Michigan Curriculum Framework · ii Mi hi C i l F k The framework emphasizes the importance of:...

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The Michigan Curriculum Framework is a resource forhelping Michigan’s public and private schools design,implement, and assess their core content area curricula.

The content standards identified in this document arepresented as models for the development of local districtcurriculum by the Michigan State Board of Education and theMichigan Department of Education. They represent rigorousexpectations for student performance, and describe theknowledge and abilities needed to be successful in today’ssociety. When content, instruction, and local and stateassessments are aligned, they become powerful forces thatcontribute to the success of student achievement.

The framework presents a content and a process for developingcurriculum that enables schools to realize Michigan’s vision forK-12 education:

Michigan’s K-12 education will ensure that allstudents will develop their potential in order to leadproductive and satisfying lives. All students willengage in challenging and purposeful learning thatblends their experiences with content knowledgeand real-world applications in preparation for theiradult roles, which include becoming:

literate individuals

healthy and fit people

responsible family members

productive workers

involved citizens

self-directed, lifelong learners

The intent of this document is to provide useful resources todistricts as they strive to implement a program which ensuresthat all students reap the benefits of a quality education andachieve the adult roles described in Michigan’s vision forK-12 education. The content standards and benchmarks serveas worthy goals for all students as they develop the knowledgeand abilities inherent in their adult roles. They represent anessential component in the process of continuous schoolimprovement, which like professional development, should befocused on improving student achievement.

INTRODUCTION

We believe that efforts toset clear, common, stateand/or community-basedacademic standards forstudents in a given schooldistrict or state arenecessary to improvestudent performance.Academic standardsclearly define whatstudents should know andbe able to do at certainpoints in their schooling tobe considered proficient inspecific academic areas.We believe that states andcommunities can benefitfrom working together totap into the nation’s bestthinking on standards andassessments.

1996 National EducationSummit Policy Statement

ii Mi hi C i l F k

The framework emphasizes the importance of:

using continuous school improvement to align alldistrict initiatives for the purpose of increasingstudent achievement;

building a curriculum based on rigorous contentstandards and benchmarks;

using student achievement data to make decisionsabout continuous school improvement, curriculum,instruction, and professional development; and,

incorporating research-supported teaching andlearning standards into daily instructional practice.

BACKGROUND

In 1993, the Michigan Department of Education, incollaboration with representatives from five state universities,was awarded federal funding from the U.S. Department ofEducation to develop curriculum framework components forEnglish language arts, mathematics, science, and geography.In addition, the Michigan Council for the Social Studies offered,and was supported by the State Board of Education, to developa curriculum framework component for social studies whichwould include history, economics, and American government,and would be complementary to the geography framework.

The Michigan Curriculum Framework brings together the workof individual content area projects to present a unified view ofcurriculum, one which addresses the educational needs of thewhole learner. The goal of the curriculum framework is toimprove student achievement by aligning classroom instructionwith core curriculum content standards and national contentstandards. It is designed to be used as a process for thedecision-making that guides continuous school improvement.It describes curriculum, instruction, and assessment andfocuses on improving program quality by aligning all theprocesses that affect a student’s achievement of rigorouscontent standards.

Framework project co-directors, university representatives, andthe Michigan Department of Education content area consultantsmet regularly with members of their content area professionalorganizations to design the components of the curriculumframework. Committees of teachers and university personnelworked together to draft the content standards, benchmarks,and performance standards for their specific content areas.

Co-directors met on a monthly basis to coordinate the efforts ofthe various content area committees in developing K-12standards and benchmarks for their subject areas. Theirpurpose was to ensure that the framework represents aconsistent view of curriculum across content areas. Theywanted to facilitate continuous school improvement byemphasizing commonalities among the content areas withregard to professional development, assessment, andinstruction.

ÒSetting high standards forour children. It’s the sinequa non for any otherreforms anyone might wantto implement. We can’tjudge the efficacy of ideasbecause we have noyardsticks by which tomeasure success or failure.”

Louis V. GerstnerChairman and CEO, IBM

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The co-directors were guided by a Joint Steering Committeecomprised of representatives from the content areas, parents,business leaders, labor leaders, house and senate staff, andeducators. Joint Steering Committee members reviewed theframework projects at each phase of their development andmade recommendations for improving their quality. Theirinsight helped the co-directors incorporate the views of all ofMichigan’s interested parties into the final frameworkdocument.

WHAT IS IN THE FRAMEWORK?

The framework includes the resources needed to develop astandards-based curriculum. Standards and benchmarks forEnglish language arts, mathematics, science, and social studiesare included in this edition of the framework. Standards andbenchmarks for arts education, career and employability skills,health education, life management education, physicaleducation, technology education, and world languages will beadded to the next edition of the framework. The processdescribed in the framework will be expanded to incorporate theadditional core content areas when they are completed. Thechart on page vi provides a list of the materials that eventuallywill be included in the framework document.

Tier I

Content Standards and BenchmarksTier I begins with a complete list of core curriculum contentstandards and benchmarks for grades K-12 in the areas ofEnglish language arts, mathematics, science and social science.The standards describe what all students should know and beable to do in each of the subject areas. The benchmarksindicate what students should know and be able to do atvarious developmental levels (i.e., early elementary school,later elementary school, middle school, and high school).

PlanningThe framework includes a planning section. It provides amodel for using the standards and benchmarks to create a localdistrict curriculum as part of continuous school improvement.It discusses the importance of involving representatives fromall stakeholders in the curriculum development process. Inaddition, it emphasizes the need for alignment among all ofthe processes that comprise continuous school improvementand focuses attention on placing student achievement at thecenter of all decision-making. It emphasizes the need forcontinuity in a K-12 curriculum. Continuity is developed byclearly defining benchmarks that establish increasinglycomplex demonstrations of rigorous standards.

Teaching and LearningThe section on teaching and learning describes standards thatare the foundation to successful learning in all content areas.The standards include deep knowledge, higher-order thinking,substantive conversation, and connections to the world beyondthe classroom. It illustrates the standards through sample

iv Michigan Curriculum Framework

teaching vignettes in each of the content areas. It discussesthe importance of incorporating strategies for technology,connecting with the learner, interdisciplinary learning, andmaking school-to-work connections into the curriculum.

Assessment SystemThe framework contains a section on assessment whichdescribes the need for developing a local assessment systemto monitor student growth and program effectiveness. Thissection of the framework is divided into three parts. The firstpart provides a rationale for why an assessment system isneeded. The second part describes how teachers can developperformance assessments based on the content standards andbenchmarks. The third part discusses important issues relatedto building an assessment system that aligns local assessmentpractices with state assessment.

