Complex Solutions of Quadratics
Lesson 4 of 8
Objective: SWBAT solve quadratic equations with real coefficients that have complex solutions. SWBAT extend polynomial identities to the complex numbers.
In the scope and sequence of my previous Algebra II course, this newly aligned Common Core curriculum corresponds nicely to what I previously taught under the framework of the Indiana Academic Standards. However, I will work to make careful note of the changes in my approach to teaching the following skills:.
- Solving quadratic equations with real coefficients that have complex solutions (N.CN.7)
- Extending polynomial identities to the complex numbers (N.CN.8)
Look for these changes in the Out with the Old and in with the New reflection in this section.
As the students enter the classroom, I have on their desks a copy of the entry activity which asks the students to solve a quadratic equation using Completing the Square and the Quadratic Formula. Rather than reteaching this skill to my students from scratch, I like to begin class by seeing what they remember.
If I find that in the first few minutes students are struggling, I take a deep breath and remind myself that the skills (although foundational) are not critical to the students mastering the standards at hand. I, not unlike many math teachers, can get sidetracked when students have difficulty recalling previously learned skills. Although it is tempting to go back and remediate, this practice is contrary to moving students forward as we expect with the new Common Core. Sure, we must make sure that the students have an entry level understanding of using the methods of Completing the Square and the Quadratic Formula, but it is important to keep the focus of the class period on creating scaffolding activities for the students to reason with, reflect upon, and practice mathematics - - NOT constantly revisit past skills.
If I find that the students are struggling with the two methods, I enter into the second phase of this lesson. If most students appear to be understanding the problems and are able to accomplish the task, I ask for them to come to the board in groups of 4-5 to illustrate the process and talk through the key points... obviously if this is the case then I forego the next section of this lesson to allow time for these groups to work.
Fresh off of solving a couple of quadratics with real solutions, the students should be ready to extend their knowledge to the world of complex numbers. I like to think of rolling this idea out to them in three phases during this workshop.
1) What's next?
Prior to doing anything, I probe the students to guess what is next in the lesson. Where could we possibly be going? How can we connect this to what we have been studying? etc. Although sometimes a little cheesy, asking questions of this nature never gets old because it forces the students to make use of structure while also making connections - the heart of the Common Core and Practice Standards! In my past experience, the students are able to guess that we are going to investigate solving quadratics with complex solutions... considering we are in the heart of a complex and imaginary numbers unit. After reaching this conclusion, I begin to lead the students into the next phase...
2) Can you create one?
Rather than just doing an example of a quadratic with a complex solution, I ask my students to see if THEY can CREATE one first. Given the nature of the solutions to quadratics, their knowledge of factoring, and our study of the operations of complex numbers, my hope is that the students can brainstorm to manufacture a quadratic by working backwards from a complex solution that they create. Although many of the students will not be successful in this endeavor (because they will fail to select complex conjugate roots) several of the students will likely accomplish the mission and realize that the solutions must be conjugates of each other for the i's to cancel, thus yielding a quadratic equation with strictly real coefficients. I make a list of successful student efforts at the board - showing both the quadratic and the roots so that the students can begin to make the connection. They will need this connection to fill in the blanks in the PowerPoint!
3) Now that you created one, can you solve one?
Now, and only now, I issue a formal example for the students to try. After working our way through it, my make sense of our solution in the grand scheme of things by completing the fill in the blank slides in the PowerPoint.
Discussion and Homework
Please see the attached video narrative that elaborates on a 2-3 minute discussion that I have with my students over polynomial identities.
The homework for this lesson has been attached as a resource. It contains 5 problems that fit the theme of today's lesson, as well as one problem that asks the students to extend their learning by extending the polynomial identities. This problem was previously discussed with the students in the last workshop so that they are clear on the expectations.