Imagine THIS!

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SWBAT identify that there is an imaginary number "i" and use it to simplify problems.

Big Idea

Introduce imaginary numbers to extend the student’s knowledge of the number system.

Entry Slip

5 minutes

Many times it is beneficial to kick off a new unit by starting with an old bag of tricks...

The entry slip I hand out as students walk through the door asks them to complete a set of problems that they ALL will be able to handle.  The first five numbers look at the nature of squaring a number, and how the result is always positive.  The next six look at real roots of numbers, with #11 being the motivation for today's lesson.

N.CN.1 states that students should know that there is a complex number i.  In order for them to truly know what "i" is, they must first understand the need for it and where it fits in the larger scheme of mathematics.  The entry slip helps set this up for the students as we enter into this new unit. 

After allowing 2-3 minutes for the students to work, I "popcorn" call on students to read the 11 answers.

The Motivation for "i"

15 minutes

Guided Practice

10 minutes


The attached Imaginary Numbers guided practice problems help build students' confidence in working with imaginary numbers.  Although the skill itself is relatively easy for the students to pick up , and no major changes as a transition to the Common Core in this standard (the existing Indiana Academic Standard is worded similarly), students may confuse using i to represent a complex number with "i" as a variable.  The six examples are designed to help with this. 

In addition, you will notice that I did not include solving equations with imaginary solutions problems, every example says "simplify" this is intentional as I want my students to independently extend their learning from simplification to solving.  It is similar to asking a player in basketball practice to dribble two basketballs at one time:  Will they ever need to do this in a game? No way! However, it is excellent practice for the skill of ball handling, and the focused drill work supports players on game day! In many ways, mathematics problems and examples are no different.  I ask students to take the wheel "on game day" in the following assignment.  Naturally, just like any good coach, I will be there to support them along the way if they have questions.  They have already seen the opponent in the scouting report; now it is just a matter of sealing the deal!




Assignment Work Time

15 minutes


After finishing the guided practice, I circulate the Imaginary Numbers Homework Set  to students and allow them to begin working.  Although many problems are concrete and centered on N.CN.1, a few require recalling prior knowledge about rationalizing a radical denominator. In class we will have touched on the fact that can not be left in the denominator and these problems require students to multiply by the reciprocal throwing them an extra little challenge in what is an introductory homework assignment.   

As an optional accommodation for special needs students, assigning just the odds or the evens is often helpful.  That can also be done for every student, and you can take the time to finish the remaining problems in some form of entry activity on the following day.  Either way that I do things in my class, I always treat the last 5 minutes of class as it is an informal Exit Ticket.  I rotate the room looking at each students first 3-4 problems to see how everyone is doing.  If I notice that a segment of the class is missing something, I take the last couple of minutes to address this issue before I send them on their way.  The important thing that I have learned is that an Exit Ticket really doesn't need to be a "ticket" at all!