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1 BOE APPROVED 2/14/12 REVISED 9/25/12 Curriculum Scope & Sequence Subject/Grade Level: MATHEMATICS/HIGH SCHOOL Course: ALGEBRA II CP/HONORS *The goals and standards addressed are the same for both levels of Algebra II; however, the depth at which they are explored will vary based on the level of the course. 16 Days Units: Students will be Properties able to Functions independently use Intercepts their learning to Writing equations Graphing model and analyze Formulas linear relationships. Systems of linear equations 1 -Linear Equations Standards: A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. A.CED.1 Create equations and inequalities in one variable and use them to solve problems. A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales A.CED.3 Represent constraints by equations or inequalities and by systems of equations or inequalities and interpret solutions as viable or nonviable options in a modeling context, For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. A.CED.4 Rearrange formulas to highlight a quantity of The symbolic language of algebra is used to communicate and generalize patterns in mathematics, which can be represented graphically, numerically, symbolically, or verbally. Mathematical models can be used to describe and quantify physical relationships. Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole. How can change be best represented mathematically? How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations? How are patterns of change related to the behavior of functions? What makes an algebraic algorithm both effective and efficient?

2 interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V=IR to highlight resistance R. A.REI.11 Explain why 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 the solutions approximately using technology to graph the functions, make tables of values to find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential and logarithmic functions. F.IF.1 Understand that a function assigns to each element of the domain exactly one element of the range. F.IF.2 Evaluate functions for inputs in their domain and interpret statements. F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of quantities and sketch graphs showing key features given a verbal description of the relationship. Key features include intercepts, intervals where the function is increasing, decreasing positive of negative, relative minimum and maximums, symmetries, end behavior and periodicity. F.IF.5 Relate the domain of a function to its graph and where applicable to the quantitative relationship it describes. For example, if the function h(n) give the number of person hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain. F.IF.8 Write a function defined by an expression in different but equivalent form.

3 F.IF.9 Compare properties of two functions each represented in a different way. 2 - Inequalities and Absolute Value 12 Days Units: One Variable Inequalities Two Variable Inequalities Absolute value Absolute value inequalities (H) Extraneous solutions System of linear inequalities Standards: A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales A.CED.3 Represent constraints by equations or inequalities and by systems of equations or inequalities and interpret solutions as viable or nonviable options in a modeling context, For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. A.SSE.1 Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. 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. F.BF.3 Identify the effect on the graph of replacing Students will be able to independently use their learning to model complex relationships where solutions may not be obvious or possible. Just as you use properties of equality to solve equations, you can use properties of inequalities to solve inequalities. An absolute value quantity is nonnegative. Since opposites have the same absolute value, an absolute value equation can have two answers. You can solve a system of inequalities in more than one way. Graphing is usually the most appropriate method. The solution is the set of points that are solutions of each inequality in the system. How is solving an inequality similar to solving an equation? What is the solution to an absolute value equation? How do you know which region on a graph is the solution? When do we encounter absolute values in our daily lives?

4 f(x) by f(x) + k, k*f(x), f(kx) and f(x+k) for specific values of k (both positive and negative) find the value of k given the graphs. A.REI.6 Solve a system of linear equations exactly and approximately focusing on pairs of linear equations in two variables. A.REI.11 Explain why 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 the solutions approximately using technology to graph the functions, make tables of values to find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential and logarithmic functions. A.REI.12 Graph the solutions to a linear inequality in two variables as a half plane and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half - plane 3- Quadratics 16 Days Units: Exponent rules Factoring Foiling Graphing Parabola properties Solving by square roots Complex numbers Completing the square Quadratic formula Systems of nonlinear functions Students will be able to independently use their learning to relate and apply all knowledge of quadratic equations to all other types of equations they have studied previously and will study in There are different methods to solving quadratic functions. Changing the parameters of quadratic function changes the graph in predictable ways. Quadratic equations are used in many other disciplines and How can you represent the same mathematical idea in different ways? What is the most efficient way to solve this problem? How does graphing an equation make it easier to draw conclusions? What can minimums

5 Standards: A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales A.CED.3 Represent constraints by equations or inequalities and by systems of equations or inequalities and interpret solutions as viable or nonviable options in a modeling context, For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. N.CN.1 Know 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 N.CN.2 Use the relation i 2 =-1 and the commutative associative and distributive properties to add, subtract, and multiply complex numbers. N.CN.7 Solve quadratic equations with real coefficients that have complex solutions N.CN.8 Extend polynomial identities to the complex numbers (i.e. factoring/foiling, etc.) N.CN.9 Know the Fundamental Theorem of Algebra; Show that it is true for quadratic polynomials. F.BF.3 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 (both positive and negative) find the value of k given the graphs. F.IF.4 For a function that models a relationship between two quantities, interpret key features of the future. real world situations. Complex numbers allow some unsolvable problems to be solved. and maximums tell us about equations? How would you use an equation and a graph to get your point across? How can we express complex numbers?

