Conceptual Category(s) Algebra Functions Domain Seeing Structure in Expressions (A-SSE) Reasoning with Equations and Inequalities (A-REI) Arithmetic with Polynomials and Rational Expressions (A-APR) Creating Equations (A-CED) Building Functions (F-BF) Cluster Interpret the structure of expressions (A-SSE.1b, 2) Understand solving equations as a process of reasoning and explain the reasoning (A-REI.2) Represent and solve equations and inequalities graphically (A-REI.11) Rewrite rational expressions (A-APR. 7) Create equations that describe numbers or relationships (A-CED.1-3) Build new functions from existing functions (F-BF.4a,b) Alignments: CCSS: See below Performance: 1.6, 3.4 Knowledge: (MA) 4 MACLE: See below NETS: 1d; 4c,d DOK: 1-3 Standards Learning Targets A-SSE.1b, 2 1. Interpret expressions that represent a quantity in terms of its context 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 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-REI.2, 11 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise Unit A: Rational Expressions, Equations, & Inequalities: Perform operations with and simplify rational expressions, create and solve rational equations and inequalities, and find inverses of rational functions Write equivalent expressions in a variety of forms by factoring Factor GCF, trinomials, difference of squares, sums & differences of cubes, and factor by grouping CCSS: A-SSE.1b,2 MACLE: AR.1.C; AR.2.B 1
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, 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 A-APR.7 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.1-3 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 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales 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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods F-BF. 4a,b 4. Find inverse functions 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. For example, f(x) =2 x 3 or f(x) = (x+1)/(x 1) for x 1 b. (+) Verify by composition that one function is the inverse of another Add and subtract rational expressions Multiply and divide rational expressions Simplify rational expressions including complex rational fractions Rewrite rational expressions in different forms CCSS: A-APR.7 MACLE: N/A Create a simple rational equation/inequality in 1 variable Solve a simple rational equation/inequality CCSS: A-CED.1; A-REI.2 MACLE: AR.1.D; AR.2.C Solve a system of functions using graphing calculator technology including linear, quadratic, and rational functions Verify the solution(s) of a system of functions Explain the solution(s) of a system of functions Isolate any variable in a rational equation CCSS: A-CED.1; A-REI.2,11 MACLE: AR.1.D; AR.2.C Determine an equation in two variables using provided data (word problems) Recognize viable solutions Understand/apply restrictions on domain CCSS: A-CED.2,3 MACLE: N/A Find the inverse of a rational function Verify inverses through function composition CCSS: F-BF.4a,b MACLE: AR.2.B 2
Instructional Strategies Lecture enhanced with: SMART Notebook PowerPoint the Internet Drill and guided practice Problem solving: Domain investigation Reflective discussion Student self-reflection Class discussion Computer assisted instruction Game: Factoring Match-It Assessments/Evaluations The students will be assessed on the concepts taught using a variety of modalities: Direct teacher observations Project: Potential Project: Light it Up Quizzes Homework assignments see pacing guide Formal common assessment Unit A test Mastery Level: 80% 3
Textbook(s): (sample copies on the bookshelf) Pearson Algebra 2 Common Core (primary source) Holt Algebra 2 Common Core Edition Houghton Mifflin On Core Mathematics: Algebra 2 Website(s): http://www.mathsisfun.com/algebra/rational-expression.html http://www.purplemath.com/modules/rtnldefs.htm http://www.youtube.com/watch?v=b4bvldghf5i Graphing calculator Algebra 2 Instructional Resources/Tools 4
