California Noteables Interactive Study Notebook (11734.0K)

Contributing AuthorDinah ZikeConsultantDouglas Fisher, Ph.D.Director of Professional DevelopmentSan Diego, CA

ContentsCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Foldables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Vocabulary Builder . . . . . . . . . . . . . . . . . . . . . . . 21-1 A Plan for Problem Solving . . . . . . . . . . . 41-2 Variables, Expressions, andProperties . . . . . . . . . . . . . . . . . . . . . . . . . 61-3 Integers and Absolute Value . . . . . . . . . 91-4 Adding Integers . . . . . . . . . . . . . . . . . . . 121-5 Subtracting Integers . . . . . . . . . . . . . . . 161-6 Multiplying and DividingIntegers . . . . . . . . . . . . . . . . . . . . . . . . . 181-7 Writing Equations . . . . . . . . . . . . . . . . . 211-8 Problem-Solving Investigation:Work Backward . . . . . . . . . . . . . . . . . . . 231-9 Solving Addition and SubtractionEquations . . . . . . . . . . . . . . . . . . . . . . . 241-10 Solving Multiplication andDivision Equations . . . . . . . . . . . . . . . . . 26Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Foldables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Vocabulary Builder . . . . . . . . . . . . . . . . . . . . . . 332-1 Rational Numbers . . . . . . . . . . . . . . . . . 352-2 Comparing and OrderingRational Numbers . . . . . . . . . . . . . . . . . 382-3 Multiplying Positive andNegative Fractions . . . . . . . . . . . . . . . . . 402-4 Dividing Positive and NegativeFractions . . . . . . . . . . . . . . . . . . . . . . . . . 422-5 Adding and SubtractingLike Fractions . . . . . . . . . . . . . . . . . . . . . 452-6 Adding and Subtracting UnlikeFractions . . . . . . . . . . . . . . . . . . . . . . . . . 472-7 Solving Equations with RationalNumbers . . . . . . . . . . . . . . . . . . . . . . . . . 492-8 Problem-Solving Investigation:Look for a Pattern . . . . . . . . . . . . . . . . 512-9 Powers and Exponents . . . . . . . . . . . . . 522-10 Scientific Notation . . . . . . . . . . . . . . . . . 54Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Foldables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Vocabulary Builder . . . . . . . . . . . . . . . . . . . . . . 623-1 Square Roots . . . . . . . . . . . . . . . . . . . . . 643-2 Estimating Square Roots . . . . . . . . . . . . 663-3 Problem-Solving Investigation:Use a Venn Diagram . . . . . . . . . . . . . . . 683-4 The Real Number System . . . . . . . . . . . 693-5 The Pythagorean Theorem . . . . . . . . . . 723-6 Using the Pythagorean Theorem . . . . . 753-7 Geometry: Distance on theCoordinate Plane . . . . . . . . . . . . . . . . . . 77Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Foldables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Vocabulary Builder . . . . . . . . . . . . . . . . . . . . . . 854-1 Ratios and Rates . . . . . . . . . . . . . . . . . . 874-2 Proportional andNonproportional Relationships . . . . . . 894-3 Solving Proportions . . . . . . . . . . . . . . . . 914-4 Problem-Solving Investigation:Draw a Diagram . . . . . . . . . . . . . . . . . . 934-5 Similar Polygons . . . . . . . . . . . . . . . . . . 944-6 Measurement: Converting Length,Weight/Mass, Capacity, and Time . . . . . 974-7 Measurement: Converting SquareUnits and Cubic Units . . . . . . . . . . . . . 1004-8 Scale Drawings and Models . . . . . . . . 1024-9 Rate of Change . . . . . . . . . . . . . . . . . . 1044-10 Constant Rate of Change . . . . . . . . . . 107Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 110Foldables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Vocabulary Builder . . . . . . . . . . . . . . . . . . . . . 1165-1 Ratios and Percents . . . . . . . . . . . . . . . 1185-2 Comparing Fractions, Decimals,and Percents . . . . . . . . . . . . . . . . . . . . 1205-3 Algebra: The PercentProportion . . . . . . . . . . . . . . . . . . . . . . 1235-4 Finding Percents Mentally . . . . . . . . . 1255-5 Problem-Solving Investigation:Reasonable Answers . . . . . . . . . . . . . . 1275-6 Percent and Estimation . . . . . . . . . . . . 1285-7 Algebra: The Percent Equation . . . . . 1315-8 Percent of Change . . . . . . . . . . . . . . . . 1335-9 Simple Interest . . . . . . . . . . . . . . . . . . . 137Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 139Foldables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Vocabulary Builder . . . . . . . . . . . . . . . . . . . . . 1446-1 Line and Angle Relationships . . . . . . . 1466-2 Problem-Solving Investigation:Use Logical Reasoning . . . . . . . . . . . . . 1496-3 Polygons and Angles . . . . . . . . . . . . . . 1506-4 Congruent Polygons . . . . . . . . . . . . . . 1526-5 Symmetry . . . . . . . . . . . . . . . . . . . . . . . 154California Mathematics Grade 7iii

Contents6-6 Reflections . . . . . . . . . . . . . . . . . . . . . . 1566-7 Translations . . . . . . . . . . . . . . . . . . . . . 159Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 162Foldables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166Vocabulary Builder . . . . . . . . . . . . . . . . . . . . . 1677-1 Circumference and Areaof Circles . . . . . . . . . . . . . . . . . . . . . . . . 1697-2 Problem-Solving Investigation:Solve a Simpler Problem . . . . . . . . . . . 1717-3 Area of Complex Figures . . . . . . . . . . . 1727-4 Three-Dimensional Figures . . . . . . . . . 1747-5 Volume of Prisms andCylinders . . . . . . . . . . . . . . . . . . . . . . . . 1777-6 Volume of Pyramids and Cones . . . . . 1807-7 Surface Area of Prisms andCylinders . . . . . . . . . . . . . . . . . . . . . . . . 1827-8 Surface Area of Pyramids . . . . . . . . . . 1857-9 Similar Solids . . . . . . . . . . . . . . . . . . . . 187Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 1909-7 Problem-Solving Investigation:Use a Graph . . . . . . . . . . . . . . . . . . . . . 2409-8 Scatter Plots . . . . . . . . . . . . . . . . . . . . . 241Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 244Foldables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248Vocabulary Builder . . . . . . . . . . . . . . . . . . . . . 24910-1 Linear and Nonlinear Functions . . . . . 25010-2 Graphing Quadratic Functions . . . . . . 25310-3 Problem-Solving Investigation:Make a Model . . . . . . . . . . . . . . . . . . . 25510-4 Graphing Cubic Functions . . . . . . . . . . 25610-5 Multiplying Monomials . . . . . . . . . . . . 25910-6 Dividing Monomials . . . . . . . . . . . . . . 26110-7 Powers of Monomials . . . . . . . . . . . . . 26410-8 Roots of Monomials . . . . . . . . . . . . . . 266Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 268Foldables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Vocabulary Builder . . . . . . . . . . . . . . . . . . . . . 1968-1 Simplifying Algebraic Expressions . . . 1978-2 Solving Two-Step Equations . . . . . . . . 2008-3 Writing Two-Step Equations . . . . . . . . 2038-4 Solving Equations with Variableson Each Side . . . . . . . . . . . . . . . . . . . . . 2068-5 Problem-Solving Investigation:Guess and Check . . . . . . . . . . . . . . . . . 2088-6 Inequalities . . . . . . . . . . . . . . . . . . . . . 2098-7 Solving Inequalities by Adding orSubtracting . . . . . . . . . . . . . . . . . . . . . 2128-8 Solving Inequalities by Multiplyingor Dividing . . . . . . . . . . . . . . . . . . . . . . 214Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 216Foldables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220Vocabulary Builder . . . . . . . . . . . . . . . . . . . . . 2219-1 Functions . . . . . . . . . . . . . . . . . . . . . . . 2239-2 Representing Linear Functions . . . . . . 2269-3 Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . 2309-4 Direct Variation . . . . . . . . . . . . . . . . . . 2339-5 Slope-Intercept Form . . . . . . . . . . . . . . 2369-6 Writing Systems of Equationsand Inequalities . . . . . . . . . . . . . . . . . . 238Foldables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272Vocabulary Builder . . . . . . . . . . . . . . . . . . . . . 27311-1 Problem-Solving Investigation:Make a Table . . . . . . . . . . . . . . . . . . . . 27511-2 Histograms . . . . . . . . . . . . . . . . . . . . . . 27611-3 Circle Graphs . . . . . . . . . . . . . . . . . . . . 27911-4 Measures of Central Tendencyand Range . . . . . . . . . . . . . . . . . . . . . . 28311-5 Measures of Variation . . . . . . . . . . . . . 28611-6 Box-and-Whisker Plots . . . . . . . . . . . . 28911-7 Stem-and-Leaf Plots . . . . . . . . . . . . . . 29211-8 Select an Appropriate Display . . . . . . 296Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 298Foldables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302Vocabulary Builder . . . . . . . . . . . . . . . . . . . . . 30312-1 Counting Outcomes . . . . . . . . . . . . . . 30512-2 Probability of CompoundExperiments . . . . . . . . . . . . . . . . . . . . . 30812-3 Experimental andTheoretical Probability . . . . . . . . . . . . 31112-4 Problem-Solving Investigation:Act It Out . . . . . . . . . . . . . . . . . . . . . . . 31512-5 Using Sampling to Predict . . . . . . . . . . 316Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 319Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.iv California Mathematics Grade 7

Organizing Your FoldablesMake this Foldable to help you organize andstore your chapter Foldables. Begin with onesheet of 11" × 17" paper.FoldFold the paper in half lengthwise. Then unfold.Fold and GlueFold the paper in half widthwise and glue all of the edges.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Glue and LabelGlue the left, right, and bottom edges of the Foldableto the inside back cover of your Noteables notebook.Reading and Taking Notes As you read and study each chapter, recordnotes in your chapter Foldable. Then store your chapter Foldables insidethis Foldable organizer.California Mathematics Grade 7v

This note-taking guide is designed to help you succeed inCalifornia Mathematics Grade 7. Each chapter includes:C H A P T E R3 Real Numbers andthe Pythagorean TheoremUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with two sheets of 8 1_ " by 11" paper.2Fold one in half fromtop to bottom. Cutalong fold from edgesto margin.Fold the other sheetin half from topto bottom. Cut alongfold between margins.The Chapter Openercontains instructions andillustrations on how to makea Foldable that will help youto organize your notes.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.A Note-Taking Tipprovides a helpfulhint you can usewhen taking notes.Insert first sheetthrough secondsheet and align folds.Label each page with Chapter 3Real Numbersa lesson number and title.and thePythagoreanTheoremNOTE-TAKING TIP: When you take notes, clarifyterms, record concepts, and write examples foreach lesson. You may also want to list ways inwhich the new concepts can be used in yourdaily life.The Build Your Vocabularytable allows you to writedefinitions and examplesof important vocabularyterms together in oneconvenient place.California Mathematics Grade 7 61Chapter 3C H A P T E R3BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 3.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary Termabscissa[ab-SIH-suh]conversecoordinate planehypotenuseirrational numberlegsordered pairordinate[OR-din-it]originperfect squareFoundon PageDefinitionDescription orExampleWithin each chapter,Build Your Vocabularyboxes will remind youto fill in this table.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.62 California Mathematics Grade 7vi California Mathematics Grade 7

3–6 Using the Pythagorean TheoremMAIN IDEA• Solve problems usingthe PythagoreanTheorem.Standard7MG3.3 Know andunderstand thePythagorean theoremand its converse and useit to find the length of themissing side of a righttriangle and the lengthsof other line segmentsand, in some situations,empirically verify thePythagorean theorem bydirect measurement.EXAMPLE Use the Pythagorean TheoremRAMPS A ramp to a newlyconstructed building mustbe built according to theguidelines stated in theAmericans with DisabilitiesAct. If the ramp is 24.1feet long and the top of the ramp is 2 feet off theground, how far is the bottomof the ramp from the baseof the building?Notice the problem involves a right triangle. Use thePythagorean Theorem.24.1 2 = a 2 + 2 2 Replace c with 24.1 andb with 2.= a 2 + Evaluate 24. 1 2 and 2 2 .Lessons cover the content of thelessons in your textbook. As yourteacher discusses each example,follow along and completethe fill-in boxes. Take notes asappropriate.Each lesson iscorrelated to theCalilfornia Standards.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ORGANIZE ITOn Lesson 3-6 of yourFoldable, explain thePythagorean Theorem inyour own words and givean example of how itmight be used in areal-life situation.- = a 2 = - Subtract from each side.≈ aThe end of the ramp is aboutthe building.= a 2 Simplify.= a Defi nition of square rootSimplify.from the base ofCheck Your Progress If a truck ramp is 32 feet long andthe top of the ramp is 10 feet off the ground, how far is the endof the ramp from the truck?ORGANIZE ITOn Lesson 3-1 of yourFoldable, explain howto find the square rootof a number and give anexample.Chapter 3Real Numbersand thePythagoreanTheoremCheck Your Progress Find each square root.a. √ 643–1Examples parallel theexamples in your textbook.b. - √ 25_144c. ± √ 2.25Chapter 3Real Numbersand thePythagoreanTheoremFoldables featurereminds you to takenotes in your Foldable.California Mathematics Grade 7 75EXAMPLE Use an Equation to Solve a ProblemMUSIC The art work of the square picture in a compactdisc case is approximately 14,161 mm 2 in area. Find thelength of each side of the square.The area is equal to the square of the length of a side.Let A = the area and let s = the length of the side A = s 214,161 = s 2 Write the equation.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.C H A P T E R3Use your Chapter 3 Foldableto help you study for yourchapter test.3-1Square RootsBRINGING IT ALL TOGETHERSTUDY GUIDEVOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 3, go toglencoe.comComplete each sentence.1. The principle square root is the square rootof a number.2. To solve an equation in which one side of the square is a squaredterm, you can take theof each side of theequation.Find each square root.3. √ 900 4. - √ 36_495. - √ 625 6. √ 25_1213-2Estimating Square RootsDetermine between which two consecutive whole numberseach value is located.7. √ 23 8. √ 599. √ 27 10. √ 18BUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 62–63) to help you solvethe puzzle.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:= √ s 2Take the square root of each side.The length of a side of a compact disc case is aboutmillimeters since distance cannot be negative.Check Your Progress A piece of art is a square picture thatis approximately 11,025 square inches in area. Find the lengthof each side of the square picture.Check Your ProgressExercises allow you tosolve similar exerciseson your own.California Mathematics Grade 7 65Bringing It All TogetherStudy Guide reviewsthe main ideas and keyconcepts from each lesson.80 California Mathematics Grade 7California Mathematics Grade 7vii

NOTE-TAKING TIPSYour notes are a reminder of what you learned in class. Taking good notescan help you succeed in mathematics. The following tips will help you takebetter classroom notes.• Before class, ask what your teacher will be discussing in class. Reviewmentally what you already know about the concept.• Be an active listener. Focus on what your teacher is saying. Listen forimportant concepts. Pay attention to words, examples, and/or diagramsyour teacher emphasizes.• Write your notes as clear and concise as possible. The following symbolsand abbreviations may be helpful in your note-taking.Word or PhraseSymbol orAbbreviationWord or PhraseSymbol orAbbreviationfor example e.g. not equal ≠such as i.e. approximately ≈with w/ therefore ∴without w/o versus vsand + angle ∠• Use a symbol such as a star (★) or an asterisk (*) to emphasize importantconcepts. Place a question mark (?) next to anything that you do notunderstand.• Ask questions and participate in class discussion.• Draw and label pictures or diagrams to help clarify a concept.• When working out an example, write what you are doing to solve theproblem next to each step. Be sure to use your own words.• Review your notes as soon as possible after class. During this time, organizeand summarize new concepts and clarify misunderstandings.Note-Taking Don’ts• Don’t write every word. Concentrate on the main ideas and concepts.• Don’t use someone else’s notes as they may not make sense.• Don’t doodle. It distracts you from listening actively.• Don’t lose focus or you will become lost in your note-taking.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.viii California Mathematics Grade 7

C H A P T E R1Algebra: IntegersChapter 1Use the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with a plain piece of 11" × 17" paper.Fold the paper in sixths lengthwiseOpen and Fold a 4" tab along theshort side. Then fold the rest in half.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Label Draw lines along the folds and labelas shown. NOTE-TAKING TIP: When taking notes, it may behelpful to explain each idea in words and give oneor more examples.California Mathematics Grade 7 1

C H A P T E R1BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 1.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExampleabsolute valueadditive inversealgebraalgebraic expression[AL-juh-BRAY-ihk]conjecturecoordinatecounterexampledefine a variableequation[ih-KWAY-zhuhn]evaluateinequalityCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.2 California Mathematics Grade 7

Chapter 1 BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExampleinteger[IHN-tih-juhr]inverse operationsnegative numbernumerical expressionoppositesorder of operationsCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.positive numberpowerspropertysolutionsolvevariableCalifornia Mathematics Grade 7 3

1–1A Plan for Problem SolvingMAIN IDEA• Solve problems usingthe four-step plan.BUILD YOUR VOCABULARY (pages 2–3)Some problem-solving strategies require you to make anor conjecture.EXAMPLES Use the Four-Step PlanORGANIZE ITSummarize the four-stepproblem-solving planin words and symbols.Include an example ofhow you have used thisplan to solve a problem.HOME IMPROVEMENT The Vorhees family plans to paintthe walls in their family room. They need to cover 512square feet with two coats of paint. If a 1-gallon can ofpaint covers 220 square feet, how many 1-gallon cans ofpaint do they need?EXPLORE Since they will be usingcoats of paint, we mustthe area to be painted.PLAN They will be covering × square feetStandard 7MR1.1Analyze problemsby identifyingrelationships,distinguishing relevantfrom irrelevantinformation, identifyingmissing information,sequencing and prioritizinginformation, and observingpatterns. Reinforcementof Standard 6AF2.3Solve problems involvingrates, average speed,distance, and time.or square feet. Next, divide byare needed.SOLVE ÷ ≈CHECKto determine how many cans of paintSince they will purchase a whole number of cans ofpaint, round to .They will need to purchasecans of paint.Check Your Progress Jocelyn plans to paint her bedroom.She needs to cover 400 square feet with three coats of paint.If a 1-gallon can of paint covers 350 square feet, how many1-gallon cans of paint does she need?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.4 California Mathematics Grade 7

1–1REMEMBER ITAlways check tomake sure your answeris reasonable. You cansolve the problem againif you think your answeris not correct.GEOGRAPHY Study the table. The five largest states intotal area, which includes land and water, are shown.Of the five states shown, which one has the smallestarea of water?Largest States in AreaState Land Area (mi 2 ) Total Area (mi 2 )Alaska 570,374 615,230Texas 261,914 267,277California 155,973 158,869Montana 145,556 147,046New Mexico 121,364 121,598Source: U.S. Census BureauEXPLORE What do you know? You are given the total area andthe land area for five states. What are you trying tofind? You need to find the water area.PLAN To determine the water area, thefrom thefor eachstate.SOLVE Alaska = 615,230 - 570,374 =Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:CHECKsquare miles.Texas = 267,277 - 261,914 =California = 158,869 - 155,973 =Montana = 147,046 - 145,556 =New Mexico = 121,598 - 121,364 =Compare the water area for each state to determinewhich state has the least water area.has the least water area withCheck Your Progress Refer to Example 2. How many timeslarger is the land area of Alaska than the land area of Montana?California Mathematics Grade 7 5

1–2 Variables, Expressions, and PropertiesMAIN IDEA• Solve problems usingthe four-step plan.BUILD YOUR VOCABULARY (pages 2–3)A variable is a, usually a letter, used torepresent a .Standard 7AF1.2Use the correctorder of operationsto evaluate algebraicexpressions such as3(2x + 5 ) 2 .Standard 7AF1.3Simplify numericalexpressions byapplying propertiesof rational numbers(e.g. identity, inverse,distributive, associative,commutative) andjustify the process used.Standard 7AF1.4 Usealgebraic terminology(e.g. variable, equation,term, coeffi cient, inequality,expression, constant)correctly.KEY CONCEPTOrder of Operations1. Do all operationswithin groupingsymbols first; startwith the innermostgrouping symbols.2. Evaluate all powersbefore otheroperations.3. Multiply and dividein order from left toright.4. Add and subtract inorder from left toright.An algebraic expression contains a, anumber, and at least onesymbol.When you substitute a number for the, analgebraic expression becomes a numeric expression.To evaluate an expression means to find itsvalue.To avoid confusion, mathematicians have agreed on acalled the order of operations.EXAMPLES Evaluate Algebraic ExpressionsEvaluate each expression if q = 5, r = 6, and s = 3.4 (r - s) 24 (r - s) 2= 4 ( - ) 2 Replace with 6 andwith 3.= 4 ( ) 2 Perform operations in thefi rst.= 4 Evaluate the .= Simplify.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.6 California Mathematics Grade 7

1–2BUILD YOUR VOCABULARY (pages 2–3)Expressions such as 7 2 and 2 3 are called powers andrepresent repeated .q 2 - 4r - 1q 2 - 4r - 1 =4- 4 - 1 Replace with 5 andwith 6.= - 4(6) - 1= 25 - - 1Evaluateother operations..before= -=Add and subtract in orderfrom left to right..Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.6q_5sThe fraction bar is a grouping symbol. Evaluate the expressionsin the numerator and denominator separately before dividing._ 6q5s = _ 6 (5)5 (3)= _ 3015Replace with 5 and with 3.Do all= .Check Your Progress Evaluate each expression.a. 5p - 3s + 2 if p = 2 and s = 1b. b 2 + 3c - 5 if b = 4 and c = 2c._ 3sif q = 2 and s = 4q + 4fi rst.California Mathematics Grade 7 7

1–2BUILD YOUR VOCABULARY (pages 2–3)A mathematical sentence that contains ansign (=) is called an equation.An equation that contains asentence.Properties areany numbers.is an opensentences that are true forA counterexample is an example that shows that aconjecture is .REMEMBER ITCommutativePropertya + b = b + aa · b = b · aAssociative PropertyEXAMPLES Identify PropertiesName the property shown by 12 · 1 = 12.Multiplying by 1 does not change the number.This is theProperty.a + (b + c) = (a + b) + ca · (b · c) = (a · b) · cDistributive Propertya (b + c) = ab + aca (b - c) = ab - acIdentity Propertya + 0 = aa · 1 = aHOMEWORKASSIGNMENTPage(s):Exercises:Check Your Progress3 · 2 = 2 · 3.EXAMPLES Find a CounterexampleName the property shown byState whether the following conjecture is true or false.If false, provide a counterexample.The sum of an odd number and an even number isalways odd.This conjecture is .Check Your Progress State whether the followingconjecture is true or false. If false, provide a counterexample.Division of whole numbers is associative.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.8 California Mathematics Grade 7

1–3 Integers and Absolute ValuesStandard 7NS2.5 Understand the meaning of the absolute value of a number;interpret the absolute value as the distance of the number from zero on a numberline; and determine the absolute value of real numbers.MAIN IDEA• Graph integers on anumber line and findabsolute valueBUILD YOUR VOCABULARY (pages 2–3)A negative number is a numberthan zero.numbers, positive numbers, andare members of the set of integers.EXAMPLE Compare Two IntegersReplace the with < or > to make -2 -1 atrue sentence. The number line shows that -2 isthan -1, since itlies to the of -1. So, write -2 -1.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Check Your Progress Replace each with < or > tomake a true sentence.a. -3 2b. -4 -6BUILD YOUR VOCABULARY (pages 2–3)The that corresponds to a iscalled the coordinate of that point.A sentence thattwo different numbersof quantities is called an inequality.California Mathematics Grade 7 9

1–3BUILD YOUR VOCABULARY (pages 2–3)The absolute value of a number is the distance thenumber is fromon the number line.REMEMBER ITThe absolute valueof a number is not thesame as the opposite ofa number. Rememberthat the absolute valueof a number cannot benegative.EXAMPLES Expressions with Absolute ValueEvaluate each expression.⎪5⎥ - ⎪5⎥ The graph of 5 isunits from 0 on the number line.So, ⎪5⎥ =. Then subtract 5 units.Thus, ⎪ 5⎥ - ⎪5⎥ =⎪6⎥ - ⎪-5⎥⎪6⎥ - ⎪-5⎥ = - ⎪-5⎥ The absolute value of 6 is .= 6 - ⎪-5⎥ == Simplify.Evaluate ⎪6 - 9⎥ - ⎪5 - 3⎥ .⎪6 - 9⎥ - ⎪5 - 3⎥ = ⎪ ⎥ - ⎪ ⎥Simplify the absolutevalue expressions.= - ⎪2⎥ The absolute value of-3 is .= 3 - The absolute value of2 is .= Simplify.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.10 California Mathematics Grade 7

1–3Evaluate ⎪x⎥ + 13 if x = -4.⎪x⎥ + 13 = ⎪ ⎥ + 13 Replace x with .= + 13 ⎪-4⎥ == Simplify.Check Your Progress Evaluate each expression.a. ⎪-3⎥ - ⎪3⎥ b. ⎪9⎥ - ⎪-6 ⎥ c. ⎪4 - 7⎥ - ⎪11 - 6⎥d. Evaluate ⎪x⎥ + 7 if x = -2.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:California Mathematics Grade 7 11

1–4Adding IntegersEXAMPLE Add Integers with the Same SignMAIN IDEA• Add integers.Standard 7NS1.2Add, subtract,multiply, anddivide rational numbers(integers, fractions, andterminating decimals)and take positive rationalnumbers to whole-numberpowers.Standard 7AF1.3Simplify numericalexpressions byapplying propertiesof rational numbers(e.g. identity, inverse,distributive, associative,commutative) and justifythe process used.KEY CONCEPTAdding Integers withthe Same Sign To addintegers with the samesign, add their absolutevalues. Give the result thesame sign as the integers.Add -8 + (-4).Use a number line.Start at zero.Move units to the left.From there, move 4 units . So, -8 + (-4) = .Check Your Progress Add using a number lineor counters.a. -3 + (-6)b. -13 + (-12)Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.12 California Mathematics Grade 7

1–4EXAMPLES Add Integers with Different SignsORGANIZE ITExplain and giveexamples of how to addintegers with the samesign and how to addintegers with a differentsigns.Find 4 + (-6).Use a number line.Start at .Move 4 units . From there, move units left. So, 4 + (-6) = .Find -5 + 9.Use a number line.Start at .Move units .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.KEY CONCEPTSAdding Integers withDifferent Signs To addintegers with differentsigns, subtract theirabsolute values. Givethe result the same signas the integer with thegreater absolute value.From there, move units left . So, -5 + 9 = .Find -33 + 16.-33 + 16 = To fi nd -33 + 16, subtract⎪16⎥ from ⎪-33⎥ .The sum isbecause ⎪-33⎥ > ⎪16⎥ .California Mathematics Grade 7 13

1–4Check Your Progressa. 3 + (-5)Add.b. -6 + 8c. 25 + (-15) .BUILD YOUR VOCABULARY (pages 2–3)Two numbers with the samebutdifferent signs are called opposites.An integer and itsare also calledadditive inverses.EXAMPLE Add Three or More IntegersFind the sum 2 + (-5) + (-3).2 + (-5) + (-3) = 2 + [ + (-3)] Associative Property= 2 + Order of operations.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.= Simplify.14 California Mathematics Grade 7

1–4Check Your Progress Find each sum.a. 3 + (-6) + (-2) b. -10 + 5 + 10 + 7EXAMPLE Add Three or More IntegersSTOCKS An investor owns 50 shares in a video-gamemanufacturer. A broker purchases 30 shares more forthe client on Tuesday. On Friday, the investor asks thebroker to sell 65 shares. How many shares of this stockwill the client own after these trades are completed?Selling a stock decreases the number of shares, so the integerfor selling is .Purchasing new stock increases the number of shares, so theinteger for buying is .Add these integers to the starting number of shares to find thenew number of shares.50 + + ( )= (50 + ) + ( ) Associative PropertyCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:= + (-65) 50 + == Simplify.Check Your Progress MONEY Jaime gets an allowanceof $5. She spends $2 on video games and $1 on lunch. Her bestfriend repays a $2 loan and she buys a $3 pair of socks. Howmuch money does Jaime have left?California Mathematics Grade 7 15

1–5WRITE ITExplain why -b does notnecessarily mean that thevalue of -b is negative.EXAMPLES Evaluate Algebraic ExpressionsEvaluate each expression if p = 6, q = -3, and r = -7.12 - r12 - r = 12 - Replace r with .= 12 + To subtract add .= Add.q - p 2q - p 2 = -3 - 6 2 Replace q with andp with .Simplify 6 2 .= -3 + To subtract , add .= Add.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:Check Your Progress Evaluate each expression ifa = 3, b = -6, and c = 2.a. 10 - c b. b - aCalifornia Mathematics Grade 7 17

1–6Multiplying and Dividing IntegersStandard 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers,fractions, and terminating decimals)…whole-number powers. Standard 7AF1.3Simplify numerical expressions by applying properties of rational numbers (e.g.identity, inverse, distributive, associative, commutative) and justify the process used.EXAMPLE Multiply Integers with Different SignsMAIN IDEA• Multiply and divideintegers.Find 8 (-4) .8 (-4) =The factors havesigns. The product is.KEY CONCEPTSMultiplying Two IntegersThe product of twointegers with differentsigns is negative.EXAMPLE Multiply Integers with the Same SignFind -12 (-12) .-12 (-12) =The product of twointegers with the samesign is positive.Dividing Integers Thequotient of two integerswith different signs isnegative.The quotient of twointegers with the samesign is positive.REMEMBER ITDecide on the signof the product beforemultiplying. If thenumber of negativesis even the product ispositive. If the numberof negatives is odd theproduct is negative.The factors have the sign. The productis .EXAMPLE Multiply More Than Two IntegersFind 6 (-2)(-4).6 (-2)(-4) = [6(-2)] Property= -12 6 (-2) == -12(-4) =Check Your Progress Multiply.a. 6 (-3) b. -2(6)c. -8(-8) d. 5 (-3)(-2)Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.18 California Mathematics Grade 7

1–6EXAMPLE Divide IntegersFind 30 ÷ (-5) .30 ÷ (-5) =The dividend and the divisor havesigns.The quotient is .ORGANIZE ITDescribe why theproduct or quotient oftwo integers with thesame sign is positiveand the product orquotient of two integerswith different signs isnegative. Check Your Progress Divide.a. 36 ÷ (-6) b. _ -305EXAMPLE Evaluate Algebraic ExpressionsEvaluate 3x - (-4y) if x = -10 and y = -4.3x - (-4y)= 3 ( ) - [-4 ( ) ] Replace x withand y with .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.= - 3 (-10) = and-4(-4) == -30 + To subtract , add.= Add.Check Your Progress Evaluate 2a - (-3b) if a = -6 andb = -4.California Mathematics Grade 7 19

1–6EXAMPLE Find the Mean of a Set of IntegersWEATHER The tableshows the lowtemperature for eachmonth in McGrath,Alaska. Find themean (average) of all12 temperatures.To find the mean of a set ofnumbers, find the sum of thenumbers. Then divide theresult by how many numbersthere are in the set.Average Low TemperaturesMonth Temp. (°C)Jan. -27Feb. -26March -19April -9May 1June 7July 9Aug. 7Sept. 2Oct. -8Nov. -19Dec. -26Source: weather.com_____________-27 + (-26) + (-19) + (-9) + 1 + 7 + 9 + 7 + 2 + (-8) + (-19) + (-26)12= __12HOMEWORKASSIGNMENTPage(s):Exercises:=Check Your ProgressThe table shows a set of recordlow temperatures. Find themean (average) of all12 temperatures.Average Low TemperaturesMonth Temp. (°C)Jan. -20Feb. -15March -5April 10May 25June 31July 41Aug. 38Sept. 34Oct. 19Nov. 3Dec. -15Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Source: weather.com20 California Mathematics Grade 7

1–7 Writing EquationsMAIN IDEA• Write algebraicexpressions andequations from verbalphrases and sentences.BUILD YOUR VOCABULARY (pages 2–3)When you choose a variable and an unknown quantity forthe variable to represent, this is called defining the variable.EXAMPLE Write an Algebraic EquationCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Standard 7AF1.1Use variablesand appropriateoperations to write anexpression, an equation,an inequality, or a systemof equations or inequalitiesthat represents a verbaldescription (e.g. threeless than a number, half aslarge as area A.Standard 7AF1.4 Usealgebraic terminology(e.g. variable, equation,term, coeffi cient, inequality,expression, constant)correctly.REMEMBER ITIt is often helpful toselect letters that caneasily be connectedto the quantity theyrepresent. For example,age = a.CONSUMER ISSUES The cost of a book purchased onlineplus $5 shipping and handling comes to a total of $29.Write an equation to model this situation.WordsVariableEquationThe equation is .The price of a book plus $5 shipping is $29.Let b represent the price of the book.The priceof a book plus $5 shipping is $29.+ = 29Check Your Progress Write the price of a toy plus $6shipping is $35 as an algebraic equation.EXAMPLE Write an Equation to Solve a ProblemNUTRITION A box of oatmeal contains 10 individualpackages. If the box contains 30 grams of fiber, writean equation to find the amount of fiber in one packageof oatmeal.WordsVariableEquationTen packages of oatmeal contain 30 gramsof fi ber.Let f represent the grams of fi ber perpackage.Ten packages30 gramsof oatmeal contain of fi ber.= 30California Mathematics Grade 7 21

1–7REVIEW ITExplain why it isimportant to read a wordproblem more than oncebefore attempting tosolve it.Check Your Progress A particular box of cookies contains10 servings. If the box contains 1,200 calories, write an equationto find the number of calories in one serving of cookies.EXAMPLESTANDARDS EXAMPLE The eighth grade has $35 less inits treasury than the seventh grade has. Given s, thenumber of dollars in the seventh grade treasury, whichequation can be used to find e, the number of dollars inthe eighth grade treasury?A e = 35 - sB e = s - 35C e = s ÷ 35D e = 35sRead the Test ItemThe phrase $35 less . . . than the seventh grade indicates.HOMEWORKASSIGNMENTPage(s):Exercises:Solve the Test ItemThe amount of money inthe eighth grade treasury isthe amount of money inthe seventh grade treasury less $35.e = s - 35The solution is .Check Your Progress Helena and her friends ordered 3bags of popcorn and 4 drinks from the snack stand. Whichequation could be used to find c, the total cost if p representsthe cost of a bag of popcorn and d represents the cost of a drink?F c = 7 (p + d)H c = 3p + 4dG c = 7 (p - d)J c = 7p + 7dCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.22 California Mathematics Grade 7

1–8 Problem-Solving Investigation:Work BackwardEXAMPLEMAIN IDEA• Solve problems byworking backward.Standard 7MR1.1Analyze problemsby identifyingrelationships,distinguishing relevantfrom irrelevant information,identifying missinginformation, sequencingand prioritizing information,and observing patterns.Standard 7NS1.2 Add,subtract, multiply, anddivide rational numbers(integers, fractions, andterminating decimals)and take positive rationalnumbers to whole-numberpowers.SCHEDULING Wendie is meeting some friends for amovie and a dinner. She needs to be finished with dinnerby 7:30 P.M. to make it home by 8:00 P.M. The movie runsfor 90 minutes, and she wants to have at least 1 hour fordinner. If it takes 20 minutes to get from the theater tothe restaurant, what is the latest starting time she canchoose for the movie she wants to see?EXPLORE You know what time Wendie needs to head home.You know the time it takes for each event. You needto determinePLAN Start with the and work backward.SOLVE Finish dinner 7:30 P.M.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:Go back 1 hour for dinner.Go back for travel. 6:10 P.M.Go back 90 minutes for the movie.CHECK Assume the movie starts at Workfoward, adding the time for each event.The latest starting time for the movie isCheck Your Progress SHOPPING Mia spent $9.50 ata fruit stand, then spent three times that amount at thegrocery store. She had $7.80 left. How much money did shehave initially?California Mathematics Grade 7 23

1–9 Solving Addition and Subtraction EquationsReinforcement of Standard 6AF1.1 Write and solve one-step linear equations in onevariable.MAIN IDEA• Solve equations usingthe Subtraction andAddition Properties ofEquality.BUILD YOUR VOCABULARY (pages 2–3)When you solve an equation, you are trying to find thevalues of the variable that makes the equation .A solution is the value of thethat makes theequation true.EXAMPLE Solve an Addition EquationKEY CONCEPTSSubtraction Property ofEquality If you subtractthe same number fromeach side of an equation,the two sides remainequal.Addition Property ofEquality If you add thesame number to eachside of an equation, thetwo sides remain equal.Solve 7 = 15 + c.METHOD 1 Vertical Method7 = 15 + c Write the equation.7 = 15 + c Subtract from each side._______-15 = -15= c 7 - 15 = ; 15 - 15 =METHOD 2 Horizontal Method7 = 15 + c Write the equation.7 - = 15 + c - Subtract from each side.= c 7 - 15 = and- 15 = 0Check Your Progress Solve 6 = 11 + a.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.24 California Mathematics Grade 7

