Algebra and More
Course 2
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interactive student edition
Algebra and More
Course 2 Developed by Education Development Center, Inc. Principal Investigators Investigators:: Faye Nisonoff Ruopp, E. Paul Goldenberg Senior Project Director: Cynthia J. Orrell Senior Curriculum Developers: Michelle Manes, Susan Janssen, Sydney Foster,
Daniel Lynn Watt, Nina Arshavsky, Ricky Carter, Joan Lukas, Charles Lovitt Curriculum Developers: Phil Lewis, Debbie Winkler
Thealgebracontentfor Impact Mathematics wasadaptedfromtheseries, Access to Algebra, by NevilleGrace,JayneJohnston, Johnston,BarryKissane,IanLowe,andSueWillis. Willis.Permissiontoadapt adapt thismaterialwasobtainedfromthepublisher,CurriculumCorporationofLevel5,2 LonsdaleStreet, Melbourne,Australia. Copyright©2004byTheMcGraw-HillCompanies,Inc. Allrightsreserved.Printedinthe United States of America. Except Except as permitted underthe United States CopyrightAct, no part of this publication may be reproduced reproduced or distributed inany form or by any means, orstored in a database or retrievalsystem, without prior written permission fromthe publisher. Sendallinquiriesto: Glencoe/McGraw-Hill 8787OrionPlace Columbus, OH 43240-4027 ISBN 0-07-860920-8 2 3 4 56 7 8 910 058/055 058/05 5 13 12 11 10 0908 07 06 05 04
Impact Mathematics Project Reviewers Education Development Development Center appreciates all the feedback feedback from the curriculum specialists specialists and teachers teachers who participated in review and testing. Special thanks to: Peter Braunfeld Professor Prof essor of Mathe Mathematic maticss Eme Emeritus ritus University Univ ersity of Illino Illinois is
Sherry L. Me Sherry Meier ier Assistant Professor of Mathematics Illinois State University
Judith Roitman Professo Prof essorr of Math Mathemati ematics cs University Univ ersity of Kansa Kansass
Marcie Abramson Thurston Middle School Boston, Massachusetts
Alan Dallman Amherst Middle School Amherst, Massachusetts
Steven J.J. Fox Bendle Middle School Burton, Burto n, Michi Michigan gan
Denise Airola Fayetteville Fay etteville Public Schools Fayetteville, Fay etteville, Arizona
Sharon DeCarlo Sudbury Sudbu ry Publi Publicc Scho Schools ols Sudbury, Massachusetts
Kenneth L. Goodwin Jr. Jr. Middletown Middle School Middletown, Delawar Delawaree
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David P. P. DeLeon Preston Area School Lakewood, Pennsylv Pennsylvania ania
Fred E. Gros Fred Grosss Sudbury Sudbu ry Publi Publicc Schoo Schools ls Sudbury, Massachusetts
Jeanne A. Arnold Mead Junior Junior High Elk Grov Grovee Village, Illino Illinois is
Jacob J.J. Dick Cedar Grove School Cedarr Grov Ceda Grove, e, Wisco Wisconsin nsin
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Joanne J.J. Astin Lincoln Middle School Forrest For rest City City,, Arkan Arkansas sas
Sharon Ann Dudek Holabird Middle School Baltimore, Maryland
Jean Hawkins James River Day School Lynchburg, Virginia
Jack Beard Urbana Junior High Urbana, Urban a, Ohio
Cheryl Elisara Centennial Middle School Spokane, Washin Washington gton
Robert Kalac Robert Butlerr Juni Butle Junior or High Frombell, Pennsylv Pennsylvania ania
Chad Cluver Maroa-Forsyth Maroa-F orsyth Junior Junior High Maroa, Mar oa, Illino Illinois is
Patricia Elsroth Wayne Highlands Highlands Middle School Honesdale, Pennsylv Pennsylvania ania
Robin S. Kalder Robin Somers High School Somers, Some rs, New York
Robert C. Bieri Robert Bieringer nger Patchogue-Medford Patchogue-M edford School Dist. Center Cent er Mori Moriches ches,, New York
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Darrin Kamps Lucille Umbarge Elementary Burlington, Washin Washington gton
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Pat Kin Pat Kingg Holmes Junior High Davis, California
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Kim Lazarus San Diego Jewish Academy La Jolla, California
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Kathy L. Te Terwelp rwelp Summit Public Schools Summit, Summ it, New Jers Jersey ey
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Mary Lund Lundquist quist Farmington High School Farmington, Connecticut
Sue Saunders Abell Jr. High School Midland, Texa Texass
Ellen McDonald-Kni McDonald-Knight ght San Diego Unified School District San Diego, Diego, California
Ivy Schr Schram am Massachusetts Department of Youth Servic Services es Massachusetts
