Complex Numbers Solutions
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This lesson provided by:
Author: Morgan Boyd
Organization: Retirement

General Lesson Information
Lesson Plan ID:
35678
Title:

Complex Numbers Solutions
Overview/Annotation:

This lesson is designed to teach the students that some quadratic equations will have imaginary solutions. The lesson will examine the concept of complex numbers in terms i . The student will use the quadratic formula to solve the equations and write the the solutions in the form a +bi.

This lesson results from the ALEX Resource Gap Project.

Associated Standards and Objectives
Content Standard(s):
Mathematics MA2015 (2016) Grade: 9-12 Algebra II 5 ) (+) Extend polynomial identities to the complex numbers.

Example: Rewrite x ^{2} + 4 as (x + 2i )(x - 2i ). [N-CN8]

Mathematics MA2015 (2016) Grade: 9-12 Algebra II with Trigonometry 5 ) (+) Extend polynomial identities to the complex numbers. [N-CN8]

Example: Rewrite x ^{2} + 4 as (x + 2i )(x - 2i ).

Local/National Standards:

Primary Learning Objective(s):

The student will be able to solve quadratic equations using the quadratic formula.

The student will be able to write complex solutions in the form a + bi.

Additional Learning Objective(s):

Preparation Information
Total Duration:

31 to 60 Minutes

Materials and Resources:

Technology Resources Needed:

Background/Preparation:

Teacher

The teacher will need to preview the website, https://www.desmos.com/calculator . The teacher will use the website to show the difference between real and non-real solutions to quadratic equations. If the parabola crosses or touches the x-axis, then the equation has real solution(s). If the parabola does not cross or touch the x-axis, then the equation has imaginary or complex solutions.

Student

The student should remember that the square root of -1 is i . Therefore, the student will be asked to simplify square roots of negative numbers. The student will need to be able to use the quadratic formula with quadratic equations. The student will need to be able to simplify square roots. The student needs to be able to graph quadratic equations.

Procedures/Activities:
Before

As the students enter the classroom, the teacher will have the bell ringer, Bell Ringer with Complex Numbers , displayed on the interactive whiteboard. The directions are on the sheet. The students can use their technology devices or regular graph paper, listed in the attachments. The answers are hidden in the color, white. The teacher will highlight under the word, answers, and change the color to black. The teacher will click on the website link to open the graphs. The teacher will ask students to comment on the second part of the bell ringer.

During

The teacher will use the bell ringer graphs to show the students that when the graph does not cross the x-axis, then the equation does not have real solutions. The teacher will say, “Solutions to the quadratic equation can be either real or imaginary/complex.” The teacher will show the students that when the graph crosses or touches the x-axis, then the solutions are real solutions.
The teacher will write on the interactive white board, “The square root of -1 is ____”. The students will turn and talk. The teacher will give one minute for the students to discuss their answers. The teacher will call on two or three students to fill in the blank. The answer is i , which means imaginary. The teacher can do an informal assessment.
The teacher will introduce the video from YouTube. The teacher will discuss the idea that the quadratic formula may have a possible negative square root. The equations will have imaginary roots or complex solutions.
The teacher will show the video, https://www.youtube.com/watch?v=jU_aLT2YMjA .
The teacher will ask for questions or comments.
The teacher will show the next video, https://www.youtube.com/watch?v=H5AM1bzqCQw . Stop the video when the presenter has the equation on the screen. The teacher will ask the students to work with a partner and solve the equation. The teacher will walk around the room and monitor student behavior and engagement. The teacher will give one-on-one instruction if needed.
After three or four minutes, the teacher will continue playing the video. The teacher will do an informal assessment while the students are checking their work.
As the video is ending, the teacher will pass out the worksheet, Quadratics with Complex Solutions, in the attachments. The teacher will place three students in each group. The intervention will be to allow the students to use the website in the materials section. (https://www.mathpapa.com/quadratic-formula )
The teacher will call on students to write their work and answers on the interactive whiteboard.
For the accelerated students, the teacher will give them the worksheet, Accelerated Complex Numbers .
After

The teacher will pass out the exit slip form the attachments, Exit Slip Complex Solutions . The teacher will use the exit slip as the formal assessment. The students will turn in the exit slip as they leave the classroom.

Attachments: **Some files will display in a new window. Others will prompt you to download.

Assessment
Assessment Strategies

Informal

The teacher will ask students questions during the lesson as well as monitor the students work during the group assignment.

Formal

The teacher will use the exit slip as the formal assessment.

Acceleration:

The accelerated students will have a worksheet to complete called Accelerated Complex Numbers .

Intervention:

The students will work in groups. A peer-tutor will be assigned by the teacher. The webiste, https://www.mathpapa.com/quadratic-formula , can be used with the devices. The teacher will do one-on-one with students that are still struggling with the formula or simplifying the radicals.

View the

Special Education resources for
instructional guidance in providing modifications and adaptations
for students with significant cognitive disabilities who qualify for the Alabama Alternate Assessment.