Standard 12 Standard 28 Standard 310 Standard 421 Standard

Standard 1...........................................................................2

Standard 2...........................................................................8

Standard 3..........................................................................10

Standard 4..........................................................................21

Standard 5..........................................................................27

Standard 6.........................................................................36

Standard 7.........................................................................40

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Students will use mathematicalanalysis, scientific inquiry, andengineering design, as appro-priate, to pose questions, seekanswers, and develop solutions.

&STANDARD

1

ENGINEERING DESIGN

1 Engineering design is an iterativeprocess involving modeling and opti -mization finding the best solutionwithin given constraints which isused to develop technological solutionsto problems within given constraints.

• describe objects, imaginary or real,that might be modeled or made dif-ferently and suggest ways in whichthe objects can be changed, fixed, orimproved

• investigate prior solutions and ideasfrom books, magazines, family,friends, neighbors, and communitymembers

• generate ideas for possible solu-tions, individually and throughgroup activity; apply age-appropri-ate mathematics and science skills;evaluate the ideas and determinethe best solution; and explain rea-sons for the choices

SCIENTIFIC INQUIRY

1 The central purpose of scientificinquiry is to develop explanations ofnatural phenomena in a continuing,creative process.

• ask “why” questions in attempts toseek greater understanding concern-ing objects and events they haveobserved and heard about

• question the explanations they hearfrom others and read about, seekingclarification and comparing themwith their own observations andunderstandings

• develop relationships among obser-vations to construct descriptions ofobjects and events and to form theirown tentative explanations of whatthey have observed use of whatthey have observed

2 Beyond the use of reasoning and con -sensus, scientific inquiry involves thetesting of proposed explanationsinvolving the conventional

Elementary Students:

MATHEMATICAL ANALYSIS

1 Abstraction and symbolic representa -tion are used to communicate mathe -matically.

• use special mathematical notationand symbolism to communicate inmathematics and to compare anddescribe quantities, express relation-ships, and relate mathematics totheir immediate environments

2 Deductive and inductive reasoningare used to reach mathematical con -clusions.

• use simple logical reasoning todevelop conclusions, recognizingthat patterns and relationships pre-sent in the environment assist themin reaching these conclusions

3 Critical thinking skills are used inthe solution of mathematical prob -lems.

Continued.

ENGINEERING DESIGN

• plan and build, under supervision, amodel of the solution using familiarmaterials, processes, and hand tools

• discuss how best to test the solution;perform the test under teachersupervision; record and portrayresults through numerical andgraphic means; discuss orally whythings worked or didn’t work; andsummarize results in writing, sug-gesting ways to make the solutionbetter

SCIENTIFIC INQUIRY

conventionaltechniques and proceduresand usually requiring considerable inge -nuity.

• develop written plans for exploringphenomena or for evaluating expla-nations guided by questions or pro-posed explanations they havehelped formulate

• share their research plans with oth-ers and revise them based on theirsuggestions

• carry out their plans for exploringphenomena through direct observa-tion and through the use of simpleinstruments that permit measure-ments of quantities (e.g., length,mass, volume, temperature, andtime)

3 The observations made while testingproposed explanations, when ana -lyzed using conventional and invent -ed methods, provide new insightsinto phenomena.

• organize observations and measure-ments of objects and events throughclassification and the preparation ofsimple charts and tables

• interpret organized observationsand measurements, recognizingsimple patterns, sequences, andrelationships

• share their findings with others andactively seek their interpretationsand ideas

• adjust their explanations and under-standings of objects and eventsbased on their findings and newideas



• explore and solve problems generat-ed from school, home, and commu-nity situations, using concreteobjects or manipulative materialswhen possible

Analysis,Inquiry, and

Design

STANDARD

ENGINEERING DESIGN


• identify needs and opportunities fortechnical solutions from an investi-gation of situations of general orsocial interest

• locate and utilize a range of printed,electronic, and human informationresources to obtain ideas

• consider constraints and generateseveral ideas for alternative solu-tions, using group and individualideation techniques (group discus-sion, brainstorming, forced connec-tions, role play); defer judgmentuntil a number of ideas have beengenerated; evaluate (critique) ideas;and explain why the chosen solu-tion is optimal

• develop plans, including drawingswith measurements and details ofconstruction, and construct a modelof the solution, exhibiting a degreeof craftsmanship

SCIENTIFIC INQUIRY


• formulate questions independentlywith the aid of references appropri-ate for guiding the search for expla-nations of everyday observations

• construct explanations independent-ly for natural phenomena, especiallyby proposing preliminary visualmodels of phenomena

• represent, present, and defend theirproposed explanations of everydayobservations so that they can beunderstood and assessed by others

• seek to clarify, to assess critically,and to reconcile with their ownthinking the ideas presented by oth-ers, including peers, teachers,authors, and scientists

2 Beyond the use of reasoning and con -sensus, scientific inquiry involves thetesting of proposed explanationsinvolving the use of conventionaltechniques and procedures and usual -ly requiring considerable ingenuity.

Intemediate Students:



• extend mathematical notation andsymbolism to include variables andalgebraic expressions in order todescribe and compare quantitiesand express mathematical relation-ships


• use inductive reasoning to con-struct, evaluate, and validate conjec-tures and arguments, recognizingthat patterns and relationships canassist in explaining and extendingmathematical phenomena


• apply mathematical knowledge tosolve real-world problems and prob-lems that arise from the investiga-tion of mathematical ideas, usingrepresentations such as pictures,charts, and tables

Continued.


Design

STANDARD

ENGINEERING DESIGN

• in a group setting, test their solutionagainst design specifications, pre-sent and evaluate results, describehow the solution might have beenmodified for different or betterresults, and discuss tradeoffs thatmight have to be made

SCIENTIFIC INQUIRY

• use conventional techniques andthose of their own design to makefurther observations and refine theirexplanations, guided by a need formore information

• develop, present, and defend formalresearch proposals for testing theirown explanations of common phe-nomena, including ways of obtain-ing needed observations and waysof conducting simple controlledexperiments

• carry out their research proposals,recording observations and mea-surements (e.g., lab notes, audiotape, computer disk, video tape) tohelp assess the explanation

3 The observations made while test-ing proposed explanations, whenanalyzed using conventional andinvented methods, provide newinsights into phenomena.

