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Chapter Chapter 3Chapter 3

Contents 1Copyright © 2010 Nelson Education Ltd.

Contents

OVERVIEW

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Curriculum across Grades 5 to 7: Number . . . . . . . . . . . . 2Math Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Planning for Instruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

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

Chapter 3 Planning Chart . . . . . . . . . . . . . . . . . . . . . . . . . 4Chapter 3 Assessment Summary . . . . . . . . . . . . . . . . . . . . 6

TEACHING NOTES

Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Getting Started: Banner Design . . . . . . . . . . . . . . . . . . . . . 9Lesson 1: Identifying Factors . . . . . . . . . . . . . . . . . . . . . . 13Lesson 2: Identifying Multiples . . . . . . . . . . . . . . . . . . . . 18Curious Math: String Art . . . . . . . . . . . . . . . . . . . . . . . . . 22Lesson 3: Prime and Composite Numbers . . . . . . . . . . . . 24Math Game: Colouring Factors . . . . . . . . . . . . . . . . . . . . 29Lesson 4: Identifying Factors by Dividing . . . . . . . . . . . . 31Lesson 5: Creating Composite Numbers . . . . . . . . . . . . . 35Mid-Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Lesson 6: Solving Problems Using an Organized List . . . . . 43Lesson 7: Representing Integers . . . . . . . . . . . . . . . . . . . . 47Curious Math: Countdown Clock . . . . . . . . . . . . . . . . . . 51Lesson 8: Comparing and Ordering Integers . . . . . . . . . . 53Lesson 9: Order of Operations . . . . . . . . . . . . . . . . . . . . . 58Math Game: Four in a Row . . . . . . . . . . . . . . . . . . . . . . . 62

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Chapter Task: A Block Dropping Game . . . . . . . . . . . . . 69Chapters 1–3 Cumulative Review . . . . . . . . . . . . . . . . . . 72

CHAPTER 3 BLACKLINE MASTERS

Family Letter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Scaffolding for Getting Started . . . . . . . . . . . . . . . . . 75–76Scaffolding for Lesson 2, Question 3 . . . . . . . . . . . . . . . . 77String Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Mid-Chapter Review—Frequently Asked Questions . . . . 79Four in a Row Game Board . . . . . . . . . . . . . . . . . . . . . . . 80Calculation Cards . . . . . . . . . . . . . . . . . . . . . . . . . . . 81–82Chapter Review—Frequently Asked Questions . . . . . . . . 83Chapter 3 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84–86Chapter 3 Task: A Block Dropping Game . . . . . . . . . 87–88Answers for Chapter 3 Masters . . . . . . . . . . . . . . . . . 89–91From Masters BookletReview of Essential Skills: Chapter 3 . . . . . . . . . . . . . . . . . 51 cm Grid Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 cm Grid Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23100 Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Number Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Initial Assessment Summary . . . . . . . . . . . . . . . . . . . . . . 57Assessment Rubrics for Mathematical Processes . . . . . 58–61Chapter Checklist: Chapter 3 . . . . . . . . . . . . . . . . . . . . . 64Self-Assessment: Chapter 3 Lesson Goals . . . . . . . . . . . . . 75Self-Assessment: Mathematical Processes . . . . . . . . . . . . . 84Self-Assessment: What I Like . . . . . . . . . . . . . . . . . . . . . . 85Self-Assessment: How I Learn . . . . . . . . . . . . . . . . . . . . . 85

IntroductionThis chapter provides students with opportunities to usetheir understanding of number relationships to identifyfactors and multiples, to determine whether a number isprime or composite, to compare and order integers, and touse the rules for order of operations to calculate the value of an expression. They will build upon the mental mathematicsstrategies developed in Grade 5 to determine factors andmultiples.

Throughout the chapter, students use concrete andpictorial models to help develop an understanding of newconcepts before attempting to use mental mathematicsstrategies.

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.

Number Relationships

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Copyright © 2010 Nelson Education Ltd.2 Chapter 3: Number Relationships

Curriculum across Grades 5 to 7: Number The Grade 6 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 5 Grade 6 Grade 7

Strand: NumberGeneral Outcome: Develop number sense.

Specific OutcomeN3. 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 factsto 81 and related division facts.[C, CN, ME, R, V]

Specific Outcomes N3. Demonstrate an understanding of factors and multiples by

• determining multiples and factors of numbers less than 100• identifying prime and composite numbers• solving problems involving multiples. (1, 2, CM1, 3, MG1,

4, 5, 6)[PS, R, V]Achievement Indicators• Identify multiples for a given number and explain the

strategy used to identify them. (2, CM1, 6)• Determine all the whole-number factors of a given

number using arrays. (1, MG1)• Identify the factors for a given number and explain the

strategy used, e.g., concrete or visual representations,repeated division by prime numbers, or factor trees. (1, MG1, 4, 5, 6)

• Provide an example of a prime number and explain why itis a prime number. (3, 4, 5)

• Provide an example of a composite number and explainwhy it is a composite number. (3, 4, 5)

• Sort a given set of numbers as prime and composite. (3)• Solve a given problem involving factors or multiples.

(1, 2, CM1, 3, 6)• Explain why 0 and 1 are neither prime nor composite. (3)

N7. Demonstrate an understanding of integers, concretely,pictorially, and symbolically. (7, CM2, 8)[C, CN, R, V]Achievement Indicators• Extend a given number line by adding numbers less than

zero and explain the pattern on each side of zero. (7, CM2)• Place given integers on a number line and explain how

integers are ordered. (8)• Describe contexts in which integers are used, e.g., on a

thermometer. (7, CM2)• Compare two integers; represent their relationship using

the symbols <, >, and =, and verify using a number line. (8)• Order given integers in ascending or descending order. (8)

N9. Explain and apply the order of operations, excludingexponents, with and without technology (limited to wholenumbers). (9 MG2)[CN, ME, PS, T]Achievement Indicators• Demonstrate and explain with examples why there is a

need to have a standardized order of operations. (9)• Apply the order of operations to solve multi-step

problems with or without technology, e.g., computer,calculator. (9, MG2)

Specific Outcomes N1. Determine and explain why a number is

divisible by 2, 3, 4, 5, 6, 8, 9, or 10, andwhy a number cannot be divided by 0.[C, R]

N6. Demonstrate an understanding ofaddition and subtraction of integers,concretely, pictorially, and symbolically.[C, CN, PS, R, V]

Mathematical Processes: C Communication, CN Connections, ME Mental Mathematics and Estimation, PS Problem Solving, R Reasoning, T Technology, V Visualization Features: CM1 (Curious Math: String Art), MG1 (Math Game: Colouring Factors), CM2 (Curious Math: Countdown Clock), MG2 (Math Game: Four in a Row)


Overview 3Copyright © 2010 Nelson Education Ltd.

Planning for Instruction

Problem SolvingIn Lesson 6, students solve problems by using an organizedlist. Students will also solve a variety of problems throughoutthe chapter as they apply their understanding of factors,multiples, and integers.

Assign a Problem of the Week from the selection below orfrom your own collection.1. A number has nine different factors. Two of its multiples

are 72 and 108. What is the number? (36: factors are 1, 2,3, 4, 6, 9, 12, 18, and 36; 2 � 36 � 72; 3 � 36 � 108)

2. The temperature on Monday was �11 °C. The temperatureon Tuesday was �15 °C. The temperature on Wednesdaywas �13 °C. On Thursday, it was colder than Monday butwarmer than Wednesday. What was the temperature onThursday? (�12 °C: �13 � �12 � �11)

3. Maddy copied down a number sentence in math class,but she forgot to write the brackets. Where should Maddyplace the brackets to make the number sentence true?3 � 2 � 6 � 12 � 8 � 6 (Maddy should place thebrackets around the addition and subtraction.3 � (2 � 6) � (12 � 8)� 3 � 8 � 4� 24 � 4� 6)

Reading StrategiesThe reading strategies highlighted in this chapter areMonitoring Comprehension (Mid-Chapter Review) andFinding Important Information (Lesson 6). To reinforce theuse of these strategies, you may apply them to other questionsthroughout the lessons as opportunities present themselves.

Connections to LiteratureExpand your classroom library or math centre with booksrelated to the math in this chapter. For example:• Frasier, Debra. On the Day You Were Born. Harcourt

Children’s Books, 1991.

Math Background An understanding of number relationships is essential tofunctioning in daily life. Students gain this understandingby exploring factors, multiples, and integers directly. Studentsalso gain an intuitive understanding about numbers by relatingnumbers to a variety of real-world contexts. For example,students use reasoning to solve number problems in the realworld. In addition, visualizing number patterns andrelationships allows students to make connections and identifynumber relationships, further developing number sense.

Throughout the chapter, students are encouraged to usemental math to determine factors and multiples and to solve

• Merrill, Jean. The Toothpaste Millionaire. Houghton Mifflin,2006.

• Murphy, Stuart. Less Than Zero. HarperTrophy, 2003.

Connections to Other Math StrandsPatterns and Algebra: In the Getting Started activity,students will identify the pattern in a banner design. InLesson 2, students will use number patterns as a way toidentify multiples.

Shape and Space: In the Chapter Task, students will describehow squares and rectangles with different dimensions can beused to fill a large square.

Measurement: In Lesson 8, students will use their knowledgeof integers to compare and order temperatures.

Connections to Other CurriculaArt: In Curious Math: String Art, students will use a modifiedversion of string art to represent multiples of numbers.

Science: In Lesson 2, students will use multiples to determinethe years in which the comet Kojima will likely be visiblefrom Earth. In Lesson 8, students will compare and orderpositive and negative temperatures. In Lesson 9, students willuse formulas to calculate heart rate and lung capacity.

Connections to Home and Community• Have students use everyday situations to order and compare

numbers, identify factors and multiples, and use the order ofoperations.

• Send home Family Letter p. 74, which contains suggestionsfor a variety of activities related to the math in this chapterthat students can do at home.

• Have students complete the Nelson Math Focus 6 Workbookpages for this chapter at home.

• Use the suggestions for at-home activities in Follow-Up andPreparation for Next Class in various lessons.

complex expressions using the order of operations. It isimportant for students to demonstrate computationalmath skills as well as flexibility with numbers. Students are encouraged to use reasoning to check their answers, to analyze and evaluate their thinking, and to listen and learn from the strategies of others.

See PRIME (Professional Resources and Instruction forMathematics Educators): Number and Operations by MarianSmall (Thomson Nelson, 2005) for additional mathbackground and teaching strategies.



Chapter 3 Planning Chart

Key Concepts*Number and Operations• Numbers tell how many or how much.• Classifying numbers provides information about the characteristics

of the numbers.• There are different, but equivalent, representations for a number.• Benchmark numbers are useful for relating and estimating numbers.

Key Principles• A number can be described as the product of its factors.• Describing a number as a multiple suggests thinking of it in terms of a unit

other than 1; for example, since 6 is a multiple of 3, it is two 3s.• Knowing that a number is prime or composite gives you information about

how many factors it has, as well as about how it can be represented as anarray.

• Integers include the natural numbers and their opposites, as well as zero.They describe amounts above, below, and including the zero benchmark.

• Integers can be compared by using their positions relative to the zerobenchmark.

• Order of operations rules are used to ensure that everyone reading anexpression interprets it the same way.

Student Book Section Lesson GoalGrade 6 Outcomes

Pacing13 Days Prerequisite Skills/Concepts

Getting Started Banner Designspp. 68–69 (TR pp. 9–12)

Activate knowledge aboutnumber relationships.

1 day • Recall multiplication facts and related division facts to 81.• Identify and extend number patterns.

Lesson 1Identifying Factorspp. 70–73 (TR pp. 13–17)

Identify factors to solveproblems.

N3 1 day • Calculate products and quotients using mental math.• Divide a two-digit number by a one-digit number.• Understand the meaning of the term factor.• Use arrays to multiply and divide numbers.

Lesson 2Identifying Multiplespp. 74–76 (TR pp. 18–21)

Identify multiples to solveproblems.

N3 1 day • Identify factors of whole numbers.• Extend a number pattern by multiplying or adding whole numbers.

Lesson 3Prime and Composite Numberspp. 78–80 (TR pp. 24–28)

Identify prime andcomposite numbers.

N3 1 day • Identify factors and multiples of whole numbers.

Lesson 4Identifying Factors by Dividingpp. 82–84 (TR pp. 31–34)

Identify factors by dividingcomposite numbers byprimes.

N3 1 day • Identify prime and composite numbers.• Identify factors of whole numbers.

Lesson 5Creating Composite Numbersp. 85 (TR pp. 35–38)

Multiply combinations offactors to create compositenumbers.

N3 1 day • Multiply and divide combinations of one-digit and two-digit numbers.• Identify prime and composite numbers.

Lesson 6Solving Problems Using an OrganizedList, pp. 88–89 (TR pp. 43–46)

Use an organized list tosolve problems that involvenumber relationships.

N3 1 day • Identify factors and multiples of whole numbers.• Identify prime and composite numbers.

Lesson 7Representing Integers pp. 90–92 (TR pp. 47–50)

Use integers to describesituations.

N7 1 day • Locate numbers on a number line.

Lesson 8Comparing and Ordering Integers pp. 94–97 (TR pp. 53–57)

Use a number line tocompare and orderintegers.

N7 1 day • Locate integers on a number line.• Use the symbols <, >, and � to compare numbers.

Lesson 9Order of Operationspp. 98–100 (TR pp. 58–61)

Apply the rules for order ofoperations with wholenumbers.

N9 1 day • Use mental math to add, subtract, multiply, and divide whole numbers.

Curious Math 1 p. 77 (TR pp. 22–23)Math Game 1 p. 81 (TR pp. 29–30)Mid-Chapter Review pp. 86–87 (TR pp. 39–42)Curious Math 2 p. 93 (TR pp.51–52)Math Game 2 p. 101 (TR pp. 62–63)Chapter Review, pp. 102–104 (TR pp. 64–68)Chapter Task, p. 105 (TR pp. 69–71)Chapters 1–3 Cumulative Review pp. 106–107 (TR pp. 72–73)

3 days

*PRIME (Professional Resources and Instruction for Math Educators): Number andOperations by Marian Small (Thomson Nelson, 2005)



Chapter Goals• Identify prime numbers, composite numbers, factors, and multiples.• Determine the factors of a composite number.• Use an organized list to solve problems.• Represent, order, and compare integers.• Explain and apply the order of operations with whole numbers.

Materials Masters Extra Practice in the Student Book and Workbook

• pencil crayons • 2 cm Grid Paper, Masters Booklet p. 23• Optional: Scaffolding for Getting Started pp. 75–76• Optional: Review of Essential Skills: Chapter 3, Masters Booklet p. 5• Optional: Initial Assessment Summary, Masters Booklet p. 57

• Optional: counters• Optional: linking

cubes

• Optional: 1 cm Grid Paper, Masters Booklet p. 22• Optional: Chapter Checklist: Chapter 3, Masters Booklet p. 64

Mid-Chapter Review Questions 1 & 2Chapter Review Questions 1, 2, & 3Workbook, p. 17

• rulers• Optional: counters

• Number Lines, Masters Booklet p. 33• Optional: Scaffolding for Lesson 2, Question 3 p. 77

Mid-Chapter Review Questions 3 & 4Chapter Review Questions 4 & 5Workbook, p. 18

• counters • 100 Chart, Masters Booklet p. 30• 2 cm Grid Paper, Masters Booklet p. 23

Mid-Chapter Review Question 5Chapter Review Questions 6 & 7Workbook p. 19

• number cards 40 to 50 Mid-Chapter Review Questions 6 & 7Chapter Review Question 8Workbook, p. 20

• Optional: chart paperand markers

• Optional: 100 Chart, Masters Booklet p. 30• Optional: 1 cm Grid Paper, Masters Booklet p. 23

Workbook p. 21

• Optional: 100 Chart, Masters Booklet p. 30 Chapter Review Question 9Workbook, p. 22

• Number Lines, Masters Booklet p. 33 Chapter Review Question 10Workbook p. 23

• Number Lines, Masters Booklet p. 33 Chapter Review Questions 11 & 12Workbook, p. 24

• calculators Chapter Review Questions 13 & 14Workbook p. 25

For materials and masters for features, reviews, and the Chapter Task, see the TR section. Workbook p. 26



Chapter 3 Assessment Summary

Opportunities for Feedback: Assessment for Learning

Student Book Section Chart Key Question Grade 6 Outcomes Mathematical Process Focus for Key Question

Lesson 1Identifying Factors pp. 70–73

TR p. 17 5, model, written answer N3. Demonstrate an understanding offactors and multiples by• determining multiples and factors of

numbers less than 100• identifying prime and composite numbers• solving problems involving multiples. [PS, R, V]

Reasoning, Visualization

Lesson 2Identifying Multiples pp. 74–76

TR p. 21 5, short answer, written answer

N3 Problem Solving, Visualization

Curious MathString Art p. 77

TR p. 23 N3 Problem Solving, Reasoning, Visualization

Lesson 3Prime and Composite Numbers pp. 78–80

TR p. 28 4, written answer N3 Reasoning, Visualization

Math GameColouring Factors p. 81

TR p. 30 N3 Reasoning

Lesson 4Identifying Factors by Dividing pp. 82–84

TR p. 34 4, written answer N3 Reasoning

Lesson 5Creating Composite Numbers p. 85

TR p. 38 entire exploration,investigation

N3 Problem Solving, Reasoning

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.

Mid-Chapter Review pp. 86–87 TR p. 41 1, model, written answer N3 Visualization

2, short answer N3 Reasoning

3, short answer N3 Problem Solving

4, short answer, written answer N3 Reasoning


6, short answer N3 Reasoning, Visualization

7, written answer N3 Reasoning

Lesson 6Solving Problems Using anOrganized List pp. 88–89

TR p. 46 6, written answer N3 Problem Solving, Reasoning

Lesson 7Representing Integers pp. 90–92

TR p. 50 4, short answer, model N7. Demonstrate an understanding ofintegers, concretely, pictorially, andsymbolically.[C, CN, R, V]

Reasoning, Visualization

Curious MathCountdown Clock p. 93

TR p. 52 N7 Connections, Reasoning

Lesson 8Comparing and Ordering Integerspp. 94–97

TR p. 57 6, model, written answer N7 Communication, Connection, Visualization

Lesson 9Order of Operations pp. 98–100

TR p. 61 4, short answer N9. Explain and apply the order ofoperations, excluding exponents, with andwithout technology (limited to wholenumbers).[CN, ME, PS, T]

Connections, Mental Mathematics andEstimation, Problem Solving, Technology

Math GameCountdown Clock p. 101

TR p. 63 N9 Mental Mathematics and Estimation

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



Assessment of Learning

Student Book Section Chart Question Grade 6 Outcome Mathematical Process Focus for Question

Mid-Chapter Reviewpp. 86–87

TR pp. 41–42 1, model, written answer N3 Visualization





6, short answer N3 Visualization


Chapter Reviewpp. 102–104andChapter Test(TR pp. 84–86)

TR pp. 66–68 1, written answer N3 Visualization


3, 4, short answer N3 Reasoning

5, written answer N3 Problem Solving




9, short answer N3 Problem Solving, Reasoning

10, written answer N7 Communication

11, written answer, model N7 Visualization

12, short answer, model N7 Communication, Visualization

13, short answer N9 Mental Mathematics and Estimation

14, short answer, written answer N9 Problem Solving

Chapter TaskA Block Dropping Game, p. 105

TR p. 71 entire task, investigation N3 Problem Solving, Reasoning, Visualization

Assessment as Learning

Student Book Section Student Self-Assessment Masters

Mid-Chapter Reviewpp. 86–87

Chapter 3 Lesson Goals, Masters Booklet p. 75Self-Assessment: Mathematical Processes, Masters Booklet p. 84Self-Assessment: What I Like, Masters Booklet p. 85Self-Assessment: How I Learn, Masters Booklet p. 85

Chapter Reviewpp. 102–104

Chapter 3 Lesson Goals, Masters Booklet p. 75Self-Assessment: Mathematical Processes, Masters Booklet p. 84Self-Assessment: What I Like, Masters Booklet p. 85Self-Assessment: How I Learn, Masters Booklet p. 85



8 Copyright © 2010 Nelson Education Ltd.

Chapter Opener STUDENT BOOK PAGES 66–67

Using the Chapter OpenerDraw students’ attention to the photograph on Student Bookpages 66 and 67. Tell students that the Craik Eco-Centre isan energy-efficient building that uses renewable energy.Together, read the opening task. Record and discuss students’responses.

