Reflection: Discourse and Questioning Graphing y=1/x - Section 2: Investigation and New Learning


When I first started teaching and my students created graphs that were horribly wrong, I panicked and corrected them. I thought that I must have done something wrong if they were doing crazy things. For instance, in this lesson, a "crazy" approach would be to plug in all the integers from -10 to 10 and then to plot these points and connect them. This obviously results in a graph that looks nothing like it should.

I have made a lot of progress in my understanding of the learning process, and I realize now that this "crazy" approach makes a lot of sense based on what students have previously learned. When graphing most functions, this can be a good strategy to get a sense of the graph's shape. 

Now, when I see something like this, I like to ask questions: 

*What strategy did you use? Why did that seem like a good strategy? 

*Is there a way you could check this? How confident are you that the graph looks this way?

*Do you think there are any important points on this graph that you may have missed?

*My graph actually looks way different than yours. Can you investigate more thoroughly?

Hopefully these questions will get students to ask more questions and to start wondering what is going on here. If they need more guidance, this is when I might ask them to consider some inputs between -1 and 1.

The important thing in these situations is to respond to their work as though it is a great starting point--they have already done a lot of great work to get to this point, and this is the beginning of the dialogue about this graph.

At the end of class, I use this "misconception" to start a conversation--because even if students have successfully graphed the function, I still want them to understand how this misconception comes about--even thought it is "wrong" there are many correct things about it. I try as much as possible to use misconceptions as the beginning of conversations.

  Using Misconceptions
  Discourse and Questioning: Using Misconceptions
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Graphing y=1/x

Unit 6: Rational Functions
Lesson 3 of 10

Objective: Students will graph the function y=1/x. Students will use the concept of infinity to describe the behavior of this function and identify the asymptotes of this function.

Big Idea: Use silly, exaggerated division problems to understand the behavior of the function y = 1/x. Make arguments involving 0 and infinity.

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Math, Precalculus and Calculus, rational function
  70 minutes
graph misconception
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