NEW JERSEY'S MATHEMATICS STANDARDS*
Descriptive Statements and Cumulative Progress Indicators
STANDARD 1* 
All students will develop the ability to pose and solve
mathematical problems in mathematics, other disciplines, and
everyday experiences. 

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 problemsolving skills,
students will come to realize the potential usefulness of mathematics
in their lives.
Cumulative Progress Indicators
By the end of Grade 4, students:
 Use discoveryoriented, inquirybased, and problemcentered
approaches to investigate and understand mathematical content
appropriate to early elementary grades.
 Recognize, formulate, and solve problems arising from
mathematical situations and everyday experiences.
 Construct and use concrete, pictorial, symbolic, and graphical
models to represent problem situations.
 Pose, explore, and solve a variety of problems, including
nonroutine problems and openended problems with several solutions
and/or solution strategies.
 Construct, explain, justify, and apply a variety of
problemsolving strategies in both cooperative and independent
learning environments.
 Verify the correctness and reasonableness of results and
interpret them in the context of the problems being solved.
 Know when to select and how to use gradeappropriate
mathematical tools and methods (including manipulatives, calculators
and computers, as well as mental math and paperandpencil techniques)
as a natural and routine part of the problemsolving process.
 Determine, collect, organize, and analyze data needed to solve
problems.
 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:
 Use discoveryoriented, inquirybased, and problemcentered
approaches to investigate and understand mathematical content
appropriate to the middle grades.
 Recognize, formulate, and solve problems arising from
mathematical situations, everyday experiences, and applications to
other disciplines.
 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.
 Recognize that there may be multiple ways to solve a
problem, weigh their relative merits, and select and use appropriate
problemsolving strategies.
 Persevere in developing alternative problemsolving
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:
 Use discoveryoriented, inquirybased, and problemcentered
approaches to investigate and understand the mathematical content
appropriate to the high school grades.
 Recognize, formulate, and solve problems arising from
mathematical situations, everyday experiences, applications to other
disciplines, and career applications.
 Monitor their own progress toward problem solutions.
 Explore the validity and efficiency of various
problemposing and problemsolving strategies, and develop alternative
strategies and generalizations as needed.
STANDARD 2 
All students will communicate mathematically through
written, oral, symbolic, and visual forms of expression. 

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:
 Discuss, listen, represent, read, and write as vital activities
in their learning and use of mathematics.
 Identify and explain key mathematical concepts, and model
situations using oral, written, concrete, pictorial, and graphical
methods.
 Represent and communicate mathematical ideas through the use of
learning tools such as calculators, computers, and manipulatives.
 Engage in mathematical brainstorming and discussions by asking
questions, making conjectures, and suggesting strategies for solving
problems.
 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:
 Identify and explain key mathematical concepts and model
situations using geometric and algebraic methods.
 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.
 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:
 Formulate questions, conjectures, and generalizations about
data, information, and problem situations.
 Reflect on and clarify their thinking so as to present
convincing arguments for their conclusions.
STANDARD 3 
All students will connect mathematics to other learning
by understanding the interrelationships of mathematical idea and the
roles that mathematics and mathematical modeling play in other
disciplines and in life. 

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:
 View mathematics as an integrated whole rather than as a series
of disconnected topics and rules.
 Relate mathematical procedures to their underlying concepts.
 Use models, calculators, and other mathematical tools to
demonstrate the connections among various equivalent graphical,
concrete, and verbal representations of mathematical concepts.
 Explore problems and describe and confirm results using various
representations.
 Use one mathematical idea to extend understanding of
another.
 Recognize the connections between mathematics and other
disciplines, and apply mathematical thinking and problem solving in
those areas.
 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:
 Recognize and apply unifying concepts and processes which are
woven throughout mathematics.
 Use the process of mathematical modeling in mathematics and
other disciplines, and demonstrate understanding of its methodology,
strengths, and limitations.
 Apply mathematics in their daily lives and in careerbased
contexts.
 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:
 Recognize how mathematics responds to the changing needs of
society, through the study of the history of mathematics.
STANDARD 4 
All student will develop reasoning ability and will
become selfreliant, independent mathematical thinkers. 

