Running and the Domain of Middle Earth: Modeling a Run and a Hobbit's Journey through Piece-wise Functions
Lesson 7 of 10
Objective: SWBAT interpret key features (domain,rates of change) of graphs of piece-wise functions; SWBAT model and critique models of everyday situations by creating piece-wise functions.
During the Entry Ticket students practice solving equations and inequalities. During this unit, I want students to become fluent at solving equations and inequalities. I use problems similar to those used for homework the night before from the Kuta Software site. I might put up 2-4 equations to solve and an additional 2-3 inequalities for students to solve for this entry ticket.
After the entry ticket is complete, I tell students they are doing a great job with solving equation and we are now shifting to a slightly different skill set.
Students then begin work on the Class Activity: Modeling Everyday Scenarios Using Piece-Wise Functions. I read through the first scenario of Jim running with the class to try and be sure everyone has a concrete understanding of the context before diving into the work.
I then have students work in pairs (or small groups) on the Class Activity. The activity also lets students practice graphing key features of functions and continue their work on determining the rate of change over different intervals as well as interpreting what the rate of change means based on the context of the scenario.
I really like giving students the opportunity to work with piece-wise functions. They quickly discover that many times, a piece-wise function provides a better model than a linear function. And, if the pieces are themselves linear, the work does not require extensive knowledge of more complicated functions.
While students work, I rotate among the pairs of students, asking questions and checking in. If I notice a number of students are struggling with the same section of the task, I might regroup the class and have a short discussion on that part of the task. the reason being is I want students to struggle with the math, but not so much so that they give up. This is a delicate balance to achieve, and is one of the biggest challenges and thrills of teaching.
After students have completed the work for the first scenario on Jim's run, I have them complete an Idea Organizer to respond to the two writing challenges. The writing challenges are meant to have students compare and contrast linear and exponential models. I also want to make sure that students are clear that linear functions always change at a constant rate.
After students complete the Idea Organizer, I suggest having those students type up their responses in complete paragraphs. This provides more practice with the writing process.
The class then reads through the scenario in the Class Activity about Frodo's journey to open the section. This section is very language heavy, so depending on the reading comprehension level of your class, you may want to put in additional support when reading through Frodo's journey. For example, have students rotate reading the steps out loud as a class so everyone is following along.
The intent of this exercise is to give students a chance to create their own piece-wise graph to model Frodo's journey, which has many bumps and bruises along the way. Again, I like using piece-wise functions as they provide a chance to show that real-life scenarios are not always one directional and definitely not constant!
In this 20 minutes, many students can complete the graphing portion of the task and at least begin or get through most of the analysis questions that follow. I like to have students work in pairs during this task, and I see myself as more of a facilitator of learning than a teacher during this segment of the class.
For the remaining 15 minutes of class I have students continue working on the Frodo Scenario in pairs.
For this time, I would expect most students to wrap up the analysis questions and begin and/or finish writing the piece-wise function as equations before the end of class.
That way, I assign two writing prompts based on Frodo's journey for students to complete for homework. I assign any remaining parts of the class activity for most other students, but for students who are really struggling I might have them focus on finishing writing the equations rather than complete the writing assignment.
For more advanced students, I would also assign them the task of creating their own scenario that could be modeled using piece-wise functions. In addition I would ask the students to write the equations for the scenario and graph the scenario (see the section of this lesson called Graphing Piece-wise Functions for an excellent tutorial from Texas Instruments on how to complete that task).
The video below is a great resource that takes the user through step-by-step directions for graphing piece-wise functions with the Texas Instruments NSpire Teacher software: