transitioning to the common core - connected...

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Transitioning to the

Common CoreState Standards

Lappan, Fey, Fitzgerald, Friel, Phillips

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Grade Eight

Thinking With Mathematical ModelsLinear and Inverse Variation

Looking for PythagorasThe Pythagorean Theorem

Growing, Growing,GrowingExponential Relationships

Frogs, Fleas, and Painted CubesQuadratic Relationships

Kaleidoscopes, Hubcaps,and MirrorsSymmetry and Transformations

Say It With SymbolsMaking Sense of Symbols

The Shapes of AlgebraLinear Systems and Inequalities

Samples and PopulationsData and Statistics

Grade Eight Units

Classroom tested, proven effective!Before work began on CMP2, mathematics teachers in more than18 school districts—that's over 80 teachers—reviewed ConnectedMathematics. More than 100 classroom teachers tried outConnected Mathematics 2 at 49 schools all across the country. This classroom testing allowed the authors to carefully study andrevise the program to make sure the materials help math studentslike you every day, in every classroom.

Lappan, Fey, Fitzgerald, Friel, Phillips

GRADE EIGHT

Grade Eight

Thinking With Mathematical ModelsLinear and Inverse Variation

Looking for PythagorasThe Pythagorean Theorem

Growing, Growing,GrowingExponential Relationships

Frogs, Fleas, and Painted CubesQuadratic Relationships

Kaleidoscopes, Hubcaps,and MirrorsSymmetry and Transformations

Say It With SymbolsMaking Sense of Symbols

The Shapes of AlgebraLinear Systems and Inequalities

Samples and PopulationsData and Statistics

Grade Eight Units

Classroom tested, proven effective!Before work began on CMP2, mathematics teachers in more than18 school districts—that's over 80 teachers—reviewed ConnectedMathematics. More than 100 classroom teachers tried outConnected Mathematics 2 at 49 schools all across the country. This classroom testing allowed the authors to carefully study andrevise the program to make sure the materials help math studentslike you every day, in every classroom.

Overview of Common Core State Standards for Mathematics . . . . . . . . . . . . . . ii

The Standards of Mathematical Practices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Common Core State Standards Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

CMP2 Pacing Guide for CCSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Common Core Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1CC-1 Exponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1CC-2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7CC-3 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15CC-4 Geometry Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25CC-5 Categorical Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Teaching Notes and Resources for Common Core Investigations . . . . . . . . . . . 41CC-1 Exponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41CC-2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51CC-3 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61CC-4 Geometry Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71CC-5 Categorical Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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ii Common Core State Standards for Mathematics

The Common Core State Standards for Mathematics represent the collaborative efforts ofmathematicians, researchers, educators, and state education officials from 48 states and the District ofColumbia to develop a single set of rigorous and internationally-benchmarked standards for K-12mathematics. These standards set clear expectations for what students are to know and be able to doat each grade level from Kindergarten through High School.

Two guiding principles framed the development of these standards for the writing group: that they bebased on evidence and research and build on the strengths of current state standards and that at eachgrade level, the standards be fewer in number, but clearer and more rigorous. The reduced number ofstandards at each grade level challenged the writing team to articulate a progression of concepts andfrom that progression, to identify critical areas for each grade level. The Overview that followspresents the main areas of emphasis in Grade 8, new concepts or approaches that these standardsadvance, and the progression of the concepts through Grade 8.

Main Areas of Emphasis in Grade 8

The three main areas of emphasis in Grade 8 are:

• expressions and equations, including systems oflinear equations

• functions to describe quantitative relationships

• analysis of two- and three-dimensional spaceand figures and the Pythagorean Theorem

In Grade 8, students engage in formal study ofalgebraic expressions and linear equations. Theywrite and use linear equations, linear functions, andsystems of linear equations to represent, analyze, andsolve problems. Students recognize that linearequations in the form y = mx + b representproportional relationships, and can explain slope as aconstant of proportionality. Students use a linearmodel to describe the relationship between twoquantities in a bivariate data set (e.g., arm span toheight) and informally describe the fit of the modelto the data.

Students choose efficient and generalizablemethods to solve linear equations in one variable,applying properties of operations and of equality.

They solve systems of two linear equations in twovariables and relate the systems to pairs of lines inthe plane.

Students begin their study of functions in Grade 8.They explain the concept and uses of functions anddescribe functions in different representations(graphic, tabular, algebraic). Further, they analyzefunctions to describe how elements of the functionare shown in each representation.

Students’ study of geometry formalizes concepts ofcongruence and similarity through the exploration oftransformations. They explore the behavior of shapesunder translations, rotations, reflections, anddilations, and relate these behaviors to concepts ofcongruence and similarity. Students explore anglerelationships in shapes (triangles) and on parallellines cut by a transversal. Students are able toexplain the Pythagorean Theorem and its converse,and why it holds. They apply the PythagoreanTheorem to solve problems. Students complete theirstudy of volume by finding the volumes of cones,cylinders, and spheres.

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Common Core State Standards for Mathematics iii

New Approaches to Content in Grade 8

The Common Core State Standards consistentlypromote a more conceptual and analytical approachto mathematics instruction. In the elementaryyears, the focus is on interpreting operations to helpstudents make sense of operations conceptually andto apply this conceptual understanding in differentcontexts with different forms of numbers. In themiddle school years, the focus is on strengtheningstudents’ multiplicative reasoning skills and onbuilding a solid foundation of algebraic conceptsand skills.

The development of algebraic concepts and skillsgrows from students’ understanding of arithmeticoperations. Students consistently apply theirknowledge of place value, properties of operations,and the inverse relationships between operations(addition and subtraction; multiplication anddivision) to write and solve algebraic equations ofvarying complexities. Students manipulate parts ofthe expression and explore the meaning of theexpression when it is rewritten in different forms.This analytic focus helps students to look more fullyat the equations and expressions so that they begin

to see patterns in the structure. These patterns willbe useful when students explore more complexalgebraic concepts.

In Grade 8, students synthesize their knowledge ofoperations, proportional relationships, and algebraicequations to deepen their understanding of linearityand linear equations. They explain straight lines,model linear relationships and represent slopealgebraically, numerically in tables, and graphically.Grade 8 students work with these linear relationshipsin their study of statistics. They interpret scatter plotsand two-way tables and use linear models to interpretslope and intercept.

Grade 8 students begin a study of congruence andsimilarity based on transformations. They usephysical models or geometry software to explore thebehaviors of shapes when transformed. They explaincongruence and similarity based on transformations.They also study the Pythagorean Theorem. Theyexplain a proof of the theorem and its converse.From that understanding, they apply the theorem tosolve real-world and mathematical problems.

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iv Common Core State Standards for Mathematics

Beginning with counting and cardinality inKindergarten, students repeatedly extend theirconception of number and number systems. InGrades 6 and 7, students study rational numbers; inGrade 8, students augment the rational numberswith the irrational numbers to form the realnumbers. In high school, students augment the realnumbers with the imaginary numbers to form thecomplex numbers.

Grade 6 students evaluated algebraic expressionsand manipulated the parts of the expressions. InGrade 7, they extended their analysis of expressionsto explain the relationships among the quantitiesthat are revealed when an expression is rewritten.They expanded their understanding of expressions tosolving multi-step algebraic equations of inequalities,solving inequalities and graphing the solutions onnumber lines. In Grade 8, students solve linearequations with more than two steps as well as pairsof linear equations. They draw from theirunderstanding of proportional relationships andtheir exploration of unit rate as a measure ofsteepness to define more formally the slope of linearequations. They solve linear equations in onevariable and analyze and solve pairs of simultaneouslinear equations. In Algebra I, students extend theirstudy to include nonlinear equations, formulas,matrices, and graphical solutions of inequalities.

Students are introduced to the study of functions inGrade 8. Students explore functions presented indifferent ways (algebraically, numerically in tables,and graphically) and compare the properties ofdifferent functions presented in different ways. Theydescribe the functional relationship between twoquantities and represent the relationship graphically.

