chapter 2: fraction operations - · pdf filechapter 2: fraction operations ... number . . . ....

Teacher’s Resource

Chapter 2: Fraction Operations

SAMPLE CHAPTER

Chapter Chapter 2Chapter 2

Contents

OVERVIEW

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Curriculum across Grades 7 to 9: Number . . . . . . . . . . . . .2Math Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Planning for Instruction . . . . . . . . . . . . . . . . . . . . . . . . . . .3

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

Chapter 2 Planning Chart . . . . . . . . . . . . . . . . . . . . . . . . . .4Chapter 2 Assessment Summary . . . . . . . . . . . . . . . . . . . . .8

TEACHING NOTES

Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Getting Started: Pattern Block Designs . . . . . . . . . . . . . . .11Lesson 2.1: Multiplying a Whole Number by

a Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15Lesson 2.2: Exploring Calculating a Fraction

of a Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21Lesson 2.3: Multiplying Fractions . . . . . . . . . . . . . . . . . . .24Lesson 2.4: Exploring Estimating Fraction Products . . . . .29Lesson 2.5: Multiplying Fractions Greater than 1 . . . . . . .33Mid-Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39Lesson 2.6: Dividing Fractions by Whole Numbers . . . . . .45Lesson 2.7: Estimating Fraction Quotients . . . . . . . . . . . .50Lesson 2.8: Dividing Fractions by Measuring . . . . . . . . . . .55Curious Math: It Is Just Like Multiplying! . . . . . . . . . . . . .61Lesson 2.9: Dividing Fractions Using a Related

Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63Math Game: Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . .682

3

Lesson 2.10: Order of Operations . . . . . . . . . . . . . . . . . . .70Lesson 2.11: Communicating about Multiplication

and Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75Chapter Self-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82Chapter Task: Computer Gizmos . . . . . . . . . . . . . . . . . . .89

CHAPTER 2 BLACKLINE MASTERS

Family Letter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92Scaffolding for Getting Started Activity . . . . . . . . . . . . . . .93Mid-Chapter Review—Frequently Asked Questions . . . . .97Mid-Chapter Review—Study Guide . . . . . . . . . . . . . . . . .98Fractions and Operations Cards I . . . . . . . . . . . . . . . . . . .99Fractions and Operations Cards II . . . . . . . . . . . . . . . . . .100Chapter Review—Frequently Asked Questions . . . . . . . .101Chapter Review—Study Guide . . . . . . . . . . . . . . . . . . . .102Chapter 2 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103Chapter 2 Task: Computer Gizmos . . . . . . . . . . . . . . . . .106Answers for Chapter 2 Masters . . . . . . . . . . . . . . . . . . . . .108From Masters BookletReview of Essential Skills: Chapter 2 . . . . . . . . . . . . . . . .3–51 cm Grid Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .322 cm Grid Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34Number Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39Fraction Strips Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45Fraction Spinner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46Assessment Rubrics for Mathematical Processes . . . . . .47–52Chapter Checklist: Chapter 2 . . . . . . . . . . . . . . . . . . . . . .54Self-Assessment: Chapter 2 Lesson Goals:

Fraction Operations . . . . . . . . . . . . . . . . . . . . . . . . . . .65Self-Assessment: Mathematical Processes . . . . . . . . . . . . . .75Self-Assessment: What I Like . . . . . . . . . . . . . . . . . . . . . . .76Self-Assessment: How I Learn . . . . . . . . . . . . . . . . . . . . . .76

IntroductionThis chapter provides students with opportunities to usetheir understanding of fractions and their proficiency withwhole-number operations to multiply and divide positivefractions and mixed numbers. They will use theirunderstanding of equivalent fractions developed in Grade 5,their ability to relate improper fractions and mixed numbersdeveloped in Grade 6, and their proficiency with fractionaddition, subtraction, and comparison developed in Grade 7.Throughout the chapter, new concepts are introduced usingconcrete models and then linked to pictorial representationssuch as drawings and grids. Students are then presented withopportunities to work abstractly with symbols.

Answers and Solutions

Answers 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.

Fraction Operations

1ContentsCopyright © 2008 Nelson Education Ltd.

2 Chapter 2: Fraction Operations Copyright © 2008 Nelson Education Ltd.

Curriculum across Grades 7 to 9: NumberThe Grade 8 outcomes and achievement indicators listed below are addressed in this chapter. When the outcome or indicator isthe focus of a lesson or feature, the lesson number or feature is indicated in brackets.

Grade 7 Grade 8 Grade 9

2. FRACTIONSN6. Demonstrate an understanding of multiplyingand dividing positive fractions and mixed numbers, concretely, pictorially, and symbolically.[C, CN, ME, PS] • Identify the operation required to solve a given

problem involving positive fractions. (8, 11)• Provide a context that requires the multiplying of

two given positive fractions. (3, 5)• Provide a context that requires the dividing of two

given positive fractions. (8, 9, 11)• Estimate the product of two given positive proper

fractions to determine if the product will be closer to 0, , or 1. (4)

• Estimate the quotient of two given positive fractionsand compare the estimate to whole numberbenchmarks. (7)

• Express a given positive mixed number as animproper fraction and a given positive improperfraction as a mixed number. (1, 5)

• Model multiplication of a positive fraction by a wholenumber concretely or pictorially and record theprocess. (1, 11)

• Model multiplication of a positive fraction by apositive fraction concretely or pictorially using anarea model and record the process. (2, 3, 5)

• Model division of a positive proper fraction by awhole number concretely or pictorially and record theprocess. (6)

• Model division of a positive proper fraction by apositive proper fraction pictorially and record theprocess. (7, 8, 9)

• Generalize and apply rules for multiplying anddividing positive fractions, including mixed numbers.(1, 3, 5, 6, 8, CM, 9, MG, 11)

• Solve a given problem involving positivefractions taking into consideration order of operations(limited to problems with positive solutions). (10, 11)

12

Strand: NumberGeneral Outcome: Develop number sense.

2. FRACTIONSN5. Demonstrate an understanding of adding and subtractingpositive fractions and mixed numbers, with like and unlikedenominators, concretely, pictorially, and symbolically (limited topositive sums and differences). [C, CN, ME, PS, R, V] • Model addition and subtraction of a given positive fraction or a given

mixed number using concrete representations, and record symbolically. • Determine the sum of two given positive fractions or mixed numbers

with like denominators. • Determine the difference of two given positive fractions or mixed

numbers with like denominators. • Determine a common denominator for a given set of positive fractions

or mixed numbers. • Determine the sum of two given positive fractions or mixed numbers

with unlike denominators. • Determine the difference of two given positive fractions or mixed

numbers with unlike denominators. • Simplify a given positive fraction or mixed number by identifying the

common factor between the numerator and denominator.• Simplify the solution to a given problem involving the sum or difference

of two positive fractions or mixed numbers. • Solve a given problem involving the addition or subtraction of

positive fractions or mixed numbers and determine if the solutionis reasonable.

N7. Compare and order positive fractions, positive decimals (to thousandths), and whole numbers by using: • benchmarks • place value • equivalent fractions and/or decimals. [CN, R, V] • Order the numbers of a given set that includes positive fractions,

positive decimals, and/or whole numbers in ascending or descendingorder, and verify the result using a variety of strategies.

• Identify a number that would be between two given numbers in anordered sequence or on a number line.

• Identify incorrectly placed numbers in an ordered sequence or on anumber line.

• Position fractions with like and unlike denominators from a given set ona number line and explain strategies used to determine order.

• Order the numbers of a given set by placing them on a number line thatcontains benchmarks, such as 0 and 1 or 0 and 5.

• Position a given set of positive fractions, including mixed numbers andimproper fractions, on a number line and explain strategies used todetermine position.

1. RATIONAL NUMBERSN3. Demonstrate an understanding ofrational numbers by: • comparing and ordering rational

numbers • solving problems that involve arithmetic

operations on rational numbers. [C, CN, PS, R, T, V]

• Order a given set of rational numbers, infraction and decimal form, by placing them ona number line, e.g., , -0.666, 0.5, .

• Identify a rational number that is betweentwo given rational numbers.

• Solve a given problem involving operations on rational numbers in fraction form anddecimal form.

N4. Explain and apply the order ofoperations, including exponents, with and without technology. [PS, T] • Solve a given problem by applying the order of

operations without the use of technology. • Solve a given problem by applying the order of

operations with the use of technology. • Identify the error in applying the order of

operations in a given incorrect solution.

-58

35

Math Background Number sense refers to a person’s general understanding ofnumber and operations, along with the ability to use thisunderstanding in flexible ways to make mathematicaljudgments and to develop useful strategies for solvingcomplex problems. Students have already developed whole-number sense through exploration, leading up to anunderstanding of traditional algorithms. By applying thiswhole-number sense to fractions, students can build theirunderstanding of fraction operations.

Previous grades have focused on developing basic fractionconcepts, such as what fractions are, how they relate to wholenumbers, and how they can be represented. In Grade 7,students built upon this understanding to compare and orderfractions and to add and subtract fractions. Now studentsconnect and extend that knowledge to develop fluency withfraction multiplication and division.

As in Grade 7, this chapter guides students along aconcrete-to-symbolic continuum. Students will begin by

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

3OverviewCopyright © 2008 Nelson Education Ltd.

using concrete and pictorial models to represent fractionmultiplication and fraction division before learningmathematical procedures and using symbolic representation.For example, the use of fraction strips can help studentsvisualize the relationship between dividend and divisor.In addition to more traditional procedures, students willapply strategies such as multiplication as repeated additionand fraction division using a common denominator.Understanding these alternate strategies helps studentsto make sense of fraction calculations.

Fraction strips and grids and counters are the primarymodels in this chapter, but many other materials, includingnumber lines and pattern blocks, are appropriate for workwith fractions, and students will benefit from experience witha wide assortment of models. In general, when working withmodels, use fractions whose denominators are 12ths or less tokeep the models manageable.

Throughout the chapter, students are encouraged toconsider whether they can use mental math to calculate andsolve problems, either in whole or in part; to analyze andevaluate their thinking; to connect concepts to previouslearning; to communicate their reasoning; and to listen toand 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.

Planning for Instruction

Problem SolvingAssign a Problem of the Week from the selection below (seesample answers on p. 108) or from your own collection. 1. Andrea’s yard is hectares in area. On Monday, she

mowed of the lawn. On Tuesday, she mowed as much of the lawn as she did on Monday. On Wednesday, shemowed as much of the lawn as she did on Tuesday.What area did Andrea mow on Wednesday?

2. Robbie saw a movie that was h long, and he calculated that the end credits lasted for of the movie. Carolyn saw a movie that was h long and she calculated that the end credits lasted for of the movie. Express the length of the end credits at Robbie’s movie asa fraction of the length of the end credits at Carolyn’smovie.

3. Tony read of one book in h. If he reads another book of the same length half as quickly, how long will ittake him to read the other book?

CommunicationLesson 2.11 provides opportunities for students to developand refine their communication skills by describing situationsthat involve multiplying and dividing mixed numbers andfractions. Students will also explain why a situation isappropriate for given numbers and operations.

423

712

415

134

320

112

14

13

12

58

Reading StrategiesThe reading strategies highlighted in this chapter includeQuestioning (Lesson 2.1) and Visualizing (Lesson 2.5). Toreinforce the use of these strategies, you may apply theseexamples to other questions throughout the lessons asopportunities present themselves.

Connections to LiteratureMaking connections to literature in middle school involvesworking with situations drawn from age-appropriate novelsand looking at Grades 7–9 concepts embedded in storybooksfor younger readers. Sources for ideas and lesson plans include

Mathematics Teaching in the Middle School, NationalCouncil of Teachers of Mathematics

Math and Literature: Grades 6 to 8 ( Jennifer M. Bay-Williams and Sherrie L. Martini, Math Solutions, 2004)

The Math Forum at Drexel University (website) You can also expand your classroom library or math centrewith books related to the math in this chapter. For example,

The Warlord’s Puzzle (Virginia Pilegard, Pelican Publishing,2000)

The Number Devil: A Mathematical Adventure (Hans Magnus Enzensberger; Henry Holt & Company Inc.; 2000)

Connections to Other Math StrandsShape and Space: When working with fraction strips,number lines, and grids, students use linear measurementconcepts such as longer than, shorter than, and total length.In Lesson 2.5, students use an area model to multiplyfractions greater than 1.

Connections to Other CurriculaInformation and Communication Technology: Studentssolve problems involving computer file sizes and memorycapacity.Social Studies: Students solve problems involving fractionsof populations and geographical areas.

Connections to Home and Community• Have students identify situations in which fractions are

multiplied and divided in their daily life and in the worldaround them.

• Have students measure items around their home, expressthe measurements as fractions of a whole (for example, thelength of the room could be the whole), and divide toexpress the length of one item as a fraction of the other.

• Send home Family Letter p. 92, which containssuggestions for a variety of activities related to the math inthis chapter that students can do at home.

• Have students complete the Nelson Math Focus 8 Workbookpages for this chapter at home.

• Use the suggestions for at-home activities in Follow-Upand Preparation for Next Class in Lessons 2.3 and 2.11.

Chapter 2 Planning Chart

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

division are intrinsically related.• There are many algorithms for performing a given operation.


Getting Started Activate knowledge about 1 day • Model addition and subtraction of positive fractions Pattern Block Designs adding and subtracting and mixed numbers.pp. 44–45 (TR pp. 11–14) fractions. • Determine the sum of two positive fractions or mixed numbers

with like denominators.• Determine the difference of two positive fractions or mixed

numbers with like denominators.• Determine a common denominator for a given set of positive

fractions or mixed numbers.• Determine the sum of two positive fractions or mixed numbers

with unlike denominators.• Determine the difference of two positive fractions or mixed

numbers with unlike denominators.• Simplify a positive fraction or mixed number. • Simplify the solution to a given problem involving the sum or

difference of two positive fractions or mixed numbers.• Solve a problem involving the addition or subtraction of positive

fractions or mixed numbers.

Lesson 2.1 Use repeated addition to N6 1 day • Rename mixed numbers as improper fractions and vice versa.Multiplying a Whole Number by a Fraction multiply fractions by whole • Perform multiplication as repeated addition.pp. 46–50 (TR pp. 15–20) numbers.

Lesson 2.2 Represent one fraction as part N6 1 day • Understand proper fractions as parts of wholes.Exploring Calculating a Fraction of a Fraction of another fraction. • Determine equivalent fractions.p. 51 (TR pp. 21–23)

Lesson 2.3 Multiply two fractions less N6 1 day • Determine equivalent fractions.Multiplying Fractions than 1. • Relate multiplication to calculating the area of a rectangle.pp. 52–56 (TR pp. 24–28) • Multiply decimal tenths.

• Multiply whole numbers by fractions.

Lesson 2.4 Estimate to predict whether a N6 1 day • Determine equivalent fractions.Exploring Estimating Fraction Products fraction product is closer to 0, • Compare fractions.p. 57 (TR pp. 29–32) , or 1.

Lesson 2.5 Multiply mixed numbers and N6 1 day • Relate multiplication to calculating the area of a rectangle.Multiplying Fractions Greater than 1 improper fractions. • Multiply proper fractions.pp. 58–63 (TR pp. 33–38) • Understand the distributive principle.

• Rename improper fractions as mixed numbers and vice versa.• Write fractions in lowest terms.

Lesson 2.6 Use a sharing model to N6 1 day • Name fractions as parts of sets and wholes.Dividing Fractions by Whole Numbers represent the quotient of a • Multiply and divide whole numbers.pp. 68–71 (TR pp. 45–49) fraction divided by a whole • Understand division as sharing.

number.

Lesson 2.7 Interpret and estimate the N6 1 day • Model fractions with fraction strips.Estimating Fraction Quotients quotient of fractions less than 1.pp. 72–75 (TR pp. 50–54)

Lesson 2.8 Divide fractions using models N6 1 day • Use a measurement model of division.Dividing Fractions by Measuring and using equivalent fractions • Determine equivalent fractions.pp. 76–80 (TR pp. 55–60) with a common denominator.

12

Grade 8 PacingStudent Book Section Lesson Goal Outcome 15 Days Prerequisite Skills/Concepts

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

Key Principles• Multiplication can represent repeated addition; this idea is useful when multiplying a whole

number by a fraction or mixed number. Multiplication can also represent the area of arectangle; this idea is useful when multiplying two fractions or mixed numbers.

• Division can represent calculating the size of a share when a total is shared; this idea isuseful when dividing a fraction or mixed number by a whole number. Division can alsorepresent the number of times one fraction fits into another one; this idea is useful whendividing two fractions or mixed numbers.

• A quotient can also be determined by considering a related multiplication.• When multiplying or dividing fractions, it is often, but not always, easier to use improper

fraction equivalents.

(Continued on page 6)


• pattern blocks, 3 of each block per student • Optional: Scaffolding for Getting Started Activity pp. 93–96• Optional: Review of Essential Skills: Chapter 2, Masters Booklet pp. 3–5

• counters, 50/student • 2 cm Grid Paper, Masters Booklet p. 34 Mid-Chapter Review Questions 1, 2, & 3• scissors, 1/student • Fraction Strips Tower, Masters Booklet p. 45 Chapter Review Questions 1 & 2

• Number Lines, Masters Booklet p. 39 Workbook p. 10• Optional: Chapter Checklist: Chapter 2, Masters Booklet p. 54

• scissors, 1/student • Fraction Strips Tower, Masters Booklet p. 45 Mid-Chapter Review Questions 4 & 5• coloured pencils, 1/student Chapter Review Questions 3 & 4• chart paper, 1/group Workbook p. 11• markers, 1/group

• scissors, 1/student • Fraction Strips Tower, Masters Booklet p. 45 Mid-Chapter Review Questions 7, 8, & 9• coloured pencils, 1/student • 2 cm Grid Paper, Masters Booklet p. 34 Chapter Review Questions 6 & 7

Workbook p. 12

• Optional: scissors, 1/group • Fractions Spinner, Masters Booklet p. 46 Mid-Chapter Review Question 10• Fraction Strips Tower, Masters Booklet p. 45 Chapter Review Question 8• 2 cm Grid Paper, Masters Booklet p. 34 Workbook p. 13• Number Lines, Masters Booklet p. 39

• scissors, 1/student • 1 cm Grid Paper, Masters Booklet p. 32 Mid-Chapter Review Questions 11 & 12Chapter Review Questions 9, 10, & 11Workbook p. 14

• counters, 50/group • Fraction Strips Tower, Masters Booklet p. 45 Chapter Review Questions 12, 13, & 14• scissors, 1/group • 2 cm Grid Paper, Masters Booklet p. 34 Workbook p. 15

• scissors, 1/student • Fraction Strips Tower, Masters Booklet p. 45 Chapter Review Questions 15 & 16Workbook p. 16

• scissors, 1/student • Fraction Strips Tower, Masters Booklet p. 45 Chapter Review Questions 17, 18, 19, & 20Workbook p. 17

Extra Practice in the Student Book Materials Masters and Workbook

Chapter Goals• Multiply and divide fractions by whole numbers, other fractions, and mixed numbers using models, drawings, and symbols. • Estimate products and quotients of whole numbers, fractions, and mixed numbers.• Solve and create problems using fraction operations.• Calculate the value of expressions involving fractions, using the proper order of operations.

(Continued on page 7)


Lesson 2.9 Divide fractions using a related N6 1 day • Multiply fractions.Dividing Fractions Using a Related Multiplication multiplication. • Rename improper fractions as mixed numbers and vice versa.pp. 82–86 (TR pp. 63–67)

Lesson 2.10 Use the order of operations in N6 1 day • Apply the order of operations for whole numbers and integers.Order of Operations calculations involving fractions.pp. 88–91 (TR pp. 70–74)

Lesson 2.11 Describe situations involving N6 1 day • Multiply decimal tenths and hundredths.Communicating about Multiplication and Division multiplying and dividing • Understand the meanings of division.pp. 92–95 (TR pp. 75–79) fractions and mixed numbers. • Understand the meanings of multiplication.

• Determine equivalent fractions.

Mid-Chapter Review pp. 64–67 (TR pp. 39–44) 3 daysCurious Math p. 81 (TR pp. 61–62)Math Game p. 87 (TR pp. 68–69)Chapter Self-Test p. 96 (TR pp. 80–81)Chapter Review pp. 97–100 (TR pp. 82–88)Chapter Task p. 101 (TR pp. 89–91)

Grade 8 PacingStudent Book Section Lesson Goal Outcome 15 Days Prerequisite Skills/Concepts


• scissors, 1/student • Fraction Strips Tower, Masters Booklet p. 45 Chapter Review Questions 21 Workbook p. 18

• scissors, 1/student • Fractions and Operations Cards I p. 99 Chapter Review Questions 22 & 23• Fractions and Operations Cards II p. 100 Workbook p. 19

Chapter Review Question 24Workbook p. 20

For materials and masters for features, reviews, and the Chapter Task, see the TR section. Workbook pp. 21–22

Extra Practice in the Student Book Materials Masters and Workbook


Chapter 2 Assessment Summary

Lesson 2.1 TR p. 20 3, short answer N6. Demonstrate an understanding of multiplying ConnectionsMultiplying a Whole Number by a and dividing positive fractions and mixed numbers, Fraction, pp. 46–50 concretely, pictorially, and symbolically.

[C, CN, ME, PS]

Lesson 2.2 TR p. 23 entire exploration, N6 Communication, Problem Solving,Exploring Calculating a Fraction of investigation Connectionsa Fraction, p. 51

Lesson 2.3 TR p. 28 7, short answer, model N6 Communication, ConnectionsMultiplying Fractions, pp. 52–56

Lesson 2.4 TR p. 32 entire exploration, N6 EstimationExploring Estimating Fraction investigationProducts, p. 57

Lesson 2.5 TR p. 38 5, model N6 ConnectionsMultiplying Fractions Greater than 1, pp. 58–63 10, short answer N6 Communication, Problem Solving

Mid-Chapter Review TR 1, short answer N6 Connections, Reasoningpp. 64–67 pp. 41–42 2, short answer N6 Reasoning

3, short answer N6 Connections, Problem Solving, Reasoning

4, model N6 Reasoning, Visualization

5, short answer N6 Connections, Reasoning

6, models, short answer N6 Reasoning, Visualization

7, short answer N6 Connections, Reasoning, Communication,Problem Solving

8, written answer N6 Communication, Reasoning

9, written answer N6 Communication, Problem Solving

10, written answer N6 Estimation, Reasoning

11, short answer N6 Reasoning


Lesson 2.6 TR p. 49 9, written answer, N6 Connections, Problem SolvingDividing Fractions by Whole short answerNumbers, pp. 68–71

Lesson 2.7 TR p. 54 6, short answer N6 EstimationEstimating Fraction Quotients, pp. 72–75

Lesson 2.8 TR 6, written answer, N6 Communication, ConnectionsDividing Fractions by Measuring, pp. 76–80 pp. 59–60 short answer

Curious Math TR p. 62 N6 Connections, ReasoningIt Is Just Like Multiplying!, p. 81

Lesson 2.9 TR p. 67 5, written answer, N6 Communication, Connections, Dividing Fractions Using a Related short answer Problem SolvingMultiplication, pp. 82–86

Math Game TR p. 69 N6 Mental Mathematics and EstimationTarget , p. 87

Lesson 2.10 TR p. 74 9, short answer N6 Connections, Problem SolvingOrder of Operations, pp. 88–91

Lesson 2.11 TR p. 79 8, written answer N6 CommunicationCommunicating about Multiplication and Division, pp. 92–95

23

Opportunities for Feedback: Assessment for Learning

Mathematical Process FocusStudent Book Lesson Chart Key Question Grade 8 Outcome for Key Question

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

These charts list references to the many assessmentopportunities in the chapter. Formative assessment (Assessmentfor Learning) provides information about students’understanding of concepts and helps you adapt instruction tostudents’ needs. A key question in each lesson links to the lessongoal. Initial or diagnostic assessment ideas (also part of

Assessment for Learning) are provided in Getting Started.Summative assessment (Assessment of Learning) opportunitiesare provided in the Mid-Chapter Review, Chapter Review,and Chapter Task. Have students self-assess their learning(Assessment as Learning) using Self-Test and one of the fourself-assessment tools provided in the Masters Booklet.


Assessment of Learning

Student Book Section Chart Question Grade 8 Outcome Mathematical Process Focus for Question

Mid-Chapter Review TR pp. 42–44 1, short answer N6 Connections, Reasoningpp. 64–67 2, short answer N6 Reasoning

3, short answer N6 Connections, Problem Solving, Reasoning

4, written answer N6 Reasoning, Visualization

5, short answer N6 Connections, Reasoning

6, models, short answer N6 Reasoning, Visualization

7, short answer N6 Connections, Reasoning, Communication, Problem Solving

8, written answer N6 Communication, Reasoning


10, written answer N6 Estimation, Reasoning



Chapter Review pp. 97–100 TR pp. 84–88 1, model N6 Connectionsand 2, short answer N6 Connections, EstimationChapter Test

3, short answer N6 Reasoning(TR pp. 103–105)4, short answer N6 Problem Solving

5, model N6 Connections

6, short answer N6 Mental Mathematics and Estimation

7, short answer N6 Problem Solving

8, written answer N6 Estimation, Connections

9, models N6 Connections



12, diagram N6 Connections

13, short answer N6 Mental Mathematics and Estimation

14, written answer N6 Communication

15, written answer N6 Estimation

16, short answer N6 Estimation

17, model N6 Connections

18, written answer N6 Communication


20, short answer N6 Connections


22, written answer, short answer N6 Communication, Mental Mathematics


24, written answer N6 Communication, Connections

Chapter Task TR p. 91 entire task, investigation N6 Communication, Connections, Mental Computer Gizmos, p. 101 Mathematics and Estimation, Problem Solving

Assessment as Learning

Student Book Section Student Self-Assessment Masters

Mid-Chapter Review pp. 64–67 Self-Assessment: Chapter 2 Lesson Goals: Fraction Operations, Masters Booklet p. 65Self-Assessment: Mathematical Processes, Masters Booklet p. 75Self-Assessment: What I Like, Masters Booklet p. 76Self-Assessment: How I Learn, Masters Booklet p. 76

Chapter Self-Test p. 96 Self-Assessment: Chapter 2 Lesson Goals: Fraction Operations, Masters Booklet p. 65Self-Assessment: Mathematical Processes, Masters Booklet p. 75Self-Assessment: What I Like, Masters Booklet p. 76Self-Assessment: How I Learn, Masters Booklet p. 76

Chapter 2Chapter Chapter 2Chapter 2


Using the Chapter OpenerDraw students’ attention to the photograph on Student Bookpages 42 and 43. As the water level rises, there is more massof water vibrating, because the glass is in contact with thewater. This increased mass decreases the frequency.

