Applying Reasoning and Proof
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Reflect on each of the following questions about the Enveloping Functions problem that you've seen in Ms. Davis's class, and select "Show Answer" to see our response.

 Question: How did Ms. Davis ask students to explaining their reasoning and what does this accomplish? Show Answer
 Our Answer: By first using terms like "why?" or "how do you know?" to respond to student answers rather than immediately correcting wrong answers or confirming correct ones. Such questions creates a student environment where students can explain and correct their own reasoning. It also sets up an expectation that right answers must be backed up with logical reasoning.
 Question: How does this activity build on the understanding of symmetry around the x-axis, y-axis, and the origin? Show Answer
 Our Answer: Ms. Davis asks students to assess whether the pairs of functions and enveloping functions have symmetry and, if so, what type. She later builds understanding of asymptotes in a similar fashion. In both cases she checks for student understanding of the terms and their ability to apply this knowledge to the problem at hand.
 Question: How is technology used in this lesson? Show Answer
 Our Answer: After students have worked with the material as a large group, they work collaboratively with one another to graph a function and make a conjecture about the related enveloping functions. Then Ms. Davis asks students how they could draw a function without the calculator, which checks for understanding.
 Question: Ms. Davis uses groups of mixed ability both for fairness and to help achieve her pedagogical goals. Do you organize groups this way as well? Why or why not? Show Answer
 Our Answer: Answers will vary. There is no one way to organize groups, ability level, gender, and mathematical background, all must be factored in. Groups should be productive for all members, not just for some, and it is likely that one organizational approach will not work for the entire semester or term. Ms. Davis' point that the groups must be fair in her eyes and in her students' eyes is an important one, otherwise some students may sense that they are doing a disproportionate amount of work. Another risk, that students may fall silent in context of students who are much stronger mathematically, or perhaps just much more verbal, must be balanced against this. Teachers must pay attention to the content of the group discussion as well as the dynamic when working in this way.

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