Course: Algebra II Year: 2017-18 Teacher: Various Unit 1: RIGHT TRIANGLE TRIGONOMETRY Standards Essential Questions Enduring Understandings G-SRT.C.8 Use 1) How are the The concept of trigonometric ratios trigonometric ratios similarity enables us and the Pythagorean related to to explore geometric Theorem to solve right similarity? relationships and triangles in applied apply trigonometric problems.* 2) How is ratios to solve real G-SRT.C.6 trigonometry used world problems. Understand that by to find unknown similarity, side ratios values and solve in right triangles are real world properties of the problems? angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. Approximate Time Frame: 3-4 Weeks Skills Content Vocabulary Determine trigonometric ratios for a given angle in a right triangle. Use a scientific calculator appropriately to find missing values in a right triangle. Find the missing side lengths and angle measures in a right triangle. Solve real world problems using right triangle trigonometry. Trigonometric Ratios Solving Right Triangles Applications of Right Triangle Trigonometry Alpha Adjacent Side Angle of Depression Angle of Elevation Complementary Angles Cosine Degree Mode Hypotenuse Inverse Cosine, Sine, Tangent Leg Opposite Side Pythagorean Theorem Ratio Sine Tangent Theta Unit 2: FUNCTIONS Standards Essential Questions Enduring Understandings A.REI.D.10 1) What is a A function is a Understand that the function and how special kind of graph of an equation can it be relation where each in two variables is the represented? member of the set of all its solutions domain of the plotted in the 2) What is the function is coordinate plane, meaning of the associated with Approximate Time Frame: 8-9 Weeks Skills Content Vocabulary Determine if a given relation is a function or not. Given a function, determine its domain and range. Relations and Functions Parent Functions and Transformations Piecewise Functions Absolute value function Asymptote Axis of symmetry Boundary line Constant Function Constant interval Constraint
often forming a curve (which could be a line). A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F.IF.5 Relate the domain of a function to its graph and, where domain and range of a function? 3) What is a family of functions? 4) What is the effect of a transformation on the dependent and independent variables of a function? 5) How can functions be used to model real world situations, make predictions, and solve problems? exactly one member of the range of the function. Different types of functions can be transformed in similar ways. Represent a function by an equation, table, graph, or verbal description and move comfortably from one representation to another. Determine the reasonableness of the domain of a function in a realistic context. Provide examples of functions that are not smooth and in one piece, such as piecewise functions and step functions. Be able to graph piecewise and step functions, and to describe their properties. Represent a verbal description of a function transformation symbolically. Understand the difference between a transformation of an independent variable and a dependent variable. Cubic function Decreasing interval Dependent variable Domain Family of functions Function Function notation Horizontal and vertical axes Horizontal shift/translation Increasing interval Independent variable Input and output of a function Inside change Interval notation Line symmetry Maximum/minimum Ordered pair Origin Outside change Parabola Parent function Piecewise function Quadratic function Range Rational Function Reflection Relation Root functions Slope Step function Transformation Vertex Vertical stretch/shirnk Vertical shift/translation
applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. 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. F.IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For Given a function f(x), be able to describe the effects of the transformations f(x) + k, f(x + k), and kf(x) for a constant k. Given a graph of a function and a transformation of that function, be able to determine the transformation that is represented in the graph. Given a graph of a function, be able to graph (by hand) a transformation of that function. Explain the meaning of the value of an output from a function. Use functions to answer questions in realistic contexts.
