Quadratic Functions: Complex Numbers


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1 Algebra II, Quarter 1, Unit 1.3 Quadratic Functions: Complex Numbers Overview Number of instruction days: (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Develop a conceptual understanding of the meaning of i. Perform arithmetic operations (addition, subtraction, and multiplication only) on the set of complex numbers. Factor quadratic expressions. Solve quadratic equations with real coefficients that have complex solutions. Solve a system of equations consisting of a linear equation and a quadratic equation in two variables both algebraically and graphically. Use the process of factoring and completing the square to analyze quadratic functions with real and complex solutions. Create quadratic equations and inequalities in one variable and use them to solve problems. 2 Reason abstractly and quantitatively Use the quadratic formula to solve for complex solutions. Compare functions represented in different ways 5 Use appropriate tools strategically. Use a graphing calculator to verify algebraic solutions to systems of equations. 6 Attend to precision Solve quadratic equations using several methods. 7 Look for and make use of structure Look for patterns of quadratics to help with factoring. Understand and make use of the patterns in the powers of i in order to perform arithmetic operations involving complex numbers. Providence Public Schools D16
2 Quadratic Functions: Complex Numbers (1214 days) Algebra II, Quarter 1, Unit 1.3 Essential Questions What do the zeros of a quadratic function represent? How can quadratic equations with real root coefficients have complex solutions What are the various methods for solving quadratic equations? How do you determine the best method for solving a system of equations? How do you interpret the intersection of two graphs in the context of a problem? What do complex numbers represent? What are the similarities and differences between the real number system and the complex number system? Standards Common Core State Standards for Mathematical Content Number and Quantity Quantities NQ Reason quantitatively and use units to solve problems. NQ.2 Define appropriate quantities for the purpose of descriptive modeling. The Complex Number System NCN Perform arithmetic operations with complex numbers. NCN.1 NCN.2 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. Use the relation i 2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Use complex numbers in polynomial identities and equations. [Polynomials with real coefficients] NCN.7 Solve quadratic equations with real coefficients that have complex solutions. Providence Public Schools D17
3 Algebra II, Quarter 1, Unit 1.3 Quadratic Function: Complex Numbers (1214 days) Algebra Creating Equations * ACED Create equations that describe numbers or relationships [Equations using all available types of expressions, including simple root functions] ACED.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. * Reasoning with Equations and Inequalities AREI Understand solving equations as a process of reasoning and explain the reasoning AREI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Solve equations and inequalities in one variable [Quadratics with real and complex solutions] AREI.4 Solve quadratic equations in one variable. b. 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 them as a ± bi for real numbers a and b. Solve systems of equations [Linearquadratic] AREI.7 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 x 2 + y 2 = 3.] Functions Interpreting Functions FIF Analyze functions using different representations [Linear, exponential, quadratic, absolute value, step, piecewisedefined] FIF.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 both a quadratic and an algebraic function, say which has the larger maximum. D18 Providence Public Schools
4 Quadratic Functions: Complex Numbers (1214 days) Algebra II, Quarter 1, Unit 1.3 Common Core State Standards for Mathematical Practice 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Providence Public Schools D19
5 Algebra II, Quarter 1, Unit 1.3 Quadratic Function: Complex Numbers (1214 days) 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning Starting in Grade 3, students applied properties of operations to multiply and divide. In Grade 8, students learned to solve an equation of the form x 2 = p, and they learned that some square roots of positive numbers are irrational. In Algebra I, students learned to factor quadratic expressions to reveal zeros, and they solved quadratic equations with real roots. In grade 6, students wrote, interpreted, and used expressions and equations. In grade 7, students used properties of operations to generate equivalent expressions. They also solved reallife problems using numerical and algebraic expressions and equations and used properties of operations to generate equivalent expressions. Grade 8 students worked with functions and learned that functions are a set of ordered pairs. Students in Algebra I used technology to graph quadratic functions. They identified the key features of these functions from tables and graphs. Students analyzed graphs of quadratic functions that modeled realworld problems, and they identified appropriate domains of functions in relation to their graphs. Algebra 1 was the introduction to factoring quadratics and completing the square. Quadratic expressions and functions were considered critical areas in Algebra I. Students worked intensively to master quadratic functions, both from an algebraic and formal perspective as well as in the context of modeling. The work that students did with quadratic functions was connected with and reinforced their work in quadratic equations, polynomial arithmetic and seeing structure in expressions. Students learned how to reexpress quadratic functions in equivalent forms to reveal different properties. They compared the properties of quadratic functions from multiple representations. Students wrote a quadratic function that described a quantitative relationship by determining an explicit expression, a recursive process, or steps for calculation. In Geometry, students used quadratic equations to solve problems involving area and special right triangles. Current Learning D20 Providence Public Schools
