**Learning Standards for
**
**Mathematics, Science, and Technology at Three Levels**

Students will use mathematical analysis, scientific inquiry, and engineering design, as appropriate, to pose questions, seek answers, and develop solutions.

Students will access, generate, process, and transfer information using appropriate technologies.

Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry.

Students will understand and apply scientific concepts, principles, and theories pertaining to the physical setting and living environment and recognize the historical development of ideas in science.

Students will apply technological knowledge and skills to design, construct, use, and evaluate products and systems to satisfy human and environmental needs.

Students will understand the relationships and common themes that connect mathematics, science, and technology and apply the themes to these and other areas of learning.

Students will apply the knowledge and thinking skills of mathematics, science, and technology to address real-life problems and make informed decisions.

**Standard 1-Analysis, Inquiry, and Design
**Elementary

Students will use mathematical analysis, scientific inquiry, and engineering design, as appropriate, to pose questions, seek answers, and develop solutions.

**Mathematical Analysis
**

1. Abstraction and symbolic representation are used to communicate mathematically.

Students:

- use special mathematical notation and symbolism to communicate in mathematics and to compare and describe quantities, express relationships, and relate mathematics to their immediate environments.

This is evident, for example, when students:

- describe their ages as an inequality such as 7 < £ < 10.

2. Deductive and inductive reasoning are used to reach mathematical conclusions.

Students:

- use simple logical reasoning to develop conclusions, recognizing that patterns and relationships present in the environment assist them in reaching these conclusions.

3. Critical thinking skills are used in the solution of mathematical problems.

Students:

- explore and solve problems generated from school, home, and community situations, using concrete objects or manipulative materials when possible.

**Scientific Inquiry
**

1. The central purpose of scientific inquiry is to develop explanations of natural phenomena in a continuing, creative process.

Students:

- ask "why" questions in attempts to seek greater understanding concerning objects and events they have observed and heard about.
- question the explanations they hear from others and read about, seeking clarification and comparing them with their own observations and understandings.
- develop relationships among observations to construct descriptions of objects and events and to form their own tentative explanations of what they have observed.

This is evident, for example, when students:

- observe a variety of objects that either sink or float when placed in a container of water. Working in groups, they propose an explanation of why objects sink or float. After sharing and discussing their proposed explanation, they refine it and submit it for assessment. The explanation is rated on clarity and plausibility.

2. Beyond the use of reasoning and consensus, scientific inquiry involves the testing of proposed explanations involving the use of conventional techniques and procedures and usually requiring considerable ingenuity.

Students:

- develop written plans for exploring phenomena or for evaluating explanations guided by questions or proposed explanations they have helped formulate.
- share their research plans with others and revise them based on their suggestions.
- carry out their plans for exploring phenomena through direct observation and through the use of simple instruments that permit measurements of quantities (e.g., length, mass, volume, temperature, and time).

This is evident, for example, when students:

- are asked to develop a way of testing their explanation of why objects sink or float when placed in a container of water. They tell what procedures and materials they will use and indicate what results will support their explanation. Their plan is critiqued by others, they revise it, and submit it for assessment. The plan is rated on clarity, soundness in addressing the issue, and feasibility. After the teacher suggests modifications, the plan is carried out.

3. The observations made while testing proposed explanations, when analyzed using conventional and invented methods, provide new insights into phenomena.

Students:

- organize observations and measurements of objects and events through classification and the preparation of simple charts and tables.
- interpret organized observations and measurements, recognizing simple patterns, sequences, and relationships.
- share their findings with others and actively seek their interpretations and ideas.
- adjust their explanations and understandings of objects and events based on their findings and new ideas.

This is evident, for example, when students:

- prepare tables or other representations of their observations and look for evidence which supports or refutes their explanation of why objects sink or float when placed in a container of water. After sharing and discussing their results with other groups, they prepare a brief research report that includes methods, findings, and conclusions. The report is rated on its clarity, care in carrying out the plan, and presentation of evidence supporting the conclusions.

**
Engineering Design
**

1. Engineering design is an iterative process involving modeling and optimization finding the best solution within given constraints which is used to develop technological solutions to problems within given constraints.

Students engage in the following steps in a design process:

- describe objects, imaginary or real, that might be modeled or made differently and suggest ways in which the objects can be changed, fixed, or improved.
- investigate prior solutions and ideas from books, magazines, family, friends, neighbors, and community members.
- generate ideas for possible solutions, individually and through group activity; apply age-appropriate mathematics and science skills; evaluate the ideas and determine the best solution; and explain reasons for the choices.
- plan and build, under supervision, a model of the solution using familiar materials, processes, and hand tools.
- discuss how best to test the solution; perform the test under teacher supervision; record and portray results through numerical and graphic means; discuss orally why things worked or didn’t work; and summarize results in writing, suggesting ways to make the solution better.

This is evident, for example, when students:

- read a story called Humpty’s Big Day wherein the readers visit the place where Humpty Dumpty had his accident, and are asked to design and model a way to get to the top of the wall and down again safely.
- generate, draw, and model ideas for a space station that includes a pleasant living and working environment.
- design and model footwear that they could use to walk on a cold, sandy surface.

**Standard 1-Analysis, Inquiry, and Design
**Intermediate

Students will use mathematical analysis, scientific inquiry, and engineering design, as appropriate, to pose questions, seek answers, and develop solutions.

**Mathematical
Analysis
**

1. Abstraction and symbolic representation are used to communicate mathematically.

Students:

- extend mathematical notation and symbolism to include variables and algebraic expressions in order to describe and compare quantities and express mathematical relationships.

2. Deductive and inductive reasoning are used to reach mathematical conclusions.

Students:

- use inductive reasoning to construct, evaluate, and validate conjectures and arguments, recognizing that patterns and relationships can assist in explaining and extending mathematical phenomena.

This is evident, for example, when students:

- predict the next triangular number by examining the pattern 1, 3, 6, 10, r.

3. Critical thinking skills are used in the solution of mathematical problems.

Students:

- apply mathematical knowledge to solve real-world problems and problems that arise from the investigation of mathematical ideas, using representations such as pictures, charts, and tables.

**Scientific Inquiry
**

1. The central purpose of scientific inquiry is to develop explanations of natural phenomena in a continuing, creative process.

Students:

- formulate questions independently with the aid of references appropriate for guiding the search for explanations of everyday observations.
- construct explanations independently for natural phenomena, especially by proposing preliminary visual models of phenomena.
- represent, present, and defend their proposed explanations of everyday observations so that they can be understood and assessed by others.
- seek to clarify, to assess critically, and to reconcile with their own thinking the ideas presented by others, including peers, teachers, authors, and scientists.

This is evident, for example, when students:

- After being shown the disparity between the amount of solid waste which is recycled and which could be recycled, students working in small groups are asked to explain why this disparity exists. They develop a set of possible explanations and to select one for intensive study. After their explanation is critiqued by other groups, it is refined and submitted for assessment. The explanation is rated on clarity, plausibility, and appropriateness for intensive study using research methods.

2. Beyond the use of reasoning and consensus, scientific inquiry involves the testing of proposed explanations involving the use of conventional techniques and procedures and usually requiring considerable ingenuity.

Students:

- use conventional techniques and those of their own design to make further observations and refine their explanations, guided by a need for more information.
- develop, present, and defend formal research proposals for testing their own explanations of common phenomena, including ways of obtaining needed observations and ways of conducting simple controlled experiments.
- carry out their research proposals, recording observations and measurements (e.g., lab notes, audio tape, computer disk, video tape) to help assess the explanation.

This is evident, for example, when students:

- develop a research plan for studying the accuracy of their explanation of the disparity between the amount of solid waste that is recycled and that could be recycled.* After their tentative plan is critiqued, they refine it and submit it for assessment. The research proposal is rated on clarity, feasibility and soundness as a method of studying the explanations’ accuracy. They carry out the plan, with teacher suggested modifications. This work is rated by the teacher while it is in progress.

3. The observations made while testing proposed explanations, when analyzed using conventional and invented methods, provide new insights into phenomena.

Students:

- design charts, tables, graphs and other representations of observations in conventional and creative ways to help them address their research question or hypothesis.
- interpret the organized data to answer the research question or hypothesis and to gain insight into the problem.
- modify their personal understanding of phenomena based on evaluation of their hypothesis.

This is evident, for example, when students:

- carry out their plan making appropriate observations and measurements. They analyze the data, reach conclusions regarding their explanation of the disparity between the amount of solid waste which is recycled and which could be recycled., and prepare a tentative report which is critiqued by other groups, refined, and submitted for assessment. The report is rated on clarity, quality of presentation of data and analyses, and soundness of conclusions.

**
Engineering Design
**

1. Engineering design is an iterative process involving modeling and optimization finding the best solution within given constraints which is used to develop technological solutions to problems within given constraints.

Students engage in the following steps in a design process:

- identify needs and opportunities for technical solutions from an investigation of situations of general or social interest.
- locate and utilize a range of printed, electronic, and human information resources to obtain ideas.
- consider constraints and generate several ideas for alternative solutions, using group and individual ideation techniques (group discussion, brainstorming, forced connections, role play); defer judgment until a number of ideas have been generated; evaluate (critique) ideas; and explain why the chosen solution is optimal.
- develop plans, including drawings with measurements and details of construction, and construct a model of the solution, exhibiting a degree of craftsmanship.
- in a group setting, test their solution against design specifications, present and evaluate results, describe how the solution might have been modified for different or better results, and discuss tradeoffs that might have to be made.

This is evident, for example, when students:

- reflect on the need for alternative growing systems in desert environments and design and model a hydroponic greenhouse for growing vegetables without soil.
- brainstorm and evaluate alternative ideas for an adaptive device that will make life easier for a person with a disability, such as a device to pick up objects from the floor.
- design a model vehicle (with a safety belt restraint system and crush zones to absorb impact) to carry a raw egg as a passenger down a ramp and into a barrier without damage to the egg.
- assess the performance of a solution against various design criteria, enter the scores on a spreadsheet, and see how varying the solution might have affected total score.

**Standard 1-Analysis, Inquiry, and Design
**Commencement

**Mathematical
Analysis
**

1. Abstraction and symbolic representation are used to communicate mathematically.

Students:

- use algebraic and geometric representations to describe and compare data.

2. Deductive and inductive reasoning are used to reach mathematical conclusions.

Students:

- use deductive reasoning to construct and evaluate conjectures and arguments, recognizing that patterns and relationships in mathematics assist them in arriving at these conjectures and arguments.

3. Critical thinking skills are used in the solution of mathematical problems.

Students:

- apply algebraic and geometric concepts and skills to the solution of problems.

**Scientific Inquiry
**

1. The central purpose of scientific inquiry is to develop explanations of natural phenomena in a continuing, creative process.

Students:

- elaborate on basic scientific and personal explanations of natural phenomena, and develop extended visual models and mathematical formulations to represent their thinking.
- hone ideas through reasoning, library research, and discussion with others, including experts.
- work toward reconciling competing explanations; clarifying points of agreement and disagreement.
- coordinate explanations at different levels of scale, points of focus, and degrees of complexity and specificity and recognize the need for such alternative representations of the natural world.

This is evident, for example, when students:

- in small groups, are asked to explain why a cactus plant requires much less water to survive than many other plants. They are asked to develop, through research, a set of explanations for the differences and to select at least one for study. After the proposed explanation is critiqued by others, they refine it by formulating a hypothesis which is rated on clarity, plausibility, and researchability.

2. Beyond the use of reasoning and consensus, scientific inquiry involves the testing of proposed explanations involving the use of conventional techniques and procedures and usually requiring considerable ingenuity.

