The New Jersey Math Standards - April 95

The New Jersey Mathematics Standards

1. All students will be expected to demonstrate high levels of mathematical thought and will have opportunities for further study in mathematics at all grade levels, including courses beyond traditional geometry and algebra in high school, so that they are continuously challenged to maximize their achievement.

2. All students' mathematical learning will embody the notion that mathematics is a dynamic human activity where decision making, risk taking, cooperative work, perseverance, self-assessment, and self-confidence are frequently keys to success.

3. Computers, calculators, manipulatives, and other tools of learning will be regularly and routinely used by all students as an integral part of both instructional and assessment activities to enhance their understanding and command of mathematics.

4. A broad diversity of assessment tools and strategies will be used to provide multiple indicators both of the quality of every student's mathematical learning and of overall program effectiveness.

5. Posing and solving problems will be a major focus of all students' mathematical activity so that through working with interesting, engaging, and intellectually stimulating situations, they come to understand mathematics and use it effectively.

6. Communication will be a major focus of all students' mathematical activity so that through speaking, listening, writing, reading, and visually representing, they come to understand mathematics and use it effectively.

7. Making connections will be a major focus of all students' mathematical activity so they come to develop an understanding and appreciation for the interrelationship of mathematical ideas and of the powerful roles that mathematics and mathematical modeling play in other disciplines.

8. Reasoning will be a major focus of all students' mathematical activity so that they come to develop higher-order thinking skills and mathematical habits of the mind, and become confident, self-reliant mathematical thinkers.

9. All students will develop their number sense through experiences which enable them to investigate the characteristics and relationships of numbers, represent numbers in a variety of forms, and use numbers in diverse situations.

10. All students will develop their spatial sense through experiences which enable them to recognize, visualize, represent, and transform geometric shapes and to apply their knowledge of geometric properties, relationships and models to other areas of mathematics and to the physical world.

11. All students will develop their understanding of numerical operations through experiences which enable them to construct, explain, select, and apply various methods of computation including mental math, estimation, and the use of calculators, with a reduced role for paper-and-pencil techniques.

12. All students will develop their understanding of measurement and systems of measurement through experiences which enable them to use a variety of techniques, tools, and units of measurement to describe and analyze quantifiable phenomena.

13. All students will develop their understanding of estimation through experiences which enable them to recognize many different situations in which estimation is appropriate and to use a variety of effective strategies.

14. All students will develop their understanding of patterns, relationships, and functions through experiences which enable them to discover, analyze, extend, and create a wide variety of patterns and to use pattern-based thinking to understand and represent mathematical and other real-world phenomena.

15. All students will develop their understanding of probability and statistics through experiences which enable them to systematically collect, organize, and describe sets of data, to use probability to model situations involving random events, and to make inferences and arguments based on analysis of data and mathematical probabilities.

16. All students will develop their understanding of algebraic concepts and processes through experiences which enable them to describe, represent, and analyze relationships among variable quantities and to apply algebraic methods to solve meaningful problems.

17. All students will develop their understanding of the concepts and applications of discrete mathematics through experiences which enable them to use a variety of tools of contemporary mathematics to explore and model a variety of real-world situations.

18. All students will develop their understanding of the conceptual underpinnings of calculus through experiences which enable them to describe and analyze how various quantities change, to build informal concepts of infinity and limits, and to use these concepts to model, describe, and analyze natural phenomena.