In statistics, the focus of study in Grade 8 is onbivariate data. Students investigate patterns withbivariate data. They look at graphical representationsof the data (scatter plots) to describe the patternshown (if any). They use the equation of a linearmodel to solve problems related to bivariate data,interpreting the slope and intercept.

One focus of study in geometry is on congruenceand similarity through transformations. Studentsverify the properties of transformations (reflections,translations, rotations, dilations) and describe theeffect of each on two-dimensional shapes usingcoordinates. Students explain that shapes arecongruent or similar based on a sequence oftransformations, and conversely they describe aseries of transformations that will illustrate thattwo shapes are congruent.

A second focus in geometry is the PythagoreanTheorem. Students explain the theorem and itsconverse, and apply the theorem to solve real-worldand mathematical problems. In addition, studentscontinue their work with angles (angles createdwhen parallel lines are cut by a transversal) andvolume of three-dimensional solids (cones, cylinders,and spheres).

Work in geometry and measurement throughGrade 8 becomes more formalized in a geometrycourse in which students prove properties studiedand give informal arguments to verify variousmeasurement formulas learned in earlier grades.Trigonometry and other work with similar figures inthe geometry classroom are built on eighth-gradework with angles, triangles, and the Pythagoreantheorem.

Important Progressions across Grades for the Content at Grade 8

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Common Core State Standards for Mathematics v

The Common Core State Standards identify alimited number of topics at each grade level,allowing enough time for students to achieve fluency,if not mastery of these concepts. The subsequent yearof study builds on the concepts of the previous year.While some review of topics from earlier grades isappropriate and encouraged, the CCSS writers arguethat re-teaching of these topics should not be needed.

The Common Core State Standards for Grade 8 arealgebra-focused. Students work with linearequations, functions and applications involvingfunctions, and geometric concepts that include thePythagorean theorem and transformations. At the

end of Grade 8, students are well prepared for highschool algebra course.

Certain topics that have often been part of theGrade 8 curriculum are not included in the CCSS. Ingeneral, these topics are not included becausestudents are expected to have achieved fluency withthem in earlier grades. These topics include rationalnumbers and operations with rational numbers;ratios, rates, and proportional relationships;properties and attributes of two- and three-dimensional shapes; and measures of center andvariability. The study of probability concepts in K-8is limited to Grade 7.

What’s Different?

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vi CCSS Mathematical Practices

The Common Core State Standards (CCSS) articulate Standards of Mathematical Practices thathave been central to the development of the Connected Mathematics Project (CMP) materials fromtheir inception. CMP focuses on developing mathematical situations that give students opportunitiesto develop mathematical proficiency through the development of these mathematical practices asthey work through the problem situations in the CMP2 program.

The following highlights the opportunities these materials create to make the Mathematical Practicesa reality for students. It explains how CMP supports the development of mathematical proficiency instudents, citing some examples of how each Standard for Mathematical Practices is embedded in theCMP materials.

1. Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking forentry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjecturesabout the form and meaning of the solution and plan a solution pathway rather than simply jumping into asolution attempt. They consider analogous problems, and try special cases and simpler forms of the originalproblem in order to gain insight into its solution. They monitor and evaluate their progress and change courseif necessary. Older students might, depending on the context of the problem, transform algebraic expressionsor change the viewing window on their graphing calculator to get the information they need. Mathematicallyproficient students can explain correspondences between equations, verbal descriptions, tables, and graphs ordraw diagrams of important features and relationships, graph data, and search for regularity or trends. Youngerstudents might rely on using concrete objects or pictures to help conceptualize and solve a problem.Mathematically proficient students check their answers to problems using a different method, and theycontinually ask themselves, “Does this make sense?” They can understand the approaches of others to solvingcomplex problems and identify correspondences between different approaches.

This goal is fundamental to the CMP approach. CMPis a problem-centered curriculum. To be effective,problems must not just embody critical concepts andskills, but must also have the potential to engagestudents in making sense of problem situations andmathematics.

A growing body of evidence from the cognitivesciences suggests that students make sense ofmathematics when concepts and skills are embeddedwithin a context or problem. This research is one ofthe cornerstones for developing the problemsituations in the CMP2 program. These student-centered problem situations engage students inarticulating the knowns in a problem situation anddetermining a logical solution pathway. The student-student and student-teacher dialogues, anotherhallmark of the program, help students to not justmake sense of the problems, but also persevere in

finding appropriate strategies to solve them. Thesuggested questions in the Teacher Guides canprovide the metacognitive scaffolding to helpstudents monitor and refine their problem-solvingstrategies.

The Applications, Connections, and Extensionsproblems assure that all students are givenopportunities to make sense of a new problemsituation and persevere in finding the appropriatesolution.

Throughout program; for examples see: Looking ForPythagoras (Inv. 2); Say It With Symbols (Inv. 3);Kaleidoscopes, Hubcaps, and Mirrors (Inv. 3, 4);Thinking With Mathematical Models (Inv. 1, 2);Growing, Growing, Growing (Inv. 4); The Shapes ofAlgebra (Inv. 4); Samples and Populations (Inv. 1);Frogs, Fleas, and Painted Cubes (Inv. 1)

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CCSS Mathematical Practices vii

2. Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations.They bring two complementary abilities to bear on problems involving quantitative relationships: the ability todecontextualize—to abstract a given situation and represent it symbolically and manipulate the representingsymbols as if they have a life of their own, without necessarily attending to their referents—and the ability tocontextualize, to pause as needed during the manipulation process in order to probe into the referents for thesymbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problemat hand; considering the units involved; attending to the meaning of quantities, not just how to compute them;and knowing and flexibly using different properties of operations and objects.

In the CMP classroom, students are supported intheir acquisition of mathematical language andmathematical ways of reasoning, both of which areunderpinnings of abstract and quantitativereasoning. The problem situations in CMP2 aredesigned to support the development of students’mathematical reasoning abilities. As studentsexplore a set of connected problems within aninvestigation, they look to understand the quantitiesin the problem and the relationship among thesequantities. Students are frequently expected totranslate a problem situation in an expression orequation and to then manipulate the equation to finda solution, or in other words, to “decontextualize.”

Conversely, throughout the problem-solving process,students are encouraged to translate from anequation or expression back to the problem situationto verify that the equation accurately represents thesituation, with particular attention to the units andquantities of the problems.

Throughout program; for examples see: LookingFor Pythagoras (Inv. 3); Say It With Symbols (Inv. 5);Kaleidoscopes, Hubcaps, and Mirrors (Inv. 3, 4);Thinking With Mathematical Models (Inv. 1);Growing, Growing, Growing (Inv. 2); The Shapesof Algebra (Inv. 5); Samples and Populations(Inv. 2); Frogs, Fleas, and Painted Cubes (Inv. 4)

3. Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previouslyestablished results in constructing arguments. They make conjectures and build a logical progression ofstatements to explore the truth of their conjectures. They are able to analyze situations by breaking them intocases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others,and respond to the arguments of others. They reason inductively about data, making plausible arguments thattake into account the context from which the data arose. Mathematically proficient students are also able tocompare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which isflawed, and—if there is a flaw in an argument—explain what it is. Elementary students can constructarguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can makesense and be correct, even though they are not generalized or made formal until later grades. Later, studentslearn to determine domains to which an argument applies. Students at all grades can listen or read thearguments of others, decide whether they make sense, and ask useful questions to clarify or improve thearguments.

Reasoning and justification are central to theprogram at all levels. In the CMP classroom, studentsroutinely participate in mathematics discourse asthey explain their thinking about a problem situationand their reasoning for a solution pathway. Theproblems that students encounter in the programoffer opportunities to construct mathematicalarguments and to critique other students’ solutionsand strategies.