1–9BUILD YOUR VOCABULARY (pages 2–3)Addition and subtraction are called inverse operationsbecause they “undo” each other.EXAMPLE Solve an Addition EquationORGANIZE ITCompare how to solvean equation involvingwhole numbers andan equation involvingintegers. OCEANOGRAPHY At high tide, the top of a coralformation is 2 feet above the surface of the water.This represents a change of -6 feet from the height ofthe coral at low tide. Write and solve an equation todetermine h, the height of the coral at low tide.WordsVariableEquationh + (-6) = 2The height at low tide plus the change is theheight at high tide.Let h represent the height at low tide.h + (-6) = 2Write the equation.h + (-6) - = 2 - Subtract from eachside.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:h =Simplify.The height of the coral at low tide is 8 feet.EXAMPLE Solve a Subtraction EquationSolve -5 = z - 16.Use the horizontal method.-5 = z - 16 Write the equation.-5 + = z - 16 + Add to each side.= z -16 + 16 = and+ 16 = 11.Check Your Progress Solve x - 12 = -6.California Mathematics Grade 7 25

1–10 Solving Multiplication and Division EquationsReinforcement of Standard 6AF1.1 Write and solve one-step linear equations in onevariable.EXAMPLE Solve a Multiplication EquationMAIN IDEA• Solve equations byusing the Divisionand MultiplicationProperties of Equality.Solve 7z = -49.7z = -49 Write the equation.__ 7z -49= __ each side by .z = 7 ÷ 7 = , -49 ÷ 7 =KEY CONCEPTS= Identity Property; 1z =Division Property ofEquality If you divideeach side of an equationby the same nonzeronumber, the two sidesremain equal.Multiplication Propertyof Equality If youmultiply each side of anequation by the samenumber, the two sidesremain equal.EXAMPLE Solve a Division EquationSolve _ c 9 = -6.c_9c_9= -6 Write the equation.= -6 Multiply each side by .c = -6 =EXAMPLE Use an Equation to Solve a ProblemSURVEYING English mathematician Edmund Gunterlived around 1600. He invented the chain, which wasused as a unit of measure for land and deeds. One chainequals 66 feet. If the south side of a property measures330 feet, how many chains long is it?WordsVariableEquationOne chain equals 66 feet.Let c = the number of chains infeet.Measurement66 times theof property is number of chains330 =Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.26 California Mathematics Grade 7

1–10ORGANIZE ITOn your Foldable table,explain how to solvemultiplication equationsusing the multiplicationproperties of equality. Solve the equation.330 = 66c Write the equation.__ 330 66c= __ Divide each side by .= 330 ÷ =The number of chains in 330 feet is .Check Your Progressa. Solve 8a = -64. b. Solve x _5 = -10.c. Most horses are measured in hands. One hand equals4 inches. If a horse measures 60 inches, how manyhands is it?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:California Mathematics Grade 7 27

C H A P T E R1BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 1 Foldableto help you study for yourchapter test.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 1, go to:glencoe.comBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 2–3) to help you solvethe puzzle.1-1A Plan for Problem SolvingUse the four-step plan to solve the problem.1. Lisa plans to redecorate her bedroom. Each wall is 120 squarefeet. Three walls need a single coat of paint and the fourth wallneeds a double coat. If each can of paint will cover 200 square feet,how many gallons of paint does Lisa need?1-2Variables, Expressions and Properties2. Number the operations in the correct order for simplifying2 + 4 (9 - 6 ÷ 3) .additionmultiplicationsubtractiondivision3. Describe how the expressions 2 + 5 and 5 + 2 are different.Then determine whether the two expressions are equal to eachother. If the expressions are equal, name the property that saysthey are equal.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.28 California Mathematics Grade 7

Chapter 1 BRINGING IT ALL TOGETHER1-3Integers and Absolute ValuesComplete each sentence with either left or right to make a truesentence. Then write a statement comparing the two numberswith < or >.4. -45 lies to the of 0 on a number line.5. 72 lies to the of 0 on a number line.6. -3 lies to the of -95 on a number line.7. 6 lies to the of -7 on a number line.1-4Adding IntegersDetermine whether you add or subtract the absolute values ofthe numbers to find the sum. Give reasons for your answers.8. 4 + 8Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.9. -3 + 510. 9 + (-12)11. -23 + (-16)1-5Subtracting IntegersRewrite each difference as a sum. Then find the sum.12. 2 - 913. -3 - 814. 10 - (-12)15. -5 - (-16)California Mathematics Grade 7 29

Chapter 1 BRINGING IT ALL TOGETHER1-6Multiplying and Dividing IntegersFind each product or quotient.16. 9 (-2) 17. -6(-7)18. 12 ÷ (-4) 19. -35 ÷ (-7)1-7Writing Expressions and EquationsDetermine whether each situation requires addition,subtraction, multiplication or division.20. Find the difference in the cost of a gallon of premium gasoline andthe cost of a gallon of regular gasoline.21. Find the flight time after the time has been increased by 15minutes.1-8Problem Solving Investigation: Work Backward22. LOANS Alonso bought supplies for a camping trip. He has about$2 left. He spent $15.98 at the grocery store, then spent $21.91 atthe sporting goods store. He also spent a third of his money for adeposit on the campsite. About how much money did Alonso haveoriginally?1-9Solving Addition and Subtraction EquationsSolve each equation.23. x + 6 = 9 24. s - 5 = 14 25. 11 + m = 331-10Solving Multiplication and Division EquationsCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Solve each equation.26. 8r = 32 27. 3 = x_728. -9 = -9g30 California Mathematics Grade 7

C H A P T E R1ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without using mynotes or asking for help.Visit glencoe.com to accessyour textbook, moreexamples, self-check quizzes,and practice tests to helpyou study the concepts inChapter 1.• You are probably ready for the Chapter Test.• You may want take the Chapter 1 Practice Test onpage 79 of your textbook as a final check.I used my Foldable or Study Notebook to complete the review ofall or most lessons.• You should complete the Chapter 1 Study Guide and Review onpages 74–78 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 1 Practice Test onpage 79 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 1 Foldable.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.• Then complete the Chapter 1 Study Guide and Review onpages 74–78 of your textbook.• If you are unsure of any concepts or skills, refer back to the specificlesson(s).• You may also want to take the Chapter 1 Practice Test onpage 79 of your textbook.Student SignatureParent/Guardian SignatureTeacher SignatureCalifornia Mathematics Grade 7 31

C H A P T E R2Algebra: Rational NumbersUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with five sheets of 8 _ 1 " × 11" paper.2Place 5 sheets of paper3_ inch apart.4Roll up bottom edges.All tabs should be thesame size.Staple along the fold.Label the tabs with thelesson numbers.NOTE-TAKING TIP: As you study a lesson, writedown questions you have, comments andreactions, short summaries of the lesson, and keypoints that are highlighted and underlined.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.32 California Mathematics Grade 7

C H A P T E R2BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 2.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExamplebar notationChapter 2baseCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.dimensional analysisexponentlike fractionsmultiplicative inverses(continued on the next page)California Mathematics Grade 7 33

Chapter 2 BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExamplepowerrational numberreciprocalsrepeating decimalscientific notationterminating decimalunlike fractionsCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.34 California Mathematics Grade 7

2–1 Rational NumbersStandard 7NS1.3 Convert fractions to decimals and percents and use theserepresentations in estimations, computations, and applications. Standard 7NS1.5Know that every rational number is either a terminating or repeating decimal andbe able to convert terminating decimals into reduced fractions.MAIN IDEA• Express rationalnumbers as decimalsand decimals asfractions.BUILD YOUR VOCABULARY (pages 33–34)A rational number is any number that can be expressed inthe form a_ where a and b are and b ≠ 0.bA decimal like 0.0625 is a terminating decimal becausethe division ends, or terminates, when theis 0.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.KEY CONCEPTRational NumbersA rational number isany number that can beexpressed in the forma_ , where a and b arebintegers and b ≠ 0.EXAMPLE Write a Fraction as a DecimalWrite3_ as a decimal.163_ means 31616.0.187516 3.0000 Divide 3 by 16.__ 16140__ 128120__ 11280__ 80 Division ends when the is 0.0You can also use a calculator.The fraction _ 3 can be written as16.Check Your Progress Write1_ as a decimal.16California Mathematics Grade 7 35

2–1BUILD YOUR VOCABULARY (pages 33–34)Alike 1.6666 . . . is called a repeating decimal.Since it is not possible to show all of the, youcan use bar notation to show that the 6 .WRITE ITExplain how you decidewhere the bar is placedwhen you use barnotation for a repeatingdecimal.EXAMPLE Write a Mixed Number as a DecimalWrite -3_2 as a decimal.11You can write -3 2_ as _-3511 11 or _ 35-11decimal, find or .. To change -32_11 to aORGANIZE ITUnder the tab forLesson 2–1, explain inyour own words how toexpress rational numbersas decimals and decimalsas fractions.-11 35.0000__ -3320__ -1190__ -8820__ -1190__ -882 The remainder after each step is 2 or 9.The mixed number -3 2_ can be written as11.Check Your Progress Write 51_ as a decimal.9Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.36 California Mathematics Grade 7

2–1EXAMPLE Write a Terminating Decimal as a FractionWrite 0.32 as a fraction.0.32 =__ 320.32 is 32 .= Simplify. Divide by the greatestcommon factor of 32 and 100, .The decimal 0.32 can be written as .Check Your ProgressWrite 0.16 as a fraction.EXAMPLE Write a Repeating Decimal as a FractionALGEBRA Write 2. − 7 as a mixed number.Let N = 2. − 7 or 2.777 . . . . Then 10N = .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:Multiply N bybecause 1 digit repeats.Subtract N = 2.777 . . . to eliminate the0.777 . . . .10N = 27.777 . . .________-1N = 2.777 . . . N = 1NN == 25 10N - 1N == Divide each side by .Check Your ProgressSimplify.Write 1.− 7 as a mixed number.part,California Mathematics Grade 7 37

2–2Comparing and Ordering Rational NumbersStandard 7NS1.1 Read, write, and compare rational numbers in scientific notation(positive and negative powers of 10), compare rational numbers in general.MAIN IDEACompare and orderrational numbers.EXAMPLE Compare Positive Rational NumbersReplace with , or = to make _ 3 7 _ 813 atrue sentence.Write the fractions with the same denominator.For _ 3 7 and _ 8 , the least common denominator is 91.133_78_13==3 ·__7 ·8 ·__13 ·= _91= _91Since _91 < _91 , _ 3 78_13 .ORGANIZE ITUnder the tab forLesson 2–2, explain howyou can compare twonumbers by expressingthem as decimals andcomparing the decimals.EXAMPLE Compare Using DecimalsReplace with , or = to make 0.7 7 _11 atrue sentence.0.7 7 _11So, 0.7 Express 7_ as a decimal.11In the tenths place, 7 > 6.7_11 .Check Your Progressmake a true sentence.a. 2_3 _ 3 5Replace each with , or = tob. 4_9 0.5Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.38 California Mathematics Grade 7

2–2EXAMPLE Order Rational NumbersREMEMBER ITOn a number line,a number to the leftis always less than anumber to the right.CHEMISTRY The values forthe approximate densitiesof various substances areshown in the table. Orderthe densities from leastto greatest.Write each fraction asa decimal.1 4_5 =2 1_4 =2 3 _5 =SubstanceDensity(g/ cm 3 )aluminum 2.7beryllium 1.87brick 1 4_5crown glass 2 1_4fused silica 2. − 2marble2 _ 3 5nylon 1.1pyrex glass 2.32rubber neoprene 1. − 3Source: CRC Handbook of Chemistryand PhysicsFrom the least to the greatest, the densities are1.1, 1. − 3 , 1 4_5 , 1.87, 2. − 2 , 2 1_4 , 2.32, 2 _ 3 , and 2.7. So, the is5Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:the least dense, andCheck Your ProgressThe ride times for fiveamusement parkattractions are shown inthe table. Order the lengthsfrom least to greatest.is the most dense.CoasterRide Time(min)Big Dipper1 _ 3 4Double Loop 1.5Mind Eraser 1.8Serial Thriller 2 1_12X–Flight 2. − 3Source: www.coasterglobe.comCalifornia Mathematics Grade 7 39

2–3Multiplying Positive and Negative FractionsMAIN IDEA• Multiply fractions.BUILD YOUR VOCABULARY (pages 33–34)Dimensional analysis is the process of including units ofwhen you .KEY CONCEPTMultiply FractionsTo multiply fractions,multiply the numeratorsand multiply thedenominators.Standard7NS1.2 Add,subtract, multiply,and divide rationalnumbers (integers,fractions, and terminatingdecimals) and take positiverational numbers to wholenumberpowers.Standard 7MG1.3Use measures expressedas rates (e.g. speed,density) and measuresexpressed as products(e.g. person-days) to solveproblems; check theunits of the solutions;and use dimensionalanalysis to check thereasonableness of theanswer.EXAMPLE Multiply FractionsFind _ 3 7 · _ 8 . Write in simplest form.93_7 · 8 _9 = 31_7 · _ 8 93= ___= 8 _21Divide 3 and 9 by their GCF, .Multiply the numerators.Multiply the denominators.Simplify.EXAMPLE Multiply Negative FractionsFind -_3 4 · _ 7 . Write in simplest form.12-3_4 · 7 _112 = - _ 34 · _ 7124= ___Divide -3 and 12 by their GCF, .Multiply the numerators.Multiply the denominators.= - _ The factors have different signs,so the product is negative.EXAMPLE Multiply Mixed NumbersFind 3 _ 1 5 · 1 _ 3 . Write in simplest form.43 1_5 · 1 3 _4 = · 3 1_5 = , 1 3 _4 =Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.40 California Mathematics Grade 7

ORGANIZE ITUnder the tab forLesson 2–3, explain inyour own words howto multiply rationalnumbers.= 164_5 · _ 7 41= ___5 · 12–3Divide 16 and 4 by theirGCF, .Multiply the numerators.Multiply the denominators.= , or 5 Simplify.Check Your ProgressMultiply. Write in simplest form.2_a. -15 · _ 5 9b. 3 2_5 · 2 2_9EXAMPLECopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:VOLUNTEER WORK Last summer the 7th gradersperformed a total of 250 hours of community service.If the 8th graders spent 1 _ 1 this much time volunteering,5how many hours of community service did the 8thgraders perform?The 8 graders spent 1 1_ times the amount of time as the57th graders on community service.1 1_5 · 250 = ·= _ 1,500 or5The 8th graders didlast summer.of community serviceCheck Your Progress VOLUNTEER WORK Lastsummer the 5th graders performed a total of 150 hoursof community service. If the 6th graders spent 1 1_3 thismuch time volunteering, how many hours of communityservice did the 6th graders perform?California Mathematics Grade 7 41

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.2–4Dividing Positive and Negative FractionsStandard 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers,fractions, and terminating decimals). . . Standard 7MG1.3 Use measures expressed. . . as products (e.g. person-days) to solve problems; check the units of the solutions;and use dimensional analysis to check the reasonableness of the answer.MAIN IDEA• Divide fractions.BUILD YOUR VOCABULARY (pages 33–34)Two numbers whose product is one are multiplicativeinverses.The numbers 4 and 1_4 areor reciprocals of each other.KEY CONCEPTSInverse Property ofMultiplication Theproduct of a rationalnumber and itsmultiplicative inverse is 1.Dividing FractionsTo divide by a fraction,multiply by itsmultiplicative inverse.EXAMPLE Find a Multiplicative InverseWrite the multiplicative inverse of -2_4 7 .-2 4_ = Write -24_as an improper fraction.7 77_Since -18_7 (- = , the multiplicative inverse18)of -2 4_7 is .Check Your Progressa. Write the multiplicative inverse of -1 5 _6 .ORGANIZE ITOn the tab for Lesson2–4, explain in your ownwords how to dividerational numbers.EXAMPLE Divide Negative Fractions9)Find _ 2 7 ÷ ( - _ 82_7 ÷ 8_(- =9) 2_ 7. Write in simplest form.· Multiply by the multiplicativeinverse of -8_ which is .9= 21_7 · _( -9Divide 2 and 8 by their GCF, .8 )4= The fractions have different signs,so the quotient is negative.42 California Mathematics Grade 7

2–4EXAMPLE Divide Mixed NumbersFind 3 _ 1 4 ÷ ( -2 _ 1 . Write in simplest form.8)3 1_4 ÷ (-2 1_8) ( ) = ÷ 3 1_4 = ,8_-2 1_8 == · (-17 ) The multiplicative= 13 _4128_8_inverse of is -17 .· (- ) Divide 4 and 8 by their17GCF, ._26= - or Simplify.17Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.WRITE ITExplain how you woulddivide a fraction by awhole number.Check Your Progresssimplest form.a. -3_5 ÷ 9 _10Find each quotient. Write inb. 2 1_3 ÷ (-1 1_9)California Mathematics Grade 7 43

2–4EXAMPLEPAINTING It took the five members of the Johnson family10 _ 1 days to paint the 7 rooms in their house. How long2will it take the four members of the Reyes family tocomplete a similar task in their house assuming theywork at the same rate?If persons of the Johnson family each workeddays, the project required 5 × 10 1_ person-days of work. Divide2this number by persons to find the number of days it willtake the Reyes family to complete their task.5 10 1_2 person-days ÷ 4=5 × 10____1_2 person-days×1__ 14 personsMultiply by themultiplicative inverseof 4, which is .HOMEWORKASSIGNMENTPage(s):Exercises:= _ 52.5 or Simplify.4It will take the Reyes familysimilar painting task in their house.days to complete aCheck Your Progress LAWN CARE The 6 employees ofLove Your Lawn each worked 8 1_ days to mow 20 lawns. How2long will it take the 5 members of Mow Over to complete asimilar mowing task assuming they work at the same rate?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.44 California Mathematics Grade 7

2–5Adding and Subtracting Like FractionsStandard 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers,fractions, and terminating decimals) and take positive rational numbers to whole-numberpowers.MAIN IDEA• Add and subtractfractions with likedenominatorsBUILD YOUR VOCABULARY (pages 33–34)Fractions with likeare calledlike fractions.EXAMPLE Add Like FractionsFind3_3_16 + ( - _ 15 . Write in simplest form.16)_16 + 15(-16) =+ ( )____16= _-12or Simplify.16Add the numerators.The denominatorsare the same.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.KEY CONCEPTSAdding Like FractionsTo add fractions withlike denominators, addthe numerators andwrite the sum over thedenominator.Subtracting LikeFractions To subtractfractions with likedenominators, subtractthe numerators and writethe difference over thedenominator.EXAMPLE Subtract Like FractionsFind -_710 - _ 9 . Write in simplest form.107_-10 - _ 9Subtract the numerators.The denominatorsare the same.16or Rename -_10 as -1 _ 61010 = ___10= _ -1610or .Check Your Progress Find each difference. Write insimplest form.a. 2_9 + 8_(-b. -7_9)8 - _ 5 8California Mathematics Grade 7 45

2–5ORGANIZE ITUnder the tab forLesson 2–5, recordmodels illustrating theaddition and subtractionof like fractions.EXAMPLE Add Mixed NumbersFind 2 _ 5 8 + 6 _ 1 . Write in simplest form.82 5 _8 + 6 1_8 = ( + ) + ( 5 _8 + 1_8) Add the wholenumbers andfractions separately.= + _ 5 + 18= or Simplify.Add the numerators.EXAMPLE Subtract Mixed NumbersHEIGHTS In the United States, the average heightof a 9-year-old girl is 53 _ 4 inches. The average height of a516-year-old girl is 64 _ 1 inches. How much does an5average girl grow from age 9 to age 16?64 1_ - 534_5 5 = __ - __5 5Write the mixed numbersas improper fractions.HOMEWORKASSIGNMENTPage(s):Exercises:=-____5= 52 _5Subtract the numerators.The denominators arethe same.52or Rename _5 as .The average girl grows inches from age 9 to age 16.Check Your Progressa. Find 3 _ 310 + 4 1_ . Write in simplest form.10b. Ainsley was 42 1_ inches tall when she was 4 years old. When7she was 10 years old, she was 50 _ 3 inches tall. How much did7she grow between the ages of 4 and 10?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.46 California Mathematics Grade 7

2–6Adding and Subtracting Unlike FractionsStandard 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers,fractions, and terminating decimals) and take positive rational numbers to whole-numberpowers. Standard 7NS2.2 Add and subtract fractions by using factoring to findcommon denominators.MAIN IDEA• Add and subtractfractions with unlikedenominators.BUILD YOUR VOCABULARY (pages 33–34)Fractions withdenominators are calledunlike fractions.EXAMPLES Add and Subtract Unlike FractionsAdd or subtract. Write in simplest form.5_4)8 + (- 3 _5_8 + (-3_4) = 5 _3_8 + (- · The LCD is 2 · 2 · 2 or 8.4)= + Rename the fractionsusing the LCD.= Add the numerators.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.KEY CONCEPTAdding and SubtractingUnlike Fractions To findthe sum or difference oftwo fractions with unlikedenominators, renamethe fractions with acommon denominator.Then add or subtract andsimplify, if necessary.7_-96 - ( - 15_128)7_= Simplify.7_-96 - 15_(- = -128)96 · + · _ 3 3= _384 + _384=__ -28 + 4596 = 2 · 3, 128 = 2The LCD is 2 7 · 3 or.Rename using the LCD.Add the numerators.= _384Simplify.California Mathematics Grade 7 47

2–6ORGANIZE ITUnder the tab forLesson 2–6, record thedifferences betweenadding and subtractinglike and unlike fractions.Check Your Progresssimplest form.a. _ 5 6 + 2_(-Add or subtract. Write in3) b. 1_ 3 - 3_(-EXAMPLE Add Mixed NumbersFind -4_1 8 + 2 _ 5 . Write in simplest form.125)-4 1_8 + 2 _ 5 = + Write the mixed numbers12as fractions.= -33_8 · _ 3 3 + _ 2912 · 2_ The LCD is 2 · 2 · 2 · 32or .= + Rename each fractionusing the LCD.= ____24Add the numerators.HOMEWORKASSIGNMENTPage(s):Exercises:= or -1 Simplify.Check Your Progress Find -51_6 + 3 _ 5 . Write in8simplest form.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.48 California Mathematics Grade 7

2–7 Solving Equations with Rational NumbersEXAMPLES Solve by Using Addition or SubtractionCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.MAIN IDEA• Solve equationsinvolving rationalnumbers.Standard 7AF1.1Use variablesand appropriateoperations to write anexpression, an equation,an inequality, or a systemof equations or inequalitiesthat represents a verbaldescription (e.g. threeless than a number, half aslarge as area A.Standard 7NS1.2Add, subtract, multiply, anddivide rational numbers(integers, fractions, andterminating decimals)and take positive rationalnumbers to whole-numberpowers.ORGANIZE ITUnder the tab forLesson 2–7, summarizein your own words whatyou have learned aboutsolving equations withrational numbers.Solve g + 2.84 = 3.62.g + = 3.62 Write the equation.g + 2.84 - = 3.62 - Subtract fromeach side.g =Solve -_4 5 = s - _ 2 3 .- 4_5 = s - 2_3Simplify.Write the equation.- 4_5 + = s - 2_ + Add to each side.3- 4_5 + = s Simplify.+ _ 10 = s Rename each fraction15using the LCD.= s Simplify.EXAMPLES Solve by Using Multiplication or DivisionSolve7_11 c = -21.7_ c = -2111Write the equation.(_711 c ) = (-21) Multiply each side by .c =Simplify.California Mathematics Grade 7 49

2–7REVIEW ITWhat is a mathematicalsentence containingequals sign called?(Lesson 1–7)Solve 9.7t = -67.9.9.7t = -67.9 Write the equation.__ 9.7t= __ -67.9Divide each side by .t =Simplify.Check Your ProgressSolve each equation.a. h + 2.65 = 5.73 b. -2_5 = x - 3 _4 .c. _ 3 x = -27 d. 3.4t = -27.25EXAMPLE Write an Equation to Solve a ProblemHOMEWORKASSIGNMENTPage(s):Exercises:PHYSICS You can determine the rate an object istraveling by dividing the distance it travels by the timeit takes to cover the distance (r = _ d t). If an object travelsat a rate of 14.3 meters per second for 17 seconds, howfar does it travel?r = _ d t__ d14.3 =(14.3) = 17(__ d)= d Simplify.Write the equation.Multiply each side by .Check Your Progress If an object travels at a rate of73 miles per hour for 5.2 hours, how far does it travel?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.50 California Mathematics Grade 7

2–8 Problem-Solving Investigation:Look for a PatternEXAMPLEMAIN IDEA• Solve problems bylooking for a pattern.Standard 7MR2.4Make and testconjectures byusing both inductive anddeductive reasoning.Standard 7NS1.2Add, subtract, multiply,and divide rationalnumbers (integers,fractions, and terminatingdecimals) and takepositive rational numbersto whole-number powers.INTEREST The table below shows the amount ofinterest $3,000 would earn after 7 years at variousinterest rates. How much interest would $3,000 earnat 6 percent interest?Interest Rate(%)Interest Earned($)1 $2102 $4203 $6304 $8405 $1,050EXPLORE You know the amount of interest earned at interestrates of 1%, 2%, 3%, 4%, and 5%. You want to knowthe amount of interest earned at 6%.PLANLook for a pattern in the amounts of interestearned. Then continue the pattern to find theCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:SOLVECHECKamount of interest earned at a rate of .For each increase in interest rate, the amount ofinterest earned increases by $210. So for an interestrate of 6%, the amount of interest earned would be$1,050 + $210 = .Check your pattern to make sure the answer iscorrect.Check Your Progress INTEREST The table belowshows the amount of interest $5,000 would earn after 3 yearsat various interest rates. How much interest would $5,000 earnat 7 percent interest?Interest Rate(%)Interest Earned($)1 $1502 $3003 $4504 $6005 $750California Mathematics Grade 7 51

2–9 Powers and ExponentsMAIN IDEA• Use powers andexponents inexpressions.BUILD YOUR VOCABULARY (pages 33–34)The base is the number that is .The exponent tells how many times theisused as a .The number that is expressed using ancalled a power.isKEY CONCEPTZero and NegativeExponents Any nonzeronumber to the zeropower is 1. Any nonzeronumber to the negativen power is 1 divided bythe number to the nthpower.Standard7NS1.2 Add,subtract, multiply, anddivide rational numbers(integers, fractions, andterminating decimals) andtake positive rationalnumbers to wholenumberpowers.Standard 7NS2.1Understand negativewhole-numberexponents. Multiplyand divide expressionsinvolving exponents with acommon base.Standard 7AF2.1Interpret positive wholenumberpowers asrepeated multiplicationand negative wholenumberpowers asrepeated division ormultiplication by themultiplicative inverse.Simplify and evaluateexpressions that includeexponents.EXAMPLES Write Expressions Using PowersWrite _ 1 3 · _ 1 3 · _ 1 · 7 · 7 using exponents.31_3 · 1_3 · 1_ · 7 · 7 = · Associative Property3= Defi nition ofexponentsWrite p · p · p · q · p · q · q using exponents.p · p · p · q · p · q · q= p · p · p · p · q · q · q Property= (p · p · p · p) · (q · q · q) Property= · Defi nition of exponentsCheck Your Progress Write each expressionusing exponents.a. 2 · 2 · 2 · 2 · 5 · 5 · 5 b. x · y · x · x · y · y · yCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.52 California Mathematics Grade 7

2–9ORGANIZE ITOn the tab forLesson 2–9, comparehow to evaluate anexpression with positiveexponents and one withnegative exponents.EXAMPLES Evaluate PowersEvaluate ( 3 _4) 5 .( 3 _4) 5 = Defi nition of exponents= _ 2431,024Simplify.Evaluate 3 -7 .3 -7 ==__ 1__ 1Defi nition of negative exponentsSimplify.ALGEBRA Evaluate x 3 · y 5 if x = 4 and y = 2.x 3 · y 5 =3·5Replace x withandy with .= ( ) · ( )Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:Write the powers as products.= 64 · 32 Simplify.= Simplify.Check Your Progress Evaluate each expression.a. 6 5 b. 2 -5c. Evaluate x 2 · y 4 if x = 3 and y = 4.California Mathematics Grade 7 53

2–10 Scientific NotationStandard 7NS1.1 Read, write, and compare rational numbers in scientific notation(positive and negative powers of 10), compare rational numbers in general.MAIN IDEA• Express numbers inscientific notationBUILD YOUR VOCABULARY (pages 33–34)A number is expressed in scientific notation when it iswritten as aof a factor and aof 10.EXAMPLES Express Numbers in Standard FormKEY CONCEPTScientific NotationA number is expressedin scientific notationwhen it is written as theproduct of a factor anda power of 10. The factormust be greater than orequal to 1 and lessthan 10.9.62 × 10 5 in standard form.9.62 × 10 5 = 962000 The decimal place moves=places to the right.Write 2.85 × 10 -6 in standard form.2.85 × 10 -6 = 0.00000285 The decimal point moves 6places to the left.=Check Your Progressstandard form.a. 5.32 × 10 4b. 3.81 × 1 0 -4Write each number inCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.54 California Mathematics Grade 7

2–10ORGANIZE ITUnder the tab forLesson 2–10, collectand record examplesof numbers youencounter in your dailylife and write them inscientific notation.EXAMPLES Write Numbers in Scientific NotationWrite 931,500,000 in scientific notation.931500000 = 9.315 × 100,000,000 The decimal point moves8 places.= The exponent is positive.Write 0.00443 in scientific notation.0.00443 = × 0.001 The decimal point movesplaces.= 4.43 × The exponent is .EXAMPLE Compare Numbers in Scientific NotationCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.PLANETS The followingtable lists the averageradius at the equator forplanets in our solar system.Order the planets accordingto radius from largest tosmallest.First order the numbersaccording to their exponents.Then order the numbers withthe same exponents bycomparing the factors.STEP 1Jupiter, Neptune,Saturn, UranusPlanetRadius (km)Earth 6.38 × 10 3Jupiter 7.14 × 10 4Mars 3.40 × 10 3Mercury 2.44 × 10 3Neptune 2.43 × 10 4Saturn 6.0 × 10 4Uranus 2.54 × 10 4Venus 6.05 × 10 3Source: CRC Handbook of Chemistryand PhysicsEarth, Mars,Mercury, Venus× 10 4 6.38 × 10 32.43 × 10 4 3.40 × 10 36.0 × 10 4 > 2.44 × 10 32.54 × 10 4 × 10 3California Mathematics Grade 7 55

2–10STEP 27.14 × 10 4 > 6.0 × 10 4 > 2.54 × 10 4 > 2.43 × 10 4Jupiter Saturn Uranus Neptune6.38 × 1 0 3 > 6.05 × 1 0 3 > 3.40 × 1 0 3 > 2.44 × 1 0 3Earth Venus Mars MercuryThe order from largest to smallest is, Saturn,Uranus, Neptune, Earth, Venus, Mars, and Mercury.Check Your Progress Write each number inscientific notation.a. 35,600,000 b. 0.000653HOMEWORKASSIGNMENTPage(s):Exercises:c. The table lists the mass foreach of the planets in oursolar system. Order theplanets according to massfrom largest to smallest.PlanetMass(in tons)Mercury 3.64 × 10 20Venus 5.37 × 10 21Earth 6.58 × 10 21Mars 7.08 × 10 20Jupiter 2.09 × 10 24Saturn 6.25 × 10 23Uranus 9.57 × 10 22Neptune 1.13 × 10 23Source: nssdc.gsfc.nasa.govCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.56 California Mathematics Grade 7

C H A P T E R2BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 2 Foldableto help you study for yourchapter test.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 2, go to:glencoe.comBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 33–34) to help you solvethe puzzle.2-1Fractions and DecimalsWrite each fraction or mixed number as a decimal.1. -3_2. 3 1_3. -7 2_465Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Write each decimal as a fraction or mixed number in simplest form.4. 9.5 5. 0. − 6 6. 8.1252-2Comparing and Ordering Rational NumbersUse , or = to make each sentence true.7. -4_5-2_38. 4.4 4 2_59. 2.93 2.93Graph each pair of rational numbers on a number line.10. 1_5 , 1_311. -4_5 , - 9_10California Mathematics Grade 7 57

Chapter 2 BRINGING IT ALL TOGETHER2-3Multiplying Rational NumbersComplete each sentence.12. The greatest common factor of two numbers is thenumber that is aof both numbers.13. Numerators and denominators are by theirgreatest common factors tofractions.Multiply. Write in simplest form.14. -7_12 · _ 3 415. 4 2_3 · 5 1_82-4Dividing Rational NumbersWrite the multiplicative inverse for each mixed number.16. 2 1_5Complete the sentence.17. -1 3 _819. To divide by a , multiply by itsinverse.18. 3 4_720. To a number by 2 1_5 , multiply by 5 _11 .2-5Adding and Subtracting Like FractionsDetermine whether each pair of fractions are like fractions.21. _ 3 5 , _ 3 722. _ 5 8 , _ 7 823. 4_7 , - 5_7Add or subtract. Write in simplest form.24. _ 5 9 , - 2_3Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.25. _ 5 9 - 2_926. _ 5 8 + _ 7 827. 4_7 - _ 5 758 California Mathematics Grade 7

Chapter 2 BRINGING IT ALL TOGETHER2-6Adding and Subtracting Unlike FractionsAdd or subtract. Write in simplest form.28. _ 58 - _ 71229. _ 3 5 + _ 3 730. -2_3 + 5 _92-7Solving Equations with Rational NumbersMatch the method of solving with the appropriate equation.31. 25a = 3.75 a. Subtract _ 3 from each side.532. _ 3 5 m + _ 71033. r - 1.25 = 4.534. _ 3 5 + f = 1_22-8b. Multiply each side by 5 _3 .c. Divide each side by 3.75.d. Add 1.25 to each side.e. Divide each side by 1.25.Problem Solving Investigation: Look for a PatternCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.35. LIFE SCIENCE The table shows Outsideabout how many times a firefly Temperatureflashes at different temperatures. (˚C)About how many times will afirefly flash when thetemperature is 36°C?2-9Powers and ExponentsEvaluate each expression.36. 5 4 37. 6 3 38. 2 82-10Scientific NotationFlashes perMinute16 820 924 1128 14Write each number in scientific notation.39. 8,790,000 40. 0.0000125California Mathematics Grade 7 59

C H A P T E R2ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study are given witheach item.I completed the review of all or most lessons without using my notesor asking for help.Visit glencoe.com to accessyour textbook, moreexamples, self-check quizzes,and practice tests to helpyou study the concepts inChapter 2.• You are probably ready for the Chapter Test.• You may want take the Chapter 2 Practice Test onpage 139 of your textbook as a final check.I used my Foldable or Study Notebook to complete the review ofall or most lessons.• You should complete the Chapter 2 Study Guide and Review onpages 134–138 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 2 Practice Test onpage 139 of your text book.I asked for help from someone else to complete the review of all ormost lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 2 Foldable.• Then complete the Chapter 2 Study Guide and Review on pages134–138 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 2 Practice Test onpage 139 of your textbook.Student SignatureParent/Guardian SignatureCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Teacher Signature60 California Mathematics Grade 7

C H A P T E R3 Real Numbers andthe Pythagorean TheoremUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with two sheets of 8 _ 1 " by 11" paper.2Fold one in half fromtop to bottom. Cutalong fold from edgesto margin.Fold the other sheetin half from topto bottom. Cut alongfold between margins.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Insert first sheetthrough secondsheet and align folds.Label each page with Chapter 3a lesson number and title.Real Numbersand thePythagoreanTheoremNOTE-TAKING TIP: When you take notes, clarifyterms, record concepts, and write examples foreach lesson. You may also want to list ways inwhich the new concepts can be used in yourdaily life.Chapter 3California Mathematics Grade 7 61

C H A P T E R3BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 3.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExampleabscissa[ab-SIH-suh]conversecoordinate planehypotenuseirrational numberlegsordered pairordinate[OR-din-it]originCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.perfect square62 California Mathematics Grade 7

Chapter 3 BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExamplePythagorean Theoremquadrantsradical signreal numberCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.square rootx-axisx-coordinatey-axisy-coordinateCalifornia Mathematics Grade 7 63

3–1 Square RootsStandard 7NS2.4 Use the inverse relationship between raising to a power andextracting the root of a perfect square; for an integer that is not square, determinewithout a calculator the two integers between which its square root lies and explain why.MAIN IDEA• Find square roots ofperfect squares.BUILD YOUR VOCABULARY (pages 62–63)Numbers such as 1, 4, 9, and 25 are called perfect squaresbecause they are squares ofnumbers.Theof squaring a number is finding asquare root.The symbol √⎯⎯ is called a radical sign and is used toindicate the positive .Asquare root is called the principalsquare root.EXAMPLES Find Square RootsKEY CONCEPTSquare Root A squareroot of a number is oneof its two equal factors.Find each square root.√ 81√ 81 indicates the square root of 81.Since_= 81, √ 81 = .- √ 1681- √ _ 1681 indicates the square root of _ 1681 .Since± √ 1.44± √ 1.44 indicates bothsquare roots of 1.44.= _ 1681 , - √ _ 1681 = .Since = 1.44 and = 1.44, ± √ 1.44 = ±1.2,Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.or .64 California Mathematics Grade 7

