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Teacher’s Resource

Chapter 6: Multiplication

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Chapter 7Chapter 7Chapter Chapter 6Chapter 6

Contents 1Copyright © 2008 Nelson Education Ltd.

Contents

OVERVIEW

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Curriculum across Grades 4 to 6: Number,

Patterns and Relations . . . . . . . . . . . . . . . . . . . . . . . . . 2Math Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Planning for Instruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Reading Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Connections to Literature . . . . . . . . . . . . . . . . . . . . . . . 3Connections to Other Math Strands . . . . . . . . . . . . . . . 3Connections to Other Curricula . . . . . . . . . . . . . . . . . . 3Connections to Home and Community . . . . . . . . . . . . 3

Chapter 6 Planning Chart . . . . . . . . . . . . . . . . . . . . . . . . . 4Chapter 6 Assessment Summary . . . . . . . . . . . . . . . . . . . . 6

TEACHING NOTES

Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Getting Started: Making Dream Catchers . . . . . . . . . . . . . 9Lesson 1: Multiplication Strategies . . . . . . . . . . . . . . . . . . 12Lesson 2: Special Products . . . . . . . . . . . . . . . . . . . . . . . . 16Lesson 3: Relating Multiplication Facts . . . . . . . . . . . . . . 20 Lesson 4: Multiplying by Tens, Hundreds,

and Thousands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Lesson 5: Halving and Doubling to Multiply . . . . . . . . . 27Mid-Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Lesson 6: Multiplying Numbers Close to Tens . . . . . . . . . 34Lesson 7: Estimating Products . . . . . . . . . . . . . . . . . . . . . 38Lesson 8: Multiplying Two-Digit Numbers . . . . . . . . . . . 42Lesson 9: Multiplying with Base Ten Blocks . . . . . . . . . . 46Lesson 10: Multiplying with Arrays . . . . . . . . . . . . . . . . . 50Lesson 11: Communicating about Multiplication

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Curious Math: Lattice Multiplication . . . . . . . . . . . . . . . 58Math Game: Rolling Products . . . . . . . . . . . . . . . . . . . . . 59Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Chapter Task: Homework Blogs . . . . . . . . . . . . . . . . . . . 65

CHAPTER 6 BLACKLINE MASTERS

Family Letter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Scaffolding for Getting Started . . . . . . . . . . . . . . . . . . . . 69Scaffolding for Lesson 1, Question 11 . . . . . . . . . . . . . . . 70Scaffolding for Lesson 5, Question 4 . . . . . . . . . . . . . . . . 71Mid-Chapter Review—Frequently Asked Questions . . . . 72Scaffolding for Lesson 7, Question 8 . . . . . . . . . . . . . . . . 73Scaffolding for Lesson 11, Questions 2, 3, & 4 . . . . . 74–75Chapter Review—Frequently Asked Questions . . . . . . . . 76Chapter 6 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77–79Chapter 6 Task: Homework Blogs . . . . . . . . . . . . . . . 80–81Answers for Chapter 6 Masters . . . . . . . . . . . . . . . . . 82–84From Masters BookletReview of Essential Skills: Chapter 6 . . . . . . . . . . . . . 10–111 cm Grid Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Multiplication Table 0–9 . . . . . . . . . . . . . . . . . . . . . . . . . 33Base Ten Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40–42Spinners/Fraction Circles 2 . . . . . . . . . . . . . . . . . . . . . . . 52Initial Assessment Summary . . . . . . . . . . . . . . . . . . . . . . 56Assessment Rubrics for Mathematical Processes . . . . 57–60Chapter Checklist: Chapter 6 . . . . . . . . . . . . . . . . . . . . . 66Self-Assessment: Chapter 6 Lesson Goals . . . . . . . . . . . . 77Self-Assessment: Mathematical Processes . . . . . . . . . . . . . 83Self-Assessment: What I Like . . . . . . . . . . . . . . . . . . . . . . 84Self-Assessment: How I Learn . . . . . . . . . . . . . . . . . . . . . 84

IntroductionThis chapter extends previous work in multiplication byproviding opportunities for students to explore strategies formultiplication, including halving and doubling, skipcounting from a known fact, repeated addition orsubtraction, and visualizing a multiplication expressionthrough use of base ten blocks and sketches. In this chapter,students will multiply increasingly greater numbers throughstrategies practised on lesser numbers. They also solvemultiplication problems by estimating or calculating. Thischapter presents several strategies for multiplication thatstudents can extend through multi-digit multiplication.These strategies will prove helpful for understanding divisionin Chapter 9.

Answers and SolutionsAnswers to all numbered questions are provided in theStudent Book. Full solutions are provided in the SolutionsManual. Selected answers are provided in the Teacher’sResource lesson notes.

Multiplication

Copyright © 2008 Nelson Education Ltd.2 Chapter 6: Multiplication

Curriculum across Grades 4 to 6: Number, Patterns and RelationsThe Grade 5 outcomes and achievement indicators listed below are addressed in this chapter. When the outcome or indicator is the focus of a lesson or feature, the lesson number or feature is indicated in brackets.

Grade 4 Grade 5 Grade 6

Strand: NumberGeneral Outcome: Develop number sense.

Specific OutcomesN4. Explain the properties

of 0 and 1 formultiplication, andthe property of 1 fordivision. [C, CN, R]

N5. Describe and applymental mathematicsstrategies, such as• skip counting from

a known fact • using doubling or

halving • using doubling or

halving and addingor subtracting onemore group

• using patterns inthe 9s facts

• using repeateddoubling

to determine basicmultiplication facts to9 � 9 and relateddivision facts.[C, CN, ME, PS, R]

N6. Demonstrate anunderstanding ofmultiplication (two-or three-digit by one-digit) to solveproblems by• using personal

strategies formultiplication withand withoutconcrete materials

• using arrays torepresentmultiplication

• connectingconcreterepresentationsto symbolicrepresentations

• estimatingproducts

[C, CN, ME, PS, R, V]

Specific OutcomesN2. Use estimation strategies including

• front-end rounding • compensation • compatible numbersin problem-solving contexts. (7, 9, 11, MG) [C, CN, ME, PS, R, V]Achievement Indicators• Provide a context for when estimation is used to

- make predictions- check reasonableness of an answer - determine approximate answers (7, 9)

• Describe contexts in which overestimating is important. (7) • Determine the approximate solution to a given problem not requiring an exact answer. (7, 11) • Estimate a sum or product using compatible numbers. (7, 9) • Estimate the solution to a given problem using compensation and explain the reason for compensation. (7, 9) • Select and use an estimation strategy for a given problem. (7, 11)• Apply front-end rounding to estimate

- products

N3. Apply mental mathematics strategies and number properties, such as• skip counting from a known fact • using doubling or halving • using patterns in the 9s facts • using repeated doubling or halving to determine answers for basic multiplication facts to 81 and related division facts. (1, 2, 3) [C, CN, ME, R, V] Achievement Indicators• Describe the mental mathematics strategy used to determine a given basic fact, such as

- skip count up by one or two groups from a known fact - skip count down by one or two groups from a known fact- doubling - patterns when multiplying by 9 - repeated doubling (1, 2, 3)

• Explain why multiplying by zero produces a product of zero. (1) • Recall multiplication facts to 81 and related division facts. (1, 2, 3)

N4. Apply mental mathematics strategies for multiplication, such as• annexing then adding zero • halving and doubling • using the distributive property. (4, 5, 6, 8) [C, ME, R]Achievement Indicators• Determine the products when one factor is a multiple of 10, 100, or 1000 by annexing zero or adding zeros (4, 5, 6, 8) • Apply halving and doubling when determining a given product (5) • Apply the distributive property to determine a given product involving multiplying factors that are close to

multiples of 10 (6, 8)

N5. Demonstrate an understanding of multiplication (two-digit by two-digit) to solve problems. (8, 9, 10, 11, CM) [C, CN, PS, V]Achievement Indicators• Illustrate partial products in expanded notation for both factors (9, 10)• Represent both two-digit factors in expanded notation to illustrate the distributive property (9, 10) • Model the steps for multiplying two-digit factors using an array and base ten blocks, and record the

process symbolically. (9, 10) • Describe a solution procedure for determining the product of two given two-digit factors using a pictorial

representation, such as an area model. (10) • Solve a given multiplication problem in context using personal strategies and record the process. (8, 9, 10, 11)

Specific Outcomes N2. Solve problems

involving largenumbers, usingtechnology.[ME, PS, T]

N8. Demonstrate anunderstanding ofmultiplication anddivision of decimals(one-digit wholenumber multipliersand one-digit naturalnumber divisors). [C, CN, ME, PS, R, V]

Strand: Patterns and Relations (Variables and Equations)General Outcome: Represent algebraic expressions in multiple ways.

Specific OutcomePR6. Solve one-step

equations involvinga symbol torepresent anunknown number. [C, CN, PS, R, V]

Specific OutcomePR2. Solve problems involving single-variable, one-step equations with whole number coefficients and whole

number solutions. (4, 5) [C, CN, PS, R] Achievement Indicators• Express a given problem in context as an equation where the unknown is represented by a letter variable. (4, 5)• Solve a given single-variable equation with the unknown in any of the terms (4, 5)

Specific Outcome PR3. Represent

generalizations arisingfrom number relation-ships using equationswith letter variables.[C, CN, PS, R, V]

Mathematical Processes: C Communication, CN Connections, ME Mental Mathematics and Estimation, PS Problem Solving, R Reasoning, T Technology, V VisualizationFeatures: CM Curious Math, MG Math Game

Overview 3

Connections to LiteratureExpand your classroom library or math centre with booksrelated to the math in this chapter. For example:

Bourgeois, Paulette; Slavin, Bill. Too Many Chickens. KidsCan Press, 1998

Justin, Norton. The Phantom Tollbooth. Yearling, 1988

Morgan, Rowland, In the Next Three Seconds, Reed Books, 1977.

Schwartz, David M. If You Hopped Like a Frog. ScholasticPress, 1999

Tang, Gregory.The Best of Times. Schoslatic Press, 2002

Connections to Other Math StrandsPatterns: In Lesson 1, students use skip counting as amultiplication strategy.Measurement: In Lesson 4, students count the number of times adragonfly beats its wings in a given amount of time. In Lesson 9,students use area to calculate products.

Connections to Other CurriculaSocial Studies: In Lesson 2, students learn about the importanceof embroidery in Ukrainian life. In Lesson 7, students learn aCree counting game.

Connections to Home and Community• Have students identify situations in which multiplication is

used in daily life. Focus on one- and two-digitmultiplication.

• Send home Family Letter p. 68.• Have students complete the Nelson Math Focus 5 Workbook

pages for this chapter at home.• Use the suggestions for at-home activities in Follow-Up

and Preparation for Next Class in various lessonsthroughout the chapter.

Math Background Multiplication is a necessary skill for everyday life. Mentalmultiplication is used in purchasing items, calculating pay,and planning events. The skill of estimating is practical ineveryday situations and allows students to check theiranswers. Determining multiplication facts by using strategieswill help students solve increasingly difficult problems andwill ensure that students can recall facts when needed.

Multiplication concepts and skills are developed through the useof concrete and pictorial representations, including base ten blocks

and arrays. Students are encouraged to use their own strategies forcalculating, and to explain these to others. When students have avariety of skills and strategies for multiplying, they will be able tochoose the most efficient strategy for a particular situation.

See PRIME (Professional Resources and Instructions forMathematics Educators: Number and Operations by Marian Small(Thomson Nelson, 2005) for additional math background andteaching strategies.

Planning for Instruction

Problem SolvingAssign a Problem of the Week from the selection below orfrom your own collection. 1. Fran’s home is 4 km from school. She travels to and from

school 5 days a week. How far has Fran travelled to andfrom school in 1 week? in 4 weeks? (Fran travels 8 kmeach day. There are 5 school days in 1 week so I must solve8 � 5. I can skip count by 5s or I can use doubling to solve.Because 8 � 10 � 80, then 8 � 5 � 40. For 4 weeks,I just multiply 40 � 4. Because 4 � 4 � 16, then40 � 4 � 160.)

2. In 1990, about 52 tonnes of material was recycled eachhour in Canada. About how many tonnes of materialwas recycled in one full day? Explain your answer. (Thereare 24 hours in a day so I need to calculate 24 � 52.Because the problem uses the word “about” I can estimate.24 � 52 is about 24 � 50. I can use halving anddoubling: 12 � 100 � 1200. So, about 1200 tonnes ofmaterial was recycled in one day.)

3. Find two numbers whose tens and ones digits arereversed, (e.g., 64 and 46) and whose product is closestto 400. Explain how you might use estimation to narrowyour choices. (I’ll try 82 � 28 because I can estimate easilyusing 80 � 30. 80 � 30 � 1800, so this number is toolarge. I’ll try 12 � 21. 10 � 20 is 200. This is too low.Trying 13 � 31, the estimation is 13 � 30 � 390, whichis close to 400. 13 � 31 � 403)

CommunicationLesson 11 provides opportunities for students to develop andrefine their communication skills by explaining strategies usedto estimate or calculate multiplication expressions. Students areencouraged to use both pictorial models and words in theirexplanations.

Reading StrategyThe reading strategy highlighted in this chapter is Questioning(Lesson 2). To reinforce the use of this strategy, you may applyit to other questions throughout the lessons as opportunitiespresent themselves.

Copyright © 2008 Nelson Education Ltd.

4 Chapter 6: Multiplication

Chapter 6 Planning Chart

Key Concepts*Operations• Multiplication and division are extensions of addition and subtraction. Multiplication and

division are intrinsically related.• There are many algorithms for performing a given operation.Number• There are different, but equivalent, representations for a number.• Benchmark numbers are useful for relating and estimating numbers.Patterns and Relations• Relationships between quantities can be described using rules involving variables.

Key Principles• A variety of strategies can be used to calculate any multiplication fact.• Mental math strategies can be used to multiply by 10, 100, 1000 and numbers close

to them.• Mental math strategies can be used to multiply certain pairs of numbers, including two

single-digit multiples of 10.• A variety of strategies can be used to estimate a product.• The product of 2 two-digit numbers can be calculated using four partial products, based on

the distributive principle.

Student Book Section Lesson GoalGrade 5 Outcomes

Pacing16 Days Prerequisite Skills/Concepts

Getting Started Making Dreamcatcherspp. 174–175 (TR pp. 9–11)

Activate knowledge aboutmultiplication.

1 day • Solve problems involving multiplication of two-digit numbers byone-digit numbers.

• Estimate products.

Lesson 1Multiplication Strategiespp. 176–179 (TR pp. 12–15)

Multiply one-digit numbersusing mental mathstrategies.

N3 1 day • Use mental math strategies such as skip counting or doubling tofigure out multiplication facts to 81.

Lesson 2Special Productspp. 180–182 (TR pp. 16–19)

Use special strategies tomultiply by 8 and 9.

N3 1 day • Multiply one-digit numbers by 10.• Multiply with one-digit numbers by doubling.

Lesson 3Relating Multiplication Factsp. 183 (TR pp. 20–22)

Describe how multiplicationfacts are related.

N3 1 day • Use mental math strategies such as skip counting or doubling tofigure out multiplication facts to 81.

Lesson 4Multiplying by Tens, Hundreds, and Thousandspp. 184–187 (TR pp. 23–26)

Calculate products withmultiples of tens, hundreds,or thousands using mentalmath.

N4, PR2 1 day • Use the relationships 10 tens = 100 and 10 hundreds = 1000.• Multiply single-digit numbers by 10 or 100.

Lesson 5Halving and Doubling to Multiplypp. 188–191 (TR pp. 27–30)

Multiply by halving anddoubling.

N4, PR2 1 day • Double two-digit numbers.• Use the half/double strategy for multiplication facts.• Multiply by 10, 100, or 1000 or multiples of 10 using mental math.• Use letters to represent unknown numbers.

Lesson 6Multiplying Numbers Close to Tenspp. 194–197 (TR pp. 34–37)

Multiply using a simpler,related question.

N4 1 day • Multiply by 10 and multiples of 10 using mental math.• Add and subtract two-digit and three-digit numbers.

Lesson 7Estimating Productspp. 198–200 (TR pp. 38–41)

Estimate to solve problems. N2 1 day • Round to the nearest tens or hundreds.• Multiply by 10, 100, and multiples of 10 using mental math.

Lesson 8Multiplying Two-Digit Numbersp. 201 (TR pp. 42–45)

Multiply two-digit numbersusing your choice ofstrategies.

N4, N5 1 day • Divide two-digit numbers (multiples of 3) by 3.• Multiply two-digit and three-digit numbers by one-digit numbers.• Multiply by 10 or 100.

Lesson 9Multiplying with Base Ten Blockspp. 202–205 (TR pp. 46–49)

Represent the products oftwo-digit numbers.

N5 2 days • Multiply multiples of 10.• Multiply two-digit numbers by one-digit numbers.• Represent multiplication using an array.

Lesson 10Multiplying with Arrayspp. 206–207 (TR pp. 50–53)

Multiply two-digit numbersusing arrays.

N5 1 day • Multiply multiples of 10.• Multiply two-digit numbers by one-digit numbers.• Represent multiplication using an array.

Lesson 11Communicating about Multiplication Methodspp. 208–209 (TR pp. 54–57)

Explain your calculationmethod when solving aproblem.

N2, N5 1 day • Estimate products of two-digit numbers.• Multiply two-digit numbers.

Curious Math p. 210 (TR p. 58)Math Game p. 211 (TR p. 59)Mid-Chapter Review pp. 192–193 (TR pp. 31–33)Chapter Review pp. 212–214 (TR pp. 60–64)Chapter Task p. 215 (TR pp. 65–67)

3 days

*PRIME (Professional Resources and Instruction for Mathematics Educators) by Marian Small (Thomson Nelson, 2005)


Overview 5

Chapter Goals • Recall multiplication facts to 81 using a variety of strategies.• Multiply by 10, 100, and 1000 using mental math.• Estimate and calculate products of two-digit numbers using a variety of strategies.• Create and solve multiplication problems.• Communicate about choosing a multiplication strategy.

Manipulatives Substitute• Base Ten Blocks, Masters Booklet pp. 40–42.

Materials Masters Extra Practice in the Student Book and Workbook

• base ten blocks • Review of Essential Skills: Chapter 6, Masters Booklet p. 10• Optional: Scaffolding for Getting Started p. 69• Optional: Review of Essential Skills: Chapter 6, Masters Booklet p. 10• Optional: Initial Assessment Summary, Masters Booklet p. 56

• counters • Multiplication Table 0–9, Masters Booklet p. 33• Optional: Scaffolding for Lesson 1 Question 11, p. 70• Optional: Chapter Checklist: Chapter 6, Masters Booklet p. 68

Mid-Chapter Review Questions 1 & 2Chapter Review Question 1Workbook p. 43

• Optional: counters Mid-Chapter Review Questions 3 & 4Chapter Review Question 2Workbook p. 44

• paper clips• Optional: chart paper and

markers• Optional: counters

• Spinners/Fraction Circles 2, Masters Booklet p. 52 Workbook p. 45

• base ten blocks • Manipulatives Substitute: Base Ten Blocks, Masters Booklet pp. 40–42 Mid-Chapter Review Questions 5–7Chapter Review Question 3Workbook p. 46

• Optional: counters • Optional: Scaffolding for Lesson 5, Question 4 p. 71 Mid-Chapter Review Questions 8 & 9Chapter Review Question 4Workbook p. 47

• base ten blocks • Manipulatives Substitute: Base Ten Blocks, Masters Booklet pp. 40–42 Chapter Review Question 5Workbook p. 48

• Optional: counters • Optional: Scaffolding for Lesson 7, Question 8 p. 73 Chapter Review Question 6Workbook p. 49

• Optional: chart paper andmarkers

• Optional: base ten blocks

Chapter Review Question 7Workbook p. 50

• base ten blocks • Manipulatives Substitute: Base Ten Blocks, Masters Booklet pp. 40–42 Chapter Review Questions 8–11Workbook p. 51

• 1 cm Grid Paper, Masters Booklet p. 22 Chapter Review Questions 12 & 13Workbook p. 52

• Optional:: Scaffolding for Lesson 11, Questions 2, 3, & 4 pp. 74–75 Chapter Review Question 14Workbook p. 53

For materials and masters for features, reviews, and the Chapter Task, see the TR section. Workbook p. 54



These charts list references to the many assessmentopportunities in the chapter. Formative assessment(Assessment for Learning) provides information aboutstudents’ understanding of concepts and helps you adaptinstruction to students’ needs. A key question in each lessonlinks to the lesson goal. Initial or diagnostic assessment ideas

(also part of Assessment for Learning) are provided in GettingStarted. Summative assessment (Assessment of Learning)opportunities are provided in the Mid-Chapter Review,Chapter Review, and Chapter Task. Have students self-assesstheir learning (Assessment as Learning) using one of the self-assessment tools provided in the Masters Booklet.

Chapter 6 Assessment Summary

Opportunities for Feedback: Assessment for Learning

Student Book Lesson Chart Key Question Grade 5 Outcomes Mathematical Process Focus for Key Question

Lesson 1Multiplication Strategiespp. 176–179

TR p. 15 8, written answer N3 Mental Mathematics and Estimation, Reasoning

Lesson 2Special Productspp. 180–182

TR p. 19 4, written answer N3 Mental Mathematics and Estimation, Visualization

Lesson 3Relating Multiplication Factsp. 183

TR p. 22 entire exploration, investigation

N3 Mental Mathematics and Estimation, Connections

Lesson 4Multiplying by Tens, Hundreds, and Thousandspp. 184–187

TR p. 26 5, short answer N4 Mental Mathematics and Estimation

Lesson 5Halving and Doubling to Multiplypp. 188–191

TR p. 30 6, written answer N4, PR2 Mental Mathematics and Estimation, Reasoning

Mid-Chapter Reviewpp. 192–193

TR p. 32 1, short answer N3 Mental Mathematics and Estimation, Reasoning

2, short answer N3 Connections

3, written answer N3 Mental Mathematics and Estimation, Communication

4, short answer N3 Mental Mathematics and Estimation

5, drawing N4 Mental Mathematics and Estimation, Visualization


7, short answer N4 Mental Mathematics and Estimation, Reasoning



Lesson 6Multiplying Numbers Close to Tenspp. 194–197

TR p. 37 5, written answer N4 Mental Mathematics and Estimation, Reasoning

Lesson 7Estimating Productspp. 198–200

TR p. 41 4, written answer N2 Mental Mathematics and Estimation, Problem Solving

Lesson 8Multiplying Two-Digit Numbersp. 201

TR p. 45 entire exploration, investigation

N4, N5 Communication, Problem Solving

Lesson 9Multiplying with Base Ten Blockspp. 202–205

TR p. 49 8, written answer N5 Problem Solving, Visualization

Lesson 10Multiplying with Arrayspp. 206–207

TR p. 53 3, drawing N5 Communication, Connections, Visualization

Lesson 11Communicating about Multiplication Methodspp. 208–209

TR p. 57 2, written answer N2, N5 Communication, Problem Solving

Curious MathLattice Multiplicationp. 210

TR p. 58 N5 Connections, Visualization

Math GameRolling Products p. 211

TR p. 59 N2 Mental Mathematics and Estimation

Mathematical Processes: C Communication, CN Connections, ME Mental Mathematics and Estimation, PS Problem Solving, R Reasoning, T Technology, V Visualization


Overview 7

Assessment of Learning

Student Book Section Chart Question Grade 5 Outcomes Mathematical Process Focus for Question


TR p. 33 1, short answer N3 Mental Mathematics and Estimation, Reasoning


3, written answer N3 Mental Mathematics and Estimation,Communication




7, written answer N4 Mental Mathematics and Estimation



Assessment as Learning

Student Book Section Student Self-Assessment Masters


Chapter 6 Lesson Goals, Masters Booklet p. 77Self-Assessment: Mathematical Processes, Masters Booklet p. 83Self-Assessment: What I Like, Masters Booklet p. 84Self-Assessment: How I Learn, Masters Booklet p. 84

Chapter Reviewpp. 212–214

Chapter 6 Lesson Goals, Masters Booklet, p. 77Self-Assessment: Mathematical Processes, Masters Booklet p. 83Self-Assessment: What I Like, Masters Booklet p. 84Self-Assessment: How I Learn, Masters Booklet p. 84

Chapter Reviewpp. 212–214andChapter Test(TR pp. 77–79)

TR pp.62–64

1, written answer N3 Connections


3, model, drawing N4 Mental Mathematics and Estimation, Visualization


5, written answer N4 Mental Mathematics and Estimation

6, written answer N2 Problem Solving

7, short answer N5 Problem Solving


9, short answer N5 Problem Solving

10, model, written answer N5 Connections, Visualization

11, short answer N5 Visualization

12, drawing, short answer N5 Connections, Problem Solving


14, written answer N4, N5 Communication

Chapter TaskHomework Blogsp. 215

TR p. 67 entire task, investigation N2, N4, N5 Communication, Mental Mathematics andEstimation, Problem Solving


Chapter 6

Chapter 6: Multiplication8

Using the Chapter OpenerDraw students’ attention to the photograph on Student Book pages 172 and 173.

Sample Discourse“Are there usually more than 10 or less than 10 songs on a CD?”• more than 10“About how many songs would you say are on a CD?”• about 14“Suppose there are 10 songs on 1 CD, how many songs are on 2 CDs?”• There are 20 songs on 2 CDs.“Did you get your answer by adding or multiplying?”• I added 10 � 10 to get 20.• I multiplied 2 � 10 to get 20.

Have students think of other addition problems that can alsobe written as multiplication problems.

Discuss the five goals of the chapter. Have students discussany personal strategies for recalling the multiplication facts to81, such as number patterns for the nines or skip counting.Discuss the benefits of estimating, especially when dealingwith greater numbers.

Ask students to record in their journals their thoughtsabout one of the goals, using a prompt such as, “I like tomultiply by 10 because …” At the end of the chapter, youcan ask students to complete the same prompt. Then, theycan compare their ideas and reflect on what they havelearned.

At this point, it would be appropriate to• send home Family Letter p. 68• ask students to look through the chapter and add math

word cards to your classroom word wall. Here are someterms related to this chapter:

Chapter Chapter 6Chapter 6


halve

double

product

multiple

68 Chapter 6: Multiplication Copyright © 2008 Nelson Education Ltd.

Chapter Chapter Chapter 76

Family Letter

Dear Parent/Caregiver:

Over the next four weeks, your child will be learning multiplication strategies. He orshe will also learn to estimate the products of one-digit and two-digit numbers.Models will be used to represent multiplication, and your child will be able to usevarious strategies to solve multiplication problems.

To reinforce the concepts your child is learning at school, you and your child can workon some at-home activities such as these:

• Play a game with two people and a deck of cards (remove the face cards). Eachplayer takes a card and places it face up. The first player to state the multiplicationfact wins a point. You may choose to count to three before answering to allow yourchild a fairer game.

• Find items to multiply in daily life. For example, if a snack costs $2, determine thetotal price if a snack was purchased for several friends.

• Discuss when estimates are appropriate and when they are not. Is it necessary toknow the exact price of your grocery items before you reach the register, or is anestimate sufficient? Would you rather your estimate be too high or too low whenbuying groceries?

You may want to visit the Nelson website at www.nelson.com/mathfocus for moresuggestions to help your child learn mathematics and develop a positive attitudetoward learning mathematics. As well, you can check the Nelson website for links to other websites that provide online tutorials, math problems, brainteasers, and challenges.

Family Letter p. 68

Chapter Opener STUDENT BOOK PAGES 172–173

42 Multi-Purpose Masters: Base Ten Blocks: Ones or Thousandths Copyright © 2008 Nelson Education Ltd.

41Multi-Purpose Masters: Base Ten Blocks: Tens or HundredthsCopyright © 2008 Nelson Education Ltd.


PREREQUISITE SKILLS/CONCEPTS

• Solve problems involving multiplication of two-digitnumbers by one-digit numbers.

• Estimate products.

Manipulatives Substitute: Base Ten Blocks, MastersBooklet pp. 40–42

Scaffolding for Getting StartedSTUDENT BOOK PAGE 174

Making Dream Catchers

Maya and her grandmother are making eight dream catchers as gifts. They need 65 cm of willow to make each dream catcher.

? How much willow do Maya and her grandmother need to make eight dream catchers?

A. How much willow do Maya and her grandmother need to make two dream catchers?

They need 65 cm � 65 cm � cm of willow.

This is the same as 2 � cm � cm.

B. How do you know that they need more than 400 cm of willow to make eight dream catchers?

Two dream catchers will use more than 100 cm of willow.

Four dream catchers will use more than cm of willow.

Eight dream catchers will use more than cm of willow.

C. How can you use base ten blocks to show the amount of willow they need to make eight dream catchers?

How many sets of 65 base ten blocks do you need?

Will you be able to regroup any of the blocks? Explain.

D. How can you solve the problem without using base ten blocks?

I can calculate �

E. How much willow do Maya and her grandmother need?

69Blackline Masters

Name: Date:


STUDENT BOOK PAGES 174–175

9Getting Started: Making Dream Catchers

Getting Started:Making Dream Catchers


GOALActivate knowledge about multiplication.

Pacing 30–40 min Activity10–20 min What Do You Think?10–15 min Consolidation

Materials • base ten blocks

Masters • Optional: Scaffolding for Getting Started p. 69

• Optional: Review of Essential Skills: Chapter 6, Masters Booklet p. 10

• Optional: Initial Assessment Summary,Masters Booklet p. 56

• Manipulatives Substitute: Base Ten Blocks, Masters Booklet pp. 40–42

Nelson Website Visit www.nelson.com/mathfocus and follow the links to Nelson Math Focus 5, Chapter 6.

Preparation and Planning

Optional: Scaffolding for Getting Started p. 69

56 Assessment Master: Initial Assessment Summary Copyright © 2008 Nelson Education Ltd.

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Optional: InitialAssessment Summary,Masters Booklet p. 56

10 Review of Essential Skills: Chapter 6

Chapter 6: Multiplication

1. Calculate each product.

a) 6 � 2 � g) 5 � 7 �

b) 5 � 5 � h) 6 � 6 �

c) 3 � 6 � i) 6 � 4 �

d) 4 � 4 � j) 6 � 8 �

e) 3 � 7 � k) 7 � 9 �

f) 4 � 7 � l) 9 � 8 �

2. Calculate.

a) 7 � 10 � g) 3 � 100 �

b) 7 � 60 � h) 7 � 100 �

c) 4 � 90 � i) 9 � 500 �

d) 50 � 8 � j) 700 � 7 �

e) 30 � 9 � k) 300 � 8 �

f) 20 � 8 � l) 600 � 2 �

Name: Date:


Optional: Review ofEssential Skills: Chapter 6,Masters Booklet p. 10

Using the Activity (Whole Class/Individual) ➧ 30–40 min

Use this activity to activate knowledge of multiplication andas an opportunity for initial assessment.

Together, read Student Book page 174. Tell students thatdream catchers were first made by the Ojibwe First Nationsand are now made by many Aboriginal artists. A dreamcatcher was thought to protect a sleeping person fromnegative dreams while letting positive dreams come through.Read the central question. Be sure that students understandthey will be determining an estimate of the amount of willowneeded to make dream catchers.

If extra support is required, guide those students andprovide copies of Scaffolding for Getting Started p. 69.

Answers to the ActivityA. 130 cm; 2 � 65 � 2 � 60 � 2 � 5

� 120 � 10 � 130

B. For example, 8 � 4 � 2. Two dream catchers use more than100 cm, so 4 � 2 dream catchers need more than 400 cm.

C. For example, I can show 8 sets of 65. I would trade everygroup of 10 ones for 1 ten and every group of 10 tens for 1 hundred.

D. For example, I can multiply 8 by 65 to solve the problem.E. 520 cm

40 Multi-Purpose Masters: Base Ten Blocks: Hundreds or Tenths Copyright © 2008 Nelson Education Ltd.


Using What Do You Think? (Small Groups/Whole Class) ➧ 10–20 min

Use this anticipation guide to activate knowledge andunderstanding of multiplication strategies. Explain tostudents that the statements involve math concepts or skillsthey will learn about in the chapter and they are not expectedto know the answers at this point. Ask students to read thestatements, think about each one for a few seconds, anddecide whether they agree or disagree with each. Havevolunteers explain the reasons for their choices. Students canexchange their thoughts in small groups, in groups where allagree or disagree, or in a general class discussion. Tellstudents they can revisit their ideas at the end of the chapter.


Possible Responses for What Do You Think?Correct responses are indicated with an asterisk (*). Studentsshould be able to give correct responses by the end of thechapter.1. *For example, agree; if I double something, I multiply by

2. If I double again, I multiply by 4.For example, disagree; if I double something, I onlymultiply by 2.

2. *For example, agree; The number I multiply 10 by tellsme the number of tens. So, 9 � 10 means 9 tens or 90.For example, disagree; if I am multiplying by 10, I am using the 10 facts.

3. *For example, agree; I can multiply by adding somethingto itself over and over. I also need to add if I multiplyparts of a number and then put them together.For example, disagree; if I know my multiplication facts,I can just use them without adding.

4. For example, agree; 14 is 10 more than 4, so I wouldmultiply by 10.*For example, disagree; 10 times as much as 4 � 28would be 40 � 28, not 14 � 28.

Initial Assessment: Assessment for Learning

What you will see students doing

When students understand If students misunderstand

Prompts A, C, & D• Students can write repeated addition as multiplication and are modelling

numbers using base ten blocks.• Students cannot explain why 65 � 65 is equivalent to 2 � 65. (See 2 or 3

below.)

Prompt B

• Students use rounding or front-end estimating to determine estimates. • Students may not be able to make any estimate to determine if more than 400 cm of willow is needed. (See 2 below.)

Prompt E

• Students use multiplication or repeated addition to figure out an amount. • Students are unable to chose an algorithm needed to solve the problem. (See 2 or 3 below.)

Differentiating Instruction: How you can respond

SUPPORTING STUDENTS WHO ARE ALMOST THERE

1. Use Scaffolding for Getting Started p. 69.

2. Use Review of Essential Skills: Chapter 6, Masters Booklet p. 10, to activate students’ skills.

3. Have students practise with counters to see that multiplication is repeatedaddition. Ask students to make 4 groups with 5 counters in each group. Then,have students skip count by 5s until they arrive at 20. Have students restate the situation by saying 5 � 5 � 5 � 5.

SUPPORTING STUDENTS WHO ARE NOT READY

This chapter assumes that students are already comfortable with relating addition to multiplication and the lower multiplication facts.

In some lessons, suggestions for adapting the lesson to deal with students who are in a lower developmental phase can be found at the end of theOpportunities for Feedback: Assessment for Learning chart.

For this activity

• Have students alter the problem so that each dream catcher requires 50 cm ofwillow. Remind students that they can also think of 50 cm as 5 groups of 10,rather than 50. (Point out that some dream catchers are fashioned into jewelleryso even 10 cm is an appropriate amount to work with.) Have students answerPrompts A and C through E.

11Getting Started: Making Dream CatchersCopyright © 2008 Nelson Education Ltd.

Chapter 6 1111

PREREQUISITE SKILL/CONCEPT

• Use mental math strategies such as skip counting ordoubling to figure out multiplication facts to 81.

SPECIFIC OUTCOME

N3. Apply mental mathematics strategies and numberproperties, such as• skip counting from a known fact• using doubling or halving• using patterns in the 9s facts• using repeated doubling or halving to determine answers for basic multiplication facts to 81and related division facts.[C, CN, ME, R, V]

Achievement Indicators• Describe the mental mathematics strategy used to

determine a given basic fact, such as– skip count down by one or two groups from a

known fact, e.g., if 8 � 8 � 64, then 7 � 8 isequal to 64 � 8 and 6 � 8 is equal to 64 � 8 � 8

– doubling, e.g., for 8 � 3 think 4 � 3 � 12, and 8 � 3 � 12 � 12

• Explain why multiplying by zero produces a productof zero.

• Recall multiplication facts to 81 and related division facts.

GOALMultiply one-digit numbers using mental math strategies.

Pacing 5–10 min Introduction15–20 min Teaching and Learning20–30 min Consolidation

Materials • counters, about 30 per student orpair of students

Masters • Multiplication Table 0–9, MastersBooklet p. 33

• Optional: Scaffolding for Lesson 1,Question 11 p. 70

• Optional: Chapter Checklist: Chapter 6,Masters Booklet p. 68

Recommended Questions 3, 6, 8, 11, 12, & 14 Practising Questions

Key Question Question 8

Extra Practice Mid-Chapter Review Questions 1 and 2Chapter Review Question 1Workbook p. 43

Mathematical ME (Mental Mathematics and Estimation) andProcess Focus R (Reasoning)



Math BackgroundStudents should quickly recall one-digit multiplicationfacts from 0 to 9. To facilitate this, it is important thatstudents are aware of possible strategies to use and theconnections that can be made from one multiplicationfact to another one. Some mental math strategies thatstudents will use include• skip counting by one of the factors (e.g., to determine

6 � 2, a student skip counts up by 2s (2, 4, 6, 8, 10, 12))

• adding the same addend repeatedly (e.g., to determine 7 � 4, a student adds the number 4 seven times (4 � 4 � 4 � 4 � 4 � 4 � 4 � 28))

• making a connection with another known multipli-cation fact (e.g., to determine 6 � 9, a student firstconnects 6 � 9 with 3 � 9 � 27, then doubles 27 to get 54).

Students use reasoning to understand that all of the abovestrategies will yield the correct answer.

Multiplication Strategies STUDENT BOOK PAGES 176–179


33Multi-Purpose Masters: Multiplication Table 0 to 9Copyright © 2008 Nelson Education Ltd.

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Name: Date:

Scaffolding for Lesson 1, Question 11STUDENT BOOK PAGE 179

11. a) How much more is 6 � 9 than 3 � 9? How do you know?

Use these steps.

How many groups of 9 are in 6 � 9?


How many more groups of 9 are there in 6 � 9 than in 3 � 9?

How much more is 6 � 9 than 3 � 9?

b) How much more is 7 � 7 than 5 � 7? How do you know?

Use these steps.






Optional: Scaffolding for Lesson 1, Question 11p. 70

68 Assessment Masters: Chapter Checklist: Chapter 8 Copyright © 2008 Nelson Education Ltd.

Chapter 8: Measurement ChecklistThroughout the chapter, observe individual students for evidence that they understand key knowledge and can perform key skills.

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Optional: ChapterChecklist: Chapter 6,Masters Booklet p. 68

Introduction (Small Groups) ➧ 5–10 min

Provide counters to each group of students. Ask each group tomake 3 groups of 6 counters. Keeping the first arrangement ofcounters, ask each group to make 4 groups of 6 counters andthen make 5 groups of 6 counters. Have students statemultiplication expressions for each arrangement of counters.

Sample Discourse“What is the multiplication expression that represents 3groups of 6 counters?”• The multiplication expression is 3 � 6.“What is the difference between the 3 � 6 group and the 4 � 6 group? Explain your answer.”• There are 6 more counters in the 4 � 6 group because there is

1 more group of 6.“What is the difference between the 3 � 6 group and the 5 � 6 group?”• There are 12 more counters in the 5 � 6 group because there

are 2 more groups of 6, and 6 � 6 � 12.“Explain how you could determine the number of counters in a7 � 6 group using what you know about the 3 � 6 group.”• I know the 7 � 6 group has 4 more groups of 6, and 4 � 6

is 24, so the 7 � 6 group will have 18 � 24 � 42 counters.• Because there are 4 more groups of 6, I can just add 6 four

times to the number of counters in 3 � 6. So, 18 � 6 � 6 � 6 � 6 � 42 counters.

Teaching and Learning(Whole Class) ➧ 15–20 min

With students, read about Owen and his swimming scheduleon Student Book page 176. Discuss the number of days ineach week and the number of days Owen will be swimmingeach week. Be sure all students agree that the multiplicationexpression that represents four weeks of six days swimmingper week will be 4 � 6.

As a class, work through each strategy shown. Providecounters or ones blocks for students to model Ami’s Strategy.Ask students to choose the strategy they would most likelyhave used and have them explain why. Discuss the benefitsand drawbacks for each method.

Sample Discourse“Whose method, Owen’s, Ami’s, or Justine’s requires knowinganother multiplication fact to determine the answer?”• Ami’s method, because she needed to know what 2 � 6 was

before she determined her answer • Justine’s method, because she needed to know 5 � 6“How can you use doubling to determine 8 � 6?”• I can start with 2 � 6 � 12 and double that to determine

4 � 6 � 24. Then, I would double 24 to determine 8 � 6 � 48 because 24 � 24 � 48.

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13Lesson 1: Multiplication StrategiesCopyright © 2008 Nelson Education Ltd.

Practising (Individual)

4. & 7. Provide students with counters or ones blocks tomodel the facts and visualize an array or begin a sketch.

11. If extra support is required, guide those students andprovide copies of Scaffolding p. 70.

13. Provide copies of Multiplication Table, MastersBooklet p. 33. Remind students that the order of thefactors doesn’t matter. For example, 4 � 1 is equal to 1 � 4. Therefore, this fact will fill in two places in themultiplication chart.

Answer to Key Question8. 56 days; for example, I needed to calculate 8 � 7. I took

10 � 7 and subtracted two groups of 7. 7 � 8 � 10 � 7 – 7 – 7

� 70 – 14 � 60 – 4 � 56

Closing (Whole Class)

Question 14 allows students to reflect on and consolidatetheir learning for this lesson. Allow students to use thechalkboard or interactive whiteboard to explain threemethods for a chosen fact.

