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Algebra 2 | Unit B: Quadratic Functions Unit Overview Major Focus: Students will understand the various properties of quadratic functions and will use quadratic functions to model real-world phenomena. Tasks: Identify key features of quadratic functions such as vertex (max/min), intercepts (including zeroes), focus, and directrix, and use those features to sketch a graph. Solve quadratic equations by using inspection, taking square roots, completing the square, factoring, and using the quadratic formula. Model real-world phenomena using quadratic functions. Solve systems of equations that include a quadratic equation. Perform operations with complex numbers and solve quadratic equations that have complex solutions. Compare properties of functions that are represented differently (i.e. algebraically and graphically). Textbook Resources Mathematics Formative Assessment System Tasks Pearson Prentice Hall Algebra 2 copyright 2011 Sections: 4-1, 4-2, CC1, CC2, 4-5, 4-6, CC3, 4-7, 4-8, CC4 The system includes tasks or problems that teachers can implement with their students, and rubrics that help the teacher interpret students' responses. Teachers using MFAS ask students to perform mathematical tasks, explain their reasoning, and justify their solutions. Rubrics for interpreting and evaluating student responses are included so that teachers can differentiate instruction based on students' strategies instead of relying solely on correct or incorrect answers. The objective is to understand student thinking so that teaching can be adapted to improve student achievement of mathematical goals related to the standards. Like all formative assessment, MFAS is a process rather than a test. Research suggests that welldesigned and implemented formative assessment is an effective strategy for enhancing student learning. http://www.cpalms.org/resource/mfas.aspx MUST ADD: Algebra 2 Common Core Additional Lessons: CC-1: Equation of a Parabola (using focus and directrix) CC-2: Modeling with Quadratic Functions CC-3: Quadratic Expressions and Functions (Graphing by Factoring and Completing the Square) CC-4: Systems of Linear and Quadratic Equations Available under Teacher Resources of the Algebra 2 content at www.pearsonsuccessnet.com. Section to consider for remediation: 4-4 This a working document that will continue to be revised and improved taking your feedback into consideration. Math Florida Standards Content Standards (bold are directly assessed) MAFS.912.A-REI.2.4 MAFS.912.A-REI.3.7 MAFS.912.A-SSE.2.3a,b MAFS.912.F-BF.2.3 MAFS.912.F-IF.3.7a MAFS.912.F-IF.3.8a MAFS.912.F-IF.3.9 MAFS.912.G-GPE.1.2 MAFS.912.N-CN.1.1 MAFS.912.N-CN.1.2 MAFS.912.N-CN.3.7 Standards for Mathematical Practice MAFS.K12.MP.2.1 MAFS.K12.MP.7.1 Other Resources Kuta Software Purple Math Algebra Nation Online Graphing Calculator National Library of Virtual Manipulatives Geogebra Virtual Nerd YouTube Khan Academy—Math Engage NY TI Nspired Resource Center for Educators Pasco County Schools, 2014-2015 Algebra 2 | Unit B: Quadratic Functions Tasks: Interactive Digital Paths and Dynamic Activities – Pearson Online a. section 4-1 Quadratic Functions and Transformations Pearson Prentice online text teacher resources digital path dynamic activity allows manipulation of h, k and a to yield various changes in the graph b. section 4-2 Standard form of a Quadratic Function Pearson Prentice online text teacher resources digital path dynamic activity allows manipulation of a, b and c when the equation is in polynomial form c. section 4-4 Factoring Quadratic Expressions Pearson Prentice online text teacher resources digital path dynamic activity allows one to model then factor the polynomial d. section 4-5 Quadratic Equaitons Pearson Prentice online text teacher resources digital path 2nd dynamic activity allows one to model then factor the polynomial that has a>2 e. section 4-5 Quadratic Equations Pearson Prentice online text teacher resources digital path dynamic activity allows one to graph the factored form the quadratic and observe how changes in a, r1 and r2 affect the graph f. section 4-6 Completing the Square Pearson Prentice online text teacher resources digital path dynamic activity allows one to complete the square with electronic Algeblocks g. section 4-7 Quadratic Formula Pearson Prentice online text teacher resources digital path dynamic activity allows one change a, b and c on the quadratic to give a visual of how that affects the zeros of the equation. Glencoe McGraw-Hill “Precalculus” copyright 2011 student Study Notebook sections 7-1 and 7-2 This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015 Algebra 2 | Unit B: Quadratic Functions Unit Scale (Multidimensional) (MDS) The multidimensional, unit scale is a curricular organizer for PLCs to use to begin unpacking the unit. The MDS should not be used directly with students and is not for measurement purposes. This is not a scoring rubric. Since the MDS provides a preliminary unpacking of each focus standard, it should prompt PLCs to further explore question #1, “What do we expect all students to learn?” Notice that all standards are placed at a 3.0 on the scale, regardless of their complexity. A 4.0 extends beyond 3.0 content and helps students to acquire deeper understanding/thinking at a higher taxonomy level than represented in the standard (3.0). It is important to note that a level 4.0 is not a goal for the academically advanced, but rather a goal for ALL students to work toward. A 2.0 on the scale represents a “lightly” unpacked explanation of what is needed, procedural and declarative knowledge i.e. key vocabulary, to move students towards proficiency of the standards. 