Learning Environment Standards and Expectations

Experiences will be such that all students:
A	have the opportunity to study a core curriculum containing challenging ideas and tasks rather than one that dwells on repetitive and low-level cognitive activities.
B	are challenged by rich, open-ended problems which require them to use mathematics in meaningful ways and which provide them with exciting and interesting mathematical experiences.
C	learn mathematics in classrooms which reflect the vision of the New Jersey Mathematics Standards.
D	see mathematics as integral to the development of all cultures and civilizations, and in particular to that of our own society.
E	understand the important role that mathematics plays in enhancing their own success, no matter what direction their careers take.
F	frequently interact with parents and other members of their communities, both men and women from a variety of cultural backgrounds, who use mathematics in their daily lives and occupations.
G	receive encouragement and reinforcement for their actual mathematical accomplishments, rather than discouragement and separation resulting from their not achieving mastery at the same rate as their peers.
H	are provided with services that help them understand the mathematical skills and concepts necessary to assure success in the core curriculum.
I	receive equitable treatment without regard to gender, ethnicity, or predetermined expectations for success.
J	learn mathematics in classes which reflect the diversity of the total school student population.
K	learn mathematics by working with well prepared mathematics faculty whose composition and assignments reflect the gender and ethnic diversity of New Jersey.
L	are provided with opportunities at all grade levels for further study of mathematics
M	are challenged to maximize their mathematical achievements at all grade levels.
N	actively experience a full program of meaningful mathematics so that they can pursue post-secondary education.
O	are provided with opportunities to study mathematical topics beyond traditional geometry and algebra in high school, including calculus, discrete mathematics, probability, statistics, and calculus beyond the expectations in the New Jersey Mathematics Standards.

Experiences will be such that all students:
A	take intellectual risks, being willing to clarify and defend positions without the fear of being incorrect.
B	gain confidence through successful experiences in mathematics and thereby develop a positive disposition toward doing mathematics.
C	regularly self-assess to determine the effectiveness of their strategies, the correctness of their results, and to monitor their attitudes toward learning mathematics.
D	learn to accept responsibility for their own learning.
E	use time effectively and efficiently in order to further their understanding of mathematics.
F	actively participate and recognize that their participation is an integral part of the learning process.
G	interact with others in a variety of settings, including cooperative learning groups, to enhance understanding.
H	respond to open-ended questions, freely exchange ideas and problem-solving strategies with their classmates, and participate in discussions that illustrate their higher-order thinking.
I	exhibit learning styles which are accommodated through a variety of instructional methods.
J	recognize related ideas within mathematics as well as those in other subject areas.
K	make conjectures, pose their own problems, devise their own approaches to problem-solving, and use their results for informed decision-making.
L	spend the time and use the tools needed for mathematical exploration and discovery.
M	understand and appreciate the history of mathematics and that people are continually developing new mathematics and new ways of learning mathematics.

Experiences will be such that all students:
A	learn to select and use different calculators, software, manipulatives, and other tools based on their capabilities and limitations and on the problem situation.
B	use physical objects and manipulatives at all grade levels to model problem situations and to develop and explain mathematical concepts involving number, space, and data.
C	use a variety of tools to measure characteristics and attributes of mathematical and physical objects in the world around them.
D	have access to all these tools as much as possible, for instructional and assessment activities in school, and for work at home.
E	use a variety of technologies to explore number patterns, develop number sense, explore geometric concepts, formulate and solve problems, evaluate and validate solutions, explore and analyze data, and investigate properties of functions through their graphs.
F	use computer spreadsheets and graphing programs to organize and display quantitative information and to investigate properties of functions.
G	program calculators and computers so that they can be used effectively and efficiently to assist in applying mathematical concepts and principles to various types of problems.
H	use emerging technology, as a result of regular evaluation and upgrading of equipment.

Experiences will be such that all students:
A	demonstrate competency through varied assessment alternatives including, but not limited to, individual and group tests, authentic performance tasks, portfolios, seminars, interviews, and extended projects.
B	regularly self-assess to determine the effectiveness of their strategies and the reasonableness of their results.
C	reflect upon and communicate their attitude toward learning mathematics.
D	understand expectations for, and criteria for evaluation of, performance.
E	use errors as a springboard for further development.
F	develop the ability to give and receive constructive criticism.
G	use computational tools effectively during assessment.

All students will experience a mathematics program in which:
H	the main function of assessment is to improve learning.
I	all aspects of mathematics are assessed.
J	rich, challenging problems from mathematics and other disciplines form the basis for assessment.
K	measures used for program evaluation, such as standardized tests, New Jersey statewide assessments, and other external tests, reflect these standards.

