Algebra II. Key Resources: Page 3


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1 Algebra II Course This course includes the study of a variety of functions (linear, quadratic higher order polynomials, exponential, absolute value, logarithmic and rational) learning to graph, compare, perform operations and manipulate them in order to solve, analyze and apply to problems. Students will use probability and statistics to evaluate outcomes of decisions. Students develop rigorous problem solving skills, logical reasoning and mathematical communication skills required for success in higher math courses and real life experiences. *Graphing calculators are required. Missouri Learning Standards Extend and use the relationship between rational exponents and radicals Use complex numbers Define and use logarithms Solve equations and inequalities Solve general systems of equations and inequalities Perform operations on polynomials and rational expressions Use and interpret functions Create new functions from existing functions Use functions to model realworld problems Make inferences and justify conclusions Fit a data set to a normal distribution Key Resources: Page 3
2 Scope And Sequence Timeframe Unit Instructional Topics 9 Day(s) Functions 1. Parent Functions, Transformations, Domain, and Range 2. Even and Odd Functions, Function definition, and different representations of functions. 3. Function notation and evaluation, including piecewise. 4. Piecewise graphs including step functions 5. Function Operations and Composition 6. Inverse Functions 11 Day(s) Equations and Inequalities 1. Applications of Functions 2. Literal Equation 3. Systems of Equations and Inequalities 4. Linear Programming 5. Average Rate of Change 9 Day(s) Radical Functions & Rational Exponents 1. Graphing Radical Functions 2. Simplifying Radicals and Solve by Taking Roots 3. Rationalizing the Denominator 4. Solving Radical Equations 5. Applications of Radical Functions 8 Day(s) Quadratics 1. Graphing Parabolas 2. Change between Vertex & Standard Forms 3. Introduction to Complex Numbers 4. Solving Quadratic Equations 5. Quadratic Inequalities 9 Day(s) Polynomials 1. Definitions and Binomial Theorem 2. Graphing Polynomials 3. Long Division and Synthetic Division 4. Solve Polynomials by Factoring 5. Use Zeros to Write Polynomial Functions 10 Day(s) Logarithmic and Exponential Functions 1. Graph Exponential Functions 2. Applications of Exponential Functions 3. Definition of Logarithms and Inverses 4. Graph Logarithmic Functions 5. Properties of Logarithms 6. Solve Logarithmic and Exponential Equations 7. Geometric Series 7 Day(s) Rational Functions 1. Multiplying & Dividing Rational Expressions 2. Adding & Subtracting Rational Expressions 3. Simplifying Complex Fractions 4. Solving Rational Equations 5. Applying Rational Functions 5 Day(s) Statistics 1. Summarize, represent, and interpret data on a single count or measurement variable 2. Understand and evaluate random processes underlying statistical experiments 3. Make inferences and justify conclusions from sample surveys, experiments, and observational studies. 4. (+) Use probability to evaluate outcomes of decisions. 12 Day(s) Trigonometry 1. Right Triangle Review and Radian Measure 2. Building the Unit Circle 3. Pythagorean Identities 4. Graphing and Applications Course Rationale In alignment with Common Core State Standards, the Park Hill School District's Mathematics courses provide students with a solid foundation in number sense while building to the application of more demanding math concepts and procedures. The courses focus on procedural skills and conceptual understandings to ensure coherence and depth in mathematical practices and application to real world issues and challenges. Enduring Understandings 1. Mathematical understanding is built through problem solving and reasoning. 2. There is a direct relationship between Logarithmic and Exponential Functions. 3. Statistics and probability are tools to solve many mathematical situations and make predictions in the real world. 4. Parent functions are the foundation of identifying, analyzing, constructing, and transforming graphs. 5. Appropriate mathematical tools help to investigate, solve and explain real world situations. 6. Problems are solved using both visual (including a graphing calculator) and analytical means. Page 3
