It Took Me 2 Years to Get This Approach to Imaginary Numbers

This past week the NCTM annual conference was held in Boston, and what an enriching epxerience! What made it so special this time around was meeting and hearing from my PLC of Twitter friends, many of whom I had admired from afar for some time. I’ll discuss the power of the MTBoS (Math-Twitter-Blog O’Spehere) in a later post.  Today I want to focus on a powerful session I attended in Boston, and how a new persepective developed – even after a 2 year delay.

The story starts 2 years ago at Twitter Math Camp in Philadelphia.  At that conference, I participated in an Algebra 2 small group, facilitated by the super-creative Max Ray, from the Math Forum. Splitting into smaller groups, I worked with a team to think about rational expressions – a unit which is often dry as sand in Alg 2 courses, and where I thought we could make some head-way. While we worked on our slightly-less dry, yet safe lessons, Max and a small group were discussing complex numbers on the board. There were mysterious circles, transformations, and discussions I didn’t understand.  I suppose I was taught about complex numbers the “traditional” way – we need them to solve certain quadratics and memrize some wierd rules about their behavior. We perform strange operations on them, and we definitely don’t ask why. I suppose I could have simply wandered over to the group and found out more, but the mathematical intimidation factor was high – I’m sometimes too proud to admit what I don’t know.

Fast forward 2 years, and I see Max is presenting a session with Michael Pershan. This is a must-attend. Two engaging speakers whom I appreciate for their ability to use students’ natural curiosity to facilitate math conversations.

Here’s the set-up: Michael finds a handful of volunteers to stand at the front of the room, standing on a hypothetical number line (Max stands at zero). The participants are then asked to consider the following transformations to their value, and move accordingly, returning after each move to their original position.

  • Add 2 to your value – participants all move to the right 2 spaces.
  • Multiply your value by 3 – participants all move to the left or right accordingly, depending on whether their original value is positive or negative.
  • Multiply your value by -1. OK, now the plot thickens.  While we can find our new position, Michael does a materful job in having participatins reflect upon the transformation. The first two moves required left and right shifts; here we need to consider a rotation about the origin. This rotation provides a rule for multiplication by a negative.

Photo Apr 16, 12 36 08 PM

The table has been set, the silverware polished. and now we need some new volunteers. We have a new number line, and some new transformations to think about.  BUT this time around we want to complete our movement by using the same transformation twice.  Let’s roll!

  • First, add 4 by using the same transformation twice.  This is a nice appetizer – let’s move 2, then 2 more.
  • Next, multiply by 9. This is a little trickier, as some folks almost crashed into the next presentation room. But two multiply by 3 moves do the job.
  • Now, multiply by 5. Oooh….we have an entry point into radicals. Some quick discussion, and we two moves – multiplying by a little mroe than 2 each time.
  • Finally, multiply by -1…in two moves…..

WAIT!  This is the stuff Max was talking about 2 years ago that I didn’t get.  The bulbs have gone off.  I GET this now!  We do a 180 degree rotation do perform a multiplication by -1, so now we need two 90 degree rotations.  And now we have an entry point into imaginary numbers, without the scary-sounding term.

What I appreciate most here is that we don’t need to wait until deep into algebra 2 to think about the imaginary unit.  These concepts are accessible to younger students, and we have a responsibility to achieve some conceptual buy-in before just thrusting abstract ideas in front of our students. You can find Michael and Max’s shared files here on their Teaching Complex Numbers page.

I get it now…I think….and I’m not ashamed to say it took me 2 years.

UPDATE: You need to immediately run to check out the fun summary Ashli has provided of this session. Her notebook sketches are unreal (in the non-numbr sense)!


By Bob Lochel

HS Math Teacher. Hatboro-Horsham School District, Horsham, PA.

16 replies on “It Took Me 2 Years to Get This Approach to Imaginary Numbers”

A post I read in February got me looking into this stuff about root(-1). Here is what I wrote at the time:
It looks at rigid motions of the plane, in particular the rotations, and after seeing that two rotations of 180 (corresponding to x(-1) each) gets you back to “square 1” (actually the square of -1) the rest followed. Too heavy an approach for first intro, but no need to wait for Alg 2 (no matrices!). My style is somewhat cryptic, but if you think it’s worth fleshing out I could have a go.

Bob – what a great description! I was in that small group in Philadelphia, and truth be told, the idea was hovering just on the periphery of my comprehension! Your post clarified it for me – learning something in 2 years is better than never learning it at all!

Bob, I was in your small group in Philly working on rational functions…and I shared similar feelings. I remember overhearing Max and others at the front of the room discussing imaginary numbers, and I too felt intimidated by their knowledge. Sadly, I haven’t been to a math conference since that Twitter math camp as I’ve began my master’s degree at Harvard. I do believe I saw you at the Red Sox game on Friday night though – You were in the row directly in front of me. I would’ve said hi, but figured you wouldn’t remember me from such a long time ago. Thanks for sharing this post, and I’ll have to check out their presentation at some point regarding this topic!

Fantastic! Please say hello next time you see me at a ball game….or where-ever. I do have an awful memory of faces and names, but tend to remember when we chat about math. Glad to hear I wasn’t the only one uber-intimidated.

Thanks for this post, Bob. Clarified and refreshed a lot of what happened in there before I forgot the nitty-gritty. (I was in the same talk, too.) And thanks for being honest about sometimes being intimidated. I was trained as an engineer. I can DO math, but I wasn’t trained to think so deeply about it, and too often I just don’t. Hearing others’ more-educated insights is both inspiring and intimidating. I always want to learn more, but it’s a fine line to dance up to, not feeling dumb but still learning. And that, in turn, brings me back to my own students. This feeling of wanting to learn but not wanting to feel dumb for not knowing is really a fantastic idea to be mindful of we teach our students, who haven’t ever seen much of “this stuff” before. I’m rambling. But I have so much to chew on! Thank you.

Eg.f(z) = z2 then = [ ]
z1 z1
1. Evaluate (a)along the line y =
(b)along the real axis from 0 to 3 and then vertically to 3+i
© along the parabola x =3y2

= = x2 – y2+ i2xy
Therefore u = x2 – y2 and v = 2xy
Thus ux = 2x , uy = -2y , Vx= 2y ,Vy= 2x
Ux = Vy and Uy = – Vx .Therefore f(z) is analytic
Therefore the value of the integral is independent of the path of integration.and prof dr mircea orasanu


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