Professional DevelopmentThe section on professional development lists standards forthe context, content, and process of professional developmentexperiences. It includes a process for designing professionaldevelopment which aligns with school improvement,curriculum content, student learning, and assessment needs.A vignette of one teacher’s personal, professional developmentexperiences is provided to illustrate Michigan’s Standards forProfessional Development.

Executive Summaries and GlossaryThe appendices of the framework contain executivesummaries of important resources that will aid a district as itdevelops, implements, and monitors its local curriculum. Aglossary of framework terms is also provided.

Tier II

ToolkitsTier II contains a collection of toolkits designed to helpdistricts with specific tasks such as conducting discrepancyanalyses. There are additional toolkits to guide districts inincorporating principles associated with connecting with thelearner, technology, curriculum integration, and makingschool-to-work connections. There are toolkits on planningsubject area instructional units, designing classroomassessments, and planning a district assessment system.(Some of the above mentioned toolkits are still underdevelopment.)

Tier III

ResourcesTier III contains content-area specific resources that helpclarify the curriculum development process described in theframework. These include resources such as the ScienceEducation Guidebook, the Mathematics Teaching and LearningSample Activities, Guidelines for the Professional Developmentof Teachers of English Language Arts, and Powerful &Authentic Social Studies Standards for Teaching. It also

Section I • Introduction v

contains a guidebook written specifically for parents and thebusiness community explaining the elements of the framework.

HOW TO USE THE FRAMEWORK

District school improvement committees and curriculumdevelopment committees will find the framework and itstoolkits very useful as they begin the process of creating astandards-based curriculum. Reading and discussing thecontents of the framework will help school improvementcommittee members gain a clearer understanding of thecurriculum development process. The toolkits will helpsubcommittees develop techniques for creating and aligningcurriculum, assessment, and instruction. They will also helpdistricts make decisions about the professional developmentstrategies which will most effectively help their students reachtargeted achievement goals.

The first step in using the framework is to make sure that allinterested parties are familiar with its content. Then ananalysis to determine what needs to be done should becompleted. Once the district identifies the tasks that need tobe completed, a plan for structuring committees and a time-linefor completing the tasks should be designed.

The framework is intended for use by all districts. While thewriters used the structure of a middle-sized district as a frameof reference, the content and processes it describes are equallyimportant for large and small districts. Although privateschools are not bound by the core curriculum requirements ofthe Michigan School Code, they may find the framework usefulas a tool for curriculum development. Large districts, smalldistricts, private schools, and public school academies maychoose to modify the process to reflect their organizationalstructures. The number and size of committees needed toimplement the framework will vary from district to district, butthe task will remain the same: to align curriculum, instruction,assessment, and professional development for the purpose ofincreasing student achievement of rigorous content standards.

Assessment System

Introduction

• Guidelines for the Professional Development ofTeachers of English Language Arts

• Mathematics Professional Development Component• Powerful and Authentic Social Studies Professional

Development Package• Powerful and Authentic Social Studies Standards

for Teaching

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TIER I

These documents are toolkits designedto help districts achieve alignment whiledeveloping curriculum, instruction, andassessment consistent with their standardsand benchmarks. (For availability,see http://www.mde.state.mi.us)

These resources are specific to each content area andhelp clarify and strengthen the curriculumdevelopment processes described in the first two tiers.(For availability, see http:/www.mde.state.mi.us)

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TIER II TIER III

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This document introduces the frameworkstandards and describes the componentsand processes needed to develop K-12curricula.

Discrepancy Analysis1. Analysis of Curriculum2. Analysis of Instruction3. Analysis of Assessment4. Analysis of Professional Development5. Analysis of School Operations

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• Michigan Geography Framework Poster• Analysis of Mathematics Instructional and

Assessment Materials• Mathematics Research Component• Social Studies Curriculum Planning Guide

Connecting with the LearnerTechnologyCurriculum IntegrationConnecting School-To-Work

Local Assessment System

Parent and Business LeaderGuides (under development)

Professional Development

Planning

Standards & Benchmarks

Teaching and Learning

Michigan Curriculum Framework

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Executive Summaries ofToolkits and Other Resources

• Science Education Guidebook• New Directions Science Teaching Units• Profiles of Early Literacy Instruction in Primary

Classrooms• A Collection of English Language Arts Vignettes• Readings from the Demonstration Projects• Mathematics Teaching and Learning Sample

Activities• Standards Based Geography Units• Powerful & Authentic Social Studies: Standards

for Teaching and Learning

• Mathematics Assessment Framework• The Model Assessment Items Resource Book

(Science)• Michigan Assessment Prototypes for Geography• Plan for Statewide Assessment of Social Studies• Social Studies Assessment Guide

Section II:

Content Standards& Draft Benchmarks

A t its July 19, 1995, meeting, the Michigan State Board ofEducation unanimously adopted the model contentstandards for curriculum. The content standards provide

descriptions of what students should know and be able to do inthe subject areas of English language arts, social studies,mathematics and science. In addition, benchmarks in each ofthe content areas were drafted to further clarify the contentstandards. The standards and benchmarks are not a statecurriculum, but are specifically designed to be used by localdistricts as they develop their curricula.

The model content standards for curriculum and accompanyingdraft benchmarks will assist in the development of qualitycomprehensive local curricula, foster local diversity inestablishing high quality learning expectations, and giveparents, as customers within an education marketplace, anaccountability tool. In addition, they will serve as a basis forrevisions and new test development for the MEAP and HighSchool Proficiency Tests. They will provide a commondenominator to determine how well students are performingand will assure that all students are measured on the sameknowledge and skills using the same method of assessment.

Model Content Standards for CurriculumThe model content standards for curriculum were revised, basedupon public input, by writing teams in each of thecontent areas. The Curriculum Framework Joint SteeringCommittee which includes representatives from business,education, government, professional organizations, and laborwas extensively consulted in the development and revision ofthe standards. The State Board also made revisions to thestandards during its July 19, 1995, meeting.

BenchmarksThe draft benchmarks provide indicators of studentexpectations at various developmental levels includingelementary, middle school, and high school. The working draftbenchmarks are the most current versions and represent theefforts made by teams of subject area specialists with inputfrom over 2,000 Michigan citizens.

Field reviews on the benchmarks were held to gather commentsfrom teachers, parents, administrators, and communitymembers. Additionally, the benchmarks have been reviewed forconsistency with the model content standards for curriculum.The department continues the process of field testing thebenchmarks at the Curriculum Framework projects’ schooldemonstration sites.

DRAFT BENCHMARKSONTENT STANDARDS &C

Because the benchmarks are continuously being revised tofurther clarify the standards and reflect the learning needs ofMichigan’s students, districts should consult electronic versionsin order to ensure that they are working with the most currentrevisions. The model content standards for curriculum and theworking draft benchmarks are available through the Internet onthe Department of Education’s gopher server (gopher://gopher.mde.state.mi.us), and through the World Wide Web(http://cdp.mde.state.mi.us).

