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**Standards**:

- CN-A.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 - CN-A.2. Use the relation
*i*^{2}= –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. - CN-A.3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

**Technology-enhanced investigations:**

**Key vocabulary for this domain**. Use with CN-A,
CN-B, and CN-C:

Interactive Mathematics: Complex numbers vocabulary

Mathwords.com: Math Dictionary:

Art of Problem Solving:
Introduction to Algebra Videos: These are the videos that go with
each of the sections within the 13-chapter textbook, *Introduction to
Algebra*. Ch. 12 is on complex numbers (imaginary numbers
introduction, powers of *i*, what are complex numbers; and
adding, subtracting, and multiplying complex numbers).

CK-12: Trigonometry: Concepts, practice problems for learners, and videos included in the following:

- Complex Numbers: Aligns with CN-A.1.
- Quadratic Formula and Complex Sums: There are two lessons in this group, the first of which includes addition and subtraction of complex numbers. Aligns with CN-C.2 and CN-C.7.
- Products and Quotients of Complex Numbers: Aligns with CN-A.2 and CN-A.3.

Dave's Short Course on Complex Numbers provides an introduction to complex numbers. Aligns with CN-A.1, CN-A.2, and CN-A.3.

Study.com: Algebra: Complex Numbers: This is a series of five video lessons on complex numbers that also include sample problems and a short quiz with each video. Learn about imaginary numbers, how to perform operations with complex numbers, graph complex numbers, and solve quadratics with complex numbers. There is a chapter multiple choice exam following the last video in the chapter. Aligns with CN-A.1, CN-A.2 and CN-A.3 and CN-B.4, and CN-C.7.

LearnZillion:

- Lesson set:
Understand complex numbers:
Three video lessons: How to determine if a square root is real
or imaginary, how to write the square root of a negative number using imaginary
*i*, how to determine if a complex number is real or imaginary. Aligns with CN-A.1. - Lesson set (additional):
Understand complex numbers:
Five additional video lessons: Define number systems, understand the number
*i*, simplify radicals using*i*, extend the real number system to understand the complex number system, evaluate higher powers of*i*. Aligns with CN-A.1. - Lesson set:
Add,
subtract, and multiply complex numbers:
Five video lessons: adding and subtracting complex numbers by combining like
terms, multiplying complex numbers, and simplifying positive integer
powers of
*i*. Aligns with CN-A.2. - Lesson set: Use mathematical properties to add, subtract, and multiply complex numbers: Five video lessons: Apply real number properties to complex numbers; add, subtract complex numbers; multiply by imaginary numbers, multiply two complex numbers. Aligns with CN-A.2.

Math Warehouse: Algebra: Complex and Imaginary Numbers: Under the algebra section, you will find video tutorials and online step by step demonstrations for how to simplify imaginary numbers, how to multiply imaginary numbers, how to identify and graph complex numbers, and how to add, subtract, multiply, and divide complex numbers. Online practice problems included within each tutorial. A complex number calculator is available. There is a link to free pdf worksheets with constructed response questions at MathWorksheetsGo.com.

Purple Math: Complex Numbers: Introduction: Lesson and examples. The module also includes operations with complex numbers and using the quadratic formula yielding answers that are complex numbers.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 9 manipulatives addressing CN-A.1, 11 manipulatives addressing CN-A.2, 1 manipulative for CN-A.3. Among those are:

- Complex number: basic elements of representations of a complex number in a two-dimensional Cartesian coordinate system or in a polar coordinate system. Aligns with CN-A.1, CN-B.4.
- Argand diagram: The representation of a complex number as a point in the complex plane. Aligns with CN-A.1, CN-B.4.
- Roots of a complex number: The nth roots of a complex number with n>2 form a regular polygon of n sides with the center at the origin. Aligns with CN-A.1, CN-A.2, CN-B.4, CN-B.5.
- Powers of complex roots: Aligns with CN-A.1, CN-A.2, CN-B.4, CN-B.5.
- Roots of unity: Complex solutions to the equation x
^{n }- 1 = 0. Aligns with CN-A.1, CN-A.2, CN-B.4, CN-B.5. -
*n*th roots of a complex number: Aligns with CN-A.1, CN-A.2, CN-B.4, CN-B.5. - Complex multiplication: Visualization on the plane using magnitude and argument. Aligns with CN-A.2, CN-B.4, CN-B.5.
- Complex addition: Visualization on the plane. Aligns with CN-A.2, CN-B.4, CN-B.5.
- Multiplying complex numbers: Calculations. Aligns with CN-A.2.
- Complex zeroes of quadratic functions: Aligns with CN-A.3, CN-B.4, CN-C.7.

**Multiple Choice:**

ACT Academy and OpenEd:

- Introduction to complex numbers CN.1: 5 multiple choice questions. Aligns with CN-A.1.
- Introduction to complex numbers 2 CN.2: 4 multiple choice questions. Aligns with CN-A.2.
- Introduction to complex numbers 3 CN.3: 4 multiple choice questions. Aligns with CN-A.3.

