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 develop the ability to communicate
mathematically through experiences which involve a variety of written,
oral, symbolic, and visual forms of expression of mathematical
ideas. |
4.3 |
All students will develop the ability to connect mathematics to
other learning, through experiences which focus on 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
self-reliant, independent mathemathematical thinkers through
experiences which reinforce and extend their mathematical and logical
thinking skills. |
4.5 |
All students will regularly and routinely use calculators,
computers, manipulatives, and other mathematical tools in both
instructional and assessment activities in order to enhance their
mathematical thinking, understanding, and power. |
4.6 |
All students will develop number sense through experiences which
enable them to investigate the characteristics and relationships of
numbers, represent numbers in a variety of forms, and use numbers in
diverse situations. |
4.7 |
All students will develop spatial sense through experiences which
enable them to recognize, visualize, represent, and transform
geometric shapes, and to apply their knowledge of geometric
properties, relationships, and models to other areas of mathematics
and to the physical world. |
4.8 |
All students will develop an understanding of numerical operations
throug experiences which enable them to construct, explain, select,
and apply various methods of computation, including mental math,
estimation, and the use of calculators, with less emphasis
on paper-and-pencil techniques. |
4.9 |
All students will develop an understanding of measurement and
systems of measurement through experiences which enable them to use a
variety of techniques, tools, and units of measurement to describe and
analyze quantifiable phenomena. |
4.10 |
All students will develop an understanding of estimation through
experiences which enable them to recognize situations in which
estimation is appropriate, and to use a variety of estimation
strategies. |
4.11 |
All students will develop an understanding of patterns,
relationships, and functions through experiences which enable them to
discover, analyze, extend, and create a variety of patterns, and to
use pattern-based thinking to understand and represent mathematical
and other real-world phenomena. |
4.12 |
All students will develop an understanding of probability and
statistics through experiences which enable them to systematically
collect, organize, and describe sets of data, to use probability to
model situations, and to make appropriate inferences and arguments. |
4.13 |
All students will develop an understanding of algebraic concepts
and processes throughexperiences which enable them to describe,
represent, and analyze relationships among variablequantities, and to
apply algebraic methods to solve problems. |
4.14 |
All students will develop an understanding of the concepts and
applications of discrete mathematics, through experiences which enable
them to use a variety of tools of contemporary mathematics to explore
and model a variety of practical situations. |
4.15 |
All students will develop an understanding of the conceptual
building blocks of calculus, through experiences which enable them to
describe and analyze how various quantities change, to build informal
concepts of infinity and limits, and to use these concepts to model,
describe, and analyze natural phenomena. |
4.16 |
All students will demonstrate high levels of mathematical thought
through experiences which extend beyond traditional computation,
algebra, and geometry. |
|
Descriptive Statement: Problem-solving and posing involve examining situations that arise in mathematics and other disciplines and in common experiences, describing these situations mathematically, formulating appropriate mathematical questions, and using a variety of strategies to find solutions. By developing their problem-solving skills, students will come to realize the potential usefulness of mathematics in their lives.
Cumulative Progress Indicators
By the end of Grade 4, students:
| |
1. |
Use discovery-oriented, inquiry-based, and problem-centered
approaches to investigate and understand mathematical content
appropriate to early elementary grades. |
2. |
Recognize, formulate, and solve problems arising from
mathematical situations and everyday experiences. |
3. |
Construct and use concrete, pictorial, symbolic, and
graphical models to represent problem situations. |
4. |
Pose, explore, and solve a variety of problems, including
non-routine problems and open-ended problems with several solutions
and/or solution strategies. |
5. |
Construct, explain, justify, and apply a variety of
problem-solving strategies in both cooperative and independent
learning environments. |
6. |
Verify the correctness and reasonableness of results and
interpret them in the context of the problems being solved. |
7. |
Know when to select and how to use grade-appropriate
mathematical tools and methods (including manipulatives, calculators
and computers, as well as mental math and paper-and-pencil
techniques) as a natural and routine part of the problem-solving process. |
8. |
Determine, collect, organize, and analyze data needed to
solve problems. |
9. |
Recognize that there may be multiple ways to solve a problem.
