DOCUMENT RESUME - ERICDOCUMENT RESUME ED 287 718 SE 048 678 TITLE Mathematics: Common Curriculum...

DOCUMENT RESUME

ED 287 718 SE 048 678

TITLE Mathematics: Common Curriculum Goals.INSTITUTION Oregon State Dept. of Education, Salem.PUB DATE Oct 87NOTE 79p.; For the comprehensive curriculum goals, see SE

048 679.PUB TYPk, Guides - Non-Classroom Use (055)

EDRS PRICE MF01/PC04 Plus Postage.DESCRIPTORS Academic Achievement; Communication Skills;

Curriculum Development; Educational Objectives;*Elementary School Mathematics; Elementary SecondaryEducation; *Mathematics Curriculum; *Secondary SchoolMathematics; State Curriculum Guides; *StateStandards; *Student Educational Objectives; StudentEvaluation; Study Skill.

IDENTIFIERS *Oregon

ABSTRACTThis document defines what are considered to be the

essentials in a strong mathematics program for the state of Oregonfor grades K-12. The common curriculum goals are organized into ninecontent strands: (1) number and numeration; (2) appropriatecomputational skills; (3) problem solving; (4) geometry andvisualization skills; (5) measurement; (6) statistics andprobability; (7) mathematical relationships; (8) oral and writtencommunication skills; and (9) appropriate study skills. Each strandcontains knowledge and skills to be developed with expected outcomesfor students to achieve by grades 3, 5, 8, and 11. (RH)

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MATHEMATICS

Common Curriculum Goals

October 1987

Oregon Department of Education700 Pringle Parkway SESalem, OR 97310-0290

Verne A. DuncanState Superintendent of P 4blic Instruction

Oregon Schools A Tradition of Excellence

Adopted by the State Board of Education September 18, 1987

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4)U S DEPARTMENT Of EDUCATION

Offtce of Educational Research and Improvement

EDUCATIONAL RESOURCES INFORMATION

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Tent has been reproduced aseceived from the person or organization

onomat.ng itMinor changes have been made to improvereproduction quality

Points of view or opinions stated in this documerit do not necessarily represent officialOE RI posit,on or policy

"PERMISSION TO REPiii4DUCE THISMATERIAL11 AS BE 0, ANTED BY

TO THE EDUCATIONAL RESOURCESINFORMATION CENTER (ERIC)"

Af,3

FOREWORD

In June 1984 the State Board of Education adopted the Oregon Action Plan forExcellence which established the direction for school improvement in the state overthe next decade. The Action Plan drew upon the insights of teachers, administrators,school board members and community and business leaders.

A central concept of the Action Plan is that while the state will determine WHAT mustbe taught in public schools, the schools will determine HOW it wil! be taught. Thisdocurnant is intended to provide the essential information which local districts need tomerge state curriculum expectations with their own local determinations for Mathe-matics.

All who have joined in the spirit of the Action Plan for Excellence have she red acommitment to high-quality performance. We are continuing to learn about how toprovide children with the very best in public education, and we welcome your com-ments and questions. For further information about this guide, contact the Specialistfor Mathematics Education, 378-3566.

Verne A. DuncanState Superintendentof Public Instruction

It is the policy of th,-. State Board of Education and a priority of the Oregon Department ofEducation that there will be no discrimination or harassment on the grounds of race, color,sex, marital status, religion, national origin, age or handicap in any educational programs,activities or employment. Persons having questions about equal opportunity and non-discrimination should contact the State Superintendent of Public Instruction at the OregonDepartment of Education.

3205619874000

ACKNOWLEDGMENTS

Grateful acknowledgment is made to the following people for themany hours they have contributed to developing ideas and writingmaterials for this guide. They studied the literature, then share°their thinking and experiences as classroom teachers and asteachers of teachers at numerous inservice and preservice teachereducation programs. School districts, colleges and math projectsreleased these people for work on this publication. Special thanksgoes to chief writer Judy Johnson, a middle school teacher andmath specialist, for putting it all together. Fond appreciation isacknowledged to Meredith and Oscar Schaaf for their career ofteamed help, advice and consulting with the mathematics educa-tion community.

COMMON CURRICULUM GOALS FOR MATHEMATICS OREGONDEPARTMENT OF EDUCATION STEERING/WRITING TEAM

Phil Bartsch, Carus School District 29Jack Hopper, Portland School District 1JJudy Johnson, Eugene School District 4JDonna McBride, Portland School District 1JDiane (Price) Stone. Philomath School District 17JOscar Schaaf, University of OregonTom Stone, Eugene School District 4JRon Zaraza, Portland School District 1J

CHIEF WRITERJudy Johnson, Eugene School District 4J

HELPER /ADVISOR /CONSULTANT

Meredith Schaaf, Lane ESDOscar Schaaf, University of Oregon

Two major curriculum documents served as the primary basis forthe development of this publication: (1) Essential Learning Killsand (2) The Oregon Mathematics Concept Papers. The conceptpapers were developed as an Oregon Mathematics EducationCouncil (OMEC) curriculum project. Nearly 100 volunteers workedon their own time for about two years in study, debate, presenta-tions and writing. Their conclusions present "forward-looking"

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ideas for school mathematic:,. Their names and schools at the timethe OMEC Curnculum Project was launched are listed below.

OREGON MATHEMATICS CONCEPT PAPERSOMEC Writing Teams

OMEC PRESIDENTS/PROJECT FACILITATORSMike Morgan, Linn-Benton Community CollegeRon Niorgali, Western Oregon State CollegeGary Musser, Oregon State UniversityRon Smit, University of Portland

OREGON MATHEMATICS CONCEPT PAPER NO. 1:ELEMENTARY MATHEMATICS CURRICULUM, K-5

Jill Board, Oakridge Elementary SchoolDouglas Cruikshank, Linfield CollegeBarbara Fuhrer, Clear Lake Elementary SchoolJan Heaton, Oak Elementary SchoolDennis Jones, Lincoln County Schoo' DistrictSusan Knapp, Chapman Elementary SchoolDiane (Price) Stone, Philomath Elementary SchoolKathryn Warrior, Yaquina View Elementary School

OREGON MATHEMATICS CONCEPT PAPER NO. 2:MIDDLE SCHOOL MATHEMATICS, 6-8

Fred Board, Westridge Junior High SchoolDianne Erickson, Talent Junior High SchoolArlene Foss, Ainsworth Elementary Su;-nolJudy Johnson, Kennedy Middle SchoolScott McFadden, Henry D. Sheldon High SchoolDon McNair, McLoughlin Junior High SchoolJulie (Nakata) Keener, Beaumont Middle SchoolOscar Schaaf, University of OregonLarry Sleeman, Philomath High SchoolJim Specti, Neil Armstrong Middle SchoolMarcia Swanson, Calapooia Middle SchoolKathryn Warrior, Yaquina View Elementary SchoolJoanne Wilkie, Tubman Middle School

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OREGON MATHEMATICS CONCEPT PAPER NO. 3:GEOMETRY, K-12

William F. Burger, Oregon State UniversityBarbara Culpepper, Franklin High SchoolDennis Dedrick, Medford Mid High SchoolJean Eversole, McKay High SchoolTony Gerlicz, Catlin Gabel SchoolVirginia O'Donnell, Binnsmead Middle SchoolDiane (Price) Stone, Philomath Elementary SchoolLarry Sleeman, Philomath Middle School

OREGON MATHEMATICS CONCEPT PAPER NO. 4:STATISTICS AND PROBABILITY, K-12

Mildred Bennett, Portland State UniversityDavid Bohlmann, Hood River Valley High SchoolCindy Brown, Elmira High SchoolTim Hahn, Creston Elementary SchoolKaren Higgins, Oak lea Middle SchoolRon Morgali, Western Oregon State CollegeLeona Pittenger, Oak lea Middle SchoolFred Rectanus, Portland School District 1JMarcia Swanson, Calapooia Middle SchoolJoanne Wilkie, Tubman Middle School

OREGON MATHEMATICS CONCEPT PAPER NO. 5:AN INTEGRATED MATHEMATICS CURRICULUM FORHIGH SCHOOL

Charles Barnhart, Obsidian Junior High SchoolDavid Erickson, Mt. View Senior High SchoolMike Gardner, Redmond High SchoolDaniel Gossack, North Salem High SchoolWendell Hall, North Eugene ugh SchoolFloyd Halvorsen, Churchill High SchoolSeymour Hanfling, Graduate Student at U of 0

8 iv

Karen Higgins, Oaklea Middle SchoolKathie Hledik, Willamette High SchoolDon Leslie, North Eugene High Schoo.Judi Mathis, Graduate Student at U of 0Carolyn Miller, South Salem High SchoolKathy Pfaendler, Mt. View Junior High SchoolRick Purn, Pilot Butte Junior High SchoolOscar Schaaf, University of OregonRon Steffani, Southern Oregon State collegeRick Thomas, South Eugene High SchoolLynne Tracy, Henry D. Sheldon High SchoolJohn Vogt, Springfield High School

OREGON MATHEMATICS CONCEPT PAPER NO. 6: OREGONGRADUATE REQUIREMENTS: MATHEMATICS

Mike Bolduan, Catlin Gabel SchoolJack Hopper, Portland School District 1JSue McGraw, Lake Oswego High SchoolKatie Miller, Columbia High SchoolRichard L. Moon, Crescent Valley High SchoolHoward Wilson, Oregon State UniversityRon Zaraza, Wilson High School

OREGON MATHEMATICS CONCEPT PAPER NO. 7:VOCATIONAL AND TECHNICAL MATHEMATICS

Bob Holly, Oakridge High SchoolRichard L. Moon, Crescent Valley High SchoolJudy Paulus, Central High SchoolDoug Stensrud, Wilson High SchoolRon Zaraza, Wilson High School

OMEC PAPER: ENRICHMENT/ACCELERATION (Unpublished)Jim Stewart, Oregon Institute of Technology

TABLE OF CONTENTS

Page

Foreword by the State Superintendent i

Acknowledgements 'I'

Introduction 1

The Oregon Action Plan for ExcellenceEssential Learning Skills

+Common Knowledge and Skills=Common Curriculum Goals

Philosophy/Rationale Underlying This Curriculum 2

Common Curriculum Goals 5

v

INTRODUCTION

THE OREGON ACTION PLAN FOR EXCELLENCE

The Action Plan identified seven areas of improvement, one ofwhich called for a statewide definition of what students shouldlearn:

The Oregon Department of Education, working with localschool districts and higher education institutions, shall definethe required common curriculum goals for elementary andsecondary schools in terms of the learning skills and knowl-edge students are expected to possess as a result of theirschooling experience.

Local school districts, with assistance from the OregonDepartment of Education, shall be responsible for organizingthe curriculum and delivering instruction to achieve the com-mon ::urrkulum goals.

Common Curriculum Goals

The first stage in defining the Common Curriculum Goals was todevelop the Essential Learning Skills the basic skill and per-formance expectations for all students in the areas of reading,writing, speaking, listening, mathematics, reasoning and studyskills. The second and present state is to develop Common Knowl-edge and Skills in individual subject areas. Together with theEssential Learning Skills, they form the Common CurriculumGoals for all students.

A. Essential Learning Skills

The Essential Learning Skills are considered basic to all stu-dents' learning, and all teachers are expected to provideinstruction in these skills. Only to the degree that studentsdevelop these skills and form the habit of using them, caninstruction in subject matter areas be successful. The skills arenot specific to any one discipline but provide a link across alldisciplines. Furthermore, the skills do not grow in isolation

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from content; they are strengthened through practice and usein all subject areas.

B. Common Knowledge and Skills

Looking beyond the Essential Learning Skills, this documentdefines more fully what are considered to be the essentials in :1strong Mathematics program. Each district will want to extendand elaborate upon this base in order to create its own unique,comprehensive mathematics curriculum. Students shouldhave the opportunity to demonstrate their achievement in avariety of ways. Equal opportunity to learn and the specialneeds of students are primary considerations in determiningacceptable performance levels.

State Standards

The Com,non Curriculum Goals as presented in this documentreceive their authority from the Oregon State Standards for PublicSchools, OAR 581-22-420 and 581-22-425. These rules wereamended by the State Board of Education in January 1986

PHILOSOPHY/RATIONALE UNDERLYINGTHIS CURRICULUM

The 1980s have been a time of educate nal reform in the nation atlarge. There has been a strong and pervasive quest for excellenceand equity in education in general, and mathematics education inspecific. The Oregon Action Plan for Excellence establishea ihedirection for school improvement in the state and the E tonna,Learning Skills document outlined the common skills across allprogram areas for elementary and secondary education. This doc-ument, Mathematics: Common Curriculum Goals, is written inrelationship to the preceding documents specifically, as well as tothe reform effort generally.

Although the underlying principles of mathematics are constant,the optimum structure for the presentation and use of mathematicshas been shifting in response to the rapidly expanding importanceof technology in solving problems. Today's world demands theability to think and use mathematical ideas to solve problems andto make decisions. The time our pupils spend learning mathemat-ics can no longer be limited to practicing long, repetitive or tediousprocedures which are more efficiently done with calculators. Theimpact of technology and its implications for mathematics educa-tion are reflected in this document.

The increasingly common uses of calculators underlines the needfor mental computations and estimations. Although the develop-ment of these skills has always been implicit in mathematicsinstruction, they have not always been taught systematically andfully. Deliberate and thorough development of the ability to esti-mate and do menta! arithmetic is a regular part of the computa-tional strand at all school levels in the curriculum outlined in thisdocument.

Even though the calculator is becoming an increasingly availabletool for problem solving, its use is of little value if the user hasinadequate conceptual understandings of number and operation.To help students understand "which buttons to push," the curricu-lum must place significant emphasis on the concepts which5nderliebasic mathematical skills. Increasingly, research supports the useof concrete materials as models for number concepts and foroperations. The curriculum outlined in this document provides thata broad range of manipulatives be used to introduce new conceptsat all levels. In addition to going from concrete to the abstract,opportunities are provided for finding concrete representations forabstract concepts and their symbolic representations. The use ofmanipulatives as tools for increasing understandings extends nat-urally to their use as tools to assist in learning problem-solvingskills. The use of concrete materials as problem-solving tools isincorporated into a variety of curriculum content areas as outlinedin this document.

As hardware becomes both more sophisticated and less expen-sive and as software development continues to expand andbecome more "user friendly," computers will inevitably become

2

major tools for mathematics education. The value of computers increating geometric displays, organizing and graphing data, simu-lating real-life situations, and generating numerical sequences isrecognized in this document.

Mathematics is about making sense of the world. The mathematicsoutlined in this document is consistent with the nature of the sub-ject. This means that pupils are learning mathematics in ways thatallow them to explore relationships and to develop understand-ings. The fundamental pit rnise on which this document is based isthat every aspect of school mathematics that pupils encountershould enhance their understanding of mathematical ideas andpromote the growth of thinking.

Organization

In order to provide a curriculum consistent with the philosophyoutlined above, the common curriculum goals for math have beenorganized into nine content strands. They are:

1.0 Number and Numeration. Students demonstrate an under-standing of number and numeration concepts and use theseunderstandings to interpret and solve problems.

2.0 Appropriate Computational Skills. Students select and usethe most appropriate form of computation manipulative,mental, paper/pencil, estimation or calculator usage to solveproblems and will c'ieck all computations for reasonability.

3.0 Problem Solving. Students use problem-solving skills andstrategies to solve routine and nonroutine problems.

4 0 Geometry and Visualization Skills. Students recognize geo-meti-ic patterns and relationships and apply them in solvingproblems and making predictions.

5.0 Measurement. Students measure quantities and use meas-urements to keep records, solve problems and ma1 3 predic-tions.

.0 ,

6.0 Statistics and Probability. Students collect, organize, recordand interpret data, and predict probable outcomes based oncollected data.

7.0 Mathematical Relationships. Students recognize and usenumber patterns, relationships and logical thinking skills tf)make predictions and to solve problems.