Ann Miller Castle Rock Rock Middle School School Castle Rock, Colorado
Robert Sega Robert Segall ll Windham Public Schools Willimantic, Connecticut
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Karen Smith East Middle School School Braintree, Massachusetts
Maria Peterson Chenery Chen ery Middle School Belmont, Massachusetts
Kim Spillane Oxford Central School Oxford, Oxfo rd, New Jers Jersey ey
Lonnie Pilar Tri-County Middle School Howard City, Michigan
Carol Struchtemeyer Lexington R-5 Schools Lexington, Missouri
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Marcia Uhls Truesdale Middle School Wichita, Kansas Vendula Vogel Westridge School School for Girls Pasadena,, California Pasadena Judith A. Webbe Webberr Grand Blanc Middle School Grand Blanc, Michigan Sandy Weisha Weishaar ar Woodland Junior Junior High Fayetteville, Fay etteville, Arkansas Tamara L. Weiss Tamara Forest Hills Middle School School Forest For est Hills Hills,, Michi Michigan gan Kerrin Wertz Haverford Haverfor d Middle School Havertown, Pennsylv Pennsylvania ania Anthony Williams Jackie Robinson Middle School Brooklyn, Broo klyn, New York Deborah Winkler The Baker Scho School ol Brookline, Massachusetts Lucy Zizka Best Middle School School Ferndale, Michigan
T Under Und ersta standi nding ng Expr Expres essio sions ns . . . . . . 2
Geometry Geom etry in in Three Three Dimens Dimensions ions . . . 76
Lesson 1.1: Variables Variables and Expressions Expressions . . . . . . . . . . . 4 Investig Inve stigatio ationn 1: Express Expressions ions . . . . . . . . . . . . . . . 5 Investigation Investigati on 2: Writing Writing Expressions Expressions . . . . . . . . . 10 Investigation Investigati on 3: Evaluating Evaluating Expressions Expressions . . . . . . . 13 Investigation Investigati on 4: Using Flowcharts Flowcharts . . . . . . . . . . . 18 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . . 22
Lesson 2.1: Block Block Patterns . . . . . . . . . . . . . . . . . . 78 Investig Inve stigatio ationn 1: A Staircase Staircase Patte Pattern rn . . . . . . . . . 79 Investigation Investigati on 2: Other Other Block Patterns . . . . . . . . 80 Investig Inve stigatio ationn 3: Buildin Buildingg Block Block Patterns Patterns . . . . . . 82 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . . 84
Lesson 1.2: Expression Expressionss and Formulas Formulas . . . . . . . . . . 32 Investigation Investigati on 1: What’s What’s the Variable? Variable? . . . . . . . . 34 Investigation Investigati on 2: Using Using Formulas Formulas . . . . . . . . . . . . 37 Lab Investiga Investigation tion:: Formulas Formulas and Spreadshe Spreadsheets ets . . 42 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . . 46 Lesson 1.3: The Lesson The Distrib Distributiv utivee Property Property . . . . . . . . . . 52 Investig Inve stigatio ationn 1: Groupi Grouping ng Bags Bags and Bloc Blocks ks . . . . 54 Investig Inve stigatio ationn 2: The Same Same and Differen Differentt . . . . . . 56 Investigation 3: Grouping with the Distributive Property Prop erty . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Investigation 4: Removing and Inserting Parentheses Parenthes es . . . . . . . . . . . . . . . . . . . . . . . . 64 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . . 68 Review & Self-Assessmen Self-Assessmentt . . . . . . . . . . . . . . . . . . 74
Lesson 2.2: Visualizing and Measuring Blockk Structur Bloc Structures es . . . . . . . . . . . . . . . . . . . . . . . 91 Investigation Investigati on 1: Seeing Seeing All the the Angles . . . . . . . . 92 Investig Inve stigatio ationn 2: Diffe Different rent Vie Views ws . . . . . . . . . . . 94 Investig Inve stigatio ationn 3: Creating Creating Engin Engineeri eering ng Views Views . . . 97 Investigation Investigati on 4: Measuring Block Block Structures Structures . . . . 98 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 100 Lesson 2.3: Lesson 2.3: Surface Surface Area Area and and Volume Volume . . . . . . . . . 109 Investigation 1: Finding Volumes of Block Structur Structures es . . . . . . . . . . . . . . . . . . 110 Investigation Investigati on 2: Finding Finding Other Volu Volumes mes . . . . . . 112 Investigation 3: Modeling with Block Structures Structures . . . . . . . . . . . . . . . . . . . . 116 Lab Investi Investigati gation: on: The The Soft Drin Drinkk Promoti Promotion on . . 119 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 122 Lesson 2.4: Lesson 2.4: Nets and Solids Solids . . . . . . . . . . . . . . . . 129 Investig Inve stigatio ationn 1: Will Will It Fold? Fold? . . . . . . . . . . . . . 130 Investigation 2: Using Nets to Investigate Solids Soli ds . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Investigation 3: Is Today’s Soft Drink Can the Best Shape? . . . . . . . . . . . . . . . . . . . . 134 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 136 Review & Self-Assessmen Self-Assessmentt . . . . . . . . . . . . . . . . . 141
