Standard: 2 Understands and applies basic and advanced properties of the concepts of numbers

2-1 Understands the properties (e.g., relative magnitude, density, absolute value) of the real number system and its subsystems (e.g., irrational numbers, natural numbers, integers, rational numbers)

2-2 Understands the properties and basic theorems of roots, exponents (e.g., [bm][bn] = bm+n), and logarithms

2-3 Understands that mathematical systems that appear to be very different may have the same structural underpinnings (e.g., binary multiplication, a series electrical circuit, and the logical operation "and" have the equivalent roles of "0," "off," and "false," as well as of "1," "on," and "true," respectively)

2-4 Uses number theory concepts (e.g., divisibility and remainders, factors, multiples, prime, relatively prime) to solve problems

2-5 Uses discrete structures (e.g., finite graphs, matrices, sequences) to represent and to solve problems

Standard: 3 Uses basic and advanced procedures while performing the processes of computation

3-1 Adds, subtracts, multiplies, divides, and simplifies rational expressions

3-2 Adds, subtracts, multiplies, divides, and simplifies radical expressions containing positive rational numbers

3-3 Understands various sources of discrepancy between an estimate and a calculated answer

3-4 Uses a variety of operations (e.g., finding a reciprocal, raising to a power, taking a root, taking a logarithm) on expressions containing real numbers

3-5 Understands basic applications of and operations on matrices

3-6 Uses recurrence relations (i.e., formulas expressing each term as a function of one or more of the previous terms, such as the Fibonacci sequence or the compound interest equation) to model and to solve real-world problems (e.g., home mortgages, annuities)

3-7 Understands counting procedures and reasoning (e.g., use of the Addition Counting Principle to find the number of ways of arranging objects in a set, the use of permutations and combinations to solve counting problems)

Standard: 4 Understands and applies basic and advanced properties of the concepts of measurement

4-1 Solves problems involving rate as a measure (e.g., velocity, acceleration)

4-2 Understands the concepts of absolute and relative errors in measurement

4-3 Selects and uses an appropriate direct or indirect method of measurement in a given situation (e.g., uses properties of similar triangles to measure indirectly the height of an inaccessible object)

4-4 Solves real-world problems involving three-dimensional measures (e.g., volume, surface area)

Standard: 5 Understands and applies basic and advanced properties of the concepts of geometry

5-1 Understands that objects and relations in geometry correspond directly to objects and relations in algebra (e.g., a line in geometry corresponds to a set of ordered pairs satisfying an equation of the form ax + by = c)

5-2 Uses the Pythagorean Theorem and its converse and properties of special right triangles (e.g., 30ø-60ø-90ø triangle) to solve mathematical and real-world problems

5-3 Uses synthetic (i.e., pictorial) representations and analytic (i.e., coordinate) methods to solve problems involving symmetry and transformations of figures (e.g., problems involving distance, midpoint, and slope; determination of symmetry with respect to a point or line)

5-4 Understands the characteristics and uses of vectors (e.g., representations of velocity and force)

5-5 Uses geometric constructions (e.g., the parallel to a line through a given point not on the line, line segment congruent to a given line segment) to complete simple proofs, to model, and to solve mathematical and real-world problems

5-6 Uses basic operations on vectors (e.g., vector addition, scalar multiplication)

5-7 Understands the basic concepts of right triangle trigonometry (e.g., basic trigonometric ratios such as sine, cosine, and tangent)

5-8 Uses trigonometric ratio methods to solve mathematical and real-world problems (e.g., determination of the angle of depression between two markers on a contour map with different elevations)

5-9 Understands the basic properties and uses of polar coordinates

5-10 Uses inductive and deductive reasoning to make observations about and to verify properties of and relationships among figures (e.g., the relationship among interior angles of parallel lines cut by a transversal)

5-11 Uses properties of and relationships among figures to solve mathematical and real-world problems (e.g., uses the property that the sum of the angles in a quadrilateral is equal to 360 degrees to square up the frame for a building; uses understanding of arc, chord, tangents, and properties of circles to determine the radius given a circular edge of a circle without the center)

Standard: 6 Understands and applies basic and advanced concepts of statistics and data analysis

6-1 Selects and uses the best method of representing and describing a set of data (e.g., scatter plot, line graph, two-way table)

6-2 Understands measures of central tendency and variability (e.g., standard deviation, range, quartile deviation) and their applications to specific situations

6-3 Understands the concept of correlation (e.g., the difference between a "true" correlation and a "believable" correlation; when two variables are correlated)

