Algebra II Mathematics N-CN The Complex Number System

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1 GRADE HS Algebra II Mathematics N-CN The Complex Number System K Perform arithmetic operations with complex numbers. N.CN.1 N.CN.2 There is a complex number i such that i 2 = -1, and every complex number has the form a + bi with a and b real. The Properties of Commutative, Associative, and Distributive, and the relation i 2 = -1 can be used to perform operations on complex numbers. Explain imaginary and complex numbers. Summarize why complex numbers are used where real numbers cannot solve problems. Define the complex number i. Demonstrate that every complex number has the form a + bi where a and b are real numbers. Add, subtract, and multiply complex numbers using commutative, associative, and distributive properties. Use complex numbers in polynomial identities and equations. N.CN.7 Quadratic equations with real coefficients have complex solutions. Solve quadratic equations with real coefficients that have complex solutions. Explain why in a quadratic equation, imaginary roots occur only in conjugate pairs.

2 GRADE HS Mathematics A-SSE Seeing Structure in Expressions Interpret the structure of expressions. A.SSE.1 A.SSE.2 Write expressions in equivalent forms to solve problems. A.SSE.4 The laws of exponents for whole number exponents follow from an understanding of exponents as indicating repeated multiplication, and from the associative property of multiplication. The interpretation of zero, fractional and negative exponents follows from extending the laws of exponents to those values. Complex expressions can be interpreted by chunking : temporarily viewing a part of the expression as a single entity. The structure of an expression is used to identify ways to rewrite it. A formula can be derived for the sum of finite geometric series when the common ratio is not 1 and use the formula to solve problems. Interpret parts of an expression, such as terms, factors, variables, and coefficients. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r) n as the product of P and a factor not depending on P. Rewrite an expression in order to simplify it or help solve the problem. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). Apply the formula for finding n th term of a geometric sequence Derive the formula for the sum of the first n terms of a geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.

3 GRADE HS Mathematics A-APR Arithmetic with Polynomials and Rational Expressions. Perform arithmetic operations on polynomials. A.APR.1 Polynomials form a system similar to the integers, closed under the operations of addition, subtraction, and multiplication. Compare addition, subtraction, and multiplication of integers with the same operations with polynomials. Explain that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication. Add and subtract polynomials. Multiply and divide polynomials. Understand the relationship between zeros and factors of polynomials. A.APR.2 A.APR.3 The Remainder Theorem states: For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a) = 0 if and only if (x a) is a factor of p(x). When suitable factorizations are available, identify the zeros of polynomials and use the zeros to construct a rough graph of the function defined by the polynomial. Explain the relationship between zeros and factors of polynomials. Explain the fundamental Theorem of Algebra. Explain the Remainder Theorem of polynomials. For example, for a polynomial p(x) and a number a, the remainder on division by x-a is p(a), so p(a) =0 if and only if (x-a) is a factor of p(x). State and apply the Remainder Theorem of polynomials. Identify zeros of polynomials when suitable factorizations are available. Use the zeros of a polynomial to construct a rough graph of the function defined by the polynomial.

4 GRADE HS Mathematics A-APR Arithmetic with Polynomials and Rational Expressions. Use polynomial identities to solve problems. A.APR.4 Once polynomial identities are proven, they can be used to describe numerical relationships. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x 2 + y 2 ) 2 = (x 2 - y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples. Rewrite rational expressions. A.APR.6 A.APR.7 Simple rational expressions can be rewritten in different forms. Rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a non-zero rational expression. (+). Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x),and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. Add, subtract, multiply and divide rational expressions. Simplify expressions with polynomials in the numerator and denominator. Use long division to divide polynomials. Use synthetic substitution to evaluate a function.

