01-16 MAC White-866867File Chapter Resource Masters contain different levels and formats for all tests, as well as intermediate opportunities for assessment. Standardized Test Practice,

In today’s technology-driven society,mathematical skills and understandingare more important than ever. The need to use mathematics with fluency

and comfort occurs daily—not just for thosein the scientific and technical community,but in the workplace and in everyday situations. Those who understand and canuse mathematics will have significantlyenhanced opportunities and options forshaping their careers and futures. The pasttwo decades have seen an increased recogni-tion of the importance of mathematics forevery student and accompanying need for creating uniform national standards in mathematics education. The NationalCouncil of Teachers of Mathematics (NCTM)has led this reform movement from itsbeginning.

NCTM is the world’s largest mathematicseducation organization, with more than100,000 members and 250 affiliates through-out the United States and Canada. Between1989 and 1995, NCTM released a trio of pub-lications on curriculum and evaluation,assessment, and professional standards toarticulate goals for mathematics teachersand policymakers. Since the release of thesepublications, they have given focus, organi-zation, and fresh ideas to efforts to improvemathematics education.

In 2000, NCTM released its most compre-hensive project—the Principles and Standardsfor School Mathematics. The Principles andStandards for School Mathematics representsthe culmination of five years of developmentby the NCTM’s Commission on the Future ofthe Standards and the Standards 2000 Writing

The NCTMPrinciples andStandards forSchoolMathematics

MATHEMATICSWITH UNDERSTANDING

History of the Mathematics Education Reform Movement1989 National Council of Teachers of Mathematics (NCTM) publishes Curriculum and Evaluation Standards for

School Mathematics1991 NCTM publishes Professional Standards for Teaching Mathematics1995 NCTM publishes Assessment Standards for School Mathematics1995 NCTM appoints the Commission on the Future of Standards to oversee the Standards 2000 project1997 The Commission on the Future of Standards appoints the Standards 2000 Writing Group and the Standards

2000 Electronic Format Group1997 to 1999 The Standards 2000 Writing Group, with input from Association Review Groups, the NCTM Research

Advisory Committee, the National Research Council, and more than 650 individuals and 70 groups, writes thePrinciples and Standards for School Mathematics

2000 NCTM publishes the Principles and Standards for School Mathematics

All Standards documents are available at www.nctm.org.

Group. The Standards 2000 Writing Groupincluded teachers, teacher educators, admin-istrators, researchers, and mathematicianswith a wide range of expertise. The first draftwas released in 1998. Over 650 individualsand more than 70 groups, including a com-mittee of experts from the National ResearchCouncil, provided assistance and feedback,and the final version of the Principles andStandards for School Mathematics was releasedin 2000.

The Principles set forth important overallcharacteristics of mathematics programs, andthe Standards describe the mathematical con-tent that students should learn. Together, thePrinciples and Standards for School Mathematicsconstitutes a vision to guide educators asthey strive for the continual improvement of mathematics education in classrooms,schools, and educational systems. The Princi-ples and Standards for School Mathematics areconsistent with the best and most recent evi-dence on teaching and learning mathe-matics; they are chosen through a complexprocess that involves past practice, researchfindings, societal expectations, and the visionof the professional field (Heibert, 1999).

The vision for mathematics educationdescribed in the Principles and Standards forSchool Mathematics is highly ambitious.Achieving this vision requires committed,competent, and knowledgeable teacherswho can integrate instruction with assess-ment, administrative policies that support

learning and access to technology, and solidmathematics curricula.

ACHIEVING THE NCTM PRINCIPLES AND STANDARDS WITH GLENCOEMATHEMATICS: APPLICATIONS ANDCONCEPTSRealizing the Principles and Standards forSchool Mathematics requires raising expecta-tions for students’ learning, developingeffective methods of supporting the learningof mathematics by all students, and provid-ing students and teachers with the resourcesand curricula they need. A school’s or dis-trict’s choice of mathematics curriculum canbe a strong determinant of what studentshave an opportunity to learn.

Glencoe/McGraw-Hill, one of the nation’slargest textbook developers, has risen to thechallenge set by the Principles and Standardsfor School Mathematics and developed theMathematics: Applications and Concepts series.This series was specifically designed withseveral key characteristics recommended bythe Principles and Standards for School Mathe-matics for effective curricula.

■ Different topical strands, such as algebra,geometry, and statistics, that are highlyinterconnected;

■ Central mathematical ideas that are organ-ized and integrated, so that students cansee how the ideas build on, or connectwith, other ideas;

NCTM Principles for School MathematicsEquity Excellence in mathematics education requires equity—high expectations and strong support for all students.

Curriculum A curriculum is more than a collection of activities; it must be coherent, focused on important mathematics, andwell articulated across the grades.

Teaching Effective mathematics teaching requires understanding what students know and need to learn and then chal-lenging and supporting them to learn it well.

Learning Students must learn mathematics with understanding, actively building new knowledge from experience andprior knowledge.

Assessment Assessment should support the learning of important mathematics and furnish useful information to bothteachers and students.

Technology Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught andenhances students’ learning.

Principles and Standards for School Mathematics is available at www.nctm.org.

■ Foundational ideas such as equivalence,proportionality, functions, and rate ofchange;

■ Activities to facilitate development ofmathematical thinking and reasoningskills, including making conjectures anddeveloping sound deductive arguments;

■ Opportunities for experiences thatdemonstrate mathematics’ usefulness in modeling and predicting real-worldphenomena;

■ Guidance for teachers on the depth ofstudy warranted at particular times andwhen closure is expected for particularskills or concepts;

■ Emphasis on the mathematics processesand skills that support the quantitativeliteracy of students, such as judgingclaims, finding fallacies, evaluating risks,and weighing evidence.

PRINCIPLESThe Glencoe Mathematics: Applications andConcepts series was designed to meet all sixof the Principles set forth in the Principles andStandards for School Mathematics.

■ Equity The Glencoe Mathematics:Applications and Concepts encourages highachievement at all academic levels.Prerequisite Skills, at the beginning ofeach chapter and in each lesson in theStudent Edition help teachers assess stu-dent readiness for the upcoming con-cepts. Daily Intervention features in theTeacher Wraparound Edition providesuggestions for addressing various learn-ing styles and helping students who arehaving difficulty. Numerous teacher sup-port materials provide activities for dif-ferentiated instruction, promotion ofreading and writing skills, pacing forindividual levels of achievement, andother intervention opportunities.

■ Curriculum The Glencoe Mathematics:Applications and Concepts program wasdeveloped with a philosophy and scopeand sequence to ensure a continuum ofmathematical learning that builds onprior knowledge and extends conceptstoward more advanced mathematicalthinking. The texts offer a balancedapproach of real-world applications,

hands-on labs, writing exercises, andpractice that enables students to developboth conceptual understanding and pro-cedural knowledge.

■ Teaching Glencoe recognizes that theteacher is a vital factor in students’ aca-demic achievement. The Glencoe Mathe-matics: Applications and Concepts serieswas designed with Teacher WraparoundEditions and multiple supplementaryresources to provide teachers with exten-sive support for excellence in teaching.The comprehensive Teacher WraparoundEditions provide Mathematical Contentand Teaching Strategies at the beginningof each chapter to summarize mathe-matical content and skills in each lesson.The Continuity of Instruction sectiondescribes prerequisite student knowl-edge, the chapter content, and futureconnections to more advanced concepts.Teacher to Teacher presents suggestionsfrom experienced teachers, and Tips forNew Teachers throughout the chaptersalert new teachers to potential difficultiesand suggests instructional strategies.