Conceptual Category(s) Domain Cluster Alignments: CCSS: See below Performance: 3.4 Knowledge: (MA) 4 MACLE: See below NETS: 3d; 5b; 6b DOK: 1-3 Algebra Number and Quantity Functions Arithmetic with Polynomials and Rational Expressions (A-APR) Seeing Structures of Expressions (A-SSE) Creating Equations (A-CED) Reasoning with Equations and Inequalities (A-REI) Building Functions (F-BF) Interpreting Functions (F-IF) Linear, Quadratic, and Exponential Models (F-LE) Complex Number System (N-CN) Perform arithmetic operations on polynomials (A-APR.1) Understand the relationship between zeros and factors of polynomials (A-APR.2, 3) Use polynomial identities to solve problems (A-APR.5) Rewrite rational expressions (A-APR.6) Interpret the structures of expressions (A-SSE.1, 2) Create equations that describe numbers or relationships (A-CED.2, 3) Represent and solve equations and inequalities graphically (A-REI.11) Build new functions from existing functions (F-BF.3) Analyze functions using different representations (F-IF.7) Construct and compare linear, quadratic, and exponential models and solve problems (F-LE.3) Use complex numbers in polynomial identities and equations (N-CN.8) 5
Standards A-APR.1-3, 5 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 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) 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 5. Know and apply the Binomial Theorem for the expansion of (x + y) n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle 6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system A-SSE.1, 2 1. Interpret expressions that represent a quantity in terms of its context 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-CED.2, 3 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales Learning Targets Unit B: Polynomial Expressions & Functions: Simplify polynomial expressions, solve problems with polynomials, and graph and identify characteristics of polynomial functions Add, subtract, and multiply polynomials Classify and graph polynomials Use properties of end behavior to analyze, describe, and graph polynomial functions Identify maxima and minima of polynomial functions (turning points) Determine the intervals when a polynomial function is increasing or decreasing Identify even and odd functions from a graph and/or algebraic expression CCSS: A-APR.1; F-IF.7; F-LE.3 MACLE: AR.2.A Use the Factor Theorem to determine factors and zeros of a polynomial Identify the multiplicity of zeros Write a polynomial function from its zeros CCSS: A-APR.3; A-SSE.2 MACLE: AR.2.A Find all the zeros of a polynomial by factoring (GCF, quadratic trinomials, difference of squares, grouping, sum or difference of cubes) Find all the real zeros of a polynomial by graphing CCSS: A-SSE.2; A-APR.3 MACLE: AR.2.A Use long division and synthetic division to divide polynomials. 6
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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods A-REI.11 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, 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 F-BF.3 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 an odd functions from their graphs and algebraic expressions for them F-IF. 7 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-LE. 3 3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function N-CN. 8 8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x 2 + 4 as (x + 2i)(x 2i) Know and apply the Remainder Theorem CCSS: A-APR.2 MACLE: NO.1.B Use the Rational Root Theorem and the Conjugate Root Theorem to solve polynomial equations CCSS: N-NC.8 MACLE: AR.2.C Use the Fundamental Theorem of Algebra to solve polynomial equations with complex solutions CCSS: N.NC.8; A-APR.2,3 MACLE: NO.1.B Use binomial expansion to expand binomial expressions that are raised to positive integer powers Determine the coefficients of a binomial expansion by Pascal s Triangle CCSS: A-APR.1,5 MACLE: AR.1.C Use technology to find polynomial models for a given set of data (linear, quadratic, cubic, quartic) CCSS: A-CED.3; F-IF.7; A-CED.2; A-SSE.1 MACLE: DP.2.C Apply transformations to graphs of polynomials Use technology to find the intersection of graphs (linear vs. polynomial) CCSS: F-BF.3; F-IF.7; A-REI.11 MACLE: GSR.3.B 7
Instructional Strategies Lecture enhanced with: SMART Notebook PowerPoint the Internet Drill and guided practice Demonstrations: Investigate the graphs and characteristics of f(x) = xn) Problem solving: Build a box of maximum volume Reflective discussion Student self-reflection Class discussion Technology enhanced (TI graphing calculator) Assessments/Evaluations The students will be assessed on the concepts taught using a variety of modalities: Direct teacher observations Quizzes Homework assignments see pacing guide Formal common assessment Unit B test Mastery Level: 80% Instructional Resources/Tools Textbook(s): (sample copies on bookshelf) Pearson Algebra 2 Common Core (primary source) Holt Algebra 2 Common Core Edition Houghton Mifflin On Core Mathematics: Algebra 2 Graphing calculator 8