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 problemsolving 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:
 Make educated guesses and test them for correctness.
 Draw logical conclusions and make generalizations.
 Use models, known facts, properties, and relationships to
explain their thinking.
 Justify answers and solution processes in a variety of
problems.
 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:
 Make conjectures based on observation and information, and test
mathematical conjectures and arguments.
 Justify, in clear and organized form, answers and solution
processes in a variety of problems.
 Follow and construct logical arguments, and judge their
validity.
 Recognize and use deductive and inductive reasoning in all areas
of mathematics.
 Utilize mathematical reasoning skills in other disciplines
and in their lives.
 Use reasoning rather than relying on an answerkey 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:
 Make conjectures based on observation and information, and
test mathematical conjectures, arguments, and proofs.
 Formulate counterexamples to disprove an argument.
STANDARD 5 
All students will regularly and routinely use
calculators, computers, manipulatives, and other mathematical tools to
enhance mathematical thinking, understanding, and power. 

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 paperandpencil 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
realworld problems.
Cumulative Progress Indicators
By the end of Grade 4, students:
 Select and use calculators, software, manipulatives, and other
tools based on their utility and limitations and on the problem
situation.
 Use physical objects and manipulatives to model problem
situations, and to develop and explain mathematical concepts involving
number, space, and data.
 Use a variety of technologies to discover number patterns,
demonstrate number sense, and visualize geometric objects and
concepts.
 Use a variety of tools to measure mathematical and physical
objects in the world around them.
 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:
 Use a variety of technologies to evaluate and validate problem
solutions, and to investigate the properties of functions and their
graphs.
 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:
 Use calculators and computers effectively and efficiently in
applying mathematical concepts and principles to various types of
problems.
STANDARD 6 
All students will develop number sense and an ability to
represent numbers in a variety of forms and use numbers in diverse
situations. 

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:
 Use reallife experiences, physical materials, and technology to
construct meanings for whole numbers, commonly used fractions, and
decimals.
 Develop an understanding of place value concepts and numeration
in relationship to counting and grouping.
 See patterns in number sequences, and use patternbased thinking
to understand extensions of the number system.
 Develop a sense of the magnitudes of whole numbers, commonly
used fractions, and decimals.
 Understand the various uses of numbers including counting,
measuring, labeling, and indicating location.
 Count and perform simple computations with money.
 Use models to relate whole numbers, commonly used fractions, and
decimals to each other, and to represent equivalent forms of the same
number.
 Compare and order whole numbers, commonly used fractions, and
decimals.
 Explore reallife settings which give rise to negative
numbers.
Building upon knowledge and skills gained in the preceding grades,
by the end of Grade 8, students:
 Understand money notations, count and compute money, and
recognize the decimal nature of United States currency.
 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.
 Develop number sense necessary for estimation.
 Expand the sense of magnitudes of different number types to
include integers, rational numbers, and roots.
 Understand and apply ratios, proportions, and percents in a
variety of situations.
 Develop and use order relations for integers and rational
numbers.
 Recognize and describe patterns in both finite and infinite
number sequences involving whole numbers, rational numbers, and
integers.
 Develop and apply number theory concepts, such as primes,
factors, and multiples, in realworld and mathematical problem
situations.
 Investigate the relationships among fractions, decimals, and
percents, and use all of them appropriately.
 Identify, derive, and compare properties of numbers.
Building upon knowledge and skills gained in the preceding grades,
by the end of Grade 12, students:
 Extend their understanding of the number system to include
real numbers and an awareness of other number systems.
 Develop conjectures and informal proofs of properties of
number systems and sets of numbers.
 Extend their intuitive grasp of number relationships, uses,
and interpretations, and develop an ability to work with rational and
irrational numbers.
 Explore a variety of infinite sequences and informally
evaluate their limits.
STANDARD 7 
All students will develop spatial sense and an ability to
use geometric properties and relationships to solve problems in
mathematics and in everyday life. 