|
Building upon knowledge and skills gained in the preceding grades, and
demonstrating continued progress in Indicators 4, 5, 6, 7, and 8
above, by the end of Grade 8, students:
| |
10. |
Use discovery-oriented, inquiry-based, and problem-centered
approaches investigate and understand mathematical content appropriate
to the middle grades. |
11. |
Recognize, formulate, and solve problems arising from
mathematical situations, everyday experiences, and applications to
other disciplines. |
12. |
Construct and use concrete, pictorial, symbolic, and
graphical models to represent problem situations and effectively apply
processes of mathematical modeling in mathematics and other areas. |
13. |
Recognize that there may be multiple ways to solve a problem,
weigh their relative merits, and select and use appropriate
problem-solving strategies. |
14. |
Perservere in developing alternative problem-solving
strategies if initially selected approaches do not work.
|
Building upon knowledge and skills gained in the preceding grades, and
demonstrating continued progress in Indicators 4, 5, 6, 7, 8, 12, and
14 above, by the end of Grade 12, students:
| |
15. |
Use discovery-oriented, inquiry-based, and problem-centered
approaches to investigate and understand the mathematical content
appropriate to the high school grades. |
16. |
Recognize, formulate, and solve problems arising from
mathematical situations, everyday experiences, applications to other
disciplines, and career applications. |
17. |
Monitor their own progress toward problem solutions. |
18. |
Explore the validity and efficiency of various problem-posing
and problem-solving strategies, and develop alternative strategies and
generalizations as needed. |
|
Descriptive Statement: Communication of mathematical ideas will help students clarify and solidify their understanding of mathematics. By sharing their mathematical understandings in written and oral form with their classmates, teachers, and parents, students develop confidence in themselves as mathematics learners and enable teachers to better monitor their progress.
Cumulative Progress Indicators
By the end of Grade 4, students:
| |
1. |
Discuss, listen, represent, read, and write as vital
activities in their learning and use of mathematics. |
2. |
Identify and explain key mathematical concepts, and model
situations using oral, written, concrete, pictorial, and graphical methods. |
3. |
Represent and communicate mathematical ideas through the use
of learning tools such as calculators, computers, and manipulatives. |
4. |
Engage in mathematical brainstorming and discussions by asking
questions, making conjectures, and suggesting strategies for solving
problems. |
5. |
Explain their own mathematical work to others, and justify
their reasoning and conclusions.
|
Building upon knowledge and skills gained in the preceding grades, and
demonstrating continued progress in Indicators 1, 3, 4, and 5 above,
by the end of Grade 8, students:
| |
6. |
Identify and explain key mathematical concepts and model
situations using oral, written, concrete, pictorial, graphical,
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. |
Understand, explain, analyze, and evaluate mathematical
arguments and conclusions presented by others.
|
Building upon knowledge and skills gained in the preceding grades, and
demonstrating continued progress in Indicators 1, 3, 4, 5, 6, 7, and 8
above, by the end of Grade 12, students:
| |
9. |
Formulate questions, conjectures, definitions, 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 studentscan 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 verb representations of mathematical concepts. |
4. |
Explore problems and describe and confirm results using various
representations. |
5. |
Use one mathematical idea to extend understanding of another. |
6. |
Recognize the connections between mathematics and other
disciplines, and apply mathematical thinking and problem-solving in
those areas. |
7. |
Recognize the role of mathematics in their daily lives and in society.
