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The algebra content for IMPACT Mathematics was adapted from the series Access to Algebra, by Neville Grace, Jayne Johnston, Barry Kissane, Ian Lowe, and Sue Willis. Permission to adapt this material was obtained from the publisher, Curriculum Corporation of Level 5, 2 Lonsdale Street, Melbourne, Australia.

Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such materials be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with the Glencoe/McGraw-Hill IMPACT Mathematics program. Any other reproduction, for sale or other use, is expressly prohibited.

Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 978-0-07-888704-8MHID: 0-07-888704-6

Printed in the United States of America.

1 2 3 4 5 6 7 8 9 10 055/079 17 16 15 14 13 12 11 10 09 08

These materials include work supported in part by the National Science Foundation under Grant No. ESI-9726403 to MARS (Mathematics Assessment Resource Service). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding agencies. For more information on MARS, visit http://www.nottingham.ac.uk/education/MARS.

About the Cover

Hammers can be used to drive nails. They can also be used to shape metal that is struck against an anvil.

In ancient times, the renowned Greek mathematician Pythagoras noticed the exceptional sounds of hammers striking anvils. He determined that the weights of different-sized hammers were proportional, with weights of 12, 9, 8, and 6 pounds. Upon further examination, he determined that the harmonious sounds were a result of the numerical relationships among the hammers’ weights and discovered a connection among numbers, ratios, and music.

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G7-FP3

G7-FP3

G7-FP3

G7-FP3

G7-FP2, G7-FP4C

G7-FP6C, G7-FP7C

G7-FP3, G7-FP5C

G7-FP1, G7-FP3,

G7-FP5C

G7-FP3, G7-FP5C

G7-FP1, G7-FP4C

Expressions ............................................................................................ 2

Exponents ............................................................................................ 72

Signed Numbers ....................................................................... 124

Magnitude of Numbers ................................................... 172

Geometry in Three Dimensions ............................ 210

Data and Probability .......................................................... 260

Real Numbers .............................................................................. 318

Linear Relationships .......................................................... 366

Equations ............................................................................................434

Proportional Reasoning and Percents ........ 492

Contents in Briefand ConnectionsSee pages vi and vii for key.

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Principal InvestigatorFaye Nisonoff RuoppBrandeis UniversityWaltham, Massachusetts

Consultants and DevelopersConsultants

Frances Basich WhitneyProject Director, Mathematics K–12Santa Cruz County Office of EducationSanta Cruz, California

Dr. Selina Vásquez MirelesAssociate Professor of MathematicsTexas State University—San MarcosSan Marcos, Texas

Robyn SilbeyMathematics Content CoachMontgomery County Public SchoolsGaithersburg, Maryland

Teri WillardAssistant ProfessorCentral Washington UniversityEllensburg, Washington

Special thanks to:Peter BraunfeldProfessor of Mathematics EmeritusUniversity of Illinois

Sherry L. MeierAssistant Professor of MathematicsIllinois State University

Judith RoitmanProfessor of Mathematics University of Kansas

DevelopersSenior Project DirectorCynthia J. Orrell

Senior Curriculum DevelopersMichele Manes, Sydney Foster, Daniel Lynn Watt, Ricky Carter, Joan Lukas, Kristen Herbert

Curriculum DevelopersHaim Eshach, Phil Lewis, Melanie Palma, Peter Braunfeld, Amy Gluckman, Paula Pace

Special ContributorsElizabeth D. Bjork, E. Paul Goldenberg

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Project ReviewersGlencoe and Education Development Center would like to thank the curriculum specialists, teachers, and schools who participated in the review and testing of the first edition of IMPACT Mathematics. The results of their efforts were the foundation for this second edition. In addition, we appreciate all of the feedback from the curriculum specialists and teachers who participated in review and testing of this edition.