Conceptual Category(s) Algebra Functions Domain The Real Number System (N-RN) Building Functions (F-BF) Reasoning with Equations and Inequalities (A-REI) Creating Equations (A-CED) Interpreting Functions (F-IF) Cluster Extend the properties of exponents to rational exponents (N-RN.2) Build a function that models a relationship between two quantities (F-BF.1) Build new functions from existing functions (F-BF.3, 4) Understand solving equations as a process of reasoning and explain the reasoning (A-REI.2) Represent and solve equations and inequalities graphically (A-REI.11) Create equations that describe numbers or relationships (A-CED.2, 3) Analyze functions using different representations (F-IF.7) Alignments: CCSS: See below Performance: 1.4, 1.6, 2.7, 3.4, 3.6, 3.7 Knowledge: (MA) 1,4,5 MACLE: See below NETS: 1a; 3d; 5c DOK: 1-3 Standards Learning Targets N-RN.2: 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents F-BF.1c, 3, 4a,b,d 1. Write a function that describes a relationship between two quantities c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time Unit C: Radical Expressions & Functions: Perform operations with and simplify radical expressions; create and solve radical equations and inequalities; graph and transform radical functions; find inverses and the composition of radical functions Change radical expressions to expressions with fractional exponents Change expressions with fractional exponents to radical expressions Evaluate expressions with rational exponents or nth roots CCSS: N-RN.2 MACLE: AR.2.A; NO.3.D 9
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. 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 4. Find inverse functions 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. For example, f(x) =2 x 3 or f(x) = (x+1)/(x 1) for x 1 b. (+) Verify by composition that one function is the inverse of another d. (+) Produce an invertible function from a non-invertible function by restricting the domain A-REI.2, 11 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise 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, 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 A-CED.2, 3 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales 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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods Solve equations involving nth roots Solve radical equations that may include extraneous solutions CCSS: A-REI.2 MACLE: AR.2.A; NO.3.D Perform operations with functions (add, subtract, multiply, & divide) Perform composition on given functions (ex: f(g(x)) ) CCSS: F-BF.1c MACLE: AR.1.B; AR.2.B Perform composition on given functions (ex: f(g(x)) ) Determine the domain of a composition CCSS: F-BF.1c; A-CED.3 MACLE: AR.2.B Find the inverse of a function, including power functions Determine whether two functions are inverses Determine if a function is one to one Write an inverse function of a non-invertible function by restricting the domain CCSS: F-BF.4a,b,d MACLE: AR.2.B Graph square root functions by hand and using a calculator Identify and describe transformations of f(x) including horizontal/vertical reflection, horizontal/vertical stretch and shrink, horizontal/vertical shift Determine the domain and range of square root functions Identify even and odd functions from graph and algebraic expressions CCSS: F-BF.3; F-IF.7; A-CED.3 MACLE: AR.2.A 10
F-IF.7b 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 Graph cube root functions by hand and using a calculator Identify and describe transformations of f(x) including horizontal/vertical reflection, horizontal/vertical stretch and shrink, horizontal/vertical shift Determine the domain and range of cube root functions Identify even and odd functions from graph and algebraic expressions CCSS: F-BF.3; F-IF.7; A-CED.3 MACLE: AR.2.A Solve a system of functions (by hand and using a calculator) Verify the solution(s) Explain the meaning of the solution(s) Determine a radical equation in two variables using provided data (word problems) Graph equations on a coordinate plane including labels and scales on axes Identify an appropriate domain and range for a given model involving radical functions including real world application problems Instructional Strategies CCSS: A-REI.11; A-CED.2,3 MACLE: AR.1.C,E; AR.2.C; AR.3.A Lecture enhanced with: SMART Notebook PowerPoint the Internet Drill and guided practice Demonstrations: Use the graphing calculator to investigate square root and cube root transformations Reflective discussion Student self-reflection Class discussion Computer assisted instruction 11