Process Standards and Expectations

Experiences will be such that all students:
A	use discovery-oriented, inquiry-based, and problem-centered approaches to investigate and understand mathematics.
B	recognize, formulate, clarify, and engage in solving problems arising from mathematical situations, everyday experiences, applications to other disciplines, and real-world applications.
C	construct and use concrete, pictorial, symbolic, and graphical models to represent problem situations.
D	determine, collect, organize, and analyze data needed to solve problems.
E	effectively apply processes of mathematical modeling in mathematics and other areas.
F	construct, explain, justify, and apply a variety of problem-solving strategies in both independent and cooperative learning environments.
G	choose and apply appropriate mathematical tools, including technology, as a natural and routine part of the problem-solving process.
H	recognize that there may be many ways to solve a problem, weigh their relative merits, and select and use appropriate ones.
I	pose, explore, persist at, and solve a rich variety of problems, including non-routine problems, multi-step "story" problems, and open-ended problems with several solutions and/or solution strategies.
J	develop flexibility, persistence, and the ability to self-monitor progress toward problem solution.
K	explore the validity and efficiency of various problem-posing and problem-solving strategies and develop alternative strategies and generalizations as needed.
L	verify the correctness and reasonableness of results and interpret them in the context of the problem being solved.

Experiences will be such that all students:
A	understand and appreciate that discussing, listening, representing, reading, and writing are vital parts of learning and using mathematics.
B	model situations and discuss mathematical ideas using oral, written, concrete, pictorial, and graphical, and algebraic methods.
C	use a variety of technologies, such as computers, calculators, video, CD-ROM, and laser disc, to represent and communicate their mathematical ideas.
D	identify and explain key mathematical concepts using speech, writing, pictures, diagrams, and physical representations.
E	use mathematical language and symbols to represent problem situations and explain what they have done.
F	understand and appreciate the economy and power of mathematical symbolism and its role in the development of mathematics.
G	engage regularly in mathematical brainstorming and discussions to empower them to ask questions, make conjectures, and suggest strategies for solving problems.
H	explain their conclusions, thought processes, and strategies in problem-solving situations.
I	understand, explain, analyze, and evaluate mathematical arguments and conclusions presented by others.
J	formulate questions, conjectures, definitions, and generalizations about data, information, and problem situations.
K	reflect on and clarify their thinking so that they can present convincing arguments for their conclusions.

Experiences will be such that all students:
A	view mathematics as an integrated whole.
B	link conceptual and procedural knowledge.
C	understand and use various representations of concepts, and connect them to one another.
D	explore problems and describe and confirm results using various representations including graphical, physical, and verbal mathematical models.
E	use models, calculators and other technologies to connect different representations of mathematical concepts.
F	recognize and apply unifying concepts and processes which are woven throughout mathematics.
G	use one mathematical idea to extend their understanding of another mathematical idea.
H	recognize the connections between mathematics and other disciplines and apply mathematical thinking and problem-solving in those areas.
I	recognize situations in other disciplines in which probabilistic, statistical, geometric, algebraic, optimization, discrete mathematical, and analytic models may be applicable, and apply appropriate models to those situations.
J	use the process of mathematical modeling appropriately, and are aware of the methodology, strengths, and limitations of mathematical modeling.
K	recognize the role of mathematics in their daily lives, in careers, and in society.
L	apply mathematics in their daily lives and in career-based contexts.
M	recognize the evolutionary, dynamic, and human nature of mathematics and how it responds to the changing needs of society.

Experiences will be such that all students:
A	formulate and test mathematical conjectures and arguments.
B	draw logical conclusions and make generalizations.
C	use models, known facts, properties, and relationships to explain their thinking.
D	justify, in clear and organized form, answers and solution processes in a variety of problems.
E	follow logical arguments, construct simple, valid arguments, and judge the validity of arguments.
F	analyze mathematical situations by recognizing and using patterns and relationships.
G	recognize and apply deductive and inductive reasoning.
H	appreciate the pervasive use and power of reasoning as a part of mathematics.
I	develop the habit of monitoring and validating their own thinking.
J	utilize their mathematical reasoning skills in other disciplines in their lives.

