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3#11125 (i2180)—Algebra Readiness, Teacher Resource Guide

Table of Contents

Program Research and SupportActive Algebra Foreword. . . . . . . . . . . . . . . . . . . 4

Algebra Readiness Research Base. . . . . . . . . . . . 7

References Cited . . . . . . . . . . . . . . . . . . . . . . . . 13

Correlation to NCTM Standards . . . . . . . . . . . . 14

Program Organization and StructureAlgebra Readiness Table of Contents . . . . . . . . 22

How to Use This Program . . . . . . . . . . . . . . . . . 25

Instructional Planning and SupportPlanning Your Algebra Readiness Program . . . . 28

Algebra Readiness Time Line . . . . . . . . . . . . . . 32

Alternative Instructional Settings . . . . . . . . . . . 33

Algebra Readiness Action Plan . . . . . . . . . . . . . 35

Classroom ManagementInvolving Parents . . . . . . . . . . . . . . . . . . . . . . . 36

Common Classroom Management Issues . . . . . 38

Management of Classroom Manipulatives . . . . 40

Strategies for Universal AccessGrouping Strategies for the Classroom. . . . . . . 47

Differentiated Learning . . . . . . . . . . . . . . . . . . . 50

Vocabulary Development for English Language Learners . . . . . . . . . . . . . . . . . . . . 54

Developing Mathematical Reasoning Through Problem Solving . . . . . . . . . . . . . . . 75

Strategies to Engage Students . . . . . . . . . . . . . 77

Maximizing Instructional Minutes. . . . . . . . . . . 81

AssessmentTypes of Assessments . . . . . . . . . . . . . . . . . . . . 83

Grading Procedures. . . . . . . . . . . . . . . . . . . . . . 85

Data-Driven Instruction. . . . . . . . . . . . . . . . . . . 93

Diagnostic Test Item Analysis . . . . . . . . . . . . . 100

TechnologyTeacher Resource CD. . . . . . . . . . . . . . . . . . . . 112

Graphing Calculator Methods. . . . . . . . . . . . . 115

Games and ManipulativesDirections for the Games . . . . . . . . . . . . . . . . 124

Games and Manipulatives Matrix . . . . . . . . . . 128

Preparing Cups and Chips. . . . . . . . . . . . . . . . 128

AppendicesAppendix A: CD Filenames . . . . . . . . . . . . . . . 129

Appendix B: Contents of the Professional Development DVD . . . . . . . . . 133

Appendix C: Mathematical Terms Glossary . . . 138

Appendix D: Index. . . . . . . . . . . . . . . . . . . . . . 142

22

Program Organization and Structure

Algebra Readiness Table of ContentsUnit 1: Number Sense and Operations

Lesson 1: Algebra Readiness Introduction and Pretest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Lesson 2: Understanding Whole Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Lesson 3: Understanding Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Lesson 4: Adding and Subtracting Whole Numbers and Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Lesson 5: Multiplying and Dividing Whole Numbers and Decimals. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Lesson 6: Understanding Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Lesson 7: Multiplying and Dividing Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Lesson 8: Adding and Subtracting Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Lesson 9: Using the Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Lesson 10: Understanding Equals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Lesson 11: Number Sense and Operations Unit Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Problem-Solving Lesson 1: Using Simpler Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Unit 2: Algebraic Expressions and IntegersLesson 12: Writing Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Lesson 13: Properties and Order of Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Lesson 14: Order of Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Lesson 15: Adding Integers with Chips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Lesson 16: Multiplying and Dividing Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Lesson 17: Mixed Integers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Lesson 18: Integers Review 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Lesson 19: Integers Review 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Lesson 20: Collecting Like Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Lesson 21: Distributing and Collecting 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Lesson 22: Distributing and Collecting 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Lesson 23: Distributing and Collecting Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Lesson 24: Writing One-Variable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Lesson 25: Writing Expressions and Equations 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Lesson 26: Writing Expressions and Equations 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Lesson 27: Integers Unit Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Problem-Solving Lesson 2: Working Backwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Unit 3: Linear EquationsLesson 28: Solving Linear Equations with Cups and Chips 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Lesson 29: Solving Linear Equations with Cups and Chips 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Lesson 30: Solving Equations 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Lesson 31: Solving Equations 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Lesson 32: Solving Equations with Fractions 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Lesson 33: Solving Equations with Fractions 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

#11125 (i2180)—Algebra Readiness, Teacher Resource Guide

23

Unit 3: Linear Equations (cont.)

