Algebra 2 CP Unit 3: Quadratics

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1 Algebra 2 CP Unit 3: Quadratics Unit #: APSDO Duration: 8.0 Week(s) Date(s): Team: Jodi Kryzanski (Author), Tracy Andreana, Sally degozzaldi, Jennifer Greene, Jeanine LaBrosse, Jaclyn Lawlor, Melinda Litke, Ben Lukowicz, Jennifer Miller, Matthew Mooney, James Murray, Marlaina Napoli, Andrew Riddle, Steven Rivoira Grades: 10, 11 Subjects: Mathematics Unit Focus In this unit, students will graph quadratic functions and determine key values. They will understand the meaning of complex numbers and their application to a quadratic function. They will also apply quadratic functions to real world problems. Students will find optimal solutions for applications. Summative assessments may include projects, labs and test. Stage 1: Desired Results - Key Understandings Common Core Mathematics: 10 Established Goals 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. CCSS.MATH.CONTENT.HSA.CED.A.1 Factor a quadratic expression to reveal the zeros of the function it defines. CCSS.MATH.CONTENT.HSA.SSE.B.3.A Graph linear and quadratic functions and show intercepts, maxima, and minima. CCSS.MATH.CONTENT.HSF.IF.C.7.A Know there is a complex number i such Transfer T1 (T50) Based on an understanding of any problem, initiate a plan, execute it and evaluate the reasonableness of the solution. T2 (T53) Articulate how mathematical concepts relate to one another in the context of a problem or in the theoretical sense. T3 (T51) Examine alternate methods to accurately and efficiently solve problems. T4 (T52) Use appropriate tools strategically to deepen understanding of mathematical concepts. T5 (T21) Perform operations in a conventional order within the real and complex number system. T6 (T22) Describe and/or solve problems using algebraic expressions, equations, inequalities, and functions. T7 (T23) Use functions or equations to model relationships among quantities. Understandings Meaning Essential Questions Page 1 of 5

2 that i2 = -1, and every complex number has the form a + bi with a and b real. CCSS.MATH.CONTENT.HSN.CN.A.1 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. CCSS.MATH.CONTENT.HSF.IF.C.8.A Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. CCSS.MATH.CONTENT.HSA.SSE.B.3.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.B.4.B Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. CCSS.MATH.CONTENT.HSN.CN.A.2 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. CCSS.MATH.CONTENT.HSN.CN.A.3 Fit a linear function for a scatter plot that suggests a linear association. CCSS.MATH.CONTENT.HSS.ID.B.6.C 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 U1 (U531) Models can distort or reveal patterns; therefore it is essential to recognize the appropriate representation. U2 (U540) The choice of a mathematical tool depends upon the information you have and the information you want. U3 (U541) The accuracy of a solution depends upon the proper selection and effective use of a mathematical tool. U4 (U561) Recognition of patterns and structures fosters efficiency in solving problems. U5 (U201) The same value can be represented in multiple ways. U6 (U206) A function can represent how quantities in the real world relate to one another. U7 (U207) Recognition of predictable mathematical patterns supports the analysis of functional relationships and the prediction of data. U8 (U562) Mastery of basic facts and rules maximizes conceptual and procedural fluency. Acquisition of Knowledge and Skill Knowledge Q1 (Q532) Which model best represents this problem? Q2 (Q533) How do I use the model to solve other problems? Q3 (Q540) What tool(s) is appropriate for use with this model? Q4 (Q541) How do I use tools to solve problems? Q5 (Q563) How does being fluent with basic facts and rules help me solve a complex problem? Q6 (Q572) How does understanding the pattern/structure help me solve the problem? Q7 (Q205) How can I represent this relationship as a function or equation? (Gr. 6-12) Q8 (Q206) How do I evaluate this function or solve the equation? (Gr. 6-12) Q9 (Q208) What function best models the data? How do its characteristics help me make predictions? (Gr. 8-12) S1 Skills Graph quadratic functions by hand S2 Solve equations by graphing and factoring S3 Completing the square to write in vertex form S4 Page 2 of 5

3 functions from their graphs and algebraic expressions for them. CCSS.MATH.CONTENT.HSF.BF.B.3 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. CCSS.MATH.CONTENT.HSN.CN.B.5 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3. CCSS.MATH.CONTENT.HSA.REI.C.7 Solve quadratic equations with real coefficients that have complex solutions. CCSS.MATH.CONTENT.HSN.CN.C.7 Look for and express regularity in repeated reasoning. CCSS.MATH.MP.8 Look for and make use of structure. CCSS.MATH.MP.7 Model with mathematics. CCSS.MATH.MP.4 Use appropriate tools strategically. CCSS.MATH.MP.5 Use a graphical device to identify the solutions, max/min, y-intercept S5 Graph quadratic equations and inequalities in vertex form using transformations S6 Solve quadratic inequalities and express solution in interval notation S7 Find a regression model for a linear or quadratic set of data S8 Solve a system involving linear and quadratic functions using a graphical device and algebraically S9 Model and solve quadratic functions involving vertical motion and optimization (area, price/revenue/quantity) S10 Understand how the roots of a quadratic relate to the factored form of the function S11 Understand when a linear or quadratic model is appropriate for a set of data S12 Understand relationship between a quadratic equation and it s graph (different forms of the equation) Page 3 of 5

4 S13 Calculate and use the discriminant to describe the nature and number of solutions S14 Simplify radicals with negative radicands and perform operations with complex numbers (add, subtract, multiply, divide, rationalize denominator) S15 Use quadratic formula to find exact solutions (including imaginary) S16 Write the equation of a quadratic function in vertex form/standard form, and given: vertex and point, two roots (product-sum), three points (using system and graphical device) S17 Understand complex numbers Stage 3: Learning Plan Coding Code Description of Learning Activity LA1 Duration: 2.0 Day(s) Learning Activity Learning Activity Solving Quadratic Inequalities: Warm up to connect to prior knowledge and introduce new topic Warm Up: Mr. Dunford will be launching a rocket in his Physics class. Its height, in feet, after t seconds is given Page 4 of 5

5 by: h(t) = - 16t t Graph the quadratic. Review. Use Graphing calculator to explore properties. See attached notes sheet for more details and example problems. Resources RES1 Solving Quadratic Inequalities Video Clip YouTube Video Solving Quadratic Inequalities RES2 Lesson Notes Lesson Notes Download File RES3 Homework Download File RES4 Practice - Application Problems Download File Page 5 of 5