If students have trouble getting started, have them use 12linking cubes and make as many different rectangular prismsas they can. Encourage students to arrange prisms in multiplelayers, such as 2 � 2 � 3, as well as single layers, such as 1 � 2 � 6. Review how the length, width, and thickness of a prism can be used to identify factors of 12. As studentsassemble model walls with 36 linking cubes, encourage them to build walls with layers as well. Students might identifydifferent numbers of walls depending on whether or not they distinguish between the order of the dimensions. Forexample, they might consider a 2 � 18 and an 18 � 2 wallto be equivalent walls.

Sample Discourse“Suppose a wall has a thickness of 1 cube. How manydifferent walls can you make with 36 cubes?”• Five: 36 cubes long and 1 cube high, 18 cubes long and

2 cubes high, 12 cubes long and 3 cubes high, 9 cubes longand 4 cubes high, and 6 cubes long and 6 cubes high

• If you know two factors that multiply together to make 36,these factors represent the length and height of a wall.

“Suppose a wall has a thickness of 2 cubes. How manydifferent walls can you make with 36 cubes?”• Three: 18 cubes long and 1 cube high, 9 cubes long and

2 cubes high, and 6 cubes long and 3 cubes high“What other wall can you make with 36 linking cubes?• I can make a wall 4 cubes thick, 3 cubes long, and

3 cubes high.• I can make a wall 2 cubes thick, 9 cubes long, and

2 cubes high.• I can make a wall 6 cubes thick, 3 cubes long, and

2 cubes high.

Read and discuss the five goals of the chapter. Ask studentsto suggest different ways they can determine the factors of anumber. Have students record in their journals their thoughtsabout one of the goals, using a prompt such as “Examples ofsituations where I would need to identify the factors of anumber are….” At the end of the chapter, you can askstudents to complete the same prompt. Then they cancompare their ideas with the ones recorded at the beginningof the chapter and reflect on what they have learned.

At this point, it would be appropriate to• send home Family Letter p. 74• 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 3: Number Relationships

prime number

multiple

factor

product

composite number

Family Letter p. 74

integer

opposite integers

rules for order of operations

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9Getting Started: Banner DesignsCopyright © 2010 Nelson Education Ltd.

Getting StartedBanner Designs

STUDENT BOOK PAGES 68–69

PREREQUISITE SKILLS/CONCEPTS

• Recall multiplication facts and related division facts to 81.• Identify and extend number patterns.

GOALActivate knowledge about number relationships.

Preparation and PlanningPacing 30–40 min Activity

10–20 min What Do You Think?

Materials • pencil crayons

Masters • 2 cm Grid Paper, Masters Booklet p. 23• Optional: Scaffolding for Getting

Started pp. 75–76• Optional: Review of Essential Skills:

Chapter 3, Masters Booklet p. 5• Optional: Initial Assessment Summary,

Masters Booklet p. 57

Nelson Website Visit www.nelson.com/mathfocus and followthe links to Nelson Math Focus 6, Chapter 3.

Math BackgroundThe Getting Started activity helps students activateknowledge of number relationships and principles learnedin earlier grades. Specifically, students will use numberpatterns, skip counting, and multiplication to determinemultiples of two whole numbers. Students need a firmunderstanding of multiplication and division facts to helpthem identify both multiples and factors of wholenumbers.

2 cm Grid Paper, MastersBooklet p. 23

Optional: Scaffolding forGetting Started p. 75–76

Optional: Review ofEssential Skills: Chapter 3,Masters Booklet p. 5

Optional: InitialAssessment Summary,Masters Booklet p. 57


Chapter 3: Number Relationships Copyright © 2010 Nelson Education Ltd.10

Using the Activity(Whole Class/Pairs/Small Groups)

� 30–40 min

Use this activity to activate knowledge of factors andmultiples and number patterns and as an opportunity forinitial assessment.

Together, read about Daniel’s Heritage Day banner andthen read the central question on Student Book page 68.Distribute grid paper to students. Have students work in pairsor small groups to answer Prompts A to C. Students havingdifficulty sketching may prefer writing the letter E for eagleinstead of drawing the symbol. Discuss the answers to theseprompts as a class. Have students work in groups to answerPrompt D. Have volunteers share their banners with the class.If extra support is required, guide these students and providecopies of Scaffolding for Getting Started pp. 75–76.

Answers to the ActivityA. For example,

B. For example, I saw the pattern 6, 12, 18.

6 12 18

The pattern shows skip counting by 6s. So the nextsquare with an eagle should be the 24th square because18 � 6 = 24.

C. For example, I can multiply 1, 2, and 3 by 6 to get 6 � 1 = 6, 6 � 2 = 12, and 6 � 3 = 18, which are the numbers of the first three red squares that have aneagle. So I can solve the equation � 6 = 30 to figureout the number of red squares with an eagle.I can divide by 6 to solve the problem. There are 30squares and 30 � 6 = 5, so 5 red squares will have an eagle.

D. For example, I’ll create a banner with 100 squares. I’llcolour every second square yellow. Every fifth square willhave the symbol for a horse. I’ll figure out how manyyellow squares will have a horse.

Every 10th square has a horse in a yellow square. So Ipredict that the number of yellow squares with a horse in100 squares is 100 � 10 = 10.

10horse horse


Getting Started: Banner DesignsCopyright © 2010 Nelson Education Ltd. 11

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

Use this anticipation guide to activate knowledge andunderstanding of factors and multiples. Explain to studentsthat the statements involve math concepts or skills they willlearn about in the chapter—they are not expected to knowthe answers at this point. Ask students to read the statements,think about each one for a few seconds, and decide whetherthey agree or disagree. Have volunteers explain the reasonsfor their choices. Students can exchange their thoughts insmall groups, in groups where all agree or disagree, or in ageneral class discussion. Tell students they can revisit theirideas 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 you multiply 5 by 6, you get 30.

You can also multiply 1 and 30 to get 30, and there areother factors of 30 too. So when you multiply two wholenumbers, the product has more than two factors.*For example, disagree. When you multiply 1 by 1, youget 1, and 1 is the only factor.

2. For example, agree. The last digit is 0 so when youmultiply numbers like 10 and 20, you get a 0 in the onesdigit of the product.*For example, disagree. 8 � 25 = 200 and neither factorhas 0 as the ones digit.

3. *For example, agree. If you extend the first pattern byadding 5 and the second pattern by adding 7, you get 35 on both lists. Then if you keep adding 5 and 7, youwill get 70 as the next number on both lists. So if youcontinue adding both 5 and 7, you will get lots of thesame numbers on both lists.For example, disagree. The three numbers in each list aredifferent. One list of numbers goes up by 5s and theother list goes up by 7s. So you will not get many of thesame numbers.

4. For example, agree. 3 has two factors, 1 and 3. 2 � 3 = 6. 6 has four factors: 1, 2, 3, and 6. So multiplying 3 by 2 doubled the number of factors.*For example, disagree. 4 has three factors: 1, 2, and 4; 2 � 4 = 8. 8 has four factors: 1, 2, 4 and 8. So whenyou multiply 4 by 2, you do not get double the numberof factors.



Initial Assessment: Assessment for Learning

What you will see students doing

When students understand If students misunderstand

Prompt B

• Students explain how to use a number pattern to predict the next red bannersquare that will have an eagle symbol.

Prompt C

• Students explain how to use a multiplication equation to figure out how manyred squares will have an eagle.

Prompts C & D

• Students determine the number of coloured squares that will have a symboland explain their method.

• Students may not recognize that every sixth square has both characteristics(eagle, red) and cannot extend the pattern 6, 12, 18, … beyond 18. (See 3below.)

• Students may not be able to connect multiplication facts with determining thenumber of red squares that will have an eagle. (See 4 below.)

• Students may not connect determining the number of squares with numberpatterns or multiplication facts. (See 3 and 5 below.)

SUPPORTING STUDENTS WHO ARE ALMOST THERE

1. Use Scaffolding for Getting Started pp. 75–76.

2. Use Review of Essential Skills: Chapter 3, Masters Booklet p. 5 toactivate students’ skills.

3. Have students number the squares from left to right and note the numbers of the coloured squares that have a symbol. Discuss the pattern in thenumbers (6, 12, 18, …) and discuss the strategies that students might use to determine the next number in the pattern, for example, skip counting by 6.

4. Remind students that a multiplication fact is another way to represent skipcounting. For example, to complete the multiplication sentence � 6 = 18,students can skip count by 6s until they reach 18, and count the number ofskips. There are three skips, so 3 � 6 = 18. Suggest students use the samethinking for patterns that reach greater numbers.

Differentiating Instruction: How you can respond

5. Remind students that in a multiplication fact, two factors are multiplied togive a product. Help students understand that one of the factors is thenumber of squares from one coloured square with a symbol to the next, andthe product is the total number of squares in the banner. The unknown factoris the number of coloured squares with a symbol that will be in the banner.For example, if there is a coloured square with a symbol every 5 squares anda total of 50 squares, students can use the multiplication fact � 5 = 50to calculate the number of coloured squares with a symbol that will be inthe banner.

SUPPORTING STUDENTS WHO ARE NOT READY

This chapter assumes that students are already comfortable identifying andextending number patterns and calculating the missing factor in a multiplicationequation.

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

For this activity:

• You may want to focus on working with number patterns and eliminateconsideration of multiplication equations.


Chapter 3 1111

13Lesson 1: Identifying FactorsCopyright © 2010 Nelson Education Ltd.

Identifying Factors STUDENT BOOK PAGES 70–73


• Calculate products and quotients using mental math.• Divide a two-digit number by a one-digit number.• Understand the meaning of the term factor.• Use arrays to multiply and divide numbers.

SPECIFIC OUTCOME

N3. Demonstrate an understanding of factors and multiples by• determining multiples and factors of numbers less

than 100• identifying prime and composite numbers• solving problems involving multiples. [PS, R, V]

Achievement Indicators• Determine all the whole-number factors of a given

number using arrays.• Identify the factors and multiples for a given number

and explain the strategy used, e.g., concrete or visualrepresentations, repeated division by prime numbers, orfactor trees.

• Solve a given problem involving factors or multiples.

GOALIdentify factors to solve problems.

Preparation and PlanningPacing 5–10 min Introduction

15–20 min Teaching and Learning20–30 min Consolidation

Materials • Optional: counters• Optional: linking cubes

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

• Optional: Chapter Checklist: Chapter 3, Masters Booklet p. 64

Recommended Questions 3, 4, 5, 6, 7, 8, & 13Practising Questions

Key Question Question 5

Extra Practice Mid-Chapter Review Questions 1 & 2Chapter Review Questions 1, 2, & 3Workbook p. 17

Mathematical R (Reasoning) and V (Visualization) Process Focus


Math BackgroundStudents should be familiar with the relationship betweenfactors of a number and division of that number. Forexample, 2 is a factor of 10 because the quotient (5) is awhole number and the remainder is 0. To identify all ofthe factors of a number and to help them visualize thosefactors, students can use arrays.

An array is a pictorial or concrete model of a number in which the rows and columns of the array representfactors of the number. For example, a 4-by-5 array showsthat 4 and 5 are factors of 20 because the array has 4 rows, 5 columns, and a total of 20 elements.

As students use reasoning and mental math to identifythe factors of a number, they can show the factors in afactor rainbow. A factor rainbow lists all of a number’sfactors in a row and pictorially links the factors that canbe multiplied together to result in that number. It isimportant to list the factors systematically so none areforgotten.

Optional: 1 cm Grid Paper,Masters Booklet, p. 22

Optional: ChapterChecklist: Chapter 3,Masters Booklet p. 64



Introduction(Small Groups/Whole Class)

� 5–10 min

Distribute 12 counters to each group. Have students formthe counters into an array. Alternatively, have them colourarrays of 12 on grid paper. Ask volunteers to share theirarrays with the class. Try to elicit all of the possible arrays forthe number 12: 1-by-12, 2-by-6, and 3-by-4. Some studentsmay also suggest reversing the order of rows and columns, forexample, 12-by-1. Accept these answers but make surestudents realize that the factors are still the same.

Sample Discourse“How did you decide how many counters would go in eachrow of your array?”• I tried to make rows that were all the same size without

having any counters left over. I then counted the number ofcounters in each row to determine one factor.

• I chose a number of rows that is a factor of 12, and then putthe counters into that number of rows.

“Can you make an array with five rows?”• No, because there will be two counters left over.• No, because 5 is not a factor of 12.• No, because 5 does not divide evenly into 12.

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

Together, read about the Earth Day project and then read thecentral question on Student Book page 70. Work throughMai’s Arrays together. Students may represent the arrays withsymbols as Mai did, or they may use counters or grid paper.Some students may use pairs of factors to identify two arraysrather than one array.

Work through Jason’s Factor Rainbow with students toshow how to systematically record all the factors of 18. Forexample, students may reverse the rows and columns to get 6 arrays for 18 seedlings: 1-by-18 and 18-by-1, 2-by-9 and 9-by-2, and 3-by-6 and 6-by-3. Tell students they can solve theproblem either way as long as they list the number of arraysthe same way for each number in the chart. They should alsonote that the factors 1, 2, 3, 6, 9, and 18 remain the same.

Have students work in pairs to complete Prompts A to C.When students have completed the activity, discuss theanswers as a class.

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Lesson 1: Identifying FactorsCopyright © 2010 Nelson Education Ltd. 15

Answers to PromptsA. For example, I used a factor rainbow to record the

number of factors and the number of arrays for eachnumber of seedlings.The factors of 25 are 1, 5, and 25. So 25 seedlings can beplanted in 2 arrays:

1-by-25, 5-by-5

The factors of 29 are 1 and 29. So 29 seedlings can beplanted in 1 array:

1-by-29

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. So36 seedlings can be planted in 5 arrays:

1-by-36, 2-by-18,3-by-12, 4-by-9, 6-by-6

1 32 64 129

Grade 4

18 36

1

Grade 3

29

1

Grade 2

5 25

The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. So 48 seedlings can be planted in 5 arrays:

1-by-48, 2-by-24,3-by-16, 4-by-12, 6-by-8

The factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56. So 56seedlings can be planted in 4 arrays:

1-by-56, 2-by-28,4-by-14, 7-by-8

B. For example, I chose Jason’s method because I can usemental math to figure out the factors of a number. Thefactor rainbow helps me keep track of the factors I havefigured out. I didn’t use Mai’s method because it takestoo long to draw all the arrays for each number.

C. Both 36 and 48 seedlings can be planted in 5 arrays.

Reflecting (Whole Class)

Here students compare and contrast arrays with factor rainbowsas methods for identifying the factors of a number. Studentsalso explain how they know when they have identified all ofthe factors, using each method. Ensure students understandthat arrays can be used to identify factors, while factorrainbows are primarily a method for recording the factors.Students should also connect the dimensions of the arrayswith the factors listed in the factor rainbow.

1 7 842 2814

Grade 6

56

1 32 6 84 1612

Grade 5

24 48



Answers to Reflecting QuestionsD. For example, they are the same in that each of the

dimensions of Mai’s arrays matches a factor pair in Jason’sfactor rainbow. They are different because Mai has todraw rectangles to list arrays while Jason uses mentalmath to list factors.

E. For example, Mai drew arrays with 1, 2, and 3 rows. Sheknew that 4 and 5 aren’t factors of 18, so she couldn’tplant 18 seedlings in 4 or 5 rows. She knew that a 3-by-6array can be arranged in either 3 rows of 6 or 6 rows of 3.She knew that 7 and 8 aren’t factors of 18, so shecouldn’t plant 18 seedlings in 7 or 8 rows. She knew thata 2-by-9 array can be arranged in either 2 rows of 9 or 9 rows of 2. So she knew that there are no other possiblearrays for 18 seedlings.Jason’s factor rainbow shows he identified the matchingfactors of 1 and 18, 2 and 9, and 3 and 6. He only hadto see if 4 or 5 is a factor because he had already figuredout factors of 18 that are 6 or greater. Because 4 and 5are not factors, he knew he had identified all factors of 18.

Consolidation � 20–30 min

Checking (Pairs)

Students can use either arrays or factor rainbows to identifythe factors. Refer students to Mai’s and Jason’s methods forguidance. Have counters and grid paper available for studentsto use to model arrays.

Practising (Individual)

These questions provide students with practice in usingarrays and factor rainbows to identify and record factors.Provide counters or grid paper to students to help themmodel the arrays.6. Students should recognize that the number of coins can

only be divided by 1 and itself. In Lesson 3, students willformalize this understanding as they learn about primeand composite numbers.

Answers to Key Question5. a)

1-by-24

2-by-12

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3-by-8

4-by-6b) For example, each number of rows and columns in an

array represents a factor of 24. So the factors of 24 are1, 2, 3, 4, 6, 8, 12, and 24.

Closing (Whole Class)

Question 13 allows students to reflect on and consolidatetheir learning for this lesson as they connect the numbers ofrows and columns in an array to the factors of the number.

Answer to Closing Question13. For example, if you want to identify the factors of 26,

you can draw arrays.

The numbers of rows and columns of the arrays are thefactors of 26. So 1, 2, 13, and 26 are factors of 26.You can also use mental math to identify the factors anduse a rainbow to help you keep track.

Follow-Up and Preparation for Next ClassHave students follow up on the lesson at home using a groupof small items such as toothpicks. Suggest that studentsarrange the group into an array. Using the array, studentsshould identify factors of the number used in the array.

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Lesson 1: Identifying FactorsCopyright © 2010 Nelson Education Ltd. 17




• Students use arrays and/or factor rainbows to identify the factors of a number.

Key Question 5 (Reasoning, Visualization)• Students draw all of the possible arrays for the number 24 and explain how

the dimensions of the arrays relate to the factors of 24.

• Students may not identify all of the factors. (See Extra Support 1.)

• Students may not connect the numbers of rows and columns with the factorsof the number. (See Extra Support 2.)

EXTRA SUPPORT

1. Help students understand how they can use a factor rainbow to organizetheir work. Have students write the numbers 1 to 16 in a row. Tell studentsto look at each number in the row and use mental math or arrays to decide ifit is a factor of 16. If it is a factor, have students circle the number; if it isnot a factor, have students cross out the number. Finally, have students drawarches to connect the numbers that can be multiplied together to give aproduct of 16. For square numbers (16 = 4 � 4), suggest that studentssimply draw an arch from the 4 to itself.