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
threedimensional space, such as paperfolding, 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:
 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.
 Explore relationships among shapes, such as congruence,
symmetry, similarity, and selfsimilarity.
 Explore properties of three and twodimensional shapes using
concrete objects, drawings, and computer graphics.
 Use properties of three and twodimensional shapes to identify,
classify, and describe shapes.
 Investigate and predict the results of combining, subdividing,
and changing shapes.
 Use tessellations to explore properties of geometric shapes and
their relationships to the concepts of area and perimeter.
 Explore geometric transformations such as rotations (turns),
reflections (flips), and translations (slides).
 Develop the concepts of coordinates and paths, using maps,
tables, and grids.
 Understand the variety of ways in which geometric shapes and
objects can be measured.
 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:
 Relate twodimensional and threedimensional geometry using
shadows, perspectives, projections and maps.
 Understand and apply the concepts of symmetry, similarity
and congruence.
 Identify, describe, compare, and classify plane and solid
geometric figures.
 Understand the properties of lines and planes, including
parallel and perpendicular lines and planes, and intersecting lines
and planes and their angles of incidence.
 Explore the relationships among geometric transformations
(translations, reflections, rotations, and dilations), tessellations
(tilings), and congruence and similarity.
 Develop, understand, and apply a variety of strategies for
determining perimeter, area, surface area, angle measure, and
volume.
 Understand and apply the Pythagorean Theorem.
 Explore patterns produced by processes of geometric change,
relating iteration, approximation, and fractals.
 Investigate, explore, and describe geometry in nature and
realworld 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:
 Understand and apply properties involving angles, parallel
lines, and perpendicular lines.
 Analyze properties of threedimensional shapes by
constructing models and by drawing and interpreting twodimensional
representations of them.
 Use transformations, coordinates, and vectors to solve
problems in Euclidean geometry.
 Use basic trigonometric ratios to solve problems involving
indirect measurement.
 Solve realworld and mathematical problems using geometric
models.
 Use inductive and deductive reasoning to solve problems and
to present reasonable explanations of and justifications for the
solutions.
 Analyze patterns produced by processes of geometric change,
and express them in terms of iteration, approximation, limits,
selfsimilarity, and fractals.
 Explore applications of other geometries in realworld
contexts.
STANDARD 8 
All students will understand, select, and apply various
methods of performing numerical operations. 

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,
paperandpencil 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 paperandpencil skills.
Cumulative Progress Indicators
By the end of Grade 4, students:
 Develop meaning for the four basic arithmetic operations by
modeling and discussing a variety of problems.
 Develop proficiency with and memorize basic number facts using a
variety of fact strategies (such as "counting on" and
"doubles").
 Construct, use, and explain procedures for performing whole
number calculations in the various methods of computation.
 Use models to explore operations with fractions and
decimals.
 Use a variety of mental computation and estimation
techniques.
 Select and use appropriate computational methods from mental
math, estimation, paperandpencil, and calculator methods, and check
the reasonableness of results.
 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:
 Extend their understanding and use of arithmetic operations to
fractions, decimals, integers, and rational numbers.
 Extend their understanding of basic arithmetic operations on
whole numbers to include powers and roots.
 Develop, apply, and explain procedures for computation and
estimation with whole numbers, fractions, decimals, integers, and
rational numbers.
 Develop, apply, and explain methods for solving problems
involving proportions and percents.
 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:
 Extend their understanding and use of operations to real
numbers and algebraic procedures.
 Develop, apply, and explain methods for solving problems
involving factorials, exponents, and matrices.
STANDARD 9 
All students will develop an understanding of and will
use measurement to describe and analyze phenomena. 

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:
 Use and describe measures of length, distance, capacity, weight,
area, volume, time, and temperature.
 Compare and order objects according to some measurable
attribute.
 Recognize the need for a uniform unit of measure.
 Develop and use personal referents for standard units of measure
(such as the width of a finger to approximate a centimeter).
 Select and use appropriate standard and nonstandard units of
measurement to solve reallife problems.
 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:
 Use estimated and actual measurements to describe and compare
phenomena.
 Read and interpret various scales, including those based on
number lines and maps.
 Determine the degree of accuracy needed in a given situation and
choose units accordingly.
 Understand that all measurements of continuous quantities
are approximate.
 Develop formulas and procedures for solving problems related
to measurement.
 Explore situations involving quantities which cannot be
measured directly or conveniently.
 Convert measurement units from one form to another, and
carry out calculations that involve various units of measurement.
 Understand and apply measurement in their own lives and in
other subject areas.
 Understand and explain the impact of the change of an
object's linear dimensions on its perimeter, area, or volume.
 Apply their knowledge of measurement to the construction of
a variety of two and threedimensional figures.
Building upon knowledge and skills gained in the preceding grades,
by the end of Grade 12, students:
 Use techniques of algebra, geometry, and trigonometry to
measure quantities indirectly.
 Use measurement appropriately in other subject areas and
careerbased contexts.
 Choose appropriate techniques and tools to measure
quantities in order to achieve specified degrees of precision,
accuracy, and error (or tolerance) of measurements.
STANDARD 10 
All students will use a variety of estimation strategies
and recognize situations in which estimation is appropriate. 