■ Learning The Glencoe Mathematics:Applications and Concepts series offersextensive support to help all studentsachieve success in mathematics. KeyConcept and Concept Summary boxes inthe Student Editions help students iden-tify main concepts, and Study Tips in themargins help students understand newmaterial. Foldables™ Study Organizershelp students focus on the organizationand analysis of main ideas and vocabu-lary. How to . . . Study Skills and How toUse Your Math Textbook help studentsorganize, understand, and retain courseinformation to get the most from theStudent Editions.

■ Assessment The Glencoe Mathematics:Applications and Concepts series offers mul-tiple strategies for the teacher to assessstudent learning, as well as student self-assessment. Mid-Chapter Practice Testsand Chapter Practice Tests provide waysfor students to check their own progress.Online study tools, such as Self-CheckQuizzes, offer a unique way for studentsto use the Internet to monitor theirprogress. The assessment tools in the Fast

File Chapter Resource Masters containdifferent levels and formats for all tests,as well as intermediate opportunities forassessment. Standardized Test Practice,aligned with standardized tests, is alsowoven throughout exercises and chaptersto offer consistent opportunities to learn,practice, and apply test-taking skills.

■ Technology The Glencoe Mathematics:Applications and Concepts Student Editionsoffer extensive opportunities to utilizegraphing calculators, spreadsheets, andgeometry software in the exploration ofalgebra and geometry concepts. Calcula-tor tips are included in the Study Tipsfeature in the Student Edition margins,and Graphing Calculator Investigationsand Spreadsheet Investigations areincluded in each course. The TeacherWraparound Editions offer teaching tipson using technology, and the GraphingCalculator and Computer Masters offeradditional activities. In addition,Glencoe’s Web site is constantly updatedto offer innovative uses of technology inmathematics lessons.

STANDARDSThe Glencoe Mathematics: Applications andConcepts series was also designed to meet all

of the Principles and Standards for SchoolMathematics’ Content Standards. The ContentStandards state that instructional programsfrom pre-kindergarten through grade 12should enable all students to master specificskills in ten content areas.

RESEARCH-BASED INSTRUCTIONALSTRATEGIES USED IN GLENCOE MATHEMATICS: APPLICATIONS AND CONCEPTSIn addition to responding to the goals set by the Principles and Standards for SchoolMathematics, extensive efforts were under-taken to ensure that the latest research onbest practices in mathematics education was used in the development of the Glencoe Mathematics: Applications and Con-cepts series. Educational research serves as abasis for many of the assertions madethroughout about what is possible for stu-dents to learn about certain content areas atcertain levels and under certain pedagogicalconditions.

Glencoe Mathematics: Applications and Con-cepts was specifically designed to utilize sev-eral important research-based instructionalstrategies that reinforce the Principles andStandards for School Mathematics.

Content Area and Specific Standards from NCTM Principles and Standards for

School Mathematics

Examples from Glencoe Mathematics: Applications and Concepts (page numbers)

Numbers and Operations• Understand numbers, ways of representing numbers, relationships

among numbers, and number systems• Understand the meaning of operations and how they relate to

each other• Compute fluently and make reasonable estimates

Course 1: 6–41, 75–89, 102–124, 134–164, 177–209, 218–247, 256–285,294–323, 333–369, 380–417, 428–453, 465–497, 506–517, 546–578Course 2: 6–21, 24–27, 30–36, 38–41, 43–45, 69–72, 76–79, 106–115,120–124, 128–131, 134–141, 150–152, 156–163, 166–169, 172–175, 177–185,197–200, 203–213, 216–231, 240–251, 254–261, 264–273, 278–280, 288–295,297–300, 304–308, 312–321, 323–325, 334–337, 340–343, 345–347,350–360, 370–383, 387–390, 393–396, 398–401, 413–415, 418–425, 428–431,434–437, 440–443, 446–454, 456–459, 470–473, 475–477, 479–485,489–495, 498–503, 514–517, 520–522, 524, 527, 532–535, 538–545Course 3: 11–15, 17–21, 23–31, 34–38, 45–53, 62–80, 82–85, 88–95,98–101, 104–107, 116–122, 125–129, 132–140, 142–145, 156–164, 166–173,178–182, 184–191, 194–197, 206–214, 216–223, 228–244, 256–260,262–265, 267–270, 272–275, 279–282, 286–294, 296–303, 314–323,326–329, 331–339, 342–345, 347–355, 358–362, 374–377, 380–391,396–403, 406–409, 420–424, 426–433, 435–438, 442–457, 469–481,484–487, 492–504, 511–515, 517–520, 522–529, 533–536, 539–542,544–551, 560–563, 565–568, 570–577, 580–587, 590–592

How Glencoe Meets the Standards

Data Analysis and Probability• Formulate questions that can be addressed with data and collect,

organize and display relevant data to answer them• Select and use appropriate statistical methods to analyze data• Develop and evaluate inferences and predictions that are based on

data• Understand and apply basic concepts of probability

Course 1: 50–89, 386–389, 415–417, 428–431, 433–436, 438–441,444–447, 450–453Course 2: 54–57, 60–72, 76–83, 85–89, 92–95, 345–347Course 3: 62–80, 104–107, 374–377, 380–391, 396–403, 406–409,420–424, 426–433, 435–438, 442–457

Algebra• Understand patterns, relations, and functions• Represent and analyze mathematical situations and structures using

algebraic symbols• Use mathematical models to represent and understand quantitative

relationships• Analyze change in various contexts

Course 1: 18–21, 28–41, 121–124, 135–164, 219–247, 259–285, 310–323,333–369, 428–453, 564–566, 570–573Course 2: 14–21, 24–27, 30–36, 43–45, 106–115, 120–124, 128–131,134–141, 150–152, 156–163, 166–169, 177–185, 197–200, 203–213, 216–231,258–261, 264–266, 270–273, 275–277, 297–300, 304–308, 312–315,319–321, 323–325, 340–343, 345–347, 350–360, 378–383, 387–390,413–415, 418–425, 428–431, 434–437, 440–443, 446–454, 456–459,470–473, 475–477, 479–485, 489–495, 498–503, 514–517, 520–522,524–527, 532–535, 538–545Course 3: 11–15, 17–21, 28–31, 34–42, 45–53, 62–75, 82–85, 88–95,98–101, 116–119, 132–140, 142–145, 160–164, 166–173, 178–182, 184–191,194–197, 210–214, 216–219, 232–244, 256–260, 262–265, 267–270,272–275, 279–282, 286–294, 296–303, 314–323, 326–329, 331–339,342–345, 347–355, 358–362, 400–403, 420–424, 426–433, 435–438,442–457, 469–481, 484–487, 492–504, 517–520, 522–529, 533–536,539–542, 544–551, 565–568, 570–573, 584–587, 590–592

Geometry• Analyze characteristics and properties of two- and three-dimensional

geometric shapes and develop mathematical arguments about geo-metric relationships

• Specify locations and describe spatial relationships using coordinategeometry and other representational systems

