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Algebra II Level 3 Curriculum Unit A - Rebuilding Algebra Skills Overview This brief unit is a review of Pre-Algebra and Algebra 1 topics that are integral for succeeding in Algebra 2. Although some procedure is necessary, students may really struggle with solving equations, and care should be given to teach each topic in a way that illuminates the reasons behind the methodology. For example, students should understand why, when terms are on the same side of an equations, they are combined, but when on opposite sides of the equation, we need to add the opposite to eliminate a term from one side, or why absolute value equations may have two solutions, one, or no solutions at all. 21st Century Capacities: Analyzing Stage 1 - Desired Results ESTABLISHED GOALS/ STANDARDS Transfer: Students will be able to independently use their learning in new situations to... MP 1 Make sense sense of problems and persevere in solving them MP2 Reason abstractly and quantitatively MP7 Look for and make use of structure CCSS.MATH.CONTENT.HSN.Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. CCSS.MATH.CONTENT.HSN.Q.A.2 Define appropriate quantities for the purpose of descriptive modeling. Interpret the structure of expressions. CCSS.MATH.CONTENT.HSA.SSE.A.1 Interpret expressions that represent a quantity in terms of its context.* 1. Manipulate equations/expressions or objects to create order and establish relationships. 2. Represent and interpret patterns in numbers, data and objects. 3. Make sense of a problem, initiate a plan, execute it, and evaluate the reasonableness of the solution. (Analyzing) Meaning: UNDERSTANDINGS: Students will understand that: ESSENTIAL QUESTIONS: Students will explore & address these recurring questions: 1. Mathematicians understand that placing a problem in a category gives one a familiar approach to solving it. 2. Mathematicians examine the impact of operations and how they relate to one another. 3. Mathematicians flexibly use different tools, strategies, and operations to build conceptual knowledge or solve problems. A. What have I seen in the past that might help me now? B. How do operations relate to one another? C. What does the solution tell me? D. How else might I solve this problem? E. What is another to represent this number? Madison Public Schools | July 2016 1 Algebra II Level 3 Curriculum CCSS.MATH.CONTENT.HSA.SSE.A.1.A Interpret parts of an expression, such as terms, factors, and coefficients. CCSS.MATH.CONTENT.HSA.SSE.A.1.B Interpret complicated expressions by viewing one or more of their parts as a single entity. CCSS.MATH.CONTENT.HSA.SSE.A.2 Use the structure of an expression to identify ways to rewrite it. CCSS.MATH.CONTENT.HSA.CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Acquisition: Students will know… Students will be skilled at… 1. Unless told otherwise, answers should be given as an exact answer, so a repeating decimal should be indicated or left as a decimal. 2. Why there are two solutions to an absolute equation 3. The meaning of the word solution, and how to tell if an equation has one, multiple, infinite, or no solutions. 4. Vocabulary: rational, irrational, terms, variables, base, exponents, equation, expression, cube, squared, product, inequality, absolute value 1. Simplifying expressions using the order of operations 2. Identifying properties of commutative, inverse and distributive 3. Rewrite an expression using exponents 4. Evaluating an expression using substitution 5. Write algebraic expressions from words 6. Solving equations - including literal 7. Checking solutions to equations 8. Modeling with equations and inequalities 9. Solving inequalities 10. Solving absolute value equations CCSS.MATH.CONTENT.HSA.CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. CCSS.MATH.CONTENT.HSA.REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method CCSS.MATH.CONTENT.HSA.REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. CCSS.MATH.CONTENT.HSF.BF.A.1 Write a function that describes a relationship between two quantities.* Madison Public Schools | July 2016 2 Algebra II Level 3 Curriculum Unit B - Equations on the Coordinate Plane Overview The purpose of this unit is to use math to analyze situations in which the rate of change is constant and to model those situations using linear equations. Students should make a connection between tabular, algebraic, and graphic representations of relations. In later units students will use the concepts and skills from this unit to work with quadratic and exponential functions. 