Parking and Pencils: Step Functions and Piecewise Functions
Objective
Big Idea
Launch
Today's lesson builds on the students' experience with the The Function Game. In the game students were exposed to greatest integer and piecewise functions. To begin class today, ask students to work in groups to brainstorm a list of every type of function they can name. Five minutes should be sufficient.
After each group generates a list, ask students to share function types to make a class list. Let the students know that they will be referencing this list throughout the year. I keep this list displayed in the class as reference for students.
Here is a list of all of the functions that will be appropriate for my class. You may have additions or deletions for your specific class. While adding them to your class list, it is a good idea to see if students can name some of the important characteristics.
Linear
Quadratic
Cubic
Polynomial
Rational
Exponential
Logarithmic
Piecewise
Greatest integer
Square Root
Trigonometric
Explore
After compiling the list of functions, let students know that they will be using some of these functions to model certain situations. The attached worksheet (Tasks - Step and Piecewise Functions Applications) should be given to students to see if they can work with these functions. I would give students 10-15 minutes to work on these equations with their table groups.
A big part of this assignment is being able to recognize what type of function to use. If a group is completely stuck, reference the list of functions and ask which one would be appropriate in the situation. It might also be prudent to have them generate a table to see if they can picture what the function would look like.
After they have found an answer, make sure that they test specific values to see if the function makes sense. Suggest that students trade functions with a neighbor and test certain inputs. This is especially important in the parking lot problem since they can reason through to see if they are getting the correct outputs.
Heads up: For the parking lot problem, many students may not understand what it means when parking is charged "for each hour or fraction of an hour." If you have this discussion at the beginning with the entire class, it might be too leading and everyone will know that they have to use the least integer function. Thus, it might be better to check in with individual groups to see if they need clarity.
Share and Summarize
When discussing the Parking Garage problem, I plan to focus on how you can get the function to "round up" even though you are using a greatest integer function that normally "rounds down." Adding 1 to the number of hours parked and then performing the greatest integer function will essentially produce a rounding up function. Some students may have been wondering if there is a least integer function, but after this discussion they can see that it is not completely necessary since you can manipulate the greatest integer function to suite the needs of the context. (See my Deciding between Two Functions reflection for more on this conversation.)
For the piecewise problem, students may wonder whether the domain should be all real numbers or discrete positive integers. This is a good conversation to have - how there is a domain for the function but also a relevant domain for the context of the problem.
An important thread running through both of these problems is that writing both functions allows the costs to be calculated algorithmically without "thinking." A calculator or computer can figure out the correct cost given only the input, no other classification or reasoning is needed. This is an important mathematical concept that has ramifications in computer programming and other fields.
Here are two questions I would use to close this discussion:
- What were the characteristics of the problem that signaled the need for a piecewise function? A greatest integer function?
- How are these two functions different than other functions you have studied in the past? (An important difference is continuity - be sure to touch on this!)