MathsIsFun.com: Complex Numbers: Content defines the complex number and illustrates addition, subtraction, multiplication, the conjugate and division of complex numbers. Following the lesson are 10 multiple choice problems. Aligns with CN-A.

Khan Academy: Practice questions with videos.

- The imaginary unit and complex numbers: Aligns with CN-A.1.

McDougal Littell ClassZone, Algebra 2, 2011: Ch. 4.6 Quiz: Perform Operations with Complex Numbers

Study.com: Algebra: Complex numbers multiple choice chapter exam: This exam should be completed after learners view a series of five video lessons on complex numbers in which they learn about imaginary numbers, how to perform operations with complex numbers, graph complex numbers, and solve quadratics with complex numbers. Aligns with CN-A.1, CN-A.2 and CN-A.3 and CN-B.4, and CN-C.7.

**Constructed-response:**

CK-12 Flexbook: Math Analysis: Ch.4, section 5: Operations on Complex Numbers. Learners work with roots of quadratic equations by using the discriminant, add and subtract complex numbers, find the complex conjugate, multiply and divide complex numbers, and use their calculator to perform operations. Practice problems (use paper-pencil) and answers are provided.

Interactive Mathematics: Basic Operations with Complex Numbers: Lesson and practice problems. Use with CN-A.2 and CN-A.3.

Khan Academy: Practice questions with videos.

- Adding and subtracting complex numbers: Aligns with CN-A.2.
- Multiplying complex numbers: Aligns with CN-A.2.
- Dividing complex numbers: Aligns with CN-A.3.
- Absolute value of complex numbers: Aligns with CN-A.3.

MATHguide.com: Complex numbers: Lessons and videos on performing operations with complex numbers. Each lesson is followed by a quizmaster. Fill-in and check answers. Aligns with CN-A.

MathWarehouse.com: Complex Numbers: Video tutorials, calculators, and worksheets. Worksheets include simplify imaginary numbers; add, subtract, multiply, divide complex numbers, and an end of unit review sheet. Each worksheet has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

**Performance tasks:**

Illustrative Mathematics: Number and Quantity: Computations with Complex Numbers: This task aligns with standards CN-A.1 and CN-A.2.

Mathematics Vision Project, Secondary 2 Student Edition:

- Module 3: Quadratic Equations: This module contains 10 classroom tasks. Complex number standards addressed in Module 3 include CN-A.1, CN-A.2 within tasks 9-10, and CN-C.7, CN-C.8, and CN-C.9 within tasks 8-9.

**Standards**:

CN-B.4. (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

CN-B.5. (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

*For example, (1 - √3i)*^{3}= 8 because (1 - √3i) has modulus 2 and argument 120˚.CN-B.6. (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

**Technology-enhanced investigations:**

CK-12: Trigonometry: Concepts, practice problems for learners, and videos included in the following:

- Polar Form of a Complex Number: Aligns with CN-B.4.
- De Moivre's Theorem: Simplify the calculation of powers of complex numbers. Aligns with CN-B.5.

Thinking Mathematics: Graphing Complex Numbers: This is a YouTube video on graphing complex numbers on a plane.

Walter Fendt: Math Applets: Computing with Complex Numbers: This is an applet that can display Cartesian or Polar Coordinates. Students manipulate complex numbers on the plane and choose to add, subtract, multiply, or divide them and see the result on the applet.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 20 manipulatives addressing CN-B.4, and 10 manipulatives addressing CN-B.5. Among those are:

- Complex number: basic elements of representations of a complex number in a two-dimensional Cartesian coordinate system or in a polar coordinate system. Aligns with CN-A.1, CN-B.4.
- Argand diagram: The representation of a complex number as a point in the complex plane. Aligns with CN-A.1, CN-B.4.
- Roots of a complex number: The nth roots of a complex number with n>2 form a regular polygon of n sides with the center at the origin. Aligns with CN-A.1, CN-A.2, CN-B.4, CN-B.5.
- Powers of complex roots: Aligns with CN-A.1, CN-A.2, CN-B.4, CN-B.5.
- RootRoots of unity: Complex solutions to the equation x
^{n }- 1 = 0. Aligns with CN-A.1, CN-A.2, CN-B.4, CN-B.5. -
*n*th roots of a complex number: Aligns with CN-A.1, CN-A.2, CN-B.4, CN-B.5. - Complex multiplication: Visualization on the plane using magnitude and argument. Aligns with CN-A.2, CN-B.4, CN-B.5.
- Complex addition: Visualization on the plane. Aligns with CN-A.2, CN-B.4, CN-B.5.
- Complex zeroes of quadratic functions: Aligns with CN-A.3, CN-B.4, CN-C.7.
- Complex numbers in rectangular and polar form: Aligns with CN-B.4.
- Polar and rectangular coordinates: Aligns with CN-B.4.
- Polar coordinates: Aligns with CN-B.4.

**Multiple Choice:**

ACT Academy and OpenEd:

- Complex plane CN.4: 4 multiple choice questions. Includes related review resources. Aligns with CN-B.4.
- Complex plane 2 CN.5: 4 multiple choice questions. Includes related review resources. Aligns with CN-B.5.
- Distance in the complex plane CN.6: 5 multiple choice questions. Includes related review resources. Aligns with CN-B.6.

