IM Commentary
The goal of this task is to motivate the definition of a function by carefully analyzing some different relationships. In some of these relationships, one quantity can be determined in terms of the other while in others this is not possible. In this way, students are led to see what is special about a function, namely that to each input there corresponds one and only one output. The statement of the task is ambiguous for examples (ii) and (iii). There are at least two valid approaches open to students:
 Not entering any value in the table if there is not a single, well determined value,
 Entering all appropriate values.
In some ways, the latter approach is more informative because then the table is seen as a different way of presenting the relationship under examination. The variable in the second column of a table is called a function of the variable in the first column when there is exactly one entry in the second column for each entry in the first column.
Below is an example the teacher may wish to use where it is not feasible to give a complete list in one of the tables:
City 
State 
Baton Rouge 

Seattle 

Honolulu 

State 
City 
Massachusetts 

Iowa 

Texas 

The first table can be filled out by listing the states in which these three cities are found (Louisiana, Washington, Hawaii). For different city names, such as Springfield or Jackson, there would be many states having a city with that name. The second table cannot be filled out readily as we would need to know every city in these states.
The teacher may wish to point out that scenario (i) is very different from (ii) and (iii) in the sense that there is a mathematical rule which relates the number of dimes to the number of minutes in an unambiguous way: for $x$ dimes, we get $6x$ minutes of parking and for $y$ minutes of parking (where $y$ is a multiple of 6) we needed to insert $\frac{y}{6}$ dimes.
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