Please note: The standards and benchmarks have been codedso that districts can more easily refer to them in theircurriculum, instruction, assessment, and professionaldevelopment activities. The numbering system will be usefulas districts conduct discrepancy analyses as part of thecontinuous school improvement decision-making process. Onesystem for numbering has been applied to all of the contentstandards and benchmarks in an attempt to provideconsistency and facilitate curriculum alignment.

The numbering system begins with the subject area. Englishlanguage arts is assigned the code of ELA; Mathematics, MAT;Social Studies, SOC; and Science, SCI. The first numeral in thecode is a Roman numeral; it identifies the content area strand.The second numeral is an Arabic numeral; it identifies acontent standard. The letters that follow the content standardsignify cluster levels such as: E (elementary), EE (earlyelementary), LE (later elementary), MS (middle school), and HS(high school). The third numeral is another Arabicnumeral; it identifies a benchmark.

The coding system has been used to identify standards andbenchmarks in the sections on assessment and teaching andlearning. Please note that although one coding system is used,each set of content areas and benchmarks has some uniquecharacteristics. For instance, the English language artsstandards and benchmarks do not identify strands. It is veryimportant to study the standards and benchmarks carefully sothey can be used to their full advantage.

SOC.II.1.LE.1

SOC. Social Studies(Subject Area)

II. Geographic Perspective(Content Strand)

1. All students willdescribe, compare, andexplain the locations andcharacteristics of places,cultures, and settlements.(People, Places andCultures) (ContentStandard)

LE. Later Elementary

1. Locate and describecultures and compare thesimilarities and differencesamong the roles of women,men and families. (LaterElementary Benchmark)

Mathematics

VisionStatement

Mathematics is the science of patterns and relationships. It isthe language and logic of our technological world.Mathematical power is the ability to explore, to conjecture, toreason logically and to use a variety of mathematical methodseffectively to solve problems. The ultimate goal of mathematicseducation is for all students to develop mathematical power toparticipate fully as a citizen and worker in our contemporaryworld.

A mathematically powerful individual shouldbe able to:

J reason mathematically;

J communicate mathematically;

J problem solve using mathematics; and,

J make connections within mathematics and betweenmathematics and other fields.

The fifteen content standards have beencategorized into the following six strands:

I. Patterns, Relationships, and Functions

II. Geometry and Measurement

III. Data Analysis and Statistics

IV. Number Sense and Numeration

V. Numerical and Algebraic Operations andAnalytical Thinking

VI. Probability and Discrete Mathematics

OVERVIEW OF THE MATHEMATICS CONTENTSTRANDS

Strand I. Patterns, Relationships, and FunctionsPatterns, relationships and functions comprise one of the mostimportant themes in the study of mathematics. Mathematicalthinking begins with the recognition of similarities amongobjects or events, proceeds to generalization and abstraction,and culminates in the ability to understand, explain and makepredictions. Contexts that exhibit structure and regularityprovide rich opportunities for describing the physical world,studying mathematics and solving problems.

Standard I.1 PatternsStudents recognize similarities and generalize patterns, usepatterns to create models and make predictions, describe thenature of patterns and relationships, and constructrepresentations of mathematical relationships.

Wherever there is mathematics there are patterns, andwherever there are patterns there is mathematics. Patternsare regularities or similarities that characterize sets ofnumbers, shapes, graphs, tables or other mathematicalobjects. Mathematicians look for patterns in everything theydo; thus, mathematics is frequently defined as the science ofpatterns. In studying mathematics, students learn torecognize, describe, analyze and create patterns, to extend andgeneralize patterns, to create mathematical models based onobserved patterns, and to predict the behavior of real-worldphenomena based on such observed patterns. They learn tocommunicate the nature of mathematical patterns andrelationships in various ways including words, physical models,diagrams, tables, charts, graphs, and equations. Since eachrepresentation highlights different aspects of the patterns andrelationships, students must be able to construct multiplerepresentations of mathematical relationships and to translateamong them.

Standard I.2 Variability and ChangeStudents describe the relationships among variables, predictwhat will happen to one variable as another variable ischanged, analyze natural variation and sources of variability,and compare patterns of change.

Variability and change are as fundamental to mathematics asthey are to the physical world, and an understanding of theconcept of a variable is essential to mathematical thinking.Students must be able to describe the relationships amongvariables, to predict what will happen to one variable asanother variable is changed, and to compare different patternsof change. The study of variability and change provides a basisfor making sense of the world and of mathematical ideas.

Strand II. Geometry and MeasurementWe live in a three-dimensional world. In order to interpret andmake sense of that world, students need both analytical andspatial abilities. Geometry and measurement, which involvenotions of shape, size, position, and dimension, are usedextensively to describe and understand the world around us.

Standard II.1 Shape and Shape RelationshipsStudents develop spatial sense, use shape as an analytic anddescriptive tool, identify characteristics and define shapes,identify properties and describe relationships among shapes.

Spatial sense is developed when students recognize, draw,construct, visualize, compare, classify and transform geometricshapes in both two and three dimensions. They learn toidentify those characteristics that are necessary to define agiven shape, and they can differentiate one shape fromanother. Students also develop an awareness of the propertiesof a shape and of the relationships among shapes. This

includes hierarchical classifications of shapes (e.g., all squaresare rhombuses), relationships among components of a shape(e.g., opposite sides of a rectangle are parallel), symmetries ofa shape, congruence and similarity.

Standard II.2 PositionStudents identify locations of objects, identify location relativeto other objects, and describe the effects of transformations(e.g., sliding, flipping, turning, enlarging, reducing) on anobject.

Position refers to the location of physical objects or points inspace as well as to the relative locations and positions ofobjects, points, lines, planes and other geometric elements. Itincludes such notions as betweenness, collinearity andcoordinates in two and three dimensions, as well as the locus ofa point as it moves through space and the location of specialpoints.

Standard II.3 MeasurementStudents compare attributes of two objects or of one objectwith a standard (unit), and analyze situations to determinewhat measurement(s) should be made and to what level ofprecision.

Measurement reflects the usefulness and practicality ofmathematics and puts students in touch with the world aroundthem. Measurement requires the comparison of an attribute(distance, surface, capacity, mass, time, temperature) betweentwo objects or to a known standard, the assignment of anumber to represent the comparison, and the interpretation ofthe results. Measurement also introduces students to theimportant concepts of precision, approximation, tolerance,error and dimension.

Strand III. Data Analysis and StatisticsWe live in a sea of information. In order not to drown in thedata that inundate our lives every day, we must be able toprocess and transform data into useful knowledge. The abilityto interpret data and to make predictions and decisions basedon data is an essential basic skill for every individual.