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T Explor Exp lorin ingg Expon Exponent entss . . . . . . . . . 144
Workin Wo rkingg with with Signed Signed Numbe Numbers rs . . 216
Lesson 3.1: 3.1: Stretching Stretching and Shrinking Shrinking Machines Machines . . . 146 Investiga Inves tigation tion 1: Machine Machine Hook Hookups ups . . . . . . . . . 147 Investiga Inves tigation tion 2: Repeater Repeater Mach Machine iness . . . . . . . . 149 Investigationn 3: Strings Investigatio Strings of Machines Machines . . . . . . . . 152 Investigationn 4: More Strings Investigatio Strings of Machines Machines . . . . 156 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 160
Lesson 4.1: Adding and Subtracting with Negative Nega tive Numbe Numbers rs . . . . . . . . . . . . . . . . . . . . 218 Lab Investi Investigati gation: on: Walki Walking ng the the Plank Plank . . . . . . . 220 Investigationn 1: Walking Investigatio Walking the the Number Line . . . . 222 Investiga Inves tigation tion 2: Equival Equivalent ent Operat Operations ions . . . . . . 228 Investigationn 3: Taking Investigatio Taking It Further Further . . . . . . . . . . 231 Investigation 4: Predicting Signs of Sums and Di Diffe ffere rence ncess . . . . . . . . . . . . . . . . . . . . 234 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 236
Lesson 3.2: 3.2: Shrinking Shrinking and and Super Machines . . . . . . 164 Investigationn 1: A New Shrinking Investigatio Shrinking Machine Machine . . . . 164 Investiga Inves tigation tion 2: Dividin Dividingg and Expone Exponents nts . . . . . 166 Investigationn 3: Super Machines Investigatio Machines . . . . . . . . . . . 168 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 172 Lesson 3.3: Lesson 3.3: Growing Growing Exponent Exponentiall iallyy . . . . . . . . . . . 176 Investigationn 1: Telling Investigatio Telling the the Difference Difference . . . . . . . 176 Investigation 2: Exponential Increase and Decrease . . . . . . . . . . . . . . . . . . . . . . 179 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 183 Lesson 3.4: 3.4: Describing Describing Large Numbers . . . . . . . . . 190 Investigationn 1: Millions Investigatio Millions and Billions Billions . . . . . . . . 191 Investiga Inves tigation tion 2: Powers Powers of 10 10 . . . . . . . . . . . . 192 Investigationn 3: Scientific Investigatio Scientific Notation Notation . . . . . . . . . 196 Investigation 4: Scientific Notation on Your Your Calc Calculat ulator or . . . . . . . . . . . . . . . . . . 200 Lab Investigation: Investigation: The Tower Tower of Hanoi . . . . . . . 203 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 206 Review & Self-Assessmen Self-Assessmentt . . . . . . . . . . . . . . . . . 213
Lesson 4.2: Multiplying and Dividing with Negative Nega tive Numbe Numbers rs . . . . . . . . . . . . . . . . . . . . 242 Investigation 1: The Product of a Positive and a Negative . . . . . . . . . . . . . . . . . . . . . . . . . 243 Investigation 2: The Product of Two Negatives Negat ives . . . . . . . . . . . . . . . . . . . . . . . . 245 Investigation 3: Dividing with Negative Numbers Numbe rs . . . . . . . . . . . . . . . . . . . . . . . . . 248 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 250 Lesson 4.3: Plotting Plotting Points in Four Four Quadrants . . . . 254 Investigation 1: Plotting Points with Negative Coordinates . . . . . . . . . . . . . . . . . . . . . . . 255 Investigation 2: Parts of the Coordinate Planee . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Plan Investigation 3: Representing Operations on the Coo Coordi rdinat natee Plane Plane . . . . . . . . . . . . . . 262 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 264 Lesson 4.4: 4.4: Finding Distances . . . . . . . . . . . . . . . 