Give students about 5 minutes to work on this question (piecewise_functions_open.doc) by themselves before sharing their ideas with a partner. During that initial five minutes, have students write down some ideas about how they are going to solve each part of this problem (MP1).
Once students begin working with their partners, monitor student progress and take notes on various student approaches to solving the problem. Encourage students to use as much mathematical language as possible when expressing their ideas verbally and in writing. This will help to make the student work and explanation as precise as possible (MP3).
Typically, when students are investigating a rate of change problem like this, they will look to the lattice points on the graph to help them determine the rate. Many students may be able to come up with the fact that $60 was raised in two hours but will be unsure how to write that as a rate of change. Some students may be savvy enough to see that after 1 hour $30 was made and use repeated reasoning to see that this rate continues. Other students may also go right to the last point and see that $240 was raised over all 8 hours so the rate must be $30/hr.
When discussing this problem with students, begin by finding out how students thought about finding the unit rate. In this problem, most of the approaches mentioned in the previous section were unique and interesting and so sequencing may not be an issue. Try to move from students who will give a less sophisticated explanation to those who will provide more insight. You can also use Think-Pair-Share between students to help them process other student's ways of thinking. If students simply listen to one another without being able to process another students response they will not be able to make meaning and develop flexibility in their thinking.
Next, determine how to graphically and numerically show that the "amount was raised at twice the hourly rate during the next four hours." Again, take ideas from the class and either show student work using a document camera or sketch students ideas on the board. Push students to justify their way of thinking about the problem when they are sharing their ideas (MP3).
Lastly, we want to be able to write a function to describe this situation (a piecewise function!). Guide students towards understanding that the first function f(x)=30x is defined on the interval from [0,8]. The second function is defined on the interval from (8,12] but is a much more difficult function to name. Students will see that the rate of change (slope) should be 60 but will have trouble finding the y-intercept. Show them that by extending the line the y-intercept will fall well below the x-axis. Then use the idea of inputs and outputs to find the y-intercept. If after 9 hours $300 dollars should be raised, buy 9*60=540. We need to adjust our number by -240 (which ends up being the y-intercept). If students have trouble following this, that is OK at this point. Much more time will be spent on this concept during the linear functions unit. For now, show students how to write this situation as a piecewise function. They will be able to see that if they are using inputs of 0-8 they would use the function f(x)=30x+0. If they are using input values from 8 to 12 (not including 8) they would use the function g(x)=60x-240. This would also be a good place to discuss why the number 8 is in the domain of the first function using the context of the problem to support the answer.
This ticket out (piecewise_functions_close) may be used as guided practice rather than independent practice. The purpose here is to allow students to work with piecewise functions that are more abstract (MP2). However, if you feel that students are not ready to attempt something like this on their own you could certainly walk them through the first question and have them attempt the second. This concept will also be addressed in a future lesson so this ticket out could give you a baseline of the student's ability to work with piecewise functions.