• Apply transformations and use symmetry to analyze mathematical situations

• Use visualization, spatial reasoning, and geometric modeling to solveproblems

Course 1: 39–41, 158–164, 240–247, 256–284, 366–369, 506–537Course 2: 112–115, 177–185, 270–273, 275–277, 479–485, 489–495,498–503, 524–527Course 3: 132–140, 142–145, 166–169, 178–182, 184–191, 194–197,256–260, 262–265, 267–270, 272–275, 279–282, 286–294, 296–303,314–323, 326–329, 331–339, 342–345, 347–355, 358–362, 522–529,533–536, 539–542, 544–551

Measurement• Understand measurable attributes of objects and the units, systems

and processes of measurement• Apply appropriate techniques, tools and formulas to determine

measurements

Course 1: 158–164, 240–247, 380–383, 391–393, 465–468, 470–473,476–479, 484–487, 490–497, 506–512, 515–517Course 2: 38–41, 80–83, 244–251, 254–257, 267–273, 275–277, 288–295,297–300, 304–308, 370–383, 483–485, 489–495, 498–503Course 3: 71–80, 132–140, 142–145, 178–182, 184–191, 256–260,262–265, 267–270, 272–275, 279–282, 286–294, 296–303, 314–323,326–329, 331–339, 342–345, 347–355, 358–362, 420–424, 426–433,


School Mathematics


Problem Solving• Build new mathematical knowledge through problem solving• Solve problems that arise in mathematics and in other contexts• Apply and adapt a variety of appropriate strategies to solve problems• Monitor and reflect on the process of mathematical problem solving

Course 1: 6–13, 24–27, 34–37, 39–41, 66–69, 72–75, 80–83, 86–89,102–105, 108–113, 116–119, 121–124, 135–138, 141–147, 152–155, 158–164,177–180, 182–189, 194–209, 219–225, 228–231, 235–238, 240–247,256–258, 261–267, 272–279, 282–284, 294–298, 300–303, 316–319,333–336, 339–342, 344–347, 350–353, 355–357, 362–369, 380–383,386–389, 391–393, 395–397, 400–406, 409–412, 415–417, 428–431,433–436, 438–441, 444–447, 450–453, 465–468, 470–473, 476–479,484–487, 490–497, 506–512, 515–517, 522–525, 528–531, 534–536,546–549, 551–554, 556–559, 564–566, 570–573, 575–578

Reasoning and Proof• Recognize reasoning and proof as fundamental aspects of

mathematics• Make and investigate mathematical conjectures• Develop and evaluate mathematical arguments and proofs• Select and use various types of reasoning and methods of proof

Course 1: 6–13, 66–69, 102–105, 108–113, 116–119, 121–124, 177–180,182–189, 194–209, 256–258, 261–267, 272–279, 282–284, 294–298,300–303, 333–336, 339–342, 344–347, 350–353, 355–357, 362–369,380–383, 386–389, 391–393, 395–397, 400–406, 409–412, 415–417,428–431, 433–436, 438–441, 444–447, 450–453, 465–468, 470–473,476–479, 484–487, 490–497, 506–512, 515–517, 522–525, 528–531,534–536, 546–549, 551–554, 556–559, 564–566, 570–573, 575–578Course 2: 6–21, 24–27, 30–36, 38–41, 43–45, 60–63, 85–89, 92–95,106–115, 120–124, 128–131, 134–141, 150–152, 156–163, 166–169, 172–175,177–185, 240–251, 254–261, 264–273, 275–277, 334–337, 340–343,345–347, 350–360, 470–473, 475–477, 479–485, 489–495, 498–503,514–517, 520–522, 524–527, 532–535, 538–545Course 3: 6–15, 116–122, 125–129, 132–140, 142–145, 156–164, 166–173,178–182, 184–191, 194–197, 206–214, 216–223, 228–244, 256–260,262–265, 267–270, 272–275, 279–282, 286–294, 296–303, 314–323,326–329, 331–339, 342–345, 347–355, 358–362, 374–377, 380–391,396–403, 406–409. 420–424, 426–433, 435–438, 442–457, 469–481,484–487, 492–504, 511–515, 517–520, 522–529, 533–536, 539–542,544–551, 560–563, 565–568, 570–577, 580–587, 590–592


School Mathematics


Problem Solving (cont.)

Communication• Organize and consolidate their mathematical thinking through

communication• Communicate their mathematical thinking coherently and clearly to

peers, teachers, and others• Analyze and evaluate the mathematical thinking and strategies of others• Use the language of mathematics to express mathematical ideas

precisely

Course 1: 34–37, 50–75, 86–89, 198–201, 366–369, 380–383, 386–389,391–393, 395–397, 400–406, 409–412, 415–417, 465–468, 470–473,476–479, 484–487, 490–497, 522–525, 528–531, 534–536

Course 2: 6–21, 24–27, 30–36, 38–41, 43–45, 106–115, 120–124, 128–131,134–141, 150–152, 156–163, 166–169, 172–175, 177–185, 240–251, 254–261,264–273, 275–277, 334–337, 340–343, 345–347, 350–360, 514–517,520–522, 524–527, 532–535, 538–545

Course 3: 39–42, 116–122, 125–129, 132–140, 142–145, 156–164, 166–173,178–182, 184–191, 194–197, 380–391, 396–403, 406–409

Connections• Recognize and use connections among mathematical ideas

• Understand how mathematical ideas build on one another to producea coherent whole

• Recognize and apply mathematics in contexts outside of mathematics

Course 1: 14–27, 34–37, 56–59, 66–69, 72–75, 80–83, 86–89, 102–105,108–113, 116–119, 121–124, 135–138, 141–147, 152–155, 158–164, 219–225,228–231, 235–238, 240–247, 256–258, 261–267, 272–279, 282–284,304–307, 310–313, 316–319, 339–342, 344–347, 350–353, 355–357, 362–369,380–383, 386–389, 391–393, 395–397, 400–406, 409–412, 415–417,428–431, 433–436, 438–441, 444–447, 450–453, 465–468, 470–473,476–479, 484–487, 490–497, 506–512, 515–517, 522–525, 528–531,534–536, 546–549, 551–554, 556–559, 564–566, 570–573, 575–578

Course 2: 6–21, 24–27, 30–36, 38–41, 43–45, 60–63, 106–115, 120–124,128–131, 134–141, 150–152, 156–163, 166–169, 172–175, 177–185, 197–200,203–213, 216–231, 240–251, 254–261, 264–273, 275–277, 292–295, 297–300,304–308, 312–315, 319–321, 323–325, 334–337, 340–343, 345–347, 350–360,374–383, 387–390, 398–401, 413–415, 418–425, 428–431, 434–437,440–443, 446–454, 456–459, 470–473, 475–477, 479–485, 489–495,498–503, 514–517, 520–522, 524–527, 532–535, 538–545Course 3: 6–10, 45–49, 71–80, 88–95, 104–107, 116–122, 125–129, 132–140,142–145, 156–164, 166–173, 178–182, 184–191, 194–197, 206–214, 216–223,228–244, 256–260, 262–265, 267–270, 272–275, 279–282, 286–294,296–303, 314–323, 326–329, 331–339, 342–345, 347–355, 358–362,374–377, 380–391, 396–403, 406–409, 420–424, 426–433, 435–438,442–457, 469–481, 484–487, 492–504, 511–515, 517–520, 522–529, 533–536,539–542, 544–551, 560–563, 565–568, 570–577, 580–587, 590–592