8.0 Oral and Written Communication Skills. Students use vocab-ulary, speech, numerals and other symbol systems essentialfor effective individual and group problem solving, and foreffective oral and written communication of mathematicalconcepts, problem-solving processes and results.

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9.0 Appropriate Study Skills. Students will be able to select anduse appropriate study skills in order to accomplish mathe-mati;a1 learning tasks.

The content outlined in each content strand includes both theessential learning skills deemed appropriate for mathematicsinstruction and the common curriculum outcomes unique to math-ematics. It is very important to remember that the student out-comes appearing in the columns headed Grade 3, Grade 5, Grade8 and Grade 11 are expectancies for students to reach by the endof these grade levels. An expectancy appearing in the Grade 3column, for example, represents a goal to be achieved as a resultof four years of learning. Another Oregon Department of Educationpublication, "A Model Comprehensive Mathematics Program," willsuggest developmental stages for each grade.

COMMON CURRICULUM GOALS

The content of mathematics programs is organized under nine content strands. Thissection presents a brief summary .ind rationale followed by the common curriculumgoals for each strand.

Number and Numeration: Basic Concepts, Principles and Meanings

The mastery of a particular bit or area of knowledge at a level that makes it genuinelyfunctional in one's life requires that it be understood thoroughly, that it be connected torelated bodies of thought, and that it be integrated with other knowledgejattitudes/perceptions the student holds.

Specifically applied to mathematics, this means that students need to apply in aconscious way the basics of number and numeration if they are to be masterful usersof the algorithms they are learning. To assist in this objective the mathematics contentprovides opportunities to build concepts and demonstrate understandings through theuse of concrete models for whole number, fraction and decimal numerations.

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1.0: Number and Numeration. Students demonstrate an understanding of number and numerationconcepts and use these understandings to interpret and solve problems.

KNOWLEDGE/SKILLS GRADE 3 GRADE 5 GRADE 8 GRADE 11

Students will be able to:

1.1 READ, WRITE, ORDER,COMPARE AND USE NUM-BERS* (ELS 1.4)**

'Read and write whole numbersto 1,000, commonly used frac-tions (1/2,1/3,5/1o), and decimals intenths and hundredths (using amoney model)

'Read and write whole numbers 'Read and write numbers, includ-to one million; commonly used ing decimals, commonly usedproper fractions, mixed num- fractions and percentsbers, and improper fractions;and decimals to thousandths

'Demonstrate the counting 'Demonstrate skip countingskills of skip counting by 2, 5 skillsand 10; "counting on" andcounting backwards to andfrom 100

bExpress large numbers inexpanded exponential nota-tion

'Read, write and order numbersincluding decimals, commonly usedfractions, percents and numbers inscientific notation

bExpress large and small numbersin expanded exponential nota-tion

'Order signed numbers and 'Order signed numbers and com-commonly used fractions, monly used fractions, decimalsdecimals and percents and percents

dExpress large whole numbersin scientific notation

The additional outcomes for mathematics, both in the skill column and the grade level expectancies, are it bold print to distinguish them from the Essential I earning Skills

** Learning outcomes drawn from the ODE Essential Learning Skills document are cross-referenced by citing their original identifying number in parentheses. Any modification in theoriginal wording is indicated by bold print.

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1 j


1.2 USE CONCRETE AND PIC-TORIAL MODELS TO DEM-ONSTRATE NUMBER ANDNUMERATION CONCEPTS(ELS 1.4)

'Order, compare and model(demonstrate comprehension byuse of objects or a drawing)place values to IWO, commonlyused fractions and decimals(using money models) intenths and hundredths

"Identify the number of ones,tens and hundreds in numbersless than 1000 and use con-crete models to demonstrateunderstanding

'Order, compare and modelplace values to one million; com-monly used fractions and deci-mals to thousandths, and useconcrete and pictorial modelsto demonstrate an under-standing of the above

identify the number of ones,tens, hundreds and thousandsin numbers less than one mil-lion, and tenths, hundredthsand thousandths in numbersless than one

'Order, compare and modelcommonly used fractions, deci-mals, percents and signed num-bers, and give examples ofpositive and negat:ve quan-tities (e.g., temperature, foot-ball, bank balances, altitude)

"Identify the number of ones,tens, hundreds, thousands,ten-thousands, and hundred-thousands in numbers lessthan one million, and tenths,hundredths and thousandthsin numbers less than one

'Order, compare and model com-monly used fractions, decimals, per-cents and signed numbers, andgive examples of positive andnegative quantities (e.g., tem-perature, football, bank bal-ances, altitude)

"Identify the number of ones,tens, hundreds, thousands, ten-thousands, and hundred-thou-sands it numbers less than onemillion, and tenths, hundredthsand thousandths in numbers lessthan one

1.3 RECOGNIZE AND USENUMBER PROPERTY CON-CEPTS

'Demonstrate, use and applythe properties of addition andsubtraction

'Demonstrate, use and applythe properties of addition,subtraction, multiplicationand division with whole num-bers

'Use and apply the properties 'Use and apply operational prop-of addition, subtraction, mul- ertiestiplication and division withwhole numbers and positiverational numbers, and dem-onstrate the properties of oneand zero (including nondivi-sion by zero), and the closureproperty

'Recognize and use mathe- 'Recognize and use mathe-matical terms matical terms

'Explain the reasons for therules for order of operationsand use of grouping symbols

`Use and apply order of opera-tions rules as appropriate formental, paper/pencil and cal-culator usage

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2.0

Appropriate Computational Skills

Mastery of computational skills is a necessary outcome of a mathematics program.Computational skills need to be related to real-world situations and seen as a means ofenhancing a person's ability to use mathematics in daily living. Memorization of all one-digit basic facts at the quick recall level is imperative. Computational skills includemental arithmetic, estimation and calculator use as well as paper/pencil computation.Instruction should include opportunities to select which mode of computation is moreappropriate and to determine whether or not answers are reasonable. FurthermorP,the instructional content provides opportunities to demonstrate conceptual under-standings and reasonability of answers through the use of concrete models andmaterials.

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2.0: Appropriate Computational Skills. Students select ay d use the most appropriate form ofcomputation manipulative, mental, piper /pencil, estimation or calculator usage to solveproblems and check all computations for reasonability.



2.1 USE MENTAL, PAPER ANDPENCIL, ESTIMATION ANDCALCULATOR COMPUTA-TIONS TO SOLVE APPRO-PRIATE PROBLEMS (ELS 1.4and 1 7)

r!A.

'Use mental, manual or calculatorprocesses to perform grade-levelarithmetic operations

bSelect the most appropriatemethod of computation(manipulative, mental,paper/pencil, calculator) touse in a given situation

Ùse estimating skills, such asrounding, to make approxi-mate whole number com-putations

°Apply acquired strategiesincluding modeling patterns(such as "countino an," "dou-bles," "neighbors," etc) andproperties (commutativityand associativity), to aid inquick recall of addition, sub-traction and multiplicationfacts

"Solve mentally, appropriateaddition and subtractionproblems involving placevalue understanding, e.g.,add or subtract 10 e; 100 to(from) any 3-digit number;add or subtract multiples of10 or 100

'Use mensal, manual or calculatorprocesses to perform grade-levelarithmetic operations

bSelect the most appropriatemethod(s) of computation(manipulative, mental,paper/pencil, calculator) touse in a given situation

Ùse rounding and other tech-niques useful in mental com-putation to estimate andmake approximate wholenumber, fraction and decimalcomputations

°Apply acquired strategies toaid in quick recall of all basicfacts

,-Solve mentally, appropriatewhole number, fraction anddecimal problems, e.g., 10 X64; 60 X 20; 14,000 7,000;5,000 + 261; 3,000 X 7; 1/4r 3/4; 5/8 4/8; 3 0.5

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'Use mental, manual, calculatorand computer processes to per-form mathematical operations

bSelect the most appropriatemethod(s) of computation(mental, paper/pencil, cal-culator) to use in a given sit-uation

Ùse rounding and other tech-nilues useful in mental com-putation to estimate andmake approximate wholenumber, fraction, decimaland percent computations

