# Algebra 2

Students develop their conceptual understanding, fluency, and ability to apply advanced functions, draw connections between function types, and build functions that model contextual situations.

## Course Summary

What is Algebra 2 all about?

Algebra 2 develops students’ conceptual understanding, fluency, and ability to apply advanced functions. Students draw connections between function types. In particular, students apply skills learned early in the year with linear, quadratic, and polynomial functions to inform their understanding later in the year when they study rational, radical, and trigonometric functions. Students choose appropriate functions and restrictions, based in solid understanding of the features of the functions, to build functions that model contextual situations. Fluency is an important part of Algebra 2, as the ability to perform procedures quickly and easily allows students to more deeply understand concepts.

How did we order the units?

In Unit 1, Linear Functions and Applications, students review the features of functions through the study of inverse functions, modeling contextual situations, and operating with functions, systems of functions, and piecewise functions. Students will increase their fluency in identifying and analyzing features of linear functions through algebraic, graphic, contextual, and tabular representations. Students will use these features to effectively model and draw conclusions about contextual situations. The skills students develop in this unit will be applied and extended to other function types throughout the year, including quadratic, polynomial, rational, exponential, logarithmic, and trigonometric functions.

In Unit 2, Quadratics, students will revisit concepts learned in Algebra 1, such as features of quadratic equations, transformation of quadratic functions, systems of quadratic functions, and moving from one equation form to another (e.g., vertex form to standard form, standard form to intercept form). Increased fluency with quadratic equations and functions provides a strong base for studying polynomials, rational functions, and trigonometric identities. In this unit, students will also be introduced to a new type of number system, the imaginary numbers, and will identify and operate with imaginary solutions. As with Unit 1, students will apply quadratic equations to contextual situations, to systems of functions, and when translating between representations. Graphing calculators are introduced heavily in this unit and will be used for the remainder of the year.

In Unit 3, Polynomials, students will apply skills from the first two units to develop an understanding of the features of polynomial functions. Analysis of polynomial functions for degree, end behavior, and number and type of solutions builds on the work done in Unit 2; these are advanced topics that will be applied to future function types. Students will write polynomial functions to reveal features of the functions, find solutions to systems, and apply transformations, building from Units 1 and 2. Students will be introduced to the idea of an “identity” in this unit as well as operate with polynomials. Division of polynomials is introduced in this unit and will be explored through the concepts of remainder theorem as well as a prerequisite to rational functions.

In Unit 4, Rational and Radical Functions, students will extend their understanding of inverse functions to functions with a degree higher than 1. Alongside this concept, students will factor and simplify rational expressions and functions to reveal domain restrictions and asymptotes. Students will become fluent in operating with rational and radical expressions and use the structure to model contextual situations. In this unit, students will also revisit the concept of an extraneous solution, first introduced in Unit 1, through the solution of radical and rational equations.

In Unit 5, Exponential Modeling and Logarithms, students will model with exponential growth and decay, including use of the continuous compounding base, e, to solve contextual problems in finance, biology, and other situations. Students will learn that logarithms are the inverse of exponentials and operate with and graph logarithms fluently. Students will discover the strength of logarithms to identify solutions, features, and patterns in functions. Students will use exponential functions and logarithmic functions as part of a system of functions in modeling contexts.

In Unit 6, The Unit Circle and Trigonometric Functions, students will review geometric trigonometry as an introduction to trigonometric functions. Students will use sketches of the trigonometric functions of sine and cosine to develop understanding of the reciprocal trig functions, inverse trig functions, and transformational identities of trig functions. Features of trigonometric functions represented graphically will be translated to algebraic representations, and the features unique to trig functions will be explored and used in mathematical and application problems. Students will be introduced to the unit circle and will be expected to derive this easily. The Pythagorean identity will be used heavily in this unit, and students will be expected to know this identity and derive other forms of the identity for use in problems. This unit concludes the formal study of transformation, inverse, systems, features of functions, and using different functions to model contexts that began in Unit 1.

In Unit 7, Trigonometric Identities and Equations, students will develop a foundation for calculus concepts by expanding their conception of trigonometric functions and looking at connections between trigonometric functions. Reasoning flexibly about trigonometric functions and seeing that expressions that look different on the surface can actually act the same on certain domains sets the stage for a study of differentiation and integration, where periodic functions have many useful properties and act as useful tools to study calculus. Students will also apply algebraic techniques to trigonometry, helping them better understand trigonometric functions graphically and through the unit circle, as well as see the power of algebraic manipulation and structure in expressions.

In Unit 8, Probability and Statistical Inference, students explore experimental and conditional probability in an experimental context. An emphasis on conditional probability helps students to reason about cause and effect and serves as an introduction to principles of experimental analysis. Students will also explore making inferences, with a focus on normal distributions and understanding the outcomes of random processes when they are repeated over time. Lastly, students will use distributions to make inferences about populations based on samples and apply an understanding of variability to reason about the relationship between samples and populations.

The final unit in this course, Unit 9, Limits and Continuity, serves as an introduction to Calculus. This unit includes topics of limits, continuity, and derivatives, and it provides a foundation in the essential calculus skill of thinking and reasoning about the infinitely small and the infinitely large while also arguing logically based on definitions and theorems. Students will work with piecewise functions, find finite and infinite limits of various types of functions graphically and algebraically, and define continuity. This unit also deepens knowledge of various types of functions, which is essential foundational knowledge for calculus.

This course follows the 2017 Massachusetts Curriculum Frameworks and incorporates foundational material from Algebra 1 where it is supportive of the current standards.