Answer to Closing Question 14. Agree; for example, 3 � 6. One way is to double 3 � 3.

Another way is to start with 3 � 5 and add 3. Anotherway is to start with 4 � 6 � 24 and count back 6 from24: 24 � 6 � 18.

Reflecting (Whole Class)

Students reflect on the different multiplication strategies andhow multiplication facts relate. For Prompt B, discussmultiplication facts that may not be easily calculated by doubling.

Sample Discourse“What method would you use to determine 7 � 9?”• I know 5 � 9 � 45, so I can skip count up by 9s from 45:

45, 54, 63.• I know 3 � 9 � 27, so I can determine 6 � 9 by doubling

27: 27 � 27 � 54. Then, I can skip count up 9 more: 54, 63.

Answers to Reflecting QuestionsA. For example, I can count up from 3 � 6 � 18 by adding

another 6. 18 � 6 � 24.B. For example, I can calculate any 4� fact by doubling the

2� fact. For example, 4 � 7 � double 2 � 7 (or 28).

Consolidation ➧ 20–30 min

Checking (Pairs)

1. a) Refer students to Owen’s Strategy if help is neededwith skip counting from a known fact. Ensure thatthey use more than one strategy.

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Follow-Up and Preparation for Next ClassHave students use items from home to practise figuring outmultiplication facts. For example,• There are four muffins in a package. If you have five

packages, how many muffins are there?

• A toy costs $3. There are four toys in each set. How muchdoes the whole set cost?

Encourage students to write examples they might seeoutside of school.




• Students select a mental math strategy to determine a multiplication fact using a known fact.

• Students explain whether they used skip counting from a known fact, doubling, or counting back to determine answers for basic multiplication facts.

• Students use manipulatives or sketches to explain how doubling and skipcounting are used to determine a multiplication fact.

Key Question 8 (Mental Mathematics, Reasoning)• Students choose a strategy (skip counting from a known fact or doubling) to

calculate the solution to the problem.

• Students may have difficulty choosing a strategy to relate a knownmultiplication fact to another multiplication fact. (See Extra Support 1.)

• Students may have difficulty explaining the mental math strategy used todetermine a product using a known fact. (See Extra Support 3.)

• Students may not be able to model or sketch a representation of one digitmultiplication. (See Extra Support 2.)

• Students may not be able to identify a strategy to calculate the solution.(See Extra Support 1, 2, or 3.)


EXTRA SUPPORT

1. Have students write a multiplication expression to figure out. Tell studentsthat, beside the problem, they should write a math fact containing one of the factors in the original problem. For example, to solve 4 � 9, a student maywrite 2 � 9 � 18. After a mental math strategy has been decided upon andthe answer determined, have students write SC (skip count) or D (double) nextto the fact, 4 � 9 � 36.

2. To assist students having difficulty modelling problems, rephrase a problemusing “groups” in the wording. For 5 � 4, say “5 groups of 4”, for 7 � 2, “7 groups of 2.”

3. To assist students who have difficulty explaining the mental math strategy they used to determine a math fact, ask the following questions:

• Did you use addition to determine your answer? If so, what number did youadd? How many times did you add the number?

• Did you double the answer to one fact to determine another answer?• Did you begin with an answer greater than your final answer and count

back? Why did you choose that strategy?

EXTRA CHALLENGE

• Have students choose a number between 0 and 9. Challenge students to figureout two multiplication facts whose difference of products is the number theyselected. For example, if a student chooses 8, one solution can be (4 � 4) � (4 � 2), which is 16 � 8 � 8. Another solution can be (8 � 9) � (8 � 8), which is 72 � 64 � 8. See how many distinct productdifferences they calculate for each number they select.

SUPPORTING DEVELOPMENTAL DIFFERENCES

• Some learners may not be ready to use a variety of number sense strategies to determine multiplication facts. Work with students to complete a blankmultiplication table (Masters Booklet p. 33) using skip counting or repeatedaddition. After the chart is complete, discuss with students how doubling can also be used to determine one fact from another fact. For example, todetermine 3 � 6, have students find the entry for 3 � 3 and then double the answer. 3 � 3 � 9 and 9 doubled � 18

SUPPORTING LEARNING STYLE DIFFERENCES

• Kinesthetic learners may benefit from using counters or other manipulatives toform the groups for each multiplication expression. Have students arrange thecounters in rows and columns so they can see the number of groups and thenumber of counters in each group. Then, show students that a differentmultiplication fact can be modelled by adding another row or column.

• Auditory learners may benefit from skip counting aloud. To provide practice inskip counting aloud, tell students the number to begin with and how to count.For example, “Start at 20 and skip count up by 5.”

15Lesson 1: Multiplication StrategiesCopyright © 2008 Nelson Education Ltd.

Chapter 6 212122 Special Products STUDENT BOOK PAGES 180–182


• Multiply one-digit numbers by 10.• Multiply with one-digit numbers by doubling.

SPECIFIC OUTCOME

N3. Apply mental mathematics strategies and numberproperties, such as• skip counting from a known fact• using doubling or halving• using patterns in the 9s facts• using repeated doubling or halving to determine answers for basic multiplication facts to 81and related division facts.[C, CN, ME, R, V]


determine a given basic fact, such as– doubling, e.g., for 8 � 3 think 4 � 3 � 12, and

8 � 3 � 12 � 12 – patterns when multiplying by 9, e.g., for 9 � 6,

think 10 � 6 � 60, and 60 � 6 � 54; for 7 � 9,think 7 � 10 � 70, and 70 � 7 � 63

– repeated doubling, e.g., if 2 � 6 is equal to 12,then 4 � 6 is equal to 24 and 8 � 6 is equal to 48.

• Recall multiplication facts to 81 and related divisionfacts.

GOALUse special strategies to multiply by 8 and 9.

Pacing 5–10 min Introduction(allow 5 min for 15–20 min Teaching and Learningprevious homework) 15–25 min Consolidation

Materials • Optional: counters

Recommended Questions 4, 5, 7, 9, & 10Practising Questions


Extra Practice Mid-Chapter Review Questions 3 & 4Chapter Review Question 2Workbook p. 44

Mathematical ME (Mental Mathematics) Process Focus



Math BackgroundStudents will use their knowledge of multiplicationstrategies to apply special strategies for multiplying by 8 or 9. The special strategies involving 8 use halving ordoubling. An expression such as 8 � 7 can be calculatedby halving the 8 to 4 � 7 and then using mental mathto double the answer. Strategies involving 9 build on thefact that 9 � 10 � 1. For example, to calculate 9 � 7,calculate 10 � 7 and then subtract 1 group of 7. Tovisualize this method, students can use the tens andones from base ten blocks to show subtraction oraddition of the extra group.

The introductory problem in this lesson draws onUkrainian heritage. The Ukrainian heritage has a majorinfluence in Western Canadian history. As of 2007, inWestern Canada, nearly 10 000 students who live in the Prairie provinces of Alberta, Manitoba, andSaskatchewan have enrolled in English-Ukrainianbilingual schools.

Ask students if any of them have Ukrainian traces intheir heritage or if they have any personal items they canshare with the class that have Slavic characteristics.

Reading StrategyQuestioningLearning to ask clarifying questions is a reading strategythat helps students better understand information intext. In mathematics, when students ask clarifyingquestions, they are more actively engaged in searchingfor meaning. Students may discover that answers to thequestions are provided as they read on—in pictures,graphs, or symbolic representations.

Have students open their Student Books to page 182.Remind students that, when we read, we often askquestions to clarify our understanding. Explicitly teachstudents how to ask questions, and provide prompts asneeded (e.g. How many/much …? Does __ alwaysapply? Are all ideas the same?) Let students readQuestion 4 independently.

Create a two-column chart on grid paper or thechalkboard with the headings “Questions,” and“Answers.” As a class, have students generate questions to clarify their understanding of the problem.

Review the questions with the class. Have studentsanswer each question orally. Reinforce the idea that goodquestions can clarify their understanding of math problems.


Teaching and Learning(Whole Class/Pairs) ➧ 15–20 min

With students, read about the Ukrainian pillowcase onStudent Book page 180. Tell students that, when Ukrainiansettlers came to Canada many years ago, they brought the artof embroidery. Shirts, blouses, table linens, and pillows aredecorated with traditional Ukrainian embroidery to celebratespecial occasions in the communities.

Together, read the central question and work throughMaya’s Strategy. This lesson focuses on the connectionbetween the 8� multiplication facts and the 2� facts, aswell as the connection between the 9� facts and the 10�facts. Make sure that students have a general understandingof the 2� and the 10� multiplication facts beforeproceeding. Students can then complete Prompts A to D inpairs.

Sample discourse“What multiplication expression is the same as doubling 2?”• 2 � 2“What multiplication expression is the same as doubling 4?”• 2 � 4“So, what multiplication expression represents doubling 2twice?”• 2 � 2 � 2

Introduction (Whole Class/Pairs) ➧ 5–10 min

Review some doubling strategies involving two-digit numberswith students to prepare them for the upcoming lesson.Encourage students to use mental math strategies wheneverpossible. You may want to review some addition strategies aswell for students that have difficulty adding two-digitnumbers to two-digit numbers.

Sample discourse“What does it mean to double a number?”• It means to multiply a number by 2.• It means to add the same number to itself.“So, how would you double 32?”• I would multiply 32 by 2 to get 64.• I would add 32 � 32 � 64.“What is 28 doubled?”• 56, because 2 � 28 � 56.• 56, because 28 � 28 � 56.“Suppose you have problems doubling 28 using mentalmath, what is another way you could do this?”• I could double 20 to get 40, then double 8 to get 16. I can

then add 40 � 16 to get 56.• I could add the tens first to get 20 + 20 = 40. I can then add

the ones to get 8 � 8 � 16. And 40 � 16 � 56.

Have students practise doubling strategies by doubling thefollowing numbers: 18, 24, 27, and 36. Have students sharetheir strategies with the rest of the class.

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17Lesson 2: Special ProductsCopyright © 2008 Nelson Education Ltd.

4 � 6 � 4 � 5 � 4 � 20 � 4 � 24, then double to get 8 � 6 � 24 � 24 � 40 � 8 � 48. 64 legs on eight spiders – 48 legs on eight ants � 16.

b) 18; For example, one spider has 8 – 6 � 2 more legsthan one ant. If eight spiders have 16 more legs thaneight ants, then nine spiders would have 16 � 2 � 18more legs than nine ants.


Question 10 allows students to reflect on and consolidatetheir learning for this lesson. Ask students to share theirvarious strategies for calculating the products.

Answers to Closing Question 10. a) For example, I know 10 groups of 4 or 10 � 4 is 40.

So, 1 less group of 4 is 40 – 4 � 36. I also know that9 � 2 � 18, so I could double it to get 18 � 18 � 20 � 16 � 36.

b) For example, I could skip count up by 5s eight times:5, 10, 15, 20, 25, 30, 35, 40. I can also double 5 � 4to get 5 � 8. Since 5 � 4 is 20, and 20 � 20 � 40,then 5 � 8 is 40.

Follow-Up and Preparation for Next ClassHave students review all multiplication strategies learned sofar. Ask students to explain when they might choose to use acertain strategy.

Answers to PromptsA. For example, I can double 6 � 2 to get 6 � 4, and then

double 6 � 4 to get 6 � 8.B. For example, the diagram shows 6 groups of 10 (60),

which is changed into 6 groups of 9 by covering one outof each 10. If I cover 6, I subtract 6 from 60.

C. For example, I can subtract 6 from 54.D. 48 flowers


Students reflect on the strategy of doubling more than onceto determine a multiplication fact. Be sure that all studentsunderstand that doubling is simply multiplying by 2.Doubling a second time is multiplying by 2 � 2 or 4.

Sample discourse“Doubling is the same as multiplying by what number?”• 2“So, doubling a second time is the same as multiplying bywhat number?”• 2 � 2, or 4

Answers to Reflecting QuestionsE. For, example, I can double 8 to get 2 � 8 � 16.

Then, I can double 16 to get 4 � 8 � 16 � 16 � 32.Then, I can double 32 to get 8 � 8 � 32 � 32 � 64.

F. For example, 5 � 9 � 50 – 5 � 45 8 � 9 � 80 – 8 � 727 � 9 � 70 – 7 � 63When I multiply a one-digit number by 9, the tens digitis always one less than the one-digit number, and theones digit is always 10 minus the one-digit number.


Checking (Pairs)

2. Encourage students to draw a picture, or use counters, ifthey need to model or visualize the mathematicalexpression.


4. Explain to students that there is more than one way tosolve this problem. They can calculate the differencebetween the number of legs on one spider and one ant,then multiply that difference by 8. Alternatively, studentscan subtract the total number of legs on ants from thetotal number of legs on spiders.

Answers to Key Question4. a) 16; For example, eight spiders have 8 � 8 legs.

You can start with 2 � 8 � 16, and double to get 4� 8 � 16 � 16 � 20 � 12 � 32. Double again to get 8 � 8 � 32 � 32 � 60 � 4 � 64.Eight ants have 8 � 6 legs. You can start with

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What you will see students doingWhen students understand If students misunderstand

• Students can use a multiple of 10 to determine a multiple of 9.

• Students can use repeated doubling to figure out a multiple of 8.

Key Question 4 (Mental Mathematics, Visualization) • Students solve the problem by calculating the total number of legs for spiders

and total number of legs for ants to calculate the difference, or they use thedifference of 2 legs multiplied by 8 and calculate the difference in legs.

• Students may not recognize or be able to use the pattern in the multiples of 9.(See Extra Support 1 or 2.)

• Students may not recognize which facts to double to multiply by 8. (See Extra Support 3.)

• Students may misinterpret the problem and think the answer is 2 and may not infer that 2 must be multiplied by 8 to get the total difference. (See Extra Support 4.)

Differentiating Instruction: How you can respondEXTRA SUPPORT

1. Kinesthetic learners may value this strategy to multiply 9 by a number between 1 and 10. Have students display their hands and number their fingers as shown. To multiply 9 � 3, have students bring down finger number 3.Then, count the number of fingers to the left of the third finger to determine thetens digit (2) and the number of fingers to the right of the third finger to determinethe ones digit (7). So, 9 � 3 is 27. Have them try this for multiplying 9 by othernumbers 1 to 10.

2. Have students write out the first 10 multiples of 9 separately (not in themultiplication chart). Then, have them separate the tens digits from the onesdigits. Ask students to state the pattern for the tens and the ones digits. Next,have students add the tens and ones digits from each multiple to see that thesum of the digits is always 9.

3. Use patterns to show students how repeated doubling can be used to figure out8� facts from other facts. For example,

2 � 6 � 124 � 6 � 248 � 6 � 48

4. Have students use the Questioning reading strategy to help them understandand consider all the relevant information in the problem.

EXTRA CHALLENGE

• Challenge students to use skip counting to continue writing multiples of 9 up to 9 � 20. Have students check to see if there is a pattern for the tens digits,ones digits, and the sum of the digits, not including 9 � 11.

• Challenge students to develop a multiplication strategy for multiplying by 16 ormultiplying by 19.

For example, I can multiply 16 and 5 by multiplying 2 and 5 and then doubling 3 times, because 2 � 2 � 2 � 2 � 16. So, 2 � 5 � 10, double to get 20,double to get 40, then double to get 80. So, 16 � 5 � 80.

For example, I can multiply 19 � 8 by multiplying 20 � 8 and then subtract agroup of 8, because 20 � 1 � 19. So, 20 � 8 � 160 and 160 � 8 � 152. So, 19 � 8 � 152.


• Some students may have problems with repeated doubling and halving.Encourage these students to continue to use skip counting or repeated addition to determine the multiplication facts. Emphasize the fact that both strategies yield the correct answer.

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19Lesson 2: Special ProductsCopyright © 2008 Nelson Education Ltd.

313133Chapter 6 Relating MultiplicationFacts

STUDENT BOOK PAGE 183


• Use mental math strategies such as skip counting ordoubling to figure out multiplication facts to 81.

SPECIFIC OUTCOME

N3. Apply mental mathematics strategies and numberproperties, such as• skip counting from a known fact• using doubling or halving• using patterns in the 9s facts• using repeated doubling or halvingto determine answers for basic multiplication facts to 81and related division facts.[C, CN, ME, R, V]


determine a given basic fact, such as– skip count up by one or two groups from a known

fact, e.g., if 5 � 7 � 35, then 6 � 7 is equal to35 � 7 and 7 � 7 is equal to 35 � 7 � 7

– skip count down by one or two groups from aknown fact, e.g., if 8 � 8 � 64, then 7 � 8 is equalto 64 – 8 and 6 � 8 is equal to 64 � 8 � 8

– doubling, e.g., for 8 � 3 think 4 � 3 � 12, and 8 � 3 � 12 � 12

– patterns when multiplying by 9, e.g., for 9 � 6,think 10 � 6 � 60, and 60 � 6 � 54; for 7 � 9,think 7 � 10 � 70, and 70 � 7 � 63

– repeated doubling, e.g., if 2 � 6 is equal to 12,then 4 � 6 is equal to 24 and 8 � 6 is equal to 48

– repeated halving, e.g., for 60 � 4, think 60 � 2 � 30 and 30 � 2 � 15.

• Recall multiplication facts to 81 and related division facts.

GOALDescribe how multiplication facts are related.


Materials • paper clips• Optional: chart paper and markers• Optional: counters

Masters • Spinners/Fraction Circles 2, Masters Booklet p. 52

Key Question entire exploration

Extra Practice Workbook p. 45

Mathematical ME (Mental Mathematics and Estimation) and Process Focus CN (Connections)



Math BackgroundDeveloping a sense of number properties frommultiplication facts will give students a greaterunderstanding of multiplication. The written and oralexplanation by each student will develop mathematicalcommunication skills. Students may use a variety ofmethods to show how to calculate one multiplicationfact from another. For example, students can relate 2 � 7 to 8 � 6 because 2 � 7 � 14 and it can bedoubled to show 4 � 7 � 28. Students can double againto calculate 8 � 7 � 56. Because 8 � 7 has one moregroup of 8 than 8 � 6, the extra group of 8 must besubtracted. Thus, 56 � 8 � 48 is the solution to theexpression 8 � 6.

The connections that students make when relatingmultiplication facts can help their understanding ofnumber sense as well as give them a method to solvemore difficult problems.

52 Multi-Purpose Masters: Spinners/Fraction Circles 2 Copyright © 2008 Nelson Education Ltd.

Spinners/Fraction Circles 2,Masters Booklet p. 52


Circulate and observe students as they work. Ask questionssuch as• What strategies are you using most often? Doubling and

halving? Adding or subtracting groups? • What strategies are you most comfortable working with? • What facts are you having the most difficulty relating?

What is it that makes them difficult?• What facts are the easiest to relate? What is it that makes

them easy?

Possible Solutions

Sample Solution 1:We showed how we used two different facts to get two others.5 � 5 to get 4 � 7• If you know that 5 � 5 � 25, then you know that

4 � 5 � 20, since it is one less 5.• If you know that 4 � 5 � 20, then for 4 � 7, you need

two extra 4s. So, 4 � 7 must be 28.4 � 9 to get 6 � 7• If you know that 4 � 9 � 36, then 2 � 9 is half as much

or 18.• If you add 4 � 9 and 2 � 9, you get 6 � 9, so

6 � 9 � 36 � 18 � 54.• If you know that 6 � 9 � 54, you know that 6 � 7 has

two less in each of the six groups. So, think: 54 – 2 – 2 – 2 – 2 – 2 – 2 by counting back: 52, 50, 48, 46, 44, 42. 6 � 7 � 42

Introduction (Whole Class)

➧ 5–10 min

Review with students how multiplication facts can becalculated from other facts. On a transparency or interactivewhiteboard, write the following table. Ask students todescribe how to calculate the facts in the left side of thecolumn using the facts from the right side. Remind studentsthat it may take more than one or two steps to calculate afact from another fact.

Sample Discourse“How can you calculate 7 � 6 using 5 � 6?”• Since 5 � 6 is 30, I can add two more groups of 6 to get

7 � 6. So, 30 � 6 � 6 � 42. “To calculate 8 � 9 from 4 � 5, you might first determine 8 � 5 from 4 � 5. How do you do that?”• I can double 4 � 5 to get 8 � 5. Since 4 � 5 is 20 then

8 � 5 is 40.“Now how can you calculate 8 � 9 from 8 � 5?”• I can double 8 � 5 to get 8 � 10, which is 40 � 40 � 80.

I can then subtract one 8 to get 8 � 9 � 80 – 8 � 72.

Teaching and Learning (Whole Class/Pairs/Small Groups)

➧ 20–30 min

Present the game on Student Book page 183 to students.Have students make spinners using the circle with 10 equalsections on Spinners/Fraction Circles, Masters Bookletp. 52 and writing the numbers 0 to 9 in the sections. Providepaper clips.