Have students consider the opening question. Discuss thefractions represented by the different amounts of liquid inthe glasses and how those fractions are related.

Sample Discourse“What fraction of each glass is full?”• 1, , , , , , ,

• , , , , , , , “How do the fractions compare with each other?”• Each fraction is greater than the fraction to its right.• The fractions become greater from right to left.• is twice as much as ; is twice as much as .“Which fractions with a denominator of 8 can be renamedin lower terms?”• can be renamed as .

• can be renamed as .

• can be renamed as .“Which fractions can be added to give a sum of 1?”• + = 1

• + = 1

• + = 1“Which fractions have a difference of ?”• - =

• - =

• - =

• - =

“Look at the two glasses holding yellow liquid. How manytimes would you pour the amount in the less-full glass intoa glass to match the amount in the more-full glass?”• 5 times“If you poured out the full blue glass into empty glasses,pouring the same amount as in the small green glass intoeach new glass, how many glasses would you need?”• 4 glasses

Pass out chart paper and markers to groups of students andhave students write out as many fractions and conceptsdealing with fractions as the picture could represent. Sharecharts with the rest of the class and discuss the fractions.

Discuss the five goals of the chapter. Ask for a volunteerto demonstrate adding + using fraction strips, grids, andnumber lines. Tell students that they will be using similarmodels for multiplication and division.

14

14

12

18

68

12

28

68

12

38

78

12

48

88

12

58

38

68

28

78

18

34

68

12

48

14

28

38

68

28

48

18

18

28

38

48

58

68

78

88

18

14

38

12

58

34

78

Chapter Opener

reciprocal

benchmark fraction

mixed number

improper fraction

product

quotient

STUDENT BOOK PAGES 42–43


Family Letter

Dear Parent/Caregiver:

Over the next three weeks, your child will be learning how to multiply and dividefractions and mixed numbers. Several models will be used to build understandingbefore the introduction of formal procedures. Your child will learn number strategiesthat he/she can apply to solving real-world problems that are relevant to his/her life.

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

• Involve your child in planning a family meal recipe. Your child can determine theamounts of ingredients that would be needed to make two or three times the recipe,or to make half the recipe.

• Have your child examine use of time daily or weekly. For example, ask your child todetermine in fractions how much of the day was spent at school, sleeping, practisingsports and/or music, doing homework, doing chores, etc. Your child can thencalculate how much time is spent monthly or annually on each of these tasks or, forexample, determine how many days or weeks it will take to spend 100 hours oneach task.

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

Chapter Chapter Chapter 72

Family Letter p. 92

Ask students to record in their journals their thoughts aboutone of the goals, using a prompt such as, “Examplesof situations in which I would need to multiply or dividefractions include....” At the end of the chapter, you canask students to complete the same prompt. Then they cancompare their ideas and reflect on what they have learned.

At this point, it would be appropriate to:• send home Family Letter p. 92• ask students to look through the chapter and add math

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

Name: Date:

Scaffolding for Getting Started Activity Page 4STUDENT BOOK PAGES 44–45

a) area: red and blue

equation:

b) area: how much more is green than blue

equation:

c) area: how much more is yellow and red than green and blue

equation:

d) area: how much more is red than green

equation:

G. a) Is it possible to create a design where the yellow area is 1units greater than the blue area?

How many yellow blocks? _____

How many blue blocks? _____

___ � ___ � 1

b) Is it possible to create a design where the blue and red area, together, is unit greater than the green area?


How many red blocks? _____

How many green blocks? _____

___ � ___ �

___ � ___ � 16

16

13

13


Name: Date:

95Blackline MastersCopyright © 2008 Nelson Education Ltd.


c) Write an equation with fractions to describe how much is yellow and red.

___ � ___ �

Write an equation with fractions to describe how much is green and blue.

___ � ___ �

Write an equation with fractions to describe how much more is yellow and red than green and blue.

___ � ___ �

d) Write an equation with fractions to describe how much more is red than green.

___ � ___ �

E. Write three other fraction equations that describe areas in Alison’s design.

a) area:

equation:

b) area:

equation:

c) area:

equation:

F. Make your own design below using

• yellow, red, blue, and green pattern blocks

• a total of eight blocks

• at least two yellow blocks

• at least one block of each other colour

Name: Date:


yellow

yellow

green

green

yellow

green

blue

red


C. The equation 3 � � 2 describes how much more of the design is one colour than

another.

a) Which colour block has a total area of ? _________

b) How many more blocks would you need to have a total area of 3? _________

c) Which colour block represents the total area of 3?_________

d) So, 3 � � 2 shows the difference between which two colours of the design?

D. Write equations with fractions and/or mixed numbers to describe each area, using the units in part A. Solve the equations. Show your work.

a) the red area and the blue area

• Colour in the hexagon to show the blue blocks and the red blocks from Allison’s design.

• How many sixths are filled?

red � blue � ______

Write an equation with fractions to describe the area of the red and blue parts.

___ � ____ �

b) How much more is green than blue?

• How many sixths are blue? _______

• How many sixths are green? _______

• green � blue � ______

• Write an equation with fractions to describe how much more is green than blue.

___ � ___ �

12

12

12

12

12

5Review of Essential Skills Masters

Chapter 2: Fraction Operations Page 3

Mixed Numbers

Write a mixed number as an improper fraction by multiplying the whole number by the

denominator of the fraction and adding that number to the numerator.

For example, 2 � because 2 � 4 � 8 and 1 � 8 � 9.

Then add the improper fractions as usual.

6. Write the mixed numbers as improper fractions and then add or subtract.

a) 1 � 2 b) 3 � 1

Renaming Fractions You rename fractions as equivalents in lower terms by dividing the numerator and denominatorby the same amount.

7. Rewrite in lowest terms.

a) b) c) 246

1610

1220

35

14

38

13

94

14

Name: Date:

Copyright © 2008 Nelson Education Ltd.

3 4

1216

=

÷ 4

÷ 4

4 Review of Essential Skills Masters


Adding and Subtracting Fractions with Like Denominators

Add or subtract fractions with like denominators by adding or subtracting the numerators. For example, � � .

3. Calculate.

a) � b) � c) � d) �

Adding and Subtracting Fractions with Unlike Denominators

You can create equivalent fractions by multiplying or dividing the numerator and denominator

by the same amount.

4. Complete.

a) � b) � c) �

You can add or subtract fractions with unlike denominators by creating equivalent fractions. For example, to add � , use a common denominator of 15.

� � � � �

5. Calculate.

a) � b) � c) � 15

23

12

45

38

34

1715

5 � 1215

1215

515

1215

45

515

13

45

13

■

2034

■

1523

■

1025

512

1012

48

78

39

29

35

15

710

410

310

Name: Date:


312

14

=

× 3

× 3


11Getting Started: Pattern Block DesignsCopyright © 2008 Nelson Education Ltd.

PREREQUISITE SKILLS/CONCEPTS

• Model addition and subtraction of positive fractions andmixed numbers.

• Determine the sum of two positive fractions or mixednumbers with like denominators.

• Determine the difference of two positive fractions or mixednumbers with like denominators.

• Determine a common denominator for a given set ofpositive fractions or mixed numbers.

• Determine the sum of two positive fractions or mixednumbers with unlike denominators.

• Determine the difference of two positive fractions or mixednumbers with unlike denominators.

• Simplify a positive fraction or mixed number.• Simplify the solution to a given problem involving the sum

or difference of two positive fractions or mixed numbers.• Solve a problem involving the addition or subtraction of

positive fractions or mixed numbers.

Getting StartedPattern Block Designs


GOALActivate knowledge about adding and subtracting fractions.

Preparation and PlanningPacing 25–40 min Activity

15–20 min What Do You Think?

Materials • pattern blocks, 3 of each block per student

Masters • Optional: Scaffolding for Getting StartedActivity pp. 93–96

• Optional: Review of Essential Skills: Chapter 2,Masters Booklet pp. 3–5

Nelson Website Visit www.nelson.com/mathfocus and followthe links to Nelson Math Focus 8, Chapter 2.

Math BackgroundThe Getting Started Activity gives students anopportunity to recall and review fraction principlesintroduced in earlier grades:• To add or subtract fractions with like denominators,

add or subtract the numerators.• To add or subtract fractions with unlike

denominators, use a common denominator.• Fractions have more than one name.• A fraction can be written in lower terms by

identifying a common factor of the numeratorand denominator.

Ensure students understand that the denominator tells the number of parts in the whole and the numerator tells how many of the parts the fraction represents, thatfractions with numerators greater than their denominatorsare greater than 1, and that a mixed number can berenamed as an improper fraction and vice versa.

Students need a solid understanding of these conceptsand principles before they begin to multiply or dividefractions.

Optional: Review of Essential Skills: Chapter 2, Masters Booklet pp. 3–5

3Review of Essential Skills Masters


Modelling Addition and Subtraction of FractionsYou can use fraction strips or grids and counters to model addition and subtraction of fractions and mixed numbers.

For example, these fraction strips show that � � .

1. Use fraction strips to calculate � .

For example, these counters show that � � .

2. Use grids and counters to calculate � .12

45

112

23

34

34

23

512

16

1

4

Name: Date:


Name: Date:


yellow

yellow

green

green

yellow

green

blue

red


A. The area of one yellow block is 1 unit.1 yellow block � 1

a) Which colour block fills of the area of

the yellow block?

1 _______ block �

b) Which colour block fills of the area of the yellow block?

1 _______ block �

c) Which colour block fills of the area of the yellow block?

1 _______ block �

B. The equation 3 � � 3 tells the sum of the areas

of two of the colours.

Which are the two colours?

Which colour block has a total area of 3?

_______ blocks � 3

Shade the area.

Which colour block has a total area of ?

_______ blocks �

Shade the area.

What is the total of the area you have shaded? ______

How can you show that this area is also 3 ?

___________________________________________________

___________________________________________________

12

36

36

12

36

13

13

16

16

12

12

Chapter 2

Optional: Scaffolding for Getting Started Activity pp. 93–96

c) For example, - = 2 describes howmuch greater the yellow and red areas are than thegreen and blue areas.

d) For example, 1 - = 1 describes how much more isred than green.

For Prompt E, for example, 2 + describes the yellowand blue areas, – describes how much more is red than blue, and describes how muchgreater the yellow and green areas are than the red andblue areas.

G. a) Yes; for example,

b) Yes; for example,

A2 +26 B - A32 +

13 B

13

32

13

16

26

12

56A13 +

26 BA2 +

32 B


Using the Activity (Whole Class/Individual) ➧ 25–40 min

Use this activity to activate knowledge of fractions, mixednumbers, adding fractions and mixed numbers, andsubtracting fractions and mixed numbers, and as anopportunity for initial assessment.

Begin the activity by having students look at Alison’sdesign on Student Book page 44. If necessary, review therelationships between the pattern blocks before beginning theactivity. Have students answer the prompts individually andthen discuss the answers as a class.

Sample Discourse“How can you determine the area of each colour block?”• I can place the different colour blocks on the yellow block to

see how many will fit.“In Prompt D, how do you know whether to add or subtract?”• The word “and” in “red and blue parts” tells me to add

because I need to find what their area is altogether.• “How much more” tells me to subtract because I need to find

the difference between the two areas.

If extra support is required, guide those students and providecopies of Scaffolding for Getting Started Activity pp. 93–96.

Answers to the Activity

A. a) unit b) unit c) unit

B. Yellow and green; the area of each yellow block is 1 unit, and there are 3 yellow blocks; the area of each green block is unit, and there are 3 greens; 3 + + + = 3 + = 3 + = 3 .

C. Yellow and red; the area of 3 yellow blocks is 3 units; thearea of 1 red block is unit; 3 - = 2 .

D. • + =

• - =

• 3 - + = 2

• - = 0

E. For example, 3 + describes the yellow and blue areas, - describes how much more is red than blue,and 3 - - - describes how much more is yellow than any other colour.

F.

For Prompt D, a) For example, 1 + = 1 describes the red and blue

areas.b) For example, - = 0 describes how much more is

green than blue.

13

26

56

13

12

13

36

12

13

12

13

36

12

23

13 BA361

2

16

13

36

56

13

12

12

12

12

12

12

36

16

16

16

16

16

13

12

13Getting Started: Pattern Block DesignsCopyright © 2008 Nelson Education Ltd.

Using What Do You Think? (Individual/Small Groups/Whole Class)

➧ 15–20 min

Use this anticipation guide to activate knowledge andunderstanding of fraction operations. Explain to studentsthat the statements involve math concepts or skills theywill learn about in the chapter—they are not expected toknow the answers at this point. Ask students to read thestatements, think about each one for a few seconds, anddecide whether they agree or disagree with each. Havevolunteers explain the reasons for their choices. Studentscan exchange their thoughts in small groups, in groupswhere all agree or disagree, or in a general class discussion.Tell students they can revisit their ideas at the end of thechapter.

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. Agree; for example, + = , but + + 7 1 (here the

student is only thinking of unit fractions). *Disagree; for example, + is greater than 1.

2. Agree; for example, - = or .*Disagree; if the two fractions are really close, thedifference is less than both.

3. Agree; for example, when you multiply numbers, theproduct is always more.*Disagree; for example, 0.5 * 0.5 = 0.25.

4. Agree; for example, when you divide, you take a greateramount and share it out into smaller amounts, but whenyou multiply, you have many groups, so it’s more thanyou started with.*Disagree; for example, 2 , 0.5 = 4, but 2 * 0.5 = 1.

5. *Agree; for example, “how many groups” is a meaningof division.Disagree; for example, you can’t calculate a , b, becauseyou don’t know how much a and b are worth.

56 -

26 =

36

12

14

34

34

34

12

12

12

12

14

14


SUPPORTING STUDENTS WHO ARE NOT READY

This chapter assumes that students already understand fractions as parts ofa whole and are comfortable determining equivalent fractions, writing mixednumbers as improper fractions and vice versa, and adding and subtractingfractions. They are already operating at a minimum of Phase 3 and may bein Phase 4 on the PRIME Number development curriculum.

In selected lessons you will find suggestions for adapting the lesson to dealwith students who are at a lower developmental phase.

For this activity:

1. Some students may not be ready to add and subtract mixed numbers.Provide a design using only red, green, and blue blocks and have studentscomplete Prompts A to E based on this new design.

2. Some students may benefit from having the context of the problem removed.Provide these students with the equations to solve for Prompts D and E toavoid any confusion that may arise from having to determine the fractionalareas represented by each colour.

SUPPORTING STUDENTS WHO ARE ALMOST THERE

1. Use Scaffolding for Getting Started Activity pp. 93–96.

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

3. Have students place enough red blocks on 1 yellow block to completelycover the yellow block. Ask students, “How many red blocks will fit on ayellow block?” Point out that since the yellow block represents 1 unit, eachred block represents unit. Repeat with the other colours of blocks.

4. Have students divide the addends into unit fractions. For example,. Then ask students, “What colour

represents 1 unit?” (yellow) “What colour represents unit?” (green).16

3 +36 = 1 + 1 + 1 +

16 +

16 +

16

12

5. If students are having difficulty using fractions and mixed numbers tosymbolize the areas, ask them to first use unit fractions to symbolize thearea of each colour. For example, 1 green block has an area of , so the area ofthe green blocks in Alison’s design is . Remind students to add thenumerators to find the total area of the green blocks: .

6. Ask students to think about which situations with numbers suggest addition(combining) and which suggest subtraction (taking away, comparing, orfinding a missing part).

7. For the first design, students might use one blue block, decide how muchyellow to use, and then add other necessary blocks. For the second design,students might choose a number of greens to use and then a number ofblues. They then make the reds fit the rule and then add yellows.

16 +

16 +

16 =

36

16 +

16 +

16

16

Differentiating Instruction: How you can respond

Initial Assessment: Assessment for Learning

What you will see students doing

When students understand If students misunderstand

Prompt A• Students identify the area of each colour block based on the area of the

yellow block.

Prompts B & C• Students identify the colours represented by given fractions and mixed

numbers.

Prompts D, E & F• Students write addition and subtraction equations with fractions and mixed

numbers to represent relationships among different-coloured areas in thedesign.

Prompt G• Students create their own designs with pattern blocks according to given

rules.

• Students may not be able to recognize the relationships between the yellowblock and the other blocks. (See Extra Support 3.)

• Students may not be able to make the connection between the given mixednumbers and fractions and the corresponding pattern blocks. (See ExtraSupport 4.)

• Students may not understand how to use fractions and mixed numbers tosymbolize concrete models. (See Extra Support 5.)

• Students may not be able to translate a phrase into an addition or subtractionsentence. (See Extra Support 6.)

• Students may not be able to perform addition or subtraction of fractions ormixed numbers with unlike denominators. (See Extra Support 2.)

• Students may not be able to determine the number of each colour block touse in order to create a design that follows the given rules. (See ExtraSupport 7.)

15Lesson 2.1: Multiplying a Whole Number by a FractionCopyright © 2008 Nelson Education Ltd.


Preparation and PlanningPacing 5–10 min Introduction

15–20 min Teaching and Learning 20–30 min Consolidation

Materials • counters, 50 per student• scissors, 1 per student

Masters • 2 cm Grid Paper, Masters Booklet p. 34• Fraction Strips Tower, Masters Booklet p. 45• Number Lines, Masters Booklet p. 39• Optional: Chapter Checklist: Chapter 2,

Masters Booklet p. 54

Recommended Questions 3, 5, 6, 8, 13, & 16Practising Questions

Key Question Question 3

Extra Practice Mid-Chapter Review Questions 1, 2, & 3Chapter Review Questions 1 & 2Workbook p. 10

Mathematical CN (Connections)Process Focus

Vocabulary/Symbols improper fraction, mixed number


34 2 cm Grid Paper Copyright © 2008 Nelson Education Ltd.

2 cm Grid Paper, Masters Booklet p. 34

112

14

15

18

19

110

111

112

13

16

16

16

16

16

17

17

17

17

17

17

18

18

18

18

18

18

18

19

19

19

19

19

19

19

19

110

110

110

110

110

110

110

110

110

111

111

111

111

111

111

111

111

111

111

112

112

112

112

112

112

112

112

112

112

112

12

13

13

14

14

14

15

15

15

15

16

17

45Fraction Strips TowerCopyright © 2008 Nelson Education Ltd.

Fraction Strips Tower, Masters Booklet p. 45

Chapter 2 Checklist: Fraction OperationsThroughout the chapter, observe individual students for evidence that they understand key knowledge and can perform key skills.

Use

repe

ated

add

ition

to m

ultip

ly fr

actio

ns b

yw

hole

num

bers

.

Repr

esen

t one

frac

tion

as p

art o

f ano

ther

fract

ion.

Mul

tiply

two

fract

ions

less

than

1.

Estim

ate

to p

redi

ct w

heth

er a

frac

tion

prod

uct

is c

lose

r to

0,

, or 1

.1 2

Mul

tiply

mix

ed n

umbe

rs a

nd

impr

oper

frac

tions

.

Use

a sh

arin

g m

odel

to re

pres

ent t

he q

uotie

ntof

a fr

actio

n di

vide

d by

a w

hole

num

ber.

Inte

rpre

t and

est

imat

e th

e qu

otie

nt o

f fra

ctio

nsle

ss th

an 1

.

Divid

e fra

ctio

ns u

sing

mod

els

and

usin

geq

uiva

lent

frac

tions

with

a c

omm

on d

enom

inat

or.

Divi

de fr

actio

ns u

sing

a re

late

d m

ultip

licat

ion.

Use

the

orde

r of o

pera

tions

in c

alcu

latio

nsin

volv

ing

fract

ions

.

Desc

ribe

situ

atio

ns in

volv

ing

mul

tiply

ing

and

divi

ding

frac

tions

and

mix

ed n

umbe

rs.

Student

54 Assessment Masters Copyright © 2008 Nelson Education Ltd.

Optional: ChapterChecklist: Chapter 2,Masters Booklet p. 54

Number Lines, Masters Booklet p. 39

39Number LinesCopyright © 2008 Nelson Education Ltd.

GOALUse repeated addition to multiply fractions by whole numbers.

Multiplying a WholeNumber by a Fraction


• Rename mixed numbers as improper fractions and vice versa.• Perform multiplication as repeated addition.

SPECIFIC OUTCOME

N6. Demonstrate an understanding of multiplying anddividing positive fractions and mixed numbers, concretely,pictorially, and symbolically.[C, CN, ME, PS]

Achievement Indicators• Express a given positive mixed number as an

improper fraction and a given positive improperfraction as a mixed number.

• Model multiplication of a positive fraction by awhole number concretely or pictorially and recordthe process.

• Generalize and apply rules for multiplying anddividing positive fractions, including mixed numbers.

Math BackgroundThis lesson provides students with several different hands-on models for multiplying a fraction by a whole numberand asks them to make a conjecture about what a processfor multiplying a fraction by a whole number might be.Multiple models are used in order to trigger connectionsamong the different models.

Using grids and fraction strips, students model a properfraction (a fraction where the numerator is less than thedenominator). Using the idea of multiplication as repeatedaddition, they calculate the product by repeating the modelthe required number of times. Then they can determine theproduct by finding “wholes” within the model and formingmixed number products when there are extra “parts.”Multiplying on a number line is a bit less concrete. To dothis, students start at zero and then make a whole numberof jumps of fractional size. They must be able to accuratelyread and place fractions on a number line and make equaljumps of fractional size.

In many of the Practising exercises, it is not specified whichstrategy (grid, fraction strip, or number line) should be used.Although each student will probably have a “preferred”method, encourage fluency with all three models.

2.1


Introduction (Whole Class) ➧ 5–10 min

Review the fundamentals of fractions. Review the terms“numerator” and “denominator,” reminding students that thedenominator gives the number of equal parts into which awhole is divided, and the numerator gives the number ofparts being named. Ask the class to suggest ways to model .Guide them to name a variety of model types, such as circlemodels, grid models, number strips, and number lines.Sketch several different models.

Sample Discourse

“What does the fraction mean?”• Three of four equal parts“How could you use a rectangle to model ?”• Divide the rectangle into four equal parts. Then you can colour

in three of the parts. Three fourths of the rectangle is coloured.

34

34

34

1

2

3

4

5

6

7

8

Reading StrategiesQuestioningLearning to ask clarifying questions during reading is astrategy that helps students break down the facts andbetter understand the information provided in the text. In mathematics, when students learn to ask clarifyingquestions while they are reading, they are more activelyengaged in searching for meaning.

Have students open their Student Books to page 49 andread Question 1 independently. Create a two-column charton chart paper, a transparency, the board, or interactivewhiteboard, with the headings “Questions” and “Answers.”

Encourage students to ask questions about the problemand provide the answer as they walk through the problemstep by step. Provide prompts as needed (e.g., How muchwater would be in the pot after Jennifer pours twice?).

Have students work with a partner and write threequestions that could help solve the problem. Review thequestions with the class, providing answers whereappropriate. Reinforce the reading strategy—point outhow good questioning techniques can help clarify text andsupport comprehension of problems in math.

Students are prompted to use the Questioning strategyin Question 1.

Questions Answers

“How could you use fraction strips to model ?”• Three fourths is like using three parts.“How could you model on a number line?” • Draw a number line from 0 to 1. Divide the space between

0 and 1 into four equal parts by marking the number line.The marks are apart. Three fourths is the mark closest to 1.

“How can you model using a grid and counters?”• I can use a grid that holds 4 counters and place 1 counter in

3 of the squares.

Teaching and Learning ➧ 15–20 min

LEARN ABOUT the Math (Whole Class/Individual/Pairs)

Together with students, read the problem and centralquestion on page 46 of the Student Book. Distribute a pieceof grid paper and about 20 counters to each student or pair.Ask students what operation is needed to solve the problem.Some students may suggest using addition to determine thenumber of pitchers that can be filled. Acknowledge that this

1

2

3

4

5

6

7

8

34

14

34

14

34


Answers to Learn about the MathA. For example, the lemonade can fill 3 whole pitchers

because each pitcher is less than half full and 6 halvesis 3 wholes.

B. For example,

C. For example,

, 2 , or 2D. For example, there are 6 groups of , and adding a

number 6 times is the same as multiplying the numberby 6.

E. For example,

Reflecting (Pairs/Whole Class)

These questions help students reflect on how the model canlead them to generalizations about multiplying a fraction by awhole number.

038 8 8 8 8 8

6 9 12 15 181 2 3

38

14

28

188

is a valid strategy and point out that each of the addends is .Draw from students that multiplication is a way to performrepeated addition of addends, so another way to solve theproblem is to solve 6 � . Work through the problemtogether, with each student or pair modelling the problem,step by step. Students communicate their reasoning whenthey explain how they made their estimate. Using the model,they visualize multiplication of a whole number and afraction, and make the connection that multiplication is thesame as repeated addition. Students can complete PromptsA to E to solve the central question (in pairs or individually).

Sample Discourse“How can you use the benchmark fraction to estimate theproduct of 6 and ?”

• is a bit less than , which is . If each of the 6 pitchers werehalf full, I would have the same as 3 full pitchers. So, myanswer should be a bit less than 3 full pitchers.

“Once you have modelled six times, what do you do withthe counters to calculate the product?”• I move the counters around so that I make as many full grids

as I can. The leftover counters become the fraction part of themixed number.

“What other models could you use to represent the problem?”• I could use an eighths strip with 3 eighths marked for each

pitcher and place them side by side to get the total length ofall the strips.

38

12

48

38

38

12

38

38


Sample Discourse“When you add fractions with the same denominator, whatis the denominator of the sum?”• The same as the denominator of the addends“In this problem, we used as an addend 6 times. So,the denominator of the sum should be 8. Another wayto describe the problem is that we multiplied three eighths6 times. Since multiplication is the same as repeatedaddition, what should the denominator of the product be?”• 8“We multiplied three eighths 6 times. What is the numeratorof the product?”• 18“How did you get 18?”• I multiplied 6 by 3.

