# Study Guide

## Overview and Test Objectives

Field 022: Mathematics (Secondary)

### Test Overview

Format | Computer-based test (CBT) |
---|---|

Number of Questions | 80 multiple-choice questions |

Time | 2 hours 30 minutes* |

Passing Score | 220 |

*Does not include 15-minute CBT tutorial

### Test Objectives

Subarea | Range of Objectives | Approximate Percentage of Questions on Test | |
---|---|---|---|

I | Mathematical Processes and Number Concepts | 001–004 | 22% |

II | Patterns, Algebraic Relationships, and Functions | 005–009 | 28% |

III | Measurement and Geometry | 010–013 | 22% |

IV | Data Analysis, Statistics, Probability, and Discrete Mathematics | 014–018 | 28% |

#### Subarea I—MATHEMATICAL PROCESSES AND NUMBER CONCEPTS

##### Objective 001—Understand principles of mathematical reasoning and techniques for communicating mathematical ideas.

Includes:

- analyzing the nature and purpose of axiomatic systems
- using inductive and deductive logic to develop and validate conjectures
- applying the laws of deductive logic to draw valid conclusions
- developing counterexamples to a conjecture
- developing and evaluating direct and indirect proofs
- using appropriate mathematical terminology
- translating common language into symbols and vice versa
- using a variety of numeric, symbolic, and graphic methods to communicate mathematical ideas and concepts
- making connections among numeric, symbolic, graphic, and verbal representations

##### Objective 002—Understand problem-solving strategies, connections among different mathematical ideas, and the use of mathematics in other fields.

Includes:

- devising, carrying out, and evaluating a problem-solving plan
- applying a range of strategies (e.g., drawing a diagram, working backwards, creating a simpler problem) to solve problems
- analyzing problems that have multiple solutions
- selecting an appropriate tool or technology to solve a given problem
- recognizing connections among two or more mathematical concepts (e.g., Fibonacci numbers and the golden rectangle symmetry and group theory)
- exploring the relationship between geometry and algebra
- applying mathematics across the curriculum and in everyday contexts

##### Objective 003—Understand number systems and equivalent ways of representing numbers.

Includes:

- identifying characteristics and relationships among natural, whole, integer, rational, irrational, real, imaginary, and complex numbers (e.g., 1 half = 0.5 = 50% = square root of one fourth )
- applying properties of number operations (e.g., commutative, distributive)
- applying order relations to numbers
- using set operations (e.g., union, intersection, complement)
- using manipulatives, verbal expressions, and geometric models to represent numbers

##### Objective 004—Understand number theory and operations on number systems.

Includes:

- analyzing properties of prime numbers, factors, multiples, and divisibility
- extending the relationships of primes, factors, multiples, and divisibility in an algebraic setting
- using scientific notation to compute with very large and very small numbers
- comparing and contrasting models of operations across number systems (e.g., using a rectangular array to model multiplication of whole numbers and fractions)
- solving problems involving ratios and proportional reasoning
- using manipulatives, verbal expressions, and geometric models to represent number operations
- applying and evaluating estimation strategies
- analyzing standard and nonstandard computational algorithms
- solving a variety of problems using number operations
- performing operations with complex numbers (e.g., conjugates, products, roots)
- using rectangular, polar, matrix, and vector representations to solve problems

#### Subarea II—PATTERNS, ALGEBRAIC RELATIONSHIPS, AND FUNCTIONS

##### Objective 005—Describe, analyze, and generalize mathematical patterns.

Includes:

- recognizing and extending numerical and geometric patterns
- constructing, representing, and recording patterns using charts, tables, graphs, matrices, and vectors
- exploring and describing symmetric and spatial patterns (e.g., fractals, tessellations)
- analyzing and generalizing sequences, series, and recursive patterns
- using patterns to make inferences, predictions, and decisions

##### Objective 006—Use symbolic expressions to describe and analyze patterns of change and functional relationships.

Includes:

- exploring patterns of change characteristic of families of functions (e.g., quadratic, exponential, periodic)
- translating among verbal, graphic, tabular, and symbolic representations of functions
- distinguishing between relations and functions
- analyzing functions in terms of range, domain, and intercepts
- exploring function operations [e.g., f(
*x*) + g(*x*)], composition [e.g., f(g(*x*))], and inverses - using piecewise- and recursively defined functions
- analyzing the relationship among the graphs of f(
*x*) and transformations such as f(*x*±*c*),*c*f(*x*), and 1 over f(*x*) - using graphing calculators and utilities to analyze properties of functions

##### Objective 007—Understand properties and applications of linear and quadratic functions, and solve related equations and inequalities.

Includes:

- analyzing linear relationships
- modeling and solving problems using linear equations and inequalities
- investigating the relationship between a linear equation and its graph
- modeling and solving problems using linear systems (e.g., using matrices, using graphs)
- solving quadratic equations, inequalities, and systems using a variety of methods (e.g., graphical, analytic)
- using graphing calculators to solve systems of equations
- analyzing how changing the coefficients of a quadratic function changes its graph
- using quadratic functions to model and solve problems, including maximum and minimum problems

##### Objective 008—Understand properties and applications of polynomial, rational, radical, exponential, logarithmic, and trigonometric functions, and solve related equations and inequalities.