Standard III.1 Collection, Organization and Presentationof DataStudents collect and explore data, organize data into a usefulform, and develop skill in representing and reading datadisplayed in different formats.

Knowing what data to collect and where and how to collectthem is the starting point of quantitative literacy. Themathematics curriculum should capitalize on students’ naturalcuriosity about themselves and their surroundings to motivatethem to collect and explore interesting statistics andmeasurements derived from both real and simulated situations.Once the data are gathered, they must be organized into auseful form, including tables, graphs, charts and pictorialrepresentations. Since different representations highlightdifferent patterns within the data, students should develop skillin representing and reading data displayed in different

formats, and they should discern when one particularrepresentation is more desirable than another.

Standard III.2 Description and InterpretationStudents examine data and describe characteristics of adistribution, relate data to the situation from which they arose,and use data to answer questions convincingly andpersuasively.

Students must be able to examine data and describe salientcharacteristics of the distribution. They also must be able torelate the data to the physical situation from which they arose.Students should use the data to answer key questions and toconvince and persuade.

Standard III.3 Inference and PredictionStudents draw defensible inferences about unknown outcomes,make predictions, and identify the degree of confidence theyhave in their predictions.

Based on known data, students should be able to drawdefensible inferences about unknown outcomes. They shouldbe able to make predictions and to identify the degree ofconfidence that they place in their predictions.

Strand IV. Number Sense and NumerationNumber sense is to mathematics what vocabulary is tolanguage. Students must learn to quantify and measure,concretely at first and increasingly more abstractly as theymature. They also must develop an understanding ofnumeration systems and of the structure of such systems.They must learn to estimate mathematical quantities and torepresent and communicate mathematical ideas in thelanguage of mathematics.

Standard IV.1 Concepts and Properties of NumbersStudents experience counting and measuring activities todevelop intuitive sense about numbers, develop understandingabout properties of numbers, understand the need for andexistence of different sets of numbers, and investigateproperties of special numbers.

Fundamental questions like “What is a number?” or “What isthree?” can be deceptively difficult to answer. Students requireextensive involvement with concrete experiences of countingand measuring in order to develop an intuitive sense aboutnumber. Through both informal and formal means, studentsdevelop understanding about important properties of numberssuch as even vs. odd, whole number vs. fraction, positive vs.negative. They understand the existence of different sets ofnumbers (whole numbers, integers, rationals, reals, …) and theproperties of special numbers such as 0, 1, π, or the inverse ofa number.

Standard IV.2 Representation and Uses of NumbersStudents recognize that numbers are used in different wayssuch as counting, measuring, ordering and estimating,understand and produce multiple representations of a number,and translate among equivalent representations.

Students recognize that numbers are used in different wayssuch as to answer the questions “How many?” (counting), “Howmuch?” (measuring), and “Which one?” (ordering). Theyunderstand that a numerical quantity can be represented inmany different ways, and they can produce multiplerepresentations of numbers (e.g., fractions, decimals, andpercents in the middle grades; vectors and coordinaterepresentations in later years) and they can translate easilyamong equivalent representations. As students mature fromthe middle school on, they develop a solid understanding ofboth linearity and proportionality.

Standard IV.3 Number RelationshipsStudents investigate relationships such as equality, inequality,inverses, factors and multiples, and represent and comparevery large and very small numbers.

Students develop understanding of important relationshipsamong numbers including the relationships of (=, ≠) and (<,␣ >);of opposites (additive inverses) and reciprocals (multiplicativeinverses); of factors and multiples; of primes, composites, andrelatively prime numbers; of powers and roots. Theyunderstand and can represent very large and very smallnumbers and can compare the orders of magnitude ofnumbers.

Strand V. Numerical and Algebraic Operationsand Analytical ThinkingThe ability to represent quantitative situations with algebraicsymbolism, numerical operations and algebraic thinking isessential to solving problems in significant contexts andapplications. The concepts of number and variable and theirsymbolic representation and manipulation are central to theunderstanding of arithmetic and its generalization in algebra.The contemporary applications of mathematics in virtuallyevery field of work and study rely on algebraic and analyticthinking and communication as fundamental tools.

Standard V.1 Operations and Their PropertiesStudents understand and use various types of operations (e.g.,addition, subtraction, multiplication, division) to solveproblems.

The ultimate reason for mastering the operations of arithmeticand algebra is to solve problems. To that end, understandingthe basic computational operations and their algorithms isessential for competence in mathematics, but the emphasismust be on understanding and using the operations, not onmemorizing algorithms. In computation, understanding andaccuracy are always more important than speed.Understanding the operations requires the concomitantunderstanding and application of the properties of thoseoperations, and it involves knowing what operations to use in aparticular situation. There is no one way to perform acalculation. Students must be competent in performingcalculations, but they need not have a rigid adherence to onealgorithm. Methods of computation include proficiency withmental calculation, paper and pencil, and calculators; the

ability to represent computations with manipulatives andgeometric models; and the discernment of which computationalmethod to use in a given situation. Computational methodsalso involve estimating and assessing the reasonableness of theresults of a computation.

Standard V.2 Algebraic and Analytic ThinkingStudents analyze problems to determine an appropriate processfor solution, and use algebraic notations to model or representproblems.

Mathematical representations allow us to visualize andunderstand problems. These representations may benumerical, literal, symbolic, graphical, pictorial or physical.Facility with multiple representations of numerical andalgebraic concepts and relationships is essential tomathematical competence. This includes the development of“symbol sense” as well as “number sense” and theunderstanding that the notion of solution involves a process aswell as a product. Thus, the solution of a mathematicalproblem requires both an understanding of the question forwhich an answer is sought and the development of a strategyto obtain that answer. The context of the problem determinesthe nature and the degree of precision of the required solution.The increasing use of quantitative methods in all disciplines hasmade algebra the fundamental tool for mathematicalapplications. Algebraic thinking is learned most effectivelywhen it is studied in the context of applications, bothmathematical and real-world, that reveal the power of algebrato model real problems and to generalize to new situations.Students should use algebraic techniques to analyze anddescribe relationships, to model problem situations, and toexamine the structure of mathematical relationships. Thealgebra curriculum should employ contemporary technology,including spreadsheets and graphical analysis, to emphasizeconceptual understanding of algebra and analytic thinking assophisticated means of representation and as powerfulproblem-solving tools.

Strand VI. Probability and DiscreteMathematicsContemporary uses of mathematics demand that studentslearn to deal with uncertainty, to make informed decisionsbased on evidence and expectations, to exercise criticaljudgment about conclusions drawn from data, and to applymathematical models to real-world phenomena. Thetechnological world in which we live also depends uponinformation and the communication of information and uponapplications of systems with separate (discrete) entities. Topicsof discrete mathematics such as counting and permutationproblems, matrix operations, vertex-edge networks, andrelationships among finite sets have significant real-worldapplications that students will encounter in diverse fields ofwork and study.