Make sure that they understand the game rules bymodelling a round. For example, if someone spins 5 and 4and then 6 and 8 they must use the fact 5 � 4 � 20 tofigure out 6 � 8. Have students spin again if they land on 0,to avoid connecting math facts to facts with 0.

If students need support, you might ask other students tosuggest a possibility for this pair of facts. For example, theymight think: 5 � 8 is double 5 � 4, so it is 40.6 � 8 is 5 � 8 and an extra 8, so it is 40 � 8 � 48.

Together, work through Jay’s strategy for relating 3 � 8 to 5 � 4 as shown on the student page. Encourage students totalk through many strategies for relating one fact to another.

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Calculate this fact Using this fact

7 � 6 5 � 6

8 � 9 4 � 5

21Lesson 3: Relating Multiplication FactsCopyright © 2008 Nelson Education Ltd.

44Consolidation ➧ 10–15 min


Provide the opportunity for students to share and communicatethe strategies they used to relate multiplication facts as theyplayed the game. Students could summarize their strategies onchart paper.

Ask students who are listening to comment on what theyliked about the strategies presented and what confused them.The presenters can invite questions from other students andattempt to answer the questions. Encourage students toidentify similarities and differences among their strategies.

Follow-Up and Preparation for Next ClassExplain to students that knowledge of related multiplicationfacts will give students mental strategies to solve morecomplicated problems. The next lessons will cover two-,three-, and four-digit multiplication.

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• Students will choose an appropriate strategy (skip counting from a known fact, doubling or halving, patterns in the 9s, or repeated doubling or halving) to relate multiplication facts.

• Students realize that one strategy can be quicker than the others, but allstrategies will yield the same result.

• Students may be unable to relate the simpler multiplication facts (e.g., 2 � 3and 3 � 3, or 2 � 5 and 4 � 5) using skip counting from a known fact ordoubling. (See Extra Support 1 or 2.)

• Students are not able to relate facts when more than one strategy is necessary.(See Extra Support 1 or 2.)


1. When facts must be related with more than one strategy, give students themiddle step to help them see the direction. For example, when relating 5 � 4 to 10 � 3, tell students to first double the 5 in 5 � 4 to get 10 � 4.Then, students can relate 10 � 4 to 10 � 3 by subtracting 10. Encouragestudents to look for a way to relate one of the factors, and then use a strategy to relate the second factor.

2. Alter the game by spinning one number and then multiplying two different factors by that number. For example, if a player spins 5, ask the player to explain how to relate 3 � 5 to 5 � 5. Keep one factor the same throughout the game so students can practise relating simpler facts by doubling or repeated addition. Alternatively, allow students to choose pairs of facts theywork with rather than spinning them.

EXTRA CHALLENGE

• Challenge students to use at least three different strategies per turn when playing the game to relate the multiplication facts.


• Kinesthetic and visual learners may benefit by using manipulatives when attempting to relate facts.

Sample Solution 2:We showed how we used two different facts to get two others.3 � 5 to get 6 � 6• If you know that 3 � 5 � 15, then you know that 6 � 5 is twice

as much, so 6 � 5 � 15 � 15 � 30.• 6 � 6 is 6 more than 6 � 5, since each of the 6 groups has

an extra 1, so 6 � 6 � 36.3 � 4 to get 8 � 6• If you know that 3 � 4 � 12, then 6 � 4 is twice as

much: 12 � 12 � 24. • 6 � 8 is twice as much as 6 � 4: 24 � 24 � 48• 8 � 6 is the same as 6 � 8; so, it is 48.

We realized that you could always change one number byeither doubling or taking half or adding a little at a time, andthen you could change the second number the same way.


44Chapter 6 44 Multiplying by Tens,Hundreds, and Thousands


GOAL

Calculate products with multiples of tens, hundreds, or thousands using mental math.


• Use the relationships 10 tens � 100 and 10 hundreds � 1000.

• Multiply single-digit numbers by 10 or 100.

SPECIFIC OUTCOMES

N4. Apply mental mathematics strategies formultiplication, such as• annexing then adding zero• halving and doubling• using the distributive property[C, ME, R]

Achievement Indicators• Determine the products when one factor is a multiple

of 10, 100, or 1000 by annexing zero or adding zeros,e.g., for 3 � 200 think 3 � 2 and then add two zeros.

PR2. Solve problems involving single-variable, one-stepequations with whole number coefficients and wholenumber solutions. [C, CN, PS, R]

Achievement Indicator• Solve a given single-variable equation with the

unknown in any of the terms.



Masters • Manipulatives Substitute: Base Ten Blocks,Masters Booklet pp. 40–42



Extra Practice Mid-Chapter Review Questions 5, 6, 7Chapter Review Question 3Workbook p. 46

Mathematical ME (Mental Mathematics and Estimation)Process Focus



Math BackgroundIn this lesson, students use their knowledge ofmultiplication facts to multiply numbers that aremultiples of 10, 100, or 1000. Words are used withnumbers so that students make the connection betweensingle-digit multiplication and multiplying by thesegreater numbers. For example, 40 is written as 4 tens.So, 3 � 40 is calculated by thinking of 3 groups of4 tens, which is 12 tens. And 12 tens are equal to 120.This method allows students to use mental math whenmultiplying by multiples of ten. Be sure that studentsunderstand that 40 � 30 can be written as 4 tens times3 tens, which would be 12 hundreds.

23Lesson 4: Multiplying by Tens, Hundreds, and ThousandsCopyright © 2008 Nelson Education Ltd.





Teaching and Learning (Whole Class/Pairs) ➧ 20–30 min

Together with students, read the information and the centralquestion on Student Book page 184. Draw students’ attention tothe Communication Tip about the symbol for seconds andclarify if needed. Discuss Ami’s Strategy with students. If needed,further clarify why Ami’s expression is 20 � 30. Students cancomplete Prompts A through E in pairs. Provide base ten blocksfor students to model multiplying by tens.

Sample Discourse“In Ami’s solution, what does the 20 represent in 20 � 30?”

• 20 seconds“What does the 30 represent?”• 30 beats in 1 second

Answers to PromptsA. 30 � 3 tensB. 60 tens since 20 � 3 � 60; 600 timesC. 600; 20 � 30 is 20 � 3 tens � 60 tens � 600 since

10 tens � 100D. 20 � 2 hundredsE. 20 � 200 � 4000, since

20 � 2 hundreds � 40 hundreds, andeach 10 hundreds is 1000

Introduction (Whole Class/Individual) ➧ 5–10 min

Ask students to imagine a number of groups of ten and tellhow many items there would be in total. Ask for severalanswers. Then, ask students what the answers all have incommon.

Sample Discourse“Choose any number. Imagine that many groups of ten. Howmany items would there be?”• 20, 50, 70, 100…“What do all of the numbers have in common?”• They all end in zero. Repeat for groups of hundreds.

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Students reflect on the different ways to represent a multipleof ten using words, models, and numbers, therebysimplifying multiplication strategies. Allow students to statethe strategy in their own words to help with understanding.Draw students' attention to the definition for multiples.

Answers to Reflecting QuestionsF. For example, there are 20 rows and 30 small cubes in

each row.20 � 30 � 2 � 3 hundreds

� 6 hundreds� 600

G. This is the number of beats in 2 seconds.H. For example, I always use 2 � 3 � 6, but the 6 represents

tens, hundreds, or thousands. 2 � 30 � 2 � 3 tens, so it’s 6 tens, or 60; 2 � 300 � 2 � 3 hundreds, so it’s 6 hundreds, or 600; 2 � 3000 � 2 � 3 thousands, so it’s6 thousands, or 6000; 20 � 30 is20 � 3 tens � 2 � 3 tens multiplied by 10, so it’s 600.


Checking (Pairs)

1. Allow students to begin writing a pattern to help themwith writing the multiplication statement. For example,1 s 40 times So, 1 � 40 � 402 s 80 times So, 2 � 40 � 803 s 120 times So, 3 � 40 � 120and so on. Ask students to skip to writing the expressionfor the number of beats in 20 s.


Encourage students to use several strategies for each problem sothey may determine which is easiest in a particular situation.Provide base ten blocks.6. Draw students' attention to the Communication Tip

about the symbol for minutes.7. Emphasize using the word form to determine the products

since 6 � 500 can be read "6 times 5 hundreds". Therefore,6 � 5 hundreds � 30 hundreds or 3000.

9. Ask students why the tens digits are the important onesin solving for the missing number.

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25Lesson 4: Multiplying by Tens, Hundreds, and ThousandsCopyright © 2008 Nelson Education Ltd.

55Answer to Closing Question 15. For example, when I multiply a number by 40, 400, or

4000, I just have to figure out the place value of eachdigit when multiplying the number by 4. 6 � 4 � 24.So, the digit 2 in 24 will have a place value of hundred,thousand, or ten thousand when 6 is multiplied by 40,400, and 4000.

Follow-Up and Preparation for Next ClassRevisit Question 9 in Practising and point out to studentsthat, when one factor is halved, the other is doubled. Thisrewriting of equivalent expressions is emphasized in the nextlesson.

Answers to Key Question5. a) 3200; For example, I imagined 4 rows of 8 hundreds

blocks. That would be 40 rows of 80.b) 8100; For example, I imagined 9 rows of 9 hundreds

blocks. That would be 90 rows of 90.c) 12 000; For example, I imagined 6 groups of 2 thousands

blocks. That would be 12 thousands, or 12 000.d) 3500; For example, I imagined 5 sets of 7 hundreds

blocks. That would be 35 hundreds blocks, or 3500.


Question 15 allows students to reflect on and consolidatetheir learning for this lesson. Emphasize the practicality ofthinking in words to simplify mental multiplication.




• Students can annex a zero or zeros to simplify multiplication, and arrive at the correct solution.

• Students recognize how to represent the product of multiples of 10 as anarray of hundreds.

Key Question 5 (Mental Mathematics)

• Students multiply multiples of tens, hundreds, and thousands correctly andexplain their strategies.

• Students may not be able to connect words to numbers and see that 30 tens isthe same as 3 hundreds. (See Extra Support 1.)

• Students imagine the expression 40 � 80 as 4 tens blocks and 8 tens blocks,not realizing an array of the blocks must be created. (See Extra Support 2.)

• Students may calculate incorrectly. (See Extra Support 2.)


EXTRA SUPPORT

1. Have students write the words and their numerical meanings when solvingproblems. For example, to write 30 tens, have students write “30” and then“10.” Then, ask them to write 3 hundreds as “3” and “100.” Show students that both phrases have the same digits and the same number of zeros.

2. Have students use ones blocks to make arrays to show multiplication facts,such as 3 � 5. Have them read this as 3 rows of 5. Relate this to 30 � 50represented as an array of 3 hundreds by 5 hundreds, read “30 rows of 50” or“3 rows of 5 hundreds.”

EXTRA CHALLENGE

• Have students observe that 40 � 60 is a product with 2 zeros at the end (2400). Challenge students to come up with all two-digit by two-digit productsthat will produce 3 zeros at the end. For example, 50 � 60 � 3000.


• Some students will not be ready to work with multiples of 10. Those studentsshould focus on work with facts and multiplying single-digit numbers bymultiples of 10.


• Have kinesthetic learners rewrite 30 � 40 using sticky notes as:

Students can then physically move the zeros to the answer. Make sure thatstudents understand that by moving the zeros to the answer they are actuallyforming hundreds from two tens because 10 � 10 � 100. As they are movingthe sticky notes have them verbalize the multiplication: “3 tens times 4 tens is12 hundreds.”

3 0

4 0�

1 2 0 0


5555Chapter 6 Halving and Doubling to Multiply



• Double two-digit numbers.• Know the half/double strategy for multiplication facts.• Multiply by 10, 100, or 1000 or multiples of 10 using

mental math.• Know how to use letters to represent unknown numbers.

SPECIFIC OUTCOMES




• Apply halving and doubling when determining agiven product, e.g., 32 � 5 is the same as 16 � 10.

PR2. Solve problems involving single-variable, one-stepequations with whole number coefficients and wholenumber solutions.[C, CN, PS, R]

Achievement Indicator• Solve a given single-variable equation with the

unknown in any of the terms.

GOALMultiply by halving and doubling.

Math BackgroundThe multiplication strategies used in the previous lessonsfor multiplying one-digit numbers are now used forgreater numbers. Students will reason that, just as 6 � 5could be halved to 3 � 5, so can 18 � 5 be halved to 9 � 5. But rather than double the answer, students willdouble the other factor. So, 18 � 5 is equivalent to 9 � 10, and this new expression can be calculated usingmental math. Encourage students to see that halving anddoubling is most useful when you can create an equivalentexpression that is simpler to solve.



Masters • Optional: Scaffolding for Lesson 5, Question 4 p. 71



Extra Practice Mid-Chapter Review Questions 8 & 9Chapter Review Question 4Workbook p. 47

Mathematical ME (Mental Mathematics) and R (Reasoning)Process Focus




4. Calculate each product using the half/double strategy. Look for numbers that can be halved or doubled to make 10, 100, or 1000.

a) 5 � 12

Which factor of 5 � 12 can be doubled to 10?

Double the 5 and halve the 12: �

5 � 12 �

b) 9 � 200

Which factor of 9 � 200 can be halved to 100?

Double the and halve the : �

9 � 200 �

c) 500 � 14



500 � 14 �

d) 50 � 24


50 � 24 �

e) 200 � 18 � 100 �

200 � 18 �

f) 18 � 500 � �

18 � 500 �

71Blackline Masters

Name: Date:


Optional: Scaffolding for Lesson 5, Question 4 p. 71

27Lesson 5: Halving and Doubling to MultiplyCopyright © 2008 Nelson Education Ltd.

Sample Discourse“How many counters are in 4 equal groups?”• 6 counters“How many counters are in 2 equal groups?”• 12 counters“When we halved the number of groups from 4 to 2, what happenedto the number of counters in each group?”• They doubled from 6 to 12.“How many counters are in 6 equal groups of counters?”• 4 counters“How many counters are in 3 equal groups?”• 8 counters“When we halved the number of groups from 6 to 3, whathappened to the number of counters in each group?”• They doubled from 4 to 8.“So, let’s say I have a product of two factors. If I take half ofone factor from a product, what can I do the other factor, if Iwant the same product in return?”• I can double it.

Teaching and Learning(Whole Class/Individual) ➧ 20–30 min

Together with students, read the information and the centralquestion on Student Book page 188. Discuss Justine’s Solutionand why she decided to multiply rather than add. Moststudents will immediately visualize 16 � 50 as 16 groups of 50.After students complete Prompts A and B, guide them intoregrouping 16 groups of 50 as 8 groups of 100, afterdetermining that 2 packs of 50 CDs equals 100 CDs.

Sample Discourse“Suppose there were also packs of CDs that came in 100s.How many packs of 50 CDs would you have to buy to equala package of 100 CDs?”• 2“So, 4 packs of 50 CDs would be the same as how manypacks of 100 CDs?”• 2, because 4 � 50 is the same as 2 � 100“Then, 16 packs of 50 CDs would be the same as how manypacks of 100 CDs?”• 8, because 16 � 50 is the same as 8 � 100

Have students decide whether they can always rewrite amultiplication expression by doubling one of the numbersand halving the other. Then, transition from this discussionto the definition of the half/double strategy. Students cancomplete Prompts C and D individually.

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Introduction(Whole Class/Small Groups)

➧ 5–10 min

Provide groups of students with 24 counters or other items pergroup. Ask students to arrange these counters into 4 equalgroups. Ask how many counters are in each group. Then, askstudents to arrange the counters into half as many equal groups.Ask how many groups they are going to have and how manycounters will be in each group. Repeat this process starting with6 groups of 4 counters. Discuss the results with students.

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Answers to PromptsA. There are 100 CDs in two packs.B. There are 16 packs altogether. That’s 8 groups of 2 packs.

There are 100 CDs in every two packs; 8 � 100 � 800. So, 16 � 50 � 800.

C. Eight groups of two packs doubled is equal to the totalnumber of CDs. She bought 800 CDs, since 8 � 100 � 800.


Students reflect on how halving and doubling can be used asa multiplication strategy with greater numbers. Remindstudents to use halving only on even numbers.

Answers to Reflecting QuestionsE. For example, 16 can be easily halved to 8, and 50 can be

easily doubled to 100. You can then multiply 8 � 100easily using mental math.

F. For example, the half/double strategy is useful when the numbers that are doubled or halved result in a multiple of 10.


Checking (Pairs)

Have students discuss in pairs the two possible ways to halveand double the problem in Question 1 a). One way will resultin multiplication by 100, the other will result in multiplicationby 400. Point out multiplying by 100 is simpler.


Remind students it is probably easier to halve only the evennumbers. Encourage several approaches to see which willresult in the quickest computation.4. If extra support is required, guide those students and

provide copies of Scaffolding p. 71.8. Point out that repeated doubling can be used to make

one factor a 10, 100, or 1000.

Answers to Key Question6. a) 12 � 10 � w; w � 120; For example, half of 24 is

12, and double 5 is 10.b) x � 100 � 7; x � 700; For example, I knew the double of

50 is 100. Since 14 was even, I took half of it to get 7. 100 � 7 � 700.

c) 4 � 1000 � y; y � 4000; For example, half of 8 is 4,and double 500 is 1000.

d) z � 1000 � 9; z � 9000; For example, double 500 is1000, and half of 18 is 9.

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29Lesson 5: Halving and Doubling to MultiplyCopyright © 2008 Nelson Education Ltd.

Follow-Up and Preparation for Next ClassAsk students to search through advertisements at home tofind a multiplication expression that can be solved throughhalving and doubling.

Next class is the Mid-Chapter Review. Ask students to gothrough Lessons 1 to 5 and note any questions or problemsthey have.


Question 14 allows students to reflect on and consolidatetheir learning for this lesson.

Answer to Closing Question 14. For example, I would tell him to take half of 48 to get 24

and double 50 to get 100. Then, you multiply 24 � 100, so it is just 2400.




• Students understand that, although the halving/doubling procedure will yield acorrect answer for any multiplication expression, only certain expressions willbecome simpler. These expressions include those that will result in a multipleof 10 as a factor.

Key Question 6 (Mental Mathematics and Reasoning) • Students understand that doubling the factor that results in a multiple of 10

results in the simplest expression.

• Students may not be able to explain why halving one factor and doubling theother factor will not change the product. (See Extra Support 3.)

• Students may not easily determine which factor to halve and which to double.(See Extra Support 2.)


EXTRA SUPPORT

1. Use Scaffolding for Lesson 5, Question 4 p. 71.

2. Have students practise halving and doubling several numbers. Write thefollowing numbers on the board: 2, 3, 4, 5, 8, 11, 15, 18, 22, 24, 29. Askstudents to halve and double each one. Discuss which numbers cannot be halved (and still have a whole number). Elicit from students that allnumbers can be doubled.

3. Have students draw a picture like this:

Help them see that the 3 groups of 8 can be thought of as 6 groups of 4 bydrawing a line down the middle. There are twice as many groups since eachsingle group became two groups. Although the grouping of the items changed,the total number didn’t.

EXTRA CHALLENGE

• Challenge students to create problems in which quadrupling one factor and dividing the other by 4 will result in a simpler equation. For example, 48 � 250 could be written as 12 � 1000.

• Some students might be ready to think about using this strategy with prices tosimplify calculations. Ask students to figure out how much 8 items at $2.50would cost and why the half double strategy might make it easy to figure that out.


• Have visual learners break the method into steps by first circling the numberthat will be halved and drawing a box around the number that will be doubled.

For example, have students write 4 � 50 as 4 � 50 . The circle should promptstudents to halve the number inside from 4 to 2 and the square should promptstudents to double 50 to 100 to get 2 � 100.

Have students write the new expression directly under the original expressionso they can see the process on paper.

Chapter 6

xxxx xxxxxxxx xxxxxxxx xxxx


Using the Mid-Chapter ReviewThis review provides an opportunity for students to monitor their progress with the chapter skills and concepts(Assessment as Learning), as well as for you to monitor theprogress of the class and see where re-teaching may be required(Assessment for Learning). You may also use it to assessindividual student achievement (Assessment of Learning).

SPECIFIC OUTCOMES

N3. Apply mental mathematics strategies and numberproperties, such as• skip counting from a known fact• using doubling or halving• using patterns in the 9s facts• using repeated doubling or halvingto determine answers for basic multiplication facts to 81and related division facts.[C, CN, ME, R, V]


determine a given basic fact, such as:– skip count up by one or two groups from a

known fact – skip count down by one or two groups from a

known fact– doubling– patterns when multiplying by 9– repeated doubling– repeated halving

• Recall multiplication facts to 81 and related divisionfacts.



of 10, 100 or 1000 by annexing zero or adding zeros. • Apply halving and doubling when determining a

given product.