|
Building upon knowledge and skills gained in the preceding grades, and
demonstrating continued progress in Indicators 1, 2, 3, and 4 above,
by the end of Grade 8, students:
| |
8. |
Recognize and apply unifying concepts and processes which are
woven throughout mathematics. |
9. |
Use the process of mathematical modeling in mathematics and
other disciplines, and demonstrate understanding of its methodology,
strengths, and limitations. |
10. |
Apply mathematics in their daily lives and in career-based contexts. |
11. |
Recognize situations in other disciplines in which mathematical
models may be applicable, 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, and 10
above, by the end of Grade 12, students:
| |
12. |
Recognize situations in other disciplines in which probability,
statistical, geometric, algebraic, optimization, discrete
mathematical, and analytic models may be applicable, and use
appropriate models, mathematical reasoning, and problem-solving. |
13. |
Recognize how mathematics responds to the changing needs of
society, through the study of the history of mathematics. |
|
Descriptive Statement: Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions, and recognize how those solutions can be applied. Mathematical reasoners are able to reflect on solutions to problems and determine whether or not they make sense. They appreciate the pervasive use and power of reasoning as a part of mathematics.
Cumulative Progress Indicators
By the end of Grade 4, students:
| |
1. |
Make educated guesses and test them for correctness. |
2. |
Draw logical conclusions and make generalizations. |
3. |
Use models, known facts, properties, and relationships to
explain their thinking. |
4. |
Justify answers and solution processes in a variety of problems. |
5. |
Analyze mathematical situations by recognizing and using
patterns and relationships.
|
Building upon knowledge and skills gained in the preceding grades, and
demonstrating continued progress in Indicators 2, 3, and 5 above, by
the end of Grade 8, students:
| |
6. |
Make conjectures based on observation and information, and
test mathematical conjectures and arguments. |
7. |
Justify, in clear and organized form, answers and solution
processes in a variety of problems. |
8. |
Follow logical arguments, construct simple, valid arguments,
and judge the validity of arguments. |
9. |
Recognize and use deductive and inductive reasoning in all
areas of mathematics. |
10. |
Utilize mathematical reasoning skills in other disciplines
and in their lives. |
11. |
Use reasoning rather than relying on an answer-key to check
the correctness of solutions to problems.
|
Building upon knowledge and skills gained in the preceding grades, and
demonstrating continued progress in Indicators 2, 5, 8, 9, 10, and 11
above, by the end of Grade 12, students:
| |
12. |
Make conjectures based on observation and information, and
test mathematical conjectures, arguments, and proofs. |
13. |
Formulate counter-examples to disprove an argument. |
|
Descriptive Statement: Calculators, computers, manipulatives, and other mathematic tools need to be used by students, not to replace mental math and paper-and-pencil computational skills, but to enhance both their understanding of mathematics and their power to use mathematics. Historically, people have developed and used manipulatives (such as fingers, base ten blocks, geoboards, and algebra tiles) and mathematical devices (such as protractors, coordinate systems, and calculators) to help them understand and develop mathematics. Students should explore both new and familiar concepts with calculators and computers, but should also become proficient in using technology as it is used by adults, that is, for assistance in solving real-world problems.
Cumulative Progress Indicators
By the end of Grade 4, students:
| |
1. |
Select and use calculators, software, manipulatives, and
other tools based on their utility and limitations and on the problem
situation. |
2. |
Use physical objects and manipulatives to model problem
situations, and to develop and explain mathematical concepts involving
number, space, and data. |
3. |
Use a variety of technologies to discover number patterns,
demonstrate number sense (an intuitive, common sense approach to using
numbers), visualize geometric concepts, and explore and analyze
numerical data. |
4. |
Use a variety of tools to measure mathematical and physical
objects in the world around them. |
5. |
Use technology to gather data and to express and display
mathematical information.
|
Building upon knowledge and skills gained in the preceding grades, and
demonstrating continuedprogress in Indicators 1, 2, 3, 4, and 5 above,
by the end of Grade 8, students:
| |
6. |
Use a variety of technologies to evaluate and validate
problem solutions, and to investigate the properties of functions and
their graphs. |
7. |
Use computer spreadsheets and graphing programs to organize
and display quantitative information and to investigate properties of
functions.
|
Building upon knowledge and skills gained in the preceding grades, and
demonstrating continued progress in Indicators 1, 2, 3, 5, and 7
above, by the end of Grade 12, students:
| |
8. |
Use a variety of technologies to visualize and display
geometric objects and concepts. |
9. |
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 comes from comfort with what numbers represent, how different types of numbers, such as fractions and decimals, are related to each other, and how they can best be used to describe a particular situation. Number sense is an attribute of all successful users of mathematics.