• design charts, tables, graphs andother representations of observa-tions in conventional and creativeways to help them address theirresearch question or hypothesis

• interpret the organized data toanswer the research question orhypothesis and to gain insight intothe problem

• modify their personal understand-ing of phenomena based on evalua-tion of their hypothesis

Intermediate Students:


Design

STANDARD

ENGINEERING DESIGN


• initiate and carry out a thoroughinvestigation of an unfamiliar situa-tion and identify needs and oppor-tunities for technological inventionor innovation

• identify, locate, and use a widerange of information resources, anddocument through notes andsketches how findings relate to theproblem

• generate creative solutions, breakideas into significant functional ele-ments, and explore possible refine-ments; predict possible outcomesusing mathematical and functionalmodeling techniques; choose theoptimal solution to the problem,clearly documenting ideas againstdesign criteria and constraints; andexplain how human understands,economics, ergonomics, and envi-ronmental considerations haveinfluenced the solution

SCIENTIFIC INQUIRY


• elaborate on basic scientific and per-sonal explanations of natural phe-nomena, and develop extendedvisual models and mathematical for-mulations to represent their think-ing

• hone ideas through reasoning,library research, and discussionwith others, including experts

• work toward reconciling competingexplanations; clarifying points ofagreement and disagreement

• coordinate explanations at differentlevels of scale, points of focus, anddegrees of complexity and specifici-ty and recognize the need for suchalternative representations of thenatural world

2 Beyond the use of reasoning and con -sensus, scientific inquiry involves thetesting of proposed explanationsinvolving the use of conventionaltechniques and procedures and usual -ly requiring considerable ingenuity.

• devise ways of making observationsto test proposed explanations.

• refine their research ideas throughlibrary investigations, includingelectronic information retrieval andreviews of the literature, andthrough peer feedback obtainedfrom review and discussion

Commencement Students:



• use algebraic and geometric repre-sentations to describe and comparedata


• use deductive reasoning to constructand evaluate conjectures and argu-ments, recognizing that patterns andrelationships in mathematics assistthem in arriving at these conjecturesand arguments


• apply algebraic and geometric con-cepts and skills to the solution ofproblems

Continued.


Design

STANDARD

ENGINEERING DESIGN

• develop work schedules and work-ing plans which include optimal useand cost of materials, processes,time, and expertise; construct amodel of the solution, incorporatingdevelopmental modifications whileworking to a high degree of quality(craftsmanship)

• devise a test of the solution accord-ing to the design criteria and per-form the test; record, portray, andlogically evaluate performance testresults through quantitative, graph-ic, and verbal means. Use a varietyof creative verbal and graphic tech-niques effectively and persuasivelyto present conclusions, predictimpacts and new problems, andsuggest and pursue modifications

SCIENTIFIC INQUIRY

• develop and present proposalsincluding formal hypotheses to testtheir explanations, i.e., they predictwhat should be observed underspecified conditions if the explana-tion is true

• carry out their research plan for test-ing explanations, including selectingand developing techniques, acquir-ing and building apparatus, andrecording observations as necessary

3 The observations made while testingproposed explanations, when ana -lyzed using conventional and invent -ed methods, provide new insightsinto phenomena.

• use various means of representingand organizing observations (e.g.,diagrams, tables, charts, graphs,equations, matrices) and insightfullyinterpret the organized data

• apply statistical analysis techniqueswhen appropriate to test if chancealone explains the result

• assess correspondence between thepredicted result contained in thehypothesis and the actual result andreach a conclusion as to whether ornot the explanation on which theprediction was based is supported

• based on the results of the test andthrough public discussion, theyrevise the explanation and contem-plate additional research

• develop a written report for publicscrutiny that describes their pro-posed explanation, including a liter-ature review, the research they car-ried out, its result, and suggestionsfor further research



Design

STANDARD

STANDARD

Students will access, generate,process, and transfer informationusing appropriate technologies.

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COMMENCEMENT

1 Information technology is used toretrieve, process, and communicateinformation and as a tool to enhancelearning.

• understand and use the moreadvanced features of word process-ing, spreadsheets, and data-basesoftware

• prepare multimedia presentationsdemonstrating a clear sense of audi-ence and purpose

• access, select, collate, and analyzeinformation obtained from a widerange of sources such as researchdata bases, foundations, organiza-tions, national libraries, and elec-tronic communication networks,including the Internet

• students receive news reports fromabroad and work in groups to pro-duce newspapers reflecting the per-spectives of different countries

INTERMEDIATE


• use a range of equipment and soft-ware to integrate several forms ofinformation in order to create goodquality audio, video, graphic, andtext-based presentations

• use spreadsheets and data-base soft-ware to collect, process, display, andanalyze information. Studentsaccess needed information fromelectronic data bases and on-linetelecommunication services

• systematically obtain accurate andrelevant information pertaining to aparticular topic from a range ofsources, including local and nationalmedia, libraries, museums, govern-mental agencies, industries, andindividuals

ELEMENTARY


• use a variety of equipment and soft-ware packages to enter, process, dis-play, and communicate informationin different forms using text, tables,pictures, and sound

• telecommunicate a message to a dis-tant location with teacher help

• access needed information fromprinted media, electronic data bases,and community resources

.

2 Knowledge of the impacts and limita -tions of information systems is essen -tial to its effective and ethical use.

• describe the uses of information sys-tems in homes, schools, and busi-nesses

Continued.

COMMENCEMENT

• utilize electronic networks to shareinformation

• model solutions to a range of prob-lems in mathematics, science, andtechnology using computer simula-tion software


• explain the impact of the use andabuse of electronically generatedinformation on individuals and fam-ilies

• evaluate software packages relativeto their suitability to a particularapplication and their ease of use

• discuss the ethical and social issuesraised by the use and abuse of infor-mation systems

3 Information technology can have pos -itive and negative impacts on society,depending upon how it is used.

• work with a virtual community toconduct a project or solve a problemusing the network

• discuss how applications of infor-mation technology can addresssome major global problems andissues

• discuss the environmental, ethical,moral, and social issues raised bythe use and abuse of informationtechnology

INTERMEDIATE

• collect data from probes to measureevents and phenomena

• use simple modeling programs tomake predictions


• understand the need to question theaccuracy of information displayedon a computer because the resultsproduced by a computer may beaffected by incorrect data entry

• identify advantages and limitationsof data-handling programs andgraphics programs

• understand why electronicallystored personal information hasgreater potential for misuse thanrecords kept in conventional form


• use graphical, statistical, and pre-sentation software to presents pro-ject to fellow classmates

• describe applications of informationtechnology in mathematics, science,and other technologies that addressneeds and solve problems in thecommunity

• explain the impact of the use andabuse of electronically generatedinformation on individuals and fam-ilies

ELEMENTARY

• understand that computers are usedto store personal information

• demonstrate ability to evaluateinformation


• describe the uses of information sys-tems in homes and schools

• demonstrate ability to evaluateinformation critically

STANDARD

InformationSystems

STANDARD

MathematicsStudents will understand mathematicsand become mathematically confidentby communicating and reasoningmathematically, by applying mathe-matics in real-world settings, and bysolving problems through the inte-grated study of number systems,geometry, algebra, data analysis,probability, and trigonometry.