Assessments/Evaluations The students will be assessed on the concepts taught using a variety of modalities: Direct teacher observations Quizzes Homework assignments see pacing guide Formal common assessment Unit C test Mastery Level: 80% Instructional Resources/Tools Textbook(s): (sample copies on bookshelf) Pearson Algebra 2 Common Core (primary source) Holt Algebra 2 Common Core Edition Houghton Mifflin On Core Mathematics: Algebra 2 Website(s): http://www.purplemath.com/modules/graphrad.htm Graphing calculator 12
Conceptual Category(s) Number & Quantity Algebra Functions Domain The Real Number System (N-RN) Seeing Structure in Expressions (A-SSE) Creating Equations (A-CED) Reasoning with Equations and Inequalities (A-REI) Interpreting Functions (F-IF) Building Functions (F-BF) Linear, Quadratic, and Exponential Models (F-LE) Cluster Extend the properties of exponents to rational exponents (N-RN.1, 2) Interpret the structure of expressions (A-SSE.1a,b, 2) Write expressions in equivalent forms to solve problems (A-SSE.3c) Create equations that describe numbers or relationships (A-CED.1-3) Represent and solve equations and inequalities graphically (A-REI.10, 11) Understand the concept of a function and use function notation (F-IF.1, 2) Interpret functions that arise in applications in terms of the context (F-IF.4-6) Analyze functions using different representations (F-IF.7e, 8b) Build a function that models a relationship between two quantities (F-BF.1a,b) Build new functions from existing functions (F-BF.3) Construct and compare linear, quadratic, and exponential models and solve problems (F-LE.1a,c, 2) Interpret expressions for functions in terms of the situation they model (F-LE.5) Alignments: CCSS: See below Performance: 1.6, 1.8, 1.10 Knowledge: (MA) 4 MACLE: See below NETS: 1d; 4a; 6a DOK: 1-3 13
Standards N-RN.1, 2: 1. Explain how the definition of the meaning of rational exponents follow from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 to hold, so (5 1/3 ) 3 must equal 5 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents A-SSE.1a,b, 3c: 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 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression c. Use properties of exponents to transform expressions for exponential functions. For example, the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t to reveal he approximate equivalent monthly interest rate if the annual rate is 15% A-CED.1-3: 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 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales Learning Targets Unit D: Exponential Functions: Write, solve, and graph exponential functions Review properties of exponents to rewrite expressions involving radicals and rational exponents Identify parts of an exponential expression/equation Identify exponential equations as growth, decay, or neither Prove that exponential functions grow by equal factors over equal intervals CCSS: N-RN.1,2; F-IF.6,8b; A-SSE.1a,b; F-LE.1a MACLE: AR.2.A; AR.4.A Graph exponential growth functions Identify asymptotes, domain/range, intercepts, end behavior, increasing/decreasing & transformations Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane CCSS: A-REI.10; A-CED.2,3: F-IF.4,7e; F-BF.3 MACLE: AR.4.A Graph exponential decay functions Identify asymptotes, domain/range, intercepts, end behavior, increasing/decreasing & transformations Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another CCSS: A-REI.10;A-CED.2,3: F-IF.4, F-IF.5,7e; F-BF.3; F-LE.1c MACLE: AR.4.A 14
3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods A-REI.10: 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) F-IF.1, 2, 4-7e, 8b: 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) 2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context 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 periodicit 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) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the functions 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 from a graph Write and solve exponential growth and decay problems from a set of points Construct exponential functions given a graph or two input-output pairs CCSS: A-SSE.3c; A-CED.1-3; F-LE.2; F-IF.1,2 MACLE: AR.3.A Use and solve exponential growth and decay equations and inequalities to model real-life situations Construct exponential functions given a description of a relationship Interpret the parameters of an exponential function in terms of context CCSS: F-BF.1b; F.LE.2,5; A-CED.1,3 MACLE: AR.4.A 15