3–1ORGANIZE ITOn Lesson 3-1 of yourFoldable, explain howto find the square rootof a number and give anexample.Chapter 3Real Numbersand thePythagoreanTheoremCheck Your Progressa. √ 64b. - √ _ 25144Find each square root.c. ± √ 2.25EXAMPLE Use an Equation to Solve a ProblemCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:MUSIC The art work of the square picture in a compactdisc case is approximately 14,161 mm 2 in area. Find thelength of each side of the square.The area is equal to the square of the length of a side.Let s = the length of the side.14,161 = s 2 Write the equation.= √ s 2Take the square root of each side.The length of a side of a compact disc case is aboutmillimeters since distance cannot be negative.Check Your Progress A piece of art is a square picture thatis approximately 11,025 square inches in area. Find the lengthof each side of the square picture.California Mathematics Grade 7 65

3–2Estimating Square RootsStandard 7NS2.4 Use the inverse relationship between raising to a power and extractingthe root of a perfect square; for an integer that is not square, determine without acalculator the two integers between which its square root lies and explain why.EXAMPLES Estimate Square RootsMAIN IDEA• Estimate square roots.Estimate √ 54 to the nearest whole number.The first perfect square less than 54 is .The first perfect square greater than 54 is .49 < 54 < 64 Write an inequality.< 54 < 49 = and 64 =√ 7 2 < √ 54 < √ 8 2 Take the square root ofeach number.7 < √ 54 < 8 Simplify.So, √ 54 is between and . Since 54 is closer to 49than 64, the best whole number estimate for √ 54 is .Estimate √ 41.3 to the nearest whole number.• The first perfect square less than 41.3 is 36.• The first perfect square greater than 41.3 is 49.Plot each square rooton a number line.Then plot √ 41.3 . 36 < 41.3 < 49 Write an inequality.< 41.3 < 36 = and 49 =√ 6 2 < √ 41.3 < √ 7 2 Find the square root of each number.< √ 41.3 < Simplify. So, √ 41.3 is between and . Since 41.3 is closer to 36than 49, the best whole number estimate for √ 41.3 is .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.66 California Mathematics Grade 7

3–2EXAMPLE Estimate Square RootsORGANIZE ITOn Lesson 3-2 of yourFoldable, explain how toestimate square roots.Chapter 3Real Numbersand thePythagoreanTheoremFINANCE If you were to invest $100 in a bank account fortwo years, your investment would earn interest dailyand be worth more when you withdrew it. If you had$120 after two years, the interest rate, written as adecimal, would be found using the expression ___( √ 120 - 10).10Estimate the value.First estimate the value of √ 120 .100 < 120 < 121 and areperfect squares.10 2 < 120 < 11 2 100 = and 121 =< √ 120 < Take the square root of each number.Since 120 is closer tothan 100, the best wholenumber estimate for √ 120 is . Use this to evaluatethe expression.( √ 120 - 10)___10( - 10)= ___ or10Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:The approximate interest rate is 0.10 or .Check Your Progressa. Estimate √ 65 to the nearest whole number.b. If you were to invest $100 in a bank account for two years,your money would earn interest daily and be worth morewhen you withdrew it. If you had $250 after two years, theinterest rate, written as a decimal, would be found using theexpression ___( √ 150 - 10).10California Mathematics Grade 7 67

3–3 Problem-Solving Investigation:Use a Venn DiagramEXAMPLEMAIN IDEA• Solve problems byusing a Venn diagram.Standard 7MR2.5Use a variety ofmethods, such aswords, numbers, symbols,charts, graphs, tables,diagrams, and models,to explain mathematicalreasoning.Standard 7NS1.2 Add,subtract, multiply, anddivide rational numbers(integers, fractions, andterminating decimals)and take positive rationalnumbers to whole-numberpowers.HOMEWORKASSIGNMENTPage(s):Exercises:LANGUAGES Of the 40 foreign exchange studentsattending a middle school, 20 speak French, 23speak Spanish, and 22 speak Italian. Nine studentsspeak French and Spanish, but not Italian. Sixstudents speak French and Italian, but not Spanish.Ten students speak Spanish and Italian, but not French.Only 4 students speak all three languages. Use a Venndiagram to find how many exchange students do notspeak any of these languages.EXPLORE You know how many students speak each ofthe different languages. You want to organizethe information.PLAN Make a Venn Diagram toorganize the information.SOLVECHECKSince 4 students speakall three languages,place a three in the section that representsall three languages. Fillin the other sectionsas appropriate.Add the numbers in each region of the diagram:1 + 9 + 6 + 4 + 10 + 2 =Since there are 40 exchange students altogether,40 - 32 = of them do not speak French,Spanish, or Italian.Check each circle to see if the appropriate numberof students is represented.Check Your Progress SPORTS Of the 30 students in Mr.Hall’s gym class, 14 play basketball, 9 play soccer, and 11 playvolleyball. Three students play basketball and soccer, but notvolleyball. One student plays soccer and volleyball, but notbasketball. Six students play basketball and volleyball, butnot soccer. Only 2 students play all three sports. Use a Venndiagram to find how many students in the class do not play anyof these sports.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.68 California Mathematics Grade 7

3–4 The Real Number SystemStandard 7NS1.4 Differentiate between rational and irrational numbers.MAIN IDEA• Identify and classifynumbers in the realnumber system.BUILD YOUR VOCABULARY (pages 62–63)Numbers that are notare calledirrational numbers.The set of rational numbers and the set ofnumbers together make up the set of real numbers.EXAMPLES Classify NumbersCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.KEY CONCEPTIrrational Number Anirrational number is anumber that cannot beexpressed as a_b , wherea and b are integersand b ≠ 0.Name all sets of numbers to which each real numberbelongs.0.090909 . . .The decimal ends in apattern.It is a number because it is equivalent to .√ 25Since √ 25 = , it is a number, an, and a rational number.- √ 12- √ 12 ≈ -3.46101615 . . . Since the decimal does not repeat or, it is an number.Check Your Progress Name all sets of numbers towhich each real number belongs.a. 0.1010101010...b. √ 64c. √ 13California Mathematics Grade 7 69

3–4EXAMPLES Graph Real NumbersORGANIZE ITOn Lesson 3-4 of yourFoldable, summarize theproperties of the realnumber system.Chapter 3Real Numbersand thePythagoreanTheoremEstimate √ 8 and - √ 2 to the nearest tenth. Thengraph √ 8 and - √ 2 on a number line.Use a calculator to determine the approximate decimal values.√ 8 ≈- √ 2 ≈Locate these points on a number line. √ 8 ≈ and - √ 2 ≈ .Check Your Progress Estimate √ 3 and - √ 6 to the nearesttenth. Then graph √ 3 and - √ 6 on a number line.REMEMBER ITAlways simplifynumbers beforeclassifying them.EXAMPLES Compare Real NumbersReplace each with , or = to make a true sentence.3 _ 7 8 √ 15Write each number as a decimal.3 _ 7 8 = √ 15 =Since is greater than ,3 7 _8 = √ 15 .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.70 California Mathematics Grade 7

3–4WRITE ITExplain why you candetermine that - √ 2 isless than 1.2 withoutcomputation.3. − 2 √ 10.4Write √ 10.4 as a decimal.√ 10.4 ≈Since 3. − 2 is than 3.224903099...,3. − 2 √ 10.4 .Check Your Progress Replace each with , or = tomake a true sentence.a. 3 _ 3 8 √ 14 b. 1. − 5 √ 2.25EXAMPLECopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:BASEBALL The time in seconds that it takes an object tofall d feet is 0.25 √ d . How many seconds would it take fora baseball that is hit 250 feet straight up in the air to fallfrom its highest point to the ground?Use a calculator to approximate the time it will take for thebaseball to fall to the ground.0.25 √ d = 0.25 Replace d with .≈ 3.95 or aboutIt will take aboutthe ground.Use a calculator.for the baseball to fall toCheck Your Progress The time in seconds that it takes anobject to fall d feet is 0.25 √ d . How many seconds would it takefor a baseball that is hit 450 feet straight up in the air to fallfrom its highest point to the ground?California Mathematics Grade 7 71

3–5 The Pythagorean TheoremMAIN IDEA• Use the PythagoreanTheorem.BUILD YOUR VOCABULARY (pages 62–63)A right triangle is a triangle with one right angle of 90°.The sides that form the right angle are called legs.The hypotenuse is the side opposite the right angle.The Pythagorean Theorem describes the relationshipbetween the lengths of the legs and the hypotenuse forany right triangle.KEY CONCEPTPythagorean TheoremIn a right triangle, thesquare of the length ofthe hypotenuse is equalto the sum of the squaresof the lengths of the legs.EXAMPLES Find the Length of a SideWrite an equation you could use to find the length of themissing side of the right triangle. Then find the missinglength. Round to the nearest tenth if necessary.12 in.cStandard7MG3.3 Know andunderstand thePythagorean theoremand its converse and useit to find the length of themissing side of a righttriangle and the lengthsof other line segmentsand, in some situations,empirically verify thePythagorean theorem bydirect measurement.Standard 7MR3.2 Notethe method ofderiving the solutionand demonstratea conceptualunderstanding of thederivation by solvingsimilar problems.c 2 = a 2 + b 216 in.Pythagorean Theoremc 2 = 12 2 + Replace a with and b with .c 2 = + Evaluate 12 2 and 16 2 .c 2 = Add 144 and 256.c = ± √ 400 Defi nition of square rootc = or Simplify.The equation has two solutions, and .However, the length of a side must be positive. So, thehypotenuse isinches long.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.72 California Mathematics Grade 7

Check Your Progress3–5Write an equation you could use to find the length of themissing side of the right triangle. Then find the missing length.Round to the nearest tenth if necessary.8 cmcORGANIZE ITOn Lesson 3-5 of yourFoldable, explain howto use the PythagoreanTheorem to find themissing length of a sideof a right triangle.Chapter 3Real Numbersand thePythagoreanTheorem15 cmEXAMPLE Find the Length of a SideThe hypotenuse of a right triangle is 33 centimeterslong and one of its legs is 28 centimeters. What is a, thelength of the other leg?c 2 = a 2 + b 2 Pythagorean Theorem2= a 2 +2Replace the variables.1,089 = a 2 + 784 Evaluate each power.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.REMEMBER ITThe longest side of aright triangle is thehypotenuse. Therefore, crepresents the length ofthe longest side.- = a 2 + - Subtract.= a 2 Simplify.± √ 305 = a Defi nition of square root≈ aThe length of the other leg is aboutUse a calculator.centimeters.Check Your Progress The hypotenuse of a right triangle is26 centimeters long and one of its legs is 17 centimeters. Findthe length of the other leg.California Mathematics Grade 7 73

3–5KEY CONCEPTConverse of thePythagorean TheoremIf the sides of a trianglehave lengths a, b, and cunits such thatc 2 = a 2 + b 2 , thenthe triangle is a righttriangle.EXAMPLE Identify a Right TriangleThe measures of three sides of a triangle are 24 inches,7 inches, and 25 inches. Determine whether the triangleis a right triangle.c 2 = a 2 + b 2Pythagorean Theorem25 2 7 2 + 24 2 c = 25, a = 7, b = 24625 + 576 Evaluate 25 2 , 7 2 , and 24 2 .= 625 Simplify. The triangle is aright triangle.Check Your Progressa. The base of a 12-foot ladder is 5 feet from the wall. How highcan the ladder reach?b. The measures of three sides of a triangle are 13 inches,5 inches, and 12 inches. Determine whether the triangleis a right triangle.HOMEWORKASSIGNMENTPage(s):Exercises:Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.74 California Mathematics Grade 7

3–6 Using the Pythagorean TheoremEXAMPLE Use the Pythagorean TheoremMAIN IDEA• Solve problems usingthe PythagoreanTheorem.Standard7MG3.3 Know andunderstand thePythagorean theoremand its converse and useit to find the length of themissing side of a righttriangle and the lengthsof other line segmentsand, in some situations,empirically verify thePythagorean theorem bydirect measurement.RAMPS A ramp to a newlyconstructed building mustbe built according to theguidelines stated in theAmericans with DisabilitiesAct. If the ramp is 24.1feet long and the top of the ramp is 2 feet off theground, how far is the bottomof the ramp from the baseof the building?Notice the problem involves a right triangle. Use thePythagorean Theorem.24.1 2 = a 2 + 2 2 Replace c with 24.1 andb with 2.= a 2 + Evaluate 24. 1 2 and 2 2 .- = a 2 = - Subtract from each side.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ORGANIZE ITOn Lesson 3-6 of yourFoldable, explain thePythagorean Theorem inyour own words and givean example of how itmight be used in areal-life situation.= a 2 Simplify.= a Defi nition of square root≈ aThe end of the ramp is aboutthe building.Simplify.from the base ofCheck Your Progress If a truck ramp is 32 feet long andthe top of the ramp is 10 feet off the ground, how far is the endof the ramp from the truck?Chapter 3Real Numbersand thePythagoreanTheoremCalifornia Mathematics Grade 7 75

3–6BUILD YOUR VOCABULARY (pages 62–63)Whole numbers such as 3, 4, and 5, which satisfy the, are calledPythagorean triples.EXAMPLESTANDARDS EXAMPLE The cross-sectionof a camping tent is shown. Find thewidth of the base of the tent.A 6 ftC 10 ftB 8 ftD 12 ftRead the Test ItemFrom the diagram, you know that the tent forms two congruentright triangles. Let a = 1_ x represent half the base of the tent.2Solve the Test ItemUse the Pythagorean Theorem.c 2 = a 2 + b 2Pythagorean TheoremHOMEWORKASSIGNMENTPage(s):Exercises:= a 2 + c = , b == a 2 + Evaluate 10 2 and 8 2 .100 - 64 = a 2 + 64 - 64 Subtract 64 from each side.= a 2 Simplify.= a Defi nition of square root= a SimplifyThe width of the base of the tent is a + a or + =feet. Therefore, choiceCheck Your ProgressThe diagram shows the crosssectionof a roof. How long iseach rafter, r?F 15 ft H 20 ftG 18 ft J 22 ftis correct.r32 ft12 ftrCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.76 California Mathematics Grade 7

3–7Geometry: Distance on the Coordinate PlaneStandard 7MG3.2 Understand and use coordinate graphs to plot simple figures,determine lengths and areas related to them, and determine their image undertranslations and refl ections.MAIN IDEA• Find the distancebetween points on thecoordinate plane.BUILD YOUR VOCABULARY (pages 62–63)A coordinate plane is formed by two number lines thatform right angles and intersect at theirpoints.The point of intersection of the two number lines is theorigin.TheThenumber line is the y-axis.number line is the x-axis.The number lines separate the coordinate plane intosections called quadrants.Any point on the coordinate plane can be graphed by usingan ordered pair of numbers.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ORGANIZE ITOn Lesson 3-7 of yourFoldable, explain inwriting how to useordered pairs to find thedistance between twopoints.Chapter 3Real Numbersand thePythagoreanTheoremThenumber in the ordered pair is called thex-coordinate.Thenumber of an ordered pair is they-coordinate.Another name for theis abscissa.Another name for theis ordinate.EXAMPLE Name an Ordered PairName the ordered pair for point A.• Start at the origin.• Move right to find theof point A, which is .yAx(continued on the next page)California Mathematics Grade 7 77

3–7• Move up to find the ,which is .So, the ordered pair for point A is .Check Your Progress Name the orderedypair for point A.AxEXAMPLES Graphing Ordered PairsGraph and label each point on the same coordinateplane.J (-3, 2.75)• Start atand moveunits to the. ThenJ(3, 2.75)ymove• Draw a dot and label itK ( 4, -1 1 _4).units.• Start at and move units to the .Then moveunits.• Draw a dot and label it .Check Your ProgressGraph and label eachpoint on the samecoordinate plane.a. J (-2.5, 3.5)b. K ( 2, -2 1_2)xK ( 14, 1 ) 4Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.78 California Mathematics Grade 7

3–7EXAMPLE Find the Distance on the Coordinate PlaneGraph the ordered pairs (0, -6)and (5, -1) . Then find the distancebetween the points.Oyx(5, 1)(0, 6)Let c = distance between the two points, a = 5, and b = 5.c 2 = a 2 + b 2Pythagorean Theoremc 2 = + Replace a with and b with .c 2 = + =√ c 2 =Defi nition ofc ≈The points are aboutSimplify.apart.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.REMEMBER ITYou can use thePythagorean Theoremto find the distancebetween two points ona coordinate plane.HOMEWORKASSIGNMENTPage(s):Exercises:Check Your Progress Graph the ordered pairs (0, -3) and( 2, -6). Then find the distance between the points.California Mathematics Grade 7 79

C H A P T E R3BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 3 Foldableto help you study for yourchapter test.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 3, go toglencoe.comBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 62–63) to help you solvethe puzzle.3-1Square RootsComplete each sentence.1. The principle square root is the square rootof a number.2. To solve an equation in which one side of the square is a squaredterm, you can take theequation.of each side of theFind each square root.3. √ 900 4. - √ _ 36495. - √ 625 6. √ _ 251213-2Estimating Square RootsDetermine between which two consecutive whole numberseach value is located.7. √ 23 8. √ 599. √ 27 10. √ 18Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.80 California Mathematics Grade 7

Chapter 3 BRINGING IT ALL TOGETHER3-3Problem-Solving Investigation: Use a Venn Diagram11. NUMBER THEORY A subset is a part of a set. The symbol ⊂ means“is a subset of.” Consider the following two statements.integers ⊂ rational numbersrational numbers ⊂ integersAre both statements true? Draw a Venn diagram to justifyyour answer.3-4The Real Number SystemCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Match the property of real numbers with the algebraicexample.a. (x + y) + z = x + (y + z)12. Commutativeb. pq = qp13. Associativec. h + 0 = h14. Distributive15. Identity16. Multiplicative Inverse3-5The Pythagorean Theoremd. c + (-c) = 0e. x (y + z) = xy + xzf. a _b · b _a = 1Use the Pythagorean Theorem to determine whether each ofthe following measures of the sides of a triangle are the sidesof a right triangle.17. 4, 5, 6 18. 9, 12, 1519. 10, 24, 26 20. 5, 7, 9California Mathematics Grade 7 81

Chapter 3 BRINGING IT ALL TOGETHER3-6Using the Pythagorean Theorem21. The triple 8-15-17 is a Pythagorean Triple. Complete the table tofind more Pythagorean triples.a b c Check: c 2 = a 2 + b 2original 8 15 17 289 = 64 + 225× 2× 3× 5× 10Determine whether each of the following is a Pythagoreantriple.22. 13-84-85 23. 11-60-6124. 21-23-29 25. 12-25-373-7Geometry: Distance on the Coordinate PlaneMatch each term of the coordinate plane with its description.26. ordinate a. one of four sections of thecoordinate plane27. y-axis b. x-coordinate28. origin c. y-coordinate29. abscissa d. vertical number line30. x-axis e. horizontal number linef. point where number lines meetCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.82 California Mathematics Grade 7

C H A P T E R3ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study are givenwith each item.I completed the review of all or most lessons without using mynotes or asking for help.Visit glencoe.com to accessyour textbook, moreexamples, self-check quizzes,and practice tests to helpyou study the concepts inChapter 3.• You are probably ready for the Chapter Test.• You may want to take the Chapter 3 Practice Test onpage 183 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 3 Study Guide and Reviewon pages 179–182 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 3 Practice Test onpage 183 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 3 Foldable.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.• Then complete the Chapter 3 Study Guide and Review onpages 179–182 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 3 Practice Test onpage 183 of your textbook.Student SignatureParent/Guardian SignatureTeacher SignatureCalifornia Mathematics Grade 7 83

C H A P T E R4 Proportions and SimilarityUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with a plain sheet of 11" by 17" paper.Fold in thirds widthwise.Open and fold the bottomto form a pocket.Glue edges.Label each pocket. Placeindex cards in each pocket.NOTE-TAKING TIP: When you take notes, definenew vocabulary words, describe new ideas, andwrite examples that help you remember themeanings of the words and ideas.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.84 California Mathematics Grade 7

C H A P T E R4BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 4.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExamplecongruentconstant of proportionalitycorresponding partsCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.cross productsequivalent ratiosnonproportionalpolygonproportion(continued on the next page)Chapter 4California Mathematics Grade 7 85

Chapter 4 BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExampleproportionalraterate of changeratioscalescale drawingscale factorscale modelsimilarunit rateCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.unit ratio86 California Mathematics Grade 7

4–1Ratios and RatesMAIN IDEA• Express ratios asfractions in simplestform and determineunit rates.BUILD YOUR VOCABULARY (pages 85–86)A ratio is a comparison of two numbers by .A rate is a special kind of. It is a comparisonof two quantities with different types of units.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Standard7AF4.2 Solvemultistep problemsinvolving rate, averagespeed, distance, andtime or a direct variation.Standard 7MG1.3Use measures expressedas rates (e.g. speed,density) and measuresexpressed as products(e.g. person-days)to solve problems;check the units of thesolutions; and usedimensional analysis tocheck the reasonablenessof the answer.When a rate is, it is called a unit rate.EXAMPLE Write Ratios in Simplest Formso it has a denominator ofExpress 12 blue marbles out of 18 marbles insimplest form.Divide the numerator and denominator__12 marbles18 marbles = _The ratio of blue marbles to marbles isout of .EXAMPLE Find a Unit Rateby the greatest common factor, .Divide out common units.READING Yi-Mei reads 141 pages in 3 hours. How manypages does she read per hour?Write the rate that expresses the comparison of pages to hours.Then find the unit rate.__ 141 pages3 hours= _ pagesorDivide the numerator and denominatorhour by to get a denominator of 1.Yi-Mei reads an average of pages per .California Mathematics Grade 7 87

4–1REVIEW ITWhat is the greatestcommon factor of two ormore numbers? How canyou find it?(Prerequisite Skill)Check Your Progress Express each ratio insimplest form.a. 5 blue marbles out of 20 marblesb. 14 inches to 2 feetc. On a trip from Columbus, Ohio, to Myrtle Beach, SouthCarolina, Lee drove 864 miles in 14 hours. What was Lee’saverage speed in miles per hour?ORGANIZE ITWrite the definitions ofrate and unit rate on anindex card. Then on theother side of the card,write examples of howto find and compare unitrates. Include these cardsin your Foldable.HOMEWORKASSIGNMENTPage(s):Exercises:EXAMPLE Compare Unit RatesSHOPPING Alex spends $12.50 for 2 pounds of almondsand $23.85 for 5 pounds of jellybeans. Which item costsless per pound? By how much?For each item, write a rate that compares the cost to theamount. Then find the unit rates.Almonds:__ $12.502 pounds = __1 poundJellybeans: __ $23.855 pounds = __1 poundThe almonds costper pound and the jellybeanscost per pound. So, the jellybeans cost -orper pound less than the almonds.Check Your Progress Cameron spends $22.50 for 2 poundsof macadamia nuts and $31.05 for 3 pounds of cashews. Whichitem costs less per pound? By how much?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.88 California Mathematics Grade 7

4–2 Proportional and Nonproportional RelationshipsPreparation for 7AF3.4 Key Plot the values of quantities whose ratios are always thesame (e.g., cost to the number of an item, feet to inches, circumference to diameter of acircle). Fit a line to the plot and understand that the slope of the line equals the quantities.MAIN IDEA• Identify proportionaland nonproportionalrelationships.BUILD YOUR VOCABULARY (pages 85–86)If two quantities are, then they have aratio.For ratios in which this ratio is, the twoquantities are said to be .EXAMPLES Identify Proportional RelationshipsCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.KEY CONCEPTSProportional A statementof equality of two ratioswith a constant ratio.Nonproportional Arelationship in which twoquantities do not have acommon ratio.HOUSE CLEANING A house-cleaning service charges$45 for the first hour and $30 per hour for eachadditional hour. The service works for 4 hours. Is the feeproportional to the number of hours worked? Make atable of values to explain your reasoning.Find the cost for 1, 2, 3, and 4 hours and make a table todisplay hours and cost.Hours Worked 1 2 3 4Cost ($)For each number of hours, write the relationship of the cost andnumber of hours as a ratio in simplest form.___ costhours worked45_1 or 75_2 or _ 1053 or _ 1354 orSince the ratios of the two quantities are ,the cost isto the number of hoursworked. The relationship is .California Mathematics Grade 7 89

4–2BAKING A recipe for jelly frosting calls for _ 1 cup of jelly3and 1 egg white. Is the number of egg whites usedproportional to the cups of jelly used? Make a table ofvalues to explain your reasoning.Find the amount of jelly and egg whites needed for differentnumbers of servings and make a table to show these measures.Cups of JellyEgg whites 1 2 3 4For each number of cups of jelly, write the relationship of theratio in simplest form.to the1_2_13_1 or 3_1_2 or _ 34 oras aSince the ratios between the two quantities are all equalto , the amount of jelly used is to thenumber of egg whites used.HOMEWORKASSIGNMENTPage(s):Exercises:Check Your Progressa. PLUMBING A plumbing company charges $50 for the firsthour and $40 for each additional hour. Suppose a service callis estimated to last 4 hours. Is the fee proportional to thenumber of hours worked?b. COOKING Among other ingredients, a chocolate chipcookie recipe calls for 2.5 cups of flour for every 1 cupof sugar and every 2 eggs. Is the amount of flour usedproportional to the number of eggs used?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.90 California Mathematics Grade 7

4–3 Solving ProportionsStandard 7AF4.2 Solve multistep problems involving rate, average speed,distance, and time or a direct variation.MAIN IDEA• Use proportions tosolve problems.BUILD YOUR VOCABULARY (pages 85–86)In a proportion, two are .In a proportion, the cross products are .KEY CONCEPTSProportion A proportionis an equation statingthat two ratios areequivalent.Property of ProportionsThe cross products of aproportion are equal.Be sure toinclude this definitionand property in yourFoldable.EXAMPLE Write and Solve a Proportion.COOKING A recipe serves 10 people and calls for 3 cupsof flour. If you want to make the recipe for 15 people,how many cups of flour should you use?cups of fl ourtotal people served3_10 = _ n15cups of fl ourtotal people served= Find the crossproducts.45 = 10n Multiply.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.You will need15 people.45_ = _ 10n Divide eachside by .= n Simplify.cups of flour to make the recipe forCheck Your Progress COOKING A recipe serves 12people and calls for 5 cups of sugar. If you want to make therecipe for 18 people, how many cups of sugar should you use?California Mathematics Grade 7 91

4–3EXAMPLEFOOD Haley bought 4 pounds of tomatoes for $11.96.Write an equation relating the cost to the number ofpounds of tomatoes. How much would Haley pay for6 pounds at this same rate? for 10 pounds?Find the constant of proportionality between cost and pounds.____cost in dollarspounds of tomatoes = _ 11.96 or 2.99 The cost is $2.99 per pound.4WordsVariablesEquationThe cost is $2.99 times the number of pounds.Let c represent the cost.Let p represent the number of pounds.c = 2.99 · pUse this same equation to find the cost for 6 and 10 pounds oftomatoes sold at the same rate.c = 2.99p Write the equation. c = 2.99pc = 2.99Replace p with thenumber of pounds.c = 2.99c = Multiply. c =HOMEWORKASSIGNMENTPage(s):Exercises:The cost for 6 pounds of tomatoes is10 pounds is .and forCheck Your Progress FOOD Cameron bought 3 poundsof apples for $11.37. Write an equation relating the cost to thenumber of pounds of apples. How much would Cameron pay for5 pounds at this same rate?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.92 California Mathematics Grade 7

4–4 Problem-Solving Investigation:Draw a DiagramEXAMPLEMAIN IDEA• Solve problems bydrawing a diagram.Standard 7MR2.5Use a variety ofmethods, such aswords, numbers, symbols,charts, graphs, tables,diagrams, and models,to explain mathematicalreasoning.Standard 7AF4.2Solve multistepproblems involving rate,average speed, distance,and time or a directvariation.VOLUME A bathtub is being filled with water. After4 minutes, _ 1 of the bathtub is filled. How much longer5will it take to completely fill the bathtub assuming thewater rate is constant?EXPLORE After 4 minutes, the bathtub is 1_ of the way filled.5How many more minutes will it take to fill thebathtub?PLANDraw a diagram showing the water level after every4 minutes.SOLVE The bathtub will be filled after 4-minuteperiods. This is a total of 5 × 4 or .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:CHECKThe question asks how much longer will it taketo completely fill the bathtub after the initial 4minutes. Since the total time needed is 20 minutes,it will take or tocompletely fill the bathtub.Check Your Progress VOLUME A swimming pool is beingfilled with water. After 3 hours, 1_ of the pool is filled. How4much longer will it take to completely fill the swimming poolassuming the water rate is constant?California Mathematics Grade 7 93

4–5Similar PolygonsReinforcement of Standard 6NS1.3 Use proportions to solve problems. Use crossmultiplication as a method for solving such problems, understanding it as themultiplication of both sides of an equation by a multiplicative inverse.MAIN IDEA• Identify similarpolygons and findmissing measures ofsimilar polygons.KEY CONCEPTSimilar Polygons If twopolygons are similar, then• their correspondingangles are congruent,or have the samemeasure, and• their correspondingsides are proportional.BUILD YOUR VOCABULARY (pages 85–86)A polygon is a simple closed figure in a plane formedbyline segments.Polygons that have theshape are called similarpolygons.The parts offigures that “match” are calledcorresponding parts.Congruent means to have themeasure.EXAMPLE Identify Similar PolygonsDetermine whether triangle DEF is similar to triangleHJK. Explain your reasoning.EJ4D35F5 6.25H3.75First, check to see if corresponding angles are congruent.∠D ∠H,

4–5Check Your ProgressDetermine whethertriangle ABC is similar totriangle TRI. Explainyour reasoning.A3C45BT4.5I67.5RBUILD YOUR VOCABULARY (pages 85–86)Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ORGANIZE ITMake vocabularycards for each term inthis lesson. Be sure toplace the cards in yourFoldable.Theof the lengths of twosides of two similar polygons is called the scale factor.EXAMPLE Finding Missing Measures Given that rectangleLMNO ∼ rectangle GHIJ, find the missing measure. METHOD 1 Write a proportion.The missing measure n is the length of −−− NO . Write a proportioninvolving NO that relates corresponding sides of the tworectangles.rectangle GHIJrectangle LMNO2_3 = 4_n=GJ = , LO = , IJ = , and NO =· n = · 4 Find the cross products.= Multiply.= Divide each side by 2.rectangle GHIJrectangle LMNOMETHOD 2 Use the scale factor to write an equation.Find the scale factor from rectangle GHIJ to rectangle LMNOby finding the ratio of corresponding sides with known lengths.scale factor: _ GJLO =The scale factor is the constant ofproportionality.(continued on the next page)California Mathematics Grade 7 95

4–5WordsA length on rectangle GHIJ istimes as longas a corresponding length on rectangle .VariablesLet represent the measure of .Equation4 = 2_ n Write the equation.34 · = · 2_ n Multiply each side by .3= Simplify.HOMEWORKASSIGNMENTPage(s):Exercises:Check Your Progress Given that rectangle ABCD ∼rectangle WXYZ, write a proportion to find the measure of −− ZY .Then solve. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.96 California Mathematics Grade 7

4–6Measurement: Converting Length,Weight/Mass, Capacity, and TimeMAIN IDEA• Convert customaryand metric units oflength, weight or mass,capacity, and time.BUILD YOUR VOCABULARY (pages 85–86)Ais a ratio in which the denominator is1 unit.Standard 7MG1.1Compare weights,capacities,geometric measures,times, and temperatureswithin and betweenmeasurement systems(e.g. miles per hourand feet per second,cubic inches to cubiccentimeters).EXAMPLESConvert 2 _ 1 pounds to ounces.2Since2 1_ lb = 21_2 2pound = 16 ounces, the unit ratio is __ .1 lb= 2 1_2lb ·16 ozby _1 lb .16 ozlb ·_l lbDivide out common units,leaving the desired unit,Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.REMEMBER ITThe prefix kilomeans1,000. Theprefix centi- means0.01. The prefix millimeans0.001.= 2 1_ · 16 oz or2oz Multiply.2 1_ pounds is equal to2ounces.SHIPPING Cristos is shipping a package of model trainparts. The sum of the weights of the parts is 900 grams.What is the weight of the parts in kilograms?900 g = 900 g ·= g ·__ 1 kg1,000 g__ 1 kgl,000 g= kg or 0.9 kg Multiply..Since 1 kilogram =grams,1 kgmultiply by __1,000 g .Divide out commonunits, leaving the desiredunit, .The weight of the train parts iskilogram.California Mathematics Grade 7 97

4–6Convert 10 miles to kilometers. Round to the nearesthundredth.METHOD 1Use 1 km ≈ 0.621 mi.The unit ratio is .10 mi ≈ mi ·__ 1 km0.621 miSince 1km ≈ 0.621mi,1 kmmultiply by __0.621 mi . Divideout common units, leavingthe desired unit, kilometer.≈__ 10 kmor km Multiply.0.621METHOD 2Use 1 mi ≈ 1.609 km.The unit ratio is .10 mi ≈ mi ·≈ 10 · 1.609 or__ 1.609 km1 mi__1.609 kmMultiply by . Divide1 miout common units, leavingthe desired unit, kilometer.km Multiply.10 miles is equal to about kilometers.Check Your Progress Convert each measurement.Round to the nearest hundredth if necessary.a. 7 km = mb. 72 in. = ftc. 12 fl oz = mLCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.98 California Mathematics Grade 7

4–6EXAMPLEHEALTH To replenish water lost during moderateactivity, the average adult should consume 35 millilitersof fluid per kilogram of body weight. How many fluidounces is this per pound?To convert milliliters per kilogram to fluid ounces perpound, use conversion factors relating milliliters to fluid ouncesand kilograms to pounds.__ · ___ 1 fl oz· __ 0.454 kgkgmL29.574 1=__ 35 mL·kg= ___29.574 lb__ 1 fl oz29.574 mLf l oz0.454 kg ·__1 lboutcommon units.Multiply.= ___ f l ozDivide.1Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:The average adult should consume aboutper pound.fluid ouncesCheck Your Progress HORSES A Clydesdale horse drinksabout 114 liters of water per day. How many gallons is this perweek?California Mathematics Grade 7 99

4–7 Measurement: Converting Square Unitsand Cubic UnitsMAIN IDEAEXAMPLES Convert Units of Area• Convert square and Complete each conversion.cubic units of length,weight or mass,capacity, and time in 5.8 m 2 = c m 2both customary andmetric systems. cm cmStandard 7MG1.1Compare weights,capacities,geometric measures,times, and temperatureswithin and betweenmeasurement systems(e.g. miles per hourand feet per second,cubic inches to cubiccentimeters).Standard 7MG2.4 Relatethe changes inmeasurement with achange of scale to theunits used (e.g., squareinches, cubic feet) andto conversions betweenunits (1 square foot =144 square inchesor [1 f t 2 ] = [144 i n 2 ],1 cubic inch isapproximately 16.38cubic centimeters or[1 i n 3 ] = [16.38 c m 3 ]).5.8 m 2 = 5.8 × m × m × __1 m= c m 21,296 in . 2 = f t 21,296 in . 2 = 1,296 × in. × in. ×= f t 2× __1 m__ 1 ft×__ 1 ftCheck Your Progress Complete each conversion.a. 3.5 y d 2 = f t 2 b. 12,500 c m 2 = m 2EXAMPLE Convert Units of VolumeAIR QUALITY Morgan is using an air purifier in a roomthat he estimates holds about 27 cubic yards of air. Howmany cubic feet of air will the air purifier need to clean?To convert from cubic yards toin.in.by the unit100 cmratio, __1 m .Multiplyby theunit ratio,, use the unit.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ratio .100 California Mathematics Grade 7

4–7The unit ratio is .27 y d 3 = × yd × yd × yd × _ 3 ft1 yd × _ 3 ft1 yd × _ 3 ft1 yd= f t 3The air purifier will need to cleancubic feet of air.Check Your Progress POOLS A swimming pool holds2,500 cubic feet of water. How many cubic yards is this?EXAMPLE Convert Between SystemsConvert 90 cubic feet to cubic meters.The unit ratio is .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:90 f t 3 = 90 × ft × ft × ft × __1 ft≈ m 3Check Your Progresscentimeters.m m m× __1 ft× __1 ftConvert 64 square inches to squareCalifornia Mathematics Grade 7 101

