I begin this lesson with a simple quadratic on the board (like x^2-2x+4) that doesn't factor easily. I challenge my students to find the zeros and walk around listening while they work on it. (MP1) I expect at least a few will try to find them by graphing, to no avail while others will try to factor, again with limited success. I hope that a few remember the quadratic formula, but if not, I refresh their memories. This will result in an answer, (1+- sqrt of -3) but I don't let my students rest there. I ask them "How can you have a sqrt of -3? What does it mean?!" This has resulted in all sorts of interesting explanations in previous classes, but my favorite so far was the student who said "It doesn't have to mean anything, it's just math." After a few moments of discussion I tell my students that I want to introduce them to a new player on their math team - enter imaginary numbers. I explain that mathmaticians created "i" to represent sqrt of -1 so they could talk about things like factors of (x^2-2x+4). I share the definition of a complex number and show the format for writing them (a + bi).
You will need copies of the Complex Numbers handout for this section of the lesson. I tell my students that today they get to practice working with and classifying real, imaginary and complex numbers. I explain that they will be working independently to complete a handout about complex numbers but before they begin I'm going to show them a few examples. I walk them through several examples of complex numbers, identifying the real and imaginary components and simplifying whenever possible. You can see how I do this in my educreations video, which I make available to my students to use after the lesson for reference/review as needed. I don't show them how to simplify i^n by dividing n by 4, because I want them to work that out for themselves if possible, but I've included it in the educreations video for those who need extra help. I ask if there are any questions, then distribute the handout and let my students get to work. (MP1) While they're working, I walk around offering encouragement and redirection as needed. When everyone is finished, or after about 25 minutes I tell my students that they need to be ready to share in about five minutes. I've already observed which students are struggling and/or aren't going to finish in time and made a note to connect with them later to make sure they're on track. I randomly call on students to share their answers with the class, having them explain how they got the answer and accept questions and critique from their classmates. (MP3) I don't intercede unless an answer is incorrect and none of my students catches it, so that they become more confident about their own work. While we're going through the problems, I encourage my students to self-check their answers and also to look for more elegant, interesting, or simple ways to solve the problems. I explain my reasons for this last directive in my that's complex video. I'm sure your students are familiar with checking their own work but if they don't have experience looking for different ways to solve problems, you might try some of the following suggestions. Ask them to rearrange the parts of the problem to see if they fit together differently, like solving a puzzle from borders in or from the picture on the box. Ask them to try to find one step in their solution that can be eliminated without chnaging the answer, like taking out most of the "that"s in a paragraph. Or ask them to try working backward from their answer to the original problem and see if they follow the same path.
To close this lesson I ask my students to summarize in their notes how to identify the real and imaginary components of complex numbers in writing and with an example of their own creation. By asking them to write out their summary I'm helping them strengthen their communication skills, while the example gives them a visual reference for future study.