Students are called on to defend their solutions andtheir strategies for solving the problems, and toexplain their reasoning that led to the solution they

put forth. The Teacher Guides offer questions thatsupport the development of a classroom culture thatfocuses on argument and critique as a part of solvingmathematical problems.

Throughout program; for examples see: LookingFor Pythagoras (Inv. 2 p. 30); Say It With Symbols(Inv. 4 p. 71); Kaleidoscopes, Hubcaps, and Mirrors(Inv. 2, p. 47); Thinking With Mathematical Models(Inv. 1); Growing, Growing, Growing (Inv. 5 p. 58);The Shapes of Algebra (Inv. 3); Samples andPopulations (Inv. 2 p. 46); Frogs, Fleas, and PaintedCubes (Inv. 3, 4)

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viii CCSS Mathematical Practices

Mathematical modeling begins in Grade 6, andcontinues to grow in sophistication throughout theprogram. CMP engages students in learning toconstruct, make inferences from, and interpretconcrete, symbolic, graphic, verbal, and algorithmicmodels of mathematical relationships in problemsituations. Students are also asked to translateinformation from one model to another.

Throughout the program, students apply conceptsand skills to particular problem situations providedas well as to problem situations of their own creationor design. The regular and dynamic application ofmathematical concepts to solve real-world problemsis yet another hallmark of the Connected

Mathematics Project. Students are well-equipped torespond to these challenges as they regularlyconstruct and analyze a range of visual, graphicaland algebraic models. As students develop fluencywith these models, they realize the applicability ofthese models in different problem situations.

Throughout program; for examples see Looking ForPythagoras (Inv. 4); Say It With Symbols (Inv. 1);Kaleidoscopes, Hubcaps, and Mirrors (Inv. 1, 3);Thinking With Mathematical Models (Inv. 1, 3);Growing, Growing, Growing (Inv. 2); The Shapes ofAlgebra (Inv. 1); Samples and Populations (Inv. 3);Frogs, Fleas, and Painted Cubes (Inv. 4)

4. Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising ineveryday life, society, and the workplace. In early grades, this might be as simple as writing an additionequation to describe a situation. In middle grades, a student might apply proportional reasoning to plan aschool event or analyze a problem in the community. By high school, a student might use geometry to solve adesign problem or use a function to describe how one quantity of interest depends on another. Mathematicallyproficient students who can apply what they know are comfortable making assumptions and approximationsto simplify a complicated situation, realizing that these may need revision later. They are able to identifyimportant quantities in a practical situation and map their relationships using such tools as diagrams, two-waytables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to drawconclusions. They routinely interpret their mathematical results in the context of the situation and reflect onwhether the results make sense, possibly improving the model if it has not served its purpose.

5. Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. Thesetools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, acomputer algebra system, a statistical package, or dynamic geometry software. Proficient students aresufficiently familiar with tools appropriate for their grade or course to make sound decisions about when eachof these tools might be helpful, recognizing both the insight to be gained and their limitations. For example,mathematically proficient high school students analyze graphs of functions and solutions generated using agraphing calculator. They detect possible errors by strategically using estimation and other mathematicalknowledge. When making mathematical models, they know that technology can enable them to visualize theresults of varying assumptions, explore consequences, and compare predictions with data. Mathematicallyproficient students at various grade levels are able to identify relevant external mathematical resources, suchas digital content located on a website, and use them to pose or solve problems. They are able to usetechnological tools to explore and deepen their understanding of concepts.

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CCSS Mathematical Practices ix

In the CMP program, students work with a small setof tools as the primary vehicles for exploringproblem situations. Once students gain familiaritywith these tools, they make decisions about whichtools are most appropriate for a given problemsituation. Students can describe uses of differenttools; for example, they realize that calculators canbe used to compute, to check their thinking, toexplore possibilities, to see whether an approachmakes sense, and to use the graphing capability toexamine functions to see how they behave. Studentsbecome facile with graphing tools as a way “see intoa problem situation” and to find solutions to problems.

Students come to recognize that tools such aspolystrips can be used to explore concepts such asthe rigidity of triangle forms and the lack of rigidityof square forms and plastic two-dimensional shapesto explore the question of what shape has thegreatest area when built from a given number ofsquares.

Throughout program; for examples see: ThinkingWith Mathematical Models (Inv. 1, 2, 3); The Shapesof Algebra (Inv. 2, 3); Growing, Growing, Growing(Inv. 1, 5); Say It With Symbols (Unit Projectpp. 85–86); Looking For Pythagoras (Inv. 1, 3)

6. Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions indiscussion with others and in their own reasoning. They state the meaning of the symbols they choose,including using the equal sign consistently and appropriately. They are careful about specifying units ofmeasure, and labeling axes to clarify the correspondence with quantities in a problem. They calculateaccurately and efficiently, express numerical answers with a degree of precision appropriate for the problemcontext. In the elementary grades, students give carefully formulated explanations to each other. By the timethey reach high school they have learned to examine claims and make explicit use of definitions.

A key goal of CMP is helping students to learn to“talk” mathematics using precise terms anddefinitions, arguing that the clarity of a student’sthinking is reflected in the student’s use of precisemathematical language. The key mathematical goalsin each unit identify the important mathematicalterms, definitions, and ways of thinking andreasoning.

Student books include mathematical definitions thatare student-friendly while being mathematicallyaccurate. The goal of presenting definitions instudent-friendly language is to develop students’facility in talking mathematics at an appropriatelevel of mathematical maturity.

In addition to supporting the development of preciseuse of mathematical language, CMP supports

students in developing precision in theirpresentation of arguments. The series of questions ina problem push students to articulate more clearlytheir solutions and the processes by which they havereached these solutions.

A regular feature of the CMP student materials,the Mathematical Reflections (MR) pages thatoccur at the end of each investigation, also helpsstudents develop precision in their thinking andcommunicating of mathematical ideas. The MRpages consist of a set of questions that help studentssynthesize and organize their understandings ofimportant concepts and strategies.

Throughout program; for examples see: Samples andPopulations (Inv. 2, 3); Looking For Pythagoras (Inv. 2,4); Kaleidoscopes, Hubcaps, and Mirrors (Inv. 3)

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x CCSS Mathematical Practices

7. Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example,might notice that three and seven more is the same amount as seven and three more, or they may sort acollection of shapes according to how many sides the shapes have. Later, students will see 7 � 8 equals thewell remembered 7 � 5 � 7 � 3, in preparation for learning about the distributive property. In the expressionx2 + 9x + 14, older students can see the 14 as 2 � 7 and the 9 as 2 � 7. They recognize the significance of anexisting line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems.They also can step back for an overview and shift perspective. They can see complicated things, such as somealgebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 � 3(x � y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be morethan 5 for any real numbers x and y.

The CMP materials were designed to help studentsbuild mathematical understandings in ways thatilluminate and make use of mathematical structure.In Grade 6, for example, students look for patterns inthe data presented in data tables, and they analyzenumbers to determine their prime structure.

As Grade 7 students examine proportional reasoningsituations of various kinds, they begin to see patternsin proportions from which they draw generalizationsabout strategies for solving proportions. Theyexamine the structure of algebraic expressions tounderstand patterns in algebraic operations.

Grade 8 students examine graphical representationsof linear, exponential, and quadratic relationships sothey can begin to see the structure of theserelationships and functions. Although it is unusual

for students to examine quadratic and exponentialfunctions in middle school, the benefit is that theybegin to see the attributes of these differentrelationships, the structure of functions, in particular,what is revealed about the function through itsstructure.

In all grades, students see structure in measurement.They examine formulas, create algorithms forcomputation with rational numbers, and comparealgorithms for scope of use and efficiency.

Throughout program; for examples see: Looking ForPythagoras (Inv. 3); Say It With Symbols (Inv. 1);Kaleidoscopes, Hubcaps, and Mirrors (Inv. 1, 2, 3, 4,5); Growing, Growing, Growing (Inv. 1, 2, 3, 4, 5);Samples and Populations (Inv. 1, 2, 3, 4); Frogs, Fleas,and Painted Cubes (Inv. 1, 3, 4)

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8. Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods andfor shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the samecalculations over and over again, and conclude they have a repeating decimal. By paying attention to thecalculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middleschool students might abstract the equation (y � 2)/(x � 1) � 3. Noticing the regularity in the way termscancel when expanding (x � 1)(x � 1), (x � 1)(x2 � x � 1), and (x � 1)(x3 � x2 � x � 1) might lead them tothe general formula for the sum of a geometric series. As they work to solve a problem, mathematicallyproficient students maintain oversight of the process, while attending to the details. They continually evaluatethe reasonableness of their intermediate results.

One of the guiding principles of the CMP2curriculum is helping students see regularity inmathematics. As students investigate problems ateach grade level, they are encouraged to look forconnections to previous problems and previoussolution strategies. Students are aided in seeingopportunities to use strategies previously used tosolve a problem in order to solve a new problemthat looks on the surface to be very different. Thiskind of thinking and reasoning about solvingproblems promotes a view of mathematics asconnected in many different ways, rather than asan endless set of problems to be solved andforgotten.

The CMP classroom promotes student-to-studentdiscourse around mathematics. The problems arewritten to be engaging to students in the middlegrades and to encourage the development ofmathematical thinking and reasoning. Even the titlesof the materials express the importance the authorsplace on making connections–all kinds ofmathematics connections. Noting such connections isfundamental in seeing mathematics as a connectedwhole rather than an endless string of algorithms orprocesses to be learned.

Throughout program; for examples see:Kaleidoscopes, Hubcaps, and Mirrors (Inv. 1, 2, 3, 4, 5)

CCSS Mathematical Practices xi

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xii Common Core State Standards Correlations

COMMON CORE STATE STANDARDS GRADE 8 CMP2 UNITS CONTENT

The Number System

Know that there are numbers that are not rational, and approximate them by rational numbers.

8.NS.1Understand informally that every number has a decimal expansion; the rational numbers are those with decimal expansions that terminate in 0s or eventually repeat. Know that other numbers are called irrational.

Looking For Pythagoras

Inv. 4: Using the Pythagorean Theorem

8.NS.2Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2).For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.



Expressions and Equations

Use properties of operations to generate equivalent expressions.

8.EE.1Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.

Growing, Growing, Growing

Inv. 5: Patterns With Exponents

8.EE.2Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.


CC Transition Kit

Inv. 2: Squaring Off

Inv. 3: The Pythagorean Theorem


CC Inv. 1: Exponents

Connected Mathematics (CMP) is a field-tested and research-validated program that focuses on afew big ideas at each grade level. Students explore these ideas in depth, thereby developing deepunderstanding of key ideas that they carry from one grade to the next. The sequencing of topicswithin a grade and from grade to grade, the result of lengthy field-testing and validation, helps toensure the development of students' deep mathematical understanding and strong problem-solvingskills. By the end of grade 8, CMP students will have studied all of the content and skills in theCommon Core State Standards (CCSS) for middle grades (Grades 6–8).

The sequence of content and skills in CMP2 varies in some instances from that in the CCSS, so incollaboration with the CMP2 authors, Pearson has created a set of investigations for each grade levelto further support and fully develop students' understanding of the CCSS. The authors are confidentthat the CMP2 curriculum supplemented with the additional investigations at each grade level willaddress all of the content and skills of the CCSS, but even more, will contribute significantly toadvancing students’ mathematical proficiency as described in the Mathematical Practices of theCCSS. Through the in-depth exploration of concepts, students become confident in solving a varietyof problems with flexibility, skill, and insightfulness, and are able to communicate their reasoning andunderstanding in a variety of ways.

The following alignment of the Common Core State Standards for Mathematics (June 2, 2010release) to Pearson’s Connected Mathematics 2 (CMP2) ©2009 program includes the supplementalinvestigations that complete the CMP2 program.

CMP2_G8_IN_FM_xii-xvii.qxd 11/18/10 9:03 AM Page xii

Common Core State Standards Correlations xiii


8.EE.3Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.


Inv. 1: ACE 39–40

Inv. 2: ACE 15–17

Inv. 4: ACE 8

Inv. 5: ACE 56–60

8.EE.4Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.


Inv. 5: ACE 56–57, 60

Understand the connections between proportional relationships, lines, and linear equations.

8.EE.5Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

Thinking With Mathematical Models

CC Transition Kit

Inv. 2: Linear Models and Equations

CC Inv. 2: Functions

8.EE.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.


CC Transition Kit


CC Inv. 2: Functions

Analyze and solve linear equations and airs of simultaneous linear equations.

8.EE.7Solve linear equations in one variable.


Say It With Symbols


Inv. 1: Equivalent Expressions

Inv. 2: Combining Expressions

Inv. 3: Solving Equations

8.EE.7.aGive examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a,or a = b results (where a and b are different numbers).

CC Transition Kit CC Inv. 2: Functions

8.EE.7.bSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.


Say It With Symbols


Inv. 1: Equivalent Expressions


Inv. 3: Solving Equations

Inv. 4: Looking Back at Functions

8.EE.8Analyze and solve pairs of simultaneous linear equations.

The Shapes of Algebra Inv. 2: Linear Equations and Inequalities

Inv. 3: Equations With Two or More Variables

Inv. 4: Solving Systems of Linear Equations Symbolically

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xiv Common Core State Standards Correlations


8.EE.8.aUnderstand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.




8.EE.8.bSolve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

The Shapes of Algebra Inv. 1: ACE 56–57

Inv. 2: Linear Equations and Inequalities



8.EE.8.cSolve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.




Functions

Define, evaluate, and compare functions.

8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.NOTE Function notation is not required in Grade 8.

CC Transition Kit CC Inv. 2: Functions

8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.



Frogs, Fleas and Painted Cubes

Say It With Symbols

Inv. 1: Exploring Data Patterns

Inv. 1: ACE 25–26, 38, 47

Inv. 2: Quadratic Expressions

Inv. 3: Quadratic Patterns of Change

Inv. 4: What Is a Quadratic Function?


8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.



The Shapes of Algebra

Say It With Symbols


Inv. 3: Inverse Variation

Inv. 5: Patterns With Exponents




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Common Core State Standards Correlations xv


Use functions to model relationships between quantities.

8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.



Say It With Symbols

Inv. 1: Exploring Data Patterns





8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.



Frogs, Fleas, and Painted Cubes

Say It With Symbols


Inv. 1 Exponential Growth

Inv. 2 Examining Growth Patterns

Inv. 3 Growth Factors and Growth Rates

Inv. 4 Exponential Decay

Inv. 1 Introduction to Quadratic Relationships

Inv. 2 Quadratic Expressions

Inv. 3 Quadratic Patterns of Change

Inv. 4 What Is a Quadratic Function?

Inv. 4 Looking Back at Functions

Geometry

Understand congruence and similarity using physical models, transparencies, or geometry software.

8.G.1 Verify experimentally the properties of rotations, reflections, and translations:

Kaleidoscopes, Hubcaps, and Mirrors

CC Transition Kit

Inv. 1: Three Types of Symmetry

Inv. 2: Symmetry Transformations

Inv. 3: Exploring Congruence

Inv. 4: Applying Congruence and Symmetry

Inv. 5: Transforming Coordinates

CC Inv. 3: Transformations

CMP2_G8_IN_FM_xii-xvii.qxd 11/18/10 9:03 AM Page xv


8.G.1.aLines are taken to lines, and line segments to line segments of the same length.

Kaleidoscopes,Hubcaps, and Mirrors

CC Transition Kit







8.G.1.bAngles are taken to angles of the same measure.


CC Transition Kit







8.G.1.cParallel lines are taken to parallel lines.


CC Transition Kit







8.G.2Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.



8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.


CC Transition Kit

Inv. 2: ACE 24–25, 32



8.G.4Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two dimensional figures, describe a sequence that exhibits the similarity between them.

