Descriptive Statement

Problem posing and problem solving involve examining situations that arise in mathematics and other disciplines and in common experiences, describing these situations mathematically, formulating appropriate mathematical questions, and using a variety of strategies to find solutions. By developing their problem-solving skills, students will come to realize the potential usefulness of mathematics in their lives.

Cumulative Progress Indicators

By the end of Grade 4, students:

1. Use discovery-oriented, inquiry-based, and problem-centered approaches to investigate and understand mathematical content appropriate to early elementary grades.

2. Recognize, formulate, and solve problems arising from mathematical situations and everyday experiences.

3. Construct and use concrete, pictorial, symbolic, and graphical models to represent problem situations.

4. Pose, explore, and solve a variety of problems, including non-routine problems and open-ended problems with several solutions and/or solution strategies.

5. Construct, explain, justify, and apply a variety of problem-solving strategies in both cooperative and independent learning environments.

6. Verify the correctness and reasonableness of results and interpret them in the context of the problems being solved.

7. Know when to select and how to use grade-appropriate mathematical tools and methods (including manipulatives, calculators and computers, as well as mental math and paper-and-pencil techniques) as a natural and routine part of the problem-solving process.

8. Determine, collect, organize, and analyze data needed to solve problems.

9. Recognize that there may be multiple ways to solve a problem.

Building upon knowledge and skills gained in the preceding grades, and demonstrating continued progress in Indicators 4, 5, 6, 7, and 8 above, by the end of Grade 8, students:

10. Use discovery-oriented, inquiry-based, and problem-centered approaches to investigate and understand mathematical content appropriate to the middle grades.

11. Recognize, formulate, and solve problems arising from mathematical situations, everyday experiences, and applications to other disciplines.

12. Construct and use concrete, pictorial, symbolic, and graphical models to represent problem situations and effectively apply processes of mathematical modeling in mathematics and other areas.

13. Recognize that there may be multiple ways to solve a problem, weigh their relative merits, and select and use appropriate problem-solving strategies.

14. Persevere in developing alternative problem-solving strategies if initially selected approaches do not work.

Building upon knowledge and skills gained in the preceding grades, and demonstrating continued progress in Indicators 4, 5, 6, 7, 8, 12, and 14 above, by the end of Grade 12, students:

15. Use discovery-oriented, inquiry-based, and problem-centered approaches to investigate and understand the mathematical content appropriate to the high school grades.

16. Recognize, formulate, and solve problems arising from mathematical situations, everyday experiences, applications to other disciplines, and career applications.

17. Monitor their own progress toward problem solutions.

18. Explore the validity and efficiency of various problem-posing and problem-solving strategies, and develop alternative strategies and generalizations as needed.

Descriptive Statement

Communication of mathematical ideas will help students clarify and solidify their understanding of mathematics. By sharing their mathematical understandings in written and oral form with their classmates, teachers, and parents, students develop confidence in themselves as mathematics learners and enable teachers to better monitor their progress.

Cumulative Progress Indicators

By the end of Grade 4, students:

1. Discuss, listen, represent, read, and write as vital activities in their learning and use of mathematics.

2. Identify and explain key mathematical concepts, and model situations using oral, written, concrete, pictorial, and graphical methods.

3. Represent and communicate mathematical ideas through the use of learning tools such as calculators, computers, and manipulatives.

4. Engage in mathematical brainstorming and discussions by asking questions, making conjectures, and suggesting strategies for solving problems.

5. Explain their own mathematical work to others, and justify their reasoning and conclusions.

Building upon knowledge and skills gained in the preceding grades, and demonstrating continued progress in Indicators 1, 2, 3, 4, and 5 above, by the end of Grade 8, students:

6. Identify and explain key mathematical concepts and model situations using geometric and algebraic methods.

7. Use mathematical language and symbols to represent problem situations, and recognize the economy and power of mathematical symbolism and its role in the development of mathematics.

8. Analyze, evaluate, and explain mathematical arguments and conclusions presented by others.

Building upon knowledge and skills gained in the preceding grades, and demonstrating continued progress in Indicators 1, 2, 3, 4, 5, 6, 7, and 8 above, by the end of Grade 12, students:

9. Formulate questions, conjectures, and generalizations about data, information, and problem situations.

10. Reflect on and clarify their thinking so as to present convincing arguments for their conclusions.

Descriptive Statement

Making connections enables students to see relationships between different topics, and to draw on those relationships in future study. This applies within mathematics, so that students can translate readily between fractions and decimals, or between algebra and geometry; to other content areas, so that students understand how mathematics is used in the sciences, the social sciences, and the arts; and to the everyday world, so that students can connect school mathematics to daily life.

