Real World Applications of Piecewise Functions

For some of my students, this entire lesson (and yesterday's lesson) was just a little bit too slow. I had some idea of who these students would be, but I used today's warm-up to help suss them out. I asked my students who were already a bit advanced with respect to the day's investigation to:

1. Work through the warm-up up to Problem (4) 
2. Show me that they already fully understood the key ideas of the lesson

Then I game them some laptops and asked them to create these graphs: Extension Graphs.

I wanted the task to be challenging, so I didn't give them any guidance on the math concepts behind the graphs. I did clarify some of the details about how to use the brackets and the notation in Desmos.com, but other than that I didn't give them much coaching.

This was an easy way to make this lesson way more effective, because these students were highly engaged in a task that was basically self-checking (the computers showed them the graphs, so they didn't need me to check their work).

Even though it was only 4 students in each of my classes, several things were accomplished as a result: (1) the students who took longer to understand the material couldn't just turn to their advanced peers and ask them for the answers and (2) the students who might have finished quickly didn't have the chance to distract others. 

These exit ticket questions shift the focus of the day from the concrete to the abstract (MP2): Real World Applications of Piecewise Functions Exit Ticket. Students will be introduced to the phrase piecewise function. If they haven’t already understood the meaning of this concept, you can ask them what they think it means and how they think it relates to the functions that they explored in class today.

The goal is for students to be able to explain that the inequalities come from the domain restrictions that tell us when to use each input rule to find the desired outputs. We can find the outputs the usual way, by successfully applying the function rule. Even if students are not able to explain this right now, they can start thinking about this question.

Finally, this closing also introduces the concept of continuity or continuous piecewise functions. This is a another thing just thrown in the at the end of the lesson, but I like to do this just so that students are exposed to the word and the idea. Obviously we will keep coming back to it in future lessons. I tell them this just to make sure that they don’t worry if they don’t fully understand this so far.