Standard VI.1 ProbabilityStudents develop an understanding of the notion of certaintyand of probability as a measure of the degree of likelihood that

can be assigned to a given event based on the knowledgeavailable, and make critical judgments about claims that aremade in probabilistic situations.

Dealing with uncertainty and making predictions and decisionsin the face of uncertainty are essential skills for coping with themodern world. Students must develop an understanding of thenotion of uncertainty and of probability as a measure of thedegree of likelihood that can be assigned to a given event basedon the knowledge available. They also must be able to makecritical judgments about claims that are made in probabilisticsituations.

Standard VI.2 Discrete MathematicsStudents investigate practical situations such as scheduling,routing, sequencing, networking, organizing and classifying,and analyze ideas like recurrence relations, induction, iteration,and algorithm design.

Discrete (discontinuous) mathematics has grown in significancein recent years and today has applications in many importantpractical situations such as scheduling, routing, sequencing,networking, organizing and classifying. Important ideas likerecurrence relations, induction and algorithm design also havepractical applications in a variety of fields. Computers, whichare finite, discrete machines, require an understanding ofdiscrete mathematics for the solution of problems usingcomputer methods.

Section II • Michigan Content Standards and Draft Benchmarks 53

MATHEMATICSCONTENT STANDARDS AND WORKING DRAFT BENCHMARKS

I. Patterns, Relationships and Functions

Content Standard 1: Students recognize similarities and generalize patterns, use patterns tocreate models and make predictions, describe the nature of patterns and relationships, andconstruct representations of mathematical relationships. (Patterns)

Elementary Middle School High School

1. Recognize, describe and extendnumerical and geometricpatterns.

1. Describe, analyze andgeneralize patterns arising in avariety of contexts and expressthem in general terms.

1. Analyze and generalizemathematical patterns includingsequences, series and recursivepatterns.

2. Represent and record patternsand relationships in a variety ofways including tables, chartsand pictures.

2. Represent and record patternsin a variety of ways includingtables, charts and graphs, andtranslate between variousrepresentations.

2. Analyze, interpret and translateamong representations ofpatterns including tables,charts, graphs, matrices andvectors.

3. Use patterns to describe real-world phenomena.

3. Use patterns and theirgeneralizations to make andjustify inferences andpredictions.

3. Study and employ mathematicalmodels of patterns to makeinferences, predictions anddecisions.

4. Explore various types ofnumeric and geometric patterns(repeating, growing, shrinking).

4. Explore and describe visual andnumeric patterns, includinglinear expressions, near-linearpatterns and symmetric andspatial patterns.

4. Explore patterns (graphic,numeric, etc.) characteristic offamilies of functions; explorestructural patterns withinsystems of objects, operationsor relations.

5. Apply their experiences withpatterns to help solve problemsand explore new content.

5. Use patterns andgeneralizations to solveproblems and explore newcontent.

5. Use patterns and reasoning tosolve problems and explore newcontent.

Content Standard 2: Students describe the relationships among variables, predict what will happento one variable as another variable is changed, analyze natural variation and sources of variability,and compare patterns of change. (Variability and Change)


1. Recognize change andvariability when it occurs in avariety of settings.

1. Identify and describe the natureof change; recognize change inmore abstract and complexsituations and explore differentkinds of change and patterns ofvariation.

1. Identify and describe the natureof change and begin to use themore formal language such asrate of change, continuity, limit,distribution and deviation.

2. Recognize that change is oftenpredictable, but variable, andthat patterns emerge that helpto describe the change.

2. Connect an initial state to a finalstate and generalize a rule thatdescribes a pattern of change.

2. Develop a mathematicalconcept of function andrecognize that functions displaycharacteristic patterns ofchange (e.g., linear, quadratic,exponential).

54 Section II • Michigan Content Standards and Draft Benchmarks

3. Explore change, and realize thatchanges are frequentlyinterdependent.

3. Begin to investigate applicationsin bivariate data and linearrelationships and explorequestions of what will happen toone quantity if another variableis changed.

3. Expand their understanding offunction to include non-linearfunctions, composition offunctions, inverses of functions,and piecewise- and recursively-defined functions.

4. Use tables, charts, opensentences and hands-on modelsto represent change andvariability.

4. Represent variability or changeby ordered pairs, tables, graphsand equations.

4. Represent functions usingsymbolism such as matrices,vectors and functionalrepresentation (f(x)).

5. Begin to describe anddifferentiate between types ofrelationships, especiallyrepeating, growing andshrinking patterns.

5. Differentiate between functionsand relationships such as linearvs. not linear or continuous vs.non-continuous.

5. Differentiate and analyzeclasses of functions includinglinear, power, quadratic,exponential, circular andtrigonometric functions, andrealize that many differentsituations can be modeled by aparticular type of function.

6. Explore variability and changein a variety of contexts,investigations and problems.

6. Continue to explorerelationships arising frominteresting contexts and usevariables and relationships tosolve mathematical problems.

6. Increase their use of functionsand mathematical models tosolve problems in context.

II. Geometry and Measurement

Content Standard 1: Students develop spatial sense, use shape as an analytic and descriptive tool,identify characteristics and define shapes, identify properties and describe relationships amongshapes. (Shape and Shape Relationships)


1. Recognize and name familiarshapes in one, two and threedimensions such as lines,rectangles and spheres andinformally discuss the shape of agraph.

1. Distinguish among shapes anddifferentiate between examplesand non-examples of shapesbased on their properties;generalize about shapes ofgraphs and data distributions.

1. Use shape to identify plane andsolid figures, graphs, loci,functions and data distributions.

2. Describe the attributes offamiliar shapes.

2. Generalize the characteristicsof shapes and apply theirgeneralizations to classes ofshapes.

2. Determine necessary andsufficient conditions for theexistence of a particular shapeand apply those conditions toanalyze shapes.

3. Compare, sort and classifyfamiliar shapes.

3. Derive generalizations aboutshapes and apply thosegeneralizations to developclassifications of familiarshapes.

3. Use transformational,coordinate or synthetic methodsto verify (prove) thegeneralizations they have madeabout properties of classes ofshapes.

4. Draw and build familiar shapes. 4. Construct familiar shapes usingcoordinates, appropriate tools(including technology),sketching and drawing two- andthree-dimensional shapes.

4. Draw and construct shapes intwo and three dimensions andanalyze and justify the steps oftheir constructions.


5. Explore ways to combine,dissect and transform shapes.

5. Combine, dissect and transformshapes.

5. Study transformations of shapesusing isometries, sizetransformations and coordinatemappings.

6. Recognize parallel andperpendicular line segments andfigures that have similarityand/or congruence.

6. Generalize about the commonproperties of similar, congruent,parallel and perpendicularshapes and verify theirgeneralizations informally.