Debra AllredMath TeacherWiley Middle SchoolLeander, Texas

Tricia S. BiesmannRetired TeacherSisters Middle SchoolSisters, Oregon

Kathryn Blizzard BallinSecondary Math SupervisorNewark Public SchoolsNewark, New Jersey

Linda A. BohnyDistrict Supervisor of MathematicsMahwah Township School DistrictMahwah, New Jersey

Julia A. ButlerTeacher of MathematicsRichfield Public School AcademyFlint, Michigan

April ChauvetteSecondary Mathematics FacilitatorLeander ISDLeander, Texas

Amy L. ChazarettaMath Teacher/Math Department ChairWayside Middle School, EM-S ISDFort Worth, Texas

Franco A. DiPasquaDirector of K–12 MathematicsWest Seneca CentralWest Seneca, New York

Mark J. ForzleyJunior High School Math TeacherWestmont Junior High SchoolWestmont, Illinois

Virginia G. HarrellEducation ConsultantBrandon, Florida

Lynn HurtDirectorWayne County SchoolsWayne, West Virginia

Andrea D. Kent7th Grade Math & Pre-AlgebraDodge Middle School, TUSDTucson, Arizona

Russ Lush6th Grade Teacher & Math Dept. ChairNew Augusta—NorthIndianapolis, Indiana

Katherine V. Martinez De MarchenaDirector of Education 7–12Bloomfield Public SchoolsBloomfield, New Jersey

Marcy MyersMath FacilitatorSouthwest Middle SchoolCharlotte, North Carolina

Joyce B. McClainMiddle School Mathematics ConsultantHillsborough County SchoolsTampa, Florida

Suzanne D. ObuchowskiMath TeacherProctor SchoolTopsfield, Massachusetts

Michele K. OlderMathematics InstructorEdward A. Fulton Jr. HighO’Fallon, Illinois

Jill PlattnerMath Program Developer (Retired)Bend La Pine School DistrictBend, Oregon

E. Elaine RaffertyRetired Math CoordinatorSummerville, South Carolina

Karen L. ReedMath Teacher—Pre-APChisholm Trail IntermediateFort Worth, Texas

Robyn L. RiceMath Department ChairMaricopa Wells Middle SchoolMaricopa, Arizona

Brian StilesMath TeacherGlen Crest Middle SchoolGlen Ellyn, Illinois

Nimisha Tejani, M.Ed.Mathematics TeacherKino Jr. HighMesa, Arizona

Stefanie TurnageMiddle School MathematicsGrand Blanc AcademyGrand Blanc, Michigan

Kimberly WaltersMath TeacherCollinsville Middle SchoolCollinsville, Illinois

Susan WessonMath Teacher/ConsultantPilot Butte Middle SchoolBend, Oregon

Tonya Lynnae WilliamsTeacherEdison Preparatory SchoolTulsa, Oklahoma

Kim C. WrightenberryMath TeacherCane Creek Middle SchoolAsheville, North Carolina

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The Curriculum Focal Points identify key mathematical ideas for this grade. They are not discrete topics or a checklist to be mastered; rather, they provide a framework for the majority of instruction at a particular grade level and the foundation for future mathematics study. The complete document may be viewed at www.nctm.org/focalpoints.

G7-FP1 Number and Operations and Algebra and Geometry: Developing an understanding of and applying proportionality, including similarityStudents extend their work with ratios to develop an understanding of proportionality that they apply to solve single and multistep problems in numerous contexts. They use ratio and proportionality to solve a wide variety of percent problems, including problems involving discounts, interest, taxes, tips, and percent increase or decrease. They also solve problems about similar objects (including figures) by using scale factors that relate corresponding lengths of the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and identify the unit rate as the slope of the related line. They distinguish proportional relationships (

y _ x = k, or y = kx) from other

relationships, including inverse proportionality (xy = k, or y = k _ x ).

G7-FP2 Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areas and volumes of three-dimensional shapesBy decomposing two- and three-dimensional shapes into smaller, component shapes, students find surface areas and develop and justify formulas for the surface areas and volumes of prisms and cylinders. As students decompose prisms and cylinders by slicing them, they develop and understand formulas for their volumes (Volume = Area of base × Height). They apply these formulas in problem solving to determine volumes of prisms and cylinders. Students see that the formula for the area of a circle is plausible by decomposing a circle into a number of wedges and rearranging them into a shape that approximates a parallelogram. They select appropriate two- and three dimensional shapes to model real-world situations and solve a variety of problems (including multistep problems) involving surface areas, areas and circumferences of circles, and volumes of prisms and cylinders.