3 Board Approval Date Unit: Functions Course Details Unit Overview Students will understand the concept of a function and how to graph and manipulate them. Duration: 9 Day(s) Enduring Understandings Parent functions are the foundation of identifying, analyzing, constructing, and transforming graphs. Basic mathematical operations can be performed on given functions resulting in an infinite number of possible new functions. A function must pass the horizontal line test to have an inverse. Essential Questions When might it be useful to combine functions instead of working with two separate functions? How can the horizontal line test be used to determine if a function has an inverse? How are parent functions used to graph transformations? Example Assessment Items Write a composition of two given functions. Determine if a given graph has an inverse. Given a function, use transformations to graph. Academic Vocabulary composition of functions domain range function inverse function piecewise function even function odd function Topic: Parent Functions, Transformations, Domain, and Range Duration: 1 Day(s) The student will 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. Experiment with cases and illustrate an explanation of the effects on the graph using technology. The student will recognize the graph of an algebraic function. Alg2IF.A.1 Alg2IF.A.2 Alg2NQ.A.2 Topic: Even and Odd Functions, Function definition, and different representations of functions. Duration: 1 Day(s) Even and odd functions with graphs and symmetry. Function definition including vertical line test. Different representations of functions such as algebraic, table of values, graph, and verbal. The student will 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). The student will 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. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Including recognizing even and odd functions from their graphs and algebraic expressions for them. The student will, 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. Alg2BF.A.1 Alg2BF.A.2 Alg2BF.A.3 Page 3
4 Topic: Function notation and evaluation, including piecewise. Duration: 1 Day(s) The student will be able to evaluate functions using function notation, such as f(2) and f(x+5). Alg2IF.A.2 Topic: Piecewise graphs including step functions Duration: 1 Day(s) Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions. Alg2FM.A.1 Topic: Function Operations and Composition Duration: 1 Day(s) The student will be able to perform function operations such as add, subtract, multiply, and divide including composed functions. Alg2APR.A.4 Alg2BFA.1 Topic: Inverse Functions Duration: 2 Day(s) Focus on developing the idea of inverses to be generalized so you can revisit it in upcoming units that will be covered. Show different representations (graphical, tabular, word problems, multiple function types) so that the understanding piece is the focus, not the procedural algebra steps. The student will read values of an inverse function from a graph or a table, given that the function has an inverse. The student will solve an equation of the form f(x) for a simple function f that has an inverse and write an expression for the inverse. The student will verify by composition that one function is the inverse of another. Alg2BF.A.2 Unit: Equations and Inequalities Duration: 11 Day(s) Topic: Applications of Functions Duration: 3 Day(s) Page 3
5 Given one representation the students will be able to write a function that best models it, also be able to switch between different representations (i.e. verbal, tabular, graphical, symbolic) Alg2REI.A.1 Alg2REI.B.1 Topic: Literal Equation Duration: 1 Day(s) The students will be able to solve formulas for a specified variable. Alg2NQ.A.1 Alg2NQ.A.3 Alg2NQ.A.4 Topic: Systems of Equations and Inequalities Duration: 2 Day(s) Note: 3x3 systems are no longer solved by hand. Focus on realworld problems with 3 variables and using technology to solve. The student will be able to understand that the solution(s) of a system represents intersection points of curves (linear, polynomial, rational, absolute value, exponential, logarithmic). Emphasize using graphing technology for curves not yet explicitly taught. Alg2REI.A.2 Topic: Linear Programming Duration: 2 Day(s) Topic: Average Rate of Change Duration: 1 Day(s) Page 4
6 The student will be able to find the average rate of change with multiple representations (symbolic, tabular, graphical). Unit: Radical Functions & Rational Exponents Duration: 9 Day(s) Topic: Graphing Radical Functions Duration: 2 Day(s) Given one representation of a radical function (square roots and cube roots), students will be able to represent it in any of the other representations (graphical, equation, tabular, written description). The student will be able to identify and interpret the key features of the graph (intercepts, where function is increasing and decreasing, domain, and range). The student will be able to find the inverse of a radical function, including domain restrictions. (For example, the inverse of the square root function will be a quadratic function, but you can't keep the whole quadratic function because it fails the horizontal line test). Topic: Simplifying Radicals and Solve by Taking Roots Duration: 1 Day(s) The student will be able to simplify radical expressions using absolute values properly with even roots. The student will be able to extend the idea of absolute values for even roots to solve basic equations by taking roots and using the positive/negative sign (example: x^2=4 is x =2, therefore x=2, 2). Topic: Rationalizing the Denominator Duration: 1 Day(s) The student will be able to rationalize the denominator of expressions using conjugates and expressions with