Comparing Piecewise and Absolute Value Functions

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Objective

SWBAT compare and contrast absolute value and piecewise functions and to justify their claims with evidence.

Big Idea

How do piecewise functions and distance functions compare? Use true and false statements to get students thinking more deeply about these two types of functions.

Warm-Up

30 minutes

By the end of the week, I want students to master of the skills and ideas behind these types of problems. I find that the end of the week is a good day to ask students to really push themselves to make progress, or to solidify their understanding, so as they are working, I circulate and ask them to make sure that they push themselves to understand as much as possible. I also ask students, "Would you like me to have somebody explain _______ to you?" Often they would love their peers to help them understand problems, but they don't know who to ask. So I move students around to have them learn from each other. 

Students are usually able to work and stay focused on these warm-up problems (absolute value and piecewise functions warm-up) for 30-40 minutes, and I like to really push them to use all of this time. 

Investigation

40 minutes

Closing

5 minutes

The closing is a great time for a "think aloud" to really model MP3 for students. I like to create an example of a justification that highlights some common misconceptions about mathematical justification. For instance, students may claim that the difference between two numbers is the same as the distance between two numbers by using one example in which these two quantities are equal. I may show this justification and ask students to discuss whether or not this justification is adequate. Hopefully some students will suggest that only one example is not a sufficient justification, and we can talk about how to improve it together. 

Even though these are true or false questions, I like to treat them as more open-ended questions and encourage the students to defend both true or false answers. For instance, the statement "Distance functions are kind of like parabolas," is vague enough that either a true or a false answer can be defended. For the closing, I ask students to swap papers with another student and to look for one or two places where their partner's justifications could be improved. 

I can then ask students to show me one of their highest quality justifications on their way out the door, so that I have the chance to give them some quick feedback.