G7-FP3 Number and Operations and Algebra: Developing an understanding of operations on all rational numbers and solving linear equationsStudents extend understandings of addition, subtraction, multiplication, and division, together with their properties, to all rational numbers, including negative integers. By applying properties of arithmetic and considering negative numbers in everyday contexts (e.g., situations of owing money or measuring elevations above and below sea level), students explain why the rules for adding, subtracting, multiplying, and dividing with negative numbers make sense. They use the arithmetic of rational numbers as they formulate and solve linear equations in one variable and use these equations to solve problems. Students make strategic choices of procedures to solve linear equations in one variable and implement them efficiently, understanding that when they use the properties of equality to express an equation in a new way, solutions that they obtain for the new equation also solve the original equation.

KEYG7-FP1

Grade 7 Focal Point 1

G7-FP2 Grade 7 Focal Point 2

G7-FP3 Grade 7 Focal Point 3

G7-FP4C Grade 7 Focal Point 4

Connection


Connection


Connection


Connection

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http://www.nctm.org/focalpoint

Connections to the Focal Points

G7-FP4C Measurement and Geometry: Students connect their work on proportionality with their work on area and volume by investigating similar objects. They understand that if a scale factor describes how corresponding lengths in two similar objects are related, then the square of the scale factor describes how corresponding areas are related, and the cube of the scale factor describes how corresponding volumes are related. Students apply their work on proportionality to measurement in different contexts, including converting among different units of measurement to solve problems involving rates such as motion at a constant speed. They also apply proportionality when they work with the circumference, radius, and diameter of a circle; when they find the area of a sector of a circle; and when they make scale drawings.

G7-FP5C Number and Operations: In grade 4, students used equivalent fractions to determine the decimal representations of fractions that they could represent with terminating decimals. Students now use division to express any fraction as a decimal, including fractions that they must represent with infinite decimals. They find this method useful when working with proportions, especially those involving percents. Students connect their work with dividing fractions to solving equations of the form ax = b, where a and b are fractions. Students continue to develop their understanding of multiplication and division and the structure of numbers by determining if a counting number greater than 1 is a prime, and if it is not, by factoring it into a product of primes.

G7-FP6C Data Analysis: Students use proportions to make estimates relating to a population on the basis of a sample. They apply percentages to make and interpret histograms and circle graphs.

G7-FP7C Probability: Students understand that when all outcomes of an experiment are equally likely, the theoretical probability of an event is the fraction of outcomes in which the event occurs. Students use theoretical probability and proportions to make approximate predictions.

Reprinted with permission from Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence, copyright 2006, by the National Council of Teachers of Mathematics. All rights reserved.

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Table of ContentsExpressions ............................................................................. 2Lesson 1.1 Variables and Expressions ........................ 4 Investigation 1: Sequences and Rules ........................................ 5

Investigation 2: Write Expressions ............................................... 9

Investigation 3: Evaluate Expressions ........................................12

Investigation 4: Flowcharts ..........................................................17

On Your Own Exercises .............................................................21

Lesson 1.2 Expressions and Formulas ..................... 30 Investigation 1: What’s the Variable? ........................................ 32

Investigation 2: Formulas ............................................................ 35

Inquiry Investigation 3: Formulas and Spreadsheets ..... 40

On Your Own Exercises ............................................................ 43

Lesson 1.3 The Distributive Property ........................ 49 Investigation 1: Bags and Blocks ............................................... 51

Investigation 2: The Same and Different ................................. 53

Investigation 3: Use Parentheses .............................................. 56

Investigation 4: Combine Like Terms ....................................... 60


Review & Self-Assessment ...................................................... 69

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and ConnectionsSee pages vi and vii for key.

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Exponents ................................................................................. 72Lesson 2.1 Factors and Multiples .................................. 74 Investigation 1: Factors ....................................................................75

Investigation 2: Prime Numbers .................................................. 77

Investigation 3: Common Factors ................................................ 80

Investigation 4: Multiples .............................................................. 83

On Your Own Exercises ............................................................... 86

Lesson 2.2 Exponent Machines ...................................... 92 Investigation 1: Model Exponents ............................................... 93

Investigation 2: Multiply Expressions with the Same Base ..... 96

Investigation 3: Multiply Expressions with the Same Exponent ................................................ 100

On Your Own Exercises ................................................................. 104

Lesson 2.3 More Exponent Machines ..................... 107 Investigation 1: Division Machines with Exponents .............. 107

Investigation 2: Divide Expressions with Exponents .............. 109

Investigation 3: Power Law of Exponents ............................... 111

Inquiry Investigation 4: The Tower of Hanoi ...................... 114


Review & Self-Assessment ..................................................... 120

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3 Signed Numbers ............................................................ 124Lesson 3.1 Add and Subtract with Negative

Numbers ............................................................. 126 Investigation 1: The Two-Color Chip Model ............................. 128

Inquiry Investigation 2: A Number Line Model .................. 131

Investigation 3: Add and Subtract on the Number Line ........ 134

Investigation 4: Equivalent Operations ...................................... 140

Investigation 5: Inequalities and Negative Numbers .............. 143

Investigation 6: Predict Signs of Sums and Differences ........ 146

On Your Own Exercises ............................................................. 148

Lesson 3.2 Multiply and Divide with Negative Numbers ............................................................. 154

Investigation 1: Multiply a Positive and a Negative ................ 155

Investigation 2: Multiply Two Negatives ....................................157

Investigation 3: Divide with Negative Numbers ...................... 160

Investigation 4: Signed Numbers and Data .............................. 162



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4 Magnitude of Numbers ...................................... 172Lesson 4.1 Scientific Notation ........................................ 174 Investigation 1: Powers of 10 ..................................................... 175

Investigation 2: Work with Scientific Notation ......................... 178

Investigation 3: Scientific Notation on a Calculator ................ 182

Inquiry Investigation 4: Relative Error ................................... 185


Lesson 4.2 Negative Exponents .................................... 194 Investigation 1: Model Negative Exponents ............................. 194

Investigation 2: Evaluate Expressions with Negative Exponents ........................................................... 198

Investigation 3: Laws of Exponents and Scientific Notation .............................................................. 200



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5 Geometry in Three Dimensions ............. 210Lesson 5.1 Surface Area and Volume ...................... 212 Investigation 1: Measure Prisms ................................................. 213

Investigation 2: Volume of Prisms and Cylinders ................... 216

Inquiry Investigation 3: Package Design .............................. 220


Lesson 5.2 Nets and Solids .............................................. 228 Investigation 1: Use a Net ........................................................... 229

Investigation 2: Use Nets to Investigate Solids ........................ 230

Investigation 3: Is Today’s Beverage Can the Best Shape? ... 233


Lesson 5.3 Mass and Weight .......................................... 240 Investigation 1: Measure Mass .................................................... 241

Investigation 2: Estimate Mass .................................................... 244

Investigation 3: Measure Weight ................................................ 247



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6 Data and Probability .............................................. 260Lesson 6.1 Dependence .................................................... 262 Investigation 1: Combinations and Probability ........................ 262

Investigation 2: Heads or Tails? .................................................. 264

Investigation 3: Is It Fair? ............................................................. 268

Inquiry Investigation 4: The Hidden Prize ............................ 270


Lesson 6.2 Make Predictions .......................................... 278 Investigation 1: Samples and Predictions ................................. 279

Investigation 2: Sample Sizes ...................................................... 280

Investigation 3: Representative Samples .................................. 283


Lesson 6.3 Data Graphs ..................................................... 294 Investigation 1: Double Bar and Double Line Graphs ........... 295

Investigation 2: Make a Circle Graph ......................................... 299