2. Have students use grid paper and shade in as many rectangles as possiblethat have a total area of 24 grid squares. Then have students label eachrectangle with the number of rows and the number of columns that areshaded, for example, “4-by-6.” Guide students to understand that “4-by-6”means “4 multiplied by 6.” Since the area of the rectangle is 24, 4 and 6 arefactors of 24. Repeat the exercise, using counters in an array in place of thegrid paper, and guide students to connect the numbers of rows and columnsin the arrays with the factors of 24.

EXTRA CHALLENGE

• Challenge students to identify the number between 1 and 50 that can bemodelled with the greatest number of arrays. Encourage students to developstrategies to help them eliminate some numbers, rather than drawing thearrays for each number. For example, students might eliminate any numberthat can only be drawn in an array with one row.

SUPPORTING DEVELOPMENTAL DIFFERENCES

• Provide students with an array and have them work together or individually toidentify the factors. Then ask students to create another array with the samenumber of counters. This exercise will give students an opportunity to explorefactors and products without identifying all of the factors of a particularnumber.

SUPPORTING LEARNING STYLE DIFFERENCES

• Kinesthetic learners will benefit from creating their arrays with countersrather than just drawing them.


Chapter 3 2222


Identifying Multiples STUDENT BOOK PAGES 74–76


• Identify factors of whole numbers.• Extend a number pattern by multiplying or adding whole

numbers.

SPECIFIC OUTCOME

N3. Demonstrate an understanding of factors andmultiples by• determining multiples and factors of numbers less

than 100• identifying prime and composite numbers• solving problems involving multiples.[PS, R, V]

Achievement Indicators• Identify multiples for a given number and explain the

strategy used to identify them.• Solve a given problem involving factors or multiples.

GOALIdentify multiples to solve problems.


Preparation and PlanningPacing 5–10 min Introduction(allow 5 min for 10–15 min Teaching and Learningprevious homework) 20–30 min Consolidation

Materials • rulers• Optional: counters

Masters • Number Lines, Masters Booklet p. 33• Optional: Scaffolding for Lesson 2,

Question 3 p. 77

Recommended Questions 2, 3, 5, 8, & 9Practising Questions


Extra Practice Mid-Chapter Review Questions 3 & 4Chapter Review Questions 4 & 5Workbook p. 18

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

Vocabulary/Symbols multiple


Math BackgroundIn previous grades, students have multiplied factors tocalculate a product. In this lesson, students will approachmultiplication from a different perspective as theycalculate multiples of a number using knownmultiplication facts and skip counting. Students willmultiply a given number by sequential whole numbers tobuild a list of multiples. For example, to build a list ofmultiples of 6, students will multiply 6 by 1, 2, 3, 4, … toget the multiples 6, 12, 18, 24, …. To use skip counting,students will count in units of the given number. Forexample, to build a list of multiples of 5, students willcount by 5s to get the multiples 5, 10, 15, 20, and so on.Students use a number line to help them visualize thepattern in the list of multiples. Students will apply theseskills in various problem-solving contexts.

Number Lines, MastersBooklet p. 33

Optional: Scaffolding forLesson 2, Question 3 p. 77


Lesson 2: Identifying MultiplesCopyright © 2010 Nelson Education Ltd. 19

Introduction(Whole Class) � 5–10 min

Briefly review some mental math strategies that students havelearned for multiplication. On the board, on a transparency,or on an interactive whiteboard, write the followingmultiplication expressions:

4 � 8 6 � 7 8 � 5

Ask volunteers to share their strategies for calculating eachproduct. Try to elicit a variety of strategies.

Sample Discourse“How can you calculate the product of 4 and 8?”• I used doubling. I know 2 � 8 = 16,

so 4 � 8 = 16 � 16, which is 32.• I used doubling. I know 4 � 4 = 16,

so 4 � 8 = 16 � 16, which is 32.“How can you calculate the product of 6 and 7?”• I skip counted up. I know 6 � 6 = 36,

so 6 � 7 = 36 � 6, which is 42.• I skip counted down. I know 7 � 7 = 49,

so 6 � 7 = 49 � 7, which is 42.“How can you calculate the product of 8 and 5?”• I used doubling. I know 2 � 5 = 10, so 4 � 5 = 10 � 10,

which is 20, and 8 � 5 is 20 � 20, or 40.• I skip counted down. I know 10 � 5 = 50, so 9 � 5 =

50 � 5, which is 45, and 8 � 5 = 45 � 5, which is 40.

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Teaching and Learning(Whole Class/Small Groups)

� 10–15 min

Before reading, remind students that a comet is a small bodythat orbits the Sun, and it is only visible from Earth atcertain points in its orbit. Comets that appear regularly arereferred to as periodic comets. Together, read about thecomets and then read the central question on Student Bookpage 74. Have students set up Oleh’s List and retrace hissteps to show the first multiples of 7. Then direct them toLéa’s Number Line. Tell students to use their rulers to drawan open number line with two arrows. Ask them to point outwhich number Léa starts with on the number line and howshe gets to the next number. When students have becomecomfortable with Léa’s method, have them work throughPrompts A to C in small groups. You may want to discuss thetwo methods as a group and have volunteers explain whichmethod they prefer.

Sample Discourse“Which math operations did Oleh use in his method? How isOleh’s method different from Léa’s method?”• Oleh used multiplication to determine the multiples of 7 and

addition to calculate the years the comet would be seen fromEarth. Léa only used addition to figure out the years after2000 the comet would be seen.

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“Which method is easier for you to use? Explain.”• Oleh’s method is easier because multiplying to determine the

multiples is faster than adding, and I only have to replace thelast digits of 2000 with the multiples of 7 to get the years.

• Léa’s method is easier because I like adding better thanmultiplying.

Answers to PromptsA. For example, I multiplied 7 by 3 to get 21.B. For example, I added 7 to 2014 to get the year 2021.C. For example, I listed the multiples of 7 until I got to 70.

I stopped at 70 because I know 2000 � 70 = 2070 ispast 2067.7, 14, 21, 28, 35, 42, 49, 56, 63, 70, …I added these multiples to 2000 to get these years inwhich the comet will likely be seen from Earth: 2007,2014, 2021, 2028, 2035, 2042, 2049, 2056, and 2063.


Here students reflect on the relationship between factors andmultiples. Students should recognize that a multiple is theproduct of a factor and a counting number.

Answers to Reflecting QuestionsD. For example, you create a multiple of 7 by multiplying

7 by a counting number. So any multiple is 7 times acounting number and 7 must be a factor.

E. For example, any factor of 9 has to be 9 or less, so thereare only 9 possible numbers. But multiples of 9 arecreated by continually adding 9s and you can add 9sforever.


Checking (Pairs)

Draw students’ attention to the Communication Tip. Ensurethat they are comfortable with the notation “…,” which iscalled an ellipsis. If students require additional guidance, referthem to Oleh’s and Léa’s methods in the example. You maywant to distribute number lines to students; however, studentsdo not need to use scaled number lines; rather, they cansketch empty number lines.


These questions give students opportunities to practisecalculating multiples. Students will also explain connectionsbetween factors and multiples. Encourage students to usemental math strategies in their calculations. Encouragestudents to use number lines as visualization tools.2. Ensure students understand that the “first five multiples”

can be calculated by multiplying by the first fivecounting numbers, 1, 2, 3, 4, and 5, or by repeatedlyadding the number to itself until five multiples are listed.

3. If extra support is required, guide these students andprovide copies of Scaffolding for Lesson 2, Question 3 p. 77.

7. Students create lists of multiples of two numbers andthen identify the numbers that appear in both lists. Inlater grades, students will formalize this understanding as they learn about common multiples.

Answers to Key Question5. a) 8, 16, 24, 32, 40, 48, 56, 64, 72, 80 plates

b) For example, 10 packages; Pauline needs to buy platesfor 80 people, and 80 plates are in 10 packages.

c) 12, 24, 36, 48, 60, 72, 84, 96, 108, 120d) For example, 7 packages; Pauline needs to buy at least

80 glasses, and 6 packages have 72 glasses, which istoo little, but 7 packages have 84 glasses, which isenough.

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Lesson 2: Identifying MultiplesCopyright © 2010 Nelson Education Ltd. 21




• Students multiply by counting numbers and/or use skip counting to calculatemultiples.

Key Question 5 (Problem Solving, Visualization)• Students use mental math to calculate multiples of 8 up to 80 and multiples

of 12 up to 120 and use their calculations to solve a problem.

• Students may have difficulty using mental math to calculate multiples. (SeeExtra Support 1.)

• Students may have difficulty choosing the correct number of plates and cups.(See Extra Support 2.)

• Students may have difficulty using mental math to calculate multiples. (SeeExtra Support 1.)

EXTRA SUPPORT

1. Discuss mental math strategies for multiplying by counting numbers:

Doubling: Students can multiply a known factor by 2. For example, since 2 � 6 = 12, then doubling the counting number will result in 4 � 6 = 24.Doubling can be repeated. For example, 8 � 6 = 48.

Skip counting: Students can skip count from a known factor. For example,since 5 � 6 = 30, then 6 � 6 = 30 � 6, which is 36. Students can also skip count down. For example, since 5 � 6 = 30, then 4 � 6 = 30 � 6,which is 24.


2. Guide students to skip count by 8s using a 100 chart until they reach anumber between 70 and 80, circling each multiple of 8. Repeat with 12s,circling each multiple of 12 with a different colour.

EXTRA CHALLENGE

• Have students research and write a problem about an event that occurs everynumber of years, for example, the Olympics or leap years. Then have studentsexchange their problems with a partner and solve the problems.

Example:

a) The summer and winter Olympics both occur every four years. Calculatethe years for the next five Olympic Summer Games, starting with 2008.Then calculate the years for the next five Olympic Winter Games, startingwith 2006.

b) Andrea’s 21st birthday is in the year 2016, and she wants to know if thesame year will have an Olympic Games. Which Olympic Games, if any, isoccurring that year?


• Some students may be able to determine the multiples but have difficultyadding them to a first number, like to the year 2007. The addition componentmight be eliminated for these students.

• Other students might have difficulty calculating multiples without concretesupport. Provide counters to help students create equal groups to determinemultiples.


• Some students may benefit from comparing visual representations of differentsets of multiples. For example, on a 100 chart, they can colour the multiplesof 6, 8, and 9 in different colours to see how the multiples of 9 are morespread out than the multiples of 6 or 8.


Question 9 allows students to reflect on and consolidate theirlearning for this lesson as they think about multiples of anumber.

Answer to Closing Question9. Disagree; for example, numbers like 1, 10, 19, and

28 are 9 apart but none are multiples of 9. The listwould have to start at 0, 9, or a multiple of 9 for thenumbers to be all multiples of 9.

Follow-Up and Preparation for Next ClassHave students list the multiples of 2 from 2 to 48. Challengestudents to explain which of the numbers they listed has themost factors.




STUDENT BOOK PAGE 77

PREREQUISITE SKILL/CONCEPT

• Identify multiples of whole numbers.

SPECIFIC OUTCOME




strategy used to identify them.• Solve a given problem involving factors or multiples.

Using Curious MathIn this activity, students are exposed to another visualrepresentation of factors and multiples as they create andinterpret a string art design. Students will identify thepatterns formed in the string art and determine whichnumbers should be connected with each colour. Encouragestudents to explain how they are completing their project,using the terms factors and multiples in their explanations.Encourage students to see that all numbers connected by thesame colour string have at least one factor in common.Students may draw conclusions about the numbers joined bytwo or more colours.

Answers to Curious Math1. For example, the multiples of 12 up to 48 are connected

by blue lines.2. 12, 24, 36, 483. 12, 24, 36, 484. For example, if I use yellow to connect multiples of 4,

I predict there will be 4 lines at 12, 24, 36 and 48.


Curious MathString Art

Preparation and PlanningMaterials • pencil crayons

• rulers

Masters • String Art Circle p. 78

Mathematical PS (Problem Solving), R (Reasoning), and Process Focus V (Visualization)


Math BackgroundString art, or curve stitching, is a technique that uses linesegments to produce apparent curves. Collectively, thelines form an approximation of a curve. In this activity,students will draw line segments of different lengths toconnect multiples of various numbers. The frame for thestring art consists of dots arranged in a circle andnumbered from 2 to 48. Students will use differentcolours to draw the lines for various multiples. Theresulting design is a visual representation of factors andmultiples. Students will use reasoning to identify factorsbased on the colours of lines joined at the number.

String Art p. 78


Curious Math: String ArtCopyright © 2010 Nelson Education Ltd. 23




• Students draw lines to connect multiples of different numbers.

• Students make the connection between lines that will be joined at 48 and thefactors of 48.

• Students may have difficulty calculating multiples of the different numbers.(See Extra Support 1.)

• Students may have difficulty identifying other numbers that can be connectedin lines that end at 48. (See Extra Support 2.)

EXTRA SUPPORT

1. Have students talk about some of the mental math strategies that they canuse to calculate multiples of the different numbers. For example, studentsmight skip count by 3 to identify the multiples of 3, or connect all of the evennumbers to identify the multiples of 2.


2. Have students talk about what the numbers 2, 3, 4, and 12 have in common.Students might mention that they are all factors of the same numbers, suchas 12 and 24. Guide students to understand that the lines connecting themultiples of these numbers end at 48 because they are all factors of 48.Discuss how students might find other factors of 48.

EXTRA CHALLENGE

• Challenge students to create string art with a different shape and a differentnumber of dots. For example, students may use a hexagon shape with anumber such as 36, and connect multiples of 2, 3, 4, 6, 9, 12, and 18.


• Some students may not be able to calculate factors and multiples usingmental math. Provide these students with 48 counters and have them formgroups or arrays to assist with their calculations.


• Some students may enjoy experimenting with different colours. Provide thesestudents with multiple copies of the String Art Circle blackline master andencourage them to create a variety of designs.


Chapter 3 3333 Prime and CompositeNumbers


GOALIdentify prime and composite numbers.



• Identify factors and multiples of whole numbers.

SPECIFIC OUTCOME



Achievement Indicators• Provide an example of a prime number and explain

why it is a prime number.• Provide an example of a composite number and

explain why it is a composite number.• Sort a given set of numbers as prime and composite.• Solve a given problem involving factors or multiples.• Explain why 0 and 1 are neither prime nor composite.



Materials • counters

Masters • 100 Chart, Masters Booklet p. 30• 2 cm Grid Paper, Masters Booklet p. 23

Recommended Questions 2, 3, 4, & 8Practising Questions


Extra Practice Mid-Chapter Review Question 5Chapter Review Questions 6 & 7Workbook p. 19

Mathematical R (Reasoning) and V (Visualization)Process Focus

Vocabulary/Symbols prime number, composite number


Math BackgroundIn Lesson 1, students used arrays to determine the factorsof numbers and investigated the different arrays in whichnumbers could be arranged. In this lesson, studentsformalize their understanding of prime and compositenumbers as they use reasoning to identify numbers thatcan be arranged in only one array. By arranging countersin arrays, students are able to visualize numbers that canbe arranged in only one row or column; these numbersare prime, as their only factors are 1 and themselves.Numbers that can be arranged in more than one array arecomposite; each array represents two factors.

A 100 chart is used to identify prime and compositenumbers to 100, using a procedure called the Sieve ofEratosthenes (er-uh-tos-thuh-neez), which was developedand named for the ancient Greek mathematicianEratosthenes. In this procedure, the smallest primenumber on the chart is circled and then each of itsmultiples is crossed off. This is repeated until all of thecomposite numbers have been crossed off, leaving onlythe prime numbers.

100 Chart, Masters Booklet p. 30

2 cm Grid Paper, Masters Booklet p. 23


Lesson 3: Prime and Composite NumbersCopyright © 2010 Nelson Education Ltd. 25

Introduction(Whole Class/Small Groups)

� 5–10 min

Distribute various numbers of counters to each small group, andhave them arrange their counters into as many arrays as possiblewith no counters left over. Ask a volunteer from each group tosay the number of counters they had and describe the differentarrays they were able to make. On the board, on a transparency,or on an interactive whiteboard, record the number of countersand the number of arrays for each group. As a class, talk aboutwhat the rows and columns in an array represent.

Sample Discourse“How can you use arrays to find the factors of a number?”• The numbers of rows and columns in an array are factors of

the number.• I can arrange counters in rows and columns to find the factors.

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� 20–25 min

Together, read about Robin’s batteries and then read thecentral question on Student Book page 78. Distribute 10 counters to each small group and have students formarrays for 2, 3, and 4 and relate these to the packages ofbatteries. Have them continue to make as many arrays asthey can for the numbers 5 to 10. Talk about which numberscan be arranged in only one row or column (2, 3, 5, and 7)and which numbers can be arranged in more than one way (4, 6, 8, 9, and 10). Draw students’ attention to the margindefinitions and ensure they understand the differencebetween prime numbers and composite numbers.

Distribute 100 charts and work through Robin’s Chart onpage 79 together. Have students work through Prompts A toD in groups. Talk about the answer to Prompt C as a class.Students should realize that after they cross off the multiplesof 7, only prime numbers will remain in the chart. Askvolunteers to share their solutions to Prompt D to ensurethat each group correctly identified the prime numbers to 50.

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B. For example, 5 is the next prime number because 4 iscomposite and 5 has only two different factors, 1 and 5.

C. For example, when I crossed off multiples of 2, 3, and 5,then 14, 28, 35, and 42 were crossed off, so only 49 isleft. It was not crossed off because 49 is a multiple of 7 but not a multiple of 2, 3, or 5.

D. The prime numbers to 50 are 2, 3, 5, 7, 11, 13, 17, 19,23, 29, 31, 37, 41, 43, and 47.


Students explain how they used Robin’s Chart to identify allof the prime numbers to 50.

Answers to Reflecting QuestionsE. For example, each multiple of 11 has been crossed off as a

multiple of 2 or 3. All multiples of primes greater than 11have either been crossed off.

F. 1 is the only number that isn’t circled or crossed off.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

Answers to PromptsA. For example, you get every third number in the chart by

skip counting by 3. So every third number in the chart isa multiple of 3 and can be divided by 3. Each multipleof 3 greater than 3 has at least 1, 3, and the number itselfas a factor. So multiples of 3 greater than 3 have morethan two different factors and are composite numbers.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50




Question 6 allows students to reflect on and consolidate theirlearning for this lesson as they explain the connection betweenthe number of arrays that can be used to represent a numberand whether that number is prime or composite.

Answer to Closing Question6. For example, if you can arrange the counters in only one

array, the number of counters is a prime number. Sevencounters can only be arranged as a 1-by-7 array. So it is aprime number.

1 row of 7If you can arrange the counters in more than one

array, the number of counters is a composite numberbecause it means the number has more than twodifferent factors. Six counters can be arranged as 1-by-6and 2-by-3 arrays. So it is a composite number.

1 row of 6

2 rows of 3

Follow-Up and Preparation for Next ClassStudents can review how to determine factors of numbersfrom 1 to 100 at home. They can use small objects such asmarbles or building blocks as counters to set up differentarrays of a particular number. Encourage students to explainto a friend or family member what they are doing.


Checking (Pairs)

Students complete their 100 chart as they apply the Sieve ofEratosthenes to the numbers from 50 to 100. Discuss theanswer as a class to ensure that students have identified all ofthe prime numbers.


These questions provide students with opportunities topractise identifying prime and composite numbers in avariety of contexts.2. Students use reasoning to determine whether numbers

are prime or composite. You may want to make countersavailable.

3. Students may have difficulty communicating theiranswers for parts b) and c). Ask them how their answerssupport the definitions of a prime number and acomposite number.