Cumulative Progress Indicators

By the end of Grade 4, students:

1. View mathematics as an integrated whole rather than as a series of disconnected topics and rules.

2. Relate mathematical procedures to their underlying concepts.

3. Use models, calculators, and other mathematical tools to demonstrate the connections among various equivalent graphical, concrete, and verbal representations of mathematical concepts.

4. Explore problems and describe and confirm results using various representations.

5. Use one mathematical idea to extend understanding of another.

6. Recognize the connections between mathematics and other disciplines, and apply mathematical thinking and problem solving in those areas.

7. Recognize the role of mathematics in their daily lives and in society.

Building upon knowledge and skills gained in the preceding grades, and demonstrating continued progress in Indicators 1, 2, 3, and 4 above, by the end of Grade 8, students:

8. Recognize and apply unifying concepts and processes which are woven throughout mathematics.

9. Use the process of mathematical modeling in mathematics and other disciplines, and demonstrate understanding of its methodology, strengths, and limitations.

10. Apply mathematics in their daily lives and in career-based contexts.

11. Recognize situations in other disciplines in which mathematical models may be applicable, andapply appropriate models, mathematical reasoning, and problem solving to those situations.

Building upon knowledge and skills gained in the preceding grades, and demonstrating continued progress in Indicators 1, 2, 3, 8, 9, 10 and 11 above, by the end of Grade 12, students:

12. Recognize how mathematics responds to the changing needs of society, through the study of the history of mathematics.

Descriptive Statement

Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions, and recognize how those solutions can be applied. Mathematical reasoners are able to reflect on solutions to problems and determine whether or not they make sense. They appreciate the pervasive use and power of reasoning as a part of mathematics.

Cumulative Progress Indicators

By the end of Grade 4, students:

1. Make educated guesses and test them for correctness.

2. Draw logical conclusions and make generalizations.

3. Use models, known facts, properties, and relationships to explain their thinking.

4. Justify answers and solution processes in a variety of problems.

5. Analyze mathematical situations by recognizing and using patterns and relationships.

Building upon knowledge and skills gained in the preceding grades, and demonstrating continued progress in Indicators 2, 3, and 5 above, by the end of Grade 8, students:

6. Make conjectures based on observation and information, and test mathematical conjectures and arguments.

7. Justify, in clear and organized form, answers and solution processes in a variety of problems.

8. Follow and construct logical arguments, and judge their validity.

9. Recognize and use deductive and inductive reasoning in all areas of mathematics.

10. Utilize mathematical reasoning skills in other disciplines and in their lives.

11. Use reasoning rather than relying on an answer-key to check the correctness of solutions to problems.

Building upon knowledge and skills gained in the preceding grades, and especially demonstrating continued progress in Indicators 2, 5, 8, 9, 10, and 11 above, by the end of Grade 12, students:

12. Make conjectures based on observation and information, and test mathematical conjectures, arguments, and proofs.

13. Formulate counter-examples to disprove an argument.

Descriptive Statement

Calculators, computers, manipulatives, and other mathematical tools need to be used by students in both instructional and assessment activities. These tools should be used, not to replace mental math and paper-and-pencil computational skills, but to enhance understanding of mathematics and the power to use mathematics. Historically, people have developed and used manipulatives (such as fingers, base ten blocks, geoboards, and algebra tiles) and mathematical devices (such as protractors, coordinate systems, and calculators) to help them understand and develop mathematics. Students should explore both new and familiar concepts with calculators and computers, but should also become proficient in using technology as it is used by adults, that is, for assistance in solving real-world problems.

Cumulative Progress Indicators

By the end of Grade 4, students:

1. Select and use calculators, software, manipulatives, and other tools based on their utility and limitations and on the problem situation.

2. Use physical objects and manipulatives to model problem situations, and to develop and explain mathematical concepts involving number, space, and data.

3. Use a variety of technologies to discover number patterns, demonstrate number sense, and visualize geometric objects and concepts.