Students:

- devise ways of making observations to test proposed explanations.
- refine their research ideas through library investigations, including electronic information retrieval and reviews of the literature, and through peer feedback obtained from review and discussion.
- develop and present proposals including formal hypotheses to test their explanations, i.e., they predict what should be observed under specified conditions if the explanation is true.
- carry out their research plan for testing explanations, including selecting and developing techniques, acquiring and building apparatus, and recording observations as necessary.

This is evident, for example, when students:

- develop, through research, a proposal to test their hypothesis of why a cactus plant requires much less water to survive than many other plants. After their proposal is critiqued, it is refined and submitted for assessment by a panel of students. The proposal is rated on clarity, appropriateness, and feasibility. Upon approval, students complete the research. Progress is rated holistically by the teacher

3. The observations made while testing proposed explanations, when analyzed using conventional and invented methods, provide new insights into phenomena.

Students:

- use various means of representing and organizing observations (e.g., diagrams, tables, charts, graphs, equations, matrices) and insightfully interpret the organized data.
- apply statistical analysis techniques when appropriate to test if chance alone explains the result.
- assess correspondence between the predicted result contained in the hypothesis and the actual result and reach a conclusion as to whether or not the explanation on which the prediction was based is supported.
- based on the results of the test and through public discussion, they revise the explanation and contemplate additional research.
- develop a written report for public scrutiny that describes their proposed explanation, including a literature review, the research they carried out, its result, and suggestions for further research.

This is evident, for example, when students:

- carry out a research plan, including keeping a lab book, to test their hypothesis of why a cactus plant requires much less water to survive than many other plants. After completion, a paper is presented describing the research. Based on the class critique, the paper is rewritten and submitted with the lab book for separate assessment or as part of a portfolio of their science work. It is rated for clarity, thoroughness, soundness of conclusions, and quality of integration with existing literature.

Engineering Design

1. Engineering design is an iterative process involving modeling and optimization finding the best solution within given constraints which is used to develop technological solutions to problems within given constraints.

Students engage in the following steps in a design process:

- initiate and carry out a thorough investigation of an unfamiliar situation and identify needs and opportunities for technological invention or innovation.
- identify, locate, and use a wide range of information resources, and document through notes and sketches how findings relate to the problem.
- generate creative solutions, break ideas into significant functional elements, and explore possible refinements; predict possible outcomes using mathematical and functional modeling techniques; choose the optimal solution to the problem, clearly documenting ideas against design criteria and constraints; and explain how human understands, economics, ergonomics, and environmental considerations have influenced the solution.
- develop work schedules and working plans which include optimal use and cost of materials, processes, time, and expertise; construct a model of the solution, incorporating developmental modifications while working to a high degree of quality (craftsmanship).
- devise a test of the solution according to the design criteria and perform the test; record, portray, and logically evaluate performance test results through quantitative, graphic, and verbal means. Use a variety of creative verbal and graphic techniques effectively and persuasively to present conclusions, predict impacts and new problems, and suggest and pursue modifications.

This is evident, for example, when students:

- search the Internet for world wide web sites dealing with renewable energy and sustainable living and research the development and design of an energy efficient home.
- develop plans, diagrams, and working drawings for the construction of a computer-controlled marble sorting system that simulates how parts on an assembly line are sorted by color.
- design and model a portable emergency shelter that could be heated by a person’s body to a life sustaining temperature when the outside temperature is 20
^{o}F.

**Standard 2-Information Systems**

Elementary

Students will access, generate, process, and transfer information using appropriate technologies.

Information Systems

1. Information technology is used to retrieve, process, and communicate information and as a tool to enhance learning.

Students:

- use a variety of equipment and software packages to enter, process, display, and communicate information in different forms using text, tables, pictures, and sound.
- telecommunicate a message to a distant location with teacher help.
- access needed information from printed media, electronic data bases, and community resources.

This is evident, for example, when students:

- use the newspaper or magazine index in a library to find information on a particular topic.
- invite local experts to the school to share their expertise.

2. Knowledge of the impacts and limitations of information systems is essential to its effective and ethical use.

Students:

- describe the uses of information systems in homes, schools, and businesses.
- understand that computers are used to store personal information.
- demonstrate ability to evaluate information.

This is evident, for example, when students:

- look for differences among species of bugs collected on the school grounds, and classify them according to preferred habitat.

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

Students:

- describe the uses of information systems in homes and schools.
- demonstrate ability to evaluate information critically.

**Standard 2-Information Systems
**Intermediate

Students will access, generate, process, and transfer information using appropriate technologies.

Information Systems

1. Information technology is used to retrieve, process, and communicate information and as a tool to enhance learning.

Students:

- use a range of equipment and software to integrate several forms of information in order to create good quality audio, video, graphic, and text-based presentations.
- use spreadsheets and data-base software to collect, process, display, and analyze information. Students access needed information from electronic data bases and on-line telecommunication services.
- systematically obtain accurate and relevant information pertaining to a particular topic from a range of sources, including local and national media, libraries, museums, governmental agencies, industries, and individuals.
- collect data from probes to measure events and phenomena.
- simple modeling programs to make predictions.

This is evident, for example, when students:

- compose letters on a word processor and send them to representatives of industry, governmental agencies, museums, or laboratories seeking information pertaining to a student project.
- acquire data from weather stations.
- use a software package, such as Science Tool Kit, to monitor the acceleration of a model car traveling down a given distance on a ramp.
- use computer software to model how plants grow plants under different conditions.

2. Knowledge of the impacts and limitations of information systems is essential to its effective and ethical use.

Students:

- understand the need to question the accuracy of information displayed on a computer because the results produced by a computer may be affected by incorrect data entry.
- identify advantages and limitations of data-handling programs and graphics programs.
- understand why electronically stored personal information has greater potential for misuse than records kept in conventional form.

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

Students:

- use graphical, statistical, and presentation software to presents project to fellow classmates.
- describe applications of information technology in mathematics, science, and other technologies that address needs and solve problems in the community.
- explain the impact of the use and abuse of electronically generated information on individuals and families.

**Standard 2-Information Systems
**Commencement

Students will access, generate, process, and transfer information using appropriate technologies.

Information Systems

1. Information technology is used to retrieve, process, and communicate information and as a tool to enhance learning.

Students:

- understand and use the more advanced features of word processing, spreadsheets, and data-base software.
- prepare multimedia presentations demonstrating a clear sense of audience and purpose.
- access, select, collate, and analyze information obtained from a wide range of sources such as research data bases, foundations, organizations, national libraries, and electronic communication networks, including the Internet.
- students receive news reports from abroad and work in groups to produce newspapers reflecting the perspectives of different countries.
- utilize electronic networks to share information.
- model solutions to a range of problems in mathematics, science, and technology using computer simulation software.

This is evident, for example, when students:

- collect and amend quantitative and qualitative information for a particular purpose and enter it into a data-handling package for processing and analysis.
- visit businesses, laboratories, environmental areas, and universities to obtain on-site information
- receive news reports from abroad, and work in groups to produce newspapers reflecting the perspectives of different countries.
- join a list serve and send electronic mail to other persons sharing mutual concerns and interests.
- use computer software to simulate and graph the motion of an object.
- study a system in a dangerous setting (e.g., a nuclear power plant).

2. Knowledge of the impacts and limitations of information systems is essential to its effective and ethical use.

Students:

- explain the impact of the use and abuse of electronically generated information on individuals and families.
- evaluate software packages relative to their suitability to a particular application and their ease of use.
- discuss the ethical and social issues raised by the use and abuse of information systems.

This is evident, for example, when students:

- discuss how unauthorized people might gain access to information about their interests and way of life.

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

Students:

- work with a virtual community to conduct a project or solve a problem using the network.
- discuss how applications of information technology can address some major global problems and issues.
- discuss the environmental, ethical, moral, and social issues raised by the use and abuse of information technology.

**Standard 3-Mathematics
**Elementary

Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry.

**Mathematical Reasoning
**

1. Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument.

Students:

- use models, facts, and relationships to draw conclusions about mathematics and explain their thinking.
- use patterns and relationships to analyze mathematical situations.
- justify their answers and solution processes.
- use logical reasoning to reach simple conclusions.

This is evident, for example, when students:

- build geometric figures out of straws.
- find patterns in sequences of numbers, such as the triangular numbers 1, 3, 6, 10, . . . .
- explore number relationships with a calculator (e.g., 12 + 6 = 18, 11 + 7 = 18, etc.) and draw conclusions.

**Number and Numeration**

2. Students use number sense and numeration to develop an understanding of the multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and the use of numbers in the development of mathematical ideas.

Students:

- use whole numbers and fractions to identify locations, quantify groups of objects, and measure distances.
- use concrete materials to model numbers and number relationships for whole numbers and common fractions, including decimal fractions.
- relate counting to grouping and to place-value.
- recognize the order of whole numbers and commonly used fractions and decimals.
- demonstrate the concept of percent through problems related to actual situations.

This is evident, for example, when students:

- count out 15 small cubes and exchange ten of the cubes for a rod ten cubes long.
- use the number line to show the position of 1/4.
- figure the tax on $4.00 knowing that taxes are 7 cents per $1.00.

Operations

3. Students use mathematical operations and relationships among them to understand mathematics.

Students:

- add, subtract, multiply, and divide whole numbers.
- develop strategies for selecting the appropriate computational and operational method in problem solving situations.
- know single digit addition, subtraction, multiplication, and division facts.
- understand the commutative and associative properties.

This is evident, for example, when students:

- use the fact that multiplication is commutative (e.g., 2 x 7 = 7 x 2), to assist them with their memorizing of the basic facts.
- solve multiple-step problems that require at least two different operations.
- progress from base ten blocks to concrete models and then to paper and pencil algorithms.

Modeling/Multiple Representation

4. Students use mathematical modeling/multiple representation to provide a means of presenting, interpreting, communicating, and connecting mathematical information and relationships.

Students:

- use concrete materials to model spatial relationships.
- construct tables, charts, and graphs to display and analyze real-world data.
- use multiple representations (simulations, manipulative materials, pictures, and diagrams) as tools to explain the operation of everyday procedures.
- use variables such as height, weight, and hand size to predict changes over time.
- use physical materials, pictures, and diagrams to explain mathematical ideas and processes and to demonstrate geometric concepts.

This is evident, for example, when students:

- build a 3 x 3 x 3 cube out of blocks.
- use square tiles to model various rectangles with an area of 24 square units.
- read a bar graph of population trends and write an explanation of the information it contains.

**Measurement**

5. Students use measurement in both metric and English measure to provide a major link between the abstractions of mathematics and the real world in order to describe and compare objects and data.

Students:

- understand that measurement is approximate, never exact.
- select appropriate standard and nonstandard measurement tools in measurement activities.
- understand the attributes of area, length, capacity, weight, volume, time, temperature, and angle.
- estimate and find measures such as length, perimeter, area, and volume using both nonstandard and standard units.
- collect and display data.
- use statistical methods such as graphs, tables, and charts to interpret data.

This is evident, for example, when students:

- measure with paper clips or finger width.
- estimate, then calculate, how much paint would be needed to cover one wall.
- create a chart to display the results of a survey conducted among the classes in the school, or graph the amounts of survey responses by grade level.

Uncertainty

6. Students use ideas of uncertainty to illustrate that mathematics involves more than exactness when dealing with everyday situations.

Students:

- make estimates to compare to actual results of both formal and informal measurement.
- make estimates to compare to actual results of computations.
- recognize situations where only an estimate is required.
- develop a wide variety of estimation skills and strategies.
- determine the reasonableness of results.
- predict experimental probabilities. make predictions using unbiased random samples.
- determine probabilities of simple events.

This is evident, for example, when students:

- estimate the length of the room before measuring.
- predict the average number of red candies in a bag before opening a group of bags, counting the candies, and then averaging the number that were red.
- determine the probability of picking an even numbered slip from a hat containing slips of paper numbered 1, 2, 3, 4, 5, and 6.

Patterns/Functions

7. Students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics, and construct generalizations that describe patterns simply and efficiently.

