##
* *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*

# Graphing y=1/x

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.

*70 minutes*

#### Closing

*10 min*

At this point, once all students have completed a graph of the reciprocal function and can explain its behavior, I will give them some information about the two concepts above. The document below introduces two key questions about the graph, which will turn into approach statements. These approach statements are a precursor to limits, which is geared towards preparing students for calculus.

I use Exit Ticket #3. I let students guess at the approach statements first. They will probably guess very incorrectly, because these are really abstract and new for them. This is their first early exposure to it, so the point is just to get them started thinking about it. (Don’t be worried if none of them seem to understand these at this point. They will understand them over time.) For today, the key questions are what is essential. So, I take 5-10 minutes to go over the new inputs on this page, then have students answer the exit ticket questions.

During this segment of the lesson, you can choose whether or not you want to provide direct instruction about approach statements. At most, I would present to students the answers to the questions in the approach statements document, and take the time to show how the answers appear in the graphs and in the data tables. Eventually, these will connect to the asymptotes and students will make generalizations about them, but for now just looking at the graph and data table will be concrete and clear to most students.

*expand content*

- UNIT 1: Linear and Nonlinear Functions
- UNIT 2: Piecewise Functions
- UNIT 3: Absolute Value Functions and More Piecewise Functions
- UNIT 4: Introduction to Quadratic Functions through Applications
- UNIT 5: More Abstract Work with Quadratic Functions
- UNIT 6: Rational Functions
- UNIT 7: Polynomial Functions
- UNIT 8: Exponential Functions
- UNIT 9: Ferris Wheels
- UNIT 10: Circles
- UNIT 11: Radical Functions
- UNIT 12: Cubic Functions

- LESSON 1: Introduction to Rational Functions with Real-World Applications
- LESSON 2: More Applications of Rational Functions
- LESSON 3: Graphing y=1/x
- LESSON 4: Transformations of y=1/x
- LESSON 5: Graphing y=a/(x-b)+c
- LESSON 6: Writing Approach Statements from Graphs
- LESSON 7: Matching Graph Transformations to Equations
- LESSON 8: Compare and Contrast Graphs of Rational Functions
- LESSON 9: Rational Function Review
- LESSON 10: Rational Function Summative Assessment and Portfolio