Although the standards adopted by the Board are substantially the same as the previous version, the standards have been renumbered; new standards #14 were previously #58, new standard #5 was previously #3, new standards #615 were previously #918, and new standard #16 was previously #1.
4.1 
All students will develop the ability to pose and solve mathematical
problems in mathematics, other disciplines, and everyday experiences. 
4.2 
All students will communicate mathematically through written, oral,
symbolic, and visual forms of expression. 
4.3 
All students will connect mathematics to other learning by
understanding the interrelationships of mathematical ideas and the
roles that mathematics and mathematical modeling play in other
disciplines and in life. 
4.4 
All students will develop reasoning ability and will become
selfreliant, independent mathematical thinkers. 
4.5 
All students will regularly and routinely use calculators, computers,
manipulatives, and other mathematical tools to enhance mathematical
thinking, understanding, and power. 
4.6 
All students will develop number sense and an ability to represent
numbers in a variety of forms and use numbers in diverse situations. 
4.7 
All students will develop spatial sense and an ability to use
geometric properties and relationships to solve problems in
mathematics and in everyday life. 
4.8 
All students will understand, select, and apply various methods of
performing numerical operations. 
4.9 
All students will develop an understanding of and will use measurement
to describe and analyze phenomena. 
4.10 
All students will use a variety of estimation strategies and recognize
situations in which estimation is appropriate. 
4.11 
All students will develop an understanding of patterns, relationships,
and functions and will use them to represent and explain realworld
phenomena. 
4.12 
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. 
4.13 
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. 
4.14 
All students will apply the concepts and methods of discrete
mathematics to model and explore a variety of practical situations. 
4.15 
All students will develop an understanding of the conceptual building
blocks of calculus and will use them to model and analyze natural
phenomena. 
4.16 
All students will demonstrate high levels of mathematical thought
through experiences which extend beyond traditional computation,
algebra, and geometry. 

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:
 
1. 
Use discoveryoriented, inquirybased, and problemcentered
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
nonroutine problems and
openended problems with several solutions and/or solution
strategies.

5. 
Construct, explain, justify, and apply a variety of
problemsolving 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 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 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 discoveryoriented, inquirybased, and problemcentered 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 problemsolving
strategies.

14. 
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:
 
15. 
Use discoveryoriented, inquirybased, and problemcentered 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 problemposing and
problemsolving 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 careerbased contexts.

11. 
Recognize situations in other disciplines in which mathematical models
may be applicable, and apply 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 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:
 
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 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:
 
12. 
Make conjectures based on observation and information, and test mathematical
conjectures, arguments, and proofs.

13. 
Formulate counterexamples 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 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:
 
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 continued progress 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 reallife 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 patternbased 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 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:
 
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 realworld 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 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:
 
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 selfsimilarity.

3. 
Explore properties of three and twodimensional shapes using concrete objects,
drawings, and computer graphics.

4. 
Use properties of three and twodimensional 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 twodimensional and threedimensional 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 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:
 
20. 
Understand and apply properties involving angles, parallel lines, and
perpendicular lines.

21. 
Analyze properties of threedimensional shapes by constructing models
and by drawing and interpreting twodimensional 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 realworld 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,
selfsimilarity, and fractals.

27. 
Explore applications of other geometries in realworld 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, paperandpencil techniques, and the use of calculators. Students must understand how to add, subtract, multiply, and divide whole numbers, fractions, and others 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:
 
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 andpencil, 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 nonstandard units of
measurement to solve reallife 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 threedimensional 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 careerbased
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 with measurement.

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 realworld 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 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:
 
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 realworld 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 realworld 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 realworld 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 approach theoretical 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 reallife 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 reallife 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 paperandpencil 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 paperandpencil
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 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:
 
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 nonterminating 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, lowlevel cognitive activities.

2. 
Work at rich, openended 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.
