## Image- Complex Numbers Day 2 Exit Ticket - Section 3: Exit Ticket

# Complex Numbers Day 2 of 2

Lesson 6 of 12

## Objective: Students will be able to multiply and divide complex numbers.

*50 minutes*

#### Warm up and Homework Review

*10 min*

I include **Warm ups** with a **Rubric** as part of my daily routine. My goal is to allow students to work on **Math Practice 3** each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. This lesson’s Warm Up- Complex Numbers Day 2, asks students to explain the pattern found in difference of squares. This will provide a lead in to the division of complex numbers using the conjugate.

I also use this time to correct and record the previous day's Homework.

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#### Complex Numbers

*37 min*

If you have access to a computer lab or laptops that can run Geogebra , it will be useful in this lesson. If not, that section can be done as a teacher demonstration. In the previous lesson, students looked at the graphic representation of the addition and subtraction of complex numbers. The geometric versions of multiplication and division are more involved and a bit beyond the scope of this class. I have crafted this lesson to give them a conceptual base in the complex plane while their skill base will be more algebraic.

We begin by looking at the multiplication of a complex to a real number. This extends the length of the vector made by the complex number to the origin by the scale of the real number. The students explore and come to this conclusion as a class. We then look at multiplying a complex number by *i*. This is a bit more challenging conceptually. Fortunately, it was a topic of the video shown in the previous lesson. The students explore and discuss a the fact the i rotates the vector created by a complex number by 90^{o}.

We won’t be hand drawing the graphic version of a complex number multiplied by a complex number. This is covered in the 12^{th} grade course. Instead, we will be exploring this concept using Geogebra. I have the students log on and go to this website: http://www.geogebratube.org/student/m14795

Please watch the Video Narrative for more information on using this Geogebra activity. I give them some time to change the complex numbers in Geogebra to discover the pattern behind the multiplication and division (**Math Pattern 8**). It can be helpful to ignore the values of the complex numbers themselves and focus on their geometric structure instead (**Math Practice 7**). I give the students the opportunity to explore in pairs (**Math Practice 1**) and then we discuss their findings as a class. The conclusions need to include the fact that the length of vectors created from the complex numbers multiply in length for a product and divide in length for a quotient. The other important concept is that the angles add for multiplication and subtract for division.

The remainder of the lesson practices the algebraic skills of multiplying and dividing through a guided practice. Conjugates are a new concept for my students, at least this year, so I spend extra time on this idea.

Detailed presentation notes are located in the PowerPoint.

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#### Exit Ticket

*3 min*

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

Today's Exit Ticket asks students to multiply two complex numbers.

#### Resources

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This homework reinforces the algebraic skills of complex number multiplication and division. I have also included an extension question on the difference of squares identity involving imaginary numbers.

This lesson was created with Kuta Software. I highly recommend this product to any mathematics teacher.

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
- UNIT 9: Trigonometric Functions
- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review

- LESSON 1: The Real Number System
- LESSON 2: Simplifying Square Roots
- LESSON 3: Imaginary Numbers Day 1 of 2
- LESSON 4: Imaginary Numbers Day 2 of 2
- LESSON 5: Complex Numbers Day 1 of 2
- LESSON 6: Complex Numbers Day 2 of 2
- LESSON 7: Completing the Square Day 1 of 2
- LESSON 8: Completing the Square Day 2 of 2
- LESSON 9: Real and Complex Number System Quiz
- LESSON 10: Quadratic Formula
- LESSON 11: Complex Numbers and Quadratic Equations Review
- LESSON 12: Complex Numbers and Quadratic Equations Test