It seems that when we teach functions we always have a particular 
function defined by a "formula" on the chalkboard.  When we talk about 
where a function is increasing or decreasing, we always have a given 
function before us.  And heaven forbid that we discuss derivatives 
without calculating one. The best way around these difficulties is to 
choose a simple example from the "real world" and attempt to model it.

   We can start by asking, "What are the seasons?" We can lead our 
students to the following (nonstandard) definition of the seasons. 
   "Winter is when, in a given 24-hour period, the dark time is 
      longer than the light time and each light time is longer than 
      the last one. 

    Spring is when the light time is longer than the dark time and 
      each light time is longer than the last one. 
   
    Summer is when the light time is longer than the dark time and 
      each light time is shorter than the last one.

    Finally, Autumn is when the light time is shorter than the dark 
      time and each light time is shorter than the last one."

  Once we arrive at this formulation for the seasons, it becomes clear 
that we should be able to describe the seasons completely in terms of 
the amount of light time in a given one day (4-hour) period.  
We shall do so with a function.

  We let t be the time, in days, since some arbitrary starting point 
occurred. Then our function f will give the total light time between 
one-half day before t and one-half day after t less one-half day. 
If we insist on a formula,
f(t) =total time of light in days during [t-1/2, t+1/2] – 1/2 day.

  It is then easy to characterize the seasons:

  We have a "natural function" for which it makes sense to discuss zeros, 
increase, decrease, and critical values.  The zero between summer and 
autumn is approximately the autumnal equinox, and the other zero is 
approximately the vernal equinox.  The critical value between spring and 
summer approximately the summer solstice, and the other is approximately 
the winter solstice.
  
  Finally, we observe that f is periodic with fundamental period 365.