Students:

- recognize, describe, extend, and create a wide variety of patterns.
- represent and describe mathematical relationships.
- explore and express relationships using variables and open sentences.
- solve for an unknown using manipulative materials.
- use a variety of manipulative materials and technologies to explore patterns.
- interpret graphs.
- explore and develop relationships among two- and three-dimensional geometric shapes.
- discover patterns in nature, art, music, and literature.

This is evident, for example, when students:

- represent three more than a number is equal to nine as n + 3 = 9.
- draw leaves, simple wallpaper patterns, or write number sequences to illustrate recurring patterns.
- write generalizations or conclusions from display data in charts or graphs.

**Standard 3-Mathematics
**Intermediate

Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry.

Mathematical Reasoning

1. Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument.

Students:

- apply a variety of reasoning strategies.
- make and evaluate conjectures and arguments using appropriate language.
- make conclusions based on inductive reasoning.
- justify conclusions involving simple and compound (i.e., and/or) statements.

This is evident, for example, when students:

- use trial and error and work backwards to solve a problem.
- identify patterns in a number sequence.
- are asked to find numbers that satisfy two conditions, such as n > -4 and n
__<__6.

Number and Numeration

2. Students use number sense and numeration to develop an understanding of the multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and the use of numbers in the development of mathematical ideas.

Students:

- understand, represent, and use numbers in a variety of equivalent forms (integer, fraction, decimal, percent, exponential, expanded and scientific notation).
- understand and apply ratios, proportions, and percents through a wide variety of hands-on explorations.
- develop an understanding of number theory (primes, factors, and multiples).
- recognize order relations for decimals, integers, and rational numbers.

This is evident, for example, when students:

- use prime factors of a group of denominators to determine the least common denominator.
- select two pairs from a number of ratios and prove that they are in proportion.
- demonstrate the concept that a number can be symbolized by many different numerals as in:

1/4 = 3/12 = 25/100 = 0.25 = 25%

Operations

3. Students use mathematical operations and relationships among them to understand mathematics.

Students:

- add, subtract, multiply, and divide fractions, decimals, and integers.
- explore and use the operations dealing with roots and powers.
- use grouping symbols (parentheses) to clarify the intended order of operations.
- apply the associative, commutative, distributive, inverse, and identity properties.
- demonstrate an understanding of operational algorithms (procedures for adding, subtracting, etc.).
- develop appropriate proficiency with facts and algorithms.
- apply concepts of ratio and proportion to solve problems.

This is evident, for example, when students:

- create area models to help in understanding fractions, decimals, and percents.
- find the missing number in a proportion in which three of the numbers are known, and letters are used as place holders.
- arrange a set of fractions in order, from the smallest to the largest:

3/4 , 1/5 , 2/3 , 1/2 , 1/4

- illustrate the distributive property for multiplication over addition, such as 2(a + 3) = 2a + 6.

Modeling/Multiple Representation

4. Students use mathematical modeling/multiple representation to provide a means of presenting, interpreting, communicating, and connecting mathematical information and relationships.

Students:

- visualize, represent, and transform two- and three-dimensional shapes.
- use maps and scale drawings to represent real objects or places.
- use the coordinate plane to explore geometric ideas.
- represent numerical relationships in one- and two-dimensional graphs.
- use variables to represent relationships.
- use concrete materials and diagrams to describe the operation of real world processes and systems.
- develop and explore models that do and do not rely on chance.
- investigate both two- and three-dimensional transformations.
- use appropriate tools to construct and verify geometric relationships.
- develop procedures for basic geometric constructions.

This is evident, for example, when students:

- build a city skyline to demonstrate skill in linear measurements, scale drawing, ratio, fractions, angles, and geometric shapes.
- bisect an angle using a straight edge and compass.
- draw a complex of geometric figures to illustrate that the intersection of a plane and a sphere is a circle or point.

Measurement

5. Students use measurement in both metric and English measure to provide a major link between the abstractions of mathematics and the real world in order to describe and compare objects and data.

Students:

- estimate, make, and use measurements in real-world situations.
- select appropriate standard and nonstandard measurement units and tools to measure to a desired degree of accuracy.
- develop measurement skills and informally derive and apply formulas in direct measurement activities.
- use statistical methods and measures of central tendencies to display, describe, and compare data.
- explore and produce graphic representations of data using calculators/computers.
- develop critical judgment for the reasonableness of measurement.

This is evident, for example, when students:

- use box plots or stem and leaf graphs to display a set of test scores.
- estimate and measure the surface areas of a set of gift boxes in order to determine how much wrapping paper will be required.
- explain when to use mean, median, or mode for a group of data.

Uncertainty

6. Students use ideas of uncertainty to illustrate that mathematics involves more than exactness when dealing with everyday situations.

Students:

- use estimation to check the reasonableness of results obtained by computation, algorithms, or the use of technology.
- use estimation to solve problems for which exact answers are inappropriate.
- estimate the probability of events.
- use simulation techniques to estimate probabilities.
- determine probabilities of independent and mutually exclusive events.

This is evident, for example, when students:

- construct spinners to represent random choice of four possible selections.
- perform probability experiments with independent events (e.g., the probability that the head of a coin will turn up, or that a 6 will appear on a die toss).
- estimate the number of students who might chose to eat hot dogs at a picnic.

Patterns/Functions

7. Students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics, and construct generalizations that describe patterns simply and efficiently.

Students:

- recognize, describe, and generalize a wide variety of patterns and functions.
- describe and represent patterns and functional relationships using tables, charts and graphs, algebraic expressions, rules, and verbal descriptions.
- develop methods to solve basic linear and quadratic equations.
- develop an understanding of functions and functional relationships: that a change in one quantity (variable) results in change in another.
- verify results of substituting variables.
- apply the concept of similarity in relevant situations.
- use properties of polygons to classify them.
- explore relationships involving points, lines, angles, and planes.
- develop and apply the Pythagorean principle in the solution of problems.
- explore and develop basic concepts of right triangle trigonometry.
- use patterns and functions to represent and solve problems.

This is evident, for example, when students:

- find the height of a building when a 20-foot ladder reaches the top of the building when its base is 12 feet away from the structure.
- investigate number patterns through palindromes (pick a 2-digit number, reverse it and add the two–repeat the process until a palindrome appears) 42 86 +24 +68 palindrome 66 154 +451 605 +506 palindrome 1111
- solve linear equations, such as 2(x + 3) = x + 5 by several methods

**Standard 3-Mathematics
**Commencement

Mathematical Reasoning

1. Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument.

Students:

- construct simple logical arguments.
- follow and judge the validity of logical arguments.
- use symbolic logic in the construction of valid arguments.
- construct proofs based on deductive reasoning.

This is evident, for example, when students:

- prove that an altitude of an isosceles triangle, drawn to the base, is perpendicular to that base.
- determine whether or not a given logical sentence is a tautology.
- show that the triangle having vertex coordinates of (0,6), (0,0), and (5,0) is a right triangle.

Number and Numeration

2. Students use number sense and numeration to develop an understanding of the multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and the use of numbers in the development of mathematical ideas.

Students:

- understand and use rational and irrational numbers.
- recognize the order of the real numbers.
- apply the properties of the real numbers to various subsets of numbers.

This is evident, for example, when students:

- determine from the discriminate of a quadratic equation whether the roots are rational or irrational.
- give rational approximations of irrational numbers to a specific degree of accuracy.
- determine for which value of x the expression
__2x + 6__is undefined.

x- 7

Operations

3. Students use mathematical operations and relationships among them to understand mathematics.

Students:

- use addition, subtraction, multiplication, division, and exponentiation with real numbers and algebraic expressions.
- develop an understanding of and use the composition of functions and transformations.
- explore and use negative exponents on integers and algebraic expressions.
- use field properties to justify mathematical procedures.
- use transformations on figures and functions in the coordinate plane.

This is evident, for example, when students:

- determine the coordinates of triangle A(2,5), B(9,8), and C(3,6) after a translation (x,y) --> (x + 3, y - 1).
- evaluate the binary operation defined as x * y = x
^{-2}+ (y + x)^{2}for 3 * 4. - identify the field properties used in solving the equation 2(x - 5) + 3 = x + 7.

Modeling/Multiple Representation

4. Students use mathematical modeling/multiple representation to provide a means of presenting, interpreting, communicating, and connecting mathematical information and relationships.

Students:

- represent problem situations symbolically by using algebraic expressions, sequences, tree diagrams, geometric figures, and graphs.
- manipulate symbolic representations to explore concepts at an abstract level.
- choose appropriate representations to facilitate the solving of a problem.
- use learning technologies to make and verify geometric conjectures .
- justify the procedures for basic geometric constructions.
- investigate transformations in the coordinate plane.
- develop meaning for basic conic sections.
- develop and apply the concept of basic loci to compound loci.
- use graphing utilities to create and explore geometric and algebraic models.
- model real-world problems with systems of equations and inequalities.

This is evident, for example, when students:

- determine the locus of points equidistant from two parallel lines.
- explain why the basic construction of bisecting a line is valid.
- describe the various conics produced when the equation ax
^{2}+ by^{2}= c^{2}is graphed for various values of a, b, and c.

Measurement

5. Students use measurement in both metric and English measure to provide a major link between the abstractions of mathematics and the real world in order to describe and compare objects and data.

Students:

- derive and apply formulas to find measures such as length, area, volume, weight, time, and angle in real-world contexts.
- choose the appropriate tools for measurement.
- use dimensional analysis techniques.
- use statistical methods including measures of central tendency to describe and compare data.
- use trigonometry as a method to measure indirectly.
- apply proportions to scale drawings, computer-assisted design blueprints, and direct variation in order to compute indirect measurements.
- relate absolute value, distance between two points, and the slope of a line to the coordinate plane.
- understand error in measurement and its consequence on subsequent calculations.
- use geometric relationships in relevant measurement problems involving geometric concepts.

This is evident, for example, when students:

- change mph to ft/sec.
- use the tangent ratio to determine the height of a tree.
- determine the distance between two points in the coordinate plane.

Uncertainty

6. Students use ideas of uncertainty to illustrate that mathematics involves more than exactness when dealing with everyday situations.

Students:

- judge the reasonableness of results obtained from applications in algebra, geometry, trigonometry, probability, and statistics.
- judge the reasonableness of a graph produced by a calculator or computer.
- use experimental or theoretical probability to represent and solve problems involving uncertainty.
- use the concept of random variable in computing probabilities.
- determine probabilities using permutations and combinations.

This is evident, for example, when students:

- construct a tree diagram or sample space for a compound event.
- calculate the probability of winning the New York State Lottery.
- develop simulations for probability problems for which they do not have theoretical solutions.

Patterns/Functions

7. Students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics, and construct generalizations that describe patterns simply and efficiently.

Students:

- use function vocabulary and notation.
- represent and analyze functions using verbal descriptions, tables, equations, and graphs.
- translate among the verbal descriptions, tables, equations and graphic forms of functions.
- analyze the effect of parametric changes on the graphs of functions.
- apply linear, exponential, and quadratic functions in the solution of problems.
- apply and interpret transformations to functions.
- model real-world situations with the appropriate function.
- apply axiomatic structure to algebra and geometry.
- use computers and graphing calculators to analyze mathematical phenomena.

This is evident, for example, when students:

- determine, in more than one way, whether or not a specific relation is a function.
- explain the relationship between the roots of a quadratic equation and the intercepts of its corresponding graph.
- use transformations to determine the inverse of a function.

**Standard 3-Mathematics
**Four year sequence in mathematics

Measurement

Students:

- construct indirect proofs or proofs using mathematical induction.
- investigate and compare the axiomatic structures of various geometries.

This is evident, for example, when students:

- prove indirectly that: if n 2 is even, n is even.
- prove using mathematical induction that: 1 + 3 + 5 + . . . + (2n - 1) = n
^{2}. - explain the axiomatic differences between plane and spherical geometries.