Lesson 34: Algebra Applications with Angles 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Lesson 35: Solving Equations with Fractions 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Lesson 36: Algebra Applications with Angles 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Lesson 37: Linear Equations Unit Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Problem-Solving Lesson 3: Using Logical Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Unit 4: Probability, Percent, and ProportionLesson 38: Percent of Region 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Lesson 39: Percent of Region 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Lesson 40: Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Lesson 41: Fraction-to-Decimal Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Lesson 42: Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Lesson 43: Percent and Probability 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Lesson 44: Percent and Probability 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Lesson 45: Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Lesson 46: Unit Review 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Lesson 47: Unit Review 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Lesson 48: Probability, Percent, and Proportion Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Problem-Solving Lesson 4: Creating a Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Unit 5: GraphingLesson 49: Plotting Points on a Coordinate Plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Lesson 50: Calculating Lengths on a Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Lesson 51: Coordinate Plane Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Lesson 52: Progress Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Lesson 53: Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Lesson 54: Equations as Relations 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Lesson 55: Equations as Relations 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Lesson 56: Lines and Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Lesson 57: Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Lesson 58: Writing Equations from Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Lesson 59: Mid-Unit Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Lesson 60: Slope 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Lesson 61: Slope 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Lesson 62: Slope 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Lesson 63: Scatter Plots and Lines of Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Lesson 64: Slope-Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Lesson 65: Slope Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Lesson 66: Graphing Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63



Algebra Readiness Table of Contents (cont.)

24

Unit 5: Graphing (cont.)

Lesson 67: Working with Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Lesson 68: Graphing Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Lesson 69: Graphing Unit Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Problem-Solving Lesson 5: Acting It Out or Using Concrete Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Unit 6: Inequalities and SystemsLesson 70: Graphing Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Lesson 71: Solving Multistep Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Lesson 72: Union and Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Lesson 73: Compound Inequalities 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Lesson 74 Compound Inequalities 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Lesson 75: Compound Inequalities 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Lesson 76: Absolute Value Inequalities 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Lesson 77: Absolute Value Inequalities 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Lesson 78: Comparing Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Lesson 79: Writing Two-Variable Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Lesson 80: Writing Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Lesson 81: Inequalities and Systems Unit Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Problem-Solving Lesson 6: Analyzing and Investigating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Unit 7: Symbolic NotationLesson 82: Understanding Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Lesson 83: Multiplying Monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Lesson 84: Dividing Monomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Lesson 85: Mixed Operations with Monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Lesson 86: Adding Polynomials with Algebra Tiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Lesson 87: Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Lesson 88: Using the Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Lesson 89: Pythagorean Triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Lesson 90: Simplifying Radical Expressions 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Lesson 91: Simplifying Radical Expressions 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Lesson 92: Simplifying Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Lesson 93: Symbolic Notation Unit Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Lesson 94: End-of-Year Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Lesson 95: Diagnostic Posttest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Problem-Solving Lesson 7: Creating a Tree Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55



Algebra Readiness Table of Contents (cont.)

27#11125 (i2180)—Algebra Readiness, Teacher Resource Guide

How to Use This Program (cont.)

Components of the Program


Teacher Resource GuideProgram research and support • Strategies for universal access • Assessmentsuggestions • Data-driven instruction charts • Classroom managementsuggestions • Outline of lessons for entire course • Graphing calculatorinformation • Steps for preparing games and manipulatives • Contents of the Teacher Resource CD • Contents of the Professional Development DVD

Teacher EditionsLesson plans • Mathematics content standards • Specific materials list • Step-by-step procedure • Notes and practice problems • Review • Reteaching suggestions • Teacher tips • Assessment appendix • Games appendix • Answer keys

Guided Practice Book (Student Edition)• All necessary activity sheets for the students are provided in the student

Guided Practice Book. There are page references to this book within thelessons. The activity sheets are also provided on the Teacher Resource CD.