21st Century Capacities: Analyzing, Presentation Stage 1 - Desired Results ESTABLISHED GOALS/ STANDARDS Transfer: Students will be able to independently use their learning in new situations to... MP4 Model with Mathematics MP7 Look for and make use of structure MP8 Look for and express regularity in repeated reasoning CCSS.MATH.CONTENT.HSN.Q.A.2 Define appropriate quantities for the purpose of descriptive modeling. CCSS.MATH.CONTENT.HSA.SSE.A.1.B Interpret complicated expressions by viewing one or more of their parts as a single entity. CCSS.MATH.CONTENT.HSA.CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CCSS.MATH.CONTENT.HSA.CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. CCSS.MATH.CONTENT.HSA.REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. CCSS.MATH.CONTENT.HSA.REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). CCSS.MATH.CONTENT.HSA.REI.D.11 Explain why the x- 1. Model relationships among quantities. (Analyzing) 2. Represent and interpret patterns in numbers, data and objects. (Analyzing and Presentation) 3. Draw conclusions about graphs, shapes, equations, or objects. (Analyzing and Presentation) Meaning: UNDERSTANDINGS: Students will understand that: 1. Mathematicians apply the mathematics they know to solve problems occurring in everyday life. 2. Mathematicians examine relationships to discern a pattern, generalizations, or structure. 3. Mathematicians can describe patterns, relations, and/or functions to access strategies to solve problems. ESSENTIAL QUESTIONS: Students will explore & address these recurring questions: A. How can change be described? B. How can a variable/ expression / equation/graph tell a story? C. How do you express and describe a pattern and use it to make predictions and solve a problem? D. How can the solutions to an equation or inequality be represented? Madison Public Schools | July 2016 3 Algebra II Level 3 Curriculum 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, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases wheref(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* CCSS.MATH.CONTENT.HSA.REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. CCSS.MATH.CONTENT.HSF.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.* CCSS.MATH.CONTENT.HSF.IF.C.7.A Graph linear and quadratic functions and show intercepts, maxima, and minima. CCSS.MATH.CONTENT.HSF.BF.A.1 Write a function that describes a relationship between two quantities.* CCSS.MATH.CONTENT.HSF.BF.A.1.A Determine an explicit expression, a recursive process, or steps for calculation from a context. CCSS.MATH.CONTENT.HSF.LE.A.1.B Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. CCSS.MATH.CONTENT.HSF.LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). CCSS.MATH.CONTENT.HSF.LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context. CCSS.MATH.CONTENT.HSG.GPE.B.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). CCSS.MATH.CONTENT.HSS.ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Acquisition: Students will know… Students will be skilled at… 1. That situations that have a constant rate of change can be modeled with a linear equation 2. Slope of a line can tell you what the function of a graph looks like in graphical form 3. Point-slope and slope - y intercept form of a line 4. The meaning of the slope and intercepts of a function in context 5. Any point on a graph is a solution to the equation or inequality….conversely, any point not on the graph is not a solution 6. When using linear programing vertex points are often key points in solving the problem 7. Vocabulary: vertical, horizontal, slope, rise, run, parallel, perpendicular, intercept, solution, system, intersection, vertex 1. Graphing a linear equation using a table, slope and y-intercept, and/or intercepts 2. Finding the slope of a function from an equation, graph or table 3. Comparing functions in table, equation, or graphical form 4. Using linear models to predict 5. Graphing linear inequalities 6. Graphing absolute value functions 7. Finding the solution to a system of equations by graphing, elimination algorithm and by substitution 8. Modeling situations using a system of equations 9. Solving systems of inequalities by graphing 10. Using linear programing to model and find solutions to problems Madison Public Schools | July 2016 4 Algebra II Level 3 Curriculum Unit C - Quadratic Equations and Parabolas Overview The purpose of this unit is to move beyond linear functions and to learn strategies to solve quadratic equations. Students should understand that the power of 2 creates a specific shaped graph (parabola). Students should also learn the importance of the complex number system, and should be taught about the history of complex numbers not being all that different from the history of negative numbers. 21st Century Capacities: Analyzing Stage 1 - Desired Results ESTABLISHED GOALS/ STANDARDS Transfer: Students will be able to independently use their learning in new situations to... MP 1 Make sense of problems and persevere in solving them MP2 Reason abstractly and quantitatively MP6 Attend to precision MP7 Look for and make use of structure CCSS.MATH.CONTENT.HSN.Q.A.2 Define appropriate quantities for the purpose of descriptive modeling. Perform arithmetic operations with complex numbers. CCSS.MATH.CONTENT.HSN.CN.A.1 Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real. CCSS.MATH.CONTENT.HSN.CN.A.2 Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. CCSS.MATH.CONTENT.HSN.CN.C.7 Solve quadratic equations with real coefficients that have complex solutions. CCSS.MATH.CONTENT.HSA.SSE.A.1.B Interpret complicated expressions by viewing one or more of their parts as a single entity. CCSS.MATH.CONTENT.HSA.SSE.A.2 Use the structure of an expression to identify ways to rewrite it. CCSS.MATH.CONTENT.HSA.SSE.B.3 1. Manipulate equations/expressions or objects to create order and establish relationships. (Analyzing) 2. Draw conclusions about graphs, shapes, equations, or objects. (Analyzing) Meaning: UNDERSTANDINGS: Students will understand that: 1. Mathematicians examine relationships to discern a pattern, generalizations, or structure. 