Integrated Mathematics 3 Volume 1 Pdf

Unformatted text preview: Carnegie Learning Integrated Math III © Carnegie Learning Teacher's Implementation Guide 451451_IM3_TIG_FM_Vol1.indd 1 Volume 1 09/12/13 2:07 PM 437 Grant St., Suite 918 Pittsburgh, PA 15219 Phone 412.690.2442 Customer Service Phone 877.401.2527 Fax 412.690.2444 1 2 3 4 5 6 7 8 © © © © © © © © iStockphoto.com/marekuliasz; iStockphoto.com/Cristian Baitg; David Rivera/Aquarium of the Pacific; istockphoto.com/Alistair Forrester Shankie; istockphoto.com/Steve Maehl; istockphoto.com/Carolina K. Smith, M.D.; istockphoto.com/Ryan Kelly; istockphoto.com/Florin Tirlea; Copyright © 2013 by Carnegie Learning, Inc. All rights reserved. Carnegie Learning, Cognitive Tutor, SchoolCare, Software, and Learning by Doing are all registered marks of Carnegie Learning, Inc. All other company and product names mentioned are used for identification purposes only and may be trademarks of their respective owners. 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Any other usage or reproduction in any form is prohibited without the expressed consent of the publisher. © Carnegie Learning Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter ISBN: 978-1-60972-237-1 Teacher's Implementation Guide, Integrated Math III, Volume 1 Printed in the United States of America 1-12/2013 HPS 451451_IM3_TIG_FM_Vol1.indd 2 09/12/13 2:07 PM Carnegie Learning Integrated Math III © Carnegie Learning Teacher's Implementation Guide 451451_IM3_TIG_FM_Vol2.indd 1 Volume 2 09/12/13 2:09 PM 437 Grant St., Suite 918 Pittsburgh, PA 15219 Phone 412.690.2442 Customer Service Phone 877.401.2527 Fax 412.690.2444 9 © istockphoto.com/samxmeg; 10 © istockphoto.com/blackred; 11 © iStockphoto.com/EpicStockMedia; 12 © iStockphoto.com/bartvdd; 13 © iStockphoto.com/Ivan Bliznetsov; 14 © iStockphoto.com/greatpapa; 15 © iStockphoto.com/mbbirdy; 16 © Luca Parmitano/ISS Expedition 36, Volare; Copyright © 2013 by Carnegie Learning, Inc. All rights reserved. Carnegie Learning, Cognitive Tutor, SchoolCare, Software, and Learning by Doing are all registered marks of Carnegie Learning, Inc. All other company and product names mentioned are used for identification purposes only and may be trademarks of their respective owners. Permission is granted for photocopying rights within licensed sites only. Any other usage or reproduction in any form is prohibited without the expressed consent of the publisher. © Carnegie Learning Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter ISBN: 978-1-60972-237-1 Teacher's Implementation Guide, Integrated Math III, Volume 2 Printed in the United States of America 1-12/2013 HPS 451451_IM3_TIG_FM_Vol2.indd 2 09/12/13 2:39 PM Dear Teacher, It is our goal to provide you with instructional materials to support your implementation of the Common Core State Standards for Mathematics (CCSS) and the Standards for Mathematical Practice (SMP). At Carnegie Learning, we analyzed the CCSS and coupled it with the best academic research on teaching and learning practices. The results: a text that introduces and develops mathematical concepts with coherence, combats common student misconceptions, and accommodates all students as well as a variety of classroom implementations. The CCSS and SMP are a call for change. It is our responsibility as educators to create a safe environment to learn, provide appropriate instructional materials, and believe that all students can achieve academic excellence and become productive mathematical thinkers. To produce successful learners, we must support students' effective communication These through dialogue and discussion of different strategies. This textbook resources are encourages active engagement through a student-centered classroom designed to align teaching to environment, which inspires students to learn from each other. It is our learning. intent that students become knowledgeable and independent learners. © Carnegie Learning We realize that students enter your classroom with varying degrees of mathematical experience and success. Prior knowledge that is fragmented or based on memorization rather than a deep conceptual understanding is an unstable foundation for developing mathematical relationships and concepts. This text is intentionally designed to help students make connections, develop a conceptual understanding of mathematics, and Learn by Doing™. Key formative assessment questions geared toward student comprehension are embedded throughout each lesson. It is our recommendation that you take the time at the beginning of each chapter to do the math yourself. This will provide you the first-hand experience necessary to make informed instructional decisions about which parts of the lesson will drive your mathematical goals. Yours in Education, The Carnegie Learning ®Curriculum Development Team 451451_IM3_TIG_FM_Vol1.indd 3 09/12/13 2:07 PM Acknowledgments Carnegie Learning Authoring Team • John Fitsioris • Sandy Bartle Senior Academic Officer Curriculum Developer • David Dengler • Danielle Kandrack Sr. Director, Curriculum Development • Michael Amick Math Editor • Beth Karambelkar Math Editor Curriculum Developer • Allison Dockter • David "Augie" Rivera Math Editor Math Editor Acknowledgments • Joshua Fisher • Lezlee Ross Math Editor Curriculum Developer Contributing Author • Dr. Mary Lou Metz • Jaclyn Snyder Vendors • Bradford & Bigelow • Mind Over Media • Lapiz • eInstruction Special Thanks • Carnegie Learning Managers of School Partnerships for their content review. • Teacher reviewers and students for their input and review of lesson content. • Carnegie