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Reason NumberOperations

Model

Measure

Uncertain

Pat t e r n

COMMENCEMENT

• derive and apply formulas to findmeasures such as length, area, vol-ume, weight, time, and angle inreal-world contexts

• choose the appropriate tools formeasurement

• use dimensional analysis techniques

• use statistical methods includingmeasures of central tendency todescribe and compare data

• use trigonometry as a method tomeasure indirectly

• apply proportions to scale draw-ings, computer-assisted design blue-prints, and direct variation in orderto compute indirect measurements

• relate absolute value, distancebetween two points, and the slopeof a line to the coordinate plane

• understand error in measurementand its consequence on subsequentcalculations

• use geometric relationships in relevant measurement problemsinvolving geometric concepts

INTERMEDIATE

• apply a variety of reasoning strate-gies

• make and evaluate conjectures andarguments using appropriate lan-guage

• make conclusions based on induc-tive reasoning

• justify conclusions involving simpleand compound (i.e., and/or) state-ments

Students use mathematical reasoning to analyze mathematicalsituations, make conjectures, gather evidence, and construct anargument. Students will:

ELEMENTARY

• use models, facts, and relationshipsto draw conclusions about mathe-matics and explain their thinking

• use patterns and relationships toanalyze mathematical situations

• justify their answers and solutionprocesses

• use logical reasoning to reach sim-ple conclusions

MATHEMATICAL REASONING

MathematicsSTANDARD 3

FOUR-YEAR SEQUENCE

• construct indirect proofs or proofs using mathematical induction

• investigate and compare the axiomatic structures of various geometries

Reason

COMMENCEMENT

• judge the reasonableness of resultsobtained from applications in alge-bra, geometry, trigonometry, proba-bility, and statistics

• judge the reasonableness of a graphproduced by a calculator or comput-er

• use experimental or theoreticalprobability to represent and solveproblems involving uncertainty

• use the concept of random variablein computing probabilities

• determine probabilities using per-mutations and combinations

INTERMEDIATE

• understand, represent, and usenumbers in a variety of equivalentforms (integer, fraction, decimal,percent, exponential, expanded andscientific notation)

• understand and apply ratios, pro-portions, and percents through awide variety of hands-on explo-rations

• develop an understanding of num-ber theory (primes, factors, andmultiples)

• recognize order relations for deci-mals, integers, and rational numbers

Students use number sense and numeration to develop anunderstanding of multiple uses of numbers in the real world, useof numbers to communicate mathematically, and use of num-bers in the development of mathematical ideas. Students will:

ELEMENTARY

• use whole numbers and fractions toidentify locations, quantify groupsof objects, and measure distances

• use concrete materials to modelnumbers and number relationshipsfor whole numbers and commonfractions, including decimal frac-tions

• relate counting to grouping and toplace-value

• recognize the order of whole num-bers and commonly used fractionsand decimals

• demonstrate the concept of percentthrough problems related to actualsituations

NUMBER & NUMERATION

Mathematics

FOUR-YEAR SEQUENCE

• understand the concept of infinity.

• recognize the hierarchy of the complex number system.

• model the structure of the complex number system.

• recognize when to use and how to apply the field properties.

Number

STANDARD 3

COMMENCEMENT

• use function vocabulary and nota-tion

• represent and analyze functionsusing verbal descriptions, tables,equations, and graphs

• translate among the verbal descrip-tions, tables, equations and graphicforms of functions

• analyze the effect of parametricchanges on the graphs of functions.

• apply linear, exponential, and qua-dratic functions in the solution ofproblems

• apply and interpret transformationsto functions

• model real-world situations with theappropriate function

• apply axiomatic structure to algebraand geometry

• use computers and graphing calcu-lators to analyze mathematical phe-nomena

INTERMEDIATE

• add, subtract, multiply, and dividefractions, decimals, and integers

• explore and use the operations deal-ing with roots and powers

• use grouping symbols (parentheses)to clarify the intended order of oper-ations

• apply the associative, commutative,distributive, inverse, and identityproperties

• demonstrate an understanding ofoperational algorithms (proceduresfor adding, subtracting, etc.)

• develop appropriate proficiencywith facts and algorithms

• apply concepts of ratio and propor-tion to solve problems

Students use mathematical operations and relationships amongthem to understand mathematics. Students will:

ELEMENTARY

• add, subtract, multiply, and dividewhole numbers

• develop strategies for selecting theappropriate computational andoperational method in problem-solving situations

• know single digit addition, subtrac-tion, multiplication, and divisionfacts

• understand the commutative andassociative properties

OPERATION

Mathematics

FOUR-YEAR SEQUENCE

• use appropriate techniques, including graphing utilities, to perform basicoperations on matrices.

• use rational exponents on real numbers and all operations on complex num-bers.

• combine functions using the basic operations and the composition of twofunctions.

Operations

STANDARD 3

COMMENCEMENT

• represent problem situations sym-bolically by using algebraic expres-sions, sequences, tree diagrams, geo-metric figures, and graphs

• manipulate symbolic representa-tions to explore concepts at anabstract level

• choose appropriate representationsto facilitate the solving of a problem

• use learning technologies to makeand verify geometric conjectures

• justify the procedures for basic geo-metric constructions

• investigate transformations in thecoordinate plane

• develop meaning for basic conic sec-tions

• develop and apply the concept ofbasic loci to compound loci

• use graphing utilities to create andexplore geometric and algebraicmodels

• model real-world problems withsystems of equations and inequali-ties

INTERMEDIATE

• visualize, represent, and transformtwo- and three-dimensional shapes

• use maps and scale drawings to rep-resent real objects or places

• use the coordinate plane to exploregeometric ideas

• represent numerical relationships inone- and two-dimensional graphs

• use variables to represent relation-ships

• use concrete materials and diagramsto describe the operation of realworld processes and systems

• develop and explore models that doand do not rely on chance

• investigate both two- and three-dimensional transformations

• use appropriate tools to constructand verify geometric relationships

• develop procedures for basic geo-metric constructions

Students use mathematical modeling/multiple representation toprovide a means of presenting, interpreting, communicating, andconnecting mathematical information and relationships.Students will:

ELEMENTARY

• use concrete materials to model spa-tial relationships

• construct tables, charts, and graphsto display and analyze real-worlddata

• use multiple representations (simu-lations, manipulative materials, pic-tures, and diagrams) as tools toexplain the operation of everydayprocedures

• use variables such as height, weight,and hand size to predict changesover time

• use physical materials, pictures, anddiagrams to explain mathematicalideas and processes and to demon-strate geometric concepts

MODELING/MULTIPLE REPRESENTATION

Mathematics

Continued.