6-4 Understands different methods of curve-fitting (e.g., median-fit line, regression line) and various applications (e.g., making predictions)

6-5 Understands how outliers may affect various representations of data (e.g., a regression line might be strongly influenced by a few aberrant points, whereas the scatter plot for the same data might suggest that the aberrant points represent mistakes)

6-6 Understands how the reader's bias, measurement error, and display distortion can affect the interpretation of data

6-8 Understands sampling distributions, the central limit theorem, and confidence intervals

6-9 Understands how concepts of representativeness, randomness, and bias in sampling can affect experimental outcomes and statistical interpretations

6-10 Understands that making an inference about a population from a sample always involves uncertainty and the role of statistics is to estimate the size of that uncertainty

Standard: 7 Understands and applies basic and advanced concepts of probability

7-1 Understands the concept of a random variable

7-2 Understands the concepts of independent and dependent events and how they are related to compound events and conditional probability

7-3 Uses a variety of experimental, simulation, and theoretical methods (e.g., counting procedures, trees, formulas for permutations and combinations, Monte Carlo simulations, statistical experiments) to determine probabilities

7-4 Understands the differences among experimental, simulation, and theoretical probability techniques and the advantages and disadvantages of each

7-5 Understands the properties of the normal curve (i.e., the graph used to approximate the distribution of data for many real-world phenomena) and how the normal curve can be used to answer questions about sets of data

7-6 Understands the concept of discrete probability distribution

Standard: 8 Understands and applies basic and advanced properties of functions and algebra

8-1 Understands appropriate terminology and notation used to define functions and their properties (e.g., domain, range, function composition, inverses)

8-2 Uses expressions, equations, inequalities, and matrices to represent situations that involve variable quantities and translates among these representations

8-3 Understands characteristics and uses of basic trigonometric functions (e.g., the sine and cosine functions as models of periodic real-world phenomena)

8-4 Understands properties of graphs and the relationship between a graph and its corresponding expression (e.g., maximum and minimum points)

8-5 Understands basic concepts (e.g., roots) and applications (e.g., determining cost, revenue, and profit situations) of polynomial equations

8-6 Understands the concept of a function as the correspondences between the elements of two sets (e.g., in algebra, functions are relationships between variables that represent numbers; in geometry, functions relate sets of points to their images under motions such as flips, slides, and turns; in the "real-world," functions are mathematical representations of many input-output situations)

8-7 Uses a variety of models (e.g., written statement, algebraic formula, table of input-output values, graph) to represent functions, patterns, and relationships

8-8 Understands the general properties and characteristics of many types of functions (e.g., direct and inverse variation, general polynomial, radical, step, exponential, logarithmic, sinusoidal)

8-9 Understands the effects of parameter changes on functions and their graphs

8-10 Understands the basic concept of inverse function and the corresponding graph

8-11 Uses a variety of methods (e.g., with graphs, algebraic methods, and matrices) to solve systems of equations and inequalities

8-12 Understands formal notation (e.g., sigma notation, factorial representation) and various applications (e.g., compound interest) of sequences and series

8-13 Uses a variety of methods (e.g., approximate solutions, such as bisection, sign changes, and successive approximation) to solve complex equations (e.g., polynomial equations with real roots)

Standard: 9 Understands the general nature and uses of mathematics

9-1 Understands that mathematics is the study of any pattern or relationship, but natural science is the study of those patterns that are relevant to the observable world

9-2 Understands that mathematics began long ago to help solve practical problems; however, it soon focused on abstractions drawn from the world and then on abstract relationships among those abstractions

9-3 Understands that in mathematics, as in other sciences, simplicity is one of the highest values; some mathematicians try to identify the smallest set of rules from which many other propositions can be logically derived

9-4 Understands that theories in mathematics are greatly influenced by practical issues; real-world problems sometimes result in new mathematical theories and pure mathematical theories sometimes have highly practical applications

9-5 Understands that new mathematics continues to be invented even today, along with new connections between various components of mathematics

9-6 Understands that science and mathematics operate under common principles: belief in order, ideals of honesty and openness, the importance of review by colleagues, and the importance of imagination

9-7 Understands that mathematics provides a precise system to describe objects, events, and relationships and to construct logical arguments

9-8 Understands that the development of computers has opened many new doors to mathematics just as other advances in technology can open up new areas to mathematics

9-9 Understands that mathematics often stimulates innovations in science and technology

9-10 Understands that mathematicians commonly operate by choosing an interesting set of rules and then playing according to those rules; the only limit to those rules is that they should not contradict each other