Learning Software Development Team for their contributions to research and content. © Carnegie Learning • Cenveo® Publisher Services • Mathematical Expressions • Bookmasters, Inc. • Hess Print Solutions • William S. Hadley for being a mentor to the development team, his leadership, and his pedagogical pioneering in mathematics education. • Amy Jones Lewis for her review of content. • Colleen Wolfe for project management. FM-4 Acknowledgments 451451_IM3_TIG_FM_Vol1.indd 4 09/12/13 2:07 PM Table of Contents 1 Interpreting Data in Normal Distributions 1.1 1 Recharge It! Normal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 #I'mOnline The Empirical Rule for Normal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Below the Line, Above the Line, and Between the Lines Acknowledgments Table of Contents Z-Scores and Percentiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.4 You Make the Call Normal Distributions and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Chapter 1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2 Making Inferences and Justifying Conclusions 2.1 47 For Real? Sample Surveys, Observational Studies, and Experiments . . . . . . . . . . . . . . . . . 49 2.2 Circle Up Sampling Methods and Randomization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.3 Sleep Tight 2.4 How Much Different? Using Statistical Significance to Make Inferences About Populations . . . . . . . . . 85 2.5 DIY Designing a Study and Analyzing the Results . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Chapter 2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 FM-6 © Carnegie Learning Using Confidence Intervals to Estimate Unknown Population Means . . . . . . . . . 71 Table of Contents 451451_IM3_TIG_FM_Vol1.indd 6 09/12/13 2:07 PM 3 Searching for Patterns 3.1 109 Patterns: They're Grrrrrowing! Exploring and Analyzing Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.2 Are They Saying the Same Thing? Using Patterns to Generate Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . 117 3.3 Are All Functions Created Equal? Comparing Multiple Representations of Functions . . . . . . . . . . . . . . . . . . . . . . . 133 3.4 Water Under the Bridge Modeling with Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 3.5 I've Created a Monster, m(x) Analyzing Graphs to Build New Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Chapter 3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Quadratic Functions 4.1 Table of Contents 4 195 Shape and Structure Forms of Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 4.2 Function Sense Translating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 4.3 Up and Down Vertical Dilations of Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 4.4 Side to Side Horizontal Dilations of Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 4.5 What's the Point? Deriving Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 © Carnegie Learning 4.6 Now It's Getting Complex . . . But It's Really Not Difficult! Complex Number Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 4.7 You Can't Spell "Fundamental Theorem of Algebra" without F-U-N! Quadratics and Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Chapter 4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Table of Contents 451451_IM3_TIG_FM_Vol1.indd 7 FM-7 09/12/13 2:07 PM 5 Polynomial Functions 5.1 313 Planting the Seeds Exploring Cubic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 5.2 Polynomial Power Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 5.3 Function Makeover Transformations and Symmetry of Polynomial Functions . . . . . . . . . . . . . . . . . 347 5.4 Polynomial DNA Key Characteristics of Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 5.5 That Graph Looks a Little Sketchy Building Cubic and Quartic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 5.6 Closing Time Acknowledgments Table of Contents The Closure Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Chapter 5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 6 Polynomial Expressions and Equations 6.1 423 Don't Take This Out of Context Analyzing Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 6.2 The Great Polynomial Divide Polynomial Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 6.3 The Factors of Life The Factor Theorem and Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 451 Break It Down Factoring Higher Order Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 6.5 Getting to the Root of It All Rational Root Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 6.6 Identity Theft Exploring Polynomial Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 6.7 © Carnegie Learning 6.4 The Curious Case of Pascal's Triangle Pascal's Triangle and the Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 Chapter 6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 FM-8 Table of Contents 451451_IM3_TIG_FM_Vol1.indd 8 09/12/13 2:07 PM 7 Polynomial Models 7.1 511 Unequal Equals Solving Polynomial Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 7.2 America's Next Top Polynomial Model Modeling with Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 7.3 Connecting Pieces Piecewise Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 7.4 Modeling Gig Modeling Polynomial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 7.5 The Choice Is Yours Comparing Polynomials in Different Representations . . . . . . . . . . . . . . . . . . . . . 