Materials • Optional: base ten blocks

Masters • Mid-Chapter Review—Frequently Asked Questions p. 72

• Optional: Multiplication Table, Masters Booklet p. 33

• Manipulatives Substitute: Base Ten Blocks, Masters Booklet pp. 40–42




Name: Date:

Mid-Chapter Review—Frequently Asked QuestionsSTUDENT BOOK PAGE 192

Q: What strategies can you use to multiply one-digit numbers?

A:

Q: How can you multiply by multiples of tens, hundreds, or thousands?

A:

Q: How can you simplify a calculation using the half/double strategy?

A:


Mid-Chapter Review—Frequently AskedQuestions p. 72


31Mid-Chapter ReviewCopyright © 2008 Nelson Education Ltd.

Mid-Chapter Review STUDENT BOOK PAGES 192–193





Frequently Asked Questions(Whole Class)

Have students keep their Student Books closed. Write theFrequently Asked Questions on Student Book p. 192 on thechalkboard or interactive whiteboard, or use Mid-ChapterReview—Frequently Asked Questions p. 72. (Distributethe master or display it using an overhead transparency orinteractive whiteboard.) Use the discussion to draw out whatthe class thinks is the best answer to each question. Then,have students compare the class answers with the answers inthe Student Book. Have students summarize the answers intheir own words as a way of reflecting on the concepts.

Students can refer to the answers to the Frequently AskedQuestions as they work through the Practice questions. Atthis time, you can also discuss any other questions related toLessons 1 to 5 that students may have.

Practice (Individual)

Students should be able to complete Questions 1 to 5 inclass. Assign the rest for homework. Encourage students toidentify which questions they found easy and which morechallenging. Ask them what they can do to become proficientat these questions. The review questions are organized bylesson. Students can go back to the lesson indicated to reviewthe concepts for the question.3. Encourage students to describe a strategy even if they

have the math fact committed to memory.5. Encourage students to model with base ten blocks before

sketching.



Questions 1, 3, 4, 5, 6, 7, 8, & 9 (Mental Mathematics, Reasoning)• Students will calculate products using strategies that they can fully explain.

Question 2 (Connections)• Students apply multiplication strategies to new contexts.

• Students may only use one strategy to calculate products and be unable to fullyexplain the strategy.

• Students may not be able to identify or carry out the correct calculations.

Question 5 (Visualization)• Students will sketch a model of the multiplication expression using base ten

models or groupings of objects.• Students may draw only a partial representation of the model or draw the model

incorrectly.

Differentiating Instruction: How you can respondRefer to the Differentiating Instruction in Lessons 1 to 5.


Assessment of Learning—What to look for in student workQuestion 1, short answer Specific Outcome and Process Focus: N3 [ME, R]• Calculate each product using a different strategy. Explain your strategy for one answer.

a) 3 � 5 b) 7 � 6 c) 4 � 9 d) 7 � 9

(Score 1 point for each correct answer, and 1 point for an appropriate strategy; for a total out of 5.)

Question 2, short answer Specific Outcome and Process Focus: N3 [CN]• Juice drinks are sold in packs of three. How many juice drinks are in six packs?(Score 1 point for correct answer.)

Question 3, written answer Specific Outcome and Process Focus: N3 [ME, C]• Calculate each product and describe the strategy you used.

a) 8 � 9 b) 5 � 8 c) 7 � 8 d) 7 � 4

(Score 1 point for each appropriate strategy and 1 point for each correct answer; for a total out of 8.)

Question 4, short answer Specific Outcome and Process Focus: N3 [ME]• Ami bought eight plates of sushi like the one shown. How many pieces of sushi did she buy?

(Score 1 point for correct answer.)

Question 5, drawing Specific Outcome and Process Focus: N4 [ME, V]• Show that 40 � 60 = 2400.

Question 6, short answer Specific Outcome and Process Focus: N4 [ME]• Calculate

a) 30 � 60 d) 8 � 2000b) 5 � 800 e) 3 � 900 c) 40 � 90 f) 10 � 300

(Score 1 point for each correct answer; for a total out of 6.)

Question 7, short answer Specific Outcome and Process Focus: N4 [ME, R]• A bank teller has 18 $100 bills in a cash drawer. What is the value of the bills?


Question 8, short answer Specific Outcome and Process Focus: N4 [ME, R]• Rewrite each equation using the half/double strategy. Then, calculate.

a) 50 � 12 = m c) p = 16 � 500b) n = 5 � 36 d) 25 � 16 = q

(Score 1 point for each rewritten equation and 1 point for each correct answer; for a total of 8.)

Question 9, short answer Specific Outcome and Process Focus: N4 [ME, R]• Carolyn babysat for 38 hours over the summer. She earned $5 an hour. How much did she earn over the summer?


Work meets standard of excellence

Work meets standard of proficiency

Work meets acceptable standard

Work does not yet meet acceptable standard

• uses visual representationsinsightfully to demonstrate a thorough understanding of multiplying by tens

• uses visual representationsmeaningfully to demonstrate areasonable understanding ofmultiplying by tens

• uses visual representations simply to demonstrate a basic understandingof multiplying by tens

• uses visual representations poorly to demonstrate an incompleteunderstanding of multiplying by tens

33Mid-Chapter ReviewCopyright © 2008 Nelson Education Ltd.


• Multiply by 10 and multiples of 10 using mental math.• Add and subtract two-digit and three-digit numbers.

SPECIFIC OUTCOME




• Apply the distributive property to determine a givenproduct involving multiplying factors that are close tomultiples of 10, e.g., 98 � 7 � (100 � 7) – (2 � 7).

6666Chapter 6

GOALMultiply using a simpler, related question.



Masters • Manipulatives Substitute: Base Ten Blocks,Masters Booklet pp. 40–42

Recommended Questions 3, 4, 5, 10, 11, 12, & 17Practising Questions


Extra Practice Chapter Review Question 5 Workbook p. 48

Mathematical ME (Mental Mathematics) and R (Reasoning)Process Focus



Math BackgroundThis lesson continues using multiplication strategiespreviously learned, but with numbers that are close to tens.One of those methods required students to use mentalmath to show how 7 � 9 is related to 7 � 10. Studentsunderstood that the second expression contains one moregroup of 7. Thus, 7 � 9 is is calculated using 7 � 10 � 7.Using reasoning, students will see that 7 � 99 also hasone less group of 7 than 7 � 100. So, 7 � 99 is calculatedby solving 7 � 100 and subtracting 7.


Multiplying NumbersClose to Tens







Tell students that you have several packages of paper clips inyour desk. The large paper clips come with 50 clips per box.The smaller paper clips come with 73 clips per box. Tellstudents to think about the following question: If you have 8boxes total of each kind, is it easier to find the number of largepaper clips or the number of small paper clips? Point out tostudents that it is easier to multiply by a multiple of 10 thanby another two-digit number that is not a multiple of 10,because more than one calculation is needed. But using thestrategies they have learned so far, they will learn how tosimplify multiplying by numbers that are close to a multiple of 10.

Discuss with students some of the math strategies theyhave used thus far.

Sample Discourse“How can 3 � 4 be related to 2 � 4?”• I know that 3 � 4 is 12 and, because this expression has one

more group of 4, I can subtract 4 to find the answer to 2 � 4.Since 12 � 4 is 8, I know that 2 � 4 is 8.

“How can 5 � 9 be related to 10 � 9?”• I know that 5 � 9 is 45 and that 5 doubled is 10. So, this

means 10 � 9 must be 45 doubled or 90.Explain to students that these same strategies can be used tomultiply numbers close to tens or hundreds.

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With students, read about the hotel and the central questionon Student Book page 194. Provide pairs of students withbase ten blocks. Work through Brandon’s Solution as a class.Have students explain why 7 must be subtracted from the 7groups of 40. Explain to students that Brandon’s strategyincludes wanting to make the multiplication easier. Remindthem that this is the goal of any multiplication strategy.

Sample Discourse“Since 1 was added to 39 to get 40, why is 7 (instead of 1)subtracted from 280?”• Because 1 window was added to each of the 7 floors• Because 1 group of 7 was added and not 1 group of 1


Students reflect on how to use simpler numbers and addition or subtraction as a multiplication strategy withgreater numbers. Have students explain when they will need to add and when they will need to subtract.

Answers to Reflecting QuestionsA. For example, because 39 is close to 40, 7 � 40 is easier

to calculate than 7 � 39.B. For example, for 7 � 38 he could use 7 � 40 and

subtract 14; for 7 � 41 he could use 7 � 40 and add 7.

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35Lesson 6: Multiplying Numbers Close to TensCopyright © 2008 Nelson Education Ltd.


Checking (Pairs)

1. Students should be able to solve this using Brandon’smethod, which is effective when the factor differs from amultiple of 10 or 100 by only 1 or 2.

Practising (Individual/Pairs)

6. Encourage students to share answers in pairs.7. Have students think in terms of groups when answering

this question. How many more (or fewer) groups of 7 arethere when compared to 7 � 80?

8. Guide students to use words like “increasing” and“decreasing” when describing the patterns.

Answers to Key Question

5. a) 28 � 9 � 28 � 10 � 28� 280 – 28� 252 km

b) For example, there are 31 days in March, so 3 daysmore than February. She would walk 3 � 9 � 27 kmmore in March, or 252 km � 27 km.


Question 17 allows students to reflect on and consolidate theirlearning for this lesson. Invite students to share strategies aswell as compare answers. Allow students to choose the strategythey like best after hearing different possibilities.

Answer to Closing Question 17. 6 � 98 is only 12 less than 6 � 100; you can subtract

12 from 600 using mental math by subtracting 10 and then 2.

Follow-Up and Preparation for Next ClassHave students search for a real-life problem that utilizes thisstrategy for multiplying numbers close to tens. Ask them tocount windows in a real house, or parking spaces in one rowof a parking lot, or the number of books on one shelf. Then,have students share with the class and calculate the totalnumber of windows on nine of the same houses, or thenumber of parking spaces in nine rows, or the number of books on eight shelves.

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• Students use a variety of multiplication strategies (e.g., halving/doubling,repeated addition/subtraction, annexing/adding zeros) to simplify themultiplication process.

• Students are able to look at expressions and be able to state which expressionis greater by comparing individual factors.

Key Question 5 (Mental Mathematics, Reasoning)• Students use effective and efficient strategies relating multiplication

expressions to solve problems.

• Students may have difficulty choosing a multiplication strategy to simplify aproblem. (See Extra Support 1.)

• Students may not understand that the expressions 7 � 39 and 7 � 40 differ bya group of 7. (See Extra Support 2.)

• Students may use inefficient methods to solve the problem, rather than relatingone calculation to another. (See Extra Support 1 or 2.)


EXTRA SUPPORT

1. Remind students about the strategy of relating 8� facts and 9� facts tomultiplying by 10, as in Lesson 2. Have them relate this to multiplying bynumbers close to 10 and 100.

2. Have students write patterns with smaller factors to help them understand how to compare expressions by comparing factors. For example, to see how 7 � 39 and 7 � 40 differ, have students begin with 7 � 1 � 7, 7 � 2 � 14, 7 � 3 � 21, and point out that each differs from the one below it by 7. So, 7 � 39 will differ from 7 � 40 by 7.

EXTRA CHALLENGE

• Challenge students to summarize this lesson’s strategy into a writtenparagraph. Have students include in their paragraph– for which numbers the strategy works best– the general concept of the strategy and how it works– examples using the strategyHave students share their summaries with the rest of the class.

• Challenge students to use the strategy with numbers close to hundreds. For example, have students solve 8 � 298 (2384) or 5 � 302 (1510).

37Lesson 6: Multiplying Numbers Close to TensCopyright © 2008 Nelson Education Ltd.

7777Chapter 6 Estimating Products STUDENT BOOK PAGES 198–200


• Round to the nearest tens or hundreds.• Multiply by 10, 100, and multiples of 10 using mental

math.

SPECIFIC OUTCOME

N2. Use estimation strategies including• front-end rounding• compensation• compatible numbersin problem-solving contexts.[C, CN, ME, PS, R, V]

Achievement Indicators• Provide a context for when estimation is used to

– make predictions – check reasonableness of an answer– determine approximate answers

• Describe contexts in which overestimating isimportant.

• Determine the approximate solution to a givenproblem not requiring an exact answer.

• Estimate a sum or product using compatible numbers. • Select and use an estimation strategy for a given

problem. • Apply front-end rounding to estimate

– products, e.g., the product of 23 � 24 is greaterthan 20 � 20 (400) and less than 25 � 25 (625)

GOALEstimate to solve problems.



Masters • Optional: Scaffolding for Lesson 7, Question 8 p. 73

Recommended Questions 2, 3, 4, 6, 10, & 11Practising Questions


Extra Practice Chapter Review Question 6Workbook p. 49

Mathematical ME (Estimation) and PS (Problem Solving)Process Focus



Math BackgroundCertain problem-solving situations may not requireexact answers, and, in those cases, students will useestimates of numbers to simplify the multiplicationprocess. Explain to students that estimates cannot beincorrect, but some estimates are better than others.Emphasize the importance of communicating the mathwhen students support their estimates by explaining thestrategies used.

73Blackline Masters

Name: Date:



8. A class of 36 students is having a bridge-building contest. Each group of 4 students has 35 straws to make a bridge. The straws come in bagsof 50. Calculate the number of bags needed for the class by following the steps below.

How many groups of 4 are in 36? 36 � 4 � groups

How many straws does each group have? straws

How can you estimate the total number of straws needed for all groups?

�

How many bags of straws does the class need? Explain.

Optional: Scaffolding forLesson 7, Question 8 p. 73


Teaching and Learning (Whole Class/Pairs) ➧ 20–30 min

Together, read about the Cree First Nations game of countingsticks on Student Book page 198. Discuss with students why itis better to play the game with an odd number of sticks (39)rather than an even number. (You cannot divide an odd numberof sticks into two even groups, so one bundle will always have anodd number of sticks, and the other bundle will always have aneven number of sticks. Therefore, there will always be one correctbundle and one incorrect bundle to choose from.) Remind studentsto read through the information presented in the problemcarefully, and to use only the relevant information. Discuss whyan estimate can be used to answer the central question ratherthan an exact answer.

Sample Discourse“Owen decided to estimate instead of calculating the exactproduct. What led Owen to decide an exact product was not necessary in this case?”• He only needed to know how many boxes were going to be

enough for the whole class, not the exact number of sticks.

Have students work through Prompts A through D in pairs.Then, have students share their responses to Prompt D withthe class.

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39Lesson 7: Estimating ProductsCopyright © 2008 Nelson Education Ltd.

Introduction(Individual/Small Groups) ➧ 5–10 min

Revisit the concept of underestimating and overestimating.Discuss with students some situations in which anoverestimate or an underestimate is appropriate. Provide pairsof factors to multiply and ask students to suggest how tooverestimate and underestimate. For example, 47 � 57 canbe estimated using 40 � 50 or 50 � 60. Repeat with pairssuch as 62 � 88 and 53 � 12. Note that front-end roundingalways results in an underestimate, and rounding up alwaysresults in an overestimate.

Sample Discourse“Write the numbers 47 and 57. What rounded numbers willresult in an underestimate when multiplied? Explain.”• 40 and 50; because both numbers are less than the actual

numbers, so the product will be less “What rounded numbers will result in an overestimate whenmultiplied? Explain.”• 50 and 60; because both numbers are greater than the actual

numbers, so the product will be greater


Checking (Pairs)

Have students count the number of students so that all areworking with the same number. Compare answers with otherpairs of students to see how estimates might differ.


2. & 3. Have students compare answers in pairs toencourage discussion of the different strategies.Encourage students to use appropriate math vocabulary.

8. If extra support is required, guide those students andprovide copies of Scaffolding p. 73.

Answer to Key Question4. For example, if she uses 4 � 90 as an estimate for

4 � 84, her estimate will be greater than the actualproduct. Then, she’ll be sure she has enough money topay for the blankets.

Answers to PromptsA. 12 is between 10 and 20 and 39 is between 30 and 40,

so 12 � 39 is between the lesser product and the greater product.

B. For example, 12 is close to 10 and 39 is close to 40, so10 � 40 is close to 12 � 39. It’s also probably closer tothe actual answer, since I decreased one number andincreased the other to make up for it.

C. For example, there are 24 students, and 24 is close to 25.25 � 40 is easy to do since 25 � 4 � 100. You have totake half, since students play the game in pairs.

D. For example, my first estimate was 400, so the classwould need 3 boxes (2 boxes is only 300). My secondestimate was 500, so the class would need 4 boxes since 3 � 150 � 450 and that’s not enough. I would probablyget 4 boxes; it’s better to estimate high and have enough.


Students consider situations where it is possible to estimate ratherthan calculate an exact answer. Students also consider the benefitsand drawbacks to an estimate being too high or too low.

Sample Discourse“What may happen if the estimate is low?”• Some students will not have enough sticks.“What happens if the estimate is high?”• All students will have enough sticks and there might be some

left over.

Answers to Reflecting QuestionsE. For example, I would pick C since I think the numbers

are closest to the actual amounts, but the calculation isstill easy to do.

F. For example, you want to make sure that you don’t haveany students who can’t play the game.


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• Students choose an appropriate strategy for estimating products and explaintheir procedures for estimating.

• Students explain the reasonableness of their estimates.

Key Question 4 (Mental Mathematics and Problem Solving) • Students are able to explain that, when purchasing an item, a person must

have at least the cost of the item. If the estimate is high, the purchaser is sureto cover the cost.

• Students may not understand that the context of a problem should indicatewhich type of estimate is needed to solve it.

• Students may not be able to explain why an estimate might be greater or lessthan the actual answer. (See Extra Support 2.)

• Students may not be able to explain why using 4 � 90 to estimate 4 � 84 willresult in a high estimate. (See Extra Support 2.)


EXTRA SUPPORT

1. Use Scaffolding for Lesson 7, Question 8 p. 73 to provide steps for solving the problem.

2. Have students calculate 4 � 39 and 4 � 40. Students should see that theanswers differ by 4. Have students calculate 6 � 50 and 6 � 52. Point outthat 6 � 52 is greater by 2 groups of 6, or 12. Repeat with other problems.Then, check students' understanding by asking them to estimate 7 � 57 andasking if their estimate will be greater or less than the actual amount.

EXTRA CHALLENGE

• Challenge students to determine two or more estimates for each problem inQuestion 2, then decide which is closest to the actual answer beforecalculating the exact answer. Make sure that students can justify how theyknow which one is closest. Students can also predict which estimates areunderestimates or overestimates, following the same logic.


• Modify the situation that leads to the central question by having only 8 students in Ami's class, rather than 24.

• In Practising Question 2, allow students to choose their own products toestimate.

• Modify some of the questions so that only one factor is a two-digit number.


• Visual learners may benefit from writing the original expression and thenwriting the rounded numbers directly below. This will help students keep trackof their methods when sharing strategies.


Question 11 allows students to reflect on and consolidatetheir learning for this lesson. Point out that students probablyalready do this type of estimating without even thinking. Forexample, if you are bringing in cookies that come in boxes of12 to share with your class and you want each student tohave 2 cookies, you would probably estimate to ensure thatyou bring enough boxes rather than calculate the exactamount. You can then calculate to see how many cookies willbe left over. Ask students to pay attention to situations whenestimating is used outside of the classroom and encouragethem to share these with the class.

Answers to Closing Question 11. a) For example, I want to know about how much money

to take to the store to buy 4 CDs, so I would estimateto predict the product.

b) For example, if I solved a problem and the answerlooked like it was too much, I would estimate to check.

c) For example, I might want to know how many daysof school I have each month. I don’t need to knowthe exact number, so I would estimate it.

41Lesson 7: Estimating ProductsCopyright © 2008 Nelson Education Ltd.

8888Chapter 6

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Multiplying Two-DigitNumbers

STUDENT BOOK PAGE 201


• Divide two-digit numbers (multiples of 3) by 3.• Multiply two-digit and three-digit numbers by

one-digit numbers.• Multiply by 10 or 100.

SPECIFIC OUTCOMES

N4. Apply mental mathematics strategies formultiplication, such as• annexing then adding zero• halving and doubling • using the distributive property. [C, ME, R]



• Apply the distributive property to determine a givenproduct involving multiplying factors that are close tomultiples of 10, e.g., 98 � 7 � (100 � 7) � (2 � 7).

N5. Demonstrate an understanding of multiplication(two-digit by two-digit) to solve problems.[C, CN, PS, V]

Achievement Indicators• Solve a given multiplication problem in context using

personal strategies and record the process.

GOALMultiply two-digit numbers using your choice of strategies.


Materials • Optional: chart paper and markers• Optional: base ten blocks

Key Question entire exploration


Mathematical C (Communication) and PS (Problem Solving)Process Focus



Math BackgroundThis exploratory lesson provides students with anopportunity to solve a rich problem using personalstrategies. By setting only one problem, time is assuredfor deep engagement with the problem.