Representation• Create and use representations to organize, record, and communicate

mathematical ideas• Select, apply, and translate among mathematical representations to

solve problems• Use representations to model and interpret physical, social, and math-

ematical phenomena

Course 1: 6–41, 50–89, 102–105, 108–113, 116–119, 121–124, 135–138,141–147, 152–155, 158–164, 177–180, 182–189, 194–209, 219–225, 228–231,235–238, 240–247, 256–258, 261–267, 272–279, 282–284, 294–298,300–307, 310–313, 316–323, 333–336, 339–342, 344–347, 350–353,355–357, 362–369, 380–383, 386–389, 391–393, 395–397, 400–406,409–412, 415–417, 428–431, 433–436, 438–441, 444–447, 450–453,465–468, 470–473, 476–479, 484–487, 490–497, 506–512, 515–517,522–525, 528–531, 534–536, 546–549, 551–554, 556–559, 564–566,570–573, 575–578Course 2: 6–21, 24–27, 30–36, 38–41, 43–45, 54–57, 60–72, 76–83,85–89, 92–95, 106–115, 120–124, 128–131, 134–141, 150–152, 156–163,166–169, 172–175, 177–185, 197–200, 203–213, 216–231, 240–251, 254–261,264–273, 275–277, 288–295, 297–300, 304–308, 312–321, 323–325,334–337, 340–343, 345–347, 350–360, 370–383, 387–390, 393–396,398–401, 413–415, 418–425, 428–431, 434–437, 440–443, 446–454,456–459, 470–473, 475–477, 479–485, 489–495, 498–503, 514–517,520–522, 524–527, 532–535, 538–545Course 3: 6–15, 17–21, 23–31, 45–49, 62–80, 98–101, 104–107, 116–122,125–129, 132–140, 142–145, 156–164, 166–173, 178–182, 184–191, 194–197,206–214, 216–223, 228–244, 256–260, 262–265, 267–270, 272–275,279–282, 286–294, 296–303, 314–323, 326–329, 331–339, 342–345,347–355, 358–362, 374–377, 380–391, 396–403, 406–409, 420–424,426–433, 435–438, 442–457, 469–481, 484–487, 492–504, 511–515,517–520, 522–529, 533–536, 539–542, 544–551, 560–563, 565–568,570–577, 580–587, 590–592


School Mathematics


Connections (cont.) Course 2: 6–21, 24–27, 30–36, 38–41, 43–45, 80–83, 92–95, 106–115,120–124, 128–131, 134–141, 150–152, 156–163, 166–169, 172–175, 177–185,197–200, 203–213, 216–231, 240–251, 254–261, 264–273, 275–277,288–295, 297–300, 304–308, 312–321, 323–325, 334–337, 340–343,345–347, 350–360, 370–383, 387–390, 393–396, 398–401, 413–415,418–425, 428–431, 434–437, 440–443, 446–454, 456–459, 470–473,475–477, 479–485, 489–495, 498–503, 514–517, 520–522, 524–527,532–535, 538–545Course 3: 17–21, 39–42, 116–122, 125–129, 132–140, 142–145, 156–164,166–173, 178–182, 184–191, 194–197, 216–219, 228–244, 262–265,267–270, 272–275, 279–282, 286–294, 296–303, 319–323, 326–329,331–339, 342–345, 347–355, 358–362, 374–377, 380–391, 396–403,406–409, 469–481, 484–487, 492–504, 522–525, 533–536, 539–542,544–551, 570–573, 580–587, 590–592

1. Balancing implicit and explicit learning

Research shows that teachers cannot sim-ply transfer knowledge to students by lec-turing. Students have to take an active role intheir own learning, and to accomplish this,mathematics programs must include ampleopportunity to explore, question, discuss,and discover. This is not to say that teachersare removed from the educational process.Rather, the learning experience shouldinclude a balance of implicit and explicitinstruction. Implicit instruction occurs whenstudents figure out for themselves how tograpple with problems and construct con-ceptual knowledge (Pressley, Harris, &Marks, 1992; Shulman & Keislar, 1996).Explicit instruction occurs when teachersand textbooks clearly explain problem-solving strategies to students in a direct,low-inference fashion (Duffy, 2002).

The Glencoe Mathematics: Applications andConcepts series offers a balanced approach ofreal-world applications, hands-on labs, writ-ing exercises, and practice (see Figure 1).Application problems give students theopportunity to use the skills they havelearned in a real-world setting, and CriticalThinking exercises require students toexplain, make conjectures, and prove mathe-

matical relationships. Calculator and Spread-sheet Investigations use technology to promote the discovery of patterns and rela-tionships, and Modeling, Speaking, andWriting activities in the Assess sections of theTeacher Wraparound Editions require stu-dents to summarize what they have learnedby responding to open-ended prompts.

2. Using prior knowledge to learn new information

Prior knowledge strategies help studentsretrieve information stored in their long-term memories to learn new, related infor-mation. These strategies include recallingremembered information, asking questions,and elaborating on textbook and teacherinformation, and referring students to thetextbook and other meaningful information(including use of analogies). The GlencoeMathematics: Applications and Concepts seriesintertwines concepts and continuously refersto material in previous chapters and in students’ personal experiences to makemathematics more relevant.

Asking students to use prior knowledgelocated in a text may remind them of information already in their long-term mem-ory that, for some reason, is not easily

Mean

A number that helps describe all of the data in a data set is an

, or a . One of the most

common measures of central tendency is the mean.measure of central tendency

average

Find Mean

MONEY The cost of fifteen

different backpacks is

shown. Find the mean.

mean �

← sum of the data

← number of data items

� �51150� or 34

The mean cost of the backpacks is $34.

19 � 25 � 30 � ... � 22��

15

76 Chapter 2 Statistics and Graphs

Work with a partner.

Suppose the table at the right shows your

scores for five quizzes.

• Place pennies in each cup to represent

each score.

• Move the pennies from one cup to another cup so that each

cup has the same number of pennies.

1. How many pennies are in each cup?

2. For the five quizzes, your average score was points.

3. Suppose your teacher gave you another quiz and you scored

14 points. How many pennies would be in each cup?

?

8 7 9 6 10

• 40 pennies

• 5 plastic cups

Quiz Score

1 8

2 7

3 9

4 6

5 10

Mean

Words The of a set of data is the sum of the data divided by

the number of pieces of data.

Example data set: 8, 7, 9, 6, 10 → mean: � �

450� or 88 � 7 � 9 � 6 � 10

��5

mean

Backpack Costs (S| )

194522

254035

305045

304649

272522

Find the mean of a set

of data.

averagemeasure of central

tendency

meanoutlier

Lesson 2-6 Mean 77F. Stuart Westmorland/Photo Researchers

1. Explain how to find the mean of a set of data.

2. OPEN ENDED Write a set of data that has an outlier.

3. DATA SENSE Choose the correct value for n in the data set 40, 45, 48, n,42, 41 that makes each sentence true.

a. The mean is 44. b. The mean is 45.

Find the mean for each set of data.

4. 25, 30, 33, 23, 27, 31, 27 5. 38, 52, 54, 48, 40, 32

GEOGRAPHY For Exercises 6–8, use the table at the right. It lists the average depths of the oceans.