4–8 Scale Drawings and ModelsStandard 7MG1.2 Construct and read drawings and models made to scale.MAIN IDEA• Solve problemsinvolving scaledrawings.BUILD YOUR VOCABULARY (pages 85–86)A scale drawing or a scale model is used to represent anobject that is too or too to be drawnor built at actual size.The scale is determined by theof given lengthon ato the corresponding actuallength of the object.EXAMPLE Find a Missing MeasurementRECREATION Use the map to find the actual distancefrom Bingston to Alanton.BingstonDolifREMEMBER ITScales and scalefactors are usuallywritten so that thedrawing length comesfirst in the ratio.TribunetAlantonScale: 1 in. = 5 miUse an inch ruler to measure the map distance.The map distance is about 1.5 inches.METHOD 1 Write and solve a proportion.mapactual_ 1 in.5 mi == Find the cross products.x =METHOD 2 Write and solve an equation.Write the scale asper inch.which meansmapactualSimplify.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.102 California Mathematics Grade 7

4–8WordsVariablesThe actual distance isper inch ofmap distance.Let a represent the actual distance in miles.Let m represent the map distance in inches.Equationa =Write the equation.a = 5 Replace m with .a =Multiply.The actual distance from Bingston to Alanton is .EXAMPLE Find the ScaleCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ORGANIZE ITWrite definitions of scale,scale drawing, and scalemodel on cards and giveyour own examples. Besure to explain how tocreate a scale for a scaledrawing or model.HOMEWORKASSIGNMENTPage(s):Exercises:SCALE DRAWINGS A wall in a room is 15 feet long. On ascale drawing it is shown as 6 inches. What is the scaleof the drawing?Write and solve a proportion to find the scale of the drawing.Length of Roomscale drawing lengthactual length_ 6 in.15 ft = _ 1 in.x ft=x =So, the scale is 1 inch = .Scalescale drawing lengthactual lengthFind the crossproducts. Multiply.Then divide eachside by 6.Simplify.Check Your Progress The length of a garage is 24 feet. Ona scale drawing the length of the garage is 10 inches. What isthe scale of the drawing?California Mathematics Grade 7 103

4–9 Rates of ChangePreparation for Standard 7AF3.4 Plot the values of quantities whose ratios are alwaysthe same (e.g., cost to the number of an item, feet to inches, circumference to diameterof a circle). Fit a line to the plot and understand that the slope of the line equals thequantities.MAIN IDEA• Find rates of change.BUILD YOUR VOCABULARY (pages 85–86)A rate of change is a rate that describes how one quantityinto another.EXAMPLE Find a Rate of ChangeDOGS The table below shows the weight of a dog inpounds between 4 and 12 months old. Find the rateof change in the dog’s weight between 8 and 12 monthsof age.Age (mo) 4 8 12Weight (lb) 15 28 43REMEMBER ITRate of change isalways expressed as aunit rate.___change in weight (43 - ) pounds= ____change in age( - 8) months==pounds___monthspounds____monthThe dog grew from28 to 43 pounds fromages 8 to 12 months.Subtract to fi nd thechange in weightsand ages.Express this rate asa .The dog grew an average of pounds per .Check Your Progress The table below shows Julia’s heightin inches between the ages of 6 and 11. Find the rate of changein her height between ages 6 and 9.Age (yr) 6 9 11Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Weight (in.) 52 58 60104 California Mathematics Grade 7

4–9EXAMPLE Find a Negative Rate of ChangeKEY CONCEPTRate of Change To findthe rate of change,divide the difference inthe y-coordinate by thedifference in thex-coordinate.Record thisconcept on one side ofan index card. Write anexample on the otherside of the card.SCHOOLS The graph showsthe number of students inthe seventh grade between2000 and 2004. Find therate of change between2002 and 2004. Use the data to write a rate comparing the change in studentsto the change in time.-____change in students= ____change in time-The number ofstudents changedfrom 485 to 459 from2002 to 2004.= __ Simplify.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.REMEMBER ITAlways read graphsfrom left to right.= __ Express as a unit rate.The rate of change is students per .Check Your Progress The graph below shows the numberof students in the 6th grade between 1999 and 2005. Find therate of change between 2003 and 2005. California Mathematics Grade 7 105

4–9EXAMPLES Compare Rates of ChangeTEMPERATURE The graphshows the temperaturemeasured on each hourfrom 10 A.M. to 3 P.M.During which 1-hour periodwas the rate of change intemperature the greatest?Find the rates of change for each1-hour period. Use the ratio____change in temperature.change in time55° - 54°10 A.M. to 11 A.M. ___11 A.M. - 10 A.M. = 11 A.M. to 12 P.M.12 P.M. to 1 P.M.1 P.M. to 2 P.M.___59° - 55°12 P.M. - 11 A.M.___60° - 59°2 P.M. - 12 P.M.___60° - 60°2 P.M. - 1 P.M.===2 P.M. to 3 P.M.___62° - 60°3 P.M. - 2 P.M.=HOMEWORKASSIGNMENTPage(s):Exercises:The greatest rate of change in temperature isbetweenCheck Your ProgressThe graph shows thetemperature measuredeach hour from 10 a.m.to 4 p.m. Find the 1-hourtime period in whichthe rate of change intemperature was thegreatest. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.106 California Mathematics Grade 7

4–10Constant Rate of ChangePreparation for Standard 7AF3.4 Plot the values of quantities whose ratios are alwaysthe same (e.g., cost to the number of an item, feet to inches, circumference to diameterof a circle). Fit a line to the plot and understand that the slope of the line equals thequantities.MAIN IDEA• Identify proportionaland nonproportionalrelationships by findinga constant rate ofchange.BUILD YOUR VOCABULARY (pages 85–86)A relationship that has a straight-line graph is called a. The rate of change between anytwo points of a is .EXAMPLE Identify linear RelationshipsBABY-SITTING The amounta baby-sitter charges isshown. Is the relationshipbetween the number ofhours and the amountcharged linear? If so,find the constant rateof change. If not, explainyour reasoning.Number ofHoursAmountEarned1 $82 $163 $244 $32Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Examine the change in the number of hours worked and in theamount earned.+1+1+1Number ofHoursSince the rate of changeAmountEarned1 $82 $163 $244 $32. The, this isis _ 8 or . This means that the babysitter earns1.+8+8+8California Mathematics Grade 7 107

4–10Check Your ProgressBABY-SITTING The amounta baby-sitter charges is shown.Is the relationship betweenthe number of hours and theamount charged linear? If so,find the constant rate ofchange.Number ofHoursAmountEarned1 $122 $243 $364 $48EXAMPLE Find a Constant Rate of ChangeTRAVEL Find the constantrate of change for the hourstraveled and miles traveled.Interpret its meaning.Choose any two points on theline and find the rate of changebetween them.(2, 60)Miles300240180120600Miles and Hours Traveledy24 6 8Hoursx(4, 120)___change in mileschange in time == Subtract.The amount of milesfrom 60 to 120between hours 2and 4.= Express as a unit rate.The rate of speed is .Check Your ProgressTRAVEL Find the constant rateof change for the hours traveledand miles traveled. Interpretits meaning.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.108 California Mathematics Grade 7

4–10EXAMPLETAXIS Use the graph todetermine if there is aproportional linearrelationship between themiles driven and thecharge for a ride.Explain your reasoning.Since the graph of the dataforms a line, the relationshipbetween the two scales is linear.Charge$24$20$16$12$8$40Cost of a Taxi5 10 15 20MilesThis can also be seen in the table of values created using thepoints on the graph.Charge ($) 4 8 12 16 20Miles 0 5 10 15 20+4 +4 +4 +4 Constant Rate of Change___change in chargechange in miles =+5 +5 +5 +5To determine if the two scales are proportional, express therelationship between the charges for several miles as a ratio.__ chargemiles8_5 = 12_10 = 16_15 ≈Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:Since the ratios areisCheck Your ProgressMOVIES Use the graph todetermine if there is aproportional linear relationshipbetween the number ofmovies rented and the totalcost. Explain your reasoning., the total chargeto the number of miles driven. California Mathematics Grade 7 109

C H A P T E R4BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 4 Foldableto help you study for yourchapter test.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 4, go to:glencoe.comBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 85–86) to help you solvethe puzzle.4-1Ratios and RatesMatch each phrase with the term they describe.1. a comparison of two numbersa. unit rate2. a comparison of two quantitieswith different types of units3. a rate that is simplified so ithas a denominator of 14. Express 12 wins to 14 losses in simplest form.5. Express 6 inches of rain in 4 hours as a unit rate.4-2b. numeratorc. ratiod. rateProportional and Nonproportional RelationshipsDetermine whether each relationship is proportional.6. Side length (ft) 1 2 3 4 5Perimeter (ft) 4 8 12 16 207. Time (hr) 1 2 3 4 5Rental Fee ($) 10.00 12.50 15.00 17.50 20.00Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.110 California Mathematics Grade 7

Chapter 4 BRINGING IT ALL TOGETHER4-3Solving Proportions8. Do the ratios _ a b and _ c always form a proportion? Why or why not?dSolve each proportion.9. _ 7 b = _ 35510. _ a16 = _ 3 811. 4_13 = 3 _c4-4Problem-Solving Investigation: Draw a Diagram12. FAMILY At Willow’s family reunion, 4_ of the people are 18 years5of age or older. Half of the remaining people are under 12 yearsold. If 20 children are under 12 years old, how many people areat the reunion?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.4-5Similar Polygons13. If two polygons have corresponding angles that are congruent,does that mean that the polygons are similar? Why or why not?14. Rectangle ABCD has side lengths of 30 and 5. Rectangle EFGHhas side lengths of 15 and 3. Determine whether the rectanglesare similar.California Mathematics Grade 7 111

Chapter 4 BRINGING IT ALL TOGETHER4-6Measurement: Converting Length,Weight/Mass, Capacity, and Time15. ANIMALS A blue whale calf measures about 23 feet in length.How many yards is this?Complete each conversion. Round to the nearest hundredthif necessary.16. 6 qt ≈ L 17. 20 cm ≈ in.18. 100 gal/h ≈ L/min 19. 5 km/min ≈ mi/h4-7Measurement: Converting Square Units and Cubic UnitsComplete each conversion. Round to the nearest hundredthif necessary.20. 4 f t 2 ≈ in . 2 21. 150 f t 3 ≈ y d 322. 320 c m 2 ≈ in . 2 23. 3 m i 3 ≈ k m 34-8Scale Drawings and Models24. The scale on a map is 1 inch = 20 miles.Find the actual distance for the map distance of 5 _8 inch.25. What is the scale factor for a model if part of the model that is4 inches corresponds to a real-life object that is 16 inches?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.112 California Mathematics Grade 7

Chapter 4 BRINGING IT ALL TOGETHER4-9Rate of ChangeUse the table shown to answer each question.26. Find the rate of change in the numberof bicycles sold between weeks 2 and 4.27. Between which weeks is the rate ofchange negative?4-10Constant Rate of ChangeFind the constant rate of change for each graph and interpretits meaning.28. Week Bicycles Sold2 24 146 148 12Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 29. yScoops1086420 2 4 6 8 10ServingsxCalifornia Mathematics Grade 7 113

C H A P T E R4ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.Visit glencoe.com to accessyour textbook, moreexamples, self-check quizzes,and practice tests to helpyou study the concepts inChapter 4.• You are probably ready for the Chapter Test.• You may want to take the Chapter 4 Practice Test onpage 247 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 4 Study Guide and Reviewon pages 242–246 of your textbook.• If you are unsure of any concepts or skills, refer to the specificlesson(s).• You may also want to take the Chapter 4 Practice Test onpage 247.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in yourStudy Notebook and Chapter 4 Foldable.• Then complete the Chapter 4 Study Guide and Review onpages 242–246 of your textbook.• If you are unsure of any concepts or skills, refer to thespecific lesson(s).• You may also want to take the Chapter 4 Practice Test onpage 247.Student SignatureParent/Guardian SignatureCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Teacher Signature114 California Mathematics Grade 7

C H A P T E R5 PercentChapter 5Use the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with 4 sheets of 8 _ 1 " × 11" paper.2Draw a large circle onone of the sheetsof paper.Stack the sheets of paper.Place the one with thecircle on top. Cut all foursheets in the shape ofa circle.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Staple the circles on theleft side. Write the chaptertitle and the first four lessonnumbers on each circle.Turn the circles to the backside so that the staples arestill on the left. Write thelast four lesson titles onthe front and right pagesof the journal.NOTE-TAKING TIP: When you take notes, it mayhelp to create a visual representation, such as adrawing or a chart, to organize the informationyou learn. When you use a visual, be sure toclearly label it.California Mathematics Grade 7 115

C H A P T E R5BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 5.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExamplecompatible numbersdiscountinterestmarkuppercentCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.116 California Mathematics Grade 7

Chapter 5 BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExamplepercent equationpercent of changepercent of decreaseCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.percent of increasepercent proportionprincipalselling priceCalifornia Mathematics Grade 7 117

5–1Ratios and PercentsStandard 7NS1.3 Convert fractions to decimals and percents and use theserepresentations in estimations, computations, and applications.MAIN IDEA• Write ratios as percentsand vice versa.BUILD YOUR VOCABULARY (pages 116–117)written as percents.such as 27 out of 100 or 8 out of 25 can beKEY CONCEPTPercent A percent is aratio that compares anumber to 100.EXAMPLES Write Ratios as PercentsPOPULATION According to a recent census, 13 out ofevery 100 people living in Delaware were 65 or older.Write this ratio as a percent.13 out of every = 13%BASEBALL Through 2005, Manny Ramirez has gottenon base 40.9 times for every 100 times at bat. Write thisratio as a percent.40.9 out of = 40.9%Check Your Progress Write each ratio as a percent.a. 59 out of 100 b. 68 out of 100EXAMPLES Write Ratios and Fractions as PercentsTRANSPORTATION About 4 out of 5 commuters in theUnited States drive or carpool to work. Write this ratioas a percent.×4__5 = 80100× Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.So, out of equals .118 California Mathematics Grade 7

ORGANIZE ITWrite in words andsymbols what you’velearned about expressingratios as percents.5–1INTERNET In 2000, about3_ of the population in Peru200used the Internet. Write this fraction as a percent.÷3_200 = 1.5100÷_So, out of equals .Check Your Progressas a percent.Write each ratio or fraction_a. 3 out of 5 b. 122200 teensEXAMPLE Write Percents as FractionsCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:SCHEDULE The circle graph shows an estimate of thepercent of his day that Peter spends on each activity.Write the percents for eating and sleeping as fractions insimplest form.How Peter Spends His DayEating: 5% =Sleeping: 35% =ororCheck Your ProgressThe circle graph shows anestimate of the percent of hisday that Leon spends on eachactivity. Write the percents forschool and television asfractions in simplest form.15%Other15%Television5%Eat30%School35%Sleep California Mathematics Grade 7 119

5–2 Comparing Fractions, Decimals, and PercentsStandard 7NS1.1 Read, write, and compare rational numbers in scientific notation(positive and negative powers of 10), compare rational numbers in general. Standard7NS1.3 Convert fractions to decimals and percents and use these representations inestimations, computations, and applications.EXAMPLES Percents as DecimalsMAIN IDEA• Write percents asfractions and decimalsand vice versa.Write each percent as a decimal.52%52% = 52% Divide by .= Remove the percent symbol.KEY CONCEPTSDecimals and PercentsTo write a percent as adecimal, divide by 100and remove the percentsymbol.To write a decimal asa percent, multiply by100 and add the percentsymbol.245%245% = 245% Divide by .= Remove the percent symbol.Check Your Progress Write each percent as a decimal.a. 28% b. 135%EXAMPLES Decimals as PercentsWrite each decimal as a percent.0.30.3 = 0.30 Multiply by .= Add the percent symbol.0.710.71 = 0.71 Multiply by .= Add the percent symbol.Check Your Progress Write each decimal as a percent.a. 0.91 b. 1.65Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.120 California Mathematics Grade 7

5–2Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.REVIEW ITShow an example ofhow to write fractions asdecimals. (Lesson 2-1)EXAMPLES Fractions as PercentsWrite _ 3 as a percent.4METHOD 1METHOD 2Use a proportion.First write as a decimal.3_4 = x_Then write as a percent.1003_3 · 100 =4 = 0.75 0.754 3.00= _ 2820300 =_ 200== xSo, _ 3 4 can be written as .Write _ 1 as a percent.6METHOD 1Use a proportion.1_6 = x_100= 6 · x= 6x== xSo, 1_ can be written as .6Check Your Progressa. 1_4METHOD 2First write as a decimal.Then write as a percent.1_6 = 0.16 − 6 0.166...6 1.0000_ 6=40_ 3640_ 364Write each fraction as a percent.b. 1_9California Mathematics Grade 7 121

5–2ORGANIZE ITWrite in words andsymbols what you havelearned about therelationship betweenpercents, decimals, andfractions.EXAMPLE Compare NumbersPOLITICS In Sun City, 0.45 of voters are Democrats. InMoon Town, 48% of voters are Democrats. In which townis there a greater portion of Democrats?Write 0.45 as a percent.0.45 = and addthesymbol.Since is less than , there areDemocrats in Moon Town.Check Your Progress In Star City,3_ of voters are20Republicans. In Meteorville, 13% of voters are Republicans.In which town is there a greater proportion of Republicans?HOMEWORKASSIGNMENTPage(s):Exercises:Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.122 California Mathematics Grade 7

5–3 Algebra: The Percent ProportionStandard 7NS1.3 Convert fractions to decimals and percents and use theserepresentations in estimations, computations, and applications.MAIN IDEA• Solving problems usingthe percent proportion.BUILD YOUR VOCABULARY (pages 116–117)In a percent proportion,of the numbers, calledthe part, is being compared to thequantity,also called the base. The other ratio is the percent, writtenKEY CONCEPTPercent Proportion_ partwhole = __ percent100as a fraction, whose base is .EXAMPLE Find the Percent34 is what percent of 136?Since 34 is being compared to 136, is part and isthe whole. You need to find the percent. Let n represent thepercent.partwhole34_136 = n_100Write the percent proportion.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.· = · n Find the cross products.= Multiply.= Divide each side by .= Simplify.So, 34 is of 136.Check Your Progress 63 is what percent of 210?California Mathematics Grade 7 123

5–3ORGANIZE ITBe sure to explain howto find the percent, thepart, and the base of apercent proportion. Youalso may want to showthe ideas in a chart likethe Concept Summary inyour text.EXAMPLE Find the PartWhat number is 70% of 600?The percent is 70, and the whole is 600. You need to find thepart. Let n represent the part.partwholen_600 = _ 70100n · 100 = 600 · 70100n =___ 100n100 = 42,000100Write the percent proportion.Find the cross products.Multiply.Divide each side by .n =Simplify.So, is 70% of 600.Check Your Progress What number is 40% of 400?EXAMPLE Find the BaseHOMEWORKASSIGNMENTPage(s):Exercises:BASEBALL From 1999 to 2001, Derek Jeter had 11 hitswith the bases loaded. This was about 30% of his atbats with the bases loaded. How many times was he atbat with the bases loaded?The percent is 30, and the part is 11 hits. You need to find thewhole number of hits.partwhole11_n = _ 30100 percent Write the percentproportion.11 · = n · Find the cross products.He had about= Multiply.≈ n Divide each side by 30.at bats with the bases loaded.Check Your Progress BASEBALL In 2005, AlexRodriguez had 194 hits. This was about 32% of his at bats.How many times was he at bat?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.124 California Mathematics Grade 7

5–4Finding Percents MentallyStandard 7NS1.3 Convert fractions to decimals and percents and use theserepresentations in estimations, computations, and applications.EXAMPLES Use Fractions to Compute MentallyMAIN IDEA• Compute mentally withpercents.Compute mentally.40% of 8040% of 80 = of 80 or Use the fraction form of66 _ 2 % of 75340%, which is .66 2_ % of 75 = of 75 or . Use the fraction form of366 2_ %, which is .3EXAMPLES Use Decimals to Compute MentallyCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.WRITE ITExplain how you canmove the decimal pointto mentally multiply 0.1by 1.1.Compute mentally.10% of 6510% of 65 = of 65 or1% of 3041% of 304 = of 304 orCheck Your Progress Compute mentally.a. 20% of 60 b. 66 2_ % of 3003c. 10% of 13 d. 1% of 244California Mathematics Grade 7 125

5–4ORGANIZE ITIn your Foldable, be sureto include examples thatshow how to estimatepercents of numbers.EXAMPLE Use Mental Math to Solve a ProblemTECHNOLOGY A company produces 2,500 of aparticular printer. They later discover that 25% ofthe printers have defects. How many printers fromthis group have defects?METHOD 1 Use a fraction.25% of 2,500 = of 2,500THINK1_ of 2,000 is and1_of 500 is .4 4So, of 2,500 is + or .METHOD 2 Use a decimal.25% of 2,500 = of 2,500THINK 0.5 of 2,500 is .So, 0.25 of 2,500 is · or .HOMEWORKASSIGNMENTPage(s):Exercises:There wereprinters that had defects.Check Your Progress A company produces 1,400 ofa particular monitor. They later discover that 20% of themonitors have defects. How many monitors from this grouphave defects?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.126 California Mathematics Grade 7

5–5 Problem-Solving Investigation:Reasonable AnswersEXAMPLEMAIN IDEA• Determine whetheranswers arereasonable.Standard 7MR3.1Evaluate thereasonablenessof the solution in thecontext of the originalsituation.Standard 7NS1.3 Convertfractions to decimalsand percents and usethese representationsin estimations,computations, andapplications.SHOPPING Cara sees an advertisement for a pair ofshoes. One pair costs $34.99 plus 5 percent tax. Shewants to buy a black pair and a brown pair. Cara has$75 saved in her clothing budget. Can she afford bothpairs of shoes?EXPLORE You know the cost of the shoes and the sales taxrate. You want to know if two pairs of shoes plussales tax will be or than .PLAN Use to determine a reasonableSOLVEanswer.THINK $34.99 × 2 ≈10% of $70 = $7, so 5% of $70 =The total cost will be about $70 + $3.50 =Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:. Since Cara has $75, she will haveenough to buy .CHECK Find the of the two pairs of shoes.Then compute the sales tax and compare the sumto $75.Check Your Progress SHOPPING David wants to buy aCD for $11.99 and a pack of batteries for $3.99. The sales taxrate is 5 percent. If David has $17 in his wallet, will he haveenough to buy the CD and batteries?California Mathematics Grade 7 127

5–6Percent and EstimationStandard 7NS1.3 Convert fractions to decimals and percents and use theserepresentations in estimations, computations, and applications.MAIN IDEA• Estimate by usingequivalent fractions,decimals, and percents.BUILD YOUR VOCABULARY (pages 116–117)Compatible numbers are two numbers that are easy to add,subtract, multiply, or divide mentally.EXAMPLES Estimate Percents of NumbersEstimate.48% of 7048% is about or . and 70 arecompatible numbers.of 70 is .So, 48% of 70 is about .12% of 8112% is about 12.5% or , and areand 81 is about .of is .So, 12% of 81 is about .23% of 8223% is about 1_4 , and 82 is about . 1_41_ of is .4So, 23% of 82 is about .compatible numbers.and arecompatible numbers.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.128 California Mathematics Grade 7

5–6Check Your Progress Estimate.a. 51% of 60 b. 25% of 33 c. 34% of 59EXAMPLEPOPULATION About 9% of the population of Texas lives inthe city of Houston. If there are about 22 million peoplein the state of Texas, estimate the population of Houston.9% of 22 million ≈ or of 22 million 9% is about= × 22 =So, the population of Houston is about .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ORGANIZE ITInclude the meaning ofthe symbol “≈.” Youmay wish to include anexample of estimatinga percent in which thesymbol ≈ is used.Check Your Progress LEFT-HANDEDNESS About 11%of the population is left-handed. If there are about 17 millionpeople in Florida, about how many Florida residents areleft-handed?EXAMPLES Estimate PercentsEstimate each percent.12 out of 4712_ ≈ or1_47 41_4 = %47 is about .So, 12 out of 47 is about .California Mathematics Grade 7 129

5–641 out of 20041_200 ≈ or 1_ 541 is about .1_5 =So, 41 out of 200 is about .58 out of 71_ 5871 ≈ or _ 5 658 is about , and 71 is about .5_6 = %So, 58 out of 71 is about .Check Your Progressa. 15 out of 76Estimate each percent.HOMEWORKASSIGNMENTPage(s):Exercises:b. 14 out of 47c. 58 out of 121Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.130 California Mathematics Grade 7

5–7Algebra: The Percent EquationStandard 7NS1.3 Convert fractions to decimals and percents and use theserepresentations in estimations, computations, and applications. Standard7NS1.7 Solve problems that involve discounts, markups, commissions, and profitand compute simple and compound interest.MAIN IDEA• Solve problems usingthe percent equation.BUILD YOUR VOCABULARY (pages 116–117)The percent equation is an equivalent form of the percentproportion in which theis written as a.REVIEW ITExplain how to write adecimal as a percent.(Lesson 5-2)EXAMPLE Find the PartFind 30% of 450.Estimate 10% of 450 is 45. So, 30% of 450 is 3 · 45 or 135.The percent is . The whole is . You need to find thepart. Let n represent the part.part = percent · wholen = · Write the percent equation.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.n =So, 30% of 450 is .EXAMPLE Find the Percent102 is what percent of 150?Estimate _ 102150 ≈ _ 100150or 662_3 %Multiply.The part is . The whole is . You need to find thepercent. Let n represent the percent.part = percent · whole= n · Write the percent equation._ 102150 = _ 150nDivide each side by 150.150= n Simplify.Since = %, 102 is % of 150.California Mathematics Grade 7 131

5–7ORGANIZE ITWrite the percentequation in words andsymbols. Explain whythe rate in a percentequation is usuallywritten as a decimal.EXAMPLE Find the Base144 is 45% of what number?Estimate 144 is 50% of 288.The part is . The percent is . You need to find thewhole. Let n represent the whole.part = percent · whole= · n Write the percent equation._ 1440.45 = _ 0.45n0.45Divide each side by 0.45.= n Simplify.So, 144 is 45% of .Check Your Progress Find the part, percent, or base.a. Find 20% of 315. b. 135 is what percent of 250?c. 186 is 30% of what number?HOMEWORKASSIGNMENTPage(s):Exercises:EXAMPLE Solve a Real-Life ProblemSALES TAX The price of a sweater is $75. The sales tax is5 _ 3 %. What is the total price of the sweater?4You need to find what amount is 5 _ 3 % of $75.4Let t = the amount of tax.t = · Write the equation.t =The amount of tax isis $75 + or .Simplify.. The total cost of the sweaterCheck Your Progress The price of a pair of shoes is $60.The sales tax is 5 percent. What is the total price of the shoes?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.132 California Mathematics Grade 7

5–8Percent of ChangeStandard 7NS1.6 Calculate the percentage of increases and decreases of aquantity. Standard 7NS1.7 Solve problems that involve discounts, markups,commissions, and profi t and compute simple and compound interest.MAIN IDEA• Find and use thepercent of increase ordecrease.BUILD YOUR VOCABULARY (pages 116–117)A percent of change is athat compares thechange in quantity to the original amount. When the newamount isthan the original, the percent ofchange is called a percent of increase.When the new amount isthan the original, thepercent of change is called a percent of decrease.EXAMPLE Find the Percent of IncreaseCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.KEY CONCEPTPercent of Change Apercent of change is aratio that compares thechange in quantity to theoriginal amount.HOMES The Neitos bought a house several years agofor $120,000. This year, they sold it for $150,000. Findthe percent of change. State whether the change is anincrease or decrease.Step 1 The amount of change is 150,000 - 120,000 =Step 2 Percent of change =____amount of changeoriginal amount= ___Defi nition ofpercent ofchange= 0.25 Divide.Step 3 The decimal 0.25 written as a percent ispercent of change is .The new amount isof is 25%.. So, thethan the original. The percentCheck Your Progress CLUBS Last year Cedar Park SwimClub had 340 members. This year they have 391 members.Find the percent increase.California Mathematics Grade 7 133

5–8ORGANIZE ITBe sure to includean explanation andexamples showing thedifference betweenpercent of increase andpercent of decrease.EXAMPLE Find the Percent of ChangeSCHOOLS Johnson Middle School had 240 students lastyear. This year, there are 192 students. Find the percentof change. State whether the percent of change is anincrease or a decrease.Step 1 The amount of change is 240 - 192 = .Step 2 Percent of change =____amount of changeoriginal amount= __= 0.20 Divide.Step 3 The decimal 0.20 written as a percent is .The percent of change is. Since the new amount isthan the original, it is a percent of .Check Your Progress CARS Meagan bought a new carseveral years ago for $14,000. This year she sold the car for$9,100. Find the percent of change. State whether the percentof change is an increase or a decrease.BUILD YOUR VOCABULARY (pages 116–117)The markup is the amount the price of an item isfor the item.above the price the storeThe selling price is the amount thepays.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.The amount by which aisis called the discount.134 California Mathematics Grade 7

5–8EXAMPLE Find the Selling PriceREMEMBER ITThere may be morethan one way to solve aproblem. See pages 286and 287 of your textbookfor other methods youcan use to solve Examples3 and 4.MARKUP Shirts bought by a sporting goods store costthem $20 per shirt. They want to mark them up 40%.What will be the selling price?METHOD 1 Find the amount of the markup first.The whole is . The percent is . You need to find theamount of the markup, or the part. Let m represent the amountof the markup.part = percent · wholem = · Write the equation.m =Multiply.Add the markupto the cost of each shirt to find the sellingprice. + =Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.METHOD 2 Find the total percent first.The customer will pay 100% of the store’s cost plus an extra40% of the cost. Find 100% + 40% or 140% of the store’s cost.Let p represent the price.part = percent · wholep = · Write the equation.p = Multiply.The selling price of the shirts for the customer is .Check Your Progress Silk flowers bought by a craft storecost them $10 per yard. They want to mark them up 35 percent.What will be the selling price?California Mathematics Grade 7 135

5–8EXAMPLE Find the Sale PriceSHOPPING A computer usually sells for $1,200. Thisweek, it is on sale for 30% off. What is the sale price?METHOD 1 Find the amount of the discount first.The percent is , and the whole is . We need tofind the amount of the discount, or the part. Let d represent theamount of discount.part = percent · wholed = · Write the equation.d =Multiply.Subtract the amount of the discount from the original price tofind the sale price.- =HOMEWORKASSIGNMENTPage(s):Exercises:METHOD 2 Find the percent paid first.If the amount of the discount is 30%, the percent paid is100% - 30% or 70%. Find 70% of $1,200. Let s representthe sale price.part = percent · wholes = · Write the equation.s = Multiply.The sale price of the computer is .Check Your Progress A DVD sells for $28. This week it ison sale for 20% off. What is the sale price?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.136 California Mathematics Grade 7

5–9 Simple InterestStandard 7NS1.7 Solve problems that involve discounts, markups, commissions,and profi t and compute simple and compound interest.MAIN IDEA• Solve problemsinvolving simpleinterest.BUILD YOUR VOCABULARY (pages 116–117)Interest is the amount of money paid orfor theuse of money.Principal is the amount of moneyor borrowed.EXAMPLE Find Simple InterestFind the simple interest for $2,000 invested at 5.5%for 4 years.I = prtWrite the simple interest formula.I = · · Replace p with , rwith , and t with .I = The simple interest is .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.REMEMBER ITThe t in the simpleinterest formularepresents time inyears. If time is givenin months, weeks, ordays, the time must bechanged to time in years.EXAMPLE Find the Total AmountSTANDARDS EXAMPLE Find the total dollar amount inan account where $80 is invested at a simple annualinterest rate of 6% for 6 months.A. $41.20 B. $82.40 C. $84.80 D. $108.80Read the Test ItemYou need to find the total amount in an account. The time isgiven in months. Six months is _ 6 or year.12Solve the Test ItemI = prtI = · ·I =The amount in the account is $80 + or .The correct answer is choice .California Mathematics Grade 7 137

5–9ORGANIZE ITExplain what you havelearned about computingsimple interest. Be sureto include the simpleinterest formula.Check Your Progressa. Find the simple interest for $1,500 invested at 5% for3 years.b. Find the total amount of money in an account where $60 isinvested at 8% for 3 months.EXAMPLE Find the Interest RateLOANS Gerardo borrowed $4,500 from his bank for homeimprovements. He will repay the loan by paying $120a month for the next four years. Find the simple interestrate of the loan.Use the formula I = prt. To find I, first find the total amount ofmoney Gerardo will pay.$120 · 48 = .HOMEWORKASSIGNMENTPage(s):Exercises:He will pay - $4,500 or in interest.So I = 1,260.The principle is $4,500. So, p = 4,500. The loan will be for48 months or 4 years. So, t = 4.I = p · r · t= · r ·= Simplify.= Divide each side by 18,000.= r Simplify.The simple interest rate is .Check Your Progress Jocelyn borrowed $3,600 from herbank for home improvements. She will repay the loan by paying$90 a month for the next 5 years. Find the simple interest rateof the loan.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.138 California Mathematics Grade 7

C H A P T E R5BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 5 Foldableto help you study for yourchapter test.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 5, go to:glencoe.comBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 116–117) to help yousolve the puzzle.5-1Ratios and PercentsWrite each ratio or fraction as a percent.1. 21 out of 100 2. 4:10 3. 9 _25Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Write each percent as a fraction in simplest form.4. 27% 5. 50% 6. 80%5-2Fractions, Decimals, and PercentsWrite each percent as a decimal.7. 29% 8. 376% 9. 5%Write each decimal or fraction as a percent.10. 3.9 11. _ 7 12. 1_83California Mathematics Grade 7 139

Chapter 5 BRINGING IT ALL TOGETHER5-3The Percent ProportionSolve.13. What percent of 48 is 6? 14. 14 is 20% of what number?5-4Finding Percents MentallyComplete each statement.15. 40% of 25 = of 25 or 16. of 36 = 1_ of 36 or417. 66 2_ % of 48 = of 48 or 18. of 89 = 0.1 of 89 or35-5Problem-Solving Investigation: Reasonable Answers19. AGRICULTURE An orange grower harvested 1,260 pounds oforanges from one grove, 874 pounds from another, and 602 poundsfrom a third. What is a reasonable number of crates to have onhand if each crate holds 14 pounds of oranges?5-6Percent and Estimation20. Are 1_ and 56 compatible numbers? Explain.821. Describe how to estimate 65% of 64 using compatible numbers.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.140 California Mathematics Grade 7

Chapter 5 BRINGING IT ALL TOGETHER5-7The Percent EquationWrite each percent proportion as a percent equation.22. _ 1664 = _ 2510023. _ a14 = 2_10024. _ 96b = _ 4810025. _ 13100 = p_6755-8Percent of ChangeFind the percent of change. Round to the nearest tenth ifnecessary. State whether the change is an increase or decrease.29. Original: 29 30. Original: 51New: 64 New: 42Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.31. Find the selling price for the sweater.Cost to store: $15Mark up: 35%5-9Simple InterestWrite interest or principal to complete each sentence.32. is the amount of money paid or earned for the useof money.33. equals times rate times time.34. Find the total amount in the account where $560 is invested at5.6% for 6 months.First, find the earned. Then, add the earnedand theto find the total amount in the account. Whatis the total amount for $560 at 5.6% for 6 months?California Mathematics Grade 7 141

C H A P T E R5ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study are givenwith each item.I completed the review of all or most lessons without using mynotes or asking for help.Visit glencoe.com to accessyour text book, moreexamples, self-check quizzes,and practice tests to helpyou study the concepts inChapter 5.• You are probably ready for the Chapter Test.• You may want to take the Chapter 5 Practice Test onpage 299 of your textbook as a final check.I used my Foldable or Study Notebook to complete the review ofall or most lessons.• You should complete the Chapter 5 Study Guide and Reviewon pages 295–298 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may want to take the Chapter 5 Practice Test on page 299.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 5 Foldable.• Then complete the Chapter 5 Study Guide and Review onpages 295–298 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may want to take the Chapter 5 Practice Test onpage 299.Student SignatureParent/Guardian SignatureCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Teacher Signature142 California Mathematics Grade 7

C H A P T E R6Geometry and Spatial ReasoningUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with 7 sheets of 8 _ 1 " × 11" paper.2Fold a sheet ofpaper in halflengthwise. Cut a1" tab along the leftedge through onethickness.Chapter 6Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Glue the 1" tab down.Write the lesson titleon the front tab.Repeat Steps 1–2 for theremaining sheets of paper.Staple together to form abooklet.NOTE-TAKING TIP: When you read and learnnew concepts, help yourself remember theseconcepts by taking notes, writing definitions andexplanations, and draw models as needed.California Mathematics Grade 7 143

C H A P T E R6BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 6.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExampleanglecollinearcomplementary anglescongruent anglescongruent polygonequiangularequilateralinterior anglelineline of reflectionCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.line of symmetry144 California Mathematics Grade 7

Chapter 6 BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExampleline symmetryplanepointrayreflectionCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.regular polygonrotational symmetrysupplementary anglestransformationtranslationvertical anglesCalifornia Mathematics Grade 7 145

