Mathematical Reasoning and Communication; Number Theory and Concepts

Algebra, Analytic Geometry, and Calculus

Geometry and Trigonometry

Data Analysis, Probability, Statistics, and Discrete Mathematics

SUBAREA I—MATHEMATICAL REASONING AND COMMUNICATION

Test Objective 0001

Understand reasoning processes, including inductive and deductive logic and symbolic logic.

For example:

analyzing mathematical situations by gathering evidence, making conjectures, formulating counterexamples, and constructing and evaluating arguments.inquiry and engineering design

analyzing the nature and purpose of axiomatic systems (including those of the various geometries)

analyzing and interpreting the truth value of simple and compound statements (e.g., negations, disjunctions, conditionals) in truth tables and Ven diagrams

using laws of inference to draw conclusions and to test the validity of conclusions

applying the principle of mathematical induction to prove theorems

Using a variety of software (e.g., spreadsheets, graphing utilities, statistical packages, simulations) and infonnation technologies to model and solve problems in mathematics, science, and technology

Test Objective 0002

Understand the meaning of mathematical concepts and symbols and how to communicate mathematical ideas in writing.

For example:

translating among algebraic, graphic, numeric, and written modes of presenting mathematical ideas

converting between mathematical language, notation, and symbols and standard English language

deducing the assumptions inherent in a given mathematical statement, expression, or definition

evaluating the precision or accuracy of a mathematical statement

Test Objective 0003

Understand mathematical modeling and apply multiple mathematical representations to connect ,athematical ideas and solve problems.

For example:

evaluating mathematical models (e.g., graphs, equations, physical and pictorial representations) and analyzing their appropriateness, efficiency, and accuracy in solving a given problem

analyzing techniques of estimation and identifying situations in which estimation is appropriate

using estimation to evaluate the reasonableness of a solution to a problem

representing problem situations using a variety of mathematical models (e.g., algebraic expressions, sequences, diagrams, geometric figures, graphs)

analyzing the use of software (e.g., simulations, spreadsheets) to model and solve problems

SUBAREA 11—ALGEBRA

Test Objective 0004

Understand principles and properties of the set of complex numbers and its subsets.

For example:

applying principles of number theory (e.g., prime numbers, divisibility) to solve problems

applying number concepts (e.g., fractions, percents, exponents) to solve problems

applying knowledge of real numbers to arithmetic and algebraic operations.

justifying the need for the extension of a given number system.

using multiple representations of the complex numbers and their operations (e.g., polar form; algebraic and geometric interpretations of the sum, difference, and product of complex numbers)

analyzing and applying the properties of vectors, groups, and fields to the complex numbers and its subsets

Test Objective 0005

Understand the principles and properties of patterns and algebraic operations and relations.

For example:

determining algebraic expressions that best represent patterns among data presented in tables, graphic, and diagrams

generalizing patterns using explicitly defined and recursively defined functions

performing and analyzing basic operations on numbers and algebraic expressions

distinguishing the algebraic model that best represents a given situation and analyzing the strengths and weaknesses of that model

applying algebraic concepts of relation and function (e.g., range, domain, inverse) to analyze mathematical relationships

analyzing the results of transformations (e.g., translations, dilations, reflections, rotations) on the graphs of functions

Test Objective 0006

Understand the principles and properties of linear functions and relations.

For example:

analyzing a linear equation in terms of slope and intercepts

determining the linear function that best models a set of data

solving systems of linear equations and inequalities using a variety of techniques (e.g., algebraic, graphic, matrix)

applying properties of linear equations and inequalities to model and solve a variety of real-world problems

using techniques of linear programming to model and solve real-world problems

determining connections among proportions, linear functions, and constant rates of change

Test Objective 0007

Understand the properties of quadratic and higher-order functions and relations.

For example:

analyzing the characteristics of the roots of quadratic and higher-order polynomial functions

solving systems of quadratic equations or inequalities using a variety of techniques (e.g., factoring, graphing, completing the square, quadratic formaula)

modeling and solving a variety of mathematical and real-world problems involving quadratic functions and relations

analyzing symbolic, graphic, or tabular representations of a given quadratic and higher-order polynomial relation

analyzing the results of changing parameters on the graphs of quadratic and higher-order polynomial functions

applying properties of polynomials to model and solve problems

Test Objective 0008

Understand the properties of rational, radical, and absolute value functions and relations.

For example:

analyzing the properties of a given function (e.g., range, domain, asymtotes)

solving problems involving rational, radical, and absolute value functions using various algebraic techniques

modeling and solving problems graphically using systems of equations and inequalities involving rational, radical, and absolute value relations (including the use of graphing calculators)

interpreting and analyzing the effects of transformations on the graph of a function

applying the properties of rational, radical, and absolute value functions to model and solve problems

Test Objective 0009

Understand the properties of exponential and logarithmic functions.

For example:

applying the relationship between logaruthmic and exponential functions

applying the laws of logarithms to manipulate and simplify expressions

analyzing the graph of a logarithmic or exponential function or relation

Solving problems involving exponential growth and decay

modeling and solving problems analytically or graphically using exponential and logarithmic relations

SUBAREA III—TRIGONOMETRY AND CALCULUS

Test Objective 0010

Understand principles, properties, and relationships involving trigonometric functions and their associated geometric representations.

For example:

using degree and radian measures

analyzing connections and identities among right traingle ratios, trigonometric functions, and the unit circle

analyzing characteristics of the graph of a trigonometric function (e.g., frequency, period, amplitude, phase shift)

determining connections between trigonometric functions and power series

using the complex exponential function to explore properties of trigonometric functions

Test Objective 0011

Apply the principles and techniques of trigonometry to model and solve problems.

