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Topic: Teaching the Concept of Functions


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Subject:   RE: Teaching the Concept of Functions
Author: lanius
Date: Sep 23 2004
On Sep 23 2004, Susan wrote:
> Every year, I try to get the students not to say, "Oh, you just
> cross out f(x) and write y...what's the big deal."  They can recite
> the definition of a function from their text--for each x-coordinate
> there is exactly one corresponding y-coordinate, but they have no
> idea what this means or how it applies in the real world.  

Susan, you have raised such a great question. With this definition, students are
thinking of functions as their graphical representations, so it is reasonable
for them to think x,y.

Function notation -- f(x), g(q) or m(b)-- should become more expedient when
they start working with functions as rules, rather than just graphical
representations, especially if you give them non-numeric examples.

What if the input is letters of the alphabet? Then you could have a bunch of
rules on the alphabet. Rule 1: make all possible English words. So input letters
a, t, b. Output -- at, tab, bat, and tat. Not a function.  Job for them - Find
a three-letter input that would yield a function. Rule 2 make all possible
English words but don't allow the letter to be used more than once. ETC. So if
you create a whole system of rules that you need to talk about and work with,
wouldn't it be nice if you could symbolize the system with some notation? Voila!
So now I could talk about f, g, and h of (b) where b is letters of the alphabet.
Then if rule 1 above is f, f(a, t, b)=at, tab, bat, and tat.

Another system of rules could be family relationships. Input is people. Rules
are relationships (parents, cousins, etc)

Also, think of a lemonade stand. You want to optimize your profits.
Profits depend on  sales and expenses. Sales depend on temperature and traffic.
Create a system of functions that reflect the given. What are some rules to
reflect this system? What are the inputs on the various rules? What are the
outcomes?

You gave me some neat things to think about. Thanks...


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