6–1 Line and Angle RelationshipsStandard 7MG3.1 Identify and construct basic elements of geometric figures (e.g.,altitudes, midpoints, diagonals, angle bisectors, and perpendicular bisectors; centralangles, radii, diameters, and chords of circles) by using a compass and straightedge.MAIN IDEA• Identify specialpairs of angles andrelationships ofangles formed by twoparallel lines cut by atransversal.BUILD YOUR VOCABULARY (pages 144–145)A is simply a location that has neither shape norsize. A is made up of points and has no thicknessor width. Ais a flat surface made up of pointsthat extends infinitely in all directions.KEY CONCEPTSSpecial Pairs of AnglesVertical angles areopposite angles formedby intersecting lines.Vertical angles arecongruent.The sum of the measuresof supplementary anglesis 180°.The sum of the measuresof complementary anglesis 90°.EXAMPLES Name Lines and PlanesUse the figure to name each of the following.a line containing point LThere are points on the line. Any of the pointscan be used to name the line.JK ,, KJ ,, LK , .The line can also be named as line .a plane containing point LThe plane can be named as plane . You can also use theletters of any three noncollinear points to name the plane.plane , plane , planeCheck Your Progress Use thefigure at the right to name eachof the following.a. a line containing point P•SP Q Ra• • •JMbK LTBCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.b. a plane containing point P146 California Mathematics Grade 7

ORGANIZE ITUse sketches and wordsto define the lines andangles discussed in thislesson. Try to showrelationships amongdifferent lines andangles. Write this in yourFoldable.EXAMPLESEWING The drawing showsa piece of fabric marked forcutting. If the edges of thefabric meet at right anglesand m∠1 = 15°, classify therelationship between ∠1 and∠2. Then find m∠2.∠1 and ∠2 areangle measures is .m∠1 + m∠2 = 90126–1angles. The sum of theirWrite the equation.+ x = 90 m∠1 = and m∠2 = x-15 = -15 Subtract from each side.x = 75Simplify.So, the measure of ∠2 is .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Check Your Progress SEWINGThe drawing shows two piecesof fabric that are being sewntogether to use in a quilt. If theedges of the fabric meet at astraight angle and m∠2 = 75°,classify the relationship between∠1 and ∠2. Then find m∠1.1 2California Mathematics Grade 7 147

6–1EXAMPLE Find a Missing Angle MeasureFind the value of x in the figure.Use the two vertical angles to solvefor x.68x+ x = Write an equation.___________–68 –68 Subtract 68 from each side.x =Simplify.Check Your ProgressFind the value of x in the figure.70˚x˚HOMEWORKASSIGNMENTPage(s):Exercises:Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.148 California Mathematics Grade 7

6–2 Problem-Solving Investigation:Use Logical ReasoningEXAMPLE Use Logical ReasoningMAIN IDEA• Solve problems byusing logical reasoning.Standard 7MR1.2Formulate andjustify mathematicalconjectures based ona general descriptionof the mathematicalquestion or problemposed.Standard 7NS1.3 Convertfractions to decimalsand percents and usethese representations inestimations, computations,and applications.FOOD Mona, Sharon, Pat, and Dena each have a favoritefood. One likes pizza, another fish and chips, anotherchicken, and another hamburgers. From the given clues,give each person’s favorite food.• Pat does not like pizza, hamburgers, or fish and chips.• Neither Mona nor Dena likes hamburgers.• Mona does not like to eat fried food.EXPLORE You know that each of the four students has aparticular favorite food. Use the clues given andlogical reasoning to determine the favorite food ofeach student.PLANSOLVERead each clue and deduce what you know aboutthe favorite foods of the students.According to the first clue, Pat does not like pizza,hamburgers, or fish and chips. The only otheroption is , so Pat likes .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:CHECKSince neither Mona nor Dena likes hamburgers,that means thatmust like hamburgers.Finally, there are two students left, Mona andDena, and two food choices left, pizza and fish andchips. Since Mona does not like, shemust like . Dena likes .Read each clue again and make sure the answersseem reasonable.Check Your Progress SPORTS Craig, Amy, Julia, andRonaldo each have a favorite sport. One likes soccer, anotherbasketball, another tennis, and another skateboarding. Fromthe given clues, give each person’s favorite sport.• Amy does not like soccer, basketball, or skateboarding.• Neither Craig nor Ronaldo likes playing soccer.• Craig prefers individual sports as opposed to team sports.California Mathematics Grade 7 149

6–3 Polygons and AnglesEXAMPLE Find the Sum of Interior Angle MeasuresMAIN IDEAS• Find the sum of anglemeasures of a polygon.• Find the measure ofan interior angle of apolygon.KEY CONCEPTInterior Angle Sum of aPolygonThe sum of the measuresof the interior angles ofa polygon is (n - 2) 180,where n is the numberof interior angles in thepolygon.Find the sum of the measures of the interior anglesof a hexagon.A hexagon hasS = (n - 2) 180sides.Write an equation.S = ( - 2) 180 Replace n with .S = (4) 180 orSimplify.The sum of the measures of the interior angles of a hexagonis .Check Your Progress Find the sum of the measures of theinterior angles of a heptagon (7-sided figure).Standard7MR3.3 Developgeneralizationsof the results obtainedand the strategies usedand apply them to newproblem situations.Standard 7AF1.1 Usevariables andappropriate operationsto write an expression,an equation, an inequality,or a system of equationsor inequalities thatrepresents a verbaldescription (e.g. threeless than a number, half aslarge as area A.EXAMPLE Find the Measure of an Interior AngleDESIGN A designer is creating a new logo for a bank.The logo consists of a regular pentagon surrounded byisosceles triangles. Find the measure of an interior angleof a pentagon.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.A pentagon hassides.150 California Mathematics Grade 7

6–3Step 1Find the sum of the measures of the angles.S = (n - 2) 180Write an equation.S = ( - 2) 180 Replace n with .S = (3) 180 orSimplify.The sum of the measures of the interior angles ofa regular pentagon is .Step 2 Divide 540 by , the number of interior angles, tofind the measure of one interior angle. So, themeasure of one interior angle of a regular pentagon is÷ or .Check Your Progress DESIGN Michelle is designing anew logo for the math club. She wants to use a regular nonagonas part of the logo. Find the measure of an interior angle of anonagon.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:California Mathematics Grade 7 151

6–4 Congruent PolygonsStandard 7MG3.4 Demonstrate an understanding of conditions that indicatetwo geometrical fi gures are congruent and what congruence means about therelationships between the sides and angles of the two figures.EXAMPLE Identify Congruent PolygonsMAIN IDEA• Identify congruentpolygons.KEY CONCEPTCongruent PolygonsIf two polygons arecongruent, theircorresponding sides arecongruent and theircorresponding angles arecongruent.Determine whether the trapezoids shown are congruent.If so, name the corresponding parts and write acongruence statement.R4 cm8 cmS2 cmT6 cmThe arcs indicate that ∠S ∠G,Q6 cmEH2 cmG∠T ∠H, ∠Q ∠E, and .The side measures indicate that ST −− GH −−− ,8 cm4 cmF−−−TQ −−− HE , −−− QR −− EF , and .Sincepairs of corresponding angles and sides are, the two trapezoids are .One congruence statement is trapezoidEFGH trapezoid .Check Your Progress Determine whether the trianglesshown are congruent. If so, name the corresponding parts andwrite a congruence statement.BA3555FC5535DECopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.152 California Mathematics Grade 7

6–4EXAMPLES Find Missing MeasuresIn the figure, △FGH △QRSF 6 cmG25˚9 cm35˚ HRQSFind m∠S.According to the congruence statement, ∠H and ∠S arecorresponding angles. So, .Since m∠H = , m∠S = .Find QR.−−−FG corresponds to . So, .Since FG = centimeters, QR = centimeters.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:Check Your Progress In the figure, △ABC △LMN.MB L753 in.50 55A5 in.CNa. Find m∠ N. b. Find LN.California Mathematics Grade 7 153

6–5 SymmetryStandard 7MG3.2 Understand and use coordinate graphs to plot simple figures,determine lengths and areas related to them, and determine their image undertranslations and refl ections.MAIN IDEA• Identify line symmetryand rotationalsymmetry.BUILD YOUR VOCABULARY (pages 144–145)A figure has line symmetry if it can be folded over a line sothat one half of the figurethe other half.EXAMPLES Identify Line SymmetryDetermine whether each figure has line symmetry. If itdoes, draw all lines of symmetry. If not, write none.This figure hasline of symmetry.This figure hasline of symmetry.WRITE ITHow many degrees doesone complete turn of afigure measure? Why is itthis number of degrees?Check Your ProgressDetermine whether theleaf has line symmetry.If it does, draw all lines ofsymmetry. If not, write none.EXAMPLE Identify Rotational SymmetryFLOWERS Determine whether the flower design hasrotational symmetry. Write yes or no. If yes, name itsangle(s) of rotation.0˚Yes, this figure hassymmetry. It will match itself after beingrotated 90°, 180°, and .180˚90˚Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.270˚154 California Mathematics Grade 7

6–5ORGANIZE ITUse sketches andwords to show linesof symmetry and linesymmetry. Write this inyour Foldable.Check Your Progress Determine whether each flowerdesign has rotational symmetry. Write yes or no. If yes,name its angle(s) of rotation.a. b.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:ARCHITECTURE A rosette is a paintedor sculptured ornament, usuallycircular, having designs that radiatesymmetrically from the center. Copy andcomplete the picture of the rosetteshown so that the completed figure hasrotational symmetry with 90°, 180°, and270° as its angles of rotation.Use the procedure described above and the points indicated torotate the figure 90°, 180°, and 270° counterclockwise. Use a90° rotation clockwise to produce the same rotation as a 270°rotation counterclockwise.counterclockwisecounterclockwiseCheck Your Progress DESIGN Copyand complete the figure so that the completeddesign has rotational symmetry with 90°, 180°,and 270° as its angles of rotation.clockwiseCalifornia Mathematics Grade 7 155

6–6 ReflectionsStandard 7MG3.2 Understand and use coordinate graphs to plot simple figures,determine lengths and areas related to them, and determine their image undertranslations and refl ections.MAIN IDEA• Graph reflections on acoordinate plane.BUILD YOUR VOCABULARY (pages 144–145)A reflection (sometimes called a flip) is a transformation inwhich aimage is produced bya figure over a line.EXAMPLE Draw a ReflectionKEY CONCEPTProperties of Reflections1. Every point on areflection is the samedistance from the lineof reflection as thecorresponding pointon the original figure.2. The image is congruentto the original figure,but the orientation ofthe image is differentfrom that of theoriginal figure.Draw the image of trapezoid STUV after a reflectionover the given line.Step 1 Count the number of unitsS' Sbetween each vertex andthe line of .Step 2 Plot a point for each vertexthedistanceaway from the line on theother side.Step 3 Connect the newT'U'V' Vto form theimage of trapezoid STUV, trapezoid S'T'U'V'.Check Your Progress Draw the image of trapezoid TRAPafter a reflection over the given line.TRATUCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.P156 California Mathematics Grade 7

6–6ORGANIZE ITDraw a triangle or simplequadrilateral on graphpaper. Reflect your figureover the x-axis. Add yourwork to your Foldable.EXAMPLE Reflect a Figure over the x-axisGraph quadrilateral EFGH with verticles E (-4, 4) ,F (3, 3) , G (4, 2) , and H (-2, 1) . Then graph the image ofEFGH after a reflection over the x-axis and write thecoordinates of its vertices.EyFHH'OGxG'E'F'The coordinates of the verticles of the image are E' ,F' , G' and H' .sameoppositesE (-4, 4) E' (-4, -4)F (3, 3) F' (3, -3)G (4, 2)Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.H (-2, 1)Notice that the y-coordinate of a point reflected over the x-axisis theof the y-coordinate of the original point.Check Your Progress Graph quadrilateral QUAD withvertices Q (2, 4) , U (4, 1) , A (-1, 1) , and D (-3, 3) . Then graphthe image of QUAD after a reflection over the x-axis, and writethe coordinates of its vertices.OyxCalifornia Mathematics Grade 7 157

6–6EXAMPLE Reflect a Figure over the y-axisGraph trapezoid ABCD with vertices A (1, 3) , B (4, 0) ,C (3, -4) , and D (1, -2) . Then graph the image of ABCDafter a reflection over the y-axis, and write thecoordinates of its vertices.A'yAB'OD' DBxC'CThe coordinates of the vertices of the image are A' ,B' , C' , and D' .oppositessameA (1, 3) A' (-1, 3)B (4, 0) B' (-4, 0)C (3, -4)HOMEWORKASSIGNMENTPage(s):Exercises:D (1, -2)Notice that the x-coordinate of a point reflected over the y-axisis the opposite of the x-coordinate of thepoint.Check Your Progress Graph quadrilateral ABCD withvertices A (2, 2) , B (5, 0) , C (4, -2), and D (2, -1). Then graphthe image of ABCD after a reflection over the y-axis, and writethe coordinates of its vertices.OyxCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.158 California Mathematics Grade 7

6–7 TranslationsStandard 7MG3.2 Understand and use coordinate graphs to plot simple figures,determine lengths and areas related to them, and determine their image undertranslations and refl ections.MAIN IDEA• Graph translations on acoordinate plane.BUILD YOUR VOCABULARY (pages 144–145)A translation (sometimes called a slide) is theof a figure from one position to anotherturning it.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.KEY CONCEPTProperties of Translations1. Every point on theoriginal figure ismoved the samedistance and in thesame direction.2. The image is congruentto the original figure,and the orientation ofthe image is the sameas that of the originalfigure.EXAMPLE Draw a TranslationDraw the image of △EFG after a translation of 3 unitsright and 2 units up.Step 1 Move each vertex of the triangleandFEunits up.F'GE'G'units rightStep 2 Connect the new vertices to form the .Check Your Progress Draw the image of △ABC after atranslation of 2 units right and 4 units down.ACBCalifornia Mathematics Grade 7 159

6–7ORGANIZE ITDraw a triangle or simplequadrilateral on graphpaper. Then draw atranslation. Show howyou determined thepoints needed to graphthe translated figure.Put your work in yourFoldable.EXAMPLE Translation in the Coordinate PlaneGraph △ABC with vertices A (-2, 2) , B (3, 4) , andC (4, 1) . Then graph the image of △ABC after atranslation of 2 units left and 5 units down. Write thecoordinates of its vertices.OyxThe coordinates of the vertices of the image areA' , B' , and C' . Notice thatthese vertices can also be found by addingto thex-coordinates and to the y-coordinates, or (-2, -5) .Original Add (-2, -5) ImageA (-2, 2) (-2 + (-2) , 2 + (-5) )B (3, 4) (3 + (-2) , 4 + (-5) )C (4, 1) (4 + (-2) , 1 + (-5) )Check Your Progress Graph △PQR with vertices P (-1, 3) ,Q (2, 4) , and R (3, 2) . Then graph the image of △PQR aftera translation of 2 units right and 3 units down. Write thecoordinates of its vertices.OyxCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.160 California Mathematics Grade 7

6–7EXAMPLESTANDARDS EXAMPLE If triangleRST is translated 4 units rightand 3 units up, what are thecoordinates of point T′?A (0, 3) C (2, 1)B (1, 2) D (1, 1)TRSOyxRead the Test ItemYou are asked to find the coordinates of point T′ after theoriginal figure has been translated 4 units right and 3 units up.Solve the Test ItemYou can answer this question without translating the entiretriangle.The coordinates of point T areOriginal fi gure.The x-coordinate of T′ is, so the x-coordinateof T′ is + 4Translating 4 units right isthe same asto the x-coordinate.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:or .The y-coordinate of T is ,so the y-coordinate of T′ is+ 3 or .The coordinates of T′ areThe answer is ..Check Your ProgressIf triangle LMN is translated 4 unitsleft and 2 units up, what are thecoordinates of point L′?F (0, -1) H (-1, -4)G (-3, 2) J (-2, 3)Translating 3 units up is thesame as addingy-coordinate.OyNLto thexMCalifornia Mathematics Grade 7 161

C H A P T E R6BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 6 Foldableto help you study for yourchapter test.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 6, go to:glencoe.comBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 144–145) to help yousolve the puzzle.6-1Line and Angle RelationshipsFor Questions 1–4, use the figure at the right.1. Classify the relationshipbetween ∠5 and ∠6.c2. Classify the relationshipbetween ∠5 and ∠8.3. Find m∠3 if m∠2 = 60°.4. Find m∠4 if m∠2 = 60°.6-25 67 81 23 4Problem-Solving Investigation: Use Logical Reasoning5. BASKETBALL Juan, Dallas, and Scott play guard, forward, andcenter on a team, but not necessarily in that order. Juan and thecenter drove Scott to practice on Saturday. Juan does not playguard. Who is the guard?abCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.162 California Mathematics Grade 7

Chapter 6 BRINGING IT ALL TOGETHER6-3Polygons and AnglesFind the sum of the measures of the interior angles of eachpolygon.6. heptagon 7. nonagon 8. 15-gonFind the measure of one interior angle in each regularpolygon.9. hexagon 10. decagon 11. 18-gon6-4Congruent Polygons12. Complete the sentence. Two polygons are congruent if theirsides are congruent and the correspondingangles are .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.△ABC △EDF. m∠A = 40° and m∠B = 50°.∠E ∠A and ∠F ∠C.13. What is m∠C? 14. What is m∠D?6-5SymmetryWrite whether each sentence is true or false. If false, replacethe underlined words to make a true sentence.15. A figure has line symmetryif it can be folded over a lineso that one half of the figurematches the other half.16. To rotate a figure meansto turn the figure from itscenter.17. A figure has rotationalsymmetry if it first matchesitself after being rotatedexactly 360°.California Mathematics Grade 7 163

Chapter 6 BRINGING IT ALL TOGETHER6-6Reflections18. Complete. A reflection is a image of a figureproduced by flipping the figure over a line.19. If you graphed quadrilateral HIJKreflected over the y-axis, what wouldbe the coordinates of these vertices:H′ ( ) J′ ( )yHOIJKx6-7Translations20. Complete. A translation is the movement of a figure from oneposition to anotherturning it.21. If you graphed the image of y Equadrilateral DEFG afterDFa translation 3 units right andG4 units down, what would bethe coordinates of these vertices:OD′ ( ) F′ ( )xCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.164 California Mathematics Grade 7

C H A P T E R6ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without using mynotes or asking for help.Visit glencoe.com to accessyour textbook, moreexamples, self-check quizzes,and practice tests to helpyou study the concepts inChapter 6.• You are probably ready for the Chapter Test.• You may want take the Chapter 6 Practice Test onpage 347 of your textbook as a final check.I used my Foldable or Study Notebook to complete the review ofall or most lessons.• You should complete the Chapter 6 Study Guide and Review onpages 342–346 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 6 Practice Test onpage 347.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 6 Foldable.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.• Then complete the Chapter 6 Study Guide and Review on pages342–346 of your textbook.• If you are unsure of any concepts or skills, refer back to the specificlesson(s).• You may also want to take the Chapter 6 Practice Test onpage 347.Student SignatureParent/Guardian SignatureTeacher SignatureCalifornia Mathematics Grade 7 165

C H A P T E R7Geometry: Measurement: Area and VolumeUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with a plain piece of 8 _ 1 " × 11" paper.2Fold in half widthwise.Open and fold thebottom to form a pocket.Glue edges.Label each pocket.Place several indexcards in each pocket.NOTE-TAKING TIP: As you read and learn a newconcept, such as how to measure area or volume,write examples and explanations showing themain ideas of the concept.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.166 California Mathematics Grade 7

C H A P T E R7BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 7.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExamplebasecentercircumferencechordCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.complex figureconecylinderdiameteredgefacelateral faceChapter 7lateral surface area(continued on the next page)California Mathematics Grade 7 167

Chapter 7 BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExamplenetparallelpiprismpyramidradiusregular pyramidsimilar solidsslant heighttotal surface areavertexCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.volume168 California Mathematics Grade 7

7–1Circumference and Area of CirclesMAIN IDEA• Find the circumferenceand the area of circles.BUILD YOUR VOCABULARY (pages 167–168)The radius of a circle is the distance from theto any point the circle. A is any segment withendpoints on the circle.The diameter of a circle is thecircle through the center.theThe circumference of a circle is thethe circle.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.KEY CONCEPTSCircumference of a CircleThe circumference C ofa circle is equal to itsdiameter d times π, or 2times its radius r times π.Area of a Circle The areaA of a circle is equal to πtimes the square of theradius r.Standard 7MG2.1Use formulasroutinely for findingthe perimeter and areaof basic two-dimensionalfigures and the surfacearea and volume of basicthree-dimensional figures,including rectangles,parallelograms, trapezoids,squares, triangles, circles,prisms, and cylinders.Standard 7MG3.1Identify and constructbasic elements ofgeometric figures (e.g.,altitudes, midpoints,diagonals, angle bisectors,and perpendicularbisectors; central angles,radii, diameters, andchords of circles) byusing a compass andstraightedge.EXAMPLES Find the Circumferences of CirclesFind the circumference of each circle. Round to thenearest tenth.5 ftC =Circumference of a circleC = · Replace d with .C =Use a calculator to find 5π.5 2nd π ENTER = 15.70796327The circumference is about .3.8 mC =This is the exactcircumference.Circumference of a circleC = 2 · π · Replace r with .C ≈The circumference is about .Use a calculator.California Mathematics Grade 7 169

7–1Check Your Progress Find the circumference of eachcircle. Round to the nearest tenth.a.b.EXAMPLES Find the Areas of CirclesFind the area of each circle. Round to the nearest tenth.ORGANIZE ITOn index cards, write theformulas for finding thecircumference and areaof a circle. Sketch a circleand label its parts. Placeyour cards in the “Area”pocket of your Foldable.A =Area of a circleA =π·2Replace r with .A =π· Evaluate 3 2 .A ≈Use a calculator.The area is about .HOMEWORKASSIGNMENTPage(s):Exercises:A = π r 2A = π ·2Area of a circler = 1_ of 102A = π · Evaluate 5 2 .A ≈The area is about .Use a calculator.Check Your Progress Find the area of each circle.Round to the nearest tenth.a.b.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.170 California Mathematics Grade 7

7–2 Problem-Solving Investigation:Solve a Simpler ProblemEXAMPLEMAIN IDEA• Solve problems bysolving a simplerproblem.GARDENS A series of gardens framed by tiles is arrangedsuch that each successive garden is one tile longer thanthe previous garden. The width of the gardens is fourtiles. The first three gardens are shown below. Howmany tiles surround Garden 10?Standard 7MR1.3Determine whenand how to breaka problem into simplerparts.Standard 7MR2.2 Applystrategies and resultsfrom simpler problems tomore complex problems.Standard 7AF4.2 Solvemultistep problemsinvolving rate, averagespeed, distance, and timeor a direct variation. EXPLORE You know how many tiles surround the first threegardens. Use this information to predict how manytiles will surround Garden 10.PLANIt would take a long time to draw each of thegardens 1 through 10. Instead, find the numberof tiles surrounding the smaller gardens and lookfor a pattern.SOLVE Garden 1 2 3 4Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Surrounding Tiles 10 12 14 16+2 +2 +2For each successive garden,additional tilesare needed to surround it. The 10th garden willhave 16 + 2 + 2 + 2 + 2 + 2 + 2 orCHECK Check your answer by drawing Garden 10.tiles.Check Your Progress GAMES The figures below showthe number of tiles on a game board after the first 4 roundsof the game. Each round, the same number of tiles are addedto the board. How many tiles will be on the board after the12th round? California Mathematics Grade 7 171

7–3 Area of Complex FiguresMAIN IDEA• Find the area ofcomplex figures.BUILD YOUR VOCABULARY (pages 167–168)A complex figure is made up ofshapes.Standard 7MG2.1Use formulasroutinely for findingthe perimeter and area ofbasic two-dimensionalfigures and the surfacearea and volume of basicthree-dimensional figures,including rectangles,parallelograms,trapezoids, squares,triangles, circles, prisms,and cylinders.Standard 7MG2.2Estimate and computethe area of morecomplex or irregular twoandthree-dimensionalfigures by breaking thefigures down into morebasic geometric objects.EXAMPLES Find the Areas of a Complex FigureFind the area of the complex figure. Round to thenearest tenth if necessary.The figure can be separated into twoa .Area of one semicircle Area of rectangleA = 1_2 π r 2 A = lwA = A =A ≈ A =The area of the garden is 14.1 + + or100.2 square centimeters.andCheck Your Progress Find the area of the complex figure.Round to the nearest tenth if necessary.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.172 California Mathematics Grade 7

GARDENING The dimensions of a flower garden areshown. What is the area of the garden?7–3The garden can be separated into aand twocongruent .Area of rectangleArea of one triangleA = lw A = 1_2 bhA = A =A = A =The area of the garden is + + orsquare feet.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:Check Your Progress GARDENING The dimensionsof a flower garden are shown. What is the area ofthe garden?California Mathematics Grade 7 173

7–4 Three-Dimensional FiguresStandard 7MG3.6 Identify elements of three-dimensional geometric objects(e.g., diagonals of rectangular solids) and describe how two or more objectsare related in space (e.g., skew lines, and the possible ways three planes mightintersect).MAIN IDEA• Identify and drawthree-dimensionalfigures.BUILD YOUR VOCABULARY (pages 167–168)A polyhedron is a solid withsurfaces that are.An edge is where two planesin a line.KEY CONCEPTCommon PolyhedronsA face is asurface.A vertex is where three or more planesa point.attriangular prismA prism is a polyhedron with twofaces, or bases.A pyramid is a polyhedron with one base that is aand faces that are .rectangular prismtriangular pyramidrectangular pyramid174 California Mathematics Grade 7EXAMPLES Identify RelationshipsUse the figure at the right toidentify the following.a plane that is parallel to plane GKJPlaneis parallel to plane GKJ.a segment that is skew to −− JN−−JN and are skew because they do not andare not coplanar.two sets of points between which a diagonal can be drawnLines drawn between points G and and points and Jwould form diagonals.Check Your Progress Use thefigure at the right to identifythe following.GLKPHMJNCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

7–4a. a plane that is parallel to plane QUXTb. a segment that is skew to −−− XWc. two sets of points between which a diagonal can be drawnEXAMPLES Identify Prisms and PyramidsIdentify each solid. Name the number and shapes of thefaces. Then name the number of edges and vertices.The figure has two parallelbases that are, so it is anprism. The other faces are rectangles.It has a total of faces, edges, and vertices.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.The figure has one base that is a ,so it is a .The other faces are triangles. It has a total offaces, edges, and vertices.Check Your Progress Identify each solid. Name thenumber and shapes of the faces. Then name the numberof edges and vertices.a.California Mathematics Grade 7 175

7–4b.EXAMPLES Analyze Real-Life DrawingsARCHITECTURE The plansfor a hotel fireplace areshown at the right.Draw and label the top,front, and side views.view view viewHOMEWORKASSIGNMENTPage(s):Exercises:Check Your ProgressThe plans for a buildingare shown to the right.Draw and label the top,front, and side views.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.176 California Mathematics Grade 7

7–5Volume of Prisms and CylindersMAIN IDEA• Find the volumes ofprisms and cylinders.BUILD YOUR VOCABULARY (pages 167–168)Volume is the measure of theoccupied by asolid. Volume is measured in cubic units.EXAMPLE Find the Volume of a Rectangular PrismFind the volume of the rectangular prism.KEY CONCEPTVolume of a Prism Thevolume V of a prism isthe area of the base Btimes the height h.V = BhVolume of a prismV = ( ) h The base is a rectangle,so B = .5 in.7 in.11 in.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Standard 7MG2.1Use formulasroutinely for findingthe perimeter and areaof basic two-dimensionalfi gures and the surfacearea and volume ofbasic three-dimensionalfigures, includingrectangles, parallelograms,trapezoids, squares,triangles, circles, prisms,and cylinders.Standard 7MG2.2Estimate and compute thearea of more complex orirregular two- and threedimensionalfigures bybreaking the figuresdown into more basicgeometric objects.V = (5 · 7) 11 l = 5, w = 7, h = 11V =The volume is 385Simplify.inches.EXAMPLE Find the Volume of a Triangular PrismFind the volume of the triangular prism.V = BhVolume of a prismV = ( 1_2 · 9 · 15 ) h The base is a, soB = 1_2 · 9 · 15.V = ( 1_2 · 9 · 15 ) 4 The height of the prism is .9 ft15 ft4V =Simplify.The volume iscubic inches.California Mathematics Grade 7 177

7–5Check Your Progress Find the volume of each prism.a. b.BUILD YOUR VOCABULARY (pages 167–168)A cylinder is a solid whose bases are congruent, parallel,, connected with a side.EXAMPLE Find the Volumes of CylindersKEY CONCEPTVolume of a Cylinder Thevolume V of a cylinderwith radius r is the areaof the base B times theheight h.ORGANIZE ITOn index cards, write theformula for the volumeof a rectangular prism,a triangular prism, anda cylinder. Sketch eachfigure and label its parts.Place your cards in the“Volume” pocket of yourFoldable.Find the volume of each cylinder. Round to the nearesttenth if necessary.3 cm12 cmV = π r 2 hV = π ·V ≈The volume is about 339.32Volume of a cylinder· r = , h =Simplify.centimeters.Check Your Progress Find the volume of the cylinder.Round to the nearest tenth if necessary.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.178 California Mathematics Grade 7

7–5EXAMPLE Find the Volume of a Complex SolidTOYS A wooden block has asingle hole drilled entirelythough it. What is the volumeof the block? Round to thenearest hundredth.6 cm4 cm1 cm3 cmThe block is a rectangular prism with a cylindrical hole.To find the volume of the block,the volumeof the from the volume of the .Rectangular PrismCylinderV = V =V = (6 · 3) 4 or 72 V = π (1) 2 (3) or 9.42The volume of the box is about - orcubic centimeters.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:Check Your ProgressA small wooden cube hasbeen glued to a larger woodenblock for a whittling project.What is the volume of thewood to be whittled?California Mathematics Grade 7 179

7–6Volume of Pyramids and ConesStandard 7MG2.1 Use formulas routinely for finding the perimeter and area of basictwo-dimensional fi gures and the surface area and volume of basic three-dimensionalfigures, including rectangles, parallelograms, trapezoids, squares, triangles, circles,prisms, and cylinders.EXAMPLE Find the Volume of the Pyramid.MAIN IDEA• Find the volumes ofpyramids and cones.KEY CONCEPTVolume of a PyramidThe volume V of apyramid is one-third thearea of the base B timesthe height h.Find the volume of the pyramid.V = 1_20 cm3 Bh Volume of a pyramidV = 1_3( · ) B = · ,V = 140The volume ish =Simplify.7 cm3 cmCheck Your ProgressFind the volume of the pyramid.EXAMPLE Use Volume to Solve a ProblemSOUVENIRS A novelty souvenir company wants to makesnow “globes” shaped like pyramids. It decides that themost cost-effective maximum volume of water for thepyramids is 12 cubic inches. If a pyramid globe measures4 inches in height, find the area of its base.V = 1_ Bh Volume of a pyramid3= 1_ · B · 4 Replace V with and h with .312 = 4_3 · B Simplify.· 12 = · 4_ · B Multiply each side by .3= BCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.The area of the base of the snow globe is .180 California Mathematics Grade 7

7–6Check Your ProgressA company is designing pyramidshaped building blocks with a square base. They want thevolume of the blocks to be 18 cubic inches. If the length ofthe side of the base is 3 inches, what should be the heightof the blocks?KEY CONCEPTVolume of a ConeThe volume V of a conewith radius r is one-thirdthe area of the base Btimes the height h.BUILD YOUR VOCABULARY (pages 167–168)A cone is a three-dimensional figure with onebase. A curved surface connects the base and the .EXAMPLE Find the Volume of a ConeCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ORGANIZE ITOn index cards, write theformula for the volumeof a pyramid and a cone.Sketch each figure andlabel its parts. Place yourcards in the “Volume”pocket of your Foldable.HOMEWORKASSIGNMENTPage(s):Exercises:Find the volume of the cone. Round to the nearest tenth.V = 1_3 π r 2 h Volume of a coneV = 1_3 · π · 2· Replace r withV ≈and h with .Simplify.The volume is .Check Your Progressto the nearest tenth.3 m8 mFind the volume of the cone. RoundCalifornia Mathematics Grade 7 181

7–7 Surface Area of Prisms and CylindersMAIN IDEA• Find the surface areasof prisms and cylinders.BUILD YOUR VOCABULARY (pages 167–168)The surface area of a solid is theof theof all its, or faces.EXAMPLE Surface Area of a Rectangular PrismKEY CONCEPTSurface Area of aRectangular Prism Thesurface area S of arectangular prism withlength l, width w, andheight h is the sum of theareas of the faces.Find the lateral andtotal surface area ofthe rectangular prism.Perimeter of BaseP = 2l + 2w7 mmArea of BaseB = lwP = 2 + 2 or B = · or15 mm 9 mmStandard 7MG2.1Use formulasroutinely for findingthe perimeter and areaof basic two-dimensionalfi gures and the surfacearea and volume of basicthree-dimensionalfigures, includingrectangles, parallelograms,trapezoids, squares,triangles, circles, prisms,and cylinders.Standard 7MG3.5Construct twodimensionalpatternsfor three-dimensionalmodels, such ascylinders, prisms, andcones.Use this information to find the lateral and total surface area.Lateral Surface Area Total Surface AreaL = PhS = L + 2BL = 48 or S = + 2 · orThe lateral surface area is, andthe total surface area is .Check Your Progress Find the total surface area of therectangular prism.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.182 California Mathematics Grade 7

7–7REVIEW ITWhat is the formulafor finding the area ofa triangle? How doesthis relate to findingthe surface area of atriangular prism?(Lesson 7-1)EXAMPLE Surface Area of a Triangular PrismCAMPING A family wants to reinforce the fabric of theirtent with a waterproofing treatment. Find the totalsurface area, including the floor, of the tent below.5 ft5.8 ft6.3 ft5.8 ftA triangular prism consists of twocongruentthreefaces.faces andDraw and label a net of this prism. Find the area of each face.bottom · = 29left side · = 36.54Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ORGANIZE ITOn index cards, writethese formulas forfinding surface area.Then sketch and labeleach figure. Place thecards in the “Area”pocket of your Foldable.right side · = 36.54two bases 2 ( 1_2 · 5 · ) = 29The surface area of the tentis 29 + 36.54 + 36.54 + 29or about .Check Your Progress Julia is painting triangularprisms to use as decoration in her garden. Find the surfacearea of the prism.California Mathematics Grade 7 183

7–7EXAMPLE Surface Area of a CylinderKEY CONCEPTSurface Area of aCylinder The surfacearea S of a cylinder withheight h and radius ris the area of the twobases plus the area of thecurved surface.Find the lateral area andthe surface area of thecylinder. Round to thenearest tenth.Lateral Surface AreaTotal Surface AreaL = 2πrh S = L + 2π r 25 m2 mL = 2π S ≈ + 2πL ≈S ≈The lateral surface area is about ,and the total surface area is about .Check Your Progress Find the total surface area of thecylinder. Round to the nearest tenth.HOMEWORKASSIGNMENTPage(s):Exercises:Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.184 California Mathematics Grade 7

7–8 Surface Area of PyramidsStandard 7MG2.1 Use formulas routinely for finding the perimeter and area of basictwo-dimensional fi gures and the surface area and volume of basic three-dimensionalfigures, including rectangles, parallelograms, trapezoids, squares, triangles, circles,prisms, and cylinders.MAIN IDEA• Find the surface areasof pyramids and cones.BUILD YOUR VOCABULARY (pages 167–168)Theof a pyramid are calledlateral faces.The altitude or of each iscalled the slant height.The sum of the of the is thelateral area.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ORGANIZE ITOn a card, write theformula for findingthe surface area of apyramid. Then sketch apyramid and label theparts. Place the card inthe “Area” pocket ofyour Foldable.EXAMPLE Surface Area of a PyramidFind the lateral and total surfaceareas of the triangular pyramid.Find the lateral area and the areaof the base.Area of each lateral faceA =8 in.5 in.Area of a triangle5 in.A = 10.8 in 25 in.A = 1_ 2( )( ) or Replace b with andh with .There are 3 faces, so the lateral area is 3 (square inches.Area of baseA =The total surface area of the pyramid is +) ororsquare inches.California Mathematics Grade 7 185

7–8Check Your Progresssquare pyramid.Find the total surface area of theEXAMPLETOYS A toy block has the shape ofa regular pyramid with a squarebase. The manufacturer wants topaint the lateral surface green.How many square centimeterswill be painted green?L = 1_ Pl Lateral surface area of a pyramid2L = 1_2P = and l = 8L =Simplify.HOMEWORKASSIGNMENTPage(s):Exercises:The lateral surface area is .Check Your ProgressTOYS A toy block has theshape of a regular pyramid witha triangular base. The manufacturerwants to paint the lateral surfacegreen. How many square centimeterswill be painted green?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.186 California Mathematics Grade 7

