Grade: 10-12
Credit: 1
Prerequisite: Geometry or Advanced Geometry GT

The depth and level of understanding expected in Advanced Algebra is beyond the scope of Algebra 2. Students are not only expected to master algebraic mechanics but also to understand the underlying theory and to apply the concepts to real-world situations in a meaningful way. A thorough treatment of advanced algebraic concepts is provided through the study of functions, polynomials, rational expressions, complex numbers, matrices, sequences and series, permutations and combinations, and selected topics in discrete mathematics. Emphasis is on modeling, logic, and interpretation of results. A transformational approach to graphing is used with families of related graphs. Numerical, graphical, and algebraic solutions should be considered for all problems where appropriate. Graphing utilities, especially graphing calculators, are integral to the course.

Indicators marked with an asterisk * relate directly to the Virginia Standards of Learning for Algebra 2. All Virginia Standards of Learning for Algebra 2 are included in this course.

Benchmarks and indicators are organized by the following strands:

I. Mechanics of Algebra

II. Equations, Inequalities, and Systems

III. Relations and Functions

IV. Sequences and Series

V. Data Analysis

Benchmark
Students use problem-solving skills to operate on and simplify algebraic expressions.

1. Identify and apply field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers and its subsets, complex numbers, and matrices.*
2. Add, subtract, multiply, divide, and simplify rational expressions, including complex fractions, and use these in real-world settings.*
3. Add, subtract, multiply, divide, and simplify power, root, and radical expressions and use them to solve real-world problems.*
4. Add, subtract, multiply, divide (using both long division and synthetic division techniques), and factor polynomials.
5. Identify and factor completely expressions representing the greatest common factor, difference of squares, trinomials, and the sum or difference of cubes.*
6. Perform operations on complex numbers and express results in simplest form. Simplifying results will involve using patterns of the powers of i.*
7. Organize data into matrices and manipulate the data using addition, subtraction, scalar multiplication, matrix multiplication, and determinant evaluation and use them in real-world settings.*
8. Use the properties of common and natural logarithms to evaluate logarithmic expressions and to solve real-world problems.

Benchmark
Students develop techniques for solving equations, inequalities, and systems of equations that model real-world situations.

1. Solve linear and absolute value equations and inequalities and apply problem-solving strategies to solve real-world problems.*
2. Select, justify, and apply a technique to solve a quadratic equation over the set of complex numbers.*
3. Solve equations containing rational expressions, equations containing radical expressions, exponential equations (including e), and logarithmic equations (including ln).*
4. Represent real-world situations with a two- or three-variable system of linear equations and solve the system using an appropriate method (i.e., graphs, substitution, linear combination, or inverse matrices).*
5. Solve systems of linear inequalities and linear programming problems.*
6. Solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic.*

Benchmark
Students apply a transformational approach with graphing calculators to analyze, describe, and compare a variety of functions and relations.

1. Graph linear, quadratic, absolute value, step, piecewise, and exponential functions; convert between a graph, a table, and symbolic form and use these in real-world situations.*
2. Determine characteristics of a function, i.e., domain, range, zeros, inverse, and value(s) of a function and composition of multiple functions. Recognize the general shape of polynomial functions and sketch their graphs.*
3. Investigate and describe the relationships between the solution of an equation, zero of a function, x-intercepts of a graph, and factors of a polynomial expression graphically. Use synthetic division and apply the rational zero test.*
4. Identify the relationship between polynomial functions and their graphs and write an appropriate equation given the properties of a polynomial graph.
5. Identify conic sections (circle, ellipse, parabola, and hyperbola) from their equations, sketch their graphs given the equation in (h, k) form, and write an appropriate equation given properties of a conic.*
6. Sketch the graph of a rational function using zeros, "holes," and vertical and horizontal asymptotes.
7. Create and use models with direct and inverse variation to solve real-world problems.*
8. Identify characteristics of exponential and logarithmic functions, including their inverse relationship; graph exponential and logarithmic functions.*

Benchmark
Students investigate patterns of numbers and sums to develop an intuitive understanding of limits.

1. Investigate and apply the properties of arithmetic and geometric sequences and series to solve problems, including writing the first n terms, finding the nth term, and evaluating summation formulas, including the infinite geometric series. Notation will include å and a_n.*
2. Expand binomials having positive integral exponents through the use of the binomial theorem, combinations, and Pascal’s triangle.
3. Identify and classify recursive functions and use recursive functions to model real-world phenomena.

Benchmark
Students design and apply a variety of data analysis techniques to real-world situations.

1. Collect, organize, and analyze data to make predictions, write equations, and solve practical problems.*
2. Investigate scatter plots to determine the equation for a curve of best fit and use correlation coefficient to analyze data.
3. Find the probability of an event using probability principles and use these to answer questions about real-world problems.

All students will take the Virginia Standards of Learning Test for Algebra 2.

Last update: September 10, 1998