In the first slide of this warm-up (more_w_piecewise_open.pdf), students are going to check their understanding of the definition of a function. This understanding is imperative to working with piecewise functions. Students will examine both the table and the graph to determine if there are two output values for the same input value (MP2). Neither relation is a function. Students will then use what they have learned to make a sketch of a function and justifying why their drawing is correct. To check for understanding, I will take several student sketches and place them under the document camera and let the students justify why their sketch represents a function (MP3).
In the second slide, students will once again be justifying a solution. The difference here is that the solution is given to them. Students sometimes have difficulty with this type of question. By giving them the solution it helps them to build meaning around the process of finding the value of x that yields a given output.
This portion of the lesson begins with the most famous piecewise function: the absolute value function (more_w_piecewise_direct.pdf). Through graphing these two functions, students can see how the absolute value function has a different structure from most functions students are used to seeing (MP7).
I will have students use dry erase boards to make a table and a graph of the absolute value function in Slide #1. Try to bridge this function to the idea of a piecewise function by showing students that the absolute value function is essentially f(x) = -x for all values less than or equal to 0 and f(x) = x for all values greater than or equal to zero. Because at zero, both of these functions have the same value this relation is a function.
Put the majority of your emphasis on the first question in this section. If time permits, or if students are having difficulty graphing the absolute value function, you can also use the second example that is given. Otherwise, proceed directly to the guided practice portion of the lesson.
On Slide 2 of the attached powerpoint (more_w_piecewise_guided.pdf) we go into an in-depth explanation of how to sketch this piecewise function based on the work done in the opening activity. There is also a video explaining how this could be taught to students.
Before starting any direct instruction, allow students 2-3 minutes to discuss what they see in the piecewise function and how it looks different from other functions that they have seen so far. This processing time will allow students to make noticings about different attributes of the function of interest.
Slide 3 allows students an opportunity to practice graphing another piecewise function based on what they have just learned.
Recommended materials for this portion of the lesson would include a small dry-erase board for each student so that they can work along with you as you graph. I also like to make sure that each pair of students has two different color dry-erase markers. This helps when graphing a piecewise function that has two distinct parts (especially for more visual learners).
I have also included two additional problems on Slides 4 and 5 that can be used if time permits.
As I close this lesson I ask students to design their own piecewise function that is composed of at least two different functions (MP1) Instruct students to pay particular attention to the domain (input) values of each part of the function so that their function will pass the vertical line test. When students first start inventing piecewise graphs, they often forget that they are graphing a function. Encourage students to make a sketch of their function as they work. This should help to ensure that they are choosing appropriate domain values.