##
* *Reflection:
Real World Applications of Piecewise Functions - Section 3: Investigation and New Learning

The second situation is challenging for students because they are unsure how to interpret the times of day. The two big issues that came up were deciding what the independent variable represented and interpreting the start time. Students were most likely to use 4 as their first input to this function, which made things confusing because then they thought when the got to 1 AM the graph would somehow loop back on itself. They were confused by this. Then I asked them, "What does *x* represent in this situation?" They started to think that it would make more sense to count the number of hours that she had worked, as opposed to the time of day. This made for more confusion when it came to setting up the inequalities, because they wanted to say "Before midnight" or "after midnight," but the abstraction required them to use *x >8 * for instance, because 8 represents the number of hours she will have worked by midnight.

Some students wrote (4, 4) as the first entry in their data table. This shows two misconceptions. In addition to the misconception about what the *x*-value represented, it also shows that the student thought that the babysitter would earn $4 at the instant she arrived. Because she started working at 4 pm, at 4 pm, her total earnings would be only $0 at this point. This seemed pretty obvious to me, but many students wrote this, so this was a worthwhile conversation.

I try to have these conversations on a more individual basis, because that way I can ask student many questions along the way, and I can get a better sense of whether the ideas are making sense. I really don't like to have these conversations as a whole class, because then many students will just write things down without understanding them. I think that the best way to address these misconceptions is to wait until they arise in students' work, and then ask students to explain them. Even though this obviously takes longer and seems less efficient, it is much more effective in the long run.

*Some Challenges to Anticipate*

*Some Challenges to Anticipate*

# Real World Applications of Piecewise Functions

Lesson 2 of 12

## Objective: SWBAT create piecewise functions to describe real-world situations.

*85 minutes*

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:

- Work through the warm-up up to Problem (4)
- 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.

#### Resources

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#### Closing

*10 min*

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.

*expand content*

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- UNIT 1: Linear and Nonlinear Functions
- UNIT 2: Piecewise Functions
- UNIT 3: Absolute Value Functions and More Piecewise Functions
- UNIT 4: Introduction to Quadratic Functions through Applications
- UNIT 5: More Abstract Work with Quadratic Functions
- UNIT 6: Rational Functions
- UNIT 7: Polynomial Functions
- UNIT 8: Exponential Functions
- UNIT 9: Ferris Wheels
- UNIT 10: Circles
- UNIT 11: Radical Functions
- UNIT 12: Cubic Functions

- LESSON 1: The Overtime Problem
- LESSON 2: Real World Applications of Piecewise Functions
- LESSON 3: Graphing and Writing Equations for Piecewise Functions
- LESSON 4: Progressive Income Taxes and Piecewise Functions
- LESSON 5: Compare and Contrast Graphs of Piecewise Functions
- LESSON 6: Write Piecewise Functions to Match Graphs
- LESSON 7: Piecewise Functions Level Review
- LESSON 8: Make Piecewise Functions Continuous
- LESSON 9: Write Absolute Value Functions as Piecewise Functions
- LESSON 10: More Absolute Value Graphs
- LESSON 11: Piecewise Functions and Absolute Value Functions Portfolio and Review
- LESSON 12: Piecewise and Absolute Value Functions Summative Assessment