Answers to Reflecting F. For example, I was adding eighths, so the denominator

would stay 8.G. For example, multiply the whole number by the

numerator. Then write a fraction with the productas the new numerator and the original denominator.I think it will work since the product is the quantityin each group (the fraction) multiplied by the numberof groups (the whole number).

38

Consolidation ➧ 20–30 min

WORK WITH the Math (Pairs)

Have students read Example 1 in pairs, using grids andcounters to work through the steps in the example. Havestudents read Example 2, using fraction strips to workthrough the steps. Be sure students understand that thestrips are lined up end to end and matched to full strips.In Example 3, provide students with a number line markedin halves and wholes. Have students make the 5 jumps of

on their number lines.

Checking (Pairs/Whole Class)

Provide each pair of students with grid paper, counters, andfraction strips. Have students work in pairs to solve eachproblem and then discuss the results as a class. For eachproblem, encourage students to estimate before calculatingso that they can check to see if their answers are reasonable.1. Have students use the reading strategy Questioning with

this question. Tell students they can use whichever modelthey like to determine the answer. Ask for volunteers toexplain their solutions.

2. Remind students that multiplication is repeated addition.The first factor gives the number of addends; the secondfactor names the addends. So, they will use as anaddend 5 times.

Practising (Individual)

Provide students with grid paper, counters, and fractionstrips. These questions give students opportunities to practisemultiplying fractions by whole numbers using grids,counters, fraction strips, and number lines as visualizationtools. Students will apply reasoning skills to determine boththe structure of the grid and how to represent the product asa whole or mixed number. They have opportunities to solveproblems in varying contexts involving fractions and wholenumbers and their products, and to communicate theirunderstanding of strategies for multiplying whole numbersand fractions by explaining their thinking.3. Remind students that a fraction is greater than 1 when its

numerator is greater than its denominator. A fraction isequal to 1 when its numerator and denominator are equal.

4. Students can use benchmark fractions to estimate. Forexample, in part d), is between and 1, so the productof will be between the products and .

7. Remind students that the fraction is often read as “onequarter.”

10. To help them write 2.3 as a mixed number, remindstudents that the decimal number 2.3 is read “two andthree tenths.” They can use the mixed number toform the improper fraction , and then multiply. Or,they can multiply the whole number and fraction part ofthe mixed number separately and sum the two products10 � .15

10 BA

2310

2 310

14

7 * 17 *127 *

710

12

710

34

112

1

2

3

4

5

6

7

8


Answers to Key Question3. For example,

a)

b) Show 5 sets of by putting counters in 3 out of 5 squares in each 1 � 5 rectangle.

c) 6 *

3

8=

6 * 3

8=

18

8= 2

2

8 or 2

1

4

5 *

35

=

5 * 3

5=

15

5= 3

038 8 8 8 8 8

6 9 12 15 181 2

35

2 *

13

=

2 * 1

3=

2

3

d)

e) Show 3 sets of by putting counters in each of6 squares in each of three 1 * 6 rectangles; then addthree extra counters into a fourth 1 * 6 rectangle.

f )

Closing (Whole Class)

Have students summarize what they have learned by askingthem how they get the numerator and denominator of theproduct when they multiply a whole number and a fraction.Then ask them to respond to Question 16 to consolidatetheir learning for this lesson. Students might find thequestion easier to answer if they are allowed to do the workof calculating each product first.

Answers to Closing Question 16. a) The numerator of the product of a whole number and

a fraction is the product of the whole number and thenumerator of the fraction. In each case, the wholenumber is 5 and the numerator of the fraction is 2.So, the numerator of each product is 10.

b) The denominator of the product of a whole numberand a fraction is the denominator of the fraction. Thedenominators of the fractions in each product aredifferent.

Follow-Up and Preparation for Next ClassAsk students to work with their parents to model multiplyinga fraction by a whole number using measuring cups. Forexample, they can solve by filling a -cup measuring cup 4 times, each time pouring the contents into a 2-cupglass measuring cup. Then they can read the markings on thecup to show the product.

134 *

13

8 *

42

=

8 * 42

=

322

= 16

3 *

76

=

3 * 76

=

216

= 336

or 312

76

4 *

2

5=

4 * 25

=

85

= 135


EXTRA CHALLENGE

• Challenge students to determine possible whole number * fractioncalculations that result in a product of 2 . For example, 4 *

58 = 2 12, 5 *

12 = 2 12, and 8 *

516 = 2 12.

12

EXTRA SUPPORT

1. Help students understand the relationship between addition andmultiplication using whole numbers. Show students 4 groups of 3 counterseach and ask them to identify the total number of counters. Ask themto name the operation needed, and show them that they can either add(3 + 3 + 3 + 3) or multiply (4 * 3). Next, show them 8 red trapezoid pattern blocks and remind them that each one represents half a yellowhexagon. Tell them that the 8 trapezoids can be used to represent the sum

or the product . Show them that both problems lead to the 8 halves joining to form 4 wholes.

2. Help students understand how counters can be rearranged from one gridto another without changing the value of the model. On a transparency,board, or electronic whiteboard, display 4 hexagons each with 6 equal parts,5 of which are shaded.

8 *12

12 +

12 +

12 +

12 +

12 +

12 +

12 +

12

Ask, “How many triangular pieces are shaded?” (20)

Ask students how they would rearrange the triangular pieces to form as manywholes as possible just as they rearranged the counters on the grids. Afterthey rearrange the triangles, ask, “How many triangular pieces are shaded?”(20) Point out that the total number of green triangles did not change, so thetwo models represent the same number.

Ask, “Which improper fraction and which mixed number does the modelrepresent?” A 20

6 = 326 B


SUPPORTING DEVELOPMENTAL DIFFERENCES

• Some students may not be ready to multiply a whole number by an improperfraction because they have difficulty modelling the improper fraction. Changethe question in Example 3 and Questions 3 c), 3 f), 4 a), 4 c), and 4 f) so thatthe second factor is a proper fraction rather than an improper fraction.




• Students model the fraction and then repeat the model the appropriatenumber of times to show multiplication.

• Students recognize that the counters in the collected models can be movedaround to form wholes and that leftover counters must be included in theproduct as the fraction part of a mixed number.

• Students use an algorithm to multiply a whole number by a proper fraction.They determine products and express them in the desired form (mixed number or improper fraction).

Key Question 3 (Connections)• Students make connections between models and algorithms. They can

multiply a whole number by a fraction using grids and counters, fractionstrips, number lines, or an algorithm.

• Students may not understand the relationship between addition andmultiplication. They may not understand that a multiplication problem can be solved using addition. (See Extra Support 1.)

• Students may not understand how counters can be rearranged from one gridto another without changing the value of the model. (See Extra Support 2.)

• Students may not understand how to apply the algorithm to multiply a wholenumber by a proper fraction. (See Extra Support 1.)

• Students may not understand the connection between models and thealgorithm used to multiply. (See Extra Support 1.)

21Lesson 2.2: Exploring Calculating a Fraction of a FractionCopyright © 2008 Nelson Education Ltd.


• Understand proper fractions as parts of wholes.• Determine equivalent fractions.

SPECIFIC OUTCOME


Achievement Indicator• Model multiplication of a positive fraction by a

positive fraction concretely or pictorially using anarea model and record the process.

Exploring Calculating aFraction of a Fraction

STUDENT BOOK PAGE 51

GOALRepresent one fraction as part of another fraction.

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

Connections are made as students apply the fractionstrips model to a new context, finding a fraction of afraction. Students develop mathematical communicationskills as they present their solutions to their classmatesand respond to classmates’ questions. Listening skills aredeveloped as students listen to classmates’ strategies andare asked to comment on them. When reviewing oneanother’s solutions, students will realize that there aremultiple ways to solve this problem, and they will learnto value a variety of approaches.

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

Materials • scissors, 1 per student• coloured pencils, 1 per student• chart paper, 1 per group• markers, 1 per group

Masters • Fraction Strips Tower, Masters Booklet p. 45

Key Question entire exploration

Extra Practice Mid-Chapter Review Questions 4 & 5Chapter Review Questions 3 & 4Workbook p. 11

Mathematical PS (Problem Solving), C (Communication), and Process Focus CN (Connections)


112

14

15

18

19

110

111

112

13

16

16

16

16

16

17

17

17

17

17

17

18

18

18

18

18

18

18

19

19

19

19

19

19

19

19

110

110

110

110

110

110

110

110

110

111

111

111

111

111

111

111

111

111

111

112

112

112

112

112

112

112

112

112

112

112

12

13

13

14

14

14

15

15

15

15

16

17



2.2



Have 12 students stand in a line at the front of theclassroom. Discuss different ways to represent the number of students using fractions. Include a discussion aboutfractions of fractions of the group.

Sample Discourse“If there are 12 students in the whole group, what fraction

does each student represent?”•“How many students are in half of the group?”• 6“What fraction of the whole group is made up of 3 students?”•


EXPLORE the Math (Whole Class/Pairs/Small Groups)

Present the problem and central question on Student Bookpage 51 to students. Provide each pair or group with avariety of materials to work with, including Fraction StripsTower, Masters Booklet p. 45, coloured pencils, scissors, andalso chart paper and markers. Explain that students are to• make any decisions they feel are necessary to solve the

problem;• write out the main points in their solution on chart paper;

and• be prepared to communicate their solution process to the

rest of the class.

No one approach to the problem should be suggested;encourage students to choose their own strategies. Theymust take the information given and work toward what theproblem is asking for. Students may look for a method thatthey think you would want to see. Encourage them to useany method they want to come up with the solution.

Circulate and observe students as they work. Ask questionssuch as those in the following sample discourse.

Sample Discourse“How does the fraction tower help you?”• The fraction tower shows me the relationship between different

fraction strips.“Why might you show as half of a fraction on the tower,but not a third of a fraction?” • Two fifths is half of , which is on the tower, but 3 groups of

goes past the end of the tower.“What do you notice about the relationship between the twofractions on the card and the fraction you are trying to cover?”• The fractions on the card are greater than the fraction that I

am trying to cover.

25

45

25

1

2

3

4

5

6

7

8

14

112

1

2

3

4

5

6

7

8

Possible Solutions to Explore the Math

Possible solutions to cover :of or of (and equivalent fractions such as of , of

, etc.)of or of (and equivalent fractions such as of , of , etc.)of or of (and equivalent fractions such as of , of

, etc.)of or of (and equivalent fractions such as of , of , etc.)

Possible solutions to cover :of or of (and equivalent fractions such as of , of , etc.)of or of (and equivalent fractions such as of , of , etc.)

Possible solutions to cover :of or of (and equivalent fractions such as of , of

, etc.)of or of (and equivalent fractions such as of , of

, etc.)of or of 3

778

78

37

58

610

610

58

35

58

58

35

34

36

68

12

12

34

34

12

38

610

23

35

46

23

35

35

23

810

12

45

24

12

45

45

12

25

1012

15

56

210

15

56

56

15

29

68

29

912

34

29

29

34

312

46

69

14

23

14

14

23

39

24

26

12

12

13

13

12

16

23Lesson 2.2: Exploring Calculating a Fraction of a FractionCopyright © 2008 Nelson Education Ltd.



Provide the opportunity for students to share and communicateabout their work. Have students describe to the rest of the classhow they solved the problem, using their chart paper as anorganizing tool for other students to follow. If you saw some

1

2

3

4

5

6

7

8

particularly interesting approaches when you circulated amongstudents as they were working through the problem, be sure toinvite those students to be among the presenters.

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




• Students use fraction strips to find a fraction of a fraction.

• Students rename fractions to determine if a product is equivalent to a givenfraction.

• Students may choose the wrong fraction strips to model a particular fraction.(See Extra Support 1.)

• Students are unable to determine what parts of fraction strips to use. (SeeExtra Support 1 and 2.)

• Students may not understand that fractions can have more than one name.(See Extra Support 3 and 4.)

• Students are unable to write fractions in lowest terms. (See Extra Support 4.)

EXTRA SUPPORT

1. Review the fraction strips model. Remind students that the number of partson a fraction strip represents the denominator and the number of shadedparts represents the numerator.

2. Tell students to use the fraction tower as a visual aid. For example, to coverof , first colour three sections on a fifths strip, then fold the coloured part

in half and compare the half to the different rows in the fraction tower todetermine a part of another fraction strip that is the same length.

3. Have students take a sheet of paper and fold it in half. Tell them to shadeone half of the sheet. Next have students fold the sheet in half again. Ask

35

12


EXTRA CHALLENGE

• Starting with , cover each unit fraction in the fraction strip tower using eachof the unit fractions that are greater than the given fraction. For example,what fraction of will cover ? What fraction of will cover ? What fractionof will cover ?1

514

15

13

15

12

13


• For students who are not ready to think of one fraction in relation to another,change the problem to one where students compare fractions instead, forexample, by determining fractions greater or less than given ones from as manyrows as possible.

SUPPORTING LEARNING STYLE DIFFERENCES

• This exploration is best suited to visual and kinesthetic learners. Those whorespond well to interpersonal situations will benefit from the small-groupdiscussion, so ensure that students are working co-operatively in groups anddiscussing the various solutions.

students how many parts the sheet has (4) and how many are shaded (2).Write “ ” on the board, a transparency, or an interactive whiteboard.Have students continue to fold the sheet into smaller and smaller parts,naming the fraction that is shaded with each number of parts.

4. Remind students that if a fraction is written in lowest terms, the numeratorand denominator have no common factors. Check to see if the numeratorand denominator have any common factors, and then divide both numbersby that factor.

12 =

24

• Some students may not be ready to work with equivalent fractions. Providelists of fractions that are equivalent to , , and , so students will be able torecognize when they have found a solution.

38

25

16



• Determine equivalent fractions.• Relate multiplication to calculating the area of a rectangle.• Multiply decimal tenths.• Multiply whole numbers by fractions.

SPECIFIC OUTCOME


Achievement Indicators• Provide a context that requires the multiplying of two

given positive fractions.• Model multiplication of a positive fraction by a



Multiplying Fractions STUDENT BOOK PAGES 52–56


15–20 min Teaching and Learning20–25 min Consolidation

Materials • scissors, 1 per student• coloured pencils, 1 per student

Masters • Fraction Strips Tower, Masters Booklet p. 45• 2 cm Grid Paper, Masters Booklet p. 34

Recommended Questions 5, 7, 8, 10, 11, 18, & 19Practising Questions


Extra Practice Mid-Chapter Review Questions 7, 8, & 9Chapter Review Questions 6 & 7Workbook p. 12

Mathematical C (Communication) and CN (Connections)Process Focus


Math BackgroundThe multiplication expression 4 � 3 can be modelledby determining the area of a rectangle with four rowsand three columns. The dimensions of the rectangle arethe factors of the multiplication. The idea behind thislesson is that the expressions involving fractions, suchas , can also be modelled using areas of rectangleswith five rows and ten columns. The dimensions of therectangle represents the denominators of the factors.

Using concrete and pictorial models helps studentsvisualize multiplying a fraction by a fraction andreinforces the fact that, unlike multiplying wholenumbers, multiplying two fractions less than 1 willgive a product that is less than either factor. Bycommunicating their strategies for calculating afraction of a fraction, students will be able to identifymisconceptions held by themselves and other students.They will make connections between the concrete andpictorial models and the symbolic representation offraction multiplication.

25 *

110

112

14

15

18

19

110

111

112

13

16

16

16

16

16

17

17

17

17

17

17

18

18

18

18

18

18

18

19

19

19

19

19

19

19

19

110

110

110

110

110

110

110

110

110

111

111

111

111

111

111

111

111

111

111

112

112

112

112

112

112

112

112

112

112

112

12

13

13

14

14

14

15

15

15

15

16

17


Fraction Strips Tower,Masters Booklet p. 45



GOALMultiply two fractions less than 1.

2.3

25Lesson 2.3: Multiplying FractionsCopyright © 2008 Nelson Education Ltd.


Draw a circle on the board, a transparency, or an interactivewhiteboard. Shade one half of the circle. Tell students thatthree friends are equally sharing half of a pizza, and theshaded part of the circle represents the pizza. Ask how muchof the whole pizza each share is.

Sample Discourse“What fraction of the half of the pizza will each friend get?”• Each friend will get one third of one half of the pizza.“How can you determine what fraction of the whole pizzaeach friend will get?”• I can cut the shaded portion of the circle into three equal pieces.

Each friend gets one of those pieces. I can cut the unshadedportion of the circle into three equal pieces and count the totalnumber of pieces. There are six equal pieces in the whole pizza.Each friend will get one sixth of the whole pizza.


LEARN ABOUT the Math(Pairs/Whole Class)

Together, read the problem presented on Student Bookpage 52. Distribute enough fraction strips and grid paperto students to use throughout the lesson. Have studentswork through Examples 1 and 2, replicating the modelsused and explaining the steps to one another. Discuss whythe models work and encourage students to suggest otherpossible solution strategies.


These questions encourage students to reflect on how usinga model can help them understand how the product of afraction multiplication is related to its factors. You couldhave students discuss the questions in pairs and then hold aclass discussion.

Sample Discourse“In what way is multiplying two fractions like multiplying afraction by a whole number?”• You can use fraction strips to model both types of multiplication.• To multiply a fraction by a whole number, you can determine

the new numerator by multiplying the whole number by thenumerator of the fraction. To multiply two fractions, you candetermine the new numerator by multiplying the twonumerators together.

“In what way is multiplying two fractions different frommultiplying a fraction by a whole number?”• You can use repeated addition to multiply a fraction by a

whole number, but not to multiply a fraction by a fraction.• The product of fraction multiplication will be less than either

factor, but the product of multiplying a whole number by afraction will be greater than the fraction.

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

• To multiply two fractions, you can calculate the newdenominator by multiplying the two denominators together.

“What size grid would you use to multiply ?”• I would use a grid with five rows and nine columns.

Answers to Reflecting A. Nikita used a 5 � 10 grid to model the multiplication

of . The model shows that of is , because 2squares of the grid are shaded, which represents , or .If you turn the model to make a 10 5 rectangle, youcan also see that she shaded in of .

B. For example, with the grid model, the total number ofsquares in the grid is the product of the denominators,since one denominator is represented by the rows andthe other is represented by the columns. The numeratoris represented by the number of squares that are shaded.For example, to model , the grid will have 5 � 7 �35 squares. Therefore, the new denominator is 35. Thereare two columns out of seven that are shaded, and threerows of those two columns are shaded. Therefore, 3 � 2squares are shaded. The new numerator is 3 � 2 � 6.

C. Multiply the two numerators and multiply the twodenominators. The product of the numerators is thenew numerator, and the product of the denominators isthe new denominator. I think this works because I cansee that if I make a grid, I multiply the denominators toget the total number of squares and the numerators to getthe total number of shaded squares.

35 *

27

25

110

*

125

250

125

110

25

25 *

110

35 *

29


Consolidation➧ 20–25 min


In pairs, have students work through the two solutions forExample 3, using fraction strips and grid paper. Have thestudents explain the steps to each other.

Checking (Individuals/Pairs)

Have students answer questions individually and then checktheir solutions in pairs. Discuss with students the basicstrategy behind each type of model; sections on fraction stripsare divided into smaller sections, while rows and columns ina grid represent the denominators of the two fractions.Students can refer to Examples 1 and 3 a) for assistance withQuestion 1 a); Examples 2 and 3 b) for assistance withQuestion 1 b); and Examples 1, 2, and 3 for assistance withQuestions 2 and 3.1. a) Remind students that the second strip is separating

each fourth from the first strip into three equalsections.

1

2

3

4

5

6

7

8


These questions provide students with practice in usingconcrete and pictorial models in multiplying two fractions.Ensure that students have enough fraction strips and gridpaper before beginning the Practice questions.6. Students need to rename the products to get the answers

in the box.15. Students need to be able to subtract fractions with

different denominators to answer this question. Remindthem that they need to find a common denominator.

16. Students must connect with previous learning in orderto do decimal multiplication.

17. Students who are struggling may find this question toodifficult.

18. Students must communicate their understanding offraction multiplication.

27Lesson 2.3: Multiplying FractionsCopyright © 2008 Nelson Education Ltd.

Answers to Key Question7. a) For example, the rectangle has total dimensions of

5 by 8, and the area is 40 units. The part of therectangle that is shaded is 2 units by 3 units, for anarea of 6 units out of the 40 total. So, .

b) For example, and


Question 19 allows students to reflect on and consolidatetheir learning for this lesson. The product of two fractionsless than 1 is less than either of the factors.

35 *

28

15 *

68

25 *

38 =

640

Answers to Closing Question 19. a) For example, the denominator of the product is a

multiple of 5.b) For example, the numerator of the product is less

than the denominator of the product.

Follow-Up and Preparation for Next ClassHave students look around at home for an example of asituation that involves multiplication of fractions, and thenuse one of the models discussed today to calculate the product.





• Students use fraction strips and area models to model the multiplication oftwo fractions less than 1.

• Students identify the fractions modelled by fraction strips or an area model.

Key Question 7 (Communication, Connections)• Students draw fraction strips or an area model to represent the equation

and explain their reasoning.

• Students choose two fractions whose numerators have a product of 6 andwhose denominators have a product of 40.

• Students may have difficulty choosing a second fraction strip that divides thesections of the first fraction strip into smaller sections. (See Extra Support 1and 2.)

• Students may only be able to identify one of the fractions of a fraction modelthat uses strips. (See Extra Support 2.)

• Students may only be able to associate an area model with whole numbersand not fractions. (See Extra Support 3.)

• Students may be able to draw and on separate fraction strips but may notbe able to connect the two to model the product. (See Extra Support 2.)

• Students may not be able to divide 6 and/or 40 into other factors to generatean equivalent product. (See Extra Support 4.)

38

25

EXTRA SUPPORT

1. Provide additional examples for students. Provide one or two examples usingunit fractions, such as , where you start with a and separate eachfourth into three equal sections, and , where you start with andseparate each third into four equal sections. Then provide one or twoexamples using one unit fraction and one other fraction, such as (startwith and separate each half into five equal sections) and (start with

and separate each fifth into two equal sections).

2. Have student use scissors to separate one of the factors into parts. Forexample, to model , tell students to first model the factor by cuttingone fraction strip into three equal pieces, and separating one piece from theother two.

Next, have students imagine each third as a new fraction strip and modeleighths using all three fraction strips. (colour, not cutting)

23

58 *

23

35

12 *

35

12

35 *

12

13

14 *

13

14

13 *

14


EXTRA CHALLENGE

• Have students devise a way to multiply three fractions less than 1 usingfraction strips and illustrate their method with at least two examples.


• If students are not yet ready to multiply fractions by fractions, you may wantto focus on multiple representations of a fraction. For example, have thesestudents express some of the fractions in the questions assigned as theproduct of various fractions and whole numbers, for example, as 3 * , 3 * or 9 * , and so on.1

152

10

15

35


• The teaching and learning in this lesson are best suited for a visual learner.Some students may benefit from an approach that also supports kinestheticlearning. By recreating the models shown in the lesson for Examples 1, 2, and3, kinesthetic learners will gain a better understanding of the models and bebetter able to apply the strategy to practice questions. Kinesthetic learners

Finally, ask students why they should only colour 5 of the 8 sections in only two of the pieces and not all three. (because I only use of the pieces)

Have students count all the pieces from all three strips (24) and then countthe number of coloured pieces (10) to form their fraction , or .

3. Students may find it helpful to see a very simple example of multiplyingfractions. Have students begin with problems that multiply unit fractions thatcan be easily modelled, such as , which can be modelled by folding apiece of paper in half vertically and then horizontally. Then have them tryexamples with other fractions multiplied by .

4. Have students concentrate on the 5 � 8 grid. How else could they colour in6 squares on the grid? Guide them to make the squares contiguous andinclude the top left square. For example, 6 squares in the top row, startingwith the top left square. What multiplication do the squares represent? Arow is , each column is , so � � . How else could you place thesquares on the grid?

640

68

15

18

15

12

12 *

12

512

1024

23

can also use household items such as egg cartons to model multiplication offractions involving halves, thirds, fourths, sixths, and twelfths. See if studentscan find other objects that have equal divisions that can easily be used tomodel fractions.

29Lesson 2.4: Exploring Estimating Fraction ProductsCopyright © 2008 Nelson Education Ltd.

PREREQUISITE SKILLS/CONCEPT

• Determine equivalent fractions.• Compare fractions.

SPECIFIC OUTCOME


Achievement Indicator• Estimate the product of two given positive proper

fractions to determine if the product will be closer to0, , or 1.1

2



Materials • Optional: scissors, 1 per group

Masters • Fractions Spinner, Masters Booklet p. 46• Fraction Strips Tower, Masters Booklet p. 45• 2 cm Grid Paper, Masters Booklet p. 34• Number Lines, Masters Booklet p. 39

Key Question entire exploration

Extra Practice Mid-Chapter Review Question 10Chapter Review Question 8Workbook p. 13

Mathematical ME (Mental Mathematics and Estimation)Process Focus


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

This lesson consolidates students’ understanding ofthe strategies they have learned for multiplying fractions.Connections are made with content from prior gradesas students use estimation to compare fractions.

Students develop mathematical communication skillsas they present their solutions to their classmates andrespond to classmates’ questions. Listening skills aredeveloped as students listen to classmates’ strategies andare asked to comment on them. When reviewing oneanother’s solutions, students will realize that there aremultiple ways to solve this problem, and they will learnto value a variety of approaches.

910

34

15

23

46 Fractions Spinner Copyright © 2008 Nelson Education Ltd.

Fractions Spinner,Masters Booklet p. 46

112

14

15

18

19

110

111

112

13

16

16

16

16

16

17

17

17

17

17

17

18

18

18

18

18

18

18

19

19

19

19

19

19

19

19

110

110

110

110

110

110

110

110

110

111

111

111

111

111

111

111

111

111

111

112

112

112

112

112

112

112

112

112

112

112

12

13

13

14

14

14

15

15

15

15

16

17




2 cm Grid Paper,Masters Booklet p. 34

GOAL

Estimate to predict whether a fraction product is closer to 0, , or 1.1

2

Exploring EstimatingFraction Products2.4

Number Lines, Masters Booklet p. 39

39Number LinesCopyright © 2008 Nelson Education Ltd.