Cumulative Progress Indicators
By the end of Grade 4, students:
| |
1. |
Use real-life experiences, physical materials, and
technology to construct meanings for whole numbers, commonly used
fractions, and decimals. |
2. |
Develop an understanding of place value concepts and
numeration in relationship to counting and grouping. |
3. |
See patterns in number sequences, and use pattern-based
thinking to understand extensions of the number system. |
4. |
Develop a sense of the magnitudes of whole numbers, commonly
used fractions, and decimals. |
5. |
Understand the various uses of numbers including counting,
measuring, labeling, and indicating location. |
6. |
Count and perform simple computations with money. |
7. |
Use models to relate whole numbers, commonly used fractions,
and decimals to each other, and to represent equivalent forms of the
same number. |
8. |
Compare and order whole numbers, commonly used fractions,
and decimals. |
9. |
Explore real-life settings which give rise to negative numbers.
|
Building upon knowledge and skills gained in the preceding grades, by
the end of Grade 8, students:
| |
10. |
Understand money notations, count and compute money, and
recognize the decimal nature of United States currency. |
11. |
Extend their understanding of the number system by
constructing meanings for integers, rational numbers, percents,
exponents, roots, absolute values, and numbers represented in
scientific notation. |
12. |
Develop number sense necessary for estimation. |
13. |
Expand the sense of magnitudes of different number types to
include integers, rational numbers, and roots. |
14. |
Understand and apply ratios, proportions, and percents in a
variety of situations. |
15. |
Develop and use order relations for integers and rational numbers. |
16. |
Recognize and describe patterns in both finite and infinite
number sequences involving whole numbers, rational numbers, and integers. |
17. |
Develop and apply number theory concepts, such as primes,
factors, and multiples, in real-world and mathematical problem situations. |
18. |
Investigate the relationships among fractions, decimals, and
percents, and use all of them appropriately. |
19. |
Identify, derive, and compare properties of numbers. |
20. |
Establish an intuitive grasp of number relationships, uses,
and interpretations.
|
Building upon knowledge and skills gained in the preceding grades, by
the end of Grade 12, students:
| |
21. |
Extend their understanding of the number system to include
real numbers and an awareness of other number systems. |
22. |
Develop conjectures and informal proofs of properties of
number systems and sets of numbers. |
23. |
Extend their intuitive grasp of number relationships, uses,
and interpretations, and develop an ability to work with rational and
irrational numbers. |
24. |
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 both the concepts of traditional geometry and other, less formal ways of looking at two- and three-dimensional space, such as paper-folding, transformations, tessellations, and projections. Geometry is all around us in art, nature, and the things we make. Students of geometry can apply their spatial sense and knowledge of the properties of shapes and space to the real world.
Cumulative Progress Indicators
By the end of Grade 4, students:
| |
1. |
Explore spatial relationships such as the direction,
orientation, and perspective of objects in space, their relative
shapes and sizes, and the relations between objects and their shadows
or projections. |
2. |
Explore relationships among shapes, such as congruence,
symmetry, similarity, and self-similarity. |
3. |
Explore properties of three- and two-dimensional shapes using
concrete objects, drawings, and computer graphics. |
4. |
Use properties of three- and two-dimensional shapes to
identify, classify, and describe shapes. |
5. |
Investigate and predict the results of combining, subdividing, and changing shapes. |
6. |
Use tessellations to explore properties of geometric shapes
and their relationships to the concepts of area and perimeter. |
7. |
Explore geometric transformations such as rotations (turns),
reflections (flips), and translations (slides). |
8. |
Develop the concepts of coordinates and paths, using maps,
tables, and grids. |
9. |
Understand the variety of ways in which geometric shapes and
objects can be measured. |
10. |
Investigate the occurrence of geometry in nature, art, and
other areas.
