...not just on the CAS version, but on the numerical version too.

...it's not disabled in test mode.

...it works well. (The TI-84 has an

*i*button, but in my experience it's unreliable.)

...oh *&^%.

First, I thought, "the children must never know about this!"

Yeah. Right.

If there is a button I will find the button. |

There it is. |

Here are some things it can do:

So after I threw out every assessment I used to use for this unit, I settled into the place of "What the *&^% do I do now?"

But it's good. Good! Good, I say.

It made me give some super-serious thought to what the complex number system is good for.

It's good for solving equations that don't have real solutions. Hello there, quadratics! We meet again. A little earlier this year.

It stops the Fundamental Theorem of Algebra from breaking.

It's full of

*numbers*that represent

*two*dimensions.

I can work with that.

It makes all conics the same.

ReplyDeleteIt makes trigonometric functions the same as exponential functions.

Yeah, but what about games that involve complex numbers, and you have to find the total score?

ReplyDeletehttp://blog.mrmeyer.com/?p=8854

FYI: the TI89 and any other calc with a computer algebra system have been handling complex numbers quite well.

ReplyDeleteBut can't you just assess their ability to manipulate complex numbers on a calculator-free quiz?

I think you're right to step away from the worship of complex numbers as a topic. I agree its good for solving quadratics...

I like to emphasize that numbers are human inventions. We invented negative numbers because when we tried to say 5 + __ = 3, it didn't work with positives. Rationals and Reals are similar inventions to satisfy problems with how we wanted operations to work for us.

Complex numbers are an invention. Its first use was to satisfy the square root of -1. But the ability to summarize and work with 2D information all at once has proven powerful.

Well for one complex numbers are widely used in fourier transforms, which are applied in the context of AM/FM wave modulations (ie like the radio). Just sharing. Peace.

ReplyDeleteScott - Apologies in advance that I'm going to sound like a total bitch right now, but it's early and I'm a little cranky.

ReplyDeleteI'd like to address your points one at a time:

1. I know

2. I know

3. I know

4. I know

If I had a nickel for every time a boy came around this blog trying to tell me a bunch of shit that it is a little insulting they think I don't know...I'd have like two dollars.

I'd like to address Scott's points a little differently: why are you so sure that numbers are invented? Is red a human invention? How about cold?

ReplyDeleteIf redness as a concept can be abstracted away from the collection of red things, and coldness as a concept can be abstracted away from the collection of cold things, then why can't threeness as a concept be abstracted away from the collection of collections of three things, without suddenly calling it "human invention"? And if you're about to pull that old dirty Kronecker trick, why is negative three an invention instead of a natural property just because it's a subtler concept that isn't quite as apparent as three or red?

Hi Kate, thanks for all the tips on your blog, do you use Geogebra much? you can do some cool stuff with complex numbers...

ReplyDeletehttp://mrcrookes.wikispaces.com/Complex+numbers

e.g. a pendulum

http://www.geogebratube.org/student/m419

i've only just scratched the surface

Kate, sorry, I meant no offense. Since you didn't know about the Nspire's complex mode, I figured I'd let you know about the other calculators' capabilities.

ReplyDeleteI was confused when you said that you threw out every assessment you used for the [complex] unit, based on the fact that the Nspire can solve the questions. So, I was curious as to what assessments you are going to do now? Or, what is your unit going to be like now?

drmathochist:

Interesting points. Perhaps "invention" is not quite the right word.

So, when we speak about math are we talking about the math that has been used by humans? Or the fuzzy concepts out there waiting to be discovered?

Is ours the only system of mathematics?

Scott! I have been feeling like such a jerk for writing such a snippy comment. I know you didn't intend anything negative. My capacity for giving people the benefit of the doubt is seriously a function of how much coffee I have consumed that day.

ReplyDeleteYou mentioned just quizzing them without allowing a calculator, but I find that that doesn't really work. They will just take the hit on one quiz rather than learn how to do something they know the calculator can do. Did I mention I don't really teach honors kids?

I have been coming up with alternative questions to see what they know about the purpose/use/definitions in the complex numbers unit. Here are some examples: 1. Show that the product of a complex number and its conjugate is a real number. 2. Explain what happens graphically on the complex plane when a + bi is multiplied by -2. when a+bi is multiplied by i. 3. Find all the roots of x^4 - 49 = 0 and state whether each is in the set N, Z, Q, R, or C. 4. Find two complex numbers whose sum is 10 and product is 34. 5. Find a quadratic equation whose roots are 8+i and 8-i.

I have been borrowing heavily from the textbook CME Project Algebra 2, since it does an excellent job of developing complex numbers. My textbook (PH Algebra 2 with Trigonometry) does a TERRIBLE job - everything having to do with i is in one section of one chapter, no development of the ideas, it's RIDICULOUS.

Mark I have used a GSP sketch that was programmed to allow you to see graphically all the operations on complex numbers. I tried to make something analogous on nspire but I'm not sure it's possible. There's supposed to be a way to share nspire files interactive-java-magically. I'm trying to get it to work and post it.

ReplyDeleteok but I love opensource software, its a lot easier to share and hack

ReplyDeleteYeah I love it too but I have to live with what my district selected.

ReplyDeletesounds like your situation is similar to the UK, big business and private monopolies seem to be playing a bigger and bigger role in education (as well as everything else)...oh well, gonna try to get my school to buy some Arduinos! I think opensource will win in the end!

ReplyDeletethanks for all your posts, they're great to read as a new teacher.

Yep. My school district prefers to pay through the nose for closed systems, for which free, better-working options are available. I don't understand it either.

ReplyDeletea least u don't have a queen

ReplyDeleteScott: there's not even only one system of mathematics that we use, but many of the systems we do use actually come with some really interesting uniqueness properties.

ReplyDeleteConsider natural-number arithmetic, for instance. Any system having a first number (zero) and a successor (n->n+1) that doesn't ever repeat itself turns out to always contain the natural numbers. And so any culture at all, anywhere in the universe, that comes up with even this rudimentary sense of counting must come up with the same system as we have.

Late to this party, sorry.

ReplyDeleteHopefully this allows you to change the focus of a complex unit to involve the graphical representation of complex numbers in the plane.

Almost all the connections to other topics are through this representation, and almost all its uses in college and careers are through this representation.

It's also a chance to get kids working with right triangle trig again before you hit them over the head with circle trig later.

(Anyway, you probably know all this, since you're stealing from a cool book. Hee hee.)

I also think that if all you're doing with complex numbers is add, subtract, multiply, and divide, there is no real point / upshot to it, so might as well bail.

Scott: the first use of complex numbers was not to satisfy the square root of -1, it was to solve cubic equations. Dudes in Italy kept finding stuff like (2 + sqrt(-121) + 2 - sqrt(-121)) in their formulas. They basically said "Wellllll, I don't like this number, but it looks like it's going to vanish, so I'll just go with it." This is what led to the derogatory term "imaginary number".

In general these new systems evolve from a desire to solve problems in the current number system.

Rock on!