Students apply strategies used with one-digit and two-digit numbers or their own personal strategies to multiply 2 two-digit numbers. Because this problem solving requires a substantial amount ofplanning, students may need to take more time in setting up the problem. Encourage students to see thatdivision is also taking place because they can determinehow many 3s are in 45. Students may have severaldifferent strategies to arrive at the correct answer. Havestudents explain their methods to other students to foster communication in math.


Introduction (Whole Class) ➧ 5–10 min

If possible, obtain a copy of Rowland Morgan’s book In theNext Three Seconds. The book is a collection of predictionsabout everyday and not-so-everyday events that will takeplace in the next three seconds, the next three minutes, allthe way up to the next three million years. Discuss withstudents how we use past events to make future predictions.Discuss the fact that, although a prediction is an estimate,students will not be estimating their answers; rather, theycalculate the numbers from the prediction.

One of the most useful mathematical concepts is thecapability of predicting future events by recognizing a patternof similar past events. Have students use multiplication tomake some simple predictions. Read the following and havestudents share solutions with the rest of the class.• Your uncle puts 8 L of gasoline in his motor scooter each

week. How many litres will he put into his scooter over thenext four weeks?

• You are reading a book. You notice that it has taken you10 min to read 4 pages. How many pages do you thinkyou could read in the next 30 min?

Sample Discourse“How many times will your uncle have to put 8 L of gasolinein his scooter in 4 weeks?”• Four times“What do you have to multiply to get the total number oflitres of gasoline?”• 4 � 8, or 32 litres“How many times can you multiply 10 minutes to get 30 minutes?”• 3 times, because 10 � 3 � 30“How many pages can you read if you have 3 times as manyminutes?”• I can read 3 times as many pages or 4 � 3 � 12 pages.

Teaching and Learning (Whole Class/Small Groups)

➧ 20–30 min

Present the problem and central question on Student Bookpage 201 to students. Have available materials such as baseten blocks for students to work with as well as chart paperand markers. Make sure students understand that they are to perform three calculations: one for 45 seconds, one for 75 seconds, and one for 99 seconds.

Draw attention to the fact that the data they have is forthree seconds, and not for one second. Ask for ideas on howto use the three seconds to solve the problem and point outthat they can look for the number of groups of 3 in 45.

On their chart paper, they are to explain their thinking oneach of the three problems they are solving.

Explain that they are to• make any decisions they feel are necessary to solve the

problem• write out the main points in their solution on chart paper• be prepared to communicate their solution process to the

rest of the class

No one approach to the problem should be suggested;encourage students to choose their own strategies. Studentsmay look for a method they think you would want to see,but should be encouraged to use any methods that work.

Circulate and observe students as they work. Ask questionssuch as• Why might it be useful to figure out the number for

30 seconds?• How did you use what you know about 45 seconds to help

you with 75 seconds?• Why could you not just multiply 95 by 100 and then

subtract 95 to get the number for 99 seconds?• Which calculation was easiest for you? Explain.

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43Lesson 8: Multiplying Two-Digit NumbersCopyright © 2008 Nelson Education Ltd.

Possible Solutions

Sample Solution 1:For 45 seconds, we figured out that we wanted to multiply95 � 15 since there are 15 threes in 45. To do that, wedecided to multiply 100 � 15, since that is easy, and thenwe subtracted 5 � 15.100 � 15 � 1500

5 � 15 � 50 � 25 � 75

So, for 45 seconds, it would be 1500 � 75 � 1425 planes.For 75 seconds, we decided to add the number for

30 seconds to the number for 45 seconds. The number for30 seconds is 10 times as much as for 3 seconds.

10 � 3 � 3095 � 10 � 950

1425 � 950 � 2375 planesFor 99 seconds, we first figured out the number of

take-offs in 9 seconds. Since 9 � 3 � 3, then the number for9 seconds is 3 � 95.3 � 95 � 3 � 100 � 3 � 5

� 300 � 15 � 285

Then, in 90 seconds, there would be 10 times as many.10 � 285 � 2850So, for 99 seconds, it would be 285 � 2850 � 3135 planes.

Sample Solution 2:45 seconds is 15 groups of 3 seconds, we need 15 � 95.That means I need 15 rows of 95.I used base ten blocks.

I replaced every 10 tens blocks with 1 hundreds block.

There was a total of 900 � 400 � 50 � 50 � 25 � 1425planes.


Closing (Whole Class/Pairs)

Provide the opportunity for students to share and communicateabout their work. Have students describe to the rest of the classhow they solved the problem, using their chart paper as anorganizing tool. If you saw some particularly interestingapproaches when you circulated among students as they wereworking through the problem, invite those students to beamong the presenters.

Ask students who are listening to comment on what theyliked about the approach presented and what confused them.Some students may think of the strategy of using four partialproducts, For example, for 15 � 45, they might multiply 10 � 40 � 10 � 5 � 5 � 40 � 5 � 5. Others will use othernumber relationships. The presenters can invite questions fromother students and attempt to answer the questions. Encouragestudents to identify similarities and differences among theirmethods.

Follow-Up and Preparation for Next ClassChange the activity to focus on the number of events in3 minutes. Events can include the number of times a person blinks in 3 minutes, the number of breaths in3 minutes, or the number of heartbeats in 3 minutes. Have students calculate total numbers for 45 minutes,75 minutes and 99 minutes.

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• Students understand the problem and choose an appropriate solution usingmultiplication strategies they know.

• Students may not be able to relate math facts to each other to determine asolution. (See Extra Support 1 and 2.)


EXTRA SUPPORT

1. Remind students of how they calculated 9� a number by calculating 10�the number and subtracting the number. Help them see how they could use a similar idea, for example, in calculating 15 � 95, they could start with15 � 100 and then ask how much they need to add or take away.

2. Some students may benefit from setting up base ten blocks. Allow them time to count the blocks to verify their solutions.

EXTRA CHALLENGE

• Challenge students to work backwards to solve for the number of secondsrather than finding the number of take-offs. For example, after how manyseconds will 380 airplanes have taken off? (12 s, because 95 � 4 � 380so, 3 � 4 � 12 s) After how many seconds will about 9500 airplanes have taken off? (300 s, because 95 � 100 � 9500, so 3 � 100 � 300) If necessary,encourage students to use a guess-and-check method to determine the answers.


• 95 planes in 3 seconds is a little more than 30 planes every second. Allow students to work with 30 planes every second to calculate the number ofplanes in 10� seconds, such as 40, 70, and 100 seconds.

• Allow students to choose the number of seconds they use instead of 45, 75, and 99.


• Creative learners may benefit from obtaining their own facts to use formaking predictions. For example, students can count the number of timestheir heart beats in 1 minute to predict how many times their heart wouldbeat in 30 minutes, 45 minutes or 99 minutes. They could also use facts suchas the number of times they blink in 3 minutes or the number of glasses ofmilk they drink in a week.

45Lesson 8: Multiplying Two-Digit NumbersCopyright © 2008 Nelson Education Ltd.

9999Chapter 6 Multiplying with BaseTen Blocks


Multiply one-digit numbers using mental math strategies.


• Multiply multiples of 10.• Multiply two-digit numbers by one-digit numbers.• Represent multiplication using an array.

SPECIFIC OUTCOME


Achievement Indicators• Illustrate partial products in expanded notation for

both factors, e.g., for 36 � 42, determine the partialproducts for (30 � 6) � (40 � 2).

• Represent both two-digit factors in expandednotation to illustrate the distributive property, e.g., todetermine the partial products of 36 � 42, (30 � 6) � (40 � 2) � 30 � 40 � 30 � 2 � 6 � 40 � 6 � 2 � 1200 � 60 � 240 � 12 � 1512.

• Model the steps for multiplying two-digit factorsusing an array and base ten blocks, and record theprocess symbolically.

• Solve a given multiplication problem in context usingpersonal strategies and record the process.

GOALRepresent the products of two-digit numbers.



Masters • Manipulatives Substitute: Base Ten Blocks, Masters Booklet pp. 40–42

Recommended Questions 3, 5, 8,10, & 13Practising Questions


Extra Practice Chapter Review Questions 8, 9, and 10Workbook p. 51

Mathematical PS (Problem Solving) and V (Visualization))Process Focus



Math BackgroundIn this lesson, base ten blocks are used to visualize themultiplication of two-digit numbers. Students will use theblocks to represent the multiplication and then solve theproblem. Allow students time to arrange the base ten blocksinto arrays and then check to be sure that it is the productthey want to represent. Have them count the number ofblocks both down and across to verify the numbers beingmultiplied. Use the blocks to show that four products willresult, and these products must be added to arrive at theanswer. Estimation can be used to check answers.







Write the following expressions on the board: 3 � 2, 3 � 20,3 � 200. Ask for a show of hands to vote for the easiestproblem. Lead a discussion as some students will say they areall equally easy. Explain that numbers can be expanded sothat multiplication is simpler.

Tell students that writing numbers in expanded form willhelp them simplify multiplication calculations by using thedistributive property.

Sample Discourse“How many tens and how many ones are in the number 34?”• There are 3 tens and 4 ones in 34.“What is the value of 3 tens?”• The value of 3 tens is 30.“What is the expanded form of 34?”• 30 � 4“What is the expanded form of 28?”• 20 � 8

Continue with questions about expanded form if studentsneed more practice. Explain to students that using theexpanded form will give them more strategies for multiplying2 two-digit numbers.

Teaching and Learning(Whole Class/Small Groups)

➧ 15–20 min

Discuss with students the information they might want tocollect to put in an address book. Examples could includename, address, home phone, cell phone, e-mail address, etc.With students, come up with 11 in all. If time permits,complete Rebecca’s chart on a transparency or interactivewhiteboard, showing all 11 cells written side by side. Withstudents, read about Rebecca and the central question onStudent Book page 202.

Sample Discourse“How many cells will one entry have?”• 11“How many cells will two entries have? • 22“What multiplication facts could represent the number ofcells in two entries?”• 2 � 11 or 11 � 2

Provide groups with base ten blocks to model Rebecca’sSolution. Discuss Rebecca’s Solution and why she set it up asa rectangle.

Students may also benefit from seeing an additional visual torepresent Rebecca’s method horizontally without the use of baseten blocks. Start by writing 23 � 11 as (20 � 3) � (10 � 1).Relate each step in the symbolic representation to each part ofthe array of base ten blocks.

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47Lesson 9: Multiplying with Base Ten BlocksCopyright © 2008 Nelson Education Ltd.


Students reflect on how two-digit multiplication can bemodelled with base ten blocks.

Answers to Reflecting QuestionsA. For example, this is a shorter way of recording her work.

Rebecca might use an array with 1 row of 23 squares and10 rows of 23 squares since she can use mental math tomultiply 23 times 1 and 23 times 10.

B. For example, modelling an array with four parts allowedRebecca to work with easier numbers to multiply: one-digit numbers or numbers that end in 0.


Checking (Pairs/Small Groups)

Have students refer to Rebecca’s Solution in the StudentBook. Have base ten blocks available and allow students towork in pairs or small groups to model the arrays. Encouragestudents to count across and down the array to ensure thatthe correct multiplication expression is being represented.


4. & 5. Encourage students to read each array as rows andof squares, as in Question 3. For example, Question 4 a)shows 21 rows of 42 squares.

11. Point out that Ami uses 40 � 40 and subtracts, ratherthan adds, partial products. Also point out that, since thefirst number (40) is not expanded, it will not have thefour products students are familiar with seeing; rather, ithas only two products.

Answer to Key Question8. 168 tiles; For example, I can think of 12 rows of 14 tiles

as 12 � 14. I can rewrite this as (10 � 2) � (10 � 4). Ican use arrays to show 10 � 10, 10 � 4, 2 � 10, and 2 � 4. There are 100 � 40 � 20 � 8 � 168 tiles in all.

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8Copyright © 2008 Nelson Education Ltd.48 Chapter 6: Multiplication


Question 13 allows students to reflect on and consolidatetheir learning for this lesson. Point out that students haveseveral strategies for calculating the product. These includethe current lesson strategy as well as rounding up and thenusing repeated subtraction.

Answer to Closing Question 13. For example, Strategy 1: I made an array that is 30 � 9

down and 50 � 9 across and added the four parts (30 � 50 � 30 � 9 � 9 � 50 � 9 � 9).Strategy 2: I made an array that is 40 � 59 and thensubtracted 1 � 59.




• Students create models using base ten blocks to represent the partial productswhen multiplying a two-digit number by a two-digit number.

Key Question 8 (Problem Solving, Visualization)• Students clearly explain the strategy used to multiply a two-digit number by a

two-digit number, with or without the use of a model.

• Students may not calculate all four partial products when multiplying a two-digit number by a two-digit number. (See Extra Support 1.)

• Students may show an incomplete model or partial products that result in anincorrect solution. (See Extra Support 1.)


EXTRA SUPPORT

1. Have students create their model and use the following checklist to helpthem check the model:

• Count the units across the top. Does this match one of the numbers beingmultiplied?

• Count the units going down the side of the model. Does this number matchthe other number being multiplied?

• Does your model form a rectangle?

Next, guide students through calculating the product for the upper left region,the upper right region, the lower left region, and the lower right region. Then, they add all four of these partial products.

EXTRA CHALLENGE

• Have students investigate these products: 19 � 21, 29 � 31, 39 � 41, and so on.Ask them what they notice. (The product is always 1 less than a multiple of 100.)


• Begin by setting up an array for a multiplication fact. Model the array 6 � 3.Encourage students to read this as “six rows of three.” Next, have them count all the blocks for a total of 18.

50 2

30 30 � 50 30 � 2

7 7 � 50 7 � 2

49Lesson 9: Multiplying with Base Ten BlocksCopyright © 2008 Nelson Education Ltd.

1010101010Chapter 6


• Multiply multiples of 10.• Multiply two-digit numbers by one-digit numbers.• Represent multiplication using an array.

SPECIFIC OUTCOME




• Represent both two-digit factors in expandednotation to illustrate the distributive property, e.g., todetermine the partial products of 36 � 42, (30 � 6) � (40 � 2) � 30 � 40 � 30 � 2 � 6 � 40 � 6 � 2 � 1200 � 60 � 240 � 12 � 1512.

• Model the steps for multiplying two-digit factorsusing an array and base ten blocks, and record theprocess symbolically.

• Describe a solution procedure for determining theproduct of two given two-digit factors using apictorial representation, such as an area model.

• Solve a given multiplication problem in context usingpersonal strategies and record the process.

Multiplying withArrays


Masters • 1 cm Grid Paper, Masters Booklet p. 22



Extra Practice Chapter Review Questions 11, 12, & 13Workbook p. 52

Mathematical C (Communication), CN (Connections),Process Focus V (Visualization)



Math BackgroundIn the previous lesson, students multiplied two-digitnumbers by determining four partial products and addingto determine the final product. Rather than using base tenblocks, students will sketch arrays. Students can beencouraged to make the connection from the base tenblocks to the array by identifying and separating the fourpartial products. Have students communicate math byexplaining their array and how it relates to the problem inorder to solidify their understanding. Some students mayprefer this strategy of drawing rectangular arrays,separating them into four unique areas, and adding themtogether to traditional and more formal methods ofmultiplication.

22 Multi-Purpose Masters: 1 cm Grid Paper Copyright © 2008 Nelson Education Ltd.

1 cm Grid Paper, MastersBooklet p. 22



GOALMultiply two-digit numbers using arrays.


Ask students for examples of games or puzzles that might bean array. Examples can include crossword puzzles, wordsearches, and checkerboards. Explain that arrays can bebroken down into smaller arrays, and these smaller arrays canbe used to simplify multiplication.

Review the various strategies for multiplication byexamining a two-digit number multiplied by a one-digitnumber. Remind students that a two-digit number can bewritten in expanded notation for multiplication. Forexample, when multiplying 4 � 83, we can write 4 � (80 � 3) or (4 � 80) � (4 � 3).

Sample Discourse“What is 4 � 80? Explain the strategy you used.”• I calculated 4 � 80 as 4 � 8 tens.

4 � 8 tens � 32 tens� 320

“How can you calculate 4 � 83?”• Because 4 � 80 is 320 and 4 � 3 � 12, 4 � 83 is

320 � 12 or 332.

Repeat with several other expressions if needed.

Teaching and Learning(Whole Class/Pairs) ➧ 15–20 min

Distribute several sheets of grid paper to students to be usedthroughout the lesson. Together with students read thecentral question and work through Brandon’s Strategytogether. Remind students that even the squares that havebeen darkened have to be counted in determining the totalnumber of squares. Discuss with students why Brandon doesnot try to divide the puzzle into equal-sized squares. Studentscan complete Prompts A to C in pairs.

Sample Discourse“Why does Brandon make the top left square of the puzzlethe largest?”• Because it is 10 rows long and 10 columns high, like a base

ten block.• He makes the first block a hundreds block.“Which of the four parts represent the number of tens blocks?”• The top right and the bottom left represent the number of

tens blocks.

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51Lesson 10: Multiplying with ArraysCopyright © 2008 Nelson Education Ltd.

Answer to Key Question5. For example, the model shows an array of 42 rows of 53.

The sides show these numbers broken into parts:40 � 2 and 50 � 3, so the product of length and widthcan be 42 � 53 or (40 � 2) � (50 � 3). The foursections of the rectangle show the four arrays that can beadded to get the product, which are 40 � 50 and 2 � 50 and 40 � 3 and 2 � 3.


Question 6 allows students to reflect on and consolidate theirlearning for this lesson. Discuss the multiplication strategies learnedin previous lessons and how they are used in this current lesson.

Answer to Closing Question 6. For example, Brandon broke up the rectangles into arrays

that either had sides that were single-digit numbers orsides that were multiples of 10. So, you only need tocalculate with multiples of 10 to figure out the amountsin the rectangles with those kinds of dimensions. UsingBrandon’s strategy, I would break 24 � 36 into 20 � 30 � 4 � 30 � 20 � 6 � 4 � 6. Then, I onlyneed to calculate with multiples of 10 and single-digitnumbers.

Follow-Up and Preparation for Next Class

Explain to students that the methods they have learned can beused to multiply any 2 two-digit numbers. To practise at home,have students use a deck of cards with face cards removed.Students pull two cards to represent the first two-digit numberand then select two more cards to represent the second two-digit number. Encourage students to think about whether theywould prefer to estimate or to calculate the answers.

Answers to PromptsA. top left part: 10 rows and 10 columns

top right part: 10 rows and 5 columnsbottom left part: 5 rows and 10 columnsbottom right part: 5 rows and 5 columns

B. top left part: 10 � 10 � 100 squarestop right part: 10 � 5 � 50 squaresbottom left part: 5 � 10 � 50 squaresbottom right part: 5 � 5 � 25 squares

C. 225 squares; 100 � 50 � 50 � 25 � 225


Students reflect that multiplication of 2 two-digit numberscan be broken into the multiplication of tens and ones; thefour partial products that are formed are calculated and thenadded together. This will often make calculations easier.

Answers to Reflecting Questions

D. For example, the sections Brandon created all involveeasier numbers. His puzzle shows 25 � 25 as 20 � 20 � 5 � 20 � 20 � 5 � 5 � 5. These are easynumbers to multiply and add.


Checking (Pairs)

1. Provide students with grid paper so they can sketch a 36 � 36 array. If needed, guide students into firstdividing the array into a 30 � 30 block.


2. One of the oldest Ukrainian traditions called a rushnyk,or an embroidered towel, is referenced. It is consideredan art form shared by many Slavic cultures.

Remind students how they used the base ten blocks to buildarrays in the last lesson. Encourage them to begin sketchingtheir models for these problems and then look at dividing themodels into partial products. Remind students they can usethe halving and doubling strategy or estimation strategies tocheck their answers.

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• Students are able to sketch arrays and divide the sketches into easilycalculated rectangle areas.

• Students sketch the correct array to represent the multiplication expression.

Key Question 5 (Communication, Connections, Visualization) • Students explain how two equations relate to an array.

• Students may not understand how to divide the array into rectangles. (See Extra Support 1.)

• Students arrays do not match the multiplication expression. (See Extra Support 2 or 3.)

• Students may understand only one (or none) of the equations and how they canbe shown to be true. (See Extra Support 1 or 2.)


EXTRA SUPPORT

1. Have students use grid paper where every 10th line is dark. This will helpthem more easily see the sections that they need.

2. Redistribute base ten blocks for students for students to visualize the arraysthey are to draw on grid paper.

3. Guide students by first setting up the base ten blocks for an expression and askstudents to divide the array into parts. After students become proficient individing your arrays, have them set up their own arrays and divide them intopartial products before dividing.

EXTRA CHALLENGE

• Challenge students to use a grid to multiply a three-digit number by a two-digit number. For example, 72 � 548 is shown here. The solution is 39 456.


