Teaching the concept of a function.

I am doing a class for at risk high school math students on the concept of a function. I have seen all the Internet lesson plans and different differentiated instruction plans. The idea of a function as a machine has always sat well with me, so I was thinking of playing off that. Are there any "out of the box" ideas that perhaps someone used or saw or knows that might hit home?

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Look at how successful the toaster analogy was at: everybodyisageniusblog.blogspot.com/2012/12/… – Amzoti Sep 12 '13 at 0:51
So many examples of relations in computer programming... Particularly databases. math.stackexchange.com/questions/98973/… – dls Sep 12 '13 at 0:52
Are a lot of kids doing computer programming that are at risk in math? – Eleven-Eleven Sep 12 '13 at 1:03
Well, these kids are likely burning videos and using double sided dvds, so they will say, wait, isn't this a two-sided function cause I can fit two movies, one on each side! :-) – Amzoti Sep 12 '13 at 1:29
I would make two points @ChristopherErnst... 1) No single analogy will work for every student. You need to be able to talk about computers, Magic the Gathering, baseball statistics, whatever interests the student. If they're already "at risk", they don't want to hear another pointless application in which they have no interest. 2) No, not every child is familiar with computer programming. But they're mostly all familiar with computers or smart phones. And there are plenty of examples of functions/relations to be had, just maybe not at the level of programming. – dls Sep 12 '13 at 2:16

Some everyday concepts could help. Such as

In a restaurant menu (f=food item, p=price of item):

Is f a function of p? Is p a function of f?

On back of a mailed envelop (s=street address, z=5-digit zip code):

Is s a function of z? Is z a function of s?

In a teacher's grade book (n=name of student who took a test, g= grade of student)

Is n a function of g? Is g a function of n?

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This is great. A function is a machine with one (or more) mouth(s). You stick something in the mouth(s), and something special and unique comes out the other end.

Now we should note that if you stick one thing into the mouth machine, and two different things come out the other end, then your machine is not a function.

Christopher you are a saint (by virtue deciding to deliver to your chosen audience). I think I have some pretty stellar ideas on fascinating delivery of these things, but I really think the machine with the mouth and the other end is the best in this case. I can't top your idea. Now of course you are a professional, but when you start talking about putting something in a mouth and something coming out the other end, you should get giggles (you of course smiling and ignoring the innuendo). I like your style, always trying to wow, engage, and excite the audience! Best of luck.

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Uh, I'll probably stay away from mentioning mouths and what goes in them. Perhaps at the collegiate level! :) – Eleven-Eleven Sep 12 '13 at 2:17
It is all about delivery. I understand that you have minors in your audience. You do not have to directly draw the connection to the human digestive cycle. In any case, If you want to engage an audience and have them soak in what you are delivering, one trick is to deliver your critical information while they are smiling. It's all about delivery. – J. W. Perry Sep 12 '13 at 2:20
You absolutely have to appear oblivious to that connection by the way. The most entertaining part would be that you appear to not see some parallel like what you are thinking about at all (somewhat self deprecating). It is outside the box man, loosen up, get them smiling, and them blow them away with your delivery. – J. W. Perry Sep 12 '13 at 2:29
No, I totally understand where your coming from, and indeed if this becomes a permanent gig I may throw that in when a level of comfort sits in... The principal will be there though and I don't have the confidence to pull that off all non chalant.... I do appreciate the help though.... – Eleven-Eleven Sep 12 '13 at 2:31
Ah, that changes everything. Do not step too far out of your box on this occasion unless you know exactly where you are going. I now understand where you are coming from much more. I guess mother functions and asymptotes are out of the question here too. – J. W. Perry Sep 12 '13 at 2:36

Functions are not about programming or machines or mouths or inputs and outputs or any other metaphors that do not reveal what we are actually dealing with.

The word "function" is a generic term like "animal." It refers to the fact that the world if full of quantities that are related to each other. The circumference and area of a circle. The diameter and circumference of a circle. The area and circumference of a circle. A volume and weight of water. Altitude and air pressure. Age and life expectancy. Amount of food consumed and weight. The amount of powder in a bullet and its exit velocity. And millions of others.

The question for mathematics is how to express a relationship between two quantities. Perhaps the most obvious way is by a table but that is limited to the entries. An ingenious way is to express operations on one of the quantities that produce the other. This object is called an "expression" and the explicit quantity is called a "variable." These may not be the most descriptive names but they are totally entrenched. There are various instances of expressions. Expressions (functions) have PROPERTIES and a good deal of the study of functions is the exploration of their properties. Then we have the "function equation," the purpose of which is to enable us to have "implicit functions" where BOTH quantities are represented by variables and to have systems of function equations in multiple variables. These various representations are called different "forms" of a function. Graphical form is another totally ingenious form that makes some properties visual. We also have arithmetic and algebraic sums etc. of expressions, and composite functions and functions like x to the x power that are none of the above. There are other graphical forms and infinite series of expressions and there are named functions expressed by other non-algebraic operations that cannot be expressed in closed algebraic form.

Functions are an entire ingenious universe within mathematics that is terribly short changed by metaphors and memorized rules and emphasis on proof.

All of mathematics is completely purposeful and sensible and those aspects should be made perfectly clear from the very beginning and all along.

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I agree with what you are saying and my presentation will deal with the fact that these things are relations. But for at risk math students who don't see the relations abstractly as math provides, we need to use other ways to get the point across of what a function is. This also helps with notation, which kids who aren't strong at math have a big issue with. SO it is important to provide analogous ideas that are not mathematically presented. – Eleven-Eleven Sep 13 '13 at 13:31
It is a mistake to consider mathematics to be abstract. I did not use the word "relation." The question is do you want to give them something to wonder about and think about and question and work with or do you want to condition them? "Mathematical ideas that are not mathematically presented" is an oxymoron. Mathematics is a concrete physical science that suffers terribly from an ancient erroneous misrepresentation that seems to have no end of believers. – George Frank Sep 14 '13 at 10:06
Okay, so my question on this post was "teaching the concept of a function", and while your philosophy varies differently from the rest of the pack, what would be you methods to reach the kids who are falling behind now? I'm not asking how to systematically reteach the subject, I'm asking in what ways would you present the ideas to a group of failing math students in 45 minutes to help bring them back to pasture? And while you didn't use the word relation, you did in fact say "It refers to the fact that the world if full of quantities that are $\mathbf{related}$ to each other." – Eleven-Eleven Sep 14 '13 at 12:20
You also said "The question for mathematics is how to express a $\mathbf{relation}$ship between two quantities." And mathematics is very abstract...just look at category theory. Or even the topic of abstract algebra...where we deal with things like groups and the operations they employ, which could be a variety of different binary relations. There is a reason the brilliant minds who thought of and created the maths we work with refer to this stuff themselves as abstract. I want to be able to reach kids who are falling behind so what you you do to teach a class on functions? – Eleven-Eleven Sep 14 '13 at 12:29
You have the usual preconception of mathematics as abstract. There is nothing abstract about wanting to relate quantities like the height at which an object is dropped and the time to hit the ground. Or the age of an egg and the porousness of its shell. Or your age and the length of your hair if you never cut it. And so on. A "function" is a way to express such a relationship. There is no "pure" and "applied" mathematics. The applications of mathematics are not about mathematics. And neither are lemma, definition, theorem, proof, and corollaries. Mathematics is about achieving, constructing. – George Frank Sep 14 '13 at 14:25