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Lesson Plan Title: Imaginary Numbers Grade: 11th Subject: Algebra 2 Trigonometry Duration of Lesson: 86 minutes Lesson Summary/Overview: Today’s lesson will introduce the concept of imaginary numbers. Students will learn the definition of imaginary numbers, the value of i, how to express the square root of negative numbers using i, how to determine the value of i to any power including powers greater than 3, and how to solve quadratic equations with real coefficients that have complex solutions. Common Core Learning Standards: N.CN.1 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. N.CN.7 Solve quadratic equations with real coefficients that have complex solutions Objective: Given practice problems involving imaginary numbers, students will demonstrate the ability to correctly and completely express the square root of a negative number as the product of a real number multiplied by the imaginary unit i. Given examples of i to powers greater than 3, students will demonstrate the ability to correctly and completely represent the imaginary number as 1, i, -1, or –i. Given quadratic equations with real coefficients with complex solutions, students will demonstrate the ability to correctly and completely solve for the roots and express them in a + bi form. Scope and Sequence: Students must have prerequisite knowledge of division, reducing radicals, and solving quadratic equations (using completing the square and the quadratic formula) with real solutions. Following this lesson, students will be able to do the same with complex numbers. Materials: Notes 2.4/Practice 2.4 Calculators Routines (5 minutes): As students enter: Have Power Point on Screen detailing what to pick up on front table, Today’s HW Assignment (Practice 2.4 evens), Upcoming Tests and Quizzes Take attendance and pass out calculators Have Students clear desks except last night’s homework, pencil and calculator Quiz/Check Homework (20 minutes): Give out quiz – Factoring and Radicals Check/grade Homework – p. 196 (3,6,9,12,13,15,19-23) Answer Questions on Prior Night’s Homework (15 minutes) The Lesson: Imaginary Numbers (15 minutes): Have Students turn to Notes 2.4 Explain Imaginary Numbers o Up until now you’ve been told that you can’t have a negative under the radical – can but not in real number system o Can be written as a real number multiplied by the imaginary unit i, which is defined by the property i2 = -1 o Imaginary numbers have a variety of essential, concrete applications in science and engineering Solve x2 + 1 = 0 on Elmo, explain solution, check o Explain i to powers 0 through 3 o Explain definition of square root of negative k when k>0. Express as i times square root of k o Have students pair up Give students 4 problems from Notes 2.4 involving a negative square root Have them work together to rewrite using real components and i o Bring group back together and discuss answers and questions Exponents of i greater than three (10 minutes): Explain process of solving by dividing exponent by 4 o Remind students that since i2 = -1, i4 = 1 (every 4 i’s is 1) o Have students pair up Give students 4 problems from Notes 2.4 involving i to powers higher than 3 Have them work together to determine solution, 1, i, -1, or –i. o Bring group back together and discuss answers and questions Solving Quadratics with real coefficients and complex solutions (15 minutes): o Explain – can use same processes we used with problems that have real solutions to solve problems with complex solutions. o Have students pair up Give students quadratic equations from Notes 2.4 Have them solve using completing the square and/or the quadratic formula o Bring group back together and discuss answers and questions Closing (5 minutes): Check for any remaining questions Show of hands – Got It All, Got Most of It, Totally Confused Collect Calculators Equity/Accessibility: Each student can work at own pace – work completed will be dependent on time, not amount of work Problems of varying difficulty Students can get assistance from teachers and peers Ask and answer questions throughout the lesson Assessment: During pair work “tour” room to assess understanding – ask clarifying and guiding questions Quiz Grade Homework References: Gantert, A. X. (2009). Algebra 2 trigonometry. New York, NY: AMSCO School Publications, Inc. Higgins, Julie. Cooperating Teacher. Guilderland High School. Albany, NY.