6. What is the mean of the data?

7. Which depth is an outlier? Explain.

8. How does this outlier affect the mean of the data?

Exercise 1

In statistics, a set of data may contain a value much higher or lowerthan the other values. This value is called an . Outliers cansignificantly affect the mean.

outlier

Determine How Outliers Affect Mean

WEATHER Identify the outlier in the temperature data. Then find the mean with and without the outlier. Describe how the outlier affects the mean of the data.

Compared to the other values, 40°F is extremely low. So, it is an outlier.

mean with outlier mean without outlier

mean � mean �

� �36

50

� or 72 � �32

40

� or 80

With the outlier, the mean is less than all but one of the datavalues. Without the outlier, the mean better represents the valuesin the data set.

80 � 81 � 77 � 82��

480 � 81 � 40 � 77 � 82��

5

Day Temp. (°F)

Monday 80Tuesday 81Wednesday 40Thursday 77Friday 82

Pacific 15,215Atlantic 12,881Indian 13,002Arctic 3,953Southern 14,749

Depths of World’s Oceans

Ocean Depth (ft)

Source: www.enchantedlearning.com

msmath1.net/extra_examples78 Chapter 2 Statistics and GraphsDavid Weintraub/Stock Boston

20. MULTIPLE CHOICE Which piece of data in the data set 98, 103, 96,147, 100, 85, 546, 120, 98 is an outlier?103 120 147 54621. MULTIPLE CHOICE Benito scored 72 points in 6 games. What was the

mean number of points he scored per game?68

121322. SCHOOL The ages of the teachers at Fairview Middle School

are shown in the stem-and-leaf plot. Into what intervals do most of the data fall? (Lesson 2-5)

23. Which type of graph is best used to make predictions over time? (Lesson 2-4)

IH

GF

DC

BA

Find the mean for each set of data.9. 13, 15, 17, 12, 1310. 28, 30, 32, 21, 29, 28, 28

11. 76, 82, 75, 8712. 13, 17, 14, 16, 16, 14, 16, 14

13.14.

NATURE For Exercises 15–17, use the table that shows the approximate heights of some of the tallest U.S. trees.15. Find the mean of the data.16. Identify the outlier.

17. Find the mean if the Coast Redwood is not included in the data set.18. BABY-SITTING Danielle earned $15, $20, $10, $12, $20, $16, $18, and $25 baby-sitting. What is the mean of the amounts she earned?19. CRITICAL THINKING Write a set of data in which the mean is affected by an outlier.

Extra Practice See pages 598, 625.

For Exercises9–14, 15,

17–1816, 19

See Examples1

2

Price Tally Frequency$25 II 2$50 IIII 4$60 I 1$70 III 3

Stem Leaf7 0 2 5 68 0 0 0 0 29 1 3 6

10 3 4 8 93 � 93

Stem Leaf2 3 83 8 94 1 4 5 6 7 95 0 0 5 86 3 41 � 41 years

PREREQUISITE SKILL Subtract. (Page 589)24. 125 � 76 25. 236 � 89 26. 175 � 106 27. 224 � 156

Source: The World Almanac

Largest Trees in U.S.Tree Height (ft)

Western Red Cedar 160Coast Redwood 320Monterey Cypress 100California Laurel 110Sitka Spruce 200Port-Orford-Cedar 220

msmath1.net/self_check_quiz

Figure 1

Course 1, pages 76–78

remembered (Bransford, 1979; Pressley &McCormick, 1995). In the Teacher Wrap-around Editions, the Building on PriorKnowledge section in Continuity of Instruc-tion summarizes the skills and knowledgestudents should have mastered prior to eachchapter, and Mathematical Background pro-vides an overview of the mathematics ineach lesson and links to prior knowledgeand future concepts (see Figure 2).

3. Practicing important tasks and skills

Providing students with practice onimportant tasks has long been considered asuccessful strategy to improve understand-ing and memory. Practicing helps studentsacquire additional information as theysearch and productively struggle, withteacher guidance, for understanding andapplication of mathematical information.Research shows that mastering a skillrequires focused practice. During practice,students adapt and shape what they havelearned. In doing so, they increase their con-

ceptual understanding of the skill(Clement, Lockhead, & Mink, 1979;Davis, R.B., 1984; MathematicalScience Education Board, 1990;Romberg & Carpenter, 1986).

In Glencoe Mathematics: Applica-tions and Concepts, Practice andApplication Exercises correspondto Guided Practice exercises, whichare structured so that studentspractice the same concepts whetherthey are assigned odd- or even-numbered problems. HomeworkHelp refers students to examples asthey complete these exercises. Eachlesson also contains StandardizedTest problems correlated to contentin the lesson or to content in previ-ous chapters. This enables studentsto practice skills in different formsand to gain experience in takingstandardized tests. (See Figure 3.)

4. Note-Taking

In the process of note-taking,students identify the importantitems from reading and write thatinformation in an organized for-mat. While writing and drawing

Figure 3


472 Chapter 10 Algebra: More Equations and Inequalities

1. Define like terms.

2. OPEN ENDED Write an expression that has four terms and simplifies to 3n � 2. Identify the coefficient(s) and constant(s) in your expression.

3. Which One Doesn’t Belong? Identify the expression that is not equivalent to the other three. Explain your reasoning.

5x - 36 + 5x - 95(x - 3)x - 3 + 4x

Use the Distributive Property to rewrite each expression.

4. 5(x � 4) 5. �3(a � 9) 6. �6(g � 2)

Identify the terms, like terms, coefficients, and constants in eachexpression.

7. 8a � 4 � 6a 8. 7 � 3d � 8 � d 9. 5n � n � 3 � 2n

Simplify each expression.

10. 5x � 2x 11. 8n � n 12. 10y � 17y

13. 12c � c 14. 4p � 7 � 6p 15. 11x � 12 � 6x � 9

Exercises 1 & 3

Use the Distributive Property to rewrite each expression.

16. 3(x � 8) 17. 7(m � 6) 18. �8(b � 5)

19. �7(n � 2) 20. �4(k � 8) 21. (c � 8)(�8)

22. �5(a � 9) 23. (x � 6)(�4) 24. 2(a � b)

25. 4(x � y) 26. 3(2y � 1) 27. �4(3x � 5)

GEOMETRY Write two equivalent expressions for the area of each figure.

28. 29. 30. 31.

Identify the terms, like terms, coefficients, and constants in eachexpression.

32. 2 � 3a � 9a 33. 7 � 5x � 1 34. 4 � 5y � 6y � y

35. n � 4n � 7n � 1 36. �3d � 8 � d � 2 37. 9 � z � 3 � 2z

Simplify each expression.

38. 4y � 7y 39. n � 5n 40. 12x � 5x 41. 4k � 7k

42. 10k � k 43. 5x � 4 � 9x 44. 2 � 3d � d 45. 6 � 4c � c

46. 2m � 5 � 8m 47. 3r � 7 � 3r 48. 9y � 4 � 11y � 7 49. 3x � 2 � 10 � 3x

x � 3

18x � 4

16

12

x � 7x � 5

10

Extra Practice See pages 640, 657.

