# How Do I...?

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## Objective

SWBAT use arithmetic properties to manipulate complex equations.

#### Big Idea

Back to the basics...your students will feel confident as they use well honed arithmetic skills to simplify complex expressions in a challenge card activity

## Set the Stage

15 minutes

I begin this class by putting the roots (1+- sqrt of -3) of yesterday's quadrilateral  on my board.  Since my students have already seen the quadratic equation these came from, I hope to make the connection between imaginary roots and complex number arithmetic a bit more meaningful.  I challenge them to multiply these, be ready to share their result and to show how they got it. (MP1) While my students are working on that challenge, I walk around offering encouragement and redirection as needed.  When everyone is done I ask for volunteers to put their calculations and answer on the board (I can fit three or four students at one time).  While the volunteers are working I suggest to the rest of the class that they can check their result by making it a quadratic equation (y=...) and then graphing their equation to see if it matches the one from yesterday.  When all my volunteers are done posting their work on the board I ask the class to review the work and ask questions.  After all the questions have been answered I throw out an additional challenge.  I ask my students to consider how they would add and subtract complex numbers by pair-sharing with their left-shoulder partner.  After a minute or two, I randomly select students to tell the class what they discussed.  I summarize the suggestions for adding and subtracting, then work through a few examples of each.  I include at least one example of finding the absolute value of a complex number, not by giving my students the formula, but by working out an example for them and then asking them to generalize the result. (MP8) {|a + bi| = sqrt(a^2 + b^2)} This gives them some ownership of the process rather than just memorizing a "shortcut".  When there are no more questions about doing arithmetic with complex numbers, I tell my students that they now get to work with their partner to explore what they've just learned.

## Put it into Action

35 minutes

For this part of the lesson you will want a copy of the Complex Arithmetic worksheet.  You can use it just as a worksheet or you can make the problems into flash cards with the answers on the back.  I hand-write these on 3x5 notecards cut in half.  If you use flash cards you may also want to have your students use whiteboards and markers or scratch paper.  I explain to my students that they get to explore adding, subtracting, and multiplying complex numbers by challenging each other with flash cards.  I say that they can either use whiteboards or scratch paper and will be taking turns working solutions then checking each other.  I ask if there are any questions, then distribute the card sets.  While my students are working I walk around offering encouragement and redirection as needed. (MP1) Using flash card sets, I've found that my students will go through the problems more than once and are more willing to work to build their skills with complex numbers than if I just had them complete the same problems as a worksheet.

## Wrap it Up

5 minutes

To close this lesson I challenge my students to create one additional complex number arithmetic problem of their choice to "stump the teacher".  I say that the problem has to fit all the rules they've learned about complex numbers and must be legible.  I give them each a notecard to write their problem on and tell them that I'll return card with answer tomorrow if I can figure it out! This really gives my students incentive to think about complex numbers as they try to come up with a way to stump me. (MP2)  I explain more about why I use this for closing the lesson in my  video.