4. Use a variety of tools to measure mathematical and physical objects in the world around them.

5. Use technology to gather, analyze, and display mathematical data and information.

Building upon knowledge and skills gained in the preceding grades, and demonstrating continuedprogress in Indicators 1, 2, 3, 4, and 5 above, by the end of Grade 8, students:

6. Use a variety of technologies to evaluate and validate problem solutions, and to investigate the properties of functions and their graphs.

7. Use computer spreadsheets and graphing programs to organize and display quantitative information and to investigate properties of functions.

Building upon knowledge and skills gained in the preceding grades, and demonstrating continued progress in Indicators 1, 2, 3, 5, and 7 above, by the end of Grade 12, students:

8. Use calculators and computers effectively and efficiently in applying mathematical concepts and principles to various types of problems.

Descriptive Statement

Number sense is defined as an intuitive feel for numbers and a common sense approach to using them. It is a comfort with what numbers represent, coming from investigating their characteristics and using them in diverse situations. It involves an understanding of how different types of numbers, such as fractions and decimals, are related to each other, and how they can best be used to describe a particular situation. Number sense is an attribute of all successful users of mathematics.

Cumulative Progress Indicators

By the end of Grade 4, students:

1. Use real-life experiences, physical materials, and technology to construct meanings for whole numbers, commonly used fractions, and decimals.

2. Develop an understanding of place value concepts and numeration in relationship to counting and grouping.

3. See patterns in number sequences, and use pattern-based thinking to understand extensions of the number system.

4. Develop a sense of the magnitudes of whole numbers, commonly used fractions, and decimals.

5. Understand the various uses of numbers including counting, measuring, labeling, and indicating location.

6. Count and perform simple computations with money.

7. Use models to relate whole numbers, commonly used fractions, and decimals to each other, and to represent equivalent forms of the same number.

8. Compare and order whole numbers, commonly used fractions, and decimals.

9. Explore real-life settings which give rise to negative numbers.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 8, students:

10. Understand money notations, count and compute money, and recognize the decimal nature of United States currency.

11. Extend their understanding of the number system by constructing meanings for integers, rational numbers, percents, exponents, roots, absolute values, and numbers represented in scientific notation.

12. Develop number sense necessary for estimation.

13. Expand the sense of magnitudes of different number types to include integers, rational numbers, and roots.

14. Understand and apply ratios, proportions, and percents in a variety of situations.

15. Develop and use order relations for integers and rational numbers.

16. Recognize and describe patterns in both finite and infinite number sequences involving whole numbers, rational numbers, and integers.

17. Develop and apply number theory concepts, such as primes, factors, and multiples, in real-world and mathematical problem situations.

18. Investigate the relationships among fractions, decimals, and percents, and use all of them appropriately.

19. Identify, derive, and compare properties of numbers.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 12, students:

20. Extend their understanding of the number system to include real numbers and an awareness of other number systems.

21. Develop conjectures and informal proofs of properties of number systems and sets of numbers.

22. Extend their intuitive grasp of number relationships, uses, and interpretations, and develop an ability to work with rational and irrational numbers.

23. Explore a variety of infinite sequences and informally evaluate their limits.

Descriptive Statement

Spatial sense is an intuitive feel for shape and space. It involves the concepts of traditional geometry, including an ability to recognize, visualize, represent, and transform geometric shapes. It also involves other, less formal ways of looking at two- and three-dimensional space, such as paper-folding, transformations, tessellations, and projections. Geometry is all around us in art, nature, and the things we make. Students of geometry can apply their spatial sense and knowledge of the properties of shapes and space to the real world.

Cumulative Progress Indicators

By the end of Grade 4, students:

1. Explore spatial relationships such as the direction, orientation, and perspectives of objects in space, their relative shapes and sizes, and the relations between objects and their shadows or projections.

2. Explore relationships among shapes, such as congruence, symmetry, similarity, and self-similarity.

3. Explore properties of three- and two-dimensional shapes using concrete objects, drawings, and computer graphics.

4. Use properties of three- and two-dimensional shapes to identify, classify, and describe shapes.

5. Investigate and predict the results of combining, subdividing, and changing shapes.

6. Use tessellations to explore properties of geometric shapes and their relationships to the concepts of area and perimeter.

7. Explore geometric transformations such as rotations (turns), reflections (flips), and translations (slides).

8. Develop the concepts of coordinates and paths, using maps, tables, and grids.

9. Understand the variety of ways in which geometric shapes and objects can be measured.

10. Investigate the occurrence of geometry in nature, art, and other areas.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 8, students:

11. Relate two-dimensional and three-dimensional geometry using shadows, perspectives, projections and maps.

12. Understand and apply the concepts of symmetry, similarity and congruence.

13. Identify, describe, compare, and classify plane and solid geometric figures.

14. Understand the properties of lines and planes, including parallel and perpendicular lines and planes, and intersecting lines and planes and their angles of incidence.

