# 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% |

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**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