I begin this lesson with a story about going to the candy store with my children when they were young. The candy was sold by the piece, by the dozen or by the pound depending on what and how much you chose. I ask my students to brainstorm how we might represent these costs as a function and with a graph. I could just tell them how to graph piece-wise functions, but letting them discover it for themselves encourages student ownership. I ask for a volunteer to scribe the suggestions on the whiteboard then step back (figuratively, at least) and let them work collaboratively to write the function and graph. (MP1, MP2, MP4) Once they've completed the graph I ask for an explanation of key features like the "fences" between pieces and which piece "owns" each fence. This should be review for most of my students, but I've found that there are always a few who are confused about open and closed circles or dotted and solid lines on a graph, so this is a good opportunity to clarify those things before getting into practice part of the lesson.
For this part of the lesson I tell my students they get to graph additional piecewise functions independently. I let them know that they will be graphing by hand, which is met with cheers from those who are still uncomfortable with their calculators and groans by those who have come to appreciate their calculator as a tool for efficiency. I explain that piecewise functions can be a bit tricky to do so I want them to build their skills at working with them before using a calculator.
I distribute the building fences handout, ask if there are any questions, then tell them they have about 20 minutes to complete the assignment. (MP1, MP2) While they're working I walk around offering encouragement and assistance as needed. I anticipate a few students will struggle with the x and y axes and what numbers to put where because of the amount of information given for each problem and the format the "fences" are presented in. For these students I suggest making a table of values for each part of the function instead of trying to graph it all at once.
When everyone is done or after about 20 minutes I tell them they will be working with the left-shoulder partner for the next part of the activity. I also tell them they may use calculators if they choose, but that the focus is to compare and critique each others' answers and to be able to explain what and where the intercepts and discontinuities (fences) are. (MP3)
As I explain in my video, I close this lesson by giving each student a notecard and asking them to write two things they understand better because of today's lesson and two things they're still unsure of. As a ticket-out-the door, self reflecting questions like these help my students focus on what was important in the lesson and provides a sense of continuity between what they've just learned and what they still need to work on.