|
Building upon knowledge and skills gained in the preceding grades, by
the end of Grade 8, students:
| |
11. |
Relate two-dimensional and three-dimensional geometry using
shadows, perspectives, projections and maps. |
12. |
Understand and apply the concepts of symmetry, similarity and
congruence. |
13. |
Identify and describe plane and solid geometric figures,
characterize geometric figures using a minimum set of properties,
classify geometric figures according to common properties, and develop
definitions for common geometric figures. |
14. |
Understand the properties of lines and planes, including
parallel an perpendicular lines and planes, and intersecting lines and
planes and their angles of incidence. |
15. |
Explore the relationships among geometric transformations
(translations, reflections, rotations, and dilations), tessellations
(tilings), and congruence and similarity. |
16. |
Develop, understand, and apply a variety of strategies for
determining perimeter, area, surface area, angle measure, and
volume. |
17. |
Understand and apply the Pythagorean Theorem. |
18. |
Explore patterns produced by processes of geometric change,
relating iteration, approximation, and fractals. |
19. |
Investigate, explore, and describe geometry in nature and
real-world applications. |
20. |
Use models, manipulatives, and computer graphics software to
build a conceptual understanding of geometry and its connections to
other parts of mathematics, science, and art.
|
Building upon knowledge and skills gained in the preceding grades, and
demonstrating continuedprogress in Indicators 16 and 20 above, by the
end of Grade 12, students:
| |
21. |
Understand and apply properties involving angles, parallel
lines, an perpendicular lines. |
22. |
Analyze properties of three-dimensional shapes by constructing
models an by drawing and interpreting two-dimensional representations
of them. |
23. |
Use transformations, coordinates, and vectors to solve
problems in Euclidean geometry. |
24. |
Interpret algebraic equations and inequalities geometrically,
and describe geometric objects algebraically. |
25. |
Use basic trigonometric ratios to solve problems involving
indirect measurement. |
26. |
Solve real-world and mathematical problems using geometric
models. |
27. |
Use inductive and deductive reasoning to solve problems and to
present reasonable explanations of and justifications for the
solutions. |
28. |
Analyze patterns produced by processes of geometric change,
and express them in terms of iteration, approximation, limits,
self-similarity, and fractals. |
29. |
Explore applications of other geometries in real-world
contexts. |
|
Descriptive Statement: Numerical operations are an essential part of the math curriculum. 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 meaningsand uses of the operations, and on estimation and mental skills, rather than soley on developing paper-and-pencil skills.
Cumulative Progress Indicators
By the end of Grade 4, students:
| |
1. |
Develop meaning for the four basic arithmetic operations by
modeling and discussing a variety of problems. |
2. |
Develop proficiency with and memorize basic number facts using
a variety of fact strategies (such as "counting on" and "doubles"). |
3. |
Construct, use, and explain procedures for performing whole
number calculations in the various methods of computation. |
4. |
Use models to explore operations with fractions and
decimals. |
5. |
Use a variety of mental computation and estimation
techniques. |
6. |
Select and use appropriate computational methods from mental
math, estimation, paper-and-pencil, and calculator methods, and check
the reasonableness of results. |
7. |
Understand and use relationships among operations and
properties of operations.
|
Building upon knowledge and skills gained in the preceding grades, and
demonstrating continued progress in Indicator 6 above, by the end of
Grade 8, students:
| |
8. |
Extend their understanding and use of arithmetic operations to
fractions, decimals, integers, and rational numbers. |
9. |
Extend their understanding of basic arithmetic operations on
whole numbers to include powers and roots. |
10. |
Develop, analyze, apply, and explain procedures for
computation and estimation with whole numbers, fractions, decimals,
integers, and rational numbers. |
11. |
Develop, analyze, 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, analyze, 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 a critical for students' understanding of the world around them.