6 graphs and tables in terms of quantities and sketch graphs showing key features given a verbal description of the relationship. Key features include intercepts, intervals where the function is increasing, decreasing positive of negative, relative minimum and maximums, symmetries, end behavior and periodicity. F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of quantities and sketch graphs showing key features given a verbal description of the relationship. Key features include intercepts, intervals where the function is increasing, decreasing positive of negative, relative minimum and maximums, symmetries, end behavior and periodicity. F.IF.5 Relate the domain of a function to its graph and where applicable to the quantitative relationship it describes. For example, if the function h(n) give the number of person hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain. F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change form a graph. F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F.IF.9 Compare properties to two functions each functions each represented in a different way (algebraically, graphically, numerically in tables or by verbal descriptions). For examples, given a graph

7 of one quadratic function and an algebraic expression for another, say which has the larger maximum. A.REI.11 Explain why 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 the solutions approximately using technology to graph the functions, make tables of values to find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential and logarithmic functions. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. 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 ) A.APR.3 Identify zeroes of polynomials when suitable factors are available, and use the zeroes to construct a rough graph of the function defined by the polynomial. 4- Elementary Probability 8 Days Units: Combinations Permutations Counting Dependent events Independent events Standards: S.IC.2 Decide if a specified model is consistent with results given from a data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to Students will be able to independently use their learning to analyze occurrences rationalizing the reasons why they are happening. You can use multiplication to quickly count the number of ways an event can happen. The probability of an impossible event is 0. The probability of a certain event is 1. Otherwise the probability of an event is a number between 0 and 1. How are permutations and combinations helpful? Is order important? Does probability give actual outcomes? How can multiple events change the likelihood of an occurrence?

8 question the model? S.MD.6 Use probabilities to make fair decisions (e.g. drawing by lots, using random number generators) S.MD.7 Analyze decisions ad strategies using probability concepts (e.g. product testing, medical testing, pulling a hockey goalie at the end of a game) S.CP.6 Find the conditional probability of A given B as the fraction of B s outcomes that also belong to A, and interpret the answer in terms of the model. S.CP.7 Apply the addition Rule, P(A or B) = P(A) + P(B) P(A and B) and interpret the answer in terms of the model. S.CP.8 Apply the general Multiplication rule in a uniform probability model, P(A and B) = P(A)P(B A) = P(B)P(A B), and interpret the answer in terms of the model. S.CP.9 Use permutations and combinations to compute probabilities of compound events and solve problems. F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. To find the probability of two events occurring together, you have to decide whether one event occurring affects the other events. Conditional probability exists when two events are dependent. Does order matter in analysis of multiple events?

9 5 -Data Analysis 12 Days Units: Regressions Mean Median Mode Normal Curve Standard deviation Samples Surveys Standards: A.SSE.4 - Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and sketch graphs showing key features F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change form a graph. F.BF.1.b Write a function that describes a relationship between two quantities: Combine standard function types using arithmetic operation. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. S.ID.4 Use the mean and standard deviation of the data set to fit it to normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not Students will be able to independently use their learning to rationalize trends in sets of data and understand the meanings of the trends. You can use polynomial functions to model many realworld situations. The behavior of the graphs of polynomial functions of different degrees can suggest what type of polynomial will best fit a particular set. You can describe and compare sets of data using various statistical measures depending on what characteristics would want to study. Standard deviation is a measure of how far the numbers in the set deviate from the mean. You can get good statistical information about a population by studying a sample of the population. You can use binomial probabilities in situations involving two possible outcomes. What defines a good relationship? What defines a correct decision? Why is the average so important? What is the best way to find out about a group of people? What is normal?

