# General Curriculum Study Guide

## Test Design and Framework

The test design below describes general testing information. The framework that follows is a detailed outline that explains the knowledge and skills that this test measures.

### Test Design

Format Computer-based test (CBT) 45 multiple-choice items, 1 open-response item 2.5 hours testing time for Mathematics subtest 15 minutes additional time to complete a CBT tutorial

### Mathematics Subtest Framework Subareas Range of Objectives Approximate Test Weighting
V Numbers and Operations 16–19 41%
VI Functions and Algebra 20–22 22%
VII Geometry and Measurement 23–24 18%
VIII Statistics and Probablility 25–26 9%
IX Integration of Knowledge and Understanding 27 10%

#### Subarea V–Numbers and Operations

##### Objective 0016: Understand the number system and the concept of place value.

For example:

• Analyze the structures and properties of the base-10 and other numeral systems (e.g., expanded form of a number, visual representations of place value, numeration systems of ancient cultures).
• Recognize decimal expansions.
• Use scientific notation in the real world.
• Analyze differences in written English (e.g., level of formality, how dialects are used).
• Analyze procedures (e.g., rounding, regrouping) for estimation.
• Determine the reasonableness of estimates.
• Identify subsets of the real numbers (e.g., integer, rational, irrational) and their characteristics.
##### Objective 0017: Understand integers, fractions, decimals, percents, and mixed numbers.

For example:

• Understand the meanings and models of integers, fractions, decimals, percents, and mixed numbers and apply them to the solution of word problems.
• Analyze and convert among various representations of numbers (e.g., graphic, numeric, symbolic, verbal).
• Use number lines.
• Compare, sort, order, and round numbers.
• Recognize equivalent representations of numbers (e.g., fractions, decimals, percents).
##### Objective 0018: Understand and apply principles of number theory.

For example:

• Identify prime and composite numbers and their characteristics.
• Find the prime factorization of a number and recognize its uses.
• Demonstrate knowledge of the divisibility rules and why they work.
• Find the least common multiple (LCM) and greatest common factor (GCF) of a set of numbers.
• Apply the LCM and GCF in real-world situations.
##### Objective 0019: Understand operations on numbers.

For example:

• Understand the meaning and models of operations on numbers (e.g., integers, fractions, decimals).
• Analyze and justify standard and nonstandard computational algorithms and mental math techniques (e.g., by application of the arithmetic properties, such as commutative, associative, distributive).
• Evaluate the validity of nonstandard or unfamiliar computational strategies.
• Recognize and analyze various representations (e.g., graphic, pictorial, verbal) of number operations.
• Recognize relationships among operations (e.g., addition and subtraction, addition and multiplication, multiplication and exponentiation).
• Identify and apply the arithmetic properties and the transitive properties of equality and inequality.
• Apply the order of operations.
• Apply the laws of exponents.
• Demonstrate fluency in arithmetic computation, including operations on fractions.
• Interpret the concept of absolute value.
• Apply appropriate strategies (e.g., proportional thinking, ratios) to estimate quantities in real-world situations.
• Solve problems using arithmetic operations with various representations of numbers.

#### Subarea VI–Functions and Algebra

##### Objective 0020: Understand algebra as generalized arithmetic.

For example:

• Recognize and apply the concepts of variable, function, equality, and equation to express relationships algebraically.
• Manipulate simple algebraic expressions and solve linear equations and inequalities.
• Justify algebraic manipulations by application of the properties of equality, the order of operations, the number properties, and the order properties.
• Use algebra to solve word problems involving fractions, ratios, proportions, and percents.
• Identify variables and derive algebraic expressions that represent real-world situations.
##### Objective 0021: Understand the concept of function.

For example:

• Understand the definition of function and various representations of functions (e.g., input/output machines, tables, graphs, mapping diagrams, formulas).
• Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic).
• Identify and analyze direct and inverse relationships in tables, graphs, algebraic expressions and real-world situations.
• Use qualitative graphs to represent functional relationships in the real world.
• Translate among different representations (e.g., tables, graphs, algebraic expressions, verbal descriptions) of functional relationships.
##### Objective 0022: Understand linear functions and linear equations.

For example:

• Recognize the formula and graph of a linear function.
• Distinguish between linear and nonlinear functions.
• Find a linear equation that represents a graph.
• Analyze the relationships among proportions, constant rates, and linear functions.
• Interpret the meaning of the slope and the intercepts of a linear equation that models a real-world situation.
• Select the linear equation that best models a real-world situation.

#### Subarea VII–Geometry and Measurement

##### Objective 0023: Understand and apply concepts of measurement.

For example:

• Estimate and calculate measurements using customary, metric, and nonstandard units of measurement.
• Use unit conversions and dimensional analysis to solve measurement problems.
• Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures.
• Determine how the characteristics (e.g., area, volume) of geometric figures and shapes are affected by changes in their dimensions.
• Solve a variety of measurement problems (e.g., time, temperature, rates, average rates of change) in real-world situations.
##### Objective 0024: Understand and apply concepts of geometry.

For example:

• Classify and analyze polygons using attributes of sides and angles, including real-world applications.
• Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices.
• Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry, similarity, and congruence; and use these concepts to solve problems.
• Match three-dimensional figures and their two-dimensional representations (e.g., nets, projections, perspective drawings).
• Recognize and apply connections between algebra and geometry (e.g., the use of coordinate systems, the Pythagorean theorem).

#### Subarea VIII–Statistics and Probability

##### Objective 0025: Understand descriptive statistics.

For example:

• Use measures of central tendency (e.g., mean, median, mode) and spread to describe and interpret real-world data.
• Select appropriate ways to present data and communicate statistical information (e.g., tables, graphs, line plots, Venn diagrams).
• Analyze and interpret various graphic and nongraphic data representations (e.g., frequency distributions, percentiles).
• Compare different data sets.
##### Objective 0026: Understand and apply basic concepts of probability.

For example:

• Calculate the probabilities of simple and compound events and of independent and dependent events.
• Recognize and apply the concept of conditional probability.
• Recognize the difference between experimentally and theoretically determined probabilities in real-world situations.
• Apply knowledge of combinations and permutations to the computation of probabilities.

#### Subarea IX–Integration of Knowledge and Understanding

##### Objective 0027: Apply mathematical knowledge and reasoning to communicate multiple solutions in detail to a problem involving two or more of the following subareas: Numbers and Operations, Functions and Algebra, Geometry and Measurement, and Statistics and Probability.
• (Refer to objectives 0016 through 0026 and associated descriptive statements.)