# Study Guide

## Overview and Test ObjectivesField 022: Mathematics (Secondary)

### Test Overview

Format Computer-based test (CBT) 80 multiple-choice questions 2 hours 30 minutes* 220

*Does not include 15-minute CBT tutorial

### Test Objectives

Table outlining test content and subject weighting by sub area and objective.
Subarea Range of Objectives Approximate Percentage of Questions on Test
1 Mathematical Processes and Number Concepts 001–004 22%
2 Patterns, Algebraic Relationships, and Functions 005–009 28%
3 Measurement and Geometry 010–013 22%
4 Data Analysis, Statistics, Probability, and Discrete Mathematics 014–018 28%

Hover over each subarea for details of subtest content or see table above.

Sub area 1 22%, Sub area 2 28%, Sub area 3 22%, and Sub area 4 28%.

#### Subarea I1—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 II2—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), cf(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 III3—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 IV4—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