15. Explore the relationships among geometric transformations (translations, reflections, rotations, and dilations), tessellations (tilings), and congruence and similarity.

16. Develop, understand, and apply a variety of strategies for determining perimeter, area, surface area, angle measure, and volume.

17. Understand and apply the Pythagorean Theorem.

18. Explore patterns produced by processes of geometric change, relating iteration, approximation, and fractals.

19. Investigate, explore, and describe geometry in nature and real-world applications, using models, manipulatives, and appropriate technology.

Building upon knowledge and skills gained in the preceding grades, and demonstrating continued progress in Indicators 16 and 19 above, by the end of Grade 12, students:

20. Understand and apply properties involving angles, parallel lines, and perpendicular lines.

21. Analyze properties of three-dimensional shapes by constructing models and by drawing and interpreting two-dimensional representations of them.

22. Use transformations, coordinates, and vectors to solve problems in Euclidean geometry.

23. Use basic trigonometric ratios to solve problems involving indirect measurement.

24. Solve real-world and mathematical problems using geometric models.

25. Use inductive and deductive reasoning to solve problems and to present reasonable explanations of and justifications for the solutions.

26. Analyze patterns produced by processes of geometric change, and express them in terms of iteration, approximation, limits, self-similarity, and fractals.

27. Explore applications of other geometries in real-world contexts.

Descriptive Statement

Numerical operations are an essential part of the mathematics curriculum. Students must be able to select and apply various computational methods, including mental math, estimation, paper-and-pencil techniques, and the use of calculators. Students must understand how to add, subtract, multiply, and divide whole numbers, fractions, and other kinds of numbers. With calculators that perform these operations quickly and accurately, however, the instructional emphasis now should be on understanding the meanings and uses of the operations, and on estimation and mental skills, rather than solely on developing paper-and-pencil skills.

Cumulative Progress Indicators

By the end of Grade 4, students:

1. Develop meaning for the four basic arithmetic operations by modeling and discussing a variety of problems.

2. Develop proficiency with and memorize basic number facts using a variety of fact strategies (such as "counting on" and "doubles").

3. Construct, use, and explain procedures for performing whole number calculations in the various methods of computation.

4. Use models to explore operations with fractions and decimals.

5. Use a variety of mental computation and estimation techniques.

6. Select and use appropriate computational methods from mental math, estimation, paper-and-pencil, and calculator methods, and check the reasonableness of results.

7. Understand and use relationships among operations and properties of operations.

Building upon knowledge and skills gained in the preceding grades, and demonstrating continued progress in Indicator 6 above, by the end of Grade 8, students:

8. Extend their understanding and use of arithmetic operations to fractions, decimals, integers, and rational numbers.

9. Extend their understanding of basic arithmetic operations on whole numbers to include powers and roots.

10. Develop, apply, and explain procedures for computation and estimation with whole numbers, fractions, decimals, integers, and rational numbers.

11. Develop, apply, and explain methods for solving problems involving proportions and percents.

12. Understand and apply the standard algebraic order of operations.

Building upon knowledge and skills gained in the preceding grades, and demonstrating continued progress in Indicator 6 above, by the end of Grade 12, students:

13. Extend their understanding and use of operations to real numbers and algebraic procedures.

14. Develop, apply, and explain methods for solving problems involving factorials, exponents, and matrices.

Descriptive Statement

Measurement helps describe our world using numbers. We use numbers to describe simple things like length, weight, and temperature, but also complex things such as pressure, speed, and brightness. An understanding of how we attach numbers to those phenomena, familiarity with common measurement units like inches, liters, and miles per hour, and a practical knowledge of measurement tools and techniques are critical for students' understanding of the world around them.

Cumulative Progress Indicators

By the end of Grade 4, students:

1. Use and describe measures of length, distance, capacity, weight, area, volume, time, and temperature.

2. Compare and order objects according to some measurable attribute.

3. Recognize the need for a uniform unit of measure.

4. Develop and use personal referents for standard units of measure (such as the width of a finger to approximate a centimeter).