'Mentally wholenumbers and decimals bypowers of ten and multiplesof ten

Use mental, manual, calculator andcomputer processes to performmathematical operations

bSelect the most appropriatemethod(s) of computation (men-tal, paper/pencil, calcdator) touse in a given situation

Ùse and apply estimation tech-niques


'Perform addition and sub-traction algorithms with andwithout regrouping using 1-3digit whole numbers

hUse a calculator to solveappropriate problems and tocheck approximate calcu;a-tions (e.g., real problems withlengthy calculations or largenumbers)

Use estimation and otherskills to check answers forreasonableness

Use paper/pencil to performaddition, subtraction, multi-plication o: whole numbers,1-digit division, addition andsubtraction of decimals, addi-tion and subtraction of frac-tions with like denominators

'Compute using measures oflength, weight (mass), timeand money

hUse calculator and/or com-puter to solve appropriateproblems

Use estimation and otherskills to check answers forreasonableness

'Recognize that the samearithmetic algorithms usedwith 1-3 digit numbers can beextended to multidigit com-putations

'Compute using measures

hUse calculator and/or com-puter to solve appropriateproblems

'Use estimation and othersk:Ils to check answers forreasonableness

'Recognize that the same arith-metic algorithms used with 1-3digit numbers can be extendedto multidigit computations

'Compute using measures

hUse calculator and/or computerto solve appropriate problems

'Convert mentally, manually andelectronically among decimals, per-cents and commonly used fractions

;Use estimation and other skills tocheck answers for reasonableness

2.2 DEMONSTRATE COM-PUTATIONALALGORITHMS WITH CON-CRETE MATERIALS ORREAL-WORLD EXAMPLES

Use concrete models to per-form whole number computa-tions, and demonstrate placevalue exchanges (borrowinga;id carrying) up to 1000 andto model the various mean-ings of multiplication

Use concrete materials tomodel the various meaningsof multiplication and divisionand to interpret remainders

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'Demonstrate an understand-ing of the various meanings ofmultiplication and division(including remainders) ofwhole numbers by drawingsorby referencing "real world"applications

bModel the various meaningscf addition and subtractioninvolving signed numbers,and add, subtract and multi-ply signed numbers in situa-tions meaningful to students

bUse concrete or pictor:al modelsto demonstrate addition, sub-traction and multiplication ofsigned numbers

e

eKNOWLEDGE /SKILLS GRADE 3 GRADE 5 GRADE 8 GRADE 11

Ilse models such as money ormetrics to demonstrate addi-tion, subtraction and multi-plication of decimals bywhole numbers

dUse concrete models to dem-onstrate addition and sub-traction of commonly usedfractions

cUse concrete materials or"real world" examples todemonstrate operations withdecimals and percents

dUse concrete materials or"real world' examples todemonstrate operations withcommonly used fractions

cUse concrete materials or "realworld" examples to demonstrateoperations with decimals andpercents

dUse concrete materials or "realworld" examples to demonstrateoperations with commonly usedfractions

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4 -U

Problem Solving

Problem solving has been designated as the central goal for mathematics. A problem isa perplexing situation in which an individual or group accepts the challenge of findingways to clarify or resolve the difficulties involved. Frequently, the problem can beapproached in many ways. Occasionally, the resulting investigations are nonproduc-tive. Sometimes they are so productive as to lead to many different solutions orsuggest more problems to solve.

Problem-solving skills and strategies should be explicitly emphasized and problemsolving should be incorporated frequently into the approaches used in teaching therequired topics throughout the grade levels.

3.0: Problem Solving. Students use problem-solving skills and strategies to solve routine and nonroutineproblems.



3.1 IDEN TIFY PROBLEMS ANDAPPROACH THEIR SOLUTIONIN AN ORGANIZED MANNER;ELS 6.3)

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'Identify problems that need asolution

bUse simple questioning toc!arify problems

cUse data from questioning todevelop a problem-solvingplan

dSolve problems using strategiessuch as guessing and checking,using concrete objects, making amodel, generating a pattern ordrawing a picture

identify alternative solutions to asimple problem

'Choose and apply mental, man-ual and calculator processes toproblem-solving strategy(ies)

Identify problems, recognizeinformation necessary to solveproblems, and supply additionalinformation if needed.

bUse simple questioning strat-egies to clarify problems

cUse data from the questioningprocess to develop a problem-solving plan

dSolve problems using a variety ofstrategies such as guessing andchecking, making predictionsbased upon a pattern, making adrawing or model

'Identify alternative solutionsto problems

'Choose and apply mental, man-ual and calculator processes toproblem-solving strategy(ies)

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Define a problem, choose infor-mation to solve the problem andsupply additional information, ifneeded

bUse a combination of question-ing strategies and observation toanalyze problems

cUse data from several sources todevelop a problem-solving plan

dSolve problems using appropri-ate strategies such as guessingand checking, making a system-atic list, looking for patterns,making or drawing a model,eliminating possible answers orsolving a simpler problem

'Identify alternative solutionsto problems

'Choose and apply mental, man-ual, calculator and computerprocesses to problem-solvingstrategy(ies)

gSelect and apply appropriateproblem-solving took,including computer software

Define a problem, choose informa-tion to solve the problem and supplyadditional information, if needed

bApply recognized research tech-niques to analyze problems

Design and ca.-ry out a plan for solv-ing a problem

dSolve problems using the mostappropriate tools, methodologies,processes and operations in solvinga variety of problems

identify alternative solutions toproblems

'Choose and apply mental, manual,calculator and computer processesto problem-solving stratcgy(ies)

gSelect and apply appropriateproblem-solving tools, includingcomputer software

o


hShare successful and unsuc-cessful problem-solving strat-egies

'Engage in cooperative problemsolving aod compare alternativesolution strategies

Develop new suggestions orapproaches if problem is notsolved

hShare successful and unsuc-cessful problem-solving strat-egies

'Engage in cooperative problemsolving and compare alternativesolution strategies

Use formative (in process) data tomodify or col m proLlem-v.)iv-ing plan

"Describe ooth successful andunsuccessful soluti,11 strategies


lUse summative (final) data todetermine if the problem-solvingapproach was successful, and ifnot, how it should be modified

hEvaluate problem-solving strategiesin terms of tools, methodologies,processes, operations


lAnalyze formative and summativedata to confirm or revise the pro-posed solution

3.2 CREATE AND SOLVEWORD PROBLEMS APPRO-PRIATE TO THE GRADELEVEL (ELS 6.3)

'Recognize appropriate oper-ation(s) (+, X) for solu-tions of word problems

bSolve one-step word prob-lems including those involv-ing money, measurement anddata presented in graphs,tables and charts

cCreate word problems tomatch addition, subtractionand multiplication algorithms

°2

'Recognize appropriate oper-ation(s) for solutions of wordproblems and recognizeinformation necessary tosolve word problems, andsupply reasonable additionalinformation, if needed

'Recognize appropriate oper-ation(s) for solutions of wordproblems and recognizeint.Irmation necessary tosolve word problems, andsupply reasonable additionalinformation, if needed

bSolve one-. nd two-step word 'Pose and solve multiple-stepproblems including those word problemsinvolving money, measure-ment and data presented ingraphs, tables and charts

cCreate word problems tomatch whole number, frac-tion and decimal algorithms

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'Recognize appropriate opera-tion(s) for solutions of wordproblems and recognize infor-mation necessary to solve wordproblems, and supply reasonableadditional information ifneeded

'Pose and solve word problems

cSolve and create word prob- cSolve and create problems to!ems to match exercises match exercises involving ratios,involving whole numbers, proportions and formulasfractions, decimals and per-cent

dldentify, invent or create dldentify, invent or create prob-problems that can be solved lens that raj be solved by usingby using ratio and proportion; ratio and proportion; and useand use proportion to solve proportion to solve problemsproblems


eSolve problems with morethan one possible solutionand recognize problemswhich cannot be solvedbecause they contain too littleinformation

eSolve problems with morethan one possible solutionand recognize problemswhich cannot be solvedbecause they contain too littleinformation

eSolve problems with morethan one possible solutionand recegnize problemswh;ch cannot be solvedbecause they contain too littleinformation

'Translate "real-world" prob-lems into mathematical state-ments, and mathematicalproblems and answers backinto "real-world" context

eSolve problems with more thanone possible solution and recog-nize problems which cannot besolved because they contain toolittle information

'Translate "real-world"' problemsinto mathematical statements, andmathemat.cal problems andanswers back into "real-world"'context

19

Geometry and Visualization Skills

We live on a sphere called Earth and work in a three-dimensional world. Citizens,consumers and workers require some knowledge of geometry. Much more knowledgeis needed in omupations such as plumbing, carpentry, forestry, interior decorating,architecture and engineering. In addition then to acquiring knowledge of certain geo-metric concepts and properties and their applications, students need to develop theirspatial and visualization skills.