For Exercises

16–31

32–37

38–49

50–53

See Examples

1–4

5

6–8

9

Lesson 10-1 Simplifying Algebraic Expressions 473

55. SHORT RESPONSE Write an expression in simplest form for the perimeter of the figure.

56. MULTIPLE CHOICE Dustin is 3 years younger than his oldersister. If his older sister is y years old, which expression represents the sum of their ages?

2y � 3 y � 3 y2 � 3 2y � 3

State the dimensions of each matrix. Then identify the position of thecircled element. (Lesson 9-8)

57. [3 �2] 58. � � 59. � � 60. � �

TECHNOLOGY For Exercises 61 and 62, refer to the graphs at the right. (Lesson 9-7)

61. Which graph gives the impression that the number of DVD players sold in 2001 was more than 5 times the amount sold in 1999?

62. About how many times more DVD’s were sold in 2001 than in 1999?

DVD Player Sales

Sale

s(m

illio

ns)

Year

’99 ’00 ’01

DVD Player Sales

Sale

s(m

illio

ns)

Year

1412108640

’99 ’00 ’01

121086420

Graph A Graph B

9 3 5�4 7 1

4�2

7

�4 50 2

DCBA

PREREQUISITE SKILL Solve each equation. Check your solution.(Lessons 1-8 and 1-9)

63. x � 8 � 2 64. y � 5 � �9 65. 32 � �4n 66. �3a

� � �15

6x

4x

x � 3 x � 3

For Exercises 50–53, write an expression in simplest form that representsthe total amount in each situation.

50. MOVIES You buy 2 drinks that each cost x dollars, a large bag of popcorn for $3.50, and a chocolate bar for $1.50.

51. PHYSICAL EDUCATION Each lap around the school track is a distance of y yards. You ran 2 laps on Monday, 3�

12

� laps on Wednesday, and 100 yardson Friday.

52. SHOPPING You buy x shirts that each cost $15.99, the same number ofjeans for $34.99 each, and a pair of sneakers for $58.99.

53. FUND-RAISING You have sold t tickets for a school fund-raiser. Yourfriend has sold 24 more than you.

54. CRITICAL THINKING Is 2(x � 1) � 3(x � 1) � 5(x � 1) a true statement? If so, explain your reasoning. If not, give a counterexample.

msmath3.net/self_check_quiz

510C Chapter 12

Drawing Three-DimensionalFigures

Drawing three-dimensional figures is animportant skill because it allows us to representwhat we see around us in our everyday lives.Students already have experience in drawingpolygons and circles. As they combine theseshapes to draw three-dimensional figures, makesure they use dotted lines appropriately torepresent portions of their figures that are notvisible from the outside.

Volume of Rectangular PrismsIn this lesson, students apply their knowledgeof finding the areas of rectangles to calculatethe volumes of rectangular prisms. While many students will be inclined to find volumeby multiplying the length, width, and height,encourage them to begin by determining thearea of the rectangular base and thenmultiplying by the height. Proceeding in thismanner will help them understand futureconcepts of finding the volumes of cylindersand various right prisms.

Building on Prior Knowledge Students have learned about three-dimensionalfigures in previous courses. They know how tomultiply fractions and how to multiply decimals.Students have learned to find the areas ofrectangles and the areas of circles, and they havelearned to find the circumference of a circle.Students have measured objects in previous mathcourses and in real life.

Learning in This Chapter Students learn how to classify and draw three-dimensional figures. They apply what they havelearned about finding the areas of rectangles tocalculate the volumes and surface areas ofrectangular prisms. They also apply their knowledgeof area and circumference of circles to find volumesand surface areas of cylinders. Finally, they learn tomeasure objects using significant digits.

Making Future Connections Three-dimensional figures appear in everyday life,and are studied in mathematics courses throughoutmiddle school and high school. Calculatingvolumes and surface areas of rectangular prismsand cylinders requires important skills that areessential for understanding concepts studied inhigh school geometry. Students will encountersignificant digits in science classes such as physicsand chemistry.

Bulletin BoardHave students work in pairs to design a rectangular prismor cylinder that could be used to package a fictitiousproduct. Give students the basic dimensions of theproduct and ask them to construct their packages usingpaper or cardboard. They should also find the volumeand surface area of the package and record thesemeasurements. As students complete their projects,they should attach them to the bulletin board.

In-Class SpeakerInvite an architect to speak to students about applying theconcept of volume to the design of buildings and homes.Ask the architect to discuss the procedures for calculatingthe total amount of space in each room, as well as in theentire structure. Another topic for discussion could beselecting a heating or air conditioning unit that iscommensurate with the amount of space in the structure.

Figure 2

Course 2, Teacher Wraparound Edition, page 510C

notes, students see relationships within theinformation. Notes need not be verbatim;note-taking is most valuable when studentslearn to analyze information and select theimportant points (Bretzing & Kulhary, 1979).When study skills, such as note-taking aretaught within the teaching of content, theypromote learner activity and improve meta-cognition (Hattie et al., 1996; Robinson &Kiewra, 1995).

The Glencoe Mathematics: Applications andConcepts texts include instructions for studyorganizers, called Foldables™, created byDinah Zike. These are handmade paper book-lets, folded and cut into tabs (see Figure 4).Designed to fit each chapter’s content, theFoldable™ guides students in choosing theimportant concepts and recording them inan organized format. Since students maketheir own three-dimensional Foldables™ aswell as enter the notes, they feel a sense ofownership. The Study Guide and Reviewfeature at the end of each chapter consists ofstep-by-step examples and practice exer-cises. The Lesson-by-Lesson Exercises andExamples present a clear picture of theimportant concepts in each lesson, reviewsvocabulary, and provides students with a model for how they might take notes. The How to . . . Study Skill pages includestrategies that students can use to read,study, and comprehend the mathematicsthey are learning.

5. Feedback

Providing students individual feedbackon their practice helps in monitoring andfostering their mathematical learning, andhaving students continue working on a task until they succeed can increase self-confidence (Bangert-Drowns, Kulik, Kulik &Morgan, 1991; Crooks, 1988). Feedback ismost useful when students are told how theyare performing relative to a specific learningobjective, rather than in relationship to otherstudents (Crooks, 1988), and the greatestimprovement occurs when feedback is givenimmediately after a test or activity (Bangert-Drowns, Kulik, Kulik & Morgan, 1991;Crooks, 1988). Students can also providesome of their own feedback by tracking theirperformance and comparing answers toanswer keys (Wiggins, 1993).

Glencoe Mathematics: Applications and Con-cepts includes Your Turn practice problemsdirectly following many examples (seeFigure 5). These allow students to receive

Chapter 8 Getting Started 293

Take this quiz to see whether you are ready tobegin Chapter 8. Refer to the page number inparentheses if you need more review.

Vocabulary ReviewChoose the correct term to completeeach sentence.

1. The (sum, ) of 3 and 4 is 12.(Page 590)

2. The result of dividing two numbersis called the (difference, ).(Page 591)

Prerequisite SkillsAdd. (Page 589)

3. 12 � 15 4. 3 � 4

5. 5 � 7 6. 16 � 9

7. 8 � 13 8. 5 � 17

Subtract. (Page 589)

9. 14 � 6 10. 9 � 4

11. 11 � 5 12. 8 � 3

13. 7 � 5 14. 10 � 6

Multiply. (Page 590)

15. 7 � 6 16. 10 � 2

17. 5 � 9 18. 8 � 3

19. 4 � 4 20. 6 � 8

Divide. (Page 591)

21. 32 � 4 22. 63 � 7

23. 21 � 3 24. 18 � 9

25. 72 � 9 26. 45 � 3

quotient

product

Integers Make this Foldableto help you organizeinformation about integers.Begin with a sheet of 11" � 17" unlined paper.

