Programming is The How of Functions.
There are lots of examples of functions:

• [get examples from class]
  Note that arithmetic's notation wasn't always the clearest: you've been taught "y=x+1" is a function; what is the input? What is the output? (Is "x=y+1" the same function? What about "m=n+1"?) Really, "add1" might be a better name for the function, and we could write "let add1(n) = n+1".
• The Academy Awards (takes in a list of movies, and returns (ahem) the best one. Note that the Academy is what performs the computation.
• The registrar has a function from student to library-fine. (Well, okay, at-which-time is also an implicit input to the function.)
• A web-search-engine is a function, taking in keywords, return a web-page (with links to other pages).

So functions aren't always about numbers. That said, today we'll only use functions on numbers. (We'll start enriching our world starting next time.)

Examles of computing, and generalizing to programs.
• A pizza costs $12, and has eight slices.
  If you eat 2 pieces, you owe $12*(2/8) = $3.
  If you eat 1 pieces, you owe $12*(1/8) = $1.50
  If you eat 8 pieces, you owe $12*(8/8) = $12.00
  Of course, we learn to abstract out the parts of the calcuation in common, and express it as a more general rule:
  owes(p) = $12*(p/8). Save this, on board..
• A different formula, that you once memorized in school (since it's not so obvious, if you're not taught it).
  A disc of radius 6" has an area of 36pi sq. in.
  In general: dough-area(r) = pi*r^2.
• Consider the folks who make the pizza. For a given diameter, how much of the pizza is covered with toppings? Note that there is a 1-inch border, which is crust-only.

  A pizza of diameter 14" has a radius of 7", of which 1" is crust, so the toppings cover a disc of radius 6": Draw picture.
  pizza-topping-area(14) = dough-area(6) = 36pi.

  In general: pizza-topping-area(d) = dough-area( d/2 - 1).
  Does this rule make sense? Let's try some more tests:
  pizza-topping-area(8) = dough-area(8/2 - 1) = dough-area(3).
  pizza-topping-area(2) = dough-area(2/2 - 1) = dough-area(0).

  Finally, a question: we could instead have written
  pizza-topping-area(d) = pi (d/2 - 1)^2, or even
  pizza-topping-area(d) = pi/4*d^2 - pi*d + pi.
  Do we have a preference? Yes! The first version reflects how we think about the problem, while the second makes the reader scratch their head and wonder if it's really correct. The First Law of Programming: Your code should reflect the way you think about a problem. We'll come back and find another advantage to the first phrasing, as well.

You have been programming and computing for a long time, in elementary school and junior high. In this class, we'll spend ten weeks just exploring implications of pre-algebra.

  ___
-/16

  ______________
-/ 7*7 + 24*24

         _____________
 -5 +  -/ 5*5 - 4*1*7
----------------------
       2*1

(define (owes p)
  (* 12 (/ p 8)))

(define (dough-area r)
  (* pi (* r r)))

(define (pizza-topping-area d)
  (dough-area (- (/ d 2) 1)))

The pizza industry is a tough biz; it's hard to stand out from the competition. [Not actually true in houston, but we'll ignore that.] What if we want to switch to hexagonal pizzas? What code needs to be changed? Well, certainly we change dough-area, to reflect the formula for area of a hexagon ("radius" now means to dist-from-center-to-middle-of-side). But what about pizza-topping-area? No change needed! (Imagine that we'd used the weird pi/4*d^2... version -- we'd be wading through a long list of various pizza functions, and trying to figure out which were dependent on being a circle.) Thus, by re-using dough-area, we don't have to re-write other functions!

The Second Law of Programming: Re-use existing code.
We call this a "single point of control". Other uses of this (very general management) principle:

• the same registrar information being replicated in the general announcements, the newspaper handout, departmental web pages, and advising handouts in o-week packets. Each time these are collected from a different source, you run the risk of contradictary material, and if you update one rule you need to remember to update all of the above.
• Consider a set of web pages which say "page maintained by jane@rice.edu". What happens when jane moves on, and hugo takes her place: Do you go through and replace *each* occurrence of "jane" with "hugo"? But jane might have many other mentions on other web pages! A central point of control is saying "contact webmaster@rice.edu"; this email can be redirected as the job changes.

If you ever find yourself writing the same code to do the same thing (in this case, squaring a radius and multiplying by pi), you should stop and think about how to re-use the common code.

To do before next lecture:
Start up DrScheme, and double-check your expression for

         _____________
 -5 +  -/ 5*5 - 4*1*7
----------------------
       2*1

• computing as functions;
• as algebra, substitute equal expressions until value reached
• single point of control

• This vs other comp courses:
  • comp 210, comp212
  • comp 100: intro to IT
  • comp 110: solving problems w/ mathematica (for engineers w/o computing)
  • comp 200: computer science for poets -- overview, some programming
  • caam 210: solving problmes w/ matlab; basis for engineering courses
• Your responsibilities:
  • expect to think, while in class.
  • review previous lecture, before each class; do the daily problem.
  • homework: better in small steps, than all the night before.
  • Ask questions in lecture (or communicate otherwise), or lab sections, or web feedback.
• Your rights:
  • Clear idea of what is expected for homeworks.
  • Access to class staff.
  • Explanations of grades.
  • A challenging, rewarding course!

Administrivia:
• read handout yourselves!
• homework teams, honor policy
• will sign up for lab times on web, starting this afternoon.
• exams in evenings?
• Reading: Through sect.3.1, less2.4(errors).