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)