FoldFold the short sides sothey meet in the middle.

Reading and Writing As you read and study the chapter,write examples of addition, subtraction, multiplication, anddivision problems under each tab.

Fold AgainFold the top to the bottom.

CutUnfold and cut along the second fold to make four tabs.

LabelLabel each tab as shown.

8-2

Integers

8-3

Integers

8-4

Integers

8-5

Integers

Figure 4

Course 1, page 293

Lesson 1-2 Powers and Exponents 11

1. Explain what five to the fourth power means.

2. Write 75 in words.

3. OPEN ENDED Write one number in exponential form and another number in standard form.

4. Which One Doesn’t Belong? Identify the number that cannot be writtenas a power with an exponent greater than 1. Explain your reasoning.

Write each power as a product of the same factor.

5. 62 6. 44 7. 85

Evaluate each expression.

8. 34 9. 55 10. 103

Write each product in exponential form.

11. 5 � 5 � 5 � 5 � 5 � 5 12. 1 � 1 � 1 � 1

13. Evaluate eleven to the third power.

4 9

Exercises 1 & 4

You can , or find the value of, powers by multiplying thefactors. Numbers written without exponents are in .standard form

evaluate

Numbers written with exponents are in .exponential form

Write Powers in Standard Form


25 43

25� 2 � 2 � 2 � 2 � 2 43� 4 � 4 � 4

� 32 � 64


a. 102 b. 73 c. 54

Write Numbers in Exponential Form

Write 3 � 3 � 3 � 3 in exponential form.

3 is the base. It is used as a factor 4 times. So, the exponent is 4.

3 � 3 � 3 � 3 � 34

Write each product in exponential form.

d. 5 � 5 � 5 e. 12 � 12 � 12 � 12 � 12 � 12

16 50

Technology Youcan use a calculatorto compute largeexponents. To find 45, press 4

5 . The displayshows 1024.

ENTER

msmath2.net/extra_examples

Figure 5

Course 2, page 11

immediate feedback of their understandingof concepts. In addition, students may mon-itor their own performance using theselected answers in the back of the StudentEditions. Self-Check Quizzes for every les-son are available on Glencoe’s Web site, andimmediate feedback helps students checktheir progress and find specific pages andexamples in the Student Editions.

6. Questions and Cues

Questions and cues are teaching strategiesto help students recall and use what theyalready know about a topic. Cues are hintsabout what students will do or learn; forexample, a brief description of a hands-onlab activity before students begin. Activationof prior knowledge through cues or ques-tions is critical to learning. Asking studentsquestions before a learning activity helpsthem process information (Pressley et al.,1992).

Each Unit in the Glencoe Mathematics:Applications and Concepts series begins with aUnit Opener, a brief description of what stu-dents will learn in the unit. The openingquestion and answer in each chapter providea cue, giving a sample of the applications ofmath concepts in that chapter. Many lessons

also begin with “When am I ever going touse this?” questions, which include applica-tion problems that will be solved during thatlesson (see Figure 6). Each lesson finisheswith “Getting Ready for the Next Lesson” toprompt students to recall prerequisite skillsthat will apply to the next lesson.

7. Cooperative learning

Cooperative learning occurs when stu-dents work in pairs or groups of three orfour to complete tasks. Research shows thatcooperative learning provides practice atvaluable skills, such as positive interdepend-ence, face-to-face interactions, individual andgroup accountability, interpersonal skills,and group processing (Johnson & Johnson,1999). Cooperative learning has a highly pos-itive effect when compared with strategies inwhich students compete with each other andstrategies in which students work on tasksindividually (Johnson, Maruyama, Johnson,Nelson & Skon, 1981). There needs to be abalance of cooperative learning and individ-ual learning, however, because studentsneed time to practice skills independently(Anderson, Keder, & Simon, 1997).

Glencoe Mathematics: Applications and Con-cepts texts were designed to provide a mix ofindividual and cooperative learning opportu-nities. Hands-On Labs and Mini-Labs, locatedthroughout the texts, direct students to workwith other students in carefully structuredactivities (see Figure 7 on the next page). TheDaily Intervention feature in the TeacherWraparound Editions includes FlexibleGrouping suggestions for specific activitieswithin lessons, such as think-pair-share orteams in a tournament. MindJogger Video-quizzes offer practice in an interactive,game-show format, with students workingin teams to earn points. WebQuest InternetProjects also offer the opportunity for teamprojects, in which students do research on theInternet, gather data, and make presentations.

8. Identifying similarities and differences

Comparison and classification skills arevital in mathematics and science; whenidentifying similarities and differences, stu-dents determine how the current problemand previously solved problems are alike

NEW VocabularyNEW Vocabularypercent of changepercent of increasepercent of decreasemarkupselling pricediscount

5-7 Percent of Change

MONEY MATTERS Over the years, some prices increase. Study the change in gasoline prices from 1930 to 1960.

1. How much did the price increase from 1930 to 1940?

2. Write the ratio . Then write the ratio as a percent.

3. How much did the price increase from 1940 to 1950? Write the ratio . Then write the ratio as a percent.

4. How much did the price increase from 1950 to 1960? Write the ratio . The write the ratio as a percent.

5. Compare the amount of increase for each ten-year period.

6. Compare the percents in Exercises 2–4.

7. Make a conjecture about why the amounts of increase are thesame but the percents are different.

amount of increase��

price in 1950


price in 1940


price in 1930

am I ever going to use this?

In the above application, you expressed the amount of change as apercent of the original. This percent is called the .percent of change

236 Chapter 5 Percent

Percent of Change

Words A percent of change is a ratio that compares the change inquantity to the original amount.

Symbols percent of change �

Example original: 12, new: 9

�12

1�2

9� � �

132� � �

14

� or 25%

amount of change��

original amount

Price of a Gallon of Gasoline

Year Price (¢)

1930 10

1940 15

1950 20

1960 25

Source: Senior Living

Find and use the percent of increase or decrease.

percent of changepercent of increasepercent of decreasemarkupselling pricediscount

Figure 6

Course 3, page 236

and different. Research tells us thatcomparing and classifying areeffective ways to identify similari-ties and differences (Chen, 1996;English, 1997; and Newby et al.,1995). The most effective methodsof working with similarities and dif-ferences are to have students iden-tify similarities and differences ontheir own (Chen, 1996; Mason &Sorzio, 1996), use graphic and sym-bolic representations as well aswords (Mason, 1994), and beginwith concrete examples and thenmove toward abstract knowledge(Reeves & Weisberg, 1993).

Glencoe Mathematics: Applicationsand Concepts utilizes the “Solve aSimpler Problem” feature to iden-tify a simpler problem that sharessalient characteristics with thegiven problem. In activities andassignments, students must oftenexplain the difference between tworelated concepts, such as a sequenceand a series, or comparing andcontrasting two equations or twographs. Students also read andcreate Venn diagrams and otherdiagrams showing classification(see Figure 8).

