Additional Resources 2. Reason abstractly and quantitatively

Transcription

1 CCSS for Mathematical Practice: Review Standards 1. Make sense of problems and persevere in solving them (M)ajor Content, Additional Resources 2. Reason abstractly and quantitatively (S)upporting Content Other Resources, Teacher Resources, Find the Errors, 4. Model with mathematics (A)dditional Content Chapter Projects, Performance Tasks, Activities, Games 5. Use appropriate tools strategically (+) Honors/4 th Math and Puzzles 6. Attend to precision Look for and make use of structure 8. Look for and express regularity in repeated reasoning Create your own worksheets Enriching tasks Unit 3: Quadratics Quarter 2 Suggested Instructional Days: 13 Unit Timeline Standards Learning Expectation Vocabulary Resources Quadratics 1 day (S) A.CED.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. (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. Students will identify and graph quadratic functions. Parabola Quadratic function Vertex form Axis of Symmetry Vertex Minimum value Maximum value Sec 4.1 HS FAL: Forming Quadratics Algebra II Quarter 2 p. 1

2 2 days (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.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. standard form focus of a parabola directrix focal length Sec 4.2 Sec day Students will model data with quadratic functions. Algebra II Quarter 2 p. 2

3 (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 HS Task A16: Sorting Functions HS Task E13: Sidewalk Stones Rocket Activity See Department Chair for files 2 days (M) A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). Students will find common and binomial factors of quadratic expressions. Students will factor special quadratic expressions. Factoring Greatest common factor of an expression Perfect square trinomial Difference of two squares Sec 4.3 Sec 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. Students will solve quadratic equations by factoring and graphing. Zero of a function Zero-product property Sec 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 x 2 = 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 Sec 4.7 Algebra II Quarter 2 p. 3

4 them as a ± bi for real numbers a and b. HS Task N04: Creating Equations HS FAL: Solving Quadratic Equations: Cutting Corners Aussie Fir Treehttp://schools.nyc.gov/NR/rdonlyres/48D7F470-FDD4-477F-B108-E02F7D969E93/0/NYCDOEHSAlgebraAussieFirTree_Final.pdf Parabola Project 2 days (A) N.CN.1 Know there is a complex number i such that i 2 = -1, and every complex number has the form a + bi with a and b real. (A) N.CN.2 Use the relation i 2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. (A) N.CN.7 Solve quadratic equations with real coefficients that have complex solutions. (+)N.CN.8 Extend polynomial identities to the complex numbers. For example, rewrite x as (x + 2i)(x 2i). Students will identify, graph and perform operations with complex numbers. Students will find complex number solutions of quadratic equations. Imaginary unit Imaginary number Complex number Pure imaginary number Complex number plane Absolute value of a complex number Complex conjugate Sec 4.8 (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. Students will solve and graph systems of linear and quadratic equations. Students will solve and graph systems of quadratic inequalities. Sec days Review and Test Unit 4: Polynomials Quarter 2 Suggested Instructional Days: 9 Polynomials 1 day (S) F.IF.7 c Graph functions expressed symbolically Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior Students will classify polynomials. Students will graph polynomial functions and describe end behavior. Monomial Degree of a monomial Polynomial Degree of a polynomial Polynomial function Standard form of a polynomial function Turning point Sec 5.1 Algebra II Quarter 2 p. 4

5 End behavior HS Task N07: Building Functions HS FAL: Representing Polynomials HS FAL: Sorting Equations and Identities 2 days (M) F.BF.1 Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (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 (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples. Students will analyze the factored form of the polynomial. Students will write a polynomial function from its zeros. HS Task A17: Cubic Graph 1 day (M) A.REI.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. (M) A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). Students will solve polynomial equations by factoring and graphing. Factor theorem Multiple zero Multiplicity Relative maximum Relative minimum Sum of cubes Difference of cubes Sec 5.2 Sec 5.3 Algebra II Quarter 2 p. 5

6 2 days A.APR.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. (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 1 day (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. (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 Students will divide polynomials using long division and synthetic division. Synthetic division Remainder theorem Sec Students will solve equations using the Rational Root Theorem. Students will use the Conjugate Root Theorem. Sec 5.9 Rational root theorem Conjugate root theorem Descartes rule of signs Algebra II Quarter 2 p. 6

7 2 days Review and Quarter 2 Test NOTES/REFLECTIONS: 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). Algebra II Quarter 2 p. 7