Answers to Key Question4. a) For example, every number of candles that is prime

can be arranged in only one row or in one column.I will be 12 next month. So when I am 13 or 17,I can arrange the number of candles on a birthdaycake in only one array. For all other ages up to 18, Ican arrange the number of candles in more than onearray.

b) For example, I know prime numbers have only twofactors and one of the factors has to be 1. So you canrepresent the numbers in only one row or one column.Composite numbers have more than two factors soyou can arrange them in more than one array. So Ijust had to identify the prime numbers from 12 to 18to answer part a).

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Lesson 3: Prime and Composite NumbersCopyright © 2010 Nelson Education Ltd. 27






• Students identify prime numbers and composite numbers.

Key Question 4 (Reasoning, Visualization)• Students use their understanding of prime and composite numbers to identify

the ages from 12 to 18 for which the candles on a cake can be arranged inonly one array.

• Students may have difficulty crossing off multiples using a 100 chart toidentify prime numbers. (See Extra Support 1.)

• Students may not realize that prime numbers can be arranged in only onearray and composite numbers can be arranged in more than one array. (SeeExtra Support 2.)


EXTRA SUPPORT

EXTRA CHALLENGE

• Have students create a game involving prime numbers and compositenumbers, using a pair of dice and a 100 chart. For example, a player rolls thedice and determines the sum of the numbers on the dice. Then the playercrosses the number off the chart. If the sum is prime, the player can alsocross off the multiples of the number. The first player to cross off all thenumbers to 50 on his or her chart is the winner.

• Have students predict the number of factors of a product when two differentprime numbers are multiplied together.

1. Have students practise identifying multiples of a number, such as 3, by skipcounting or by counting 123 123 123, etc.

2. Have students talk about what the rows and columns in an array represent.Elicit from students that the number of rows and the number of columns arethe factors of a number; that is, if they multiply the number of rows by thenumber of columns, the product will be the number. Distribute counters and

have students record the factors in the different arrays that they can buildfor each number from 12 to 18. Talk about which numbers can only berepresented with one array (13 and 17) and talk about the factors for thosearrays (1 and 13; 1 and 17). Elicit from students that if a number can only berepresented by one array, its only factors are 1 and itself, and it is a primenumber (except for the number 1).


• Ask students to use counters to create equal groups to show all the multiples(other than the number itself) of 2 from 2 to 50, all the multiples of 3 from 3 to50, etc. Then have them mark off those products on a 100 chart. Explain thatthe leftover numbers (other than 1) are the primes and the crossed-offnumbers are the composites.


• Encourage students to use a variety of colours to complete their 100 charts.For example, students can use a different colour to circle each prime numberand then use the same colour to shade in the squares that are multiples ofthat prime number.



29Math Game: Colouring FactorsCopyright © 2010 Nelson Education Ltd.

Math GameColouring Factors



• Identify factors of whole numbers.

SPECIFIC OUTCOME



Achievement Indicators• Determine all the whole-number factors of a given

number using arrays.• Identify the factors for a given number and explain the

strategy used.

Preparation and PlanningNumber of Players 2

Materials • pencil crayons• Optional: counters

Masters • 100 Chart, Masters Booklet p. 30

Mathematical R (Reasoning)Process Focus


Math BackgroundThis math game helps students consolidate theirunderstanding of factors, prime numbers, and compositenumbers. Students will apply their reasoning skills toselect the numbers with the fewest factors and toidentify the factors of the numbers selected by theirpartners.

100 Chart, Masters Booklet p. 30




Proficient players Less-proficient players

• Students use reasoning to identify numbers with as few factors as possible tolimit their opponents’ scores.

• Students identify all the factors of the number selected for them to maximizetheir scores.

• Students may make poor choices about which numbers to colour. (See ExtraSupport 1 and 3.)

• Students may skip over a factor of the chosen number in the chart. (See ExtraSupport 2 and 3.)


EXTRA SUPPORT

EXTRA CHALLENGE

• Have students play using the entire 100 chart, rather than just the numbersfrom 1 to 50. This will make for a longer game that requires more complicatedcalculations.

1. Have students work with a partner against whom they will not be playingand give them a short planning period before the game begins. Encouragestudents to talk about the numbers with the most factors and the numberswith the fewest factors. Then have them talk about which numbers theyshould choose when it is their turn to colour a number.

2. Remind students to think of the factor rainbows they have created fornumbers. They should be colouring all the factors of a number as if theywere creating a factor rainbow.

3. Have less-proficient students play the game with numbers from 1 to 20,gradually working up to 50.


• Some students may have difficulty developing their own strategies. Providecounters. Allow students to try to quickly rearrange the counters into arrays tohelp them decide which numbers to colour or which numbers are factors ofthe other player’s number.


Using the Math GameProvide each pair of students with a 100 chart and pencilcrayons. Have students cut off the numbers from 51 to 100on their 100 chart. The game of colouring factors can givestudents an opportunity to apply what they have learnedabout numbers and their factors. Make sure all studentsunderstand the rules of the game. Allow time for students todiscuss the strategies they applied while playing the game.

When to PlayStudents can play the game after they demonstrate anunderstanding of identifying the factors of a number. While understanding of prime and composite numbers is not essential to playing the game, it will allow for moresophisticated strategies.

StrategiesHave students discuss the strategies to colour a number. Tominimize the number of factors their opponent can colour,students should choose prime numbers. To minimize theiropponent’s total score, students should choose numbers withfew factors. To maximize their own score, students shouldchoose large prime numbers.

DiscussAsk students to share effective strategies with the rest of theclass to encourage students to learn from one another.


Chapter 3 4444

31Lesson 4: Identifying Factors by DividingCopyright © 2010 Nelson Education Ltd.

Identifying Factorsby Dividing


GOALIdentify factors by dividing composite numbers by primes.



Materials • number cards 40 to 50

Recommended Questions 2, 4, 6, & 7Practising Questions


Extra Practice Mid-Chapter Review Questions 6 & 7Chapter Review Question 8Workbook p. 20

Mathematical R (Reasoning)Process Focus



• Identify prime and composite numbers.• Identify factors of whole numbers.

SPECIFIC OUTCOME



Achievement Indicator• Identify the factors for a given number and explain

the strategy used, e.g., concrete or visualrepresentations, repeated division by prime numbers,or factor trees.

Math BackgroundIn this lesson, students use their reasoning ability toidentify a prime number that is a factor of a givennumber. They will divide the given number by the primefactor. This can be done by using repeated division orfactor trees. Both techniques help students to identify thefactors of the number, including factors that are prime.Students do not need to complete the division or factortree, but they should try starting with a prime numberwhen they divide.




Remind students that they have written factor pairs fornumbers, for example, 3 � 6 for 18. To encourage studentsto think of a number as the product of three factors, remindstudents that 6 can be written as 2 � 3. So 18 can be writtenas 3 � 2 � 3.

Next, ask students to write 24 as the product of threefactors other than 1. Have students share their answers withthe class. Students will see that there are various solutions.For example, students may write 2 � 2 � 6 � 24 or 4 � 2 � 3 � 24.

If time permits, repeat the activity for 30 and 75.Recall the definitions of prime and composite numbers

with students. Write the equation 24 � 3 � 2 � 4 on theboard, on a transparency, or on an interactive whiteboard.

Sample Discourse“What are the two least prime numbers?”• 2 and 3“What is 24 divided by 2?”• 24 divided by 2 is 12.“What is 12 divided by 3?”• 12 divided by 3 is 4.“What type of numbers did you divide by each time, primeor composite?• Each time, I divided by a prime number, either 2 or 3.

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� 15–25 min

Together, read about Daniel and Léa’s card game withcomposite numbers. Read the central question on StudentBook page 82. Work through Léa’s Repeated Divisiontogether. Point out that Léa started by dividing by 5, but she could have started with other prime numbers. Discusshow to identify other factors of 45 from Léa’s division (e.g., 9 and 15). Then have students repeat Léa’s processstarting with 45 and dividing by 3.

Then direct students to Daniel’s method. Tell students thatDaniel starts by dividing 40 by 2, and then continues todivide by prime numbers. Have students work throughDaniel’s method with the number 45 to see if they get thesame results as Léa.

Sample Discourse“Is there another pair of factors Daniel can start his factortree with other than 2 and 20?”• He can start with 5 and 8, because 5 � 8 is 40.“If Daniel uses 5 and 8, which number will he continueto factor?”• 8, because 5 is prime, but 8 is composite.

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Lesson 4: Identifying Factors by DividingCopyright © 2010 Nelson Education Ltd.


Checking (Pairs)

Remind students to factor each number using repeateddivision or factor trees before determining different factorsthat are prime. Point out that they are not finding thenumber of factors that are prime, rather they are finding thenumber of different or distinct factors that are prime.


These questions provide students with opportunities topractise identifying the factors of a number. Remind studentsthat they can use other appropriate strategies besides thefactor tree.4. a) Ask students how Manon got 32, and then have them

work backward to get the top number.

Answers to Key Question4. a) 96; for example, she divided a number by 3 and got

32. So the number must be 3 � 32 � 96.b) For example, once you get to 16, you can divide only

by 2.c) For example, if I divide 16 by 2, I get another factor, 8.


Question 7 allows students to reflect on and consolidate theirlearning for this lesson. Students should familiarize themselveswith using repeated division or factor trees to determine factorsof a composite number. Encourage students to determine thefactors of various composite numbers. Observe the numberof factors that are prime in each composite number.

Answer to Closing Question7. For example, not always. 16 is greater than 12. But 2 is

the only prime number that is a factor of 16, while 2 and3 are two different factors that are prime for 12.

Follow-Up and Preparation for Next ClassAt home, students can practise factoring two-digit numberswith the help of a parent or siblings. Students can presenttheir factor trees on poster paper and bring the poster to classto display on the wall.

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Answers to PromptsA. 3, 5, 9, and 45B. 2, 4, 5, 20, and 40C. No; for example, 2 is the only prime number that is a

factor of the quotient 4 in his factor tree and he alreadydivided by 2.

D. 42: 3 points; 44: 2 points; 45: 2 points; 46: 2 points;48: 2 points; 49: 1 point; 50: 2 points; 42 has thehighest score.


Remind students that a factor of a number is any numberthat can be divided into that number and leave no remainder.

Answers to Reflecting QuestionsE. For example, 2 is prime and a factor of every even

number. So you score at least 1 point.F. No. For example, when I kept dividing 32 by 2, I kept

getting numbers that I could divide by 2. So I onlyscored 1 point.

2 16

2 8

2 4

32

33






• Students divide any composite number less than 100 by a prime number.

Key Question 4 (Reasoning)• Students use a factor tree to determine factors of a composite number.

• Students may not know how to select a prime number to use as a divisor of acomposite number. (See Extra Support 1 or 2.)

• Students may not recognize when a number in the factor tree can be factoredfurther. (See Extra Support 3.)


EXTRA SUPPORT

EXTRA CHALLENGE

• Mai conjectures that all even composite numbers will have more factorsthat are prime (repeated or non-repeated) than odd composite numbers, ifboth numbers have the same tens digit. Challenge students to exploreMai’s conjecture. For example, 45 has more factors that are prime, (3, 3,and 5) than 46 (2 and 23).

1. Have students look at the first 10 multiples of 2 and see what they notice.They can talk about how that might help them recognize other multiples of 2.Do the same with the first 10 multiples of 5.

2. Ensure students notice that multiples of 3 are 3 apart. For example, how dothey know that 3 is not a factor of 85? Since they know that 90 is a multipleof 3, they could count back by 3s to see that 87 and 84 are multiples of 3.Thus, 3 is a factor of 87 and 84, but not of 85.

3. Students might benefit from using square tiles to see if a particular numberof square tiles can be rearranged into a rectangle that is not 1 unit wide.This would mean that some of the factors on the tree can be factoredfurther.


• Encourage students to use counters to model the factors of each number from10 to 20. To determine the factors of 18, for example, a student may begin byarranging 18 counters into an array of 6 rows of 3 counters each. Have themwrite 6 and 3 as factors of 18. Next have them try to arrange 3 counters into

an array other than 1 row of 3 or 3 rows of 1 and determine that 3 is prime.Since 6 counters can be arranged in an array of 2 rows of 3 counters each, 6 is a composite number with factors 2 and 3 that are prime. Thus, studentsdetermine that 2 and 3 are factors of 18 that are prime.


Chapter 3 5555

35Lesson 5: Creating Composite NumbersCopyright © 2010 Nelson Education Ltd.

Creating CompositeNumbers



• Multiply and divide combinations of one-digit and two-digit numbers.

• Identify prime and composite numbers.

SPECIFIC OUTCOME


than 100• identifying prime and composite numbers• solving problems involving multiples[PS, R, V]

Achievement Indicators• Identify the factors for a given number and explain the

strategy used.• Provide an example of a prime number and explain

why it is a prime number.• Provide an example of a composite number and

explain why it is a composite number.• Solve a given problem involving factors or multiples.

GOAL

Multiply combinations of factors to create composite numbers.


Materials • Optional: chart paper and markers

Masters • Optional: 100 Chart, Masters Booklet p. 30• Optional: 1 cm Grid Paper, Masters Booklet

p. 22

Key Question entire exploration

Extra Practice Workbook p. 21

Mathematical PS (Problem Solving) and R (Reasoning)Process Focus


Math BackgroundStudents will begin this activity using reasoning todetermine the prime numbers from 1 to 50. Students havealready learned that prime numbers have only two factorsand that composite numbers have more than two factors.Now they will explore the implications of that distinctionin the context of solving a problem. They will have theopportunity to recognize that prime numbers can bemultiplied to make any composite number from 2 to 50.

Optional: 100 Chart, Masters Booklet p. 30

Optional: 1 cm Grid Paper, Masters Booklet p. 22




Review with students the various ways they can identifyfactors of numbers. These include forming arrays of counters,factor rainbows, repeated division, and factor trees.

Sample Discourse“How would you identify the factors of 24?”• I would start by listing 1 and 24 because 1 � 24 � 24.

I know 2 � 12 � 24 so 2 and 12 are factors. 3 � 8 � 24 so3 and 8 are factors. 4 � 6 � 24 so 4 and 6 are factors. Thefactors of 24 in order are 1, 2, 3, 4, 6, 8, 12, and 24.

“How can you use your list of factors of 24 to tell whether 24is a prime or composite number?”• It’s not a prime number because it has more than two different

factors.“In your list of factors of 24, which factors are primenumbers?”• 2 and 3 are the only two prime numbers that are factors of 24.“Can you multiply combinations of only 2 and 3 to get 24?”• Yes; if you calculate 2 � 2 � 2 � 3, you get 24.

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Teaching and Learning(Pairs/Small Groups) � 15–25 min

With students, read about Oleh’s licorice-stretching machineon Student Book page 85. Clarify that the buttons on Oleh’smachine can be pressed more than once to stretch thelicorice. Ask them why a 1 button wouldn’t be needed on themachine. Ensure they understand that pressing the 2, 3, and5 buttons means 2 � 3 � 5 � 30, which produces a length30 times as long as the original. Also, pressing the 2 buttonthree times means 2 � 2 � 2 � 8.

Read the central question and have students work in smallgroups to answer it. Have available 100 charts (or at least thenumbers from 1 to 50); grid paper; chart paper; and markers.

Explain that students are to• write out the main points in their solution on chart paper• be prepared to communicate their solution process to the

rest of the class• describe the buttons needed to stretch the licorice using

multiple stretches from 2 to 50No one approach to the problem should be suggested.

Encourage students to choose their own methods. They musttake the information given and work toward a solution.Circulate and observe students as they work.

Sample Discourse“What buttons can you include to stretch the licorice 12 times as long as the original licorice?”• I can use a 6 and a 2.• I can use a 12 and a 1.• I can use a 2 and a 2 and a 3.“How does the length of the licorice increase when you pressa 2 button or a 3 button three or four times?”• If I press a 2 button three times, I have 2 � 2 � 2 � 8. If I

press a 2 button four times, I have 2 � 2 � 2 � 2 � 16.• If I press a 3 button three times, I have 3 � 3 � 3 � 27.“What button do you need to stretch the licorice 17 times aslong as the original licorice?”• I need a 17 button.“Why do you only need buttons that are prime numbers?”• I only need buttons that are prime numbers because pressing

prime-number buttons once will give the prime numbers neededand pressing combinations of prime-number buttons will giveall the composite numbers needed.

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Lesson 5: Creating Composite NumbersCopyright © 2010 Nelson Education Ltd. 37

Possible SolutionsSample Solution 1:We used the numbers from 2 to 50 to keep track of buttonsthat aren’t needed. Then we made a chart to show one waythe needed buttons can be used to stretch licorice from 2 to50 times. The order of pressing the buttons does not matter.

The only buttons we need to include are the prime numbersfrom 2 to 47. You can stretch the licorice from 2 to 50 timesby pressing combinations of the prime-number buttons.

24 Press 2 three times,then 3

Press 2, then 13

34 Press 2, then 17

Need

37 Need

Need 43 Need

31 Need

21 Press 3, then 7

18 Press 2, then 3 twice

15 Press 3, then 5

12 Press 2 twice, then 3

9 Press 3 twice

6 Press 2, then 3

Need3

25 Press 5 twice

Press 2 twice, then 728

Press 2 three times,then 5

40

Press 2, then 2346

Press 7 twice49


Press 2, then 7

Press 2 three times

Need

Need

Need

Need

Need

27 Press 3 three times

30 Press 2, then 3,then 5

Press 2 five times

Press 5, then 7

Press 2, then 19

Need


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50 Press 2, then 5 twice

22 Press 2, then 11

19

16 Press 2 four times

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Press 2 twice

7

Press 2, then 5

Need

Need

Need

4

33 Press 3, then 11

36 Press 2 twice, then3 twice

39 Press 3, then 13

42 Press 2, then 3,then 7

45 Press 3 twice, then 5

48 Press 2 four times,then 3

Sample Solution 2:We discovered that we can stretch licorice a compositenumber of times from 2 to 50 by using only the prime-number buttons. We made a chart. The checkmarks showwhat buttons you must push. Sometimes you have to pressthe same button more than once.

CompositeNumber

x2

x3

x5

x7

x11

x13

x17

x19

x23

4 ¸

6

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30

¸ ¸

¸

¸

¸ ¸

¸ ¸

¸ ¸

¸ ¸

¸

¸ ¸

¸

¸

¸ ¸ ¸

32 ¸

33 ¸ ¸

34 ¸ ¸

38 ¸ ¸

46 ¸ ¸

39 ¸ ¸

40 ¸ ¸

50 ¸ ¸

44 ¸ ¸

35 ¸ ¸

36 ¸ ¸

45 ¸ ¸

49 ¸

48 ¸ ¸

42 ¸ ¸ ¸

¸

¸

¸

¸ ¸

¸

¸ ¸

¸

¸ ¸

¸





Provide an opportunity for students to share and communicateabout their work. Have students describe to the rest of theclass how they solved the problem, using chart paper as anorganizing tool for students to follow. Ask students tocomment on the approach presented. The presenters may

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invite questions from other students and attempt to answerthe questions. Encourage students to identify similarities anddifferences among their methods.

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




• Students use different combinations of prime numbers to calculate as manyproducts as possible. Students then use reasoning to discover that all theprime numbers from 2 to 47 will yield every number needed.

• Students may not identify the combinations of prime numbers needed to formthe composite numbers. This may result in incomplete lists of numbersneeded or a list containing a mix of prime and composite numbers. (See ExtraSupport 1.)