higher roots. Topic: Solving Radical Equations Duration: 2 Day(s) The student will be able to solve equations involving radicals and rational exponents. Topic: Applications of Radical Functions Duration: 1 Day(s) The student will be able to use a radical model to solve for a given situation. Unit: Quadratics Topic: Graphing Parabolas Duration: 1 Day(s) The student will be able to graph a parabola in standard form and vertex form (simpler graphs by hand, more complicated by technology). Topic: Change between Vertex & Standard Forms Duration: 1 Day(s) The student will be able to change between standard form and vertex form and vice versa (introduction to completing the square). Topic: Introduction to Complex Numbers Duration: 1 Day(s) The student will be able recognize there is a complex number i such that i² = 1, and every complex number has the form a + bi with a and b real. Page 5
7 The student will be able to use the relation i² = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Alg2NQ.B.1 Alg2NQ.B.2 Alg2NQ.B.3 Topic: Solving Quadratic Equations Duration: 2 Day(s) The student will be able to solve quadratic equations by square roots, factoring, and quadratic formula including imaginary solutions. The student will be able to solve quadratic equations by completing the square. The student will be able to model quadratic relationships and identify and interpret key features. Topic: Quadratic Inequalities Duration: 1 Day(s) The student will be able to use technology to identify intervals of a quadratic function where the function is greater than (or equal to) or less than (or equal to) a given number. Example x^29>0 so the interval that satisfies this after graphing on a calculator (infinity, 3) U (3, infinity). Unit: Polynomials Duration: 9 Day(s) Topic: Definitions and Binomial Theorem Duration: 1 Day(s) The student will be able to interpret parts of an expression, such as terms, factors, and coefficients. The student will be able to apply the binomial theorem to multiply polynomials. The student will be able to prove polynomial identities by showing that the left and right sides are equivalent. Alg2APR.A.1 Alg2APR.A.3 Topic: Graphing Polynomials Duration: 3 Day(s) The student will be able to find zeros of polynomials, including multiplicity to show its behavior on the graph (this also includes more complicated factoring, such as sum/difference of cubes, quadratic in form). The student will be able to state end behavior from different representations (equation, graph, tabular). The student will be able to graph polynomials by hand using the ideas of zeros, multiplicity, end behavior, and intercepts. Alg2APR.A.2 Page 5
8 Topic: Long Division and Synthetic Division Duration: 1 Day(s) The student will be able to divide polynomials using long division or synthetic division. Alg2APR.A.2 Topic: Solve Polynomials by Factoring Duration: 1 Day(s) The student will be able to use division to factor and solve polynomial equations, including complex solutions. Alg2APR.A.1 Topic: Use Zeros to Write Polynomial Functions Duration: 1 Day(s) Given zeros of a function, the student will be able to write the factored or standard form of the polynomial function. Also, the student will be able to use complex numbers to factor. Example x^2+4=(x+2i)(x2i). Alg2APR.A.5 Unit: Logarithmic and Exponential Functions Duration: 10 Day(s) Topic: Graph Exponential Functions Duration: 1 Day(s) The student will be able to graph exponential decay and exponential growth functions. Topic: Applications of Exponential Functions Duration: 1 Day(s) The student will be able to use exponential decay and exponential growth models to solve problems. Topic: Definition of Logarithms and Inverses Duration: 1 Day(s) The student will be able to use the definition of logarithms to switch between logarithmic and exponential forms. The student will be able to find the inverse of an exponential function and a logarithmic function. Topic: Graph Logarithmic Functions Duration: 1 Day(s) The student will be able to graph logarithmic functions and identify key components. Alg2SSE.A.1 Alg2SSE.A.2 Alg2SSE.A.3 Alg2SSE.A.4 Topic: Properties of Logarithms Duration: 1 Day(s) The student will be able to use the properties of logarithms to attain equivalent expressions. Page 6
9 Alg2SSE.A.1 Alg2SSE.A.2 Alg2SSE.A.3 Alg2SSE.A.4 Topic: Solve Logarithmic and Exponential Equations Duration: 3 Day(s) The student will be able to solve exponential equations (including problems with matching bases on both sides AND problems with nonmatching bases on both sides). The student will be able to solve logarithmic equations (including problems where you have to switch to log form to solve, problems where you have to use a property of logarithms first before you can solve, problems where you have a common log base on both sides). The student will be able to solve application problems that involve exponentials and/or logarithms. Alg2SSE.A.1 Alg2SSE.A.2 Alg2SSE.A.3 Alg2SSE.A.4 Topic: Geometric Series Duration: 1 Day(s) This is to be done after the test over Logarithmic