Study Guide
Overview and Test Objectives
Field 089: Mathematics (Elementary)
Test Overview
Format  Computerbased test (CBT) and paperbased test (PBT) 

Number of Questions  80 multiplechoice questions 
Time 

Passing Score  220 
*Does not include 15minute CBT tutorial
Test Objectives
Subarea  Range of Objectives  Approximate Percentage of Questions on Test  

I  Mathematical Processes and Number Concepts  001–005  28% 
II  Patterns, Algebraic Relationships, and Functions  006–010  28% 
III  Measurement and Geometry  011–014  22% 
IV  Data Analysis, Statistics, Probability, and Discrete Mathematics  015–018  22% 
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 (e.g., understanding the relationships among theorems, postulates, definitions, and undefined terms)
 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 problemsolving strategies, connections among different mathematical ideas, and the use of mathematical modeling to solve realworld problems.
Includes:
 devising, carrying out, and evaluating a problemsolving plan
 evaluating the reasonableness of a solution
 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., area as a quadratic function)
 exploring the relationship between geometry and algebra
 applying mathematics across the curriculum and in everyday contexts
Objective 003—Understand and apply concepts of proportional reasoning.
Includes:
 analyzing connections between fraction concepts and ratios and proportions
 describing the relationship between proportions and direct and inverse variation
 analyzing and applying the relationship between proportions and similar figures
 applying connections among proportions, probability, and sampling
 analyzing a variety of representations of proportional relationships
 modeling and solving problems involving ratios and proportions
Objective 004—Understand number systems and equivalent ways of representing numbers.
Includes:
 applying place value concepts to numeration systems
 identifying characteristics and relationships among natural, whole, integer, rational, irrational, and real numbers
 using a variety of equivalent representations of numbers (e.g., ½ = 0.5 = 50% = √¼)
 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 005—Understand number theory and operations on number systems.
Includes:
 analyzing properties of prime numbers, factors, multiples, and divisibility
 applying number properties to manipulate and simplify algebraic expressions
 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)
 using manipulatives, verbal expressions, and geometric models to represent number operations
 applying and evaluating mental mathematics and estimation strategies
 analyzing standard and nonstandard computational algorithms
 solving a variety of problems using number operations
Subarea II—PATTERNS, ALGEBRAIC RELATIONSHIPS, AND FUNCTIONS
Objective 006—Describe, analyze, and generalize mathematical patterns.
Includes:
 recognizing and extending numerical and geometric patterns
 constructing, representing, and recording patterns using charts, tables, graphs, and matrices
 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 007—Use variables and symbolic expressions to describe and analyze patterns of change and functional relationships.
Includes:
 representing situations using variables and expressions
 exploring patterns of change characteristic of families of functions (e.g., linear, quadratic, exponential)
 translating among verbal, graphic, tabular, and symbolic representations of functions
 distinguishing between relations and functions
 analyzing functions in terms of range, domain, and intercepts
 using piecewise functions
 analyzing the relationship among the graphs of f(x) and transformations [e.g., f(x ± c), f(x) ± c, c f(x), one over f parentheses x end parentheses
 using graphing calculators and utilities to analyze properties of functions
Objective 008—Understand properties and applications of linear functions, and solve related equations and inequalities.
Includes:
 describing properties of slope and intercepts
 analyzing the relationship between a linear equation and its graph
 determining the equation of a line in a variety of situations
 modeling problems using linear equations and inequalities
 solving linear systems using a variety of methods (e.g., using substitution, using graphs, using matrices)
Objective 009—Understand properties and applications of quadratic functions, and solve related equations and inequalities.
Includes:
 solving quadratic equations, inequalities, and systems using a variety of methods (e.g., graphical, analytical)
 exploring the zeros, turning point (vertex), and symmetry of a quadratic function
 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 010—Understand properties and applications of nonlinear functions and the conceptual foundations of calculus.
Includes:
 using exponential functions to model and solve realworld problems
 recognizing the relationship between inverse variation and rational functions
 exploring the properties and graphs of polynomial, rational, radical, exponential, logarithmic, and trigonometric (i.e., sine, cosine, tangent) functions
 using graphing calculators to solve systems of equations involving these functions
 analyzing the relationships among the graph, slope of the secant line, and the derivative of a function
 recognizing the relationship between the area under a curve and integration
 describing how calculus is used to solve problems involving dynamic change
Subarea III—MEASUREMENT AND GEOMETRY
Objective 011—Understand attributes of measurement and measuring units.
Includes:
 selecting appropriate units (standard and nonstandard) 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 012—Apply measurement principles to analyze the spatial characteristics of two and threedimensional shapes.
Includes:
 deriving and applying formulas for the perimeter, area, surface area, or volume of two and threedimensional 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 threedimensional drawings of objects
 analyzing cross sections and nets of threedimensional figures
Objective 013—Apply geometric principles of points, lines, angles, planes, congruence, and similarity to analyze the formal characteristics of two and threedimensional shapes.
Includes:
 determining necessary and sufficient conditions for the existence of a particular shape
 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 twodimensional shapes to analyze threedimensional shapes
 recognizing the uses of dynamic geometry software in making conjectures and investigating properties of shapes
Objective 014—Apply properties of geometric transformations and coordinate geometry 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
 locating objects in terms of their position using rectangular 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
Subarea IV—DATA ANALYSIS, STATISTICS, PROBABILITY, AND DISCRETE MATHEMATICS
Objective 015—Understand methods of organizing, displaying, analyzing, and interpreting data.
Includes:
 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)
 evaluating the source, organization, and presentation 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
 using appropriate technology to analyze and manipulate data
 evaluating the validity of statistical arguments
Objective 016—Understand methods of collecting data and making predictions and inferences based on data.
Includes:
 applying appropriate techniques for collecting data
 analyzing factors that may affect the validity of a survey, including bias
 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
 determining conditional probabilities
 finding the probability of dependent and independent events
 calculating expected values
 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 realworld 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 finite graphs and trees to model problem situations
 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 computerrelated situations)
 using linear programming to model and solve problems