Uncertainty

Students:

- 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.

This is evident, for example, when students:

- relate the concept of infinity when graphing the tangent function.
- show that the set of complex numbers form a field under the operations of addition and multiplication.
- show that the set of complex numbers forms a field under the operations of addition and multiplication.
- represent a complex number in polar form.

Operations

3. Students use mathematical operations and relationships among them to understand mathematics.

Students:

- use appropriate techniques, including graphing utilities, to perform basic operations on matrices.
- use rational exponents on real numbers and all operations on complex numbers.
- combine functions using the basic operations and the composition of two functions.

This is evident, for example, when students:

- relate specific matrices to certain types of transformations of points on the coordinate plane.
- evaluate expressions with fractional exponents, such as 8 2/3 4 -1/2 .
- determine the value of compound functions such as (f o g) (x).

Modeling/Multiple Representation

Students:

- model vector quantities both algebraically and geometrically.
- represent graphically the sum and difference of two complex numbers.
- model and solve problems that involve absolute value, vectors, and matrices.
- model quadratic inequalities both algebraically and graphically.
- model the composition of transformations.
- determine the effects of changing parameters of the graphs of functions.
- use polynomial, rational, trigonometric, and exponential functions to model real-world relationships.
- use algebraic relationships to analyze the conic sections.
- use circular functions to study and model periodic real-world phenomena.
- illustrate spatial relationships using perspective, projections, and maps.
- represent problem situations using discrete structures such as finite graphs, matrices, sequences, and recurrence relations.
- analyze spatial relationships using the Cartesian coordinate system in three dimensions.

This is evident, for example, when students:

- determine coordinates which lie in the solution of the quadriatic inequality, such as y < x
^{2}+ 4x + 2. - find the distance between two points in a three-dimension coordinate system.
- describe what happens to the graph when b increases in the function y = x
^{2}+ bx + c.

**Measurement**

Students:

- derive and apply formulas relating angle measure and arc degree measure 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 the general solutions of triangles.
- apply the normal curve and its properties to familiar contexts.
- design a statistical experiment to study a problem and communicate the outcomes, including dispersion.
- use statistical methods, including scatter plots and lines of best fit, to make predictions.
- apply the conceptual foundation of limits, infinite sequences and series, the area under a curve, rate of change, inverse variation, and the slope of a tangent line to authentic problems in mathematics and other disciplines.
- determine optimization points on a graph.
- use derivatives to find maximum, minimum, and inflection points of a function.

This is evident, for example, when students:

- use a chi-square test to determine if one cola really tastes better than another cola.
- can illustrate the various line segments which represent the sine, cosine, and tangent of a given angle on the unit circle.
- calculate the first derivative of a function using the limit definition.

**Uncertainty**

Students:

- interpret probabilities in real-world situations.
- use a Bernoulli experiment to determine probabilities for experiments with exactly two outcomes.
- use curve fitting to predict from data.
- apply the concept of random variable to generate and interpret probability distributions.
- create and interpret applications of discrete and continuous probability distributions.
- make predictions based on interpolations and extrapolations from data.
- obtain confidence intervals and test hypotheses using appropriate statistical methods.
- approximate the roots of polynomial equations.

This is evident, for example, when students:

- verify the probabilities listed for the state lottery for second, third, and fourth prize.
- use graphing calculators to generate a curve of best fit for an array of data using linear regression.
- determine the probability of getting at least 3 heads on 6 flips of a fair coin.

Patterns/Functions

Students:

- solve equations with complex roots using a variety of algebraic and graphical methods with appropriate tools.
- understand and apply the relationship between the rectangular form and the polar form of a complex number.
- evaluate and form the composition of functions.
- use the definition of a derivative to examine the 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 the 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 patterns 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.

This is evident, for example, when students:

- transform polar coordinates into rectangular forms.
- find the maximum height of an object projects upward with a given initial velocity.
- find the limit of expressions like
__n - 2__as n goes to infinity.

3n + 5

Students will understand and apply scientific concepts, principles, and theories pertaining to the physical setting and living environment and recognize the historical development of ideas in science.

Physical Setting

1. The Earth and celestial phenomena can be described by principles of relative motion and perspective.

Students:

- describe patterns of daily, monthly, and seasonal changes in their environment.

This is evident, for example, when students:

- conduct a long-term weather investigation, such as running a weather station or collecting weather data.
- keep a journal of the phases of the moon over a one-month period. This information is collected for several different one-month periods and compared.

2. Many of the phenomena that we observe on Earth involve interactions among components of air, water, and land.

Students:

- describe the relationships among air, water, and land on Earth.

This is evident, for example, when students:

- observe a puddle of water outdoors after a rainstorm. On a return visit after the puddle has disappeared, students describe where the water came from and possible locations for it now.
- assemble rock and mineral collections based on characteristics such as erosional features or crystal size features.

3. Matter is made up of particles whose properties determine the observable characteristics of matter and its reactivity.

Students:

- observe and describe properties of materials using appropriate tools.
- describe chemical and physical changes, including changes in states of matter.

This is evident, for example, when students:

- compare the appearance of materials when seen with and without the aid of a magnifying glass.
- investigate simple physical and chemical reactions and the chemistry of household products, e.g., freezing, melting, and evaporating; a comparison of new and rusty nails; the role of baking soda in cooking.

4. Energy exists in many forms, and when these forms change energy is conserved.

Students:

- describe a variety of forms of energy (e.g., heat, chemical, light) and the changes that occur in objects when they interact with those forms of energy.
- observe the way one form of energy can be transformed into another form of energy present in common situations (e.g., mechanical to heat energy, mechanical to electrical energy, chemical to heat energy).

This is evident, for example, when students:

- investigate the interactions of liquids and powders that result in chemical reactions (e.g., vinegar and baking soda) compared to interactions that do not (e.g., water and sugar).
- in order to demonstrate the transformation of chemical to electrical energy, construct electrical cells from objects, such as lemons or potatoes, using pennies and aluminum foil inserted in slits at each end of fruits or vegetables; the penny and aluminum are attached by wires to a millimeter. Students can compare the success of a variety of these electrical cells.

5. Energy and matter interact through forces that result in changes in motion.

Students:

- describe the effects of common forces (pushes and pulls) on objects, such as those caused by gravity, magnetism, and mechanical forces.
- describe how forces can operate across distances.

This is evident, for example, when students:

- investigate simple machines and use them to perform tasks.

The Living Environment

1. Living things are both similar to and different from each other and nonliving things.

Students:

- describe the characteristics of and variations between living and nonliving things.
- describe the life processes common to all living things.

This is evident, for example, when students:

- grow a plant or observe a pet, investigating what it requires to stay alive, including evaluating the relative importance and necessity of each item.
- investigate differences in personal body characteristics, such as temperature, pulse, heart rate, blood pressure, and reaction time.

2. Organisms inherit genetic information in a variety of ways that result in continuity of structure and function between parents and offspring.

Students:

- recognize that traits of living things are both inherited and acquired or learned.
- recognize that for humans and other living things there is genetic continuity between generations.

This is evident, for example, when students:

- interact with a classroom pet, observe its behaviors, and record what they are able to teach the animal, such as navigation of a maze or performance of tricks, compared to that which remains constant, such as eye color, or number of digits on an appendage.
- use breeding records and photographs of racing horses or pedigreed animals to recognize that variations exist from generation to generation but "like begets like."

3. Individual organisms and species change over time.

Students:

- describe how the structures of plants and animals complement the environment of the plant or animal.
- observe that differences within a species may give individuals an advantage in surviving and reproducing.

This is evident, for example, when students:

- relate physical characteristics of organisms to habitat characteristics (e.g., long hair and fur color change for mammals living in cold climates).
- visit a farm or a zoo and make a written or pictorial comparison of members of a litter and identify characteristics that may provide an advantage.

4. The continuity of life is sustained through reproduction and development.

Students:

- describe the major stages in the life cycles of selected plants and animals.
- describe evidence of growth, repair, and maintenance, such as nails, hair, and bone, and the healing of cuts and bruises.

This is evident, for example, when students:

- grow bean plants or butterflies; record and describe stages of development.

5. Organisms maintain a dynamic equilibrium that sustains life.

Students:

- describe basic life functions of common living specimens (guppy, mealworm, gerbil).
- describe some survival behaviors of common living specimens.
- describe the factors that help promote good health and growth in humans.

This is evident, for example, when students:

- observe a single organism over a period of weeks and describe such life functions as moving, eating, resting, and eliminating.
- observe and demonstrate reflexes such as pupil dilation and contraction and relate such reflexes to improved survival.
- analyze the extent to which diet and exercise habits meet cardiovascular, energy, and nutrient requirements.

6. Plants and animals depend on each other and their physical environment.

Students:

- describe how plants and animals, including humans, depend upon each other and the nonliving environment.
- describe the relationship of the sun as an energy source for living and nonliving cycles.

This is evident, for example, when students:

- investigate how humans depend on their environment (neighborhood), by observing, recording, and discussing the interactions that occur in carrying out their everyday lives.
- observe the effects of sunlight on growth for a garden vegetable.

7. Human decisions and activities have had a profound impact on the physical and living environment.

Students:

- identify ways in which humans have changed their environment and the effects of those changes.

This is evident, for example, when students:

- give examples of how inventions and innovations have changed the environment; describe benefits and burdens of those changes.

**Standard 4-Science
**Intermediate

Students will understand and apply scientific concepts, principles, and theories pertaining to the physical setting and living environment and recognize the historical development of ideas in science.

Physical Setting

1. The Earth and celestial phenomena can be described by principles of relative motion and perspective.

Students:

- explain daily, monthly, and seasonal changes on earth.

This is evident, for example, when students:

- create models, drawings, or demonstrations describing the arrangement, interaction, and movement of the Earth, moon, and sun.
- plan and conduct an investigation of the night sky to describe the arrangement, interaction, and movement of celestial bodies.

2. Many of the phenomena that we observe on Earth involve interactions among components of air, water, and land.

Students:

- explain how the atmosphere (air), hydrosphere (water), and lithosphere (land) interact, evolve, and change.
- describe volcano and earthquake patterns, the rock cycle, and weather and climate changes.

This is evident, for example, when students:

- add heat to and subtract heat from water and graph the temperature changes, including the resulting phase changes.
- make a record of reported earthquakes and volcanoes and interpret the patterns formed worldwide.

3. Matter is made up of particles whose properties determine the observable characteristics of matter and its reactivity.

Students:

- observe and describe properties of materials, such as density, conductivity, and solubility.
- distinguish between chemical and physical changes.
- develop their own mental models to explain common chemical reactions and changes in states of matter.

This is evident, for example, when students:

- test and compare the properties (hardness, shape, color, etc.) of an array of materials.
- observe an ice cube as it begins to melt at temperature and construct an explanation for what happens, including sketches and written descriptions of their ideas.

4. Energy exists in many forms, and when these forms change energy is conserved.

Students:

- describe the sources and identify the transformations of energy observed in everyday life.
- observe and describe heating and cooling events.
- observe and describe energy changes as related to chemical reactions.
- observe and describe the properties of sound, light, magnetism, and electricity.
- describe situations that support the principle of conservation of energy.

This is evident, for example, when students:

- design and construct devices to transform/transfer energy.
- conduct supervised explorations of chemical reactions (not including ammonia and bleach products) for selected household products, such as hot and cold packs used to treat sport injuries.
- build an electromagnet and investigate the effects of using different types of core materials, varying thicknesses of wire, and different circuit types.

5. Energy and matter interact through forces that result in changes in motion.

Students:

- describe different patterns of motion of objects.
- observe, describe, and compare effects of forces (gravity, electric current, and magnetism) on the motion of objects.