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:
 Judge without counting whether a set of objects has less than,
more than, or the same number of objects as a reference set.
 Use personal referents, such as the width of a finger as one
centimeter, for estimations withmeasurement.
 Visually estimate length, area, volume, or angle measure.
 Explore, construct, and use a variety of estimation
strategies.
 Recognize when estimation is appropriate, and understand the
usefulness of an estimate as distinct from an exact answer.
 Determine the reasonableness of an answer by estimating the
result of operations.
 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:
 Develop, apply, and explain a variety of different estimation
strategies in problem situations involving quantities and
measurement.
 Use equivalent representations of numbers such as fractions,
decimals, and percents to facilitate estimation.
 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:
 Estimate probabilities and predict outcomes from realworld
data.
 Recognize the limitations of estimation, assess the amount
of error resulting from estimation, and determine whether the error is
within acceptable tolerance limits.
STANDARD 11 
All students will develop an understanding of patterns,
relationships, and functions and will use them to represent and
explain realworld phenomena. 

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 patternbased thinking to understand and represent
mathematical and other realworld phenomena. These explorations
present unlimited opportunities for problemsolving, making and
verifying generalizations, and building mathematical understanding and
confidence.
Cumulative Progress Indicators
By the end of Grade 4, students:
 Reproduce, extend, create, and describe patterns and sequences
using a variety of materials.
 Use tables, rules, variables, open sentences, and graphs to
describe patterns and other relationships.
 Use concrete and pictorial models to explore the basic concept
of a function.
 Observe and explain how a change in one physical quantity can
produce a corresponding change in another.
 Observe and recognize examples of patterns, relationships, and
functions in other disciplines and contexts.
 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:
 Represent and describe mathematical relationships with tables,
rules, simple equations, and graphs.
 Understand and describe the relationships among various
representations of patterns and functions.
 Use patterns, relationships, and functions to model situations
and to solve problems in mathematics and in other subject areas.
 Analyze functional relationships to explain how a change in
one quantity results in a change in another.
 Understand and describe the general behavior of
functions.
 Use patterns, relationships, and linear functions to model
situations in mathematics and in other areas.
 Develop, analyze, and explain arithmetic sequences.
Building upon knowledge and skills gained in the preceding grades,
by the end of Grade 12, students:
 Analyze and describe how a change in an independent variable
can produce a change in a dependent variable.
 Use polynomial, rational, trigonometric, and exponential
functions to model realworld phenomena.
 Recognize that a variety of phenomena can be modeled by the
same type of function.
 Analyze and explain the general properties and behavior of
functions, and use appropriate graphing technologies to represent
them.
 Analyze the effects of changes in parameters on the graphs
of functions.
 Understand the role of functions as a unifying concept in
mathematics.
STANDARD 12 
All students will develop an understanding of statistics
and probability and will use them to describe sets of data, model
situations, and support appropriate inferences and arguments. 