CC Transition Kit CC Inv. 4: Geometry Topics

8.G.5Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

CC Transition Kit CC Inv. 4: Geometry Topics

xvi Common Core State Standards Correlations

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Common Core State Standards Correlations xvii


Understand and apply the Pythagorean Theorem.

8.G.6Explain a proof of the Pythagorean Theorem and its converse.



8.G.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.




8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.


Inv. 2: Squaring Off


Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres .

8.G.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.



Say It With Symbols

CC Transition Kit

Inv. 1: ACE 47–49

Inv. 2: ACE 28

Inv. 3: ACE 24

Inv. 3: ACE 18–22, 25–26

Inv. 4: ACE 57–58

Inv. 1: ACE 55

Inv. 3: ACE 41

Inv. 4: ACE 39

CC Inv. 4: Geometry Topics

Statistics and Probability

Investigate patterns of association in bivariate data.

8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

Samples and Populations

Inv. 4: Relating Two Variables

8.SP.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

Samples and Populations


Inv. 4: Relating Two Variables


8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.



Inv. 2: Linear Equations and Inequalities



Inv. 3: Inverse Variation

8.SP.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

CC Transition Kit CC Inv. 5: Categorical Data

Common Core State Standards Correlations xvii

CMP2_G8_IN_FM_xii-xvii.qxd 11/18/10 9:03 AM Page xvii

KEY ✓ Core Content

R Review

* Extending

Thinking with Mathematical Models Standard 18 days • Block 9 days

Inv. 1 Exploring Data Patterns 8.F.2, 8.F.4, 8.F.5, 8.SP.3 ✓

Inv. 2 Linear Models and Equations 8.EE.6, 8.EE.7, 8.EE.7.b, 8.F.3, 8.F.5, 8.SP.2, 8.SP.3 ✓

Inv. 3 Inverse Variation 8.F.3 ✓

Looking for Pythagoras Standard 18 ½ days • Block 9 ½

Inv. 1 Coordinate Grids Reviews 6.NS.6 R

Inv. 2 Squaring Off 8.EE.2 ✓

Inv. 3 The Pythagorean Theorem 8.EE.2, 8.G.6, 8.G.7, 8.G.8, 8.G.9 ✓

Inv. 4 Using the Pythagorean Theorem 8.NS.1, 8.NS.2, 8.EE.2, 8.G.7, 8.G.9 ✓

Growing, Growing, Growing Standard 25 days • Block 12 ½

Inv. 1 Exponential Growth 8.EE.3, 8.F.2, 8.F.5 ✓

Inv. 2 Examining Growth Patterns 8.EE.3, 8.F.5 ✓

Inv. 3 Growth Factors and Growth Rates 8.F.5 ✓

Inv. 4 Exponential Decay 8.EE.3 ✓

Inv. 5 Patterns with Exponents 8.EE.1, 8.EE.3, 8.EE.4, 8.F.3 ✓

CC Inv. 1 Negative Exponents 8.EE.1, 8.EE.2 ✓

CC Inv. 2 Functions 8.EE.5, 8.EE.6, 8.EE.7.a, 8.F.1, 8.F.2 ✓

xviii Common Core State Standards Pacing Chart

This Pacing Chart offers pacing suggestions as you look to implement the Common Core StateStandards for Grade 8 in the CMP2 classroom. The Chart shows placement recommendations forthe Common Core Investigations provided in this supplement.

Investigations labeled as Review (R) offer timely practice of concepts from earlier grades, helping toactivate students’ prior knowledge as they are introduced to new concepts that build on theseconcepts. Investigations labeled as Extending (*) offer students the opportunity to explore conceptsin greater depth or to extend their study of concepts.

The suggested number of standard days for each unit is based on a 45-minute class period; a blockperiod is assumed to be 90 minutes of instructional time. The total pacing leaves time in the schoolyear for assessments, projects, assemblies, or other special events that vary from school to school.

CMP2_G8_IN_FM_xviii-xx.qxd 11/18/10 9:04 AM Page xviii

KEY ✓ Core Content

R Review

* Extending

Frogs, Fleas, and Painted Cubes Standard 22 ½ days • Block 11 ½

Inv. 1 Introduction to Quadratic Relationships 8.F.5 ✓

Inv. 2 Quadratic Expressions 8.F.2, 8.F.5 ✓

Inv. 3 Quadratic Patterns of Change 8.F.2, 8.F.5 ✓

Inv. 4 What Is a Quadratic Function? 8.F.2, 8.F.5 ✓

Kaleidoscopes, Hubcaps, and Mirrors Standard 32 days • Block 16

Inv. 1 Three Types of Symmetry 8.G.1, 8.G.1.a, 8.G.1.b, 8.G.1.c, 8.G.9 ✓

Inv. 2 Symmetry Transformations 8.G.1, 8.G.1.a, 8.G.1.b, 8.G.1.c, 8.G.3, 8.G.9 ✓

Inv. 3 Exploring Congruence 8.G.1, 8.G.1.a, 8.G.1.b, 8.G.1.c, 8.G.2, 8.G.9 ✓

Inv. 4 Applying Congruence and Symmetry 8.G.1, 8.G.1.a, 8.G.1.b, 8.G.1.c ✓

Inv. 5 Transforming Coordinates 8.G.1, 8.G.1.a, 8.G.1.b, 8.G.1.c, 8.G.3 ✓

CC Inv. 3 Transformations 8.GG.1, 8.GG.2, 8.GG.3 ✓

CC Inv. 4 Geometry Topics 8.GG.3, 8.GG.4, 8.GG.5, 8.GG.9 ✓

Say It With Symbols Standard 21 days • Block 10 ½

Inv. 1 Equivalent Expressions 8.EE.7, 8.EE.7.a, 8.EE.7.b, 8.G.9 ✓

Inv. 2 Combining Expressions 8.EE.7, 8.EE.7.a, 8.EE.7.b, 8.F.2 ✓

Inv. 3 Solving Equations 8.EE.7, 7.b, 8.G.9 ✓

Inv. 4 Looking Back at Functions 8.EE.7.b, 8.G.9, 8.F.3, 8.F.4, 8.F.5 ✓

Inv. 5 Reasoning With Symbols Prepares for A-SSE-3 *

Shapes of Algebra Standard 25 days • Block 12 ½

Inv. 1 Equations for Circles and Polygons 8.EE.8.b ✓

Inv. 2 Linear Equations and Inequalities 8.EE.8, 8.EE.8.a, 8.EE.8.b, 8.EE.8.c, 8.SP.3 ✓

Inv. 3 Equations With Two or More Variables 8.EE.8, 8.EE.8.a, 8.EE.8.b, 8.EE.8.c, 8.F.3, 8.SP.3 ✓

Inv. 4 Solving Systems of Linear Equations Symbolically

8.EE.8, 8.EE.8.a, 8.EE.8.b, 8.EE.8.c, 8.F.3 ✓

Inv. 5 Linear Inequalities Prepares for A-REI-12 *

Samples and Populations Standard 7 days • Block 3 ½

Inv. 1 Comparing Data Sets Reviews 6.SP.4 R

Inv. 2 Choosing a Sample From a Population Reviews 7.SP.1 R

Inv. 3 Solving Real-World Problems Reviews 7.SP.3 R

Inv. 4 Relating Two Variables 8.SP.1, 8.SP.2 ✓

CC Inv. 5 Categorical Data 8.SP.4 ✓

Common Core State Standards Pacing Chart xix

CMP2_G8_IN_FM_xviii-xx.qxd 11/18/10 9:04 AM Page xix

Geometry Topics 25

Problem 4.1

Investigation 4: Geometry Topics

A transversal is a line that intersects two or more other lines. When parallellines are intersected by a transversal, many angles are formed. They can beacute, obtuse, or have special relationships between pairs. Knowing aboutangle relationships is useful when you make designs.

The Riverside School students have been invited to propose designs for asmall park on a rectangular plot of land next to the school. The plot has twoparallel paths crossing it.

A. Juan, Marsha, and Cora decide to include a new path to connectdifferent parts of the park in their design. The path will be diagonaland cross the two existing paths. They use a pencil to represent thediagonal path on a piece of lined paper to help decide its position.Move the pencil to different positions as shown below and observethe angles.

1. a. How does &1 change in relation to &2 as you move the pencil?

b. Name another pair of angles with this same relationship.

2. a. When the pencil is moved, how does this affect the relationshipbetween &1 and &4?

b. Name another pair of angles with this same relationship.

3. Describe the relationship between &1 and &8.

4. Describe the relationship between &1 and &5.

13 4

2 1 2 1 23 4 3 4

5 67 8

5 67 8

57 8

6

Common Core State Standards: 8.G.5; 8.G.9

8cmp10se_Investigation_4.qxd 11/19/10 4:49 PM Page 25

26 Common Core Additional Investigations

B. Use a straightedge to draw a transversal like the one shown in thethird figure.

1. Measure all of the angles.

2. Do your measurements verify the angle relationships youdiscovered in Part A? Explain.

C. Juan, Marsha, and Cora are working on thedesign shown. The black paths on the designare parallel.

1. The terms below are commonly used when describing angles thatare formed when parallel lines are intersected by a transversal.

• supplementary • alternate

• interior • vertical

• exterior • corresponding

a. Use the terms and what you have observed about angle pairs tolabel each group of angles shown in the table below. Somegroups can be described using more than one term.

b. List any angles that may be missing from each group.

2. In the park design,&2 measures 65°.

a. Show how Juan, Marsha, and Cora can use angle relationshipsto find the measures of each of the other angles. List the anglemeasures and justify your reasoning.

b. Measure the angles to verify your findings.

Group 1 Group 2 Group 3

&1 and &2 &3 and &6 &1 and &8

&7 and &8 &4 and &5 &2 and &7

Group 4 Group 5 Group 6

&3 and &7 &1 and &3 &1 and &4

&2 and &6 &6 and &8 &2 and &3

&3 and &4

1 2

3 4

5 6

7 8

New Path


Geometry Topics 27

A local nursery offers to donate rose bushes for a planned rose garden inthe park. Marsha designs a rose garden shaped like a parallelogram. Therose garden will have boundaries along the paths.

A. An iron fence will surround the park. The measures of angles a, b, c,and d must be included in the blueprint for the person making thefence. How can Marsha find these angle measures without using anangle ruler or protractor?

B. What are the measures of angles a, b, c, and d?

C. The final plan includes short stone walls that join opposite corners ofthe Rose Garden.

One of the walls forms a 76° angle with New Path. The other wall will makea 28° angle with Parallel Way. Show how Juan, Marsha, and Cora can findthe measures of &e through &j.

28�

76�e f g

hij

New

Pat

h

118°

Parallel Way

a b

New

Pat

h

d118° c

Rose Garden

Parallel Way

Problem 4.2



Problem 4.3

Juan drew this plan to show two straight rows of bushes that will cross andconnect two parallel paths, New Path and East Path. Two triangles areformed by the bushes and paths.

A. 1. What is the relationship between angle y and the angle with ameasure of 47°?

2. What is the measure of angle y?

3. What is the relationship between angle x and the angle with ameasure of 72°?

4. What is the measure of angle x?

5. What is the relationship between angles w and z?

B. Juan is trying to determine if the triangular sections are similar.Corresponding angles of similar triangles have equal measures.

1. Can Juan decide whether the triangles formed by the bushes andpaths are similar knowing only the given angles and the measures ofangles x and y? Explain why or why not.

2. Describe how to find the measure of angle w using the angle sum oftriangles.

3. Describe how to find the measure of angle z using properties oftransversals of parallel lines.

4. How many pairs of angles of the triangles need to be congruent todetermine that the triangles are similar?

C. Juan drew the location of a sprinkler line that runs parallel to NewPath. The new line forms a smaller triangle within the larger triangle.Are the triangles similar? Explain why or why not.

72�

47�

x z

ywN

ew P

ath

East

Pat

h

47�

xa

bwN

ew P

ath


Geometry Topics 29

Problem 4.4

Getting Ready for Problem 4.4

Use these formulas to find the volumes of cones, cylinders, and spheres.

Cylinder Cone Sphere

V = πr2h V = πr2h V = πr3

A. Cora proposes adding a circular pool with a fountain to the rosegarden. The budget for the park will allow for a fountain pump thatoperates best in a pool with a maximum volume of 750 ft3.

1. Find the exact volume of a circular pool with a radius of 5 ft and adepth of 3 ft. Approximate the pool's volume by using the value3.14 for π.

2. What is the greatest radius the pool can have if its depth is at most2 ft?

3. Cora wants the pool to be between 1 ft and 2 ft deep. What

dimension should the pool have? Explain your choice.

B. Cora is designing a sculpture for the garden. The height of the conewill be the same as the diameter of the sphere.

1. To make the sculpture stable, Cora wants the volumesof the two pieces to be the same. If the height of thecone is 30 cm, what does its radius need to be?

2. Cora decides that the sculpture will look better if theradius of the cone matches the radius of the sphere.How tall should the cone be to keep the volumes of thetwo pieces the same?

13

43

r

r

rhh

Volume is the amount of space enclosed in a solid figure. Volume isexpressed in cubic units, such as cm3 or ft3. The exact volume of a cone,cylinder, or sphere includes the value π.

12



ExercisesUse the figure below for Exercises 1–5. Lines L1 and L2 are parallel.

1. Name two pairs of alternate interior angles.

2. Name two pairs of vertical angles.

3. Name two pairs of corresponding angles.

4. Name two angles that are supplementary to &4.

5. The measure of &1 is 80°. Find the measures of the other angles.

6. Multiple Choice What is the reason that &5 and &6 do not form a pairof alternate interior angles?

A. They are not alternate angles.

B. They are not interior angles.

C. They share the same vertex.

D. They are not supplementary.

12

34

56

78

L

t

1 L2


Geometry Topics 31

7. Rose Avenue is parallel to Acorn Avenue and perpendicular to Smith Street.

Explain why Smith Street must be perpendicular to Acorn Avenue.

8. Find the value of x.

Use parallelogram ABCD for Exercises 9–10.

9. Name the pair of line segments for which &1 and &4 form a pair ofalternate interior angles.

10. The measure of &2 is 28°. Find the measures of any other angles in thefigure that are possible to find.

A B21

3 4

D C

2x�x�

Rose Avenue

Acorn Avenue

Smith Street



11. The figure shows two parallel lines intersected by two transversals.

Find the value of x.

12. A surveyor has been hired to lay out the boundaries of a trapezoid-shaped park. The park’s designer has specified external angles of140° and 110°, as shown below.

Find the measures of &k,&m,&n, and &p.

13. The figure shows four of the runways at Metropolitan Airport.

Find the measures of &x and &y.

155�

parallel

105�

x

y

140� 110�p

k m

n

Maple Avenue

x�35� 72�


Geometry Topics 33

14. Two parallel lines are cut by a transversal.

Find the measure of angle a.

15. The figure shows the design of the flag of Jamaica.

Find n and the measures of the two angles.

16. In the figure, the two horizontal lines are parallel.

Find the measure of &a.

17. Angle 1 is supplementary to angle 2. Angle 2 is vertical to angle 3.Angle 3 is an alternate exterior angle to angle 4. Angle 4 issupplementary to angle 5. How is angle 1 related to angle 5?

100�

40�

a

(n � 90)°

(2n � 70)°

(2x � 15)�

(3x � 10)�a



For Exercises 18–21, determine whether #ABC is similar to #XYZ.