• Call 888-333-4551 or visit http://www.tcmpub.com to order more copies of this consumable product.

Teacher Resource CD Templates for teacher resource sheets • PowerPointTM slide shows • Foundational skills practice sheets • Copies of Guided Practice Book pages •Targeted standards review sheets • Form A and Form B of assessments •Preparation materials for games

Professional Development DVDDemonstrations for how to complete lessons with manipulatives or games

30 #10824 (i1631)—Algebra Readiness, Teacher Resource Guide

Instructional Planning and Support

Algebra Readiness Time Line Program Type: _____________________________________________________________________

Times and Days of Instruction: _______________________________________________________

Program Length: ___________________________________________________________________

Day/Date Lesson Title Pages Adaptations or Notes

Lesson 2: Writing AlgebraicExpressions

13–15 Give your own assignment ashomework.

Example

Day 29/06/2006

filename: timeline.doc

Unit 1: Number Sense and Operations

Name _____________________________________

Understanding Whole Numbers1. Write the value of the underlined number.

5,222,2222 _________________ 222,225 _________________

252,222 _________________ 22,252 _________________

2,522 _________________ 5,222 _________________

2. Explain how you know the value of each 5.

__________________________________________________________

__________________________________________________________

Directions: Fill in each blank with the correct period name for the following numbers.

3. 5,246,842 =

5 ____________ 246 ______________ 842 ____________

4. 900,735,215 =

900___________ 735 ______________ 215 ____________

Directions: Write, in words, the name of the place value of the underlined digit. Then, identify the value of that digit.

14 #11000—Algebra Readiness, Guided Practice Book

5. 34,527

Place Value: _______________

Value of Digit: _____________

6. 832,576,104

Place Value: _______________

_________________________


7. 912,620,215

Place Value: _______________

_________________________


8. 264,173,003

Place Value: _______________

_________________________


Teacher Tips

Notes

Schedule

Assessment

Practice

Assignment

CooperativeGroups

Transparencyin a CD File

CD File

Lesson Plan Icons The followingicons are usedthroughout thelessons to guideteachers in theirplanning.

MathematicsGame

Lesson

35 Solving Equations with Fractions 3Linear Equations Unit

34

Materials• Guided Practice Book—Mixed Fractions (pages 76–77;

linear09.pdf)

• Transparencies folder on the CD—Writing Expressions andEquations 9 (trans15.pdf)

ProcedureStep 1 Assign the Mixed Fractions activity sheet.

• Complete the first ten problems with the students.

Step 2 Go over the Writing Expressions and Equations 9 transparency.

• Set up and solve the problems.

• Write a therefore statement to answer each question. Use thetherefore symbol (∴).

• Do this for all word problems throughout the year. It will helpstudents check for reasonableness and answer each problemfor the correct object.

Step 3 Review solving equations with fractions. Have students work ingroups of three to complete these problems. Then, as a class, goover the answers and address any questions or concerns. Answerscan be left as improper fractions.

a. x + 5 = 10 x =

b. 6p – (–8) = –4 p = –2

#10824 (i1626)—Algebra Readiness, Unit 3

Step 115 min.

Step 330 min.

Step 215 min.

Step 430 min.

152

23

CD File

Assignment

Cooperative

Groups

California Mathematics ObjectivesGrade 7, Algebra and Functions 4.1—Solve two-step linear equationsand inequalities in one variable over the rational numbers, interpret thesolution or solutions in the context from which they arose, and verify thereasonableness of the results.

Algebra I 5.0—Students solve multistep problems, including wordproblems, involving linear equations and linear inequalities in one variableand provide justification for each step.

Manipulatives Set• Any manipulatives used within the lessons are included in the manipulatives

set. There is a class set for 30 students if used as directed in the lessons.

• Call 888-333-4551 or visit http://www.tcmpub.com to order moremanipulatives sets, if needed.