EXTRA SUPPORT

EXTRA CHALLENGE

• Oleh believes that his machine can also stretch the licorice for all compositevalues from 51 to 100 times without adding any additional buttons.Challenge students to show whether Oleh’s belief is correct or incorrect.

• Have students work in pairs to answer questions such as the following:

If the licorice-stretching machine works for 9, but not for 12, what button isbroken?If the licorice-stretching machine works for 16, but not for 28, what button isbroken?Challenge students to formulate similar questions for a partner to answerand explain.

1. Remind students that factors can be repeated when calculating a product.Point out that this will allow them to use fewer buttons, but still arrive at the correct product. For example, have students use repeated division of 16to see that only the 2 button is needed. Help students see that whenever acomposite button is left, it could be replaced by other buttons, e.g., 6 by 2and 3 or 10 by 2 and 5. They may use repeated division by primes or factortrees to determine the prime numbers needed and use a chart to organizetheir findings.


• For some students, the abstractness of the context may be a problem. Allowthese students to continue to explore the concept of a number being primeusing a more concrete model. For example, tell students that they are tryingto create paper strips of all the lengths from 2 cm to 50 cm using as fewstrip lengths as possible. Have them use paper strips of lengths 2 cm, 3 cm,5 cm, and 7 cm. Ask them to use each strip more than once, but as manytimes as they want to try to make a total length. For example, three 2 cmstrips can be used to make 6 cm. They can record which lengths they areable to make and explore why these are composite numbers (since they aregroups of another number).


• Rather than presenting their work to the whole class, some students mightprefer presenting to a smaller group.



39Mid-Chapter ReviewCopyright © 2010 Nelson Education Ltd.

Mid-Chapter Review STUDENT BOOK PAGES 86–87

SPECIFIC OUTCOME




strategy used to identify them.• Determine all the whole-number factors of a given

number using arrays.• Identify the factors for a given number and explain the

strategy used, e.g., concrete or visual representations,repeated division by prime numbers, or factor trees.

• Provide an example of a prime number and explainwhy it is a prime number.

• Provide an example of a composite number andexplain why it is a composite number.

Preparation and PlanningMaterials • Optional: counters

Masters • Mid-Chapter Review—Frequently AskedQuestions p. 79


Mid-Chapter Review—Frequently AskedQuestions p. 79

Reading StrategyMonitoring Comprehension is a strategy that readersuse when what they are reading does not make sense.Effective readers try several approaches to find meaningwhen they have trouble understanding something theyare reading. Often they look at the context for clues tofigure out unknown words. In mathematics, studentsmight encounter new vocabulary or a challengingprocedure that affects comprehension. When thisoccurs, students need to call on other known strategiesor a combination of strategies such as visualizing,questioning, predicting, summarizing, inferring, andrereading to regain comprehension.

Use a self-questioning strategy with the class. Havestudents use key words from the first half of the chapterand use a check mark to signify their level ofunderstanding for each term.

Have students share their responses in pairs and telleach other what they know about each term. Use theglossary at the back of the Student Book to check.

Students are prompted to use a MonitoringComprehension strategy in the Practice questions of this Mid-Chapter Review.

Key Word Lots! Some Not Much

factor

factor rainbow

multiple



Frequently Asked Questions (Whole Class)

Have students keep their Student Books closed. Write theFrequently Asked Questions on Student Book page 86 on theboard, or use Mid-Chapter Review—Frequently AskedQuestions p. 79. (Distribute the master or display it usingan overhead transparency.) Use the discussion to draw outwhat the class thinks is the best answer to each question.Then have students compare the class answers with theanswers in the Student Book. Have students summarize theanswers in their own words as a way of reflecting on theconcepts. Students can refer to the answers to the FrequentlyAsked Questions as they work through the Practice questions.

At this time, you can also discuss any other questionsrelated to Lessons 1 to 5 that students may have.

Practice (Individual)

Students should be able to complete all the questions in class.For Question 5, encourage students to identify all thepossible two-digit numbers that can be spun by making a listor chart, e.g., 22, 23, 24, 25, 32, 33, 34, 35, and so on.

Encourage students to identify which questions they foundeasy and which more challenging. Ask them what they cando to become more proficient at questions they foundchallenging. The review questions are organized by lesson.Students can go back to the lesson indicated to review theconcepts for the question.

Using the Mid-Chapter ReviewThis review provides an opportunity for students to monitortheir progress with the chapter skills and concepts (Assessmentas Learning), as well as for you to monitor the progress of theclass and see where re-teaching may be required (Assessmentfor Learning). You may also use it to assess individual studentachievement (Assessment of Learning).


Mid-Chapter ReviewCopyright © 2010 Nelson Education Ltd. 41

(Continued on next page)


Refer to the Differentiating Instruction ideas in Lessons 1 to 5.

Question 6 (Reasoning, Visualization)• Students identify factors from a factor tree.

Question 7 (Reasoning)• Students identify three possible numbers that have three different prime

numbers as factors.

• Students may not be able to interpret the factor tree.

• Students may not be able to identify one or more numbers that have threeprime numbers as factors.




Question 1 (Visualization)• Students use arrays to identify the factors of 40 and connect the dimensions

of the arrays with the factors of 40.

Question 2 (Reasoning)• Students calculate factors of numbers.

Question 3 (Problem Solving)• Students use multiples to calculate the years after 2007 in which the

Women’s World Cup will occur.

Question 4 (Reasoning)• Students calculate multiples of numbers and explain their reasoning clearly

and concisely.

Question 5 (Problem Solving)• Students identify the prime numbers and composite numbers that can be

formed with the digits 2, 3, 4, and 5.

• Students may have difficulty identifying all of the possible arrays for 40. Theymay not connect the numbers of rows and columns with the pairs of factorsthat multiply to 40.

• Students may not identify all of the factors.

• Students may have difficulty identifying the numbers to multiply together orthe number by which to skip count to calculate the years in which theWomen’s World Cup will be played.

• Students may have difficulty identifying the numbers to multiply together orthe number by which to skip count. Students may arrive at correct answersbut not be able to explain their thinking.

• Students may be confused by the numbers in a new context (forming two-digit numbers using a spinner) and not recognize a simple problem in whichthey must identify prime and composite numbers.

Work meets standard Work meets standard Work meets Work does not yet meet of excellence of proficiency acceptable standard acceptable standard• uses visual representations

insightfully to demonstrate athorough understanding of factors

• uses visual representationsmeaningfully to demonstrate areasonable understanding of factors

• uses visual representations simply todemonstrate a basic understanding offactors

• uses visual representations poorly todemonstrate an incompleteunderstanding of factors

Question 1, written answer, model Specific Outcome and Process Focus: N3 [V]• A veterinarian has 40 indoor dog kennels.

a) What arrays can she form with 40 kennels?b) How can you use the arrays in part a) to identify all the factors of 40?

Assessment of Learning—What to look for in student work

Question 2, short answer Specific Outcome and Process Focus: N3 [R]• Identify the factors of each number.

a) 14 b) 45 c) 54 d) 75

(Score 1 point for each correct answer for a total out of 4.)


Work meets standard Work meets standard Work meets Work does not yet meet of excellence of proficiency acceptable standard acceptable standard• chooses efficient and effective

strategies to identify possibleages

• chooses workable and reasonablestrategies to identify possible ages

• chooses partially appropriate andworkable strategies to identifypossible ages

• chooses inappropriate and/orunworkable strategies to identifypossible ages

Work meets standard Work meets standard Work meets Work does not yet meet of excellence of proficiency acceptable standard acceptable standard• demonstrates an insightful

understanding of the problem

• differentiates between relevantand irrelevant information

• develops a thorough plan forsolving the problem

• chooses an efficient and effectivestrategy; may demonstratecreativity and innovation inhis/her approach

• demonstrates a completeunderstanding of the problem

• identifies relevant information

• develops a workable plan forsolving the problem

• chooses an appropriate andworkable strategy

• demonstrates a basic understandingof the problem

• identifies some relevant information

• develops a basic plan for solving theproblem

• chooses a simplistic and/or routinestrategy

• demonstrates a limitedunderstanding of the problem

• has difficulty discerning relevantfrom irrelevant information

• develops a minimal and/or flawedplan for solving the problem

• chooses an inappropriate orunworkable strategy


Question 5, short answer Specific Outcome and Process Focus: N3 [PS]• You can form a two-digit number by spinning the spinner twice. The first number spun is the tens digit.

The second number spun is the ones digit. How many more composite numbers than prime numbers can be spun?

Question 6, short answer Specific Outcome and Process Focus: N3 [R, V]• What factors of 48 can you identify from the factor tree at the left?

(Score 1 point for all factors listed for a total out of 5.)

Question 7, written answer Specific Outcome and Process Focus: N3 [R]• Pablo found that his uncle’s age can be divided by three different prime numbers.

What are three possible ages for his uncle? Show your work.

Work meets standard Work meets standard Work meets Work does not yet meet of excellence of proficiency acceptable standard acceptable standard• chooses efficient and effective

strategies to identify multiples• chooses workable and reasonable

strategies to identify multiples• chooses partially appropriate and

workable strategies to identifymultiples

• chooses inappropriate and/orunworkable strategies to identifymultiples



• develops a thorough plan forsolving the problem


• develops a workable plan forsolving the problem


• develops a basic plan for solving theproblem


• develops a minimal and/or flawedplan for solving the problem


Question 4, short answer, written answer Specific Outcome and Process Focus: N3 [R]• Identify the first five multiples of each number. Explain what you did for one number.

a) 11 b) 22 c) 20 d) 35

Question 3, short answer Specific Outcome and Process Focus: N3 [PS]• The Women’s World Cup of soccer is held every four years. The World Cup was played in China in 2007.

In what years will the five World Cups after China be played?

(Score 1 point for each correct year for a total out of 5.)


Chapter 3 6666

43Lesson 6: Solving Problems Using an Organized ListCopyright © 2010 Nelson Education Ltd.

Solving Problems Usingan Organized List



• Identify factors and multiples of whole numbers.• Identify prime and composite numbers.

SPECIFIC OUTCOME




strategy used to identify them.

GOAL

Use an organized list to solve problems that involvenumber relationships.



Masters • Optional: 100 Chart, Masters Booklet p. 30

Recommended Questions 2, 6, & 7Practising Questions


Extra Practice Chapter Review Question 9Workbook p. 22

Mathematical PS (Problem Solving) and R (Reasoning)Process Focus


Reading StrategyFinding Important Information is a reading strategythat students use to focus their attention on useful parts of the text and ignore irrelevant information. Inmathematics, students identify the question being asked,decide the most relevant information needed to answerthe question, and categorize the rest of the informationas useful or not useful. Knowing essential informationmakes problem solving manageable.

Students are prompted to use a Finding ImportantInformation strategy in Question 1. As you discuss theproblem with students, ask them to identify the factsgiven in the problem. Then have them identify whichfacts are not necessary for solving the problem. Ask themto state in their own words what the problem asks themto find out, and discuss strategies for solving the problem.

Math BackgroundWhen there is more than one condition to be satisfied inorder to solve a problem, making an organized list is anappropriate strategy. An organized list can be written tosatisfy the initial condition and then the list can benarrowed down, based on additional conditions. Thisproblem-solving strategy allows students to reason thatno possible solution has been overlooked or eliminated inerror. The conditions that students will work with in thislesson involve multiples, prime and composite numbers,and factors. Students will apply what they have learned inprevious lessons about these concepts to arrive at a solutionto each problem. For example, students will identify amultiple of two different numbers by listing multiples ofthe first number and then identifying multiples of thesecond number in the same list.

Optional: 100 Chart, Masters Booklet p. 30

• Identify the factors for a given number and explain thestrategy used, e.g., concrete or visual representations,repeated division by prime numbers, or factor trees.





To prepare students for making organized lists, use skillslearned in previous lessons to play “What’s my number?” Tellstudents that you are thinking of a number from 10 to 16whose factors include 1, 2, 3, 4, and 6. Ask students to writetheir answers on a piece of paper. When everyone is finished,have students hold up their answers. Ask several studentswhat method they used to determine the number.

Sample Discourse“What method did you use to determine the number?”• I tried each even number because 2 is a factor.• I eliminated the prime numbers 11 and 13 first.• I wrote down each number from 10 to 16 and tested to see

if it was the number.“Why did you write the numbers down?”• It was a good way to keep track of each number as I tested

whether or not it was the solution.“How did you keep track of the numbers as you testedthem?”• I crossed off the numbers that did not have all the factors.“What is my number?”• Your number is 12.Repeat the activity with each clue below.• I am thinking of a prime number between 20 and 28. (23)• I am thinking of a number between 16 and 26 that is a

multiple of 9. (18)

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

Together, read the information about cones for Sage’s jingledress and then read the central question on page 88 of theStudent Book. Discuss what information can be used to solvethe problem. Together, read Mai’s understanding of theproblem. Point out that Mai has stated what she needs todetermine and the conditions that must be met to answer thequestion. Work through the rest of Mai’s Solution together.

Sample Discourse“After Mai understands the information given in theproblem, how does she plan to solve the problem?”• She plans to make a list of possible numbers of cones, starting

with multiples of 4 between 20 and 50.• She lists the multiples of 4, starting with 24 and ending at 48.“As Mai carries out her plan, she must consider moreinformation about the number of cones. What else does sheknow about the number of cones?”• The number of cones is a multiple of 3.• The cones can be arranged in three equal rows with none

left over.“Which multiples of 4 are also multiples of 3?”• The numbers 24, 36, and 48 are also multiples of 3.“How could you check that Mai’s answer of 24, 36, or48 cones meets all the conditions given in the problem?”• Each number is between 20 and 50, each number has 4 as

a factor, and each number has 3 as a factor.

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Lesson 6: Solving Problems Using an Organized ListCopyright © 2010 Nelson Education Ltd. 45


Students reflect on why using an organized list is a goodproblem-solving strategy for this problem.

Answer to Reflecting QuestionA. For example, an organized list was a good strategy for

Mai to use because it allowed her to list all possibleanswers. She could list numbers based on one of theclues. Then she could use the other clues to eliminatesome of the numbers she listed for the first clue.


Checking (Pairs)

Have students identify the conditions that must be satisfiedin the problem. Encourage students to restate theseconditions in their own words as Mai did in the Understandpart of her problem-solving plan. Have student pairs make aplan to solve the problem and list the steps they will take tocarry out the plan. The plan they carry out must address allinformation given about the number of cones, so it isimportant that students identify that information correctly.Remind students that an array represents a pair of factors of anumber.


Tell students that an organized list can give them a good startto the problem. Suggest that they use the information in each problem to list all the possible answers and then useadditional information to add to and/or narrow down the list.2. Students should begin by listing the prime numbers

between 20 and 50.6. Remind students that all two-digit numbers formed

using the spinner will yield numbers from 11 to 99inclusive. None of the numbers will have 0 as the onesdigit, because there is no 0 on the spinner.

Answer to Key Question6. Natalie. For example, use the problem-solving process.

Understand: Since the spinner contains the numbers 1 through 9 and each girl spins the spinner twice, it ispossible to create any two-digit number between 11 and99 that doesn’t have 0 as the ones digit. I need todetermine even multiples of 7 and odd multiples of 9.Make a Plan: I will list all the two-digit numbersbetween 11 and 99 that are multiples of 7 and 9 anddon’t have 0 as the ones digit. Then, I will circle the even multiples of 7 and odd multiples of 9.

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Carry Out the Plan: This is the list of two-digit numbersbetween 11 and 99 that are multiples of 7 or multiples of9 and don’t have 0 as the ones digit.

The circled numbers are the even multiples of 7 and theodd multiples of 9.

There are six even multiples of 7 between 11 and 99 thatdon’t have 0 as the ones digit, so Natalie has seven waysto score 1 point. There are five odd multiples of 9 between11 and 99, so Gwen has five ways to score 1 point.Natalie has more ways to score 1 point.Look Back: I checked all the circled numbers to see ifthey match the conditions.

14 � 7 � 2 28 � 7 � 4 42 � 7 � 656 � 7 � 8 84 � 7 � 12 98 � 7 � 14 �27 � 9 � 3 45 � 9 � 5 63 � 9 � 7 �81 � 9 � 9 99 � 9 � 11My solutions are reasonable.


Question 7 allows students to reflect on and consolidate theirlearning for this lesson. Ask students to remember to findsomething the numbers 42, 45, and 48 have in commonbefore they begin writing the problem.

Answer to Closing Question7. For example, I created this problem:

Shaun has between 40 and 50 model cars in hiscollection.The number of cars is a multiple of 3.What are the possible numbers of cars in Shaun’scollection?I solved the problem by listing the multiples of 3 from40 to 50: 42, 45, and 48. Shaun has 42, 45, or 48 cars inhis collection.

Follow-Up and Preparation for Next ClassHave students research the locations of the warmesttemperatures (in Celsius) in Canada using the Internet,newspapers, or magazines. Tell them to organize their findingsin a table and bring it in for discussion in the next class.

Multiples of 7: 14 21 28 35 42 49 56 63 77 84 91 98

Multiples of 9: 18 27 36 45 54 63 72 81 99

Multiples of 7: 14 21 28 35 42 49 56 63 77 84 91 98

Multiples of 9: 18 27 36 45 54 63 72 81 99






• Students use an organized list as a problem-solving strategy.

Key Question 6 (Problem Solving, Reasoning)• Students make an organized list to determine each girl’s chances of scoring

1 point and then use reasoning to determine who has more chances to scorepoints.

• Students may not consider all the information given in the problem. (See ExtraSupport 1.)

• Students may not understand that each girl’s situation is an individualproblem to be solved first. Then the results must be compared to solve theproblem. (See Extra Support 2.)


EXTRA SUPPORT

EXTRA CHALLENGE

• Challenge students to use an organized list to determine the possiblenumbers of packages of hot dogs and numbers of packages of hot dog rollsfor a crowd of between 50 and 100 people. Hot dogs come in packages offour and rolls come in packages of six. Plan to provide two hot dogs for eachperson. Students should be prepared to explain their solution to the class.

1. Point out to students that there is a connection between the informationthey state in the Understand part of the problem-solving plan and the stepstaken to carry out the plan. Suggest that students restate each conditionfrom the problem in one column on their paper and write the step taken toaddress that condition next to it.

2. Remind students that since both girls will spin the same spinner, thepossible range of numbers for each girl is the same. Tell students that afterthey determine the possible range of numbers, they should consider eachgirl’s ways of scoring in separate problems. Lastly, they should decide whichplayer has more ways to score points by comparing the results of eachseparate problem they solved.


• Some students may have difficulty making an organized list. Allow thesestudents to use a model to look for possible solutions. For example, to modelthe multiples of 3 and multiples of 4 given in the opening question, studentscan make a 3-by-4 array. One array has 12 counters.

Ask students if 12 counters are a possible solution. Because 12 is less than20, 12 is not a possible solution.

Have students extend the array by forming another 3-by-4 array and countingthe total number of counters. Emphasize that they can keep extending the 3-by-4 array because this array is already in multiples 3 and multiples of 4. Thiswill result in the following sets of arrays having multiples of 3 and multiples of 4.

This array has 24 counters. Since 24 is between 20 and 50, and it’s still amultiple of 3 and 4, 24 is one of the solutions.

Extend the array by repeating the process.

This array has 36 counters. Since 36 is between 20 and 50, and it’s still amultiple of 3 and 4, 36 is another solution.

Extend the array by repeating the process.

This array has 48 counters. Since 48 is between 20 and 50, and it’s still amultiple of 3 and 4, 48 is another solution.