Cumulative Progress Indicators
By the end of Grade 4, students:
| |
1. |
Use and describe measures of length, distance, capacity,
weight, area, volume, time, and temperature. |
2. |
Compare and order objects according to some measurable
attribute. |
3. |
Recognize the need for a uniform unit of measure. |
4. |
Develop and use personal referents for standard units of
measure (such as the width of a finger to approximate a
centimeter). |
5. |
Select and use appropriate standard and non-standard units of
measurement to solve real-life problems. |
6. |
Understand and incorporate estimation and repeated measures in
measurement activities. |
7. |
Use measurement in subjects other than mathematics.
|
Building upon knowledge and skills gained in the preceding grades, by
the end of Grade 8, students:
| |
8. |
Use estimated and actual measurements to describe and compare
phenomena. |
9. |
Read and interpret various scales, including those based on
number lines and maps. |
10. |
Determine the degree of accuracy needed in a given situation
and choose units accordingly. |
11. |
Understand that all measurements of continuous quantities are
approximate. |
12. |
Develop formulas and procedures for solving problems related
to measurement. |
13. |
Explore situations involving quantities which cannot be
measured directly or conveniently. |
14. |
Convert measurement units from one form to another, and carry
out calculations that involve various units of measurement. |
15. |
Understand and apply measurement in their own lives and in
other subject areas. |
16. |
Understand and explain the impact of the change of an object's
linear dimensions on its perimeter, area, or volume. |
17. |
Apply their knowledge of measurement to the construction of a
variety two- and three-dimensional figures.
|
Building upon knowledge and skills gained in the preceding grades, by
the end of Grade 12, students:
| |
18. |
Determine the degree of accuracy of a measurement, for example
by understanding and using significant digits. |
19. |
Develop and use the concept of indirect measurement, and use
techniques of algebra, geometry, and trigonometry to measure
quantities indirectly. |
20. |
Use measurement appropriately in other subject areas and
career-based contexts. |
21. |
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 real-world
data. |
12. |
Recognize the limitations of estimation, and assess the amount
of error resulting from estimation. |
13. |
Determine whether errors resulting from estimation are 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. These explorations present unlimited opportunities for problem-solving, making and verifying generalizations, and building mathematical understanding and confidence.
Cumulative Progress Indicators
By the end of Grade 4, students:
| |
1. |
Reproduce, extend, create, and describe patterns and sequences
using a variety of materials. |
2. |
Use tables, rules, variables, open sentences, and graphs to
describe patterns and other relationships. |
3. |
Use concrete and pictorial models to explore the basic concept
of a function. |
4. |
Observe and explain how a change in one physical quantity can
produce corresponding change in another. |
5. |
Observe and appreciate 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,simple
equations, and graphs. |
8. |
Understand and describe the relationships among various
representations of patterns and functions. |
9. |
Use patterns, relationships, and functions to represent and
solve problems. |
10. |
Analyze functional relationships to explain how a change in
one quantity results in a change in another. |
11. |
Understand and describe the general behavior of functions. |
12. |
Use patterns, relationships, and linear functions to model
situations in mathematics and in other areas. |
13. |
Develop, analyze, and explain arithmetic sequences.