This is evident, for example, when students:

- investigate physics in everyday life, such as at an amusement park or a playground.
- use simple machines made of pulleys and levers to lift objects and describe how each machine transforms the force applied to it.
- build "Rube Goldberg" type devices and describe the energy transformations evident in them.

The Living Environment

1. Living things are both similar to and different from each other and nonliving things.

Students:

- compare and contrast the parts of plants, animals, and one-celled organisms.
- explain the functioning of the major human organ systems and their interactions.

This is evident, for example, when students:

- conduct a survey of the school grounds and develop appropriate classification keys to group plants and animals by shared characteristics.
- use spring-type clothespins to investigate muscle fatigue or rulers to determine the effect of amount of sleep on hand-eye coordination.

2. Organisms inherit genetic information in a variety of ways that result in continuity of structure and function between parents and offspring.

Students:

- describe sexual and asexual mechanisms for passing genetic materials from generation to generation.
- describe simple mechanisms related to the inheritance of some physical traits in offspring.

This is evident, for example, when students:

- contrast dominance and blending as models for explaining inheritance of traits.
- trace patterns of inheritance for selected human traits.

3. Individual organisms and species change over time.

Students:

- describe sources of variation in organisms and their structures and relate the variations to survival.
- describe factors responsible for competition within species and the significance of that competition.

This is evident, for example, when students:

- conduct a long-term investigation of plant or animal communities.
- investigate the acquired effects of industrialization on tree trunk color and those effects on different insect species.

4. The continuity of life is sustained through reproduction and development.

Students:

- observe and describe the variations in reproductive patterns of organisms, including asexual and sexual reproduction.
- explain the role of sperm and egg cells in sexual reproduction.
- observe and describe developmental patterns in selected plants and animals (e.g., insects, frogs, humans, seed-bearing plants).
- observe and describe cell division at the microscopic level and its macroscopic effects.

This is evident, for example, when students:

- apply a model of the genetic code as an analogue for the role of the genetic code in human populations.

5. Organisms maintain a dynamic equilibrium that sustains life.

Students:

- compare the way a variety of living specimens carry out basic life functions and maintain dynamic equilibrium.
- describe the importance of major nutrients, vitamins, and minerals in maintaining health and promoting growth and explain the need for a constant input of energy for living organisms.

This is evident, for example, when students:

- record and compare the behaviors of animals in their natural habitats and relate how these behaviors are important to the animals.
- design and conduct a survey of personal nutrition and exercise habits, and analyze and critique the results of that survey.

6. Plants and animals depend on each other and their physical environment.

Students:

- describe the flow of energy and matter through food chains and food webs.
- provide evidence that green plants make food and explain the significance of this process to other organisms.

This is evident, for example, when students:

- construct a food web for a community of organisms and explore how elimination of a particular part of a chain affects the rest of the chain and web.

7. Human decisions and activities have had a profound impact on the physical and living environment.

Students:

- describe how living things, including humans, depend upon the living and nonliving environment for their survival.
- describe the effects of environmental changes on humans and other populations.

This is evident, for example, when students:

- conduct an extended investigation of a local environment affected by human actions, (e.g., a pond, stream, forest, empty lot).

**Standard 4-Science
**Commencement

Physical Setting

1. The Earth and celestial phenomena can be described by principles of relative motion and perspective.

Students:

- explain complex phenomena, such as tides, variations in day length, solar insulation, apparent motion of the planets, and annual traverse of the constellations.
- describe current theories about the origin of the universe and solar system.

This is evident, for example, when students:

- create models, drawings, or demonstrations to explain changes in day length, solar insulation, and the apparent motion of planets.

2. Many of the phenomena that we observe on Earth involve interactions among components of air, water, and land.

Students:

- use the concepts of density and heat energy to explain observations of weather patterns, seasonal changes, and the movements of the Earth’s plates.
- explain how incoming solar radiations, ocean currents, and land masses affect weather and climate.

This is evident, for example, when students:

- use diagrams of ocean currents at different latitudes to develop explanations for the patterns present.

3. Matter is made up of particles whose properties determine the observable characteristics of matter and its reactivity.

Students:

- explain the properties of materials in terms of the arrangement and properties of the atoms that compose them.
- use atomic and molecular models to explain common chemical reactions.
- apply the principle of conservation of mass to chemical reactions.
- use kinetic molecular theory to explain rates of reactions and the relationships among temperature, pressure, and volume of a substance.

This is evident, for example, when students:

- use the atomic theory of elements to justify their choice of an element for use as a lighter than air gas for a launch vehicle.
- represent common chemical reactions using three-dimensional models of the molecules involved.
- discuss and explain a variety of everyday phenomena involving rates of chemical reactions, in terms of the kinetic molecular theory (e.g., use of refrigeration to keep food from spoiling, ripening of fruit in a bowl, use of kindling wood to start a fire, different types of flames that come from a Bunsen burner).

4. Energy exists in many forms, and when these forms change energy is conserved.

Students:

- observe and describe transmission of various forms of energy.
- explain heat in terms of kinetic molecular theory.
- explain variations in wavelength and frequency in terms of the source of the vibrations that produce them, e.g., molecules, electrons, and nuclear particles.
- explain the uses and hazards of radioactivity.

This is evident, for example, when students:

- demonstrate through drawings, models, and diagrams how the potential energy that exists in the chemical bonds of fossil fuels can be converted to electrical energy in a power plant (potential energy a heat energy a mechanical energy a electrical energy).
- investigate the sources of radioactive emissions in their environment and the dangers and benefits they pose for humans.

5. Energy and matter interact through forces that result in changes in motion.

Students:

- explain and predict different patterns of motion of objects (e.g., linear and angular motion, velocity and acceleration, momentum and inertia).
- explain chemical bonding in terms of the motion of electrons.
- compare energy relationships within an atom’s nucleus to those outside the nucleus.

This is evident, for example, when students:

- construct drawings, models, and diagrams representing several different types of chemical bonds to demonstrate the basis of the bond, the strength of the bond, and the type of electrical attraction that exists.

The Living Environment

1. Living things are both similar to and different from each other and nonliving things.

Students:

- explain how diversity of populations within ecosystems relates to the stability of ecosystems.
- describe and explain the structures and functions of the human body at different organizational levels (e.g., systems, tissues, cells, organelles).
- explain how a one-celled organism is able to function despite lacking the levels of organization present in more complex organisms.

2. Organisms inherit genetic information in a variety of ways that result in continuity of structure and function between parents and offspring.

Students:

- explain how the structure and replication of genetic material result in offspring that resemble their parents.
- explain how the technology of genetic engineering allows humans to alter the genetic makeup of organisms.

This is evident, for example, when students:

- record outward characteristics of fruit flies and then breed them to determine patterns of inheritance.

3. Individual organisms and species change over time.

Students:

- explain the mechanisms and patterns of evolution.

This is evident, for example, when students:

- determine characteristics of the environment that affect a hypothetical organism and explore how different characteristics of the species give it a selective advantage.

4. The continuity of life is sustained through reproduction and development.

Students:

- explain how organisms, including humans, reproduce their own kind.

This is evident, for example, when students:

- observe the development of fruit flies or rapidly maturing plants, from fertilized egg to mature adult, relating embryological development and structural adaptations to the propagation of the species.

5. Organisms maintain a dynamic equilibrium that sustains life.

Students:

- explain the basic biochemical processes in living organisms and their importance in maintaining dynamic equilibrium.
- explain disease as a failure of homeostasis.
- relate processes at the system level to the cellular level in order to explain dynamic equilibrium in multicelled organisms.

This is evident, for example, when students:

- investigate the biochemical processes of the immune system, and its relationship to maintaining mental and physical health.

6. Plants and animals depend on each other and their physical environment.

Students:

- explain factors that limit growth of individuals and populations.
- explain the importance of preserving diversity of species and habitats.
- explain how the living and nonliving environments change over time and respond to disturbances.

This is evident, for example, when students:

- conduct a long-term investigation of a local ecosystem.

7. Human decisions and activities have had a profound impact on the physical and living environment.

Students:

- describe the range of interrelationships of humans with the living and nonliving environment.
- explain the impact of technological development and growth in the human population on the living and non-living environment.
- explain how individual choices and societal actions can contribute to improving the environment.

This is evident, for example, when students:

- compile a case study of a technological development that has had a significant impact on the environment.

**Standard 5-Technology
**Elementary

**
Students will apply technological knowledge and skills to design, construct, use, and evaluate products and systems to satisfy human and environmental needs.
**

Engineering Design

1. Engineering design is an iterative process involving modeling and optimization used to develop technological solutions to problems within given constraints.

Students:

- describe objects, imaginary or real, that might be modeled or made differently and suggest ways in which the objects can be changed, fixed, or improved.
- investigate prior solutions and ideas from books, magazines, family, friends, neighbors, and community members.
- generate ideas for possible solutions, individually and through group activity; apply age-appropriate mathematics and science skills; evaluate the ideas and determine the best solution; and explain reasons for the choices.
- plan and build, under supervision, a model of the solution using familiar materials, processes, and hand tools.
- discuss how best to test the solution; perform the test under teacher supervision; record and portray results through numerical and graphic means; discuss orally why things worked or didn’t work; and summarize results in writing, suggesting ways to make the solution better.

This is evident, for example, when students:

- read a story called Humpty’s Big Day wherein the readers visit the place where Humpty Dumpty had his accident, and are asked to design and model a way to get to the top of the wall and down again safely.
- generate and draw ideas for a space station that includes a pleasant living and working environment
- design and model footwear that they could use to walk on a cold, sandy surface.

Tools, Resources, and Technological Process

2. Technological tools, materials, and other resources should be selected on the basis of safety, cost, availability, appropriateness, and environmental impact; technological processes change energy, information, and material resources into more useful forms.

Students:

- explore, use, and process a variety of materials and energy sources to design and construct things.
- understand the importance of safety, cost, ease of use, and availability in selecting tools and resources for a specific purpose.
- develop basic skill in the use of hand tools.
- use simple manufacturing processes (e.g., assembly, multiple stages of production, quality control) to produce a product.
- use appropriate graphic and electronic tools and techniques to process information.

This is evident, for example, when students:

- explore and use materials, joining them with the use of adhesives and mechanical fasteners to make a cardboard marionette with moving parts.
- explore materials and use forming processes to heat and bend plastic into a shape that can hold napkins.
- explore energy sources by making a simple motor that uses electrical energy to produce continuous mechanical motion
- develop skill with a variety of hand tools and use them to make or fix things.
- process information electronically such as using a video system to advertise a product or service.
- process information graphically such as taking photos and developing and printing the pictures.

Computer Technology

3. Computers, as tools for design, modeling, information processing, communication, and system control, have greatly increased human productivity and knowledge.

Students:

- identify and describe the function of the major components of a computer system.
- use the computer as a tool for generating and drawing ideas.
- control computerized devices and systems through programming.
- model and simulate the design of a complex environment by giving direct commands.

This is evident, for example, when students:

- control the operation of a toy or household appliance by programming it to perform a task.
- execute a computer program, such as SimCity, Theme Park, or The Factory to model and simulate an environment.
- model and simulate a system using construction modeling software, such as The Incredible Machine.

Technological Systems

4. Technological systems are designed to achieve specific results and produce outputs, such as products, structures, services, energy, or other systems.

Students:

- identify familiar examples of technological systems that are used to satisfy human needs and wants, and select them on the basis of safety, cost, and function.
- assemble and operate simple technological systems, including those with interconnecting mechanisms to achieve different kinds of movement.
- understand that larger systems are made up of smaller component subsystems.

This is evident, for example, when students:

- assemble and operate a system made up from a battery, switch, and doorbell connected in a series circuit.
- assemble a system with interconnecting mechanisms, such as a jack-in-the-box that pops up from a box with a hinged lid.
- model a community-based transportation system which includes subsystems such as roadways, rails, vehicles, and traffic controls.

