## 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

Unit 2: Piecewise Functions
Lesson 2 of 12

## Big Idea: Use simple real world situations to generate piecewise functions.

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Standards:
Subject(s):
Math, piecewise-defined function, Inequalities (Algebra), Precalculus and Calculus, multiple representations, Differentiated Instruction, function, PreCalculus, absolute value, piecewise-defined functions
85 minutes

### Hilary Yamtich

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