5 GRADE HS Mathematics A-CED Creating Equations Create equations that describe numbers or relationships. A.CED.1 A.CED.2 A relation between two quantities can be represented by an equation in variables representing coordinates for the quantities; by a graph on a pair of axes marked with units for the quantities; and by a table of coordinate pairs from the relation. The graph and the table show pairs that are solutions to the equation. Create equations and inequalities in one variable and use them to solve real-world scenarios. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create a quadratic equation to represent relationships between quantities and graph it on a coordinate plane. Graph quadratic functions y = ax 2 +bx+c, where a 0, identifying the y intercept, axis of symmetry, and x coordinate of the vertex. Write and graph quadratic function in vertex form.

6 GRADE HS Mathematics A-REI Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning. A.REI.2 Extraneous solutions may arise from solving simple rational and radical equations in one variable. Solve simple rational and radical equations in one variable. Explain extraneous solutions. Give examples of equations with extraneous solutions. Represent and solve equations and inequalities graphically. A.REI.11 The x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x). Find solutions to equations and inequalities using technology to graph the functions, making tables of values, or finding successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Explain the concept of the solution for a system of equations or system of inequalities. Find the approximate solutions to a system of equations (include combination of linear, polynomial, rational, radical, absolute value, and exponential functions).

7 GRADE HS Mathematics F-IF Interpreting Functions Interpret functions that arise in application in terms of the context. F.IF.4 F.IF.5 F.IF.6 Functions model a relationship between two quantities, where key features of graphs and tables in terms of the quantities are interpreted and key features given a verbal description of the relationship are sketched as a graph. The domain of a function relates to its graph and at times, to the quantitative relationship that it describes. The average rate of change of a function (presented symbolically or as a table) over a specified interval can be calculated and interpreted. Rate of change can be estimated from a graph. Interpret key features of graphs of linear, exponential, and quadratic functions. Key features include: intercepts; intervals where the function is increasing, decreasing, positive or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Sketch graphs of functions showing key features given a verbal description of the relationships between two quantities. Understand the relationship between domain and independent variable. Relate the domain of a function to its graph. Interpret domain of a function in a realworld context (including quadratic functions). Interpret the average rate of change of a function over a specified interval. Estimate the rate of change from a graph. Analyze functions using different representations. F.IF.7 F.IF.8 F.IF.9 Functions expressed symbolically can be graphed to show key features of the graph, by hand for simple cases and using technology for complicated cases. Graphs of linear and quadratic functions show intercepts, maxima, and minima. Functions that are expressed symbolically can be graphed, such Identify equations of functions and common graphic representations of functions. For example, y = mx + b (linear), y = ax 2 + bx + c (quadratic), and y = Ab x (exponential). Distinguish between linear and non-linear functions. Identify the shape and key features of different types of functions such as linear, quadratic, exponential, logarithmic, step functions, absolute value functions. Graph functions using technology.

8 GRADE HS Mathematics F-IF Interpreting Functions as square root, cube root, and piecewise-defined functions. Square root function is a function that maps the set of non-negative real numbers onto itself. The domain and range are the nonnegative real numbers. Cube root function is an odd function, symmetric with respect to its origin. The domain of f consists of the entire real numbers. Step function is a function on the real numbers and a piecewise constant function having only finitely many pieces. Absolute value function is an even function where the domain is the real numbers and the range is non-negative real numbers. Identify zeros when suitable factorizations are available and show end behavior when graphing polynomial functions. Graphs of exponential and logarithmic functions can show intercepts and end behavior. Graphs of trigonometric functions can show period, midline, and amplitude. Functions defined by an expression can be written in Graph linear functions. Graph quadratic functions and identify the maxima or minima of the quadratic functions. Show the intercept of linear and quadratic functions. Graph root functions. Graph cube root functions. Graph piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions by identifying zeros (when able to factor) and show end behavior. Graph exponential and logarithmic functions by showing intercepts and end behavior. Identify exponential growth and decay. Convert exponential expressions to the equivalent logarithmic expression and vice versa. Graph trigonometric functions (sine and cosine functions) showing period, midline, and amplitude. Employ the process of factoring in a quadratic function to show zeros, maxima, minima, and symmetry of the graph. Interpret features of the graphs of quadratic equations in terms of a real-world context. Identify percent rate of change in functions such as y = (1.02) t, y = (0.97) t, y = (1.01) 12t, y = (1.2) t/10, and classify them as representing exponential growth or decay.

9 GRADE HS Mathematics F-IF Interpreting Functions different but equivalent forms to reveal and explain different properties of the function. In a quadratic function, the process of factoring and completing the square will show zeros, extreme values, and symmetry of the graph, and interpretation of these in terms of the context. Properties of exponents can be used to interpret expressions for exponential functions. Properties of two functions can be compared as each is represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare two linear, two exponential or two quadratic functions each represented in a different way (algebraically, graphically, in tables, or by verbal description).

10 GRADE HS Mathematics F-BF Building Functions Build a function that models a relationship between two quantities. F.BF.1 A function can be written to describe a relationship between two quantities by determining an explicit expression, a recursive process, or steps for calculation from a context. Using arithmetic operations, standard function types can be combined. Write a function that describes a relationship between two quantities focusing on more complex situations, including quadratic and trigonometric functions. Determine an explicit expression for a function in a real-world context. Combine standard function types using arithmetic operations. Build new functions from existing functions. F.BF.3 F.BF.4 New functions can be obtained by composing functions, using arithmetic combinations and by making changes to either the input or output of a function. An inverse function is a function that undoes another function. Identify the effect on the graph of replacing f(x) by f(x) + k for specific values of k, and find the value of k given the graphs focusing on quadratic functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k, and find the value of k given the graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k, and find the value of k given the graphs focusing on quadratic functions and using technology. Recognize even and odd functions from their graphs. Determine algebraically whether a function is even or odd or neither. Find the inverse of a linear function. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For

11 GRADE HS Mathematics F-BF Building Functions example, f(x) = 2 x 3 or f(x) = (x+1)/(x-1) for x 1.

12 GRADE HS Mathematics F-LE Linear, Quadratic, and Exponential Models Construct and compare linear, quadratic, and exponential models and solve problems. F.LE.4 Exponential models can be expressed as a logarithm the solution to ab ct = d where a, c, and d are numbers and the base b is 2, 10, or e. The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. The logarithm of a product is the sum of the logarithms of the factors: log b(xy) = log b(x)+log b(y). Explain the relationship between properties of exponents and properties of logarithms. Evaluate logarithms with a base of 2, 10, or e. Evaluate logarithms using technology. Demonstrate that multiplying two numbers and taking the log is the same as taking their logs and adding [ log b (xy) = log b x+log b y].

13 GRADE HS Mathematics F-TF Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. F.TF.1 F.TF.2 Model periodic phenomena with trigonometric functions. F.TF.5 F.TF.7 The radian measure of an angle is the length of the arc on the unit circle subtended by the angle. The unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. Trigonometric functions can model periodic phenomena with specified amplitude, frequency, and midline. Trigonometric functions that arise in modeling contexts can be solved using inverse functions. (+) Define radian measure of an angle. Measure an angle in degrees and radians. Convert from radian measure to degrees and vice versa. Explain the concept of the unit circle and how it extends the domain of trigonometric functions to all real numbers. Derive the trigonometric identity sin 2 (θ) + cos 2 (θ) = 1. Choose a trigonometric function to model periodic phenomenon. Identify the amplitude, frequency, and midline of a trigonometric function. Use inverse operations to solve trigonometric equations that arise in modeling contexts and evaluate the solutions using technology and interpret them in terms of the context. (+) Prove and apply trigonometric identities. F.TF.8 The Pythagorean Theorem states that sin 2 (θ) + cos 2 (θ) = 1. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1. Use the Pythagorean identity to find sin(θ), cos(θ), or tan(θ), given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

14 GRADE HS Mathematics S-ID Interpreting Categorical and Quantitative Data Summarize, represent, and interpret data on a single count or measurement variable. S.ID.4 The mean and standard deviation of a data set can fit into a normal distribution and estimate population percentages. There are data sets for which such a procedure is not appropriate. Calculators, spreadsheets, and tables can be used to estimate areas under the normal curve. Calculate the mean and standard deviation of a data set. Use the mean and standard deviation to estimate population percentages. Explain a normal distribution of a data set and compare actual distributions of realworld data sets to a normal bell curve. Identify the characteristics of a normal distribution. Recognize that only some data are welldescribed by a normal distribution. Identify data sets for which using the mean and standard deviation to estimate population percentages may not be appropriate. Use calculators, graphs, or tables to estimate areas under a normal curve.