Briefly discuss the use of grids, counters, and fractionstrips to multiply fractions. Demonstrate the modellingof multiplication such as using halves and sixthsfraction strips.

Tell students that in this lesson they will be estimatingfraction products and comparing the products with 0, ,and 1. On the board, overhead transparency, or interactivewhiteboard, write a number line similar to the one belowfrom 0 to 1, marking the point.

Place your finger on 0 and start moving it slowly to the right.

Sample Discourse“At what point will I be closer to than to 0?”• When you pass the halfway point between 0 and “What is of ?” •“So, if I have a product that is less than of a half, I wouldbe closer to 0 than to . (Mark on the number line andcontinue moving to the right.) At what point would I becloser to 1 than to ?”• When you pass the halfway point between and 1, which is “So, if I have a product that is greater than of a whole, Iwould be closer to 1 than to . (Mark on the number line.)Between which two points would I be closer to than toboth 0 or 1?”• If the point is between and


EXPLORE the Math(Whole Class/Pairs/Small Groups)

Present the problem and central question on Student Bookpage 57 to students. Provide each pair or group with a varietyof materials to work with, including Fraction Strips Tower,Masters Booklet p. 45, 2 cm Grid Paper, Masters Bookletp. 34, Fractions Spinner, Masters Booklet p. 46, counters,and also chart paper and markers. Explain that they are to• make any decisions they feel are necessary to solve the

problem;• write out the main points in their solution on chart paper;

and• be prepared to communicate their solution process to the

rest of the class.

1

2

3

4

5

6

7

8

34

14

12

34

12

34

34

12

12

14

12

12

14

12

12

12

12

12 10

12

12

13 *

12

1

2

3

4

5

6

7

8

No one approach to the problem should be suggested;encourage students to choose their own strategies. Theymust take the information given and work toward whatthe problem is asking for. Encourage them to use a varietyof methods to come up with the solution.

For this activity, students need to realize that closer to than0 means between and 1. They must use reasoning to realizethat a product will score 2 points if it is between and .

Sample DiscourseCirculate and observe students as they work. Ask questionssuch as:“How many points would you get if your product were ?”• Since is greater than , only 1 point is scored.“What if the product were ? What if it were ?”• Zero points are scored for the product , and 2 points are

scored for the product .“What would happen if you multiplied twice if it were onthe spinner?” • The product is . Because is the same distance from as it

is from 0, zero points are scored.“What would happen if you multiplied two proper fractionsclose to 1?”• The product is less than either factor, but still close to 1, and

so 1 point is scored.“What would happen if you multiplied two fractions close to 0?”• The product is less than either factor, so the product is closer to

0 than either factor, and zero points are scored.

12

14

14

12

38

18

38

18

34

78

78

34

14

14

12

31Lesson 2.4: Exploring Estimating Fraction ProductsCopyright © 2008 Nelson Education Ltd.

Possible Solutions to Explore the Math

Sample Solution 1:There are 4 numbers you can spin.We figured out what happened when you combined eachof those numbers with each other possibility.Suppose you spin first.The next spin could be either , , , or .

* =

We drew a hundredths grid to see that * = .

is closer to than 0, so we got 1 point.But it’s not closer to than 1, so we don’t get a second point.

We knew that * was since we multiplied thenumerators and multiplied the denominators.

is closer to 0 than since is and 9 is closer to0 than 25. So we don’t even get 1 point.

* is almost , but not quite .So we figured it’s closer to than 0 and also closer to thanto 1, so you get 2 points for this one.

* is just a bit more, but not much more, than * ,so you also get 2 points with those spins.

Next we figured out what would happen with a first spin ofWe didn’t check with a second spin of , since we already

did that one.For * , the product is only . That’s closer to 0 than , soyou don’t get a point.

For * , we drew fraction strips and got . That’s closer to0 than , so there are 0 points.

* wouldn’t be much different than * , so we decidedthere would be 0 points.

The only other combinations we had to try were * , * ,and * .

For * , we used a rectangle and got . That’s close to , sowe got 1 point, and it’s a little less than , so it’s closer to than 1, so we got 2 points.

For * , we knew that we were using 2 of the 3 fourths, soit would be . Since = , we got 2 points.

For * , we knew the product was . is just a little morethan , which is , so we also got 2 points.1

2816

916

916

34

34

12

24

24

34

23

12

12

12

49

23

23

34

34

34

23

23

23

23

15

34

15

12

215

23

15

12

125

15

15

910

15.

23

910

34

910

12

12

23

23

23

910

2550

12

12

950

950

15

910

12

12

81100

81100

910

910

81100

910

910

34

23

15

910

910

So, you get 2 points with * , * , * , * , and * .

Sample Solution 2:We figured out that to get the first point, the fraction had tobe between and 1.To get the second point, the fraction had to be less than but more than .So we figured out all the products and tested to see if theywere between and .

The combinations we tried were:*

*

*

*

*

*

*

*

* and*

We knew that we could just multiply numerators andmultiply denominators. So the products were

* =

* =

* = =

* =

* =

* =

* =

* =

* = = and* =

We used equivalent fractions to compare.= so * doesn’t work; it’s too big.= ; that’s less than ,which is , so that doesn’t work.is a bit more than half, so we definitely get 1 point. =

and since is less, then we get 2 points with * .is less than , so it’s less than . It’s also more than ,

which is , so we get 2 points with * .is way too small—0 points.is also way too small—0 points.is less than , which is . That’s too small—0 points.

= which is more than and less than , so it’s2 points for * .

is definitely 2 points since it’s closer to than anything.is more than , which is and less than , which is , so

it’s 2 points.So, you get 2 points with * , * , * , * , and

* .34

34

34

23

23

23

34

910

23

910

34

1216

416

14

916

12

12

23

23

A1836 B1

2A 936 B1

41636

49

15

420

320

215

125

34

910

2040

12

34

3040

2740

23

910

915

912

34

915

14

25100

18100

950

910

910

75100,3

4

916

34

34

12

612

34

23

49

23

23

320

34

15

215

23

15

125

15

15

2740

34

910

915

1830

23

910

950

15

910

81100

910

910

34

34

34

23

23

23

34

15

23

15

15

15

34

910

23

910

15

910

910

910

34

14

14

34

14

34

34

34

23

23

23

34

910

23

910





• Students use pictorial models to multiply fractions.

• Students use pictorial, concrete, or symbolic strategies to compare fractions.

• Students determine equivalent fractions to compare fractions.

• Students may have difficulty determining the number of rows and columns ina grid model. (See Extra Support 1.)

• Students may have difficulty determining the number of counters to use in agrid model. (See Extra Support 1.)

• Students compare models of factors to 0, , and 1, rather than comparingmodels of the products. (See Extra Support 2.)

• Students are unable to find equivalent fractions. (See Extra Support 3 and 4.)

• Students are unable to determine common denominators. (See Extra Support 4.)

12

EXTRA SUPPORT

1. Provide students with additional examples of fraction multiplications tomodel. The grid to find the product has 10 rows and 4 columns (the denominators). In 9 rows, 3 squares are shaded (the numerators).

2. Remind students that the goal of the game is to get a product that is close to . Discuss the fraction strips and grid models, focusing on what parts ofthe model represent the product. (For fraction strips, the product is theshaded part of the second strip; for grid models, the total number of shadedsquares on the grid is the numerator of the product; the total number ofsquares on the grid is the denominator of the product.)

12

910

*

34

=

2740

910 *

34

3. Remind students that they can find equivalent fractions by multiplying ordividing the numerator or denominator by the same number. Provide somesimple examples, such as and , and have students find two equivalentfractions for each.

4. Have students find multiples of each denominator and then compare thelists. Whichever multiple appears on both lists is a common denominator.Multiply the numerator of each fraction by the same factor by which theymultiplied the denominator to get the common multiple.

812

23


EXTRA CHALLENGE

• Have students design a four-section spinner made up of four differentfractions whose products score 2 points all the time and another spinnerwhose products score zero points all the time. If students cannot design anyof the spinners described above, have them explain why.


• For students still struggling with the concept of multiplying fractions, youmight change the problem to a situation where they are adding fractions andget 1 point for a sum between and 1 and 2 points for a sum between 1 and1 , perhaps using concrete materials.1

2

12


• Spatial learners will benefit most from using visual models to comparefractions.

• Have students design a four-section spinner made up of four differentfractions, two greater than one half and two less than one half, whoseproducts score 2 points all the time and another spinner whose productsscore 0 points all the time. If students cannot design any of the spinnersdescribed above, have them explain why.



Provide the opportunity for students to share and communicateabout their work. Have students describe to the rest of the classhow they solved the problem, using their chart paper as anorganizing tool for other students to follow. If you saw someparticularly interesting approaches when you circulated among

1

2

3

4

5

6

7

8

students as they were working through the problem, be sure toinvite those students to be among the presenters.

Ask students who are listening to comment on what they liked about the approach presented and what confusedthem. The presenters can invite questions from otherstudents and attempt to answer the questions. Encouragestudents to identify similarities and differences among theirmethods.

33Lesson 2.5: Multiplying Fractions Greater than 1Copyright © 2008 Nelson Education Ltd.


• Relate multiplication to calculating the area of a rectangle.• Multiply proper fractions.• Understand the distributive principle.• Rename improper fractions as mixed numbers and vice versa.• Write fractions in lowest terms.

SPECIFIC OUTCOME


Achievement Indicators• Provide a context that requires the multiplying of two

given positive fractions.• Express a given positive mixed number as an

improper fraction and a given positive improperfraction as a mixed number.

• Model multiplication of a positive fraction by apositive fraction concretely or pictorially using anarea model and record the process.



GOALMultiply mixed numbers and improper fractions.

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

Materials • scissors, 1 per student

Masters • 1 cm Grid Paper, Masters Booklet p. 32

Recommended Questions 5, 7, 8, 10, 12, 14, & 16Practising Questions

Key Questions Questions 5 & 10

Extra Practice Mid-Chapter Review Questions 11 & 12Chapter Review Questions 9, 10, & 11 Workbook p. 14

Mathematical CN (Connections), C (Communication), and Process Focus PS (Problem Solving)


Math BackgroundIn this lesson, students apply their knowledge of thedistributive property and of renaming mixed numbersas improper fractions to multiply mixed numbers andimproper fractions. Recall that the distributive propertylets you multiply a sum by multiplying each addendseparately and then adding the products; for example, a(b � c) � ab � ac. The distributive property is usedin Example 1 to calculate the area.

Other examples require the renaming of mixed numbersas improper fractions. To rename a mixed number,students must understand the nature of mixed numbers,which are made of wholes and a fraction. The number of parts in a whole is given by the denominator, so thenumber of parts represented by the mixed number is thenumber of wholes multiplied by the number of parts in a whole (the denominator), plus the parts represented bythe numerator of the fraction.

This lesson is a continuation of the learning in Lesson 2.3,in which students multiplied two fractions less than 1. In thislesson, students expand their knowledge of multiplyingfractions by multiplying fractions that are greater than 1.Students will use concrete and pictorial models to visualizethe different sizes of each fraction or mixed number, andconnections will be made to the process that was developedfor multiplying fractions less than 1. Students use reasoningto determine the most appropriate size of grid to use tomultiply two fractions. Students also apply their problem-solving skills in contexts related to multiplying fractions.

Reading StrategiesVisualizing is a reading strategy that helps students createmental pictures from text, drawing on personal experiencesand prior knowledge. It provides a concrete representationfor abstract ideas or information, while allowing readers tobecome involved with the text. In mathematics, mentalpictures help students see mathematical ideas more clearlyand remember them more easily.

Read the question to the class and ask students howthey could create mental pictures to represent the ideas in the problem. Have them work with a partner andrepresent the information from the problem in a diagramor sketch. Use the sketch to solve the problem. Havestudents share the results with another group and examinethe two ways of visually representing the information.Prompt with questions such as, “How are therepresentations similar? How are they different?”

Students are prompted to use the Visualizing strategyin Question 12.

Multiplying FractionsGreater than 12.5



On the board, a transparency, or an interactive whiteboard,write the multiplication expressions and . Ask students to predict which expression will have a productthat is less than the factors and which will have a productthat is greater than the factors. Ask them to recall Lesson 2.3and explain why the first expression has a product that isless than the factors.

Sample Discourse“Will the product of be less than or greater than ? ?”• Less than“Why will the product be less than either factor?”• Since part of is less than and part of is less than “Will the product of be less than or greater thaneither of the factors?”• Greater than“Why will the product be greater than either factor? Estimate.”• and is about 2 * 2


LEARN ABOUT the Math(Whole Class/Pairs)

Together with students, read the problem on Student Bookpage 58. Have students discuss the methods used inExamples 1 and 2, explaining the steps to one another.Discuss which methods students prefer and why. Encouragestudents to suggest other solution methods and discuss whythey might or might not work.

Sample Discourse“Why did Aaron set up the rectangle that way?”• He drew the rectangle with side lengths of and because

the area of the rectangle is equal to the product.• It is difficult to multiply mixed numbers, so Aaron set up the

rectangle so that he could use what he already knew how todo—multiply whole numbers and fractions.

“Why did Misa rename the fractions?”• Misa renamed the mixed numbers as improper fractions so

that she could multiply numerators and denominators.

2 121

12

1

2

3

4

5

6

7

8

1 342

12

212 * 13

4

25

25

34

34

25

34

34 *

25

212 * 13

434 *

25

1

2

3

4

5

6

7

8


The reflecting questions encourage students to reflect on howthe methods learned in Lesson 2.3 can be applied to multiplyingfractions greater than 1. You could have students discuss thequestions in pairs and then hold a class discussion.

Sample Discourse“In what way is multiplying fractions greater than 1 likemultiplying fractions less than 1?”• You can use an area model to model both types of

multiplication.• You can multiply numerator by numerator and denominator

by denominator for both types of multiplication.“In what way is multiplying fractions greater than 1 differentfrom multiplying fractions less than 1?”• The product of two fractions less than 1 is less than either

factor. The product of two fractions greater than 1 is greaterthan either factor.

• When you use an area model to multiply fractions greaterthan 1, you need to determine partial products and add them.When you use an area model to multiply fractions less than 1,there is only one area to consider.

“What are the benefits of each method shown in this lesson?”• When you use an area model, you do not have to convert a

mixed number to an improper fraction.• When you use the procedure for multiplying fractions, if

you have improper fractions, you only have to perform onemultiplication and you do not need to do any addition.


the distributive property is used to calculate the area. ForExample 3 Solution B, the diagram below shows in moredetail the 5 halves of the whole and the 10 thirds of thewhole model.

Checking (Pairs)

Have students work in pairs to solve the problems. 1. Discuss the basic strategy behind estimating products;

choose a whole number that is close to the fraction andmultiply to estimate the product.

2. Discuss how each type of model can be used to multiplyfractions greater than 1; multiply the components ofmixed numbers to determine the four partial areas, andthen add or extend a grid to represent the two improperfractions. Fraction strips can be used to multiply afraction less than 1 by a fraction greater than 1. Studentscan refer to Examples 1, 2, and 3 for assistance.

13

13

13

13

13

13

13

13

13

13

1 21 2

1 21 2

1 2 1 whole

13

12

16

of =

Answers to Reflecting A. For example, I would use Aaron’s method, since the

picture makes sense to me.B. For Aaron’s model, I would make a rectangle like this:

The total area is .Using Misa’s method, I would write and .I would multiply



Have pairs of students work through the two models usedto solve Example 3, explaining the steps to each other. ForExample 3 Solution A, encourage students to identify how

1

2

3

4

5

6

7

8

= 816

=496

72 *

73 =

7 * 72 * 3

213 =

7331

2 =72

816

2

1

6

3

1

1

3

12

16



These questions provide students with practice in usingmodels and general rules to multiply two fractions.Distribute grid paper to students before beginning thepractice problems.4. & 6. Students must connect with previous learning in

order to multiply a fraction less than 1 by a fractiongreater than 1.

Answers to Key Questions5. For example,

234 * 41

3 = 8 +812 +

312 +

3612 = 8 +

1112 + 3 = 1111

12

8

8

36

3

10. a) Estimating would not help him realize his mistakebecause most likely his estimate would be 3 � 4 � 12.Therefore, his answer would not be far from hisestimate.

b) For example, you could use a model of .

Tai has calculated only the areas of the shadedrectangles. He needs to calculate the areas of allof the rectangles for the correct answer.

3 x 4

x

313 * 43

8





• Students use area models and grids to model the multiplication of twofractions greater than 1.

• Students apply the generalizations formed for multiplying fractions less than1 to multiply fractions greater than 1.

Key Question 5 (Connections)• Students draw an area model or grid to represent the equation.

Key Question 10 (Communication, Problem Solving)• Students communicate that estimation is limited in its ability to identify

calculation errors.

• Students calculate the correct product of and use reasoning toidentify the error in Tai’s method.

3 13 * 4 3

8

• Students may not know what numbers to multiply to determine partial areas.(See Extra Support 1.)

• Students may not convert mixed numbers to improper fractions beforemultiplying. (See Extra Support 2 and 3.)

• Students may multiply the wrong numbers together to determine the product.(See Extra Support 1 and 4.)

• Students may do the first point above for this question.

• Students may do the second point above for this question (See Extra Support 1.)

• Students may not recognize that the method described is not the correct wayto multiply two mixed numbers. (See Extra Support 1 and 4.)

(Continued on next page.)


Question 16 allows students to reflect on and consolidatetheir learning for this lesson. The product of a mixednumber using thirds and a mixed number using fourthsis an improper fraction using twelfths. When the productis renamed in lowest terms, it may be a whole number, nota mixed number.

Answer to Closing Question 16. Disagree. For example, the product could be a whole

number.

Follow-Up and Preparation for Next ClassNext class is the Mid-Chapter Review. Ask students to gothrough the first five lessons in this chapter and to note anyquestions or problems they have.

223 * 21

4 =83 *

94 =

7212 = 6

Chapter 2



EXTRA SUPPORT

1. Have students multiply a whole number by a mixed number, for example,, and use an area model to determine the product. The partial

areas will be 2 � 1 and . Have students apply the distributive property, .Once students are comfortable multiplying mixed numbers by wholenumbers, they can move on to multiplying mixed numbers by mixed numbers.

2. Provide additional examples of mixed numbers for students to convert toimproper fractions. Have students use subtraction to break the mixed numberinto unit fraction parts and then add the number of parts to determine theimproper fraction. For example, for the mixed number , students shouldrepeatedly subtract , then count the number of unit fractions. Sincethere are 7 unit fractions, .13

4 =74

14

1 34

2 * 112 = 2 * A1 +

12 B = A2 * 1 B + A2 *

12 B = 2 + 1 = 3

2 *12

2 * 112


EXTRA CHALLENGE

• Challenge students to determine two mixed numbers to multiply to obtain aparticular product (e.g., 6 or 8 ).5

914


• For students who are not ready to multiply mixed numbers, you mightsubstitute questions involving whole numbers, or products of whole numbersand decimals or fractions, depending on students’ readiness.


• Have kinesthetic learners recreate the models shown in the lesson forExamples 1 and 3. This will help these students better understand the modelsand better apply the strategies to practice questions. Kinesthetic learners canalso use household items such as egg cartons to model multiplication offractions involving halves, thirds, fourths, sixths, and twelfths. See if studentscan find other objects that have equal divisions that can easily be used tomodel fractions.

3. Provide additional examples of multiplying fractions greater than 1. Beginwith improper fractions and remind students to multiply numerator bynumerator and denominator by denominator, just as with fractions lessthan 1. Continue with mixed numbers, ensuring that students convert mixednumbers to improper fractions before multiplying numerator by numeratorand denominator by denominator.

4. Students may find it helpful to relate multiplication of fractions tomultiplication of decimals. Have students begin with problems that caneasily be converted into decimal multiplication, such as . Havestudents perform the decimal multiplication using an area model andconvert the product to a fraction.

112 * 11

2


39Mid-Chapter ReviewCopyright © 2008 Nelson Education Ltd.


Preparation and PlanningMaterials • Optional: rulers

Masters • Mid-Chapter Review—Frequently AskedQuestions p. 97

• Mid-Chapter Review—Study Guide p. 98• Fraction Strips Tower, Masters Booklet p. 45• 2 cm Grid Paper, Masters Booklet p. 34• Number Lines, Masters Booklet p. 39• Self-Assessment: Chapter 2 Lesson Goals:

Fraction Operations, Masters Booklet p. 65• Self-Assessment: Mathematical Processes,

Masters Booklet p. 75• Self-Assessment: What I Like, Masters

Booklet p. 76• Self-Assessment: How I Learn, Masters

Booklet p. 76


SPECIFIC OUTCOME

N6. Demonstrate an understanding of multiplyingand dividing positive fractions and mixed numbers,concretely, pictorially, and symbolically. [C, CN, ME, PS]

Achievement Indicators• Estimate the product of two given positive proper

fractions to determine if the product will be closerto 0, , or 1.

• Express a given positive mixed number as animproper fraction and a given positive improperfraction as a mixed number.

• Model multiplication of a positive fraction by awhole number concretely or pictorially and recordthe process.



12

Mid-Chapter Review


Using the Mid-Chapter ReviewThis review provides an opportunity for students to monitortheir progress with the chapter skills and concepts, as wellas for you to monitor the progress of the class and see wherere-teaching may be required. You may also use it to assessindividual student achievement.

Frequently Asked Questions (Whole Class)

Have students keep their Student Books closed. Write theFrequently Asked Questions on Student Book page 64 onthe board, or use Mid-Chapter Review—Frequently AskedQuestions p. 97. (Distribute the master or display it using atransparency or an interactive whiteboard.) Use the discussionto draw out what the class thinks is the best answer to eachquestion. Then have students compare the class answers withthe answers in the Student Book. Have students summarizethe answers 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 questions relatedto Lessons 2.1 to 2.5 that students may have. Alternatively,you may want to use this section as more of a journallingactivity throughout Lessons 2.1 to 2.5, rather than havingstudents answer these specific questions.

Practice (Individual)

Students should be able to complete all the questions in class.Encourage students to identify which they found easy andwhich more challenging. Ask them what they can do tobecome proficient at these questions. The review questionsare organized by lesson. Students can go back to the lessonindicated to review the concepts for the question.


The Mid-Chapter Review provides opportunities forstudents to practise modelling fraction multiplication andcalculating the products of a whole number and a fraction,fractions less than 1, and fractions greater than 1.1. & 3. Provide students with fraction strips and/or number

lines.2. & 3. Provide grid paper.6. Provide grid paper.

Follow-Up and Preparation for Next Class To summarize students’ learning in Lessons 2.1 to 2.5, useMid-Chapter Review—Study Guide p. 98 to focus a classdiscussion. You may want to have students complete theguide individually, in pairs, or in small groups first.


In the Mid-Chapter Review When students understand If students misunderstand


Question 1 (Connections, Reasoning)

Question 2 (Reasoning)

Questions 3 & 8 (Connections, Problem Solving,Reasoning)

Questions 4 & 6 (Reasoning, Visualization)

Question 5 (Connections, Reasoning)

• Students will use repeated addition to representmultiplication. They will write each product asan improper fraction and a mixed number, andas an equivalent fraction in lowest terms.

• Students will use mental math to renamefractions and mixed numbers.

• Students will use the counter model to calculatethe product of a whole number and a fraction.

• Students will recognize how to apply theirfractions operations skills in a new context tosolve a problem.

• Students will model multiplication of fractionsusing fraction strips and/or grid models.

• Students will apply appropriate strategies todetermine the missing value.

• Students may have difficulty writing themultiplication as repeated addition. Theymay not be able to perform the multiplicationor division required to convert betweenmixed number and improper fraction. Studentsmay not be able to identify a common factor forthe numerator and denominator to write theproduct in lowest terms.

• Students will not connect how math facts canbe used when renaming fractions.

• Students may have difficulty determining thesize of grid to use to model the fraction.

• Students will be confused by the fractions ina new context and will not be able to makeconnections between factors and products.

• Students may have difficulty choosing the rightstrips or the right grid size to model themultiplication.

• Students may have difficulty choosing a strategyfor calculating a product or determining amissing factor.



Question 1, short answer Specific Outcome and Process Focus: N6 [CN, R]• Write as a repeated addition. Use fraction strips or a number line to add.

Write each answer as an improper fraction and as a mixed number. Write the fractions in lowest terms.

a) b) c) d)

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

5 *498 *

354 *

5126 *

15

Assessment of Learning—What to Look for in Student Work

Question 2, short answer Specific Outcome and Process Focus: N6 [R]• Use grid paper and counters to multiply.

a) b) c) d)


4 *255 *

562 *

593 *

38

Question 3, short answer Specific Outcome and Process Focus: N6 [CN, PS, R]• The product of a fraction and a whole number is . What could the fraction and whole number be?

(Score 1 point for process and 1 point for correct answer, for a total out of 2.)

245

Question 4, model Specific Outcome and Process Focus: N6 [R, V]• Draw a picture to show of .3

823

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

model when applying mathematicalknowledge and skills

• chooses workable andreasonable model when applyingmathematical knowledge andskills

• chooses partially appropriate andworkable model when applyingmathematical knowledge and skills

• chooses inappropriate and/orunworkable model when applyingmathematical knowledge and skills


In the Mid-Chapter Review When students understand If students misunderstand


Question 7 (Connections, Reasoning,Communication, Problem Solving)

Question 8 (Communication, Reasoning)

Questions 9 & 12 (Connections, Problem Solving)

Question 10 (Estimation, Reasoning)

Question 11 (Reasoning)

• Students will identify the fraction multiplicationrepresented by each model.

• Students will explain their reasoning clearlyand concisely.

• Students will recognize how to apply theirfractions operations skills in a new context tosolve a problem.

• Students will calculate products and use variousstrategies to compare the products to (e.g.,simplify fractions, rename fractions asequivalent fractions with a commondenominator).

• Students will write mixed numbers as improperfractions or use area models to calculate theproducts of fractions and mixed numbers.

12

• Students may not be able to connectdenominators with grid dimensions in the gridmodel. They may not be able to connect thedenominator of the second fraction with thenumber of sections in the second strip that arebelow the shaded section of the first strip.