History and Evolution of Technology

5. Technology has been the driving force in the evolution of society from an agricultural to an industrial to an information base.

Students:

- identify technological developments that have significantly accelerated human progress.

This is evident, for example, when students:

- construct a model of an historical or future-oriented technological device or system and describe how it has contributed or might contribute to human progress.
- make a technological timeline in the form of a hanging mobile of technological devices.
- model a variety of timekeeping devices that reflect historical and modern methods of keeping time.
- make a display contrasting early devices or tools with their modern counterparts.

Impacts of Technology

6. Technology can have positive and negative impacts on individuals, society, and the environment and humans have the capability and responsibility to constrain or promote technological development.

Students:

- describe how technology can have positive and negative effects on the environment and on the way people live and work.

This is evident, for example, when students:

- hand make an item and then participate in a line production experience where a quantity of the item is mass produced; compare the benefits and disadvantages of mass production and craft production.
- describe through example, how familiar technologies (including computers) can have positive and negative impacts on the environment and on the way people live and work.
- identify the pros and cons of several possible packaging materials for a student-made product.

Management of Technology

7. Project management is essential to ensuring that technological endeavors are profitable and that products and systems are of high quality and built safely, on schedule, and within budget.

Students:

- participate in small group projects and in structured group tasks requiring planning, financing, production, quality control, and follow-up.
- speculate on and model possible technological solutions that can improve the safety and quality of the school or community environment.

This is evident, for example, when students:

- help a group to plan and implement a school project or activity, such as a school picnic or a fund-raising event.
- plan as a group, division of tasks and construction steps needed to build a simple model of a structure or vehicle.
- redesign the work area in their classroom with an eye toward improving safety

**Standard 5-Technology
**Intermediate

Students will apply technological knowledge and skills to design, construct, use, and evaluate products and systems to satisfy human and environmental needs.

Engineering Design

1. Engineering design is an iterative process involving modeling and optimization used to develop technological solutions to problems within given constraints.

Students engage in the following steps in a design process:

- identify needs and opportunities for technical solutions from an investigation of situations of general or social interest.
- locate and utilize a range of printed, electronic, and human information resources to obtain ideas.
- consider constraints and generate several ideas for alternative solutions, using group and individual ideation techniques (group discussion, brainstorming, forced connections, role play); defer judgment until a number of ideas have been generated; evaluate (critique) ideas; and explain why the chosen solution is optimal.
- develop plans, including drawings with measurements and details of construction, and construct a model of the solution, exhibiting a degree of craftsmanship.
- in a group setting, test their solution against design specifications, present and evaluate results, describe how the solution might have been modified for different or better results, and discuss tradeoffs that might have to be made.

This is evident, for example, when students:

- reflect on the need for alternative growing systems in desert environments and design and model a hydroponic greenhouse for growing vegetables without soil.
- brainstorm and evaluate alternative ideas for an adaptive device that will make life easier for a person with a disability, such as a device to pick up objects from the floor.
- design a model vehicle (with a safety belt restraint system and crush zones to absorb impact) to carry a raw egg as a passenger down a ramp and into a barrier without damage to the egg.
- assess the performance of a solution against various design criteria, enter the scores on a spreadsheet, and see how varying the solution might have affected total score.

Tools, Resources, and Technological Process

2. Technological tools, materials, and other resources should be selected on the basis of safety, cost, r availability, appropriateness, and environmental impact; technological processes change energy, information, and material resources into more useful forms.

Students:

- choose and use resources for a particular purpose based upon an analysis and understanding of their properties, costs, availability, and environmental impact.
- use a variety of hand tools and machines to change materials into new forms through forming, separating, and combining processes, and processes which cause internal change to occur.
- combine manufacturing processes with other technological processes to produce, market, and distribute a product.
- process energy into other forms and information into more meaningful information.

This is evident, for example, when students:

- choose and use resources to make a model of a building and explain their choice of materials based upon physical properties such as tensile and compressive strength, hardness, and brittleness
- choose materials based upon their acoustic properties to make a set of wind chimes.
- use a torch to heat a steel rod to a cherry red color and cool it slowly to demonstrate how the process of annealing changes the internal structure of the steel and removes its brittleness.
- change materials into new forms using separate processes such as drilling and sawing.
- process energy into other forms such as assembling a solar cooker using a parabolic reflector to convert light energy to heat energy.
- process information into more meaningful information such as adding a music track or sound effects to an audio tape.

Computer Technology

3. Computers, as tools for design, modeling, information processing, communication, and system control, have greatly increased human productivity and knowledge.

Students:

- assemble a computer system including keyboard, central processing unit and disc drives, mouse, modem, printer, and monitor.
- use a computer system to connect to and access needed information from various Internet sites.
- use computer hardware and software to draw and dimension prototypical designs.
- use a computer as a modeling tool.
- use a computer system to monitor and control external events and/or systems.

This is evident, for example, when students:

- use computer hardware and a basic computer-aided design package to draw and dimension plans for a simple project.
- use a computer program, such as Car Builder, to model a vehicle to desired specifications.
- use temperature sensors to monitor and control the temperature of a model greenhouse.
- model a computer-controlled system, such as traffic lights, a merry-go-round, or a vehicle using Lego or other modeling hardware interfaced to a computer.

Technological Systems

4. Technological systems are designed to achieve specific results and produce outputs, such as products, structures, services, energy, or other systems.

Students:

- select appropriate technological systems on the basis of safety, function, cost, ease of operation, and quality of post purchase support.
- assemble, operate, and explain the operation of simple open- and closed-loop electrical, electronic, mechanical, and pneumatic systems.
- describe how subsystems and system elements (inputs, processes, outputs) interact within systems.
- describe how system control requires sensing information, processing it, and making changes.

This is evident, for example, when students:

- assemble an electronic kit that includes sensors and signaling devices and functions as an alarm system.
- use several open loop systems (without feedback control) such as a spray can, bubble gum machine, or wind-up toys, and compare them to closed-loop systems (with feedback control) such as an electric oven with a thermostat, or a line tracker robot.
- use a systems diagram to model a technological system, such as a model rocket, with the command inputs, resource inputs, processes, monitoring and control mechanisms, and system outputs labeled.
- provide examples of modern machines where microprocessors receive information from sensors and serve as controllers.

History and Evolution of Technology

5. Technology has been the driving force in the evolution of society from an agricultural to an industrial to an information base.

Students:

- describe how the evolution of technology led to the shift in society from an agricultural base to an industrial base to an information base.
- understand the contributions of people of different genders, races, and ethnic groups to technological development.
- describe how new technologies have evolved as a result of combining existing technologies (e.g., photography combined optics and chemistry; the airplane combined kite and glider technology with a lightweight gasoline engine).

This is evident, for example, when students:

- construct models of technological devices (e.g., the plow, the printing press, the digital computer) that have significantly affected human progress and that illustrate how the evolution of technology has shifted the economic base of the country.
- develop a display of pictures or models of technological devices invented by people from various cultural backgrounds, along with photographs and short biographies of the inventors.
- make a poster with drawings and photographs showing how an existing technology is the result of combining various technologies.

Impacts of Technology

6. Technology can have positive and negative impacts on individuals, society, and the environment and humans have the capability and responsibility to constrain or promote technological development.

Students:

- describe how outputs of a technological system can be desired, undesired, expected, or unexpected.
- describe through examples how modern technology reduces manufacturing and construction costs and produces more uniform products.

This is evident, for example, when students:

- use the automobile, for example, to explain desired (easier travel), undesired (pollution), expected (new jobs created), unexpected (crowded highways and the growth of suburbs) impacts.
- provide an example of an assembly line that produces products with interchangeable parts.
- compare the costs involved in producing a prototype of a product to the per product cost of a batch of 100.

Management of Technology

7. Project management is essential to ensuring that technological endeavors are profitable and that products and systems are of high quality and built safely, on schedule, and within budget.

Students:

- manage time and financial resources in a technological project.
- provide examples of products that are well (and poorly) designed and made, describe their positive and negative attributes, and suggest measures that can be implemented to monitor quality during production.
- assume leadership responsibilities within a structured group activity.

This is evident, for example, when students:

- make up and follow a project work plan, time schedule, budget, and a bill of materials.
- analyze a child’s toy and describe how it might have been better made at a lower cost.
- assume leadership on a team to play an audio or video communication system, and use it for an intended purpose (e.g., to inform, educate, persuade, entertain)

**Standard 5-Technology
**Commencement

Engineering Design

1. Engineering design is an iterative process involving modeling and optimization used to develop technological solutions to problems within given constraints.

Students engage in the following steps in a design process:

- initiate and carry out a thorough investigation of an unfamiliar situation and identify needs and opportunities for technological invention or innovation.
- identify, locate, and use a wide range of information resources including subject experts, library references, magazines, videotapes, films, electronic data bases and on-line services, and discuss and document through notes and sketches how findings relate to the problem.
- generate creative solution ideas, break ideas into the significant functional elements, and explore possible refinements; predict possible outcomes using mathematical and functional modeling techniques; choose the optimal solution to the problem, clearly documenting ideas against design criteria and constraints; and explain how human values, economics, ergonomics, and environmental considerations have influenced the solution.
- develop work schedules and plans which include optimal use and cost of materials, processes, time, and expertise; construct a model of the solution, incorporating developmental modifications while working to a high degree of quality (craftsmanship).
- in a group setting, devise a test of the solution relative to the design criteria and perform the test; record, portray, and logically evaluate performance test results through quantitative, graphic, and verbal means; and use a variety of creative verbal and graphic techniques effectively and persuasively to present conclusions, predict impacts and new problems, and suggest and pursue modifications.

This is evident, for example, when students:

- search the Internet for world wide web sites dealing with renewable energy and sustainable living and research the development and design of an energy efficient home.
- develop plans, diagrams, and working drawings for the construction of a computer-controlled marble sorting system that simulates how parts on an assembly line are sorted by color.
- design and model a portable emergency shelter for a homeless person that could be carried by one person and be heated by the body heat of that person to a life-sustaining temperature when the outside temperature is 20 o F.

Tools, Resources, and Technological Processes

2. Technological tools, materials, and other resources should be selected on the basis of safety, cost, availability, appropriateness, and environmental impact; technological processes change energy, information, and material resources into more useful forms.

Students:

- test, use, and describe the attributes of a range of material (including synthetic and composite materials), information, and energy resources.
- select appropriate tools, instruments, and equipment and use them correctly to process materials, energy, and information.
- explain tradeoffs made in selecting alternative resources in terms of safety, cost, properties, availability, ease of processing, and disposability.
- describe and model methods (including computer-based methods) to control system processes and monitor system outputs.

This is evident, for example, when students:

- use a range of high- tech composite or synthetic materials to make a model of a product, (e.g., ski, an airplane, earthquake-resistant building) and explain their choice of material.
- design a procedure to test the properties of synthetic and composite materials.
- select appropriate tools, materials, and processes to manufacture a product (chosen on the basis of market research) that appeals to high school students.
- select the appropriate instrument and use it to test voltage and continuity when repairing a household appliance.
- construct two forms of packaging (one from biodegradable materials, the other from any other materials), for a children’s toy and explain the tradeoffs made when choosing one or the other.
- describe and model a method to design and evaluate a system that dispenses candy and counts the number dispensed using, for example, Fischertecnik, Capsela, or Lego.
- describe how the flow, processing, and monitoring of materials is controlled in a manufacturing plant and how information processing systems provide inventory, tracking, and quality control data.

Computer Technology

3. Computers, as tools for design, modeling, information processing, communication, and system control, have greatly increased human productivity and knowledge.

