Today I begin with a problem to introduce piecewise-defined functions. I adjust this problem each year to use a recent movie. Students determine the cost of a movie ticket depending on the age of the person.
As we work I ask:
I now change the situation by telling the students to assume the computers at the movie theater go offline. The ticket person must determine the ticket price by hand. How could we organize the information so the ticket person can quickly determine the cost of a ticket? At this point most students decide to make a table (slide 2).
I state that a function like the ticket price is called a piecewise-defined function. Why do you think it is called this?
I share the book definition of a piecewise-defined function on the board for students to read. I ask the students to explain what is meant when the definition says "over a specified domain." "What is the specified domain in the ticket price function?" This is hard for many students to understand by using the bell work problem the students see how the function has specified domain values for the cost of a ticket.
I now give students a piecewise-defined function. I ask students to identify the domain variable. I then ask:"How many answers does f(2) have? What is the value(s) of f(2)?" Some students say the answer is 4 and 3. I ask the students to explain how they found their answers? The common reasoning is that the students put the 2 into both parts of the function. As we are discussing the situation some students will comment that x=2 does not fit the domain of the x^2 piece since 2 is larger than 1. This gives me the opening to ask what the student means. I have the student explain their reasoning.
If a student does not make this comment I guide the students to the realization that f(2) must have one value. I ask questions such as "What is a function? How do we know this problem is a function? If you replace the x in both expressions you have 2 answers how can that be? What is meant by x>1?" After these questions students begin to reason that one answer is needed. I then ask which of the 2 answers found is correct.
The students consider which answer is correct by discussing how each piece tells what values to use for x. I make a comment such as so x^2, x<1 is saying if x<1 then f(x)=x^2. I like to change piecewise-defined functions into if-then statements. Students have worked with if-then statements in Geometry and understand the structure of an if-then statement. We rewrite both pieces into if-then format (page 2). This helps struggling students understand how to reason with a piecewise-defined function.
We continue to find the value of f(2). I have the students determine which domain is true for f(2) and we then determine the f(2)=3.
I give the students several domain values to evaluate, including the endpoints of the domains. I have different students give the answers this allows me to informally assess student understanding.
I share another example that is a little more complicated. I ask students to evaluate the function at several domain values. The students work individually. We put the answers on the board.
I now bring the class back to the bell work problem to develop how to write a piecewise-defined function.
I ask the students how we can write an function to represent the situation. I tell students to let "a" represent age and p(a) represents the price of a ticket.
After 3-4 minutes of working on writing the function a student shares a possible answer. We look over the students' answer to determine whether the equation works. I ask the following questions about writing piecewise-defined functions:
I now put up cell phone usage problem. The students work in groups to find the function to represent the monthly cost. This type of problem has been in books for several years. I have changed the problem to be more up to date. Instead of using overage of minutes I use overage in data usage which is more common today. The problem has the data usage in 2 different units. We discuss if it is okay not to have the same units for usage. The question will be to determine how the data usage should be represented by the company. This is a time that students can do research to determine how the data usage is reported. Once students determine the units students write the function for the cost. Students share their results with the class.
I do not tell students which unit to use for the data usage. I let the students decide. When students share their equations we discuss whether different equations will give the cost. One issue I address is how important it is to identify the units the data will be reported to make sure the appropriate domain value is used. I ask students how to write their function so another person understands the units used in their function.
Identifying the units is a good example of helping student understand the importance of precision. When there are two equations that use different units for the data are shared allows me to show students that the expressions will not give the same answer. I evaluate the functions for 200 (200 mb) and show how the one function gives a unreasonable cost. Students see how the cost is unrealistic and make comment like this function is in Kb not mb so you have to change your units. I then comment that the problem doesn't tell me this how was I supposed to know that. We fix the equations to identify the units and then verify the equations are correct.
To end today, I give the students Piecewise Functions 1. Students are told to begin the worksheet and we will discuss piecewise-defined functions more tomorrow. I am expecting students to struggle with writing the piecewise-defined functions but we are practicing more tomorrow so that should help students understand the worksheet.
I use an exit slip to determine what students are understanding. I ask the students to explain when a piecewise-defined function is required for modeling a situation. I can determine if students understand the definition of a piecewise-defined function enough to write an explanation in their own words. Students not sure about this will copy the definition from the book or state they do not know when to use a piecewise-defined function.