6. Compare and analyze shapesand formally establish therelationships among them,including congruence, similarity,parallelism, perpendicularityand incidence.

7. Use shape, shape properties andshape relationships to describethe physical world and to solveproblems.



Content Standard 2: Students identify locations of objects, identify location relative to otherobjects, and describe the effects of transformations (e.g., sliding, flipping, turning, enlarging,reducing) on an object. (Position)


1. Locate and describe objects interms of their position, includingfront, back, inside, outside,right, left, over, under, next to,between and locations on thenumber line, on a coordinategraph and on a map.

1. Locate and describe objects interms of their position, includingcompass directions, Cartesiancoordinates, latitude andlongitude and midpoints.

1. Locate and describe objects interms of their position, includingpolar coordinates, three-dimensional Cartesiancoordinates, vectors and limits.

2. Locate and describe objects interms of their orientation,direction and relative position,including up, down, front, back,N- S- E- W, flipped, turned,translated; recognizesymmetrical objects andidentify their lines of symmetry.

2. Locate and describe objects interms of their orientation andrelative position, includingcoincident, collinear, parallel,perpendicular; differentiatebetween fixed (e.g., N- S- E- W)and relative (e.g., right-left)orientations; recognize anddescribe examples of bilateraland rotational symmetry.

2. Locate and describe objects interms of their orientation andrelative position, includingdisplacement (vectors), phaseshift, maxima, minima andinflection points; give precisemathematical descriptions ofsymmetries.

3. Explore what happens to thesize, shape and position of anobject after sliding, flipping,turning, enlarging or reducing it.

3. Describe translations,reflections, rotations anddilations using the language oftransformations and employtransformations to verifycongruence of figures.

3. Give precise mathematicaldescriptions of transformationsand describe the effects oftransformations on size, shape,position and orientation.

4. Locate the position of points orobjects described by two ormore conditions; locate all thepoints (locus) that satisfy agiven condition.

4. Describe the locus of a point bya rule or mathematicalexpression; trace the locus of amoving point.

5. Use concepts of position,direction and orientation todescribe the physical world andto solve problems.




Content Standard 3: Students compare attributes of two objects, or of one object with a standard(unit), and analyze situations to determine what measurement(s) should be made and to what levelof precision. (Measurement)


1. Compare attributes of objects;develop standard units ofmeasurement; and select anduse standard tools formeasurement.

1. Select and use appropriatetools; measure objects usingstandard units in both the metricand common systems andmeasure angles in degrees.

1. Select and use appropriatetools; make accuratemeasurements using both metricand common units, and measureangles in degrees and radians.

2. Identify the attribute to bemeasured and select theappropriate unit ofmeasurement for length, mass(weight), area, perimeter,capacity, time, temperature andmoney.

2. Identify the attribute to bemeasured and select theappropriate unit ofmeasurement for length, mass(weight), time, temperature,perimeter, area, volume andangle.

2. Continue to make and applymeasurements of length, mass(weight), time, temperature,area, volume, angle; classifyobjects according to theirdimensions.

3. Develop strategies forestimating measures andcompare the estimates to theresults of the measurement;decide if an estimate is “a goodestimate.”

3. Estimate measures with aspecified degree of accuracyand decide if an estimate or ameasurement is “close enough.”

3. Estimate measures with aspecified degree of accuracyand evaluate measurements foraccuracy, precision andtolerance.

4. Explain the meaning ofmeasurements and recognizethat the number of units it takesto measure an object is relatedto the size of the unit.

4. Interpret measurements andrecognize that two objects mayhave the same measurement onone attribute (e.g., area) but notnecessarily on another (e.g.,perimeter).

4. Interpret measurements andexplain how changes in onemeasure may affect othermeasures.

5. Explore scale drawings, modelsand maps and relate them tomeasurements of real objects.

5. Use proportional reasoning andindirect measurements to drawinferences.

5. Use proportional reasoning andindirect measurements,including applications oftrigonometric ratios, to measureinaccessible distances and todetermine derived measuressuch as density.

6. Apply measurement to describethe real world and to solveproblems.



III. Data Analysis and Statistics

Content Standard 1: Students collect and explore data, organize data into a useful form, anddevelop skill in representing and reading data displayed in different formats. (Collection,Organization and Presentation of Data)


1. Collect and explore datathrough counting, measuringand conducting surveys andexperiments.

1. Collect and explore datathrough observation,measurement, surveys, samplingtechniques and simulations.

1. Collect and explore datathrough observation,measurement, surveys, samplingtechniques and simulations.

2. Organize data using concreteobjects, pictures, tallies, tables,charts, diagrams and graphs.

2. Organize data using tables,charts, graphs, spreadsheetsand data bases.

2 Organize data using tables,charts, graphs, spreadsheetsand data bases.


3. Present data using a variety ofappropriate representations andexplain the meaning of the data.

3. Present data using a variety ofappropriate representations andexplain why one representationis preferred over another orhow a particular representationmay bias the presentation.

3. Present data using the mostappropriate representation andgive a rationale for their choice;show how certainrepresentations may skew thedata or bias the presentation.

4. Identify what data are needed toanswer a particular question orsolve a given problem, anddesign and implement strategiesto obtain, organize and presentthose data.

4. Identify what data are needed toanswer a particular question orsolve a given problem, anddesign and implement strategiesto obtain, organize and presentthose data.

4. Identify what data are needed toanswer a particular question orsolve a given problem anddesign and implement strategiesto obtain, organize and presentthose data.

Content Standard 2: Students examine data and describe characteristics of a distribution, relatedata to the situation from which they arose, and use data to answer questions convincingly andpersuasively. (Description and Interpretation)


1. Read and explain data they havecollected and organizedthemselves and progress toreading data from othersources.

1. Critically read data from tables,charts or graphs and explain thesource of the data and what thedata represent.

1. Critically read data from tables,charts or graphs and explain thesource of the data and what thedata represent.

2. Describe the shape of the datausing informal language.

2. Describe the shape of a datadistribution and identify thecenter, the spread, correlationsand any outliers.

2. Describe the shape of a datadistribution and determinemeasures of central tendency,variability and correlation.

3. Draw, explain and justifyconclusions, such as trendsbased on data.

3. Draw, explain and justifyconclusions based on data.

3. Use the data and theircharacteristics to draw andsupport conclusions.

4. Raise and answer questionsabout the source, collection,organization and presentation ofdata, as well as the conclusionsdrawn from the data; explorebiases in the data.

4. Critically question the sourcesof data; the techniques used tocollect, organize and presentdata; the inferences drawn fromthe data; and the possiblesources of bias in the data ortheir presentation.

4. Critically question the sourcesof data; the techniques used tocollect, organize and presentdata; the inferences drawn fromthe data; and the sources of biasand measures taken to eliminatesuch bias.