18. m&A = 30°; m&B = 90°; 19. m&A = 65°; m&C = 75°;

m&X = 30°; m&Y = 90° m&X = 65°; m&Y = 75°

20. m&B = 50°; m&C = 70°; 21. m&C = 70°; m&A = 42°;

m&X = 60°; m&Z = 70° m&Y = 68°; m&Z = 70°

22. Are the triangles shown below similar? Explain why or why not.

23. In the triangles below, AB—

is parallel to DE—

, and BC—

is parallel to EF—

.Is #ABC similar to #DEF? Explain why or why not.

24. Maya drew similar triangles for a presentation. She wants the smallertriangle to be half the size of the larger triangle. She measures thesmallest angle of the smaller triangle, and the largest angle of thelarger triangle. The smaller triangle has one angle that measures 27°.The larger triangle has one angle with a measure of 77°. What are theother two angle measures of the larger triangle?

25. Are the triangles shown below similar? Explain why or why not.

38° 38°

51°

E

B

A D C F

62°

61°

58°


Geometry Topics 35

For Exercises 26–29, use the diagram below. Lines la and lb are parallel.Lines lc, ld, and le are parallel.

26. What is the value of w?

27. What is the value of x?

28. What is the value of y?

29. What is the value of z?

30. What is the value of k in the figure below?

31. The shape shown is an equilateral triangle. What is the value of c?

32. Hiro is covering a tabletop with tiles. He sets a tile that has the shapeof an isosceles triangle in one square corner of the tabletop as shown.He needs to cut another tile to finish the corner. What is the value of p?

41°

lc

la

lb

ld le

wx

yz

41° 72°

56°

k

46°101°

c

p

36˚



33. The main path through the park will be paved with a pattern ofcongruent triangular stones. The edges of the path are parallel.

a. For the top edge of the path to be a straight line, what is the sumof the measures of &1,&2, and &3? Explain how you know.

b. What is the relationship between &1 and &4? Measure &1. Whatis the measure of &4?

c. What is the relationship between &3 and &5? Measure &3. Whatis the measure of &5?

d. What is the measure of &2?

e. Describe the relationship among &2,&4, and &5. What is thesum of their measures?

f. What is the sum of the measures of the angles of any triangle?

34. Two congruent stones have been laid along a straight path.

a. Describe the relationship among &w,&x, and &y. What is thesum of their measures?

b. For the top edge of the path to be a straight line, what is the sumof the measures of &x and &z? Explain how you know.

c. Explain why w + x + y = x + z.

d. Subtract x from each side of the equation. Explain how theequation relates the measure of an exterior angle of a triangle tothe sum of the measures of the opposite interior angles.

1 32

4 5

w

y

x z


Geometry Topics 37

For Exercises 35–38, find the exact volume of the solid.

35.

36.

37. 38.

39. A cylindrical watering can does not fit under a faucet, so Trang is usinga paper cone to fill it with water. The cone has a radius of 1 inch anda height of 4 inches. The can has a radius of 2 inches and a height of8 inches. How many full cones of water will it take to fill the can?

40. A baseball has a diameter of 2.8 in.

a. What is the volume of the baseball?

b. About many times more volume does a basketball with a radius of4.78 inches have than a baseball?

41. What is the exact volume of the largest cone that can fit into a cubewith sides of 10 inches?

12 cm

3 cm

8 cm

12 cm

4 in.

9 ft.

5 ft.


Geometry Topics 71

Investigation 4: Geometry Topics

Guided Instruction

Mathematical Goals

• Use informal arguments to establish facts about the angle sum andexterior angle of triangles.

• Use informal arguments to establish facts about the angles created whenparallel lines are cut by a transversal.

• Use informal arguments to establish facts about the angle-angle criterionfor similarity of triangles.

• Know the formulas for the volumes of cones, cylinders, and spheres, anduse them to solve problems.

Vocabulary

• supplementaryangles

• alternate angles

• interior angles

• exterior angles

• correspondingangles

• vertical angles

• volume

Materials

• lined paper

• pencil

• straightedge

• angle ruler orprotractor

• blank paper(optional)

• index card orbusiness envelope(optional)

• Labsheet Exercise 33

At a Glance

This investigation reviews concepts students studied in the CMP2 UnitsShapes and Designs and Filling and Wrapping.

In this investigation, students find relationships between the anglesformed when parallel lines are cut by a transversal, and then apply thoserelationships to find unknown angle measures. Before students begin thetopic, review the definitions of acute, right, obtuse, and straight angles, andthe fact that congruent angles have the same measure. Also make surestudents know how to correctly use the angle ruler or protractor to measureangles.

Begin by asking students to name some real-world examples of parallellines (train tracks, parallel bars) and of parallel lines cut by a transversal(parallel streets intersected by a third street). Show an example of thelatter. Without calling attention to any of the angles, ask students todescribe any apparent symmetries or patterns they see. Have them discussmethods they could use to find out whether their observations are true ingeneral for parallel lines cut by a transversal or whether they are limited tothe example you have shown.

Problem 4.1

As an alternative to using lined paper in Problem 4.1 A, students candraw a pair of parallel lines on a blank sheet of paper, using the long sidesof an index card or a business envelope as guides.

PACING 3 days

Common Core State Standards: 8.G.5; 8.G.9

08_CMP2_TRG_071-076.qxd 11/22/10 2:49 PM Page 71

As students work on Problem 4.1 A, guide them to think about how theangles are changing as they move the transversal. Make sure students focuson whether angle measures are increasing or decreasing, and how themeasures of other angles are changing at the same time, rather than whatthe exact angle measures are.

During Problem 4.1 A, ask:

• As the measure of �1 increases when you move the transversal, do anyother angle measures increase? (yes, the measures of �4, �5, and �8)

• Why do the measures of those angles increase along with the measure of �1? (They are located in positions similar to that of �1 relative to the transversal and the parallel lines.)

After Problem 4.1 B, ask: When you measured the angles, did you find anysets of congruent angles? If so, which angles were they? (Yes; �1, �4, �5,and �8 are congruent, and �2, �3, �6, and �7 are congruent.)

Before Problem 4.1 C, Part 1, have students read the bulleted terms. Askstudents if they are familiar with the everyday meaning of each term. Ask:What does the word interior mean? the word exterior? (something that isinside; outside) Then ask:

• For the angles, what do you think exterior and interior mean in relation tothe parallel lines? (interior: between the lines; exterior: outside the lines)

• What can you look for to find corresponding angles? alternate angles?(angles that are in similar positions; angle pairs where the angles are onopposite sides of a line)

• A supplement is something that is added to make something else complete.Look at angles 2, 3, and 4. Two of these angles are supplementary to angle1. Which two? Why? (�2 and �3; added to �1, each makes a straightangle.)

• The third angle forms a vertical angle with angle 1. Which one? (�4)

During Problem 4.1 C, Part 2, ask: If angle 2 measures 65�, are there anyother angles that also measure 65�? (yes; �3, �6, and �7)

Problem 4.2

Before Problem 4.2 A, have students study the figure. Ask: What do youknow about parallelograms? (Opposite sides are parallel, opposite anglesare congruent.)

During Problem 4.2 A, guide students to see that Parallel Way is atransversal that intersects two parallel sides of the rose garden. Ask: Whatcan help you find the measure of �c or �d?(�c and the 118º angle arecorresponding angles; �d and the 118º angle are supplementary angles.)

As students work on Problem 4.2 C, ask: What do you notice about thestone walls? (They form transversals to the parallel boundaries of the park.)Have students list congruent angle pairs. Then ask: How can you find themeasure of �j? of �e? (m�j � 180� � (118� � 28�); m�e � 118� � 76�)

72 Common Core Teacher’s Resource Guide


Problem 4.3

Before Problem 4.3, have students study the figure. Ask: What do younotice about each of the straight rows of bushes? (They are transversals thatcut the parallel paths, New Path and East Path.) Help students see the twotriangles formed by the rows of bushes and paths.

Before Problem 4.3 B, ask: What do you know about similar triangles?(Corresponding angles have equal measures.)

