Today students will be using a real world problem to see how piecewise-defined functions are used.
The bell work gives students a problem from the textbook. The function is given, so the students do not need to understand how to write a piecewise-defined function at first. I will ask my students to work with each other to find the answer. I will move around the room and ask questions such as:
After students have worked for several minutes we will discuss the answers as a class. I will ask students to share and justify their answers. Once we have produced a good explanation, I will be ready to move to the real world problem.
When students receive the Writing Piecewise worksheet, I will ask a student to read the first part of the worksheet to make sure the directions are clear. Then, I will have students work in groups to determine the answers to the activity.
I expect some of my students to struggle with reading the information from the bill. By answering simple questions students can analyze the information and ask questions about the bill. The activity does some scaffolding for the students.
Question 2 is designed to make the students think about how to calculate the energy charge. It can be difficult for some students to use the information that the maximum cost at each interval is the constant for the next equation. If I find that students are stuck, I will guide them with questions like, "How will you determine the cost for 500 kWh? 600kWh? 601kWh?" I plan to work with groups who are struggling by having the groups find the energy cost, and, write their process so I can discuss it with them.
Many students are able to reason numerically, before writing a more abstract equation correctly. So, once the students have a function, I will ask them to verify their equation by knowing the cost for the customer on the bill. This is a way for the students to adjust their equation (MP8).
The final question is designed to help me quickly verify if a group's equation works. As students work on it I will move around the room to assess how the students are progressing. If a group is struggling I will sit with the group and process what has been found and guide the students. Many times I have to describe how the cost is determined. When this happens I refer to the cell phone problem from yesterday. I ask ask guiding questions such:
With about 10 minutes left in class, I plan to have students share the functions they found. Since some students will simplify the expressions and others may not, we will compare the different functions and discuss if the functions are equivalent. I will also ask students if one expression is easier for a customer to understand, to encourage students to think about using mathematics to communicate. I expect there to be some differences of opinion, with some students commenting that the non-simplified expression shows what is done to find the cost of energy.
Since group work has been used for most of the lesson, it is now time for students to do some independent work. I will ask my students to work on the Piecewise Functions 1 worksheet from yesterday. I will answer student questions. I will suggest students make tables to help them with writing the equations. I will also give students specific domain values when a student is not sure how to start. For question 4 of the home work, I ask questions like" What would the cost be if the person stays 2 nights? 3 nights?"
At the end of class, students respond to following question on an exit slip:
Can a piecewise-defined function's domain restrictions ever overlap? Explain.
I want to make sure that students understand when working with functions if the restrictions on the domains overlap then the equation is not a function.