5. Formulate questions andproblems and gather andinterpret data to answer thosequestions.



Content Standard 3: Students draw defensible inferences about unknown outcomes, makepredictions, and identify the degree of confidence they have in their predictions. (Inference andPrediction)


1. Make and test hypotheses. 1. Make and test hypotheses. 1. Make and test hypotheses.

2. Conduct surveys, samplings andexperiments to solve problemsand answer questions of interestto them.

2. Design experiments to modeland solve problems usingsampling, simulations andcontrolled investigations.

2. Design investigations to modeland solve problems; also employconfidence intervals and curvefitting in analyzing the data.


3. Formulate and communicatearguments and conclusionsbased on data and evaluate theirarguments and those of others.



4. Make and explain predictionsbased on data.

4. Make predictions and decisionsbased on data, includinginterpolations andextrapolations.

4. Make predictions and decisionsbased on data, includinginterpolations andextrapolations.

5. Make predictions to answerquestions and solve problems.

5. Employ investigations,mathematical models andsimulations to make inferencesand predictions to answerquestions and solve problems.

5. Employ investigations,mathematical models, andsimulations to makeinferences and predictions toanswer questions and solveproblems.

IV. Number Sense and Numeration

Content Standard 1: Students experience counting and measuring activities to develop intuitivesense about numbers, develop understanding about properties of numbers, understand the needfor and existence of different sets of numbers, and investigate properties of special numbers.(Concepts and Properties of Numbers)


1. Develop an understanding ofwhole numbers and read, writeand count using whole numbers;investigate basic concepts offractions and decimals.

1. Develop an understanding ofintegers and rational numbersand represent rational numbersin both fraction and decimalform.

1. Develop an understanding ofirrational, real and complexnumbers.

2. Investigate and develop anunderstanding of the base-10place-value system.

2. Extend their understanding ofnumeration systems to includedecimal numeration, scientificnumeration and non-decimalnumeration systems.

2. Use the (a+bi) and polar formsof complex numbers.

3. Develop an understanding of theproperties of numbers (e.g.,order) and of the properties ofthe special numbers 0 and 1.

3. Develop an understanding of theproperties of the integer andrational number systems (e.g.,order, density) and of theproperties of special numbersincluding 0, 1 and π, and theadditive and multiplicativeinverses.

3. Develop an understanding of theproperties of the real andcomplex number systems and ofthe properties of specialnumbers including π, i, e, andconjugates.

4. Apply their understanding ofnumber systems to model andsolve problems.

4. Apply their understanding ofnumber systems to model andsolve mathematical and appliedproblems.

4. Apply their understanding ofnumber systems to model, andsolve mathematical and appliedproblems.

Content Standard 2: Students recognize that numbers are used in different ways such as counting,measuring, ordering and estimating, understand and produce multiple representations of a number, andtranslate among equivalent representations. (Representation and Uses of Numbers)


1. Represent whole numbers,fractions and decimals usingconcrete, pictorial and symbolicrepresentations.

1. Give geometric representationsof fractions, prime andcomposite numbers, triangularand square numbers, and othernumber concepts; representrational numbers and integerson the number line.

1. Give decimal representations ofrational and irrational numbersand coordinate and vectorrepresentations of complexnumbers.


2. Explore and recognize differentrepresentations for the samenumber and explain why theyare the same.

2. Recognize equivalentrepresentations of a number,especially fractions, decimalsand percents, and translatefreely among representations.

2. Develop an understanding ofmore complex representationsof numbers, includingexponential and logarithmicexpressions, and select anappropriate representation tofacilitate problem solving.

3. Investigate ways numbers areused (e.g., counting, ordering,naming, locating, measuring).

3. Distinguish between numbersthat are used for counting,numbers that are used forordering, numbers that are usedfor measuring and numbers thatare used for naming.

3. Determine when to use rationalapproximations and the exactvalues of numbers such as e, πand the irrational.

4. Develop strategies forestimating quantity and evaluatethe reasonableness of theirestimates.

4. Develop and refine strategiesfor estimating quantities,including fractional quantities,and evaluate the reasonablenessand appropriateness of theirestimates.

4. Apply estimation in increasinglycomplex situations.

5. Select appropriate numbers andrepresentations in order tosolve problems.

5. Select appropriaterepresentations for numbers,including integers and rationalnumbers, in order to simplifyand solve problems.

5. Select appropriaterepresentations for numbers,including representations ofrational and irrational numbersand coordinate and vectorrepresentations of complexnumbers, in order to simplifyand solve problems.

Content Standard 3: Students investigate relationships such as equality, inequality, inverses,factors and multiples, and represent and compare very large and very small numbers. (NumberRelationships)


1. Compare and order numbersusing “equal,” “less than” or“greater than.”

1. Compare and order integers andrational numbers using relationsof equality and inequality.

1. Compare and order realnumbers and compare rationalapproximations to exact values.

2. Use part-whole relationships toexplore numbers, developnumber concepts andunderstand computation.

2. Express numerical comparisonsas ratios and rates.

2. Express numerical comparisonsas ratios and rates.

3. Classify numbers as even or oddand explore concepts of factorsand multiples.

3. Distinguish between prime andcomposite numbers; identifyfactors, multiples, commonfactors and multiples, andrelatively prime numbers; andapply divisibility tests tonumbers.

3. Extend the relationships ofprimes, factors, multiples anddivisibility in an algebraicsetting.

4. Explain the meaning of powersand roots of numbers and usecalculators to compute powersand square roots.

4. Express number relationshipsusing positive and negativerational exponents, logarithmsand radicals.

5. Apply their understanding ofnumber relationships in solvingproblems.




V. Numerical and Algebraic Operations and Analytical Thinking

Content Standard 1: Students understand and use various types of operations(e.g., addition, subtraction, multiplication, division) to solve problems. (Operations and theirProperties).


1. Use manipulatives to modeloperations with numbers;develop their own methods ofrecording operations; and relatetheir models and recordings tostandard symbolic expressionsand algorithms.

1. Use manipulatives and diagramsto model operations and theirinverses with integers andrational numbers and relate themodels to their symbolicexpressions.

1. Present and explain geometricand symbolic models foroperations with real andcomplex numbers and algebraicexpressions.

2. Develop and apply theappropriate method ofcomputation from among mentalcomputation, estimation,paper-and-pencil or calculators;explain why they are choosing amethod and how they knowwhich operations to perform ina given situation.

2. Compute with integers, rationalnumbers and simple algebraicexpressions using mentalcomputation, estimation,calculators and paper-and-pencil; explain what they aredoing and how they know whichoperations to perform in a givensituation.

2. Compute with real numbers,complex numbers, algebraicexpressions, matrices andvectors using technology and,for simple instances, with paper-and-pencil algorithms.