268 Investiga Inves tigation tion 1: The Pythag Pythagorea oreann Theorem Theorem . . . 269 Investigationn 2: The Distance Formula . . . . . . . 273 Investigatio On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 276 Lesson 4.5: Lesson 4.5: Negative Negative Number Numberss as Exponent Exponentss . . . . 280 Investigation 1: Shrinking with Negative Exponents Expon ents . . . . . . . . . . . . . . . . . . . . . . . . 280 Investigation 2: Evaluating Expressions with Negative Negat ive Expone Exponents nts . . . . . . . . . . . . . . . . . 284 Investigationn 3: Laws of Exponents and Investigatio Scient Sci entifi ificc Not Notati ation on . . . . . . . . . . . . . . . . . . 286 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 288 Review & Self-Assessmen Self-Assessmentt . . . . . . . . . . . . . . . . . 293
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Looking at Linear Linear Relationshi Relationships ps . . 298
Solvin Sol vingg Equat Equation ionss . . . . . . . . . . . 382
Lesson 5.1: Understandi Understanding ng and Describing Describing Rates Rates . . 300 Investig Inve stigatio ationn 1: Understa Understandin ndingg Rates Rates . . . . . . . 301 Investigation Investigati on 2: Describing Describing Rates . . . . . . . . . . 305 Investig Inve stigatio ationn 3: Proportio Proportional nal Relatio Relationsh nships ips . . . 309 Lab Investigation: Investigation: Rolling Rolling Along Along . . . . . . . . . . . 312 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 314
Lesson 6.1: Two Lesson Two Solution Solution Metho Methods ds Revisite Revisitedd . . . . 384 Investig Inve stigatio ationn 1: Choosing Choosing a Solution Solution Method Method . . 385 Lab Investigation: Using a Spreadsheet to Guess-Check-and-Improv Guess-Che ck-and-Improvee . . . . . . . . . . . . . 388 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 392
Lesson 5.2: Speed and the Slope Connection Connection . . . . . 321 Investig Inve stigatio ationn 1: Walking Walking and and Jogging Jogging . . . . . . . 322 Investigation 2: Decreasing Distance with Tim Timee . . . . . . . . . . . . . . . . . . . . . . . . 326 Investig Inve stigatio ationn 3: Describi Describing ng Graphs Graphs . . . . . . . . . 328 Investigation 4: Changing the Starting Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 334 Lesson 5.3: Recogn Lesson Recognizi izing ng Linear Linear Relation Relationship shipss . . . . 344 Investigation 1: Exploring and Describing Describi ng Patterns . . . . . . . . . . . . . . . . . . 345 Investigation Investigati on 2: Graphs and Rules from Patte Patterns rns . . . . . . . . . . . . . . . . . . . . . 348 Investig Inve stigatio ationn 3: From From Rules Rules to to Graphs Graphs . . . . . . 351 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 354 Lesson 5.4: Tricks Tricks of the Trade . . . . . . . . . . . . . . 362 Investigation Investigati on 1: Finding Secret Rules Rules . . . . . . . . 362 Investigation 2: The Method of Constant Differe Diff erences nces . . . . . . . . . . . . . . . . . . . . . . . 365 Investig Inve stigatio ationn 3: Under Understan standing ding the Symbo Symbols ls . . 368 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 371 Review & Self-Assessmen Self-Assessmentt . . . . . . . . . . . . . . . . . 378
Lesson 6.2: A Model for Solving Equations Equations . . . . . . 395 Investigation 1: Making and Solving Balance Balan ce Puzzle Puzzless . . . . . . . . . . . . . . . . . . . . 396 Investigation Investigati on 2: Keeping Things Things Balanced Balanced . . . . . 398 Investigation 3: Solving Problems with Balance Balan ce Puzz Puzzles les . . . . . . . . . . . . . . . . . . . . 402 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 404 Lesson 6.3: Lesson 6.3: Thinkin Thinkingg with Symbo Symbols ls . . . . . . . . . . . 409 Investigation Investigati on 1: Writin Writingg Symbolic Symbolic Solutions Solutions . . . 410 Investigation 2: Doing the Same Thing to Both Sides . . . . . . . . . . . . . . . . . . . . . . 411 Investig Inve stigatio ationn 3: Solving Solving More More Equation Equationss . . . . . 413 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 415 Lesson 6.4: Simpl Lesson Simplifyi ifying ng Equation Equationss . . . . . . . . . . . . 419 Investigation 1: Building and Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . 419 Investigation 2: Subtracting with Parentheses Parentheses . . . . . . . . . . . . . . . . . . . 422 Investigation 3: More Practice with Parentheses Parentheses . . . . . . . . . . . . . . . . . . . 426 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 429 Lesson 6.5: Lesson 6.5: Using Using Express Expressions ions . . . . . . . . . . . . . . 434 Investig Inve stigatio ationn 1: Number Number Games Games . . . . . . . . . . . 435 Investigation 2: Simplifying Equations for Graph Graphing ing . . . . . . . . . . . . . . . . . . . . . . 437 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 440 Review Revi ew and SelfSelf-Asse Assessme ssment nt . . . . . . . . . . . . . . . 445