Students:

- understand basic computer architecture and describe the function of computer subsystems and peripheral devices.
- select a computer system that meets personal needs.
- attach a modem to a computer system and telephone line, set up and use communications software, connect to various on-line networks, including the Internet, and access needed information using e-mail, telnet, gopher, ftp, and web searches.
- use computer-aided drawing and design (CADD) software to model realistic solutions to design problems.
- develop an understanding of computer programming and attain some facility in writing computer programs.

This is evident, for example, when students:

- choose a state-of-the art computer system from computer magazines, price the system, and justify the choice of CPU, CD-ROM and floppy drives, amount of RAM, video and sound cards, modem, printer, and monitor; explain the cost benefit tradeoffs they have made.
- use a computer-aided drawing and design package to design and draw a model of their own room.
- write a computer program that works in conjunction with a bar code reader and an optical sensor to distinguish between light and dark areas of the bar code.

Technological Systems

4. Technological systems are designed to achieve specific results and produce outputs, such as products, structures, services, energy, or other systems.

Students:

- explain why making tradeoffs among characteristics, such as safety, function, cost, ease of operation, quality of post purchase support, and environmental impact, is necessary when selecting systems for specific purposes.
- model, explain, and analyze the performance of a feedback control system.
- explain how complex technological systems involve the confluence of numerous other systems.

This is evident, for example, when students:

- model, explain, and analyze how the float mechanism of a toilet tank senses water level, compares the actual level to the desired level, and controls the flow of water into the tank.
- draw a labeled system diagram which explains the performance of a system, and include several subsystems and multiple feed-back loops.
- explain how the space shuttle involves communication, transportation, biotechnical, and manufacturing systems.

History and Evolution of Technology

5. Technology has been the driving force in the evolution of society from an agricultural to an industrial to an information base.

Students:

- explain how technological inventions and innovations have caused global growth and interdependence, stimulated economic competitiveness, created new jobs, and made other jobs obsolete.

This is evident, for example, when students:

- compare qualitatively and quantitatively the performance of a contemporary manufactured product, such as a household appliance, to the comparable device or system 50-100 years ago, and present results graphically, orally, and in writing.
- describe the process that an inventor must follow to obtain a patent for an invention.
- explain through examples how some inventions are not translated into products and services with market place demand, and therefore do not become commercial successes.

Impacts of Technology

6. Technology can have positive and negative impacts on individuals, society, and the environment and humans have the capability and responsibility to constrain or promote technological development.

Students:

- explain that although technological effects are complex and difficult to predict accurately, humans can control the development and implementation of technology.
- explain how computers and automation have changed the nature of work.
- explain how national security is dependent upon both military and nonmilitary applications of technology.

This is evident, for example, when students:

- develop and implement a technological device that might be used to assist a disabled person perform a task.
- identify a technology which impacts negatively on the environment and design and model a technological fix.
- identify new or emerging technologies and use a futuring technique (e.g., futures wheel, cross impact matrix, Delphi survey) to predict what might be the second and third order impacts.

Management of Technology

7. Project management is essential to ensuring that technological endeavors are profitable and that products and systems are of high quality and built safely, on schedule, and within budget.

Students:

- develop and use computer-based scheduling and project tracking tools, such as flow charts and graphs.
- explain how statistical process control helps to assure high quality output.
- discuss the role technology has played in the operation of successful U.S. businesses and under what circumstances they are competitive with other countries.
- explain how technological inventions and innovations stimulate economic competitiveness and how, in order for an innovation to lead to commercial success, it must be translated into products and services with marketplace demand.
- describe new management techniques (e.g., computer-aided engineering, computer-integrated manufacturing, total quality management, just-in-time manufacturing), incorporate some of these in a technological endeavor, and explain how they have reduced the length of design-to- manufacture cycles, resulted in more flexible factories, and improved quality and customer satisfaction.
- help to manage a group engaged in planning, designing, implementation, and evaluation of a project to gain understanding of the management dynamics.

This is evident, for example, when students:

- design and carry out a plan to create a computer-based information system that could be used to help manage a manufacturing system (e.g., monitoring inventory, measurement of production rate, development of a safety signal).
- identify several successful companies and explain the reasons for their commercial success.
- organize and implement an innovative project, based on market research, that involves design, production, testing, marketing, and sales of a product or a service.

**Standard 6-Interconnectedness:****
Common Themes
**Elementary

Students will understand the relationships and common themes that connect mathematics, science, and technology and apply the themes to these and other areas of learning.

Systems Thinking

1. Through systems thinking, people can recognize the commonalities that exist among all systems and how parts of a system interrelate and combine to perform specific functions.

Students:

- observe and describe interactions among components of simple systems.
- identify common things that can be considered to be systems (e.g., a plant population, a subway system, human beings).

**
Models**

2. Models are simplified representations of objects, structures, or systems used in analysis, explanation, interpretation, or design.

Students:

- analyze, construct, and operate models in order to discover attributes of the real thing.
- discover that a model of something is different from the real thing but can be used to study the real thing.
- use different types of models, such as graphs, sketches, diagrams, and maps, to represent various aspects of the real world.

This is evident, for example, when students:

- compare toy cars with real automobiles in terms of size and function.
- model structures with building blocks.
- design and construct a working model of the human circulatory system to explore how varying pumping pressure might affect blood flow.
- describe the limitations of model cars, planes, or houses.
- use model vehicles or structures to illustrate how the real object functions.
- use a road map to determine distances between towns and cities.

Magnitude and Scale

3. The grouping of magnitudes of size, time, frequency, and pressures or other units of measurement into a series of relative order provides a useful way to deal with the immense range and the changes in scale that affect the behavior and design of systems.

Students:

- provide examples of natural and manufactured things that belong to the same category yet have very different sizes, weights, ages, speeds, and other measurements.
- identify the biggest and the smallest values as well as the average value of a system when given information about its characteristics and behavior.

This is evident, for example, when students:

- compare the weight of small and large animals.
- compare the speed of bicycles, cars, and planes.
- compare the life spans of insects and trees.
- collect and analyze data related to the height of the students in their class, identifying the tallest, the shortest, and the average height.
- compare the annual temperature range of their locality.

Equilibrium and Stability

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

Students:

- cite examples of systems in which some features stay the same while other features change.
- distinguish between reasons for stability–from lack of changes to changes that counterbalance one another to changes within cycles.

This is evident, for example, when students:

- record their body temperatures in different weather conditions and observe that the temperature of a healthy human being stays almost constant even though the external temperature changes.
- identify the reasons for the changing amount of fresh water in a reservoir and determine how a constant supply is maintained.

Patterns of Change

5. Identifying patterns of change is necessary for making predictions about future behavior and conditions.

Students:

- use simple instruments to measure such quantities as distance, size, and weight and look for patterns in the data.
- analyze data by making tables and graphs and looking for patterns of change.

This is evident, for example, when students:

- compare shoe size with the height of people to determine if there is a trend.
- collect data on the speed of balls rolling down ramps of different slopes and determine the relationship between speed and steepness of the ramp.
- take data they have collected and generate tables and graphs to begin the search for patterns of change.

Optimization

6. In order to arrive at the best solution that meets criteria within constraints, it is often necessary to make trade-offs.

Students:

- determine the criteria and constraints of a simple decision making problem.
- use simple quantitative methods, such as ratios, to compare costs to benefits of a decision problem.

This is evident, for example, when students:

- describe the criteria (e.g., size, color, model) and constraints (e.g., budget) used to select the best bicycle to buy.
- compare the cost of cereal to number of servings to figure out the best buy.

**Standard 6-Interconnectedness: ****Common
Themes
**Intermediate

Students will understand the relationships and common themes that connect mathematics, science, and technology and apply the themes to these and other areas of learning.

Systems Thinking

1. Through systems thinking, people can recognize the commonalities that exist among all systems and how parts of a system interrelate and combine to perform specific functions.

Students:

- describe the differences between dynamic systems and organizational systems.
- describe the differences and similarities between engineering systems, natural systems, and social systems.
- describe the differences between open- and closed-loop systems.
- describe how the output from one part of a system (which can include material, energy, or information) can become the input to other parts.

This is evident, for example, when students:

- compare systems with internal control (e.g., homeostasis in organisms or an ecological system) to systems of related components without internal control (e.g., the Dewey decimal, solar system).

Models

2. Models are simplified representations of objects, structures, or systems used in analysis, explanation, interpretation, or design.

Students:

- select an appropriate model to begin the search for answers or solutions to a question or problem.
- use models to study processes that cannot be studied directly (e.g., when the real process is too slow, too fast, or too dangerous for direct observation).
- demonstrate the effectiveness of different models to represent the same thing and the same model to represent different things.

This is evident, for example, when students:

- choose a mathematical model to predict the distance a car will travel at a given speed in a given time.
- use a computer simulation to observe the process of growing vegetables or to test the performance of cars.
- compare the relative merits of using a flat map or a globe to model where places are situated on Earth.
- use blueprints or scale models to represent room plans.

Magnitude and Scale

3. The grouping of magnitudes of size, time, frequency, and pressures or other units of measurement into a series of relative order provides a useful way to deal with the immense range and the changes in scale that affect the behavior and design of systems.

Students:

- cite examples of how different aspects of natural and designed systems change at different rates with changes in scale.
- use powers of ten notation to represent very small and very large numbers.

This is evident, for example, when students:

- demonstrate that a large container of hot water (more volume) cools off more slowly than a small container (less volume).
- compare the very low frequencies (60 Hertz AC or 6 x 10 Hertz) to the mid-range frequencies (10 Hertz-FM radio) to the higher frequencies (10 15 Hertz) of the electromagnetic spectrum.

Equilibrium and Stability

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

Students:

- describe how feedback mechanisms are used in both designed and natural systems to keep changes within desired limits.
- describe changes within equilibrium cycles in terms of frequency or cycle length and determine the highest and lowest values and when they occur.

This is evident, for example, when students:

- compare the feedback mechanisms used to keep a house at a constant temperature to those used by the human body to maintain a constant temperature.
- analyze the data for the number of hours of sunlight from the shortest day to the longest day of the year.

**Standard 6-Interconnectedness:****
Common Themes
**Intermediate

Patterns of Change

5. Identifying patterns of change is necessary for making predictions about future behavior and conditions.

Students:

- use simple linear equations to represent how a parameter changes with time.
- observe patterns of change in trends or cycles and make predictions on what might happen in the future.

This is evident, for example, when students:

- study how distance changes with time for a car traveling at a constant speed.
- use a graph of a population over time to predict future population levels.

Optimization

6. In order to arrive at the best solution that meets criteria within constraints, it is often necessary to make trade-offs.

Students:

- determine the criteria and constraints and make trade-offs to determine the best decision.
- use graphs of information for a decision making problem to determine the optimum solution.

This is evident, for example, when students:

- choose components for a home stereo system.
- determine the best dimensions for fencing in the maximum area.

**Standard 6-Interconnectedness: Common Themes
**Commencement

Systems Thinking

1. Through systems thinking, people can recognize the commonalities that exist among all systems and how parts of a system interrelate and combine to perform specific functions.

Students:

- explain how positive feedback and negative feedback have opposite effects on system outputs.
- use an input-process-output-feedback diagram to model and compare the behavior of natural and engineered systems.
- define boundary conditions when doing systems analysis to determine what influences a system and how it behaves.

This is evident, for example, when students:

- describe how negative feedback is used to control loudness automatically in a stereo system and how positive feedback from loudspeaker to microphone results in louder and louder squeals.

Models

2. Models are simplified representations of objects, structures, or systems used in analysis, explanation, interpretation, or design.

Students:

- revise a model to create a more complete or improved representation of the system.
- collect information about the behavior of a system and use modeling tools to represent the operation of the system.
- find and use mathematical models that behave in the same manner as the processes under investigation.
- compare predictions to actual observations using test models.