**Constructed-response:**

Khan Academy: Practice questions with videos.

- The complex plane: Plot complex numbers. Aligns with CN-B.4.
- Complex number polar form intuition: Plot the polar form. Aligns with CN-B.4.
- Rectangular and polar forms of complex numbers: Convert. Aligns with CN-B.4.
- Complex plane operations: Add and subtract. Aligns with CN-B.4, CN-B.5.
- Multiplying and dividing complex numbers in polar form: Aligns with CN-B.5.
- Powers of complex numbers: Use Euler's formula. Aligns with CN-B.5.
- Distance and midpoint on the complex plane: Aligns with CN-B.6.

Interactive Mathematics: Polar Form of a Complex Number: Lesson and practice problems. Use with CN-B.4.

**Performance tasks:**

Georgia Standards of Excellence Framework: Pre-Calculus Unit 7: Vectors. This unit contains ten tasks, including a culminating task: Putting It All Together. Tasks align with VM-A.1, VM-A.2, VM-A.3, VM-B.4, VM-B.5, and VM-C.11 and number standards CN-A.3, CN-B.4, CN-B.5, CN-B.6.

Illustrative Mathematics: Number and Quantity: Complex Distance: Aligns with standard CN-B.6.

**Standards**:

- CN-C.7. Solve quadratic equations with real coefficients that have complex solutions.
- CN-C.8. (+) Extend polynomial identities to the complex
numbers.
*For example, rewrite x*.^{2}+ 4 as (x + 2i)(x – 2i) - CN-C.9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

**Technology-enhanced investigations:**

CK-12: Quadratic Formula and Complex Sums: There are two lessons in this group, the second of which focuses on quadratic equations with complex solutions. Concepts, practice problems for learners, and videos included. Aligns with CN-C.7.

Khan Academy: Videos on the Fundamental Theorem of Algebra: Introduction and Verifying the theorem holds for quadratic equations. Aligns with CN-C.9.

LearnZillion:

- Lesson set: Solve quadratic equations: Four video lessons: Determine whether a number is real or imaginary by isolating the quadratic term and calculating the value of the discriminate; solving quadratic equations using the quadratic formula and by completing the square. Aligns with CN-C.7.
- Lesson set: Solve quadratic equations with real coefficients that have complex solutions: Five video lessons: Find imaginary roots of quadratic equations; find complex roots of quadratic equations by completing the square or using the quadratic formula; identify real or complex roots by evaluating the discriminant, graph quadratic functions with complex roots. Aligns with CN-C.7.

Purple Math: Complex Numbers: Introduction: Lesson and examples. The module also includes operations with complex numbers and using the quadratic formula yielding answers that are complex numbers. Aligns with CN-C.7.

Southeast Comprehensive Center of SEDL: Video: Use complex numbers in polynomial identities and equations. View online or download. Aligns with CN-C.8.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 11 manipulatives addressing CN-C.7 and 5 manipulatives addressing CN-C.9. Among those are:

- Complex zeroes of quadratic functions: Aligns with CN-A.3, CN-B.4, CN-C.7.
- Solution of quadratic equations: Use quadratic equation and see step-by-step solutions. Aligns with CN-C.7, REI-A.1, REI-B.4.
- The Fundamental Theorem of Algebra: Aligns with CN-C.9.
- Where are my roots?: This manipulative shows a plot of the graph of the polynomial (x-a)(x-b)(x-c)(x-d). Aligns with APR-B.2, APR-B.3 and CN-C.9.

**Multiple Choice:**

ACT Academy and OpenEd:

- Quadratics with complex zeroes CN.7: 4 multiple choice questions. Includes related review resources. Aligns with CN-C.7.
- Polynomials with complex numbers CN.8: 4 multiple choice questions. Includes related review resources. Aligns with CN-C.8.
- Fundamental theorem of algebra CN.9: 4 multiple choice questions. Includes related review resources. Aligns with CN-C.9.

Khan Academy: Practice questions with videos.

- Quadratic formula with complex solutions: Aligns with CN-C.7.
- Equivalent forms of expressions with complex numbers: Aligns with CN-C.8.

MathsIsFun.com: Fundamental Theorem of Algebra: Content explains the theorem. Following the lesson are 10 multiple choice problems on application of the fundamental theorem, including the nature of roots of polynomials and interpretation of roots from graphs. Aligns with CN-C.9.

McDougal Littell ClassZone, Algebra 2, 2011: Ch. 5.7 Quiz: Apply the Fundamental Theorem of Algebra Aligns with CN-C.9.

**Constructed-response:**

**Performance tasks:**

Mathematics Vision Project, Secondary 2 Student Edition:

- Module 3: Quadratic Equations: This module contains 10 classroom tasks. Complex number standards addressed in Module 3 include CN-A.1, CN-A.2, CN-C.7, CN-C.8, and CN-C.9 within three of those tasks.

Mathematics Vision Project, Secondary 3 Student Edition:

- Module 3: Polynomial Functions: This module contains 8 classroom tasks. Number standards CN-C.8, CN-C.9 are found in tasks 5-8.

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