Roots

I begin this lesson with a real-world function on my board and explain why I choose to do so in my video.  I ask my students to graph the function , which should be a review of lessons from previous classes. (MP1) I have included a graph for the teacher on the resource.

While they're working I walk around observing who is working with ease and who might be struggling.  For the struggling students I usually suggest creating a table of several points or factoring to find the roots.  Reminding them of these strategies they already know helps build their confidence for the main part of the lesson.  I do not specifically require my students to use their graphing calculators for this part of the lesson, but encourage them to do so in preparation for the next activity.

After a few minutes or when everyone is done I ask for volunteers to share their graphs with the class and explain how they created them. I anticipate at least a few of the volunteers will include key features in their explanation, like intercepts and changes in slope. (MP3) If not, I ask questions like "What parts of this graph might be particularly interesting to the builders of the roller coaster and why?" and "Where does the ride slow down to climb a slope?  How do you know?" (MP4) When we've thoroughly discussed the graph and how it relates to the roller coaster function, I let my students know they'll be practicing more graphing today.

For this section of the lesson I have my students work independently to graph and describe key features of several real-world problems using their graphing calculators.  I explain that I don't just want them to list the features, but also to describe what they represent in terms of the problem.  I distribute the graphing worksheet and ask if there are any questions then tell my students they have about 20 minutes to complete the assignment. (MP1, MP2) While they're working I walk around offering encouragement and assistance as necessary.  Some students will struggle with the descriptions because they are still uncomfortable writing about mathematics.  For them I suggest that they just write what they are thinking and worry about editing it after they've gotten the basics written.  I also suggest that they write as though they are explaining it to a fourth grader of someone else who might not understand as well as they do.   When everyone is finished or after about 20 minutes I tell my students that today they will be checking their own papers as I go through the answers.  I remind them of my standard condition for this kind of self-check - I expect them to challenge any answers I give that don't make sense, that they disagree with, or that are incomplete.  I always throw at least one or two of these kinds of answers into the activity so that my students stay alert and so that they understand that neither I nor the textbook are infallible.

I close this lesson by giving my students a real world function then asking them to create a sketch labeled with key points and descriptions.  I have included a graph for teacher use, but just give my students the function in words and symbols. This gives them additional practice at relating mathematics to real-world situations and gives me additional insight into which students might need additional support. (MP2)  It also brings the lesson back to where we started the day, graphing a real-world function and making sense of the key features.