Content Standards and Expectations

Experiences will be such that all students:
in grades K-4
A	use real-life experiences, physical materials, and technology to construct meanings for whole numbers, commonly used fractions, and decimals.
B	develop understanding of place value concepts and numeration through their relationships to counting and grouping.
C	see patterns in number sequences and use pattern-based thinking to understand extensions of the number system.
D	develop a sense of the magnitudes of whole numbers, commonly used fractions, and decimals.
E	understand the various uses of numbers including numbers used as counts, as measures, as labels, and as indicators of location.
F	use models to relate commonly used fractions, decimals, and whole numbers to each other and to represent equivalent forms of the same number.
G	compare and order whole numbers, commonly used fractions, and decimals.
H	explore real-life settings which give rise to instances of negative numbers.

building upon the K-4 expectations, in grades 5-8
I	understand money notations, can count and compute with different amounts of money, and recognize the decimal nature of United States currency.
J	extend their understanding of the number system by constructing meanings for integers, rational numbers, percents, exponents, roots, absolute value, and scientific notation.
K	develop number sense necessary for estimation.
L	expand the sense of magnitudes of different number types to include integers, rational numbers, and roots.
M	understand and apply ratios, proportions, and percents in a wide variety of situations.
N	develop and use order relations for integers and rational numbers.
O	recognize and describe patterns in both finite and infinite number sequences involving whole numbers, rational numbers, and integers.
P	develop and apply number theory concepts (e.g., primes, factors, multiples) in real-world and mathematical problem situations.
Q	investigate the relationships among fractions, decimals, and percents and use all of them appropriately.
R	identify, derive, and compare properties of numbers.
S	establish an intuitive grasp of number relationships, uses, and interpretations.

building upon the K-8 expectations, in grades 9-12
T	extend their understanding of the number system to include real numbers and an awareness of other number systems.
U	develop conjectures and intuitive proofs of properties of number systems and sets of numbers.
V	extend their intuitive grasp of number relationships, uses, and interpretations and develop an ability to work flexibly and confidently with rational and irrational numbers.
W	explore a variety of infinite sequences and informally evaluate their limits.

Experiences will be such that all students:
in grades K-4
A	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.
B	explore relationships among shapes such as congruence, symmetry, similarity, and self-similarity.
C	explore properties of three- and two-dimensional shapes using concrete objects, drawings, and computer graphics.
D	use properties of three- and two-dimensional shapes to identify, classify, and describe shapes.
E	investigate and predict the results of combining, subdividing, and changing shapes.
F	use tessellations to explore properties of geometric shapes and their relationships to the concepts of area and perimeter.
G	explore geometric transformations: rotations (turns), reflections (flips), and translations (slides).
H	develop the concepts of coordinates and paths, using maps, tables, and grids.
I	understand the variety of ways in which geometric shapes and objects can be measured.
J	investigate the occurrence of geometry in nature, art, and other areas.

building upon the K-4 expectations, in grades 5-8
K	relate two-dimensional and three-dimensional geometry using shadows, perspectives, projections and maps.
L	understand and apply the concept of symmetry, similarity and congruence.
M	identify and describe plane and solid geometric figures, characterize geometric figures using a minimum set of properties, classify geometric figures according to common properties, and develop definitions for common geometric figures.
N	understand the properties of lines and planes, including parallel lines and planes, perpendicular lines and planes, intersecting lines and planes and their angles of incidence.
O	explore the relationships among geometric transformations (translations, reflections, rotations, and dilations), tessellations (tilings), and congruence and similarity.
P	develop, understand, and apply a variety of strategies for determining perimeter, area, surface area, angle measure, and volume.
Q	understand and apply the Pythagorean Theorem.
R	explore patterns produced by processes of geometric change, intuitively relating iteration, approximation, and fractals.
S	investigate, explore, and describe the geometry in nature and real-world applications.
T	use models, manipulatives, and computers graphics software to build a strong conceptual understanding of geometry and its connections to other parts of mathematics, science, and art.

building upon the K-8 expectations, in grades 9-12
U	understand and apply properties involving angles, parallel lines, and perpendicular lines.
V	analyze properties of three-dimensional shapes by constructing models and by drawing and interpreting two-dimensional representations of them.
W	use transformations, coordinates, and vectors to solve problems in Euclidean geometry.
X	interpret algebraic equations and inequalities geometrically and describe geometric objects algebraically.
Y	extend, apply, and formalize strategies for determining perimeters, areas, volumes, and surface areas.
Z	use trigonometric ratios to solve problems involving indirect measurement.
A	solve real-world and mathematical problems using geometric models.
B	use induction or deduction to solve problems and to present reasonable explanations of and justifications for the solutions.
CC	analyze patterns produced by processes of geometric change, formally connecting iteration, approximation, limits, self-similarity, and fractals.
DD	explore applications of other geometries in real-world contexts.
EE	use manipulatives, computer graphics software, and other learning tools to demonstrate geometric concepts and connections with other parts of mathematics, science, and art.