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:
 Formulate and solve problems that involve collecting,
organizing, and analyzing data.
 Generate and analyze data obtained using chance devices such as
spinners and dice.
 Make inferences and formulate hypotheses based on data.
 Understand and informally use the concepts of range, mean, mode,
and median.
 Construct, read, and interpret displays of data such as
pictographs, bar graphs, circle graphs,tables, and lists.
 Determine the probability of a simple event, assuming equally
likely outcomes.
 Make predictions that are based on intuitive, experimental, and
theoretical probabilities.
 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:
 Generate, collect, organize, and analyze data and represent this
data in tables, charts, and graphs.
 Select and use appropriate graphical representations and
measures of central tendency (mean, mode and median) for sets of
data.
 Make inferences and formulate and evaluate arguments based
on data analysis and data displays.
 Use lines of best fit to interpolate and predict from
data.
 Determine the probability of a compound event.
 Model situations involving probability, such as genetics,
using both simulations and theoretical models.
 Use models of probability to predict events based on actual
data.
 Interpret probabilities as ratios and percents.
Building upon knowledge and skills gained in the preceding grades,
by the end of Grade 12, students:
 Estimate probabilities and predict outcomes from actual
data.
 Understand sampling and recognize its role in statistical
claims.
 Evaluate bias, accuracy, and reasonableness of data in
realworld contexts.
 Understand and apply measures of dispersion and
correlation.
 Design a statistical experiment to study a problem, conduct
the experiment, and interpret and communicate the outcomes.
 Make predictions using curve fitting and numerical
procedures to interpolate and extrapolate from known data.
 Use relative frequency and probability, as appropriate, to
represent and solve problems involving uncertainty.
 Use simulations to estimate probabilities.
 Create and interpret discrete and continuous probability
distributions, and understand their application to realworld
situations.
 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.
 Understand and use the law of large numbers (that
experimental results tend to approachtheoretical probabilities after a
large number of trials).
STANDARD 13 
All students will develop an understanding of algebraic
concepts and processes and will use them to represent and analyze
relationships among variable quantities and to solve problems. 

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 reallife situation.
Cumulative Progress Indicators
By the end of Grade 4, students:
 Understand and represent numerical situations using variables,
expressions, and number sentences.
 Represent situations and number patterns with concrete
materials, tables, graphs, verbal rules, and number sentences, and
translate from one to another.
 Understand and use properties of operations and numbers.
 Construct and solve open sentences (example: 3 + ___ = 7) that
describe reallife situations.
Building upon knowledge and skills gained in the preceding grades,
by the end of Grade 8, students:
 Understand and use variables, expressions, equations, and
inequalities.
 Represent situations and number patterns with concrete
materials, tables, graphs, verbal rules, and standard algebraic
notation.
 Use graphing techniques on a number line to model both absolute
value and arithmetic operations.
 Analyze tables and graphs to identify properties and
relationships.
 Understand and use the rectangular coordinate system.
 Solve simple linear equations using concrete, informal, and
graphical methods, as well as appropriate paperandpencil
techniques.
 Explore linear equations through the use of calculators,
computers, and other technology.
 Investigate inequalities and nonlinear equations
informally.
 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:
 Model and solve problems that involve varying quantities
using variables, expressions, equations, inequalities, absolute
values, vectors, and matrices.
 Use tables and graphs as tools to interpret expressions,
equations, and inequalities.
 Develop, explain, use, and analyze procedures for operating
on algebraic expressions and matrices.
 Solve equations and inequalities of varying degrees using
graphing calculators and computers as well as appropriate
paperandpencil techniques.
 Understand the logic and purposes of algebraic
procedures.
 Interpret algebraic equations and inequalities
geometrically, and describe geometric objects algebraically.
STANDARD 14 
All students will apply the concepts and methods of
discrete mathematics to model and explore a variety of practical
situations. 

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 decisionmakers 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:
 Explore a variety of puzzles, games, and counting problems.
 Use networks and tree diagrams to represent everyday
situations.
 Identify and investigate sequences and patterns found in nature,
art, and music.
 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.
 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:
 Use systematic listing, counting, and reasoning in a variety of
different contexts.
 Recognize common discrete mathematical models, explore their
properties, and design them for specific situations.
 Experiment with iterative and recursive processes, with the aid
of calculators and computers.
 Explore methods for storing, processing, and communicating
information.
 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:
 Understand the basic principles of iteration, recursion, and
mathematical induction.
 Use basic principles to solve combinatorial and algorithmic
problems.
 Use discrete models to represent and solve problems.
 Analyze iterative processes with the aid of calculators and
computers.
 Apply discrete methods to storing, processing, and
communicating information.
 Apply discrete methods to problems of voting, apportionment,
and allocations, and use fundamental strategies of optimization to
solve problems.
STANDARD 15 
All students will develop an understanding of the
conceptual building blocks of calculus and will use them to model and
analyze natural phenomena. 