2. Mathematicians continually evaluate their process and the reasonableness of the intermediate results. 3. Mathematicians identify relevant tools, strategies, relationships, and/or information in order to draw conclusions. ESSENTIAL QUESTIONS: Students will explore & address these recurring questions: A. How can I use symbols to communicate? B. What is another way that this problem could be solved? C. What math tools/models/strategies can I use to solve the problem? Madison Public Schools | July 2016 5 Algebra II Level 3 Curriculum Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* CCSS.MATH.CONTENT.HSA.SSE.B.3.A Factor a quadratic expression to reveal the zeros of the function it defines. CCSS.MATH.CONTENT.HSA.SSE.B.3.B Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. CCSS.MATH.CONTENT.HSA.CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CCSS.MATH.CONTENT.HSA.REI.B.4 Solve quadratic equations in one variable. CCSS.MATH.CONTENT.HSA.REI.B.4.A Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x- p)2 = q that has the same solutions. Derive the quadratic formula from this form. CCSS.MATH.CONTENT.HSA.REI.B.4.B Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. CCSS.MATH.CONTENT.HSA.REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). CCSS.MATH.CONTENT.HSF.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.* CCSS.MATH.CONTENT.HSF.IF.C.7.A Graph linear and quadratic functions and show intercepts, maxima, and minima. CCSS.MATH.CONTENT.HSF.BF.A.1 Write a function that describes a relationship between two quantities.* Acquisition: Students will know… Students will be skilled at… 1. The meaning of the imaginary number i 2. That an equation must be in standard form before attempting to use the quadratic formula to solve it 3. A parabola is symmetric 4. A quadratic equation may have 0,1 or 2 solutions (real or imaginary) 5. -b/2a gives the x value of the vertex of a parabola 6. The leading coefficient of a quadratic tells you if the graph opens up or down 7. Quadratic equations model parabolic situations 8. Vocabulary: roots, zeros, solutions, imaginary, vertex 1. Simplifying square roots 2. Combining complex numbers 3. Solving quadratic equations using isolating x2 ,factoring, graphing and the quadratic formula 4. Multiplying binomials 5. Factoring trinomials where a=1 6. Sketching graphs of quadratic equations 7. Working with quadratic functions in context Madison Public Schools | July 2016 6 Algebra II Level 3 Curriculum Unit D - Functions Overview The focus of Unit D is for students to learn what is a mathematical function and its importance in problem solving. Students will also explore and learn to use the concept of function notation. Even though function notation is awkward to learn and seems more cumbersome, it is a great tool that allows mathematicians to communicate more clearly. Students will learn to work flexibly between all representations of a relation or function (table, list, equation, graph, and mapping diagram). 21st Century Capacities: Synthesizing Stage 1 - Desired Results ESTABLISHED GOALS/ STANDARDS MP1 Make sense sense of problems and persevere in solving them MP2 Reason abstractly and quantitatively MP6 Attend to precision MP7 Look for and make use of structure CCSS.MATH.CONTENT.HSA.SSE.A.1.B Interpret complicated expressions by viewing one or more of their parts as a single entity. CCSS.MATH.CONTENT.HSA.SSE.A.2 Use the structure of an expression to identify ways to rewrite it. CCSS.MATH.CONTENT.HSA.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* CCSS.MATH.CONTENT.HSA.APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. CCSS.MATH.CONTENT.HSA.REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Transfer: Students will be able to independently use their learning in new situations to... 1. Manipulate equations/expressions or objects to create order and establish relationships. 2. Draw conclusions about graphs, shapes, equations, or objects. (Synthesizing) Meaning: UNDERSTANDINGS: Students will understand that: ESSENTIAL QUESTIONS: Students will explore & address these recurring questions: 1. Mathematicians use symbols and notations to make it easier to express themselves. 2. Mathematicians flexibly use different tools, strategies, and operations to build conceptual knowledge or solve problems. A. How can I use symbols to communicate? B. What does the function/graph tell me? Madison Public Schools | July 2016 7 Algebra II Level 3 Curriculum CCSS.MATH.CONTENT.HSF.IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. CCSS.MATH.CONTENT.HSF.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.