The instructional content emphasizes exploration, activities, informal reasoning andthe use of problem-solving skills. Much use is made of tools which aid in geometit;'explorations, including the compass, protractor, the straightedge, squared paper andthe computer.

21

0 ,'.1 U

4.0: Geometry and Visualization Skills. Students recognize geometric patterns and relationships and applythem in solving problems and making predictions.



41 RECOGNF'L AND USE GEO-METRIC PATTERNS, RELA-TIONSHIPS AND PRINCIPLESTO DESCRIBE AND CLASSIFY(ELS 1.5)

identify similar and differentattributes of two or more geometric figures

Identify properties of commongeometric figures, indudingquadrilaterals ane, geometricsolids

'Identify distinguishing prop-erties of common geometricfigures including side or anglemeasurements

'Identify and compare common two-and three-dimensional geometricshapes and solids according toattributes and properties

'Identify, sketch, model and ''Draw or model simple, common ''Sketch or build common geo- ''Model or make drawings of two- ormanipulate squares, rectangles,circles, triangles, cubes

geometrical figures with specificdimensions using ruler, tan-grams, squared paper orother concrete materials

metric solids and two-dimen-sional figures

three-dimensional shapes and solidsuseful in solving problems

'Identify symmetry and geometric 'Identify, skelL h or model inter- 'Identify, sketch or model parallel 'Recognize and apply the conceptsforms in the environment, e.g.,construction with colored

secting lines, right angles andlines of symmetry Occurring in

and intersecting lines, rightangles and lines of symmetry

of symmetry, congruency and sim-ilarity of geometrical figures as com-

tiles or cubes the environment occurring in the environment monly used in man-made objects

dCopy or extend patterns using dCopy or extend patterns using °Use drawings, models or dUse drawings, models or com-concrete models and drawing concrete models and drawing computers to demonstrate puters to demonstrate geometricpictures pictures geometric patterns and rela-

tionships such as similarityand congruence

patterns and relationships suchas similarity and congruence

'Locate points of graph paper, .1 mate and give coordinates ofmaps and globes, and graph points on graph paper, maps,coordinates (emphasize globes and other chartsexamples from the environ-ment)

23

'locate points and lines and deter-mine distance and area in a rec-tangular coordinate system(emphasize examples from theenvironment and computerapplications)

0 (21 L.)


'Demonstrate by variousmeans the relationshipsamong radius, diameter andcircumference of a circle, andthe common r;ght trianglerelationships

'Apply and use circle and trianglerelationships in solving problems

4.2 MAKE AND USE GEO-METRIC DRAWINGS ANDMODELS, INCLUDINGTESSELLATIONS

'Create simple constructionswith tiles and copy onsquared pa .r

'Find and use congruent poly-gons which will cover a sur-face without overlapping

°Sketch top and side views ofrectangular solids

'Use cubes to build structuressuggested by pictures andmake 3-D shapes from paperpatterns

'Explain why a flat surface canbe completely covered with-out overlap, by congruent tri-angles, rectangles or squares

°Ilse protractor, compass,ruler, computer and otherinstruments to make commongeometric constructions

Draw the net (2 dimentionalpattern) for common geo-metric solids (e.g., cube, rec-tangular prism, cylinder)

°Use protractor, compass, ruler,computer and other instrumentsto make common geometric con-structions

'Draw the net (2 dimentional pat-tern) for common geometric sol-ids

4.3 UNDERSTAND AND USEPERIMETER, AREA ANDVOLUME CONCEPTS

'Develop an understanding ofperimeter, area and volumeusing concrete objects

bUse common objects to esti-mate perimeter, area and vol-ume

e.1 3

'Develop an understanding ofperimeter, area and volumeusing concrete objects

t-Estimate and determineperimeter and area of rec-tangles, and volume of rec-tangular solids, by meansother than formula

24

'Demonstrate, other than byusing a formula, way, of find-ing perimeter and area ofgeneral triangles, circles, par-allelograms and trapezoids

°Estimate and determineperimeter, area and volumeof common geometric figures

Use and apply perimeter, areaand volume concepts

°Estimate perimeter, area andvolume

'1 u

e


`Use a formula for finding cCalculate perimeter, area andperimeter, area, and volume volumeof rectangular solids

dCalculate surface areas of dCalculate surface areas of regu-regularly shaped solids (e.g., lady shaped solids (e.g., cubes,cubes, cylinders, rectangular cylinders, rectangular boxes)boxes)

25

Measurement

Measurement concepts surround us. We use an understanding of measures to quan-tify and interpret the world. Modem technology is totally dependent upon measure-ment. Hidden in most of humanity's spectacular accomplishments are innumerablemeasurements, each related to or dependent upon a myriad of other measurements.

Measurement can be taught more successfully if estimation is a content objective. Toteach measurement we should be concerned with more than a system of measures.We must teach a "doing" kind of mathematics. Activity gives meaning to measuringskills, makes the resultant learning personally satisfying to the pupil and begins thedevelopment of a process that will be used throughout life,

27

43

S

5.0: Measurement. Students measure quantities and use measurements to keep records, solve problemsand make predictions.



5.1 POSE AND SOLVE PROB-LEMS THAT INVOLVE TIMEAND MONEY

al dent If y and order by value;make chtrif using U.S. coins

bRead time using standard and-igital clocks and ordermonths, seasons and days ofweek

'Identify and order by value,and make change using U.S.currency

bEstimate elapsed time forgiven activities

'Create and solve problemswhich involve time or money

Create and solve problemswhich involve time or moneyincluding consumer and wageearner situations of interestto students

Create and solve problems whichinvolve time and money, includingconsumer and wage earner sit-uations of interest to students

5.2 SELECT AND USE APPRO- 'Estimate and determine 'Recognize and use meters, 'Determine the most appropri- 'Determine the most appropriatePRIATE INSTRUMENTS length and weight (mass) centimeters, feet, yards and ate unit and instrument for a unit and instrument for a meas-AND UNITS TO ESTIMATE using nonstandard, metric or inc!tes to measure, and select measurement task urement taskAND MEASURE LENGTH- English (U.S. Customary) units the most appropriate instru-WEIGHT; VOLUME AND of measure and select and ment and unit for a measure-CAPACITY; AND TEM-PERATURE (ELS 1.7)

appropriate instrument and unitfor a measurement task

ment task

29

44

bEstimate and directly measuredistances, angles and otherquantities, and indicate in someway the precision of the me s-urement. using metric and Eng-lish (U.S. Customary)

bEstimate and directly measurelength, area, volume, time,weight (mass), etc., with reat.on-able accuracy, and/or round ameasurement to a given unit

45

KNOWLEDGE/SKILLS GRADE 3 GRADE 5 GRADE 8

gExplore the concepts ofweight (mass) using a balancescale and common classroomobjects

hEstimate, read and recordtemperature in C° and F°

'Use squared paper, trans-parent grids, or other mate-rial to estimate area/perimeter of irregular closedfigures

gEstimate and determine theweight (mass) of commonclassroom objects in metricand English (U.S. Customary)units

hEstimate, read and recordtemperature in C° and F°

'Measure by some directmeans the area of a polygonor some 2-D region withcurves as boundaries, and thevolume of 3-D objects

eGive examples of the impor-tance of congruence and pre-cision in society

'Explain why all measurementsare approximations and whyresults of all computationswith measurements areapproximations

gEstimate and determine theweight (mass) of commonobjects using metric and Eng-lish (U.S. Customary) units

hEstimate, read and recordtemperature in C° and F° inreal and given situations

GRADE 11

'Explain why all measurementsare approximations and whyresults of all computations withmeasurements are approxima-tions

5.3 DETERMINE INDIRECTMEASUREMENTS (ELS 1.7i

'Locate points, give coordi-nates of point on maps andestimate distances betweenplaces represented by points onmaps

"Make scale drawings anddetermine actual distancesfrom scale drawings, blue-prints, maps and globes

hFind and record measurementsusing proportions and formulas

'Apply ratio and proportion con-cepts in mak;0 and using scaledrawings , models, and insolving pr, ,ems

bFind and record measurementsusing proportions and formulas

30

4 oi , 4

Statistics and Probability

We live in a society in which we are confronted daily with quantitative information ordata. Statistics is the science or study of data. Quantitative literacy is a requirement forall educated individuals who want to be informed citizens and hold jobs in technicalbusinesses and industries. Throughout life, decisions about health, citizenship, parent-hood, employment, financial concerns and sports are based upon quantitative infor-mation. Statistics is most often concerned with using information in the face ofuncertainty. Probability gives us a way to measure uncertainty.

