Study Guide
Overview and Test Objectives
Field 089: Mathematics (Elementary)
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 | |
|---|---|---|---|
| 1 | Mathematical Processes and Number Concepts | 001–005 | 28% |
| 2 | Patterns, Algebraic Relationships, and Functions | 006–010 | 28% |
| 3 | Measurement and Geometry | 011–014 | 22% |
| 4 | Data Analysis, Statistics, Probability, and Discrete Mathematics | 015–018 | 22% |
Hover over each subarea for details of subtest content or see table above.
Sub area 1 28%, Sub area 2 28%, Sub area 3 22%, and Sub area 4 22%.
Subarea 1—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 problem-solving strategies, connections among different mathematical ideas, and the use of mathematical modeling to solve real-world problems.
Includes:
- devising, carrying out, and evaluating a problem-solving 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 2—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 real-world 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 3—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 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 013—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
- 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 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 4—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 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 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 computer-related situations)
- using linear programming to model and solve problems