• For students having difficulties with the concept of partial products, have themform smaller arrays first, in order to better understand the concept. Havestudents use arrays to determine the product of 8 � 5. Many students willalready know the fact is 40. Have them draw an 8-by-5 array on grid paper andthen separate it into four rectangles by drawing one vertical and one horizontalline. Have students write the dimensions of each rectangle they formed anddetermine the area of each rectangle. For example, 8 can be written as (4 � 4)and 5 can be written as (3 � 2). The resulting array can be similar to thediagram below:

Have students write out the multiplication as (4 � 4) � (3 � 2) = (4 � 3) � (4 � 2) � (4 � 3) � (4 � 2) = 12 � 8 � 12 � 8 = 40This will reinforce the concept that any product can be determined by calculating partial products.


• Verbal/linguistic learners may find it helpful to explain sketches that anotherstudent has drawn. Allow students to practise using this method before theyattempt to draw on their own.

500 40 8

70 500 � 70 40 � 70 8 � 70

2 500 � 2 40 � 2 8 � 2

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53Lesson 10: Multiplying with ArraysCopyright © 2008 Nelson Education Ltd.

Scaffolding for Lesson 11, Questions 2, 3, & 4 Page 2STUDENT BOOK PAGE 209

4. Sebastian can walk 47 cm in 2 steps. How far can he walk in 50 steps?

What can you multiply to solve the problem?

�

Are you going to estimate or calculate?

How did you choose this method?

Solve the problem.

75Blackline Masters

Name: Date:


1111111111Chapter 6


• Estimate products of two-digit numbers.• Multiply two-digit numbers.

SPECIFIC OUTCOMES

N2. Use estimation strategies including• front-end rounding• compensation• compatible numbersin problem-solving contexts.[C, CN, ME, PS, R, V]

Achievement Indicators• Provide a context for when estimation is used to

– make predictions – check reasonableness of an answer – determine approximate answers

• Determine the approximate solution to a givenproblem not requiring an exact answer.

• Select and use an estimation strategy for a given problem.




• Represent both two-digit factors in expanded. • Solve a given multiplication problem in context using



Masters • Optional: Scaffolding for Lesson 11, Questions 2, 3, & 4 pp. 74–75

Recommended Questions 2, 3, & 5Practising Questions



Mathematical C (Communication) and PS (Problem Solving)Process Focus



Math BackgroundIn this lesson, students use personal strategies to multiplynumbers. Part of the problem-solving method requiresstudents to determine if an estimate or an exact answer is needed. Students then communicate the solutionstrategies used. Students will see that communication of acomplete solution will help them ensure a correct answeror find an error in the solution process.


Name: Date:


For Questions 2 to 4, decide whether you should estimate or calculate to solve each problem. Then, solve each problem. Show your thinking as completely as possible.

2. How many months old will you be on your 14th birthday?

How many months are in a year?


�


Why did you choose this method?

Solve the problem.

3. A full tour bus holds 48 passengers. There are 15 tour buses that take visitors to the Royal Tyrrell Museum in Drumheller. Will these tour buses be able to take 600 visitors?


�



Solve the problem.


Optional: Scaffolding forLesson 11, Questions 2, 3,& 4 pp. 74–75


GOALExplain your calculation method when solving a problem.

Communicatingabout MultiplicationMethods



Revisit with students some situations that require exactanswers and some that require estimates. Have students writea short scenario where an exact amount is required andanother in which an estimate will suffice. Students shouldprepare to share their answers, as well as their reasoning, withthe class. Discuss some situations before students preparetheir own.

Sample Discourse“Suppose you start a new job. Do you need to know theexact hourly pay rate or an estimate of the hourly pay rate?Explain.” • It should be an exact number, because it is very important to

know exactly how much I am going to be paid before takingthe job.

• I only need to know an estimate. Then, I can figure out abouthow much I can make in a day or a week a lot faster.

“The transportation department is planning to build roads.Do you think the department needs to know the exactnumber of cars in a province or an estimate of the number ofcars? Why?”• It can be an estimate, because the exact number of cars would

be difficult to determine and can change from day to day.

Tell students that, although they may arrive at differentestimates for the same problem, it is important to explain thestrategy used.


Discuss with students the amount of water that might beused for regular activities, such as taking a shower, washingdishes, or bathing a pet. With students, read about Jay’s wateruse and the central question on Student Book page 208.

Sample Discourse“How much water is used in 1 minute to water the lawn?”• 17 L“How much water is used in 2 minutes?”• 34 L

Work through Jay’s Solution and the questions that Mayaasked to improve Jay's Solution. Discuss with students thereasons that Maya asked her particular questions and howanswering those questions will improve the solution.

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55Lesson 11: Communicating about Multiplication MethodsCopyright © 2008 Nelson Education Ltd.

Answers to PromptsA. For example, I need to figure out the number of litres

sprayed by the sprinkler in 22 min. Each minute, thesprinkler sprays about 17 L. I think I should multiply17 L by 22 min because it’s the same number eachminute, so that’s like equal groups. I won’t estimatebecause the question doesn’t ask for an estimate. I’ll showthe multiplication in parts to make the calculationseasier.17 � 22 � 10 � 20 � 7 � 20 � 10 � 2 � 7 � 2

� 200 � 140 � 20 � 14� 374

The sprinkler sprays 374 L in 22 min.


Students reflect on the importance of communicating themethods they use and the reasons behind them when writinga solution.

Answer to Reflecting QuestionB. For example, it is important to communicate clearly so

that others can know why I solved the problem the way Idid, showing all of my steps, in case I made an error, andusing correct math language so that my answer can beunderstood.


Checking (Pairs)

Encourage students to think about how they have useddiagrams in multiplication, for example, sketches of base tenblocks or arrays.


Encourage students to use personal strategies to solve theproblems. Those strategies include deciding whether toestimate or calculate an answer and justifying the methodsused in their explanations. Have students work in pairs, readtheir partner’s answers, and provide feedback regardingcompleteness of answers.2. Explain to students that much of this answer is a written

response. If extra support is required, guide thosestudents and provide copies of Scaffolding pp. 74–75.

Answer to Key Question2. For example, on my birthday, I’ll be 14 years old. Since

each year has 12 months, I am adding 12 months 14 times, or 14 � 12.

To multiply, I’ll make an array that is 14 units one wayand 12 units the other way and figure out how much thearray represents.

The total for 14 � 12� (10 � 10) � (10 � 4) � (2 � 10) � (2 � 4)� 100 � 40 � 20 � 8� 168I’ll be 168 months old on my 14th birthday.


Question 5 allows students to reflect on and consolidate theirlearning for this lesson. Point out the importance of clearlyexplaining a strategy, whether it is estimating or calculatingan exact answer.

Answer to Closing Question 5. For example, a soccer field has 23 bleachers. Each

bleacher has 42 seats. How many people can sit in thebleachers at once?

Solution:I’ll calculate an exact answer in case I need to sell an exactnumber of tickets to fill the bleachers. I can multiply becausethere are 23 bleachers with 42 seats in each.23 � 42 � 23 � 40 � 23 � 2

� 920 � 46� 966

966 people can sit in the bleachers at once.

Follow-Up and Preparation for Next Class

Have students determine the number of kilometres theytravel to and from school. Ask them to calculate their totalschool travel for a week or a month. Next, add all of thestudents' travel distances in a single day and calculate thetotal kilometres travelled in one week for the class. Askstudents to explain their strategies and discuss whether anestimate or an exact answer is appropriate.

Next class is the Chapter Review. Ask students to gothrough Lessons 1 to 11 and note any questions or problemsthey have.

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• Students explain their reasoning, as well as how they arrived at the solutionwhen answering questions.

Key Question 2 (Communication, Problem Solving) • Students explain clearly why they decided to estimate or calculate and clearly

explain how they arrived at their result.

• Students may not know what constitutes a complete explanation. (See ExtraSupport 2.)

• Students may give a numerical answer only. (See Extra Support 2.)

• Students may not be able to justify the reasoning behind their answer. (See Extra Support 1.)


EXTRA SUPPORT

1. Use Scaffolding for Lesson 11, Questions 2, 3, & 4 pp. 74–75. 2. Write a checklist on the board or post it somewhere in the room to remindstudents of the components of a good explanation. The checklist shouldcontain: Why did you solve it this way? How did you solve it? (Show yourwork.) Did you explain using correct mathematical terms?

EXTRA CHALLENGE

• Have students discover a pattern for multiplying two-digit numbers by 11. For example, to find 11 � 23, add the tens and ones digits of the 23. So, 2 � 3 � 5. Place the 5 between the 2 and 3 and the answer to 11 � 23 is 253. Have students solve 11 � 11, 11 � 12, 11 � 13, 11 � 14, etc., to helpthem discover the pattern.

They can also multiply greater numbers that need to be regrouped. For example, 11 � 68 uses 6 � 8 � 14. Because 14 is two digits, place the 4 between 6 and 8, then add 1 to the 6. Then, have them explain the pattern inwords or pictures and discuss why the pattern makes sense to them.


• Some students would benefit from practising their communication skills withsimpler numbers. The focus would be more on explaining their strategy than on deciding to estimate or calculate.


• Some students may benefit from reading their explanations aloud. Oftentimes,an auditory learner will believe the written answer is clear and concise. Butafter reading it aloud, discrepancies in logic may be found.

57Lesson 11: Communicating about Multiplication MethodsCopyright © 2008 Nelson Education Ltd.

Chapter 6Chapter Chapter 6Chapter 6 Chapter 6

Chapter 6: Multiplication Copyright © 2008 Nelson Education Ltd.58


• Recall multiplication facts for 0 to 9.

SPECIFIC OUTCOME


Using Curious MathThis method of multiplication was introduced in Europe in theearly 1200s by a mathematician by the name of Fibonacci. Itseparates the multiplication steps so that all the multiplication isdone first, then the regrouping, and then the addition. It isespecially useful when multiplying numbers with more thantwo digits. Inform students that lattice multiplication will workfor all multiplication problems.

Answers to Curious Math1. a) 2530

b) 1357c) 4225d) 6806

Mathematical C (Connections) and V (Visualization)Process Focus






• Students will place the tens and ones digits correctly in the lattice.

• Students will add the numbers in the diagonals.

• Students may not be clear on how to set up the numbers to be multiplied on the outside of the grid. (See Extra Support 1.)

• Students may not know what to do with regrouping using the lattice method.(See Extra Support 1.)


1. To work on learning lattice multiplication step by step, begin by havingstudents fill in the lattice by themselves, then add the diagonals as a class. Be sure to point out that if a diagonal sums to 10 or greater, they need to regroupby recording the appropriate digit in the next diagonal. Next, have studentscomplete the entire process but stop at various steps to check for accuracy.

EXTRA CHALLENGE

• Have students create a lattice for solving 421 � 568. The lattice should looklike the following.

STUDENT BOOK PAGE 210 Curious MathLattice Multiplication

Chapter Chapter 6Chapter 6Chapter Chapter 6Chapter 6


• Round numbers to the nearest tens.• Multiply 2 two-digit numbers.

SPECIFIC OUTCOME

N2. Use estimation strategies including• front-end rounding • compensation • compatible numbers in problem-solving contexts.[C, CN, ME, PS, R, V]

Using the Math GameIn this game, students estimate the product of two numbers byselecting a range in which the product will result. Any playerwhose estimate is in the correct interval scores a point. Readthrough Rebecca’s Estimate before beginning the game. Be surethat students choose estimates independently of other players sono one relies on someone else’s estimate.

When to PlayPlay can begin after all lessons have been completed.

Number of Players 2 to 4

Materials • dice• counters

Mathematical ME (Estimation)Process Focus





• Students estimate products of two-digit numbers. • Students may be unsure how to estimate when both numbers are two-digit. (See Extra Support 1.)


1. Students can play variations of this game by multiplying a two-digit by a one-digit number. In this case, change the intervals to 0–99, 100–199,200–299, and 300–399.

EXTRA CHALLENGE

• Challenge students to use intervals of 100 or 500 (e.g., 0–99, or 0–499) whenmaking their estimates, rather than the intervals of 1000 that are given.

• Challenge students to use a spinner numbered from 0 to 9 to generate their two-digit numbers, and increase the number of intervals that can be used, upto and including 9000–9999. Students can re-spin if they land on a zero.

StrategiesRemind students to think about whether they are roundingboth numbers up or both numbers down to get an estimate.A closer estimate might be obtained by rounding onenumber up and one number down.

59Math Game: Rolling ProductsCopyright © 2008 Nelson Education Ltd.

STUDENT BOOK PAGE 211 Math GameRolling Products

79Blackline Masters

Name: Date:


Chapter 6 Test Page 3

10. How can you use base ten blocks to show that 22 � 17 is greater than 24 � 15?

11. What multiplication does this array represent? Calculate the product.

12. Sketch an array to model each multiplication. Calculate each product.

a) 32 � 47 b) 58 � 23

13. Calculate.

a) 22 � 66 b) 43 � 91

14. Describe two strategies for calculating 48 � 25.

60 7

40

8


6. One boat can hold 15 passengers to ferry them across a lake. About how many passengers can 21 boats hold? Describe your estimation strategy.

7. About 14 cars pass over a small bridge every 6 hours. About how many cars pass over the bridge in a week?

8. Calculate.

a) 22 � 17 b) 15 � 15

9. Josie’s heart beats 73 times in a minute.

a) How many times will her heart beat in 12 min?

b) How many times will her heart beat in 35 min?

c) How many times will her heart beat in 41 min?


Name: Date:


SPECIFIC OUTCOMES


N3. Apply mental mathematics strategies and numberproperties, such as• skip counting from a known fact • using doubling or halving • using patterns in the 9s facts • using repeated doubling or halving to determine answers for basic multiplication facts to 81and related division facts.[C, CN, ME, R, V]

N4. Apply mental mathematics strategies formultiplication, such as• annexing then adding zero • halving and doubling • using the distributive property.[C, ME, R]



Name: Date:

Chapter Review—Frequently Asked QuestionsSTUDENT BOOK PAGE 212

Q: How can you multiply when one of the numbers is close to a ten?

A:

Q: How can you estimate a product of 2 two-digit numbers?

A:

Q: How can you multiply 2 two-digit numbers?

A:


Chapter Review—Frequently AskedQuestions p. 76

22 Multi-Purpose Masters: 1 cm Grid Paper Copyright © 2008 Nelson Education Ltd.

Optional: 1 cm Grid Paper,Masters Booklet p. 22

77Blackline Masters

Name: Date:


1. List three multiplication facts that you can use to help you calculate 5 � 8. How can you use each fact?

2. Joe fills a box with six muffins.

a) How many muffins does he need to fill eight boxes?

b) How would the number change if he fills nine boxes?

3. How can you use base ten blocks to show that 30 � 40 � 1200? Sketch your model.

4. Use a sketch to show that 18 � 50 � 9 � 100.

5. How much greater is 4 � 96 than 4 � 90? How do you know?


Chapter 6 Test pp. 77–79

60


Chapter 6: Multiplication Copyright © 2008 Nelson Education Ltd.

Materials • Optional: base ten blocks

Masters • Chapter Review—Frequently Asked Questions p. 76

• Chapter 6 Test pp. 77–79• Optional: 1 cm Grid Paper, Masters Booklet p. 22• Manipulatives Substitute: Base Ten Blocks,

Masters Booklet pp. 40–42

Extra Practice Workbook p. 54

Nelson Website Visit www.nelson.com/mathfocus and followthe links to Nelson Math Focus 5, Chapter 6.


Chapter Review STUDENT BOOK PAGES 212–214

Practice (Individual)

Most students will be able to complete Questions 1 to 6 inclass. Assign the rest for homework. 5. Remind students to look at the difference in the number

of groups of 7 to determine how much greater oneexpression is than the other.

What Do You Think Now? (Individual/Whole Class)

Revisit the What Do You Think? guide for Chapter 6 onStudent Book page 175. Have students look back to see if theirdecisions and/or explanations have changed. As well, ifstudents recorded their initial thoughts about one of thechapter goals, they can complete the same prompt again, thencompare their ideas and reflect on what they have learned.

Using the Chapter ReviewUse these pages to consolidate and assess students’understanding of the concepts developed in the chapter. ThePractice questions can be used for assessment of learning.Refer to the Assessment of Learning chart on pages 62–64for the details of each question.

Alternatively, use the Practice questions as a practice test,and then administer Chapter 6 Test pp. 77–79. The scoringguides and rubrics provided for the Practice questions canalso be used for the test questions: each question on the testcorresponds to the Practice question of the same number.

Frequently Asked Questions(Individual/Groups)

Have students read the Frequently Asked Questions (FAQs)on Student Book page 212 and create a new example foreach question in their own notes. Then, have studentssummarize the answers to the FAQs in their own words, as away of reflecting on the concepts.

Alternatively, have students complete Chapter Review—Frequently Asked Questions p. 76 with their StudentBooks closed. Discuss students' answers, and then comparethese answers with those in the Student Book. Students canrefer to the answers to the FAQs as they work through thePractice questions.

61Chapter ReviewCopyright © 2008 Nelson Education Ltd.

Assessment of Learning—What to look for in student workQuestion 1, written answer Specific Outcome and Process Focus: N3 [CN]• List three multiplication facts that you can use to help you calculate 6 � 7. How can you use each fact?





• makes insightful connections betweenmultiplication facts and 6 � 7

• makes meaningful connections between multiplication facts and 6 � 7

• makes simple connections betweenmultiplication facts and 6 � 7

• makes minimal or weak connectionsbetween multiplication facts and 6 � 7

Question 2, short answer Specific Outcome and Process Focus: N3 [ME, R]• Mireille made bracelets for her friends. She put nine beads on each bracelet.

a) How many beads did she use for seven bracelets?b) If Mereille had put only eight beads on each bracelet, how many beads would she have used?

(Score 1 point for each correct answer, for a total out of 2.)

Question 3, model, drawing Specific Outcome and Process Focus: N3 [ME, V]• How can you use base ten blocks to show that 20 � 50 � 1000? Sketch your model.





• uses visual representationsinsightfully to demonstrate a thorough understanding that 20 � 50 � 1000

• uses visual representationsmeaningfully to demonstrate areasonable understanding that 20 � 50 � 1000

• uses visual representations simply todemonstrate a basic understandingthat 20 � 50 � 1000

• uses visual representations poorlyto demonstrate an incompleteunderstanding that 20 � 50 � 1000

Question 4, drawing Specific Outcome and Process Focus: N4 [ME, V]• Use a sketch to show that 12 � 50 � 6 � 100.





• demonstrates computational fluency(i.e., applying halving and doubling tomultiply) that is efficient and flexible

• demonstrates computational fluency (i.e.,applying halving and doubling to multiply)that is workable and understood

• demonstrates computational fluency(i.e., applying halving and doubling tomultiply) that is routine and familiar

• has difficulty demonstratingcomputational fluency and must work through procedures

• uses visual representationsinsightfully to demonstrate athorough understanding that12 � 50 � 6 � 100

• uses visual representationsmeaningfully to demonstrate areasonable understanding that 12 � 50 � 6 � 100

• uses visual representations simply todemonstrate a basic understandingthat 12 � 50 � 6 � 100

• uses visual representations poorly demonstrate an incompleteunderstanding that 12 � 50 � 6 � 100

Question 5, written answer Specific Outcome and Process Focus: N4 [ME]• How much greater is 8 � 103 than 8 � 100? How do you know?





• demonstrates computational fluency(i.e., using the distributive property tomultiply) that is efficient and flexible

• demonstrates computational fluency(i.e., using the distributive property to multiply) that is workable andunderstood

• demonstrates computational fluency(i.e., using the distributive property to multiply) that is routine andfamiliar

• has difficulty demonstratingcomputational fluency and must work through procedures


Assessment of Learning—What to look for in student workQuestion 6, written answer Specific Outcome and Process Focus: N2 [PS]One bus holds 48 students. About how many students will 16 buses hold? Describe your estimation strategy.





• chooses an efficient and effectivestrategy to estimate how manystudents will fit in all 16 buses

• chooses an appropriate and workablestrategy to estimate how manystudents will fit in all 16 buses

• chooses a simplistic and/or routinestrategy to estimate how manystudents will fit in all 16 buses

• chooses an inappropriate and/orunworkable strategy to estimate how many students will fit in all 16 buses

Question 7, short answer Specific Outcome and Process Focus: N5 [PS]• About 56 babies are born every 12 h in Alberta. About how many babies are born in a week?

(Score 1 point for an appropriate strategy and 1 point for correct answer.)

Question 8, short answer Specific Outcome and Process Focus: N5 [CN]• Calculate.

a) 18 � 19b) 24 � 24


Question 9, short answer Specific Outcome and Process Focus: N5 [PS]• Annie types 37 words a minute.

a) How many words can she type in 15 min?b) How many words can she type in 25 min?c) How many words can she type in 37 min?