553 Chapter 7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 Sequences and Series 8.1 Table of Contents 8 571 Sequence—Not Just Another Glittery Accessory Arithmetic and Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 8.2 This Is Series(ous) Business Finite Arithmetic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 8.3 I Am Having a Series Craving (For Some Math)! Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 8.4 These Series Just Go On . . . And On . . . And On . . . Infinite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 8.5 The Power of Interest (It's a Curious Thing) © Carnegie Learning Geometric Series Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 8.6 A Series of Fortunate Events Applications of Arithmetic and Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . 633 Chapter 8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 Table of Contents 451451_IM3_TIG_FM_Vol1.indd 9 FM-9 31/03/14 5:22 PM Table of Contents 9 Rational Functions 9.1 647 A Rational Existence Introduction to Rational Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 9.2 A Rational Shift in Behavior Translating Rational Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 9.3 A Rational Approach Exploring Rational Functions Graphically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 Acknowledgments Table of Contents 9.4 There's a Hole In My Function, Dear Liza Graphical Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677 9.5 The Breaking Point Using Rational Functions to Solve Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 Chapter 9 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 10 Solving Rational Equations 709 10.1 There Must Be a Rational Explanation Adding and Subtracting Rational Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . 711 10.2 Different Client, Same Deal Multiplying and Dividing Rational Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . 721 10.3 Things Are Not Always as They Appear 10.4 Get to Work, Mix It Up, Go the Distance, and Lower the Cost! Using Rational Equations to Solve Real-World Problems . . . . . . . . . . . . . . . . . . 749 Chapter 10 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 FM-6 © Carnegie Learning Solving Rational Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 Table of Contents 451451_IM3_TIG_FM_Vol2.indd 6 09/12/13 2:09 PM 11 Radical Functions 769 11.1 With Great Power . . . Inverses of Power Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771 11.2 The Root of the Matter Radical Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783 11.3 Making Waves Transformations of Radical Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793 11.4 Keepin' It Real Extracting Roots and Rewriting Radicals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801 11.5 Time to Operate! Multiplying, Dividing, Adding, and Subtracting Radicals. . . . . . . . . . . . . . . . . . . 817 11.6 Look to the Horizon Solving Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 12 Graphing Exponential and Logarithmic Functions Table of Contents Chapter 11 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839 845 12.1 Small Investment, Big Reward Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847 12.2 We Have Liftoff! Properties of Exponential Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857 12.3 I Like to Move It Transformations of Exponential Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871 © Carnegie Learning 12.4 I Feel the Earth Move Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881 12.5 More Than Meets the Eye Transformations of Logarithmic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895 Chapter 12 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909 Table of Contents 451451_IM3_TIG_FM_Vol2.indd 7 FM-7 09/12/13 2:09 PM 13 Exponential and Logarithmic Equations 917 13.1 All the Pieces of the Puzzle Exponential and Logarithmic Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919 13.2 Mad Props Properties of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933 13.3 What's Your Strategy? Solving Exponential Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 941 13.4 Logging On Solving Logarithmic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953 13.5 So When Will I Use This? Applications of Exponential and Logarithmic Equations . . . . . . . . . . . . . . . . . . . 971 Acknowledgments Table of Contents Chapter 13 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983 14 Modeling with Functions 989 14.1 It's Not New, It's Recycled Composition of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 991 14.2 Paint by Numbers Art and Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013 14.3 Make the Most of It Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023 14.4 A Graph Is Worth a Thousand Words Interpreting Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029 14.5 This Is the Title of This Lesson 14.6 Grab Bag Choosing Functions to Model Situations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 Chapter 14 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 FM-8 © Carnegie Learning Fractals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037 Table of Contents 451451_IM3_TIG_FM_Vol2.indd 8 09/12/13 2:09 PM 15 Trigonometric Functions 1067 15.1 A Sense of Déjà Vu Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 15.2 Two Pi Radii Radian Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079 15.3 Triangulation The Sine and Cosine Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089 15.4 Pump Up the Amplitude Transformations of Sine and Cosine Functions . . . . . . . . . . . . . . . . . . . . . . . . . 1101 15.5 Farmer's Tan The Tangent Function. . . . . . . . . . . . . . . . . ...
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