Lesson

7Multiplying and Dividing FractionsNumber Sense and Operations Unit

49

Materials• PowerPoint folder on the CD—Multiplying and Dividing

Fractions (lessn07.ppt) (optional)

• Guided Practice Book—Multiplying and Dividing Fractions 1(page 23; numbrs08.pdf)

ProcedureStep 1 Review students’ understanding of fractions by having them solve

the following problems with partners.

a. The stock of the HIPO company trades on the New YorkStock Exchange. In October, the stock was 145 . InNovember, it was 145 , and in December, it was 145 . In which month was the stock’s value the greatest? Answer: December

b. Mei cut an apple pie into 5 pieces, and Bill ate 4 pieces.Laura cut an apple pie into 6 pieces, and Dayu ate 4 pieces.What part of a pie does each of them have left, and whohas the most pie left? Challenge: If they put the piecestogether, what fraction of a pie do they have altogether?Answer: Mei has of a pie, and Laura has of a pie. Laurahas more pie. Challenge: The total amount is of a pie.

Step 2 Explain to students that one-third of three-fourths means · .Model this by drawing a rectangle divided into four parts andshade in three of the parts. Draw a second rectangle with linesdividing the rectangle into three parts. Then, shade the partrepresenting of . (See the top of the next page for an exampleof this model.)

34

13

34

13

815

13

15

46

18

13


Lesson Description• Adds and subtracts fractions with unlike denominators; multiplies

and divides fractions. (McREL Mathematics Standard)

• Students will use models and the two different algorithms tomultiply and divide fractions. Students will simplify fractionexpressions.

Steps 1–430 min.

Steps 6–720 min.

Step 520 min.

Step 8 20 min.

Practice

Practice

Lesson

7 Multiplying and Dividing FractionsNumber Sense and Operations Unit

50

Procedure (cont.)

Step 2 (cont.)

• Ask the students to describe the answer from the secondrectangle ( ) or ( ). Ask them how to shade rectangles toshow · . Answer: Shade the top two rows.

Step 3 Present the Notes on Multiplying Fractions (pages 50–51).

• Model the steps as shown in problem c.• These notes are provided as part of the lesson’s PowerPoint

slide show on the CD (lessn07.ppt).

34

23

14

312


Notes on Multiplying Fractions

c. · 1

1 = =

Method 1 (No Canceling)

· =

72 = 2 · 2 · 2 · 3 · 3240 = 2 · 2 · 2 · 2 · 3 · 5

GCF = 2 · 2 · 2 · 3 = 24

= 310

72 ÷ 24240 ÷ 24

72240

2415

316

2415

(15 · 1 + 9)15

915

915

316 • Change 1 to an improper

fraction.

915

• Multiply numerator timesnumerator and denominator timesdenominator.

• Find the greatest common factor(GCF) of 72 and 240.

• Divide the numerator anddenominator by the GCF.

orPractice

Lesson


51

Step 4 Group the students into pairs. Have each student complete two ofthe problems below using Method 1 and two with Method 2 whilethe other person uses the other method on each. Have themcompare their methods and answers.

d. 1 · = = 3

e. · =

f. · =

g. · = = 1

Step 5 Present the Notes on Dividing Fractions (page 52).

• Model problems h and i, using the given steps. • Have volunteers work the problems j–l for the class.• These notes are provided as part of the lesson’s PowerPoint


180

8180

1825

4532

724

2164

5663

59

2539

2630

17

227

4016

935


Procedure (cont.)

Notes on Multiplying Fractions (cont.)

Method 2 (Canceling)

· = ·

· = 310

35

12

35

12

2415

316

Cooperative

Groups

• Look for the GCF of anynumerator and any denominator.Cancel (or reduce by) thecommon factors.

• Multiply numerator timesnumerator and denominatortimes denominator.

31

52

Lesson


52 #11125 (i2173)—Algebra Readiness, Unit 1

Notes on Dividing Fractions• How could you use a fraction to rewrite the problem 6 ÷ 2?

Answer: of 6

• So, 6 ÷ 2 = 6 ·

h. ÷ =

÷ = · = = 3

· = = = = 3

i. 3 ÷ 1

÷ = ·

· = or = =

= 2

• Have students solve these problems.

j. ÷ =

k. ÷ 1 =

l. 1 ÷ = = 5 128

14128

29

542

3235

38

4435

1325

4032

1320

710

2710

2710

108 ÷ 440 ÷ 4

27 · 48 · 5

2710

45

278

45

278

54

278

14

38

31

24 ÷ 88 ÷ 8

248

81

38

31

81

38

18

38

18

38

12

12

2

1

• Remember, 2 is the same as and that

2 and are reciprocals.12

21

• Take the reciprocal of thedivisor. Cancel and thenmultiply.