5. Select and use appropriate standard and non-standard units of measurement to solve real-life problems.

6. Understand and incorporate estimation and repeated measures in measurement activities.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 8, students:

7. Use estimated and actual measurements to describe and compare phenomena.

8. Read and interpret various scales, including those based on number lines and maps.

9. Determine the degree of accuracy needed in a given situation and choose units accordingly.

10. Understand that all measurements of continuous quantities are approximate.

11. Develop formulas and procedures for solving problems related to measurement.

12. Explore situations involving quantities which cannot be measured directly or conveniently.

13. Convert measurement units from one form to another, and carry out calculations that involve various units of measurement.

14. Understand and apply measurement in their own lives and in other subject areas.

15. Understand and explain the impact of the change of an object's linear dimensions on its perimeter, area, or volume.

16. Apply their knowledge of measurement to the construction of a variety of two- and three-dimensional figures.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 12, students:

17. Use techniques of algebra, geometry, and trigonometry to measure quantities indirectly.

18. Use measurement appropriately in other subject areas and career-based contexts.

19. Choose appropriate techniques and tools to measure quantities in order to achieve specified degrees of precision, accuracy, and error (or tolerance) of measurements.

Descriptive Statement

Estimation is a process that is used constantly by mathematically capable adults, and that can be mastered easily by children. It involves an educated guess about a quantity or a measure, or an intelligent prediction of the outcome of a computation. The growing use of calculators makes it more important than ever that students know when a computed answer is reasonable; the best way to make that decision is through estimation. Equally important is an awareness of the many situations in which an approximate answer is as good as, or even preferable to, an exact answer.

Cumulative Progress Indicators

By the end of Grade 4, students:

1. Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set.

2. Use personal referents, such as the width of a finger as one centimeter, for estimations withmeasurement.

3. Visually estimate length, area, volume, or angle measure.

4. Explore, construct, and use a variety of estimation strategies.

5. Recognize when estimation is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.

6. Determine the reasonableness of an answer by estimating the result of operations.

7. Apply estimation in working with quantities, measurement, time, computation, and problem solving.

Building upon knowledge and skills gained in the preceding grades, and demonstrating continued progress in Indicators 5 and 6 above, by the end of Grade 8, students:

8. Develop, apply, and explain a variety of different estimation strategies in problem situations involving quantities and measurement.

9. Use equivalent representations of numbers such as fractions, decimals, and percents to facilitate estimation.

10. Determine whether a given estimate is an overestimate or an underestimate.

Building upon knowledge and skills gained in the preceding grades, and demonstrating continued progress in Indicator 6 above, by the end of Grade 12, students:

11. Estimate probabilities and predict outcomes from real-world data.

12. Recognize the limitations of estimation, assess the amount of error resulting from estimation, and determine whether the error is within acceptable tolerance limits.

Descriptive Statement

Patterns, relationships, and functions constitute a unifying theme of mathematics. From the earliest age, students should be encouraged to investigate the patterns that they find in numbers, shapes, and expressions, and, by doing so, to make mathematical discoveries. They should have opportunities to analyze, extend, and create a variety of patterns and to use pattern-based thinking to understand and represent mathematical and other real-world phenomena. These explorations present unlimited opportunities for problem-solving, making and verifying generalizations, and building mathematical understanding and confidence.

Cumulative Progress Indicators

By the end of Grade 4, students:

1. Reproduce, extend, create, and describe patterns and sequences using a variety of materials.

2. Use tables, rules, variables, open sentences, and graphs to describe patterns and other relationships.

3. Use concrete and pictorial models to explore the basic concept of a function.

4. Observe and explain how a change in one physical quantity can produce a corresponding change in another.

5. Observe and recognize examples of patterns, relationships, and functions in other disciplines and contexts.

6. Form and verify generalizations based on observations of patterns and relationships.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 8, students:

7. Represent and describe mathematical relationships with tables, rules, simple equations, and graphs.

8. Understand and describe the relationships among various representations of patterns and functions.

9. Use patterns, relationships, and functions to model situations and to solve problems in mathematics and in other subject areas.

10. Analyze functional relationships to explain how a change in one quantity results in a change in another.

11. Understand and describe the general behavior of functions.

12. Use patterns, relationships, and linear functions to model situations in mathematics and in other areas.

13. Develop, analyze, and explain arithmetic sequences.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 12, students:

14. Analyze and describe how a change in an independent variable can produce a change in a dependent variable.

15. Use polynomial, rational, trigonometric, and exponential functions to model real-world phenomena.

16. Recognize that a variety of phenomena can be modeled by the same type of function.

17. Analyze and explain the general properties and behavior of functions, and use appropriate graphing technologies to represent them.

