## Student Grouping - Section 1: Warm Ups

# Imaginary Numbers Day 1 of 2

Lesson 3 of 12

## Objective: Students will be able to define imaginary numbers and use them to perform operations.

*50 minutes*

#### Warm Ups

*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- Imaginary Numbers Day 1, asks students to describe what makes a number real as well as how they would know that a number isn't real.

I also use this time to correct and record any past Homework.

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#### Imaginary Numbers

*37 min*

I am beginning this lesson with an area model just like the previous lesson. We start by discussing the side length of a square with area 1cm^{2} and then talk about another with an area of -1cm^{2}. This discussion leads us to the definition of imaginary numbers, both i^{2} = -1 and i = √-1.

We then extent this to i^{3} and i^{4}. After students have discussed how they found equivalent expressions for these cases, we extend the pattern to i^{15} and then i^{123 }(**Math Practice 8**). This will be a challenge (**Math Practice 1**). As students work, I walk around writing the variations I see on a small whiteboard which I then transfer to the full whiteboard. The students then discuss the variations until we come up with an solution that everyone can agree with.

Next, I ask students to write a summary which will help them formalize their thoughts on this repeated reasoning activity. Once the students have written their summary, I group them into fours(usually by seating) and have them share summaries with each other. They then pick their favorite of the four summaries and these will be shared with the class.

These are several additional examples that can be included or left out depending on the proficiency of the class.

#### Resources

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The goal of this short assignment is to ensure that students have a solid understanding of imaginary number pattern explored in class and included an extension problem that I talk about in my video narrative.

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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 imaginary numbers which specifically checks whether students truly understand that that i^{2} = -1.

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*Responding to Amelia Jamison*

Okay phew, I thought I was going crazy. It's a testament to my faith in you that I had to ask despite what the numbers were telling me!

| 2 years ago | Reply*Responding to Daniella Jaeger*

Hello Daniella, I'm so glad you caught this! Any even number (including multiples of 4) would ensure that i^2n is equivalent to 1. I'll update this in my narrative.

| 2 years ago | Reply

Hi Amelia,

I have a question about your extension question in the homework. In your video narrative you say that n would need to be any even number that is not a multiple of 4. I'm not following the multiple of 4 caveat. Could you please explain? Thanks so much!

| 2 years ago | Reply*expand comments*

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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