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Subject:   RE: function notaion
Author: Mathman
Date: Jun 17 2005
On Jun 17 2005, djw wrote:
> Math instruction:

Introducing function notation initially vs.
> learning y = mx + b format and then ATTEMPTING to get students to
> change to function notation.

Please advise.

As others suggest, it should be used when the study is more general. However, a
primary interest is in seeing that an expression IS a function of a variable,
and another different expression is another distinct function of the variable.
It is a matter of language, and language is based on familiarity.

You might consider writing [and you MUST expand on this idea with lots of
examples] simply the expressions of "x", such as

3x+ 2   and 5x - 3 ...or higher degree so long as they are already familiar with
those also.

First, you would give some introductory explanation as to why you are doing
this... Looking at a set of specific examples [a list on the board, on paper or
whatever] you'd show that all are expressions of "x", so why bring "y" into the
pcture in the first place?  Then write a few without "y =", saying that you are
simply listing the expressions of"x".  Students like to be fed.  They
particularly like being fed lots of marks to make up for that last really bad
test.  If they complete the list, they get the marks for doing so when collected
at the end of class with their name on it.

Next, have them look at the list, and suggest that if you want to talk about one
or the other function, you'd better give each a name, so everyone knows which
you are talking about, AND to keep it simple.  It's much simpler to talk about
"f(x)" than about the function "x^9 - 3x^8 - 7x^7 [etc]", and much more
complicated expresions.  So, one can be f(x), "A function of x"; or F(x),
"another function of x"; another g(x) which is another funciton of x to
distinguish it form the first two.  Get them to attach names such as p(x) to
each.  Those are for marks.

Hand out a common list now, of functions with names, and a list of tasks.  The
first might be to graph f(x) and g(x) on the same sheet of graph paper.  Define
f(x) + g(x) as the sum of the expressions, simplified as necessary.

E.G.  f(x) = 5x - 3 and g(x) = 3x + 2
Define f(x) + g(x) = (5x - 3) + (3x + 2)
and have them simplify and graph the result.

They don't need to go much further to become familiar with the use, but then you
must use it consistently thereafter.

...and a whole bunch more as you think of it. Simply stop using "y=", and use
only stand-alone expresions, or Q(x) = etc..

This is about lesson plans I think, rather than some tool.  It is the classroom
dialogue that will make the difference while presenting the material. You will
have to present lists and ask which are functions of x and which are not ...not
tricky, like the difference between function and relation, but in their face
obvious, one a function of x, another a function of z, an so on.  They are
learning a language some think to be more complex than a completely foreign
spoken language; go slowly with easy development ...and speak the language
constantly, at their level.

I hope this helps.


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