7–9 Similar SolidsEXAMPLE Find Missing Linear MeasuresMAIN IDEA• Find dimensions,surface area, andvolume of similarsolids.These cones are similar.What is the radius of Cone Ato the nearest tenth?Since the two cones are similar,the ratios of their correspondinglinear measures are proportional. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.KEY CONCEPTIf the scale factor of thelinear measures of twosimilar solids is a_b , thenthe scale factor of theirsurface areas is ( a_b) 2 andthe scale factor of theirvolumes is ( a_b) 3 .Standard 7MG2.3Compute the lengthof the perimeter, thesurface area of the faces,and the volume of athree-dimensional objectbuilt from rectangularsolids. Understandthat when the lengthsof all dimensions aremultiplied by a scalefactor, the surface area ismultiplied by the squareof the scale factor andthe volume is multipliedby the cube of the scalefactor.WordsVariableEquationr · 12 =___radius cone Aradius cone Bis proportional to ___height cone Aheight cone BLet r represent the radius of cone A.== Write the proportion.12r = 56Find the cross products.Multiply._ 12r 56= _ Divide each side by .r ≈Simplify.The radius of cone A is about .Check Your ProgressThese cones are similar. What isthe height of Cone B to thenearest tenth? California Mathematics Grade 7 187

7–9EXAMPLE Find Surface Area of a Similar SolidThese rectangular prisms are similar. Find the totalsurface area of Prism A. The ratio of the measures ofPrism A to Prism B is 12_8 or 3 _2 .____surface area of prism Asurface area of prism B = _( a b ) 2Write a proportion.= Substitute theknown values.= Simplify.· = · Find the cross products.= Divide each sideby .S =Simplify.The surface area of Prism A is .Check Your Progress These square pyramids are similar.Find the total surface area of Prism A.Pyramid A12 cm 18 cmPyramid BS 1,188 cm 2Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.188 California Mathematics Grade 7

7–9EXAMPLESTANDARDS EXAMPLE Atriangular prism has a volumeof 12 cubic centimeters. Supposethe dimensions are tripled. Whatis the volume of the new prism?A 36 cm 3 C 324 cm 3B 96 cm 3 D 1,728 cm 3Read the Test ItemYou know that the prisms are similar, the ratio of the sidelengths is , and the volume of the smaller prism is12 cubic centimeters.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:Solve the Test ItemSince the volumes of similar solids have a ratio of _( a b ) 3 anda_b = 1_3 , replace a with and b with in _( a b ) 3 ._____volume of smaller prismvolume of larger prism= _( a b ) 3 Write a proportion.= ( 1_3) 3 Substitute known values.· = · Find the cross products.= V Simplify.So, the volume of the larger prism is. The answer is .Check Your Progress STANDARDS EXAMPLE Ahexagonal prism has a volume of 25 cubic inches. Suppose thedimensions are tripled. What is the volume of the new prism?A 75 in. 3 C 200 in. 3B 120 in. 3 D 675 in. 3California Mathematics Grade 7 189

C H A P T E R7BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 7 Foldableto help you study for yourchapter test.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 7, go to:glencoe.comBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 167–168) to help yousolve the puzzle.7-1Circumference and Area of CirclesComplete.1. The distance from the center of a circle to any point on thecircle is called the, while the distance around thecircle is called the .Find the circumference and area of each circle. Round to thenearest tenth.2. The radius is 14 miles. 3. The diameter is 17.4 in 2 .7-2Problem-Solving Investigation: Solve a Simpler Problem4. LANDSCAPING Laura is helpingher father make a circular walkwayaround a flower bed as shown.What is the area, in square feet,of the walkway?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.190 California Mathematics Grade 7

Chapter 7 BRINGING IT ALL TOGETHER7-3Area of Complex Figures5. What is a complex figure?6. What is the first step in finding the area of a complex figure?7. Explain how to divide up the figure shown.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.7-4Three-Dimensional FiguresMatch each description with the word it describes.8. a flat surface9. a polyhedron with one base that is a polygon and facesthat are triangles10. where three or more planes intersect at a pointa. vertexb. edgec. faced. base11. where two planes intersect in a line12. a polyhedron with two parallel, congruent facese. prismf. pyramidCalifornia Mathematics Grade 7 191

Chapter 7 BRINGING IT ALL TOGETHER7-5Volume of Prisms and CylindersFind the volume of each solid. Round to the nearest tenthif necessary.13.14.15.14 mm31.2 m5 cm12.1 mm37 mm15.1 m10.0 m9 cm14 mm7-6Volume of Pyramids and Cones16. Fill in the table about what you know from the diagram. Thencomplete the volume of the pyramid.length of rectangle11 in.width of rectangle6 in.7-78 in.area of baseheight of pyramidvolume of pyramidSurface Area of Prisms and Cylinders17. Complete the sentence with the correct numbers. When youdraw a net of a triangular prism, there aretriangular faces andrectangular faces.18. If you unroll a cylinder, what does the net look like?congruent19. Find the surface area of the cylinder. Round the nearest tenth.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.192 California Mathematics Grade 7

Chapter 7 BRINGING IT ALL TOGETHER7-8Surface Area of Pyramids and Cones20. Complete the steps in finding the surface area of a squarepyramid.Area of each lateral faceA = 1_2 bhA = 1_2 ( 9) (16)A = 72There are faces, so the lateral area is 4 (72) =square inches.Area of baseA = s 2A = 9 2 or 81Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.The surface area of the square pyramid is +orsquare inches.21. What two areas are needed to calculate the surface area of a cone?7-9Similar SolidsFind the missing measure for each pair of similar solids.Round to the nearest tenth if necessary.22.23. California Mathematics Grade 7 193

C H A P T E R7ChecklistARE YOU READY FOR THECHAPTER TEST?Check the one that applies. Suggestions to help you study are givenwith each item.I completed the review of all or most lessons without using mynotes or asking for help.Visit glencoe.com to accessyour textbook, moreexamples, self-check quizzes,and practice tests to helpyou study the concepts inChapter 7.• You are probably ready for the Chapter Test.• You may want to take the Chapter 7 Practice Test on page 409of your textbook as a final check.I used my Foldable or Study Notebook to complete the review ofall or most lessons.• You should complete the Chapter 7 Study Guide and Review onpages 405–408 of your textbook.• If you are unsure of any concepts or skills, refer to thespecific lesson(s).• You may also want to take the Chapter 7 Practice Test onpage 409.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 7 Foldable.• Then complete the Chapter 7 Study Guide and Review onpages 405–408 of your textbook.• If you are unsure of any concepts or skills, refer to thespecific lesson(s).• You may also want to take the Chapter 7 Practice Test onpage 409.Student SignatureParent/Guardian SignatureCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Teacher Signature194 California Mathematics Grade 7

C H A P T E R8 Algebra: More Equations and InequalitiesUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with a plain sheet of 11" × 17" paper.Fold in half lengthwise.Fold again from topto bottom.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Open and cutalong the secondfold top maketwo tabs.Label each tab as shown.NOTE-TAKING TIP: When you take notes, definenew terms and write about the new concepts youare learning in your own words. Write your ownexamples that use the new terms and concepts.Chapter 8California Mathematics Grade 7 195

C H A P T E R8BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 8.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExamplecoefficientconstantequivalent expressionslike termssimplest formsimplifying theexpressiontermtwo-step equationCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.196 California Mathematics Grade 7

8–1Simplifying Algebraic ExpressionsMAIN IDEA• Use the DistributiveProperty to simplifyalgebraic expressions.BUILD YOUR VOCABULARY (page 196)Equivalent expressions are expressions that have theregardless of the value of the variable.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Standard7AF1.1 Usevariables andappropriate operationsto write an expression,an equation, an inequality,or a system of equationsor inequalities thatrepresents a verbaldescription (e.g. threeless than a number, half aslarge as area A.Standard 7AF1.3Simplify numericalexpressions by applyingproperties of rationalnumbers (e.g., identity,inverse, distributive,associative, commutative)and justify the processused.Standard 7AF1.4 Usealgebraic terminology(e.g. variable, equation,term, coefficient,inequality, expression,constant) correctly.REVIEW ITWhat is the sign ofthe product when youmultiply two integerswith different signs?with the same sign?(Lesson 1-6)EXAMPLE Write Equivalent ExpressionsUse the Distributive Property to rewrite 3 (x + 5) .3 (x + 5) = 3(x) + 3 (5)= 3x + Simplify.Check Your Progress Use the Distributive Property torewrite each expression.a. 2 (x + 6) b. (a + 6) 3EXAMPLES Write Expressions with SubtractionUse the Distributive Property to rewrite eachexpression.(q - 3) 9(q - 3) 9 = [q + (-3)]9 Rewrite q - 3 as q + (-3)-3 (z - 7)= ( ) 9 + ( ) 9 Distributive Property.= + ( ) Simplify.= - Defi nition of subtraction.-3(z - 7) = -3[z + (-7)] Rewrite z - 7 as z + (-7.)= -3(z) + (-3)(-7) Distributive Property= -3z + Simplify.California Mathematics Grade 7 197

8–1Check Your Progress Use the Distributive Property torewrite each expression.a. (q - 2) 8 b. -2(z - 4)BUILD YOUR VOCABULARY (page 196)When a plus sign separates an algebraic expression intoparts, each part is called a term.The numeric factor of a term that contains ais called the coefficient of the variable.Like terms are terms that contain thevariable.A term without ais called a constant.EXAMPLE Identify Parts of an ExpressionIdentify the terms, like terms, coefficients, and constantsin 3x - 5 + 2x - x.3x - 5 + 2x - x= 3x + ( ) + 2x + ( ) Defi nition of Subtraction= 3x + (-5) + 2x + (-1x) Identity Property; -x = -1xThe terms are 3x, , 2x, and -x. The like terms are 3x,2x, and . The coefficients are 3, , and -1. Theconstant is .Check Your Progress Identify the terms, like terms,coefficients, and constants in 6x - 2 + x - 4x.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.198 California Mathematics Grade 7

8–1BUILD YOUR VOCABULARY (page 196)An algebraic expression is in simplest form if it has noand no .When you use properties tolike terms, youare simplifying the expression.EXAMPLES Simplify Algebraic ExpressionsSimplify each expression.6n - n6n and n are terms.6n - n = 6n - Identity Property; n == (6 - 1) n Distributive Property= Simplify.8z + z - 5 - 9z + 2Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:8z, z, and are like terms. -5 and are also like terms.8z + z - 5 - 9z + 2= 8z + z + ( ) + ( ) + 2 Defi nition of subtraction.= 8z + z + (-9z ) + (-5 ) + 2 Commutative Property= [8 + 1 + (-9)] + [(-5 ) + 2] Distributive Property= 0z + Simplify.=Check Your Progress Simplify each expression.a. 7n + n b. 6s + 2 - 10sc. 6z + z - 2 - 8z + 2California Mathematics Grade 7 199

8–2 Solving Two-Step EquationsStandard 7AF4.1 Solve two-step linear equations and inequalities in onevariable over the rational numbers, interpret the solution or solutions in thecontext from which they arose, and verify the reasonableness of the results.MAIN IDEA• Solve two-stepequations.BUILD YOUR VOCABULARY (page 196)A two-step equation contains .REMEMBER ITTwo-step equationscan also be solved usingmodels. Refer topage 534 of yourtextbook.EXAMPLES Solve Two-Step EquationsSolve 5y + 1 = 26.Use the Subtraction Property of Equality.5y + 1 = 26Write the equation.Subtract from each side.−−−−−−−−−5y = 25Use the Division Property of Equality.5y = 25_ 5y=25_ Divide each side by .y =Solve -4 = 1 _3 z + 2.Simplify.-4 = 1_ z + 2 Write the equation.3-4 - = 1_ z + 2 - Subtract from each side.3= 1_3 z Simplify.(-6) = · 1_ z Multiply each side by .3= z Simplify.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.200 California Mathematics Grade 7

8–2ORGANIZE ITUnder the “Equations”tab, include examples ofhow to solve a two stepequation. You can useyour notes later to tellsomeone else what youlearned in this lesson.Check Your Progressa. 3x + 2 = 20b. -5 = 1_2 z + 8Solve each equation.EXAMPLE Equations with Negative CoefficientsSolve 8 - 3x = 14.8 - 3x = 14 Write the equation.8 + ( ) = 14 Defi nition of subtraction.8 - 8 + ( ) = 14 - 8 Subtract 8 from each side.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.REMEMBER ITWhen you are solvingan equation, watchfor the negativesigns. In Example 3,the coefficient of thevariable, x, is -3, not +3.So, divide each side by-3 to solve for x.-3x = 6 Simplify.-3x __ 6Divide each side by .__ =x = -2SimplifyCheck Your Progress Solve 5 - 2x = 11.California Mathematics Grade 7 201

8–2EXAMPLE Combine Like Terms FirstREVIEW ITSimplify -c + 4c.Solve 14 = -k + 3k - 2.14 = -k + 3k - 2 Write the equation.14 = -1k + 3k - 2 Property; -k = 1k14 = - 2 Combine like terms;-1k + 3k = (-1 + 3)k or 2k.14 + = 2k - 2 + Add to each side.16 = 2k Simplify._ 16 2k= _ Divide each side by .8 = k Simplify.Check Your Progress Solve 10 = -n + 4n - 5.HOMEWORKASSIGNMENTPage(s):Exercises:Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.202 California Mathematics Grade 7

8–3 Writing Two-Step EquationsStandard 7AF1.1 Use variables and appropriate operations to write anexpression, an equation, an inequality, or a system of equations or inequalities thatrepresents a verbal description (e.g. three less than a number, half as large as area A.EXAMPLES Translate Sentences Into EquationsMAIN IDEA• Write two-stepequations thatrepresent real-lifesituations.REVIEW ITWhat are at least twowords that will tell youthat a sentence can bewritten as an equation?(Lesson 1-7)Translate each sentence into an equation.SentenceEquationThree more than half1_2 n + = 15a number is 15.Nineteen is two more than 19 = + 2five times a number.Eight less that twice a - 8 = -35number is -35.EXAMPLE Write and Solve a Two-Step EquationTRANSPORTATION A taxi ride costs $3.50 plus $2 for eachmile traveled. If Jan pays $11.50 for the ride, how manymiles did she travel?Words$3.50 plus $2 per mile equals $11.50.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ORGANIZE ITRecord the main ideas,definitions of vocabularywords, and other notesas you learn how to writetwo-step equations.Write your notes underthe “Equations” tab.VariablesEquationLet m represent the miles driven.3.50 + 2m = 11.50+ = 11.50 Write the equation.3.50 - + 2m = 11.50 - Subtract fromeach side.2m = 8Simplify.__ = __ Divide each side by .m =Simplify.Jan traveledmiles.California Mathematics Grade 7 203

8–3Check Your Progress Translate each sentence intoan equation.a. Five more than one third a number is 7.b. Fifteen is three more than six times a number.c. Six less that three times a number is -22.d. A rental car costs $100 plus $0.25 for each mile traveled.If Kaya pays $162.50 for the car, how many miles didshe travel?EXAMPLEDINING You and your friend spent a total of $33 fordinner. Your dinner cost $5 less than your friend’s. Howmuch did you spend for dinner?WordsVariablesEquationYour friend’s dinner plus your dinnerequals $33.Let f represent the cost of your friend’s dinner.f + f - 5 = 33= 33 Write the equation.- 5 = 33 Combine like terms.2f - 5 + 5 = 33 + 52f =Add 5 to both sides.Simplify.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.(continued on the next page)204 California Mathematics Grade 7

8–3= Divide each side by .f =Simplify.Your friend spenton dinner. So you spenton dinner.Check Your Progress DINING You and your friend spenta total of $48 for dinner. Your dinner cost $4 more than yourfriend’s. How much did you spend for dinner?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:California Mathematics Grade 7 205

8–4 Solving Equations with Variables on Each SideEXAMPLE Equations with Variables on Each SideMAIN IDEA• Solve equations withvariables on each side.Solve 7x + 4 = 9x.7x + 4 = 9xWrite the equation.7x - + 4 = 9x - Subtract from each side.= Simplify by combining like terms.= Divide each side by .ORGANIZE ITDescribe in your ownwords the steps to followwhen you solve anequation with variableson both sides. Writean example of such anequation and solve it.Check Your Progress Solve 3x + 6 = x.EXAMPLE Equations with Variables on Each SideStandard7AF1.1 Usevariables andappropriate operationsto write an expression,an equation, an inequality,or a system of equationsor inequalities thatrepresents a verbaldescription (e.g. threeless than a number, half aslarge as area A.Standard 7AF4.1Solve two-step linearequations and inequalitiesin one variable overthe rational numbers,interpret the solutionor solutions in thecontext from which theyarose, and verify thereasonableness of theresults.Solve 3x - 2 = 8x + 13.3x - 2 = 8x + 13Write the equation.3x - - 2 = 8x - + 13 Subtract from eachside.-5x - 2 = 13 Simplify.-5x - 2 + = 13 + Add to each side.= Simplify.x = Divide each side by .Check Your Progress Solve 4x - 3 = 5x + 7.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.206 California Mathematics Grade 7

8–4EXAMPLEGEOMETRY The measure of an angle is 8 degrees morethan its complement. If x represents the measure ofthe angle and 90 - x represents the measure of itscomplement, what is the measure of the angle?WordsVariablesEquation8 less than the measure of an angle equalsthe measure of its complementLet x and 90 - x represent the measuresof the anglesx - 8 = 90 - x= Write the equation.x - 8 = 90 - x Add to each side.x = 98 - xSimplify.x + = 98 - x Add to each side.= 98 Simplify.= Divide each side by .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:x =Simplify.The measure of the angle is .Check Your Progress GEOMETRY The measure of anangle is 12 degrees less than its complement. If x representsthe measure of the angle and 90 - x represents the measure ofits complement, what is the measure of the angle?California Mathematics Grade 7 207

8–5Problem-Solving Investigation:Guess and CheckEXAMPLEMAIN IDEA• Solve problems byguessing and checking.THEATER 120 tickets were sold for the school play. Adulttickets cost $8 each and child tickets cost $5 each. Thetotal earned from ticket sales was $840. How manytickets of each type were sold?Standard 7MR2.8Make precisecalculations andcheck the validity of theresults from the contextof the problem.Standard 7AF1.1Use variables andappropriate operationsto write an expression,an equation, an inequality,or a system of equationsor inequalities thatrepresents a verbaldescription (e.g. three lessthan a number, half aslarge as area A.EXPLORE You know the cost of each type of ticket, the totalnumber of tickets sold, and the total income fromticket sales.PLANSOLVEUse a systematic guess and check method to findthe number of each type of ticket.Find the combination that gives 120 total ticketsand $840 in sales. In the list, a represents adulttickets sold, and c represents child tickets sold.a c 8a + 5c Check50 70 8 (50) + 5 (70) = 750 too low60 8 (60) + =HOMEWORKASSIGNMENTPage(s):Exercises:CHECKSo adult tickets and child ticketswere sold.80 + 40 = 120 tickets sold. Since$8 (80) + $5 (40) = $840, the answer is correct.Check Your Progress THEATER 150 tickets were sold forthe school play. Adult tickets were sold for $7.50 each, and childtickets were sold for $4 each. The total earned from ticket saleswas $915. How many tickets of each type were sold?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.208 California Mathematics Grade 7

8–6 InequalitiesStandard 7AF1.1 Use variables and appropriate operations to write anexpression, an equation, an inequality, or a system of equations or inequalities thatrepresents a verbal description (e.g. three less than a number, half as large as area A.EXAMPLES Write Inequalities with < or >.MAIN IDEA• Write and graphinequalities.Write an inequality for each sentence.SPORTS Members of the little league team must beunder 14 years old.Let a = person’s age.a 14CONSTRUCTION The ladder must be over 30 feet tall toreach the top of the building.Let h = ladder’s height.h 30Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Check Your Progress Write an inequality for eachsentence.a. Members of the peewee b. The new building mustfootball team must be be over 300 feet tall.under 10 years old.EXAMPLES Write Inequalities with ≤ or ≥Write an equality for each sentence.POLITICS The president of the United States must be atleast 35 years old.Let a = president’s age.a 35CAPACITY A theater can hold a maximum of 300 people.Let p = theater’s capacity.p 300California Mathematics Grade 7 209

8–6ORGANIZE ITRecord the main ideasabout how to writeinequalities. Includeexamples to helpyou remember. Writeyour notes under the“Inequalities” tab.Check Your Progress Write an inequality for eachsentence.a. To vote, you must be at b. A football stadium canleast 18 years old.hold a maximum of10,000 people.EXAMPLES Determine the Truth of an InequalityFor the given value, state whether the inequality is trueor false.x - 4 < 6, x = 0x - 4 < 6Write the inequality.- 4 6 Replace x with .< 6 Simplify.Since is less than , < is .WRITE ITWrite in words what thesymbols , ≤, and ≥mean.3x ≥ 4, x = 13x ≥ 4Write the inequality.3 4 Replace x with 1.Since≱ 4is .Simplify.is not greater than or equal to 4, the sentenceCheck Your Progress For the given value, statewhether the inequality is true or false.a. x - 5 < 8, x = 16b. 2x ≥ 9, x = 5Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.210 California Mathematics Grade 7

8–6EXAMPLES Graph an InequalityGraph each inequality on a number line.n ≤ -1Place acircle at -1. Then draw a line and anarrow to the .The closed circle means the number-1 is included in the graph.3 2 1 0 1 2 3n > -1Place ancircle at -1. Then draw a line and anarrow to the .The open circle means -1 is notincluded in the graph.3 2 1 0 1 2 3Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:Check Your Progress Graph each inequality on anumber line.a. n ≤ -3 b. n > -3 California Mathematics Grade 7 211

8–7 Solving Inequalities by Adding or SubtractingEXAMPLES Solving InequalitiesMAIN IDEA• Solve inequalities byusing the Addition orSubtraction Propertiesof Inequality.Solve -21 ≥ d - 8.-21 ≥ d - 8 Write the inequality.-21 + ≥ d - 8 + Add to each side.Standard7AF1.1 Usevariables andappropriate operationsto write an expression, anequation, an inequality,or a system of equationsor inequalities thatrepresents a verbaldescription (e.g. threeless than a number, half aslarge as area A.Standard 7AF4.1Solve two-steplinear equations andinequalities in onevariable over the rationalnumbers, interpret thesolution or solutions inthe context from whichthey arose, and verify thereasonableness of theresults.Solve y + 5 > 11.≥ d or d ≤y + 5 > 11Simplify.Write the inequality.y + 5 - > 11 - Subtract from each side.y >Simplify.Check Your Progress Solve each inequality.a. x - 7 < 3 b. -6 ≥ z + 10EXAMPLESTANDARDS EXAMPLE Kayta took $12 to the bowlingalley. Shoe rental costs $3.75. What is the most he couldspend on games and snacks?Read the Test ItemSince we want to find the most he could spend, use less than orequal to.Solve the Test ItemLet x = the amount Kayta could spend on games and snacks.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Estimate: $12 - $4 = $212 California Mathematics Grade 7

8–7shoe rental plus games andsnacksis less than orequal to$12$3.75 + x ≤ $12 Write the inequality.$3.75 - + x ≤ $12 - Subtractfrom each side.x ≤Kayta could spend no more thanSimplify.on games and snacks.HOMEWORKASSIGNMENTPage(s):Exercises:Check Your Progress Monique took $20 to thebookstore. She spent $2.25 on a snack at the library café.What is the most she could spend on books?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.California Mathematics Grade 7 213

8–8 Solving Inequalities by Multiplying or DividingEXAMPLES Solve Inequalities by Multiplying or DividingMAIN IDEA• Solve inequalities byusing the Multiplicationor Division Propertiesof Inequality.Solve 6x < -30.6x < -30Write the inequality._ 6x< _ -30Divide each side by .Standard7AF1.1 Usevariables andappropriate operationsto write an expression, anequation, an inequality,or a system of equationsor inequalities thatrepresents a verbaldescription (e.g. threeless than a number, half aslarge as area A.Standard AF4.1Solve two-steplinear equations andinequalities in onevariable over the rationalnumbers, interpret thesolution or solutions inthe context from whichthey arose, and verify thereasonableness of theresults.x -5EXAMPLES Multiply or Divide by a Negative NumberSolveb_-4 ≤ 5.b_ ≤ 5 Write the inequality.-4( ) ( b_-4) ( ) ( 5) Multiply each side byand reverse the symbol.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.b ≥Simplify.214 California Mathematics Grade 7

8–8ORGANIZE ITDescribe in your ownwords the steps to followwhen you solve aninequality by multiplyingor dividing by a negativenumber.Solve -4n > -60.-4n > -60 Write the inequality.__ -4n -60< __Divide each side byand reverse symbol.n

C H A P T E R8BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 8 Foldableto help you study for yourchapter test.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 8, go to:glencoe.comBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(page 196) to help yousolve the puzzle.8-1Simplifying Algebraic Expressions1. Simplify the expression 3x - 4 - 8x + 2 by writing the missinginformation:and are like terms. and are also like terms.3x - 4 - 8x + 2 = 3x + + (-8x) + 2 Defi nition of subtraction= 3x + + (-4) + 2 Commutative Property= x + (-4) + 2 Distributive Property= Simplify.8-2Solving Two-Step Equations2. Define two-step equation.What is the first step in solving each equation?3. 3y - 2 = 16 4. 5 - 6x = -19 5. 32 = 4b + 6 - bCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.216 California Mathematics Grade 7

Chapter 8 BRINGING IT ALL TOGETHER8-3Writing Two-Step EquationsWrite each sentence as an algebraic equation.6. Four less than six times a number is -40.7. The quotient of a number and 9, decreased by 3 is equal to 24.8. Jennifer bought 3 CDs, each having the same price. Her total forthe purchase was $51.84, which included $3.84 in sales tax. Findthe price of each CD.Let p representEquation: Price of 3 CDs + =+ = 51.843p + 3.84 - = 51.84 -Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.8-4=p == 48 _3Solving Equations with Variables on Each SideSolve each equation.9. 3x + 2 = 2x + 5 10. 6x - 2 = 3x 11. 7x - 2 = 9x + 6California Mathematics Grade 7 217

Chapter 8 BRINGING IT ALL TOGETHER8-5Problem-Solving Investigation: Guess and Check12. PROMOTIONS A sports drink company is offering free mountainbikes to people who collect enough points by buying bottles of thedrink. You earn 5 points when you buy a 20-ounce bottle, and youearn 10 points when you buy a 32-ounce bottle. To get the bike,you need to have 915 points. What is the least number of bottles ofsports drink you would have to buy in order to get the bike?13. NUMBER THEORY The product of a number and its next twoconsecutive whole numbers is 60. What are the numbers?8-6InequalitiesWrite an inequality for each sentence using the symbol , ≤, or ≥.14. Children under the age of 2 fly free.15. You must be at least 12 years old to go on the rocket ride.Write the solution shown by each graph.16. 4 3 2 1 0 1 2 3 417. 6 5 4 3 2 1 0 1 28-7Solving Inequalities by Adding or SubtractingSolve each inequality. Check your solution.18. 8 + x > 12 19. n - 3 ≤ -5 20. 1 < g - 68-8Solving Inequalities by Multiplying or DividingSolve each inequality. Check your solution.21. 7m ≥ 77 22. x_ > -3 23. -12b ≤ 485Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.218 California Mathematics Grade 7

C H A P T E R8ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.Visit glencoe.com to accessyour textbook, moreexamples, self-check quizzes,and practice tests to helpyou study the concepts inChapter 8.• You are probably ready for the Chapter Test.• You may want to take the Chapter 8 Practice Test onpage 459 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 8 Study Guide and Reviewon pages 454–458 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 8 Practice Test onpage 459.I asked for help from someone else to complete the reviewof all or most lessons.• You should review the examples and concepts in yourStudy Notebook and Chapter 8 Foldable.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.• Then complete the Chapter 8 Study Guide and Review onpages 454–458 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 8 Practice Test onpage 459.Student SignatureParent/Guardian SignatureTeacher SignatureCalifornia Mathematics Grade 7 219

C H A P T E R9Algebra: Linear FunctionsUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with seven sheets of 8 _ 1 " × 11" paper.2Fold a sheet of paperin half lengthwise. Cuta 1" tab along the leftedge through onethickness.Glue the 1" tab down.Write the title of thelesson on the front tab.Repeat Steps 1–2 forthe remaining sheetsof paper. Staple togetherto form a booklet.NOTE-TAKING TIP: When you begin studyinga chapter in a textbook, first skim through thechapter to become familiar with the topics. Asyou skim, write questions about what you don’tunderstand and what you’d like to know. Then,as you read the chapter, write answers to yourquestions.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.220 California Mathematics Grade 7

C H A P T E R9BUILD YOUR VOCABULARYChapter 9This is an alphabetical list of new vocabulary terms you will learn in Chapter 9.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExampleconstant of variationdirect variationdomainCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.functionfunction tableline of fitlinear function(continued on the next page)California Mathematics Grade 7 221

Chapter 11 BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExamplerangeriserunscatter plotslopeslope-intercept formsystem of equationssystem of inequalitiesy-interceptCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.222 California Mathematics Grade 7

9–1FunctionsMAIN IDEA• Complete functiontables.BUILD YOUR VOCABULARY (pages 221–222)Awhere one thinganother is called a function.Preparation forStandard 7AF3.3Graph linearfunctions, noting that thevertical change (changein y-value) per unit ofhorizontal change (changein x-value) is always thesame and know that theratio (“rise over run”) iscalled the slope of a graph.Standard 7MR2.5 Usea variety of methods,such as words, numbers,symbols, charts, graphs,tables, diagrams,and models, to explainmathematical reasoning.EXAMPLE Find a Function ValueFind each function value.f (4) if f (x) = x - 8f (x) = x - 8Write the function.f ( ) = - 8 Substitute for x intothe function rule.= Simplify.So, f (4) = .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ORGANIZE ITIn your Foldable, writehow you would find thevalue of a function. Youmay wish to include anexample.f (-6) if f (x) = 3x + 4f (x) = 3x + 4Write the function.f ( ) = 3 ( ) + 4 Substitute for x intof ( ) = + 4 Multiply.the function rule.= Simplify.So, f (-6) = .Check Your Progress Find each function value.a. f (2) if f (x) = x - 7 b. f (-2) if f (x) = 2x + 6California Mathematics Grade 7 223

9–1BUILD YOUR VOCABULARY (pages 221–222)The variable for theof a function is called theindependent variable.The variable for theof a function is called thedependent variable.The set ofvalues in a function is called thedomain.The set ofvalues in a function is called therange.EXAMPLE Make a Function TableComplete the functiontable for f (x) = 4x - 1.Then state the domainand the range of thefunction.Substitute each value of x, or, into the function rule.Then simplify to find the .f (x) = 4x - 1f (-3) =f (-2) =f (-1) =f (0) =f (1) =orororororInputx-3-2-101Inputx-3-2-101Rule4x - 1Rule4x - 1Outputf (x)Outputf (x)Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.The domain is .The range is .224 California Mathematics Grade 7

9–1Check Your ProgressComplete the function tablefor f (x) = 3x - 2. Then statethe domain and therange of the function.Inputx-3-2Rule3x - 2Outputf (x)-101EXAMPLE Functions with Two VariablesPARKING FEES The price for parking at a city lot is $3.00plus $2.00 per hour. Write a function to represent theprice of parking for h hours. Then determine how muchwould it cost to park at the lot for 2 hours.Words Cost of parking equals $3.00 plus $2.00 per hour.Function p = +Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:The function p =Substitutep = +p = 3 + 2 orIt will costfor h into the function rule.to park for 2 hours.represents the situation.Check Your Progress TAXI The price of a taxi ride is$5.00 plus $4.00 per hour. Write a function using two variablesto represent the price of riding a taxi for h hours. Thendetermine how much would it cost for a 3-hour taxi ride.California Mathematics Grade 7 225

9–2 Representing Linear FunctionsStandard 7AF1.5 Represent quantitative relationships graphically and interpret themeaning of a specific part of a graph in the situation represented by the graph.EXAMPLEMAIN IDEA• Graph linear functionsby using functiontables and plottingpoints.MUSIC During a clearance sale, a music store is sellingCDs for $3 and tapes for $1. Graph the function 3x + y= 6 to find how many CDs and tapes Bill can buy with $6.First, rewrite the equation by solving for y.3x + y = 6Write the equation.3x - + y = 6 - Subtract from each side.y = 6 - 3x Simplify.Choose values for x and substitute them to find y. Then graphthe ordered pairs.x y = 6 - 3x y (x, y)0 y = 6 - 31 y = 6 - 3ORGANIZE ITIn your Foldable, includea linear function and itsgraph.2 y = 6 - 3He cannot buy negative numbers of CDs or tapes, so thesolutions are CDs and tapes, CD andtapes, or CDs and tapes.Check Your Progress BAKE SALE During a bakesale, a plate of brownies is sold for $2 and a plate of cookies issold for $1. Graph the function 2x + y = 4 to find how manyplates of brownies and cookies Craig can buy with $4.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.226 California Mathematics Grade 7

2–1 9–2EXAMPLE Graph a FunctionGraph y = x - 3.Step 1 Choose some values for x. Make a function table.Include a column of ordered pairs of the form (x, y) .x x - 3 y (x, y)0 - 31 - 32 - 33 - 3Step 2 Graph each ordered pair.Draw a line that passes througheach point. Note that the orderedpair for any point on this line isa solution of y = x - 3. The lineis the complete graph of thefunction.OyxCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Check It appears from the graph that (-1 , -4) is also asolution. Check this by substitution.y = x - 3Write the function. - 3 Replace x and y.= Simplify.Check Your Progress Graph y = x - 2.OyxCalifornia Mathematics Grade 7 227

9–2BUILD YOUR VOCABULARY (pages 221–222)A function in which the graph of solutions forms ais called a linear function.The value of x where the graph crosses thecalled the x-intercept.The value of y where the graph crosses thecalled the y-intercept.isisEXAMPLESTANDARDS EXAMPLE Which line graphedbelow best represents the table of values forthe ordered pairs (x, y)?A yC yx y0 11 32 53 7OxOxByOxDRead the Test ItemYou need to decide which of the four graphs represents the datain the table.Solve the Test ItemThe values in the table represent the ordered pairs ,yO, and . Test the ordered pairs with eachxCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.graph. Graphis the only graph which contains all theseordered pairs. The answer is .228 California Mathematics Grade 7

9–2Check Your Progress Which line graphedbelow best represents the table of valuesx yfor the ordered pairs (x, y)?0 3AOyxCOyx1 02 -33 -6BDyOxCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:California Mathematics Grade 7 229

2–1 9–3 SlopeStandard 7AF3.3 Graph linear functions, noting that the vertical change(change in y-value) per unit of horizontal change (change in x-value) is always thesame and know that the ratio (“rise over run”) is called the slope of a graph.MAIN IDEA• Find the slope of aline using the slopeformula.BUILD YOUR VOCABULARY (pages 221–222)Slope is the of the rise, or change, tothe run, orchange.EXAMPLEACCESS RAMPS The access ramp from the sidewalk tothe door of a hotel rises 8 inches for every horizontalchange of 96 inches. What is the slope of the accessramp?slope =Defi nition of slope= rise = inches, run = inches= Simplify.The slope of the access ramp is .Check Your Progress ACCESS RAMPS The access rampfrom the sidewalk to the door of an office building rises14 inches for every horizontal change of 210 inches. What is theslope of the access ramp?EXAMPLE Find Slope Using a Graph yFind the slope of the line.Choose two points on the line. The verticalchange is -3 units while the horizontalchange is 2 units.OXCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.230 California Mathematics Grade 7

9–3slope = _ riserunDefi nition of slope= rise = , run =The slope of the line is .Check Your ProgressFind the slope of the line.32y32O 1 223xEXAMPLE Find Slope Using a TableThe points given in the table lieon a line. Find the slope of theline. Then graph the line.change in yslope = __change in xx -3 -1 1y -2 1 4=Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.=The slope is .Check Your Progress The points given in the table belowlie on a line. Find the slope of the line. Then graph the line.yxy2 55 78 911 11OxCalifornia Mathematics Grade 7 231

9–3EXAMPLE Positive SlopeFind the slope of the line that passes through A (3, 3)and B (2, 0) .__m = y 2 - y 1x 2- xDefi nition of slope1m = _ 0 - 3( x 1 , y 1 ) = (3, 3)2 - 3( x 2 , y 2 ) = (2, 0)m = _ -3 or 3 Simplify.-1yA(3, 3)B(2, 0)OxEXAMPLE Negative SlopeFind the slope of the line that passes through X (-2, 3)and Y (3, 0) .m = __y 2 - y 1Defi nition of slopex 2- x 1m = ___ ( x 1 , y 1 ) = (-2, 3)m = _ -35 or - 3_5( x 2 , y 2 ) = (3, 0)Simplify.HOMEWORKASSIGNMENTPage(s):Exercises:Check Your Progress Find the slope of the line thatpasses through each pair of points.a. C(1, 2) and D(2, 6)b. E(-3, -4) and F(0, -2)c. G(-2, 5) and H(4, -7)d. J(0, 8) and K(4, -2)Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.232 California Mathematics Grade 7

