This week in my precalculus class I used two worksheets that seemed to engage my students more than other worksheets have in the past. The topic was piecewise functions, functions formed by partitioning a domain into subdomains, or pieces, each of which piece is then associated with a dedicated function. For example, for a hypothetical parking lot, charges for the first hour could be $10 dollars, the second hour $8, and every hour after that $5, up to 24 hours (with partial hours rounded to the next hour). The function to calculate total charge for parking could be represented as follows,
10 if 0<h≤1
10+8 if 1<h≤2
10+8+5(ceiling(h)-2) if 2<h≤24
To find the charge for 4.5 hours the function would be evaluated as follows,
0<4.5≤1 is false
1<4.5≤2 is false
2<4.5≤24 is true
f(4.5) = 10+8+5(5-2) = $33
The piecewise function has two parts, the right-most part that specifies qualifying conditions and the left-most part that specifies a dedicated function to evaluate, if its associated condition is met. That is, the unique condition in the right-hand part, that evaluates to true for the input to the function, identifies its corresponding dedicated function in the left-hand part as the appropriate function to evaluate the input. In this case, as shown above, the condition that is true given 4.5 as an input is the condition in the last line of the function specification. Therefore, the function is evaluated with input 4.5 using (10+8+5(ceiling(h)-2)) as the relevant dedicated function, returning an answer of $33.
The details really are not important, except as a setting for my reflection.
The first worksheet presented several definitions of piecewise functions, similar to what was just shown. For each of the functions the students needed to graph the piecewise function and then evaluate the function for several input values, both algebraically and graphically. We spent approximately 15 minutes before the exercise going over how to graph piecewise functions on a TI-84 calculator so the students were testing that new knowledge at the same time as they were testing their understanding of the evaluation of piecewise functions. All of the students participated, some in small groups, some alone. I wandered the classroom looking over their shoulders and answering questions. They appeared to all be in search of answers to the worksheet problems. From the questions I received from some of the students, it was clear that not all of the students had really understood my explanations of piecewise functions, even with examples still on the board. This has become a common enough result that I am convinced that in my classroom of 40 students, from so many different backgrounds, there will always be some students that will not understand my lecture in the way that I think they should. More individual attention will apparently always be needed for some students. The in-class worksheet approach seems to provide a good way to identify at least some of the students in need.
The second worksheet presented two application problems, one about age-based pricing at a buffet restaurant and the other about the time (in minutes) that a person may safely scuba dive at a certain depth without having to decompress while surfacing—depth in feet. The second worksheet was an in-class project on the day after the first worksheet project. I lectured on some examples of application problems before distributing the second worksheet. The students worked with interest but again I received questions that showed a lack of understanding of what I had explained in the lecture. To be sure, most students had no problem establishing the correct form of the piecewise functions for both problems, but almost all of them had some trouble in accurately completing each form so that each result was a piecewise function and not just a piecewise relation. What that exactly means is not important here.
In summary, I think that each worksheet was a good, active learning experience for both the students and for me. And again, the in-class worksheets made me confront the reality that no matter what information I think I am transmitting in my lectures, not all students are receiving the intended message.