Experience First

In this lesson students use a pricing structure for smartphones to reason about piecewise functions. Students interpret a scenario (verbal representation) and then move on to creating a graph and ultimately writing a rule (equation). To make sense of the pricing structure, students first look at two concrete scenarios (question 2). 

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If students ask, clarify that you don’t need to pay for the entire extra GB if you don’t use it, and thus portions of a GB are acceptable. This may be different from their own actual phone plan. Although the goal of today’s lesson is not to talk about step functions, this could be an interesting extension for groups that are ready for it. A possible way to ask this could be: “How would the graph look different if you had to pay for a whole extra GB each time you surpassed your previous limit?” or “What if they said “$10 per additional GB or portion thereof?”

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In question 4, we purposefully use the word “rule” because we don’t expect students to know the formal notation for piecewise functions. Since our number of gigabytes is variable, the rule needs to work for all possible values of x. Students should be able to communicate informally that if x is between 0 and 2, the equation y=30 applies, and if x is greater than 2, y=30+10(x-2) applies. Nevertheless, this is still considered one singular rule, just like a piecewise function is still a singular function defined in multiple ways depending on the domain.

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Formalize Later

You may wish to consult with the Algebra 2 teachers at your school to determine students’ previous exposure to piecewise functions. At our school, this topic has been recently removed from the Algebra 2 curriculum (which explains the confused faces I saw when I asked what they remember about them from last year!). This will determine how much time you spend debriefing and practicing. 

In our experience, piecewise functions can be challenging for students, mainly because of the notation. Remind students that a piecewise function is not consistently defined by one equation so the first step when evaluating a piecewise function is always to determine which of the two or more equations applies. Students should never be inputting the same input into multiple branches of the function

Students may have various strategies for writing the equation of the line when x>2. Some may extend the line to find the y-intercept whereas others might be able to reason through point-slope form and explain how the $10 is added for each additional gigabyte after the two, thus explaining the (x-2) portion of the equation. Invite students to see how either method will give equivalent equations.