Experiences will be such that all students:
in grades K-4
A	develop meaning for the four basic operations by modeling and discussing a variety of problems.
B	develop proficiency with basic facts through the use of a variety of strategies.
C	construct, use, and explain procedures for performing whole number calculations in the various methods of computation.
D	use models to explore operations with fractions and decimals.
E	select and use an appropriate method for computing from among mental math, estimation, paper-and-pencil, and calculator methods and check the reasonableness of results.
F	use a variety of mental computation and estimation techniques.
G	understand and use relationships among operations and properties of operations.

building upon the K-4 expectations, in grades 5-8
H	select and use an appropriate method for computing from among mental math, estimation, paper-and-pencil, and calculator methods and check the reasonableness of results.
I	extend their understanding and use of arithmetic operations to fractions, decimals, integers, and rational numbers.
J	extend their understanding of basic arithmetic operations on whole numbers to include powers and roots.
K	develop, analyze, apply, and explain procedures for computation and estimation with whole numbers, fractions, decimals, integers, and rational numbers.
L	develop, analyze, apply, and explain methods for solving problems involving proportions and percents.
M	develop, analyze, and explain arithmetic sequences.
N	understand and apply the standard algebraic order of operations.

building upon the K-8 expectations, in grades 9-12
O	select and use an appropriate method for computing from among mental math, estimation, paper-and-pencil, and calculator methods and check the reasonableness of results.
P	extend their understanding and use of operations to real numbers and algebraic procedures.
Q	develop, analyze, apply, and explain methods for solving problems involving factorials, exponents, and matrices.

Experiences will be such that all students:
in grades K-4,
A	use and describe measures of length, distance, capacity, weight, area, volume, time, and temperature.
B	compare and order objects according to some measurable attribute.
C	recognize the need for a uniform unit of measure.
D	develop and use personal referents for standard units of measure (e.g., width of a finger is approximately one centimeter).
E	select and use appropriate standard and non-standard units of measurement to solve real-life problems.
F	understand and incorporate estimation and repeated measures in measurement activities.
G	integrate measurement activities across the curriculum.

building upon the K-4 expectations, in grades 5-8
H	estimate, make, and use measurements to describe and compare phenomena.
I	read and interpret various scales including those based on number lines and maps.
J	determine the degree of accuracy needed in a given situation and choose units accordingly.
K	understand that all measurement of continuous quantities is approximate.
L	develop formulas and procedures for solving problems related to measurement.
M	explore situations involving quantities which cannot be measured directly or conveniently.
N	convert measurement units from one form to another and carry out calculations that involve various units of measurement.
O	understand and apply measurement in their own lives and in interdisciplinary situations.
P	understand and explain the impact of the change of an object's linear dimensions on its perimeter, area, or volume.

building upon the K-8 expectations, in grades 9-12
Q	apply their knowledge of measurement to the construction of a variety of two- and three-dimensional figures.
R	determine the degree of accuracy of a measurement, for example by understanding and using significant digits.
S	develop and use the concept of indirect measurement, and use techniques of algebra, geometry, and trigonometry to measure quantities indirectly.
T	use measurement appropriately in other subject areas and career-based contexts.
U	choose appropriate techniques and tools to measure quantities in order to achieve specified degrees of precision, accuracy, and error (or tolerance) of measurements.

Experiences will be such that all students:
in grades K-4
A	judge, without counting, whether a set of objects has less than, more than, or the same number of objects as a reference set.
B	use personal referents, such as the width of a finger being about one centimeter, for estimations with measurement.
C	visually estimate length, area, volume, or angle measure.
D	explore, construct, and use a variety of estimation strategies.
E	recognize when estimation is appropriate and understand the usefulness of an estimate as distinct from an exact answer.
F	determine the reasonableness of an answer by estimating the result of operations.
G	apply estimation in working with quantities, measurement, computation, and problem-solving.