In addition, studying statistics and probability can help in the development of students'critical thinking skills. In carrying out experiments, students develop ways to cope withuncertainty as they are searching for truth in a situation and learning to report 'tfaithfully. Approaching situations statistically can help make students face up to theirown prejudices, think more consistently about arguments, and justify their thinkingwith numerical information.

31

46)

6.0: Statistics and Probability. Students collect, organize, record and interpret data and predict probableoutcomes based on collected data.

KNOWLEDGE/SKILLS GRADE 3 GRADE 5 GRADE & GRADE 11


6.1 RECOGNIZE AND USE MATH-EMATICAL PATTERNS, RELA-TIONSHIPS AND PRINCIPLESTO QUANTIFY PROBLEMS ORMAKE PREDICTIONS (ELS 1.6)

'Apply intuitive probability con-cepts; e.g., make predictions ingames by using terms such as"more likely," "less likely,'"fair," etc.

°Collect and record data frompicture graphs, bar graphs andcharts to draw conclusions andmake predictions

'Make and use picture graphsand bar graphs

'Generate, record and interpretdata from probability experi-ments and predict chances of anoutcome

bRecognize the concept of fairor unfair ;n game situations

cRecognize certain (1) andimpossible (0) probabilities

dUse charts, tables and lists toorganize all possible out-comes of an experiment

eRead, interpret, construct bargraphs, line graphs, tables andcharts and make predictionsbased on them

'Organize information intotables/charts and diagramsgiven appropriate scale, e.g.,box plots, box-and-whiskers,line plots, Venn diagrams

33

'Determine the number of possi-ble events and the probability of-71 outcome in a probabilityexperiment

bldentify and demonstrate sit-uations in which probabilityor chance of an event occur-ring is likely, unlikely, equallylikely; and whether a game is"fair"

cUnderstand the meaning ofprobabilities of 0 (impossible)and 1 (certain)

diJse charts, tables and lists toorganize all possible out-comes of an experiment

eRead and interpret graphs,tables and charts and makepredictions based upon them

'Make line and circle graphsfor data meaningful to stu-dents

'Interpret everyday uses of proba-bility such as weather predictions,election forecasts or chances of win-ning a lottery

bldentify and demonstrate situa-tions in which probability orchance of an event occurring islikely, unlikely, equally likely;and whether a game is "fair"

ellc2J and inter,:ret graphs,t.,:es and char's, avid male pre-dictions based upon them

'Organize and display data usingtables, charts, graphs and dia-grams


/Collect random samples /Use data gathering pro- /Use data gathering procedurescedures which will aid in which will aid in answeringanswering questions of inter- questions of interest (conductingest (conducting polls, sam- polls, sampling schemes)piing schemes)

"Understand the relationship ''Understand the relationshipbetween size of sample and between size of sample anddegree of certainty degree of certainty

'Identify misleading or incor- 'Identify misleading or incorrectrect methods of displaying or methods of displaying or inter-interpreting data preting data

'Read and interpret computer Use computer software to 'Use computer software to gener-generated graphs and tables generate graphs and tables ate graphs and tables

"Show the relationship among 'Show the relationship among van-variables using tables, graphs, abl?s using tables, graphs, formulasformulas and iiodels and models

'Predict simple possible future 'Predict probable future out- 'Defend conclusions from mforma-outcomes or actions comes or actions lion given

"Determine, interpret and corn- "'Collect, display, interpret statisticalpare advantages and disadvan- data using mean, mode median,tages of mean, median and mode range and percentile

34

t) I

KNOWLEDGE/SKILLS GRADE 3 GRADES GRADE 8 GRADE 11

6.2 GENERATE AND TEST INTER- Identify facts that support an Identify I rts of an explanation 'Interpret different s b..tweenPR ETATIONS, E XPL ANA explanation and a prediction and a prediction not supported two explanationsTIONS, PREDICTIONS AND by factHYPOTHESES (ELS 6.2)

'Identify factors that may influ- 'Predict what influence differentence a behavior or a result factors will have on a behavior or

result

Critic-"y analyze explanation andinterp, .ation to confirm or validatethem

'Develop a hypothesis from 'Develop a hypothesis using dataobserved data from a variety of sources

Identify ways to determine 'Follow directions to conduct an 'Gather data that confirms orwhether a euplicate of an even- expz 7:,ent and identify the negates a hypothesisment will produce the same hypothesis usedresults

'Design and conduct a test of ahypothesis and report the results

35

/

Mathematical Relationships

Topics in this content strand are used to show how one thing changes as anotherchanges. Formulas, tables and graphs often have ciose relationships to the "realworld" and frequently can be used in making useful predictions. Ratio, proportion andoercent are also used extensively in problem-solving situations. Program contentemphasizes patterning, using logical thinking skills, modeling of concepts and realworld applications.

37

Iil., ...=1

5

7.0: Mathematical Relationships. Students recognize and use number patterns, relationships and logicalthinking skills to make predictions and to solve problems.


Siadents will be able to:

7.1 SORT AND CLASSIFY; USELOGICAL THINKING (ELS5.2)

'Sort and classify objects by 'Classify objects and simple 'Classify geometrical figures 'Classify geometrical figures andattributes geometrical figures by and sets of numbers sets of numbers

attributes

"Use evidence such as illustra- "Use evidence from verifiable "Use evidence from verifiabletions, examples, and verifiable sources to support own ideas sources to support own ideas andsources to problem solve and concepts in problem solv- concepts in problem solving

ing

Select and organize details to Select and use details, examples, 'Select and use details, examples,problem solve illustrations, evidence and logic illustrations, evidence and logic to

to problem solve problem solve

dOrgamze information or datausing formats such as outlining,making maps, tables, charts andgraphs

dOrganize information or data dOrganize information or data usingusing formats such as outlining, formats such as outlining, makingmaking maps, tables, charts, maps, tables, charts, graphs; andgraphs; and computer spread computer spread sheetssheets

7.2 COMPREHEND MEANINGSOr WRITTEN, ORAL ANDVISUAL COMMUNICATIOASINVOLVING NUMBER PAT-TERNS AND RELA-TIONSHIPS (ELS 3.1)

F.;

'State relationships usingterms such as "greater than,""less than," and "equal to"

'State relationships using Use equality and inequality 'Use equality and inequality con-terms such as "greater than," cencepts and symbols cepts and symbols"less than," and "equal to"and the symbols <, >,

bRelate new information to bRelate new information to pre- bRelate new information to pre- "Relate new information to previousprevious knowledge vious knowledge vious knowledge knowledge

Draw logical conclusions from Draw logical conclusions from Draw logical conclusions from (Synthesize information andinformation presented information presented information presented draw r.:,.nelusions