4.0 In addition to displaying a 3.0 performance, the student must demonstrate in-depth inferences and applications that go beyond what was taught within these standards. Examples: 3.0 Given a modeling context resulting in a system of quadratic equations, solve a system of quadratic equations with up to 4 solutions, interpret the results, and determine which solutions are viable for the problem. The Student will: Solve quadratic equations in one variable. o 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. o 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. (MAFS.912.A-REI.2.4) Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. (MAFS.912.A-REI.3.7) Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. o Factor a quadratic expression to reveal the zeros of the function it defines. o Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. (MAFS.912.A-SSE.2.3) 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. (MAFS.912.F-BF.2.3) Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. o Graph linear and quadratic functions and show intercepts, maxima, and minima. (MAFS.912.F-IF.3.7a) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. o 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. (MAFS.912.F-IF.3.8a) 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. (MAFS.912.F-IF.3.9) Derive the equation of a parabola given a focus and directrix. (MAFS.912.G-GPE.1.2) Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real. (MAFS.912.N-CN.1.1) Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. (MAFS.912.N-CN.1.2) This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015 Algebra 2 | Unit B: Quadratic Functions 2.0 Solve quadratic equations with real coefficients that have complex solutions. (MAFS.912.N-CN.3.7) The student will recognize or recall specific vocabulary, such as: Key vocabulary: completing the square, square root, quadratic formula, factor, vertex form, standard form, maximum, minimum, intercepts, zeroes, extreme values, symmetry, parabola, focus, directrix, complex number The student will perform basic processes, such as: Identify the best solution method given a quadratic equation Determine whether a given solution satisfies a given system of equations (including one quadratic) Rewrite perfect square trinomials Translate functions on a graph given a verbal description Find the vertex of a parabola given a quadratic equation in standard form Factor simple trinomials Compare two functions that are represented using the same way (i.e. both are graphs) Find the focus of a parabola. Add and subtract complex numbers 1.0 With help, partial success at 2.0 content but not at score 3.0 content This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015 Algebra 2 | Unit B: Quadratic Functions Unpacking the Standard: What do we want students to Know, Understand and Do (KUD): The purpose of creating a Know, Understand, and Do Map (KUD) is to further the unwrapping of a standard beyond what the MDS provides and assist PLCs in answering question #1, “What do we expect all students to learn?” It is important for PLCs to study the focus standards in the unit to ensure that all members have a mutual understanding of what student learning will look and sound like when the standards are achieved. Additionally, collectively unwrapping the standard will help with the creation of the uni-dimensional scale (for use with students). When creating a KUD, it is important to consider the standard under study within a K-12 progression and identify the prerequisite skills that are essential for mastery. Domain: Algebra: Reasoning with Equations & Inequalities Cluster: Solve equations and inequalities in one variable. (Supporting) Standard: MAFS.912.A-REI.2.4: (Solve) quadratic equations in one variable. Understand “Essential understandings,” or generalizations, represent ideas that are transferable to other contexts. Understand that there are two solutions to every quadratic equation, that the solutions may or may not be unique and may or may not be real, and that the solutions are always obtainable using either factoring, completing the square, or the quadratic formula. Know Declarative knowledge: Facts, vocab., information Vocab: Quadratic equations, solutions to quadratic equations Do Procedural knowledge: Skills, strategies and processes that are transferrable to other contexts. Solve quadratic equations in one variable. Prerequisite skills: What prior knowledge (foundational skills) do students need to have mastered to be