Question 10, model, written answer Specific Outcome and Process Focus: N5 [CN, V]• How can you use base ten blocks to show that 35 � 47 is less than 37 � 45?





• demonstrates a sophisticated abilityto transfer knowledge and skills (i.e., modelling the multiplication of 2-digit factors using base tenmodel) to new contexts (i.e.,comparing products)

• demonstrates a consistent ability totransfer knowledge and skills (i.e., modelling the multiplication of 2-digit factors using base ten model) to new contexts (i.e., comparingproducts)

• demonstrates some ability to transfer knowledge and skills (i.e., modelling the multiplication of 2-digit factors using base tenmodel) to new contexts (i.e.,comparing products)

• demonstrates a limited ability totransfer knowledge and skills (i.e., modelling the multiplication of 2-digit factors using base ten model)to new contexts (i.e., comparingproducts)

• uses visual representations (i.e., base ten model) insightfully to demonstrate a thoroughunderstanding of multiplication of 2-digit factors

• uses visual representations (i.e., base ten model) meaningfully to demonstrate a reasonableunderstanding of multiplication of 2-digit factors

• uses visual representations (i.e., base ten model) simply todemonstrate a basic understanding of multiplication of 2-digit factors

• uses visual representations (i.e., base ten model) poorly todemonstrate an incompleteunderstanding of multiplication of 2-digit factors

63Chapter ReviewCopyright © 2008 Nelson Education Ltd.

Chapter 6Assessment of Learning—What to look for in student workQuestion 11, short answer Specific Outcome and Process Focus: N5 [V]• What multiplication does this array represent? Calculate the product.

(Score 1 point for correct multiplication, 1 point for correct product, for a total out of 2.)

Question 12, drawing, short answer Specific Outcome and Process Focus: N5 [CN, V]• Sketch an array to model each multiplication. Calculate the product.

a) 21 � 53 b) 47 � 48

(Score 1 point for each correct model, 1 point for each correct product, for a total out of 4.)

Question 13, short answer Specific Outcome and Process Focus: N5 [CN]• Calculate.

a) 35 � 48 b) 62 � 53


Question 14, written answer Specific Outcome and Process Focus: N4, N5 [C]• Describe two strategies for calculating 32 � 25.





• provides a precise and insightfuldescription of two strategies forcalculating 32 � 25

• provides a clear and logical descriptionof two strategies for calculating 32 � 25

• provides a partially clear descriptionof two strategies for calculating 32 � 25

• provides a vague and/or inaccuratedescription of two strategies forcalculating 32 � 25

• uses effective and specificmathematical language andconventions to enhance communication

• uses appropriate and correctmathematical language andconventions to support communication

• uses mathematical language andconventions to partially supportcommunication

• uses mathematical language andconventions incorrectly and/orinconsistently


81Blackline Masters

Name: Date:


Chapter 6 Task Page 2

D. How many hits do you think there might have been during October and during December? Explain.

E. When do you think the blog reached 2000 hits? Explain your reasoning.


SPECIFIC OUTCOMES


N4. Apply mental mathematics strategies formultiplication, such as• annexing then adding zero• halving and doubling• using the distributive property. [C, ME, R]


Achievement Indicator• Solve a given multiplication problem in context using


Using the Chapter TaskUse this task as an opportunity to assess students’understanding of the concepts developed in the chapter andtheir ability to apply them in a rich problem-solvingsituation. Refer to the assessment chart on page 67 for thedetails of each part of the task.


Have several students read each piece of information aboutthe number of hits out loud. Discuss the meaning of eachfact and note how students can misinterpret them. Forexample, October mentions 6 hits each night for 7 nightsonly. November mentions 200 hits for the whole month.Point out to students that the facts cannot be compared as is,because they refer to different amounts over a differentnumber of days.

Pacing 5–10 min Introduction35–50 min Using the Task

Masters • Optional: Chapter 6 Task pp. 80–81



Chapter 6 Task Page 1STUDENT BOOK PAGE 215

Homework Blogs

Maya and Jay posted a homework blog and kept track of its use during certain periods of time.

? On what date do you think the blog reached 2000 hits?

A. How many hits were there, in total, from October 1 to October 7?

B. How do you know that there were some nights in November with more hits than on October 2?

C. How many hits were there in January?


Name: Date:

October 1–7:

6 hits each night November 30:200 hits for the month

December 8:70 hits last weekJanuary 31:

18 hits each night

in January

✔ Did you explainyour thinking?

✔ Did you check yourcalculations?

✔ Did you estimatewhen appropriate?

✔ Did you use goodmathematicalreasoning?

Task Checklist


Optional: Chapter 6 Taskpp. 80–81

65Chapter Task: Homework BlogsCopyright © 2008 Nelson Education Ltd.

Chapter TaskHomework Blogs

STUDENT BOOK PAGES 215

Using the Task (Whole Class/Individual) ➧ 35–50 min

Together, read all the information on Student Bookpage 215, including the central question.

Students should work through the task independently.Remind students to use the Task Checklist as a way to helpthem produce an excellent solution. Some students may beable to work through the task as it is described on thestudent page; however, most will benefit from usingChapter 6 Task pp. 80–81 to plan and record work.

As students work through the task, observe and/orinterview individuals to see how they are interpreting andcarrying out the task.

Possible Solution to Chapter TaskA. 42, since 6 � 7 � 42B. November has 30 days and 6 hits � 30 days is only

180 hits, so some nights must have had more than 6 hits. C. 558 hits;

18 � 31 � (10 � 30) � (10 � 1) � (8 � 30) � (8 � 1)� 300 � 10 � 240 � 8� 558

D. If the number of hits each night in October was 6, therewould be 6 � 31 � 186 hits in October.If the number of hits each week in December was 70,that would be 10 a night, so there would be 10 � 31 � 310 hits in December.

E. About March 14; If there were 186 hits in October, 200 hits in November, 310 hits in December, and 558 hits in January, the total number of hits would be 558 � 186 � 200 � 310 � 1254.The site would need 746 more hits to reach 2000because 2000 – 1254 � 746.If the number of hits in February was the same asJanuary, that would be another 18 � 28 hits, which is 504 hits at the end of February. Since 746 – 504 � 242,there would need about another 14 days to reach 2000hits at 18 hits per day, because 18 � 14 � 252 hits,which is close to 242. So, around March 14, the site willreach 2000 hits.

Adapting the TaskYou can adapt the task in the Student Book to suit the needsof your students. For example:• Use Chapter 6 Task pp. 80–81.• As a class, work through each post to find the number of

hits each night, each week, and each month. Allowstudents to use this information to work through theprompts.

• Have students come up with an estimate for the entireyear. Point out that the information in the task gives thehits for 4 months out of the year. Tell students to use thisinformation to estimate the yearly total.


• provides a vague and/orinaccurate explanation ofthinking

• provides a partially clearexplanation of thinking

• provides a clear and logicalexplanation of thinking

• provides a precise andinsightful explanation ofthinking

Prompts A to EN2. Use estimation strategiesincluding• front-end rounding• compensation• compatible numbersin problem-solving contexts.

[C, CN, ME, PS, R, V]

Work does not yet meetacceptable standard

N4. Apply mental mathematicsstrategies for multiplication, such as• annexing then adding zero• halving and doubling• using the distributive property.

[C, ME, R]

N5. Demonstrate an understanding of multiplication (two-digit by two-digit) to solve problems.

[C, CN, PS, V]

Assessment of Learning—What to look for in student work

OutcomesWork meets standard of excellence



• demonstratescomputational fluency that is efficient andflexible

• demonstratescomputational fluency that is workable andunderstood

• demonstratescomputational fluency that is routine and familiar

• has difficulty indemonstratingcomputational fluency and must work throughprocedures

• demonstrates an insightful understanding of the problem

• differentiates betweenrelevant and irrelevantinformation

• demonstrates a completeunderstanding of theproblem

• identifies relevantinformation

• demonstrates a basicunderstanding of theproblem

• identifies some relevantinformation

• demonstrates a limitedunderstanding of theproblem

• has difficulty discerningrelevant from irrelevantinformation

• chooses an efficient andeffective strategy toestimate a solution

• chooses a workable andreasonable strategy toestimate a solution

• chooses a familiar strategy to estimate asolution, even though itmight not be the mostappropriate

• chooses a random orinappropriate strategy toestimate a solution

• shows flexibility andinsight when solving theproblem, adapting ifnecessary

• shows thoughtfulness whensolving the problem

• shows understanding whensolving the problem

• attempts to solve theproblem

67Chapter Task: Homework BlogsCopyright © 2008 Nelson Education Ltd.


Chapter Chapter Chapter 76

Family Letter

Dear Parent/Caregiver:

Over the next four weeks, your child will be learning multiplication strategies. He orshe will also learn to estimate the products of one-digit and two-digit numbers.Models will be used to represent multiplication, and your child will be able to usevarious strategies to solve multiplication problems.

To reinforce the concepts your child is learning at school, you and your child can workon some at-home activities such as these:

• Play a game with two people and a deck of cards (remove the face cards). Eachplayer takes a card and places it face up. The first player to state the multiplicationfact wins a point. You may choose to count to three before answering to allow yourchild a fairer game.

• Find items to multiply in daily life. For example, if a snack costs $2, determine thetotal price if a snack was purchased for several friends.

• Discuss when estimates are appropriate and when they are not. Is it necessary toknow the exact price of your grocery items before you reach the register, or is anestimate sufficient? Would you rather your estimate be too high or too low whenbuying groceries?

You may want to visit the Nelson website at www.nelson.com/mathfocus for moresuggestions to help your child learn mathematics and develop a positive attitudetoward learning mathematics. As well, you can check the Nelson website for links to other websites that provide online tutorials, math problems, brainteasers, and challenges.

Scaffolding for Getting StartedSTUDENT BOOK PAGE 174

Making Dream Catchers

Maya and her grandmother are making eight dream catchers as gifts. They need 65 cm of willow to make each dream catcher.

? How much willow do Maya and her grandmother need to make eight dream catchers?

A. How much willow do Maya and her grandmother need to make two dream catchers?

They need 65 cm � 65 cm � cm of willow.

This is the same as 2 � cm � cm.

B. How do you know that they need more than 400 cm of willow to make eight dream catchers?

Two dream catchers will use more than 100 cm of willow.

Four dream catchers will use more than cm of willow.

Eight dream catchers will use more than cm of willow.

C. How can you use base ten blocks to show the amount of willow they need to make eight dream catchers?

How many sets of 65 base ten blocks do you need?

Will you be able to regroup any of the blocks? Explain.

D. How can you solve the problem without using base ten blocks?

I can calculate �

E. How much willow do Maya and her grandmother need?

69Blackline Masters

Name: Date:



Name: Date:


11. a) How much more is 6 � 9 than 3 � 9? How do you know?

Use these steps.





b) How much more is 7 � 7 than 5 � 7? How do you know?

Use these steps.







4. Calculate each product using the half/double strategy. Look for numbers that can be halved or doubled to make 10, 100, or 1000.

a) 5 � 12


Double the 5 and halve the 12: �

5 � 12 �

b) 9 � 200

Which factor of 9 � 200 can be halved to 100?


9 � 200 �

c) 500 � 14



500 � 14 �

d) 50 � 24


50 � 24 �

e) 200 � 18 � 100 �

200 � 18 �

f) 18 � 500 � �

18 � 500 �

71Blackline Masters

Name: Date:



Name: Date:

Mid-Chapter Review—Frequently Asked QuestionsSTUDENT BOOK PAGE 192

Q: What strategies can you use to multiply one-digit numbers?

A:

Q: How can you multiply by multiples of tens, hundreds, or thousands?

A:

Q: How can you simplify a calculation using the half/double strategy?

A:


73Blackline Masters

Name: Date:



8. A class of 36 students is having a bridge-building contest. Each group of 4 students has 35 straws to make a bridge. The straws come in bagsof 50. Calculate the number of bags needed for the class by following the steps below.

How many groups of 4 are in 36? 36 � 4 � groups

How many straws does each group have? straws

How can you estimate the total number of straws needed for all groups?

�

How many bags of straws does the class need? Explain.


Name: Date:


For Questions 2 to 4, decide whether you should estimate or calculate to solve each problem. Then, solve each problem. Show your thinking as completely as possible.

2. How many months old will you be on your 14th birthday?

How many months are in a year?


�



Solve the problem.

3. A full tour bus holds 48 passengers. There are 15 tour buses that take visitors to the Royal Tyrrell Museum in Drumheller. Will these tour buses be able to take 600 visitors?


�



Solve the problem.



4. Sebastian can walk 47 cm in 2 steps. How far can he walk in 50 steps?


�


How did you choose this method?

Solve the problem.

75Blackline Masters

Name: Date:



Name: Date:

Chapter Review—Frequently Asked QuestionsSTUDENT BOOK PAGE 212

Q: How can you multiply when one of the numbers is close to a ten?

A:

Q: How can you estimate a product of 2 two-digit numbers?

A:

Q: How can you multiply 2 two-digit numbers?

A:


77Blackline Masters

Name: Date:


1. List three multiplication facts that you can use to help you calculate 5 � 8. How can you use each fact?

2. Joe fills a box with six muffins.

a) How many muffins does he need to fill eight boxes?

b) How would the number change if he fills nine boxes?

3. How can you use base ten blocks to show that 30 � 40 � 1200? Sketch your model.

4. Use a sketch to show that 18 � 50 � 9 � 100.

5. How much greater is 4 � 96 than 4 � 90? How do you know?



6. One boat can hold 15 passengers to ferry them across a lake. About how many passengers can 21 boats hold? Describe your estimation strategy.

7. About 14 cars pass over a small bridge every 6 hours. About how many cars pass over the bridge in a week?

8. Calculate.

a) 22 � 17 b) 15 � 15

9. Josie’s heart beats 73 times in a minute.

a) How many times will her heart beat in 12 min?

b) How many times will her heart beat in 35 min?

c) How many times will her heart beat in 41 min?


Name: Date:


79Blackline Masters

Name: Date:



10. How can you use base ten blocks to show that 22 � 17 is greater than 24 � 15?

11. What multiplication does this array represent? Calculate the product.

12. Sketch an array to model each multiplication. Calculate each product.

a) 32 � 47 b) 58 � 23

13. Calculate.

a) 22 � 66 b) 43 � 91

14. Describe two strategies for calculating 48 � 25.

60 7

40

8

Chapter 6 Task Page 1STUDENT BOOK PAGE 215

Homework Blogs

Maya and Jay posted a homework blog and kept track of its use during certain periods of time.

? On what date do you think the blog reached 2000 hits?

A. How many hits were there, in total, from October 1 to October 7?

B. How do you know that there were some nights in November with more hits than on October 2?

C. How many hits were there in January?


Name: Date:

October 1–7:

6 hits each night November 30:200 hits for the month

December 8:70 hits last weekJanuary 31:

18 hits each night

in January

✔ Did you explainyour thinking?

✔ Did you check yourcalculations?

✔ Did you estimatewhen appropriate?

✔ Did you use goodmathematicalreasoning?

Task Checklist


81Blackline Masters

Name: Date:


Chapter 6 Task Page 2

D. How many hits do you think there might have been during October and during December? Explain.

E. When do you think the blog reached 2000 hits? Explain your reasoning.


Answers for Chapter 6 MastersScaffolding for Getting Started p. 69

A. 130 cm; 65 cm; 130 cm

B. 200 cm; 400 cm

C. 8; Yes, I would trade every group of 10 ones blocks for one tens block and every group of 10 tens blocks for 1 hundreds block.

D. 8 � 65

E. For example, they need about 520 cm of willow. 10 � 65 � 650 8 � 65 � 650 � 130 � 520 cm

Scaffolding for Lesson 1, Question 11 p. 70

11. a) 6; 3; 3; 3 � 9 � 27

b) 5; 7; 2; 2 � 7 � 14


4. a) 5; 10 � 6; 60

b) 200; double the 9 and halve the 200: 18 � 100; 1800

c) 500; double the 500 and halve the 14: 1000 � 7; 7000

d) double the 50 and halve the 24: 100 � 12; 120

e) 200 � 18 � 100 � 36; 3600

f) 18 � 500 � 9 � 1000; 9000


8. 9 groups; 35 straws; 10 � 35; For example, because 10 groups of 4 is 40, I know that 9 groups of 4 is 40 � 4 � 36; 9 groups can be rounded to 10; each group needs 35 straws; I can use the estimate of 10 � 35 � 350; Because there are 50 straws in each bag, and 7 � 5 � 35, I need about 7 bags of straws.

Scaffolding for Lesson 11, Questions 2, 3, & 4 pp. 74–75

2. 12 months; 12 � 14; For example, I choose calculation, because the numbers in the problem are exact, not estimates, and it is not difficult to calculate 12 � 14; halving and doubling will not make this problem simpler, so I will use 4 products to find the solution. 10 � 10 � 10 � 4 � 2 � 10 � 2 � 4 � 100 � 40 � 20 � 8 � 168

3. 15 � 48; For example, I choose estimation, because I only need to know if the buses will take 600 visitors. I don’t need an exact number. I will round 48 up to 50 and use 14 as an estimate instead of 15 to make up for the higher number. I can use halving and doubling: 14 � 50 � 7 � 100 � 700. This means the buses would be able to take 600 visitors.

4. 47 � 25; For example, I don’t think that a person takes steps that are exactly 47 cm, so I can use 50 cm to estimate. If I take 50 cm in 2 steps, then I take about 25 cm in 1 step. I need to find 50 � 25 and I can multiply 5 � 25 and then put a zero at the end. So, 5 � 25 is 125 which means 50 � 25 � 1250. I think it would be about 1250 cm.


83Blackline MastersCopyright © 2008 Nelson Education Ltd.

Chapter 6 Test pp. 77–79

1. For example, I can double 5 � 8 to get 10 � 8 � 80, and then take half of the answer. So, 80 � 2 � 40. I can add 5 to 5 � 7 to get 35 � 5 � 40. I can subtract 5from 5 � 9 to get 45 � 5 � 40.

2. a) 48

b) For example, it would add one more group of 6. So, 48 � 6 � 54 muffins.

3. 30 � 40 � 1200

4. For example,

5. 24 more, because there are 6 more groups of 4 in 4 � 96 than in 4 � 90

6. For example, about 300 people; 15 � 21 is about 15 � 20, or 15 � 2 tens. That’s 30 tens or 300.

7. For example, about 700 cars; there are 24 hours in one day and there are 4 groups of 6 in 24.The number of cars that pass over the bridge in 1 day is 4 � 24. I can estimate using 4 � 25 � 100. Because there are 7 days in one week, I can estimate that about 7 � 100 � 700 cars pass over the bridge in one week.

8. a) 200 � 20 � 140 � 14 � 374

b) 100 � 50 � 50 � 25 � 225

9. a) 876 heartbeats; 73 � 12 � (70 � 10) � (70 � 2) � (3 � 10) � (3 � 2)

� 700 � 140 � 30 � 6 � 876

b) 2555 heartbeats; 73 � 35 � (70 � 30) � (70 � 5) � (3 � 30) � (3 � 5)

� 2100 � 350 � 90 � 15 � 2555

c) 2993 heartbeats; 41 is 6 more minutes than 35 so 73 � 6 � 70 � 6 + 3 � 6 � 420 + 18 � 438 more minutes. 2555 � 438 � 2993

10. For example, I can use base hundreds blocks to show that both 22 � 17 and 24 � 15 have 2 hundreds blocks since 20 � 10 � 200. I can use tens blocks to show that 22 � 17 has 14 � 2 or 16 tens blocks. 24 � 15 has 10 � 4 or 14 tens blocks. So, 22 � 17 has 2 more tens than 25 � 15. I can use ones blocks to show that 22 � 17 has 2 � 7 or 14 ones blocks and 24 � 15 has 4 � 5 or 20 ones blocks. So, the extra ones blocks in 24 � 15 don't make up the difference for the extra tens in 22 � 17.

11. 67 � 48 � (60 � 40) � (60 � 8) � (7 � 40) � (7 � 8)

� 2400 � 480 � 280 � 56

� 3216

12. a) 32 � 47 � 30 � 40 � 30 � 7 � 2 � 40 � 2 � 7 � 1200 � 210 � 80 � 14 � 1504

b) 58 � 23 � 50 � 20 � 50 � 3 � 8 � 20 � 8 � 3 � 1000 � 150 � 160 � 24 � 1334

13. a) 1452

b) 3913

14. For example, one way to calculate 48 � 25 is to use half/double twice.48 groups of 25 is the same as 24 groups of 50.24 groups of 50 is the same as 12 groups of 100.12 � 100 � 1200.I can also find four products using an array strategy.So, 48 � 25 � (40 � 20) � (40 � 5) � (8 � 20) � (8 � 5)

� 800 � 200 � 160 � 40� 1200


40 7

30

2

320

50

8


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