|
Building upon knowledge and skills gained in the preceding grades, by
the end of Grade 12, students:
| |
14. |
Analyze and describe how a change in an independent variable
can produce a change in a dependent variable. |
15. |
Use polynomial, rational, trigonometric, and exponential
functions to model real-world phenomena. |
16. |
Recognize that a variety of phenomena can be modeled by the
same type of function. |
17. |
Analyze and explain the general properties and behavior of
functions, and use appropriate graphing technologies to represent
them. |
18. |
Analyze the effects of changes in parameters on the graphs of
functions. |
19. |
Understand the role of functions as a unifying concept in
mathematics. |
|
Descriptive Statement: Probability and statistics are the mathematics used to understand chance and to collect, organize, 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 medium) 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 compount event. |
14. |
Model situations involving probability, such as genetics,
using both simulations and theoretical models. |
15. |
Use models of probability to predict events based on actual
data. |
16. |
Interpret probabilities as ratios and percents.
|
Building upon knowledge and skills gained in the preceding grades, by
the end of Grade 12, students:
| |
17. |
Estimate probabilities and predict outcomes from actual
data. |
18. |
Understand sampling and recognize its role in statistical
claims. |
19. |
Evaluate bias, accuracy, and reasonableness of data in
real-world contexts. |
20. |
Understand and apply measures of dispersion and
correlation. |
21. |
Design a statistical experiment to study a problem, conduct
the experiment, and interpret and communicate the outcomes. |
22. |
Make predictions using curve fitting and numerical procedures
to interpolate and extrapolate from known data. |
23. |
Use relative frequency and probability, as appropriate, to
represent and solve problems involving uncertainty. |
24. |
Use simulations to estimate probabilities. |
25. |
Create and interpret discrete and continuous probability
distributions, and understand their application to real-world
situations. |
26. |
Describe the normal curve in general terms, and use its
properties to answer questions about sets of data that are assumed to
be normally distributed. |
27. |
Understand and use the law of large numbers (that experimental
results tend to 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 real-life situation.
Cumulative Progress Indicators
By the end of Grade 4, students:
| |
1. |
Understand and represent numerical situations using variables,
expressions, and number sentences. |
2. |
Represent situations and number patterns with concrete materials,
tables, graphs, verbal rules, and number sentences, and translate from
one to another. |
3. |
Understand and use properties of operations and numbers. |
4. |
Construct and solve open sentences (example: 3 + ___ = 7) that
describe real-life situations.
|
Building upon knowledge and skills gained in the preceding grades, by
the end of Grade 8, students:
| |
5. |
Understand and use variables, expressions, equations, and
inequalities. |
6. |
Represent situations and number patterns with concrete materials,
tables, graphs, verbal rules, and standard algebraic notation. |
7. |
Use graphing techniques on a number line to model both absolute
value and arithmetic operations. |
8. |
Analyze tables and graphs to identify properties and
relationships. |
9. |
Understand and use the rectangular coordinate system. |
10. |
Solve simple linear equations using concrete, informal, and
graphical methods. |
11. |
Explore linear equations through the use of calculators,
computers, and other technology. |
12. |
Investigate inequalities and nonlinear equations
informally. |
13. |
Draw freehand sketches of, and interpret, graphs which model
real phenomena.
|
Building upon knowledge and skills gained in the preceding grades, by
the end of Grade 12, students:
| |
14. |
Model and solve problems that involve varying quantities
using variables, expressions, equations, inequalities, absolute
values, vectors, and matrices. |
15. |
Use tables and graphs as tools to interpret expressions,
equations, and inequalities. |
16. |
Develop, explain, use, and analyze procedures for operating
on algebraic expressions and matrices. |
17. |
Solve equations and inequalities of varying degrees using
graphing calculators and computers as well as appropriate
paper-and-pencil techniques. |
18. |
Understand the logic and purposes of algebraic procedures. |
19. |
Interpret algebraic equations and inequalities geometrically,
and describe geometric objects algebraically. |
|
Descriptive Statement: Discrete mathematics is the branch of mathematics that deals with finite arrangements of elements. 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 elements are cities arranged on a map; how to count the number of different selections of toppings for pizzas, where the elements are combinations of toppings; how best to schedule a list of tasks to be done, where some tasks cannot begin until others end; and how computers store and retrieve arrangements of information on a screen. Increasingly, it is the mathematics used by decision-makers in our society, from workers in government to those in health care, transportation, and telecommunications. The logical and practical emphases of discrete math problems and solutions 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 -- the infinitely large or the infinitesimally small. Early explorations in these areas can broaden students' interest in and understanding of an important area of applied mathematics.