Model

STANDARD 3

Mathematics

MODELING/MULTIPLE REPRESENTATION

FOUR-YEAR SEQUENCE

• model vector quantities both algebraically and geometrically

• represent graphically the sum and difference of two complex num-bers

• model and solve problems that involve absolute value, vectors, andmatrices

• model quadratic inequalities both algebraically and graphically

• model the composition of transformations

• determine the effects of changing parameters of the graphs of func-tions

• use polynomial, rational, trigonometric, and exponential functions tomodel real-world relationships

• use algebraic relationships to analyze the conic sections

• use circular functions to study and model periodic real-world phe-nomena

• illustrate spatial relationships using perspective, projec-tions, andmaps

• represent problem situations using discrete structures such as finitegraphs, matrices, sequences, and recurrence relations

• analyze spatial relationships using the Cartesian coordinate system inthree dimensions

ModelSTANDARD 3

COMMENCEMENT

• derive and apply formulas to findmeasures such as length, area, vol-ume, weight, time, and angle inreal-world contexts

• the appropriate tools for measure-ment

• use dimensional analysis techniques

• use statistical methods includingmeasures of central tendency todescribe and compare data

• use trigonometry as a method tomeasure indirectly

• apply proportions to scale draw-ings, computer-assisted design blue-prints, and direct variation in orderto compute indirect measurements

• relate absolute value, distancebetween two points, and the slopeof a line to the coordinate plane

• understand error in measurementand its consequence on subsequentcalculations

• use geometric relationships in rele-vant measurement problems involv-ing geometric concepts

INTERMEDIATE

• estimate, make, and use measure-ments in real-world situations

• select appropriate standard andnonstandard measurement unitsand tools to measure to a desireddegree of accuracy

• develop measurement skills andinformally derive and apply formu-las in direct measurement activities

• use statistical methods and mea-sures of central tendencies to dis-play, describe, and compare data

• explore and produce graphic repre-sentations of data using calcula-tors/computers

• develop critical judgment for thereasonableness of measurement

Students use measurement in both metric and English measureto provide a major link between the abstractions of mathematicsand the real world in order to describe and compare objects anddata.. Students will:

ELEMENTARY

• understand that measurement isapproximate, never exact

• select appropriate standard andnonstandard measurement tools inmeasurement activities

• understand the attributes of area,length, capacity, weight, volume,time, temperature, and angle

• estimate and find measures such aslength, perimeter, area, and volumeusing both nonstandard and stan-dard units

• collect and display data

• use statistical methods such asgraphs, tables, and charts to inter-pret data

MEASUREMENT

Mathematics

Continued.

Measure

STANDARD 3

Mathematics

MEASUREMENT

FOUR-YEAR SEQUENCE

• derive and apply formulas relating angle measure and arc degreemeasure in a circle

• prove and apply theorems related to lengths of segments in a circle

• define the trigonometric functions in terms of the unit circle

• relate trigonometric relationships to the area of a triangle and to thegeneral solutions of triangles

• apply the normal curve and its properties to familiar contexts

• design a statistical experiment to study a problem and communicatethe outcomes, including dispersion

• use statistical methods, including scatter plots and lines of best fit, tomake predictions

• apply the conceptual foundation of limits, infinite sequences andseries, the area under a curve, rate of change, inverse variation, andthe slope of a tangent line to authentic problems in mathematics andother disciplines

• determine optimization points on a graph

• use derivatives to find maximum, minimum, and inflection points ofa function

Measure

STANDARD 3

COMMENCEMENT

• judge the reasonableness of a graphproduced by a calculator or comput-er

• use experimental or theoreticalprobability to represent and solveproblems involving uncertainty

• use the concept of random variablein computing probabilities

• determine probabilities using per-mutations and combinations

INTERMEDIATE

• use estimation to check the reason-ableness of results obtained by com-putation, algorithms, or the use oftechnology

• use estimation to solve problems forwhich exact answers are inappropri-ate

• estimate the probability of events

• use simulation techniques to esti-mate probabilities

• determine probabilities of indepen-dent and mutually exclusive events

Students use ideas of uncertainty to illustrate that mathematicsinvolves more than exactness when dealing with everyday situa-tions. Students will:

ELEMENTARY

• make estimates to compare to actualresults of both formal and informalmeasurement

• make estimates to compare to actualresults of computations

• recognize situations where only anestimate is required

• develop a wide variety of estimationskills and strategies

• determine the reasonableness ofresults

• predict experimental probabilities

• make predictions using unbiasedrandom samples

• determine probabilities of simpleevents

UNCERTAINTY

Mathematics

FOUR -YEAR SEQUENCE

• interpret probabilities in real-world situations

• use a Bernoulli experiment to determine probabilities for experiments withexactly two outcomes

• use curve fitting to predict from data

• apply the concept of random variable to generate and interpret probabilitydistributions

• create and interpret applications of discrete and continuous probability dis-tributions

• make predictions based on interpolations and extrapolations from data

• obtain confidence intervals and test hypotheses using appropriate statisticalmethods

• approximate the roots of polynomial equations

Uncertain

STANDARD 3

COMMENCEMENT

• use function vocabulary and nota-tion

• represent and analyze functionsusing verbal descriptions, tables,equations, and graphs

• translate among the verbal descrip-tions, tables, equations and graphicforms of functions

• analyze the effect of parametricchanges on the graphs of functions

• apply linear, exponential, and qua-dratic functions in the solution ofproblems

• apply and interpret transformationsto functions

• model real-world situations with theappropriate function

• apply axiomatic structure to algebraand geometry

• use computers and graphing calcu-lators to analyze mathematical phe-nomena

INTERMEDIATE

• recognize, describe, and generalize awide variety of patterns and func-tions

• describe and represent patterns andfunctional relationships using tables,charts and graphs, algebraic expres-sions, rules, and verbal descriptions

• develop methods to solve basic lin-ear and quadratic equations

• develop an understanding of func-tions and functional relationships:that a change in one quantity (vari-able) results in change in another

• verify results of substituting vari-ables

• apply the concept of similarity inrelevant situations

• use properties of polygons to classi-fy them

• explore relationships involvingpoints, lines, angles, and planes

• develop and apply the Pythagoreanprinciple in the solution of problems

• explore and develop basic conceptsof right triangle trigonometry

• use patterns and functions to repre-sent and solve problems

Students use patterns and functions to develop mathematicalpower, appreciate the true beauty of mathematics, and constructgeneralizations that describe patterns simply and efficientlyStudents will: .

ELEMENTARY

• recognize, describe, extend, and cre-ate a wide variety of patterns

• represent and describe mathematicalrelationships

• explore and express relationshipsusing variables and open sentences

• solve for an unknown using manip-ulative materials

• use a variety of manipulative mate-rials and technologies to explorepatterns

• interpret graphs

• explore and develop relationshipsamong two- and three-dimensionalgeometric shapes

• discover patterns in nature, art,music, and literature

PATTERNS/FUNCTIONS

Mathematics

Continued.

Pat t er n

STANDARD 3

FOUR-YEAR SEQUENCE

• solve equations with complex roots using a variety of algebraic and graphical methods withappropriate tools

• understand and apply the relationship between rectangular form and polar form of a com-plex number

• evaluate and form the composition of functions

• use the definition of a derivative to examine properties of a function

• solve equations involving fractions, absolute values, and radicals

• use basic transformations to demonstrate similarity and congruence of figures

• identify and differentiate between direct and indirect isometries

• analyze inverse functions using transformations

• apply ideas of symmetries in sketching and analyzing graphs of functions

• use the normal curve to answer questions about data

• develop methods to solve trigonometric equations and verify trigonometric functions

• describe patterns produced by processes of geometric change, formally connecting iteration,approximations, limits, and fractals

• extend pattern and compute the nth term in numerical and geometric sequences

• use the limiting process to analyze infinite sequences and series

• use algebraic and geometric iteration to explore patterns and solve problems

• solve optimization problems

• use linear programming and difference equations in the solution of problems

PATTERNS/FUNCTIONS

Mathematics

Pat t er n

STANDARD 3

STANDARD

Students will understand and applyscientific concepts, principles, andtheories pertaining to the physicalsetting and living environment andrecognize the historical developmentof ideas in science.