3. Explore properties of operations(e.g., commutative anddistributive properties) and giveexamples of how they use thoseproperties.

3. Describe the properties ofoperations with rationals andintegers (e.g., closure;associative, commutative anddistributive properties) and giveexamples of how they use thoseproperties.

3. Describe the properties ofoperations with numbers,algebraic expressions, vectorsand matrices, and makegeneralizations about theproperties of givenmathematical systems.

4. Apply operations efficiently andaccurately in solving problems.

4. Efficiently and accurately applyoperations with integers,rational numbers and simplealgebraic expressions in solvingproblems.

4. Efficiently and accurately applyoperations with real numbers,complex numbers, algebraicexpressions, matrices andvectors in solving problems.

Content Standard 2: Students analyze problems to determine an appropriate process for solution,and use algebraic notations to model or represent problems. (Algebraic and Analytic Thinking)


1. Write and solve open sentences(e.g., ◊ + ∆ = 5) and writestories to fit the open sentence.

1. Read and write algebraicexpressions; develop originalexamples expressed verballyand algebraically; simplifyexpressions and translatebetween verbal and algebraicexpressions; and solve linearequations and inequalities.

1. Identify important variables in acontext, symbolize them andexpress their relationshipsalgebraically.

2. Explore algebraic concepts withmanipulatives such as balancescales, tables of input andoutput, and pictorialrepresentations of problems.

2. Represent algebraic conceptswith geometric models (e.g.,algebra tiles), physical models(e.g., balance beam), tables andgraphs; and write algebraicexpressions to correspond tothe multiple representations.

2. Represent algebraic conceptsand relationships with matrices,spreadsheets, diagrams, graphs,tables, physical models, vectors,equations and inequalities; andtranslate among the variousrepresentations.


3. Find replacements for thevariable(s) in open sentences.

3. Solve linear equalities andinequalities using algebraic andgeometric methods, and use thecontext of the problem tointerpret and explain theirsolutions.

3. Solve linear equations andinequalities algebraically andnon-linear equations usinggraphing, symbol-manipulatingor spreadsheet technology; andsolve linear and non-linearsystems using appropriatemethods.

4. Use analytic thinking to describesituations and solve problems.

4. Analyze problems modeled bylinear functions, determinestrategies for solving theproblems and evaluate theadequacy of the solutions in thecontext of the problems.

4. Analyze problems that can bemodeled by functions, determinestrategies for solving theproblems and evaluate theadequacy of the solutions in thecontext of the problems.

5. Explore problems that reflectthe contemporary uses ofmathematics in significantcontexts and use the power oftechnology and algebraic andanalytic reasoning toexperience the waysmathematics is used in society.

5. Explore problems that reflectthe contemporary uses ofmathematics in significantcontexts and use the power oftechnology and algebraic andanalytic reasoning toexperience the waysmathematics is used in society.

VI. Probability and Discrete Mathematics

Content Standard 1: Students develop an understanding of the notion of certainty and ofprobability as a measure of the degree of likelihood that can be assigned to a given event based onthe knowledge available, and make critical judgments about claims that are made in probabilisticsituations. (Probability)


1. Explain the difference betweenchance and certainty and giveexamples to illustrate theirunderstanding.

1. Describe events as likely orunlikely and give qualitative andquantitative descriptions of thedegree of likelihood.

1. Develop an understanding ofrandomness and chancevariation and describe chanceand certainty in the language ofprobability.

2. Compare events and describethem as “more likely” or “lesslikely” and use the language offractions to describe simpleprobabilities.

2. Describe probability as ameasure of certainty rangingfrom 0 to 1 and conductactivities that allow them toexpress probabilities of simpleevents in mathematical terms.

2. Give a mathematical definitionof probability and determine theprobabilities of more complexevents, and generate andinterpret probabilitydistributions.

3. Conduct experiments withconcrete objects to exploreconcepts and develop anintuitive understanding of howthe conditions of the experimentcan affect the outcome.

3. Conduct experiments and giveexamples to illustrate thedifference between dependentand independent events.

3. Analyze events to determinetheir dependence orindependence and calculateprobabilities of compoundevents.

4. Conduct experiments, recordthe outcomes, examine thoseoutcomes to determine if theymake sense and search forexplanations of the outcomes.

4. Explain the difference betweenprobabilities determined fromexperiments or chance events(empirical) and probabilitiesderived mathematically(theoretical), and explain howthe empirical probabilitychanges for a large number oftrials.

4. Use sampling and simulations todetermine empiricalprobabilities and, whenappropriate, compare them tothe corresponding theoreticalprobabilities; understand andapply the law of large numbers.


5. Conduct probability experimentsand simulations to model andsolve problems.

5. Conduct probability experimentsand simulations to model andsolve problems.

5. Conduct probability experimentsand simulations, to model andsolve problems, includingcompound events.

Content Standard 2: Students investigate practical situations such as scheduling, routing,sequencing, networking, organizing and classifying, and analyze ideas like recurrence relations,induction, iteration, and algorithm design. (Discrete Mathematics)


1. Use manipulatives and diagramsto explore problems involvingcounting and arranging objects.

1. Use manipulatives, diagrams andthe fundamental theorem ofcounting to count permutationsand combinations.

1. Derive and use formulas forcalculating permutations andcombinations.

2. Explore sets and setrelationships by sorting andclassifying objects.

2. Use sets and set relationships toexplore and solve simplealgebraic and geometricproblems.

2. Use sets and set relationships torepresent algebraic andgeometric concepts.

3. Explore situations in which theymodel and trace paths usingfigures consisting of verticesconnected by edges.

3. Solve problems involvingnetworks, for example planningdelivery routes or countingpaths between points.

3. Use vertex-edge graphs to solvenetwork problems such asfinding circuits, critical paths,minimum spanning trees andadjacency matrices.

4. Explore now-next patterns. 4. Explore recurrence relationsand iterations.

4. Analyze and use discrete ideas,such as induction, iteration andrecurrence relations.

5. Explore, develop and inventtheir own algorithms toaccomplish a task or to solvenumerical problems.

5. Continue to use manipulativesand drawings to model theconcepts and procedures for thestandard arithmetic algorithms,and develop and analyze theirown and other students’algorithms to accomplish a taskor solve a mathematicalproblem.

5. Describe and analyze efficientalgorithms to accomplish a taskor solve a problem in a varietyof contexts, including practical,mathematical and computer-related situations.

6. Use discrete mathematicsconcepts as described above tomodel situations and solveproblems; and look for whetheror not there is a solution(existence problems), determinehow many solutions there are(counting problems) and decideupon a best solution(optimization problems).



Michigan Curriculum Framework · ii Mi hi C i l F k The framework emphasizes the importance of:...

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