Includes:

- exploring the properties and graphs of polynomial, rational, radical, exponential, logarithmic, and trigonometric (i.e., sine, cosine, tangent) functions
- applying these functions to develop and evaluate models of real-world situations
- modeling and solving problems using polynomial, rational, radical, exponential, logarithmic, and trigonometric equations and inequalities
- analyzing the relationship between exponential and logarithmic functions
- examining the relationship between trigonometric functions and their inverses
- examining the relationship between trigonometric functions and circular functions
- modeling periodic phenomena using trigonometric functions

##### Objective 009—Understand principles and applications of differential and integral calculus.

Includes:

- investigating limits and limiting processes
- using limits to determine continuity
- analyzing the relationships among the graph, slope of the secant line, and the derivative of a function
- using differential calculus to analyze the graph of a function
- analyzing the relationship among the area under a curve, Riemann sums, and integration
- using the principles of calculus and appropriate technology to solve a variety of theoretical and applied problems

#### Subarea III—MEASUREMENT AND GEOMETRY

##### Objective 010—Understand attributes of measurement and measuring units.

Includes:

- selecting appropriate units to estimate and record measurements of angle (degree and radian), length, area, volume, mass, temperature, and time
- identifying tools for performing measurements
- converting measurements within measurement systems
- analyzing how changes in the measurement of one attribute relate to changes in others
- using dimensional analysis to solve problems
- solving problems involving density, pressure, rates of change, and other derived units
- evaluating precision, accuracy, measurement errors, and percent error

##### Objective 011—Apply measurement principles to analyze the spatial characteristics of two- and three-dimensional shapes.

Includes:

- deriving and applying formulas for the perimeter, area, surface area, or volume of two- and three-dimensional composite figures
- exploring scale factors for the area and volume of similar figures
- applying right triangle trigonometry and the Pythagorean theorem to solve problems (e.g., problems involving indirect measurements)
- interpreting three-dimensional drawings of objects
- analyzing cross sections and nets of three-dimensional figures

##### Objective 012—Apply geometric principles of points, lines, angles, planes, congruence, and similarity to analyze the formal characteristics of two- and three-dimensional shapes.

Includes:

- determining necessary and sufficient conditions for the existence of a particular shape
- analyzing concepts (e.g., parallelism) in Euclidean and non-Euclidean geometries
- applying properties of parallel and perpendicular lines and angles to analyze shapes
- comparing and analyzing shapes and formally establishing the relationships among them (e.g., congruence, similarity)
- using geometric principles to prove theorems
- applying properties of two-dimensional shapes to analyze three-dimensional shapes
- recognizing the uses of dynamic geometry software in making conjectures and investigating properties of shapes

##### Objective 013—Apply properties of geometric transformations and coordinate and vector methods to describe geometric objects in two and three dimensions.

Includes:

- analyzing figures in terms of translations, reflections, rotations, dilations, and contractions
- applying transformations to explore the concepts of congruence and similarity
- using transformations to characterize the symmetry of an object
- representing transformations using matrices
- analyzing the composition and inverse of transformations
- describing the abstract algebraic properties of a set of transformations under composition
- locating objects in terms of their position using rectangular, polar, and three-dimensional coordinate systems
- locating and describing the locus of points that satisfy a given condition
- applying concepts of slope, distance, midpoint, and parallel and perpendicular lines to determine the geometric and algebraic properties of figures in the coordinate plane (including conic sections)
- describing the position and movement of objects using vectors

#### Subarea IV—DATA ANALYSIS, STATISTICS, PROBABILITY, AND DISCRETE MATHEMATICS

##### Objective 014—Understand methods of collecting, organizing, and displaying data.

Includes:

- formulating questions requiring data gathering and applying appropriate techniques for collecting data
- analyzing factors that may affect the validity of a survey, including bias
- organizing data using tables and spreadsheets
- creating a variety of charts to display data (e.g., pie charts, box plots, stem-and-leaf plots, scatter plots, frequency histograms)
- using appropriate technology to organize and display data
- evaluating the source, organization, and presentation of data

##### Objective 015—Understand methods of describing, analyzing, and interpreting data.

Includes:

- analyzing the shape, location, and spread of a data distribution using algebraic and geometric methods to estimate a variety of statistics
- describing the range and outlines of a set of data
- applying and interpreting measures of central tendency (e.g., mean, median, mode) and spread (e.g., range, standard deviation)
- analyzing the effects of data transformations on measures of central tendency and spread
- finding the function (e.g., linear, exponential, logarithmic) that best represents a set of data
- using appropriate technology to analyze and manipulate data
- evaluating the validity of statistical arguments

##### Objective 016—Understand methods of making predictions and inferences based on data.

Includes:

- analyzing and explaining data trends
- making and testing hypotheses
- using simulations and sampling to test inferences
- applying principles of interpolation and extrapolation
- analyzing linear regression lines and correlation coefficients
- analyzing the relationship between sample size and width of confidence interval
- employing confidence intervals in making predictions and inferences based on data

##### Objective 017—Understand the theory of probability and probability distributions.

Includes:

- enumerating the sample space of an event
- determining simple and compound probabilities
- finding the probability of dependent and independent events
- using simulations and sampling to determine experimental probabilities
- solving problems using geometric probability (e.g., ratio of two areas)
- applying probability distributions (e.g., binomial, normal) to solve problems
- modeling and solving real-world problems using probability concepts

##### Objective 018—Understand principles of discrete mathematics.

Includes:

- solving counting problems using permutations and combinations
- using sets and set relations to represent algebraic and geometric concepts
- using vertex-edge graphs to solve network problems such as finding circuits, critical paths, minimum spanning trees, and adjacency matrices
- proving statements using the principle of mathematical induction
- employing recursion and iteration methods to model problems
- describing and analyzing efficient algorithms to accomplish a task or solve a problem in a variety of contexts (e.g., practical and computer-related situations)
- using discrete mathematics concepts to model a problem, evaluate the existence of solutions, determine the number of possible solutions, and choose the optimal solution to the problem
- using linear programming to model and solve problems