Investigation 3: Stem-and-Leaf Plots ......................................... 302

Investigation 4: Misleading Statistics and Graphs ................... 306



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7 Real Numbers ................................................................... 318Lesson 7.1 Rational Numbers ........................................ 320 Investigation 1: Number Sets ...................................................... 320

Investigation 2: Order and Compare Rational Numbers ....... 324


Lesson 7.2 Irrational Numbers ...................................... 331 Investigation 1: Squares and Square Roots ............................. 331

Investigation 2: Approximate Irrational Numbers ................... 336


Lesson 7.3 The Pythagorean Theorem .................... 343 Investigation 1: Right Triangles and Squares ........................... 343

Investigation 2: Use the Pythagorean Theorem ...................... 347

Inquiry Investigation 3: The Distance Formula ................... 351



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8 Linear Relationships .............................................. 366Lesson 8.1 Rates ...................................................................... 368 Investigation 1: Understand Rates ............................................. 369

Investigation 2: Describe Rates ................................................... 373

Investigation 3: Proportional Relationships .............................. 377

Inquiry Investigation 4: Rolling Along ................................... 380


Lesson 8.2 Speed and Slope .......................................... 389 Investigation 1: Walk and Jog ..................................................... 390

Investigation 2: Distance and Time ............................................ 394

Investigation 3: Describe Graphs ................................................ 396

Investigation 4: Change the Starting Point ............................... 398


Lesson 8.3 Recognize Linear Relationships ......... 410 Investigation 1: Explore and Describe Patterns ....................... 411

Investigation 2: Graphs and Rules from Patterns .................... 414

Investigation 3: From Rules to Graphs ...................................... 417

Investigation 4: Secret Rules ....................................................... 420



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9 Equations ................................................................................ 434Lesson 9.1 Find a Solution Method ........................... 436 Investigation 1: Choose a Solution Method ............................. 437

Inquiry Investigation 2: Use a Spreadsheet to Guess-Check-and-Improve ............. 440


Lesson 9.2 A Model for Solving Equations ........... 446 Investigation 1: Balance Puzzles ................................................. 447

Investigation 2: Keep Things Balanced ..................................... 449

Investigation 3: Solve Problems with Balance Puzzles .......... 453


Lesson 9.3 Solve Equations ............................................. 460 Investigation 1: Symbolic Solutions ........................................... 461

Investigation 2: Do the Same Thing to Both Sides ................. 462

Investigation 3: Solve More Equations ...................................... 464

Investigation 4: Solve One-Variable Inequalities ..................... 466


Lesson 9.4 Solve Equations with Parentheses .... 474 Investigation 1: Build and Solve Equations .............................. 474

Investigation 2: Subtract with Parentheses .............................. 477

Investigation 3: More Practice with Parentheses .................... 481



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1010 Proportional Reasoning and Percents ..................................................................................... 492Lesson 10.1 Ratios ................................................................. 494 Investigation 1: Ratio and Rate Comparisons .......................... 495

Investigation 2: Compare and Scale Ratios .............................. 497


Lesson 10.2 Proportions and Similarity ................... 505 Investigation 1: Proportional Relationships .............................. 505

Investigation 2: Equal Ratios ....................................................... 508

Investigation 3: Solve Proportions ............................................. 510

Investigation 4: Map Scales ......................................................... 514

Investigation 5: Similarity ............................................................. 517

Inquiry Investigation 6: Estimate Heights of Tall Objects ....................................... 519


Lesson 10.3 Percents and Proportions .................... 530 Investigation 1: Find Percents ..................................................... 530

Investigation 2: Proportions Using Percents ............................ 534


Lesson 10.4 Rates ................................................................... 540 Investigation 1: Unit Prices .......................................................... 540

Investigation 2: Convert Currencies ........................................... 544



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More Than Just a Textbook - PBworkssimonbaruchcurriculum.pbworks.com/f/frontmat.pdf · More Than Just a Textbook For Students Connect to StudentWorks Plus Online which ... G7-FP3