• Or, multiply, and thendivide by the GCF.

• Change the improperfraction to a mixednumber.

or

Lesson


53

Procedure (cont.)

Step 6 Present the following Notes on Simplifying FractionExpressions 1.

• Explain to students that the same rules apply to variables. • Also, explain to students that equal signs are used in these

problems to show that the expressions are equivalent.• Model problem m using the given steps. • Have students work in pairs to solve problems n and o.• These notes are provided as part of the lesson’s PowerPoint



Notes on Simplifying FractionExpressions 1

m. ·

Method 1· =

GCF = ag

=

Method 2· = ·

· =

• Have students solve these problems.

n. · = o. ÷ = nrqz

mzpr

mnqp

mlpu

lwun

mnwp

bdm

dm

b1

dm

b1

dgam

abg

bdm

abdg ÷ aggam ÷ ag

abdggam

abdggam

dgam

abg

dgam

abg • Multiply the numerators and

denominators.

• Find the greatest commonfactor (GCF) of the numeratorand denominator.

• Divide the numerator anddenominator by the GCF.

• Look for any common factors inthe numerators and thedenominators. Cancel thecommon factors.

• Multiply the numerators anddenominators.

Lesson


54

Procedure (cont.)

Step 7 Ask students to choose numbers for m, n, a, p, z, and r. Have halfthe class substitute the values into the expression below andevaluate it. Have the other half simplify the quotient first andthen substitute. Have volunteers show the computation to theclass.

p. ÷ Example: m = 2, n = 5, p = 4, q = 3, r = 4, z = 6

Method 1

÷ = · = · = = = 1

Method 2

· = = = = = 1

Step 8 Have students work in pairs to complete the activity sheetMultiplying and Dividing Fractions 1.

• Students’ answers will vary if they choose different values tosubstitute.

• Encourage students to simplify first, and then multiply ordivide the fractions.

19

109

2018

5 · 43 · 6

nrqz

p rm z

m nq p

19

109

2018

46

53

4 · 42 · 6

2 · 53 · 4

2 · 64 · 4

2 · 53 · 4

mzpr

mnqp


Practice

Teacher Tips

LOOK OUT!Hectic Day Approaching!• Tomorrow’s lesson may take longer than 90

minutes.

• The Basic Fractions Quiz is given at theend of tomorrow’s lesson. You may want to give it at theend of Lesson 9 (page 61) or at the beginning of Lesson 10(page 70). This will give students additional time to practicefractions.

Assignment

23

Unit 1: Number Sense and Operations

Name _____________________________________

Multiplying and Dividing Fractions 1Directions: Solve. Reduce each answer to its lowest terms, but it is not necessary tochange improper fractions to mixed numbers. Show your work.

#11126—Algebra Readiness, Guided Practice Book

1. · = ____________

2. · = ____________

3. 3 · 2 = __________

4. 6 ÷ 2 =__________

5. · = ____________

6. 3 ÷ = __________

7. · = __________

8. ÷ = ________

9. · = ________

10. ÷ = ______

5027

4520

1124

1633

38

59

59

329

23

45

3245

1842

12nhk

kmp15n

72kpaz

10mkzd

abcdef

djlbop

54pd25qr

64ps30rt

Directions: Evaluate the expressions for the following values:

11. x = 5, y = 6, m = 10, and n = 1 ______________ ______________

12. x = 15, y = 2, m = 3, and n = 4 ______________ ______________

13. x = 2, y = 1, m = 8, and n = 3 ______________ ______________

Directions: Translate as a mathematical statement and then solve.

14. A.J. and Sherita are filling bottles with lemonade to sell at their school trackmeet. Each bottle holds 5 ounces of lemonade. A.J. has 120 bottles. Howmany ounces of lemonade does he need? ____________________________

16

8xy3mn

15m4y

· 8xy3mn

12x9y÷

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