Students should realize that if they repeat the process one more time, thetotal number of counters will exceed 50, which is not part of the solution. SoSage could have 24, 36, or 48 metal cones for the jingle dress.


• Some students may benefit from starting with a visual representation ofpossible solutions. Have students work with a 100 chart and start eachproblem by highlighting the range of possible solutions on the chart. Suggest that students use different colours to circle possible numbers foreach condition given in the problem. Only those values that meet all theconditions can be a solution to the problem.


Chapter 3 7777

47Lesson 7: Representing IntegersCopyright © 2010 Nelson Education Ltd.

Representing Integers STUDENT BOOK PAGES 90–92

GOALUse integers to describe situations.

Pacing 5–10 min Introduction(allow 5 min for 10–15 min Teaching and Learningprevious homework) 20–30 min Consolidation

Masters • Number Lines, Masters Booklet p. 33

Recommended Questions 3, 4, & 6Practising Questions


Extra Practice Chapter Review Question 10Workbook p. 23

Mathematical R (Reasoning) and V (Visualization)Process Focus

Vocabulary/Symbols integer, opposite integer


Preparation and Planning


• Locate numbers on a number line.

SPECIFIC OUTCOME

N7. Demonstrate an understanding of integers,concretely, pictorially, and symbolically.[C, CN, R, V]

Achievement Indicators• Extend a given number line by adding numbers

less than zero and explain the pattern on each sideof zero.

• Describe contexts in which integers are used, e.g., on a thermometer.

Math BackgroundStudents are familiar with the set of whole numbers,which includes the counting numbers and zero. In thislesson, students are introduced to the set of integers,which includes positive and negative whole numbers andzero. A number line is used as a visualization tool for theset of integers. Positive integers are integers to the right ofzero on a number line. Negative integers are integers tothe left of zero on a number line.

The purpose of this lesson is to help students reasonand understand that many contexts exist where integersare used, and to understand the relationship amongpositive numbers, negative numbers, and zero. Studentswill also be introduced to the concept of opposite integers,or integers that are the same distance from zero, but onopposite sides on a number line. For example, �6 and –6are opposite integers.

Although a positive (�) sign is not often used to denotepositive integers, students will use both the positive andnegative (�) signs throughout this lesson to solidify theirunderstanding. However, zero is never written with apositive or negative sign. This point will be formally madein the next lesson on temperatures.


01-NM6TR-C03-Interior_L07.qxd 12/1/08 10:24 PM Page 47



If students collected data about the warmest temperatures inCanada in the follow-up to the previous lesson, invite them topresent that data. On the board, on a transparency, or on aninteractive whiteboard, draw a number line from 0 to 20.Have students locate the position of one of their temperatureson the number line. Make sure everyone in the class is usingtemperatures in Celsius. Have students practise moving upand down the number line; for example, have students locatea number that is between two temperatures. Talk about howstudents can use the number line to identify numbers.

Sample Discourse“How can you identify a number that is between 10 and 15?”• I can look for marked numbers between 10 and 15, such as

11, 12, 13, and 14.• I can pick any number that is to the right of 10 and to the

left of 15.

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Teaching and Learning(Whole Class/Pairs) � 10–15 min

Together, discuss the information and central question aboutJason’s cursor on Student Book page 90. Distribute numberlines or have students sketch number lines, and work throughJason's Number Line together. Draw students’ attention tothe definition of integers. Ensure students understand thatpositive integers are to the right of zero on the number lineand negative integers are to the left of zero.

Have students work through Prompts A to C in pairs, andthen discuss the answers as a class.

Answers to PromptsA. –2B. He pressed six times to get to –6.C.

+9 +10+8+7+6+5+4+3+2+10–1–2–3–4–5–6–7–8–9–10

d

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Lesson 7: Representing IntegersCopyright © 2010 Nelson Education Ltd. 49


Draw students’ attention to the margin definition of oppositeintegers. Talk about other examples of opposites related todirections, such as east and west, right and left, and up anddown. Have students work through Prompt D individuallyand then discuss the answer as a class.

Sample Discourse“How do you know east and west are opposites?”• Because I would go in one direction to go east and in the

opposite direction to go west.“How do you know right and left are opposites?”• For example, if two people stood in the centre of the room and

one walked to the right and the other walked to the left, theywould end up on opposite sides of the room.

Answer to Reflecting QuestionD. For example, they are opposite integers because they are

both 4 units from 0, but in opposite directions. You canuse n to move the cursor 4 units from 0 to the right,but you need to use m to move the cursor 4 units from0 in the opposite direction.


Checking (Pairs)

Ask for volunteers to show solutions on the board and discussthe solutions as a class. Provide students with number lines.

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These questions provide students with opportunities to applytheir understanding of integers. Students will use reasoningto identify integers. Remind students to use number lines tohelp them visualize the relative positions of the integers.4. Tell students to look for the integers between each pair,

but not including the pair.

Answers to Key Question4. a) The number line shows the integers between –4 and

�4. The integers �3, �2, �1, 0, �1, �2, and �3are between �4 and �4.

b) The number line shows the integers between –3 and0. The integers between �3 and 0 are �2 and �1.

c) The number line shows the integers between –2 and –5. The integers between �2 and �5 are �3and �4.

d) The number line shows that there are no integersbetween 0 and –1.


Question 6 allows students to reflect on and consolidate theirlearning for this lesson. Students will explain and interpretthe use of integers in a variety of real-life situations.Encourage students to share their solutions, and try to elicit avariety of examples.

Answer to Closing Question6. For example, my birthday is on April 13. So I can write

–3 to represent 3 days before my birthday, or April 10,and I can use �3 to represent 3 days after my birthday,or April 16.For example, if a car is 10 km north, I can representthe distance as �10. If the car is 10 km south, I canrepresent the distance as –10.For example, if I take $5 out of the piggy bank, I canwrite –5 to show that the amount in the piggy bank is$5 less. But if I added $5, I can write �5 to show thatthe amount in the piggy bank is $5 more.

Follow-Up and Preparation for Next ClassHave students find additional examples of situations in theirdaily lives that can be represented by integers. Have studentswrite down the examples in their notebooks and share withtheir classmates. This will help to solidify their understandingof integers.

0 +1+2+3 +5+4–1–2–5 –4 –3

0 +1+2+3 +5+4–1–2–5 –4 –3

0 +1+2+3 +5+4–1–2–5 –4 –3

0 +1+2+3 +5+4–1–2–5 –4 –3





• Students identify integers based on clues and number patterns.

• Students interpret integers in the contexts of different situations.

Key Question 4 (Reasoning, Visualization)• Students identify the integers between two integers.

• Students may have difficulty using clues to locate negative integers on anumber line. (See Extra Support 1.)

• Students may be confused by the context and may not be able to interpretpositive and negative values. (See Extra Support 2.)

• Students may have difficulty locating the integers or may miss some of thepoints in between as they mark the number line. (See Extra Support 3.)


EXTRA SUPPORT

EXTRA CHALLENGE

• Challenge students to compose puzzle questions involving integers forclassmates to solve. Puzzles may take the form of “What integer am I?” andinclude clues as to where the integer may be found on a number line.


• Some students may have difficulty conceptualizing negative numbers.Provide these students with additional examples of situations they mayencounter that can be represented with integers. For example, an elevatorat an office building might use “G” to represent the ground floor. The floor

numbers (1, 2, 3, and so on) are positive because you go up in the elevator toreach them, and the parking levels (P1, P2, P3, and so on) are negativebecause you go down in the elevator to reach them.


• Some students may benefit from using different colours to label thepositive and negative numbers on their number lines. This will help themdiscriminate visually between values greater than and less than zero.

1. Have students begin by practising locating positive numbers on a numberline. Draw a number line from 0 to 10 on the board, on a transparency, oron an interactive whiteboard. Present students with modified versions of theclues from Question 3. For example:a) It is the same distance from 6 as 4 is from 6.b) It is between 3 and 5.c) It is the next integer to the right of 2.d) It is halfway between 0 and 10.

Talk about the strategies students used to identify the numbers described inthe clues. Point out that students can use the same strategies to work withnegative numbers.

2. Explain what 0 represents in each situation: in a rocket launch, 0 means thetime at which the rocket takes off; for days before and after your birthday,0 means the day of your birthday; for kilometres from your town, 0 means the

location of your town; and for money taken from or added to a piggy bank, 0means the number of dollars you started with. Then talk about what positivenumbers mean in each situation and what negative numbers mean in eachsituation.

3. Write the numbers �5 and �5 on the board, on a transparency, or on aninteractive whiteboard. Make sure students understand that the digit5 represents the distance from 0 on the number line, so both �5 and �5 are5 units from 0. Next, point out that the negative (�) sign in front of anumber means that it is to the left of 0 on the number line, or less than 0;the positive (�) sign in front of a number means that it is to the right of 0on the number line, or greater than 0. Display a partially completed numberline on the board with –4, –2, �2, and �4 marked for students to copy.Have students mark the missing integers between –4 and �4. Check tosee that students remember to mark the 0.




Copyright © 2010 Nelson Education Ltd.

Curious MathCountdown Clock


Using Curious MathIn Lesson 7, students were introduced to situations that canbe represented with integers. In this activity, students applytheir understanding in the context of a countdown clock.Students can work through the questions individually. Youmay want to provide copies of number lines, so that studentscan visualize the relative positions of the days in thecountdown. Talk about when the countdown clock woulddisplay a negative integer and when it would display apositive integer.

Sample Discourse“When would the countdown clock display a negativeinteger?”• It would display a negative integer before the school play

because the play is on day 0.• It would display a negative integer on days before the play

because on a number line, those days would be to the left ofthe play, which is on day 0.

“When would the countdown clock display a positive integer?”• It would display a positive integer after the school play because

the play is on day 0.• It would display a positive integer on days after the play

because on a number line, those days would be to the right ofthe play, which is on day 0.


• Locate numbers on a number line.

SPECIFIC OUTCOME


Achievement Indicators• Extend a given number line by adding numbers

less than zero and explain the pattern on each sideof zero.

• Describe contexts in which integers are used, e.g., ona thermometer.

Preparation and PlanningMasters • Optional: Number Lines, Masters

Booklet p. 33

Mathematical CN (Connections) and R (Reasoning)Process Focus


Math BackgroundA countdown clock is a real-life example of using integerswith which students are likely familiar. On a countdownclock, the time of the event is assigned 0. Time before theevent has a negative value, and time after the event has apositive value. Although many students will be familiarwith the idea of counting down to a date, most will havedone so from the perspective of today’s date rather fromthe perspective of the event date; that is, they would haveconsidered the event to be, for example, 5 days in thefuture, rather than considering today to be 5 days beforethe event.

Connections to existing knowledge are made as studentswork with positive and negative integers. Students will usethis reasoning to assign integer values to dates before andafter an event.

Lesson 7: Representing Integers 51

Optional: Number Lines,Masters Booklet p. 33


5. For example, I chose my birthday on March 21. So–3 would represent 3 days before my birthday, or March18; 0 would represent the day of my birthday; �3 wouldrepresent 3 days after my birthday, or March 24.

Answers to Curious Math1. For example, it represents 5 days before the opening

night of the school play.2. –4, –3, –2, –1, 03. 04. �3




• Students use integers to represent days before and after an event. • Students may confuse positive and negative integers. (See Extra Support 1and 2.)


EXTRA SUPPORT

EXTRA CHALLENGE

• Have students create a timeline of recent and future events, assigning today as0. Students can show the different events along the timeline, assigning thedates integer values and drawing pictures to represent the events.

1. Help students relate the countdown clock to a calendar. If possible, showstudents a calendar and ask them to assign a date to the school play, forexample, the 10th of the month. Talk about how different dates in the monthcan be expressed in terms of the date of the play. For example, if today isthe 7th, you would need to subtract 3 from the date of the play to get today’sdate, so in integer terms, today has a value of �3. If today is the 14th of themonth, you would need to add 4 to the date of the play to get today’s date,so in integer terms, today has a value of �4.

2. Students may be accustomed to thinking about an event from today’sperspective, rather than from the perspective of the event. Help studentsconnect these two perspectives. Draw a number line on the board, on atransparency, or on an interactive whiteboard. Mark “today” at 0 and“school play” at �5. Draw another number line below the first so that 0 onthe new line is aligned with �5 on the old line. Discuss what integer wouldrepresent “today” if the school play is 0. Students should see that “today” isaligned with �5 on the new line.


• Some students may have difficulty understanding the concept of why thenumber of days is 5 units before or after the opening night. Instead offocusing on real-life examples of negative integers, have students practise

labelling number lines. Have them put a counter or a small object at acertain number and then tell them to move the object along the numberline to assigned positions. As they move, encourage them to count the unitsout loud.


• Some students will benefit from sketching a number line with the differentdays and the event labelled on it. Students can use different colours toindicate days before (negative integers) and after (positive integers) theschool play.

• Use masking tape or chalk to draw a number line across the classroomfloor. Some students will better understand the concept by walking alongthe number line to an assigned position and figuring out the number ofunits from the initial position.



Chapter 3 8888

53Lesson 8: Comparing and Ordering IntegersCopyright © 2010 Nelson Education Ltd.

Comparing and OrderingIntegers


GOALUse a number line to compare and order integers.


• Locate integers on a number line.• Use the symbols �, �, and � to compare numbers.

SPECIFIC OUTCOME


Achievement Indicators• Place given integers on a number line and explain

how integers are ordered.• Compare two integers; represent their relationship

using the symbols<,>, and �, and verify using anumber line.

• Order given integers in ascending or descending order.


Masters • Number Lines, Masters Booklet p. 33

Recommended Questions 3, 5, 6, 7, 8, & 10Practising Questions


Extra Practice Chapter Review Questions 11 & 12Workbook p. 24

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


Math BackgroundIn Lesson 7, students were introduced to situations thatcan be represented with integers. In this lesson, studentsbuild upon and expand their understanding as theycompare and order integers. Here are some key ideasabout comparing integers:• Numbers become greater as you move to the right

along a number line, and smaller as you move to theleft along a number line.

• Positive numbers are greater than zero and negativenumbers are less than zero.

• Any positive number is greater than any negativenumber.Students will use numbers lines to help them visualize

the relative sizes of integers. Connections are formedbetween positive and negative integers and relative size ofinteger amounts. Students develop mathematicalcommunication skills as they explain their solutions.





On the board, on a transparency, or on an interactivewhiteboard, draw a number line from 0 to 20 but label only0 and 20. Have students locate different positive numbers onthe number line. Talk about how students can use a numberline to compare numbers. Write the following numbersentences on the board and ask volunteers to complete themwith � or �.5 12 10 7 18 9 3 11

Sample Discourse“In which direction do numbers increase on a number line?”• Numbers increase as you move to the right.“How do you know that 15 is greater than 10?”• 15 is to the right of 10 on the number line.• 10 is to the left of 15 on the number line.

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Teaching and Learning(Pairs/Whole Class) � 10–15 min

Together, read about Léa’s report and then read the centralquestion on Student Book page 94. Work through Léa’sComparison together. Point out that Léa’s number line startsat �40 and that 0 is on the right. Ask students if this setupmakes sense considering the temperatures she collected in herchart. Distribute number lines and have students copy Léa’snumber line and mark the low temperature for Iqaluit andYellowknife before working through Prompts A to E in pairs.When students have completed the activity, draw a largenumber line on the board, on a transparency, or on aninteractive whiteboard. Have volunteers mark the hightemperatures on the number line (Prompt E) and describethe strategies they used.

Answers to PromptsA. For example, if the temperature shows a positive integer,

the temperature is above the freezing point of water. If thetemperature shows a negative integer, the temperature isbelow the freezing point of water. If the integer is 0, thetemperature is 0 �C or the freezing point of water.

B. For example, the temperature �31 �C is the farthest tothe left of zero on the number line. So it is the coldesttemperature.

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8


C.

�22 �C is to the left of �21�C, so it is the coldertemperature.

D. For example, to order the temperatures from coldest towarmest, I first picked the lowest temperature, which is�31 �C. Then I chose temperatures in order that are tothe right of �31 �C.Yellowknife & Iqaluit: �31 �CWinnipeg: �23 �CWhitehorse: �22 �CRegina: �21 �CEdmonton: �19 �CVictoria: �1 �C

0 °C–40 °C –10 °C–30 °C –20 °C +20 °C+10 °C

Victoria+1

Regina–21

Edmonton–19

IqaluitYellowknife

–31

Winnipeg–23

Whitehorse–22

Freezing pointof water

E. Yellowknife: �23 �CIqaluit: �22 �CWhitehorse & Winnipeg: �13 �CRegina: �11 �CEdmonton: �8 �CVictoria: �7 �C


Here students form and articulate generalizations aboutcomparing a positive number with a negative number, apositive number with a positive number, and a negativenumber with a negative number.

Answers to Reflecting QuestionsF. For example, positive temperatures are above 0 �C, the

freezing point of water. So they can be shown on theright side of zero on a number line. Negativetemperatures are below 0 �C. They can be shown on theleft side of zero on a number line. So any positivetemperature is greater than any negative temperature.

0 °C–20 °C

NegativeTemperatures

PositiveTemperatures

–10 °C +10 °C +20 °C

55Lesson 8: Comparing and Ordering IntegersCopyright © 2010 Nelson Education Ltd.


Answer to Key Question6. For example, �5 �C is below the freezing point of water so

it is to the left of zero on a number line. Temperatures like�10 �C are to the left of �5 �C and are colder than �5 �C.Temperatures like �1 �C are to the right of �5 �C and arewarmer than �5 �C. Positive temperatures are to the rightof zero, which is to the right of �5 �C, so any positivetemperature is warmer than �5 �C.


Question 10 allows students to reflect on and consolidatetheir learning for this lesson as they articulate the connectionbetween comparing temperatures and comparing integers.

Answer to Closing Question1100.. For example, I can compare �10 and �5 by thinking of

the temperatures �10 �C and �5 �C. �10 �C is colderthan �5 �C so �10 � 5.

Follow-Up and Preparation for Next ClassHave students check the newspaper or the Internet for theweek’s forecasted temperatures and order them from coldestto warmest. Encourage students to present their findings totheir friends or family members. They can elaborate theirpresentation on a number line, compare how many degrees(how many units) apart the temperatures are for certain daysby counting up or down on the number line, and so on.

–5 –4 –3 –2 –1 0–10 –6–7–8–9

–5 °C –1 °C–10 °C

G. For example, it is the same because temperatures to theright are greater than temperatures to the left on anumber line.

�5 is to the right of �1, so �5 �C is warmer than �1 �C.

�5 is to the left of �1, so �5 �C is colder than �1 �C.


Checking (Pairs)

Encourage students to use a number line to help themvisualize the relative positions of the temperatures. You maywant to point out to students that Question 2 a) asks them toorder the temperatures from warmest to coldest, while part b)asks them to order the temperatures from coldest to warmest.


These questions provide students with practice in comparingand ordering integers. Students can use various number linesto help them visualize the relative values of the numbers.9. c) There are some exceptions to the apparent

relationship between surface temperature and averagedistance of planets from the Sun, such as Venus.Students can do research to find the typical surfacetemperatures for the planets that are not listed.

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0 +1 +2 +3 +4 +5–5 –1–2–3–4

–1 °C–5 °C

0 +1 +2 +3 +4 +5–5 –1–2–3–4

+5 °C+1 °C







• Students compare and order integers.

Key Question 6 (Communication, Connections, Visualization)

• Students use a number line to explain how they can compare integers.

• Students may compare numbers without regard to the integer sign. (See ExtraSupport 1.)