&

Continued.

5 Energy and matter interact throughforces that result in changes inmotion.

• describe the effects of commonforces (pushes and pulls) on objects,such as those caused by gravity,magnetism, and mechanical forces

• describe how forces can operateacross distances

4 Energy exists in many forms, andwhen these forms change energy isconserved.

• describe a variety of forms of energy(e.g., heat, chemical, light) and thechanges that occur in objects whenthey interact with those forms ofenergy

• observe the way one form of energycan be transformed into anotherform of energy present in commonsituations (e.g., mechanical to heatenergy, mechanical to electricalenergy, chemical to heat energy)


PHYSICAL SETTING

1 The Earth and celestial phenomenacan be described by principles of rela -tive motion and perspective.

• describe patterns of daily, monthly,and seasonal changes in their envi-ronment

2 Many of the phenomena that weobserve on Earth involve interactionsamong components of air, water, andland.

• describe the relationships among air,water, and land on Earth

3 Matter is made up of particles whoseproperties determine the observablecharacteristics of matter and its reac -tivity.

• observe and describe properties ofmaterials using appropriate tools

• describe chemical and physicalchanges, including changes in statesof matter

STANDARD Science

6 Plants and animals depend on eachother and their physical environment.

• describe how plants and animals,including humans, depend uponeach other and the nonliving envi-ronment

• describe the relationship of the sunas an energy source for living andnonliving cycles

7 Human decisions and activities havehad a profound impact on the physicaland living environment.

• identify ways in which humanshave changed their environment

and the effects of thosechanges

4 The continuity of life is sustainedthrough reproduction and develop -ment.

• describe the major stages in the lifecycles of selected plants and animals

• describe evidence of growth, repair,and maintenance, such as nails, hair,and bone, and the healing of cutsand bruises

5 Organisms maintain a dynamic equi -librium that sustains life.

• describe basic life functions of com-mon living specimens (guppy, meal-worm, gerbil)

• describe some survival behav-iors of common living speci-mens

• describe the factors thathelp promote goodhealth and growth inhumans


THE LIVING ENVIRONMENT

1 Living things are both similar to anddifferent from each other and nonliv -ing things.

• describe the characteristics of andvariations between living and non-living things

• describe the life processes commonto all living things

2 Organisms inherit genetic informa -tion in a variety of ways that resultin continuity of structure and func -tion between parents and offspring.

• recognize that traits of living thingsare both inherited and acquired orlearned

• recognize that for humans and otherliving things there is genetic conti-nuity between generations

3 Individual organisms and specieschange over time.

• describe how the structures ofplants and animals complement theenvironment of the plant or animal

• observe that differences within aspecies may give individuals anadvantage in surviving and repro-ducing

Continued.

STANDARD Science


• describe different patterns of motionof objects

• observe, describe, and compareeffects of forces (gravity, electric cur-rent, and magnetism) on the motionof objects


• describe the sources and identify thetransformations of energy observedin everyday life

• observe and describe heating andcooling events

• observe and describe energychanges as related to chemical reac-tions

• observe and describe the propertiesof sound, light, magnetism, andelectricity

• describe situations that support theprinciple of conservation of energy


PHYSICAL SETTING


• explain daily, monthly, and seasonalchanges on earth


• explain how the atmosphere (air),hydrosphere (water), and lithos-phere (land) interact, evolve, andchange

• describe volcano and earthquakepatterns, the rock cycle, and weatherand climate changes


• observe and describe properties ofmaterials, such as density, conduc-tivity, and solubility

• distinguish between chemical andphysical changes

• develop their own mental models toexplain common chemical reactionsand changes in states of matter

STANDARD Science


• describe the flow of energy andmatter through food chains andfood webs

• provide evidence that green plantsmake food and explain the signifi-cance of this process to other organ-isms

7 Human decisions and activities havehad a profound impact on the physicaland living environment.

• describe how living things, includ-ing humans, depend upon the livingand nonliving environment for theirsurvival

• describe the effects of environmentalchanges on humans and other pop-ulations


• observe and describe the variationsin reproductive patterns of organ-isms, including asexual and sexualreproduction

• explain the role of sperm and eggcells in sexual reproduction

• observe and describe developmentalpatterns in selected plants and ani-mals (e.g., insects, frogs, humans,seed-bearing plants)

• observe and describe cell division atthe microscopic level and its macro-scopic effects


• compare the way a variety of livingspecimens carry out basic life func-tions and maintain dynamic equilib-rium

• describe the importance of majornutrients, vitamins, and minerals inmaintaining health and promotinggrowth and explain the need for aconstant input of energy for livingorganisms




• compare and contrast the parts ofplants, animals, and one-celledorganisms

• explain the functioning of the majorhuman organ systems and theirinteractions


• describe sexual and asexual mecha-nisms for passing genetic materialsfrom generation to generation

• describe simple mechanisms relatedto the inheritance of some physicaltraits in offspring


• describe sources of variation inorganisms and their structures andrelate the variations to survival

• describe factors responsible for com-petition within species and the sig-nificance of that competition

Continued.

STANDARD Science


• explain and predict different pat-terns of motion of objects (e.g., lin-ear and angular motion, velocityand acceleration, momentum andinertia)

• explain chemical bonding in termsof the motion of electrons

• compare energy relationships withinan atom’s nucleus to those outsidethe nucleus


• explain the properties of materialsin terms of the arrangement andproperties of the atoms that com-pose them

• use atomic and molecular models toexplain common chemical reactions

• apply the principle of conservationof mass to chemical reactions

• use kinetic molecular theory toexplain rates of reactions and therelationships among temperature,pressure, and volume of a substance


• observe and describe transmissionof various forms of energy

• explain heat in terms of kinetic mol-ecular theory

• explain variations in wavelengthand frequency in terms of the sourceof the vibrations that produce them,e.g., molecules, electrons, andnuclear particles

• explain the uses and hazards ofradioactivity


PHYSICAL SETTING


• explain complex phenomena, suchas tides, variations in day length,solar insolation, apparent motion ofthe planets, and annual traverse ofthe constellations

• describe current theories about theorigin of the universe and solar sys-tem


• use the concepts of density and heatenergy to explain observations ofweather patterns, seasonal changes,and the movements of the Earth’splates

• explain how incoming solar radia-tions, ocean currents, and landmasses affect weather and climate

STANDARD Science


• explain factors that limit growth ofindividuals and populations

• explain the importance of preserv-ing diversity of species and habitats

• explain how the living and nonliv-ing environments change over timeand respond to disturbances

7 Human decisions and activities havehad a profound impact on the physi -cal and living environment.

• describe the range of interrelation-ships of humans with the living andnonliving environment

• explain the impact of technologicaldevelopment and growth in thehuman population on the living andnon-living environment

• explain how individual choices andsocietal actions can contribute toimproving the environment


• explain how organisms, includinghumans, reproduce their own kind


• explain the basic biochemicalprocesses in living organisms andtheir importance in maintainingdynamic equilibrium

• explain disease as a failure of home-ostasis

• relate processes at the system levelto the cellular level in order toexplain dynamic equilibrium inmulticelled organisms




• explain how diversity of popula-tions within ecosystems relates tothe stability of ecosystems

• describe and explain the structuresand functions of the human body atdifferent organizational levels (e.g.,systems, tissues, cells, organelles)

• explain how a one-celled organismis able to function despite lackingthe levels of organization present inmore complex organisms


• explain how the structure and repli-cation of genetic material result inoffspring that resemble their parents

• explain how the technology ofgenetic engineering allows humansto alter the genetic makeup oforganisms


• explain the mechanisms and pat-terns of evolution

Students will apply technologicalknowledge and skills to design,construct, use, and evaluate productsand systems to satisfy human andenvironmental needs.