building upon the K-4 expectations, in grades 5-8
H	develop, apply, and explain a variety of different estimation strategies in problem situations involving quantities and measurement.
I	develop flexibility in the use of equivalent forms of numbers to facilitate estimation.
J	use estimation to predict outcomes and determine the reasonableness of results.
K	recognize situations in which an estimate is more appropriate than an exact answer.
L	determine whether a given estimate is an overestimate or an underestimate.

building upon the K-8 expectations, in grades 9-12
M	determine the reasonableness of answers to problems solved using pencil-and-paper techniques, mental math, algebraic formulas and equations, computers, or calculators.
N	estimate probabilities and measures of central tendency and predict outcomes from real-world data.
O	recognize the limitations of estimation and assess the amount of error resulting from estimation.
P	determine whether errors resulting from estimation are within acceptable tolerance limits.

Experiences will be such that all students:
in grades K-4
A	reproduce, extend, create, and describe patterns and sequences using a variety of materials.
B	explore the use of tables, rules, variables, open sentences, and graphs to describe patterns and other relationships.
C	use concrete and pictorial models to explore the basic concept of a function.
D	observe and explain how a change in one quantity can produce a change in another.
E	observe and appreciate examples of patterns, relationships, and functions in other disciplines and contexts.
F	form and verify generalizations based on observations of patterns and relationships.

building upon the K-4 expectations, in grades 5-8
G	represent and describe mathematical relationships with tables, rules, simple equations, and graphs.
H	understand and describe the relationships among various representations of patterns and functions.
I	use patterns, relationships, and functions to represent and solve problems.
J	analyze functional relationships to explain how a change in one quantity results in a change in another.
K	understand and describe the general behavior of functions.
L	use patterns, relationships, and linear functions to model situations in mathematics and in other areas.

building upon the K-8 expectations, in grades 9-12
M	analyze and describe how a change in the independent variable can produce a change in a dependent variable.
N	use polynomial, rational, trigonometric, and exponential functions to model real-world phenomena.
O	understand and appreciate that a variety of phenomena can be modeled by the same type of function.
P	analyze and explain the general properties and behavior of functions and use appropriate graphing technologies to represent them.
Q	analyze the effects of changes in parameters on the graphs of functions.
R	understand the role of functions as a unifying concept in mathematics.

Experiences will be such that all students can:
in grades K-4
A	collect, organize, and analyze data.
B	generate and analyze data obtained using chance devices such as spinners and dice.
C	make inferences and formulate hypotheses based on data.
D	understand and informally use the concepts of range, mean, mode, and median.
E	construct, read, and interpret displays of data such as pictographs, bar, and circle graphs.
F	formulate and solve problems that involve collecting and analyzing data.
G	determine the probability of a simple event assuming equally likely outcomes.
H	make predictions that are based on intuitive, experimental, and theoretical probabilities.
I	develop intuition about the probability of various events in the real world.

building upon the K-4 expectations, in grades 5-8
J	generate, collect, organize, and analyze data and represent this data in tables, charts, and graphs.
K	understand and apply measures of central tendency.
L	select appropriate graphical representations and measures of central tendency for sets of data.
M	make inferences and formulate and evaluate arguments based on data analysis and data displays.
N	use lines of best fit to interpolate and predict from data.
O	determine the probability of a compound event.
P	model probabilistic situations, such as genetics, using both simulations and theoretical models.
Q	use probabilistic models to predict events based on real-world data.
R	interpret probabilities as ratios and percents.

building upon the K-8 expectations, in grades 9-12
S	estimate probabilities and predict outcomes from real-world data.
T	understand sampling and recognize its role in statistical claims.
U	understand and apply measures of dispersion and correlation.
V	design a statistical experiment to study a problem, conduct the experiment, and interpret and communicate the outcomes.
W	use curve fitting to interpolate and predict from data.
X	use relative frequency and probability, as appropriate, to represent and solve problems involving uncertainty.
Y	use simulations to estimate probabilities.
Z	create and interpret discrete and continuous probability distributions and understand their application to real-world situations.
AA	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.
BB	make predictions based on extrapolation and interpolation of data.
CC	understand and use the law of large numbers.