• Students may arrive at correct answers but notbe able to explain their thinking.

• Students will be confused by the fractions ina new context and not recognize a simplemultiplication problem.

• Students may have difficulty determining astrategy for comparing fractions. They maynot be able to write fractions as equivalentfractions with a common denominator ornumerator. They may focus on a single strategyinstead of the most appropriate strategy.

• Students may have difficulty writing mixednumbers as improper fractions or have difficultycreating or interpreting area models.


Assessment of Learning—What to Look for in Student WorkQuestion 5, short answer Specific Outcome and Process Focus: N6 [CN, R]• What is the missing fraction?

a) of is . b) is of . c) of is .


312

34.

45.

35.

27

14

Question 6, models, short answer Specific Outcome and Process Focus: N6 [R, V]• Draw a model for each multiplication. Use your model to determine the product.

a) b)

(Score 1 point for each correct model or product, for a total out of 4.)

37 *

45

13 *

16

Question 7, short answer Specific Outcome and Process Focus: N6 [CN, R, C, PS]• What fraction multiplication does each model represent?

a) b) c)


Question 8, written answer Specific Outcome and Process Focus: N6 [CN, R]• If you multiply by another fraction, can the denominator be 20? Explain.2

8


Question 9, written answer Specific Outcome and Process Focus: N6 [C, PS]

• About of the traditional dancers of a First Nations school are girls. About of these students are in Grade 8.

What fraction of the students who dance are Grade 8 girls?

14

34

Question 10, short answer Specific Outcome and Process Focus: N6 [ME, R]• Which products are greater than ?

a) b) c) d) 35 *

23

39 *

89

16 *

78

34 *

56

12

• provides a precise and insightfulexplanation of mathematicalconcepts and/or procedures

• chooses efficient and effectivestrategies (e.g., renaming fractions)when applying mathematicalknowledge and skills (i.e.,multiplying fractions)

• provides a clear and logicalexplanation of mathematicalconcepts and/or procedures

• chooses workable and reasonablestrategies (e.g., showing severalexamples) when applyingmathematical knowledge andskills (i.e., multiplying fractions)

• provides a partially clear explanationof mathematical concepts and/orprocedures

• chooses partially appropriate and workable strategies (e.g., trialand error) when applyingmathematical knowledge and skills (i.e., multiplying fractions)

• provides a vague and/or inaccurateexplanation of mathematical conceptsand/or procedures

• chooses inappropriate and/orunworkable strategies (e.g., trying onlyone example) when applyingmathematical knowledge and skills(i.e., multiplying fractions)

Work meets standard Work meets standard Work meets acceptable Work does not yet meetof excellence of proficiency standard acceptable standard

• demonstrates a sophisticated abilityto transfer knowledge and skills tonew contexts

• demonstrates an insightfulunderstanding of the problem

• demonstrates a consistent abilityto transfer knowledge and skillsto new contexts

• demonstrates a completeunderstanding of the problem

• demonstrates some ability totransfer knowledge and skills to newcontexts

• demonstrates a basic understandingof the problem

• demonstrates a limited ability totransfer knowledge and skills to newcontexts

• demonstrates a limited understandingof the problem



Question 11, short answer Specific Outcome and Process Focus: N6 [R]• Calculate.

a) c) e)

b) d) f)

(Score 1 point for correct answer, for total out of 6.)

315 * 6 3

856 * 6 12

2525 * 1 3

5

2 34 * 3 3

41 12 * 1 2

337 * 3 1

2


Question 12, written answer Specific Outcome and Process Focus: N6 [C, PS]• Eileen used to be on the phone times as much as her sister every day. As a New Year’s resolution, she decided to cut down to about of the time she used to

be on the phone. About how many times as much as her sister is Eileen now on the phone?

253 1

2

• demonstrates a sophisticated abilityto transfer knowledge and skills tonew contexts

• demonstrates an insightfulunderstanding of the problem

• demonstrates a consistent abilityto transfer knowledge and skillsto new contexts

• demonstrates a completeunderstanding of the problem

• demonstrates some ability totransfer knowledge and skills tonew contexts

• demonstrates a basic understandingof the problem

• demonstrates a limited ability totransfer knowledge and skills to newcontexts

• demonstrates a limited understandingof the problem


45Lesson 2.6: Dividing Fractions by Whole NumbersCopyright © 2008 Nelson Education Ltd.

Dividing Fractions by Whole Numbers


GOAL

Use a sharing model to represent the quotient of a fractiondivided by a whole number.


• Name fractions as parts of sets and wholes.• Multiply and divide whole numbers.• Understand division as sharing.

SPECIFIC OUTCOME

N6. Demonstrate an understanding of multiplying anddividing positive fractions and mixed numbers, concretely,pictorially, and symbolically. [C, CN, ME, PS]

Achievement Indicators• Model division of a positive proper fraction by a

whole number concretely or pictorially and recordthe process.



Materials • counters, 50 per group• scissors, 1 per group

Masters • Fraction Strips Tower, Masters Booklet p. 45• 2 cm Grid Paper, Masters Booklet p. 34

Recommended Questions 4, 6, 9, & 11Practising Questions


Extra Practice Chapter Review Questions 12, 13, & 14Workbook p. 15

Mathematical CN (Connections) and PS (Problem Solving)Process Focus


Math BackgroundThis lesson introduces students to dividing fractions bywhole numbers. Students will solve division problemsconcretely, pictorially, and symbolically. Using fractionstrips and counters helps students visualize the size of thefraction to be divided and of the quotient, and reinforcesthe fact that only the numerator should be divided by thewhole number.

Connections are formed between division of fractionsand division of whole numbers. Students will use mentalmath to recall division facts. Students also apply theirproblem-solving skills in contexts related to subtractingfractions. They use reasoning to rename fractions and tochoose the appropriate fraction strips and grid sizes tomodel division.



Fraction Strips Tower,Masters Booklet p. 45

112

14

15

18

19

110

111

112

13

16

16

16

16

16

17

17

17

17

17

17

18

18

18

18

18

18

18

19

19

19

19

19

19

19

19

110

110

110

110

110

110

110

110

110

111

111

111

111

111

111

111

111

111

111

112

112

112

112

112

112

112

112

112

112

112

12

13

13

14

14

14

15

15

15

15

16

17


2.6


Answers to Learn about the MathA. & B.

C.

D.E. I would change the grid to show 20ths instead of 10ths.

I’d show as so I could make two groups of three.This cannot be done if the number of possible donorsis 10.

620

310

210

210


Draw a circle on the board, a transparency, or an interactivewhiteboard. Shade one half of the circle. Remind studentsthat they saw a similar model when multiplication offractions was introduced. Tell students that half of a pizzais being divided into three equal parts for each of threefriends, and the shaded part of the circle represents thepizza to be divided. Ask how much of the whole pizza eachequal part is.

Sample Discourse“What fraction of the pizza will each friend get?”• One sixth“How did you determine what portion of the pizza eachfriend will get?”• I divided each half of the pizza into three equal parts. There

were six equal parts all together. Each part is one sixth of thepizza.

• I took one third of one half of the pizza and got one sixth.


LEARN ABOUT the Math (Whole Class/Pairs)

Present the problem on Student Book page 68 to students.Provide each pair of students with counters and grids. Drawfrom students that the solution involves dividing a fractionby a whole number. Relate this to the work they did inLesson 2.1 when they multiplied a fraction by a wholenumber. Students can work through Prompts A to E inpairs. They demonstrate their reasoning when they explainhow to solve the problem in Prompt B.

Sample Discourse“How do you know how many squares to use in the gridof the fraction?”• The number of squares is the same as the denominator of the

fraction.“How do you know how many counters to use?”• The number of counters is the same as the numerator of the

fraction.“What do you do differently to the grid model of a fractionwhen you divide it by a whole number than when youmultiply it by a whole number?”• When I divide by 2, I separate the model into 2 equal groups.

When I multiply by 2, I add the model 2 times.

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8


Reflecting (Whole Class)

Here, students explain in words how they used division tosolve the central question. They also make generalizationsabout strategies for dividing a fraction by a whole numberand make a connection between fraction multiplication andfraction division.

Sample Discourse“How can you use equivalent fractions to help you divide afraction by a whole number?”• I can rename the fraction so it has a numerator that can be

evenly divided by the whole number.“If you divide a fraction by a whole number, will the answerbe greater than or less than the fraction?”• Less“What happens to a fraction when you multiply it by ?”• The value gets smaller.• It becomes half the original fraction.• The denominator doubles, but the numerator stays the same.“What happens to a fraction when you divide it by 2?”• I get two equal groups; each is one half of the fraction.

Answers to Reflecting F. I divided the fraction by the whole number 2, .

G. I changed my strategy because 2 does not divide into 3evenly.

H. When I divide a fraction by 2, I separate the fractioninto two equal groups; each is one half of the fraction.When I multiply a fraction by one half, I also take onehalf of the fraction.



Distribute fraction strips, scissors, and grids to students.Have students work through the examples in pairs. For eachexample, students can explain one of the solutions to theirpartner. Have students duplicate the process in Example 2using fraction strips.

Checking (Pairs)

Ensure that students have enough counters and fractionstrips to complete the questions. Ask for volunteers to showsolutions on the board for Question 1. Have students sharetheir explanations for Question 3. Encourage discussionabout the different methods used.

1

2

3

4

5

6

7

8

410 , 24

10

12


These questions provide students with opportunities topractise dividing fractions by whole numbers, as well as toapply their understanding in problem-solving contexts.Students will connect division with multiplication. Providestudents with grids, counters, scissors, and fraction strips tocomplete the exercises.5. Students can communicate their reasoning when they

explain why each quotient is greater than or less than .8. Connections are made between fractions and percents.

Answers to Key Question9. For example,

a) I have of the lawn left to rake. Three of my friendsagree to share the job with me. How much do weeach have to rake?

b)

We each have to rake of the lawn.16

23 , 4 =

23 *

14 =

212 =

16

23

14



Question 11 allows students to reflect on and consolidatetheir learning for this lesson, as they will have to form ageneralization about the relationship between the whole-number divisor and the quotient. Ask for volunteers toexplain their answers.

Answer to Closing Question11. The numerator is the same as the original numerator

and the denominator is the product of the denominator of the original fraction and the whole number. Yes,

the pattern is always true; for example, if you divide aproper fraction by 5, you divide each piece into 5 partsand keep one of them. Since the original number of parts was the denominator of the fraction, you wouldhave 5 times as many parts in the denominator of theproduct.

Follow-Up and Preparation for Next ClassStudents can use some of the division sentences fromQuestion 4 and create appropriate problems and a contextfor the given fractions.





• Students model division of a fraction by a whole number using counters on a grid.

• Students calculate the quotient of a fraction divided by a whole number.

• Students solve problems involving division of a fraction by a whole number.

Key Question 9 (Connections, Problem Solving)• Students create a problem that can be solved by dividing by 4 and then

solve the problem.

23

• Students may have difficulty deciding what size grid to use or how manycounters to use. (See Extra Support 1.)

• Students may have difficulty sharing the counters into the correct numberof groups. (See Extra Support 2.)

• Students may divide both numerator and denominator by the whole number.(See Extra Support 3.)

• Students may have difficulty identifying the dividend and divisor to bedivided to solve the problem. (See Extra Support 4.)

• Students may have difficulty thinking of an appropriate context for thegiven dividend and divisor. (See Extra Support 5.)

• Students may have difficulty finding the quotient of the given dividend anddivisor. (See Extra Support 3.)

EXTRA SUPPORT

1. Draw a 5 � 2 grid on the board, a transparency, or an interactivewhiteboard. Ask students to count the number of squares in the grid andtell them that this is the denominator of the fraction. Then place a number ofcounters on the grid. Ask students to count the counters and tell them thatthe counters represent the numerator of the fraction. Write the fraction thatis represented. Repeat several times, changing the size of the grid and thenumber of counters. To ascertain transfer of training, give students fractionsand have them make the grids.

2. Draw a 4 � 5 grid on the board, a transparency, or an interactivewhiteboard. Place 12 counters on the grid. Ask a volunteer to name thefraction that is represented by the model. Ask a volunteer to sharethe counters into 3 equal groups. Ask another volunteer how many countersare in each group. Write the division problem on the board, a transparency,or an interactive whiteboard. Repeat several times, changingthe numerator and the whole-number divisor.

A 1220 , 3 =

420 B

A 1220 B


EXTRA CHALLENGE

• Have students choose three division exercises from today’s lesson and rewritethe exercises as decimal division problems. Then have students calculate thequotient as a decimal and compare the answer with the fractional answer.


• Some students will not be ready to divide fractions. You may want to usedecimal equivalents or you may set parallel problems that focus only onwhole number division, depending on the needs of the student.

• You may want to eliminate consideration of fractions whose numerators arenot evenly divisible by the whole-number divisor for students who have someideas about dividing fractions but are still struggling.

3. Ask students to say what , 3 means. Have them ask if the answer shouldbe more or less than . Have them observe what happens if they divide boththe 6 and the 9 by 3 to get to see that the answer is equivalent to ,which is what they started with.

4. Have students circle all of the numbers, both numerals and in word form,that occur in the problem. Then have them identify how the numbers areused in solving the problem.

5. Have students brainstorm examples of situations in which they encounterfractions in their lives. Have them choose two situations and write a wordproblem for each. Discuss the word problems and ask students to thinkof another situation for which the numbers in each problem would beappropriate.

69

23 BA

69

69


PREREQUISITE SKILL/CONCEPT

• Model fractions with fraction strips.

SPECIFIC OUTCOME


Achievement Indicators• Estimate the quotient of two given positive fractions

and compare the estimate to whole-numberbenchmarks.

• Model division of a positive proper fraction by apositive proper fraction pictorially and record theprocess.

Estimating FractionQuotients







Extra Practice Chapter Review Questions 15 & 16Workbook p. 16

Mathematical ME (Estimation)Process Focus

Nelson Website Visit www.nelson.com/mathfocus and follow the links to Nelson Math Focus 8, Chapter 2.

Math BackgroundThis lesson is an extension of the learning in Lesson 2.6and presents students with a variety of strategies forestimating quotients. Estimation is a valuable skillin daily life. It allows one to quickly determine anapproximate answer rather than searching for paperand pencil or a calculator to determine the exactanswer. It also provides a mechanism for students tocheck the reasonableness of their calculated answers.

Students will use reasoning to choose appropriatestrategies for estimating quotients, such as using fractionstrips to visualize the quotient. Connections are madewith previous learning as students use equivalentfractions to make comparisons. Students will alsochoose compatible numbers—numbers that areclose to the numbers in a problem but are easier to work with.

112

14

15

18

19

110

111

112

13

16

16

16

16

16

17

17

17

17

17

17

18

18

18

18

18

18

18

19

19

19

19

19

19

19

19

110

110

110

110

110

110

110

110

110

111

111

111

111

111

111

111

111

111

111

112

112

112

112

112

112

112

112

112

112

112

12

13

13

14

14

14

15

15

15

15

16

17



GOALInterpret and estimate the quotient of fractions less than 1.

2.7

51Lesson 2.7: Estimating Fraction QuotientsCopyright © 2008 Nelson Education Ltd.

Introduction (Small Groups) ➧ 5–10 min

On the board, a transparency, or the interactive whiteboard,write the following fractions: . Tell studentsthat each of these fractions can be approximated as one of thefollowing unit fractions: . Have students work insmall groups to decide which unit fraction best approximateseach of the given fractions.

Sample Discourse“Suppose you were dividing by 3. How might you estimate

to make the division easier?”• I might use to make it easy to divide by 3.“Suppose you wanted to know how many s were in . Howmight you estimate?”• is about , since is close to . I know there are two

eighths in .“Which unit fraction is the best approximation for each ofthe given fractions?”•


LEARN ABOUT the Math (Whole Class/Individuals)

Together with students, read the information and centralquestion on Student Book page 72. Discuss how the circlesrepresent the fractions and . Distribute enough fractionstrips and scissors to students to use throughout the lessonand in the Checking and Practising questions. Have studentscut out several eighths and fifths strips to use for theexamples. Work through Examples 1, 2, and 3 as a class.Have students follow the steps, using their own fractionstrips for Examples 1 and 2. You may want to discuss theword “compatible” in the context of Example 3— iscompatible with since 15 is an easy number to use with 3, in this case to divide.

Sample Discourse“How does the number of sections in each circle relate tothe fraction it shows?”• The total number of sections is the denominator, and the

number of shaded sections is the numerator.

1640

1540

25

18

1

2

3

4

5

6

7

8

421 =

. 15; 5

19 =

. 14; 9

89 =

. 110; 11

24 =

. 12; 11

30 =

. 13

14

520

421

14

421

421

18

321

421

421

12, 13, 14, 15, 1

10

421, 5

19, 989, 11

24, 1130

1

2

3

4

5

6

7

8

“When might you want to estimate a quotient instead ofcalculating the exact quotient?”• I might estimate a quotient to check if my calculated answer

is reasonable.• I might estimate a quotient to determine if it is greater than

or less than another number.• I might estimate a quotient whenever I don’t need an exact

answer.“How can you find an equivalent fraction?”• I can multiply or divide the numerator and denominator

by the same number.“What are the benefits of using fraction strips to estimatea quotient?”• I can use the fraction strips to help see the answer.• I can compare the size of the quotient strip to other fraction

strips.• I don’t have to find equivalent fractions.“What are the benefits of using equivalent fractions toestimate a quotient?”• I don’t have to cut out fraction strips.• I can easily compare the numerators of fractions with the

same denominators.



These questions encourage students to reflect on how theycan use fraction strips and equivalent fractions to estimatea fraction quotient. Students will also reflect on the differentways in which they can use fraction strips to visualize afraction quotient.

Sample Discourse“How was Brian’s solution similar to Preston’s solution?”• Brian and Preston both used fraction strips to make visual

comparisons.• Brian and Preston both estimated the quotient without using

equivalent fractions or compatible numbers.“How was Brian’s solution different from Preston’s solution?”• Brian compared the length of a fraction strip with the length

of a fraction strip. Preston counted how many times a fraction strip fitted into a fraction strip.

• Brian tripled and compared the product to . Preston compared to .

Answers to ReflectingA. For example, every division is really like a multiplication.

If you want to know if one number is triple the size ofanother, you can multiply the lower one by 3 or you candivide the greater one by 3.

B. For example, she needed only to concentrate on therelationship between the numerators.

25

18

25

18

25

18

25

38



Have students work through the example in pairs. Studentscan duplicate the fraction strips model. Ensure that they have2 fourths fraction strips and 1 ninths fraction strip. Havestudents explain one of the solutions to their partner.

Checking (Pairs)

Ensure that students have fourths, fifths, sevenths, eighths,and tenths fraction strips. Ask for volunteers to show theirsolutions to Question 2. Try to elicit a variety of solutions,including dividing with fraction strips, multiplying withfraction strips, and using equivalent fractions.


These questions provide students with opportunities topractise estimating fraction quotients, using fraction stripsand equivalent fractions. Ensure that students have sufficientfraction strips to complete the exercises. 5. Encourage students to use a variety of strategies to

estimate the quotients.7. & 12. Students communicate their reasoning when they

explain their estimate of the quotient.

1

2

3

4

5

6

7

8

53Lesson 2.7: Estimating Fraction QuotientsCopyright © 2008 Nelson Education Ltd.

Answer to Key Question6. For example, is a little greater than , so Amber will

need to use the measuring cup about twice.


To consolidate students’ learning for this lesson, discussQuestion 13 together. Encourage students to share theirdifferent methods. Ask for volunteers to explain why theyprefer a particular method. Emphasize that one methodis not “better” than another. Help students realize thatdifferent approaches work better for different questions, so it is advantageous to have many methods, strategies,and tools at their disposal.

Answers to Closing Question13. a) For example, estimate using ,

or estimate using , which is the same as � 5.

b) For example, I think the first way is easier because Ifeel more confident using whole numbers.

Follow-Up and Preparation for Next ClassHave students write at least two word problems they cansolve by dividing fractions. Have them estimate the quotientsusing at least two methods from today’s class. They willcalculate exact answers to their problems in later classes.

56 ,

16

56 ,

212

1 ,2

10 = 1 ,15 = 5

13

23

34





• Students estimate fraction quotients using fraction strips.

• Students choose reasonable compatible numbers to estimate fractionquotients.

Key Question 6 (Mental Mathematics and Estimation)• Students choose and implement an appropriate strategy (either fraction strips

or compatible numbers) to calculate the number of times Amber will need tofill the measuring cup.

• Students may have difficulty using a fraction strip to estimate the quotientof two fractions. (See Extra Support 1.)

• Students may have difficulty determining equivalents for benchmark fractions.(See Extra Support 2.)

• Students may not be able to choose an appropriate compatible number toestimate a fraction division. (See Extra Support 3.)

• Students may do the third point above for this question. (See Extra Support 3.)

EXTRA SUPPORT

1. Have students cut several fraction strips to represent the divisor and thenuse as many as necessary to cover the fraction strip that represents thedividend. Students can count the number of divisor fraction strips to findwhole-number estimates of the quotient.

2. Remind students that to find an equivalent fraction, they must multiply thenumerator and denominator by the same factor. To help students performthis multiplication, have them use a multiplication table. Students can locatethe numerator and the denominator along the top of the table and then lookdown the table to the row with the factor by which they need to multiply.


EXTRA CHALLENGE

• Challenge students to use fraction strips to model the division of by andexplain their method.

4521

2


• Some students may have difficulty using fraction strips to model the divisionof fractions with unlike denominators. Have students begin by modelling thedivision of fractions with like denominators or the division of whole numbersby unit fractions.


• Students who have a verbal/linguistic learning style may benefit fromworking through the Practice questions with a partner and discussingthe solutions.

3. Tell students to try adding 1 to the numerator of the fraction and thendecide whether the new fraction can be renamed to a simple fraction suchas or . If not, try subtracting 1 from the numerator. Next try adding 1 tothe denominator of the fraction and then subtracting 1 from the denominator.Try to rename each new fraction to see which is easiest to work with, givenother numbers in the problem.

34

12

• To challenge students, have them list one pair of fractions that will give aquotient a bit less than 2 and one pair of fractions that will give a quotient abit more than 2.

• Provide students with copies of the Fraction Strips Tower used in Lesson 2.2.Have them use the tower to estimate the quotients.

55Lesson 2.8: Dividing Fractions by MeasuringCopyright © 2008 Nelson Education Ltd.


• Use a measurement model of division.• Determine equivalent fractions.

SPECIFIC OUTCOME

N6. Demonstrate an understanding of multiplyingand dividing positive fractions and mixed numbers,concretely, pictorially, and symbolically.[C, CN, ME, PS]

Achievement Indicators• Identify the operation required to solve a given

problem involving positive fractions. • Provide a context that requires the dividing of two

given positive fractions. • Model division of a positive proper fraction by a positive

proper fraction pictorially and record the process.• Generalize and apply rules for multiplying and

dividing positive fractions, including mixed numbers.

Dividing Fractionsby Measuring


GOAL

Divide fractions using models and using equivalent fractionswith a common denominator.

Math BackgroundThis lesson is an extension of Lesson 2.7, as studentsapply techniques for estimating fraction quotients tocalculating exact fraction quotients. Students will usefraction strips to concretely measure the number oftimes a divisor will fit into a dividend, and connectthe fraction strips model with symbolic representation.

In this lesson, the common-denominator algorithmfor division of fractions is introduced: determine thecommon denominator, determine the equivalentfractions, and divide the numerators. Students will use mental math to calculate equivalent fractions anddivide numerators, and reasoning to select appropriatecommon denominators.

112

14

15

18

19

110

111

112

13

16

16

16

16

16

17

17

17

17

17

17

18

18

18

18

18

18

18

19

19

19

19

19

19

19

19

110

110

110

110

110

110

110

110

110

111

111

111

111

111

111

111

111

111

111

112

112

112

112

112

112

112

112

112

112

112

12

13

13

14

14

14

15

15

15

15

16

17








Extra Practice Chapter Review Questions 17, 18, 19, & 20Workbook p. 17

Mathematical C (Communication) and CN (Connections)Process Focus



Ask students what might mean. How might youestimate the value? Draw from students both concrete modelsand symbolic solutions. After the discussion, tell studentsthat in today’s lesson, they will learn how to apply the samestrategies to find an exact quotient.

Sample Discourse“How can you use fraction strips to model this division?”• I can use a strip that represents and compare it with a strip that

represents and a strip that represents . Then I can estimate thequotient by deciding which thirds strip the strip is closer to.

• I can create a strip for and estimate how many times itwill fit in a strip for .

“Which fraction strips method would likely give you thebetter estimate?”• If I compare the strip with the and strips, my estimate

is either 2 or 3. If I estimate how many times the strip willfit in the strip, then my answer does not have to be a wholenumber, so I think that this method is better.

“What numbers can you divide to estimate the quotient?”• Since is close to 1, I can divide 1 by .• Since is equal to , I can divide by .3

989

39

13

13

89

89

13

33

23

89

89

13

89

33

23

89

89 ,

13

1

2

3

4

5

6

7

8

2.8



LEARN ABOUT the Math (Whole Class/Individual)

Together with students, read about Misa’s exercise on StudentBook page 76. Discuss the central question and draw fromstudents that the solution will involve a concrete model usingfraction strips. Provide each student with copies of FractionStrips Tower, Masters Booklet p. 45. Have students workthrough Prompts A to E individually, and then discuss theanswers as a class.

Sample Discourse“What does each whole fraction strip represent?”• 1 hour“What do the 4 whole fraction strips together represent?”• 4 hours• The amount of time Misa wants to exercise each week“What does each fraction strip represent?”• of an hour• The amount of time Misa exercises each day“What does the quotient represent?”• The number of times Misa will have to exercise each week“How is the fraction strips model in this question similarto the model you used in the estimation lesson?”• One strip is used to represent the divisor and one is used

to represent the quotient.• The quotient is the number of times the smaller strip fits

on the larger strip.

Answers to Learn about the MathA.–C.

D. timesE. times


Have students use these questions to explain how they coulduse the fraction strips model to find the quotient of ,and connect the model to a symbolic solution.

