Writing Approach Statements from Graphs

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Objective

SWBAT write approach statements (like limits) given a graph and an equation, or given only a graph or an equation. Students will be able to explain how these approach statements relate to the function rule.

Big Idea

Students generate approach statements using multiple representations of a rational function and develop their own early intuition about limits and infinity.

Warm-Up

30 minutes

Investigation

30 minutes

The only new input students need to begin working on the approach statement questions practice is a discussion of the data tables. I will ask students to attempt to create their own data tables. Or, if I think students can figure this out on their own, I will have them try to teach each other.

I really like the idea of using data tables because some students need this scaffold to even get started, but using the scaffold requires some thinking on their part. I ask students, “What inputs can we use to figure out how to complete this approach statement?” Often, my students use the same inputs every time, such as -2, -1, 0, 1, 2, which really don’t help at all. Thus, even using the data table requires some abstract reasoning and understanding of what the approach statement is asking (MP2). Other students may quickly complete the approach statements, but it is still worth asking them to take the time to show the inputs they would use in the data tables. I don’t make them fill out the data table completely because it may be tedious for them, but I do ask them to show me which inputs they would use to illustrate this approach statement.

One aspect of setting up the approach statements that is challenging for students is using the function rule to figure out the exact location of the vertical and horizontal asymptotes on the graph. This is a great time to really use the multiple representations of the function to understand its behavior. Obviously providing the function or the graph alone would be sufficient, but providing both of them to students gives me the opportunity to talk to students about how they could use either representation to write the approach statements. They can also use the various representations (function rule, graph, data table) to attempt to justify their answers to their peers (MP3).

I use the approach statement questions to guide my conversations with students. My goal is for students to be able to generate the approach statements using any of the 3 representations of the function: the function rule, the graph or the data table. 

In my experience, some students find this lesson incredibly challenging while others grasp the concept very quickly. For those who quickly master the concept, the extension is to create a graph that matches a set of approach statements.

 

 

Closing

10 minutes

The Exit Ticket questions are designed to get students thinking in a more directed way about how to connect the approach statements to each of the representations of the function.

The first question is the most open-ended, and students are most likely to struggle to articulate this. That is totally fine. The big idea that they should all understand is that we can identify the asymptotes using the function rule and that the asymptotes can help us find the approach statements. It would be even better if they can formulate the generalization that the approach statements associated with the vertical asymptote include "y approaches infinity" and "y approaches negative infinity"  and the approach statements associated with the horizontal asymptote include "x approaches infinity" and "x approaches negative infinity".

The second and third questions are much more straightforward and students should fully understand these after a brief discussion. See the Exit Ticket 6 answers.

I use the closing of the lesson to make sure that students have good answers to each question. To ensure this, I ask them to think on their own for a few minutes and put their initial thoughts down. Then I share my thinking with them and ask them to add to this. This is the closest we get to direct instruction, so it is okay to tell them some things about how to think about these questions.

For the second question, I go and physically stand near a large number line I have posted in my classroom. Students are not used to thinking about numbers in between the whole numbers on the number line so this is a good time to talk about what happens in those intervals. After I model how to think about one of these problems, I ask students to rethink their answers to the rest of them.

I use a projector to display the graph of the function being discussed and show students how the approach statements connect to the graph. This turns out to be surprisingly difficult for some students, so don’t be worried if you have the same conversation many times. I try to explain “The vertical asymptote is where the graph shoots up to infinity” and “The horizontal asymptote is what happens as the graph shoots over to infinity.” It is also a good time to ask some students who do understand this question to explain their ideas because perhaps their way of thinking about the asymptotes will make more sense to some students.