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E Similar Simi larity ity ................. 448
RatiooandPropo Rati Proporti rtion on......... 518
Lesson7.1:AreTheythe theSame? ............. 450 Investigation Investigati on 1:Identifyin Identifyingg CongruentFigures and Angl Angles es........................ 451 Investiga Inves tigation tion2: 2:AreThey TheySimilar Similar?? ......... 454 Investigation Investigati on 3:Ratios of Corresponding Sides............................ 456 Investigation Investigati on 4:Identifyin Identifyingg Similar Polygons ......................... 461 OnYour YourOwnExercises Exercises ................. 464
Lesson8.1: Lesson 8.1:Compari Comparing ngwith withRatios Ratiosand andRates Rates ... 520 Investiga Inves tigation tion1:Thinki Thinking ngabout aboutRatios Ratios ...... 521 Investigation Investigati on 2:Comparing and Scaling Scal ingRatio Ratioss ..................... 524 Investigationn3:UsingRatio Investigatio RatioTables Tables ......... 527 Investigationn4:Comparison Investigatio ComparisonShopping Shopping ....... 529 OnYour YourOwnExercises Exercises ................. 532
Lesson7.2:PolygonSimilarity SimilarityandCongruence Congruence ... 471 Investigationn1:Sides,Sides,Sides Investigatio Sides ......... 472 Investigationn2:AndMoreSides........... 473 Investigatio Investigationn3:Angles, Investigatio Angles,Angles,Angles ...... 475 LabInvestig Investigatio ation: n:Buildin BuildinggTower Towerss ......... 476 OnYour YourOwnExercises Exercises ................. 478 Lesson7.3:AreaandPerimeter ofSimilarFigures ..................... 482 Investigation Investigati on1:DilatingTriangles Triangles andRectangles..................... 482 Investiga Inves tigation tion2:Dilati Dilating ngand andFormulas Formulas ...... 485 Investigation Investigati on3:SimilarityinMore ComplexFigures Figures .................... 489 OnYour YourOwnExercises Exercises ................. 492 Lesson7.4:VolumeandSurfaceArea ofSimilarFigures ..................... 497 Investigationn1:DilatingBlock Investigatio BlockStructures Structures..... 498 Investigation Investigati on2:Volume VolumeofSimilar BlockStructures Structures .................... 499 Investigationn3: SurfaceArea Investigatio ofSimilarObjects ................... 501 Investiga Inves tigation tion4:Giants Giants ................. 504 OnYour YourOwnExercises Exercises ................. 506 Review&Self-Assessment Self-Assessment................. 514
Lesson8.2:Using UsingProportions Proportions............... 540 Investigation Investigati on 1:Reasoning about Proportional Relationships Relations hips ...................... 541 Investigation Investigati on 2:Solving Problems Involving Equal Rati Ratios os....................... 543 Investigationn3:Solving Investigatio SolvingProportions Proportions ........ 545 Investigation Investigati on 4:Applicati Applications ons Involving Similarityy......................... 548 Similarit Lab Investigation: Estimating Heights ofTall TallObjec Objects ts ..................... 551 OnYour YourOwnExercises Exercises ................. 554 Lesson8.3:PercentagesandProportions Proportions ....... 562 Investigationn1:PercentasaCommon Investigatio CommonScale... 562 Investigation Investigati on 2:Proportions Using Percentagess ....................... 566 Percentage Investigation Investigati on 3:Percent Increase andDecrease Decrease ...................... 568 Investigation Investigati on 4:Percentages of Percentagess ....................... 572 Percentage OnYour YourOwnExercises Exercises ................. 575 Lesson8.4:InterpretingandApplying Proporti Prop ortions ons ......................... 585 Investigationn 1:Interpretin Investigatio Interpretingg Comparisons Comparisons . .. . 585 Investigation Investigati on 2:Predicting Based onSmalle SmallerrGroups Groups .................. 588 OnYour YourOwnExercises Exercises ................. 590 Review&Self-Assessment Self-Assessment................. 596