example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. F.IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the 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, or negative; relative maximums
and minimums; symmetries; end behavior; and periodicity. Unit 3: QUADRATIC FUNCTIONS EXTENDED TOPICS Standards Essential Questions Enduring Understandings A.SSE.3 Choose and 1) How can the produce an equivalent nature of the roots form of an expression of a quadratic to reveal and explain equation be properties of the determined based quantity represented on the value of the by the expression. discriminant? A.SSE.3a Factor a quadratic expression 2) What is the to reveal zeros of the definition of the function it defines. imaginary number A.REI.A.2 Solve i? simple rational and radical equations in 3) What is the one variable, and give structure of the examples showing complex number how extraneous system? solutions may arise. A.REI.4 Solve 4) What is the quadratic equations in Fundamental one variable. Theorem of A.REI.4b Solve Algebra, and what quadratic equations by is its relation to inspection (e.g., for x 2 quadratic = 49), taking square functions? roots, completing the square, the quadratic 5) How are radical formula and factoring, equations solved? as appropriate to the initial form of the The study of quadratics transcends the real number system and includes complex solutions. Dynamic software, graphing calculators, and other technology can be used to explore and deepen our understanding of mathematics. Approximate Time Frame: 4-5 Weeks Skills Content Vocabulary Identify types of numbers: natural (counting), whole, integers, rational, irrational, real, complex. Determine whether or not a number, or sum or product of numbers is rational or irrational. State the definition of i. Express the square root of a negative number in terms of i. Evaluate i n for any natural number n. Identify the real and imaginary parts of a complex number, using the a+bi notation. State the conjugate of a complex number. Complex Numbers Real and Complex Zero s Radical Equations Complex Conjugate Complex Number Discriminant Extraneous Root Extraneous Solution Imaginary Number Radical Equation Radicand Root Sets of Numbers (Natural or Counting, Whole, Integers, Rational, Irrational, Real, Complex) Square Root Equation Standard Form of a Quadratic Function System of Equations Zeros of a Function (Real & Complex)
equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. A.REI.11b 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 equations f(x) = g(x); find the solutions approximately, e.g. using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. BF.A.1 Write a function that describes a relationship between two quantities. CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations Add, subtract, multiply and divide complex numbers. Solve quadratic equations that have complex solutions in a+bi form. Determine the number of roots of a quadratic equation based on the discriminant. Demonstrate that the Fundamental Theorem of Algebra is true for any quadratic function with real coefficients. Solve radical equations algebraically and graphically. Determine if a solution obtained from an equation-solving process is extraneous. Model and solve contextual problems involving radical equations.
arising from linear and quadratic functions, and simple rational and exponential functions. CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. F.IF.C.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 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.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. (Note that only functions with real coefficients are considered in this investigation.) N.RN.3 Explain why the sum or product of two rational numbers is rational and the sum of a rational and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
Unit 4: POLYNOMIALS Standards Essential Questions Enduring Understandings A.APR.1 1) How is the If (x-a) is a factor of a Understand that division of a polynomial function, polynomials form a polynomial, P(x), then the polynomial system analogous to by a binomial of the has value 0 when the integers, namely, form (x a) x=a. they are closed under connected to the the operations of polynomial The process of addition, subtraction, function, y = P(x). dividing polynomials and multiplication; using long division is add, subtract, and 2) What is the similar to the process multiply polynomials. connection between of dividing real A.APR.2 Know and the rational zeros numbers. apply the Remainder theorem, the factor Theorem: For a theorem, and the Understand that the polynomial p(x) and a roots of a structure of a number a, the polynomial polynomial can be remainder on division function? used to factor it. 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 zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. 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 Approximate Time Frame: 4-5 Weeks Skills Content Vocabulary Divide polynomials using long and synthetic division. State the Remainder Theorem and its implications. Determine if (x a) is a factor of P(x) by either calculating the remainder using division or finding the value of P(a). Use the Factor Theorem to factor a polynomial given the x-intercepts or zeros of the polynomial function. Determine the zeros of a polynomial based on the factored form of the polynomial. Determine the exact equation of a polynomial function given its degree, its zeros (real or complex) and one other point on the graph. Use known polynomial identities to factor Polynomial division Remainder and Factor Theorems Factoring Polynomials Zeros of Polynomials, including conjugate zeros theorem and rational zeros theorem Algebraic Identity Binomial Coefficient/Leading Coefficient Complex Conjugate Complex Zero Conjugate Zeros Theorem Degree Division Algorithm Divisor Difference of Squares Factor Theorem Linear Factor Long Division Polynomial (monomial, binomial, trinomial) Possible Rational Zeros Cubic Quartic Quadratic Factor Quotient Rational Zeros Theorem Remainder Remainder Theorem Repeated Factorization Root Sum and Difference of Cubes Synthetic Division X-intercepts Zero Zero Product Property
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 more complicated examples, a computer algebra system. 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.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.IF.9 Compare higher order polynomials of similar structure.
properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions.) For example given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Unit 5: GRAPHING POLYNOMIALS Standards Essential Questions Enduring Understandings F.BF.3 Recognize 1) What are the even and odd basic features of a functions from their polynomial graphs and algebraic function based on expressions for them. the degree of the F.IF.7c Graph polynomial? polynomial functions, identifying zeros when 2) What is the suitable factorizations connection between are available, and the zeros of a showing end behavior. polynomial F.IF.7 Graph function, the x- functions expressed intercepts of the symbolically and graph of the show key features of polynomial the graph, by hand in function, and the simple cases and using If (x-a) is a factor of a polynomial function, and a is a real number, then the graph of the polynomial has (a, 0) as an x-intercept. Polynomial functions can be analyzed to determine their unique characteristic s including degree, end behavior, zeros, and number of extrema Approximate Time Frame: 4-5 Weeks Skills Content Vocabulary Determine the basic shape and end behavior of the graph of a polynomial function based on the term of highest degree of the polynomial. Determine the x- and y-intercepts of a polynomial function by inspection of the equation of the function when the polynomial is in factored form. Properties of Polynomials Applications of Polynomials Graphing Polynomials Coefficient/Leading Coefficient Degree End behavior Even function Extrema Linear Factor Mathematical Model Maximum Minimum Odd Function Polynomial (monomial, binomial, trinomial) Root Rotational Symmetry Symmetric with respect to the origin
technology for more complicated cases. F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the 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, or negative; relative maxima and minima; multiplicity of roots; symmetries; end behavior; and periodicity. A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on factors of the polynomial? 3) How can polynomial functions be applied to applications and mathematical models? Identify extrema of a polynomial function given the graphic, symbolic, or numerical representations of the polynomial. Interpret the meaning of intercepts and extrema of a polynomial in the context of a real-world problem. Determine if a graph of a polynomial function is tangent to the x-axis or crosses the x-axis depending on the multiplicity of the corresponding linear factor of the function. Determine the zeros of a polynomial based on the factored form of the polynomial. Approximate the graph of a polynomial based on the zeros of the polynomial and end behavior of the polynomial. Determine the exact equation of a polynomial function given its degree, its Symmetric with respect to the y-axis X-intercepts Y-intercepts Zero Zero of multiplicity
coordinate axes with labels and scales. F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions.) For example given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. F.IF.C.7a Graph linear and quadratic functions and show intercepts, maxima, and minima. zeros (real or complex) and one other point on the graph. Interpret the meaning of characteristics of a polynomial function in the context of a realworld problem. Solve problems based on polynomial function models. Determine whether a function is even, odd or neither based on its equation or graph. Unit 6: RATIONAL EXPRESSIONS Standards Essential Questions Enduring Understandings A.APR.6 Rewrite 1) How does the simple rational denominator of a expressions in rational function different forms; write affect the domain of a(x)/b(x) in the form the function? q(x) + r(x)/b(x), where a(x), b(x), q(x), and 2) How do you r(x) are polynomials perform operations with the degree of r(x) on and simplify The arithmetic of rational expressions uses the same rules as the arithmetic of rational numbers. Basic concepts of rational numbers can Approximate Time Frame: 3-4 Weeks Skills Content Vocabulary Determine if a rational expression is fully factored and if not, factor it. Determine the domain and/or restrictions of a rational expression. Simplifying Rational Expressions (including stating domain) Multiplying and Dividing Rational Expressions Combined Variation Complex Fraction Direct Variation Domain Equivalent expressions Excluded values Factors Indirect Variation Joint Variation
less than the degree of b(x), using inspection, long division, or, for more complicated examples, a computer algebra system. A.APR.7(+) Understand that rational expressions form a 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-CED-2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. rational expressions? be applied to rational expressions. There are some value(s) for which a rational expression may be undefined. Write a rational expression in simplified form. Determine the LCD of multiple rational expressions and rewrite the expressions if necessary. Extend the operational rules of rational numbers to rational expressions. Simplify a complex fraction. Solve direct, indirect, joint and combined variation problems Adding and Subtracting Rational Expressions Simplifying Complex Fractions Solving Variation Problems Least common denominator Rational Expression Restrictions Variation Unit 7: EXPONENTIAL FUNCTIONS Standards Essential Questions Enduring Understandings A-CED-2 Create 1) How can Some exponential equations in two or exponential equations can be more variables to equations be solved using represent relationships solved? comparable between quantities; exponential graph equations on expressions. Approximate Time Frame: 3-4 of Weeks Skills Content Vocabulary Rewrite exponential expressions to solve equations. Represent a verbal description of a Solving Exponential Equations (not requiring use of logarithms) Base Equivalent Equations Exponent Exponential Functions Exponential Growth/Decay
