climate tables.

Unit Organizer 303

Problem SolvingProblem Solving

Unit 5Lessons

NCTMStandards

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

1,6–8

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, 6–10 1–2, 6–8

1, 6–8

1, 6–10

1, 6–10

1, 6–10

1,6–8

1–2,6–10

1, 5,8–10

1, 4–10 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 5, review your Content by Strand Poster to help you seta monthly pace.

NCTM Standards

MOST CLASSROOMS

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

http://standards.nctm.org


Balanced Assessment

5◆1 Use basic facts to compute fact extensions. [Operations and Computation Goal 3]

5◆2 Determine whether a number sentence is true or false.[Patterns, Functions, and Algebra Goal 2]

5◆3 Explain use of estimation to solve addition problems.[Operations and Computation Goal 6]

5◆4 Use the Distributive Property of Multiplication over Addition in the context of the partial-products algorithm.[Patterns, Functions, and Algebra Goal 4]

5◆5 Use the partial-products algorithm. [Operations and Computation Goal 4]

5◆6 Estimate reasonable solutions to whole-number multiplication problems.[Operations and Computation Goal 6]

5◆7 Demonstrate automaticity with multiplication facts.[Operations and Computation Goal 3]

5◆8 Use extended multiplication facts in a problem solving situation.[Operations and Computation Goal 3]

5◆9 Describe numeric patterns.[Patterns, Functions, and Algebra Goal 1]

5◆10 Demonstrate automaticity with multiplication facts.[Operations and Computation Goal 3]

5◆11 Compare numbers up to 1 billion. [Number and Numeration Goal 6]

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 5◆1 Understand equivalence when multiplying ones and tens

Lesson 5◆2 Align partial products according to values of digits

Lesson 5◆2 Interpret a data table

Lesson 5◆3 Plan an extended trip

Lesson 5◆5 Express the value of a digit

Lesson 5◆6 Consider the value of each digit

Lesson 5◆8 Understand appropriateness of estimation or exact answers

Unit Organizer 305

Read and write whole numbers through millions; identifydigits and their values. [Number and Numeration Goal 1]

Write powers of 10 in exponential notation.[Number and Numeration Goal 4]

Order whole numbers through millions.[Number and Numeration Goal 6]

Solve extended multiplication facts.[Operations and Computation Goal 3]

Multiply multidigit whole numbers. [Operations and Computation Goal 4]

Make estimates for addition and multiplication problems.[Operations and Computation Goal 6]

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

Measure line segments to the nearest �14� in. and 0.5 cm.

[Measurement and Reference Frames Goal 1]

CONTENT ASSESSED Self Oral/Slate Written Open Response

ASSESSMENT ITEMS

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Periodic Assessment5◆12 Progress Check 5

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

◆ Using estimation strategy to solve a problem, Lesson 5◆2◆ Estimating sums to plan a travel route, Lesson 5◆3◆ Explaining a shortcut in solving division problems, Lesson 5◆4◆ Explaining how there can be more than one correct estimate, Lesson 5◆5◆ Interpreting a remainder, Lesson 5◆8◆ Estimating using large numbers and points of reference, Lesson 5◆8◆ Explaining intersecting and perpendicular lines, Lesson 5◆9◆ Solving a million-dollar problem, Lesson 5◆12

Assessment HandbookUnit 5 Assessment Support

◆ Grade 4 Goals, pp. 37–50 ◆ Unit 5 Open Response◆ Unit 5 Assessment Overview, pp. 84–91 • Detailed rubric, p. 88

• Sample student responses, pp. 89–91

Unit 5 Assessment Masters◆ Unit 5 Self Assessment, p. 174◆ Unit 5 Written Assessment, pp. 175–177◆ Unit 5 Open Response, p. 178◆ Unit 5 Class Checklist, pp. 264, 265, and 303◆ Unit 5 Individual Profile of Progress, pp. 262,

263, and 302

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

305, 309, and 310

Find and use rules for simple functions.[Patterns, Functions, and Algebra Goal 1]

Apply the Distributive Property of Multiplication overAddition. [Patterns, Functions, and Algebra Goal 4]

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Differentiated Instruction

Daily Lesson Support

ENGLISH LANGUAGE LEARNERS

READINESS ENRICHMENT

EXTRA PRACTICE5◆1 Building a Math Word Bank5◆3 Building a Math Word Bank5◆7 Displaying multiplication algorithms5◆9 Discussing the word power

5◆1 Playing Multiplication Top-It5◆2 Reviewing partial-sums addition 5◆3 Exploring rounding with base-10 blocks5◆4 Using curved number lines to round5◆5 Modeling multiplication5◆5 Exploring patterns in extended facts5◆6 Modeling multiplication5◆7 Exploring Fact Lattice patterns5◆8 Playing High-Number Toss5◆10 Using number lines to find halfway points5◆11 Comparing large numbers