STANDARD &

Designomputer

Systems

HistoryManage

INTERMEDIATE

• identify needs and opportunities fortechnical solutions from an investi-gation of situations of general orsocial interest

• locate and utilize a range of printed,electronic, and human informationresources to obtain ideas

• consider constraints and generateseveral ideas for alternative solu-tions, using group and individualideation techniques (group discus-sion, brainstorming, forced connec-tions, role play); defer judgmentuntil a number of ideas have beengenerated; evaluate (critique) ideas;and explain why the chosen solu-tion is optimal

• develop plans, including drawingswith measurements and details ofconstruction, and construct a modelof the solution, exhibiting a degreeof craftsmanship

• in a group setting, test their solutionagainst design specifications, pre-sent and evaluate results, describehow the solution might have beenmodified for different or betterresults, and discuss tradeoffs thatmight have to be made

Engineering design is an iterative process involving modeling andoptimization used to develop technological solutions to problemswithin given constraints. Students will:

ELEMENTARY

• describe objects, imaginary or real,that might be modeled or made dif-ferently and suggest ways in whichthe objects can be changed, fixed, orimproved

• investigate prior solutions and ideasfrom books, magazines, family,friends, neighbors, and communitymembers

• generate ideas for possible solu-tions, individually and throughgroup activity; apply age-appropri-ate mathemat-ics and science skills;evaluate the ideas and determinethe best solution; and explain rea-sons for the choices

• plan and build, under supervision, amodel of the solution using familiarmaterials, processes, and hand tools

• discuss how best to test the solution;perform the test under teachersupervision; record and portrayresults through numerical andgraphic means; discuss orally whythings worked or didn’t work; andsummarize results in writing, sug-gesting ways to make the solutionbetter

ENGINEERING DESIGN

Technology

Continued.

Design

STANDARD 5

COMMENCEMENT

• initiate and carry out a thoroughinvestigation of an unfamiliar situa-tion and identify needs and oppor-tunities for technological inventionor innovation

• identify, locate, and use a widerange of information resourcesincluding subject experts, libraryreferences, magazines, videotapes,films, electronic data bases and on-line services, and discuss and docu-ment through notes and sketcheshow findings relate to the problem

• generate creative solution ideas,break ideas into the significant func-tional elements, and explore possi-ble refinements; predict possibleoutcomes using mathematical andfunctional modeling techniques;choose the optimal solution to theproblem, clearly documenting ideasagainst design criteria and con-straints; and explain how humanvalues, economics, ergonomics, andenvironmental considerations haveinfluenced the solution

• develop work schedules and planswhich include optimal use and costof materials, processes, time, andexpertise; construct a model of thesolution, incorporating developmen-tal modifications while working to ahigh degree of quality (craftsman-ship)

• in a group setting, devise a test ofthe solution relative to the designcriteria and perform the test; record,portray, and logically evaluate per-formance test results through quan-titative, graphic, and verbal means;and use a variety of creative verbaland graphic techniques effectivelyand persuasively to present conclu-sions, predict impacts and newproblems, and suggest and pursuemodifications

Technology

Design

STANDARD 5

COMMENCEMENT

• test, use, and describe the attributesof a range of material (includingsynthetic and composite materials),information, and energy resources

• select appropriate tools, instru-ments, and equipment and use themcorrectly to process materials, ener-gy, and information

• explain tradeoffs made in selectingalternative resources in terms ofsafety, cost, properties, availability,ease of processing, and disposability

• describe and model methods(including computer-based meth-ods) to control system processes andmonitor system outputs

INTERMEDIATE

• choose and use resources for a par-ticular purpose based upon ananalysis and understanding of theirproperties, costs, availability, andenvironmental impact

• use a variety of hand tools andmachines to change materials intonew forms through forming, sepa-rating, and combining processes,and processes which cause internalchange to occur

• combine manufacturing processeswith other technological processesto produce, market, and distribute aproduct

• process energy into other forms andinformation into more meaningfulinformation

Technological tools, materials, and other resources should beselected on the basis of safety, cost, availability, appropriateness,and environmental impact; technological processes change ener-gy, information, and material resources into more useful forms.Students will:

ELEMENTARY

• explore, use, and process a varietyof materials and energy sources todesign and construct things

• understand the importance of safety,cost, ease of use, and availability inselecting tools and resources for aspecific purpose

• develop basic skill in the use ofhand tools

• use simple manufacturing processes(e.g., assembly, mul-tiple stages ofproduction, quality control) to pro-duce a product

• use appropriate graphic and elec-tronic tools and techniques toprocess information

TOOLS, RESOURCES & TECHNOLOGICAL

PROCESSES

TechnologySTANDARD 5

COMMENCEMENT

• understand basic computer architec-ture and describe the function ofcomputer subsystems and peripher-al devices

• select a computer system that meetspersonal needs

• attach a modem to a computer sys-tem and telephone line, set up anduse communications software, con-nect to various on-line networks,including the Internet, and accessneeded information using e-mail,telnet, gopher, ftp, and web searches

• use computer-aided drawing anddesign (CADD) software to modelrealistic solutions to design prob-lems

• develop an understanding of com-puter programming and attain somefacility in writing computer pro-grams

INTERMEDIATE

• assemble a computer system includ-ing keyboard, central processingunit and disc drives, mouse,modem, printer, and monitor

• use a computer system to connect toand access needed information fromvarious Internet sites

• use computer hardware and soft-ware to draw and dimension proto-typical designs

• use a computer as a modeling tool

• use a computer system to monitorand control external events and/orsystems

Computers, as tools for design, modeling, information process-ing, communication, and system control, have greatly increasedhuman productivity and knowledge. Students will:

ELEMENTARY

• identify and describe the function ofthe major components of a comput-er system

• use the computer as a tool for gener-ating and drawing ideas

• control computerized devices andsystems through programming

• model and simulate the design of acomplex environment by givingdirect commands

COMPUTER TECHNOLOGY

Technology

omputer

STANDARD 5

COMMENCEMENT

• explain why making tradeoffsamong characteristics, such as safe-ty, function, cost, ease of operation,quality of post-purchase support,and environmental impact, is neces-sary when selecting systems for spe-cific purposes