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r N Interpreting Interpr eting Graphs Graphs . . . . . . . . . . 600
Data and Proba Probabili bility ty . . . . . . . . . 664
Lesson 9.1: Graphing Graphing Change over Time Time . . . . . . . . 602 Investig Inve stigatio ationn 1: Growing Growing Up Up . . . . . . . . . . . . . 603 Investigation Investigati on 2: Filling Filling It Up . . . . . . . . . . . . . . 606 Lab Investigati Investigation: on: Filling Odd-shaped Containers . . . . . . . . . . . . . . . . . . . . . . . . 608 Investig Inve stigatio ationn 3: Walki Walking ng About About . . . . . . . . . . . 610 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 616
Lesson 10.1: Dependence Dependence . . . . . . . . . . . . . . . . . . 666 Investigation 1: Combinations and Probability . . . . . . . . . . . . . . . . . . . . . . . . 666 Investigation Investigati on 2: Heads or Tails? Tails? . . . . . . . . . . . 668 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 672
Lesson 9.2: 9.2: Graphs and Equations . . . . . . . . . . . . 624 Investig Inve stigatio ationn 1: Graphs Graphs for for Squares Squares . . . . . . . . 625 Investigation Investigati on 2: Graphs Graphs for Inverses Inverses . . . . . . . . 627 Investigation 3: Using Graphs to Estimate Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 629 Investig Inve stigatio ationn 4: Doubli Doubling ng Up . . . . . . . . . . . . . 632 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 635 Lesson 9.3: Lesson 9.3: Repeating Repeating Relat Relation ionship shipss . . . . . . . . . . 644 Investig Inve stigatio ationn 1: Repeatin Repeatingg Patterns Patterns . . . . . . . . 645 Investig Inve stigatio ationn 2: Repeati Repeating ng Pattern Patternss in Life Life . . . 648 Investigation 3: Repeating Patterns in the Weather Weather . . . . . . . . . . . . . . . . . . . . . 650 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 652 Review & Self-Assessmen Self-Assessmentt . . . . . . . . . . . . . . . . . 660
Lesson 10.2: 10.2: Applying Probability Probability . . . . . . . . . . . . 676 Investigation Investigati on 1: Is it Fair? . . . . . . . . . . . . . . . 676 Investig Inve stigatio ationn 2: What’ What’ss the Diffe Differen rence? ce? . . . . . 678 Investigation Investigati on 3: The The Hidden Prize Prize . . . . . . . . . . 680 Investigation 4: The Greatest-Number Game . . . . . . . . . . . . . . . . . . . . . . . . . . . 683 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 686 Lesson 10.3: Sampling and Making Predictions Predictio ns . . . . . . . . . . . . . . . . . . . . . . . . . . 692 Investig Inve stigatio ationn 1: What’ What’s in the Bag? Bag? . . . . . . . . 693 Investigation Investigati on 2: Exploring Exploring Sample Sizes Sizes . . . . . . 694 Investigation 3: Making Predictions from Samples . . . . . . . . . . . . . . . . . . . . . . 697 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 700 Lesson 10.4: Statistical Statistical Tools Tools and Graphs . . . . . . . 709 Investig Inve stigatio ationn 1: Box-and Box-and-Whi -Whisker sker Plots Plots . . . . . 710 Investigation 2: Choosing a Graph for Displaying Displaying Data . . . . . . . . . . . . . . . . . . 714 On Your Your Own Exercises Exercises . . . . . . . . . . . . . . . . . 718 Review & Self-Assessmen Self-Assessmentt . . . . . . . . . . . . . . . . . 724 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 Photoo Credits Phot Credits . . . . . . . . . . . . . . . . . . . . . . . . . . 743
1
Understanding Expressions Algebra in the Strangest Places
You might think that algebra is a topic found only in textbooks, but you can find find algebra all around you—in some some of the strangest places. Did you know there is a relationship between the speed at which ants crawl and the air temperature? temperature? If you were to find some some ants outside and time them as they crawled, crawled, you could actually estimate estimate the the temperature. temperature. Here is the algebraic equation that describes this relationship. relationship. Celsius temperature t ϭ
15s ϩ 3 ant speed in centimeters per second
There are many ordinary and extraordinary places where you will encounter algebra.
Think About It
What do you think is the speed of a typical ant?