6 Quadratic Functions: Complex Numbers (1214 days) Algebra II, Quarter 1, Unit 1.3 The study of functions is a critical area for Algebra II students. In this unit, students study quadratic functions and also compare quadratic and linear. Students learn to understand the graph of a quadratic function by interpreting various forms of quadratic expressions. In particular, they identify the real solutions of a quadratic equation as the zeros of a related quadratic function. Algebra II students solve quadratic equations in one variable. In this unit, students also develop conceptual understanding of the meaning of i and perform arithmetic operations with complex numbers. Students become fluent with solving quadratic functions by completing the square and factoring quadratics with complex numbers. They recognize when the quadratic formula gives complex solutions. They explain why the xcoordinates of the intersection points of two graphs are the solutions to the equation. Students use technology to graph two functions, make tables of values, and find successive approximations. They also interpret solutions as viable or nonviable options in a modeling context. According to the PARCC Model Content Frameworks for Algebra 2, ARE1.1 is classified as major content for Algebra 2. NQ.2, AREI.4b, FIF.9 and ACED.1 are defined as supporting standards. NCN.1, NCN.2, NCN.7, and AREI.7 are classified as additional standards for Algebra 2. Future Learning In Precalculus, students will perform arithmetic operations with complex numbers. They will also represent complex numbers and their operations on the complex plane. Students will also graph rational functions, identify zeros and asymptotes when suitable factorizations are available, and show end behavior. Mastery of these concepts will be required in Precalculus and AP Calculus. Additional Findings According to Principles and Standards for School Mathematics, high school students should use real numbers and learn enough about complex numbers to interpret them as solutions to quadratic equations. They should fully understand the concept of a number system, how different numbers are related, and whether the properties in one system hold to another (pp ). Electronic computation technologies provide opportunities for students to work on realistic problems and perform difficult computations. According to Principles and Standards for School Mathematics, students should be able to solve problems involving complex numbers using a pencil in some cases and technology in all cases (p. 395). Assessment When constructing an endofunit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Providence Public Schools D21
7 Algebra II, Quarter 1, Unit 1.3 Quadratic Function: Complex Numbers (1214 days) Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standardsbased mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performancebased tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Demonstrate knowledge of the meaning of complex numbers and their mathematical importance. Simplify expressions that involve square roots of negative numbers. Apply properties of complex numbers to addition, subtraction and multiplication. Connect understanding of the complex number system with the real number system. Use quadratic equations and inequalities in one variable to solve problems. Solve quadratic equations by taking square roots, completing the square, using the quadratic formula and factoring with real coefficients that have solutions of the forms a + bi and a bi. Compare properties of two quadratic functions each represented in different ways i.e. algebraically, graphically, and numerically in tables or by verbal description. Solve simple systems of equations consisting of a linear equation and a quadratic equation in 2 variables. Learning Objectives Students will be able to: Instruction Write the square root of a negative number as a pure imaginary number. Know and apply the value of the first four powers of i. Add, subtract, and multiply complex numbers. Extend operations and properties to work with complex numbers. D22 Providence Public Schools
8 Quadratic Functions: Complex Numbers (1214 days) Algebra II, Quarter 1, Unit 1.3 Solve simple systems of two equations, consisting of a linear equation and a quadratic equation in 2 variables both algebraically and graphically. Interpret the intersection of two graphs in a real world context. Recognize when the quadratic formula gives complex solutions. Solve quadratic equations by using the Quadratic Formula, factoring, taking square roots and completing the square with complex roots. Use graphing technology to find the solution(s) of quadratics. Compare properties of two functions each represented in different ways i.e. algebraically, graphically, numerically in tables or by verbal description. Create quadratic equations and inequalities in one variable and use them to solve problems. Demonstrate understanding of the concepts taught in this unit. Resources Algebra 2, Glencoe McGrawHill, 2010, Student/Teacher Editions Sections 51 through 53 (pp ) Section 54 (pp ) Sections 5.5 through 56 (pp ) Section 58 (pp ) Section 107 (pp , systems of linearquadratic) Chapter 5 Resources Masters Chapter 10 Resources Masters Glencoe McGrawHill Online Interactive Classroom CD (PowerPoint Presentations) Teacher Works CDROM Exam View Assessment Suite Forming Quadratics: TINspire Teacher Software Providence Public Schools D23
