Writing, Graphing, and Describing Piecewise Linear Functions

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SWBAT describe a piecewise function in their own words, graph it, and write equations modeling the graph.

Big Idea

In this lesson we apply function notation to piecewise defined functions and assess students' ability evaluate functions in different forms.

Warm up

10 minutes

Today's Warm up is intended to take about 10 minutes for my students to complete and for me to review with the class. My students have worked with function notation before this lesson. We begin today's lesson with a review of how to evaluate a function, f(x) when given values for x. Then, students are asked to determine the value(s) of x that corresponds to a given output from a function.

In the video below, I demonstrate the strategy I use to teach graphing piecewise functions:


Using a Piecewise Function

15 minutes

After reviewing the Warm Up with students, I hand each student an Activity called Running Errands from this website: 

(last accessed 3-22-15)

I plan to have my students work on the activity individually for about 10 minutes. Then I will ask them to compare results with a table partner. For this activity the table partners are grouped homogeneously. I find that this helps my students to maintain the opportunity to think on their own, and share ideas with confidence.  

While my students are working I am walk around the room monitoring their progress.  Most of the students will be able to draw the five line segments on the graph associated with each of the five errands.  However, I expect my students to have difficulty with the following elements of the task:

  • writing the equation of each line correctly (with the slope and the y-intercept correct)
  • understanding and using function notation correctly
  • writing the domain correctly using inequality statements

When students share their work, I will emphasize the precise use of function notation (MP6).

Here are some examples of my students' work on this task:

  1. Student One uses 0 for the y-intercept in all of the equations, but does find the slope correctly.  However writes the equations in slope intercept form, and uses function notation incorrectly by placing f of x equal to an inequality.  Students should realize that f(x) is equivalent to y.
  2. Student Two did not finish.  Student two wrote two equations horizontally that were correct.  However, the student did not list the equations vertically next to each domain, and the student also numbered within the function notation.  Again, emphasizing Math Practice six and the importance of writing function notation correctly.
  3. Student Three writes all of the equations correct except the slope and the y-intercept of the last line segment.  However, this student also shows lack of understanding of f(x) because the student puts additional y= symbols inside the brackets.

Guided Practice

20 minutes

After the Using a Piecewise Function activity, I will work through a Guided Practice with students. I will lead this segment of the lesson so that we can focus on some of the difficulties that students experienced during the first half of today's lesson.

I will begin by  working some of the problems on the worksheet while the students observe. Then, I will give them time to produce graphs on their own. Earlier, I had demonstrated how to read a function definition and graph a Piecewise Function. During the Guided Practice, I focus more on the last page of Graphing Piecewise Functions where students interpret a graph and write a piecewise function to describe it. The video below contains is an example of how I might do this with my students:

Exit Slip

10 minutes

With about ten minutes remaining in the period, I give the students an Exit Slip to check for comprehension. Some of the issues I will look for in my students' work on this informal assessment are:

  1. Did students restrict the domain appropriately or did they extend a graph(s) outside of its domain?
  2. Did students accurately plot open and a closed circles on the plot.
  3. Did students may have difficulty with slope or y-intercept.
  4. Did students incorrectly use function notation.

I have provided a copy of a correct graph in the resources.