F a m mi l l y L e t t e r Dear Student and Family Members, Our class is about to begin an exciting year of Impact Mathematics. Some of the topics we will study include negative negative numbers, exponents, threedimension dime nsional al geometry geometry, rati ratios, os, proba probabili bility ty,, and data analysis. analysis. Throughout Throughout the year, your student will also develop develop and refine refine skills in algebra. We’ll begin by looking at algebraic expressions—the combinations of numbers, numbe rs, lett letters, ers, and mathemati mathematical cal symbols symbols that form the language language of algebra. We We will learn about variables—letters or symbols that can change or that represent unknown unknown quantities. For example, example, in the expression b ϩ 2, the variable is b. Once we’re we’re familiar with variables variables and expressions, expressions, we will create flowflowcharts to match expressions and then use the flowcharts to solve equations. We will also explore formulas used in everyday everyday life, such as the formula 9 used to convert degrees Celsius to degrees Fahrenheit: F ϭ ᎏ5ᎏC ϩ 32.
Vocabulary
Along the way, we’ll be learning several several new vocabulary vocabulary terms:
algebraic expression backtracking distributive property equivalent expressions expand
exponent factor flowchart formula variable
What can you do at home? Encourage your student to explain the kinds of problems he or she is solving in class. In addition, help him or her think about common occurrences of algebraic expressions in daily life. Your Your interest in your student’ss work helps emphasize the importance of mathematics and its student’ usefulness in daily life.
impactmath.com/family_letter
3
Variables and Expressions Every day people are confronted with problems they have to solve. Many of these problems involve such quantities as the amount of spice to add to a recipe, the cost of electricity electricity,, and interest rates. In some problem situations, it helps to have have a way to record information information without using a lot of words. For For example, both boxes present present the same idea. idea. To convert a Celsius temperature to a Fahrenheit Fahrenheit temperatur temperature, e, find nine fifths of the Celsius temperature and then add 32.
9
F ϭ ᎏ5ᎏ C ϩ 32
While the statement on the left may be easier to read and understand at first, the statement on the right has has several advantages. advantages. It is shorter shorter and easier to write, it shows clearly clearly how the quantities—Celsius quantities—Celsius temperature temperature and Fahrenheit temperature—are temperature—are related, related, and it allows allows you to try different different Celsius temperatures and compute their Fahrenheit equivalents. In this lesson, lesson, you’l you’lll see that by using a few few simple rules, rules, you can write write powerful algebraic expressions and equations for a variety of situations.
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Think Discuss Shaunda, Kate Shaunda, Kate,, and Simon Simon are holding holding bags of blocks blocks.. Isabel has has just two blocks.
Suppose there were 5 blocks in each of our bags. How many blocks would there be altogether?
What if there were 11 blocks in each of our bags?
What if there were 100 blocks in each bag?
The bags would break! If you know how how many blocks are in each each bag, how can you figure figure out how many blocks there are altogether?
4
CHAPTER
1
Understanding Expressions Expressions
If you know the number of blocks blocks in each bag, it’ it’ss not hard to express the total number of blocks. For example, if there are 20 blocks in each bag, you can just add: 20
ϩ
20
20
ϩ
ϩ
2
ϭ
62
Or you can multiply and add: 3
V O C A B U L A R Y variable
20
ϫ
ϩ
2
ϭ
62
What if you don’t don’t know know the number number in each bag? First, First, notic noticee that, in this situa situation tion,, the number number of bags and the number number of loose blocks blocks don’t change, but the number number of blocks in in each bag can change. change. Quantities that can chang change, e, or var vary y, are calle called d variables. In algebra, letters are are often used to represent variables. variables. For For example, example, you can let the letter n stand for the number of blocks in each bag.
n
n
n
Now you can find the total number of blocks as you did before— by ad addi ding ng nϩnϩnϩ
Remember Multiplication can be shown in several ways: 3 n 3(n) 3 n 3*n
2
or by multiplying and adding: 3
ϫ n ϩ
2
In algebra, the multiplication multiplication symbol between a number and a variable variable is usual usually ly left left out. So 3 ϫ n ϩ 2 can be written 3 n ϩ 2.
Investigation 1
Expressions
In the bags-and-blocks bags-and-blocks situation above, above, you can think of 3 n ϩ 2 as a rule for finding the total number of blocks when you know the number of blocks in each bag. Just substitute the number in each bag for n. For example, exam ple, for 100 blocks blocks in each each bag, the total total number number of blocks blocks is 3n ϩ 2
ϭ
3
ϫ
100
LESSON
ϩ
1.1
2
ϭ
302
Variables V ariables and Expressions
5
V O C A B U L A R Y
algebraic expression
Rules written written with numbers and symbols, such as n ϩ n ϩ n 3n ϩ 2, are cal called led algebraic expressions.
ϩ
2 and
As you study algebra, you will work with with algebraic expressions expressions often. Using bags and blocks is a good way to start thinking about expressions. Imagining the variable as a bag that you can put any number of blocks into can help you see how the value of an expression changes as the value of the variable changes.