During Problem 4.3 B Part 2, ask: What is the sum of the measures of allangles of a triangle? (180°).

Problem 4.4

Before Problem 4.4, during Getting Ready, ask: How do you define acylinder and a cone? (A cylinder is a three-dimensional shape with twoparallel and congruent circles as bases; a cone has one base that is a circleand one vertex.) Guide students to understand that using the value 3.14 forπ gives an approximation of the volume. Ask: How is a volume that is foundusing 3.14 for π different than a volume that is given using the term π?(Using 3.14 gives an estimation of the volume, while using the term π givesthe exact volume.)

During Problem 4.4 A, ask: What three-dimensional shape will a circularpool have? (cylinder)

During Problem 4.4 B, Part 1, ask: How can you set the volumes of thetwo pieces of the sculpture equal to each other to write an equation that

you can use to solve this problem? (The volume of the sphere is πr3, and

the volume of the cone is πr2h. The radius of the sphere is one-half the

height of the cone, or 15 cm. Write the equation π(15)3 � πr2(30), and

solve for r.)

Summarize

To summarize the lesson, ask:

• What do you know about the angles that are formed when parallel linesare cut by a transversal? (Four congruent angles are formed, and fourcongruent angles supplementary to the first set of angles also areformed.)

• How is the measure of an exterior angle of a triangle related to thetriangle's interior angles? (The measure of an exterior angle is equal tothe sum of the opposite angles of the triangle.)

• The measure of how many corresponding angles of two triangles need tobe known to determine that the triangles are similar? (2 out of the 3)

13

43

13

43

Geometry Topics 73



Assignment Guide forInvestigation 4Problem 4.1, Exercises 1–8, 17Problem 4.2, Exercises 9–10Problem 4.3, Exercises 11–16, 18–34Problem 4.4, Exercises 35–41

Answers to Investigation 4

Problem 4.1A. 1. a. The measure of �1 increases and the

measure of �2 decreases as you movethe pencil.

b. �5 and �6

2. a. The measures of �1 and �4 remainequal as the pencil moves.

b. �5 and �8

3. They have the same measure and maintainthe same measure as the pencil moves.

4. They have the same measure and maintainthe same measure as the pencil moves.

B. 1. Check students’ drawings andmeasurements. Angles 1, 4, 5, and 8 shouldhave the same measure. Angles 2, 3, 6, and7 should have the same measure.

2. Sample answer: Yes; the measures of �1,�4, �5, and �8 are the same, so theseangles are congruent. The measures of �2,�3, �6, and �7 are the same, so theseangles are also congruent.

C. 1. a. Group 1: exterior angles or exteriorsupplementary angles;Group 2: alternate interior angles;Group 3: alternate exterior angles;Group 4: corresponding angles;Group 5: supplementary angles;Group 6: vertical angles

b. Group 4: �1 and �5, �4 and �8;Group 5: �1 and �2, �2 and �4,�7 and �8, �5 and �6, �5 and �7;Group 6: �5 and �8, �6 and �7

2. a. �3, �6, and �7 are congruent to �2,so they all measure 65�; �1 and �2 are supplementary, so m�1 �180� � 65� � 115�. �4, �5, and �8 are allcongruent to �1, so they all measure 115�.

b. Check students’ measurements.

Problem 4.2A. Each horizontal path is a transversal that

intersects two parallel sides of the rosegarden, so Marsha can use the relationshipsbetween angles formed by parallel lines andtransversals; or Marsha can use the fact that�d and the 118� angle are supplementary, som�d � 180� � 118�. Then she can use the factthat parallelograms have two pairs of oppositecongruent angles and that the sum of theangle measures is 360� to find the other anglemeasures.

B. m�d � m�b � 62�; m�a � m�c � 118�.

C. m�e � 118� � 76� � 42�; m�j � 62� � 28�

� 34�; �f is an alternate interior angle with the given angle that measures 28� som�f � 28�, which means m�g � 62� � 28�

� 34�; �h is an alternate interior angle with the given angle that measures 76� som�h � 76�; m�i � 118˚ � 76� � 42�


Problem 4.3A. 1. They are vertical angles.

2. m�y � 47°

3. They are alternate interior angles.

4. m�x � 72°

5. They are alternate interior angles.

B. 1. Yes, the corresponding angles of thetriangles are: �y and the angle measuring47°, �x and the angle measuring 72°; �wand �z, which are alternate interior anglesthat have the same measure.

2. The angle sum of a triangle is 180°;72 � 47 � m�w � 180; m�w � 61°.

3. The measures of alternate interior anglesare equal, so m�z � m�w = 61°.

4. 2

C. Yes; �x and �a are corresponding angles thathave equal measures, and �w and �b also arecorresponding angles having equal measure.Two corresponding pairs of angles are equal,so the triangles are similar.

Problem 4.4A. 1. V � πr2h � π(5)2(3) � 75π ft3; V ≈ 75(3.14)

� 236 ft3

2. V � πr2h; 750 ≈ (3.14)r2(2); r ≈ 10.9 ft

3. To have the largest radius and look largestin the garden, the pool needs to have theleast depth, 1 ft. V � πr2h; 750 ≈ (3.14)r2(1);r ≈ 15.5 ft. The pool should be 1 ft deep and have a radius of about 15.5 ft.

B. 1. r ≈ 21.2 cm

2. The height of the cone should be 4 times itsradius.

Geometry Topics 75



Exercises1. �3 and �6, �4 and �5

2. Sample answer: �1 and �4, �2 and �3

3. Sample answer: �1 and �5, �4 and �8

4. �2 and �3

5. �4, �5, and �8 measure 80�; �2, �3, �6,and �7 measure 100�.

6. C

7. The marked angle measures 90�. The exteriorangle on the same side of the transversal alsomust measure 90�. That angle is at theintersection of Smith and Acorn, so they areperpendicular.

8. x � 60�

9. and

10. m�3 � 28�

11. x � 73�

12. k � 140�, m � 110�, n � 70�, p � 40�

13. x � 25�, y � 75�

14. a � 85�

15. n � 20; each angle measures 110�.

16. a � 120�

17. They are congruent because they are eithercorresponding angles or alternate interiorangles.

18. yes

19. no

20. yes

21. yes

22. No; the missing angle of the first triangle hasa measure of 180° – 90° – 38° � 52°, so thetriangles have at least two correspondingangles with measures that are not equal.

23. Yes; AC—

is a transversal to parallel segmentsAB—

and DE—

, so angles A and D have the samemeasure; DF

—is a transversal to parallel

segments BC—

and EF—

, so angles C and F havethe same measure.

24. 76°, 27°

25. No; the unknown angle for the secondtriangle has a measure of 180° – 61° – 58° �

61°.

26. 67°

27. 72°

28. 52°

29. 72°

30. 55°

31. 120°

32. 18°

33. a. 180°; a straight angle measures 180°.

b. They are alternate interior angles. Theirmeasures are equal; m�1 � 30°; m�4 � 30°

c. They are alternate interior angles. Theirmeasures are equal; m�3 � 60°; m�5 � 60°

d. 90°

e. They are the angles of a triangle; the sum oftheir measures is 30° � 60° � 90° � 180°.

f. 180°

34. a. They are the angles of a triangle; the sum oftheir measures is 180°.

b. 180°; a straight angle measures 180°.

c. The left side of the equation, w � x � y,represents the sum of the angles of atriangle, which is 180°; the right side of theequation, x � z, represents the sum of theangles that make a straight angle, whichalso is 180°.

d. The left side of the remaining equation,w � y, represents the sum of the measuresof the opposite interior angles to �x; theright side of the remaining equation is z,which is the measure of the exterior angleto �x.

35. in.3

36. 135π ft3

37. 768π cm3

38. 36π cm3

39. 24

40. a. about 11.5 in.3

b. about 39 times

41. in.3

BCAD

256π3

250π3


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