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Mathematical Analysis Scientific Inquiry

• 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.

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

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

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

• ask “whyâ€� 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.

• 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.

• 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).

• 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

 Engineering Design

• 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.

• 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.

• 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.

• 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.

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Mathematical Analysis Scientific Inquiry

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

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

• predict the next triangular number by examining the pattern 1, 3, 6, 10, £ .

• 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.

• 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.

• 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.

• 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.

• 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.

 Engineering Design

• 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.

• 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.

• 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.

• 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.

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Mathematical Analysis Scientific Inquiry

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

• 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.

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

• 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.

• 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.

• 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.

• 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

 Engineering Design

• 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.

• 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.

• 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.

• 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.

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Information Systems

• 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.

• 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.

• 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.

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

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

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Information Systems

• 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.

• 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.

• 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.

• 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.

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Information Systems

• 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.

• 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).

• 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.

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

• 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.

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Mathematical Reasoning Number and Numeration

• 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.

• 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.

• 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.

• 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 Modeling/Multiple Representation

• 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.

• 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.

• 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.

• 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.

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Measurement Uncertainty

• 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.

• 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.

• 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.

• 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 

• 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.

• 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.

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Mathematical Reasoning Number and Numeration

• 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.

• 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.

• 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.

• 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:

• .25 = 25%

Operations Modeling/Multiple Representation

• 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.

• 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,

• 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.

• 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.

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Measurement Uncertainty

• 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.

• develop critical judgment for the reasonableness of measurement.

• 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.

• 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.

• 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.

• 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.

• 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

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Mathematical Reasoning Number and Numeration

• 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.

• 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.

• 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.

• 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   

Operations Modeling/Multiple Representation

• 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.

• 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.

• 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.

• 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 = c 2 is graphed for various values of a, b, and c.

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Measurement Uncertainty

• 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.

• 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.

• judge the reasonableness of results obtained from  

• 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.

• 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 

• 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.

• 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.
  

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Measurement Uncertainty

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

• prove indirectly that: if n 2 is even, n is even.
• prove using mathematical induction that: 1 + 3 + 5 + . . . + (2n - 1) = n2 .
• explain the axiomatic differences between plane and spherical geometries.

• 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.

• 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 Modeling/Multiple Representation

• 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.

• 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).

• 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.

• 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.

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Measurement Uncertainty

• 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.

• 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.

• 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.

• 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 

• 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.

• 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  

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Physical Setting 

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

• 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.

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

• 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.

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

• 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.

• 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).

• 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 milliammeter. Students can compare the success of a variety of these electrical cells.

• 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.

• investigate simple machines and use them to perform tasks.

The Living Environment 

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

• 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.

• 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.

• 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.â€�

• 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.

• 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.

• 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.

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

• 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.

• 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.

• 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.

• 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.

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

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

Physical Setting 

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

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

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

• 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 Goldbergâ€� type devices and describe the energy transformations evident in them.
  
  

The Living Environment 

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

• 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.

• 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.

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

• 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.

• 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.

• 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.

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

• 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.

• 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.

• 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.

• 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.

• describe how living things, including humans, depend upon the living and nonliving environment for their survival.

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

Physical Setting 

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

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

• 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.

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

• 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.

• 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).

• 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.

• 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.

• 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.

• 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 

• 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.

• 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.

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

• explain the mechanisms and patterns of evolution.

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

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

• 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.

• 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.

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

• 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.

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

• 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.

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

Engineering Design Tools, Resources, and Technological 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.

• 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.

• 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.

• 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 Technological Systems

• 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.

• 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.

• 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.

• 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.
  
  

• identify technological developments that have significantly accelerated 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.

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

• handmake 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.
  
  

• 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.

• 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
  
• .
  
  

Engineering Design Tools, Resources, and Technological 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.

• 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.

• 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.

• 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 Technological Systems

• 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.

• 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.

• 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.

• 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 Impacts of Technology

• 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).

• 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.

• 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.

• 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 

• 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.

• 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)
  
• .
  
  

Engineering Design Tools, Resources, and Technological Processes

• 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.

• 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.

• 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.

• 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 Technological Systems

• 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.

• 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.

• 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.

• 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.

• 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 Impacts of Technology

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

• 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.

• 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.

• 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 

• 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.

• 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
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Systems Thinking Models

• 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).

• 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.

• 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 Equilibrium and Stability

• 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.

• 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.

• 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.

• 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.
  

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Patterns of Change Optimization

• 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.

• 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.

• 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.

• 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.
  
  

Systems Thinking Models

• 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.

• 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).

• 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.

• 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 Equilibrium and Stability

• 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.

• 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.

• 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.

• 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.
  
  

Patterns of Change Optimization

• 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.

• 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.

• 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.

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

Systems Thinking Models

• 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.

• 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.

• 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.

• 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 Equilibrium and Stability

• 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.

• 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.

• 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.

• 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 Optimization

• 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.

• 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.

• 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.

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

Connections Strategies

• 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.

• 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 determinet whether the ratios are the same for all species.

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

• 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

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Connections Strategies

• 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.

• 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.

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

• 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

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Connections Strategies

• 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.

• 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.

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

• 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

• 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

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