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function b. Use properties of exponents to interpret expressions for exponential functions. For example, identify percent of rate of change in functions such as y = (1.02) t, y = (0.97) t, y = (1.01) 12t, y = (1.2) t/10, and classify them as representing exponential growth or decay F-BF.1b, 3: 1. Write a function that describes a relationship between two quantities b. Combine standard function types using arithmetic operations. 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 3. Identify the effect of the graph of replacing f(x) by f(x) + k, k f(x), 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 and explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them 16
F-LE.1a,c, 2, 5: 1. Distinguish between situations that can be modeled with linear functions and with exponential functions a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals c. Recognize situations in which a quantity grows or decays by constant percent rate per unit interval relative to another 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (including reading these from a table) 5. Interpret the parameters in a linear or exponential function in terms of a context Direct and explicit instruction Guided and independent practice Infuse AFL tenets in instruction Graphic organizer Exponential Growth/Decay Small-group instruction/facilitation Summarizing/synthesizing concepts Student self-reflection Teacher self-reflection Class discussion Game: Exponent Block Review Instructional Strategies 17
Assessments/Evaluations The students will be assessed on the concepts taught using a variety of modalities: Direct teacher observations Quizzes Homework assignments see pacing guide Formal common assessment Unit D test Mastery Level: 80% Instructional Resources/Tools Textbook(s): (sample copies on bookshelf) Pearson Algebra 2 Common Core (primary source) Holt Algebra 2 Common Core Edition Houghton Mifflin On Core Mathematics: Algebra 2 Website(s) such as khanacademy.org Graphing calculator Computer simulation software Exponential functions apps 18
Conceptual Category(s) Domain Cluster Alignments: CCSS: See below Performance: 3.4, 1.6 Knowledge: (MA) 4 MACLE: See below NETS: 3d; 6a,b DOK: 1-3 Algebra Functions Creating Equations (A-CED) Reasoning with Equations and Inequalities (A-REI) Interpreting Functions (F-IF) Building Functions (F-BF) Linear, Quadratic, and Exponential Models (F-LE) Create Equations that describe numbers or relationships (A-CED.2-4) Represent and solve equation and inequalities graphically (A-REI.11) Analyze functions using different representations (F-IF.7e) Build new functions from existing functions (F-BF.4a,b, 5) Construct and compare linear, quadratic, and exponential models and solve problems (F-LE.4) Standards Learning Targets A-CED.2-4 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales 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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods 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 Unit E: Logarithmic Functions: Create and solve logarithmic equations and inequalities; graph and transform log functions by hand and using a calculator; understand the inverse relationship between exponents and logs; and solve a system of equations using a variety of functions Find the inverse of a logarithmic equation and identify it as an exponential equation Convert equations to/from exponential to/from logarithmic form Verify by composition that exponential and logarithmic functions are inverses CCSS: F-BF.4a,b, 5 MACLE: AR.1.D; AR.2.A,B 19
A-REI.11 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 of the approximately; e.g., using technology to graph the functions, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions F-IF.7e 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude F-BF.4a,b, 5 4. Find inverse functions 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. For example, f(x) =2x 3 of f(x) = (x+1)/(x-1) for x 1 b. (+) Verify by composition that one function is the inverse of another 5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problem involving logarithms and exponents F-LE.4 4. For exponential models, express as a logarithm the solution to ab ct =d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology Solve logarithmic equations algebraically using properties of logarithms Solve logarithmic equations without a calculator CCSS: F-BF.4a, 5; F-LE.4 MACLE: AR.1.D; AR.2.A,B Evaluate logarithms and solve logarithmic equations with base 2, 10, and e CCSS: F-LE.4 MACLE: AR.1.D; AR.2.A Graph logarithmic functions by hand including identifying the domain, range, transformations, and asymptote including labels and scales on axes CCSS: A-CED.2; F-IF.7e MACLE: AR.1.C; AR.2.A; AR.3.A; AR.4.A; GSR.4.B Determine a logarithmic equation with two variables using provided data such as word problems Identify an appropriate domain and range for a given model involving logarithmic functions CCSS: A-CED.2,3 MACLE: AR.2.A,D; AR.3.A; GSR.4.B Isolate one variable in the logarithmic equation and solve the logarithmic equation using a calculator CCSS: A-CED.4; F-LE.4 MACLE: AR.1.D; AR.2.A,B Solve a system of functions (by calculator and by hand) Verify the solution(s), and explain the meaning of the solution(s) CCSS: A-REI.11 MACLE: AR.2.D; AR.4.A; GSR.4.B; M.2.D 20
Instructional Strategies Lecture enhanced with SMART Notebook Drill and guided practice Problem solving using application problems such as: exponential growth/decay compound/continuous interest Reflective discussion Student self-reflection Class discussion Computer assisted instruction Game: MATHO review game Assessments/Evaluations The students will be assessed on the concepts taught using a variety of modalities: Direct teacher observations Quizzes Homework assignments see pacing guide Formal common assessment Unit E test Mastery Level: 80% Instructional Resources/Tools Textbook(s): (sample copies on bookshelf) Pearson Algebra 2 Common Core (primary source) Holt Algebra 2 Common Core Edition Houghton Mifflin On Core Mathematics: Algebra 2 Website(s): http://www.sparknotes.com/math/algebra2/logarithmicfunctions/ Graphing calculator 21
Conceptual Category(s) Domain Cluster Alignments: CCSS: See below Performance: 3.4, 3.5 Knowledge: (MA) 4 MACLE: See below NETS: 1b; 3a; 4d DOK: 1-3 Standards Algebra Functions Creating Equations (A-CED) Reasoning with Equations and Inequalities (A-REI) Interpreting Functions (F-IF) Building Functions (F-BF) Represent and solve equations and inequalities graphically (A-REI.11) Analyze functions using different representations (F-IF.7b, 9) Build new functions from existing functions (F-BF.3) Create equations that describe numbers or relationships (A-CED.2, 3) Learning Targets A-REI.11 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, 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 F-IF.7b, 9 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 Unit F: Other Functions: Graph, compare, write and solve piecewise, step and absolute value functions Graph piecewise functions (by hand and on the graphing calculator) on a coordinate plane including labels and scales on axes Graph absolute value functions (by hand and on the graphing calculator) on coordinate plane including labels and scales on axes Identify and describe transformations of the functions including horizontal/vertical reflection, horizontal/vertical stretch and shrink, horizontal/vertical shift State the domain/range CCSS: F-IF.7b; F-BF.3; A-CED.2,3 MACLE: AR.1.E; AR.2.C 22
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-BF.3 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. 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 A-CED.2, 3 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales 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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods Solve real world application problems involving piecewise, step, and absolute value functions CCSS: A-CED.2,3 MACLE: AR.3.A Solve a system of functions (on a calculator and by hand) Verify the solution(s), and explain the meaning of the solutions (systems may include linear, polynomial, rational, absolute value, exponential, and logarithmic functions) CCSS: A-REI.11 MACLE: AR.1.C Compare properties of two functions presented in different formats (Formats may include algebraically, graphically, table or verbal descriptions) CCSS: F-IF.9 MACLE: AR.2.C Instructional Strategies Lecture enhanced with: SMART Notebook PowerPoint the Internet Drill and guided practice Reflective discussion Student self-reflection Class discussion Computer assisted instruction 23
Assessments/Evaluations The students will be assessed on the concepts taught using a variety of modalities: Direct teacher observations Potential project with a scoring guide: http://illuminations.nctm.org/lessondetail.aspx?id=l852 Quizzes Practice/homework assignments see pacing guide Formal common assessment Unit F test Mastery Level: 80% Instructional Resources/Tools Textbook(s): (sample copies on bookshelf) Pearson Algebra 2 Common Core (primary source) Holt Algebra 2 Common Core Edition Houghton Mifflin On Core Mathematics: Algebra 2 Website(s): www.mathsisfun.com/sets/functions-piecewise.html Graphing calculator 24
