# Algebra 2

In Algebra 2, students extend the algebra and function work done in Algebra 1. Students continue to develop their picture of the complex number system by investigating how non-real solutions arise and how non-real numbers behave. Exponential functions are considered over a domain of real numbers, necessitating work with fractional exponents. The logarithm is defined as the inverse of exponentiation and from this definition students consider the properties of logarithms in addition to using them to solve for unknown exponents. Students extend their previous work with quadratics and polynomials to achieve a more general understanding of polynomials. Work done in geometry with sin, cos, and tan as operations is extended in a study of the unit circle and sin(x), cos(x), and tan(x) as functions. Students build on their work with probability from grade 7 to admit the notions of independence and conditional probability. Finally, students do further work in statistics where they revisit and extend their understanding of variability in data and of ways to describe variability in data. The normal distribution is studied, and students explore the reasoning that allows them to draw conclusions based on data from statistical studies.

## Units

A2.1 Extending the Number System

#### Summary

•  Work with infinite decimal expansions of numbers on the number line.
•  Reason about operations with rational and irrational numbers (N-RN.B.3).
•  Extend properties of integer exponents to rational exponents and write expressions with rational exponents as radicals (N-NR.A.1, N-RN.A.2).
•  Solve equations and real-world problems involving radicals and fractional exponents (A-REI.A.2).
•  Note extraneous solutions and explain where they come from (A-REI.A.2).
•  Discover a new type of number that is outside previously known number systems (N-CN.A.1).
•  Perform operations with complex numbers (N-CN.A.2).
•  Solve quadratic equations with complex solutions (N-CN.C.7).

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A2.2 Exponential Functions 2

#### Summary

•  Create and analyze a simple exponential function arising from a real-world or mathematical context (F-LE.A.2$^\star$).
•  Evaluate and interpret exponential functions at non-integer inputs (N-RN.A.1, F-LE.A.2$^\star$).
•  Understand functions of the form $f(t) = P(1 + r/n)^{nt}$ and solve problems with different compounding intervals (A-SSE.A.1$^\star$, F-LE.A.2$^\star$).
•  Understand informally how the base $e$ is used in functions to model a quantity that compounds continuously (F-BF.A.1a$^\star$).
•  Write exponential expressions in different forms (F-LE.A.2$^\star$, F-BF.A.1$^\star$).
•  Explain what the parameters of an exponential function mean in different contexts (F-LE.B.5$^\star$).
•  Use the properties of exponents to write expressions in equivalent forms (A-SSE.B.3a$^\star$).
•  Build exponential functions to model real world contexts (F-LE.A.1$^\star$, F-LE.A.2$^\star$, F-BF.A.1$^\star$).
•  Analyze situations that involve geometric sequences and series (A-SSE.A.1$^\star$).
•  Derive the formula for the sum of a finite geometric series (A-SSE.B.4$^\star$).

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A2.3 Logarithms

#### Summary

•  Understand the definition of a logarithm as the solution to an exponential equation (F-LE.A.4$^\star$).
•  Practice evaluating logarithmic expressions and converting between the exponential form of an equation and the logarithmic form (F-LE.A.4$^\star$).
•  Solve exponential equations using logarithms (F-LE.A.4$^\star$).
•  Understand the natural logarithm as a special case (F-LE.A.4$^\star$).
•  Graph exponential and logarithmic functions, both by hand and using technology (F-IF.C.7e$^\star$, F-BF.5(+)).
•  (Optional) Understand and explain the properties of logarithms.
•  (Optional) Use properties of logarithms to solve problems.
•  (Optional) Solve problems using the properties of logarithms.

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A2.4 Polynomials and Rational Functions

#### Summary

•  Add, subtract, and multiply polynomials and express them in standard form using the properties of operations (A-APR.A.1).
•  Prove and make use of polynomial identities (A-APR.C.4).
•  Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior (A-APR.B.3$^\star$, F-IF.C.7c$^\star$).
•  Use the remainder theorem to find factors of polynomials (A-APR.B.2, A-APR.D6).
•  Use various strategies including graphing and factoring to solve problems in contexts that can be modeled by polynomials in one variable.
•  Build a rational function that describes a relationship between two quantities (F-BF.A.1).
•  Graph rational functions, interpret features of the graph in terms of a context, and use the graphs to solve problems (A-SSE.A.1a, A-REI.D.11$^\star$, F-IF.B.4, F-IF.C.7d(+)).
•  Express rational functions in different forms to see different aspects of the situation they model (A-APR.D.6).
•  Solve simple rational equations and understand why extraneous roots can arise (A-REI.A.2).

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A2.5 Trigonometric Functions

#### Summary

•  Understand some real-world situations that demonstrate periodic behavior.
•  Define coordinates on the unit circle as the sine and cosine of an angle (F-TF.A.2).
•  Graph basic trigonometric functions using radians as the x-axis scale (F-IF.C.7e$^\star$).
•  Understand the relationship between parameters in a trigonometric function and the graph (F-IF.C.7e$^\star$).
•  Model with trigonometric functions, including fitting them to data (F-TF.B.5$^\star$).
•  Prove the Pythagorean Identity $\sin^2 \theta + \cos^2\theta = 1$ (F-TF.C.8).
•  Use the unit circle to prove trigonometric identities and relate them to symmetries of the graphs of sine and cosine (F-TF.A.4(+)).
• Use the Pythagorean Identity to calculate trigonometric ratios (F-TF.C.8).

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A2.6 Probability

#### Summary

• Describe events as subsets of a sample space (the set of outcomes) using characteristics of the outcomes or as unions, intersections, or complements of other subsets (“or,” “and,” “not”) (S-CP.A.1).
• Use the Addition Rule to compute probabilities of compound events in a uniform probability model, and interpret the result in terms of the model (S-CP.B.7).
• In a uniform probability model, understand the probability of A given B as the fraction of B's outcomes that also belong to A (S-CP.B.6).
• Understand the conditional probability of event A given event B as P(A and B)/P(B) (S-CP.A.3).
• Understand that A and B are independent if P(A and B) = P(A) • P(B) (S-CP.A.2).
• Interpret independence of A and B as saying that the probability of A given B is equal to the probability of A, and the probability of B given A is equal to the probability of B, i.e. P(A|B) = P(A) and P(B|A) = P(B) (S-CP.A.5).
• Recognize independence in everyday situations and explain it in everyday language (S-CP.A.5).
• Determine whether events are independent (S-CP.A.2, S-CP.A.4).
• Use data presented in two-way frequency tables to approximate conditional probabilities (S-CP.A.4).

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A2.7 Making Inferences

#### Summary

• Understand that statistical methods are used to draw conclusions from data.
• Understand that the validity of data-based conclusions depends on the quality of the data and how the data were collected.
• Critique and evaluate data-based claims that appear in popular media.
• Distinguish between observational studies, surveys and experiments.
• Explain why random selection is important in the design of observational studies and surveys.
• Explain why random assignment is important in the design of statistical experiments.
• Calculate and interpret the standard deviation as a measure of variability.
• Use the normal distribution as a model for data distributions that are approximately symmetric and bell-shaped.
• Use the least squares regression line to model linear relationships in bivariate numerical data.
• Understand sampling variability in the context of estimating a population or a population mean.
• Use data from a random sample to estimate a population proportion.
• Use data from a random sample to estimate a population mean.
• Calculate and interpret margin of error in context.
• Understand the relationship between sample size and margin of error.
• Given data from a statistical experiment, create a randomization distribution.
• Use a randomization distribution to determine if there is a significant difference between two experimental conditions.

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