18. Analyze the effects of changes in parameters on the graphs of functions.

19. Understand the role of functions as a unifying concept in mathematics.

Descriptive Statement

Probability and statistics are the mathematics used to understand chance and to collect, organize, describe, and analyze numerical data. From weather reports to sophisticated studies of genetics, from election results to product preference surveys, probability and statistical language and concepts are increasingly present in the media and in everyday conversations. Students need this mathematics to help them judge the correctness of an argument supported by seemingly persuasive data.

Cumulative Progress Indicators

By the end of Grade 4, students:

1. Formulate and solve problems that involve collecting, organizing, and analyzing data.

2. Generate and analyze data obtained using chance devices such as spinners and dice.

3. Make inferences and formulate hypotheses based on data.

4. Understand and informally use the concepts of range, mean, mode, and median.

5. Construct, read, and interpret displays of data such as pictographs, bar graphs, circle graphs,tables, and lists.

6. Determine the probability of a simple event, assuming equally likely outcomes.

7. Make predictions that are based on intuitive, experimental, and theoretical probabilities.

8. Use concepts of certainty, fairness, and chance to discuss the probability of actual events.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 8, students:

9. Generate, collect, organize, and analyze data and represent this data in tables, charts, and graphs.

10. Select and use appropriate graphical representations and measures of central tendency (mean, mode and median) for sets of data.

11. Make inferences and formulate and evaluate arguments based on data analysis and data displays.

12. Use lines of best fit to interpolate and predict from data.

13. Determine the probability of a compound event.

14. Model situations involving probability, such as genetics, using both simulations and theoretical models.

15. Use models of probability to predict events based on actual data.

16. Interpret probabilities as ratios and percents.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 12, students:

17. Estimate probabilities and predict outcomes from actual data.

18. Understand sampling and recognize its role in statistical claims.

19. Evaluate bias, accuracy, and reasonableness of data in real-world contexts.

20. Understand and apply measures of dispersion and correlation.

21. Design a statistical experiment to study a problem, conduct the experiment, and interpret and communicate the outcomes.

22. Make predictions using curve fitting and numerical procedures to interpolate and extrapolate from known data.

23. Use relative frequency and probability, as appropriate, to represent and solve problems involving uncertainty.

24. Use simulations to estimate probabilities.

25. Create and interpret discrete and continuous probability distributions, and understand their application to real-world situations.

26. Describe the normal curve in general terms, and use its properties to answer questions about sets of data that are assumed to be normally distributed.

27. Understand and use the law of large numbers (that experimental results tend to approachtheoretical probabilities after a large number of trials).

Descriptive Statement

Algebra is a language used to express mathematical relationships. Students need to understand how quantities are related to one another, and how algebra can be used to concisely express and analyze those relationships. Modern technology provides tools for supplementing the traditional focus on algebraic techniques, such as solving equations, with a more visual perspective, with graphs of equations displayed on a screen. Students can then focus on understanding the relationship between the equation and the graph, and on what the graph represents in a real-life situation.

Cumulative Progress Indicators

By the end of Grade 4, students:

1. Understand and represent numerical situations using variables, expressions, and number sentences.

2. Represent situations and number patterns with concrete materials, tables, graphs, verbal rules, and number sentences, and translate from one to another.

3. Understand and use properties of operations and numbers.

4. Construct and solve open sentences (example: 3 + ___ = 7) that describe real-life situations.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 8, students:

5. Understand and use variables, expressions, equations, and inequalities.

6. Represent situations and number patterns with concrete materials, tables, graphs, verbal rules, and standard algebraic notation.

7. Use graphing techniques on a number line to model both absolute value and arithmetic operations.

8. Analyze tables and graphs to identify properties and relationships.

9. Understand and use the rectangular coordinate system.

10. Solve simple linear equations using concrete, informal, and graphical methods, as well as appropriate paper-and-pencil techniques.

11. Explore linear equations through the use of calculators, computers, and other technology.

12. Investigate inequalities and nonlinear equations informally.

13. Draw freehand sketches of, and interpret, graphs which model real phenomena.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 12, students:

14. Model and solve problems that involve varying quantities using variables, expressions, equations, inequalities, absolute values, vectors, and matrices.