coordinate axes with labels and scales. F-IF. 7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 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.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. 2) What is the effect of a transformation on the dependent and independent variables of an exponential function? Exponential functions can be transformed in ways similar to other functional relationships. function transformation symbolically. Understand the difference between a transformation of an independent variable and a dependent variable. Given a function f(x), be able to describe the effects of the transformations f(x) + k, f(x + k), and kf(x) for a constant k. Given a graph of a function and a transformation of that function, be able to determine the transformation that is represented in the graph. Given a graph of a function, be able to graph (by hand) a transformation of that function. Graphing Exponential Functions Transformations of Exponential Functions Horizontal Asymptote Inverse Vertical/ Horizontal Reflection Vertical/Horizontal Shifts Vertical/ Horizontal Stretches/Shrinks
Unit 8: PROBABILITY Standards Essential Questions Enduring Understandings S-ID-5 Summarize 1) What is Understand that it is categorical data for probability? not possible to predict two categories in twoway with certainty short- frequency tables. 2) How are events term behavior of Interpret relative defined? random phenomena frequencies in the but it is possible to context of the data 3) What is meant use probability to (including joint, by independent / predict long-run marginal, and dependent events? patterns. conditional relative frequencies) 4) How are Probability models Recognize possible probabilities, are useful tools for associations and including making decisions, trends in the data. compound choices, or S-CP-1. Describe probabilities, predictions. events as subsets of a calculated? sample space (the set of outcomes) using 5) What is characteristics (or conditional categories) of the probability and how outcomes, or as does conditional unions, intersections, probability apply to or complements of real-life events? other events ( or, and, not ). 6) How are twoway S-CP-2. Understand frequency that two events A and tables used to B are independent if model real-life the probability of A data? and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. Approximate Time Frame: 5-6 of Weeks Skills Content Vocabulary Given the description of a random process (or a sequence random processes), specify the sample space. Use set notation to specify an event. Determine the union and intersection of two (or more) events. Determine the complement of an event. Use probability to assess the likelihood that a particular event will occur. Use the General Addition Rule to find the probability of the union of two events. Use the Complement Rule to find the probability that an event does not occur. Create a probability model in which all simple outcomes in a Sample Spaces Theoretical and Experimental Probability Independent Events and the Multiplication Rule Conditional Probability Interpreting Two- Way Frequency Tables area probability model column variable combination conditional probability conditional relative frequencies dependent events event false negative false positive fundamental counting principle general multiplication rule grand total joint relative frequencies/percentages independent events intersection of two events marginal relative frequencies / percentages multiplication rule for independent events mutually exclusive events probability probability model random process relative frequency of an event row variable sample space true negative true positive
S-CP-3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S-CP-4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected finite sample space are equally likely. Use counting techniques to determine the number of ways an event can occur then determine the probability that the event occurs. Estimate the probability that an event will occur from the relative frequency that the event occurs in many trials. Represent a sample space and events with an area probability model and use it to find probabilities. Use simulation to estimate the probability that an event will occur. Use the Multiplication Rule for Independent Events to determine whether two events are independent or dependent. Use the Multiplication Rule for Independent Events to determine the probability that two two-way table union of two events variable
student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. S-CP-5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have S-CP-6 Find the conditional probability of A given B as a fraction of B s outcomes that also belongs 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. independent events occur simultaneously. Given two events A and B from an everyday situation, assess whether is larger than, smaller than, or equal to P(A). In situations in which outcomes are equally likely, determine P(B A) as the ratio of the number of outcomes in A Bto the number of outcomes in A. Use an area probability model for two events A and B to calculate P(A B) by determining the fraction of B s area that overlaps with A. More generally, determine P(B A) as the ratio of P( A B) to P(A). Use the General Multiplication Rule to find P( A B). Understand how the General Multiplication Rule can be used with tree diagrams.
Organize data from two survey questions into a two-way frequency table. Given data organized in a two-way frequency table, calculate the marginal relative frequencies (percentages). Given data organized in a two-way frequency table, calculate the joint relative frequencies (percentages). Given data organized in a two-way table, calculate conditional relative frequencies (percentages). Represent marginal and conditional percentages with bar graphs. Interpret marginal, joint, and conditional percentages in the context of the data. Describe associations and trends of data in two-way tables.
Determine the expected value of a probability model with numeric outcomes.