4 ,34

513

513

34

34

1

2

3

4

5

6

7

8

Sample Discourse“When might it be useful to use equivalent fractions ratherthan fraction strips to divide fractions?”• When the fractions have large numerators or denominators,

such as . It might take too long to draw the models.“Why do you need a common denominator to divideequivalent fractions?”• If the fractions have the same denominator, both wholes have

been divided into the same number of pieces, so I can justdivide the numerators.

Answers to Reflecting F. For example, each whole strip represents 1 h. Placing the

strips alongside the 4 whole strips shows you how manyh are in 4 h.

G. For example, you can think of 4 as , and determinethe equivalent fraction, . You can write a divisionexpression using these fractions: . Then you dividethe numerators (16 � 3) to find out how many 3s are in 16.

164 ,

34

164

41

34

34

5 ,1120



WORK WITH the Math (Small Groups/ Whole Class)

Read through Examples 1 to 3 together, and then havestudents work through the examples in small groups.Each student should explain one of the examples to the otherstudents in the group to solidify his or her own understandingand to help clarify any confusion the other students mighthave. Guide students to see the connection between Aaron’ssolution and Allison’s solution: Aaron needed to find acommon denominator to determine what size fraction stripsto use, and Allison needed to find a common denominator inorder to divide the numerators. Some students may want toduplicate the fraction strips used in Example 1.

Checking (Pairs)

Have students explain how they identified the fractions inQuestion 1 and how they determined what fraction strips touse in Question 2. Have students explain their reasoning forQuestion 3, in particular how they chose an appropriateequivalent fraction to use in the division.


These questions give students an opportunity to practiseusing fraction strips and equivalent fractions to findthe quotient of two fractions, as well as to apply andcommunicate their understanding in problem-solvingcontexts. Students use reasoning when they determinea common denominator and use equivalent fractions. 6. & 10. Students use reasoning when deciding on a

common denominator and determining equivalentfractions. Students communicate their reasoning whenthey explain their responses to Questions 6 d) and 10.

7. Some students may use fraction strips to solve thisproblem. Ensure that students have enough fractionstrips to do so.

1

2

3

4

5

6

7

8

9. Connections are made between fraction divisionand repeated fraction subtraction. Relate this tomultiplication as repeated addition.

13. Students communicate their reasoning when they explainwhy division of fractions is not commutative; that is,

.14. Students communicate their reasoning when they explain

why dividing by a fraction is the same as multiplying byits reciprocal.

ab ,

cd Z

cd ,

ab


Answers to Key Question6. a) For equivalent fractions, use a common denominator

of 3.

� 15 � 1

b)

c)

d)

For example, I used a denominator of 30 to work for fifthsand sixths.

I multiplied the numerator and denominator of by 6, and I multiplied the numerator and denominator of by 5.

The two fractions are renamed as and .I can ignore the denominators and divide 18 by 25. This isthe same as writing .18

25

2530

1830

56

35

=1825

35 ,

56 =

1830 ,

2530

= 623

=208 ,

38

212 ,

38 =

52 ,

38

= 2 110

=2112 ,

1012

134 ,

56 =

74 ,

56

= 15

51 ,

13 =

153 ,

13


Question 16 allows students to reflect on and consolidatetheir learning for this lesson as they form a generalizationabout dividing fractions with a common denominator.The common-denominator algorithm for division offractions is the following: determine the commondenominator, determine the equivalent fractions, anddivide the numerators.

Answer to Closing Question

16. For example, . If you divide by , you cansee that there are 2 groups of in , with leftover. The

leftover is half of , so there are groups of in .

Follow-Up and Preparation for Next ClassHave students use equivalent fractions to calculate exactanswers to the problems they wrote in the Follow-Up toLesson 2.7.

5a

2a21

22a

1a

1a

5a

2a

2a

5a =

2a +

2a +

1a





• Students use fraction strips to model fraction division.

• Students determine equivalent fractions with common denominators.

• Students rename mixed numbers and whole numbers as fractions.

• Students divide numerators of fractions with common denominators.

Key Question 6 (Connections, Communication)• Students determine equivalent fractions with a common denominator and

divide the numerators to find the quotient.

• Students may not line up the correct number of divisor fraction strips toequal the length of the dividend fraction strip. (See Extra Support 1.)

• Students may have difficulty calculating equivalent fractions with a commondenominator. (See Extra Support 2 and 3.)

• Students may add the whole number to the numerator of the fractionpart of the mixed number instead of multiplying the whole number by thedenominator and adding that product to the numerator. (See Extra Support 4.)

• Students may write the whole number as a fraction with the whole numberas the numerator and as the denominator, rather than writing a fraction withthe whole number as the numerator and 1 as the denominator. (See ExtraSupport 5.)

• Students may divide the wrong numbers in the fractions, such as numeratorby denominator. (See Extra Support 6.)

• Students may write the quotient of the numerators as a fraction with thesame denominator. (See Extra Support 6.)

• Students may do the third, fourth, fifth, sixth, and seventh points abovefor this question. (See Extra Support 2, 3, 4, 5, and 6.)


Chapter 2


EXTRA CHALLENGE

• In each empty box, write a fraction or whole number that is the quotient ofthe numbers in the two boxes on the same line before it.


• Have students divide fractions with like denominators and problems to which the solution is known, such as . Then have them try more difficultproblems, such as and , where the quotient is a whole number.Some students may not even be ready for this and may use a repeatedsubtraction model for division instead.

910 ,

310

45 ,

25

34 ,

14


EXTRA SUPPORT

1. Have students cut several fraction strips to represent the divisor, and thenuse as many as necessary to cover the fraction strip that represents thedividend. Students can count the number of divisor fraction strips to getwhole-number estimates of the quotient.

2. To determine common denominators, have students write out multiples ofeach denominator until they determine a common multiple. This will bethe common denominator.

3. Remind students that to determine an equivalent fraction, they must multiplythe numerator and denominator by the same factor. To help students performthis multiplication, have them use a multiplication table. Students can locatethe numerator and the denominator along the top of the table, and then lookdown the table to the row with the factor by which they need to multiply.


4. Have students use fraction strips to model the mixed number. For example,to model , students should use a whole fourths strip to model 1 and threesections of another fourths strip to model . Tell students that the improperfraction will have a numerator equal to the total number of sections, seven,and the same denominator as the fraction part of the whole number.

5. Remind students that the denominator of a fraction tells how many partsthe whole has been divided into. For a whole number, the whole has beendivided into 1 part, so the denominator will be 1.

6. Have students use a fraction strip model with a common denominator, asin Example 1. Guide them to see that if the denominator is the same, theyneed only count the number of times the numerator of the divisor fits intothe numerator of the dividend.

34

134

• You may want to eliminate consideration of mixed numbers and wholenumbers.


61Curious Math: It Is Just Like Multiplying!Copyright © 2008 Nelson Education Ltd.


• Multiply fractions symbolically.

SPECIFIC OUTCOME

N6. Demonstrate an understanding of multiplying anddividing positive fractions and mixed numbers, concretely,pictorially, and symbolically.[C, CN, ME, PS, R]

Achievement Indicator• Generalize and apply rules for multiplying and


Sample Discourse“How could you use fraction strips to show that reallyis ?”• I could use an eighths fraction strip and divide each section

in half, and then shade in 15 of the sections. Then I could usea fourths fraction strip and shade in 3 of the sections. I couldline up the fourths fraction strips to see how many fit withinthe shaded part of the sixteenths strip.

“What are equivalent fractions?”• Fractions that are the same when written in lowest terms“What are some equivalent fractions for ?”• For example, “What are some equivalent fractions for ?”• For example, “Could you use this method to divide by ?”• No, because the numerator of the first fraction is not

a multiple of the numerator of the second fraction.• Yes, because I could write as , and then the numerator

is divisible by 2 and the denominator is divisible by 3.

1018

59

23

59

68, 9

12, 1216, 15

20

34

46, 69, 8

12, 1015

23

54

1516 ,

34

Curious Math It Is Just Like Multiplying!


Preparation and PlanningMathematical CN (Connections) and R (Reasoning)Process Focus


Math BackgroundStudents have learned that to multiply fractions, theymultiply numerator by numerator and denominator bydenominator. In this activity, students will makeconnections between multiplying fractions and dividingfractions as they investigate whether they can dividefractions by dividing the numerators and dividing thedenominators. To divide fractions in this way, it is bestto make the numerators and denominators evenlydivisible, so students will use reasoning to chooseequivalent fractions that will allow them to use thismethod effectively. However, it is still valid, if awkward,even if the numerators and denominators are not evenlydivisible. For example,

If you multiply both numerator and denominator by, the fraction becomes , which is the correct

quotient.

986 A66 = 1 B

34

,

23

=

(3 , 2)

(4 , 3)=

112

113

Using Curious MathIn this activity, students will divide numerators and dividedenominators in order to find the quotient of two fractions.Have students work through the questions and then discussthe answers as a class.

Answers to Curious Math

1.2. For example, if you rewrite as , then you can divide

15 by 3; you can’t divide 5 by 3 and get a whole number.3. For example, 23 ,

34 =

2436 ,

34 =

24 , 336 , 4 =

89

915

35

1516 ,

34 =

1516 ,

1216 = 15 , 12 =

54

4. For example, dividing is the opposite of multiplying.Since you multiply numerators and denominators tomultiply fractions, it should be similar for dividing.

5. For example, I would most likely use this method whenthe first numerator is a multiple of the second one andthe first denominator is a multiple of the second one.





• Students will calculate fraction quotients by dividing numerators and dividingdenominators.

• Students will calculate equivalent fractions that can be used to calculatefraction quotients.

• Students will not be able to calculate whole-number quotients whendividing numerators and denominators. (See Extra Support 1.)

• Students may have difficulty writing equivalent fractions. (See ExtraSupport 2.)

• Students will not be able to choose an appropriate equivalent fraction.(See Extra Support 3.)

EXTRA SUPPORT

1. Review strategies for dividing whole numbers and provide students withadditional practice.

2. Remind students that they can calculate an equivalent fraction by multiplyingthe numerator and the denominator by the same factor. Some students maybenefit from using a multiplication table. Students can locate the numeratorand denominator of the fraction along the top of the multiplication table. Thecolumns for the numerator and denominator will contain the multiples ofthese numbers, and students can look at any row of the table to select anequivalent fraction.


EXTRA CHALLENGE

• Have students use the “dividing numerators/dividing denominators” methodto divide mixed numbers. Provide some or all of the questions below.

2 110 , 12

5310 , 51

2

336 , 21

3214 , 11

2


• The questions in this lesson are best suited to a logical/mathematicallearning style. Verbal/linguistic learners may benefit from a class discussionof the reasoning required to complete the exercises.

3. Have students write out a number of the dividend and then attempt to dividethe numerator of the divisor into each of the equivalent fractions. If thenumerator is evenly divisible, then the equivalent fraction is an appropriatechoice.

63Lesson 2.9: Dividing Fractions Using a Related MultiplicationCopyright © 2008 Nelson Education Ltd.


• Multiply fractions.• Rename improper fractions as mixed numbers and

vice versa.

SPECIFIC OUTCOME

N6. Demonstrate an understanding of multiplying and dividing positive fractions and mixed numbers,concretely, pictorially, and symbolically. [C, CN, ME, PS]

Achievement Indicators• Provide a context that requires the dividing of two

given positive fractions.• Model division of a positive proper fraction by a

positive proper fraction pictorially and record theprocess.


Dividing Fractions Using a Related Multiplication


GOALDivide fractions using a related multiplication.

Math BackgroundThis lesson provides another approach to dividingfractions and is a continuation of the learning inLesson 2.8. Students will make connections betweendivision and multiplication as they use a relatedmultiplication to solve fraction division problems.This lesson introduces the concept of the reciprocal,the fraction that gives a product of 1 when multipliedby the original fraction. Students will apply this conceptas they connect dividing by a fraction with multiplyingby the reciprocal of the fraction using a meaningfulcontext. Encourage them to apply this strategy insolving the practice questions, as it is to their advantageto be comfortable with more than one method ofperforming fraction division. Students will usereciprocals to solve problems in context, and theydevelop their communication skills as they explain the results of fraction division.




Recommended Questions 5, 8, 9, 14, & 17Practising Questions


Extra Practice Chapter Review Question 21Workbook p. 18

Mathematical C (Communication), CN (Connections),Process Focus and PS (Problem Solving)

Vocabulary/Symbols reciprocal


112

14

15

18

19

110

111

112

13

16

16

16

16

16

17

17

17

17

17

17

18

18

18

18

18

18

18

19

19

19

19

19

19

19

19

110

110

110

110

110

110

110

110

110

111

111

111

111

111

111

111

111

111

111

112

112

112

112

112

112

112

112

112

112

112

12

13

13

14

14

14

15

15

15

15

16

17



2.9



Draw a square divided into four equal sections on the board,a transparency, or an interactive whiteboard. Discuss withstudents how the size of the whole relates to the size of eachpart, and how the size of one part relates to the size of thewhole.

Sample Discourse“How can you describe one part of this square in relationto the whole?”• One part of this square is equal to one fourth of the whole

square.• One part is 4 times smaller than the whole square.“How can you describe the whole square in relation to onepart?”• The whole square is equal to 4 times one part.• The whole square is 4 times larger than one part.“By what number can you multiply the area of one part toget the area of the whole?”• 4“By what number can you divide the area of the whole toget the area of one part?”• 4“By what number can you multiply the area of the whole toget the area of one part?”•“By what number can you divide the area of one part to getthe area of the whole?”•Lead a discussion about related operations. Guide studentsto realize that just as addition is the opposite of subtraction,multiplication is the opposite of division.


LEARN ABOUT the Math (Whole Class/Small Groups)

Together with students, read the information and the centralquestion on Student Book page 82. Read the highlighteddefinition for “reciprocal” and ensure that studentsunderstand what it means. Draw from students that thesolution involves the division of fractions. Work throughExamples 1 and 2 as a class. Discuss how the solutionmethods relate to strategies that students have used inprevious lessons.

1

2

3

4

5

6

7

8

14

14

1

2

3

4

5

6

7

8

Sample Discourse“How do you know this is a division problem?”• I need to determine how many smaller cans will “fit into” the

larger can, and “fit into” means division.“What other strategies can you use to find the solutions tothe examples?”• I can use fraction strips and count how many thirds strips

fit in 2 wholes.• I can write 2 as a fraction with a denominator of 1, and then

determine equivalent fractions with a common denominatorand divide the numerators.


These questions provide students with an opportunity toreflect on the relationship between multiplication anddivision, and how they can use a related multiplication factto solve a fraction division problem. Ensure that studentshave a good understanding of reciprocal fractions.


Sample Discourse“How did Alison and Nikita know that dividing by is thesame as multiplying by 3?”• 3 can be written as , and is the reciprocal of .• 3 thirds fraction strips make up 1 whole, so the number of

thirds fraction strips in a number is equal to 3 times thatnumber.

• There are 3 thirds in 1 whole, so the number of thirds in anynumber is equal to that number times 3.

Answers to ReflectingA. For example, division is used to find out how many of

one number fit into another. Alison and Nikita weretrying to find out how many of the smaller cans fit intotheir large cans. They divided by because the smallercans hold as much as the large cans.

B. For example, the reciprocal is the number of small cansthat 1 large can would fill, and he had 2 large cans, andthe 2 large cans hold twice as much as 1 large can.

C. If the small can held twice as much, each large can wouldonly fill half as much. Since they originally multiplied2 � 3 and � 3 to calculate the number of small cansthey could fill, and they could fill half as many smallcans if the small cans were twice as large, Alison andNikita would have to calculate half the original products.If you take half, you are multiplying by .

and , so youare multiplying by the reciprocal of .2

3

78 * 3 *

12 =

78 *

322 * 3 *

12 = 2 *

32

12

78

13

13

31

13

31

13


WORK WITH the Math (Individuals/Whole Class)

Distribute fraction strips and scissors. Tell students to cut outthe fifths strips and the ones strips. Have students read Example3 on their own, duplicating the steps in the three solutions.Discuss any questions.

Checking (Pairs)

Have students solve each problem, and then discuss theresults as a class. Students will communicate their solutionstrategies. Encourage students to use related multiplicationfacts to solve the division problems, rather than usingfraction strips or determining equivalent fractions witha common denominator.


These questions give students opportunities to practisedividing fractions using a related multiplication fact, solveproblems involving fraction division, and communicatetheir strategies.5. & 9. Students communicate their reasoning as they

explain what they know about a fraction quotientwithout calculating the quotient.

11. Have students explain their solution strategies.15. Connections are made with decimal division.

1

2

3

4

5

6

7

8


Answer to Key Question5. For example, because is greater than , will fit into a

little more than once, so the quotient will be greater than1, which is greater than .


Ask a volunteer to explain the meaning of “reciprocal” andexplain how a reciprocal is used to calculate a fractiondivision. Then have students answer Question 17 to reflecton and consolidate their learning for this lesson. Encouragestudents to share their answers and solve the division in thecontext of their situation.

78

78

34

34

78

Answers to Closing Question 17. a) For example, a small glass of juice holds as much as

a large glass. How many small glasses can you fill bypouring in juice from large glasses?

b) For example, it takes glasses of juice to fill a pitcher. If you have room to fill pitchers, how many smallglasses of juice will you need to fill the pitcher?

Follow-Up and Preparation for Next ClassStudents can review the various fraction division strategiesthey have learned (fraction strips, determining equivalentfractions with a common denominator, using a relatedmultiplication) and then use the strategy they learned todayto solve one or two division problems from previous lessons.

125

223

98

23





• Students divide using a related multiplication by determining the reciprocal ofthe divisor.

• Students divide mixed numbers by fractions.

• Students use reasoning to predict a quotient.

Key Question 5 (Communication, Connections, Problem Solving)• Students explain why dividing by gives a quotient that is greater than .7

834

78

• Students do not understand that dividing by a fraction is the same asmultiplying by its reciprocal. (See Extra Support 1.)

• Students are unable to write mixed numbers as improper fractions. (See ExtraSupport 2.)

• Students are unable to apply generalizations about fraction operations tospecific situations. (See Extra Support 3.)

• Students do not understand that dividing a fraction by a smaller fraction givesa quotient that is greater than the original dividend. (See Extra Support 4.)

EXTRA SUPPORT

1. Have students cut out a ones fraction strip and 2 fourths strips. Have them usethe fraction strips to model . They will find that one full three-fourths strip will fit, plus one third of the second strip, so the quotient is . Point out that is the reciprocal of , and , so multiplying by the reciprocal gives the same answer as dividing by the original fraction.

2. Have students use fraction strips to model a mixed number. For example, tomodel , students should use a whole fourths strip to model 1 and threesections of another fourths strip to model . Tell students that the improperfraction will have a numerator equal to the total number of sections, seven,and the same denominator as the fraction part of the whole number.

34

134

1 *43 =

43

34

43

43

1 ,34


EXTRA CHALLENGE

• Have students write and solve at least three division problems in which amixed number is divided by a mixed number, and then form a generalizationabout how to divide mixed numbers.


• Have students begin by dividing whole numbers by other whole numberswritten as fractions with a denominator of 1, such as . Guide themto see that they will get the same answer if they multiply 4 by . Ifstudents continue to struggle, have them use fraction strips to modelthe division. Even with this adjustment, students may not be ready todivide fractions other than concretely determining how many of onefraction fits into another.

12

4 ,21

3. Have students predict whether the quotient of 1 divided by a fraction lessthan 1 will be less than 1, equal to 1, or greater than 1. Then have studentspredict whether the quotient of 1 divided by a fraction greater than 1 will beless than 1, equal to 1, or greater than 1.

4. Ask students to use the reciprocal to explain that dividing 1 by a fractionless than 1 will give a quotient that is greater than 1, and dividing 1 by afraction greater than 1 will give a quotient that is less than the dividend.



• Add and subtract positive fractions with like and unlikedenominators.

• Multiply and divide positive fractions with like and unlikedenominators.

• Compare fractions.

SPECIFIC OUTCOME

N6. Demonstrate an understanding of multiplying anddividing positive fractions and mixed numbers,concretely, pictorially, and symbolically. [C, CN, ME, PS]

Achievement Indicator• Generalize and apply rules for multiplying and


unlike denominators. Students will also need to be proficientin finding equivalent fractions so they can determine whenthey have found an answer of , and in comparing fractionsso they can determine which answer is closest to .

StrategiesThe numbers that will form the numerators anddenominators are based on chance, but students will improvetheir chances of winning if they consider all possiblearrangements of numerator and denominator for the twofractions and try to add, subtract, multiply, and divide asmany of these combinations as possible. Students will beable to cut down on the number of combinations of fractionsto try for each operation if they think about what kind ofanswer they will get relative to the starting fractions (forexample, adding fractions gives an answer that is greater thaneither starting fraction).

DiscussAfter the game, you might ask students the following:• How did you use equivalent fractions to play this game?• What strategies did you use to determine which operation

to use for a given pair of fractions?

23

23

Chapter Chapter 2Chapter 2 Math Game Target 2

3


Preparation and PlanningNumber of Players 2 to 4

Materials • dice, 1 pair per group

Mathematical ME (Mental Mathematics and Estimation) Process Focus


Math BackgroundThis game connects the skills and strategies students havelearned in this chapter with what they have learned inprevious grades. Students will form several differentfractions from the numbers on a pair of dice and considerthe effect of each operation on the fractions. Not all of thefractions that can be formed by the numbers on the twodice will be suited to getting an answer of or close to with all four operations. Students will use estimation andreasoning to determine the best operation to perform onany two fractions and to compare their answers to thebenchmark of to determine the winner. 2

3

23

Using the Math GameThe purpose of this game is to provide students with practiceadding, subtracting, multiplying, and dividing fractions.

When to PlayStudents can play this game after they have demonstrated theability to add, subtract, multiply, and divide fractions with

69Math Game: Target 23Copyright © 2008 Nelson Education Ltd.



Proficient players Less-proficient players

• Students calculate the sum, difference, product, or quotient of two fractionscorrectly and efficiently.

• Students use an appropriate strategy (e.g., benchmarks, estimation withcompatible numbers, common denominators) to compare fractions to .2

3

• Students may be unable to add, subtract, multiply, or divide their twofractions. (See Extra Support 1.)

• Students may only consider a single operation for a pair of fractions, or mayonly consider a single pair of fractions. (See Extra Support 2.)

• Students will be unable to determine which answer is closest to . (See Extra Support 3.)

23

EXTRA SUPPORT

1. Allow students to use fraction strips or other models to calculate theanswers to their fraction calculations.

2. Change the rules so that students get an extra point for each equation theywrite that has an answer of . This will encourage players to check manydifferent fractions with each operation.

23


EXTRA CHALLENGE

• Students can use the same format of rolling two dice to form fractions, butthey can use only improper fractions and, if a 1 is rolled, whole numberswritten as a denominator of 1.


• Some students may benefit from playing this game in teams and discussingthe possible fractions and operations to use.

3. Tell students that they can apply strategies used in other lessons to comparefractions, such as estimation with compatible numbers (see Extra Support in Lesson 2.7) and finding equivalent fractions with common denominators(see Extra Support in Lesson 2.8).



• Apply the order of operations for whole numbers andintegers.

SPECIFIC OUTCOME


Achievement Indicator• Solve a given problem involving positive fractions

taking into consideration order of operations (limitedto problems with positive solutions).

Order of Operations STUDENT BOOK PAGES 88–91

GOAL

Use the order of operations in calculations involving fractions.



Masters • Fractions and Operations Cards I p. 99• Fractions and Operations Cards II p. 100

Recommended Questions 4, 7, 9, & 13Practising Questions


Extra Practice Chapter Review Questions 22 & 23Workbook p. 19

Mathematical CN (Connections) and PS (Problem Solving)Process Focus


Math BackgroundThrough the grades, students are being graduallyintroduced to the order of operations, which dictates theorder in which parts of an expression are evaluated. 1. Evaluate any expression(s) within brackets.2. Evaluate any exponents—not addressed yet.3. Perform multiplication and division in order from

left to right.4. Perform addition and subtraction in order from left

to right.These conventions ensure that an expression will alwayshave the same unique value. In this lesson, connectionswill be made as students apply the order of operations toevaluate expressions with fractions, using the strategiesthey have already learned to add, subtract, multiply, anddivide fractions. Students will also apply the order ofoperations to solve problems.

Name: Date:


23

15

58

34

25

310

÷ ÷ ×

Fractions and Operations Cards ILesson 2.10: Order of OperationsSTUDENT BOOK PAGES 88–91

Fractions and Operations Cards I p. 99

Name: Date:


× −

+

−

(

( (

) +

Fractions and Operations Cards IILesson 2.10: Order of OperationsSTUDENT BOOK PAGES 88–91

Fractions and Operations Cards II p. 100

2.10

71Lesson 2.10: Order of OperationsCopyright © 2008 Nelson Education Ltd.


This activity provides a review of the order of operations,using whole numbers and integers. Write the followingexpression on the board, a transparency, or an interactivewhiteboard:

(6 � 9) � (8 � 3) � 25 � 7 � 4 � 2

Ask volunteers to solve the expression, one step at a time,and explain what they are doing. Ensure that students arefollowing the order of operations; the correct order is shownbelow.

(6 � 9) � (8 � 3) � 25 � 7 � 4 � 2� 15 � (8 � 3) � 25 � 7 � 4 � 2� 15 � 5 � 25 � 7 � 4 � 2� 75 � 25 � 7 � 4 � 2� 3 � 7 � 4 � 2� 3 � 7 � 8� 2


LEARN ABOUT the Math (Whole Class/Pairs)

Distribute Fractions and Operations Cards I and II pp. 99–100, along with scissors. Have students cut out thecards. Together, read the Rules for Targets and the speechbubbles on Student Book page 88. Tell students that theorder of operations they have used to evaluate expressionswith whole numbers and integers also applies to expressionswith fractions. Read the central question and ensure thatstudents understand that the solution will involve arrangingthe fractions and operations on the cards to get as close aspossible to 1. Have students work through Prompts A and Bindividually, and then discuss as a class. Elicit from studentsthe different card arrangements they made and the values ofthe arrangements. Discuss why the order of operations isnecessary.

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

Sample Discourse“Why does Alison get a different answer if she follows theorder of operations?”• If she follows the order of operations, she does the

multiplication before the addition.• If she follows the order of operations, she adds to the product

of , but if she doesn’t follow the order of operations, shemultiplies the sum of by .

