Algebraic R. II H

In Algebra II, the content is organized around families of functions, including linear, quadratic, exponential, logarithmic, radical, and rational functions. As students study each family of functions, students will learn to represent them in multiple ways – as verbal descriptions, equations, tables, and graphs. Students will also learn to model real-world situations using functions in order to solve problems arising from those situations. In addition to its algebra content, Algebra II includes lessons on probability and data analysis, as well as, numerous examples and exercises involving geometry and trigonometry. These math topics often appear on standardized tests, so maintaining students’ familiarity with them is important. This course is an important part of CAPT and SAT preparation, therefore, instruction, practice and assessments will be presented in a variety of formats – multiple choice, short answer, grid-in and open ended that students will encounter on standardized testing. There will also be extensive use of technology including the graphing calculator throughout the year.

1. Enduring Understandings (broad ideas, usually grounded in the discipline):

Linear Equations, Inequalities, Functions, and Systems

1. Equations and Inequalities – students will review the relationships between the subsets of real numbers, as well as, reviewing the properties of real numbers. They will use the properties of real numbers and the order of operations to evaluate and simplify algebraic expressions, including expressions containing exponents. Then, students will use the properties of equality to solve linear equations and to rewrite formulas and equations. They will use verbal models and problem solving strategies to solve problems. Finally, students will learn to solve and graph linear inequalities and to solve absolute value equations and inequalities.

2. Linear Equations and Functions – students learn that functions are defined as relations that map each value of the domain to a unique value of the range. Students then use slope to graph and write equations for linear functions. They also use slope to identify parallel and perpendicular lines. Students learn how many real world applications can be modeled using direct variation functions, and they learn how correlation coefficients measure how well a line fits a set of data pairs and use best-fitting lines to make predictions based on linear models. They use parent functions to graph absolute value functions. Finally, they graph and interpret solutions of linear inequalities.

3. Linear Systems and Matrices – students will work with systems of equations, systems of inequalities, and matrices. For equations, students will solve systems graphically and algebraically, including systems with many solutions and systems with no solutions. The algebraic methods students will use include the methods of substitution and elimination. For inequalities, they will solve systems by graphing using linear programming. Students will solve a system of three equations by elimination, substitution or by matrices.

Quadratic, Polynomial, and Radical Function

1. Quadratic Functions and Factoring – students will learn several sets of related skills. They will learn how to graph quadratic functions written in standard form, vertex form, or intercept form. They will learn how to factor binomials and trinomials and learn how to solve quadratic equations by factoring, finding square roots, completing the square, and using the quadratic formula. Also, students will learn how to use properties of radicals, how to simplify radicals, and how to calculate with the imaginary unit i and perform operations with complex numbers.

2. Polynomials and Polynomial Functions – students learn and apply properties of exponents as they simplify expressions involving powers and add, subtract, and multiply polynomials. They learn methods to factor and solve polynomial equations, including the Remainder and Factor Theorems. Using intercepts and other methods, they graph polynomial functions, classify the zeros of the functions, and find all real zeros. Finally, students write higher degree polynomial functions using intercepts and finite differences.

3. Rational Exponents and Radical Functions – students will learn the meaning of nth roots and rational exponents, how to interchange rational exponent notation and radical notation, and how to apply the properties of rational exponents. Next, they will learn to perform function operations, including composition. Then they will learn how to determine whether a given function has an inverse that is also a function. Finally, students will learn to graph square root and cube root functions and to solve radical equations.

Second Semester material to be covered prior to the final exam:

Other Nonlinear Functions and Relations

1.Exponential and Logarithmic Functions – students will learn to graph and use exponential growth and decay functions, including functions involving the natural base e . Next, they will learn to evaluate and graph logarithmic functions and to use the properties of logarithms to rewrite logarithmic expressions. Then, students will learn to solve exponential and logarithmic equations. Finally, students will learn to write and apply exponential and power functions.

2. Rational Functions – students learn to write and use models for inverse variation and joint variation. They learn to graph rational functions, to multiply, divide, add, and subtract rational expressions, and to simplify complex fractions. Finally, students learn to solve rational equations.

Unit 4 – Trigonometry

1. Trigonometric Ratios and Functions – student swill learn the right triangle definitions of the six trigonometric functions and how to use right triangle trigonometry. Next, they will learn to use radian measure and to evaluate trigonometric functions of any angle. The, they will learn to evaluate and use inverse trigonometric functions. Finally, students will learn to apply the law of sine sand the law of cosines to solve triangles and applied problems.

2. Trigonometric Graphs, Identities, and Equations – students are introduced to the graphs of sine, cosine, and tangent functions. They learn how to use amplitudes, periods, and asymptotes to graph these functions, as well as, translations and reflections of the functions. Then students learn how trigonometric identities can be generated from the unit circle. They apply identities to solve trigonometric equations. Next, students apply their understanding of the graphs of sine and cosine functions to write sinusoidal functions and models, including sinusoidal regression equations. Finally, students extend their understanding of trigonometric identities to problems involving trigonometric sum and difference formulas and double-angle and half-angle formulas.

Exponential and Quadratic Functions - Probability, Data Analysis, and Discrete Math

Unit 5 – Other Nonlinear Functions and Relations

1. Quadratic Relations and Conic Sections – this chapter introduces students to properties and characteristics of conic sections. Students start by applying the distance and midpoint formulas, then they learn how to graph and write equations for parabolas, circles, ellipses, and hyperbolas. They investigate translations of conic sections based on equations. Finally, students use graphing, substitution, and elimination to solve quadratic systems.

Unit 6 – Probability, Data Analysis, and Discrete Math

2. Counting Methods and Probability – students learn the fundamental counting principle and the formulas for permutations and combinations of n objects taken r at a time. They apply those ideas to problems involving counting. Also, students examine patterns of combinations found in Pascal’s triangle and apply these patterns to binomial expansions. Students then expand counting methods to the study of theoretical, experimental, and geometric probability and to odds in favor of an event or against an event. Finally, they learn how to find probabilities of compound, independent, and dependent events and they construct and interpret binomial distributions.

3. Data Analysis and Statistics – students first learn how to calculate the mean, median, mode and standard deviation and to examine the effect of outliers on a data set. They then learn what the effect is on statistics then they add a constant to the data values or multiply data values by a constant. Students next extend their understanding of probability distributions and measures of central tendency to the study of normal distributions. They learn how the area under the normal curve is related to standard deviations from a mean. They also learn to calculate z-scores using the standard normal table. The students then study sampling methods for collecting data, how to identify biased samples, and how to calculate a margin of error. In the final lesson, students apply their understanding of the forms and behavior of linear, quadratic, cubic, exponential, and power functions as they learn to choose the best model to represent a set of data. They will use a graphing calculator to find equations of the models.

4. Sequences and Series – students explore sequences and series. They define explicit rules that generate number sequences whose terms have a common difference or a common ratio, and they use summation notation to represent and find the sum of the terms of a series. They use rules for the sum of arithmetic series, finite geometric series, and infinite geometric series. Also, students define recursive rules for generating arithmetic and geometric sequences and they investigate how to use iteration to generate a sequence recursively given a function rule.