• model, explain, and analyze the per-formance of a feedback control sys-tem

• explain how complex technologicalsystems involve the confluence ofnumerous other systems

INTERMEDIATE

• select appropriate technological sys-tems on the basis of safety, function,cost, ease of operation, and qualityof post-purchase support

• assemble, operate, and explain theoperation of simple open- andclosed-loop electrical, electronic,mechanical, and pneumatic systems

• describe how subsystems and sys-tem elements (inputs, processes,outputs) interact within systems

• describe how system controlrequires sensing information, pro-cessing it, and making changes

Technological systems are designed to achieve spe-cific resultsand produce outputs, such as products, structures, services,energy, or other systems. Students will:

ELEMENTARY

• identify familiar examples of tech-nological systems that are used tosatisfy human needs and wants, andselect them on the basis of safety,cost, and function

• assemble and operate simple tech-nological systems, including thosewith interconnecting mechanisms toachieve different kinds of movement

• understand that larger systems aremade up of smaller component sub-systems

TECHNOLOGICAL SYSTEMS

Technology

Systems

STANDARD 5

COMMENCEMENT

• explain how technological inven-tions and innovations have causedglobal growth and interdependence,stimulated economic competitive-ness, created new jobs, and madeother jobs obsolete

INTERMEDIATE

• describe how the evolution of tech-nology led to the shift in societyfrom an agricultural base to anindustrial base to an informationbase

• understand the contributions ofpeople of different genders, races,and ethnic groups to technologicaldevelopment

• describe how new technologies haveevolved as a result of combiningexisting technologies (e.g., photog-raphy combined optics and chem-istry; the airplane combined kiteand glider technology with a light-weight gasoline engine)

Technology has been the driving force in the evolution of societyfrom an agricultural to an industrial to an information base.Students will:

ELEMENTARY

• identify technological developmentsthat have significantly acceleratedhuman progress

HISTORY & EVOLUTION OF TECHNOLOGY

Technology

History

STANDARD 5

COMMENCEMENT

• explain that although technologicaleffects are complex and difficult topredict accurately, humans can con-trol the development and imple-mentation of technology.

• explain how computers andautomation have changed thenature of work

• explain how national security isdependent upon both military andnonmilitary applications of technol-ogy

INTERMEDIATE

• describe how outputs of a techno-logical system can be desired, unde-sired, expected, or unexpected

• describe through examples howmodern technology reduces manu-facturing and construction costs andproduces more uniform products

Technology can have positive and negative impacts on individu-als, society, and the environment and humans have the capabili-ty and responsibility to constrain or promote technological devel-opment. Students will:

ELEMENTARY

• describe how technology can havepositive and negative effects on theenvironment and on the way peoplelive and work

IMPACTS OF TECHNOLOGY

TechnologySTANDARD 5

COMMENCEMENT

• develop and use computer-basedscheduling and project trackingtools, such as flow charts andgraphs

• explain how statistical process con-trol helps to assure high quality out-put

• discuss the role technology hasplayed in the operation of successfulU.S. businesses and under what cir-cumstances they are competitivewith other countries

• explain how technological inven-tions and innovations stimulate eco-nomic competitiveness and how, inorder for an innovation to lead tocommercial success, it must betranslated into products and ser-vices with marketplace demand

• describe new management tech-niques (e.g., computer-aided engi-neering, computer-integrated manu-facturing, total quality management,just-in-time manufacturing), incor-porate some of these in a technolog-ical endeavor, and explain how theyhave reduced the length of design-to- manufacture cycles, resulted inmore flexible factories, andimproved quality and customer sat-isfaction

• help to manage a group engaged inplanning, designing, implementa-tion, and evaluation of a project togain understanding of the manage-ment dynamics

INTERMEDIATE

• manage time and financial resourcesin a technological project

• provide examples of products thatare well (and poorly) designed andmade, describe their positive andnegative attributes, and suggestmeasures that can be implementedto monitor quality during produc-tion

• assume leadership responsibilitieswithin a structured group activity

Project management is essential to ensuring that technologicalendeavors are profitable and that products and systems are ofhigh quality and built safely, on schedule, and within budget.Students will:

ELEMENTARY

• participate in small group projectsand in structured group tasksrequiring planning, financing, pro-duction, quality control, and follow-up

• speculate on and model possibletechnological solutions that canimprove the safety and quality ofthe school or community environ-ment

MANAGEMENT OF TECHNOLOGY

Technology

ManageSTANDARD 5

STANDARD

6

Interconnectness:

CommonThemes

Students will understand therelationships and common themesthat connect mathematics, science,and technology and apply the themesto these and other areas of learning.

&

Thinking Models Scale

StableChange

Systems

STANDARD 6 Common Themes

Thinking

Systems

COMMENCEMENT

• explain how positive feedback andnegative feedback have oppositeeffects on system outputs

• use an input-process-output-feed-back diagram to model and com-pare the behavior of natural andengineered systems

• define boundary conditions whendoing systems analysis to determinewhat influences a system and how itbehaves

INTERMEDIATE

• describe the differences betweendynamic systems and organizationalsystems

• describe the differences and similar-ities between engineering systems,natural systems, and social systems

• describe the differences betweenopen- and closed-loop systems

• describe how the output from onepart of a system (which can includematerial, energy, or information) canbecome the input to other parts

Through systems thinking, people can recognize the commonali-ties that exist among all systems and how parts of a system inter-relate and combine to perform specific functions.Students will:

ELEMENTARY

• observe and describe interactionsamong components of simple sys-tems

• identify common things that can beconsidered to be systems (e.g., aplant population, a subway system,human beings)

SYSTEM THINKING

ModelsCOMMENCEMENT

• revise a model to create a more com-plete or improved representation ofthe system

• collect information about the behav-ior of a system and use modelingtools to represent the operation ofthe system

• find and use mathematical modelsthat behave in the same manner asthe processes under investigation

• compare predictions to actual obser-vations using test models

INTERMEDIATE

•select an appropriate model to beginthe search for answers or solutions toa question or problem

• use models to study processes thatcannot be studied directly (e.g.,when the real process is too slow,too fast, or too dangerous for directobservation)

• demonstrate the effectiveness of dif-ferent models to represent the samething and the same model to repre-sent different things

Through systems thinking, people can recognize the commonali-ties that exist among all systems and how parts of a system inter-relate and combine to perform specific functions. Students will:

ELEMENTARY

• analyze, construct, and operatemodels in order to discover attribut-es of the real thing

• discover that a model of somethingis different from the real thing butcan be used to study the real thing

• use different types of models, suchas graphs, sketches, diagrams, andmaps, to represent various aspectsof the real world

MODELS

Stable

Scale


COMMENCEMENT

• describe the effects of changes inscale on the functioning of physical,biological, or designed systems

• extend their use of powers of tennotation to understanding the expo-nential function and performingoperations with exponential factors

INTERMEDIATE

• cite examples of how differentaspects of natural and designed sys-tems change at different rates withchanges in scale

• use powers of ten notation to repre-sent very small and very large num-bers

The grouping of magnitudes of size, time, frequency, and pres-sures or other units of measurement into a series of relative orderprovides a useful way to deal with the immense range and thechanges in scale that affect the behavior and design of systems.Students will:

ELEMENTARY

• provide examples of natural andmanufactured things that belong tothe same category yet have very dif-ferent sizes, weights, ages, speeds,and other measurements

• identify the biggest and the smallestvalues as well as the average valueof a system when given informationabout its characteristics and behav-ior

MAGNITUDE & SCALE

COMMENCEMENT

• describe specific instances of howdisturbances might affect a system’sequilibrium, from small distur-bances that do not upset the equilib-rium to larger disturbances (thresh-old level) that cause the system tobecome unstable

• cite specific examples of howdynamic equilibrium is achieved byequality of change in opposingdirections

INTERMEDIATE

• describe how feedback mechanismsare used in both designed and nat-ural systems to keep changes withindesired limits

• describe changes within equilibriumcycles in terms of frequency or cyclelength and determine the highestand lowest values and when theyoccur

Equilibrium is a state of stability due either to a lack of changes(static equilibrium) or a balance between opposing forces(dynamic equilibrium). Students will:

ELEMENTARY

• cite examples of systems in whichsome features stay the same whileother features change

• distinguish between reasons for sta-bility—from lack of changes tochanges that counterbalance oneanother to changes within cycles

EQUILIBRIUM & STABILITY

Change


COMMENCEMENT

• use sophisticated mathematicalmodels, such as graphs and equa-tions of various algebraic or trigono-metric functions

• search for multiple trends whenanalyzing data for patterns, andidentify data that do not fit thetrends

INTERMEDIATE

• use simple linear equations to repre-sent how a parameter changes withtime

• observe patterns of change in trendsor cycles and make predictions onwhat might happen in the future

Identifying patterns of change is necessary for making predictionsabout future behavior and conditions. Students will:

ELEMENTARY

• use simple instruments to measuresuch quantities as distance, size, andweight and look for patterns in thedata

• analyze data by making tables andgraphs and looking for patterns ofchange

PATTERNS OF CHANGE

COMMENCEMENT

• use optimization techniques, such aslinear programming, to determineoptimum solutions to problems thatcan be solved using quantitativemethods

• analyze subjective decision makingproblems to explain the trade-offsthat can be made to arrive at thebest solution

INTERMEDIATE

• determine the criteria and con-straints and make trade-offs todetermine the best decision

• use graphs of information for a deci-sion making problem to determinethe optimum solution

In order to arrive at the best solution that meets criteria withinconstraints, it is often necessary to make trade-offs. Students will:

ELEMENTARY

• determine the criteria and con-straints of a simple decision makingproblem

• use simple quantitative methods,such as ratios, to compare costs tobenefits of a decision problem

OPTIMIZATION

STANDARD

7Interdiscipline

ProblemSolving

Students will apply the knowledgeand thinking skills of mathematics,science, and technology to addressreal-life problems and make informeddecisions.

&

onnections

Strategies

Skills

onnections

COMMENCEMENT

• analyze science/technology/societyproblems and issues on a communi-ty, national, or global scale and planand carry out a remedial course ofaction

• analyze and quantify consumerproduct data, understand environ-mental and economic impacts,develop a method for judging thevalue and efficacy of competingproducts, and discuss cost/benefitand risk/benefit tradeoffs made inarriving at the optimal choice

• design solutions to real-world prob-lems on a community, national, orglobal scale using a technologicaldesign process that integrates scien-tific investigation and rigorousmathematical analysis of the prob-lem and of the solution

• explain and evaluate phenomenamathematically and scientifically byformulating a testable hypothesis,demonstrating the logical connec-tions between the scientific conceptsguiding the hypothesis and thedesign of an experiment, applyingand inquiring into the mathematicalideas relating to investigation ofphenomena, and using (and if need-ed, designing) technological toolsand procedures to assist in theinvestigation and in the communica-tion of results

INTERMEDIATE

• analyze science/technology/societyproblems and issues at the locallevel and plan and carry out a reme-dial course of action

• make informed consumer decisionsby seeking answers to appropriatequestions about products, services,and systems; determining thecost/benefit and risk/benefit trade-offs; and applying this knowledgeto a potential purchase

• design solutions to real-world prob-lems of general social interest relat-ed to home, school, or communityusing scientific experimentation toinform the solution and applyingmathematical concepts and reason-ing to assist in developing a solution

• describe and explain phenomena bydesigning and conducting investiga-tions involving systematic observa-tions, accurate measurements, andthe identification and control ofvariables; by inquiring into relevantmathematical ideas; and by usingmathematical and technologicaltools and procedures to assist in theinvestigation

The knowledge and skills of mathematics, science, and technolo-gy are used together to make informed decisions and solve prob-lems, especially those relating to issues of science/technology/society, consumer decision making, design, and inquiry into phe-nomen. Students will:

ELEMENTARY

• analyze science/technology/societyproblems and issues that affect theirhome, school, orcommunity, andcarry out a remedial course of action

• make informed consumer decisionsby applying knowledge about theattributes of particular products andmaking cost/benefit tradeoffs toarrive at an optimal choice

• design solutions to problems involv-ing a familiar and real context,investigate related science conceptsto inform the solution, and usemathematics to model, quantify,measure, and compute

• observe phenomena and evaluatethem scientifically and mathemati-cally by conducting a fair test of theeffect of variables and using mathe-matical knowledge and technologi-cal tools to collect, analyze, and pre-sent data and conclusions.

CONNECTIONS

Interdisciplinary Problem SolvingSTANDARD 7

and technology are used together to make informed decisionsand solve problems, especially those relating to issues of sci-ence/technology/society, consumer decision making, design, andinquiry into phenomena. Students will:

ELEMENTARY—INTERMEDIATE—COMMENCEMENT

Students participate in an extended, culminating mathematics, science, andtechnology project. The project would require students to:

• work effectively • gather and process information • generate and analyze ideas • observe common themes • realize ideas • present results

STRATEGIES

Interdisciplinary Problem SolvingSTANDARD 7Strategies

ELEMENTARY—INTERMEDIATE—COMMENCEMENT

Working Effectively: Contributing to the work of a brainstorming group, laboratory partnership,cooperative learning group, or project team; planning procedures; identify andmanaging responsibilities of team members; and staying on task, whetherworking alone or as part of a group

Gathering and Processing Information: Accessing information from printed media, electronic data bases, and commu-nity resources and using the information to develop a definition of the prob-lem and to research possible solutions

Generating and Analyzing Ideas: Developing ideas for proposed solutions, investigating ideas, collecting data,and showing relationships and patterns in the data

Common Themes: Observing examples of common unifying themes, applying them to the prob-lem, and using them to better understand the dimensions of the problem

Realizing Ideas: Constructing components or models, arriving at a solution, and evaluating theresult

Presenting Results: Using a variety of media to present the solution and to communicate theresults

SKILLS & STRATEGIES FOR INTERDISCIPLINARY

PROBLEM SOLVINGSkills

Documents

Standard 12 Standard 28 Standard 310 Standard 421 Standard