5◆2 Judging Multiplication Wrestling5◆3 Estimating the shortest route5◆4 Finding missing numbers and digits5◆5 Solving a multiplication puzzle5◆6 Solving a multistep number story 5◆6 Completing Venn diagrams5◆7 Investigating Napier’s Rods5◆8 Estimating with large numbers5◆8 Exploring big numbers5◆9 Using a calculator for powers of 105◆10 Rounding bar graph data

5◆1 Solving multiplication/division puzzles5◆11 Playing High-Number Toss

5-Minute Math5◆4 Estimating products 5◆9 Using exponential notation5◆10 Rounding numbers

Adjusting the Activity5◆1 Recording computation strategies ELL5◆1 Using base-10 blocks to model

multiplication with arrays ELL5◆1 Focusing on multiplication of ones by

tens or tens by tens5◆2 Acting out partial-products

multiplication ELL5◆3 Recording words on the board and

discussing their meaning ELL5◆3 Using clocks for rounding time5◆4 Illustrating two meanings5◆5 Using a number model 5◆6 Organizing multiplication problems

5◆6 Explaining similarities between thepartial-products algorithm andMultiplication Wrestling

5◆7 Discussing the term lattice ELL5◆7 Modifiying the lattice method5◆9 Finding another name for 10�1

5◆10 Using number lines to visualize arounding method ELL

5◆10 Explaining the relationship betweenrounding and percent error

5◆11 Finding information about geographical measurements

5◆11 Shading grids to show percents 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

Cross-Curricular LinksSocial Studies LiteratureLesson 5◆2 Students interpret data about presidents of Lesson 5◆8 Students read How Much Is a Million?

the United States.Lesson 5◆3 Students examine a travel map.Lesson 5◆11 Students compare the heights of mountains.Lesson 5◆11 Students work on the World Tour Project.

Differentiation HandbookSee the Differentiation Handbook for materials on Unit 5.

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

Connecting Math and LiteracyTen Times Better, by Richard Michelson, Marshall Cavendish Children’s Books, 2000Moira’s Birthday, by Robert Munsch, Annick Press, 1987Lesson 5◆8 How Much Is a Million?, by David M. Schwartz,HarperTrophy, 1993 On Beyond a Million: An Amazing Math Journey, by David M. Schwartz,Dragonfly Books, 2001

Student Reference Bookpp. 4, 233, 252–255, 258, 259, 264, 271, 279–281, 297, 301,

and 304

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 5 Vocabularybillionestimationextended multiplication factsexponentlatticelattice method (for

multiplication)magnitude estimatemillionpartial productpartial-products methodpowers of 10quadrillionquintillionrough estimateroundrounding (to a certain place)scientific notationsextilliontrillion

Grade 3

Grade 4

Grade 5

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Professional Development

Teacher’s Reference Manual LinksSection

10.3.2

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1.2.1

11.2.3

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Topic

Multiplication and DivisionUse Classes

Fact Families/Fact Triangles

Games

Multiplication Algorithms

Estimates in Calculations

Approximation and Rounding

Arithmetic Operations Tools


Lesson

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Counting

Extreme Numbers

Powers and Exponents

Extreme Numbers


Linear Measures in Geography


Topic

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Materials

Lesson Masters Manipulative Kit Items Other Items

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Game Master, pp. 461 and 506 4 each of number cards 1–10; �, / Fact Triangles; Study Link Master, p.139 slate; calculator; Multiplication/DivisionTeaching Aid Master, p. 430 tape measure or ruler; Facts Table*

base-10 blocks*Study Link 5◆1 4 each of number cards 0–9 index cards*Game Master, p. 488 or a 10-sided dietransparency of Math Masters, p. 488* slateStudy Link Master, p. 140Teaching Masters, pp. 141 and 142

Study Link 5◆2 per group: 8 each of number demonstration clock*Teaching Masters, pp. 143, 145 and 146 cards 1–10; base-10 blocks;Study Link Master, p. 144 calculator*Study Link 5◆3 4 each of number cards 0–9 Teaching Masters, pp. 147, 149 and 150 or a 10-sided dieGame Master, p. 485 calculatorStudy Link Master, p. 148

Study Link 5◆4 slate erasable marker; transparent tapeTeaching Aid Masters, pp. 403 or 431 tape measure or rulerStudy Link Master, p. 151 base-10 blocksTeaching Masters, pp. 152 and 153transparencies of Math Masters, pp. 432 and 433