9–4 Direct VariationMAIN IDEA• Use direct variation tosolve problems.BUILD YOUR VOCABULARY (pages 221–222)When two variable quantities have a ,their relationship is called a direct variation.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Standard7AF3.4 Plot thevalues of quantitieswhose ratios are alwaysthe same (e.g., costto the number of anitem, feet to inches,circumference todiameter of a circle). Fita line to the plot andunderstand that theslope of the line equalsthe quantities.Standard 7AF4.2 Solvemultistep problemsinvolving rate, averagespeed, distance, and timeor a direct variation.The constant ratio is called the .EXAMPLE Find a Constant RatioEARNINGS The amount of moneySerena earns at her job variesdirectly as the number of hoursshe works. Determine the amountSerena earns per hour.Since the graph of the data forms aline, the rate of change .Use the graph to find .___amount earnedhours workedSerena earns .Check Your ProgressEARNINGS The amount of moneyElizabeth earns at her job variesdirectly as the number of hours sheworks. Determine the amountElizabeth earns per hour.Dollars Earned706050403020100 y x1 2 3 4 5 6Time (hours)California Mathematics Grade 7 233

9–4KEY CONCEPTIn a direct variation, theratio of y to x is constant.This can be stated as yvaries directly with x. Adirect variation can berepresented algebraicallyas k = _ y xor y = kx wherek ≠ 0.EXAMPLE Solve a Direct VariationSHOPPING The total cost for cans of soup varies directlyas the number of cans purchased. If 4 cans of soup cost$5, how much would it cost to buy 8 cans?METHOD 1 Use an equation.Write an equation of direct variation. Let x represent thenumber of cans and let y represent the cost.y = kxDirect variation= k y = , x =1.25 = k Simplify.y = Substitute for .Use the equation to find y when x = 8.y = 1.25xy = 1.25 x =y =Multiply.METHOD 2 Use a proportion.canscostIt would cost4_5 = 8 _ycanscost= Find the cross products.4y = 40 Multiply._ 4y4 = _ 40 Divide each side by 4.4y =Simplify.to buy 8 cans.Check Your Progress SHOPPING A grocery store sells6 apples for $2.70. How much would it cost to buy 10 apples?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.234 California Mathematics Grade 7

9–4EXAMPLES Identify Direct VariationDetermine whether each linear function is a directvariation. If so, state the constant of variation.Days, x 2 4 6 8Hours worked, y 16 32 54 72Compare the ratios to check for a common ratio._ hoursdaysThe ratios are, so the function is.Hours, x 3 6 9 12Miles, y 25.5 51 76.5 102Compare the ratios to check for a common ratio._ mileshoursCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:Since the ratios are, the function isa direct variation. The constant of variation is .Check Your Progress Determine whether the linearfunction is a direct variation. If so, state the constantof variation.a. Days, x 1 2 3 4Hours worked, y 8 16 24 32b. Hours, x 2 4 6 8Miles, y 15 25 35 45California Mathematics Grade 7 235

9–5 Slope-Intercept FormStandard 7AF3.3 Graph linear functions, noting that the vertical change(change in y-value) per unit of horizontal change (change in x-value) is always thesame and know that the ratio (“rise over run”) is called the slope of a graph.MAIN IDEA• Graph linear equationsusing the slope andy-intercept.BUILD YOUR VOCABULARY (pages 221–222)Slope-intercept form is when an equation is written in theform , where m is the and b isthe .EXAMPLES Find the Slopes and y-intercepts of GraphsState the slope and the y-intercept of the graph ofeach equation.y = 3 _4 x - 5y = 3 _4 x + ( ) Write the equation in the form y = mx + b.y = mx + b m = 3 _4 , b =The slope of the graph isis .2x + y = 82x + y = 8, and the y-interceptWrite the original equation.- -__________Subtract from each side.y =Simplify.y =Write the equation in the formy = mx + b.y = mx + b m = , b =The slope of the graph isand the y-interceptCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.is .236 California Mathematics Grade 7

9–5Check Your Progress State the slope and they-intercept of the graph of each equation.a. y = 1_ x - 2 b. 3x + y = 54EXAMPLE Graph an EquationGraph y = _ 2 x + 2 using the slope and y-intercept.3Step 1 Find the slope and y-intercept.y = 2_3 x + 2slope = 2_3y-intercept = 2Step 2 Graph the y-intercept . yStep 3 Use the slope to locate a secondpoint on the line.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:m = 2_3change in y:up 2 unitschange in x:right 3 unitsStep 4 Draw a line through the two points.Check Your Progress Graph y =1_ x + 3 using the slope3and y-intercept.OyxOxCalifornia Mathematics Grade 7 237

9–6Writing Systems of Equations and InequalitiesStandard 7AF1.1 Use variables and appropriate operations to write an expression,an equation, an inequality, or a system of equations or inequalities that represents averbal description (e.g. three less than a number, half as large as area A.MAIN IDEA• Write systems ofequations andinequalities.BUILD YOUR VOCABULARY (pages 221–222)A system ofconsists of two equations andtwo unknowns. A system ofconsists oftwo inequalities and two unknowns.EXAMPLE Writing Systems of EquationsBALLOONING At a hot air ballooning event, a blue balloonis 5 meters above the ground and rising at a rate of 20meters per minute. A red balloon is 10 meters above theground and rising at a rate of 18 meters per minute. Writea system of equations that represents this situation.Let h = the height of the balloons in meters, and let m = thenumber of minutes.Blue balloon:height ofballoonsRed balloon:height ofballoonsminus meters times numberofminutesminus meters times numberofminutesSo, the following system ofequalsequalsinitialheight ofballooninitialheight ofballoonrepresents this situation.Check Your Progress MONEY Juan has 9 bills in hiswallet. The bills are a combination of $1 bills and $5 bills.The value of the bills is $21. Write a system of equations thatrepresents the number of bills Juan has.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.238 California Mathematics Grade 7

9–6EXAMPLE Writing Systems of InequalitiesHEALTH Federico walks and jogs at least 3 miles eachday. Federico walks 4.5 miles per hour and jogs 8.5 milesper hour. He only has an hour to exercise. Write a systemof inequalities that represents this situation.Let w = the number of hours walked, and j = the number ofhours jogged.number ofhours walkedplusnumber ofhours joggedis less thanor equal tohalf hourmileswalkedtimeshourswalkedplusmilesjoggedtimeshoursjoggedisgreaterthan orequal tomileseachdaySo, the system ofis as follows.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:Check Your Progress HEALTH Taylor runs and bikesat least 7 miles each day. She runs 6 miles per hour and bikes12 miles per hour. She only has one hour to exercise each day.Write a system of inequalities that represents this situation.California Mathematics Grade 7 239

9–7Problem-Solving Investigation: Use a GraphMAIN IDEA• Solve problems byusing a graph.EXAMPLE Use a GraphThe graph shows how many boxes ofcookies were sold by five students fora school fundraiser. How many boxesdid the students sell altogether?Standard 7MR2.5Use a variety ofmethods, such aswords, numbers, symbols,charts, graphs, tables,diagrams, and models,to explain mathematicalreasoning.Standard 7SDP1.2Represent two numericalvariables on a scatterplotand informally describehow the data pointsare distributed and anyapparent relationshipthat exists betweenthe two variables (e.g.,between time spent onhomework and gradelevel).HOMEWORKASSIGNMENTPage(s):Exercises:EXPLORE The graph shows you howmany boxes were sold by eachof five students. You want toknow the total number ofboxes sold by the students.PLANUse the graph to add thenumbers of boxes sold.SOLVE + + + + =CHECKThe students soldaltogether.Look at the numbers at the top of each bar.Double check your sum.Check Your Progress PETS The graph shows how manydogs Edmond walked each day this week. How many dogs didhe walk altogether during the week? Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.240 California Mathematics Grade 7

9–8Scatter PlotsStandard 7SDP1.2 Represent two numerical variables on a scatterplot andinformally describe how the data points are distributed and any apparentrelationship that exists between the two variables (e.g., between time spent onhomework and grade level).MAIN IDEA• Construct and interpretscatter plots.BUILD YOUR VOCABULARY (pages 221–222)A scatter plot is a graph that shows thebetween sets of data.A line of fit is a line that is very close topoints in a scatter plot.of the dataEXAMPLE Identify a RelationshipExplain whether the scatter plot of the data for eachof the following shows a positive, negative, or norelationship.cups of hot chocolate sold at a concession stand and theoutside temperatureAs the temperature decreases, the number of cups of hotchocolate sold. Therefore, the scatter plot mightCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.show arelationship.birthday and number of sports playedThe number of sports played does not depend on your birthday.Therefore, the scatter plot showsrelationship.Check Your Progress Determine whether a scatterplot of the data for the following might show a positive,negative, or no relationship.a. number of cups of lemonade sold at a concession stand andthe outside temperatureb. age and the color of your hairCalifornia Mathematics Grade 7 241

9–8EXAMPLE Line of FitZOOS The table at the right shows the average andmaximum longevity of various animals in captivity.Make a scatter plot using the data.Then draw a line that best seems torepresent the data.Longevity (years)Average Maximum12 4725 5015 408 2035 7040 7741 6120 54Source: Walker’s Mammals ofthe WorldWrite an equation for this line of fit.The line passes through points at and .Use these points to find the slope of the line.m = __y 2 - y 1x 2- x 1Defi nition of slopem = ( x 1 , y 1 ) = , ( x 2 y 2 ) =m =Simplify.The slope is , and the y-intercept is .Use the slope and the y-intercept to write the equation.y = mx + bSlope-intercept formy = x + m = , b =The equation for the line of fit is .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.242 California Mathematics Grade 7

9–8Use the equation to predict the maximum longevity foran animal with an average longevity of 33 years.y = _ 3 x + 17.5 Equation for the line of fi t2y = _ 3 + 17.5 or2The maximum longevity is about .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:Check Your ProgressThe table shows the average hourly earnings of U.S.production workers since 1995.a. Make a scatter plot usingthe data.b. Write an equation for thebest-fit line using points(0, 11.43) and (5, 13.76).c. Use the equation to predictthe average hourly earningsof U.S. production workersin 2004. U.S. Production WorkersEarningsYear Since1995Average HourlyEarnings0 $11.431 $11.822 $12.283 $12.784 $13.245 $13.766 $14.32Source: The World AlmanacCalifornia Mathematics Grade 7 243

C H A P T E R9BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 9 Foldableto help you study for yourchapter test.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 9, go to:glencoe.comBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 221–222) to help yousolve the puzzle.9-1FunctionsMatch each description with the word it describes.1. an output value of a function2. the set of values of the dependent variable3. the underlined letter in f (x) = 2x + 54. Complete the function table for fx = 2 (x) + 2.Then give the domain and range.Domain:Range:9-2Graphing Linear Functions5. Complete the function table. Then graph y = -x + 2.x -x + 2 y (x,y)-201Oya. independent variableb. dependent variablec. domaind. rangex 2x + 2 f (x)-2013xCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.3244 California Mathematics Grade 7

Chapter 9 BRINGING IT ALL TOGETHER9-3SlopeFind the slope of the line that passes through each pair ofpoints.6. A (1, -2), B (4, 4) 7. C (1, 2) , D (3, -2) 8. E (-1, 2) , F (2, 2)9-4Direction VariationDetermine whether each linear function is a direct variation.If so, state the constant of variation.9. hours, x 1 2 3 4wages, y $6 $12 $18 $2410. length, x 1 3 5 7width, y 2 6 10 14Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.11. hours, x 5 6 7 8miles, y 480 415 350 2859-5Slope-Intercept Form12. minutes, x 3 6 8 12pages, y 66 132 176 264State the slope and the y-intercept for the graph of eachequation.13. y = -3x + 4 14. y = 2_ x - 7 15.1_3 2 x + y = 8California Mathematics Grade 7 245

Chapter 9 BRINGING IT ALL TOGETHER9-6Writing Systems of Equations and Inequalities16. AGE The sum of Marcus’ age plus three times Khung’s ageis 36. The difference of Khung’s age minus Marcus’ age is18. Write a system of equations that represents their ages.17. FUNDRAISER The school band is ordering two types of t-shirtsto sell for a fundraiser. They want to make a profit of morethan $600. T-shirt A sells for a profit of $4, and t-shirt B sells fora profit of $6. The band plans on selling at least 150 t-shirts.Write a system of inequalities to represent this situation.9-7Problem-Solving Investigation: Use a Graph18. SHOPPING The Buy Online Company charges $1.50 per poundplus $2 for shipping and handling. The Best Catalog Companycharges $1 per pound plus $5 for shipping and handling. Use agraph to determine the weight at which the shipping and handlingwill be the same for both companies.9-8Scatter Plots19. Complete. A scatter plot that shows a negative relationship willhave a pattern of data points that go .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Write whether a scatter plot of the data for the following mightshow a positive, negative, or no relationship.20. favorite color and type of pet246 California Mathematics Grade 7

C H A P T E R9ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study are givenwith each item.I completed the review of all or most lessons without usingmy notes or asking for help.Visit glencoe.com to accessyour textbook, moreexamples, self-check quizzes,and practice tests to helpyou study the concepts inChapter 9.• You are probably ready for the Chapter Test.• You may want to take the Chapter 9 Practice Test onpage 517 of you textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 9 Study Guide and Review onpages 512–516 of your textbook.• If you are unsure of any concepts or skills, refer to the specificlesson(s).• You may also want to take the Chapter 9 Practice Test onpage 517.I asked for help from someone else to complete the reviewof all or most lessons.• You should review the examples and concepts in yourStudy Notebook and Chapter 9 Foldable.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.• Then complete the Chapter 9 Study Guide and Review onpages 512–516 of your textbook.• If you are unsure of any concepts or skills, refer to thespecific lesson(s).• You may also want to take the Chapter 9 Practice Test onpage 517.Student SignatureParent/Guardian SignatureTeacher SignatureCalifornia Mathematics Grade 7 247

C H A P T E R10Algebra: Nonlinear Functionsand MonomialsUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with eight sheets of grid paper.Cut off one section of the gridpaper along both the long andshort edges.Cut off two sections from thesecond sheet, three sectionsfrom the third sheet, and soon to the 8th sheet.Stack the sheets from narrowestto widest.Label each of the right tabs witha lesson number.NOTE-TAKING TIP: When you take notes, definenew terms and write about the new concepts youare learning in your own words. Write your ownexamples that use the new terms and concepts.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.248 California Mathematics Grade 7

C H A P T E R10BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 10.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary Termcube rootFoundon PageDefinitionDescription orExampleChapter 10monomialCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.nonlinear functionquadratic functionCalifornia Mathematics Grade 7 249

10–1Linear and Nonlinear FunctionsPreparation for Standard AF1.5 Represent quantitative relationships graphically andinterpret the meaning of a specifi c part of a graph in the situation represented by thegraph.MAIN IDEA• Determine whethera function is linear ornonlinear.BUILD YOUR VOCABULARY (page 249)Nonlinear functions do not haverates ofchange. Therefore, their graphs are not straight lines.EXAMPLES Identify Functions Using TablesORGANIZE ITExplain how to identifylinear and nonlinearfunctions using graphs,equations, and tables onthe Lesson 10-1 section ofyour Foldable.Determine whether each table represents a linear ornonlinear function. Explain.+ 2 + 2 + 2x 2 4 6 8y 2 20 54 104+ 18 + 34 + 50As x increases by, y increases by a greater amount eachtime. The rate of change is not, so this functionis .+ 3 + 3 + 3x 1 4 7 10y 0 9 18 27+ 9 + 9 + 9As x increases by , y increases by each time. Therate of change is , so this function is .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.250 California Mathematics Grade 7

10–1Check Your Progress Determine whether each tablerepresents a linear or nonlinear function. Explain.a.x 1 3 5 7y 3 7 11 15b. x 1 2 3 4y 1 8 27 64EXAMPLES Identify Functions Using GraphsDetermine whether each graph represents a linear ornonlinear function. Explain.yyxy 22 1y x 3Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.OThe graph is a curve, not astraight line. So it representsaxfunction.OThe graph is a straightline. So it represents axfunction.Check Your Progress Determine whether each graphrepresents a linear or nonlinear function. Explain.a. yb. yOy 3 x 1xy 2x 3OxCalifornia Mathematics Grade 7 251

10–1EXAMPLES Identify Functions Using EquationsDetermine whether each equation represents a linear ornonlinear function. Explain.y = 5 x 2 + 3Since x is raised to thepower, the equationcannot be written in the form y = mx + b. So, this function is.y - 4 = 5xRewrite the equation as y =. This equation issince it is of the form y = mx + b.Check Your Progress Determine whether eachequation represents a linear or nonlinear function.Explain.a. y = x 2 - 1HOMEWORKASSIGNMENTPage(s):Exercises:b. y = xCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.252 California Mathematics Grade 7

10–2 Graphing Quadratic FunctionsStandard 7AF1.5 Represent quantitative relationships graphically and interpretthe meaning of a specific part of a graph in the situation represented by the graph.Standard 7AF3.1 Graph functions of the form y = n x 2 and y = n x 3 and use in solvingproblems.MAIN IDEA• Graph quadraticfunctions.BUILD YOUR VOCABULARY (page 249)A quadratic function is a function in which thepower of the is .EXAMPLE Graph Quadratic FunctionsCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ORGANIZE ITRecord what youlearn about graphingquadratic functions andusing the graphs to solveproblems on theLesson 10-2 section ofyour Foldable.Graph y = 5 x 2 .To graph a quadratic function, make a table of values, plot theordered pairs, and connect the points with a smooth curve.x 5 x 2 y (x, y)-2 5 (-2) 2 = (-2, )-1 5 (-1) 2 = (-1, )0 5 (0) 2 = (0, )1 5(1) 2 = (1, )2 5(2) 2 = (2, )Check Your Progress Graph y = - 3x2.OyxOyxCalifornia Mathematics Grade 7 253

10–2EXAMPLE Graph Quadratic FunctionsGraph y = 3 x 2 + 1.x 3 x 2 + 1 y (x, y)-2 3(-2) 2 + 1 = (-2, )-1 3 (-1) 2 + 1 = 4 4 (-1, 4)0 3(0) 2 + 1 = (0, )1 3 (1) 2 + 1 = 4 4 (1, 4)2 3 (2) 2 + 1 = 13 13 (2, 13)yOxCheck Your Progress Graph y = - 2x2- 1.HOMEWORKASSIGNMENTPage(s):Exercises:OyxCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.254 California Mathematics Grade 7

10–3 Problem-Solving Investigation:Make a ModelEXAMPLE Make a ModelMAIN IDEA• Solve problems bymaking a model.Standard 7MR2.5Use a variety ofmethods, such aswords, numbers, symbols,charts, graphs, tables,diagrams, and models,to explain mathematicalreasoning.Standard 7AF1.1 Usevariables and appropriateoperations to write anexpression, an equation,an inequality, or a systemof equations or inequalitiesthat represents a verbaldescription (e.g. threeless than a number, half aslarge as area A.DESKS Caitlyn is arranging desks in her classroom.There are 32 desks, and she wants to have twice asmany desks in each row as she has in each column. Usea model to determine how many desks she should put ineach row and how many rows she will need.EXPLORE You know Caitlyn has 32 desks.PLANSOLVEExperiment by arranging 32 tiles into differentrows and columns until you havemany tiles in each row as are in each column.asCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:CHECKThe correct arrangement isdesks in each row.rows withCheck to see if the arrangement meets Caitlyn’soriginal requirements.Check Your Progress TABLES Mrs. Wilson wants toarrange tables into a square that is open in the middle andhas 8 tables on each side. How many tables will she needaltogether?California Mathematics Grade 7 255

10–4Graphing Cubic FunctionsMAIN IDEA• Graph cubicfunctions.Standard 7AF3.1Graph functionsof the form y = n x 2and y = n x 3 and use insolving problems. 7AF3.2Plot the values fromthe volumes of threedimensionalshapesfor various values ofthe edge lengths (e.g.,cubes with varying edgelengths or a triangleprism with a fixed heightand an equilateral trianglebase of varying lengths).EXAMPLE Graph a Cubic Function_Graph y = - x 32 .Make a table of values.x y = -_x 32-2 - __2-1 - __20 - __21 - __22 - __2Graph the function.Check Your ProgressGraph y = 2x 3 .( ) 3( ) 3( ) 3 ( ) 3( ) 3= - _2 = - _2 = - _2 = - _2 = - _2= - ( ) ====O=yxy 32x(x, y)Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.256 California Mathematics Grade 7

10–4EXAMPLEGEOMETRY Write a function for thevolume V of the triangular prism.Graph the function. Then estimatethe dimensions of the prism thatwould give a volume of approximately40 cubic meters.x mx m(2x 8)mV = BhV = 1_2 · x · x · ( )V = (2x + 8)of a triangular prismReplace B with 1_ · x · x and h with2( ) .1_2 · x · x =V = x + 4x Distributive PropertyThe function for the volume V of the box is V = .Make a table of values to graph this function. You do not needto include negative values of x since the side length of the prismcannot be negative.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.5040302010-10V x 3 4x 21 2 3 4 5 6x V = x 3 + 4 x 2 (x, V )0 (0) 3 + 4(0) 2 =0.5 (0.5) 3 + 4(0.5) 2 ≈1 (1) 3 + 4(1) 2 =1.5 (1.5) 3 + 4(1.5) 2 ≈2 (2) 3 + 4(2) 2 =2.5 (2.5) 3 + 4(2.5) 2 ≈To obtain a volume of about 40 cubic meters, the legs of thebase are about meters, and the height is (2 · + 8)or about meters.California Mathematics Grade 7 257

10–4Check Your Progress GEOMETRY Write a function forthe volume V of the rectangular prism. Graph the function.Then estimate the dimensions of the prism that would give avolume of approximately 34 cubic inches. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.258 California Mathematics Grade 7

10–5Multiplying MonomialsEXAMPLE Multiply PowersMAIN IDEA• Multiply and dividemonomials.Find 7 6 · 7 2 . Express using exponents.7 6 · 7 2 = 7 6 + 2 The common base is .= the exponents.KEY CONCEPTProduct of Powers Tomultiply powers withthe same base, add theirexponents.In theLesson 10–5 section ofyour Foldable, record theproduct of powers rule.Check 7 6 · 7 2 = (7 · 7 · 7 · 7 · 7 · 7) · (7 · 7)= 7 · 7 · 7 · 7 · 7 · 7 · 7 · 7=Check Your Progress Find 25· 2 4 . Express usingexponents.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Standard7NS2.3 Multiply,divide, and simplifyrational numbers byusing exponent rules.Standard 7AF2.1Interpret positive wholenumberpowers asrepeated multiplicationand negative wholenumberpowers asrepeated division ormultiplication by themultiplicative inverse.Simplify and evaluateexpressions that includeexponents.Standard 7AF2.2Multiply and dividemonomials; extend theprocess of taking powersand extracting roots tomonomials when the latterresults in a monomial withan integer exponent.EXAMPLE Multiply MonomialsFind 7 x 2 ( 11 x4 ). Express using exponents.7 x 2 ( 11 x4 ) = (7 · 11) Comm. and Assoc. Properties.= ( x 2 + 4 ) The common base is .= the exponents.Check Your Progress Find 3 x2 ( -5 x5 ). Express usingexponents.California Mathematics Grade 7 259

10–5EXAMPLE Multiply Negative PowersFind 4 -8 · 4 3 . Express using positive exponents.METHOD 14 -8 · 4 3 = 4 The common base is .= 4 the exponents.= Simplify.METHOD 24 -8 · 4 3 = · 4 Write4 -8 as 1_4 8 .=_____14 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 Cancelcommonvalues.= · Simplify.Check Your Progressexponents.Find x7· x -9 . Express using positiveCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.260 California Mathematics Grade 7

10–6Dividing MonomialsEXAMPLES Divide PowersMAIN IDEA• Divide MonomialsKEY CONCEPTQuotient of Powers Todivide powers with thesame base, subtract theirexponents.Simplify. Express using exponents._ 6 126 2_ 6 126 = 612 - 2 2 The common base is .= Simplify._ a 14a 8_ a 14= a 14 - 8a 8 The common base is .= Simplify.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Standard7NS2.3 Multiply,divide, and simplifyrational numbers byusing exponent rules.Standard 7AF2.1Interpret positive wholenumberpowers asrepeated multiplicationand negative wholenumberpowers asrepeated division ormultiplication by themultiplicative inverse.Simplify and evaluateexpressions that includeexponents.Standard 7AF2.2Multiply and dividemonomials; extend theprocess of taking powersand extracting roots tomonomials when the latterresults in a monomial withan integer exponent.HOMEWORKASSIGNMENTPage(s):Exercises:Check Your Progressexponents.a. _ 3103 4b. _ x 11x 3Simplify. Express usingCalifornia Mathematics Grade 7 261

10–6EXAMPLES Use Negative ExponentsSimplify. Express using positive exponents._ 8 -58 2_ 8 -5= 8 Quotient of Powers28= 8 or Simplify._ x -9x -1_ x -9x = x -1Quotient of Powers= x Subtraction of a negative number= x or Simplify.Check Your Progressexponents.a. _ 595 2EXAMPLESimplify. Express using positiveb. _ n-3n -1_STANDARDS EXAMPLE Simplify 8 y 3. Express using9positive exponents.16 yA 2 y 6B1_2 y 6C1_2 y 3D y 6_8Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.262 California Mathematics Grade 7

10–6Read the Test ItemYou are asked to simplify the monomial.Solve the Test Item(_ 8 y 3= (_816 y 16) 9)Group terms= 1_ · y Quotient of Powers.2= 1_ · y or Simplify.2The correct answer choice is ._Check Your Progress x 4 y 5Simplify . Express using positivex 7 2exponents.yA x 3 y 3 D _ x 3y 3C _ y 3x 3D1_x 3 y 3Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.California Mathematics Grade 7 263

10–7Check Your ProgressSimplify.a. ( 4 x 6) 2 b. ( -3 a 4 b 2 ) 3EXAMPLEGEOMETRY Find the volume of a cube with sides oflength 6m n 7 as a monomial.V = s 33V = ( )of a cubeReplace s with .V = 6 m n Power of a ProductV =Simplify.The volume of the cube iscubic units.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:Check Your Progress GEOMETRY Find the volume of acube with sides of length 4 a 2 b as a monomial.California Mathematics Grade 7 265

10–8Roots of MonomialsStandard 7AF2.2 Multiply and divide monomials; extend the process of taking powersand extracting roots to monomials when the latter results in a monomial with aninteger exponent.MAIN IDEA• Find roots ofmonomials.BUILD YOUR VOCABULARY (page 249)The square root of a monomial is one of the equalfactors of the monomial. The root of a monomialis one of the three equal factors of the monomial.EXAMPLES Simplify Square RootsSimplify √ 9 k 4 .√ 9 k 4 = √9 · √ Product Property of Square Roots= 3 · 3 = ; k 2 · k 2 =Simplify √ 400 w 8 x 2 .√ 400 w 8 x 2 = √ · √ w 8 · √ x 2 Product Property ofSquare Roots= 20 · · ⎪x⎥ 20 · 20 = ;w 4 · w 4 = w ; x · x = x 2= Use absolute value toindicate the positive valueof x.Check Your ProgressSimplify.a. √ 25 d 2 b. √ 121 r 4 s 6EXAMPLES Simplify Cube RootsCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Simplify 3 √ a 6 .3√ a 6 = ( a2 ) 3 =266 California Mathematics Grade 7

10–8In theLesson 10–8 section ofyour Foldable, recordthe Product Property ofSquare Roots and theProduct Property ofCube Roots.Simplify 3 √ 343 m 12 .3√ 343 m 12 = √ 3 · √ 3 m 12√= 3 · √ 3 m 4 × m 4 × m 4Product Propertyof Cube Roots= Simplify.Check Your Progressa. √ 3 y 3Simplify.b. √ 3 64 h 9EXAMPLEGEOMETRY Find the length of one side of a cube whosevolume is 729 g 18 cubic units.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:V = s 3of a cube= s 3 Replace V with .3√ 729 g 18 = √ 3 s 3 Defi nition of root3√ 729 · √ 3 g 18 == s Simplify.The length of one side of the cube isProduct Property of Cube Rootsunits.Check Your Progress GEOMETRY Find the length of oneside of a cube whose volume is 216 x 15 cubic units.California Mathematics Grade 7 267

C H A P T E R10BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 10 Foldableto help you study for yourchapter test.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 10, go to:glencoe.comBUILD YOURVOCABULARYYou can use your completedVocabulary Builder (pages 249)to help you solve the puzzle.10-1Linear and Nonlinear FunctionsWrite linear or nonlinear to name the kind of functiondescribed.1. constant rate change 2. graph that is a curve3. power of x may be greater 4. equation has the formthan oney = mx + b5. Name the kind of function represented. Explain your reasoning.x -3 0 3 6y 10 1 10 3710-2Graphing Quadratic FunctionsDetermine whether each equation represents a quadraticfunction. Write yes or no.6. y = 3x - 5 7. y = 6 - x 2 y8.9. Explain how to graph a quadratic function.OxCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.268 California Mathematics Grade 7

Chapter 10 BRINGING IT ALL TOGETHER10-3Problem-Solving Investigation: Make a Model10. DESIGN Edu-Toys is designing a new package to hold a set of30 alphabet blocks. Each block is a cube with each side of thecube being 2 inches long. Give two possible dimensions forthe package.10-4Graphing Cubic FunctionsDetermine whether each equation represents a cubic function.Write yes or no.11. y = -3 x 2 12. y = 1_3 x 3 13. y = - x 3 + 514. Explain the difference in the graph of a quadratic function andthe graph of a cubic function.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.10-5Multiplying MonomialsComplete each sentence.15. To multiply powers with the same base, their exponents.Simplify. Express using exponents.16. 5 2 · 5 6 17. 2x 2 · 4x 3 18. ( 8 x3 )( -3 x9 )California Mathematics Grade 7 269

Chapter 10 BRINGING IT ALL TOGETHER10-6Dividing Monomials19. To divide powers with the same base, their exponents.Simplify. Express using positive exponents._20. 2 52 2_21. w 3w 8_22. 18 a 76 a 310-7Powers of Monomials23. To find the power of a power, the exponents.Simplify.24. ( 8 2 ) 3 25. ( k 4 ) 5 26. ( 4 a2b 4 ) 410-8Roots of MonomialsSimplify.27. √ n 428. √ 36 x 2 y 8 29. √ 3 27 d 930. To find the length of one side of a square when given its area,find theroot of the area.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.270 California Mathematics Grade 7

C H A P T E R10ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study are given witheach item.I completed the review of all or most lessons without using mynotes or asking for help.Visit glencoe.com to accessyour textbook, moreexamples, self-check quizzes,and practice tests to helpyou study the concepts inChapter 10.• You are probably ready for the Chapter Test.• You may want to take the Chapter 10 Practice Test onpage 561 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.0• You should complete the Chapter 10 Study Guide and Reviewon pages 557–560 of your textbook.• If you are unsure of any concepts or skills, refer to thespecific lesson(s).• You may also want to take the Chapter 10 Practice Test onpage 561.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in yourStudy Notebook and Chapter 10 Foldable.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.• Then complete the Chapter 10 Study Guide and Review onpages 557–560 of your textbook.• If you are unsure of any concepts or skills, refer to the specificlesson(s).• You may also want to take the Chapter 10 Practice Test onpage 561.Student SignatureParent/Guardian SignatureTeacher SignatureCalifornia Mathematics Grade 7 271

C H A P T E R11StatisticsUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with five pieces of 8 _ 1 " × 11" paper.2Place 4 sheets of paperinch apart.3_4Roll up bottom edges.All tabs should bethe same size.Crease and staplealong the fold.Label the tabswith the topics fromthe chapter. Label thelast tab Vocabulary.NOTE-TAKING TIP: As you take notes on a topic,it helps to write how the subject relates to yourlife. For example, as you learn about differentkinds of statistical measures and graphs, you willunderstand how to evaluate statistical informationpresented in such places as advertisements andpersuasive articles in magazines.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.272 California Mathematics Grade 7

C H A P T E R11BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 11.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExampleback-to-back stem-andleafplotbox-and-whisker plotcircle graphCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.histograminterquartile rangeleaveslower quartilemeanChapter 11(continued on the next page)California Mathematics Grade 7 273

Chapter 11 BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExamplemeasures of centraltendencymeasures of variationmedianmodeoutlierquartilesrangestem-and-leaf plotstemsCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.upper quartile274 California Mathematics Grade 7

11–1Problem-Solving Investigation:Make a TableEXAMPLE Make a TableMAIN IDEA• Solve problems bymaking a table.Standard 7MR2.5Use a variety ofmethods, such aswords, numbers, symbols,charts, graphs, tables,diagrams, and models,to explain mathematicalreasoning.Standard 7SDP1.1 Knowvarious forms of displayfor data sets, includingstem-and-leaf plot or boxand-whiskerplot; use theforms to display a singleset of data or to comparetwo sets of data.The list shows the ages of 25persons selected at randomfrom the audience of a recentshowing of a comedy movie.Make a frequency table of theages using intervals 17–24,25–32, 33–40, 41–48, and 49–56.What is the most commoninterval of attendance ages?26 42 22 26 2421 27 35 28 1819 25 46 31 2917 56 19 41 2338 20 21 25 22EXPLORE You have a list of ages. You need to know how manyages fall into each interval.PLANSOLVEMake a table toshow the frequency,or number, of agesin each interval.The greatestfrequency is agesCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:CHECK, so this isthe most commoninterval of attendanceages.Make sure the frequency table includes each agefrom the list.Check Your ProgressThe list shows the favoritesports of 25 people selectedat random. In the list, Srepresents soccer, B representsbaseball, F represents football,and V represents volleyball.Make a frequency table of thefavorite sports. What is themost popular sport?V B S F VS V F V SS F B S BB S V F SF F B S VCalifornia Mathematics Grade 7 275

11–2 HistogramsStandard 7SDP1.1 Know various forms of display for data sets, including stem-andleafplot or box-and-whisker plot; use the forms to display a single set of data or tocompare two sets of data.MAIN IDEA• Construct and interprethistograms.BUILD YOUR VOCABULARY (pages 273–274)A histogram is a type ofgraph used todisplay numerical data that have been organized intointervals.EXAMPLE Construct a HistogramFOOD The list shows the 8 47 19 34 30number of grams of caffeinein certain types of tea. Use10 58 20 39 32intervals 1–20, 21–40, 41–60, 12 4 22 40 9261–80, and 81–100 to make afrequency table. Thenconstruct a histogram.18 85 26 27Place a tally mark for each value in the appropriate interval.Then add up the tally marks to find the frequency for eachinterval.ORGANIZE ITUnder the tab forLesson 11–2, explainthe difference betweena bar graph and ahistogram. Describe atype of statistics thatcould be displayed with ahistogram.To construct a histogram, follow these steps.Step 1 Draw and label a horizontal and vertical axis.Include a title.Step 2 Show thefrom the frequency tableon theaxis.Step 3 For each caffeineinterval, draw a bar whoseheight is given by thefrequencies.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.276 California Mathematics Grade 7

11–2Check Your Progress The frequency table below showsthe amount of caffeine in certain drinks. Draw a histogram torepresent the data.Caffeine Content of Certain Types of DrinkCaffeine (mg) Tally Frequency0–50 351–100 4101–150 6151–200 7EXAMPLES Analyze and Interpret DataCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.WEATHER How many monthshad 6 or more days of rain?Three months had daysof rain, and one month haddays of rain.Therefore, + ormonths had 6 or more days of rain. WEATHER How many months had exactly 2 days of rain?This cannot be determined from the data presented in thisgraph. The histogram indicates that there werethat had 2 or 3 days of rain, but it is impossible to tell howmany months haddays of rain.California Mathematics Grade 7 277

11–2Check Your Progressa. How many months had6 or more days of snow? b. How many months had exactly 6 days of snow?HOMEWORKASSIGNMENTPage(s):Exercises:Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.278 California Mathematics Grade 7

11–3Circle GraphsStandard 7SDP1.1 Know various forms of display for data sets, including stem-andleafplot or box-and-whisker plot; use the forms to display a single set of data or tocompare two sets of data.MAIN IDEA• Construct and interprethistograms.BUILD YOUR VOCABULARY (pages 273–274)A circle graph is used to compare parts of a .The entirerepresents that whole.ORGANIZE ITUnder the tab forLesson 11–3, find anexample of a circle graphfrom a newspaper ormagazine. Explain whatthe graph shows.EXAMPLE Construct a Circle Graph from PercentsTORNADOES The table shows when tornadoes occurredin the United States from 1999 to 2001. Make a circlegraph using this information.Tornadoes in the United States, 1999–2001January–March 15%April–June 53%July–September 21%October–December 11%Source: spc.noaa.gov/Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Step 1 There arein a circle. So, multiply eachpercent by 360 to find the number of degrees for eachof the graph.Jan–Mar:15% of 360 = · 360 orApr–Jun:53% of 360 = · 360 or aboutJul–Sept:21% of 360 = · 360 or aboutOct–Dec:11% of 360 = · 360 or aboutCalifornia Mathematics Grade 7 279

11–3Step 2 Use a compass to draw a circle and a radius. Thenuse a protractor to draw a angle. This sectionrepresents January–March. From the new radius, drawthe next angle. Repeat for each of the remainingangles. Label each. Then give the grapha .Check Your ProgressHURRICANES The table showswhen hurricanes or tropical stormsoccurred in the Atlantic Oceanduring the hurricane season of2002. Make a circle graph usingthis information.Hurricanes in theUnited States, 2002Month PercentJuly 7%August 21%September 64%October 8%Source: nhc.noaa.gov/Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.280 California Mathematics Grade 7

11–3EXAMPLES Construct a Circle Graph from DataBASKETBALL Construct a circle graph using theinformation in the histogram below. Step 1 Find the total number of players.6 + + 1 + + 2 =Step 2 Find the ratio that compares the number in eachpoint range to the total number of players. Round tothe nearest hundredth.11.1 to 13 : 6 ÷ 25 =13.1 to 15 : 12 ÷ 25 =Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.15.1 to 17 : 1 ÷ 25 =17.1 to 19 : 4 ÷ 25 =19.1 to 21 : 2 ÷ 25 =Step 3 Use these ratios to find the number of degrees of eachsection. Round to the nearest degree if necessary.11.1 to 13 : · 360 = or about13.1 to 15 : · 360 = or about 17315.1 to 17 : · 360 = or about17.1 to 19 : · 360 = or about19.1 to 21 : · 360 = or about 29California Mathematics Grade 7 281

11–3Step 4 Use a compass and protractorto draw a circle and theappropriate sections. Labeleach section and give thegraph a title. Write the ratiosas percents.Use the circle graph from Example 2 to describe themakeup of the average game scores of the 25 top-scoringbasketball players.Almost _ 3 of the players had average game scores between 11.14and 15 points. Fewer than 1_ had average game scores greater4than points.HOMEWORKASSIGNMENTPage(s):Exercises:Check Your Progressa. Construct a circle graphusing the informationin the histogram at right. b. Use the graph to describe the makeup of the average gamescores of the 10 top-scoring football players.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.282 California Mathematics Grade 7

11–4Measures of Central Tendency and RangeStandard 7SDP1.3 Understand the meaning of, and be able to compute, theminimum, the lower quartile, the median, the upper quartile, and the maximum of a dataset.MAIN IDEA• Find the mean, median,mode, and range of aset of data.BUILD YOUR VOCABULARY (pages 273–274)Measures of central tendency are numbers thata set of data.The mean of a set of data is theof the datathe number of items in the data set.The median of a set of data is thenumber ofWRITE ITThe words central andmiddle have similardefinitions. If mean,median, and mode aremeasures of centraltendency, what do theymeasure?the data ordered from least to greatest, or the mean of thenumbers.The mode of a set of data is the number or numbers thatoccuroften.The range of a set of data isbetweenCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.the greatest and least numbers in a set of data.EXAMPLE Find Measures of Central TendencyThe ages, in years, of the actors in a play are 4, 16, 32,19, 27, 32. Find the mean, median, mode, and range ofthe data.MeanMedian_____ 4 + 16 + 32 + 19 + 27 + 32 =≈Arrange the numbers in order fromto .4 16 19 27 32 32+___ =(continued on the next page)California Mathematics Grade 7 283

11–4Mode The data has a mode of .Range 32 - 4 or .Check Your Progress The ages, in years, of the childrenat a daycare center are 3, 5, 3, 7, 6, 4. Find the mean, median,mode, and range of the set of data.EXAMPLE Using Appropriate MeasuresOLYMPICS Select the appropriate measure of centraltendency or range to describe the data in the table.Justify your reasoning.ORGANIZE ITUnder the tab for Lesson11–4, record how to findthe mean, median, andmode of a set of data.Explain measures ofcentral tendency, mean,median, and mode inyour own words and withexamples.EventGold Medals Won by the United States at theWinter Olympics, 1924–2002GoldMedalsEventGoldMedalsAlpine skiing 10 Luge 2Bobsleigh 6 Short track speed skating 3Cross country 0 Skeleton 3Figure skating 13 Ski jumping 0Freestyle skiing 4 Snowboarding 2Ice hockey 3 Speed skating 26Find the mean, median, mode, and range of the data.Mean_________ 10 + 6 + 0 + 13 + 4 + 3 + 2 + 3 + 3 + 0 + 2 + 26= _The mean ismedals.Median Arrange the numbers from least to greatest.0, 0, 2, 2, 3, 3, 3, 4, 6, 10, 13, 26The median is the middle number, or=medals.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.284 California Mathematics Grade 7

11–4Mode There is one mode, .Range 26 - 0 or .Check Your Progress Select the appropriate measure ofcentral tendency or range to describe the data in the table.Justify your reasoning.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:CountryGold Medals (1896–2002 Summer)United States 872Great Britain 180France 188Italy 179Sweden 136Hungary 150Australia 102Finland 101Japan 97Romania 74Brazil 12Ethiopia 12Source: infoplease.comCalifornia Mathematics Grade 7 285

11–5 Measures of VariationStandard 7SDP1.3 Understand the meaning of, and be able to compute, theminimum, the lower quartile, the median, the upper quartile, and the maximum of adata set.MAIN IDEA• Find the range andquartiles of a set ofdata.BUILD YOUR VOCABULARY (pages 273–274)Measures of variation are used to describe theof a set of data.The range indicates howthe data are.Quartiles are the values that divide the data intoKEY CONCEPTSRange The range of a setof data is the differencebetween the greatestand the least numbers inthe set.equal parts.Thelower quartile.of the lower half of a set of data is theInterquartile Range Theinterquartile range isthe range of the middlehalf of the data. It is thedifference between theupper quartile and thelower quartile.The median of theupper quartile.Data that are more thanof the set of data is thetimes the value of theinterquartile range beyond the quartiles are called outliers.EXAMPLE Find Measures of VariationBASKETBALL Find themeasures of variationfor the data in the table.The range is 109 - 91.3 or.Points Scored by Top Ten TeamsDuring the NBA Playoffs, 2002Team Points ScoredDallas 109Minnesota 102Sacramento 101.1L.A. Lakers 97.8Charlotte 96.1New Jersey 95.4Orlando 93.8Indiana 91.6Boston 91.3Portland 91.3Source: nba.comCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.286 California Mathematics Grade 7

11–5Median, Upper Quartile, and Lower QuartileArrange the numbers in order from least to greatest.lower quartile median upper quartile91.3 91.3 91.6 93.8 95.4 96.1 97.8 101.1 102 109The median is , the lower quartile is , and theCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.REMEMBER ITA small interquartilerange means that thedata in the middle ofthe set are close in value.A large interquartilerange means that thedata in the middle arespread out.upper quartile is .Interquartile Range = upper quartile - lower quartile=Check Your Progress BASEBALL Find the measures ofvariation for the data in the table.Giants Batting AverageAgainst Anaheim in the WorldSeries 2002PlayerBattingAverageRueter 0.500Bonds 0.471Snow 0.407Bell 0.304Lofton 0.290Kent 0.276Aurilia 0.250Sanders 0.238Santiago 0.231Source: infoplease.comCalifornia Mathematics Grade 7 287

11–5ORGANIZE ITUnder the tab for Lesson11–5, write what youlearn about finding therange and quartiles of aset of data.EXAMPLE Find Outliers Item Sold at Football GameConcession StandCONCESSION SALES Findany outliers for the dataItem Number Soldin the table at the right. Colas 196First arrange the numbers in Diet colas 32order from least to greatest. Water 46Then find the median, upperCoffee 18quartile, and lower quartile.Candy bars 39Hotdogs 23Hamburgers 16Chips 41Popcorn 2416 18 23 24 32 39 41 46 196__ 18 + 23= 322__ 41 + 46=2Interquartile Range = - or 23Multiply the interquartile range,23, by 1.5. × = 34.5HOMEWORKASSIGNMENTPage(s):Exercises:Find the limits for the outliers.Subtract 34.5 from the lower quartile. - 34.5 =Add 34.5 to the upper quartile. + 34.5 =The limits for the outliers are and .The only outlier is .Check Your ProgressFind any outliers for thedata in the table at right.Items Sold at School BookstoreItem Number SoldPens 35Pencils 15Erasers 20Candy bars 93Folders 17School Pennants 18Calculators 2Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.288 California Mathematics Grade 7

11–6Box-and-Whisker PlotsStandard 7SDP1.1 Know various forms of display for data sets, including stemand-leafplot or box-and-whisker plot; use the forms to display a single set of dataor to compare two sets of data.MAIN IDEA• Display and interpretdata in a box-andwhiskerplot.BUILD YOUR VOCABULARY (pages 273–274)A box-and-whisker plot uses atheof a set of data.to showCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ORGANIZE ITUnder the tab for Lesson11–6, collect data fromthe Internet, such asnumber of homerunshit by the players of abaseball team. Draw abox-and-whisker plot todisplay the data.EXAMPLE Draw a Box-and-Whisker PlotPOPULATION Use thedata in the table atthe right to constructa box-and-whisker plot.World’s Most Populous CitiesCityPopulation(millions)Tokyo 34.8New York 20.2Seoul 19.9Mexico City 19.8Sao Paulo 17.9Bombay 17.9Osaka 17.9Los Angeles 16.2Cairo 14.4Manila 13.5Source: Time AlmanacStep 1 Draw athat includes the least andgreatest number in the data.Step 2 Mark the extremes, the, and the upperand lowerabove the number line.Since the data have an outlier, mark the greatest valuethat is not an .Step 3 Draw the box and whiskers.(continued on the next page)California Mathematics Grade 7 289

11–6least valuethat is notan outlierlowerquartilemedianupperquartilegreatest value thatis not an outlieroutlier10 20 30 40Check Your ProgressUse the data in the tableat the right to draw abox-and-whisker plot.Most Populous U.S. CitiesCityPopulation (inmillions)New York 8.0Los Angeles 3.7Chicago 2.9Houston 2.0Philadelphia 1.5Phoenix 1.3San Diego 1.2Dallas 1.2Source: infoplease.comEXAMPLE Interpret DataWATERFALLS What do the lengths of the parts of thebox-and-whisker plot below tell you about the data?Data in thethe data in the quartile are more spread out thanquartile. You can see that data inCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.thewhisker isquartile are the most spread out because thethan other parts of the plot.290 California Mathematics Grade 7

11–6Check Your Progress What do the lengths of the parts ofthe box-and-whisker plot below tell you about the data? Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:California Mathematics Grade 7 291

11–7 Stem-and-Leaf PlotsStandard 7SDP1.1 Know various forms of display for data sets, including stemand-leafplot or box-and-whisker plot; use the forms to display a single set of data orto compare two sets of data.MAIN IDEA• Display data instem-and-leaf plots.Interpret data in stemand-leafplots.BUILD YOUR VOCABULARY (pages 273–274)The numerical data are listed in ascending ororder in a stem-and-leaf plot. The greatestplace value of the data are used for the. Theform the next greatest place value.EXAMPLE Draw a Stem-and-Leaf PlotFOOD Display the data inthe table in a stem-and-leafplot with or without the useof technology.Step 1 Find the least andgreatest number. Thenidentify the greatestplace value digit in eachnumber.• The least number,, has 2 inthe thousands place.• The greatestnumber,, has 3 in the thousands place.Step 2 Draw a vertical line and write the stems, 2 and 3,to theof the line.Step 3 Write the leaves to the of the line, with thecorresponding stem. For example, for 2,800, write 8 tothe right of .StemLeaf2 8 8 93 0 1 2 2 2 5Peanuts Harvested, 2005StateAmount(lb/acre)Alabama 2,800Florida 2,900Georgia 3,000New Mexico 3,200North Carolina 3,100Oklahoma 3,200South Carolina 3,200Texas 3,500Virginia 2,800Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.2 8 = 2,800 lb292 California Mathematics Grade 7

11–7Check Your ProgressBASEBALL Display the datain the table in a stem-and-leafplot with or without the use oftechnology.StemLeaf5 8 96 0 1 3 4 5 67 0 35 8 = 58 home runsMost Home Runs in aSingle SeasonPlayer Home RunsBarry Bonds 73Jimmie Foxx 58Roger Maris 61Mark McGwire 65Mark McGwire 70Babe Ruth 59Babe Ruth 60Sammy Sosa 63Sammy Sosa 64Sammy Sosa 66EXAMPLE Interpret DataMEXICO The stem-and-leaf plot lists the percent ofpeople in each state that were born in Mexico, roundedto the nearest whole number.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Stem Leaf0 0 0 0 1 1 2 2 3 4 4 5 5 5 6 6 8 8 81 0 1 4 4 72 1 2 2 3 83 1 2 3 5 5 9 94 0 1 2 3 3 3 4 6 85 2 6 66 4 67 4 3 1 = 31%a. Which interval contains the most percentages?Most of the percentages occur in theinterval.b. What is the greatest percent of people living in oneU.S. state that were born in Mexico?The greatest percent of people living in one U.S. state bornin Mexico is .California Mathematics Grade 7 293

11–7c. What is the median percent of people living in oneU.S. state that were born in Mexico?The median percent of people living in one U.S. state born inMexico is .Check Your ProgressExample 2.Refer to the stem-and-leaf plot ina. What is the range of the data? .b. What is the least percent of people living in one U.S. statethat were born in Mexico? .c. What percentages occur most often?BUILD YOUR VOCABULARY (pages 273–274)Astem-and-leaf plot can be used tocompare a set of data.EXAMPLE Compare DataAGRICULTURE The yearly production of honey inCalifornia and Florida is shown for the years 2000 to2004, in millions of pounds.California Stem Florida7 1 48 4 2 0 0 2 42 1 32 3 = 32 million lb 2 0 = 20 million lba. What state produces the most honey?California: 17 + 24 + 28 + 31 + 32 =Florida: 14 + 20 + 20 + 22 + 24 =million lbmillion lbCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.produces the most honey.294 California Mathematics Grade 7

11–7b. Which state has the most varied production? Explain.The data forare more spread out, while thedata for are clustered. So, hasthe most varied production.Check Your Progress BABY-SITTING The amount ofmoney Hanna and Jasmine earned baby-sitting in 2006 isshown in the back-to-back stem-and-leaf plot.Hanna Stem Jasmine0 0 1 0 2 3 50 0 2 2 5 5 8 2 0 0 50 3 0 24 00 2 = $20 2 5 = $25a. Who earned more money baby-sitting?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:b. Who has the most varied earnings? Explain.California Mathematics Grade 7 295

11–8Select an Appropriate DisplayStandard 7SDP1.1 Know various forms of display for data sets, including stemand-leafplot or box-and-whisker plot; use the forms to display a single set of dataor to compare two sets of data.EXAMPLES Choose an Appropriate DisplayMAIN IDEA• Choose an appropriatedisplay for a set ofdata.ORGANIZE ITUnder the tab forLesson 11–8, make atable of data from yourscience or social studiestextbook. Draw a circlegraph and bar graphdisplaying the data.Discuss which graph ismost appropriate.Choose an appropriate type of display for each situation.Then make a display.FARMS Select anappropriate displayto show the acreageof farms in Maine.Justify your answer.Farms in Maine by Size1–99 acres 46.8%100–499 acres 43.8%500–999 acres 6.9%1,000 or more acres 2.5%Source: ers.usda.govThis data deals with percents that have a sum of .Awould be a good way to show percents.SCHOOLS Select anappropriate displayto show students’favorite schoolsubjects. Justify yourreasoning. Thenconstruct the display.In this case, there are specificcategories. If you want to showthe specific number, use aor a .Favorite School SubjectmathhistoryscienceEnglishotherCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.296 California Mathematics Grade 7

11–8REMEMBER ITThere are manyways to display the samedata. However, often oneof those ways makes thedata easier to understandthan do the other ways.Check Your Progressa. Select an appropriate Favorite Type of Television Programdisplay to showsitcom 54%favorite types oftelevision programs. reality 22%Justify your answer. news 10%Then construct thegame show 8%display.cartoon 6%b. Select an appropriate display to show students’ favoritehobbies. Then construct the display.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:HobbyNumber of Studentsreading 10sports 5listening to music 10photography 7other 18California Mathematics Grade 7 297

C H A P T E R11BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 11 Foldableto help you study for yourchapter test.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 11, go to:glencoe.comBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 273–274) to help yousolve the puzzle.11-1Problem-Solving Investigation: Make a Table1. MONEY The list shows weeklyallowances for a group of 13- and14-year-olds. Organize the data in atable using intervals $2.01–$3.00,$3.01–$4.00, $4.01–$5.00, and so on.What is the most common intervalof allowance amounts?$2.50 $3.00 $3.75 $4.25 $4.25$4.50 $4.75 $4.75 $5.00 $5.00$5.00 $5.00 $5.50 $5.50 $5.75$5.80 $6.00 $6.00 $6.00 $6.50$6.75 $7.00 $8.50 $10.00 $10.00$12.00 $15.0011-2HistogramsUse the histogram at the right.2. How many months have less thantwo days of rain?3. How many months had between twoand seven days of rain? Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.298 California Mathematics Grade 7

Chapter 11 BRINGING IT ALL TOGETHER11-3Circle GraphsUse the circle graph at the right.4. What percent of her time does Luisaspend studying?5. How many degrees are in the sectionthat represents sports?11-4Measures of Central Tendency6. Name the three most common measures of central tendency.7. Which measure of central tendency best represents the data?Why? 9, 9, 20, 22, 25, 27Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.11-5Measures of VariationComplete.8. Measures of variation describe the of data.9. The of a set of data is the difference between thegreatest and the least numbers in the set.10. The range is the difference between theupper and lower quartiles.California Mathematics Grade 7 299

Chapter 11 BRINGING ALL TOGETHER11-6Box-and-Whisker Plots11. Draw a box-and-whisker plot forthe data. 1, 1, 1, 2, 3, 3, 4, 4, 511-7Stem-and-Leaf PlotsFOOTBALL For Exercises12–14, use the all-timeinterception leaders datashown at the right.12. What is the mostinterceptions by an NFLplayer through 2005?13. How many NFL playershave 57 interceptionsthrough 2005?All-Time NFL InterceptionLeaders (through 2005)Stem Leaf5 7 7 7 7 7 86 2 2 3 5 87 1 98 15 2 = 62 interceptions14. What is the median number of interceptions among the leadersrepresented in the stem-and-leaf plot?11-8Select an Appropriate DisplayChoose the letter that best matches the type of display toits use.15. Line Graph a. shows the frequency of data that hasbeen organized into equal intervals16. Bar Graph b. shows the number of items in specificcategories in the data using bars17. Histogram c. shows change over a period of timeCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.18. Line Plot d. shows how many times each numberoccurs in the data300 California Mathematics Grade 7

C H A P T E R11ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study are givenwith each item.I completed the review of all or most lessons without using mynotes or asking for help.Visit glencoe.com to accessyour textbook, moreexamples, self-check quizzes,and practice tests to helpyou study the concepts inChapter 11.• You are probably ready for the Chapter Test.• You may want to take the Chapter 11 Practice Test onpage 621 of your textbook as a final check.I used my Foldable or Study Notebook to complete the review ofall or most lessons.• You should complete the Chapter 11 Study Guide and Reviewon pages 616–620 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 11 Practice Test onpage 621.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in yourStudy Notebook and Chapter 11 Foldable.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.• Then complete the Chapter 11 Study Guide and Review onpages 616–620 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 11 Practice Test onpage 621.Student SignatureParent/Guardian SignatureTeacher SignatureCalifornia Mathematics Grade 7 301

C H A P T E R12ProbabilityUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with a plain sheet of 11" × 17" paper.Fold the sheet inhalf lengthwise.Cut along the fold.Fold each half inquarters alongthe width.Unfold each piece andtape to form onelong piece.Label each page witha key topic as shown.Refold to form a booklet.NOTE-TAKING TIP: It helps to take notes as youprogress through studying a subject. New conceptsoften build upon concepts you have just learned ina previous lesson. If you take notes as you go, youwill know what you need to know for the conceptyou are now learning.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.302 California Mathematics Grade 7

C H A P T E R12BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 12.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExamplebiased samplecompound eventconvenience sampleCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.dependent eventseventexperimentalprobabilityFundamental CountingPrincipleindependent eventsoutcome(continued on the next page)Chapter 12California Mathematics Grade 7 303

Chapter 12 BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExamplepopulationprobabilityrandomsamplesample spacesimple random samplestratified randomsamplesystematic randomsampletheoretical probabilitytree diagramunbiased sampleCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.voluntary responsesample304 California Mathematics Grade 7

12–1Counting OutcomesReinforcement of Standard 6SDP3.1 Represent all possible outcomes forcompound events in an organized way (e.g., tables, grids, tree diagrams) andexpress the theoretical probability of each outcome.MAIN IDEA• Count outcomes byusing a tree diagramor the FundamentalCounting Principle.BUILD YOUR VOCABULARY (pages 303–304)A tree diagram is a diagram used to show thenumber ofin a probabilityexperiment.The Fundamental Counting Principle usesof the number of ways each event in an experiment canoccur to find the number ofin asample space.EXAMPLE Use a Tree DiagramCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.WRITE ITHow is using a treediagram to find totalnumber of outcomes likeusing a factor tree to findprime factors? (see factortrees in Prerequisite Skillspage 664)BOOKS A flea market vendor sells new and used booksfor adults and teens. Today she has fantasy novels andpoetry collections to choose from. Draw a tree diagramto determine the number of categories of books. There are different categories. California Mathematics Grade 7 305

12–1KEY CONCEPTFundamental CountingPrinciple If event M anoccur in m ways and isfollowed by event N thatcan occur in n ways, thenthe event M followed bythe event N can occur inm · n ways.Check Your Progress A store has spring outfits on sale.You can choose either striped or solid pants. You can also choosegreen, pink, or orange shirts. Finally, you can choose eitherlong-sleeved shirts or short-sleeved shirts. Draw a tree diagramto determine the number of possible outfits.EXAMPLE Use the Fundamental Counting PrincipleRESTAURANTS A manager assigns different codes toall the tables in a restaurant to make it easier for thewait staff to identify them. Each code consists of thevowel A, E, I, O, or U, followed by two digits from 0through 9. How many codes could the managerassign using this method?number ofpossiblenumbersfor the firstplaceThere are×number ofpossiblenumbers forthe secondplace×number ofpossiblenumbers forthe thirdplace× × =possible codes.=number ofpossiblecodesCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.306 California Mathematics Grade 7

Check Your Progress A middle school assigns eachstudent a code to use for scheduling. Each code consists ofa letter, followed by two digits from 0 though 9. How manycodes are possible?12–1ORGANIZE ITUnder Tree Diagram andFundamental CountingPrinciple, write notes onwhat you learned aboutcounting outcomes byusing a tree diagram andby using the CountingPrinciple. Includeexamples of each.EXAMPLE Find ProbabilityCOMPUTERS What is the probability that Liana willguess her friend’s computer password on the first try ifall she knows is that it consists of three letters?Find the number of possible outcomes. Use the FundamentalCounting Principle.choicesfor the fistletter×choices forthe secondletter×choices forthe thirdletter=totalnumber ofoutcomesCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:× × =There arepossible outcomes. There iscorrect password. So, the probability of guessing on the firsttry is .Check Your Progress What is the probability that Shaunawill guess her friend’s locker combination on the first try if allshe knows is that it consists of three digits from 0 through 9?California Mathematics Grade 7 307

12–2Probability of Compound ExperimentsReinforcement of Standard 6SDP3.1 Represent all possible outcomes for compoundevents in an organized way (e.g., tables, grids, tree diagrams) and express thetheoretical probability of each outcome.MAIN IDEA• Find the probabilityof independent anddependent events.BUILD YOUR VOCABULARY (pages 303–304)A compound event consists ofsimpleevents.Independent events areevents inwhich the outcome of one eventoutcome of the other events.affect theKEY CONCEPTProbability of TwoIndependent EventsThe probability of twoindependent events canbe found by multiplyingthe probability ofthe first event by theprobability of thesecond event.EXAMPLE Probability of Independent EventsThe two spinners below are spun. What is theprobability that both spinners will show a numbergreater than 6?19 0 2 9 0 1 28 3 8 37 47 46 56 5P(first spinner is greater than 6) =P(second spinner is greater than 6) =P(both spinners are greater than 6) = 3 _10 · 3 _10 orCheck Your Progress The two spinners below are spun.What is the probability that both spinners will show a numberless than 4?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.308 California Mathematics Grade 7

12–2EXAMPLESTANDARDS EXAMPLE A red number cube and a whitenumber cube are rolled. The faces of both cubes arenumbered from 1 to 6. What is the probability of rollinga 3 on the red number cube and rolling the number 3 orless on the white number cube?A 1_2B 1_6C 1_9D 1_12Read the Test ItemYou are asked to find the probability of rolling a 3 on the rednumber cube and rolling a number 3 or less on the whitenumber cube. The events areone number cubebecause rollingaffect rolling the other cube.Solve the Test ItemFirst, find the probability of each event.P(rolling a 3 on the red number cube) =P(rolling 3 or less on the white number cube) =Then, find the probability of both events occurring.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.KEY CONCEPTProbability of TwoDependent EventsIf two events, A andB, are dependent,then the probability ofboth events occurringis the product of theprobability of A and theprobability of B after Aoccurs.P(3 red and 3 or less white) = · P(A and B)= P(A) · P(B)The probability is , which is .= Multiply.Check Your Progress STANDARDS EXAMPLE A whitenumber cube and a green number cube are rolled. The faces ofboth cubes are numbered from 1 to 6. What is the probability ofrolling an even number on the white number cube and rolling a3 or a 5 on the green number cube?F 1_12G 1_6H 1_3J 1_2California Mathematics Grade 7 309

12–2BUILD YOUR VOCABULARY (pages 303–304)If the outcome of one event doesanother event, the compound events are calleddependent events.the outcome ofORGANIZE ITUnder IndependentEvents and DependentEvents, write what youlearned about how tofind the probabilityof independent anddependent events.EXAMPLE Probability of Dependent EventsThere are 4 red, 8 yellow, and 6 blue socks mixed up in adrawer. Once a sock is selected, it is not replaced. Findthe probability of reaching into the drawer withoutlooking and choosing 2 blue socks.Since the first sockreplaced, the first event affectsthe second event. These are dependent events.P(first sock is blue) =number of blue sockstotal number of socksHOMEWORKASSIGNMENTPage(s):Exercises:P(second sock is blue) =P(two blue socks) =ornumber of blue socksafter one blue sock isremovedtotal number of socksafter one blue sock isremovedCheck Your Progress There are 6 green, 9 purple,and 3 orange marbles in a bag. Once a marble is selected,it is not replaced. Find the probability that two purplemarbles are chosen.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.310 California Mathematics Grade 7

12–3Experimental and Theoretical ProbabilityReinforcement of Standard 6SDP3.2 Use data to estimate the probability of futureevents (e.g., batting averages or number of accidents per mile driven).MAIN IDEA• Find experimentalprobability.BUILD YOUR VOCABULARY (pages 303–304)A probability that is based onobtainedby conducting anis called anexperimental probability.A probabililty that is based onis called a theoretical probability.EXAMPLES Experimental ProbabilityNikki is conducting an experiment to find theprobability of getting various results when threecoins are tossed. The results of her experiment aregiven in the table.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ResultNumber ofTossesall heads 6two heads 32one head 30no heads 12What is the theoretical probability of tossing all headson the next turn?The theoretical probability is = .According to the experimental probability, is Nikki morelikely to get all heads or no heads on the next toss?Based on the results so far,heads is more likely.California Mathematics Grade 7 311

12–3Check Your ProgressMarcus is conducting anexperiment to find theprobability of gettingvarious results when fourcoins are tossed. Theresults of his experimentare given in the table.ResultNumber ofTossesall heads 6three heads 12two heads 20one head 7no heads 5a. What is the theoretical probability of tossing all tails on thenext turn?b. According to the experiment probability, is Marcus morelikely to get all heads or no heads on the next toss?ORGANIZE ITUnder ExperimentalProbability, write a fewwords to compare andcontrast experimentaland theoreticalprobabilities.EXAMPLE Experimental ProbabilityMARKETING Eight hundred adults were asked whetherthey were planning to stay home for winter vacation.Of those surveyed, 560 said that they were. What is theexperimental probability that an adult planned to stayhome for winter vacation?There were people surveyed and said that theywere staying home.The experimental probability is or .Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.312 California Mathematics Grade 7

12–3Check Your Progress Five hundred adults were askedwhether they were planning to stay home for New Year’sEve. Of those surveyed, 300 said that they were. What is theexperimental probability that an adult planned to stay homefor New Year’s Eve?EXAMPLE Use Probability to PredictCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.REVIEW ITExplain what aproportion is andhow you can solve aproportion. (Lesson 4-3)MATH TEAM Over the past three years, the probabilitythat the school math team would win a meet is _ 3 5 .Is this probability experimental or theoretical? Explain.This is an experimental probability since it is based on whathappened in the .If the team wants to win 12 more meets in the next3 years, how many meets should the team enter?This problem can be solved using a proportion.3 out of 5 meetswere winsSolve the proportion.3_5 = 12_x3_5 × 12_xWrite the proportion.= Find the cross products.= Multiply.= Divide each side by .x =12 out of x meetsshould be wins.They should entermeets.California Mathematics Grade 7 313

12–3Check Your Progress Over the past three years, theprobability that the school speech and debate teamwould win a meet is _ 4 5 .a. Is this probability experimental or theoretical? Explain.b. If the team wants to win 20 more meets in the next 3 years,how many meets should the team enter?HOMEWORKASSIGNMENTPage(s):Exercises:Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.314 California Mathematics Grade 7

12–4 Problem-Solving Investigation:Act It OutEXAMPLE Act it OutMAIN IDEA• Solve problems byacting them out.Melvin paid for a $5 sandwich with a $20 bill. Thecashier has $1, $5, and $10 bills in the register. Howmany different ways can Melvin get his change?EXPLORE You know that Melvin should receive $20 – $5 orStandard 7MR2.5Use a variety ofmethods, such aswords, numbers, symbols,charts, graphs, tables,diagrams, and models,to explain mathematicalreasoning.Reinforcement ofStandard 6SDP3.2 Usedata to estimate theprobability of futureevents (e.g., battingaverages or number ofaccidents per mile driven).PLANSOLVEin change. You need to determine how manydifferent ways the cashier can make $15 in changewith $1, $5, and $10 bills.Use manipulatives such as play money to act outthe problem. Record the different ways the cashiercan make $15 in change.$1 $5 $10Method 1 1 1Method 2 1Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.HOMEWORKASSIGNMENTPage(s):Exercises:Method 3Method 4Method 5Method 6The cashier can make the change inways.differentCHECK Make sure each method adds up to in change.Check Your Progress SHOPPING Amanda paid for an $8CD with a $20 bill. The cashier has $1, $5, and $10 bills in theregister. How many different ways can Amanda get her change?California Mathematics Grade 7 315

12–5Using Sampling to PredictReinforcement of Standard 6SDP2.5 Identify claims based on statistical data and,in simple cases, evaluate the validity of the claims.MAIN IDEA• Predict the actions of alarger group by using asample.BUILD YOUR VOCABULARY (pages 303–304)A sample is aselected group chosen forthe purpose of collecting data.The population is thefrom which thesamples under consideration are taken.An unbiased sample is selected so that it isof the entire population.In a stratified random sample, the population is dividedinto, nonoverlapping groups.In a systematic random sample, the items or people areselected according to a specificor item interval.In a biased sample, one or more parts of the population areover others.EXAMPLES Determine Validity of ConclusionsDetermine whether each conclusion is valid. Justifyyour answer.To determine which school lunches students like most,the cafeteria staff surveyed every tenth student whowalk into the cafeteria. Out of 40 students surveyed,19 students stated that they liked the burgers best.The cafeteria staff concludes that about 50% of thestudents like burgers best.The conclusion isstudents of the school, the sample is a. Since the population is theCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.. It is .316 California Mathematics Grade 7

12–5To determine what sports teenagers like, Janetsurveyed the student athletes on the girls’ field hockeyteam. Of these, 65% said that they like field hockey best.Janet concluded that over half of teenagers like fieldhockey best.The conclusion is. The students surveyedprobably prefer field hockey. This is .The sample isare easily accessed.because the peopleCheck Your Progress Determine whether eachconclusion is valid. Justify your answer.a. To determine what ride is most popular, every tenth personto walk through the gates of a theme park is surveyed. Outof 290 customers, 98 stated that they prefer The Zip. Thepark manager concludes that about a third of the park’scustomers prefer The Zip.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.b. To determine whether people prefer dogs or cats, aresearcher surveys 80 people at a dog park. Of thosesurveyed, 88% said that they prefer dogs, so the researcherconcludes that most people pet owners prefer dogs.EXAMPLE Using Sampling to PredictBOOKS The student council istrying to decide what types ofbooks to sell at its annual bookfair to help raise money for theeighth-grade trip. It surveys 40students at random. The booksthey prefer are in the table.If 220 books are to be sold at thebook fair, how many shouldbe mysteries?Book TypeNumber ofStudentsmystery 12adventurenovel9sports 11shortstories8California Mathematics Grade 7 317

12–5First, determine whether the sample method is valid. Thesample issince the studentswere randomly selected. Thus, the sample ._ 12or of the students prefer mysteries. So, find400.30 × =.Aboutbooks should be mysteries.ORGANIZE ITUnder Sampling, listthe different typesof samples and howto use them to makepredictions. Giveexamples.Check Your Progress The student shop sells pens. Itsurveys 50 students at random. The pens they prefer are in thetable. If 300 pens are to be sold at the student shop, how manyshould be gel pens?TypeNumbergel pens 22ball point 8HOMEWORKASSIGNMENTPage(s):Exercises:glitter 10roller balls 10Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.318 California Mathematics Grade 7

C H A P T E R12BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 12 Foldableto help you study for yourchapter test.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 12, go to:glencoe.comBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 197–198) to help yousolve the puzzle.12-1Counting Outcomes1. Complete the tree diagram shown below for how many boys andand how many girls are likely to be in a family of three children.Child 1 Child 2 Child 3 Sample OutcomeBBBBBBBBGCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.GGGG2. Use the Fundamental Counting Principle to find the numberof possible outcomes if there are 4 true-false questions on a test.× × × =California Mathematics Grade 7 319

Chapter 12 BRINGING IT ALL TOGETHER12-2Probability of Compound Experiments3. What is a compound event?4. Are the events of spinning a spinner and rolling a number cubeindependent events? Why or why not?A number cube is rolled and a penny is tossed. Find eachprobability.5. P(4 and tails) 6. P(3 or less, heads)12-3Experimental and Theoretical ProbabilityThe table at the right shows the results of a survey.7. How many people bought balloons?8. How many people were surveyed?9. What is the experimental probability thata person surveyed preferred balloons?Item10. A bag contains 15 red marbles, 25 purple marbles, and 10 yellowmarbles. Describe an experiment that you could conduct with themarbles to find an experimental probability.Number ofPeopleballoons 75cards 15decorations 25cake 50Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.320 California Mathematics Grade 7

Chapter 12 BRINGING IT ALL TOGETHER12-4Problem-Solving Investigation: Act It Out11. SPORTS There are 32 tennis players in a tournament. If eachlosing player is eliminated from the tournament, how manytennis matches will be played during the tournament?12-5Using Sampling to Predict12. If you conduct a survey by asking ten students selected at randomfrom each grade at your school what their favorite class is, whattype of random sample have you taken?13. A grocery store owner asks the shoppers in his store wherethey prefer to shop for groceries. What type of sample has heconducted?Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.California Mathematics Grade 7 321

C H A P T E R12ChecklistARE YOU READY FOR THECHAPTER TEST?Check the one that applies. Suggestions to help you study are givenwith each item.I completed the review of all or most lessons without using mynotes or asking for help.Visit glencoe.com to accessyour textbook, moreexamples, self-check quizzes,and practice tests to helpyou study the concepts inChapter 12.• You are probably ready for the Chapter Test.• You may want to take the Chapter 12 Practice Test onpage 657 of your textbook as a final check.I used my Foldable or Study Notebook to complete the review ofall or most lessons.• You should complete the Chapter 12 Study Guide and Review onpages 653–656 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 12 Practice Test onpage 657.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 12 Foldable.• Then complete the Chapter 12 Study Guide and Review onpages 653–656 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 12 Practice Test onpage 657.Student SignatureParent/Guardian SignatureCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Teacher Signature322 California Mathematics Grade 7