Cumulative Progress Indicators
By the end of Grade 4, students:
| |
1. |
Investigate and describe patterns that continue
indefinitely. |
2. |
Investigate and describe how certain quantities change over
time. |
3. |
Experiment with approximating length, area, and volume, using
informal measurement instruments.
|
Building upon knowledge and skills gained in the preceding grades, by
the end of Grade 8, students:
| |
4. |
Recognize and express the difference between linear and
exponential growth. |
5. |
Develop an understanding of infinite sequences that arise in
natural situations. |
6. |
Investigate, represent, and use non-terminating decimals. |
7. |
Represent, analyze, and predict relations between quantities,
especially quantities changing over time. |
8. |
Approximate quantities with increasing degrees of accuracy. |
9. |
Understand and use the concept of significant digits. |
10. |
Develop informal ways of approximating the surface area and
volume of familiar objects, and discuss whether the approximations
make sense. |
11. |
Express mathematically and explain the impact of the change of
an object's linear dimensions on its surface area and volume.
|
Building upon knowledge and skills gained in the preceding grades, by
the end of Grade 12, students:
| |
12. |
Develop and use models based on sequences and series. |
13. |
Develop and apply procedures for finding the sum of finite
arithmetic series and of finite and infinite geometric series. |
14. |
Develop an informal notion of limit. |
15. |
Use linear, quadratic, trigonometric, and exponential models
to explain growth and change in the natural world. |
16. |
Recognize fundamental mathematical models (such as polynomial,
exponential, and trigonometric functions) and apply basic
translations, reflections, and dilations to their graphs. |
17. |
Develop the concept of the slope of a curve, apply slopes to
measure the steepness of curves, interpret the meaning of the slope of
a curve for a given graph, and use the slope to discuss the
information contained in the graph. |
18. |
Develop an understanding of the concept of continuity of a
function. |
19. |
Understand and apply approximation techniques to situations
involving initial portions of infinite decimals and measurement. |
|
Descriptive Statement: High expectations for all students form a critical part of the learning environment. The belief of teachers, administrators, and parents that a student can and will succeed in mathematics often makes it possible for that student to succeed. Beyond that, this standard calls for a commitment that all students will be continuously challenged and enabled to go as far mathematically as they can.
Cumulative Progress Indicators
By the end of Grade 12, students:
| |
1. |
Study a core curriculum containing challenging ideas and tasks,
rather than one limited to repetitive, low-level cognitive
activities. |
2. |
Work at rich, open-ended problems which require them to use
mathematics in meaningful ways, and which provide them with exciting
and interesting mathematical experiences. |
3. |
Recognize mathematics as integral to the development of all
cultures and civilizations, and in particular to that of our own
society. |
4. |
Understand the important role that mathematics plays in their own
success, regardless of career. |
5. |
Interact frequently with parents and other members of their
communities, including men and women from a variety of cultural
backgrounds, who use mathematics in their daily lives and
occupations. |
6. |
Receive services that help them understand the mathematical
skills and concepts necessary to assure success in the core
curriculum. |
7. |
Receive equitable treatment without regard to gender, ethnicity,
or predetermined expectations for success. |
8. |
Learn mathematics in classes which reflect the diversity of the
school's total student population. |
9. |
Be provided with opportunities at all grade levels for further
study of mathematics, especially including topics beyond traditional
computation, algebra, and geometry. |
10. |
Be challenged to maximize their mathematical achievements at all
grade levels. |
11. |
Experience a full program of meaningful mathematics so that they
can pursue post-secondary education. |