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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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?
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.
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.