• Students may be unable to use a number line to compare integers. (See ExtraSupport 2.)

• Students may be unable to connect temperatures with a number line. (SeeExtra Support 3.)


EXTRA SUPPORT

EXTRA CHALLENGE

• Have students develop a game in which they compare integers. Providedice of different colours if available; otherwise, have students use one setof dice to represent positive integers and another set to represent negativeintegers.

1. Talk about what the negative and positive signs represent. Remind studentsto first look at the integer sign to determine whether a number is positive ornegative, and then consider the digit to determine where to place the numberon a number line. Talk about positive and negative numbers relative to zero.Ensure students understand that all negative numbers are less than zero andall positive numbers are greater than zero.

2. On the board, on a transparency, or on an interactive whiteboard, use stickynotes to place the numbers �3, �4, �8, 0, and �2 on a number line, withthe �2 and �8 interchanged. Ask students to find the mistake. Help themsee why �8 has to be the left of �2 since �8 is to the right of �2.

3. Draw a magnified basic thermometer on a large piece of paper cut out to thethermometer’s size, and point out 0 �C. Tell students that 0 �C on the paperthermometer represents the same as 0 on the number line. Take the paperthermometer and place it sideways on the board. Ask volunteers to use thethermometer to place different positive and negative temperatures on thenumber line. Talk about which temperatures are greater than �5 �C andwhich are less than �5 �C.


• Some students would benefit from further opportunities to describe situationsinvolving negative numbers or locating negative integers on a number linerather than comparing them.


• Some students are able to communicate orally better than in writing. Allowthese students to work on the Practising questions in pairs. Partners can helpstudents clarify any concepts that are not clear.

57Lesson 8: Comparing and Ordering Integers


Chapter 3 9999 Order of Operations STUDENT BOOK PAGES 98–100

GOALApply the rules for order of operations with whole numbers.


• Use mental math to add, subtract, multiply, and dividewhole numbers.

SPECIFIC OUTCOME

N9. Explain and apply the order of operations, excludingexponents, with and without technology (limited towhole numbers).[CN, ME, PS, T]

Achievement Indicators• Demonstrate and explain with examples why there is

a need to have a standardized order of operations.• Apply the order of operations to solve multi-step

problems with or without technology, e.g., computer,calculator.


Materials • calculators

Recommended Questions 2, 4, 5, 6, & 9Practising Questions


Extra Practice Chapter Review Questions 13 & 14Workbook p. 25

Mathematical CN (Connections), ME (Mental Mathematics Process Focus and Estimation), PS (Problem Solving), and

T (Technology)

Vocabulary/Symbols rules for order of operations


Math BackgroundIn this lesson, students learn to use both mental mathskills and technology to calculate the answers to problemsinvolving many operations. The rules for order ofoperations tell which operation should be performed first.The purpose of the order of operations is to ensure thatthe same answer is reached regardless of who performs thecalculations. When more than one operation appears in anexpression or equation, the operations must be performedin the following order:• Do the operations in brackets first.• Then divide and multiply from left to right.• Finally, add and subtract from left to right.In this lesson, students demonstrate their understandingof the connections among operations by applying therules for order of operations in a variety of problem-solving situations. Students check to see whether theircalculator follows the rules for order of operations.Calculators may yield different results depending ontheir type.



Lesson 9: Order of OperationsCopyright © 2010 Nelson Education Ltd. 59

Introduction(Whole Class/Small Groups)

� 5–10 min

Write the following expression on the board, on a transparency,or on an interactive whiteboard.

10 � 2 � 3 � 6 2Have small groups of students calculate the value of theexpression. Ask volunteers to share their solutions on theboard. Discuss why some found different answers. Tellstudents they will learn rules for doing calculations so thateveryone always gets the same answer.

Sample Discourse“Which operation did you perform first”?• I subtracted 10 � 2 because it is the first operation.• I added 3 � 6 because addition is the easiest operation.• I multiplied 2 � 3 because I knew it was equal to 6.“What answer did you calculate?”• I did the operations in order from left to right and calculated

an answer of 15.• I did the subtraction and then the addition and calculated an

answer of 36.

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Teaching and Learning(Whole Class) � 10–15 min

Together, read Oleh’s calculation to find his minimumtraining heart rate and then read the central question onStudent Book page 98. Work through Oleh’s Solutiontogether. Draw students’ attention to the definition of rulesfor order of operations and talk about how Oleh followed therules to calculate his minimum training heart rate.

Sample Discourse“How do you know that Oleh followed the rules for order ofoperations?”• Oleh did the operations inside the brackets first, which is the

first step in the order of operations.“Why did Oleh do the division last, even though divisioncomes before addition and subtraction according to the rulesof order of operation?”• The addition and subtraction are inside the brackets, and

brackets come before division in the order of operations.

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b) For example,2 � 4 4 � 4 43 � (4 � 4 � 4) 44 � (4 � 4) 4 � 45 � (4 � 4 � 4) 4


Question 9 allows students to reflect on and consolidate theirlearning for this lesson. Ask for volunteers to share their skill-testing questions. Discuss why the different skill-testingquestions would or would not likely be solved correctlywithout using the order of operations.

Answer to Closing Question9. a) For example, some would answer mixed calculations

correctly if they can be done correctly from left toright because most people calculate in that order: 6 � 3 � 5.

b) For example, 3 � 5 � 20 5 would probably bedone incorrectly if a person calculated in order fromleft to right. 15 � 20 5 � 35 5, or 7. Thecorrect answer, however, is 19.

Follow-Up and Preparation for Next ClassNext class is the Chapter Review. Ask students to go throughLessons 1 to 9 and note any questions or problems they have.


Here students reflect on how the rules for order of operationsaffect the answer to a calculation. Draw students’ attention tothe Communication Tip. Explain to students that earlycalculators, unlike modern ones, did not use the rules fororder of operations. If possible, have students enter Oleh’scalculation into their calculators to demonstrate how theycan use brackets with a calculator.

For Prompt A, some students may notice that Oleh couldcalculate the same answer if he ignored the brackets andperformed the operations from left to right. However, sinceOleh knows the rules for order of operations, if he ignoredthe brackets he would likely perform the division first, whichwould lead to an incorrect answer.

Answers to Reflecting QuestionsA. For example, if he ignored the brackets in the formula

but used the rules for order of operations, he woulddivide 72 by 2 first to get 36. Then he could do the restof the calculations in order:220 � 12 � 36 � 208 � 36 � 244So the answer would change from 140 beats each minuteto 244 beats each minute.

B. No, it didn’t matter which operation Oleh did in thebrackets for this calculation. For example, because theoperations are addition and subtraction. You can subtract12 from 220 to get 208 and then add 72 to get 280.Or you can add 72 to 220 to get 292 and then subtract12 to get 280. The answer in the brackets is still 280.


Checking (Pairs)

Provide a sample calculation of different age and height forQuestion 1. Ask volunteers to share their solutions toQuestion 1a) with the class. Discuss different strategies thatstudents used to check the reasonableness of their answers.


These questions provide students opportunities to practiseapplying the rules for order of operations in a variety ofproblem-solving situations. Encourage students to use mentalmath strategies to perform the calculations.3. Provide calculators to students.5. Note that there are many ways to use 4s to create each

number in part b).

Answers to Key Question5. a) 4 � 4 4 � 4 � 4 � 1 � 4

� 5 � 4� 1

(4 � 4) (4 � 4) � 8 (4 � 4) � 8 8 � 1

44 44 � 1

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• Students understand and use the rules for order of operations to solveproblems with multiple operations.

• Students identify expressions that do and do not need brackets to be solvedcorrectly.

Key Question 5 (Connections, Problem Solving)• Students use the rules for order of operations to show that different

expressions have a value of 1.

• Students use the rules for order of operations to write expressions withspecific answers.

• Students may perform the operations in the order in which they appear fromleft to right. (See Extra Support 1.)

• Students may not understand the purpose of brackets in an expression withmultiple operations. (See Extra Support 2.)

• Students may perform the operations in the order in which they appear fromleft to right. (See Extra Support 1.)

• Students may be unable to create alternative expressions as specified. (SeeExtra Support 3.)


EXTRA SUPPORT

EXTRA CHALLENGE

• Challenge students to use the digits from 1 to 5, as well as addition,subtraction, multiplication, division, and brackets, to write as manyexpressions with different answers as possible.

For example, (1 � 2) � 5 (4 � 3) � 15

• Students may make up puzzles by creating a calculation and then erasing theoperations. Challenge other students to figure out the missing operations, asin Question 8.

1. Have students give examples of situations in which they need to follow rules,such as traffic lights for pedestrians and rules for any game. Talk about howrules help them know what to do and how to do it. Tell students that therules for order of operations need to be followed to make sure that everyonesolves the problem the same way and gets the same answer. You may wantto help them think of ways they can remember the order of operations.

2. Emphasize the role that brackets play in calculating an answer. Ensurestudents understand that brackets indicate that they should perform anoperation first. Review the rules for order of operations and point out tostudents that without brackets, multiplication and division are alwaysperformed before addition and subtraction. Help students understand thatbrackets are needed if the addition and subtraction are supposed to be donefirst, but not if the multiplication and division are supposed to be done first.

3. Talk about the various ways 4s can be combined to get a value of 1, usingdifferent operations and brackets. Have students try various combinations ofbrackets and operations with four 4s to get a value of 2. They might begin byguessing and testing, then use reasoning to get closer to the answer. In theprocess, they might find expressions that have a value of 3, 4, or 5 instead.Have them continue until they have an expression for each value.


• For students uncomfortable with performing calculations with multipleoperations, have them practise solving simple expressions that haveparentheses, such as (3 � 1) � 4. Students can then focus on solving theexpression in the parentheses first each time. This will allow students topractise their mental math skills and solve problems with multiple steps.


• Some students may find the task easier if different colours are associatedwith � and than with � and �.

• Some students may benefit from devising a mnemonic device to help themremember the order of operations.

Lesson 9: Order of Operations 61


Chapter Chapter 3Chapter 3 Math GameFour in a Row



• Apply the rules for order of operations with wholenumbers.

SPECIFIC OUTCOME

N9. Explain and apply the order of operations, excludingexponents, with and without technology (limited to wholenumbers). [CN, ME, PS, T]

Achievement Indicator• Apply the order of operations to solve multi-step

problems with or without technology.

Preparation and PlanningNumber of Players 2

Materials • coloured counters

Masters • Four in a Row Game Board p. 80• Calculation Cards pp. 81–82

Mathematical ME (Mental Mathematics and Estimation)Process Focus


Using the Math GameIn this game, students use mental math skills to calculate theanswers to expressions from Calculation Cards pp. 81–82.On each turn, a student chooses a card and completes thecalculation. The student then places a coloured marker onthe game board square, using Four in a Row Game Boardp. 80, that coincides with their calculated answer. If thestudent has four coloured markers in a row, column, ordiagonal, the student wins the game. Students may play untilall the cards are used up, with neither player winning.

When to PlayStudents can play the game after they demonstrate anunderstanding of how to use the order of operations.

StrategiesHave students keep track of which numbers they need toform four counters in a row. They may then quickly estimatea calculation by performing the multiplication and divisionoperations mentally before choosing a card.

DiscussAfter the game, ask students to share any successful strategiesthey used to win the game. You may also ask students toshare experiences as they applied the order of operationsusing mental math.

Four in a Row Game Board p. 80

Calculation Cards pp. 81–82



Math Game: Four in a RowCopyright © 2010 Nelson Education Ltd. 63



Proficient players Less-proficient players

• Students use reasoning and mental math to perform calculations using theorder of operations.

• Students may be unable to correctly simplify an expression using the order ofoperations. (See Extra Support 1.)


EXTRA SUPPORT

EXTRA CHALLENGE

• Have students play using the same cards, but in case of not getting thedesired answer, have them add more operations to the calculation card in order to end up with the answer they need to form four in a row. Tomaintain a competitive element, have students work within a given time limit.

1. Have students work in teams of two to verify whether a calculation wasperformed correctly. Display a visual showing the order of operations forstudents to refer to. Encourage students to discuss which cards are morelikely to yield the desired answer.


• Some students may have difficulty calculating on the spot. Allow students to work with simpler cards that include only two operations, such asmultiplication and subtraction, or division and addition.




Chapter Review STUDENT BOOK PAGES 102–104


SPECIFIC OUTCOMES




strategy used to identify them.• Determine all the whole-number factors of a given

number using arrays.• Identify the factors for a given number and explain

the strategy used, e.g., concrete or visualrepresentations, repeated division by prime numbers,or factor trees.

• Provide an example of a prime number and explainwhy it is a prime number.

• Provide an example of a composite number andexplain why it is a composite number.



Achievement Indicators• Place given integers on a number line and explain

how integers are ordered.• Compare two integers; represent their relationship

using the symbols<,>, and �, and verify using anumber line.

• Order given integers in ascending or descendingorder.


Achievement Indicator• Apply the order of operations to solve multi-step

problems with or without technology, e.g., computer,calculator.

Preparation and PlanningMaterials • Optional: counters

Masters • Chapter Review—Frequently Asked Questions p. 83

• Chapter 3 Test pp. 84–86• Optional: Number Lines, Masters Booklet p. 33

Extra Practice Workbook p. 26


Chapter Review—Frequently AskedQuestions p. 83

Optional: Number Lines,Masters Booklet p. 33

Chapter 3 Test pp. 84–86

01-NM6TR-C03-Interior_RM.qxd 12/3/08 10:14 PM Page 64

Chapter ReviewCopyright © 2010 Nelson Education Ltd. 65

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 chart for the details of each question.

Alternatively, use the Practice questions as a practice test,and then administer Chapter 3 Test pp. 84–86. 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 102 and create a new example for eachquestion in their own notes. Then have students summarize

the answers to the FAQs in their own words, as a way ofreflecting on the concepts.

Alternatively, have students complete Chapter Review—Frequently Asked Questions p. 83 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.

Practice (Individual)

Most students will be able to complete Questions 1 to 14 inclass. Assign any uncompleted questions for homework.Some students may want to use materials and/or masters thatwere used in this chapter’s lessons. Provide students withcounters, number lines, or 100 charts, as needed, to completethe questions.



Work meets standard Work meets standard Work meets Work does not yet meet of excellence of proficiency acceptable standard acceptable standard• often draws insightful and logical

conclusions using knowledge offactors

• in many situations, draws logicalconclusions using knowledge offactors

• sometimes draws simple, logicalconclusions using knowledge offactors

• rarely draws conclusions from amathematical situation usingknowledge of factors

Question 2, written answer Specific Outcome and Process Focus: N3 [R]• Which number from 10 to 20 has an odd number of factors? Explain how you identified the factors.

Question 3, short answer Specific Outcome and Process Focus: N3 [R]• Maddy listed these factors of 48: 1, 2, 4, 5, 8, 16, and 48.

a) Which number listed is not a factor of 48?b) Which factors are missing?


Work meets standard Work meets standard Work meets Work does not yet meet of excellence of proficiency acceptable standard acceptable standard• often draws insightful and logical

conclusions using knowledge offactors

• in many situations, draws logicalconclusions using knowledge offactors

• sometimes draws simple, logicalconclusions using knowledge offactors

• rarely draws conclusions from amathematical situation usingknowledge of factors

Question 1, written answer Specific Outcome and Process Focus: N3 [R, V]• How do these arrays show the factors of 16?



Work meets standard Work meets standard Work meets Work does not yet meet of excellence of proficiency acceptable standard acceptable standard• provides a precise explanation of

prime and composite numbers• provides a clear and logical

explanation of prime andcomposite numbers

• provides a partially clear explanationof prime and composite numbers

• provides a vague and/or inaccurateexplanation of prime and compositenumbers



• chooses an efficient and effectivestrategy


• chooses an appropriate andworkable strategy


• chooses a simplistic and/or routinestrategy


• chooses an inappropriate orunworkable strategy

Question 5, written answer Specific Outcome and Process Focus: N3 [PS]• Every five years, Statistics Canada conducts a census to collect data about Canadians.

A census was conducted in 2006. Will a census be conducted in 2036? Explain your thinking.


Question 6, short answer, written answer Specific Outcome and Process Focus: N3 [R]a) Write two prime numbers. How do you know that these are prime numbers?b) Write two composite numbers. How do you know that these are composite numbers?

Question 7, written answer Specific Outcome and Process Focus: N3 [R]• Is there any multiple of 6 that is a prime number? Explain your thinking.

Question 4, short answer Specific Outcome and Process Focus: N3 [R]• List the first five multiples of each number.

a) 7 b) 6 c) 9 d) 40


Question 8, short answer Specific Outcome and Process Focus: N3 [R]• Jennifer divided a number by the prime number 3. Then she divided her result by 3.

Her final answer is 3. What number did she divide by 3?


Question 9, short answer Specific Outcome and Process Focus: N3 [PS, R]• A number between 40 and 80 is a multiple of 7. Another factor of the number is 9. What is the number?


Work meets standard Work meets standard Work meets Work does not yet meet of excellence of proficiency acceptable standard acceptable standard• draws insightful and logical

conclusions when determiningwhether any multiple of 6 is aprime number

• makes an insightful generalizationwhen determining whether anymultiple of a number is a primenumber

• draws logical conclusions whendetermining whether any multipleof 6 is a prime number

• makes a logical generalizationwhen determining whether anymultiple of a number is a primenumber

• draws simple, logical conclusions whendetermining whether any multiple of anumber is a prime number

• makes a simple generalization whendetermining whether any multiple ofa number is a prime number

• does not draw conclusions whendetermining whether any multiple ofa number is a prime number

• is unable to make a generalizationwhen determining whether anymultiple of a number is a primenumber

(Continued on next page)

Chapter ReviewCopyright © 2010 Nelson Education Ltd. 67


Work meets standard Work meets standard Work meets Work does not yet meet of excellence of proficiency acceptable standard acceptable standard• shows flexibility and insight with

operations and brackets whensolving the problem, adapting ifnecessary

• shows thoughtfulness withoperations and brackets whensolving the problem

• shows understanding with operationsand brackets when solving theproblem

• attempts to solve problem

Work meets standard Work meets standard Work meets Work does not yet meet of excellence of proficiency acceptable standard acceptable standard• uses insightful visual

representations that verifywhether �5 � �3

• uses meaningful visualrepresentations that verifywhether �5 � �3

• uses simple visual representationsthat verify whether �5 � �3

• uses unclear visual representationsthat verify whether �5 � �3

Work meets standard Work meets standard Work meets Work does not yet meet of excellence of proficiency acceptable standard acceptable standard• provides a precise and insightful

explanation of the meaning ofpositive and negative integers

• provides a clear and logicalexplanation of the meaning ofpositive and negative integers

• provides a partially clear explanationof the meaning of positive andnegative integers

• provides a vague and/or inaccurateexplanation of the meaning ofpositive and negative integers

Question 10, written answer Specific Outcome and Process Focus: N7 [C]• Holly has a goal to learn 10 new French words each week. She uses integers to show whether she has learned

more or fewer words than her goal. What do you think the integers –3, 0, and �3 represent? Explain.


Question 11, written answer, model Specific Outcome and Process Focus: N7 [V]• How do you know that �5 � �3? Use a number line.

Question 12, short answer, model Specific Outcome and Process Focus: N7 [C, V]• Order these temperatures from coldest to warmest. Show your work.

(Score 1 point for the order of the temperatures for a total out of 1.)