15. Use tables and graphs as tools to interpret expressions, equations, and inequalities.

16. Develop, explain, use, and analyze procedures for operating on algebraic expressions and matrices.

17. Solve equations and inequalities of varying degrees using graphing calculators and computers as well as appropriate paper-and-pencil techniques.

18. Understand the logic and purposes of algebraic procedures.

19. Interpret algebraic equations and inequalities geometrically, and describe geometric objects algebraically.

Descriptive Statement

Discrete mathematics is the branch of mathematics that deals with arrangements of distinct objects. It includes a wide variety of topics and techniques that arise in everyday life, such as how to find the best route from one city to another, where the objects are cities arranged on a map. It also includes how to count the number of different combinations of toppings for pizzas, how best to schedule a list of tasks to be done, and how computers store and retrieve arrangements of information on a screen. Discrete mathematics is the mathematics used by decision-makers in our society, from workers in government to those in health care, transportation, and telecommunications. Its various applications help students see the relevance of mathematics in the real world.

Cumulative Progress Indicators

By the end of Grade 4, students:

1. Explore a variety of puzzles, games, and counting problems.

2. Use networks and tree diagrams to represent everyday situations.

3. Identify and investigate sequences and patterns found in nature, art, and music.

4. Investigate ways to represent and classify data according to attributes, such as shape or color, and relationships, and discuss the purpose and usefulness of such classification.

5. Follow, devise, and describe practical lists of instructions.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 8, students:

6. Use systematic listing, counting, and reasoning in a variety of different contexts.

7. Recognize common discrete mathematical models, explore their properties, and design them for specific situations.

8. Experiment with iterative and recursive processes, with the aid of calculators and computers.

9. Explore methods for storing, processing, and communicating information.

10. Devise, describe, and test algorithms for solving optimization and search problems.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 12, students:

11. Understand the basic principles of iteration, recursion, and mathematical induction.

12. Use basic principles to solve combinatorial and algorithmic problems.

13. Use discrete models to represent and solve problems.

14. Analyze iterative processes with the aid of calculators and computers.

15. Apply discrete methods to storing, processing, and communicating information.

16. Apply discrete methods to problems of voting, apportionment, and allocations, and use fundamental strategies of optimization to solve problems.

Descriptive Statement

The conceptual building blocks of calculus are important for everyone to understand. How quantities such as world population change, how fast they change, and what will happen if they keep changing at the same rate are questions that can be discussed by elementary school students. Another important topic for all mathematics students is the concept of infinity - what happens as numbers get larger and larger and what happens as patterns are continued indefinitely. Early explorations in these areas can broaden students' interest in and understanding of an important area of applied mathematics.

Cumulative Progress Indicators

By the end of Grade 4, students:

1. Investigate and describe patterns that continue indefinitely.

2. Investigate and describe how certain quantities change over time.

3. Experiment with approximating length, area, and volume, using informal measurement instruments.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 8, students:

4. Recognize and express the difference between linear and exponential growth.

5. Develop an understanding of infinite sequences that arise in natural situations.

6. Investigate, represent, and use non-terminating decimals.

7. Represent, analyze, and predict relations between quantities, especially quantities changing over time.

8. Approximate quantities with increasing degrees of accuracy.

9. Understand and use the concept of significant digits.

10. Develop informal ways of approximating the surface area and volume of familiar objects, and discuss whether the approximations make sense.

11. Express mathematically and explain the impact of the change of an object's linear dimensions on its surface area and volume.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 12, students:

12. Develop and use models based on sequences and series.

13. Develop and apply procedures for finding the sum of finite arithmetic series and of finite and infinite geometric series.

14. Develop an informal notion of limit.

15. Use linear, quadratic, trigonometric, and exponential models to explain growth and change in the natural world.

16. Recognize fundamental mathematical models (such as polynomial, exponential, and trigonometric functions) and apply basic translations, reflections, and dilations to their graphs.

17. Develop and explain the concept of the slope of a curve and use that concept to discuss the information contained in graphs.

18. Develop an understanding of the concept of continuity of a function.

19. Understand and apply approximation techniques to situations involving initial portions of infinite decimals and measurement.

Descriptive Statement

High expectations for all students form a critical part of the learning environment. The belief of teachers, administrators, and parents that a student can and will succeed in mathematics often makes it possible for that student to succeed. Beyond that, this standard calls for a commitment that all students will be continuously challenged and enabled to go as far mathematically as they can.