successful with this standard? Using the addition and multiplication properties of equality, factoring trinomials, completing the square, using the quadratic formula, simplifying radicals, using complex numbers Learning Goals: Students will solve quadratic equations in one variable. Moving Beyond: In Pre-calculus, a deep understanding of parabolas will aid students in their understanding of all conics. This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015 Algebra 2 | Unit B: Quadratic Functions Uni-Dimensional, Lesson Scale: The uni-dimensional, lesson scale unwraps the cognitive complexity of a focus standard for the unit, using student friendly language. The purpose is to articulate distinct levels of knowledge and skills relative to a specific topic and provide a roadmap for designing instruction that reflects a progression of learning. The sample performance scale shown below is just one example for PLCs to use as a springboard when creating their own scales for student-owned progress monitoring. The lesson scale should prompt teams to further explore question #2, “How will we know if and when they’ve learned it?” for each of the focus standards in the unit and make connections to Design Question 1, “Communicating Learning Goals and Feedback” (Domain 1: Classroom Strategies and Behaviors). Keep in mind that a 3.0 on the scale indicates proficiency and includes the actual standard. A level 4.0 extends the learning to a higher cognitive level. Like the multidimensional scale, the goal is for all students to strive for that higher cognitive level, not just the academically advanced. A level 2.0 outlines the basic declarative and procedural knowledge that is necessary to build towards the standard. MAFS.912.A-REI.2.4: Solve quadratic equations in one variable. Score 4.0 3.5 Learning Progression 2.0 1.0 + bx + c = 0, derive the quadratic equation by I can… Derive the quadratic equation by completing the square. 1. Given the equation ax completing the square. I can do everything at a 3.0, and I can demonstrate partial success at score 4.0. I can… Solve quadratic equations in one variable. 1. Solve: 3x 2 + 7x -10 = 0 2. Solve: x 2 - 5x + 3 = 0 3.0 2.5 Sample Tasks I can do everything at a 2.0, and I can demonstrate partial success at score 3.0. I can… Complete the square Factor a quadratic trinomial Substitute the coefficients of quadratic equations into the quadratic formula 2 1. Rewrite the following equation so that it contains a perfect square: x 2 -12x = 6 2. Factor the following trinomial: 2x -13x - 7 3. Substitute the coefficients of the following equation into the quadratic 2 formula: 3x - 4x -15 = 0 2 I need prompting and/or support to complete 2.0 tasks. This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015 Algebra 2 | Unit B: Quadratic Functions Sample High Cognitive Demand Tasks: These task/guiding questions are intended to serve as a starting point, not an exhaustive list, for the PLC and are not intended to be prescriptive. Tasks/guiding questions simply demonstrate one way to help students learn the skills described in the standards. Teachers can select from among them, modify them to meet their students’ needs, or use them as an inspiration for making their own. They are designed to generate evidence of student understanding and give teachers ideas for developing their own activities/tasks and common formative assessments. These guiding questions should prompt the PLC to begin to explore question #3, “How will we design learning experiences for our students?” and make connections to Marzano’s Design Question 2, “Helping Students Interact with New Knowledge”, Design Question 3, “Helping Students Practice and Deepen New Knowledge”, and Design Question 4, “Helping Students Generate and Test Hypotheses” (Domain 1: Classroom Strategies and Behaviors). CCSS Mathematical Content Standard(s) MAFS.912.A-REI.2.4: Solve quadratic equations in one variable. Design Question 1; Element 1 CCSS Mathematical Practice(s) MAFS.K12.MP.2.1: Reason abstractly and quantitatively. MAFS.K12.MP.7.1: Look for and make use of structure. Design Question 1; Element 1 Marzano’s Taxonomy Level 4—Knowledge Utilization: “Experimenting” Questions: What would h(0) represent? Can you defend your answer with mathematical reasoning? Which part of the parabola (intercept, vertex, etc.) would this question relate to? Misconceptions: Incorrect computations (order of operations) Solving for the incorrect zero. Differentiation: Shift the diving board higher and lower for extending students. Have step-by-step instructions for struggling students. Suppose h(t) = −5t2 + 10t + 3 is an expression giving the height of a diver above the water (in meters), t seconds after the diver leaves the springboard. Teacher Notes Task a. A. How high above the water is the springboard? Explain how you know. b. B. When does the diver hit the water? c. C. At what time on the diver's descent toward the water is the diver again at the same height as the springboard? d. D. When does the diver reach the peak of the dive? This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015