Question 13, short answer Specific Outcome and Process Focus: N9 [ME]• Calculate. Use the rules for order of operations.

a) 12 � 7 � 4 � 2 b) (100 � 50 � 2 � 1) � 76 c) (4 � 7) � 2 � 12 � 2 d) 6 � 5 � 4 � 2 � 1(Score 1 point for each correct answer for a total out of 4.)

Question 14, short answer, written answer Specific Outcome and Process Focus: N9 [PS]a) Calculate (2 � 1) � (4 � 3).b) How can you use the numbers from 1 to 4 and any operations with brackets to make an expression that equals 2?




69Chapter Task: A Block Dropping GameCopyright © 2010 Nelson Education Ltd.

Chapter TaskA Block Dropping Game


SPECIFIC OUTCOME



Achievement Indicator• Solve a given problem involving factors or multiples.

Using the Chapter TaskUse this task as an opportunity to assess students’understanding of the concepts developed in the chapterand their ability to apply them in a rich problem-solvingsituation. Refer to the assessment chart on page 71 for thedetails of each part of the task.


30–45 min Using the Task

Materials • Optional: counters

Masters • Chapter 3 Task pp. 87–88• Optional: 1 cm Grid Paper, Masters Booklet

p. 22

Mathematical PS (Problem Solving), R (Reasoning), andProcess Focus V (Visualization)


Chapter 3 Task pp. 87–88Optional: 1 cm Grid Paper,Masters Booklet p. 22


The 2-by-5 block cannot be used to make the squarebecause 5 is not a factor of 12. So the computer cannotmake rows or columns of the 2-by-5 block to fit the 12-by-12 square.

The 1-by-2 block can be used to make the squarebecause 1 and 2 are both factors of 12. So the computercan make rows and columns of copies of the 1-by-2block to fit the 12-by-12 square.

C. 1-by-1, 2-by-2, 3-by-3, 4-by-4, 6-by-6, and 12-by-12;for example, each side length is a factor of 12. So copiesof the square blocks can be used by the computer tomake the square.

D. For example, my game has rectangular and square blocksdropping from the screen and two squares: a 15-by-15square and a 12-by-12 square. The computer can makeeither square if both the length and the width of theblocks are factors of 15 or 12. So a player would choose ablock depending on whether both lengths of the sides arefactors of 15 or factors of 12. For example, a playerwould choose 3-by-5 for the 15-by-15 square, and 2-by-4 for the 12-by-12 square. A block that is 4-by-5,however, would not be chosen because both dimensionsare not factors of 15 or 12.

Adapting the TaskYou can adapt the task in the Student Book to suit the needsof your students. For example:• Use Chapter 3 Task pp. 87–88.• Have students work in pairs or small groups.• Challenge students to identify blocks that can be used to

fill a rectangular game board, such as a 16-by-24 board,rather than the square game board.

Introduction (Whole Class) � 10–15 min

In this video game, the player uses the cursor to grabrectangular blocks that the computer drops from the top ofthe screen in order to form a square on the screen. Studentsshould be comfortable with factors and multiples beforeplaying. Ask students how they could determine whether anumber is a factor of another number. You may wish toactivate existing knowledge by having students identify thefactors of a number, such as 36.

Using the Task (Individual) � 30–45 min

Together, read all the information on Student Book page 105,including the central question. For Prompts A, C, and D,encourage students to use words such as factor and multiple intheir explanations. Students may find it helpful to create afactor rainbow for 12 before working through the prompts.You may want to provide counters to assist students inidentifying the factors of 12.

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 the studentpage; however, most will benefit from using Chapter 3 Taskpp. 87–88 to plan and record work. As students workthrough the task, observe and/or interview individuals to seehow they are interpreting and carrying out the task.

Possible Solutions to Chapter TaskA. For example, I can use six copies of the 2-by-3 block to

form the top row of the 12-by-12 square because 2 is afactor of 12. Then I can make three more rows like thefirst row to have a total of four rows, because 3 is also afactor of 12.

B. For example, the 3-by-4 block can be used to make thesquare because 3 and 4 are both factors of 12. So thecomputer can make rows and columns of copies of the 3-by-4 block to fit the 12-by-12 square.

12

12

3

2



Outcome

N3. Demonstrate anunderstanding of factorsand multiples by• determining multiples

and factors of numbersless than 100

• identifying prime andcomposite numbers

• solving problemsinvolving multiples.

[PS, R, V]

Work meetsstandard ofexcellence

• often draws insightful andlogical conclusions andrecognizes inappropriatelydrawn conclusions withoutprompting

• comprehensively analyzessituations and makesinsightful generalizations

• chooses efficient andeffective strategies whenapplying knowledge ofmultiples and factors

Work meets standard ofproficiency

• in many situations, drawslogical conclusions andrecognizes inappropriatelydrawn conclusions whenprompted

• completely analyzessituations and makes logicalgeneralizations

• chooses workable andreasonable strategies whenapplying knowledge ofmultiples and factors

Work meetsacceptable standard

• sometimes draws simple,logical conclusions andsometimes recognizesinappropriately drawnconclusions when prompted

• superficially analyzessituations and makes simplegeneralizations

• chooses partially appropriateand workable strategieswhen applying knowledge ofmultiples and factors

Work does not yetmeet acceptablestandard

• rarely draws conclusionsfrom a mathematicalsituation and usually doesnot recognize inappropriatelydrawn conclusions

• is unable to analyzesituations and makegeneralizations

• chooses inappropriate and/orunworkable strategies whenapplying knowledge ofmultiples and factors


71Chapter Task: A Block Dropping GameCopyright © 2010 Nelson Education Ltd.


Chapter Chapter 3Chapter 3 Chapters 1–3 Cumulative Review

STUDENT BOOK PAGES106–107

SPECIFIC OUTCOMES

N1. Demonstrate an understanding of place value fornumbers• greater than one million• less than one thousandth.[C, CN, R, T]

N2. Solve problems involving large numbers, usingtechnology.[ME, PS, T]



N7. Demonstrate an understanding of integers,concretely, pictorially and symbolically.[C, CN, R, V]


PR1. Demonstrate an understanding of the relationshipswithin tables of values to solve problems.[C, CN, PS, R]

PR3. Represent generalizations arising from numberrelationships using equations with letter variables.[C, CN, PS, R, V]

PR4. Demonstrate and explain the meaning ofpreservation of equality concretely, pictorially andsymbolically.[C, CN, PS, R, V]

Preparation and PlanningMaterials • counters

• calculator• grid paper• chart paper

Masters • Table of Values, Chapter 1 p. 63• Balance Scales, Chapter 1 p. 68• Place Value Chart to Hundred Millions,

Masters Booklet p.42• Decimal Place Values Chart to Millionths,

Masters Booklet p.45


Tables of Values, Chapter 1p. 63

Balance Scales, Chapter 1p. 68

Decimal Place Value Chartto Millionths, MastersBooklet p. 45

Place Value Chart toHundred Mullions,Masters Booklet p. 42



Chapters 1–3: Cumulative ReviewCopyright © 2010 Nelson Education Ltd. 73

Using the Cumulative ReviewThe questions on Student Book pages 106–107 providepractice with multiple-choice questions while reviewing theconcepts developed in Chapters 1 to 3.

Question Answer Grade 6 Outcome Chapter

1 D PR1 1

2 A PR3 1

3 C PR4 1

4 B N1 2

5 D N2 2

6 A N1 2

7 C N1 2

8 D N1 2

9 B N3 3

10 N3 3

11 N3 3

12 B N3 3

13 D N7 3

14 A N9 3


D

A

Family Letter

Dear Parent/Caregiver:

Over the next three weeks, your child will be learning about identifying factors andmultiples of numbers and how to determine whether a number is prime or composite.Your child will also learn how to represent, compare, and order integers, and willperform a series of calculations using the rules for order of operations. Your child willhave many opportunities to apply knowledge of factors, multiples, and integers insolving realistic problems.

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

• Have your child model factors of numbers less than 100 by putting numbers ofitems in equal groups. Your child can also calculate multiples of smaller numbersthey encounter, such as the number of snack packages in three or four boxes.

• Your child can measure and record the daily high and low temperatures during theweek and then place the temperatures on a number line. Have your child order thetemperatures from coldest to warmest or warmest to coldest. Your child can alsocompare temperatures from different cities.

• Have your child solve any skill-testing questions found on cereal boxes or othercontest entry forms, and have your child explain how he/she applied the rules fororder of operations to arrive at the correct answer.

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 toother websites that provide online tutorials, math problems, brainteasers, andchallenges.

74 Chapter 3: Number Relationships Copyright © 2010 Nelson Education Ltd.

Chapter Chapter Chapter 3

02-NM6TR-C03-BLM.qxd 12/3/08 10:41 PM Page 74

Scaffolding for Getting Started Page 1STUDENT BOOK PAGES 68–69

Banner DesignsDaniel is making a banner for Heritage Day. It has 30 squares.He coloured every second square red to represent one of the four colours on an Aboriginal medicine wheel.

He drew a symbol to represent an eagle in every third square.

? HHooww ccaann yyoouu pprreeddiicctt hhooww mmaannyy ccoolloouurreedd ssqquuaarreess wwiillllhhaavvee aa ssyymmbbooll oonn tthheemm??

AA.. Continue Daniel’s banner to 18 squares.Colour every second square red.Sketch an eagle symbol in every third square.

BB.. Circle the red squares that have an eagle.Why does the pattern 6, 12, 18, … represent the red squares with an eagle?

How can you use a number pattern to predict the nextred square with an eagle?

75Blackline MastersCopyright © 2010 Nelson Education Ltd.

Name: Date:



Name: Date:

Scaffolding for Getting Started Page 2STUDENT BOOK PAGES 68–69

CC.. Daniel’s banner has 30 squares. Suppose you want to figure out how many red squares have an eagle. How could you skip count to figure out the number of red squares with an eagle?

How many squares are there altogether?

Use your answers above to write a multiplication equation you could use to figure out how many red squares have an eagle.

How many red squares on Daniel’s banner have an eagle symbol?

Explain what you did.

DD.. Design a banner with a different number of squares on grid paper. Use one of the symbols below and another colour from the medicine wheel.

bear drum horse fish

How can you predict the number of coloured squares that have a symbol on them?



Name: Date:

Scaffolding for Lesson 2, Question 3STUDENT BOOK PAGE 76

3. What is the same about a list of multiples of 3 and 9?What is different?

• List the multiples of 3 up to 30:

• List the multiples of 9 less than 30:

• How are the two lists of multiples the same?

• How are the two lists of multiples different?

This is the sameas skip counting

by 3s from 3to 30.



Name: Date:

SSttrriinngg AArrtt

CCuurriioouuss MMaatthh:: SSttrriinngg AArrtt


2 48 4746

4544

43

41

42

40

39

38

37

36

35

34

33

32

31

3029

2827262523 24

2221

20

19

18

17

16

15

14

13

12

11

10

9

8

7

65

43



Name: Date:

MMiidd--CChhaapptteerr RReevviieeww⎯⎯FFrreeqquueennttllyy AAsskkeedd QQuueessttiioonnssSTUDENT BOOK PAGES 86–87

QQ:: What are some ways to identify factors?

AA::

QQ:: What are some ways to identify multiples?

AA::

QQ:: How are prime and composite numbers different?

AA::



Name: Date:

FFoouurr iinn aa RRooww GGaammee BBooaarrdd

MMaatthh GGaammee:: FFoouurr iinn aa RRooww


1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25



Name: Date:

CCaallccuullaattiioonn CCaarrddss

MMaatthh GGaammee:: FFoouurr iinn aa RRooww


10 –

3 ×

312

÷ 2

– 2

× 2

2 ×

(1

+ 2

) –

316

– 2

× 6

2 ×

3 +

4 –

41

+ 2

× 3

4 ×

4 –

4 ×

23

+ 2

× 2

+ 2

(3 +

4 ×

3)

÷ 3

2 ×

1 ×

51

+ 2

× 3

+ 8

÷ 2

2 ×

8 –

2 ×

2



Name: Date:

CCaallccuullaattiioonn CCaarrddss

MMaatthh GGaammee:: FFoouurr iinn aa RRooww Page 2STUDENT BOOK PAGE 101



Name: Date:

CChhaapptteerr RReevviieeww⎯⎯FFrreeqquueennttllyy AAsskkeedd QQuueessttiioonnssSTUDENT BOOK PAGE 102

QQ:: How can you represent and compare integers?

AA::

QQ:: What are the rules for order of operations?

AA::



Name: Date:

Chapter 3 Test Page 1

11.. How do these arrays show the factors of 18?

22.. Which number from 20 to 30 has exactly three factors?

Explain how you identified the factors.

33.. Natalie listed these factors of 72: 1, 2, 6, 9, 10, 12, 18, and 36.

aa)) Which number listed is not a factor of 72?

bb)) Which factors are missing?

44.. List the first five multiples of each number.

aa)) 8

bb)) 4

cc)) 12

dd)) 15

55.. The Winter Olympics were held in 2006. If the Winter Olympics are held every four years, will they be held in 2044? Explain your thinking.



66.. aa)) Write one prime number. How do you know that this is a prime number?

bb)) Write three composite numbers. How do you know that theseare composite numbers?

77.. Andrew says that if a number is even, it is not a prime number.Is Andrew correct? Explain your thinking.

88.. Garret divided the number 30 by a prime number.Then he divided his answer by another prime number.His answer is 3. What prime numbers did he divide by?

99.. A number between 40 and 50 is a multiple of 8.12 is also a factor of that number.

What is the number?

1100.. Sam used integers to compare three math marks to his first math mark in October. What do you think the integers –5, 0, and 5 represent? Explain.


Name: Date:

30

3



Name: Date:


1111.. How do you know that –2 > –6? Use a number line.

1122.. Elements melt and freeze at different temperatures. Order these temperatures from coldest to warmest. Show your work.

MMeellttiinngg PPooiinnttss ooff EElleemmeennttss

1133.. Calculate. Use the rules for order of operations.

aa)) 20 � 3 � 5 � 6 cc)) (15 � 6) � 3 � 4 � 2

bb)) 45 � (7 � 8) � 2 dd)) 24 � 6 � 5 � 3 � 1

1144.. aa)) Calculate 2 � 2 � (2 � 2).

bb)) How can you use four 2s and any operations plus brackets to make an expression that equals 5?

Melting Point of Chlorine

Melting Pointof Helium

Melting Pointof Salt

Melting Pointof Silver

Melting Pointof Mercury

–101 �C �272 �C �98 �C �961 �C –38.72 �C


Chapter 3 Task Page 1

A Block Dropping GameSTUDENT BOOK PAGE 105

In a video game, blocks shaped like rectangles drop from the top of the screen. You grab blocks that you think could form a square. The computer copies the blocks you grab and tries to make the square.


Name: Date:

? Which blocks should you grab to make the square?

RReeaadd tthhee TTaasskk CChheecckklliisstt aabboovvee bbeeffoorree yyoouu bbeeggiinn..

AA.. How do you know copies of the 2-by-3 block can be used to make the 12-by-12 square? Use a diagram to explain.

� Did you use factorsor multiples to helpsolve the problem?

� Did you check yourcalculations?

� Did you includediagrams?

� Did you explain yourthinking clearly?

Task Checklist



Name: Date:

Chapter 3 Task Page 2

BB.. Which of these blocks can be used to make the square?

• 3-by-4

• 2-by-5

• 1-by-2

CC.. Suppose that square blocks drop from the top of the screen. Which blocks would you grab? Explain.

DD.. Design a similar video game with rectangular blocks and square blocks dropping from the top of the screen. How can a player decide which block to grab?


Answers for Chapter 3 MastersSSccaaffffoollddiinngg ffoorr GGeettttiinngg SSttaarrtteedd pppp.. 7755––7766

AA..

BB..

For example, I saw the pattern 6, 12, 18,… Every 6th square is red with an eagle, so the next square that should be red with an eagle is the 24th square.

CC.. 6, 12, 18, 24, 30

30 squares altogether

� 6 � 30

Five red squares have an eagle.

For example, there are 30 squares and every 6th square is red with an eagle.I divided to find the number of red squares with an eagle: 30 � 6 � 5.

DD.. For example, I’ll create a banner with 100 squares. I’ll colour every second square yellow.Every fifth square will have the symbol for a horse. I’ll figure out how many yellow squares will have a horse.

Every 10th square is yellow with a horse symbol. There are 100 squares, so I divided 100 by 10: 100 � 10 � 10, so there will be 10 squares that are yellow with a horse symbol.

Scaffolding for Lesson 2, Question 3, p. 77

3, 6, 9, 12, 15, 18, 21, 24, 27, 30

9, 18, 27

The numbers in both lists are multiples of 3.

Sum of the numbers in both lists are multiples of 9, but sum in the list of multiples of 3 are not multiple of 9

Chapter 3 Test pp. 84–8611.. Each array has 18 circles, and each number of rows and columns represents a factor.

The arrays show 1 � 18, 2 � 9, 3 � 6.

22.. 25. For example, I knew 23 and 29 had only two factors because they are prime.

So I used mental math to identify the factors of 20, 21, 22, 24, 25, 26, 27, 28, and 30.

20 has factors 1, 2, 4, 5, 10, and 20: 6 factors

21 has factors 1, 3, 7, and 21: 4 factors

22 has factors 1, 2, 11 and 22: 4 factors

24 has factors 1, 2, 3, 4, 6, 8, 12, and 24: 8 factors

25 has factors 1, 5, and 25: 3 factors



28 has factors 1, 2, 4, 7, 14, and 28: 6 factors.

30 has factors 1, 2, 3, 5, 6, 10, 15, and 30: 8 factors

horse horse horse

E E E E E E

E E E E E E




3. a) 10

b) 3, 4

4. a) 8, 16, 24, 32, 40

b) 4, 8, 12, 16, 20

c) 12, 24, 36, 48, 60

d) 15, 30, 45, 60, 75

5. No; The Winter Olympics were held in 2006, so they will also be held in 2010, 2014, 2018, 2022, 2026, 2030, 2034, 2038, and 2042. They will not be held in 2044.

6. a) For example, 11 is prime because it has only 2 different factors, 1 and itself.

b) For example, 10, 12, and 14 because each has more than 2 different factors.

7. Andrew is not correct. 2 is even and is a prime number.

8. 2 and 5; for example, if he started with 30 and ended with 3, he must have divided by prime numbers that multiply to 10. The only prime numbers that multiply to 10 are 2 and 5.

9. 48; for example, I wrote a list of the multiples of 8 to determine the multiples between 40 and 50:

8, 16, 24, 32, 40, 48

The only number between 40 and 50 that is also a multiple of 12 is 48.

10. For example, –5 means the test score is 5 less than his first mark; 0 means the test score is the same as his first mark; 15 means the test score is 5 more than his first mark.

11. �2 is to the right of �6 on the number line, so �2 > 26.

12. For example, I recorded the temperatures on a number line.

From coldest to warmest, the temperatures are �272 �C, �101 �C, �38.72 �C, �98 �C, �961 �C.

1133.. aa)) 20 � 3 � 5 � 6

� 20 � 15 � 6

� 5 � 6

� 11

bb)) 45 � (7 � 8) � 2

� 45 � 15 � 2

� 3 � 2

� 6

cc)) (15 � 6) � 3 � 4 � 2

� 9 � 3 � 4 � 2

� 3 � 4 � 2

� 3 � 8

� 11

–272 °C –101 °C +98° +961 °C

–38.72 °C

0 °C

Helium Chlorine

Mercury

Salt Silver

–10

–6 –2

0

30

6

3

2

5


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