39

TKNOWLEDGE/SKILLS GRADE 3 GRADE 5 GRADE 8 GRADE 11

dFind numerical patterns in chartsand tables e.g., 100-chart,addition and multiplicationtables, and use number pat-terns and relationships tomake predictions

dFind numerical patterns in100-charts and addition andmultiplication tables (e.g.,odd/even, primes, squarenumbers), and u,e patterns tocomplete simple charts andtables and to make predictions

eExplore relationships found intables of value

dFind numerical patterns anduse to complete charts andtables

'Recognize direct and indirectcause and effect relationships

*Show using models such as100-grids, number lines or ameter stick how percent canbe expressed as a fraction ordecimal, and conversely

hInterpret and use the con-cepts of ratio, percent pro-portion and commonlyoccurring rates such asgrowth, speed and sportsapplications

dFind numerical patterns and useto complete charts and tables

'Infer direct and indirect cause andeffect relationships

*Demonstrate fraction, decimal,percent relationships

''Interpret and use the concepts ofratio, percent and commonly occur-ring rates such as growth, speed,interest and cost per unit

7.3 RECOGNIZE, CONSTRUCTAND DRAW INFERENCESCONCERNING RELA-TIONSHIPS AMONG THINGSAND IDEAS (ELS 6.1)

,Identify charat.zs.ristic of simpleobjects that remain the sameeven though some changeoccu-s, e.g., cutting objects intotwo pieces

bMake a simple table of valuesgiven a specific rule andmatch a table of values to itsrule

bMake a simple table of valuesgiven a specific rule andmatch a table of values to itsrule

40

bEvaluate or make a table fortwo-variable formulas whichhave meaning to students andmatch a graph or table of val-ues to its formula

bEvaluate or make a table for two-variable formulas and match agraph or table of values to itsformula

s


cExplore relationships illus- cDescribe the nature of change cDescribe the nature of change oftrated in a table of values of each variable as suggested each variable as suggested by a

by a table of values, graph or table of values, graph or formulaformula

7.4 REFLECT UPON AND 'Describe in simple terms how a 'Describe the reasoning process 'Describe the strengths and 'Present arguments supporting theIMPROVE OWN REASONING solution was reached being used weaknesses of inductive and use of deductive or inductive rea-(ELS 6.6) deductive reasoning sonmg for a particular purpose

bAct upon suggestions forimproving reasoning capabilities

41

63

e

Oral and Written Communication Skills

Two emerging trends are reflected in "Common Curriculum Goals" for math; integra-tion of learning across content areas, and the development and use of enabling skillswhich help students learn mathematics and other disciplines. The following strand,"Oral and Written Communication Skills," offers the basic communication skillsneeded to effectively read, discuss and share mathematical ideas and problems.

43

8.0: Oral and Written Communication Skills. Students use vocabulary, speech, numerals and other symbolsystems essential for effective individual and group problem solving and for effective oral and writtencommunication of mathematical concepts, problem-solving processes and results.

KNOWLEDGE /SKILLS GRADE 3 GRADE 5 GRADE 8 GRADE 11


8.1 RECOGNIZE AND USEMATHEMATICS VOCABU-LARY COMMONLY USED INGRADE-LEVEL MATERIALS(ELS 1.1)

aUse phonetic analysis

bUse context clues in a pi-ragraphto infer correct word(s)

'Distinguish compound and plu-ral words

dUse basic mathematical terms(such as sum, total, difference,product, less than, equal, greaterthan, rectangle) to convey con-cepts of quantity, order, opera-tion, and shape

bUse context clues in a passage toinfer correct word(s)

'Distinguish affixes and rootwords

dUse mathematical terms to con-vey concepts of quantity, order,operation and shape (e.g.,product, factor, quotient,remaind,r, sum, quad-rilateral)

bUse context clues in a selectionto infer correct word(s)

'Distilguish affixes and rootwords

dUse basic mathematical termsand symbols to convey conceptsof quantity, order, operation,and shape

bUse zontext clues in a selection toinfer correct word(s)

'Distinguish affixes and root words

dUse basic mathematical terms andsymbols to -onvey concepts ofquantity, order, operation, andshape

8.2 DETERMINE MEANING OF aUse adjacent words to inferUNKNOWN WORDS COM-meaning of unknown wordsMONLY USED IN MATHE-MATICAL MATERIALS (ELS1.2)

aUse context clues to infer mean-ing of unknown words

bRecognize double meanings ofwords

aUse context clues, punctuation Use context clues, punctuation andand syntax to infer meaning of syntax to infer meaning of unknownunknown words and concepts words and concepts

bRecognize double meanings of bRecognize double meanings ofwords words

'Use knowledge of each part of a 'Use knowledge of affixes and 'Use knowledge of affixes and Use knowledge of affixes and rootcompound word to determine root words to determine word root words to determine word words to determine word meaningsmeaning meanings meanings

dUse dictionaries and glossaries in dUse context to determine correct dUse dictionaries, glossaries and dUse dictionaries, glossaries and defi-grade-level curriculum materials dictionary definition of word other reference materials to find

word meaningsnitions in footnotes to find wordmeanings

C3

45

u


8.3 SPEAK WITH STANDARD 'Produce correct basic speech 'Produce correct basic specchPRONUNCIATION, APPRO- sounds soundsPRIATE VOLUME, RATE, GES-TURES AND INFLECTIONS(ELS 1.3)

'Pronounce words according to ''Pronounce words accordingacceptable Standard English acceptdsle Standard English

to 'Pronounce words according toacceptable Standard English

'Make oral presentations thatuse verbal and nonverbalcommunication skills effec-tively

''Pronounce words according toacceptable Standard English

'Make oral presentations that useverbal and nonverbal communica-tion skills effectively

8.4 USE ORAL COMMUNICA-TION TO GIVE OR RECEIVEINFORMATION AND DIREC-TIONS (ELS 2.3)

'Paraphrase oral messages 'Paraphrase oral messages

bGive accurate oral directions bGive accurate o-al directions

'Ask questions designed to clar- 'Ask questions designed to clar-ify, gain assistance or local infor- ify, gain assistance °I local infor-mation mation

"Share ideas and information ''Share ideas and informationorally with others orally with others

'Provide accurate descriptive "Provide accurate descriptivedetail orally detail orally

'lake notes based on oral presen-tations and group discussions

'Foil ow 2-step oral instruction gFollow 3 step oral instructions

Paraphrase oral messages

bGive accurate oral directions

'Ask questions designed to clar-ify, gain assistance or local infor-mation

"Share ideas and informationorally with others

'Provide accurate descriptivedetail orally

'Take notes and prepare sum-maries based on ral presenta-tions and group discussions

gFollow multistep oral instruc-tions

'Paraphrase oral messages

bGive accurate oral directions

'Ask questions designed to clarify,gain assistance or local information

ideas and information orallywith others

'Develop accurate detail 1),.sed onoral explanations by others

'Take notes and prepare summa-ries based on oral presentationsand group discussions

gFollow multistep oral instructions

46

it 5t

KNOWLEDGE/SKILLS GRADE 3 GRADES GRADE 8 GRADE 11

8.5 DETERMINE THE SIGNIFI-CANCE AND ACCURACY OFINFORMATION AND IDEASPRESENTED IN WRITTEN,ORAL, AURAL AND VISUALCOMMUNICATIONS (ELS 4.1)

'Separate between relevant and 'Distinguish between logical andirrelevant information used to illogical conclusionsdraw concluElons

'Identify propaganda and other 'Identify propaganda and other per-persuasion techni:tics (e.g., use suasion techniques (e.g., use andand misuse of statistics) misuse of statistics)

8 6 USE ORAL C OMM UN IC A 'Ask questions and draw reason-TION TO INFLUENCE OTHERS dole conclusions from answersAND TO RESPOND TO PER-SUASION (ELS 4 2)

8.7 LISTEN, READ, VIEW ANDEVALUATE PRCSENTATIONSOF MASS MEDIA (ELS 4.4)

'Provide logical answers based 'Provide logical answers based 'Provide logical answers based upol.upon factual data upon factual data factua data

"Use multiple sources to verify "Use multiple sources to verify "Use primary and secondary sourceinformation information materials to verify irformation

'Argue opposite sides of issues

'Recognize sources of persuasionand select appropriate persuasiveresponse

Use verbal persuasion techniques ina class presentation

.Recognize persuasion tech-niques found in visual commenications (e.g., use and misuseof graphs)

.Recognize persuasion techniquesfound in visual communications(e.g., use and mi:use of graphs)