This is evident, for example, when students:

- add new parameters to an existing spreadsheet model.
- incorporate new design features in a CAD drawing.
- use computer simulation software to create a model of a system under stress, such as a city or an ecosystem.
- design and construct a prototype to test the performance of a temperature control system.
- use mathematical models for scientific laws, such as Hooke’s Law or Newton’s Laws, and relate them to the function of technological systems, such as an automotive suspension system.
- use sinusoidal functions to study systems that exhibit periodic behavior.
- compare actual populations of animals to the numbers predicted by predator/ prey computer simulations.

Magnitude and Scale

3. The grouping of magnitudes of size, time, frequency, and pressures or other units of measurement into a series of relative order provides a useful way to deal with the immense range and the changes in scale that affect the behavior and design of systems.

Students:

- describe the effects of changes in scale on the functioning of physical, biological, or designed systems.
- extend their use of powers of ten notation to understanding the exponential function and performing operations with exponential factors.

This is evident, for example, when students:

- explain that an increase in the size of an animal or a structure requires larger supports (legs or columns) because of the greater volume or weight.
- use the relationship that v=f l to determine wave length when given the frequency of an FM radio wave, such as 100.0 megahertz (1.1 x 10 8 Hertz), and velocity of light or EM waves as 3 x 10 8m/sec can.

Equilibrium and Stability

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

Students:

- describe specific instances of how disturbances might affect a system’s equilibrium, from small disturbances that do not upset the equilibrium to larger disturbances (threshold level) that cause the system to become unstable.
- cite specific examples of how dynamic equilibrium is achieved by equality of change in opposing directions.

This is evident, for example, when students:

- use mathematical models to predict under what conditions the spread of a disease will become epidemic.
- document the range of external temperatures in which warm-blooded animals can maintain a relatively constant internal temperature and identify the extremes of cold or heat that will cause death.
- experiment with chemical or biological processes when the flow of materials in one way direction is counter-balanced by the flow of materials in the opposite direction.

Patterns of Change

5. Identifying patterns of change is necessary for making predictions about future behavior and conditions.

Students:

- use sophisticated mathematical models, such as graphs and equations of various algebraic or trigonometric functions.
- search for multiple trends when analyzing data for patterns, and identify data that do not fit the trends.

This is evident, for example, when students:

- use a sine pattern to model the property of a sound or electromagnetic wave.
- use graphs or equations to model exponential growth of money or populations.
- explore historical data to determine whether the growth of a parameter is linear or exponential or both.

Optimization

6. In order to arrive at the best solution that meets criteria within constraints, it is often necessary to make trade-offs.

Students:

- use optimization techniques, such as linear programming, to determine optimum solutions to problems that can be solved using quantitative methods.
- analyze subjective decision making problems to explain the trade-offs that can be made to arrive at the best solution.

This is evident, for example, when students:

- use linear programming to figure the optimum diet for farm animals.
- evaluate alternative proposals for providing people with more access to mass transportation systems.

**Standard 7-Interdisciplinary Problem Solving
**Elementary

Students will apply the knowledge and thinking skills of mathematics, science, and technology to address real-life problems and make informed decisions.

Connections

1. The knowledge and skills of mathematics, science, and technology are used together to make informed decisions and solve problems, especially those relating to issues of science/technology/society, consumer decision making, design, and inquiry into phenomena.

Students:

- analyze science/technology/society problems and issues that affect their home, school, or community, and carry out a remedial course of action.
- make informed consumer decisions by applying knowledge about the attributes of particular products and making cost/benefit tradeoffs to arrive at an optimal choice.
- design solutions to problems involving a familiar and real context, investigate related science concepts to inform the solution, and use mathematics to model, quantify, measure, and compute.
- observe phenomena and evaluate them scientifically and mathematically by conducting a fair test of the effect of variables and using mathematical knowledge and technological tools to collect, analyze, and present data and conclusions.

This is evident, for example, when students:

- develop and implement a plan to reduce water or energy consumption in their home.
- choose paper towels based on tests of absorption quality, strength, and cost per sheet.
- design a wheeled vehicle, sketch and develop plans, test different wheel and axle designs to reduce friction, chart results, and produce a working model with correct measurements.
- collect leaves of similar size from different varieties of trees, and compare the ratios of length to width in order to determine whether the ratios are the same for all species.

Strategies

2. Solving interdisciplinary problems involves a variety of skills and strategies, including effective work habits; gathering and processing information; generating and analyzing ideas; realizing ideas; making connections among the common themes of mathematics, science, and technology; and presenting results.

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

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

This is evident, for example, when students, addressing the issue of solid waste at the school in an interdisciplinary science/technology/society project:

- use the newspaper index to find out about how solid waste is handled in their community, and interview the custodial staff to collect data about how much solid waste is generated in the school, and they make and use tables and graphs to look for patterns of change. Students work together to reach consensus on the need for recycling and on choosing a material to recycle– in this case, paper.
- investigate the types of paper that could be recycled, measure the amount (weight, volume) of this type of paper in their school during a one-week period, and calculate the cost. Students investigate the processes involved in changing used paper into a useable product and how and why those changes work as they do.
- using simple mixers, wire screens, and lint, leaves, rags, etc., students recycle used paper into useable sheets and evaluate the quality of the product. They present their results using charts, graphs, illustrations, and photographs to the principal and custodial staff.

Skills and Strategies for Interdisciplinary Problem Solving

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

**Gathering and Processing Information:** Accessing information from printed media, electronic data bases, and community resources and using the information to develop a definition of the problem 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 problem, and using them to better understand the dimensions of the problem.

**Realizing Ideas:** Constructing components or models, arriving at a solution, and evaluating the result.

**Presenting Results:** Using a variety of media to present the solution and to communicate the results.

**Standard 7-Interdisciplinary****
Problem Solving
**Intermediate

Students will apply the knowledge and thinking skills of mathematics, science, and technology to address real-life problems and make informed decisions.

Connections

1. The knowledge and skills of mathematics, science, and technology are used together to make informed decisions and solve problems, especially those relating to issues of science/technology/society, consumer decision making, design, and inquiry into phenomena.

Students:

- analyze science/technology/society problems and issues at the local level and plan and carry out a remedial course of action.
- make informed consumer decisions by seeking answers to appropriate questions about products, services, and systems; determining the cost/benefit and risk/benefit tradeoffs; and applying this knowledge to a potential purchase.
- design solutions to real-world problems of general social interest related to home, school, or community using scientific experimentation to inform the solution and applying mathematical concepts and reasoning to assist in developing a solution.
- describe and explain phenomena by designing and conducting investigations involving systematic observations, accurate measurements, and the identification and control of variables; by inquiring into relevant mathematical ideas; and by using mathematical and technological tools and procedures to assist in the investigation.

This is evident, for example, when students:

- improve a habitat for birds at a park or on school property.
- choose a telescope for home use based on diameter of the telescope, magnification, quality of optics and equatorial mount, cost, and ease of use.
- design and construct a working model of an air filtration device that filters out particles above a particular size.
- simulate population change using a simple model (e.g., different colors of paper clips to represent different species of birds). Timed removals of clips from plastic cups represents the action of predators and varying the percentage of the return of clips to cups represent differences in reproductive rates. Students apply mathematical modeling techniques to graph population growth changes and make interpretations related to resource depletion.

Strategies

2. Solving interdisciplinary problems involves a variety of skills and strategies, including effective work habits; gathering and processing information; generating and analyzing ideas; realizing ideas; making connections among the common themes of mathematics, science, and technology; and presenting results.

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

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

This is evident, for example, when students, addressing the issue of auto safety in an interdisciplinary science/technology/society project:

- use an electronic data base to obtain information on the causes of auto accidents and use e-mail to collect information from government agencies and auto safety organizations. Students gather, analyze, and chart information on the number and causes of auto accidents in their county and look for trends.
- design and construct a model vehicle with a restraint system to hold a raw egg as the passenger and evaluate the effectiveness of the restraint system by rolling the vehicle down a ramp and into a barrier; the vehicle is designed with crush zones to absorb the impact. Students analyze forces and compute acceleration using F=ma calculations. They present their results, including a videotaped segment, to a driver education class.

**Skills and Strategies for Interdisciplinary
** **
Problem Solving**

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

**Gathering and Processing Information:** Accessing information from printed media, electronic data bases, and community resources and using the information to develop a definition of the problem 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 problem, and using them to better understand the dimensions of the problem.

**Realizing Ideas: ** Constructing components or models, arriving at a solution, and evaluating the result.

**Presenting Results:** Using a variety of media to present the solution and to communicate the results.

**Standard 7-Interdisciplinary ****Problem
Solving
**Commencement

**Connections**

1. The knowledge and skills of mathematics, science, and technology are used together to make informed decisions and solve problems, especially those relating to issues of science/technology/society, consumer decision making, design, and inquiry into phenomena.

Students:

- analyze science/technology/society problems and issues on a community, national, or global scale and plan and carry out a remedial course of action.
- analyze and quantify consumer product data, understand environmental and economic impacts, develop a method for judging the value and efficacy of competing products, and discuss cost/benefit and risk/benefit tradeoffs made in arriving at the optimal choice.
- design solutions to real-world problems on a community, national, or global scale using a technological design process that integrates scientific investigation and rigorous mathematical analysis of the problem and of the solution.
- explain and evaluate phenomena mathematically and scientifically by formulating a testable hypothesis, demonstrating the logical connections between the scientific concepts guiding the hypothesis and the design of an experiment, applying and inquiring into the mathematical ideas relating to investigation of phenomena, and using (and if needed, designing) technological tools and procedures to assist in the investigation and in the communication of results.

This is evident, for example, when students:

- analyze the issues related to local energy needs and develop a viable energy generation plan for the community.
- choose whether it is better to purchase a conventional or high definition television after analyzing the differences from quantitative and qualitative points of view, considering such particulars as the number of scanning lines, bandwidth requirements and impact on the frequency spectrum, costs, and existence of international standards.
- design and produce a prototypical device using an electronic volt-age divider that can be used to power a portable cassette tape or CD player in a car by reducing the standard automotive accessory power source of approximately 14.8 volts to a lower voltage.
- investigate two similar fossils to determine if they represent a developmental change over time.

**Strategies**

2. Solving interdisciplinary problems involves a variety of skills and strategies, including effective work habits; gathering and processing information; generating and analyzing ideas; realizing ideas; making connections among the common themes of mathematics, science, and technology; and presenting results.

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

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

This is evident, for example, when students, addressing the issue of emergency preparedness in an interdisciplinary science/technology/society project:

- are given a scenario–survivors from a disaster are stranded on a mountaintop in the high peaks of the Adirondacks they are challenged to design a portable shelter that could be heated by the body heat of five survivors to a life sustaining temperature, given an outside temperature of 20
^{o}F. Since the shelter would be dropped to survivors by an aircraft, it must be capable of withstanding the impact. Students determine the kinds of data to be collected, for example, snowfall during certain months, average wind velocity, R value of insulating materials, etc. To conduct their research, students gather and analyze information from research data bases, national libraries, and electronic communication networks, including the Internet. - design and construct scale models or full-sized shelters based on engineering design criteria including wind load, snow load, and insulating properties of materials. Heat flow calculations are done to determine how body heat could be used to heat the shelter. Students evaluate the trade-offs that they make to arrive at the best solution; for example, in order to keep the temperature at 20 degrees F., the shelter may have to be small, and survivors would be very uncomfortable. Another component of the project is assembly instructions–designed so that speakers of any language could quickly install the structure on site.
- prepare a multimedia presentation about their project and present it to the school’s ski club.

Skills and Strategies for Interdisciplinary Problem Solving

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

**Gathering and Processing Information:** Accessing information from printed media, electronic data bases, and community resources and using the information to develop a definition of the problem 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 problem, and using them to better understand the dimensions of the problem.

**Realizing Ideas: ** Constructing components or models, arriving at a solution, and evaluating the result.

**Presenting Results:** Using a variety of media to present the solution and to communicate the results.