For example:

Applying trigonometric functions to solve problems involving length, area, volume, or angle measure (e.g., arcs, angles and sectors associated with a circle, unknown sides and angles of polygons, vectors)

using circular functions to model periodic phenomena

solving trigonometric equations using analytic or graphing teachniques

Modeling and solving problems involving trigonometric functions

Test Objective 0012

Demonstrate an understanding of the fundamental concepts of calculus.

For example:

analyzing the concept of limit numerically, algebraically, graphically, and in writing

interpreting the derivative as the limit of the difference quotient

interpreting the definite integral as the limit of a Riemann sum

applying the fundamental theorem of calculus

applying concepts of derivatives to interpret gradients, tangents, and slopes.

applying the concept of limit to analyze and interpret the properties of functions (e.g., continuity, asymptotes)

applying the concept of rate of change to interpret statements from science, technology, economics, and other disciplines

Test Objective 0013

Apply the principles and techniques of calculus to model and solve problems.

For example:

Using derivatives to model and solve real-world problems (e.g., rates of change, related rates, optimization).

applying properties of derivatives to analyze the graphs of functions

Using integration to model and solve problems (e.g., the area under a curve, work, applications of antiderivatives).

modeling and solving problems involving first order differential equations (e.g., separation of variables, initial value problems).

SUBAREA IV—MEASUREMENT AND GEOMETRY

Test Objective 0014

Understand and apply measurement principles.

For example:

applying appropriate measurement toold and units and converting within and between measurement system (e.g., conversion factors, dimensional analysis)

recording measurements to the appropriate degree of accuracy and analyzing the effect of uncertainity on derived measures

deriving and applying formulas to find measures such as length, angle, area, and volume for a variety of geometric figures.

using nets and cross sections to determine volume and surface area formulas of three-dimensional figures

applying the Pythagorean theorem to solve measurement problems

Test Objective 0015

Understand the principles of axiomatic (synthetic) geometries.

For example:

using the properties of lines and angles (e.g., parallelism, perpendicularity, supplementary angles, vertical angles) to characterize geometric relationships

using the concepts of similarity and congruence to analyze the properties of geometric figures (e.g., triangle, parallelogram, polygon, circle)

analyzing procedures used in geometric construction (e.g., constructing the perpendicular bisector of a given line segment)

proving theorems using an axiomatic system

analyzing, comparing, and contrasting the axiomatic structure and properties of various geometries (e.g., Euclidean, non-Euclidean, projective)

applying geometric principles to analyze three-dimensional figures

Test Objective 0016

Understand the principles and properties of coordinate geometry.

For example:

applying the principles of distance, midpoint, slope, parallelism, and perpendicularity to characterize coordinate geometric relationships

using coordinate geometry to prove theorems about geometric figures (e.g., triangle, parallelogram, circle, parabola, hyperbola)

representing two- and three-dimensional geometric figures in various coordinate systems (e.g., Cartesian, polar)

analyzing and applying transformations in the coordinate plane

applying the distance formula to derive the equation of a conic section

modeling and solving problems using conic section

Test Objective 0017

Apply mathematical principles and techniques to model and solve problems involving vector and transformational geometries.

For example:

modeling and solving problems involving vector addition and scalar multiplication (e.g., force)

applying principles of geometry to model and solve problems involving the composition of transformations (e.g., translations, reflections, dilations, rotations).

analyzing how transformational geometry and symmetry are used in art and architecture (e.g., tessellations, tilings, frieze patterns, fractals).

using multiple representations of geometric transformations (e.g., coordinate, matrix, diagrams)

proving theorems using vector and transformational mathods

SUBAREA V—DATA ANALYSIS, PROBABILITY, STATISTICS, AND DISCRETE MATHEMATICS.

Test Objective 0018

Understand the principles, properties, and techniques related to sequence, series, summation, and counting strategies and their applications to problem solving.

For example:

using first- or second-order finite differences to analyze sequences

modeling and solving problems using the properties of sequences or series (e.g., arithmetic, geometric, Fibonacci)

solving a variety of problems involving permutations and combinations

analyzing the relationship between the binomial coefficients and Pascal's triangle

Test Objective 0019

Understand principles, properties, and techniques of probability and their applications.

For example:

evaluate probability of events (e.g., joint, conditional, independent, mutual exclusive)

interpreting graphic representations of probabilities (e.g., tables, charts, Venn diagrams, tree diagrams, frequency graphs, the normal curve)

modeling and solving problems involving uncertainty using the techniques of probability (e.g., addition and multiplication rules, Bernoulli experiment).

using simulations to estimate probabilites and analyzing the relationships between probability and statistics

solving problems involving random variability and probability distributions (e.g., binomial, normal)

Test Objective 0020

Understand the principles, properties, and techniques of data analysis and statistics.

For example:

applying the measures of central tendency, dispersion, and skewness to summarize and interpret data presented in graphic, tabular, or pictorial form

evaluating the statistical claims made for a given set of data (e.g., analyzing assumptions made in the sampling, analysis, and testing of statistical hypotheses) and how measures of realibility vary by discipline

interpreting the outcomes of a given statistical test (e.g., t-test, chi-square analysis, correlation, linear regression)

using graphing calculators to analyze and interpret data from a variety of disciplines (e.g., sciences, social sciences, technology)

analyzing data using linear, logarithmic, exponential, and power regression models

Test Objective 0021

Understand how techniques of discrete mathematics (e.g., diagrams, graphs, matrices, propositional statements) are applied in the analysis, interpretation, communication, and solution of problems.

For example:

representing finite data using a variety of techniques.

representing real-world situations and relationships using sequences and recurrence relations.

modeling and solving problems using graphs and matricies

evaluating the use of computers and calculators to solve problems (e.g., developing and analyzing algorithms)