Conceptual Category(s) Algebra Functions Domain Creating Equations (A-CED) Seeing Structures in Expressions (A-SSE) Interpreting Functions (F-IF) Building Functions (F-BF) Linear, Quadratic, and Exponential Models (F-LE) Cluster Create equations that describe numbers or relationships (A-CED.4) Write expressions in equivalent forms to solve problems (A-SSE.4) Understand the concept of a function and use function notation (F-IF.3) Build a function that models a relationship between two quantities (F-BF.2) Construct and compare linear, quadratic, and exponential models and solve problems (F-LE.2) Alignments: CCSS: See below Performance: 1.6 Knowledge: (MA) 4 MACLE: See below NETS: 1c; 2d; 4c DOK: 1-3 Standards Learning Targets A-CED.4 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.4 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 Unit G: Sequences and Series: Use sequences and series to analyze and solve problems Solve formulas for a specified variable Rearrange arithmetic and geometric sequences and series CCSS: A-CED.4 MACLE: AR.2.B Derive the formula for the sum of the finite geometric series Calculate and express finite geometric series 25
F-IF.3 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n 1 F-BF.1a, 2 1. Write a function that describes a relationship between two quantities a. Determine an explicit expression, a recursive process, or steps for calculation from a context 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms F-LE.2 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) Recognize and solve real-world problems involving finite geometric series and sum of a finite geometric series CCSS: A-SSE.4 MACLE: AR.2.A Write the formula/function of the sequence Use the recursive formula to generate the next term of the sequence Distinguish between explicit and recursive formulas CCSS: F-IF.3 MACLE: AR.1.B Write explicit and recursive arithmetic sequence formulas Write explicit and recursive geometric sequence formulas Translate between the recursive and explicit forms of geometric sequences CCSS: F-BF.2 MACLE: AR.1.B,C; AR.2.A Solve real-world problems involving explicit and recursive arithmetic and geometric sequences CCSS: F-BF.1 MACLE: AR.1.B Determine if a function is linear or exponential given a sequence, a graph, a verbal description, or a table Construct a function from an arithmetic sequence, a table of values, or a description of the relationship Describe the process used to construct the linear or exponential function that passes through two given points CCSS: F-LE.2 MACLE: AR.1.B,C; AR.2.A 26
Instructional Strategies Direct and explicit instruction Guided and independent practice Infuse AFL tenets in instruction We learn through integration of instructional technology such as: SMART Notebook PowerPoint Prezi the Internet discovery-type experience Graphic organizer: Formulas for arithmetic/geometric sequences and series Small-group instruction/facilitation Summarizing/synthesizing concepts Student self-reflection Teacher self-reflection Class discussion Game: Grains of Rice on a Chess Board The students will practice assessment-like questions on sequences and series. For example, a question may be asked: When you place rice on the chess board: 1 grain on the first square, 2 grains on the second square, 4 grains on the third, and so on...... doubling the grains of rice on each square... 27
Assessments/Evaluations The students will be assessed on the concepts taught using a variety of modalities: Direct teacher observations Projects with scoring guides A Reality Series Project: Students will work with a partner to research a real-world application of arithmetic and geometric sequences and series At the conclusion of their research, they will make a poster presentation of their findings and present it to the class Quizzes Homework assignments see pacing guide Formal Common Assessment Unit G test Mastery Level: 80% Instructional Resources/Tools Textbook(s): (sample copies on bookshelf) Pearson Algebra 2 Common Core (primary source) Holt Algebra 2 Common Core Edition Houghton Mifflin On Core Mathematics: Algebra 2 Website(s): www.khanacademy.org Graphing calculator Computer simulation software Sequence and series apps 28