Cumulative Progress Indicators

By the end of Grade 12, students:

1. Study a core curriculum containing challenging ideas and tasks, rather than one limited to repetitive, low-level cognitive activities.

2. Work at rich, open-ended problems which require them to use mathematics in meaningful ways, and which provide them with exciting and interesting mathematical experiences.

3. Recognize mathematics as integral to the development of all cultures and civilizations, and in particular to that of our own society.

4. Understand the important role that mathematics plays in their own success, regardless of career.

5. Interact frequently with parents and other members of their communities, including men and women from a variety of cultural backgrounds, who use mathematics in their daily lives and occupations.

6. Receive services that help them understand the mathematical skills and concepts necessary to assure success in the core curriculum.

7. Receive equitable treatment without regard to gender, ethnicity, or predetermined expectations for success.

8. Learn mathematics in classes which reflect the diversity of the school's total student population.

9. Be provided with opportunities at all grade levels for further study of mathematics, especially including topics beyond traditional computation, algebra, and geometry.

10. Be challenged to maximize their mathematical achievements at all grade levels.

11. Experience a full program of meaningful mathematics so that they can pursue post-secondary education.

(This "learning environment standard" was developed and approved by the task force that prepared the Mathematics Standards and appears in the Introduction to the Mathematics Standards chapter of the New Jersey State Department of Education's Core Curriculum Content Standards; however, since it was not considered a "content standard," it was not presented to the New Jersey State Board of Education for adoption.)

Descriptive Statement

Engagement in mathematics should be expected of all students, and the learning environment should be one where students are actively involved in doing mathematics. Challenging problems should be posed and students should be expected to work on them individually and in groups, sometimes for extended periods of time, and sometimes on unfamiliar topics. They should be encouraged to develop traits and strategies - such as perseverance, cooperative work skills, decision-making, and risk-taking - which will be key to their success in mathematics.

Cumulative Progress Indicators

Experiences will be such that all students:

1. Demonstrate confidence as mathematical thinkers, believing that they can learn mathematics and can achieve high standards in mathematics, and accepting responsibility for their own learning of mathematics.

2. Recognize the power that comes from understanding and doing mathematics.

3. Develop and maintain a positive disposition to mathematics and to mathematical activity.

4. Participate actively in mathematical activity and discussion, freely exchanging ideas and problem-solving strategies with their classmates and teachers, and taking intellectual risks and defending positions without fear of being incorrect.

5. Work cooperatively with other students on mathematical activities, actively sharing, listening, and reflecting during group discussions, and giving and receiving constructive criticism.

6. Make conjectures, pose their own problems, and devise their own approaches to problem solving.

7. Assess their work to determine the effectiveness of their strategies, make decisions about alternate strategies to pursue, and persevere in developing and applying strategies for solving a problem in situations where the method and path to the solution are not at first apparent.

8. Assess their work to determine the correctness of their results, based on their own reasoning, rather than relying solely on external authorities.

(This "learning environment standard" was developed and approved by the task force that prepared the Mathematics Standards and appears in the Introduction to the Mathematics Standards chapter of the New Jersey State Department of Education's Core Curriculum Content Standards; however, since it was not considered a "content standard," it was not presented to the New Jersey State Board of Education for adoption.)

Descriptive Statement

A variety of assessment instruments should be used to enable the teacher to monitor students' progress in understanding mathematical concepts and in developing mathematical skills. Assessment of mathematical learning should not be confined to intermittent standardized tests. The learning environment should embody the perspective that the primary function of assessment is to improve learning.

Cumulative Progress Indicators

Experiences will be such that all students:

1. Are engaged in assessment activities that function primarily to improve learning.

2. Are engaged in assessment activities based upon rich, challenging problems from mathematics and other disciplines.

3. Are engaged in assessment activities that address the content described in all of New Jersey's Mathematics Standards.

4. Demonstrate competency through varied assessment methods including, but not limited to, individual and group tests, authentic performance tasks, portfolios, journals, interviews, seminars, and extended projects.

5. Engage in ongoing assessment of their work to determine the effectiveness of their strategies and the correctness of their results.

6. Understand and accept that the criteria used to evaluate their performance will be based on high expectations.

7. Recognize errors as part of the learning process and use them as opportunities for mathematical growth.

8. Select and use appropriate tools effectively during assessment activities.

9. Reflect upon and communicate their mathematical understanding, knowledge, and attitudes.