9 Algebra II, Quarter 1, Unit 1.3 Quadratic Function: Complex Numbers (1214 days) Education.ti.com: Complex Numbers (ID: 10887). See the Supplementary Unit Materials section of this binder for the student and teacher notes for this activity Graphing Technology Lab: Systems of Linear and Quadratic Equations. See the Supplementary Unit Materials section of the binder for this resource. From Algebra 1, (Glencoe McGraw Hill) 2010 CCSS Lesson 53: Solving Quadratic Equations by Factoring. See the Supplementary Unit Materials section of the binder for this resource. Glencoe McGrawHill Online: CCSS Supplement Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery section below for specific recommendations. Materials Rulers, graph paper, TINspire graphing calculator Instructional Considerations Key Vocabulary factored form completing the square complex number factoring imaginary number definition of i Planning for Effective Instructional Design and Delivery Reinforced vocabulary taught in previous grades or units: quadratic expression, quadratic equation, standard form, quadratic function. Students must have ample opportunity to explore imaginary numbers by performing arithmetic operations with the powers and other complex forms of i. The variablelike property of i when adding and subtracting complex numbers supports its unknown quality, but, unlike most variables, i is clearly defined. This concept is fairly abstract and difficult for students to grasp. Have students make a graphic organizer such as a foldable to organize their learning of the powers of i. Students can use an identifying similarities and differences strategy to represent the cyclical nature of imaginary numbers. They can make a comparison matrix to show how multiplying 1 by itself x number of times gives a pattern: i 0 1 D24 Providence Public Schools
10 Quadratic Functions: Complex Numbers (1214 days) Algebra II, Quarter 1, Unit 1.3 i 1 i i 2 1 i 3 i i 4 1 i 5 i The Complex Numbers (ID: 10887) activity on education.ti.com provides the opportunity for students to incorporate technology into their study of operations with imaginary numbers. However, be sure to limit students to adding, subtracting, and multiplying complex numbers. Use scaffolding techniques, additional examples, and differentiated instructional guidelines as suggested by the Glencoe resource in section 51 and 52. These sections are to be used to activate prior knowledge. Living word walls will assist all students in developing content language. Word walls should be visible to all students, focus on the current unit s vocabulary, both new and reinforced, and have pictures, examples, and/or diagrams to accompany the definitions. In section 107 focus on the sections involving linearquadratic systems. The 5minute check transparencies can be used as a cue, questions, and advance organizers strategy as students will be activating prior knowledge. Some 5minute checks may take longer than the allotted time, so consider choosing only problems that activate prior knowledge and use the rest for differentiation, to formatively assess student learning, as an exit ticket, or assigning for homework. For planning considerations read through the teacher edition for suggestions about scaffolding techniques, using additional examples, and differentiated instructional guidelines as suggested by the Glencoe resource. In the classroom, have a blog where students can write entries, explaining how they decide when to use each of the different methods for solving systems of equations. One thing they can do is write a pros and cons list for each method. See Study Notebook Chapter 5 graphic organizer (pp. 73 and 77) in the Teacher Edition. Have students work in pairs for graphing systems of equations, taking turns between graphing the equation and checking the solution set. Discuss with students when a system of equations has one solution, no solution, or infinitely many solutions. Write examples of systems on the board and have students answer: one, none, or infinite solutions. The Graphing Technology Lab: Systems of Linear and Quadratic Equations may be used to deepen content knowledge of linearquadratic systems using technology. The lesson resources for this lab have been provided in the supplementary materials section of this curriculum frameworks binder. Use section 107 in the Glencoe resource for additional problems related to solving linearquadratic systems. Providence Public Schools D25
11 Algebra II, Quarter 1, Unit 1.3 Quadratic Function: Complex Numbers (1214 days) District resources for Solving Quadratic Equations by Factoring are also available in the Supplemental Unit Materials Section of this binder. They are also accessible online on the Glencoe McGrawHill Math Algebra 2 Student and Teacher textbooks. First, select the CCSS icon on the homepage of the online textbook and then select the CCSS Supplement to access supplementary Glencoe lessons identified in the resource section. Graphing technology will assist students with modeling realworld problems involving systems of equations and inequalities as well as identifying solution(s) to system of equations. The Mathematics Assessment Project, also known as Math Shell, provides a variety of projects for formative and summative assessment that may help make knowledge and reasoning visible to students. These CCSSM aligned resources include a variety of lessons focused on concept development and problem solving. It also includes numerous tasks designed for expert, apprentice and novice learners. The Forming Quadratics lesson is intended to help teachers assess how well students are able to understand what the different algebraic forms of a quadratic function reveal about the properties of its graphical representation. In particular, the lesson will help you identify and help students who have the following difficulties: Understanding how the factored form of the function can identify a graph s roots. Understanding how the completed square form of the function can identify a graph s maximum or minimum point. Understanding how the standard form of the function can identify a graph s intercept. Use the link below to access detailed information for the Forming Quadratics lesson: The lesson resources are also provided in the supplementary materials section of this curriculum frameworks binder. Incorporate the Essential Questions as part of the daily lesson. Options include using them as a do now to activate prior knowledge of the previous day s lesson, using them as an exit ticket by having students respond to it and post it, or hand it in as they exit the classroom, or using them as other formative assessments. Essential questions should be included in the unit assessment. Notes D26 Providence Public Schools
12 Quadratic Functions: Complex Numbers (1214 days) Algebra II, Quarter 1, Unit 1.3 Providence Public Schools D27
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