15 GRADE HS Mathematics S-IC Making Inferences and Justifying Conclusions Understand and evaluate random processes underlying statistical experiments. S.IC.1 S.IC.2 Make inferences and justify conclusions from sample surveys, experiments, and observational studies. S.IC.3 S.IC.4 S.IC.5 Statistics is a process for making inferences about population parameters based on a random sample from that population. Statistical inference is the use of probability theory to make inferences about a population from sample data. Simulation can be used to decide if a specified model is consistent with results from a given data-generating process. Empirical probability of an event is an estimate that the event will happen based on how often the event occurs after collecting data or running an experiment (in a large number of trials). It is based specifically on direct observations or experiences. Theoretical probability of an event is the number of ways that the event can occur, divided by the total number of outcomes. It is finding the probability of events that come from a sample space of known equally likely outcomes. Survey sampling describes the process of selecting a sample of elements from a target population in order to conduct a survey. A survey may refer to many different types or techniques of observation, but in the context of survey sampling it most often involves a questionnaire used to measure the characteristics and/or attitudes of Distinguish between a random sample and a biased sample in a statistical experiment. Describe statistics as a process for making inferences about population parameters based on a random sample from that population. Compare theoretical and empirical results of an experiment. Decide if a specified model is consistent with results from a given data-generating process. Assess the consistency (or lack of consistency) between theoretical and observed results in terms of statistics. Identify the purposed of various real-world sample surveys, experiments, or observational studies. Explain how randomization relates to each survey or experiment. Use data from a sample survey to estimate a population mean and distribution. Determine whether a data distribution is positively skewed, negatively skewed, or

16 GRADE HS Mathematics S-IC Making Inferences and Justifying Conclusions S.IC.6 people. In an experiment, investigators apply treatments to experimental units (people, animals, plots of land, etc.) and then proceed to observe the effect of the treatments on the experimental units. In an observational study, investigators observe subjects and measure variables of interest without assigning treatments to the subjects. Randomization relates to sample surveys, experiments, and observational studies. Data from a sample survey can be us.ed to estimate a population mean or proportion and a margin of error can be developed through the use of simulation models for random sampling. Data from a randomized experiment can be used to compare two treatments. Stimulations can be used to decide if differences between parameters are significant. Report can be evaluated based on data. normally distributed. Find the margin of error for sample survey results. Analyze a margin of error. Explain what the margin of error indicates about the results. Determine the size of the survey population. Compare data from randomized experiments of two treatments. Determine if the differences in the results is significant (cannot be attributable solely to random selection in sampling or random assignment). Use margin of error or effect size to determine significance of results in a randomized experiment. Analyze reports based on data and statistics.

17 GRADE HS Mathematics S-MD Using Probability to Make Decisions Use probability to evaluate outcomes of decisions. S.MD.6 S.MD.7 Probabilities can be used to make fair decisions. (+) Decisions and strategies can be analyzed using probability concepts. (+) Drawing by lots and using a random number generator to make fair decisions with probabilities. (+) Use product testing, medical testing, and pulling a hockey goalie at the end of a game to analyze decisions and strategies with probability concepts. (+) RESOURCES Common Core State Standards for Mathematics New Jersey Core Curriculum Content Standards - Mathematics New Jersey Mathematics Curriculum Framework Principles and Standards for School Mathematics by National Council of Teachers of Mathematics (NCTM) Curriculum Focal Point LLC