Programmers are rarely handed mathematical expressions to turn into
programs. Instead they typically receive informal problem descriptions that
often contain irrelevant and sometimes ambiguous information. The
programmers' first task is to extract the relevant information and then to
formulate appropriate expressions.
Here is a typical example:
Company XYZ & Co.\ pays all its employees $12 per hour. A typical
employee works between 20 and 65 hours per week. Develop a program
that determines the wage of an employee from the number of hours of work.
The last sentence is the first to mention the actual task: to write a
program that determines one quantity based on some other quantity. More
specifically, the program consumes one quantity, the number of hours of
work, and produces another one, the wage in dollars. The first sentence
implies how to compute the result, but doesn't state it explicitly. In this
particular example, though, this poses no problem. If an employee works h
hours, the wage is
#displaymath72464#
Now that we have a rule, we can formulate a Scheme program:
<#744#>(d<#744#><#745#>efine<#745#> <#746#>(wage<#746#> <#747#>h)<#747#>
<#748#>(*<#748#> <#749#>12<#749#> <#750#>h))<#750#>
The program is called <#60392#><#754#>wage<#754#><#60392#>; its parameter <#60393#><#755#>h<#755#><#60393#> stands for
the hours an employee works; and its result is <#60394#><#756#>(*<#756#>\ <#757#>12<#757#>\ <#758#>h)<#758#><#60394#>, the
corresponding wage.
~<#760#>Discuss with your students how they teased out what they had to do and what information was superfluous. It is impossible to give guidelines on that, but practice and discussing the idea repeatedly helps a lot. The following exercises prepare students thematically and practically for the section on ``a program = many function definitions''. Use the solutions as motivations for the section on comopsing functions.<#760#>
<#763#>Exercise 2.3.1<#763#>
Utopia's tax accountants always use programs that compute income taxes even
though the tax rate is a solid, never-changing 15. Define the program
<#60395#><#766#>tax<#766#><#60395#>, which determines the tax on the grosspay.
Also define <#60396#><#767#>netpay<#767#><#60396#>. The program determines the netpay of an
employee from the number of hours worked. Assume an hourly rate of
$12.~ 
Solution<#60397#><#60397#>
<#773#>Exercise 2.3.2<#773#>
The local supermarket needs a program that can compute the value of a bag
of coins. Define the program <#60398#><#775#>sum-coins<#775#><#60398#>. It consumes four numbers:
the number of pennies, nickels, dimes, and quarters in the bag; it produces
the amount of money in the bag.~ Solution<#60399#><#60399#>
<#781#>Exercise 2.3.3<#781#>
An old-style movie theater has a simple profit function. Each customer pays
$5 per ticket. Every performance costs the theater $20, plus $.50 per
attendee. Develop the function <#60400#><#783#>total-profit<#783#><#60400#>. It consumes the number
of attendees (of a show) and produces how much income the attendees
produce.~ Solution<#60401#><#60401#>