“Why is the order of operations important?”• Everyone solves problems the same way.• It makes sure that an expression will always have the same

value.

58

23 +

15

15 *

58

23


Answers to Learn about the MathA. Alison added , and then multiplied the sum by .

B. ; this isthe closest she can get to 1.


Have students use these questions to communicate theirunderstanding of the order of operations.

Sample Discourse“What effect do brackets have?”• Brackets tell you what to do first.• Brackets let you do addition and subtraction before

multiplication and division. “How else could Alison arrange the cards to get an answerof ?”• She could arrange the cards with multiplication before

addition.• or

Answers to Reflecting

C.D. Multiply, and then add.



Have students work through the example in pairs. Encourage them to ask their partner to clarify anyconfusion they might have about the solution. Guidestudents in noting the order in which the operations wereperformed and how the brackets were used to clarify thesteps.

Checking (Pairs)

1. Ask students to explain the order in which they performedthe operations.

2. Encourage students to share their answers and determinewhich student’s arrangement has the value closest to 1.

1

2

3

4

5

6

7

8

23 +

15 *

58 =

23 +

540 =

95120

A23 +15 B *

58

58 *

15 +

23

15 *

58 +

23

1924

23 +

15 *

58 =

23 +

540 =

23 +

18 =

1624 +

324 =

1924

58

23 +

15


These questions provide students with opportunities topractise following the order of operations to add, subtract,multiply, and divide fractions. Connections are made asstudents apply various strategies for fraction operations. 6. Ask students why brackets are needed in parts a) and f ).7. b) Students communicate the effect of brackets on a

mathematical expression.

73Lesson 2.10: Order of OperationsCopyright © 2008 Nelson Education Ltd.

Answer to Key Question9.


Question 13 allows students to reflect on and consolidatetheir learning for this lesson as they communicate why theorder of operations is necessary. Using the order of operationsensures that an expression has a unique value.

2 + A14 +13 B *

37 -

25 *

38 , A 1

10 +15 B = 13

4

Answer to Closing Question 13. For example, if I use the order of operations to find the

answer, I multiply, then add. If everyone else uses thesame order, we will all get the same answer.

Follow-Up and Preparation for Next ClassHave students write a word problem that uses the fractionsand operations in Question 4.






• Students follow the order of operations.

• Students apply appropriate strategies to add, subtract, multiply, and dividefractions.

Key Question 9 (Connections, Problem Solving)• Students place brackets in an equation to make it true.

• Students may perform operations as they appear from left to right. (See ExtraSupport 1 and 2.)

• Students have difficulty solving the problems. (See Extra Support 3.)

• Students may not understand the effect of brackets on an expression. (See Extra Support 2.)

• Students are unable to apply the order of operations. (See Extra Support 1 and 2.)

EXTRA SUPPORT

1. Have students create posters that show the order of operations in a creativecontext that will help them remember.

2. Write the expression 2 + 3 * 4 - 5 on the board, a transparency, or aninteractive whiteboard. Show students how to evaluate the expressionfollowing the order of operations (multiply, then add, then subtract). Next,evaluate the expression in order from left to right (add, then multiply, thensubtract). Point out the difference in the solutions. Finally, place bracketsaround (2 + 3) and (4 - 5) and point out that the brackets change the orderin which operations are performed and that the addition and subtraction willbe performed before the multiplication.


EXTRA CHALLENGE

• Have students play Target 0.

Rules for Target 01. Pick four fraction (F) cards.

2. Pick three operation (O) cards.

3. Put them in this order: F O F O F O F.

4. Rearrange the cards to get a value as close as possible to 0.

5. The closest value gets 1 point.

6. The first player to get 5 points wins.


• Provide students with expressions for which they do not need to determinecommon denominators to perform addition or subtraction.

• Some students may benefit from duplicating the steps in the example, ratherthan simply reading through it.

3. Have students discuss some strategies they have used to add, subtract,multiply, and divide fractions (e.g., fraction strips, number lines, equivalentfractions with common denominators, using a related multiplication). Guidestudents to break down complicated mathematical expressions into smallerparts and apply an appropriate strategy to each part.

• Have students begin by using the order of operations to solve problems usingwhole numbers. Provide problems that use only the numerators of thefractions in the Practice questions.

• Encourage students to discuss the Practice questions with a partner in orderto clarify their thinking, particularly Questions 7, 10, and 12.

75Lesson 2.11: Communicating about Multiplication and DivisionCopyright © 2008 Nelson Education Ltd.

GOAL


• Multiply decimal tenths and hundredths.• Understand the meanings of division.• Understand the meanings of multiplication.• Determine equivalent fractions.

SPECIFIC OUTCOME



problem involving positive fractions. • Provide a context that requires the dividing of two

given positive fractions. • Model multiplication of a positive fraction by a



• Solve a given problem involving positive fractionstaking into consideration order of operations (limitedto problems with positive solutions).

Communicating aboutMultiplication and Division



Recommended Questions 4, 8, & 9Practising Questions


Extra Practice Chapter Review Question 24Workbook p. 20

Mathematical C (Communication) Process Focus


Math BackgroundCommunication lessons provide students with examplesof what good mathematical explanations do or do notlook like, to help students become better communicators.When students are required to communicate their thoughtprocesses, they are better able to identify misconceptionsheld both by themselves and by their peers. Exploring thesemisconceptions serves to improve their understanding ofthe processes involved in fraction calculations.

In this lesson, students are given problems involvingmultiplication and division of fractions, and are asked toexplain solution methods. Students will also write wordproblems that fit a given fraction division. These tasksrequire a depth of understanding that simple calculationtasks do not. The clarity and completeness of students’responses give you a more well-rounded view of studentthinking.

Describe situations involving multiplying and dividingfractions and mixed numbers.

2.11



Have students imagine that a person from 200 years ago wasunfrozen and now needs to learn how to function in the21st century. Ask volunteers to explain how the person from200 years ago can perform a simple task, such as makingbreakfast. They can use the board or an interactivewhiteboard to draw diagrams, if necessary. Have otherstudents pretend to be the person from the past and askquestions when the explanation is not clear. Tell studentsthat today they will be working on their mathematicalcommunication skills. Discuss how they can use what theylearned in this activity to help them give good mathematicalexplanations.

Sample Discourse“Why might a person from 200 years ago not understand anexplanation?”• I might have skipped a step in the explanation.• I might use words or talk about things that the person from

200 years ago doesn’t know.“Why is it important to explain each step?”• If the person from 200 years ago has never done something

before, he or she might be confused if I skip a step.• The person from 200 years ago doesn’t know what is going on,

so I need to explain each step.


LEARN ABOUT the Math (Whole Class/Small Groups)

Together, read Preston’s word problem on Student Bookpage 92. Tell students that in Preston’s explanation, he istrying to communicate to Aaron how he came up with thenumbers he used in the problem. Read Preston’s explanationand Aaron’s questions, and then discuss the central questionand the Communication Checklist. Students can answer thecentral question by working through Prompts A and B insmall groups.

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

Answers to Learn about the MathA. For example, I was multiplying by the fraction , and

when you multiply by a fraction less than 1, you onlytake part of the thing you are multiplying.For example, I am taking of the cups of sugar inthe original recipe. For example, 3 is the number of cups of flour in the recipe and it had nothing to do with the sugar.

B. For example, how do you know that 3 is of ? 412

23

312

23

23



Have students use these questions to help them explain whata good mathematical explanation looks like.

Answers to ReflectingC. For example, Preston partially explained his steps, but

not completely. He really didn’t justify his conclusions.He didn’t use any models. Aaron asked for moreexplanation and justifications in the steps.

D. For example, is of . I’m going to use a recipe that has cups of sugar. I’m going to make a newrecipe that uses of the sugar. I’m going to let the 3 be 3 cups of flour in the new recipe. Since I am taking of

cups of sugar, the 3 cups of flour has to be of thenumber of cups of flour in the original recipe. To findthe cups of flour in the original recipe, I divided 3 by .

3 , =

So, the original recipe has cups of flour.



Have students work through the example in pairs. Havethem discuss Misa’s solution and how well she covered theCommunication Checklist.

Checking (Individuals/Pairs)

Have students complete Question 1 individually, and then share their answers to parts b) and c) with a partner.Encourage students to ask questions about their partner’ssolution to help develop each other’s communication skills.


These questions provide additional practice in communicatingabout multiplication and division of fractions and mixednumbers.2. & 4. Students connect pictorial models with symbolic

representation of fraction multiplication.5. & 6. Students connect fraction multiplication with

decimal multiplication.

1

2

3

4

5

6

7

8

412

3 *32 =

92 = 41

223

23

2331

2

23

23

312

312

23

23 * 31

2

Answer to Key Question8. Another name for 1 is . It doesn’t matter what value

you use for n, as long as it’s not 0. When you multiply, you end up multiplying the numerator by n and

multiplying the denominator by n. Multiplying by 1does not change the value of the fraction.


Question 9 allows students to reflect on and consolidate theirlearning for this lesson. Ask students to communicate theirsolutions, apply what they learned today about goodmathematical communication, and encourage a variety ofsolutions.

ab *

nn

nn

Answer to Closing Question 9. means how many sets of are in .

means how many sets of are in .Since is exactly twice the size of , it will fit in exactlyhalf as many times.

1516

58

1516

516

1516 ,

516

1516

58

1516 ,

58

Follow-Up and Preparation for Next ClassEncourage students to practise their mathematical communication skills at home with a parent or othercaregiver.

Next class is the Chapter Review. Ask students to go throughLessons 2.1 to 2.11 and note any questions or problems theyhave.






• Students write word problems that are appropriate for the given numbers andoperations.

• Students explain why a method or calculation is correct.

Key Question 8 (Communication)• Students critique the reasoning in an explanation.

• Students may choose a situation that does not use the given numbers andoperations in an appropriate context. (See Extra Support 1 and 2.)

• Students may not understand why the method described gives the correctanswer. (See Extra Support 3.)

• Students may have difficulty using words to describe the steps in a solution.(See Extra Support 4 and 5.)

• Students may do the second and third points above for this problem. (See Extra Support 3, 4 and 5.)

EXTRA SUPPORT

1. Tell students to make a list of situations in their daily lives in which they usefractions. For example, they may use measuring cups or read part of a bookor magazine.

2. Review the different meanings of division (sharing, equal groups, etc.) andmultiplication (more than one same-sized group). Then have students make alist of situations in their daily lives in which they perform division ormultiplication.


EXTRA CHALLENGE

• Have students explain why common denominators are needed to add orsubtract fractions but not to multiply fractions.

• Have students explain why multiplying a number by a fraction less than 1gives a product that is less than the number, but dividing a number by afraction less than 1 gives a product that is greater than the number.


• You may want to eliminate consideration of problems involving more thanone operation.

• You may want to eliminate consideration of mixed numbers.


• Some students might need to practise their communication skills with apartner before sharing their explanation with the whole class. Allow studentsto work with a partner before taking up the closing question.

3. Have students use the described method to solve the problem or duplicatethe steps in a shown solution, and describe to themselves or to a partnerwhat was done in each step.

4. Remind students to use the Communication Checklist as a guide. Suggestthat students discuss the solutions with a classmate to clarify their thinking.

5. Suggest to students who are using pictorial or concrete models that theysketch their models rather than use words.

• Have students write a problem using the numbers in Question 8. Have themexplain why multiplying these numbers is appropriate for the problem.

• Provide decimal multiplication questions that use only one decimal place. Forexample, replace the multiplication in Question 6 with 4 * 0.2 = 0.8.



SPECIFIC OUTCOME

N6. Demonstrate an understanding of multiplying and dividing positive fractions and mixed numbers,concretely, pictorially, and symbolically. [C, CN, ME, PS]

Achievement Indicators• Express a given positive mixed number as an improper

fraction and a given positive improper fraction as amixed number.






Preparation and PlanningMaterials • Optional: counters

Masters • Optional: 2 cm Grid Paper, Masters Booklet p. 34

• Optional: Fraction Strips Tower, MastersBooklet p. 45

• Optional: Number Lines, Masters Booklet p. 39• Self-Assessment: Chapter 2 Lesson Goals:

Fraction Operations, Masters Booklet p. 65• Self-Assessment: Mathematical Processes,

Masters Booklet p. 75• Self-Assessment: What I Like, Masters Booklet

p. 76• Self-Assessment: How I Learn, Masters Booklet

p. 76


Using the Chapter Self-TestEncourage students to complete the Chapter Self-Test beforethe Chapter Review to help them identify any areas of need.If students are unable to complete a given question, referthem to the examples, questions, and/or Frequently AskedQuestions listed in the Study Guide on the next page and tothe Frequently Asked Questions on Student Book pages 97and 98. The examples suggested often include a number ofdifferent approaches to a given type of question.

You may also want to use the Self-Assessment Masters fromthe Masters Booklet to encourage assessment as learning.

What Do You Think Now?(Individual/Whole Class)

Revisit the What Do You Think? guide for Chapter 2 onStudent Book page 45. Have students look back to see iftheir decisions and/or explanations have changed. As well, if students recorded their initial thoughts about one of thechapter goals, they can complete the same prompt again,then compare their ideas and reflect on what they havelearned.

Chapter Self-Test

81Chapter Self-TestCopyright © 2008 Nelson Education Ltd.

Question Part Chapter Section Examples & Related Questions Student Book Page(s)

1 Lesson 2.1 Example 1: Multiplying with grids and counters p. 47Example 2: Multiplying with fraction strips p. 48Example 3: Multiplying with a number line p. 48Question 3 p. 49

Mid-Chapter Review FAQs (1st question) p. 64Questions 1, 2, & 3 p. 66

2 Lesson 2.3 Example 1: Using a fraction strip model p. 52Questions 1 & 4 a) pp. 54–55

Mid-Chapter Review FAQs (2nd question) p. 64Questions 6 & 7 c) p. 66

3 Lesson 2.3 Question 18 p. 56

4 Lesson 2.3 Example 3: Multiplying fractions less than 1 p. 54

5 Lesson 2.5 Example 1: Adding partial areas p. 59Example 2: Applying a procedure p. 59Example 3: Multiplying two mixed numbers p. 60

Mid-Chapter Review FAQs (4th question) p. 65

6 Lesson 2.5 Example 2: Applying a procedure p. 59Questions 4 & 7 p. 62

Mid-Chapter Review FAQs (4th question) p. 63Question 11 p. 67

7 a) Lesson 2.8 Example 1: Using a Model p. 77Question 2 p. 79

Chapter Review FAQs (2nd question) p. 977 b) Lesson 2.9 Example 3: Dividing a mixed number by a fraction pp. 84–85

Questions 3 a), 8 a), & 8 b) pp. 85–86definition of “reciprocal” p. 83

Chapter Review FAQs (2nd question) p. 97

8 Lesson 2.8 Example 2: Using common denominators p. 78Questions 3 a), 6 d), 11 a), 11 b), & 11 c) pp. 79–80

Lesson 2.9 Example 3: Dividing a mixed number by a fraction p. 84Questions 3 a), 8 a), 8 b), & 8 f) pp. 85–86definition of “reciprocal” p. 83

Chapter Review FAQs (2nd question) p. 978 c) Lesson 2.9 Example 2: Dividing a fraction by a fraction p. 83

9 Lesson 2.10 Example: Using the order of operations with fractions p. 89Questions 1, 6, & 7 a) pp. 90–91

Chapter Review FAQs (3rd question) p. 98

Chapter 2 Self-Test Study Guide



SPECIFIC OUTCOME

N6. Demonstrate an understanding of multiplying anddividing positive fractions and mixed numbers,concretely, pictorially, and symbolically.[C, CN, ME, PS]


problem involving positive fractions. • Estimate the product of two given positive proper

fractions to determine if the product will be closer to0, , or 1.

• Estimate the quotient of two given positive fractionsand compare the estimate to whole-numberbenchmarks.

• Express a given positive mixed number as animproper fraction and a given positive improperfraction as a mixed number.

• Model multiplication of a positive fraction by a wholenumber concretely or pictorially and record the process.


• Model division of a positive proper fraction by awhole number concretely or pictorially and record theprocess.




12

Preparation and PlanningMaterials • counters, 50 per student

• scissors, 1 per student

Masters • Chapter Review—Frequently Asked Questions p. 101

• Chapter Review—Study Guide p. 102• Chapter 2 Test pp. 103–105• Fraction Strips Tower, Masters Booklet p. 45• 2 cm Grid Paper, Masters Booklet p. 34

Extra Practice Workbook pp. 21–22


Chapter Review STUDENT BOOK PAGES 97–100

Name: Date:

Chapter 2 Test Page 3

17. Sketch a model to show � � .

18. Explain how you know that � has the same quotient as � .

19. Calculate.

a) � c) �

b) � d) �

20. What fraction calculation can you use to determine the number of quarters in $3.75?

21. Taira used of her flour to make of a batch of bannock. How much of her flour

would she need to make a whole batch?

22. Which expression has the greatest value? How do you know?

a) � � �

b) � � �

c) � � �

23. Where can you place brackets to make this equation true?

4 � � � � 24

24. Use fractions to explain why 3.2 * 0.4 = 1.28.

16

14

34

49

14

12

23

49

14ba1

223

49

14b1

2a2

3

25

34

310

58

16

23

16

35

27

67

57

57

39

59

312

14

78


Using the Chapter ReviewUse these pages to consolidate and assess students’understanding of the concepts developed in the chapter.The Practice questions can be used for assessment oflearning. Refer to the assessment chart on pages 84–88 forthe details of each question.

Alternatively, use the Practice questions as a practice test,and then administer Chapter 2 Test pp. 103–105. Thescoring guides and rubrics provided for the Practice questionscan also be used for the test questions: each question on thetest corresponds to the Practice question of the same number.

Frequently Asked Questions(Individual/Small Groups)

Have students read the Frequently Asked Questions onStudent Book pages 97 and 98 and create a new examplefor each question in their own notes. Then have studentssummarize the answers to the FAQs in their own words, as a way of reflecting on the concepts.

Alternatively, have students complete Chapter Review—Frequently Asked Questions p. 101 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.

83Chapter ReviewCopyright © 2008 Nelson Education Ltd.

Name: Date:


8. Which products are closer to than either 0 or 1?

a) � b) �

9. Draw two models to represent 2 � 3 .

10. Calculate. Express the answer as an improper fraction.

a) 1 � b) 3 � 2

11. The height of a window is 1 times its width. The window is 48 cm wide. How tall isthe window?

12. Draw a diagram to show � 4 � .

13. Calculate.

a) � 2 b) � 5

14. Explain why � 3 is a fraction that can be written with a denominator of 15.

15. Which quotients are between 2 and 3?

a) �

b) �

c) 1 �

16. What fractions might you use to estimate � ?29

1532

78

910

13

78

23

45

.

5

23

68

29

89

34

23

16

79

45

12

35

23

57

79

14

12


Name: Date:

Chapter Review—Frequently Asked QuestionsSTUDENT BOOK PAGES 97–100

Q: How can you divide a fraction by a whole number?

A:

Q: How can you divide two fractions?

A:

Q: In what order do you perform a series of fraction calculations?

A:


Chapter Review—Frequently AskedQuestions p. 101

Name: Date:

Chapter Review—Study Guide

A. Calculate: � 8 � ?

Use fraction strips.

B. Use a common denominator to calculate � .

� �

C. Use multiplication to calculate � .

� � �

D. Calculate using the rules for order of operations: � � � � �

First, evaluate the brackets: � �

Next, perform multiplication and division: � �

� �

Finally, perform addition and subtraction: � � � 3

8

516

34

12

54ba3

4

54ba3

45

1634

12

38

49

23

49

23

49

2 � 5 �

3 � 4 �

25

34

45


15

15

15

15

15

Chapter Review—Study Guide p. 102

112

14

15

18

19

110

111

112

13

16

16

16

16

16

17

17

17

17

17

17

18

18

18

18

18

18

18

19

19

19

19

19

19

19

19

110

110

110

110

110

110

110

110

110

111

111

111

111

111

111

111

111

111

111

112

112

112

112

112

112

112

112

112

112

112

12

13

13

14

14

14

15

15

15

15

16

17





Name: Date:


1. Draw a model to represent 4 � .

2. Calculate each. Express the answer as a whole or mixed number.

a) 5 � � _____ c) 10 � � _____

b) 7 � � _____ d) 8 � � _____

3. Calculate the value of each expression.

a) of � _____ b) of � _____

4. What is the missing fraction in each sentence?

a) is of _____ c) of is _____

b) is of _____ d) of is _____


6. Calculate.

a) � � _________ b) � � _________

7. In Ryan’s city, it rained of the days in September. More than 2 cm of rain fell ofthose days. What fraction of the days in September had rainfall of more than 2 cm?

14

25

34

59

16

45

1528

34

57

59

79

23

16

13

35

12

37

45

38

35

13

37

25

56

34

23


Chapter 2 Test pp. 103–105

Practice (Individual)

Most students will probably be able to complete Questions 1to 8 in class. Assign the rest for homework. Encouragestudents to calculate answers where appropriate, but makefraction strips, scissors, counters, and grids available.3. & 6. Tell students to write the answers in lowest terms.

Follow-Up and Preparation for Next ClassTo summarize students’ learning for the chapter, use ChapterReview—Study Guide p. 102 to focus a class discussion. Youmay want to have students complete the guide individually,in pairs, or in small groups first.


Assessment of Learning—What to Look for in Student WorkQuestion 1, model Specific Outcome and Process Focus: N6 [CN]• Draw a model to represent 3 * .5

6

Work meets standard Work meets standard Work meets acceptable Work does not yet meet of excellence of proficiency standard acceptable standard• makes insightful connections • makes meaningful connections • makes simple connections between • makes minimal or weak connections

between pictorial models and between pictorial models and pictorial models and fraction between pictorial models and fraction multiplication fraction multiplication multiplication fraction multiplication

Question 2, short answer Specific Outcome and Process Focus: N6 [CN, ME]• Calculate each. Express the answer as a whole or mixed number.

a) b) c) d)(Score 1 point for each correct answer, for a total out of 4.)

12 *239 *

276 *

358 *

45

Question 3, short answer Specific Outcome and Process Focus: N6 [R]• What is the value of each expression?

a) of b) of c) of d) of (Score 1 point for each correct answer, for a total out of 4.)

12

46

68

23

89

38

12

15

Question 4, short answer Specific Outcome and Process Focus: N6 [PS]• What is the missing fraction in each sentence?

a) b)(Score 1 point for each correct answer, for a total out of 2.)

. of 49 is 13

25 is 23 of .


Question 5, model Specific Outcome and Process Focus: N6 [CN]• Sketch a model for this calculation.

34 *

25 =

620

Question 6, short answer Specific Outcome and Process Focus: N6 [ME]• Calculate.

a) b) c) d)


15 *

57

58 *

23

37 *

25

29 *

27

Question 7, short answer Specific Outcome and Process Focus: N6 [PS]• About of the students in Andee’s school come by bus. About of these students are on the bus for more than an hour and a half each day. What fraction of the

students in Andee’s school are on the bus for more than an hour and a half each day?

(Score correct answer out of 2—1 point for process and 1 point for answer.)

13

23

Question 9, models Specific Outcome and Process Focus: N6 [CN]• Draw two models to represent .1 3

4 * 2 15

Question 8, written answer Specific Outcome and Process Focus: N6 [ME, CN]• Which products are closer to than to 0 or 1?

a) b) c) d) 35 *

23

39 *

89

16 *

78

34 *

56

12

Question 10, short answer Specific Outcome and Process Focus: N6 [R]• Calculate. Express the answer as an improper fraction.

a) b)


3 14 * 2 2

52 13 *

25


Work meets standard Work meets standard Work meets acceptable Work does not yet meet of excellence of proficiency standard acceptable standard• makes insightful connections • makes meaningful connections •makes simple connections between • makes minimal or weak connections

between pictorial models and between pictorial models and pictorial models and fraction between pictorial models andfraction multiplication fraction multiplication multiplication fraction multiplication

Work meets standard Work meets standard Work meets acceptable Work does not yet meet of excellence of proficiency standard acceptable standard• chooses an efficient and effective • chooses a workable and •chooses a familiar strategy • chooses a random or inappropriate

strategy (e.g., multiplying and reasonable strategy (e.g., fraction (e.g., number lines) to estimate a strategy to estimate a solutioncomparing fractions) to estimate strips) to estimate a solution solution, even though it might not be a solution the most appropriate

• makes insightful connections • makes meaningful connections • makes simple connections between • makes minimal or weak connections between estimating products and between estimating products and estimating products and comparing between estimating products andcomparing and ordering fractions comparing and ordering fractions and ordering fractions comparing and ordering fractions

Work meets standard Work meets standard Work meets acceptable Work does not yet meet of excellence of proficiency standard acceptable standard• makes insightful connections • makes meaningful connections •makes simple connections between •makes minimal or weak connections

between pictorial models and between pictorial models and pictorial models and fraction between pictorial models and fractionfraction multiplication fraction multiplication multiplication multiplication




Question 13, short answer Specific Outcome and Process Focus: N6 [ME]• Calculate.

a) b) c)


45 , 69

10 , 2910 , 3

Question 15, written answer Specific Outcome and Process Focus: N6 [ME]• Which quotients are between 4 and 6?

a) b) c) 3 12 ,

34

910 ,

18

910 ,

15

Question 16, short answer Specific Outcome and Process Focus: N6 [ME]• What fractions might you use to estimate ?7

16 ,620

Question 14, written answer Specific Outcome and Process Focus: N6 [C]• Explain why is a fraction that can be written with a denominator of 6..