10 appropriate. Use calculators, spreadsheets and tables to estimate areas under the normal curve. S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from the population. S.IC.3 Recognize the purposes of and differences among sample surveys, experiments and observational studies; explain how randomization relates to each. S.IC.4 Use data from a sample survey to estimate a population mean and proportion; develop a margin of error through the use of simulation models for random sampling. S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide is differences between parameters are significant. S.IC.6 Evaluate reports based on data. Many common statistics gathered from samples in the natural world tend to have a normal distribution about their mean. 6- Polynomials 25 Days Units: Evaluate polynomials Graph and analyze graphs of polynomials Add, subtract and multiply polynomials Factor and solve polynomials Rational zeros Fundamental Theorem of Algebra Write polynomial functions Standards: N.CN.7 Solve quadratic equations with real coefficients that have complex solutions N.CN.8 Extend polynomial identities to the complex numbers (i.e. factoring/foiling, etc.) N.CN.9 Know the Fundamental Theorem of Algebra; Students will be able to independently use their learning of simplifying, solving and graphing polynomial equations to analyze how these equations will act as a basis for further math topics and how they relate Mathematical properties over-arch onto many different topics. Polynomials are accurate models of many real life situations. There are many ways to solve problems, but some are better than others. How can you tell when/if a polynomial expression can be simplified? How can a polynomial be expressed graphically and what does each part of the graph represent? Why do polynomials have special rules for operations?

11 Show that it is true for quadratic polynomials. F.IF.5 Relate the domain of a function to its graph and where applicable to the quantitative relationship it describes. For example, if the function h(n) give the number of person hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain. F.IF.7.c Graph functions expressed symbolically and show key features of the graph, by hand in some cases and using technology for more complicated cases: Graph polynomial functions identifying zeros when suitable factorizations are available and showing end behavior. A.APR.1 Understand that polynomials for a system analogous to the integers, namely, that they are closed under the operations of addition, subtraction and multiplication: add, subtract and multiply polynomials A.APR.2 Know and apply the remainder theorem: 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) A.APR.3 Identify zeroes of polynomials when suitable factors are available, and use the zeroes to construct a rough graph of the function defined by the polynomial. A.APR.5 Know and apply the Binomial Theorem for expansion of (x + y) n in powers of x and y for positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle. A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = to the world around us. What is the best way to solve a polynomial equation? What do complex numbers mean as solutions of polynomials?

12 g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately using technology to graph the functions, make tables of values to find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential and logarithmic functions. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. 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 ) F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, f(kx) and f(x+k) for specific values of k (both positive and negative) find that value of k given the graphs. F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of quantities and sketch graphs showing key features given a verbal description of the relationship. Key features include intercepts, intervals where the function is increasing, decreasing positive of negative, relative minimum and maximums, symmetries, end behavior and periodicity. 7- Rational and Radical Equations 25 Days Units: Roots Rational exponents Composition Inverse functions Graphing root and rational functions Solving radical and rational equations Multiply and divide rational expressions Students will be able to independently use their learning to evaluate and solve rational and radical equations in order Exponent expressions have to be simplified using a specific set of properties. Ratios and proportions are essential in everyday How are algebraic rules and properties developed? In what situations is zero or a negative number an inappropriate answer

13 to further understand the different function families and their properties. Standards: A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V=IR to highlight resistance R. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. 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 ) A.APR.4 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 A.APR.6 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 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. A.APR7 Understand that rational expressions form a number system analogous to the rational numbers, closed under addition, subtraction multiplication and division by a nonzero rational expression; add, subtract, multiply and divide rational expressions. A.REI.2 Solve simple rational and radical equations in one variable and give examples showing how extraneous solutions may arise. F.BF.1.b Write a function that describes a relationship between two quantities: Combine standard function types using arithmetic operation. For example, build a function that models the temperature of a cooling body by adding a constant life especially but not limited to percentages and discounts. Corresponding to every power there is a root. You can write a radical expression in an equivalent form using a rational exponent instead of a radical sign. Solving a square root equation may require that you square each side of the equation, introducing the possibility of extraneous solutions. You can add, subtract, multiply and divide functions based on how you perform these operations for real numbers, considering the domains for the functions. The inverse of a function may or may to a problem? How do you know when to use a specific format? How can a small part affect the larger portion? Does an opposite always have the same properties? How are fractions helpful?

14 function to a decaying exponential, and relate these not be a function. functions to the model. You can use much of F.BF.4.a Solve an equation of the form f(x) = c for a what you know about simple function f that has an inverse and write an multiplying and expression for the inverse. F.IF.7.b Graph square roots and cube root functions dividing fractions to F.IF.8 Write a function defined by an expression in multiply and divide different but equivalent forms. rational expressions.

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