9. Use of high-quality visuals to communicate,organize, and reinforce mathematical learning

Visuals—such as complex diagrams andelaborate drawings—used in conjunctionwith verbal description increase students’chances of learning, understanding, andremembering relationships and properties ofmathematics concepts. Visuals are often theonly way to effectively communicate ideasthat explain central concepts needed tounderstand mathematical strands such asalgebra and geometry. Research shows thatstudents are better able to organize andgroup ideas when visuals illustrate differentand common characteristics (Hegarty, Car-penter, & Just, 1991). Also, the mental imagesthat high-quality visuals encourage are anindispensable tool for recalling information,especially compared to information pre-sented with only text or lower-quality visu-als (Willows & Houghton, 1987).


Lesson 5-2 Greatest Common Factor 203

Greatest Common Factor

INTERNET A group of friends spent time in two Internet chat rooms. The diagram shows the chat rooms Angel, Sydney, Ian, Candace, and Christine visited. The friends were able to stay in one chat room or go to the other one.

1. Who visited the Fashion Chat Room?

2. Who visited the Music Chat Room?

3. Who visited both chat rooms?


photo/art ID tag

The diagram above is called a . It uses circles to showhow elements among sets of numbers or objects are related. The regionwhere circles overlap represents items that are common to two or moresets. It shows that Ian visited both chat rooms.

Venn diagrams can also show factors that are common to two or morenumbers. The greatest of these common factors is called the

. The GCF of prime numbers is 1.common factor (GCF)greatest

Venn diagram

Find the GCF by Listing Factors

Find the GCF of 20 and 24.

First, list the factors of 20 and 24.factors of 20: 1, 2, 4, 5, 10, 20factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Notice that 1, 2, and 4 are common factors of 20 and 24. So, the GCF is 4.

Check You can draw a Venn diagram to check your answer.

Find the GCF of each pair of numbers.

a. 8 and 10 b. 6 and 12 c. 10, 17

Music ChatRoom

Fashion ChatRoom

Angel

Sydney

Candace

ChristineIan

Factors of 24Factors of 20

5

20

10

36

824

12

124

msmath2.net/extra_examples

common factors: 1, 2, 4

�

Find the greatest commonfactor of two or morenumbers.

Venn diagramgreatest common

factor (GCF)

Figure 8

Course 2, page 203

Figure 7

Lesson 11-1a Hands-On Lab: Simulations 427

Spinners can also be used in simulations.

1. Explain experimental probability.

2. How is a simulation used to find the experimental probability of an event?


The probability of the Hornets beating the Jets is 0.5. The probability of the Hornets beating the Flashes is 0.25. Find the experimental probability that the Hornets beat both the Jets and the Flashes.

A probability of 0.5 is equal to �12

�. This means that the Hornets should win 1 out of 2 games. Make a spinner as shown. Label one section “win” and the other section “lose”.

A probability of 0.25 is equal to �14

�. This means that the Hornets should win 1 out of 4 games. Make a spinner as shown. Label one section “win” and the other sections “lose”.

Spin each spinner andrecord the results in a tablelike the one shown at theright. Repeat for 100 trials.

Use the results of the trials to write the ratio

�beat b

1o0th0

teams�. The ratio

represents the experimental probability that the Hornets beat both teams.

b. The probability of rain on Monday is 0.75, and the probabilityof rain on Tuesday is 0.4. Describe a simulation you could useto explore the probability of rain on both days. Conduct yoursimulation to find the experimental probability of rain on both days.

Win Lose

LoseLose

Win Lose

Outcome

Trial Hornets Hornetsand Jets and Flashes

1 L W2 W W3...

100

426 Chapter 11 Probability

SimulationsA simulation is a way of acting out a problem situation.Simulations can be used to find probability, which is the chancesomething will happen. When you find a probability by doing an experiment, you are finding experimental probability.

To explore experimental probability using a simulation, you can use these steps.

• Choose the most appropriate manipulative to aid in simulatingthe problem. Choose among counters, number cubes, coins, orspinners.

• Act out the problem for many trials and record the results tofind an experimental probability.

A Preview of Lesson 11-1


Use cups and counters to explore the experimental probabilitythat at least two of three children in a family are girls.

Place three counters in a cup and toss them onto your desk.

Count the number of red counters. This represents thenumber of boys. The number of yellow counters representsthe number of girls.

Record the results in a table like the one shown.

Repeat Steps 1–3 for 50 trials.

Suppose 23 of the 50 trials have at least two girls. Theexperimental probability that at least two of the three children in a family are girls is �

25

30� or 0.46.

a. Describe a simulation to explore the experimental probabilitythat two of five children in a family are boys. Then conductyour experiment. What is the experimental probability?

Trial Outcome

1 B B G2 B G G3...

50

Explore experimentalprobability by conducting a simulation.

• 3 two-colored counters

• cups• spinner

Glencoe Mathematics: Applications and Con-cepts includes high-quality charts, tables,graphs, art, and photographs throughoutthe Student Editions (see Figure 9). Visualsare often accompanied by captions andideas for effective use of models and manip-ulatives. Many of the Hands-On Labs andMini-Labs help students use diagrams toexplore concepts, relationships, and patterns.In an exclusive partnership with Glencoe/McGraw-Hill, USA TODAY® Education hasprovided USA TODAY Snapshots® thatmake the Student Editions come alive withcurrent, relevant data in eye-catchinggraphs, charts, and tables.

Powers and Exponents

FAMILY Every person has 2 biological parents. Study the familytree below.

1. How many 2s are multiplied to determine the number of great grandparents?

2. How many 2s would you multiply to determine the number ofgreat-great grandparents?


98 Chapter 2 Algebra: Rational Numbers

Powers Words Repeated Factors

21 2 to the first power 222 2 to the second power or 2 squared 2 � 223 2 to the third power or 2 cubed 2 � 2 � 224 2 to the fourth power 2 � 2 � 2 � 2

2n 2 to the nth power 2 � 2 � 2 � … � 2

……… �

n factors

Write an Expression Using Powers

Write a � b � b � a � b using exponents.

a � b � b � a � b � a � a � b � b � b Commutative Property

� (a � a) � (b � b � b) Associative Property

� a2 � b3 Definition of exponents

An expression like 2 � 2 � 2 � 2 can be written as the power 24.

24

The table below shows how to write and read powers.

The is the numberthat is multiplied.

base

2 parents

2 � 2 or4 grandparents

2 � 2 � 2 or8 great grandparents

The number that is expressed usingan exponent is called a .power

The tells how manytimes the base is used as a factor.

exponent�

Use powers and exponentsin expressions.

baseexponentpower

Figure 9

Course 3, page 98

SUMMARYGlencoe/McGraw-Hill is committed to the idea that curriculashould strive to reach all of the Principles and Standards for SchoolMathematics, thereby providing road maps that help teachersguide students to increasing levels of sophistication and depthsof knowledge. The NCTM Principles and Standards for SchoolMathematics were developed to accomplish several goals, includ-ing guiding the development of curriculum frameworks, assess-ments, and other instructional materials. Attaining the vision ofthe Principles and Standards for School Mathematics will require thetalents, energy, and attention of many individuals, including stu-dents, teachers, school administrators, policy makers, teachereducators, parents, mathematicians, local communities, and cur-riculum developers. Glencoe is proud to provide the Mathe-matics: Applications and Concepts series as an informed road mapto excellence in mathematics education in the 21st century.

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