* CCSS.MATH.CONTENT.HSF.IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). CCSS.MATH.CONTENT.HSF.BF.A.1 Write a function that describes a relationship between two quantities.* CCSS.MATH.CONTENT.HSF.BF.A.1.B Combine standard function types using arithmetic operations. CCSS.MATH.CONTENT.HSF.BF.A.1.C (+) Compose functions. CCSS.MATH.CONTENT.HSF.BF.B.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. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Acquisition: Students will know… Students will be skilled at… 1. A function and its inverse are reflections over the x = y line 2. A composition can verify an inverse function 3. Components of a function can be used to visualize the function 4. Vocabulary: composition, inverse, transformation, domain, range, relation, function, composition 1. Identifying the domain and range of a function 2. Identifying if a relation is a functions 3. Finding the value of f(x) for a specific x value 4. Adding, subtracting, multiplying and dividing functions 5. Finding the composition of two functions 6. Finding the inverse of a function 7. Graphing transformations of f(x) = x2 and f(x) = |xl with vertical and/or horizontal shifts and/or vertical flips CCSS.MATH.CONTENT.HSF.BF.B.4 Find inverse functions. CCSS.MATH.CONTENT.HSF.BF.B.4.A 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. CCSS.MATH.CONTENT.HSF.BF.B.4.B (+) Verify by composition that one function is the inverse of another. CCSS.MATH.CONTENT.HSF.BF.B.4.C (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. Madison Public Schools | July 2016 8 Algebra II Level 3 Curriculum Unit E - Trigonometry Overview Students will learn the basics of right triangle trigonometry, and will be able to apply trig ratios to solve word problems. Students will learn how to measure angles using radians, how to sketch angles in standard position, etc. The goal of this unit is to expose students to enough trigonometry for them to understand its value in the real world and to be successful in higher math. 21st Century Capacities: Analyzing, Synthesizing Stage 1 - Desired Results ESTABLISHED GOALS/ STANDARDS Transfer: Students will be able to independently use their learning in new situations to... MP 1 Make sense sense of problems and persevere in solving them MP5 Use appropriate tools strategically MP7 Look for and make use of structure CCSS.MATH.CONTENT.HSN.Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. CCSS.MATH.CONTENT.HSA.SSE.A.1.B Interpret complicated expressions by viewing one or more of their parts as a single entity. CCSS.MATH.CONTENT.HSA.SSE.A.2 Use the structure of an expression to identify ways to rewrite it. CCSS.MATH.CONTENT.HSF.TF.A.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended 1. Draw conclusions about graphs, shapes, equations, or objects. (Analyzing and Synthesizing) 2. Make sense of a problem, initiate a plan, execute it, and evaluate the reasonableness of the solution. (Analyzing and Synthesizing) 3. Use appropriate tools to make reaching solutions more efficient, accessible and accurate. (Analyzing and Synthesizing) Meaning: UNDERSTANDINGS: Students will understand that: 1. Effective problem solvers work to make sense of the problem before trying to solve it. 2. Mathematicians identify relevant tools, strategies, relationships, and/or information in order to draw conclusions. 3. Mathematicians use geometric models, and spatial sense to interpret and make sense of the physical environment. 4. Mathematicians analyze characteristics and properties of geometric shapes to develop mathematical arguments about geometric relationships. ESSENTIAL QUESTIONS: Students will explore & address these recurring questions: A. B. C. What math strategies can I use to solve the problem? How does classifying bring clarity? How can I use what I know in the world? Madison Public Schools | July 2016 9 Algebra II Level 3 Curriculum Acquisition: by the angle. CCSS.MATH.CONTENT.HSF.TF.B.7 (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.* CCSS.MATH.CONTENT.HSG.SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Students will know… Students will be skilled at… 1. Definitions of sin, cos, tan trig ratios 2. The relationship between the sides of a 30-60-90 and a 45-45-90 triangle 3. The relationship between sin and cos of complementary angles 4. The definition / meaning of a radian 5. Vocabulary: opposite, adjacent, hypotenuse 1. Solving right triangles using trigonometry 2. Using a calculator to find a trig function or an inverse of a trig function 3. Converting between degrees and radians 4. Drawing anges in the coordinate plane in standard position 5. Determining if two angles are coterminal or finding coterminal angles of an angle 6. Finding sector areas 7. Finding the arc length of sectors CCSS.MATH.CONTENT.HSG.SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. CCSS.MATH.CONTENT.HSG.SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* CCSS.MATH.CONTENT.HSG.C.B.