2 , 3


between pictorial models between pictorial models and pictorial models and fraction division between pictorial models and fraction and fraction division fraction division division

Work meets standard Work meets standard Work meets acceptable Work does not yet meet of excellence of proficiency standard acceptable standard• provides a precise and insightful • provides a clear and logical • provides a partially clear explanation • provides a vague and/or inaccurate

explanation of mathematical explanation of mathematical of mathematical concepts (i.e., fraction explanation of mathematicalconcepts (i.e., fraction concepts (i.e., fraction division) division) and/or procedures concepts (i.e., fraction division) division) and/or procedures and/or procedures and/or procedures

Work meets standard Work meets standard Work meets acceptable Work does not yet meet of excellence of proficiency standard acceptable standard• chooses an efficient and effective • chooses a workable and reasonable • chooses a familiar strategy • chooses a random or inappropriate

strategy (e.g., compatible numbers) strategy to estimate a solution (e.g., pictorial models) to estimate a strategy to estimate a solutionto estimate a solution solution, even though it might not be

the most appropriate

Work meets standard Work meets standard Work meets acceptable Work does not yet meet of excellence of proficiency standard acceptable standard• chooses an efficient and effective • chooses a workable and • chooses a familiar strategy • chooses a random or inappropriate

strategy (e.g., compatible reasonable strategy to estimate (e.g., pictorial models) to estimate a strategy to estimate a solutionnumbers) to estimate a solution a solution solution, even though it might not be

the most appropriate

Question 11, short answer Specific Outcome and Process Focus: N6 [PS]• The supermarket has times as many employees just before dinnertime as in the late morning. There are 18 employees in the late morning. How many

employees are there just before dinnertime?


2 12

Question 12, diagram Specific Outcome and Process Focus: N6 [CN]• Draw a diagram to show that .6

8 , 3 =28



Question 17, model Specific Outcome and Process Focus: N6 [CN]• Sketch a model to show .5

6 ,13 = 2 1

2

Question 18, written answer Specific Outcome and Process Focus: N6 [C]• Explain how you know that has the same quotient as .4

5 ,35

46 ,

36

Question 19, short answer Specific Outcome and Process Focus: N6 [R]• Calculate.

a) b) c) d)


38 ,

29

56 ,

14

58 ,

14

56 ,

16

Question 20, short answer Specific Outcome and Process Focus: N6 [CN]• What fraction calculation can you use to determine the number of quarters in $4.50?

Question 21, short answer Specific Outcome and Process Focus: N6 [PS]• Pia used of her sugar to make of a batch of cookies. How much of her sugar would she have needed to make a whole batch?


34

23

Question 22, written answer, short answer Specific Outcome and Process Focus: N6 [C, ME]• Which expression has the greater value? How do you know?

a) b) 45 * A 23 -

15 B *

58

45 *

23 -

15 *

58


between pictorial models between pictorial models and pictorial models and fraction division between pictorial models and and fraction division fraction division fraction division


explanation of mathematical explanation of mathematical of mathematical concepts (i.e., division explanation of mathematical concepts (i.e., division of fractions concepts (i.e., division of fractions of fractions with a common concepts (i.e., division of fractions with a common denominator) with a common denominator) denominator) and procedures with a common denominator) and procedures and procedures and procedures

Work meets standard Work meets standard Work meets acceptable Work does not yet meet of excellence of proficiency standard acceptable standard• demonstrates a sophisticated • demonstrates a consistent ability • demonstrates some ability to • demonstrates a limited ability to

ability to transfer knowledge to transfer knowledge and skills transfer knowledge and skills transfer knowledge and skills and skills to new contexts to new contexts to new contexts to new contexts


explanation of mathematical explanation of mathematical of mathematical concepts (i.e., order explanation of mathematical concepts (i.e., order of operations) concepts (i.e., order of operations) of operations) and/or procedures concepts (i.e., order of operations) and/or procedures and/or procedures and/or procedures

• demonstrates computational • demonstrates computational • demonstrates computational fluency • has difficulty in demonstrating fluency that is efficient fluency that is workable that is routine and familiar computational fluency and must and flexible and understood work through procedures


Chapter 2


Question 23, short answer Specific Outcome and Process Focus: N6 [R]• Where can you place brackets to make this equation true?

(Score correct answer out of 1.)

3 *23 +

13 ,

14 = 12

Question 24, written answer Specific Outcome and Process Focus: N6 [C, CN]• Use fractions to explain why 4.5 � 0.5 equals 2.25.

Work meets standard Work meets standard Work meets acceptable Work does not yet meet of excellence of proficiency standard acceptable standard• provides a precise and • provides a clear and • provides a partially clear • provides a vague and/or

insightful explanation of logical explanation of explanation of mathematical inaccurate explanation mathematical concepts mathematical concepts concepts (i.e., fraction of mathematical concepts (i.e., fraction (i.e., fraction multiplication) and/or (i.e., fraction multiplication) multiplication) and/or multiplication) and/or procedures and/or proceduresprocedures procedures

• makes insightful • makes meaningful • makes simple connections • makes minimal or weak connections between connections between between fraction connections between fraction fraction multiplication fraction multiplication and multiplication and decimal multiplication and decimal and decimal decimal multiplication multiplication multiplicationmultiplication



89Chapter Task: Computer GizmosCopyright © 2008 Nelson Education Ltd.

SPECIFIC OUTCOME



problem involving positive fractions. • Generalize and apply rules for multiplying and

dividing positive fractions, including mixed numbers. • Solve a given problem involving positive fractions

taking into consideration order of operations (limitedto problems with positive solutions).


50–55 min Using the task

Materials • Optional: counters, 50 per student• Optional: scissors, 1 per student

Masters • Chapter 2 Task pp. 106–107• Optional: Fraction Strips Tower,

Masters Booklet p. 45• Optional: 2 cm Grid Paper, Masters

Booklet p. 34


Name: Date:


Chapter 2 Task Page 2

C. Compare the journal file to all the others by indicating what fraction of the smaller file it is.

D. How much memory is still available for the book report file compared to the science project file?

Why is there more than one way to answer this question?

E. What other fraction comparisons related to these files can you make?

Science project

Short story

Book report

Chapter 2 Task pp. 106–107

Optional:Fraction Strips Tower,Masters Booklet p. 45

112

14

15

18

19

110

111

112

13

16

16

16

16

16

17

17

17

17

17

17

18

18

18

18

18

18

18

19

19

19

19

19

19

19

19

110

110

110

110

110

110

110

110

110

111

111

111

111

111

111

111

111

111

111

112

112

112

112

112

112

112

112

112

112

112

12

13

13

14

14

14

15

15

15

15

16

17


Optional: 2 cm Grid Paper,Masters Booklet p. 34


Using the Chapter TaskUse this task as an opportunity to assess students’ under-standing of the concepts developed in the chapter and theirability to apply them in a rich problem-solving situation.Refer to the assessment chart on page 91 for the details ofeach part of the task.

Introduction ➧ 5–10 min

(Whole Class)

Discuss with students the different ways they can compare fractions. For example, to compare and , they can say that

is more than , because ; they can say that is

1 times , because ; or they can say that is of , because .1

2 ,34 =

23

34

23

12

34 ,

12 =

32

12

12

34

34 -

12 =

14

12

14

34

34

12

Chapter TaskComputer Gizmos


Name: Date:


Did you use appropriatestrategies to compare file sizes?Did you use appropriateoperations to compare file sizes?Did you use visuals, words, andsymbols to explain yourthinking?Did you explain your thinkingclearly?

Task Checklist


Computer GizmosSTUDENT BOOK PAGE 101

Brian likes to write mini-applications for his computer. One application automatically displays a bar to show what fraction of a megabyte of memory a file is using at any point in time.

? How can you describe how the sizes of Brian’s files compare?

Read the Task Checklist at the top of the pagebefore you begin.

A. Which of Brian’s files have room to be doubled before they reach the 1 MB mark? How do you know?

B. Compare the science project file to all the others by indicating what fraction of thelarger file it is.

Journal

Short story

Book report

Megabyte Minder

File name Memory (in MB)

Scienceproject

Journal

Short story

Bookreport

13782556

Using the Task ➧ 50–55 min

(Whole Class/Individual)

Together, read all the information on Student Book page 101,including the central question. Have students copy the chartinto their notebooks. Remind students to divide each memorybar into the number of parts indicated by the denominator ofthe fraction for that bar, and shade in the number of partsindicated by the numerator.

For Prompt A, students must determine which files haveroom to be doubled before they hit the 1 MB mark. A varietyof approaches—including estimating, comparing, andcalculating—are possible. For example, students may multiplythe size of each file by 2 and then compare to 1 MB, orcompare each fraction to MB.

For Prompts B and C, ensure that students are expressingone file as a fraction of another file, rather than finding thedifference in the file sizes.

For Prompt D, discuss with students the different waysin which they compared the file sizes to help them see thatmore than one comparison is possible.

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 StudentBook page; however, most will benefit from using Chapter 2Task pp. 106–107 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.

Adapting the TaskYou can adapt the task in the Student Book to suit the needsof your students. For example,• Use the master Chapter 2 Task pp. 106–107.• Have students work in pairs or small groups.• Have students use pictorial models such as fraction strips

and grids to compare fractions.• Challenge students to find the size of each file as a fraction

of the total size of all four files.• Students can look at the sizes of different files on their own

computer and write fraction comparisons related to those files.

Answers to PromptsA. The science project file and the short story file, because

and , or and , and and .

B. The science project file is of the journal file.

The science project file is of the short story file.13 ,

25 =

13 *

52 =

56

56

13 ,

78 =

824 ,

2124 =

821

821

45 6 12

5 * 2 =45

23 6 11

3 * 2 =23

25 6

12

13 6

12

12

The science project file is of the book report file.

C. The journal file is of the science project file.

The journal file is of the short story file.

The journal file is of the book report file.

D. There is MB of memory left for the science project andMB left for the book report.

So there is MB more room in the scienceproject file.

So, the book report has the amount of memory leftcompared with the science project.

So, the science project has 4 times as much memory leftas the book report does.There is more than one way to answer the questionbecause you can compare the first space as a fraction ofthe second, or the second as a fraction of the first. Youcan also subtract the amount of memory used by thebook report file from the amount of memory used bythe science project file to determine how much morespace is left for the book report.

E. For example, you could add file sizes to compare thesize of two files to one e.g., or you couldmultiply a file size to compare it to another file if it weredoubled or tripled in size.

A13 +25 B ,

78 BA

23 ,

16 =

23 *

61 =

123 = 4

14

16 ,

23 =

16 *

32 =

312 =

14

23 -

16 =

12

16

23

.

.=

105 , 5120 , 6 =

2120, or 1 1

20

56 *

.

.=

3540 =

105120

56 *

.

.=

78

1 120

78 ,

25 =

78 *

52 =

3516, or 2 3

16

2 316

78 ,

13 =

78 * 3 =

218 = 25

8

258

13 ,

56 =

13 *

65 =

615 =

25

25


91Chapter Task: Computer GizmosCopyright © 2008 Nelson Education Ltd.

Assessment of Learning—What to Look for in Student WorkWork meets standard Work meets standard Work meets acceptable Work does not yet meet

Outcomes of excellence of proficiency standard acceptable standard

• demonstrates computational • demonstrates computational • demonstrates computational • has difficulty in fluency that is efficient fluency that is workable and fluency that is routine demonstrating computationaland flexible understood and familiar fluency and must work

through procedures

• chooses an efficient and • chooses a workable and • chooses a familiar strategy • chooses a random or effective strategy to reasonable strategy to to estimate a solution, even inappropriate strategy to estimate a solution estimate a solution though it might not be the estimate a solution

most appropriate

• makes insightful connections • makes meaningful connections • makes simple connections • makes minimal or weak between real-world contexts between real-world contexts between real-world contexts connections between and fraction operations and fraction operations and fraction operations real-world contexts and

fraction operations

• demonstrates a sophisticated • demonstrates a consistent • demonstrates some ability to • demonstrates a limited ability to transfer knowledge ability to transfer knowledge transfer knowledge and skills ability to transfer knowledgeand skills to new contexts and skills to new contexts to new contexts and skills to new contexts

• provides a precise and • provides a clear and logical • provides a partially clear • provides a vague and/or insightful explanation of explanation of mathematical explanation of mathematical inaccurate explanation of mathematical concepts concepts and/or procedures concepts and/or procedures mathematical concepts and/or procedures and/or procedures

Prompt A(Mental Mathematicsand Estimation) N6. Demonstrate anunderstanding ofmultiplying and dividingpositive fractions andmixed numbers,concretely, pictorially,and symbolically. [C, CN, ME, PS]

Prompts B, C, D, & E(Connections)N6

Prompt D(Communication)N6


Family Letter

Dear Parent/Caregiver:

Over the next three weeks, your child will be learning how to multiply and dividefractions and mixed numbers. Several models will be used to build understandingbefore the introduction of formal procedures. Your child will learn number strategiesthat he/she can apply to solving real-world problems that are relevant to his/her life.

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

• Involve your child in planning a family meal recipe. Your child can determine theamounts of ingredients that would be needed to make two or three times the recipe,or to make half the recipe.

• Have your child examine use of time daily or weekly. For example, ask your child todetermine in fractions how much of the day was spent at school, sleeping, practisingsports and/or music, doing homework, doing chores, etc. Your child can thencalculate how much time is spent monthly or annually on each of these tasks or, forexample, determine how many days or weeks it will take to spend 100 hours oneach task.

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

Chapter Chapter Chapter 72

Name: Date:


yellow

yellow

green

green

yellow

green

blue

red


A. The area of one yellow block is 1 unit.1 yellow block � 1

a) Which colour block fills of the area of

the yellow block?

1 _______ block �

b) Which colour block fills of the area of the yellow block?

1 _______ block �

c) Which colour block fills of the area of the yellow block?

1 _______ block �

B. The equation 3 � � 3 tells the sum of the areas

of two of the colours.

Which are the two colours?

Which colour block has a total area of 3?

_______ blocks � 3

Shade the area.

Which colour block has a total area of ?

_______ blocks �

Shade the area.

What is the total of the area you have shaded? ______

How can you show that this area is also 3 ?

___________________________________________________

___________________________________________________

12

36

36

12

36

13

13

16

16

12

12

Chapter 2

Name: Date:


yellow

yellow

green

green

yellow

green

blue

red


C. The equation 3 � � 2 describes how much more of the design is one colour than

another.

a) Which colour block has a total area of ? _________

b) How many more blocks would you need to have a total area of 3? _________

c) Which colour block represents the total area of 3?_________

d) So, 3 � � 2 shows the difference between which two colours of the design?

D. Write equations with fractions and/or mixed numbers to describe each area, using the units in part A. Solve the equations. Show your work.

a) the red area and the blue area

• Colour in the hexagon to show the blue blocks and the red blocks from Allison’s design.

• How many sixths are filled?

red � blue � ______

Write an equation with fractions to describe the area of the red and blue parts.

___ � ____ �

b) How much more is green than blue?

• How many sixths are blue? _______

• How many sixths are green? _______

• green � blue � ______

• Write an equation with fractions to describe how much more is green than blue.

___ � ___ �

12

12

12

12

12

Name: Date:



c) Write an equation with fractions to describe how much is yellow and red.

___ � ___ �

Write an equation with fractions to describe how much is green and blue.

___ � ___ �

Write an equation with fractions to describe how much more is yellow and red than green and blue.

___ � ___ �

d) Write an equation with fractions to describe how much more is red than green.

___ � ___ �

E. Write three other fraction equations that describe areas in Alison’s design.

a) area:

equation:

b) area:

equation:

c) area:

equation:

F. Make your own design below using

• yellow, red, blue, and green pattern blocks

• a total of eight blocks

• at least two yellow blocks

• at least one block of each other colour

Name: Date:


a) area: red and blue

equation:

b) area: how much more is green than blue

equation:

c) area: how much more is yellow and red than green and blue

equation:

d) area: how much more is red than green

equation:

G. a) Is it possible to create a design where the yellow area is 1units greater than the blue area?

How many yellow blocks? _____


___ � ___ � 1

b) Is it possible to create a design where the blue and red area, together, is unit greater than the green area?


How many red blocks? _____

How many green blocks? _____

___ � ___ �

___ � ___ � 16

16

13

13


Name: Date:


Mid-Chapter Review—Frequently Asked QuestionsSTUDENT BOOK PAGES 64–67

Q: How can you multiply a fraction by a whole number?

A:

Q: How can you multiply two fractions less than 1?

A:

Q: How can you estimate the product of two fractions?

A:

Q: How can you multiply two mixed numbers?

A:

Name: Date:


1 2 3 4 5 6

15

15

15

15

15

Mid-Chapter Review—Study Guide

A. Show 4 � .

B. Show � .

C. Complete the area model to show 2 � 1 .12

14

25

13

23

Name: Date:


23

15

58

34

25

310

÷ ÷ ×

Fractions and Operations Cards ILesson 2.10: Order of OperationsSTUDENT BOOK PAGES 88–91

Name: Date:


× −

+

−

(

( (

) +

Fractions and Operations Cards IILesson 2.10: Order of OperationsSTUDENT BOOK PAGES 88–91

Name: Date:

Chapter Review—Frequently Asked QuestionsSTUDENT BOOK PAGES 97–100

Q: How can you divide a fraction by a whole number?

A:

Q: How can you divide two fractions?

A:

Q: In what order do you perform a series of fraction calculations?

A:


Name: Date:

Chapter Review—Study Guide

A. Calculate: � 8 � ?

Use fraction strips.

B. Use a common denominator to calculate � .

� �

C. Use multiplication to calculate � .

� � �

D. Calculate using the rules for order of operations: � � � � �

First, evaluate the brackets: � �

Next, perform multiplication and division: � �

� �

Finally, perform addition and subtraction: � � � 3

8

516

34

12

54ba3

4

54ba3

4516

34

12

38

49

23

49

23

49

2 � 5 �

3 � 4 �

25

34

45


15

15

15

15

15

Name: Date:


1. Draw a model to represent 4 � .

2. Calculate each. Express the answer as a whole or mixed number.

a) 5 � � _____ c) 10 � � _____

b) 7 � � _____ d) 8 � � _____

3. Calculate the value of each expression.

a) of � _____ b) of � _____

4. What is the missing fraction in each sentence?

a) is of _____ c) of is _____

b) is of _____ d) of is _____


6. Calculate.

a) � � _________ b) � � _________

7. In Ryan’s city, it rained of the days in September. More than 2 cm of rain fell ofthose days. What fraction of the days in September had rainfall of more than 2 cm?

14

25

34

59

16

45

1528

34

57

59

79

�

�

�

�

23

16

13

35

�

�

�

�

12

37

45

38

35

13

37

25

56

34

23


Name: Date:


8. Which products are closer to than either 0 or 1?

a) � b) �

9. Draw two models to represent 2 � 3 .

10. Calculate. Express the answer as an improper fraction.

a) 1 � b) 3 � 2

11. The height of a window is 1 times its width. The window is 48 cm wide. How tall isthe window?

12. Draw a diagram to show � 4 � .

13. Calculate.

a) � 2 b) � 5

14. Explain why � 3 is a fraction that can be written with a denominator of 15.

15. Which quotients are between 2 and 3?

a) �

b) �

c) 1 �

16. What fractions might you use to estimate � ?29

1532

78

910

13

78

23

45

.

5

23

68

29

89

34

23

16

79

45

12

35

23

57

79

14

12


Name: Date:



18. Explain how you know that � has the same quotient as � .

19. Calculate.

a) � c) �

b) � d) �

20. What fraction calculation can you use to determine the number of quarters in $3.75?

21. Taira used of her flour to make of a batch of bannock. How much of her flour

would she need to make a whole batch?

22. Which expression has the greatest value? How do you know?

a) � � �

b) � � �

c) � � �

23. Where can you place brackets to make this equation true?

4 � � � � 24

24. Use fractions to explain why 3.2 * 0.4 = 1.28.

16

14

34

49

14

12

23

49

14ba1

223

49

14b1

2a2

3

25

34

310

58

16

23

16

35

27

67

57

57

39

59

312

14

78


Name: Date:


Did you use appropriatestrategies to compare file sizes?Did you use appropriateoperations to compare file sizes?Did you use visuals, words, andsymbols to explain yourthinking?Did you explain your thinkingclearly?

Task Checklist


Computer GizmosSTUDENT BOOK PAGE 101

Brian likes to write mini-applications for his computer. One application automatically displays a bar to show what fraction of a megabyte of memory a file is using at any point in time.

? How can you describe how the sizes of Brian’s files compare?

Read the Task Checklist at the top of the pagebefore you begin.

A. Which of Brian’s files have room to be doubled before they reach the 1 MB mark? How do you know?

B. Compare the science project file to all the others by indicating what fraction of thelarger file it is.

Journal

Short story

Book report

Megabyte Minder

File name Memory (in MB)

Scienceproject

Journal

Short story

Bookreport

13782556

Name: Date:



C. Compare the journal file to all the others by indicating what fraction of the smaller file it is.

D. How much memory is still available for the book report file compared to the science project file?

Why is there more than one way to answer this question?

E. What other fraction comparisons related to these files can you make?

Science project

Short story

Book report


yellow

yellow

green

green

yellow

green

blue

red

0 1 2 3 4 5

15

15

15

15

15

2

1

12

14

2 x 1 = 2

2 x = 212

14

14

x 1 =

x = 212

14

15

15

15

15

15

Answers for Chapter 2 MastersProblem of the Week p. 3

1. hectare; � � � � � �

2.

1 � � � �

1 � � � �

� � � � �

3. 4 h� � � � � � 4

Scaffolding for Getting Started Activity pp. 93–96

A. a) red b) green c) blue

B. yellow blocks � 3

green blocks �

C. a) red b) 5 red blocks

c) yellow d) yellow and redD. a)

; � �

b) 2; 3; 1; � � � �

c) 3 � � 3 ; � � ;

3 � � 2 or 2

d) � � 0

E. a) For example,

area: yellow and blue

equation: 3 � � 3

b) For example,

area: how much more is red than blue

equation: � �

c) For example,

area: how much more is yellow than any other colour

equation: 3 � � � � 1 23

13

36

12

16

13

12

13

13

36

12

23

46

56

12

56

13

36

12

12

16

26

36

13

12

56

13

12

56

36

16842

12

127

143

12

a423

,

712b

2741

540820

6021

940

2160

940

2160

315

74

315

34

940

320

32

320

12

2741

5192

112

516

14

13

12

58

5192

F. For example,

a) For example, equation: 1 � � 1

b) For example, equation: � � 0

c) For example, equation:

d) For example, equation: 1 � � 1

G. a) 2; 2; 2 � � 1

b) 1; 1; 4; � � ; � �

Mid-Chapter Review—Study Guide p. 98

A.

B.

C.

Chapter Review—Study Guide p. 102

A. � 8 �

B. � � �

� �

� , or 1 78

158

820

1520

2 � 45 � 4

3 � 54 � 5

25

34

110

45

16

46

56

56

12

13

13

26

16

26

12

a2 +

32b - a1

3+

26b = 2

56

13

26

56

13

12


C. � � � � �

D. � � � �

� � � � �

� � �

�

Chapter 2 Test pp. 103–105

1. For example,

4 � � � 2

2. a) 5 � � � � � � � � 3

b) 7 � � � � � � � � � � 2

c) 10 � � � � � � � � � � � � � 8 � 8

d) 8 � � � � � � � � � � � 3

3. a) of � � � � b) of � � � �

4. a) , since of is

b) or , since of is

c) , since of is d) , since of is

5. For example,

6. a) � � � b) � � �

7. � � � ; so of days in September had more than 2 cm of rainfall.

8. a) � � b) � �

B is closer to than to either 0 or 1.12

1021

23

57

736

79

14

110

110

220

14

25

512

1536

34

59

215

430

16

45

59

79

57

57

13

35

59

59

16

14

23

14

312

37

67

12

67

310

1240

45

38

45

38

15

315

35

13

35

13

37

247

37

37

37

37

37

37

37

37

37

13

26

506

56

56

56

56

56

56

56

56

56

56

56

45

145

25

25

25

25

25

25

25

25

34

154

34

34

34

34

34

34

23

83

23

18

58

38

38

21

516

34

12

38

a34

+

54b5

1634

12

38

23

1218

32

49

23

49


62 1

3 12

35

310

95

3

9. For example,

91 squares, each with an area of , gives an area of , or 9 .

10. a) 1 � � � � � b) 3 � 2 � � � �

11. The window is 84 cm tall. 1 � 48 � � 48 � � 84

12. For example,

13. a) � 2 � b) � 5 �

14. For example, you are dividing fifths into three parts, or 15ths.15. b) and c)

16. �

17.

18. For example, to divide two fractions with a common denominator, you only need to divide the numerators.So, � � and � �

19. a) � � 6 � 2 � 3 c) � � � � � 3

b) � � � � � 4 d) � � � � � � 2

20. 3 � � � � 15

21. 1 cups

� � � � � 1

Taira needs 1 cups of flour to make a whole batch.

22. Expression C has the greatest value. I calculated each answer with a denominator of 27 and thencompared the numerators.

a) � � � � � � � b) � � � � � � �

c) � � � � � � � �

23. 4 � � � � 24

24. 3.2 � 3 �

0.4 �

3.2 � 0.4 � � � � 1.28128100

410

3210

410

3210

210

16

14ba3

4

627

29

836

436

26

49

14

12

23

227

49

14

23

49

14ba1

223

127

49

14ba1

349

14b1

2a2

3

78

78

158

52

34

25

34

78

14

154

14

34

112

2512

5024

103

58

310

58

123

61

23

16

23

35

185

61

35

16

35

27

67

53

37

57

53

39

59

14

12

215

23

38

68

3364

74

34

769

15218

83

196

23

16

75

6345

79

95

79

45

110

9110

110

12

1 2 3

92

15

21

Nelson The Teachers’ Choice for Student Success!

96 % of pilot teachers recommend Math Focus

to their colleagues.

For more information about Nelson Math Focus, please contact your Nelson Education Representative.

Here’s What They Are Saying: “I would recommend it because it is set up for both students and teachers to succeed!”

“After doing an extensive comparison of the content/layout of each of the three approved resources, Nelson Math Focus is the best of the bunch.”

“I think it is a well-designed, well-aligned textbook.”

“Good coverage of curriculum in an easy-to-follow manner. Really like the workbook for extra at-home follow-up.”

“I like the Teacher’s Resource. It is especially helpful to have possible answers given to questions as well as rating the various student responses to questions which is used to determine the level of achievement.”

“It uses plenty of real-life examples. It is easy to use and covers all of the curriculum areas.”

“Nelson Education has provided us the resources and supports needed to actively engage our teachers in professional learning connected to the new approaches to mathematics instruction as well as providing them the tools needed to actively engage the students in math.”

chapter 2: fraction operations - · pdf filechapter 2: fraction operations ... number . . . ....

Documents