Experiences will be such that all students:
in grades K-4
A	understand and represent numerical situations using variables, expressions, equations, and inequalities.
B	represent situations and number patterns with concrete materials, tables, graphs, verbal rules, and equations, and translate from one to another.
C	understand and use properties of operations and numbers.
D	construct and solve open sentences (e.g., 3 + ___ = 7) that describe real-life situations.

building upon the K-4 expectations, in grades 5-8
E	understand and use literal variables, expressions, equations, and inequalities.
F	represent situations and number patterns with concrete materials, tables, graphs, verbal rules, and standard algebraic notation.
G	use graphing techniques to show the relationship between distance on a number line and arithmetic operations and absolute value for rational numbers.
H	analyze tables and graphs to identify properties and relationships.
I	understand and use the rectangular coordinate system.
J	solve simple linear equations using concrete, informal, and graphical methods.
K	explore linear equations through the use of calculators, computers, and other technology.
L	investigate inequalities and nonlinear equations informally.
M	draw freehand sketches of and interpret graphs which model real phenomena.

building upon the K-8 expectations, in grades 9-12
N	model and solve problems that involve varying quantities with variables, expressions, equations, inequalities, absolute values, vectors, and matrices.
O	use tables and graphs as tools to interpret expressions, equations, and inequalities.
P	develop, explain, use, and analyze procedures for operating on algebraic expressions and matrices.
Q	solve equations and inequalities of varying degrees using graphing calculators and computers as well as appropriate paper-and-pencil techniques.
R	understand the logic and purposes of algebraic procedures.
S	interpret algebraic equations and inequalities geometrically and describe geometric objects algebraically.

Experiences will be such that all students:
in grades K-4
A	play and explore a variety of puzzles, games, and counting problems.
B	use graphs and other discrete mathematical models to represent everyday situations.
C	identify and investigate sequences and patterns found in nature, art, and music.
D	investigate ways to represent and classify data according to attributes like shape or color and relationships, and discuss their purpose and usefulness.
E	follow, devise, and describe practical algorithmic procedures.

building upon the K-4 expectations, in grades 5-8
F	use systematic listing, counting, and reasoning in a variety of different contexts.
G	recognize discrete mathematical models that occur frequently, explore their properties, and design them for specific situations.
H	experiment with iterative and recursive processes, with the aid of calculators and computers.
I	explore methods for storing, processing, and communicating information.
J	devise, describe, and test algorithms for solving optimization and search problems.

building upon the K-8 expectations, in grades 9-12
K	understand and use basic principles, including permutations and combinations and mathematical induction and recursion, to solve combinatorial and algorithmic problems.
L	use discrete models to represent and solve problems.
M	analyze iterative processes, with the aid of calculators and computers.
N	understand the application of discrete methods to storing, processing, and communicating information.
O	understand the application of discrete methods to problems of social choice and management, and use fundamental strategies of optimization to solve problems.

Experiences will be such that all students:
in grades K-4
A	investigate and describe patterns that continue indefinitely.
B	investigate and describe how certain quantities change over time.
C	experiment with approximating length, area, and volume using informal measurement instruments.

building upon the K-4 expectations, in grades 5-8
D	recognize and express the difference between linear and exponential growth.
E	develop an understanding of infinite sequences that arise in natural situations.
F	investigate, represent, and use non-terminating decimals.
G	represent, analyze, and predict relations between quantities, especially quantities changing over time.
H	approximate quantities with increasing degrees of accuracy.
I	understand and use the concept of significant digits.
J	develop informal ways of approximating the surface area and volume of familiar objects and discuss whether the approximations make sense.
K	express mathematically and explain the impact of the change of an object's linear dimensions on its surface area and volume.

building upon the K-8 expectations, in grades 9-12
L	develop and use models based on sequences and series.
M	develop and apply procedures for finding the sum of finite arithmetic series and of finite and infinite geometric series.
N	develop an informal notion of limit.
O	use linear, quadratic, trigonometric, and exponential models to explain growth and change in the natural world.
P	recognize fundamental mathematical models (such as polynomial, exponential, and trigonometric functions) and apply basic translations, reflections, and dilations to their graphs.
Q	develop the concept of the slope of a curve, apply slopes to measure the steepness of curves, interpret the meaning of the slope of a curve for a given graph, and use the slope to discuss the information contained in the graph.
R	develop an understanding of the concept of continuity of a function.
S	understand and apply approximation techniques to situations involving initial portions of infinite decimals and measurement.