Graphing and Writing Equations for Piecewise Functions

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Objective

SWBAT graph a piecewise function given its equation and to write the equation for a piecewise function given its graph.

Big Idea

Now that students have examined real-world examples, ask them to apply this knowledge to more abstract functions.

Warm-Up

30 minutes

This Warm-Up is set up to offer students the chance to either review prior knowledge or develop new knowledge. As students enter the room, I tell them that I want them to choose three problems to solve. I inform them that problems (1), (2) and (3) are review and problems (4), (5) and (6) are new. I tell the whole class that once they accomplished three problems of their choice, they could use a few minutes to ask questions about the two portfolio tasks they had been assigned so far. 

I quickly circulate during the first few minutes of class to make sure that students had made appropriate choices. I motivate students who had mastered the material from the previous two days to skip the review problems. I ask students who had struggled to focus on the first 3 problems.

After a few minutes I encourage students to try to figure out problems 4 through 6. Some students complete the entire warm-up, but the class is able to move on after everyone has completed at least 3 problems.

DifferentiationModified Warm-Up 3.docx
I tell all the students working on the modified warm-up to complete the entire warm-up. The previous day I had realized  that for these students, figuring out the rule for a piecewise function was a real challenge because it was so abstract, so I asked them to focus on setting up the data table and the graph. 

 

Investigation and New Learning

30 minutes

Closing

10 minutes

This lesson is a good example of “flipped lesson”. In fact, the series of lessons at the start of this unit are flipped in the following sense:

Traditional textbook lessons start with teaching skills explicitly, then give students a chance to practice.  A few applications are tucked in at the very end of the section. In this unit I reverse this order in many of the lessons:

  1. Students grapple with a real world example
  2. Students learn and practice a new skill 
  3. At the closing of this lesson, students try to write or provide the direct instruction that they were not actually given by summarizing what they learned.

For today's lesson, the closing is the time to ask students to describe their steps in writing a function definition.

One benefit of this lesson is that it gives students a chance to practice finding the equations for linear models using a data tables. For some students, this task is still not easy. For these students, the closing is an opportunity to create a list of steps that somebody could use to find the rule for any piecewise graph. Writing processes is often a good way to internalize the algorithm.

The time allotted to this closing task can shrink or expand depending upon how much time you have. You can ask students to give a rough outline of the process, or you can ask them to give very detailed steps with an explanation of why they do each step. I like to allow time for at least one round of quick feedback, so I ask students to make an outline of steps independently, and then I quickly pair them with somebody else to compare their steps. This will eventually turn into a more formal assessment.