47


8.8 SELECT APPROPRIATE FORMOF WRITING (ELS 5.3)

'Write in a variety of forms such as 'Write in a variety of forms such asreports, descriptions, or in personal essays, journals,problem posing or solving reports, descriptions, or in

problem posing or solving

bUse writing appropriate to pur-pose such as to inform, poseproblems or solve problems

bUse writing appropriate to pur-pose such as to inform, poseproblems or solve problems

"Write in a variety of forms such aspersonal essays, journals, reports,descriptions, or in problem pos-ing or solving

bUse writing appropriate to purposesuch as to inform, pose problemsor solve problems

8.9 PRESENT IDEAS IN UNDER-STANDABLE SEQUENCE ONTHE TOPIC SELECTED (ELS 5.4)

'Write complete sentences 'Write complete sentences 'Write complete sentences

bWrite multiparagraph personal bWrite multiparagraph personaljournals, reports or problem journals, reports or problemsolution strategies solution strategies

8.10 SELECT AND USE GUAGE,,GESTURES AND SYMBOLSAPPROPRIATE TO PURPOSETOPIC AND SETTING WHENMAKING ORAL PRESENTA-TIONS (ELS 5.5)

'Select words whichmeaning clear

make the 'Select words which make the Use a variety of techniques andmeaning clear figurative expressions to convey

meaning

'Employ verbal, symbolic, graphicand visual techniques to conveymfomation meaning

'Plan and make oral and visual bPlan and make oral and visual bPlan and make oral and visual pre-presentations presentations sentations

811 EVALUATE AND REVISE OWNWRITING FOP MEANING,.CLARITY AND COMPREHEN-SIVENESS (REVISING ANDREWRITING) (ELS 5.6)

'Revise own writing to enhance 'Revise own writing to enhance 'Revise own writing to enhance 'Revise own writing for correc-clarity and meaning clarity and meaning clarity and meaning tiveness and comprehensiveness

bUse descriptive terms to empha- bUse descriptive and connect- bUse descriptive and connectingsize facts and quantities ing terms to enhance mean- terms to enhance meaning, clarity

ing, clarity and precision and precision

48

o

KNOWLEDGE/SKILLS GRADF 3 GRADE 5 GRADE 8 GRADE 11

8.12 APPLY THE CONVENTIONS 'Edit for capitalization, end punc- 'Edit for complete and correct 'Edit for complete and correct 'Edit to produce a correct, legible,OF WRITING TO PRODUCE tuation and complete sentences sentences, punctuation and sentences, punctuation and effective piece of writingEFFECTIVE COMMUNICA- usage usageTION (EDITING AND PROOF-READING) (ELS 5.7) bSpell correctly bSpell correctly bSpell correctly

Produce legible final copy Produce legible final copy (man- 'Produce legible final copy (man-ual or electronic processes) ual or eiectronic processes)

e

49

Appropriate Study Skills

It seems fitting ,,iat the ninth content strand be "Appropriate Study Skills." A fullrepertoire of study skills ennliles students to learn school mathematics efficiently whilebecoming independent learners. Only by reaching this level can students become life-long consumers and users of mathematics.

9.0: Appropriate St.dy Skills. Students select and use appropriate study c :' al order to accomplishmathematical learning tasks.

KNOWLEDGE /SKILLS


9.1 IDENTIFY MAIN IDEAS, SUP-POR TING DETA;LS, ANDFACTS AND OPINIONS PRE-SENTED IN WRITTEN, ORALAND VISUAL FORMATS (ELS2.1)

GRADE 3 GRADE 5 GRADE 8 GRADE 11

Jcate facts in grade-level mate- 'Locate facts in grade-level selec- 'Locate facts in grade-level 'Locate fads in grade-level selec-ials tions selections tions

bRecall facts and supporting evi- bidentify necessary and extra- bidentify necessary and extra-dence neous fads and related sup-

porting detailsneous facts and related support-ing details

identify main idea in a problem (Identify main Idea in a problem (Identify main idea in a prob- (Identify main idea in a problemsituation situation lem situation situation

9.2 USE INSTRUCTIONAL MATE-RIALS AS BASIS FOR GAININGKNOWLEDGE AND IMPROV-ING COMPREHENSION (ELS2.2)

Use table of contents to locategeneral and specific formation

'Use guide words in a dictionaryor glossary to locate words

Use table of contents and indexto locate genera! and specificinformation

bUse supportive illustrations,detail and summations to obtaininformation

Use diacritical markings orrespellings to pronounce words

'Use table of contents, index,summaries, charts, graphs andillustrations to locate informa-tion needed

bUse organization of materials(summaries, headings and reviewquestions)

Use diacritical markings orrespellings to pronounce words

Use table of contents, index, sum-maries, charts, graphs and illustra-tions to locate information needed

bUse organization of materials (sum-maries, headings and review ques-tions)

Use diacritic..1 markings or respell-ings to pronounce words

93 CLARIFY PURPOSES OFASSIGNMENT (ELS 7 1)

'Determine general purpose of 'Determine general purpose of 'Determine general purpose of 'Determine general purpose ofassignment and ask clarification assignment and ask clarification assignment and ask clarification assignment and ask clarification onon questions if necessary on questions if necessary on questions if necessary questions if necessary

bDetermine ideas and conceptsaddressed in the assignment

bDetermine ideas, concepts andgeneralities addressed in the

bDetermine ideas, concepts, gen-eralities or principles included

bDetermine ideas, concepts, gener-alities or principles included in

assignment in assignment assignment

53

ir, 4A '1 tI


9.4 USE RESOURCES BEYONDTHE CLASSROOM (ELS 7.2)

'Locate, check-out and returnbooks and other circulatingmedia materials

bLocate and use noncirculatingreference materials

`Use library classification systemand services to locate specializedresources required to completeassignments

'Locate, check-out and returnbooks and other circulatingmedia materials

'locate and use noncirculatingreference materials

Use library classification systemand services to locate specializedresources required to completeassignments

dUse computer, e.g., databases, spread sheets

9.5 SELECT AND USE APPROPRI-ATE STUDY NIQUES (ELS7.3)

'Follow a study plan including:time management, appropriatestudy environment, processingof information

bAccomplish learning task usingappropriate study techniques(read and reread text, ask clarify-ing questions, seek help whenneeded, use memory tech-niques)

`Vary reading rate according topurpose for reading the selection

dKe") study materials organizedand accessible

O

'Follow a study plan including:time management, appropriatestudy environment, processingof information

bAccomplish learning task usingappropriate study techniques(preview and review chapters,read and reread text, ask clarify-ing questions, seek help whenneeded, use memory tech-niques, summarize, study withclassmates, use self-questioning)


dKeep study materials organizedand accessible

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'Locate, check-out and return booksand other circulating media mate-rials

bLocate and use noncircuiating refer-ence materials

Use library classification system andservices to locate specializedresources required to completeassignments

dUse computer, e.g., data bases,spread sheets

'Follow a study plan including:time management, appropriatestudy environment, prof essingof information

bAccomplish learning task usingappropriate study techniques(preview and review chapters,read and reread text, ask clarify-ing questions, seek help whenneeded, use memory tech-niques, summariu tudy withclassmates, use self-questioning)


dKeep study materials, log andrelated notes organized andaccessible

'Follow a study p.an including: timemanagement, appropriate studyenvironment, processing of infor-mation

bAccomplish learning task usingappropriate study techniques (pre-view and review chapters, read andreread text, as' clarifying questions,seek help when needed, use mem-ory techniques, summarize, studywith classmates, use self-question-ing)

Nary reading rate according to pur-pose for reading the selection

dKeep study materials, log, relatednotes and filing system organizedand accessible


gum in assignments on time

'Use appropriate test-t..ing tech-niques

Turn in assignments on time

'Use appropriate test-taking tech-niques

'Turn in assignments ( me

'Use appropriate test-taking tech-niques

Turn in assignments on time

Use appropriate test-taking tech-niques

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DOCUMENT RESUME - ERICDOCUMENT RESUME ED 287 718 SE 048 678 TITLE Mathematics: Common Curriculum...

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