Alex, what would your suggestion be for teachers when they introduce the coordinate plane? Would the convention "independent variable" and "dependent variable" be a good place to start, with perhaps different variable names for these?

I usually try to approach it from a functional standpoint like this, where something 'depends' on something else. I suppose it is when we switch independent and dependent around that we have problems....

Here is a recent problem I put up in a Calc I course:

"Let N = f(t) be the total number of cans of cola Sean has consumed by age t, where the units of t are years. Interpret the following in practical terms... (f -1)'(450) = 1/7"

Man, did I ever get some interesting answers to this!

This is hard. A teacher should have a flexibility to choose among several alternatives and intelligence and experience to adapt the choices to his/her audience.

The trouble begins with the number line which is always horizontal and positive to the right. I would draw an arbitrary line put points 0 and 1 on it and ask to estimate where 5 or -3 lie. I would pick to points in the plane and ask to form a number line with the chosen points corresponding to 0 and 1.

In the plane, similarly, I would give examples of several configurations and then ask to form configurations by three points, say, O(0, 0), P(1, 0), and Q(0, 1), with points placed randomly here and there.

>Would the convention
>"independent variable" and "dependent variable" be a good
>place to start, with perhaps different variable names for
>these?

As far as the coordinates go, there is no dependent or independent variables. These are related to functions. Perhaps, just the "first" and the "second" component, with emphasis that the two numbers are nut inherently related to a point but only through a selection of a coordinate system.

>Here is a recent problem I put up in a Calc I course:
>
>"Let N = f(t) be the total number of cans of cola Sean has
>consumed by age t, where the units of t are years.
>Interpret the following in practical terms... (f -1)'(450)
>= 1/7"
>
>Man, did I ever get some interesting answers to this!

I am curious to see some.

Oh, I read (f - 1)'(450) = 1/7.

I guess what you meant was (f^(-1))'(450) = 1/7.

This would mean that in the year when the 450th can has been consumed the derivative of the consumption equals 7, meaning that over that year the fellow drank only 7 cans of Coca-Cola.

here's one...

when 450 cans were consumed, his age was changing at a rate of 1 year per 7 cans consumed... I shall reply with more when time permits...

Please allow me to respond to your fine points and then briefly give some context.

462,518,344,297 is:

In English four hundred sixty-two billion, five hundred eighteen million,...

In Polish and Korean it IS four hundred billion, sixty billion, two billion, five hundred million, ten million, eight million, etc.

Yes, we could say it that way, as they do in Poland and Korea, but we don't. But BOTH countries are saying "how many" of some unit (billion, hundred billion, etc.)

Yes, it would be inane to codify this and parade it in front of future teachers. I NEVER use the words "ordinate" and "abscissa". But I asked this as a linguistic question, not as a general pedagogical question.

My context is teaching Liberal Arts students the names of large numbers. And by large I do not mean 462,518,344,297. I mean "four hundred twenty-seven millinillitrillion, five hundred eighty-six octdecillion, five hundred three" following the naming scheme of Conway, Wechsler, and Guy. (From The Book of Numbers.)

Here one of the entire points is how we can extend language to very powerful realms. There are many linguistic issues. There is also the pragmatic issue of separating out the different components - 427 as opposed to millinillitrillion.

Now you might object that not only my question, but my larger context is absurd. But it wasn't to Conway, Wechsler, and Guy. The names Googol and Googolplex resonate with many. AND, my liberal arts students - who care about language, scale, and metaphor more than mathematics and who often feel inferior mathematicall - are REALLY excited to be able to learn this system and use it with confidence in front of those they generally feel inferior to.

Thanks for your responses. I hope that this makes a bit more sense now and might encourage you to reconsider my rationale a bit.

Thanks.

I understand you are looking for a generic term. Off the top of my hat, I would consider

- unit'scale factor
- scale unit factor
- scaling factor

or, perhaps, one may insert or replace something with "cardinal":

- cardinal unit factor
- cardinal scale factor, etc

"factor" looks like a fixture. But, perhaps,

- cardinal scale unit
- cardinal unit
- cardinal scale

may also have a chance.

On second thought, why not to use the more straightforward

- place value unit
- place value factor

As you see, I am unaware of the existing terminology. My gut feeling is that it is indeed nonexistent.

You know, I would probably locate and write Stephen Pinker. Judging by his books, he may be the one to know. Do please let us know if there anything you come across.