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Madison Public Schools | July 2016 10 Algebra II Level 3 Curriculum Unit F - Exponential and Logarithmic Functions Overview The purpose of this unit to expose students to ways of manipulating expressions using exponents. Students are expected to have a conceptual understanding of the rules around exponents and logarithms. They should explore the logic behind the development of negative exponents, zero as an exponent, and rational exponents. These should not just be taught as rules. 21st Century Capacities: Analyzing Stage 1 - Desired Results ESTABLISHED GOALS/ STANDARDS Transfer: Students will be able to independently use their learning in new situations to... MP2 Reason abstractly and quantitatively MP7 Look for and make use of structure CCSS.MATH.CONTENT.HSN.RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. CCSS.MATH.CONTENT.HSN.RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. CCSS.MATH.CONTENT.HSN.Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. CCSS.MATH.CONTENT.HSA.SSE.A.2 Use the structure of an expression to identify ways to rewrite it. CCSS.MATH.CONTENT.HSA.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* 1. Manipulate equations/expressions or objects to create order and establish relationships. (Analyzing) 2. Demonstrate fluency with math facts, computation and concepts. Meaning: UNDERSTANDINGS: Students will understand that: ESSENTIAL QUESTIONS: Students will explore & address these recurring questions: 1. Mathematicians create or use A. How can I break a problem down into manageable parts? models to examine, describe, B. What math tools/models/strategies solve. can I use to solve the problem? 2. Mathematicians can describe patterns, relations, and/or functions C. How can I simplify the problem? D. How can I use symbols to to access strategies to solve communicate? problems. 3. Mathematicians represent and analyze mathematical situations and structures using algebraic symbols to communicate thinking. Madison Public Schools | July 2016 11 Algebra II Level 3 Curriculum CCSS.MATH.CONTENT.HSA.SSE.B.3.C Use the properties of exponents to transform expressions for exponential functions. Acquisition: Students will know… Students will be skilled at… CCSS.MATH.CONTENT.HSA.SSE.B.4 Derive the formula for the sum 1. Know the properties of exponents of a finite geometric series (when the common ratio is not 1), and use the and why they work formula to solve problems. 2. That with C> 0 if a>1 the function represents growth and if 0 < a < 1 CCSS.MATH.CONTENT.HSA.REI.A.2 Solve simple rational and the function represents decay radical equations in one variable, and give examples showing how 3. The definition of a log extraneous solutions may arise. 4. Common logarithmic function has base 10, natural log has base e CCSS.MATH.CONTENT.HSA.REI.D.10 Understand that the graph of 5. The Product, Quotient and Power an equation in two variables is the set of all its solutions plotted in the Properties of logarithms coordinate plane, often forming a curve (which could be a line). 6. Vocabulary: index, argument, logarithm, base, decay, CCSS.MATH.CONTENT.HSF.IF.A.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 xis an element of its domain, then f(x) denotes the output of fcorresponding to the input x. The graph of f is the graph of the equation y = f(x). CCSS.MATH.CONTENT.HSF.IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 1. Simplifying expressions using the properties of exponents 2. Using y = Cax type equations to graph exponential growth and decay and solve application problems 3. Using a = P(1 + r/n)nt to solve compound interest problems 4. Identifying if a relationships is exponential or linear 5. Converting between radical notation and rational exponent notation 6. Solving radical equations 7. Converting between exponential and logarithmic form 8. Sketching graphs of logarithmic functions 9. Using logarithms to solve equations and in applications CCSS.MATH.CONTENT.HSF.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.* CCSS.MATH.CONTENT.HSF.IF.C.7.E Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. CCSS.MATH.CONTENT.HSF.IF.C.8.B Use the properties of exponents to interpret expressions for exponential functions. For example, 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 Madison Public Schools | July 2016 12 Algebra II Level 3 Curriculum exponential growth or decay. CCSS.MATH.CONTENT.HSF.BF.A.1 Write a function that describes a relationship between two quantities.* CCSS.MATH.CONTENT.HSF.BF.A.1.A Determine an explicit expression, a recursive process, or steps for calculation from a context. CCSS.MATH.CONTENT.HSF.BF.B.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. CCSS.MATH.CONTENT.HSF.LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. CCSS.MATH.CONTENT.HSF.LE.A.1.A Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. CCSS.MATH.CONTENT.HSF.LE.A.1.C Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. CCSS.MATH.CONTENT.HSF.LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). CCSS.MATH.CONTENT.HSF.LE.A.4 For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. CCSS.MATH.CONTENT.HSF.LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context. Madison Public Schools | July 2016 13