Study Link 5◆5 deck of number cards 3" by 5" cards*; erasable marker; Game Master, p. 489* base-10 blocks transparent tapeTeaching Aid Masters, 388* or 389*, and 403 or 431 slateTeaching Masters, pp. 155 and 156 calculator*Study Link Master, p. 154transparencies of Math Masters, pp. 432 and 433

Study Link 5◆6 4 each of number cards 1–10 �, / Fact Triangles; Teaching Aid Masters, pp. 434 and 435 index cards*; dictionary*;transparency of Math Masters, p. 434* Multiplication/Division Facts Table;Teaching Masters, pp. 158–161 scissors; chart paper; coloredStudy Link Master, p. 157 pencils and markersGame Master, p. 506

Study Link 5◆7 calculator 1 ream of copy paper; 1 empty Teaching Masters, pp. 162, 164 and 165 1 six-sided die carton used to pack 10 reams of Teaching Aid Masters, pp. 388* or 389* paper; How Much Is a Million?Study Link Master, p. 163Game Master, p. 487*Study Link 5◆8 calculatorTeaching Aid Masters, pp. 388 or 389transparency of Math Masters, p. 166*Study Link Master, p. 167Game Masters, pp. 496 and 497Teaching Master, p. 168

Study Link 5◆9 pen or colored pencilStudy Link Master, p. 169Teaching Aid Masters, pp. 410, 414, and 416Teaching Masters, pp. 170 and 171

Study Link 5◆10 4 each of number cards 0–9; tapeTeaching Aid Masters, pp. 419–421 and 426* 1 six-sided dieStudy Link Master, p. 172 slateGame Masters, pp. 487, 492, 493, and 506

Study Link 5◆11 slateAssessment Masters, pp. 174–178Study Link Masters, pp. 173–176 Technology

Assessment Management System, Unit 5iTLG, Unit 5

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Mathematical Background

The discussion below highlights the major content ideas presented in Unit 5 and helps establish instructional priorities.

Extending Multiplication Skills(Lessons 5◆1 and 5◆2)Lesson 5-1 extends the basic multiplication facts (3 � 4) to products of onesand tens (3 � 40 or 30 � 4) and products of tens and tens (30 � 40). Theseskills, useful in their own right, are important prerequisites for theintelligent use of estimation and calculators and in the development ofmultiplication and division algorithms.

You may expect that most students will have had experience with products of ones and tens. The activities in this lesson are designed to review andsolidify this knowledge and extend it to products of tens and tens.

Lesson 5-2 introduces the basic principles of multiplication with multidigitnumbers. The key idea is that a product, such as 53 � 68, may be brokendown into sums of “partial products” (50 � 60, 50 � 8, 3 � 60, and 3 � 8).Every part of one factor is multiplied by every part of the other factor. Thepartial products are then added together to find the answer. To do thisefficiently, of course, one needs to have a quick and sure sense for extendedfacts, such as 50 � 60 � 3,000 and 50 � 8 � 400. Learning extendedmultiplication facts is the focus of Lesson 5-1.

To learn more about extending multiplication skills, see Section 11.2.3of the Teacher’s Reference Manual.

Multiplication Algorithms(Lessons 5◆5–5◆7)For each of the four basic operations, Everyday Mathematics has selected a “focus algorithm”—an algorithm that all students are expected to learn.One purpose of focus algorithms is to provide students with a commonlanguage with which to share solution strategies. The focus algorithm formultiplication is the partial-products algorithm. Although students areencouraged to use whatever algorithm they like to solve problems, theywill be asked on occasion to use the partial-products algorithm.

As explained in the lessons, when using the partial-products algorithm,one starts from the left (so that the most important products—thelargest—are attended to first and the smallest ones last). Each part of the calculation, or each partial product, is written on a separate line.Adding partial products is fairly simple and has the additional benefit of providing practice with column addition.

The partial-products algorithm exploits the ideas of the MultiplicationWrestling game (see Lesson 5-2): Every part of one factor is multiplied by every part of the other factor. It is important to “talk to oneself” or to say aloud what is going on. For example, in the problem above, it is not 2 times 6, but 20 times 60, or 20 [60s].

The lattice algorithm, which is introduced in Lesson 5-7, is a usefulalternative to the partial-products algorithm. This ancient method,invented in India, is popular among students because it relies only onsimple computations.

Section 11.2.3 of the Teacher’s Reference Manual contains moreinformation about multiplication algorithms.

Magnitude Estimates (Lessons 5◆4 and 5◆6)It is often useful simply to know a very rough “ballpark” into which an answer might fall: less than one, in the ones, or tens, or hundreds, or thousands, and so on. In science and other applications, “orders ofmagnitude” are multiples of ten, and magnitude estimates give a power-of-10 “ballpark.”