and Exponential Functions. It is required for Common Core. The student will be able to derive and use the formula for a geometric series. (Extension to be done after the Unit test) Unit: Rational Functions Duration: 7 Day(s) Topic: Multiplying & Dividing Rational Expressions Duration: 1 Day(s) The student will be able to multiply and divide rational expressions and state the domain restrictions. Topic: Adding & Subtracting Rational Expressions Duration: 1 Day(s) The student will be able to add and subtract rational expressions and state the domain restrictions. Topic: Simplifying Complex Fractions Duration: 1 Day(s) The student will be able to simplify complex fractions and state the domain restrictions. Topic: Solving Rational Equations Duration: 1 Day(s) The student will be able to solve rational equations and check for extraneous solutions against the domain restrictions. Topic: Applying Rational Functions Duration: 1 Day(s) The student will be able to apply the rules for adding, subtracting, multiplying, dividing rational expressions (including complex fractions), and solve rational equations. This includes writing equations to model realworld situations and thinking about domain in the realworld context. Page 9
10 Unit: Statistics Duration: 5 Day(s) Unit Overview Resource Websites with Lessons: map.mathshell.org (go to lessons and resources, then core math tools) Topic: Summarize, represent, and interpret data on a single count or measurement variable Duration: 1 Day(s) The student will be able to use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. Topic: Understand and evaluate random processes underlying statistical experiments Duration: 1 Day(s) Include comparing theoretical and empirical results to evaluate the effectiveness of a treatment. The student will be able to understand statistics as a process for making inferences about population parameters based on a random sample from that population. The student will be able to decide if a specified model is consistent with results from a given datagenerating 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 question a model? Alg2DS.A.1 Alg2DS.A.2 Alg2DS.A.3 Topic: Make inferences and justify conclusions from sample surveys, experiments, and observational studies. The concept of statistical significance is developed informally through simulation as meaning a result that is unlikely to have occurred solely as a result of random selection in sampling or random assignment in an experiment. The student will be able to recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. The student will be able to use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. Focus on the variability of results from experiments  that is, focus on statistics as a way of dealing with, not eliminating, inherent randomness. The student will be able to use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. Focus on the variability of results from experiments  that is, focus on statistics as a way of dealing with, not eliminating, inherent randomness. The student will be able to evaluate reports based on data. Alg2DS.B.1 Alg2DS.B.2 Alg2DS.A.4 Alg2DS.A.5 Alg2DS.A6 Alg2DS.A7 Topic: (+) Use probability to evaluate outcomes of decisions. Duration: 1 Day(s) (+) Extend to more complex probability models. Include situations such as those involving quality control, or diagnostic tests that yield both false positive and false negative results. Page 9
11 The student will be able to use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). The student will be able to analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Unit: Trigonometry Duration: 12 Day(s) Topic: Right Triangle Review and Radian Measure Duration: 4 Day(s) The student will review SOHCAHTOA, special right triangles, and solve right triangles. The student will be able to understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. The student will be able to find linear and angular speed. The student will be able to find arc length and area of a sector using radian measure. Topic: Building the Unit Circle Duration: 1 Day(s) The student will be able to explain how 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. Begin with the special triangles to build the unit circle with the students. Topic: Pythagorean Identities Duration: 1 Day(s) The student will be able to prove the Pythagorean identity sin²(theta) + cos²(theta) = 1 and use it to find sin(theta), cos(theta), or tan(theta) given sin(theta), cos(theta), or tan(theta) and the quadrant of the angle (i.e. Given a trig function, students should be able to use that and right triangles with the unit circle to find other trig functions.) Topic: Graphing and Applications Duration: 4 Day(s) The student will be able to graph sine, cosine, and tangent functions with transformations. The student will be able to define and identify the following: period, amplitude, frequency, midline, and asymptotes. The student will be able to choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Page 9
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