# Study Guide

## Overview and Test ObjectivesField 089: Mathematics (Elementary)

### 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
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 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 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 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 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 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 ofpoints, 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 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 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