Lesson 5-4 introduces a routine that will be practiced many times fromnow until the end of the year. It invites students to estimate in advancewhere the answers to problems will fall and to mark their estimates on a magnitude bar. Eventually, that explicit check-off routine will become


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Partial ProductsAlgorithm

63� 241,200 ← 20 [60s] or 20 � 60

60 ← 20 [3s] or 20 � 3240 ← 4 [60s] or 4 � 60

� 12 ← 4 [3s] or 4 � 31,512

another matter of “inner speech”: “Is the answer I’m looking for in thetens? Hundreds? Thousands?” With such intuition, students can makemany decisions and almost automatically know when they have mis-keyedthe calculator and gotten a silly result or when information they have reador heard doesn’t make sense. (For example, promoters of parades andother events often claim attendance figures so huge that each spectatorwould be confined to an impossibly small space.)

Consult Section 16.1 of the Teacher’s Reference Manual for additionalinformation on magnitude estimates.

Rounding and Making SmartEstimates (Lessons 5◆3–5◆6 and 5◆10)

More often than most people suppose, an estimate is as usefulas an exact answer. In certain situations, “exact” answersare even impossible to find. Finally, counts and measuresgiven exactly to the very last digit may be misleading—they may imply a degree of accuracy which they do not possess.

In school mathematics, a lot of attention has beengiven to rounding numbers to specified places. Whilethis is an important subskill (just as memorizingbasic addition facts is important), it is far moreimportant to develop good judgment about when toround, how accurate an estimate is needed, and howaccurate an estimate is possible, independent of how“good” one might wish an estimate to be.

To further investigate rounding and makingsmart estimates, refer to Section 16.2 in theTeacher’s Reference Manual.

Magnitude Bar

0.1s 1s 10s 100s 1,000s 10,000s 100,000s 1,000,000s

Unit Organizer 311

Place-Value Structure of Big Numbers (Lesson 5◆8)Much of the power of the familiar base-ten place-value system ofnumeration comes from the regularity of progression from place to place.Moving to the left, place values increase by multiples of ten, allowing forthe representation of whole numbers as large as necessary. Moving to theright, place values decrease by multiples of one-tenth, allowing for therepresentation of decimals as small as necessary.

In estimating with large numbers, it is important to understand theprogression by multiples of one thousand, as well as by multiples of ten. A million is a thousand thousands, a billion is a thousand millions, atrillion is a thousand billions, and so on. Many adults don’t understandthe consequences of those relationships—that a billion is not just a littlebigger than a million but a thousand times bigger—so they can make little sense of public finance, which these days runs heavily into millions,billions, and trillions of dollars. The authors hope to enlighten students inthese matters, not all at once, but through many experiences.

In Lesson 5-8, numbers up to a million (and beyond) are representedprogressively as dots on paper, reams of dot paper, cartons of reams of dotpaper, and (as a project) the enormous number of dots in a roomful of dotpaper. For example, 1 is to 1,000 as a dot is to a half-page of dots. Onethousand is to 1,000,000 as the number of dots on a half-sheet of paper isto the number of dots on a ream of dot paper. During the rest of the schoolyear, help students get a sense of numerical differences in magnitude andthe roles of multiples of 10 and 1,000.

Further information on place-value structure in large numbers can befound in the Teacher’s Reference Manual in Section 16.1.2.


Powers of 10 and ScientificNotation (Lessons 5◆8 and 5◆9)It is difficult to work very large or very small numbers in their full writtenform, and it is seldom necessary or sensible to do so. On many calculators,if the numbers in an operation exceed eight digits, they can’t be entered; if an answer exceeds eight digits, an error message is given.

In the world outside of school, large numbers are nearly always writtenwith such words as million, billion, and so on, substituted for digits. Hugenumbers, such as those astronomers use to describe the universe, areusually shown with the help of powers of 10. Very small numbers, whichare common in science, are often given in terms of such units as thenanometer (one billionth of a meter) or with negative powers of 10.

These lessons begin to deal with essential skills for learning science andfor functioning in a world of calculators and computers—skills that aremore important than paper-and-pencil methods of multiplying or dividingwith large numbers or small decimals.

See Section 10.1.2 of the Teacher’s Reference Manual for more information concerning powers of 10 and scientific notation.

Unit Organizer 313

Common Methods for Writing Extremely Large or Small Numbers

Town budget $8 million $8 � 106

World population for the year 2000 6 billion people 6 � 109 people

Light-year (distance light travels in a year) 6 trillion miles 6 � 1012 mi

Wavelength of ultraviolet radiation (shortest) 4 nanometers 4 � 10�9 m

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