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Algebra II Quarter 2 Curriculum Map 2013-2014 CCSS for Mathematical Practice: 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Unit Quadratics Additional Resources https://www.pearsonsuccessnet.com Other Resources, Teacher Resources, Find the Errors, Chapter Projects, Performance Tasks, Activities, Games and Puzzles http://www.problem-attic.com/, http://www.kutasoftware.com/ Create your own worksheets http://nrich.maths.org/frontpage Enriching tasks Unit 3: Quadratics Quarter 2 Suggested Instructional Days: 13 Timeline Standards Learning Expectation Vocabulary Resources 1 day (S) A.CED.1 Create equations and inequalities in one Students will identify and graph Parabola Algebra 2 variable and use them to solve problems. Include quadratic functions. Quadratic function Common equations arising from linear and quadratic functions, Vertex form Core and simple rational and exponential functions. Axis of Symmetry Sec 4.1 (S) F.IF.7 Graph functions expressed symbolically Vertex and show key features of the graph, by hand in simple Minimum value cases and using technology for more complicated Maximum value cases. (M) F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. (A) A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. (A) F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. 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. HS FAL: Forming Quadratics http://map.mathshell.org/materials/lessons.php?taskid=224&subpage=concept • • • • • Review Standards (M)ajor Content, (S)upporting Content (A)dditional Content (+) Honors/4th Math Algebra II Quarter 2 p. 1 Algebra II Quarter 2 Curriculum Map 2013-2014 2 days 1 day (S) F.IF.8 a Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. (S) F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (M) F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. (S) F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (M) F.BF.1 Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (M) F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (S) G.GPE.2 Derive the equation of a parabola given a focus and directrix. Students will graph quadratic functions written in standard form. Students will write the equation of a parabola and graph parabolas. (M) F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the Students will model data with quadratic functions. standard form focus of a parabola directrix focal length Algebra 2 Common Core Sec 4.2 Sec 10.2 Algebra 2 Common Core Algebra II Quarter 2 p. 2 Algebra II Quarter 2 Curriculum Map 2013-2014 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. (M) F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. HS Task A07: Functions http://map.mathshell.org/materials/tasks.php?taskid=255&subpage=apprentice HS Task A16: Sorting Functions http://map.mathshell.org/materials/tasks.php?taskid=264&subpage=apprentice HS Task E13: Sidewalk Stones http://map.mathshell.org/materials/tasks.php?taskid=285&subpage=expert Rocket Activity – See Department Chair for files 2 days (M) A.SSE.2 Use the structure of an expression to Students will find common and binomial identify ways to rewrite it. For example, see x4 – y4 factors of quadratic expressions. as (x2)2 – (y2)2, thus recognizing it as a difference of Students will factor special quadratic squares that can be factored as (x2 – y2)(x2 + y2). expressions. A.SSE.1a Interpret expressions that represent a quantity in terms of its context. Interpret parts of an expression, such as terms, factors, and coefficients. Sec 4.3 Factoring Greatest common factor of an expression Perfect square trinomial Difference of two squares Algebra 2 Common Core Sec 4.4 1 day (S) A.CED1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A.SSE.1a Interpret expressions that represent a quantity in terms of its context. Interpret parts of an expression, such as terms, factors, and coefficients. Students will solve quadratic equations by factoring and graphing. Zero of a function Zero-product property Algebra 2 Common Core Sec 4.5 1 day (S) A.REI.4 a, b Solve quadratic equations in one variable. 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. 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 Students will solve quadratic equations using the Quadratic Formula. Students will determine the number of solutions by using the discriminant. Quadratic formula Discriminant Algebra 2 Common Core Sec 4.7 Algebra II Quarter 2 p. 3 Algebra II Quarter 2 Curriculum Map 2013-2014 them as a ± bi for real numbers a and b. HS Task N04: Creating Equations http://map.mathshell.org/materials/tasks.php?taskid=292&subpage=novice HS FAL: Solving Quadratic Equations: Cutting Corners http://map.mathshell.org/materials/lessons.php?taskid=432&subpage=problem Aussie Fir Treehttp://schools.nyc.gov/NR/rdonlyres/48D7F470-FDD4-477F-B108-E02F7D969E93/0/NYCDOEHSAlgebraAussieFirTree_Final.pdf Parabola Project http://www.fallriverschools.org/Brogan-Price/CP%20Algebra%202%20Parabola%20Project%20-%20Student%20Handout.pdf 2 days (A) N.CN.1 Know there is a complex number i such Students will identify, graph and perform Imaginary unit Algebra 2 that i2 = -1, and every complex number has the form a operations with complex numbers. Imaginary number Common + bi with a and b real. Students will find complex number Complex number Core (A) N.CN.2 Use the relation i2 = -1 and the solutions of quadratic equations. Pure imaginary Sec 4.8 commutative, associative, and distributive properties number to add, subtract, and multiply complex numbers. Complex number (A) N.CN.7 Solve quadratic equations with real plane coefficients that have complex solutions. Absolute value of a (+)N.CN.8 Extend polynomial identities to the complex number complex numbers. For example, rewrite x2 + 4 as (x + Complex conjugate 2i)(x – 2i). Polynomials (1 day) (A) A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. A.CED.