Problem Set A In these problems, you will continue to explore explore the situation in which which there are are 3 bags, each contain containing ing the same same number of blocks blocks,, plus 2 extra extra blocks. n
n
n
1. Copy and complete the table. Number of Blocks in Each Bag, Total Number Number of Blocks, Blocks, 3 n
0
n
2
1
2
5
8
3
4
5
is the value value of of 3n ϩ 2? 2. If n ϭ 7, what is is the value value of of 3n ϩ 2? 3. If n ϭ 25, what is many blocks are there there 4. If there are 50 blocks in each bag, how many altogether? altogether, how many many blocks are in each 5. If there are 20 blocks altogether, bag? 6. Copy and complete the table. 10
n
3 n
2
5 17
40
25 77
100
22 23
92
128
3,143 3,
7. Compare the tables in Problems 1 and 6. Which table do you think was more difficult to complete? Why?
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CHAPTER
1
Understanding Expressions Expressions
8.
Could the total number of blocks in this situation be 18? Explain.
9.
To represent the number of blocks in 3 bags plus 2 extra blocks with the expression 3n ϩ 2, you need to assume that all the bags contain the same number of blocks. Why?
10.
The expression 3n ϩ 2 describes the total number of blocks in 3 bags, each with the same number of blocks, plus 2 extra blocks. a. Describe a bags-and-blocks situation that can be represented by
the expression 5n ϩ 6. b. Explain how the expression fits your situation.
You have spent a lot of time exploring the number of blocks in three bags plus two extra blocks. Now you’ll investigate some other bags-and-blocks situations.
Problem Set B 1.
2.
Here are 5 bags and 4 extra blocks.
a.
What is the total number of blocks if each bag contains 3 blocks? If each bag contains 10 blocks?
b.
Using n to represent the number of blocks in each bag, write an algebraic expression for the total number of blocks.
c.
Find the value of your expression for n ϭ 3 and n get the same answers you found in Part a?
ϭ
10. Do you
Now suppose you have 4 bags, each with the same number of blocks, plus 2 extra blocks. a.
Draw a picture of this situation.
b.
Write an expression for the total number of blocks.
3.
Write an expression to represent 7 bags, each with the same number of blocks, plus 5 extra blocks.
4.
Write an expression to represent 10 bags, each with the same number of blocks, plus 1 extra block.
LESSON
1.1
Variables and Expressions
7
5.
Anylettercanbeusedtostandforthenumberofblocksinabag. Matcheachexpressionbelowwitha drawing. 2c ϩ 4
4m ϩ 2
4 y ϩ 5
2 f ϩ 5
a.
b.
c.
d. .
6.
Rebeccawrotetheexpression3 b ϩ 1todescribethetotalnumberof blocksrepresentedinthispicture.
a.
Whatdoesthevariable
b.
Whatdoesthe3standfor?
c.
Whatdoesthe1standfor?
d.
Completethetable forRebecca’sexpression. b 3 b
8
CHAPTER 1
UnderstandingExpressions
1 1
2 7
b stand for in Rebecca’sexpression?
3
23 31
100 76
7. Zoe thought of a new situation: “Imagine that the total number of blocks is 2 blocks less than 3 bags’ worth. This is hard to draw, but I just described it easily in words—and I can write it algebraically as 3n Ϫ 2.” a. Describe a situation that 4n Ϫ 1 could represent. b. Describe a situation that 5 x Ϫ 7 could represent. c. Describe a situation that 14 Ϫ 3 p could represent. 8. Sascha has 1 more block than Chris, and Dean has 1 more block than Sascha. Patrick has just 2 blocks. a. If Sascha has 6 blocks, how many blocks does each boy have? How many do they have altogether? b. If Sascha has 15 blocks, how many blocks does each boy have? How many do they have altogether? c. If you know how many blocks Sascha has, how can you determine the total number of blocks without figuring out how many blocks each of the other boys has? d. If the boys have 26 blocks altogether, how many does each boy have? Explain how you arrived at your answer. e. Let s stand for the number of blocks Sascha has. Write an expression for the number each boy has. Then write an expression for the total number of blocks. f. Let c stand for the number of blocks Chris has. Write an expression for the number each boy has. Then write an expression for the total number of blocks. g. Your expressions for Parts e and f both tell how many blocks the group has, and yet the expressions are different. Explain why.
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Share
Summarize 1. Make a bags-and-blocks drawing. 2. Write an expression that describes your drawing. 3. Explain how you know your expression matches your drawing.
LESSON
1.1
Variables and Expressions
9