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:
 Investigate and describe patterns that continue
indefinitely.
 Investigate and describe how certain quantities change over
time.
 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:
 Recognize and express the difference between linear and
exponential growth.
 Develop an understanding of infinite sequences that arise in
natural situations.
 Investigate, represent, and use nonterminating decimals.
 Represent, analyze, and predict relations between quantities,
especially quantities changing over time.
 Approximate quantities with increasing degrees of accuracy.
 Understand and use the concept of significant digits.
 Develop informal ways of approximating the surface area and
volume of familiar objects, and discuss whether the approximations
make sense.
 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:
 Develop and use models based on sequences and series.
 Develop and apply procedures for finding the sum of finite
arithmetic series and of finite and infinite geometric series.
 Develop an informal notion of limit.
 Use linear, quadratic, trigonometric, and exponential models
to explain growth and change in the natural world.
 Recognize fundamental mathematical models (such as
polynomial, exponential, and trigonometric functions) and apply basic
translations, reflections, and dilations to their graphs.
 Develop and explain the concept of the slope of a curve and
use that concept to discuss the information contained in graphs.
 Develop an understanding of the concept of continuity of a
function.
 Understand and apply approximation techniques to situations
involving initial portions of infinite decimals and measurement.
STANDARD 16 
All students will demonstrate high levels of
mathematical thought through experiences which extend beyond
traditional computation, algebra, and geometry. 

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:
 Study a core curriculum containing challenging ideas and tasks,
rather than one limited to repetitive, lowlevel cognitive
activities.
 Work at rich, openended problems which require them to use
mathematics in meaningful ways, and which provide them with exciting
and interesting mathematical experiences.
 Recognize mathematics as integral to the development of all
cultures and civilizations, and in particular to that of our own
society.
 Understand the important role that mathematics plays in their
own success, regardless of career.
 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.
 Receive services that help them understand the mathematical
skills and concepts necessary to assure success in the core
curriculum.
 Receive equitable treatment without regard to gender, ethnicity,
or predetermined expectations for success.
 Learn mathematics in classes which reflect the diversity of the
school's total student population.
 Be provided with opportunities at all grade levels for further
study of mathematics, especially including topics beyond traditional
computation, algebra, and geometry.
 Be challenged to maximize their mathematical achievements at
all grade levels.
 Experience a full program of meaningful mathematics so that
they can pursue postsecondary education.
STANDARD 17 
All students' mathematical learning will embody the
concept that engagement in mathematics is essential, and that
decisionmaking, risktaking, cooperative work, perseverance,
selfassessment, and selfconfidence are frequently keys to
success. 

(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,
decisionmaking, and risktaking  which will be key to their
success in mathematics.
Cumulative Progress Indicators
Experiences will be such that all students:
 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.
 Recognize the power that comes from understanding and doing
mathematics.
 Develop and maintain a positive disposition to mathematics and to
mathematical activity.
 Participate actively in mathematical activity and discussion,
freely exchanging ideas and problemsolving strategies with their
classmates and teachers, and taking intellectual risks and defending
positions without fear of being incorrect.
 Work cooperatively with other students on mathematical
activities, actively sharing, listening, and reflecting during group
discussions, and giving and receiving constructive criticism.
 Make conjectures, pose their own problems, and devise their own
approaches to problem solving.
 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.
 Assess their work to determine the correctness of their results,
based on their own reasoning, rather than relying solely on external
authorities.
STANDARD 18 
All students will be evaluated using a diversity of
assessment tools and strategies to provide multiple indicators of the
quality of every student's mathematical learning and of overall
program effectiveness. 

(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:
 Are engaged in assessment activities that function primarily to
improve learning.
 Are engaged in assessment activities based upon rich,
challenging problems from mathematics and other disciplines.
 Are engaged in assessment activities that address the content
described in all of New Jersey's Mathematics
Standards.
 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.
 Engage in ongoing assessment of their work to determine the
effectiveness of their strategies and the correctness of their
results.
 Understand and accept that the criteria used to evaluate their
performance will be based on high expectations.
 Recognize errors as part of the learning process and use them as
opportunities for mathematical growth.
 Select and use appropriate tools effectively during assessment
activities.
 Reflect upon and communicate their mathematical understanding,
knowledge, and attitudes.
^{*} Note that in the Core Curriculum Content
Standards of the New Jersey State Department of Education, the
Mathematics Standards are numbered 4.1, 4.2, 4.3, etc., since
they are preceded by standards in three other content areas.