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. 2 days Review and Test Students will solve and graph systems of linear and quadratic equations. Students will solve and graph systems of quadratic inequalities. Algebra 2 Common Core Sec 4.9 Unit 4: Polynomials Quarter 2 Suggested Instructional Days: 9 1 day (S) F.IF.7 c Graph functions expressed symbolically Students will classify polynomials. Monomial Algebra 2 and show key features of the graph, by hand in simple Students will graph polynomial functions Degree of a Common cases and using technology for more complicated and describe end behavior. monomial Core cases. Graph polynomial functions, identifying zeros Polynomial Sec 5.1 when suitable factorizations are available, and Degree of a showing end behavior polynomial (M) F.IF.4 For a function that models a Polynomial relationship between two quantities, interpret key function features of graphs and tables in terms of the Standard form of a quantities, and sketch graphs showing key features polynomial given a verbal description of the relationship. Key function features include: intercepts; intervals where the Turning point Algebra II Quarter 2 p. 4 Algebra II Quarter 2 Curriculum Map 2013-2014 function is increasing, decreasing, positive, or End behavior negative; relative maximums and minimums; symmetries; end behavior; and periodicity. A.SSE.1a Interpret expressions that represent a quantity in terms of its context. Interpret parts of an expression, such as terms, factors, and coefficients. HS Task N07: Building Functions http://map.mathshell.org/materials/tasks.php?taskid=295&subpage=novice HS FAL: Representing Polynomials http://map.mathshell.org/materials/lessons.php?taskid=436&subpage=concept HS FAL: Sorting Equations and Identities http://map.mathshell.org/materials/lessons.php?taskid=218&subpage=concept 2 days (M) F.BF.1 Write a function that describes a Students will analyze the factored form Factor theorem relationship between two quantities. Combine of the polynomial. Multiple zero standard function types using arithmetic Students will write a polynomial function Multiplicity operations. from its zeros. Relative maximum A.SSE.1a Interpret expressions that represent a Relative minimum quantity in terms of its context. Interpret parts of an expression, such as terms, factors, and coefficients. (M) A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. (A) A.APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples. HS Task A17: Cubic Graph http://map.mathshell.org/materials/tasks.php?taskid=265&subpage=apprentice 1 day (M) A.REI.11 Explain why the x-coordinates of the Students will solve polynomial equations Sum of cubes points where the graphs of the equations y = f(x) by factoring and graphing. Difference of cubes 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. (M) A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). (M) F.IF.4 For a function that models a relationship between two quantities, interpret key Algebra 2 Common Core Sec 5.2 Algebra 2 Common Core Sec 5.3 Algebra II Quarter 2 p. 5 Algebra II Quarter 2 Curriculum Map 2013-2014 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. (S) F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 2 days A.APR.1 Understand that polynomials form a system Students will divide polynomials using Synthetic division Algebra 2 analogous to the integers, namely, they are closed long division and synthetic division. Remainder Common under the operations of addition, subtraction, and theorem Core multiplication; add, subtract, and multiply Sec 5.4 polynomials. (S) A.APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with 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. (M) A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). HS Task N03: Arithmetic with Polynomials and Rational Expressions http://map.mathshell.org/materials/tasks.php?taskid=291&subpage=novice 1 day (A) F.BF.3 Identify the effect on the graph of Students will solve equations using the Rational root Algebra 2 replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for Rational Root Theorem. theorem Common specific values of k (both positive and negative); find Students will use the Conjugate Root Conjugate root Core the value of k given the graphs. Experiment with cases Theorem. theorem Sec 5.9 and illustrate an explanation of the effects on the graph Descartes’ rule of using technology. Include recognizing even and odd signs functions from their graphs and algebraic expressions for them. (S) F.IF.8 a Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Use the process of Algebra II Quarter 2 p. 6 Algebra II Quarter 2 Curriculum Map 2013-2014 factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. (S) F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (S) F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 2 days Review and Quarter 2 Test NOTES/REFLECTIONS: Algebra II Quarter 2 p. 7