Reflection: Developing a Conceptual Understanding Graphing y=a/(x-b)+c - Section 2: Investigation and New Learning


I wanted to talk more about the b tells you the vertical asymptote and c tells you the horizontal asymptote." This is obviously accurate for functions in this form, but it is not very meaningful--almost anybody could figure this out without understanding the functions too well, after just looking at a few equations and their graphs.

I really want students to be able to explain why this generalization holds, and I have recently realized that if I don't teach them why, they won't know or be able to explain this. I created the choose inputs scaffold to help students think in different ways about how the function rule itself relates to the asymptotes. The more I tried to create scaffolds to help students think about these connections, the more complicated I realized the connections were. I wanted students to understand how to choose inputs to get really big outputs, and to explain what kinds of outputs would result from really big inputs. Then, I wanted them to identify these points on the graph, and connect these points to the asymptotes. Finally, I wanted them to be able to describe these asymptotes with approach statements.

The deeper we got into this learning target, the more scaffolds I wanted to create to help students make those connections.


  Developing a Conceptual Understanding: Building a Deeper Conceptual Understanding
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Graphing y=a/(x-b)+c

Unit 6: Rational Functions
Lesson 5 of 10

Objective: SWBAT choose inputs to graph functions in the form y=a/(x-b)+c and identify the asymptotes of these functions using their data tables. Students will be able to write approach statements to describe the behavior of these functions near their asymptotes.

Big Idea: Students develop the tools to graph rational functions by choosing inputs that reveal the behavior of vertical and horizontal asymptotes. Students describe this behavior using approach statements.

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Math, Precalculus and Calculus, rational function
  70 minutes
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